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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.15836v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">I Introduction</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">II Related Works</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S3" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">III Preliminaries</a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS1" title="In IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">IV-A Task Planning</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS2" title="In IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">IV-B Motion Planning</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS3" title="In IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">IV-C Asynchronous Execution </a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS4" title="In IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">IV-D TPG Execution</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">V Results</a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5.SS1" title="In V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">V-A Simulation Performance</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5.SS2" title="In V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">V-B System Deployment</a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S6" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">VI Limitations and Future Work</a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S7" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">VII Conclusion</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.SS1" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">-A Details of Task Planning Formulation</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.SS2" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">-B Details of Lego Manipulation Policy</a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.SS3" title="In APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">-C Numerical Results</a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document"> APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly </h1> <div class="ltx_authors"> Philip Huang∗,1, Ruixuan Liu∗,1, Shobhit Aggarwal1, Changliu Liu1 and Jiaoyang Li1 ∗Equal Contribution, 1Carnegie Mellon University </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> Compared to a single-robot workstation, a multi-robot system offers several advantages: 1) it expands the system’s workspace, 2) improves task efficiency, and more importantly, 3) enables robots to achieve significantly more complex and dexterous tasks, such as cooperative assembly. However, coordinating the tasks and motions of multiple robots is challenging due to issues, e.g., system uncertainty, task efficiency, algorithm scalability, and safety concerns. To address these challenges, this paper studies multi-robot coordination and proposes APEX-MR, an asynchronous planning and execution framework designed to safely and efficiently coordinate multiple robots to achieve cooperative assembly, e.g., LEGO assembly. In particular, APEX-MR provides a systematic approach to post-process multi-robot tasks and motion plans to enable robust asynchronous execution under uncertainty. Experimental results demonstrate that APEX-MR can significantly speed up the execution time of many long-horizon LEGO assembly tasks by 48% compared to sequential planning and 36% compared to synchronous planning on average. To further demonstrate the performance, we deploy APEX-MR to a dual-arm system to perform physical LEGO assembly. To our knowledge, this is the first robotic system capable of performing customized LEGO assembly using commercial LEGO bricks. The experiment results demonstrate that the dual-arm system, with APEX-MR, can safely coordinate robot motions, efficiently collaborate, and construct complex LEGO structures. Our project website is available at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://intelligent-control-lab.github.io/APEX-MR/" title="">https://intelligent-control-lab.github.io/APEX-MR/</a>. </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> I Introduction </h2> <div class="ltx_para" id="S1.p1"> Multi-robot manipulation is critical in robotic applications, such as industrial assembly <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib3" title="">3</a>]</cite>, material handling <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib50" title="">50</a>]</cite>, and object arrangement <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib14" title="">14</a>]</cite>, etc. Compared to a single-robot setup, a multi-robot arm system can easily expand the system’s overall reachable area. Besides, with a team of robot arms, a task can be accomplished more efficiently <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib49" title="">49</a>]</cite> by having each robot execute individual tasks parallelly. In addition to improving task efficiency, multi-robot can achieve significantly greater dexterity and is necessary in many applications that require collaborations, e.g., cooperative assembly, since certain tasks cannot be done with only one arm. </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="933" id="S1.F1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 1: Illustrations of bimanual cooperative assembly. Manipulation skills denote contact-rich operations for assembly. </figcaption> </figure> <div class="ltx_para" id="S1.p2"> LEGO assembly is an example of cooperative assembly. In a LEGO structure, bricks are assembled by forcing the top knobs of a brick into the bottom cavities of another brick. The bricks are held together passively, i.e., by the friction in the knob-to-cavity connections. Thus, the connections are not rigid, and subsequent manipulation, if performed inappropriately, can easily break the existing structure. Due to the nature of the passive connection, a multi-robot system is necessary for such cooperative assembly. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a> demonstrates example assembly operations in constructing a LEGO structure. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(a) shows the robot picking up a brick by disassembling it from the LEGO plate (i.e., pick). <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(b) illustrates the robot assembling a brick by placing it at the desired location and forcing a solid connection (i.e., place-down). One robot is sufficient for these two tasks since the manipulated object is fully supported and the operation would not collapse the existing structure. On the other hand, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(c)-(e) showcase operations that require multi-arm collaboration. In <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(c), the upper robot is assembling a LEGO brick on top of the character ‘S’. To establish a solid connection, the robot needs to press down and force the knob insertion due to the passive connection nature. However, since the existing connections are non-rigid, the place-down operation would break the existing overhanging structure. Therefore, the lower robot is necessary to support and stabilize the structure from below (i.e., support-bottom). Similarly in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(e), the lower robot assembles a brick from the bottom onto the character ‘S’ by pushing it up and forcing the connection (i.e., place-up). The place-up operation would break the existing structure, and thus, the upper robot is required to press down (i.e., support-top) in order to stabilize the structure. In addition to cooperative assembling, the multi-robot collaboration also enables more dexterous object manipulation, e.g., reorienting bricks in hand. As shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(d), the upper robot grabs a brick from its top initially. To have the robot grab the brick from its bottom, we can have the upper robot handover the brick to the lower robot. With the capability of reorienting in-hand objects, the robot can subsequently accomplish the assembly from the bottom as illustrated in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(e). Despite being a toy brand, LEGO has been widely used in entertainment, education, prototyping, etc. It is ideal for assembly benchmarking since it is a low-cost, standardized, and highly customizable assembly platform. Meanwhile, constructing LEGO structures is a challenging contact-rich manipulation problem due to the high-precision requirement and non-rigid connections. Thus, we use LEGO as our cooperative assembly benchmarking platform and the remaining discussion is in the context of LEGO assembly. </div> <div class="ltx_para" id="S1.p3"> Coordinating the tasks and motions of multiple robots to accomplish cooperative assembly is challenging for several reasons. First, a system with more than a single robot introduces overhead and algorithmic challenges in their coordination. As shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a> certain operations involve contacts with objects (i.e., pick, place-down, place-up, handover, support-bottom, and support-top) while some do not (i.e., transit and wait). There exist delays in controlling the robot and receiving sensor feedback, or even contingencies due to unexpected events when performing contact-rich operations, which would cause delays or stops to one or more robots. The uncertainty makes it challenging to safely coordinate the robots throughout the assembly process. Second, due to the collaboration need, robots often need to operate closely to each other as illustrated in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>(c)-(e). Even when executing under uncertainty in a confined shared workspace, robots must always avoid collisions. Third, it is desired that the task can be accomplished more efficiently by a multi-robot system. Thus, robots must have a comprehensive understanding of the cooperative assembly task and plan optimized motions to reduce the completion time, instead of frequently stopping to avoid collisions. Lastly, the multi-robot system should scale as the complexity of assembly design increases. It is necessary that the algorithm can scale to larger and more complex tasks. </div> <div class="ltx_para" id="S1.p4"> Many recent works have proposed methods for planning multiple robot arms for assembly <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib5" title="">5</a>]</cite>, rearrangement <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib13" title="">13</a>]</cite>, or general manipulation tasks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib45" title="">45</a>]</cite>. However, these planners often assume a sequential execution order and synchronously moving robots, where all robots start a task at the same time and wait for other robots to finish. We take a different lens and propose APEX-MR, an asynchronous planning and execution framework, to robustly, efficiently, and safely coordinate multiple robots. Given an assembly task sequence, APEX-MR leverages integer-linear programming (ILP) to distribute the tasks to the robots and generate a sequential robot plan. Most importantly, APEX-MR extends the temporal plan graph (TPG) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib20" title="">20</a>]</cite> to multi-robot-arm systems and post-processes the sequential robot plan for asynchronous execution. Specifically, the TPG captures all dependencies between different robots’ tasks and motions and generates a partial-order graph, which can significantly reduce unnecessary wait time and is robust to execution delay and uncertainty. To highlight the applicability of our proposed algorithm within a full multi-robot assembly pipeline, we deploy the proposed APEX-MR to a dual-industrial-arm system to construct complex customized LEGO structures up to 258 objects in simulation and up to 47 objects in the real world. In summary, our contributions are as follows: <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i1.p1"> We extend the TPG execution framework to multiple robot arms and show that it enables robust and safe multi-robot asynchronous execution under uncertainty. </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i2.p1"> We propose a sequential multi-robot task and motion framework that is complementary to the TPG execution. </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> • <div class="ltx_para" id="S1.I1.i3.p1"> We demonstrate algorithm deployment and system integration for bimanual LEGO assembly. To our knowledge, this is the first robotic system that can accomplish flexible (i.e., customized complex designs instead of simple pick and stack) assembly using commercial LEGO bricks. </div> </li> </ul> The rest of this paper is organized as follows: <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2" title="II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. II</a> discusses relevant works. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S3" title="III Preliminaries ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. III</a> deliberates the input that APEX-MR consumes and assumptions we have. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4" title="IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. IV</a> introduces our proposed APEX-MR, an asynchronous planning and execution framework for multi-robot cooperative assembly. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5" title="V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. V</a> shows the experiment results and demonstrates the deployment to a bimanual system for LEGO assembly. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S6" title="VI Limitations and Future Work ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. VI</a> discusses limitations and future works. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S7" title="VII Conclusion ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. VII</a> concludes the paper. </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> II Related Works </h2> <div class="ltx_para" id="S2.p1"> Multi-Agent Path Finding and Execution Multi-agent path finding (MAPF) <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib54" title="">54</a>]</cite> studies how to coordinate a large team of mobile robots in a discretized world environment, often on 2D grid worlds representing warehouse settings <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib32" title="">32</a>]</cite> or pre-defined roadmaps <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib21" title="">21</a>]</cite>. State-of-the-art MAPF algorithms can plan near-optimal collision-free paths for hundreds of robots in seconds <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib26" title="">26</a>]</cite>. </div> <div class="ltx_para" id="S2.p2"> However, these MAPF algorithms achieve such impressive efficiency at the cost of neglecting robot kinematics and execution uncertainty. As a result, there is growing research within the MAPF community focused on the efficient and robust execution of (imperfect) MAPF plans on real robots. One of the most widely adopted frameworks is the temporal plan graph (TPG) <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib20" title="">20</a>]</cite>, originally proposed to post-process MAPF plans to meet robot kinematics by enforcing passing orders at locations visited by multiple robots. TPG has since been extended to execution frameworks for MAPF under uncertainty <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib33" title="">33</a>]</cite> and mobile robot coordination in warehouses <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib59" title="">59</a>]</cite>. Recent advancements introduce bidirectional TPG <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib56" title="">56</a>]</cite> and switchable TPG <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib23" title="">23</a>]</cite>, which further enhance TPG by allowing flexible passing orders at certain locations. </div> <div class="ltx_para" id="S2.p3"> Given the success of TPG in coordinating mobile robots, we aim to explore its applicability in coordinating robotic arms. A key distinction of traditional TPG versus our use case is that the robot kinematic is more complex, and may change over time as the robot arm picks up different object, which is a key focus in this paper. </div> <div class="ltx_para" id="S2.p4"> Multi-Robot Arm Motion Planning Motivated by the rise of bimanual manipulation systems, many early works in the field study the problem of dual arm motion planning <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib51" title="">51</a>]</cite>. One naive approach that scales single-robot motion planning methods to two or more robots is to plan in the composite joint space, with methods such as RRT-Connect <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib25" title="">25</a>]</cite>, BIT* <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib11" title="">11</a>]</cite>, or graph of convex sets <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib35" title="">35</a>]</cite>. However, due to the curse of dimensionality, these methods struggle to find high-quality solutions as the number of robots increases. Other common strategies include building a composite roadmap from the Cartesian product of individual roadmaps <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib15" title="">15</a>]</cite>, coordinating the speed of individually planned motions <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib2" title="">2</a>]</cite>, or using prioritized planning <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib8" title="">8</a>]</cite>. More recently, more specialized multi-robot motion planners have been proposed that are based on roadmaps (dRRT <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib53" title="">53</a>]</cite>, dRRT* <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib48" title="">48</a>]</cite>, and CBS-MP <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib52" title="">52</a>]</cite>) or utilizing multi-agent path finding techniques <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib44" title="">44</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib45" title="">45</a>]</cite>. Some approaches also use online planning or control techniques to generate motions in real-time. <cite class="ltx_cite ltx_citemacro_citet">Zhang and Pecora [<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib61" title="">61</a>]</cite> proposes an online motion coordination technique, in which they plan all robot’s paths offline and a pairwise collision matrix between robots. Then, the speed of each robot can be efficiently planned online to avoid collisions, even in the presence of execution delays. However, their method cannot always find a feasible motion and relies on a task reallocation process to avoid deadlock. Other techniques such as a distributed model predictive controller <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib10" title="">10</a>]</cite> or a dynamical system approach <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib38" title="">38</a>]</cite> can also be used to generate multi-robot arm motions in real-time, but are not tailored towards long-horizon planning tasks. </div> <div class="ltx_para" id="S2.p5"> Multi-Robot Task and Motion Planning Beyond motion planning, multi-robot arm task and motion planning (MR-TAMP) have been studied since the 1990s, e.g., object pick and place <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib24" title="">24</a>]</cite>, and many of which are designed for a dual arm setting <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib51" title="">51</a>]</cite>. Similar to our method, these methods take a two-staged approach in which they first generate a task plan with robot assignment and grasp poses, then search for corresponding motion plans with composite-state-space or prioritized planning. The authors of dRRT* have proposed extending dRRT* to multi-modal roadmaps with given possible pick-up and hand-off configurations and searching for a task and motion plan directly <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib47" title="">47</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib46" title="">46</a>]</cite>. Given that motion planning calls tend to be more costly, another approach is to generate promising task plans first in a lazy manner and subsequently find corresponding motion plans and backtrack when required. This approach can be implemented with a greedy method <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib18" title="">18</a>]</cite>, with a mixed-integer linear program <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib31" title="">31</a>]</cite> or through a satisfiability modulo theories solver <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib42" title="">42</a>]</cite>. However, a major drawback is that task planning often assumes motions to be synchronous, which can be suboptimal in practice. Our TPG framework is designed to be complementary to these synchronous tasks and motion planners, as it can relax the synchronicity assumption with post-processing and shorten the makespan of the overall plan. Another significant concern is that many MR-TAMP methods are designed for simple environments such as object pick and place and planned robot trajectories are executed in open-loop or only in simulation. Our framework scales to more complex environments and is designed to integrate with more challenging manipulation skills such as a force-feedbacked controller. </div> <div class="ltx_para" id="S2.p6"> Multi-Robot Arm Motion Execution Most existing work executes their multi-robot on real robots in a synchronized way. Since the popular motion planning framework MoveIt <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib7" title="">7</a>]</cite> does not natively support moving multiple robot arms asynchronously, some recent work have sought to address this and enable asynchronous execution. <cite class="ltx_cite ltx_citemacro_citet">Meehan et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib36" title="">36</a>]</cite> adds a trajectory reservation component and treats the entire trajectory of a moving robot arm as a static obstacle when planning another arm’s motion, which can be too conservative and cause deadlock. <cite class="ltx_cite ltx_citemacro_citet">Stoop et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib55" title="">55</a>]</cite> uses a central scheduler to check if a new trajectory collides with previously scheduled trajectories and executes it asynchronously if there is no collision. Otherwise, the new trajectory waits for the conflicting previous trajectory to finish before it can start. However, their methods do not account for execution delays, may wait longer than necessary, and directly modify the MoveIt! software stack. In contrast, our TPG formulation is robust to arbitrary delays by design, minimizes robot wait time, and can be implemented without modification to MoveIt. </div> <div class="ltx_para" id="S2.p7"> Robotic LEGO Assembly Automating LEGO assembly using robots is challenging due to the high-precision requirement, tiny sizes of LEGO bricks, and non-rigid connections in the structure. Most of the existing works address the LEGO assembly problem in simulation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib43" title="">43</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib39" title="">39</a>]</cite>, which cannot be generalized to physical assembly due to the lack of simulators to simulate the connections between LEGO bricks. Recent works <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib30" title="">30</a>]</cite> assume the structure is fully supported and only consider placing bricks on top of others, which are limited to assembling simple structures. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib34" title="">34</a>]</cite> considers assembly using customized brick toys, which does not apply to LEGO assembly. In this paper, we apply APEX-MR to a bimanual system to construct complex customized LEGO structures beyond simple stacking. </div> <figure class="ltx_figure" id="S2.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="325" id="S2.F2.g1" src="x2.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 2: An overview of APEX-MR. On a high level, APEX-MR builds a sequential task plan given an assembly sequence, plans the motion of each task with RRT-Connect, and converts the solution to a parallel, asynchronous plan for execution with a TPG. Specifically, (a) shows the example of a task graph, (b) illustrates generating robot motion from the task plan, and (c) shows the example of a multi-modal TPG.</figcaption> </figure> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> III Preliminaries </h2> <div class="ltx_para" id="S3.p1"> To coordinate a multi-robot system to perform cooperative assembly tasks, we assume three inputs are provided. </div> <div class="ltx_para" id="S3.p2"> Environment Setup We assume the environment setup, including (a) geometries of all robots, (b) poses of objects to be manipulated <math alttext="\mathcal{B}=[b_{1},b_{2},\dots,b_{N_{b}}]" class="ltx_Math" display="inline" id="S3.p2.1.m1.4"><semantics id="S3.p2.1.m1.4a"><mrow id="S3.p2.1.m1.4.4" xref="S3.p2.1.m1.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.1.m1.4.4.5" xref="S3.p2.1.m1.4.4.5.cmml">ℬ</mi><mo id="S3.p2.1.m1.4.4.4" xref="S3.p2.1.m1.4.4.4.cmml">=</mo><mrow id="S3.p2.1.m1.4.4.3.3" xref="S3.p2.1.m1.4.4.3.4.cmml"><mo id="S3.p2.1.m1.4.4.3.3.4" stretchy="false" xref="S3.p2.1.m1.4.4.3.4.cmml">[</mo><msub id="S3.p2.1.m1.2.2.1.1.1" xref="S3.p2.1.m1.2.2.1.1.1.cmml"><mi id="S3.p2.1.m1.2.2.1.1.1.2" xref="S3.p2.1.m1.2.2.1.1.1.2.cmml">b</mi><mn id="S3.p2.1.m1.2.2.1.1.1.3" xref="S3.p2.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.p2.1.m1.4.4.3.3.5" xref="S3.p2.1.m1.4.4.3.4.cmml">,</mo><msub id="S3.p2.1.m1.3.3.2.2.2" xref="S3.p2.1.m1.3.3.2.2.2.cmml"><mi id="S3.p2.1.m1.3.3.2.2.2.2" xref="S3.p2.1.m1.3.3.2.2.2.2.cmml">b</mi><mn id="S3.p2.1.m1.3.3.2.2.2.3" xref="S3.p2.1.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S3.p2.1.m1.4.4.3.3.6" xref="S3.p2.1.m1.4.4.3.4.cmml">,</mo><mi id="S3.p2.1.m1.1.1" mathvariant="normal" xref="S3.p2.1.m1.1.1.cmml">…</mi><mo id="S3.p2.1.m1.4.4.3.3.7" xref="S3.p2.1.m1.4.4.3.4.cmml">,</mo><msub id="S3.p2.1.m1.4.4.3.3.3" xref="S3.p2.1.m1.4.4.3.3.3.cmml"><mi id="S3.p2.1.m1.4.4.3.3.3.2" xref="S3.p2.1.m1.4.4.3.3.3.2.cmml">b</mi><msub id="S3.p2.1.m1.4.4.3.3.3.3" xref="S3.p2.1.m1.4.4.3.3.3.3.cmml"><mi id="S3.p2.1.m1.4.4.3.3.3.3.2" xref="S3.p2.1.m1.4.4.3.3.3.3.2.cmml">N</mi><mi id="S3.p2.1.m1.4.4.3.3.3.3.3" xref="S3.p2.1.m1.4.4.3.3.3.3.3.cmml">b</mi></msub></msub><mo id="S3.p2.1.m1.4.4.3.3.8" stretchy="false" xref="S3.p2.1.m1.4.4.3.4.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.4b"><apply id="S3.p2.1.m1.4.4.cmml" xref="S3.p2.1.m1.4.4"><eq id="S3.p2.1.m1.4.4.4.cmml" xref="S3.p2.1.m1.4.4.4"></eq><ci id="S3.p2.1.m1.4.4.5.cmml" xref="S3.p2.1.m1.4.4.5">ℬ</ci><list id="S3.p2.1.m1.4.4.3.4.cmml" xref="S3.p2.1.m1.4.4.3.3"><apply id="S3.p2.1.m1.2.2.1.1.1.cmml" xref="S3.p2.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.2.2.1.1.1.1.cmml" xref="S3.p2.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.2.2.1.1.1.2.cmml" xref="S3.p2.1.m1.2.2.1.1.1.2">𝑏</ci><cn id="S3.p2.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.p2.1.m1.2.2.1.1.1.3">1</cn></apply><apply id="S3.p2.1.m1.3.3.2.2.2.cmml" xref="S3.p2.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.p2.1.m1.3.3.2.2.2.1.cmml" xref="S3.p2.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S3.p2.1.m1.3.3.2.2.2.2.cmml" xref="S3.p2.1.m1.3.3.2.2.2.2">𝑏</ci><cn id="S3.p2.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S3.p2.1.m1.3.3.2.2.2.3">2</cn></apply><ci id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1">…</ci><apply id="S3.p2.1.m1.4.4.3.3.3.cmml" xref="S3.p2.1.m1.4.4.3.3.3"><csymbol cd="ambiguous" id="S3.p2.1.m1.4.4.3.3.3.1.cmml" xref="S3.p2.1.m1.4.4.3.3.3">subscript</csymbol><ci id="S3.p2.1.m1.4.4.3.3.3.2.cmml" xref="S3.p2.1.m1.4.4.3.3.3.2">𝑏</ci><apply id="S3.p2.1.m1.4.4.3.3.3.3.cmml" xref="S3.p2.1.m1.4.4.3.3.3.3"><csymbol cd="ambiguous" id="S3.p2.1.m1.4.4.3.3.3.3.1.cmml" xref="S3.p2.1.m1.4.4.3.3.3.3">subscript</csymbol><ci id="S3.p2.1.m1.4.4.3.3.3.3.2.cmml" xref="S3.p2.1.m1.4.4.3.3.3.3.2">𝑁</ci><ci id="S3.p2.1.m1.4.4.3.3.3.3.3.cmml" xref="S3.p2.1.m1.4.4.3.3.3.3.3">𝑏</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.4c">\mathcal{B}=[b_{1},b_{2},\dots,b_{N_{b}}]</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.4d">caligraphic_B = [ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ]</annotation></semantics></math>, and (c) states of all obstacles, is known as shown in the input section in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a>. 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Note that since duplicate objects are common in assemblies, <math alttext="b_{i}" class="ltx_Math" display="inline" id="S3.p2.5.m5.1"><semantics id="S3.p2.5.m5.1a"><msub id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml"><mi id="S3.p2.5.m5.1.1.2" xref="S3.p2.5.m5.1.1.2.cmml">b</mi><mi id="S3.p2.5.m5.1.1.3" xref="S3.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.1b"><apply id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p2.5.m5.1.1.1.cmml" xref="S3.p2.5.m5.1.1">subscript</csymbol><ci id="S3.p2.5.m5.1.1.2.cmml" xref="S3.p2.5.m5.1.1.2">𝑏</ci><ci id="S3.p2.5.m5.1.1.3.cmml" xref="S3.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.1c">b_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="b_{j}" class="ltx_Math" display="inline" id="S3.p2.6.m6.1"><semantics id="S3.p2.6.m6.1a"><msub id="S3.p2.6.m6.1.1" xref="S3.p2.6.m6.1.1.cmml"><mi id="S3.p2.6.m6.1.1.2" xref="S3.p2.6.m6.1.1.2.cmml">b</mi><mi id="S3.p2.6.m6.1.1.3" xref="S3.p2.6.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.6.m6.1b"><apply id="S3.p2.6.m6.1.1.cmml" xref="S3.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.p2.6.m6.1.1.1.cmml" xref="S3.p2.6.m6.1.1">subscript</csymbol><ci id="S3.p2.6.m6.1.1.2.cmml" xref="S3.p2.6.m6.1.1.2">𝑏</ci><ci id="S3.p2.6.m6.1.1.3.cmml" xref="S3.p2.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m6.1c">b_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m6.1d">italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> can be identical objects, e.g., identical <math alttext="2\times 2" class="ltx_Math" display="inline" id="S3.p2.7.m7.1"><semantics id="S3.p2.7.m7.1a"><mrow id="S3.p2.7.m7.1.1" xref="S3.p2.7.m7.1.1.cmml"><mn id="S3.p2.7.m7.1.1.2" xref="S3.p2.7.m7.1.1.2.cmml">2</mn><mo id="S3.p2.7.m7.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.p2.7.m7.1.1.1.cmml">×</mo><mn id="S3.p2.7.m7.1.1.3" xref="S3.p2.7.m7.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.7.m7.1b"><apply id="S3.p2.7.m7.1.1.cmml" xref="S3.p2.7.m7.1.1"><times id="S3.p2.7.m7.1.1.1.cmml" xref="S3.p2.7.m7.1.1.1"></times><cn id="S3.p2.7.m7.1.1.2.cmml" type="integer" xref="S3.p2.7.m7.1.1.2">2</cn><cn id="S3.p2.7.m7.1.1.3.cmml" type="integer" xref="S3.p2.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m7.1c">2\times 2</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m7.1d">2 × 2</annotation></semantics></math> bricks in the character ‘S’ shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a>. </div> <div class="ltx_para" id="S3.p3"> Assembly Plan Given an assembly design with <math alttext="N_{a}" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><msub id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mi id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml">N</mi><mi id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2">𝑁</ci><ci id="S3.p3.1.m1.1.1.3.cmml" xref="S3.p3.1.m1.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">N_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> objects, we assume the assembly plan <math alttext="A=[a_{1},a_{2},\dots,a_{N_{a}}]" class="ltx_Math" display="inline" id="S3.p3.2.m2.4"><semantics id="S3.p3.2.m2.4a"><mrow id="S3.p3.2.m2.4.4" xref="S3.p3.2.m2.4.4.cmml"><mi id="S3.p3.2.m2.4.4.5" xref="S3.p3.2.m2.4.4.5.cmml">A</mi><mo id="S3.p3.2.m2.4.4.4" xref="S3.p3.2.m2.4.4.4.cmml">=</mo><mrow id="S3.p3.2.m2.4.4.3.3" xref="S3.p3.2.m2.4.4.3.4.cmml"><mo id="S3.p3.2.m2.4.4.3.3.4" stretchy="false" xref="S3.p3.2.m2.4.4.3.4.cmml">[</mo><msub id="S3.p3.2.m2.2.2.1.1.1" xref="S3.p3.2.m2.2.2.1.1.1.cmml"><mi id="S3.p3.2.m2.2.2.1.1.1.2" xref="S3.p3.2.m2.2.2.1.1.1.2.cmml">a</mi><mn id="S3.p3.2.m2.2.2.1.1.1.3" xref="S3.p3.2.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.p3.2.m2.4.4.3.3.5" xref="S3.p3.2.m2.4.4.3.4.cmml">,</mo><msub id="S3.p3.2.m2.3.3.2.2.2" xref="S3.p3.2.m2.3.3.2.2.2.cmml"><mi id="S3.p3.2.m2.3.3.2.2.2.2" xref="S3.p3.2.m2.3.3.2.2.2.2.cmml">a</mi><mn id="S3.p3.2.m2.3.3.2.2.2.3" xref="S3.p3.2.m2.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S3.p3.2.m2.4.4.3.3.6" xref="S3.p3.2.m2.4.4.3.4.cmml">,</mo><mi id="S3.p3.2.m2.1.1" mathvariant="normal" xref="S3.p3.2.m2.1.1.cmml">…</mi><mo id="S3.p3.2.m2.4.4.3.3.7" xref="S3.p3.2.m2.4.4.3.4.cmml">,</mo><msub id="S3.p3.2.m2.4.4.3.3.3" xref="S3.p3.2.m2.4.4.3.3.3.cmml"><mi id="S3.p3.2.m2.4.4.3.3.3.2" xref="S3.p3.2.m2.4.4.3.3.3.2.cmml">a</mi><msub id="S3.p3.2.m2.4.4.3.3.3.3" xref="S3.p3.2.m2.4.4.3.3.3.3.cmml"><mi id="S3.p3.2.m2.4.4.3.3.3.3.2" xref="S3.p3.2.m2.4.4.3.3.3.3.2.cmml">N</mi><mi id="S3.p3.2.m2.4.4.3.3.3.3.3" xref="S3.p3.2.m2.4.4.3.3.3.3.3.cmml">a</mi></msub></msub><mo id="S3.p3.2.m2.4.4.3.3.8" stretchy="false" xref="S3.p3.2.m2.4.4.3.4.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.4b"><apply id="S3.p3.2.m2.4.4.cmml" xref="S3.p3.2.m2.4.4"><eq id="S3.p3.2.m2.4.4.4.cmml" xref="S3.p3.2.m2.4.4.4"></eq><ci id="S3.p3.2.m2.4.4.5.cmml" xref="S3.p3.2.m2.4.4.5">𝐴</ci><list id="S3.p3.2.m2.4.4.3.4.cmml" xref="S3.p3.2.m2.4.4.3.3"><apply id="S3.p3.2.m2.2.2.1.1.1.cmml" xref="S3.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p3.2.m2.2.2.1.1.1.1.cmml" xref="S3.p3.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S3.p3.2.m2.2.2.1.1.1.2.cmml" xref="S3.p3.2.m2.2.2.1.1.1.2">𝑎</ci><cn id="S3.p3.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S3.p3.2.m2.2.2.1.1.1.3">1</cn></apply><apply id="S3.p3.2.m2.3.3.2.2.2.cmml" xref="S3.p3.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.p3.2.m2.3.3.2.2.2.1.cmml" xref="S3.p3.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S3.p3.2.m2.3.3.2.2.2.2.cmml" xref="S3.p3.2.m2.3.3.2.2.2.2">𝑎</ci><cn id="S3.p3.2.m2.3.3.2.2.2.3.cmml" type="integer" xref="S3.p3.2.m2.3.3.2.2.2.3">2</cn></apply><ci id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1">…</ci><apply id="S3.p3.2.m2.4.4.3.3.3.cmml" xref="S3.p3.2.m2.4.4.3.3.3"><csymbol cd="ambiguous" id="S3.p3.2.m2.4.4.3.3.3.1.cmml" xref="S3.p3.2.m2.4.4.3.3.3">subscript</csymbol><ci id="S3.p3.2.m2.4.4.3.3.3.2.cmml" xref="S3.p3.2.m2.4.4.3.3.3.2">𝑎</ci><apply id="S3.p3.2.m2.4.4.3.3.3.3.cmml" xref="S3.p3.2.m2.4.4.3.3.3.3"><csymbol cd="ambiguous" id="S3.p3.2.m2.4.4.3.3.3.3.1.cmml" xref="S3.p3.2.m2.4.4.3.3.3.3">subscript</csymbol><ci id="S3.p3.2.m2.4.4.3.3.3.3.2.cmml" xref="S3.p3.2.m2.4.4.3.3.3.3.2">𝑁</ci><ci id="S3.p3.2.m2.4.4.3.3.3.3.3.cmml" xref="S3.p3.2.m2.4.4.3.3.3.3.3">𝑎</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.4c">A=[a_{1},a_{2},\dots,a_{N_{a}}]</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.4d">italic_A = [ italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ]</annotation></semantics></math> is provided as shown in the input section in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a>. Each step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><msub id="S3.p3.3.m3.1.1" xref="S3.p3.3.m3.1.1.cmml"><mi id="S3.p3.3.m3.1.1.2" xref="S3.p3.3.m3.1.1.2.cmml">a</mi><mi id="S3.p3.3.m3.1.1.3" xref="S3.p3.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><apply id="S3.p3.3.m3.1.1.cmml" xref="S3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p3.3.m3.1.1.1.cmml" xref="S3.p3.3.m3.1.1">subscript</csymbol><ci id="S3.p3.3.m3.1.1.2.cmml" xref="S3.p3.3.m3.1.1.2">𝑎</ci><ci id="S3.p3.3.m3.1.1.3.cmml" xref="S3.p3.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> refers to an object, such as a <math alttext="1\times 2" class="ltx_Math" display="inline" id="S3.p3.4.m4.1"><semantics id="S3.p3.4.m4.1a"><mrow id="S3.p3.4.m4.1.1" xref="S3.p3.4.m4.1.1.cmml"><mn id="S3.p3.4.m4.1.1.2" xref="S3.p3.4.m4.1.1.2.cmml">1</mn><mo id="S3.p3.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.p3.4.m4.1.1.1.cmml">×</mo><mn id="S3.p3.4.m4.1.1.3" xref="S3.p3.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.4.m4.1b"><apply id="S3.p3.4.m4.1.1.cmml" xref="S3.p3.4.m4.1.1"><times id="S3.p3.4.m4.1.1.1.cmml" xref="S3.p3.4.m4.1.1.1"></times><cn id="S3.p3.4.m4.1.1.2.cmml" type="integer" xref="S3.p3.4.m4.1.1.2">1</cn><cn id="S3.p3.4.m4.1.1.3.cmml" type="integer" xref="S3.p3.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.1c">1\times 2</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.1d">1 × 2</annotation></semantics></math> brick. The assembly plan specifies the order in which each object should be assembled. Note that due to duplicate objects in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.p3.5.m5.1"><semantics id="S3.p3.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.1b"><ci id="S3.p3.5.m5.1.1.cmml" xref="S3.p3.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.1d">caligraphic_B</annotation></semantics></math>, there are multiple candidates <math alttext="b_{i},\dots,b_{k}" class="ltx_Math" display="inline" id="S3.p3.6.m6.3"><semantics id="S3.p3.6.m6.3a"><mrow id="S3.p3.6.m6.3.3.2" xref="S3.p3.6.m6.3.3.3.cmml"><msub id="S3.p3.6.m6.2.2.1.1" xref="S3.p3.6.m6.2.2.1.1.cmml"><mi id="S3.p3.6.m6.2.2.1.1.2" xref="S3.p3.6.m6.2.2.1.1.2.cmml">b</mi><mi id="S3.p3.6.m6.2.2.1.1.3" xref="S3.p3.6.m6.2.2.1.1.3.cmml">i</mi></msub><mo id="S3.p3.6.m6.3.3.2.3" xref="S3.p3.6.m6.3.3.3.cmml">,</mo><mi id="S3.p3.6.m6.1.1" mathvariant="normal" xref="S3.p3.6.m6.1.1.cmml">…</mi><mo id="S3.p3.6.m6.3.3.2.4" xref="S3.p3.6.m6.3.3.3.cmml">,</mo><msub id="S3.p3.6.m6.3.3.2.2" xref="S3.p3.6.m6.3.3.2.2.cmml"><mi id="S3.p3.6.m6.3.3.2.2.2" xref="S3.p3.6.m6.3.3.2.2.2.cmml">b</mi><mi id="S3.p3.6.m6.3.3.2.2.3" xref="S3.p3.6.m6.3.3.2.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.6.m6.3b"><list id="S3.p3.6.m6.3.3.3.cmml" xref="S3.p3.6.m6.3.3.2"><apply id="S3.p3.6.m6.2.2.1.1.cmml" xref="S3.p3.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S3.p3.6.m6.2.2.1.1.1.cmml" xref="S3.p3.6.m6.2.2.1.1">subscript</csymbol><ci id="S3.p3.6.m6.2.2.1.1.2.cmml" xref="S3.p3.6.m6.2.2.1.1.2">𝑏</ci><ci id="S3.p3.6.m6.2.2.1.1.3.cmml" xref="S3.p3.6.m6.2.2.1.1.3">𝑖</ci></apply><ci id="S3.p3.6.m6.1.1.cmml" xref="S3.p3.6.m6.1.1">…</ci><apply id="S3.p3.6.m6.3.3.2.2.cmml" xref="S3.p3.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S3.p3.6.m6.3.3.2.2.1.cmml" xref="S3.p3.6.m6.3.3.2.2">subscript</csymbol><ci id="S3.p3.6.m6.3.3.2.2.2.cmml" xref="S3.p3.6.m6.3.3.2.2.2">𝑏</ci><ci id="S3.p3.6.m6.3.3.2.2.3.cmml" xref="S3.p3.6.m6.3.3.2.2.3">𝑘</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.6.m6.3c">b_{i},\dots,b_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.6.m6.3d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , … , italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> that can be used to accomplish an assembly step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S3.p3.7.m7.1"><semantics id="S3.p3.7.m7.1a"><msub id="S3.p3.7.m7.1.1" xref="S3.p3.7.m7.1.1.cmml"><mi id="S3.p3.7.m7.1.1.2" xref="S3.p3.7.m7.1.1.2.cmml">a</mi><mi id="S3.p3.7.m7.1.1.3" xref="S3.p3.7.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.7.m7.1b"><apply id="S3.p3.7.m7.1.1.cmml" xref="S3.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p3.7.m7.1.1.1.cmml" xref="S3.p3.7.m7.1.1">subscript</csymbol><ci id="S3.p3.7.m7.1.1.2.cmml" xref="S3.p3.7.m7.1.1.2">𝑎</ci><ci id="S3.p3.7.m7.1.1.3.cmml" xref="S3.p3.7.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.7.m7.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.7.m7.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="A" class="ltx_Math" display="inline" id="S3.p3.8.m8.1"><semantics id="S3.p3.8.m8.1a"><mi id="S3.p3.8.m8.1.1" xref="S3.p3.8.m8.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.p3.8.m8.1b"><ci id="S3.p3.8.m8.1.1.cmml" xref="S3.p3.8.m8.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.8.m8.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.p3.8.m8.1d">italic_A</annotation></semantics></math>. Furthermore, we require the assembly plan <math alttext="A" class="ltx_Math" display="inline" id="S3.p3.9.m9.1"><semantics id="S3.p3.9.m9.1a"><mi id="S3.p3.9.m9.1.1" xref="S3.p3.9.m9.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.p3.9.m9.1b"><ci id="S3.p3.9.m9.1.1.cmml" xref="S3.p3.9.m9.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.9.m9.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.p3.9.m9.1d">italic_A</annotation></semantics></math> to be physically valid <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib58" title="">58</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib28" title="">28</a>]</cite>, ensuring that each task can be performed in reality. Specifically, the partially assembled structure after each step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S3.p3.10.m10.1"><semantics id="S3.p3.10.m10.1a"><msub id="S3.p3.10.m10.1.1" xref="S3.p3.10.m10.1.1.cmml"><mi id="S3.p3.10.m10.1.1.2" xref="S3.p3.10.m10.1.1.2.cmml">a</mi><mi id="S3.p3.10.m10.1.1.3" xref="S3.p3.10.m10.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.10.m10.1b"><apply id="S3.p3.10.m10.1.1.cmml" xref="S3.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p3.10.m10.1.1.1.cmml" xref="S3.p3.10.m10.1.1">subscript</csymbol><ci id="S3.p3.10.m10.1.1.2.cmml" xref="S3.p3.10.m10.1.1.2">𝑎</ci><ci id="S3.p3.10.m10.1.1.3.cmml" xref="S3.p3.10.m10.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.10.m10.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.10.m10.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> must be physically stable, and there exists at least one feasible grasp pose for each step. The assembly sequence also specifies whether each step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S3.p3.11.m11.1"><semantics id="S3.p3.11.m11.1a"><msub id="S3.p3.11.m11.1.1" xref="S3.p3.11.m11.1.1.cmml"><mi id="S3.p3.11.m11.1.1.2" xref="S3.p3.11.m11.1.1.2.cmml">a</mi><mi id="S3.p3.11.m11.1.1.3" xref="S3.p3.11.m11.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.11.m11.1b"><apply id="S3.p3.11.m11.1.1.cmml" xref="S3.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S3.p3.11.m11.1.1.1.cmml" xref="S3.p3.11.m11.1.1">subscript</csymbol><ci id="S3.p3.11.m11.1.1.2.cmml" xref="S3.p3.11.m11.1.1.2">𝑎</ci><ci id="S3.p3.11.m11.1.1.3.cmml" xref="S3.p3.11.m11.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.11.m11.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.11.m11.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> requires two robots for cooperative assembly and the specific type of cooperation needed (i.e., support or reorientation). Lastly, we assume <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S3.p3.12.m12.1"><semantics id="S3.p3.12.m12.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p3.12.m12.1.1" xref="S3.p3.12.m12.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.p3.12.m12.1b"><ci id="S3.p3.12.m12.1.1.cmml" xref="S3.p3.12.m12.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.12.m12.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.12.m12.1d">caligraphic_B</annotation></semantics></math> is sufficient so that each object required for <math alttext="a_{j}" class="ltx_Math" display="inline" id="S3.p3.13.m13.1"><semantics id="S3.p3.13.m13.1a"><msub id="S3.p3.13.m13.1.1" xref="S3.p3.13.m13.1.1.cmml"><mi id="S3.p3.13.m13.1.1.2" xref="S3.p3.13.m13.1.1.2.cmml">a</mi><mi id="S3.p3.13.m13.1.1.3" xref="S3.p3.13.m13.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.13.m13.1b"><apply id="S3.p3.13.m13.1.1.cmml" xref="S3.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S3.p3.13.m13.1.1.1.cmml" xref="S3.p3.13.m13.1.1">subscript</csymbol><ci id="S3.p3.13.m13.1.1.2.cmml" xref="S3.p3.13.m13.1.1.2">𝑎</ci><ci id="S3.p3.13.m13.1.1.3.cmml" xref="S3.p3.13.m13.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.13.m13.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.13.m13.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> can be found in the environment. </div> <div class="ltx_para" id="S3.p4"> Manipulation Skills We assume that the robot manipulation skills are predefined and known. We denote the skill set as <math alttext="\mathcal{S}=[s_{1},s_{2},\dots,s_{N_{s}}]" class="ltx_Math" display="inline" id="S3.p4.1.m1.4"><semantics id="S3.p4.1.m1.4a"><mrow id="S3.p4.1.m1.4.4" xref="S3.p4.1.m1.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p4.1.m1.4.4.5" xref="S3.p4.1.m1.4.4.5.cmml">𝒮</mi><mo id="S3.p4.1.m1.4.4.4" xref="S3.p4.1.m1.4.4.4.cmml">=</mo><mrow id="S3.p4.1.m1.4.4.3.3" xref="S3.p4.1.m1.4.4.3.4.cmml"><mo id="S3.p4.1.m1.4.4.3.3.4" stretchy="false" xref="S3.p4.1.m1.4.4.3.4.cmml">[</mo><msub id="S3.p4.1.m1.2.2.1.1.1" xref="S3.p4.1.m1.2.2.1.1.1.cmml"><mi id="S3.p4.1.m1.2.2.1.1.1.2" xref="S3.p4.1.m1.2.2.1.1.1.2.cmml">s</mi><mn id="S3.p4.1.m1.2.2.1.1.1.3" xref="S3.p4.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.p4.1.m1.4.4.3.3.5" xref="S3.p4.1.m1.4.4.3.4.cmml">,</mo><msub id="S3.p4.1.m1.3.3.2.2.2" xref="S3.p4.1.m1.3.3.2.2.2.cmml"><mi id="S3.p4.1.m1.3.3.2.2.2.2" xref="S3.p4.1.m1.3.3.2.2.2.2.cmml">s</mi><mn id="S3.p4.1.m1.3.3.2.2.2.3" xref="S3.p4.1.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S3.p4.1.m1.4.4.3.3.6" xref="S3.p4.1.m1.4.4.3.4.cmml">,</mo><mi id="S3.p4.1.m1.1.1" mathvariant="normal" xref="S3.p4.1.m1.1.1.cmml">…</mi><mo id="S3.p4.1.m1.4.4.3.3.7" xref="S3.p4.1.m1.4.4.3.4.cmml">,</mo><msub id="S3.p4.1.m1.4.4.3.3.3" xref="S3.p4.1.m1.4.4.3.3.3.cmml"><mi id="S3.p4.1.m1.4.4.3.3.3.2" xref="S3.p4.1.m1.4.4.3.3.3.2.cmml">s</mi><msub id="S3.p4.1.m1.4.4.3.3.3.3" xref="S3.p4.1.m1.4.4.3.3.3.3.cmml"><mi id="S3.p4.1.m1.4.4.3.3.3.3.2" xref="S3.p4.1.m1.4.4.3.3.3.3.2.cmml">N</mi><mi id="S3.p4.1.m1.4.4.3.3.3.3.3" xref="S3.p4.1.m1.4.4.3.3.3.3.3.cmml">s</mi></msub></msub><mo id="S3.p4.1.m1.4.4.3.3.8" stretchy="false" xref="S3.p4.1.m1.4.4.3.4.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.4b"><apply id="S3.p4.1.m1.4.4.cmml" xref="S3.p4.1.m1.4.4"><eq id="S3.p4.1.m1.4.4.4.cmml" xref="S3.p4.1.m1.4.4.4"></eq><ci id="S3.p4.1.m1.4.4.5.cmml" xref="S3.p4.1.m1.4.4.5">𝒮</ci><list id="S3.p4.1.m1.4.4.3.4.cmml" xref="S3.p4.1.m1.4.4.3.3"><apply id="S3.p4.1.m1.2.2.1.1.1.cmml" xref="S3.p4.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p4.1.m1.2.2.1.1.1.1.cmml" xref="S3.p4.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.p4.1.m1.2.2.1.1.1.2.cmml" xref="S3.p4.1.m1.2.2.1.1.1.2">𝑠</ci><cn id="S3.p4.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.p4.1.m1.2.2.1.1.1.3">1</cn></apply><apply id="S3.p4.1.m1.3.3.2.2.2.cmml" xref="S3.p4.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.3.3.2.2.2.1.cmml" xref="S3.p4.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S3.p4.1.m1.3.3.2.2.2.2.cmml" xref="S3.p4.1.m1.3.3.2.2.2.2">𝑠</ci><cn id="S3.p4.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S3.p4.1.m1.3.3.2.2.2.3">2</cn></apply><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">…</ci><apply id="S3.p4.1.m1.4.4.3.3.3.cmml" xref="S3.p4.1.m1.4.4.3.3.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.4.4.3.3.3.1.cmml" xref="S3.p4.1.m1.4.4.3.3.3">subscript</csymbol><ci id="S3.p4.1.m1.4.4.3.3.3.2.cmml" xref="S3.p4.1.m1.4.4.3.3.3.2">𝑠</ci><apply id="S3.p4.1.m1.4.4.3.3.3.3.cmml" xref="S3.p4.1.m1.4.4.3.3.3.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.4.4.3.3.3.3.1.cmml" xref="S3.p4.1.m1.4.4.3.3.3.3">subscript</csymbol><ci id="S3.p4.1.m1.4.4.3.3.3.3.2.cmml" xref="S3.p4.1.m1.4.4.3.3.3.3.2">𝑁</ci><ci id="S3.p4.1.m1.4.4.3.3.3.3.3.cmml" xref="S3.p4.1.m1.4.4.3.3.3.3.3">𝑠</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.4c">\mathcal{S}=[s_{1},s_{2},\dots,s_{N_{s}}]</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.4d">caligraphic_S = [ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT end_POSTSUBSCRIPT ]</annotation></semantics></math>, where <math alttext="N_{s}" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><msub id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml"><mi id="S3.p4.2.m2.1.1.2" xref="S3.p4.2.m2.1.1.2.cmml">N</mi><mi id="S3.p4.2.m2.1.1.3" xref="S3.p4.2.m2.1.1.3.cmml">s</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p4.2.m2.1.1.1.cmml" xref="S3.p4.2.m2.1.1">subscript</csymbol><ci id="S3.p4.2.m2.1.1.2.cmml" xref="S3.p4.2.m2.1.1.2">𝑁</ci><ci id="S3.p4.2.m2.1.1.3.cmml" xref="S3.p4.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">N_{s}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">italic_N start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT</annotation></semantics></math> is the number of skills an individual robot has. For instance, a skill can be, inserting a pin, fastening a screw, picking up an object, etc. In our case of LEGO assembly, each skill <math alttext="s_{i}" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><msub id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml"><mi id="S3.p4.3.m3.1.1.2" xref="S3.p4.3.m3.1.1.2.cmml">s</mi><mi id="S3.p4.3.m3.1.1.3" xref="S3.p4.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><apply id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p4.3.m3.1.1.1.cmml" xref="S3.p4.3.m3.1.1">subscript</csymbol><ci id="S3.p4.3.m3.1.1.2.cmml" xref="S3.p4.3.m3.1.1.2">𝑠</ci><ci id="S3.p4.3.m3.1.1.3.cmml" xref="S3.p4.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m3.1c">s_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m3.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a composite of multiple motions parametrized by the pose of the manipulated object, the robot end-effector, and parameters learned from demonstrations. We assume an algorithm, such as interpolation or RRT-Connect, can generate a reference robot trajectory for the purpose of computing a collision-free coordination schedule. During execution, each skill is executed with a feedback controller so that its exact execution time is unpredictable. Example manipulation skills are shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>, and more details are discussed in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5" title="V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. V</a>. </div> <div class="ltx_para" id="S3.p5"> In addition to these inputs, we assume the existence of a collision-free HOME pose for each robot that never blocks other moving robots from executing their tasks. Robots can also transit between different poses, or wait and hold their current pose in place. </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System </h2> <div class="ltx_para" id="S4.p1"> In this section, we introduce APEX-MR, a framework for asynchronously coordinating the task plan, motion, and execution of a multi-robot system to accomplish cooperative assembly tasks. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> provides an overview of the three stages of APEX-MR. The first task planning stage is discussed in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS1" title="IV-A Task Planning ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. IV-A</a>. Given the input discussed in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S3" title="III Preliminaries ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. III</a>, a sequential task plan and the corresponding task graph, as shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> (a), are generated from the assembly using ILP. In the next stage explained in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS2" title="IV-B Motion Planning ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. IV-B</a>, the motion plan for each task is generated sequentially with a single-robot motion planner (see <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> (b. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS3" title="IV-C Asynchronous Execution ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. IV-C</a> presents the last and most crucial stage, which converts the sequential plan to a multi-modal TPG (e.g. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> (c)) that can be executed safely, asynchronously, and efficiently in real robots. </div> <div class="ltx_para" id="S4.p2"> For notation, we use <math alttext="i\in[1,N]" class="ltx_Math" display="inline" id="S4.p2.1.m1.2"><semantics id="S4.p2.1.m1.2a"><mrow id="S4.p2.1.m1.2.3" xref="S4.p2.1.m1.2.3.cmml"><mi id="S4.p2.1.m1.2.3.2" xref="S4.p2.1.m1.2.3.2.cmml">i</mi><mo id="S4.p2.1.m1.2.3.1" xref="S4.p2.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.p2.1.m1.2.3.3.2" xref="S4.p2.1.m1.2.3.3.1.cmml"><mo id="S4.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.p2.1.m1.2.3.3.1.cmml">[</mo><mn id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">1</mn><mo id="S4.p2.1.m1.2.3.3.2.2" 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xref="S4.p2.2.m2.2.2.3">𝑗</ci><interval closure="closed" id="S4.p2.2.m2.2.2.1.2.cmml" xref="S4.p2.2.m2.2.2.1.1"><cn id="S4.p2.2.m2.1.1.cmml" type="integer" xref="S4.p2.2.m2.1.1">1</cn><apply id="S4.p2.2.m2.2.2.1.1.1.cmml" xref="S4.p2.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.p2.2.m2.2.2.1.1.1.1.cmml" xref="S4.p2.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S4.p2.2.m2.2.2.1.1.1.2.cmml" xref="S4.p2.2.m2.2.2.1.1.1.2">𝑁</ci><ci id="S4.p2.2.m2.2.2.1.1.1.3.cmml" xref="S4.p2.2.m2.2.2.1.1.1.3">𝑎</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.2c">j\in[1,N_{a}]</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.2d">italic_j ∈ [ 1 , italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ]</annotation></semantics></math> to index over assembly steps in <math alttext="A" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><mi id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><ci id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">italic_A</annotation></semantics></math>, <math alttext="m" class="ltx_Math" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><mi id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.p2.4.m4.1b"><ci id="S4.p2.4.m4.1.1.cmml" xref="S4.p2.4.m4.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">m</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">italic_m</annotation></semantics></math> to index over tasks for each robot, and <math alttext="k\in[1,N_{b}]" class="ltx_Math" display="inline" id="S4.p2.5.m5.2"><semantics id="S4.p2.5.m5.2a"><mrow id="S4.p2.5.m5.2.2" xref="S4.p2.5.m5.2.2.cmml"><mi id="S4.p2.5.m5.2.2.3" xref="S4.p2.5.m5.2.2.3.cmml">k</mi><mo id="S4.p2.5.m5.2.2.2" xref="S4.p2.5.m5.2.2.2.cmml">∈</mo><mrow id="S4.p2.5.m5.2.2.1.1" xref="S4.p2.5.m5.2.2.1.2.cmml"><mo id="S4.p2.5.m5.2.2.1.1.2" stretchy="false" xref="S4.p2.5.m5.2.2.1.2.cmml">[</mo><mn id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml">1</mn><mo id="S4.p2.5.m5.2.2.1.1.3" xref="S4.p2.5.m5.2.2.1.2.cmml">,</mo><msub id="S4.p2.5.m5.2.2.1.1.1" xref="S4.p2.5.m5.2.2.1.1.1.cmml"><mi id="S4.p2.5.m5.2.2.1.1.1.2" xref="S4.p2.5.m5.2.2.1.1.1.2.cmml">N</mi><mi id="S4.p2.5.m5.2.2.1.1.1.3" xref="S4.p2.5.m5.2.2.1.1.1.3.cmml">b</mi></msub><mo id="S4.p2.5.m5.2.2.1.1.4" stretchy="false" xref="S4.p2.5.m5.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.2b"><apply id="S4.p2.5.m5.2.2.cmml" xref="S4.p2.5.m5.2.2"><in id="S4.p2.5.m5.2.2.2.cmml" xref="S4.p2.5.m5.2.2.2"></in><ci id="S4.p2.5.m5.2.2.3.cmml" xref="S4.p2.5.m5.2.2.3">𝑘</ci><interval closure="closed" id="S4.p2.5.m5.2.2.1.2.cmml" xref="S4.p2.5.m5.2.2.1.1"><cn id="S4.p2.5.m5.1.1.cmml" type="integer" xref="S4.p2.5.m5.1.1">1</cn><apply id="S4.p2.5.m5.2.2.1.1.1.cmml" xref="S4.p2.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.p2.5.m5.2.2.1.1.1.1.cmml" xref="S4.p2.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S4.p2.5.m5.2.2.1.1.1.2.cmml" xref="S4.p2.5.m5.2.2.1.1.1.2">𝑁</ci><ci id="S4.p2.5.m5.2.2.1.1.1.3.cmml" xref="S4.p2.5.m5.2.2.1.1.1.3">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.2c">k\in[1,N_{b}]</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.2d">italic_k ∈ [ 1 , italic_N start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ]</annotation></semantics></math> to index over usable objects in <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S4.p2.6.m6.1"><semantics id="S4.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S4.p2.6.m6.1.1" xref="S4.p2.6.m6.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S4.p2.6.m6.1b"><ci id="S4.p2.6.m6.1.1.cmml" xref="S4.p2.6.m6.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m6.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m6.1d">caligraphic_B</annotation></semantics></math>. </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> IV-A Task Planning </h3> <div class="ltx_para" id="S4.SS1.p1"> Given the assembly sequence <math alttext="A" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.1"><semantics id="S4.SS1.p1.1.m1.1a"><mi id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><ci id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.1d">italic_A</annotation></semantics></math>, task planning aims to construct a sequential task plan that includes robot assignment, robot target pose assignment, and object assignment. Specifically, for each step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><msub id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.p1.2.m2.1.1.2" xref="S4.SS1.p1.2.m2.1.1.2.cmml">a</mi><mi id="S4.SS1.p1.2.m2.1.1.3" xref="S4.SS1.p1.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><apply id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝑎</ci><ci id="S4.SS1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> in the assembly sequence, the task planner must assign one responsible robot, a support robot if required, and an object <math alttext="b_{k}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><msub id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml"><mi id="S4.SS1.p1.3.m3.1.1.2" xref="S4.SS1.p1.3.m3.1.1.2.cmml">b</mi><mi id="S4.SS1.p1.3.m3.1.1.3" xref="S4.SS1.p1.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><apply id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.p1.3.m3.1.1.2">𝑏</ci><ci id="S4.SS1.p1.3.m3.1.1.3.cmml" xref="S4.SS1.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> of the correct type. In addition, the task planner must select the feasible grasp pose, or a composite motion of multiple poses, to perform necessary manipulation skills for each <math alttext="a_{j}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><msub id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mi id="S4.SS1.p1.4.m4.1.1.2" xref="S4.SS1.p1.4.m4.1.1.2.cmml">a</mi><mi id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2">𝑎</ci><ci id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.F3" title="In IV-A Task Planning ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 3</a> provides an overview of all the decisions made. These decisions directly affect the feasibility of motion planning, as task planning determines whether there are collision-free and deadlock-free paths. </div> <div class="ltx_para" id="S4.SS1.p2"> Task Definition We define a task <math alttext="\mathcal{T}^{i}" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><msup id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">𝒯</mi><mi id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝒯</ci><ci id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\mathcal{T}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> as a piece of work that requires robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_i</annotation></semantics></math> to either transit to a goal pose or to perform some manipulation skill <math alttext="s" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">s</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝑠</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">s</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">italic_s</annotation></semantics></math> to achieve a goal constraint. For example, a task may involve a robot arm picking up an object, supporting a structure, receiving an object handed from another robot, retracting to its HOME pose, etc. In the context of LEGO assembly, each assembly task <math alttext="a_{j}" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><msub id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml"><mi id="S4.SS1.p2.4.m4.1.1.2" xref="S4.SS1.p2.4.m4.1.1.2.cmml">a</mi><mi id="S4.SS1.p2.4.m4.1.1.3" xref="S4.SS1.p2.4.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p2.4.m4.1.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2">𝑎</ci><ci id="S4.SS1.p2.4.m4.1.1.3.cmml" xref="S4.SS1.p2.4.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is broken down into a sequence of tasks. As shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a> (a) and (b), a robot must first transit to the initial location of an object, pick the object, transit the object to the location for assembly, place down the object on the target structure, and finally transit back to the robot’s HOME. For an assembly step that requires collaborative assembly, the support robot is also assigned a sequence of tasks, such as transiting to the structure and supporting it in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a> (c) and (e). If a reorientation is required as illustrated in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a> (d), the support robot will be assigned a transit and a pick task to collect the object, and a handover task to the other robot so that the object can be placed up to the structure. </div> <div class="ltx_para" id="S4.SS1.p3"> Task Graph A task graph <math alttext="\mathcal{G}=(\mathcal{V},\mathcal{E})" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.2"><semantics id="S4.SS1.p3.1.m1.2a"><mrow id="S4.SS1.p3.1.m1.2.3" xref="S4.SS1.p3.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.2.3.2" xref="S4.SS1.p3.1.m1.2.3.2.cmml">𝒢</mi><mo id="S4.SS1.p3.1.m1.2.3.1" xref="S4.SS1.p3.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.p3.1.m1.2.3.3.2" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.p3.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">𝒱</mi><mo id="S4.SS1.p3.1.m1.2.3.3.2.2" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.2.2" xref="S4.SS1.p3.1.m1.2.2.cmml">ℰ</mi><mo id="S4.SS1.p3.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.2b"><apply id="S4.SS1.p3.1.m1.2.3.cmml" xref="S4.SS1.p3.1.m1.2.3"><eq id="S4.SS1.p3.1.m1.2.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3.1"></eq><ci id="S4.SS1.p3.1.m1.2.3.2.cmml" xref="S4.SS1.p3.1.m1.2.3.2">𝒢</ci><interval closure="open" id="S4.SS1.p3.1.m1.2.3.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3.3.2"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝒱</ci><ci id="S4.SS1.p3.1.m1.2.2.cmml" xref="S4.SS1.p3.1.m1.2.2">ℰ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.2c">\mathcal{G}=(\mathcal{V},\mathcal{E})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.2d">caligraphic_G = ( caligraphic_V , caligraphic_E )</annotation></semantics></math> is a direct acyclic graph that represents the ordered set of tasks for all robots to complete the final assembly. The task graph also implicitly represents the movement of manipulated objects, the evolving environment and collision scene, as well as the kinematic switches of the robots. A node is either a task node or an object node. A task node represents a task <math alttext="\mathcal{T}^{i}_{m}\in\mathcal{V}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mrow id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><msubsup id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.2.m2.1.1.2.2.2" xref="S4.SS1.p3.2.m2.1.1.2.2.2.cmml">𝒯</mi><mi id="S4.SS1.p3.2.m2.1.1.2.3" xref="S4.SS1.p3.2.m2.1.1.2.3.cmml">m</mi><mi id="S4.SS1.p3.2.m2.1.1.2.2.3" xref="S4.SS1.p3.2.m2.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS1.p3.2.m2.1.1.1" xref="S4.SS1.p3.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3.cmml">𝒱</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1"><in id="S4.SS1.p3.2.m2.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1.1"></in><apply id="S4.SS1.p3.2.m2.1.1.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.2.1.cmml" xref="S4.SS1.p3.2.m2.1.1.2">subscript</csymbol><apply id="S4.SS1.p3.2.m2.1.1.2.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.2.2.1.cmml" xref="S4.SS1.p3.2.m2.1.1.2">superscript</csymbol><ci id="S4.SS1.p3.2.m2.1.1.2.2.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2.2.2">𝒯</ci><ci id="S4.SS1.p3.2.m2.1.1.2.2.3.cmml" xref="S4.SS1.p3.2.m2.1.1.2.2.3">𝑖</ci></apply><ci id="S4.SS1.p3.2.m2.1.1.2.3.cmml" xref="S4.SS1.p3.2.m2.1.1.2.3">𝑚</ci></apply><ci id="S4.SS1.p3.2.m2.1.1.3.cmml" xref="S4.SS1.p3.2.m2.1.1.3">𝒱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">\mathcal{T}^{i}_{m}\in\mathcal{V}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ caligraphic_V</annotation></semantics></math> for robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_i</annotation></semantics></math>, where <math alttext="m" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><mi id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><ci id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">m</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_m</annotation></semantics></math> is the index. Each task node also contains a robot target pose for this task. An object node represents an object <math alttext="b_{k}" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><msub id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml"><mi id="S4.SS1.p3.5.m5.1.1.2" xref="S4.SS1.p3.5.m5.1.1.2.cmml">b</mi><mi id="S4.SS1.p3.5.m5.1.1.3" xref="S4.SS1.p3.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><apply id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.1.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.p3.5.m5.1.1.2.cmml" xref="S4.SS1.p3.5.m5.1.1.2">𝑏</ci><ci id="S4.SS1.p3.5.m5.1.1.3.cmml" xref="S4.SS1.p3.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and its poses. An edge from one task node to another in the graph, <math alttext="\mathcal{T}^{i}_{m}\rightarrow\mathcal{T}^{i^{\prime}}_{m^{\prime}}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><mrow id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml"><msubsup id="S4.SS1.p3.6.m6.1.1.2" xref="S4.SS1.p3.6.m6.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.6.m6.1.1.2.2.2" xref="S4.SS1.p3.6.m6.1.1.2.2.2.cmml">𝒯</mi><mi id="S4.SS1.p3.6.m6.1.1.2.3" xref="S4.SS1.p3.6.m6.1.1.2.3.cmml">m</mi><mi id="S4.SS1.p3.6.m6.1.1.2.2.3" xref="S4.SS1.p3.6.m6.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS1.p3.6.m6.1.1.1" stretchy="false" xref="S4.SS1.p3.6.m6.1.1.1.cmml">→</mo><msubsup id="S4.SS1.p3.6.m6.1.1.3" xref="S4.SS1.p3.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.6.m6.1.1.3.2.2" xref="S4.SS1.p3.6.m6.1.1.3.2.2.cmml">𝒯</mi><msup id="S4.SS1.p3.6.m6.1.1.3.3" xref="S4.SS1.p3.6.m6.1.1.3.3.cmml"><mi id="S4.SS1.p3.6.m6.1.1.3.3.2" 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id="S4.SS1.p3.6.m6.1.1.3.3.cmml" xref="S4.SS1.p3.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m6.1.1.3.3.1.cmml" xref="S4.SS1.p3.6.m6.1.1.3.3">superscript</csymbol><ci id="S4.SS1.p3.6.m6.1.1.3.3.2.cmml" xref="S4.SS1.p3.6.m6.1.1.3.3.2">𝑚</ci><ci id="S4.SS1.p3.6.m6.1.1.3.3.3.cmml" xref="S4.SS1.p3.6.m6.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">\mathcal{T}^{i}_{m}\rightarrow\mathcal{T}^{i^{\prime}}_{m^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT → caligraphic_T start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, represents a precedence constraint, that states a dependency where <math alttext="\mathcal{T}^{i}_{m}" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m7.1"><semantics id="S4.SS1.p3.7.m7.1a"><msubsup id="S4.SS1.p3.7.m7.1.1" xref="S4.SS1.p3.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.7.m7.1.1.2.2" xref="S4.SS1.p3.7.m7.1.1.2.2.cmml">𝒯</mi><mi id="S4.SS1.p3.7.m7.1.1.3" xref="S4.SS1.p3.7.m7.1.1.3.cmml">m</mi><mi id="S4.SS1.p3.7.m7.1.1.2.3" xref="S4.SS1.p3.7.m7.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m7.1b"><apply id="S4.SS1.p3.7.m7.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m7.1.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1">subscript</csymbol><apply id="S4.SS1.p3.7.m7.1.1.2.cmml" xref="S4.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m7.1.1.2.1.cmml" xref="S4.SS1.p3.7.m7.1.1">superscript</csymbol><ci id="S4.SS1.p3.7.m7.1.1.2.2.cmml" xref="S4.SS1.p3.7.m7.1.1.2.2">𝒯</ci><ci id="S4.SS1.p3.7.m7.1.1.2.3.cmml" xref="S4.SS1.p3.7.m7.1.1.2.3">𝑖</ci></apply><ci id="S4.SS1.p3.7.m7.1.1.3.cmml" xref="S4.SS1.p3.7.m7.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m7.1c">\mathcal{T}^{i}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m7.1d">caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> must finish before <math alttext="\mathcal{T}^{i^{\prime}}_{m^{\prime}}" class="ltx_Math" display="inline" id="S4.SS1.p3.8.m8.1"><semantics id="S4.SS1.p3.8.m8.1a"><msubsup id="S4.SS1.p3.8.m8.1.1" xref="S4.SS1.p3.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.8.m8.1.1.2.2" xref="S4.SS1.p3.8.m8.1.1.2.2.cmml">𝒯</mi><msup id="S4.SS1.p3.8.m8.1.1.3" xref="S4.SS1.p3.8.m8.1.1.3.cmml"><mi id="S4.SS1.p3.8.m8.1.1.3.2" xref="S4.SS1.p3.8.m8.1.1.3.2.cmml">m</mi><mo id="S4.SS1.p3.8.m8.1.1.3.3" xref="S4.SS1.p3.8.m8.1.1.3.3.cmml">′</mo></msup><msup id="S4.SS1.p3.8.m8.1.1.2.3" xref="S4.SS1.p3.8.m8.1.1.2.3.cmml"><mi id="S4.SS1.p3.8.m8.1.1.2.3.2" xref="S4.SS1.p3.8.m8.1.1.2.3.2.cmml">i</mi><mo id="S4.SS1.p3.8.m8.1.1.2.3.3" xref="S4.SS1.p3.8.m8.1.1.2.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.8.m8.1b"><apply id="S4.SS1.p3.8.m8.1.1.cmml" xref="S4.SS1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.1.1.1.cmml" xref="S4.SS1.p3.8.m8.1.1">subscript</csymbol><apply id="S4.SS1.p3.8.m8.1.1.2.cmml" xref="S4.SS1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.1.1.2.1.cmml" xref="S4.SS1.p3.8.m8.1.1">superscript</csymbol><ci id="S4.SS1.p3.8.m8.1.1.2.2.cmml" xref="S4.SS1.p3.8.m8.1.1.2.2">𝒯</ci><apply id="S4.SS1.p3.8.m8.1.1.2.3.cmml" xref="S4.SS1.p3.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.1.1.2.3.1.cmml" xref="S4.SS1.p3.8.m8.1.1.2.3">superscript</csymbol><ci id="S4.SS1.p3.8.m8.1.1.2.3.2.cmml" xref="S4.SS1.p3.8.m8.1.1.2.3.2">𝑖</ci><ci id="S4.SS1.p3.8.m8.1.1.2.3.3.cmml" xref="S4.SS1.p3.8.m8.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS1.p3.8.m8.1.1.3.cmml" xref="S4.SS1.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.8.m8.1.1.3.1.cmml" xref="S4.SS1.p3.8.m8.1.1.3">superscript</csymbol><ci id="S4.SS1.p3.8.m8.1.1.3.2.cmml" xref="S4.SS1.p3.8.m8.1.1.3.2">𝑚</ci><ci id="S4.SS1.p3.8.m8.1.1.3.3.cmml" xref="S4.SS1.p3.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.8.m8.1c">\mathcal{T}^{i^{\prime}}_{m^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.8.m8.1d">caligraphic_T start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> can start. An edge from an object node to a task node, <math alttext="b_{k}\rightarrow\mathcal{T}^{i}_{m}" class="ltx_Math" display="inline" id="S4.SS1.p3.9.m9.1"><semantics id="S4.SS1.p3.9.m9.1a"><mrow id="S4.SS1.p3.9.m9.1.1" xref="S4.SS1.p3.9.m9.1.1.cmml"><msub id="S4.SS1.p3.9.m9.1.1.2" xref="S4.SS1.p3.9.m9.1.1.2.cmml"><mi id="S4.SS1.p3.9.m9.1.1.2.2" xref="S4.SS1.p3.9.m9.1.1.2.2.cmml">b</mi><mi id="S4.SS1.p3.9.m9.1.1.2.3" xref="S4.SS1.p3.9.m9.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS1.p3.9.m9.1.1.1" stretchy="false" xref="S4.SS1.p3.9.m9.1.1.1.cmml">→</mo><msubsup id="S4.SS1.p3.9.m9.1.1.3" xref="S4.SS1.p3.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.9.m9.1.1.3.2.2" xref="S4.SS1.p3.9.m9.1.1.3.2.2.cmml">𝒯</mi><mi id="S4.SS1.p3.9.m9.1.1.3.3" xref="S4.SS1.p3.9.m9.1.1.3.3.cmml">m</mi><mi id="S4.SS1.p3.9.m9.1.1.3.2.3" xref="S4.SS1.p3.9.m9.1.1.3.2.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.9.m9.1b"><apply id="S4.SS1.p3.9.m9.1.1.cmml" xref="S4.SS1.p3.9.m9.1.1"><ci id="S4.SS1.p3.9.m9.1.1.1.cmml" xref="S4.SS1.p3.9.m9.1.1.1">→</ci><apply id="S4.SS1.p3.9.m9.1.1.2.cmml" xref="S4.SS1.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.9.m9.1.1.2.1.cmml" xref="S4.SS1.p3.9.m9.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.9.m9.1.1.2.2.cmml" xref="S4.SS1.p3.9.m9.1.1.2.2">𝑏</ci><ci id="S4.SS1.p3.9.m9.1.1.2.3.cmml" xref="S4.SS1.p3.9.m9.1.1.2.3">𝑘</ci></apply><apply id="S4.SS1.p3.9.m9.1.1.3.cmml" xref="S4.SS1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.9.m9.1.1.3.1.cmml" xref="S4.SS1.p3.9.m9.1.1.3">subscript</csymbol><apply id="S4.SS1.p3.9.m9.1.1.3.2.cmml" xref="S4.SS1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.9.m9.1.1.3.2.1.cmml" xref="S4.SS1.p3.9.m9.1.1.3">superscript</csymbol><ci id="S4.SS1.p3.9.m9.1.1.3.2.2.cmml" xref="S4.SS1.p3.9.m9.1.1.3.2.2">𝒯</ci><ci id="S4.SS1.p3.9.m9.1.1.3.2.3.cmml" xref="S4.SS1.p3.9.m9.1.1.3.2.3">𝑖</ci></apply><ci id="S4.SS1.p3.9.m9.1.1.3.3.cmml" xref="S4.SS1.p3.9.m9.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.9.m9.1c">b_{k}\rightarrow\mathcal{T}^{i}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.9.m9.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT → caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, means that the object <math alttext="b_{k}" class="ltx_Math" display="inline" id="S4.SS1.p3.10.m10.1"><semantics id="S4.SS1.p3.10.m10.1a"><msub id="S4.SS1.p3.10.m10.1.1" xref="S4.SS1.p3.10.m10.1.1.cmml"><mi id="S4.SS1.p3.10.m10.1.1.2" xref="S4.SS1.p3.10.m10.1.1.2.cmml">b</mi><mi id="S4.SS1.p3.10.m10.1.1.3" xref="S4.SS1.p3.10.m10.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.10.m10.1b"><apply id="S4.SS1.p3.10.m10.1.1.cmml" xref="S4.SS1.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.10.m10.1.1.1.cmml" xref="S4.SS1.p3.10.m10.1.1">subscript</csymbol><ci id="S4.SS1.p3.10.m10.1.1.2.cmml" xref="S4.SS1.p3.10.m10.1.1.2">𝑏</ci><ci id="S4.SS1.p3.10.m10.1.1.3.cmml" xref="S4.SS1.p3.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.10.m10.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.10.m10.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is kinematically attached to the robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.11.m11.1"><semantics id="S4.SS1.p3.11.m11.1a"><mi id="S4.SS1.p3.11.m11.1.1" xref="S4.SS1.p3.11.m11.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.11.m11.1b"><ci id="S4.SS1.p3.11.m11.1.1.cmml" xref="S4.SS1.p3.11.m11.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.11.m11.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.11.m11.1d">italic_i</annotation></semantics></math> at the beginning of this task. In contrast, an edge from a task node to an object node, <math alttext="\mathcal{T}^{i}_{m}\rightarrow b_{k}" class="ltx_Math" display="inline" id="S4.SS1.p3.12.m12.1"><semantics id="S4.SS1.p3.12.m12.1a"><mrow id="S4.SS1.p3.12.m12.1.1" xref="S4.SS1.p3.12.m12.1.1.cmml"><msubsup id="S4.SS1.p3.12.m12.1.1.2" xref="S4.SS1.p3.12.m12.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.12.m12.1.1.2.2.2" xref="S4.SS1.p3.12.m12.1.1.2.2.2.cmml">𝒯</mi><mi id="S4.SS1.p3.12.m12.1.1.2.3" xref="S4.SS1.p3.12.m12.1.1.2.3.cmml">m</mi><mi id="S4.SS1.p3.12.m12.1.1.2.2.3" xref="S4.SS1.p3.12.m12.1.1.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS1.p3.12.m12.1.1.1" stretchy="false" xref="S4.SS1.p3.12.m12.1.1.1.cmml">→</mo><msub id="S4.SS1.p3.12.m12.1.1.3" xref="S4.SS1.p3.12.m12.1.1.3.cmml"><mi id="S4.SS1.p3.12.m12.1.1.3.2" xref="S4.SS1.p3.12.m12.1.1.3.2.cmml">b</mi><mi id="S4.SS1.p3.12.m12.1.1.3.3" xref="S4.SS1.p3.12.m12.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.12.m12.1b"><apply id="S4.SS1.p3.12.m12.1.1.cmml" xref="S4.SS1.p3.12.m12.1.1"><ci id="S4.SS1.p3.12.m12.1.1.1.cmml" xref="S4.SS1.p3.12.m12.1.1.1">→</ci><apply id="S4.SS1.p3.12.m12.1.1.2.cmml" xref="S4.SS1.p3.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.12.m12.1.1.2.1.cmml" xref="S4.SS1.p3.12.m12.1.1.2">subscript</csymbol><apply id="S4.SS1.p3.12.m12.1.1.2.2.cmml" xref="S4.SS1.p3.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.12.m12.1.1.2.2.1.cmml" xref="S4.SS1.p3.12.m12.1.1.2">superscript</csymbol><ci id="S4.SS1.p3.12.m12.1.1.2.2.2.cmml" xref="S4.SS1.p3.12.m12.1.1.2.2.2">𝒯</ci><ci id="S4.SS1.p3.12.m12.1.1.2.2.3.cmml" xref="S4.SS1.p3.12.m12.1.1.2.2.3">𝑖</ci></apply><ci id="S4.SS1.p3.12.m12.1.1.2.3.cmml" xref="S4.SS1.p3.12.m12.1.1.2.3">𝑚</ci></apply><apply id="S4.SS1.p3.12.m12.1.1.3.cmml" xref="S4.SS1.p3.12.m12.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.12.m12.1.1.3.1.cmml" xref="S4.SS1.p3.12.m12.1.1.3">subscript</csymbol><ci id="S4.SS1.p3.12.m12.1.1.3.2.cmml" xref="S4.SS1.p3.12.m12.1.1.3.2">𝑏</ci><ci id="S4.SS1.p3.12.m12.1.1.3.3.cmml" xref="S4.SS1.p3.12.m12.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.12.m12.1c">\mathcal{T}^{i}_{m}\rightarrow b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.12.m12.1d">caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT → italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, represents that an object would be detached from robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.13.m13.1"><semantics id="S4.SS1.p3.13.m13.1a"><mi id="S4.SS1.p3.13.m13.1.1" xref="S4.SS1.p3.13.m13.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.13.m13.1b"><ci id="S4.SS1.p3.13.m13.1.1.cmml" xref="S4.SS1.p3.13.m13.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.13.m13.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.13.m13.1d">italic_i</annotation></semantics></math> after this task ends. Each robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.14.m14.1"><semantics id="S4.SS1.p3.14.m14.1a"><mi id="S4.SS1.p3.14.m14.1.1" xref="S4.SS1.p3.14.m14.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.14.m14.1b"><ci id="S4.SS1.p3.14.m14.1.1.cmml" xref="S4.SS1.p3.14.m14.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.14.m14.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.14.m14.1d">italic_i</annotation></semantics></math> has <math alttext="M^{i}" class="ltx_Math" display="inline" id="S4.SS1.p3.15.m15.1"><semantics id="S4.SS1.p3.15.m15.1a"><msup id="S4.SS1.p3.15.m15.1.1" xref="S4.SS1.p3.15.m15.1.1.cmml"><mi id="S4.SS1.p3.15.m15.1.1.2" xref="S4.SS1.p3.15.m15.1.1.2.cmml">M</mi><mi id="S4.SS1.p3.15.m15.1.1.3" xref="S4.SS1.p3.15.m15.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.15.m15.1b"><apply id="S4.SS1.p3.15.m15.1.1.cmml" xref="S4.SS1.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.15.m15.1.1.1.cmml" xref="S4.SS1.p3.15.m15.1.1">superscript</csymbol><ci id="S4.SS1.p3.15.m15.1.1.2.cmml" xref="S4.SS1.p3.15.m15.1.1.2">𝑀</ci><ci id="S4.SS1.p3.15.m15.1.1.3.cmml" xref="S4.SS1.p3.15.m15.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.15.m15.1c">M^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.15.m15.1d">italic_M start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> tasks. A task graph itself does not limit if the tasks must be executed sequentially, synchronously, or asynchronously. </div> <div class="ltx_para" id="S4.SS1.p4"> Approach The main idea of our approach is to find a turn-based, sequential task plan according to the assembly sequence <math alttext="A" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mi id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><ci id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">italic_A</annotation></semantics></math>, which is itself sequential. Only one robot is actively moving or executing skills at any time, while the other robot waits. Each robot returns to its HOME pose at the end of completing an assembly step. </div> <figure class="ltx_figure" id="S4.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="365" id="S4.F3.g1" src="x3.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 3: An overview of the task planning in APEX-MR. Given the assembly sequence <math alttext="A" class="ltx_Math" display="inline" id="S4.F3.3.1.m1.1"><semantics id="S4.F3.3.1.m1.1b"><mi id="S4.F3.3.1.m1.1.1" xref="S4.F3.3.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S4.F3.3.1.m1.1c"><ci id="S4.F3.3.1.m1.1.1.cmml" xref="S4.F3.3.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.3.1.m1.1d">A</annotation><annotation encoding="application/x-llamapun" id="S4.F3.3.1.m1.1e">italic_A</annotation></semantics></math>, each step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S4.F3.4.2.m2.1"><semantics id="S4.F3.4.2.m2.1b"><msub id="S4.F3.4.2.m2.1.1" xref="S4.F3.4.2.m2.1.1.cmml"><mi id="S4.F3.4.2.m2.1.1.2" xref="S4.F3.4.2.m2.1.1.2.cmml">a</mi><mi id="S4.F3.4.2.m2.1.1.3" xref="S4.F3.4.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.F3.4.2.m2.1c"><apply id="S4.F3.4.2.m2.1.1.cmml" xref="S4.F3.4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.F3.4.2.m2.1.1.1.cmml" xref="S4.F3.4.2.m2.1.1">subscript</csymbol><ci id="S4.F3.4.2.m2.1.1.2.cmml" xref="S4.F3.4.2.m2.1.1.2">𝑎</ci><ci id="S4.F3.4.2.m2.1.1.3.cmml" xref="S4.F3.4.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.4.2.m2.1d">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.F3.4.2.m2.1e">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is assigned a robot, an object, a feasible object grasp pose, and a supporting pose if necessary. </figcaption> </figure> <div class="ltx_para" id="S4.SS1.p5"> Algorithmically, an ILP jointly optimizes robot assignment, object assignment, and target robot poses. A set of binary decision variables assigns a robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><mi id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><ci id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">italic_i</annotation></semantics></math> to an assembly step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S4.SS1.p5.2.m2.1"><semantics id="S4.SS1.p5.2.m2.1a"><msub id="S4.SS1.p5.2.m2.1.1" xref="S4.SS1.p5.2.m2.1.1.cmml"><mi id="S4.SS1.p5.2.m2.1.1.2" xref="S4.SS1.p5.2.m2.1.1.2.cmml">a</mi><mi id="S4.SS1.p5.2.m2.1.1.3" xref="S4.SS1.p5.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.2.m2.1b"><apply id="S4.SS1.p5.2.m2.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.2.m2.1.1.1.cmml" xref="S4.SS1.p5.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p5.2.m2.1.1.2.cmml" xref="S4.SS1.p5.2.m2.1.1.2">𝑎</ci><ci id="S4.SS1.p5.2.m2.1.1.3.cmml" xref="S4.SS1.p5.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.2.m2.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> using the object <math alttext="b_{k}" class="ltx_Math" display="inline" id="S4.SS1.p5.3.m3.1"><semantics id="S4.SS1.p5.3.m3.1a"><msub id="S4.SS1.p5.3.m3.1.1" xref="S4.SS1.p5.3.m3.1.1.cmml"><mi id="S4.SS1.p5.3.m3.1.1.2" xref="S4.SS1.p5.3.m3.1.1.2.cmml">b</mi><mi id="S4.SS1.p5.3.m3.1.1.3" xref="S4.SS1.p5.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.3.m3.1b"><apply id="S4.SS1.p5.3.m3.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.3.m3.1.1.1.cmml" xref="S4.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.p5.3.m3.1.1.2.cmml" xref="S4.SS1.p5.3.m3.1.1.2">𝑏</ci><ci id="S4.SS1.p5.3.m3.1.1.3.cmml" xref="S4.SS1.p5.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.3.m3.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.3.m3.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and a feasible grasp pose. Another set of binary decision variables denotes the assignment of support robot and support poses. We precompute feasible robot poses for grasping all available objects at initial and assembled positions, and support poses if necessary. The cost for assigning a robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p5.4.m4.1"><semantics id="S4.SS1.p5.4.m4.1a"><mi id="S4.SS1.p5.4.m4.1.1" xref="S4.SS1.p5.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.4.m4.1b"><ci id="S4.SS1.p5.4.m4.1.1.cmml" xref="S4.SS1.p5.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.4.m4.1d">italic_i</annotation></semantics></math>, an object <math alttext="b_{k}" class="ltx_Math" display="inline" id="S4.SS1.p5.5.m5.1"><semantics id="S4.SS1.p5.5.m5.1a"><msub id="S4.SS1.p5.5.m5.1.1" xref="S4.SS1.p5.5.m5.1.1.cmml"><mi id="S4.SS1.p5.5.m5.1.1.2" xref="S4.SS1.p5.5.m5.1.1.2.cmml">b</mi><mi id="S4.SS1.p5.5.m5.1.1.3" xref="S4.SS1.p5.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.5.m5.1b"><apply id="S4.SS1.p5.5.m5.1.1.cmml" xref="S4.SS1.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.5.m5.1.1.1.cmml" xref="S4.SS1.p5.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.p5.5.m5.1.1.2.cmml" xref="S4.SS1.p5.5.m5.1.1.2">𝑏</ci><ci id="S4.SS1.p5.5.m5.1.1.3.cmml" xref="S4.SS1.p5.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.5.m5.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.5.m5.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, and a corresponding grasp pose to an assembly step <math alttext="a_{j}" class="ltx_Math" display="inline" id="S4.SS1.p5.6.m6.1"><semantics id="S4.SS1.p5.6.m6.1a"><msub id="S4.SS1.p5.6.m6.1.1" xref="S4.SS1.p5.6.m6.1.1.cmml"><mi id="S4.SS1.p5.6.m6.1.1.2" xref="S4.SS1.p5.6.m6.1.1.2.cmml">a</mi><mi id="S4.SS1.p5.6.m6.1.1.3" xref="S4.SS1.p5.6.m6.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.6.m6.1b"><apply id="S4.SS1.p5.6.m6.1.1.cmml" xref="S4.SS1.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.p5.6.m6.1.1.1.cmml" xref="S4.SS1.p5.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.p5.6.m6.1.1.2.cmml" xref="S4.SS1.p5.6.m6.1.1.2">𝑎</ci><ci id="S4.SS1.p5.6.m6.1.1.3.cmml" xref="S4.SS1.p5.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.6.m6.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.6.m6.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is estimated by the sum of transit distances necessary for this assembly step. The ILP finds the best set of assignments that minimizes the sum of costs to complete the assembly and an auxiliary term for load balancing while ensuring the object type and support robot requirements are met. More details of the ILP formulation are discussed in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.SS1" title="-A Details of Task Planning Formulation ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. -A</a>. </div> <div class="ltx_para" id="S4.SS1.p6"> Combined with the precomputed robot poses, the optimized assignment gives a complete set of robot, object, and grasp pose assignment in each step. We then construct a corresponding sequential task plan and task graph (e.g., <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> (a)) with all inter-robot task dependencies and object relationships. A sequence of tasks for the assigned robot is added for each assembly step, and two object nodes are added to the task graph and connected to the pick task and place task nodes to indicate their attachment and placement, respectively. If a collaborative assembly is required, a sequence of support tasks is added to the support robot to be completed first. Inter-robot task dependencies are added to the task graph to constrain that the support task must precede any place task, and the following task after support can only start after the place task finishes. 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alttext="a_{j}" class="ltx_Math" display="inline" id="S4.SS1.p6.4.m4.1"><semantics id="S4.SS1.p6.4.m4.1a"><msub id="S4.SS1.p6.4.m4.1.1" xref="S4.SS1.p6.4.m4.1.1.cmml"><mi id="S4.SS1.p6.4.m4.1.1.2" xref="S4.SS1.p6.4.m4.1.1.2.cmml">a</mi><mi id="S4.SS1.p6.4.m4.1.1.3" xref="S4.SS1.p6.4.m4.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p6.4.m4.1b"><apply id="S4.SS1.p6.4.m4.1.1.cmml" xref="S4.SS1.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.4.m4.1.1.1.cmml" xref="S4.SS1.p6.4.m4.1.1">subscript</csymbol><ci id="S4.SS1.p6.4.m4.1.1.2.cmml" xref="S4.SS1.p6.4.m4.1.1.2">𝑎</ci><ci id="S4.SS1.p6.4.m4.1.1.3.cmml" xref="S4.SS1.p6.4.m4.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.4.m4.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_para" id="S4.SS1.p7"> Compared to other MR-TAMP and assembly methods such as <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib5" title="">5</a>]</cite>, APEX-MR generates a sequential plan first. This has two advantages: (1) It is easy to reason about inter-robot collision for scheduling tasks and avoids expensive feasibility checks needed for parallel task execution; (2) The complexity of motion planning is significantly reduced since each task becomes a single-robot planning problem, which avoids solving a challenging and time-consuming multi-robot arm motion planning problem. It is worth noting that, while the sequential plan might seem inefficient for a multi-robot system, our TPG execution framework will post-process this plan to enable efficient parallel execution. </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> IV-B Motion Planning </h3> <div class="ltx_para" id="S4.SS2.p1"> Once the sequential task plan for each robot is determined, motion planning becomes straightforward. As illustrated in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> (b), APEX-MR iterates over each task and use a single-robot RRT-Connect algorithm to plan the trajectory for this task when other robots are waiting. This is feasible because all other robots would be at a nonblocking stationary pose, i.e., HOME. A reference trajectory is also generated for tasks executed by specific manipulation skills based on the grasp/support pose. </div> <div class="ltx_para" id="S4.SS2.p2"> The motion planner generates a planned trajectory for every robot and every task. 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id="S4.SS2.p2.6.m6.2.2.2.2.1.5.cmml" xref="S4.SS2.p2.6.m6.2.2.2.2.1.5">𝑟</ci><ci id="S4.SS2.p2.6.m6.2.2.2.2.1.6.cmml" xref="S4.SS2.p2.6.m6.2.2.2.2.1.6">𝑡</ci></apply><ci id="S4.SS2.p2.6.m6.1.1.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1.1.1">𝑚</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.2c">N^{i}_{start,m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.2d">italic_N start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="N^{i}_{end,m}" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.2"><semantics id="S4.SS2.p2.7.m7.2a"><msubsup id="S4.SS2.p2.7.m7.2.3" xref="S4.SS2.p2.7.m7.2.3.cmml"><mi id="S4.SS2.p2.7.m7.2.3.2.2" xref="S4.SS2.p2.7.m7.2.3.2.2.cmml">N</mi><mrow id="S4.SS2.p2.7.m7.2.2.2.2" xref="S4.SS2.p2.7.m7.2.2.2.3.cmml"><mrow id="S4.SS2.p2.7.m7.2.2.2.2.1" xref="S4.SS2.p2.7.m7.2.2.2.2.1.cmml"><mi id="S4.SS2.p2.7.m7.2.2.2.2.1.2" xref="S4.SS2.p2.7.m7.2.2.2.2.1.2.cmml">e</mi><mo id="S4.SS2.p2.7.m7.2.2.2.2.1.1" xref="S4.SS2.p2.7.m7.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.7.m7.2.2.2.2.1.3" xref="S4.SS2.p2.7.m7.2.2.2.2.1.3.cmml">n</mi><mo id="S4.SS2.p2.7.m7.2.2.2.2.1.1a" xref="S4.SS2.p2.7.m7.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.7.m7.2.2.2.2.1.4" xref="S4.SS2.p2.7.m7.2.2.2.2.1.4.cmml">d</mi></mrow><mo id="S4.SS2.p2.7.m7.2.2.2.2.2" xref="S4.SS2.p2.7.m7.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p2.7.m7.1.1.1.1" xref="S4.SS2.p2.7.m7.1.1.1.1.cmml">m</mi></mrow><mi id="S4.SS2.p2.7.m7.2.3.2.3" xref="S4.SS2.p2.7.m7.2.3.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.2b"><apply id="S4.SS2.p2.7.m7.2.3.cmml" xref="S4.SS2.p2.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.2.3.1.cmml" xref="S4.SS2.p2.7.m7.2.3">subscript</csymbol><apply id="S4.SS2.p2.7.m7.2.3.2.cmml" xref="S4.SS2.p2.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.2.3.2.1.cmml" xref="S4.SS2.p2.7.m7.2.3">superscript</csymbol><ci id="S4.SS2.p2.7.m7.2.3.2.2.cmml" xref="S4.SS2.p2.7.m7.2.3.2.2">𝑁</ci><ci id="S4.SS2.p2.7.m7.2.3.2.3.cmml" xref="S4.SS2.p2.7.m7.2.3.2.3">𝑖</ci></apply><list id="S4.SS2.p2.7.m7.2.2.2.3.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2"><apply id="S4.SS2.p2.7.m7.2.2.2.2.1.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2.1"><times id="S4.SS2.p2.7.m7.2.2.2.2.1.1.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2.1.1"></times><ci id="S4.SS2.p2.7.m7.2.2.2.2.1.2.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2.1.2">𝑒</ci><ci id="S4.SS2.p2.7.m7.2.2.2.2.1.3.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2.1.3">𝑛</ci><ci id="S4.SS2.p2.7.m7.2.2.2.2.1.4.cmml" xref="S4.SS2.p2.7.m7.2.2.2.2.1.4">𝑑</ci></apply><ci id="S4.SS2.p2.7.m7.1.1.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1.1.1">𝑚</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.2c">N^{i}_{end,m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.2d">italic_N start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e italic_n italic_d , italic_m end_POSTSUBSCRIPT</annotation></semantics></math> are the first and last index of the trajectory <math alttext="\tau^{i}_{m}" class="ltx_Math" display="inline" id="S4.SS2.p2.8.m8.1"><semantics id="S4.SS2.p2.8.m8.1a"><msubsup id="S4.SS2.p2.8.m8.1.1" xref="S4.SS2.p2.8.m8.1.1.cmml"><mi id="S4.SS2.p2.8.m8.1.1.2.2" xref="S4.SS2.p2.8.m8.1.1.2.2.cmml">τ</mi><mi id="S4.SS2.p2.8.m8.1.1.3" xref="S4.SS2.p2.8.m8.1.1.3.cmml">m</mi><mi id="S4.SS2.p2.8.m8.1.1.2.3" xref="S4.SS2.p2.8.m8.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.8.m8.1b"><apply id="S4.SS2.p2.8.m8.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.1.1.1.cmml" xref="S4.SS2.p2.8.m8.1.1">subscript</csymbol><apply id="S4.SS2.p2.8.m8.1.1.2.cmml" xref="S4.SS2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.1.1.2.1.cmml" xref="S4.SS2.p2.8.m8.1.1">superscript</csymbol><ci id="S4.SS2.p2.8.m8.1.1.2.2.cmml" xref="S4.SS2.p2.8.m8.1.1.2.2">𝜏</ci><ci id="S4.SS2.p2.8.m8.1.1.2.3.cmml" xref="S4.SS2.p2.8.m8.1.1.2.3">𝑖</ci></apply><ci id="S4.SS2.p2.8.m8.1.1.3.cmml" xref="S4.SS2.p2.8.m8.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.8.m8.1c">\tau^{i}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.8.m8.1d">italic_τ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="N^{i}_{start,1}=1" class="ltx_Math" display="inline" id="S4.SS2.p2.9.m9.2"><semantics id="S4.SS2.p2.9.m9.2a"><mrow id="S4.SS2.p2.9.m9.2.3" xref="S4.SS2.p2.9.m9.2.3.cmml"><msubsup id="S4.SS2.p2.9.m9.2.3.2" xref="S4.SS2.p2.9.m9.2.3.2.cmml"><mi id="S4.SS2.p2.9.m9.2.3.2.2.2" xref="S4.SS2.p2.9.m9.2.3.2.2.2.cmml">N</mi><mrow id="S4.SS2.p2.9.m9.2.2.2.2" xref="S4.SS2.p2.9.m9.2.2.2.3.cmml"><mrow id="S4.SS2.p2.9.m9.2.2.2.2.1" xref="S4.SS2.p2.9.m9.2.2.2.2.1.cmml"><mi id="S4.SS2.p2.9.m9.2.2.2.2.1.2" xref="S4.SS2.p2.9.m9.2.2.2.2.1.2.cmml">s</mi><mo id="S4.SS2.p2.9.m9.2.2.2.2.1.1" xref="S4.SS2.p2.9.m9.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.9.m9.2.2.2.2.1.3" xref="S4.SS2.p2.9.m9.2.2.2.2.1.3.cmml">t</mi><mo id="S4.SS2.p2.9.m9.2.2.2.2.1.1a" xref="S4.SS2.p2.9.m9.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.9.m9.2.2.2.2.1.4" xref="S4.SS2.p2.9.m9.2.2.2.2.1.4.cmml">a</mi><mo id="S4.SS2.p2.9.m9.2.2.2.2.1.1b" xref="S4.SS2.p2.9.m9.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.9.m9.2.2.2.2.1.5" xref="S4.SS2.p2.9.m9.2.2.2.2.1.5.cmml">r</mi><mo id="S4.SS2.p2.9.m9.2.2.2.2.1.1c" xref="S4.SS2.p2.9.m9.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.9.m9.2.2.2.2.1.6" xref="S4.SS2.p2.9.m9.2.2.2.2.1.6.cmml">t</mi></mrow><mo id="S4.SS2.p2.9.m9.2.2.2.2.2" xref="S4.SS2.p2.9.m9.2.2.2.3.cmml">,</mo><mn id="S4.SS2.p2.9.m9.1.1.1.1" xref="S4.SS2.p2.9.m9.1.1.1.1.cmml">1</mn></mrow><mi id="S4.SS2.p2.9.m9.2.3.2.2.3" xref="S4.SS2.p2.9.m9.2.3.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS2.p2.9.m9.2.3.1" xref="S4.SS2.p2.9.m9.2.3.1.cmml">=</mo><mn id="S4.SS2.p2.9.m9.2.3.3" xref="S4.SS2.p2.9.m9.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.9.m9.2b"><apply id="S4.SS2.p2.9.m9.2.3.cmml" xref="S4.SS2.p2.9.m9.2.3"><eq id="S4.SS2.p2.9.m9.2.3.1.cmml" xref="S4.SS2.p2.9.m9.2.3.1"></eq><apply id="S4.SS2.p2.9.m9.2.3.2.cmml" xref="S4.SS2.p2.9.m9.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.9.m9.2.3.2.1.cmml" xref="S4.SS2.p2.9.m9.2.3.2">subscript</csymbol><apply id="S4.SS2.p2.9.m9.2.3.2.2.cmml" xref="S4.SS2.p2.9.m9.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.9.m9.2.3.2.2.1.cmml" xref="S4.SS2.p2.9.m9.2.3.2">superscript</csymbol><ci id="S4.SS2.p2.9.m9.2.3.2.2.2.cmml" xref="S4.SS2.p2.9.m9.2.3.2.2.2">𝑁</ci><ci id="S4.SS2.p2.9.m9.2.3.2.2.3.cmml" xref="S4.SS2.p2.9.m9.2.3.2.2.3">𝑖</ci></apply><list id="S4.SS2.p2.9.m9.2.2.2.3.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2"><apply id="S4.SS2.p2.9.m9.2.2.2.2.1.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1"><times id="S4.SS2.p2.9.m9.2.2.2.2.1.1.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1.1"></times><ci id="S4.SS2.p2.9.m9.2.2.2.2.1.2.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1.2">𝑠</ci><ci id="S4.SS2.p2.9.m9.2.2.2.2.1.3.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1.3">𝑡</ci><ci id="S4.SS2.p2.9.m9.2.2.2.2.1.4.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1.4">𝑎</ci><ci id="S4.SS2.p2.9.m9.2.2.2.2.1.5.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1.5">𝑟</ci><ci id="S4.SS2.p2.9.m9.2.2.2.2.1.6.cmml" xref="S4.SS2.p2.9.m9.2.2.2.2.1.6">𝑡</ci></apply><cn id="S4.SS2.p2.9.m9.1.1.1.1.cmml" type="integer" xref="S4.SS2.p2.9.m9.1.1.1.1">1</cn></list></apply><cn id="S4.SS2.p2.9.m9.2.3.3.cmml" type="integer" xref="S4.SS2.p2.9.m9.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.9.m9.2c">N^{i}_{start,1}=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.9.m9.2d">italic_N start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, and <math alttext="N^{i}_{start,m+1}=N^{i}_{end,m}+1" class="ltx_Math" display="inline" id="S4.SS2.p2.10.m10.4"><semantics id="S4.SS2.p2.10.m10.4a"><mrow id="S4.SS2.p2.10.m10.4.5" xref="S4.SS2.p2.10.m10.4.5.cmml"><msubsup id="S4.SS2.p2.10.m10.4.5.2" xref="S4.SS2.p2.10.m10.4.5.2.cmml"><mi id="S4.SS2.p2.10.m10.4.5.2.2.2" xref="S4.SS2.p2.10.m10.4.5.2.2.2.cmml">N</mi><mrow id="S4.SS2.p2.10.m10.2.2.2.2" xref="S4.SS2.p2.10.m10.2.2.2.3.cmml"><mrow id="S4.SS2.p2.10.m10.1.1.1.1.1" xref="S4.SS2.p2.10.m10.1.1.1.1.1.cmml"><mi id="S4.SS2.p2.10.m10.1.1.1.1.1.2" xref="S4.SS2.p2.10.m10.1.1.1.1.1.2.cmml">s</mi><mo id="S4.SS2.p2.10.m10.1.1.1.1.1.1" xref="S4.SS2.p2.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.10.m10.1.1.1.1.1.3" xref="S4.SS2.p2.10.m10.1.1.1.1.1.3.cmml">t</mi><mo id="S4.SS2.p2.10.m10.1.1.1.1.1.1a" xref="S4.SS2.p2.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.10.m10.1.1.1.1.1.4" xref="S4.SS2.p2.10.m10.1.1.1.1.1.4.cmml">a</mi><mo id="S4.SS2.p2.10.m10.1.1.1.1.1.1b" xref="S4.SS2.p2.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.10.m10.1.1.1.1.1.5" xref="S4.SS2.p2.10.m10.1.1.1.1.1.5.cmml">r</mi><mo id="S4.SS2.p2.10.m10.1.1.1.1.1.1c" xref="S4.SS2.p2.10.m10.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.10.m10.1.1.1.1.1.6" xref="S4.SS2.p2.10.m10.1.1.1.1.1.6.cmml">t</mi></mrow><mo id="S4.SS2.p2.10.m10.2.2.2.2.3" xref="S4.SS2.p2.10.m10.2.2.2.3.cmml">,</mo><mrow id="S4.SS2.p2.10.m10.2.2.2.2.2" xref="S4.SS2.p2.10.m10.2.2.2.2.2.cmml"><mi id="S4.SS2.p2.10.m10.2.2.2.2.2.2" xref="S4.SS2.p2.10.m10.2.2.2.2.2.2.cmml">m</mi><mo id="S4.SS2.p2.10.m10.2.2.2.2.2.1" xref="S4.SS2.p2.10.m10.2.2.2.2.2.1.cmml">+</mo><mn id="S4.SS2.p2.10.m10.2.2.2.2.2.3" xref="S4.SS2.p2.10.m10.2.2.2.2.2.3.cmml">1</mn></mrow></mrow><mi id="S4.SS2.p2.10.m10.4.5.2.2.3" xref="S4.SS2.p2.10.m10.4.5.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS2.p2.10.m10.4.5.1" xref="S4.SS2.p2.10.m10.4.5.1.cmml">=</mo><mrow id="S4.SS2.p2.10.m10.4.5.3" xref="S4.SS2.p2.10.m10.4.5.3.cmml"><msubsup id="S4.SS2.p2.10.m10.4.5.3.2" xref="S4.SS2.p2.10.m10.4.5.3.2.cmml"><mi id="S4.SS2.p2.10.m10.4.5.3.2.2.2" xref="S4.SS2.p2.10.m10.4.5.3.2.2.2.cmml">N</mi><mrow id="S4.SS2.p2.10.m10.4.4.2.2" xref="S4.SS2.p2.10.m10.4.4.2.3.cmml"><mrow id="S4.SS2.p2.10.m10.4.4.2.2.1" xref="S4.SS2.p2.10.m10.4.4.2.2.1.cmml"><mi id="S4.SS2.p2.10.m10.4.4.2.2.1.2" xref="S4.SS2.p2.10.m10.4.4.2.2.1.2.cmml">e</mi><mo id="S4.SS2.p2.10.m10.4.4.2.2.1.1" xref="S4.SS2.p2.10.m10.4.4.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.10.m10.4.4.2.2.1.3" xref="S4.SS2.p2.10.m10.4.4.2.2.1.3.cmml">n</mi><mo id="S4.SS2.p2.10.m10.4.4.2.2.1.1a" xref="S4.SS2.p2.10.m10.4.4.2.2.1.1.cmml">⁢</mo><mi id="S4.SS2.p2.10.m10.4.4.2.2.1.4" xref="S4.SS2.p2.10.m10.4.4.2.2.1.4.cmml">d</mi></mrow><mo id="S4.SS2.p2.10.m10.4.4.2.2.2" xref="S4.SS2.p2.10.m10.4.4.2.3.cmml">,</mo><mi id="S4.SS2.p2.10.m10.3.3.1.1" xref="S4.SS2.p2.10.m10.3.3.1.1.cmml">m</mi></mrow><mi id="S4.SS2.p2.10.m10.4.5.3.2.2.3" xref="S4.SS2.p2.10.m10.4.5.3.2.2.3.cmml">i</mi></msubsup><mo id="S4.SS2.p2.10.m10.4.5.3.1" xref="S4.SS2.p2.10.m10.4.5.3.1.cmml">+</mo><mn id="S4.SS2.p2.10.m10.4.5.3.3" xref="S4.SS2.p2.10.m10.4.5.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.10.m10.4b"><apply id="S4.SS2.p2.10.m10.4.5.cmml" xref="S4.SS2.p2.10.m10.4.5"><eq id="S4.SS2.p2.10.m10.4.5.1.cmml" xref="S4.SS2.p2.10.m10.4.5.1"></eq><apply id="S4.SS2.p2.10.m10.4.5.2.cmml" xref="S4.SS2.p2.10.m10.4.5.2"><csymbol cd="ambiguous" id="S4.SS2.p2.10.m10.4.5.2.1.cmml" xref="S4.SS2.p2.10.m10.4.5.2">subscript</csymbol><apply id="S4.SS2.p2.10.m10.4.5.2.2.cmml" xref="S4.SS2.p2.10.m10.4.5.2"><csymbol cd="ambiguous" id="S4.SS2.p2.10.m10.4.5.2.2.1.cmml" xref="S4.SS2.p2.10.m10.4.5.2">superscript</csymbol><ci id="S4.SS2.p2.10.m10.4.5.2.2.2.cmml" xref="S4.SS2.p2.10.m10.4.5.2.2.2">𝑁</ci><ci id="S4.SS2.p2.10.m10.4.5.2.2.3.cmml" xref="S4.SS2.p2.10.m10.4.5.2.2.3">𝑖</ci></apply><list id="S4.SS2.p2.10.m10.2.2.2.3.cmml" xref="S4.SS2.p2.10.m10.2.2.2.2"><apply id="S4.SS2.p2.10.m10.1.1.1.1.1.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1"><times id="S4.SS2.p2.10.m10.1.1.1.1.1.1.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1.1"></times><ci id="S4.SS2.p2.10.m10.1.1.1.1.1.2.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1.2">𝑠</ci><ci id="S4.SS2.p2.10.m10.1.1.1.1.1.3.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1.3">𝑡</ci><ci id="S4.SS2.p2.10.m10.1.1.1.1.1.4.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1.4">𝑎</ci><ci id="S4.SS2.p2.10.m10.1.1.1.1.1.5.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1.5">𝑟</ci><ci id="S4.SS2.p2.10.m10.1.1.1.1.1.6.cmml" xref="S4.SS2.p2.10.m10.1.1.1.1.1.6">𝑡</ci></apply><apply id="S4.SS2.p2.10.m10.2.2.2.2.2.cmml" xref="S4.SS2.p2.10.m10.2.2.2.2.2"><plus id="S4.SS2.p2.10.m10.2.2.2.2.2.1.cmml" xref="S4.SS2.p2.10.m10.2.2.2.2.2.1"></plus><ci id="S4.SS2.p2.10.m10.2.2.2.2.2.2.cmml" xref="S4.SS2.p2.10.m10.2.2.2.2.2.2">𝑚</ci><cn id="S4.SS2.p2.10.m10.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.p2.10.m10.2.2.2.2.2.3">1</cn></apply></list></apply><apply id="S4.SS2.p2.10.m10.4.5.3.cmml" xref="S4.SS2.p2.10.m10.4.5.3"><plus id="S4.SS2.p2.10.m10.4.5.3.1.cmml" xref="S4.SS2.p2.10.m10.4.5.3.1"></plus><apply id="S4.SS2.p2.10.m10.4.5.3.2.cmml" xref="S4.SS2.p2.10.m10.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.10.m10.4.5.3.2.1.cmml" xref="S4.SS2.p2.10.m10.4.5.3.2">subscript</csymbol><apply id="S4.SS2.p2.10.m10.4.5.3.2.2.cmml" xref="S4.SS2.p2.10.m10.4.5.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.10.m10.4.5.3.2.2.1.cmml" xref="S4.SS2.p2.10.m10.4.5.3.2">superscript</csymbol><ci id="S4.SS2.p2.10.m10.4.5.3.2.2.2.cmml" xref="S4.SS2.p2.10.m10.4.5.3.2.2.2">𝑁</ci><ci id="S4.SS2.p2.10.m10.4.5.3.2.2.3.cmml" xref="S4.SS2.p2.10.m10.4.5.3.2.2.3">𝑖</ci></apply><list id="S4.SS2.p2.10.m10.4.4.2.3.cmml" xref="S4.SS2.p2.10.m10.4.4.2.2"><apply id="S4.SS2.p2.10.m10.4.4.2.2.1.cmml" xref="S4.SS2.p2.10.m10.4.4.2.2.1"><times id="S4.SS2.p2.10.m10.4.4.2.2.1.1.cmml" xref="S4.SS2.p2.10.m10.4.4.2.2.1.1"></times><ci id="S4.SS2.p2.10.m10.4.4.2.2.1.2.cmml" xref="S4.SS2.p2.10.m10.4.4.2.2.1.2">𝑒</ci><ci id="S4.SS2.p2.10.m10.4.4.2.2.1.3.cmml" xref="S4.SS2.p2.10.m10.4.4.2.2.1.3">𝑛</ci><ci id="S4.SS2.p2.10.m10.4.4.2.2.1.4.cmml" xref="S4.SS2.p2.10.m10.4.4.2.2.1.4">𝑑</ci></apply><ci id="S4.SS2.p2.10.m10.3.3.1.1.cmml" xref="S4.SS2.p2.10.m10.3.3.1.1">𝑚</ci></list></apply><cn id="S4.SS2.p2.10.m10.4.5.3.3.cmml" type="integer" xref="S4.SS2.p2.10.m10.4.5.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.10.m10.4c">N^{i}_{start,m+1}=N^{i}_{end,m}+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.10.m10.4d">italic_N start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m + 1 end_POSTSUBSCRIPT = italic_N start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e italic_n italic_d , italic_m end_POSTSUBSCRIPT + 1</annotation></semantics></math> for <math alttext="m=1,\dots M-1" class="ltx_Math" display="inline" id="S4.SS2.p2.11.m11.2"><semantics id="S4.SS2.p2.11.m11.2a"><mrow id="S4.SS2.p2.11.m11.2.2" xref="S4.SS2.p2.11.m11.2.2.cmml"><mi id="S4.SS2.p2.11.m11.2.2.3" xref="S4.SS2.p2.11.m11.2.2.3.cmml">m</mi><mo id="S4.SS2.p2.11.m11.2.2.2" xref="S4.SS2.p2.11.m11.2.2.2.cmml">=</mo><mrow id="S4.SS2.p2.11.m11.2.2.1.1" xref="S4.SS2.p2.11.m11.2.2.1.2.cmml"><mn id="S4.SS2.p2.11.m11.1.1" xref="S4.SS2.p2.11.m11.1.1.cmml">1</mn><mo id="S4.SS2.p2.11.m11.2.2.1.1.2" xref="S4.SS2.p2.11.m11.2.2.1.2.cmml">,</mo><mrow id="S4.SS2.p2.11.m11.2.2.1.1.1" xref="S4.SS2.p2.11.m11.2.2.1.1.1.cmml"><mrow id="S4.SS2.p2.11.m11.2.2.1.1.1.2" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.cmml"><mi id="S4.SS2.p2.11.m11.2.2.1.1.1.2.2" mathvariant="normal" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.2.cmml">…</mi><mo id="S4.SS2.p2.11.m11.2.2.1.1.1.2.1" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.p2.11.m11.2.2.1.1.1.2.3" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.3.cmml">M</mi></mrow><mo id="S4.SS2.p2.11.m11.2.2.1.1.1.1" xref="S4.SS2.p2.11.m11.2.2.1.1.1.1.cmml">−</mo><mn id="S4.SS2.p2.11.m11.2.2.1.1.1.3" xref="S4.SS2.p2.11.m11.2.2.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.11.m11.2b"><apply id="S4.SS2.p2.11.m11.2.2.cmml" xref="S4.SS2.p2.11.m11.2.2"><eq id="S4.SS2.p2.11.m11.2.2.2.cmml" xref="S4.SS2.p2.11.m11.2.2.2"></eq><ci id="S4.SS2.p2.11.m11.2.2.3.cmml" xref="S4.SS2.p2.11.m11.2.2.3">𝑚</ci><list id="S4.SS2.p2.11.m11.2.2.1.2.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1"><cn id="S4.SS2.p2.11.m11.1.1.cmml" type="integer" xref="S4.SS2.p2.11.m11.1.1">1</cn><apply id="S4.SS2.p2.11.m11.2.2.1.1.1.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1.1"><minus id="S4.SS2.p2.11.m11.2.2.1.1.1.1.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1.1.1"></minus><apply id="S4.SS2.p2.11.m11.2.2.1.1.1.2.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2"><times id="S4.SS2.p2.11.m11.2.2.1.1.1.2.1.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.1"></times><ci id="S4.SS2.p2.11.m11.2.2.1.1.1.2.2.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.2">…</ci><ci id="S4.SS2.p2.11.m11.2.2.1.1.1.2.3.cmml" xref="S4.SS2.p2.11.m11.2.2.1.1.1.2.3">𝑀</ci></apply><cn id="S4.SS2.p2.11.m11.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS2.p2.11.m11.2.2.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.11.m11.2c">m=1,\dots M-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.11.m11.2d">italic_m = 1 , … italic_M - 1</annotation></semantics></math>. For each task, the planner first generates a sequence of poses <math alttext="{C^{i}_{n}}" class="ltx_Math" display="inline" id="S4.SS2.p2.12.m12.1"><semantics id="S4.SS2.p2.12.m12.1a"><msubsup id="S4.SS2.p2.12.m12.1.1" xref="S4.SS2.p2.12.m12.1.1.cmml"><mi id="S4.SS2.p2.12.m12.1.1.2.2" xref="S4.SS2.p2.12.m12.1.1.2.2.cmml">C</mi><mi id="S4.SS2.p2.12.m12.1.1.3" xref="S4.SS2.p2.12.m12.1.1.3.cmml">n</mi><mi id="S4.SS2.p2.12.m12.1.1.2.3" xref="S4.SS2.p2.12.m12.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.12.m12.1b"><apply id="S4.SS2.p2.12.m12.1.1.cmml" xref="S4.SS2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.12.m12.1.1.1.cmml" xref="S4.SS2.p2.12.m12.1.1">subscript</csymbol><apply id="S4.SS2.p2.12.m12.1.1.2.cmml" xref="S4.SS2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.12.m12.1.1.2.1.cmml" xref="S4.SS2.p2.12.m12.1.1">superscript</csymbol><ci id="S4.SS2.p2.12.m12.1.1.2.2.cmml" xref="S4.SS2.p2.12.m12.1.1.2.2">𝐶</ci><ci id="S4.SS2.p2.12.m12.1.1.2.3.cmml" xref="S4.SS2.p2.12.m12.1.1.2.3">𝑖</ci></apply><ci id="S4.SS2.p2.12.m12.1.1.3.cmml" xref="S4.SS2.p2.12.m12.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.12.m12.1c">{C^{i}_{n}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.12.m12.1d">italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> for the robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS2.p2.13.m13.1"><semantics id="S4.SS2.p2.13.m13.1a"><mi id="S4.SS2.p2.13.m13.1.1" xref="S4.SS2.p2.13.m13.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.13.m13.1b"><ci id="S4.SS2.p2.13.m13.1.1.cmml" xref="S4.SS2.p2.13.m13.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.13.m13.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.13.m13.1d">italic_i</annotation></semantics></math>, then determines corresponding timesteps. To ensure that each robot takes turns to complete its task according to the task sequences, the initial timestep of each task is equal to the last timestep of the previous task in the sequential plan. Then, the timestep <math alttext="t_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS2.p2.14.m14.1"><semantics id="S4.SS2.p2.14.m14.1a"><msubsup id="S4.SS2.p2.14.m14.1.1" xref="S4.SS2.p2.14.m14.1.1.cmml"><mi id="S4.SS2.p2.14.m14.1.1.2.2" xref="S4.SS2.p2.14.m14.1.1.2.2.cmml">t</mi><mi id="S4.SS2.p2.14.m14.1.1.2.3" xref="S4.SS2.p2.14.m14.1.1.2.3.cmml">n</mi><mi id="S4.SS2.p2.14.m14.1.1.3" xref="S4.SS2.p2.14.m14.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.14.m14.1b"><apply id="S4.SS2.p2.14.m14.1.1.cmml" xref="S4.SS2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.14.m14.1.1.1.cmml" xref="S4.SS2.p2.14.m14.1.1">superscript</csymbol><apply id="S4.SS2.p2.14.m14.1.1.2.cmml" xref="S4.SS2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.14.m14.1.1.2.1.cmml" xref="S4.SS2.p2.14.m14.1.1">subscript</csymbol><ci id="S4.SS2.p2.14.m14.1.1.2.2.cmml" xref="S4.SS2.p2.14.m14.1.1.2.2">𝑡</ci><ci id="S4.SS2.p2.14.m14.1.1.2.3.cmml" xref="S4.SS2.p2.14.m14.1.1.2.3">𝑛</ci></apply><ci id="S4.SS2.p2.14.m14.1.1.3.cmml" xref="S4.SS2.p2.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.14.m14.1c">t_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.14.m14.1d">italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> for each pose of the same task is set based on the maximum robot velocity, i.e., <math alttext="t^{i}_{n}=t^{i}_{n-1}+d(C^{i}_{n},C^{i}_{n-1})/v_{max}" class="ltx_Math" display="inline" id="S4.SS2.p2.15.m15.2"><semantics id="S4.SS2.p2.15.m15.2a"><mrow id="S4.SS2.p2.15.m15.2.2" xref="S4.SS2.p2.15.m15.2.2.cmml"><msubsup id="S4.SS2.p2.15.m15.2.2.4" xref="S4.SS2.p2.15.m15.2.2.4.cmml"><mi id="S4.SS2.p2.15.m15.2.2.4.2.2" xref="S4.SS2.p2.15.m15.2.2.4.2.2.cmml">t</mi><mi id="S4.SS2.p2.15.m15.2.2.4.3" xref="S4.SS2.p2.15.m15.2.2.4.3.cmml">n</mi><mi 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start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_t start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT + italic_d ( italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_C start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) / italic_v start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT</annotation></semantics></math>. Since RRT-Connect may produce jerky and long trajectory, we use a randomized shortcutting algorithm to smooth suboptimal trajectories. The output of motion planning should be a sequence of trajectories for each task, i.e., <math alttext="\tau^{i}_{m},\;\;\forall i\in N,\;\forall m\in M^{i}" class="ltx_Math" display="inline" id="S4.SS2.p2.16.m16.2"><semantics id="S4.SS2.p2.16.m16.2a"><mrow id="S4.SS2.p2.16.m16.2.2.2" xref="S4.SS2.p2.16.m16.2.2.3.cmml"><mrow id="S4.SS2.p2.16.m16.1.1.1.1" xref="S4.SS2.p2.16.m16.1.1.1.1.cmml"><mrow id="S4.SS2.p2.16.m16.1.1.1.1.2.2" xref="S4.SS2.p2.16.m16.1.1.1.1.2.3.cmml"><msubsup id="S4.SS2.p2.16.m16.1.1.1.1.1.1.1" xref="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.2.2" xref="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.2.2.cmml">τ</mi><mi id="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.3" xref="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.3.cmml">m</mi><mi id="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.2.3" xref="S4.SS2.p2.16.m16.1.1.1.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS2.p2.16.m16.1.1.1.1.2.2.3" rspace="0.727em" xref="S4.SS2.p2.16.m16.1.1.1.1.2.3.cmml">,</mo><mrow 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italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , ∀ italic_i ∈ italic_N , ∀ italic_m ∈ italic_M start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> IV-C Asynchronous Execution </h3> <div class="ltx_para" id="S4.SS3.p1"> From the sequential task and motion plan, APEX-MR converts it to a TPG to improve the quality of the plan and support asynchronous execution. Importantly, the process to construct a TPG is not limited to a sequential plan and also applies to synchronous plans commonly seen in MR-TAMP works such as <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib42" title="">42</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib49" title="">49</a>]</cite>. </div> <div class="ltx_para" id="S4.SS3.p2"> TPG Definition APEX-MR uses a multi-modal temporal plan graph (TPG) to represent an execution schedule for the team of robot manipulators. As illustrated in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S2.F2" title="In II Related Works ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 2</a> (c), multi-modal TPG is a directed acyclic graph <math alttext="G=(V,E)" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.2"><semantics id="S4.SS3.p2.1.m1.2a"><mrow id="S4.SS3.p2.1.m1.2.3" xref="S4.SS3.p2.1.m1.2.3.cmml"><mi id="S4.SS3.p2.1.m1.2.3.2" xref="S4.SS3.p2.1.m1.2.3.2.cmml">G</mi><mo id="S4.SS3.p2.1.m1.2.3.1" xref="S4.SS3.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS3.p2.1.m1.2.3.3.2" xref="S4.SS3.p2.1.m1.2.3.3.1.cmml"><mo id="S4.SS3.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS3.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">V</mi><mo id="S4.SS3.p2.1.m1.2.3.3.2.2" xref="S4.SS3.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS3.p2.1.m1.2.2" xref="S4.SS3.p2.1.m1.2.2.cmml">E</mi><mo id="S4.SS3.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS3.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.2b"><apply id="S4.SS3.p2.1.m1.2.3.cmml" xref="S4.SS3.p2.1.m1.2.3"><eq id="S4.SS3.p2.1.m1.2.3.1.cmml" xref="S4.SS3.p2.1.m1.2.3.1"></eq><ci id="S4.SS3.p2.1.m1.2.3.2.cmml" xref="S4.SS3.p2.1.m1.2.3.2">𝐺</ci><interval closure="open" id="S4.SS3.p2.1.m1.2.3.3.1.cmml" xref="S4.SS3.p2.1.m1.2.3.3.2"><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝑉</ci><ci id="S4.SS3.p2.1.m1.2.2.cmml" xref="S4.SS3.p2.1.m1.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.2c">G=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.2d">italic_G = ( italic_V , italic_E )</annotation></semantics></math> with two types of node. A pose node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><msubsup id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml"><mi id="S4.SS3.p2.2.m2.1.1.2.2" xref="S4.SS3.p2.2.m2.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p2.2.m2.1.1.2.3" xref="S4.SS3.p2.2.m2.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p2.2.m2.1.1.3" xref="S4.SS3.p2.2.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><apply id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">superscript</csymbol><apply id="S4.SS3.p2.2.m2.1.1.2.cmml" xref="S4.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.2.m2.1.1.2.1.cmml" xref="S4.SS3.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.p2.2.m2.1.1.2.2.cmml" xref="S4.SS3.p2.2.m2.1.1.2.2">𝑣</ci><ci id="S4.SS3.p2.2.m2.1.1.2.3.cmml" xref="S4.SS3.p2.2.m2.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.2.m2.1.1.3.cmml" xref="S4.SS3.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> corresponds to a configuration <math alttext="C_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><msubsup id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml"><mi id="S4.SS3.p2.3.m3.1.1.2.2" xref="S4.SS3.p2.3.m3.1.1.2.2.cmml">C</mi><mi id="S4.SS3.p2.3.m3.1.1.2.3" xref="S4.SS3.p2.3.m3.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p2.3.m3.1.1.3" xref="S4.SS3.p2.3.m3.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><apply id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m3.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1">superscript</csymbol><apply id="S4.SS3.p2.3.m3.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m3.1.1.2.1.cmml" xref="S4.SS3.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.p2.3.m3.1.1.2.2.cmml" xref="S4.SS3.p2.3.m3.1.1.2.2">𝐶</ci><ci id="S4.SS3.p2.3.m3.1.1.2.3.cmml" xref="S4.SS3.p2.3.m3.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.3.m3.1.1.3.cmml" xref="S4.SS3.p2.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">C_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> on the trajectory of a transit task for robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m4.1"><semantics id="S4.SS3.p2.4.m4.1a"><mi id="S4.SS3.p2.4.m4.1.1" xref="S4.SS3.p2.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m4.1b"><ci id="S4.SS3.p2.4.m4.1.1.cmml" xref="S4.SS3.p2.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m4.1d">italic_i</annotation></semantics></math>. A skill node <math alttext="\tilde{v}^{i}_{n}" class="ltx_Math" display="inline" id="S4.SS3.p2.5.m5.1"><semantics id="S4.SS3.p2.5.m5.1a"><msubsup id="S4.SS3.p2.5.m5.1.1" xref="S4.SS3.p2.5.m5.1.1.cmml"><mover accent="true" id="S4.SS3.p2.5.m5.1.1.2.2" xref="S4.SS3.p2.5.m5.1.1.2.2.cmml"><mi id="S4.SS3.p2.5.m5.1.1.2.2.2" xref="S4.SS3.p2.5.m5.1.1.2.2.2.cmml">v</mi><mo id="S4.SS3.p2.5.m5.1.1.2.2.1" xref="S4.SS3.p2.5.m5.1.1.2.2.1.cmml">~</mo></mover><mi id="S4.SS3.p2.5.m5.1.1.3" xref="S4.SS3.p2.5.m5.1.1.3.cmml">n</mi><mi id="S4.SS3.p2.5.m5.1.1.2.3" xref="S4.SS3.p2.5.m5.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.5.m5.1b"><apply id="S4.SS3.p2.5.m5.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.5.m5.1.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1">subscript</csymbol><apply id="S4.SS3.p2.5.m5.1.1.2.cmml" xref="S4.SS3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.5.m5.1.1.2.1.cmml" xref="S4.SS3.p2.5.m5.1.1">superscript</csymbol><apply id="S4.SS3.p2.5.m5.1.1.2.2.cmml" xref="S4.SS3.p2.5.m5.1.1.2.2"><ci id="S4.SS3.p2.5.m5.1.1.2.2.1.cmml" xref="S4.SS3.p2.5.m5.1.1.2.2.1">~</ci><ci id="S4.SS3.p2.5.m5.1.1.2.2.2.cmml" xref="S4.SS3.p2.5.m5.1.1.2.2.2">𝑣</ci></apply><ci id="S4.SS3.p2.5.m5.1.1.2.3.cmml" xref="S4.SS3.p2.5.m5.1.1.2.3">𝑖</ci></apply><ci id="S4.SS3.p2.5.m5.1.1.3.cmml" xref="S4.SS3.p2.5.m5.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.5.m5.1c">\tilde{v}^{i}_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.5.m5.1d">over~ start_ARG italic_v end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> represents a manipulation skill (e.g., pick, handover, support, etc) that will be executed by some robot controller or policy. The skill node also contains a reference trajectory generated in motion planning. A type-1 edge <math alttext="(v_{n}^{i}\rightarrow v_{n+1}^{i})" class="ltx_Math" display="inline" id="S4.SS3.p2.6.m6.1"><semantics id="S4.SS3.p2.6.m6.1a"><mrow id="S4.SS3.p2.6.m6.1.1.1" xref="S4.SS3.p2.6.m6.1.1.1.1.cmml"><mo id="S4.SS3.p2.6.m6.1.1.1.2" stretchy="false" xref="S4.SS3.p2.6.m6.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p2.6.m6.1.1.1.1" xref="S4.SS3.p2.6.m6.1.1.1.1.cmml"><msubsup id="S4.SS3.p2.6.m6.1.1.1.1.2" xref="S4.SS3.p2.6.m6.1.1.1.1.2.cmml"><mi id="S4.SS3.p2.6.m6.1.1.1.1.2.2.2" xref="S4.SS3.p2.6.m6.1.1.1.1.2.2.2.cmml">v</mi><mi id="S4.SS3.p2.6.m6.1.1.1.1.2.2.3" xref="S4.SS3.p2.6.m6.1.1.1.1.2.2.3.cmml">n</mi><mi id="S4.SS3.p2.6.m6.1.1.1.1.2.3" xref="S4.SS3.p2.6.m6.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p2.6.m6.1.1.1.1.1" stretchy="false" xref="S4.SS3.p2.6.m6.1.1.1.1.1.cmml">→</mo><msubsup id="S4.SS3.p2.6.m6.1.1.1.1.3" xref="S4.SS3.p2.6.m6.1.1.1.1.3.cmml"><mi id="S4.SS3.p2.6.m6.1.1.1.1.3.2.2" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.2.cmml">v</mi><mrow id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.cmml"><mi id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.2" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.2.cmml">n</mi><mo id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.1" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.1.cmml">+</mo><mn id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.3" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.3.cmml">1</mn></mrow><mi id="S4.SS3.p2.6.m6.1.1.1.1.3.3" xref="S4.SS3.p2.6.m6.1.1.1.1.3.3.cmml">i</mi></msubsup></mrow><mo id="S4.SS3.p2.6.m6.1.1.1.3" stretchy="false" xref="S4.SS3.p2.6.m6.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.6.m6.1b"><apply id="S4.SS3.p2.6.m6.1.1.1.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1"><ci id="S4.SS3.p2.6.m6.1.1.1.1.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.1">→</ci><apply id="S4.SS3.p2.6.m6.1.1.1.1.2.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.6.m6.1.1.1.1.2.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2">superscript</csymbol><apply id="S4.SS3.p2.6.m6.1.1.1.1.2.2.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.6.m6.1.1.1.1.2.2.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2">subscript</csymbol><ci id="S4.SS3.p2.6.m6.1.1.1.1.2.2.2.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2.2.2">𝑣</ci><ci id="S4.SS3.p2.6.m6.1.1.1.1.2.2.3.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.6.m6.1.1.1.1.2.3.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p2.6.m6.1.1.1.1.3.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.6.m6.1.1.1.1.3.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3">superscript</csymbol><apply id="S4.SS3.p2.6.m6.1.1.1.1.3.2.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.6.m6.1.1.1.1.3.2.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.6.m6.1.1.1.1.3.2.2.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.2">𝑣</ci><apply id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3"><plus id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.1.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.1"></plus><ci id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.2.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.2">𝑛</ci><cn id="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S4.SS3.p2.6.m6.1.1.1.1.3.2.3.3">1</cn></apply></apply><ci id="S4.SS3.p2.6.m6.1.1.1.1.3.3.cmml" xref="S4.SS3.p2.6.m6.1.1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.6.m6.1c">(v_{n}^{i}\rightarrow v_{n+1}^{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.6.m6.1d">( italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → italic_v start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT )</annotation></semantics></math> connects two nodes from the same robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p2.7.m7.1"><semantics id="S4.SS3.p2.7.m7.1a"><mi id="S4.SS3.p2.7.m7.1.1" xref="S4.SS3.p2.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.7.m7.1b"><ci id="S4.SS3.p2.7.m7.1.1.cmml" xref="S4.SS3.p2.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.7.m7.1d">italic_i</annotation></semantics></math> and indicates the order between two nodes. A type-2 edge <math alttext="(v_{n}^{i}\rightarrow v_{n^{\prime}}^{i^{\prime}})" class="ltx_Math" display="inline" id="S4.SS3.p2.8.m8.1"><semantics id="S4.SS3.p2.8.m8.1a"><mrow id="S4.SS3.p2.8.m8.1.1.1" xref="S4.SS3.p2.8.m8.1.1.1.1.cmml"><mo id="S4.SS3.p2.8.m8.1.1.1.2" stretchy="false" xref="S4.SS3.p2.8.m8.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p2.8.m8.1.1.1.1" xref="S4.SS3.p2.8.m8.1.1.1.1.cmml"><msubsup id="S4.SS3.p2.8.m8.1.1.1.1.2" xref="S4.SS3.p2.8.m8.1.1.1.1.2.cmml"><mi id="S4.SS3.p2.8.m8.1.1.1.1.2.2.2" xref="S4.SS3.p2.8.m8.1.1.1.1.2.2.2.cmml">v</mi><mi id="S4.SS3.p2.8.m8.1.1.1.1.2.2.3" xref="S4.SS3.p2.8.m8.1.1.1.1.2.2.3.cmml">n</mi><mi id="S4.SS3.p2.8.m8.1.1.1.1.2.3" xref="S4.SS3.p2.8.m8.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p2.8.m8.1.1.1.1.1" stretchy="false" xref="S4.SS3.p2.8.m8.1.1.1.1.1.cmml">→</mo><msubsup id="S4.SS3.p2.8.m8.1.1.1.1.3" xref="S4.SS3.p2.8.m8.1.1.1.1.3.cmml"><mi id="S4.SS3.p2.8.m8.1.1.1.1.3.2.2" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.2.cmml">v</mi><msup id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.cmml"><mi id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.2" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.2.cmml">n</mi><mo id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.3" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p2.8.m8.1.1.1.1.3.3" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3.cmml"><mi id="S4.SS3.p2.8.m8.1.1.1.1.3.3.2" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3.2.cmml">i</mi><mo id="S4.SS3.p2.8.m8.1.1.1.1.3.3.3" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3.3.cmml">′</mo></msup></msubsup></mrow><mo id="S4.SS3.p2.8.m8.1.1.1.3" stretchy="false" xref="S4.SS3.p2.8.m8.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.8.m8.1b"><apply id="S4.SS3.p2.8.m8.1.1.1.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1"><ci id="S4.SS3.p2.8.m8.1.1.1.1.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.1">→</ci><apply id="S4.SS3.p2.8.m8.1.1.1.1.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.8.m8.1.1.1.1.2.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2">superscript</csymbol><apply id="S4.SS3.p2.8.m8.1.1.1.1.2.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p2.8.m8.1.1.1.1.2.2.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2">subscript</csymbol><ci id="S4.SS3.p2.8.m8.1.1.1.1.2.2.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2.2.2">𝑣</ci><ci id="S4.SS3.p2.8.m8.1.1.1.1.2.2.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.8.m8.1.1.1.1.2.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p2.8.m8.1.1.1.1.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.8.m8.1.1.1.1.3.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3">superscript</csymbol><apply id="S4.SS3.p2.8.m8.1.1.1.1.3.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.8.m8.1.1.1.1.3.2.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.8.m8.1.1.1.1.3.2.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.2">𝑣</ci><apply id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3">superscript</csymbol><ci id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.2">𝑛</ci><ci id="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p2.8.m8.1.1.1.1.3.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p2.8.m8.1.1.1.1.3.3.1.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3">superscript</csymbol><ci id="S4.SS3.p2.8.m8.1.1.1.1.3.3.2.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3.2">𝑖</ci><ci id="S4.SS3.p2.8.m8.1.1.1.1.3.3.3.cmml" xref="S4.SS3.p2.8.m8.1.1.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.8.m8.1c">(v_{n}^{i}\rightarrow v_{n^{\prime}}^{i^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.8.m8.1d">( italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT )</annotation></semantics></math> represents an inter-robot precedence order that constrains the robot <math alttext="i^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p2.9.m9.1"><semantics id="S4.SS3.p2.9.m9.1a"><msup id="S4.SS3.p2.9.m9.1.1" xref="S4.SS3.p2.9.m9.1.1.cmml"><mi id="S4.SS3.p2.9.m9.1.1.2" xref="S4.SS3.p2.9.m9.1.1.2.cmml">i</mi><mo id="S4.SS3.p2.9.m9.1.1.3" xref="S4.SS3.p2.9.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.9.m9.1b"><apply id="S4.SS3.p2.9.m9.1.1.cmml" xref="S4.SS3.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.9.m9.1.1.1.cmml" xref="S4.SS3.p2.9.m9.1.1">superscript</csymbol><ci id="S4.SS3.p2.9.m9.1.1.2.cmml" xref="S4.SS3.p2.9.m9.1.1.2">𝑖</ci><ci id="S4.SS3.p2.9.m9.1.1.3.cmml" xref="S4.SS3.p2.9.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.9.m9.1c">i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.9.m9.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to wait for robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p2.10.m10.1"><semantics id="S4.SS3.p2.10.m10.1a"><mi id="S4.SS3.p2.10.m10.1.1" xref="S4.SS3.p2.10.m10.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.10.m10.1b"><ci id="S4.SS3.p2.10.m10.1.1.cmml" xref="S4.SS3.p2.10.m10.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.10.m10.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.10.m10.1d">italic_i</annotation></semantics></math> to reach <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.11.m11.1"><semantics id="S4.SS3.p2.11.m11.1a"><msubsup id="S4.SS3.p2.11.m11.1.1" xref="S4.SS3.p2.11.m11.1.1.cmml"><mi id="S4.SS3.p2.11.m11.1.1.2.2" xref="S4.SS3.p2.11.m11.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p2.11.m11.1.1.2.3" xref="S4.SS3.p2.11.m11.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p2.11.m11.1.1.3" xref="S4.SS3.p2.11.m11.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.11.m11.1b"><apply id="S4.SS3.p2.11.m11.1.1.cmml" xref="S4.SS3.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.11.m11.1.1.1.cmml" xref="S4.SS3.p2.11.m11.1.1">superscript</csymbol><apply id="S4.SS3.p2.11.m11.1.1.2.cmml" xref="S4.SS3.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.11.m11.1.1.2.1.cmml" xref="S4.SS3.p2.11.m11.1.1">subscript</csymbol><ci id="S4.SS3.p2.11.m11.1.1.2.2.cmml" xref="S4.SS3.p2.11.m11.1.1.2.2">𝑣</ci><ci id="S4.SS3.p2.11.m11.1.1.2.3.cmml" xref="S4.SS3.p2.11.m11.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.11.m11.1.1.3.cmml" xref="S4.SS3.p2.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.11.m11.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.11.m11.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> (i.e., reach the pose <math alttext="C_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.12.m12.1"><semantics id="S4.SS3.p2.12.m12.1a"><msubsup id="S4.SS3.p2.12.m12.1.1" xref="S4.SS3.p2.12.m12.1.1.cmml"><mi id="S4.SS3.p2.12.m12.1.1.2.2" xref="S4.SS3.p2.12.m12.1.1.2.2.cmml">C</mi><mi id="S4.SS3.p2.12.m12.1.1.2.3" xref="S4.SS3.p2.12.m12.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p2.12.m12.1.1.3" xref="S4.SS3.p2.12.m12.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.12.m12.1b"><apply id="S4.SS3.p2.12.m12.1.1.cmml" xref="S4.SS3.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.12.m12.1.1.1.cmml" xref="S4.SS3.p2.12.m12.1.1">superscript</csymbol><apply id="S4.SS3.p2.12.m12.1.1.2.cmml" xref="S4.SS3.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.12.m12.1.1.2.1.cmml" xref="S4.SS3.p2.12.m12.1.1">subscript</csymbol><ci id="S4.SS3.p2.12.m12.1.1.2.2.cmml" xref="S4.SS3.p2.12.m12.1.1.2.2">𝐶</ci><ci id="S4.SS3.p2.12.m12.1.1.2.3.cmml" xref="S4.SS3.p2.12.m12.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.12.m12.1.1.3.cmml" xref="S4.SS3.p2.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.12.m12.1c">C_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.12.m12.1d">italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> or complete the manipulation skill) before moving to <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p2.13.m13.1"><semantics id="S4.SS3.p2.13.m13.1a"><msubsup id="S4.SS3.p2.13.m13.1.1" xref="S4.SS3.p2.13.m13.1.1.cmml"><mi id="S4.SS3.p2.13.m13.1.1.2.2" xref="S4.SS3.p2.13.m13.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p2.13.m13.1.1.2.3" xref="S4.SS3.p2.13.m13.1.1.2.3.cmml"><mi id="S4.SS3.p2.13.m13.1.1.2.3.2" xref="S4.SS3.p2.13.m13.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p2.13.m13.1.1.2.3.3" xref="S4.SS3.p2.13.m13.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p2.13.m13.1.1.3" xref="S4.SS3.p2.13.m13.1.1.3.cmml"><mi id="S4.SS3.p2.13.m13.1.1.3.2" xref="S4.SS3.p2.13.m13.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p2.13.m13.1.1.3.3" xref="S4.SS3.p2.13.m13.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.13.m13.1b"><apply id="S4.SS3.p2.13.m13.1.1.cmml" xref="S4.SS3.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.13.m13.1.1.1.cmml" xref="S4.SS3.p2.13.m13.1.1">superscript</csymbol><apply id="S4.SS3.p2.13.m13.1.1.2.cmml" xref="S4.SS3.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.13.m13.1.1.2.1.cmml" xref="S4.SS3.p2.13.m13.1.1">subscript</csymbol><ci id="S4.SS3.p2.13.m13.1.1.2.2.cmml" xref="S4.SS3.p2.13.m13.1.1.2.2">𝑣</ci><apply id="S4.SS3.p2.13.m13.1.1.2.3.cmml" xref="S4.SS3.p2.13.m13.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p2.13.m13.1.1.2.3.1.cmml" xref="S4.SS3.p2.13.m13.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p2.13.m13.1.1.2.3.2.cmml" xref="S4.SS3.p2.13.m13.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p2.13.m13.1.1.2.3.3.cmml" xref="S4.SS3.p2.13.m13.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p2.13.m13.1.1.3.cmml" xref="S4.SS3.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.13.m13.1.1.3.1.cmml" xref="S4.SS3.p2.13.m13.1.1.3">superscript</csymbol><ci id="S4.SS3.p2.13.m13.1.1.3.2.cmml" xref="S4.SS3.p2.13.m13.1.1.3.2">𝑖</ci><ci id="S4.SS3.p2.13.m13.1.1.3.3.cmml" xref="S4.SS3.p2.13.m13.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.13.m13.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.13.m13.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. A type-2 edge can be added for both task dependencies and motion dependencies. In contrast to the TPG defined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib20" title="">20</a>]</cite>, a multi-modal TPG combines a TPG with a task graph and assigns a corresponding task <math alttext="\mathcal{T}_{m}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.14.m14.1"><semantics id="S4.SS3.p2.14.m14.1a"><msubsup id="S4.SS3.p2.14.m14.1.1" xref="S4.SS3.p2.14.m14.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p2.14.m14.1.1.2.2" xref="S4.SS3.p2.14.m14.1.1.2.2.cmml">𝒯</mi><mi id="S4.SS3.p2.14.m14.1.1.2.3" xref="S4.SS3.p2.14.m14.1.1.2.3.cmml">m</mi><mi id="S4.SS3.p2.14.m14.1.1.3" xref="S4.SS3.p2.14.m14.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.14.m14.1b"><apply id="S4.SS3.p2.14.m14.1.1.cmml" xref="S4.SS3.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.14.m14.1.1.1.cmml" xref="S4.SS3.p2.14.m14.1.1">superscript</csymbol><apply id="S4.SS3.p2.14.m14.1.1.2.cmml" xref="S4.SS3.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.14.m14.1.1.2.1.cmml" xref="S4.SS3.p2.14.m14.1.1">subscript</csymbol><ci id="S4.SS3.p2.14.m14.1.1.2.2.cmml" xref="S4.SS3.p2.14.m14.1.1.2.2">𝒯</ci><ci id="S4.SS3.p2.14.m14.1.1.2.3.cmml" xref="S4.SS3.p2.14.m14.1.1.2.3">𝑚</ci></apply><ci id="S4.SS3.p2.14.m14.1.1.3.cmml" xref="S4.SS3.p2.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.14.m14.1c">\mathcal{T}_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.14.m14.1d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> to each node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.15.m15.1"><semantics id="S4.SS3.p2.15.m15.1a"><msubsup id="S4.SS3.p2.15.m15.1.1" xref="S4.SS3.p2.15.m15.1.1.cmml"><mi id="S4.SS3.p2.15.m15.1.1.2.2" xref="S4.SS3.p2.15.m15.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p2.15.m15.1.1.2.3" xref="S4.SS3.p2.15.m15.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p2.15.m15.1.1.3" xref="S4.SS3.p2.15.m15.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.15.m15.1b"><apply id="S4.SS3.p2.15.m15.1.1.cmml" xref="S4.SS3.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.15.m15.1.1.1.cmml" xref="S4.SS3.p2.15.m15.1.1">superscript</csymbol><apply id="S4.SS3.p2.15.m15.1.1.2.cmml" xref="S4.SS3.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS3.p2.15.m15.1.1.2.1.cmml" xref="S4.SS3.p2.15.m15.1.1">subscript</csymbol><ci id="S4.SS3.p2.15.m15.1.1.2.2.cmml" xref="S4.SS3.p2.15.m15.1.1.2.2">𝑣</ci><ci id="S4.SS3.p2.15.m15.1.1.2.3.cmml" xref="S4.SS3.p2.15.m15.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p2.15.m15.1.1.3.cmml" xref="S4.SS3.p2.15.m15.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.15.m15.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.15.m15.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. Since there are kinematic switches and changes to the collision environment, each node also contains the robot kinematic (i.e., attached object), which will be important for the TPG construction process. </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption">Algorithm 1 Multi-Modal TPG Construction</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <div class="ltx_listing ltx_figure_panel ltx_listing" id="alg1.5">Input: A task graph <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="alg1.1.m1.1"><semantics id="alg1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.1.m1.1.1" xref="alg1.1.m1.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="alg1.1.m1.1b"><ci id="alg1.1.m1.1.1.cmml" xref="alg1.1.m1.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.1.m1.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="alg1.1.m1.1d">caligraphic_G</annotation></semantics></math>and all robot trajectories <math alttext="\tau^{i}_{m}" class="ltx_Math" display="inline" id="alg1.2.m2.1"><semantics id="alg1.2.m2.1a"><msubsup id="alg1.2.m2.1.1" xref="alg1.2.m2.1.1.cmml"><mi id="alg1.2.m2.1.1.2.2" xref="alg1.2.m2.1.1.2.2.cmml">τ</mi><mi id="alg1.2.m2.1.1.3" xref="alg1.2.m2.1.1.3.cmml">m</mi><mi id="alg1.2.m2.1.1.2.3" xref="alg1.2.m2.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.2.m2.1b"><apply id="alg1.2.m2.1.1.cmml" xref="alg1.2.m2.1.1"><csymbol cd="ambiguous" id="alg1.2.m2.1.1.1.cmml" xref="alg1.2.m2.1.1">subscript</csymbol><apply id="alg1.2.m2.1.1.2.cmml" xref="alg1.2.m2.1.1"><csymbol cd="ambiguous" id="alg1.2.m2.1.1.2.1.cmml" xref="alg1.2.m2.1.1">superscript</csymbol><ci id="alg1.2.m2.1.1.2.2.cmml" xref="alg1.2.m2.1.1.2.2">𝜏</ci><ci id="alg1.2.m2.1.1.2.3.cmml" xref="alg1.2.m2.1.1.2.3">𝑖</ci></apply><ci id="alg1.2.m2.1.1.3.cmml" xref="alg1.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.2.m2.1c">\tau^{i}_{m}</annotation><annotation encoding="application/x-llamapun" id="alg1.2.m2.1d">italic_τ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>robot <math alttext="i=1,\dots,N" class="ltx_Math" display="inline" id="alg1.3.m3.3"><semantics id="alg1.3.m3.3a"><mrow id="alg1.3.m3.3.4" xref="alg1.3.m3.3.4.cmml"><mi id="alg1.3.m3.3.4.2" xref="alg1.3.m3.3.4.2.cmml">i</mi><mo id="alg1.3.m3.3.4.1" xref="alg1.3.m3.3.4.1.cmml">=</mo><mrow id="alg1.3.m3.3.4.3.2" xref="alg1.3.m3.3.4.3.1.cmml"><mn id="alg1.3.m3.1.1" xref="alg1.3.m3.1.1.cmml">1</mn><mo id="alg1.3.m3.3.4.3.2.1" xref="alg1.3.m3.3.4.3.1.cmml">,</mo><mi id="alg1.3.m3.2.2" mathvariant="normal" xref="alg1.3.m3.2.2.cmml">…</mi><mo id="alg1.3.m3.3.4.3.2.2" xref="alg1.3.m3.3.4.3.1.cmml">,</mo><mi id="alg1.3.m3.3.3" xref="alg1.3.m3.3.3.cmml">N</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.3.m3.3b"><apply id="alg1.3.m3.3.4.cmml" xref="alg1.3.m3.3.4"><eq id="alg1.3.m3.3.4.1.cmml" xref="alg1.3.m3.3.4.1"></eq><ci id="alg1.3.m3.3.4.2.cmml" xref="alg1.3.m3.3.4.2">𝑖</ci><list id="alg1.3.m3.3.4.3.1.cmml" xref="alg1.3.m3.3.4.3.2"><cn id="alg1.3.m3.1.1.cmml" type="integer" xref="alg1.3.m3.1.1">1</cn><ci id="alg1.3.m3.2.2.cmml" xref="alg1.3.m3.2.2">…</ci><ci id="alg1.3.m3.3.3.cmml" xref="alg1.3.m3.3.3">𝑁</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.3.m3.3c">i=1,\dots,N</annotation><annotation encoding="application/x-llamapun" id="alg1.3.m3.3d">italic_i = 1 , … , italic_N</annotation></semantics></math>task <math alttext="\mathcal{T}_{m}^{i}=\mathcal{T}^{i}_{1},\dots,\mathcal{T}_{M^{i}}^{i}" class="ltx_Math" display="inline" id="alg1.4.m4.3"><semantics id="alg1.4.m4.3a"><mrow id="alg1.4.m4.3.3" xref="alg1.4.m4.3.3.cmml"><msubsup 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xref="alg1.4.m4.3.3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.4.m4.3.3.2.2.2.2.2" xref="alg1.4.m4.3.3.2.2.2.2.2.cmml">𝒯</mi><msup id="alg1.4.m4.3.3.2.2.2.2.3" xref="alg1.4.m4.3.3.2.2.2.2.3.cmml"><mi id="alg1.4.m4.3.3.2.2.2.2.3.2" xref="alg1.4.m4.3.3.2.2.2.2.3.2.cmml">M</mi><mi id="alg1.4.m4.3.3.2.2.2.2.3.3" xref="alg1.4.m4.3.3.2.2.2.2.3.3.cmml">i</mi></msup><mi id="alg1.4.m4.3.3.2.2.2.3" xref="alg1.4.m4.3.3.2.2.2.3.cmml">i</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.4.m4.3b"><apply id="alg1.4.m4.3.3.cmml" xref="alg1.4.m4.3.3"><eq id="alg1.4.m4.3.3.3.cmml" xref="alg1.4.m4.3.3.3"></eq><apply id="alg1.4.m4.3.3.4.cmml" xref="alg1.4.m4.3.3.4"><csymbol cd="ambiguous" id="alg1.4.m4.3.3.4.1.cmml" xref="alg1.4.m4.3.3.4">superscript</csymbol><apply id="alg1.4.m4.3.3.4.2.cmml" xref="alg1.4.m4.3.3.4"><csymbol cd="ambiguous" id="alg1.4.m4.3.3.4.2.1.cmml" xref="alg1.4.m4.3.3.4">subscript</csymbol><ci id="alg1.4.m4.3.3.4.2.2.cmml" 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id="alg1.4.m4.3.3.2.2.2.1.cmml" xref="alg1.4.m4.3.3.2.2.2">superscript</csymbol><apply id="alg1.4.m4.3.3.2.2.2.2.cmml" xref="alg1.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.4.m4.3.3.2.2.2.2.1.cmml" xref="alg1.4.m4.3.3.2.2.2">subscript</csymbol><ci id="alg1.4.m4.3.3.2.2.2.2.2.cmml" xref="alg1.4.m4.3.3.2.2.2.2.2">𝒯</ci><apply id="alg1.4.m4.3.3.2.2.2.2.3.cmml" xref="alg1.4.m4.3.3.2.2.2.2.3"><csymbol cd="ambiguous" id="alg1.4.m4.3.3.2.2.2.2.3.1.cmml" xref="alg1.4.m4.3.3.2.2.2.2.3">superscript</csymbol><ci id="alg1.4.m4.3.3.2.2.2.2.3.2.cmml" xref="alg1.4.m4.3.3.2.2.2.2.3.2">𝑀</ci><ci id="alg1.4.m4.3.3.2.2.2.2.3.3.cmml" xref="alg1.4.m4.3.3.2.2.2.2.3.3">𝑖</ci></apply></apply><ci id="alg1.4.m4.3.3.2.2.2.3.cmml" xref="alg1.4.m4.3.3.2.2.2.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.4.m4.3c">\mathcal{T}_{m}^{i}=\mathcal{T}^{i}_{1},\dots,\mathcal{T}_{M^{i}}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.4.m4.3d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = caligraphic_T start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , caligraphic_T start_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>task <math alttext="\mathcal{T}_{m}^{i}" class="ltx_Math" display="inline" id="alg1.5.m5.1"><semantics id="alg1.5.m5.1a"><msubsup id="alg1.5.m5.1.1" xref="alg1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.5.m5.1.1.2.2" xref="alg1.5.m5.1.1.2.2.cmml">𝒯</mi><mi id="alg1.5.m5.1.1.2.3" xref="alg1.5.m5.1.1.2.3.cmml">m</mi><mi id="alg1.5.m5.1.1.3" xref="alg1.5.m5.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.5.m5.1b"><apply id="alg1.5.m5.1.1.cmml" xref="alg1.5.m5.1.1"><csymbol cd="ambiguous" id="alg1.5.m5.1.1.1.cmml" xref="alg1.5.m5.1.1">superscript</csymbol><apply id="alg1.5.m5.1.1.2.cmml" xref="alg1.5.m5.1.1"><csymbol cd="ambiguous" id="alg1.5.m5.1.1.2.1.cmml" xref="alg1.5.m5.1.1">subscript</csymbol><ci id="alg1.5.m5.1.1.2.2.cmml" xref="alg1.5.m5.1.1.2.2">𝒯</ci><ci id="alg1.5.m5.1.1.2.3.cmml" xref="alg1.5.m5.1.1.2.3">𝑚</ci></apply><ci id="alg1.5.m5.1.1.3.cmml" xref="alg1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.5.m5.1c">\mathcal{T}_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.5.m5.1d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>is a transit to a goal pose <div class="ltx_listingline" id="alg1.l1"> 1:Add a sequence of nodes <math alttext="\{v^{i}_{n}\}" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><mrow id="alg1.l1.m1.1.1.1" xref="alg1.l1.m1.1.1.2.cmml"><mo id="alg1.l1.m1.1.1.1.2" stretchy="false" xref="alg1.l1.m1.1.1.2.cmml">{</mo><msubsup id="alg1.l1.m1.1.1.1.1" xref="alg1.l1.m1.1.1.1.1.cmml"><mi id="alg1.l1.m1.1.1.1.1.2.2" xref="alg1.l1.m1.1.1.1.1.2.2.cmml">v</mi><mi id="alg1.l1.m1.1.1.1.1.3" xref="alg1.l1.m1.1.1.1.1.3.cmml">n</mi><mi id="alg1.l1.m1.1.1.1.1.2.3" xref="alg1.l1.m1.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="alg1.l1.m1.1.1.1.3" stretchy="false" xref="alg1.l1.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.1b"><set id="alg1.l1.m1.1.1.2.cmml" xref="alg1.l1.m1.1.1.1"><apply id="alg1.l1.m1.1.1.1.1.cmml" xref="alg1.l1.m1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l1.m1.1.1.1.1.1.cmml" xref="alg1.l1.m1.1.1.1.1">subscript</csymbol><apply id="alg1.l1.m1.1.1.1.1.2.cmml" xref="alg1.l1.m1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l1.m1.1.1.1.1.2.1.cmml" xref="alg1.l1.m1.1.1.1.1">superscript</csymbol><ci id="alg1.l1.m1.1.1.1.1.2.2.cmml" xref="alg1.l1.m1.1.1.1.1.2.2">𝑣</ci><ci id="alg1.l1.m1.1.1.1.1.2.3.cmml" xref="alg1.l1.m1.1.1.1.1.2.3">𝑖</ci></apply><ci id="alg1.l1.m1.1.1.1.1.3.cmml" xref="alg1.l1.m1.1.1.1.1.3">𝑛</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.1c">\{v^{i}_{n}\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.1d">{ italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math> from <math alttext="\tau^{i}" class="ltx_Math" display="inline" id="alg1.l1.m2.1"><semantics id="alg1.l1.m2.1a"><msup id="alg1.l1.m2.1.1" xref="alg1.l1.m2.1.1.cmml"><mi id="alg1.l1.m2.1.1.2" xref="alg1.l1.m2.1.1.2.cmml">τ</mi><mi id="alg1.l1.m2.1.1.3" xref="alg1.l1.m2.1.1.3.cmml">i</mi></msup><annotation-xml encoding="MathML-Content" id="alg1.l1.m2.1b"><apply id="alg1.l1.m2.1.1.cmml" xref="alg1.l1.m2.1.1"><csymbol cd="ambiguous" id="alg1.l1.m2.1.1.1.cmml" xref="alg1.l1.m2.1.1">superscript</csymbol><ci id="alg1.l1.m2.1.1.2.cmml" xref="alg1.l1.m2.1.1.2">𝜏</ci><ci id="alg1.l1.m2.1.1.3.cmml" xref="alg1.l1.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.1c">\tau^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.1d">italic_τ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> 2:Add type-1 edges for consecutive nodes \Else Add a skill node <math alttext="\tilde{v}^{i}_{n}" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><msubsup id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml"><mover accent="true" id="alg1.l2.m1.1.1.2.2" xref="alg1.l2.m1.1.1.2.2.cmml"><mi id="alg1.l2.m1.1.1.2.2.2" xref="alg1.l2.m1.1.1.2.2.2.cmml">v</mi><mo id="alg1.l2.m1.1.1.2.2.1" xref="alg1.l2.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="alg1.l2.m1.1.1.3" xref="alg1.l2.m1.1.1.3.cmml">n</mi><mi id="alg1.l2.m1.1.1.2.3" xref="alg1.l2.m1.1.1.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><apply id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.1.cmml" xref="alg1.l2.m1.1.1">subscript</csymbol><apply id="alg1.l2.m1.1.1.2.cmml" xref="alg1.l2.m1.1.1"><csymbol cd="ambiguous" id="alg1.l2.m1.1.1.2.1.cmml" xref="alg1.l2.m1.1.1">superscript</csymbol><apply id="alg1.l2.m1.1.1.2.2.cmml" xref="alg1.l2.m1.1.1.2.2"><ci id="alg1.l2.m1.1.1.2.2.1.cmml" xref="alg1.l2.m1.1.1.2.2.1">~</ci><ci id="alg1.l2.m1.1.1.2.2.2.cmml" xref="alg1.l2.m1.1.1.2.2.2">𝑣</ci></apply><ci id="alg1.l2.m1.1.1.2.3.cmml" xref="alg1.l2.m1.1.1.2.3">𝑖</ci></apply><ci id="alg1.l2.m1.1.1.3.cmml" xref="alg1.l2.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">\tilde{v}^{i}_{n}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">over~ start_ARG italic_v end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> \EndIf </div> <div class="ltx_listingline" id="alg1.l3"> 3:Add type-1 edge from <math alttext="v^{i}_{{end,m-1}}" class="ltx_Math" display="inline" id="alg1.l3.m1.2"><semantics id="alg1.l3.m1.2a"><msubsup id="alg1.l3.m1.2.3" xref="alg1.l3.m1.2.3.cmml"><mi id="alg1.l3.m1.2.3.2.2" xref="alg1.l3.m1.2.3.2.2.cmml">v</mi><mrow id="alg1.l3.m1.2.2.2.2" xref="alg1.l3.m1.2.2.2.3.cmml"><mrow id="alg1.l3.m1.1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.cmml"><mi id="alg1.l3.m1.1.1.1.1.1.2" xref="alg1.l3.m1.1.1.1.1.1.2.cmml">e</mi><mo id="alg1.l3.m1.1.1.1.1.1.1" xref="alg1.l3.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l3.m1.1.1.1.1.1.3" xref="alg1.l3.m1.1.1.1.1.1.3.cmml">n</mi><mo id="alg1.l3.m1.1.1.1.1.1.1a" xref="alg1.l3.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l3.m1.1.1.1.1.1.4" xref="alg1.l3.m1.1.1.1.1.1.4.cmml">d</mi></mrow><mo id="alg1.l3.m1.2.2.2.2.3" xref="alg1.l3.m1.2.2.2.3.cmml">,</mo><mrow id="alg1.l3.m1.2.2.2.2.2" xref="alg1.l3.m1.2.2.2.2.2.cmml"><mi id="alg1.l3.m1.2.2.2.2.2.2" xref="alg1.l3.m1.2.2.2.2.2.2.cmml">m</mi><mo id="alg1.l3.m1.2.2.2.2.2.1" xref="alg1.l3.m1.2.2.2.2.2.1.cmml">−</mo><mn id="alg1.l3.m1.2.2.2.2.2.3" xref="alg1.l3.m1.2.2.2.2.2.3.cmml">1</mn></mrow></mrow><mi id="alg1.l3.m1.2.3.2.3" xref="alg1.l3.m1.2.3.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.2b"><apply id="alg1.l3.m1.2.3.cmml" xref="alg1.l3.m1.2.3"><csymbol cd="ambiguous" id="alg1.l3.m1.2.3.1.cmml" xref="alg1.l3.m1.2.3">subscript</csymbol><apply id="alg1.l3.m1.2.3.2.cmml" xref="alg1.l3.m1.2.3"><csymbol cd="ambiguous" id="alg1.l3.m1.2.3.2.1.cmml" xref="alg1.l3.m1.2.3">superscript</csymbol><ci id="alg1.l3.m1.2.3.2.2.cmml" xref="alg1.l3.m1.2.3.2.2">𝑣</ci><ci id="alg1.l3.m1.2.3.2.3.cmml" xref="alg1.l3.m1.2.3.2.3">𝑖</ci></apply><list id="alg1.l3.m1.2.2.2.3.cmml" xref="alg1.l3.m1.2.2.2.2"><apply id="alg1.l3.m1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1.1"><times id="alg1.l3.m1.1.1.1.1.1.1.cmml" xref="alg1.l3.m1.1.1.1.1.1.1"></times><ci id="alg1.l3.m1.1.1.1.1.1.2.cmml" xref="alg1.l3.m1.1.1.1.1.1.2">𝑒</ci><ci id="alg1.l3.m1.1.1.1.1.1.3.cmml" xref="alg1.l3.m1.1.1.1.1.1.3">𝑛</ci><ci id="alg1.l3.m1.1.1.1.1.1.4.cmml" xref="alg1.l3.m1.1.1.1.1.1.4">𝑑</ci></apply><apply id="alg1.l3.m1.2.2.2.2.2.cmml" xref="alg1.l3.m1.2.2.2.2.2"><minus id="alg1.l3.m1.2.2.2.2.2.1.cmml" xref="alg1.l3.m1.2.2.2.2.2.1"></minus><ci id="alg1.l3.m1.2.2.2.2.2.2.cmml" xref="alg1.l3.m1.2.2.2.2.2.2">𝑚</ci><cn id="alg1.l3.m1.2.2.2.2.2.3.cmml" type="integer" xref="alg1.l3.m1.2.2.2.2.2.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.2c">v^{i}_{{end,m-1}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.2d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e italic_n italic_d , italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v^{i}_{{start,m}}" class="ltx_Math" display="inline" id="alg1.l3.m2.2"><semantics id="alg1.l3.m2.2a"><msubsup id="alg1.l3.m2.2.3" xref="alg1.l3.m2.2.3.cmml"><mi id="alg1.l3.m2.2.3.2.2" xref="alg1.l3.m2.2.3.2.2.cmml">v</mi><mrow id="alg1.l3.m2.2.2.2.2" xref="alg1.l3.m2.2.2.2.3.cmml"><mrow id="alg1.l3.m2.2.2.2.2.1" xref="alg1.l3.m2.2.2.2.2.1.cmml"><mi id="alg1.l3.m2.2.2.2.2.1.2" xref="alg1.l3.m2.2.2.2.2.1.2.cmml">s</mi><mo id="alg1.l3.m2.2.2.2.2.1.1" xref="alg1.l3.m2.2.2.2.2.1.1.cmml">⁢</mo><mi id="alg1.l3.m2.2.2.2.2.1.3" xref="alg1.l3.m2.2.2.2.2.1.3.cmml">t</mi><mo id="alg1.l3.m2.2.2.2.2.1.1a" xref="alg1.l3.m2.2.2.2.2.1.1.cmml">⁢</mo><mi id="alg1.l3.m2.2.2.2.2.1.4" xref="alg1.l3.m2.2.2.2.2.1.4.cmml">a</mi><mo id="alg1.l3.m2.2.2.2.2.1.1b" xref="alg1.l3.m2.2.2.2.2.1.1.cmml">⁢</mo><mi id="alg1.l3.m2.2.2.2.2.1.5" xref="alg1.l3.m2.2.2.2.2.1.5.cmml">r</mi><mo id="alg1.l3.m2.2.2.2.2.1.1c" xref="alg1.l3.m2.2.2.2.2.1.1.cmml">⁢</mo><mi id="alg1.l3.m2.2.2.2.2.1.6" xref="alg1.l3.m2.2.2.2.2.1.6.cmml">t</mi></mrow><mo id="alg1.l3.m2.2.2.2.2.2" xref="alg1.l3.m2.2.2.2.3.cmml">,</mo><mi id="alg1.l3.m2.1.1.1.1" xref="alg1.l3.m2.1.1.1.1.cmml">m</mi></mrow><mi id="alg1.l3.m2.2.3.2.3" xref="alg1.l3.m2.2.3.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l3.m2.2b"><apply id="alg1.l3.m2.2.3.cmml" xref="alg1.l3.m2.2.3"><csymbol cd="ambiguous" id="alg1.l3.m2.2.3.1.cmml" xref="alg1.l3.m2.2.3">subscript</csymbol><apply id="alg1.l3.m2.2.3.2.cmml" xref="alg1.l3.m2.2.3"><csymbol cd="ambiguous" id="alg1.l3.m2.2.3.2.1.cmml" xref="alg1.l3.m2.2.3">superscript</csymbol><ci id="alg1.l3.m2.2.3.2.2.cmml" xref="alg1.l3.m2.2.3.2.2">𝑣</ci><ci id="alg1.l3.m2.2.3.2.3.cmml" xref="alg1.l3.m2.2.3.2.3">𝑖</ci></apply><list id="alg1.l3.m2.2.2.2.3.cmml" xref="alg1.l3.m2.2.2.2.2"><apply id="alg1.l3.m2.2.2.2.2.1.cmml" xref="alg1.l3.m2.2.2.2.2.1"><times id="alg1.l3.m2.2.2.2.2.1.1.cmml" xref="alg1.l3.m2.2.2.2.2.1.1"></times><ci id="alg1.l3.m2.2.2.2.2.1.2.cmml" xref="alg1.l3.m2.2.2.2.2.1.2">𝑠</ci><ci id="alg1.l3.m2.2.2.2.2.1.3.cmml" xref="alg1.l3.m2.2.2.2.2.1.3">𝑡</ci><ci id="alg1.l3.m2.2.2.2.2.1.4.cmml" xref="alg1.l3.m2.2.2.2.2.1.4">𝑎</ci><ci id="alg1.l3.m2.2.2.2.2.1.5.cmml" xref="alg1.l3.m2.2.2.2.2.1.5">𝑟</ci><ci id="alg1.l3.m2.2.2.2.2.1.6.cmml" xref="alg1.l3.m2.2.2.2.2.1.6">𝑡</ci></apply><ci id="alg1.l3.m2.1.1.1.1.cmml" xref="alg1.l3.m2.1.1.1.1">𝑚</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m2.2c">v^{i}_{{start,m}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m2.2d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m end_POSTSUBSCRIPT</annotation></semantics></math> \EndFor\EndFor\Fortask dependency <math alttext="(\mathcal{T}_{m}^{i}\rightarrow\mathcal{T}_{m^{\prime}}^{i^{\prime}})\in% \mathcal{E}" class="ltx_Math" display="inline" id="alg1.l3.m3.1"><semantics id="alg1.l3.m3.1a"><mrow id="alg1.l3.m3.1.1" xref="alg1.l3.m3.1.1.cmml"><mrow id="alg1.l3.m3.1.1.1.1" xref="alg1.l3.m3.1.1.1.1.1.cmml"><mo id="alg1.l3.m3.1.1.1.1.2" stretchy="false" xref="alg1.l3.m3.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l3.m3.1.1.1.1.1" xref="alg1.l3.m3.1.1.1.1.1.cmml"><msubsup id="alg1.l3.m3.1.1.1.1.1.2" xref="alg1.l3.m3.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m3.1.1.1.1.1.2.2.2" xref="alg1.l3.m3.1.1.1.1.1.2.2.2.cmml">𝒯</mi><mi id="alg1.l3.m3.1.1.1.1.1.2.2.3" xref="alg1.l3.m3.1.1.1.1.1.2.2.3.cmml">m</mi><mi id="alg1.l3.m3.1.1.1.1.1.2.3" xref="alg1.l3.m3.1.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="alg1.l3.m3.1.1.1.1.1.1" stretchy="false" xref="alg1.l3.m3.1.1.1.1.1.1.cmml">→</mo><msubsup id="alg1.l3.m3.1.1.1.1.1.3" xref="alg1.l3.m3.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m3.1.1.1.1.1.3.2.2" xref="alg1.l3.m3.1.1.1.1.1.3.2.2.cmml">𝒯</mi><msup id="alg1.l3.m3.1.1.1.1.1.3.2.3" xref="alg1.l3.m3.1.1.1.1.1.3.2.3.cmml"><mi id="alg1.l3.m3.1.1.1.1.1.3.2.3.2" xref="alg1.l3.m3.1.1.1.1.1.3.2.3.2.cmml">m</mi><mo id="alg1.l3.m3.1.1.1.1.1.3.2.3.3" xref="alg1.l3.m3.1.1.1.1.1.3.2.3.3.cmml">′</mo></msup><msup id="alg1.l3.m3.1.1.1.1.1.3.3" xref="alg1.l3.m3.1.1.1.1.1.3.3.cmml"><mi id="alg1.l3.m3.1.1.1.1.1.3.3.2" xref="alg1.l3.m3.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="alg1.l3.m3.1.1.1.1.1.3.3.3" xref="alg1.l3.m3.1.1.1.1.1.3.3.3.cmml">′</mo></msup></msubsup></mrow><mo id="alg1.l3.m3.1.1.1.1.3" stretchy="false" xref="alg1.l3.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="alg1.l3.m3.1.1.2" xref="alg1.l3.m3.1.1.2.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m3.1.1.3" xref="alg1.l3.m3.1.1.3.cmml">ℰ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m3.1b"><apply id="alg1.l3.m3.1.1.cmml" xref="alg1.l3.m3.1.1"><in id="alg1.l3.m3.1.1.2.cmml" xref="alg1.l3.m3.1.1.2"></in><apply id="alg1.l3.m3.1.1.1.1.1.cmml" xref="alg1.l3.m3.1.1.1.1"><ci id="alg1.l3.m3.1.1.1.1.1.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.1">→</ci><apply id="alg1.l3.m3.1.1.1.1.1.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l3.m3.1.1.1.1.1.2.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.2">superscript</csymbol><apply id="alg1.l3.m3.1.1.1.1.1.2.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="alg1.l3.m3.1.1.1.1.1.2.2.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.2">subscript</csymbol><ci id="alg1.l3.m3.1.1.1.1.1.2.2.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.2.2.2">𝒯</ci><ci id="alg1.l3.m3.1.1.1.1.1.2.2.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.2.2.3">𝑚</ci></apply><ci id="alg1.l3.m3.1.1.1.1.1.2.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="alg1.l3.m3.1.1.1.1.1.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l3.m3.1.1.1.1.1.3.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.3">superscript</csymbol><apply id="alg1.l3.m3.1.1.1.1.1.3.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l3.m3.1.1.1.1.1.3.2.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.3">subscript</csymbol><ci id="alg1.l3.m3.1.1.1.1.1.3.2.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.2.2">𝒯</ci><apply id="alg1.l3.m3.1.1.1.1.1.3.2.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="alg1.l3.m3.1.1.1.1.1.3.2.3.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.2.3">superscript</csymbol><ci id="alg1.l3.m3.1.1.1.1.1.3.2.3.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.2.3.2">𝑚</ci><ci id="alg1.l3.m3.1.1.1.1.1.3.2.3.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.2.3.3">′</ci></apply></apply><apply id="alg1.l3.m3.1.1.1.1.1.3.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l3.m3.1.1.1.1.1.3.3.1.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.3">superscript</csymbol><ci id="alg1.l3.m3.1.1.1.1.1.3.3.2.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.3.2">𝑖</ci><ci id="alg1.l3.m3.1.1.1.1.1.3.3.3.cmml" xref="alg1.l3.m3.1.1.1.1.1.3.3.3">′</ci></apply></apply></apply><ci id="alg1.l3.m3.1.1.3.cmml" xref="alg1.l3.m3.1.1.3">ℰ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m3.1c">(\mathcal{T}_{m}^{i}\rightarrow\mathcal{T}_{m^{\prime}}^{i^{\prime}})\in% \mathcal{E}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m3.1d">( caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → caligraphic_T start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) ∈ caligraphic_E</annotation></semantics></math> such that <math alttext="i\neq i^{\prime}" class="ltx_Math" display="inline" id="alg1.l3.m4.1"><semantics id="alg1.l3.m4.1a"><mrow id="alg1.l3.m4.1.1" xref="alg1.l3.m4.1.1.cmml"><mi id="alg1.l3.m4.1.1.2" xref="alg1.l3.m4.1.1.2.cmml">i</mi><mo id="alg1.l3.m4.1.1.1" xref="alg1.l3.m4.1.1.1.cmml">≠</mo><msup id="alg1.l3.m4.1.1.3" xref="alg1.l3.m4.1.1.3.cmml"><mi id="alg1.l3.m4.1.1.3.2" xref="alg1.l3.m4.1.1.3.2.cmml">i</mi><mo id="alg1.l3.m4.1.1.3.3" xref="alg1.l3.m4.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m4.1b"><apply id="alg1.l3.m4.1.1.cmml" xref="alg1.l3.m4.1.1"><neq id="alg1.l3.m4.1.1.1.cmml" xref="alg1.l3.m4.1.1.1"></neq><ci id="alg1.l3.m4.1.1.2.cmml" xref="alg1.l3.m4.1.1.2">𝑖</ci><apply id="alg1.l3.m4.1.1.3.cmml" xref="alg1.l3.m4.1.1.3"><csymbol cd="ambiguous" id="alg1.l3.m4.1.1.3.1.cmml" xref="alg1.l3.m4.1.1.3">superscript</csymbol><ci id="alg1.l3.m4.1.1.3.2.cmml" xref="alg1.l3.m4.1.1.3.2">𝑖</ci><ci id="alg1.l3.m4.1.1.3.3.cmml" xref="alg1.l3.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m4.1c">i\neq i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m4.1d">italic_i ≠ italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l4"> 4:Add type-2 edge from <math alttext="v^{i}_{{start,m+1}}" class="ltx_Math" display="inline" id="alg1.l4.m1.2"><semantics id="alg1.l4.m1.2a"><msubsup id="alg1.l4.m1.2.3" xref="alg1.l4.m1.2.3.cmml"><mi id="alg1.l4.m1.2.3.2.2" xref="alg1.l4.m1.2.3.2.2.cmml">v</mi><mrow id="alg1.l4.m1.2.2.2.2" xref="alg1.l4.m1.2.2.2.3.cmml"><mrow id="alg1.l4.m1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.cmml"><mi id="alg1.l4.m1.1.1.1.1.1.2" xref="alg1.l4.m1.1.1.1.1.1.2.cmml">s</mi><mo id="alg1.l4.m1.1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.1.1.1.3" xref="alg1.l4.m1.1.1.1.1.1.3.cmml">t</mi><mo id="alg1.l4.m1.1.1.1.1.1.1a" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.1.1.1.4" xref="alg1.l4.m1.1.1.1.1.1.4.cmml">a</mi><mo id="alg1.l4.m1.1.1.1.1.1.1b" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.1.1.1.5" xref="alg1.l4.m1.1.1.1.1.1.5.cmml">r</mi><mo id="alg1.l4.m1.1.1.1.1.1.1c" xref="alg1.l4.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m1.1.1.1.1.1.6" xref="alg1.l4.m1.1.1.1.1.1.6.cmml">t</mi></mrow><mo id="alg1.l4.m1.2.2.2.2.3" xref="alg1.l4.m1.2.2.2.3.cmml">,</mo><mrow id="alg1.l4.m1.2.2.2.2.2" xref="alg1.l4.m1.2.2.2.2.2.cmml"><mi id="alg1.l4.m1.2.2.2.2.2.2" xref="alg1.l4.m1.2.2.2.2.2.2.cmml">m</mi><mo id="alg1.l4.m1.2.2.2.2.2.1" xref="alg1.l4.m1.2.2.2.2.2.1.cmml">+</mo><mn id="alg1.l4.m1.2.2.2.2.2.3" xref="alg1.l4.m1.2.2.2.2.2.3.cmml">1</mn></mrow></mrow><mi id="alg1.l4.m1.2.3.2.3" xref="alg1.l4.m1.2.3.2.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.2b"><apply id="alg1.l4.m1.2.3.cmml" xref="alg1.l4.m1.2.3"><csymbol cd="ambiguous" id="alg1.l4.m1.2.3.1.cmml" xref="alg1.l4.m1.2.3">subscript</csymbol><apply id="alg1.l4.m1.2.3.2.cmml" xref="alg1.l4.m1.2.3"><csymbol cd="ambiguous" id="alg1.l4.m1.2.3.2.1.cmml" xref="alg1.l4.m1.2.3">superscript</csymbol><ci id="alg1.l4.m1.2.3.2.2.cmml" xref="alg1.l4.m1.2.3.2.2">𝑣</ci><ci id="alg1.l4.m1.2.3.2.3.cmml" xref="alg1.l4.m1.2.3.2.3">𝑖</ci></apply><list id="alg1.l4.m1.2.2.2.3.cmml" xref="alg1.l4.m1.2.2.2.2"><apply id="alg1.l4.m1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1.1"><times id="alg1.l4.m1.1.1.1.1.1.1.cmml" xref="alg1.l4.m1.1.1.1.1.1.1"></times><ci id="alg1.l4.m1.1.1.1.1.1.2.cmml" xref="alg1.l4.m1.1.1.1.1.1.2">𝑠</ci><ci id="alg1.l4.m1.1.1.1.1.1.3.cmml" xref="alg1.l4.m1.1.1.1.1.1.3">𝑡</ci><ci id="alg1.l4.m1.1.1.1.1.1.4.cmml" xref="alg1.l4.m1.1.1.1.1.1.4">𝑎</ci><ci id="alg1.l4.m1.1.1.1.1.1.5.cmml" xref="alg1.l4.m1.1.1.1.1.1.5">𝑟</ci><ci id="alg1.l4.m1.1.1.1.1.1.6.cmml" xref="alg1.l4.m1.1.1.1.1.1.6">𝑡</ci></apply><apply id="alg1.l4.m1.2.2.2.2.2.cmml" xref="alg1.l4.m1.2.2.2.2.2"><plus id="alg1.l4.m1.2.2.2.2.2.1.cmml" xref="alg1.l4.m1.2.2.2.2.2.1"></plus><ci id="alg1.l4.m1.2.2.2.2.2.2.cmml" xref="alg1.l4.m1.2.2.2.2.2.2">𝑚</ci><cn id="alg1.l4.m1.2.2.2.2.2.3.cmml" type="integer" xref="alg1.l4.m1.2.2.2.2.2.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.2c">v^{i}_{{start,m+1}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.2d">italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m + 1 end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="v^{i^{\prime}}_{{start,m^{\prime}}}" class="ltx_Math" display="inline" id="alg1.l4.m2.2"><semantics id="alg1.l4.m2.2a"><msubsup id="alg1.l4.m2.2.3" xref="alg1.l4.m2.2.3.cmml"><mi id="alg1.l4.m2.2.3.2.2" xref="alg1.l4.m2.2.3.2.2.cmml">v</mi><mrow id="alg1.l4.m2.2.2.2.2" xref="alg1.l4.m2.2.2.2.3.cmml"><mrow id="alg1.l4.m2.1.1.1.1.1" xref="alg1.l4.m2.1.1.1.1.1.cmml"><mi id="alg1.l4.m2.1.1.1.1.1.2" xref="alg1.l4.m2.1.1.1.1.1.2.cmml">s</mi><mo id="alg1.l4.m2.1.1.1.1.1.1" xref="alg1.l4.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m2.1.1.1.1.1.3" xref="alg1.l4.m2.1.1.1.1.1.3.cmml">t</mi><mo id="alg1.l4.m2.1.1.1.1.1.1a" xref="alg1.l4.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m2.1.1.1.1.1.4" xref="alg1.l4.m2.1.1.1.1.1.4.cmml">a</mi><mo id="alg1.l4.m2.1.1.1.1.1.1b" xref="alg1.l4.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m2.1.1.1.1.1.5" xref="alg1.l4.m2.1.1.1.1.1.5.cmml">r</mi><mo id="alg1.l4.m2.1.1.1.1.1.1c" xref="alg1.l4.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m2.1.1.1.1.1.6" xref="alg1.l4.m2.1.1.1.1.1.6.cmml">t</mi></mrow><mo id="alg1.l4.m2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.3.cmml">,</mo><msup id="alg1.l4.m2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.cmml"><mi id="alg1.l4.m2.2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.2.cmml">m</mi><mo id="alg1.l4.m2.2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.2.2.3.cmml">′</mo></msup></mrow><msup id="alg1.l4.m2.2.3.2.3" xref="alg1.l4.m2.2.3.2.3.cmml"><mi id="alg1.l4.m2.2.3.2.3.2" xref="alg1.l4.m2.2.3.2.3.2.cmml">i</mi><mo id="alg1.l4.m2.2.3.2.3.3" xref="alg1.l4.m2.2.3.2.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l4.m2.2b"><apply id="alg1.l4.m2.2.3.cmml" xref="alg1.l4.m2.2.3"><csymbol cd="ambiguous" id="alg1.l4.m2.2.3.1.cmml" xref="alg1.l4.m2.2.3">subscript</csymbol><apply id="alg1.l4.m2.2.3.2.cmml" xref="alg1.l4.m2.2.3"><csymbol cd="ambiguous" id="alg1.l4.m2.2.3.2.1.cmml" xref="alg1.l4.m2.2.3">superscript</csymbol><ci id="alg1.l4.m2.2.3.2.2.cmml" xref="alg1.l4.m2.2.3.2.2">𝑣</ci><apply id="alg1.l4.m2.2.3.2.3.cmml" xref="alg1.l4.m2.2.3.2.3"><csymbol cd="ambiguous" id="alg1.l4.m2.2.3.2.3.1.cmml" xref="alg1.l4.m2.2.3.2.3">superscript</csymbol><ci id="alg1.l4.m2.2.3.2.3.2.cmml" xref="alg1.l4.m2.2.3.2.3.2">𝑖</ci><ci id="alg1.l4.m2.2.3.2.3.3.cmml" xref="alg1.l4.m2.2.3.2.3.3">′</ci></apply></apply><list id="alg1.l4.m2.2.2.2.3.cmml" xref="alg1.l4.m2.2.2.2.2"><apply id="alg1.l4.m2.1.1.1.1.1.cmml" xref="alg1.l4.m2.1.1.1.1.1"><times id="alg1.l4.m2.1.1.1.1.1.1.cmml" xref="alg1.l4.m2.1.1.1.1.1.1"></times><ci id="alg1.l4.m2.1.1.1.1.1.2.cmml" xref="alg1.l4.m2.1.1.1.1.1.2">𝑠</ci><ci id="alg1.l4.m2.1.1.1.1.1.3.cmml" xref="alg1.l4.m2.1.1.1.1.1.3">𝑡</ci><ci id="alg1.l4.m2.1.1.1.1.1.4.cmml" xref="alg1.l4.m2.1.1.1.1.1.4">𝑎</ci><ci id="alg1.l4.m2.1.1.1.1.1.5.cmml" xref="alg1.l4.m2.1.1.1.1.1.5">𝑟</ci><ci id="alg1.l4.m2.1.1.1.1.1.6.cmml" xref="alg1.l4.m2.1.1.1.1.1.6">𝑡</ci></apply><apply id="alg1.l4.m2.2.2.2.2.2.cmml" xref="alg1.l4.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l4.m2.2.2.2.2.2.1.cmml" xref="alg1.l4.m2.2.2.2.2.2">superscript</csymbol><ci id="alg1.l4.m2.2.2.2.2.2.2.cmml" xref="alg1.l4.m2.2.2.2.2.2.2">𝑚</ci><ci id="alg1.l4.m2.2.2.2.2.2.3.cmml" xref="alg1.l4.m2.2.2.2.2.2.3">′</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m2.2c">v^{i^{\prime}}_{{start,m^{\prime}}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m2.2d">italic_v start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> \EndFor\Forrobots <math alttext="(i,i^{\prime})=\{1,\cdots,N\}\otimes\{1,\cdots,N\}" class="ltx_Math" display="inline" id="alg1.l4.m3.8"><semantics id="alg1.l4.m3.8a"><mrow id="alg1.l4.m3.8.8" xref="alg1.l4.m3.8.8.cmml"><mrow id="alg1.l4.m3.8.8.1.1" xref="alg1.l4.m3.8.8.1.2.cmml"><mo id="alg1.l4.m3.8.8.1.1.2" stretchy="false" xref="alg1.l4.m3.8.8.1.2.cmml">(</mo><mi id="alg1.l4.m3.1.1" xref="alg1.l4.m3.1.1.cmml">i</mi><mo id="alg1.l4.m3.8.8.1.1.3" xref="alg1.l4.m3.8.8.1.2.cmml">,</mo><msup id="alg1.l4.m3.8.8.1.1.1" xref="alg1.l4.m3.8.8.1.1.1.cmml"><mi id="alg1.l4.m3.8.8.1.1.1.2" xref="alg1.l4.m3.8.8.1.1.1.2.cmml">i</mi><mo id="alg1.l4.m3.8.8.1.1.1.3" xref="alg1.l4.m3.8.8.1.1.1.3.cmml">′</mo></msup><mo id="alg1.l4.m3.8.8.1.1.4" stretchy="false" xref="alg1.l4.m3.8.8.1.2.cmml">)</mo></mrow><mo id="alg1.l4.m3.8.8.2" xref="alg1.l4.m3.8.8.2.cmml">=</mo><mrow id="alg1.l4.m3.8.8.3" xref="alg1.l4.m3.8.8.3.cmml"><mrow id="alg1.l4.m3.8.8.3.2.2" xref="alg1.l4.m3.8.8.3.2.1.cmml"><mo id="alg1.l4.m3.8.8.3.2.2.1" stretchy="false" xref="alg1.l4.m3.8.8.3.2.1.cmml">{</mo><mn id="alg1.l4.m3.2.2" xref="alg1.l4.m3.2.2.cmml">1</mn><mo id="alg1.l4.m3.8.8.3.2.2.2" xref="alg1.l4.m3.8.8.3.2.1.cmml">,</mo><mi id="alg1.l4.m3.3.3" mathvariant="normal" xref="alg1.l4.m3.3.3.cmml">⋯</mi><mo id="alg1.l4.m3.8.8.3.2.2.3" xref="alg1.l4.m3.8.8.3.2.1.cmml">,</mo><mi id="alg1.l4.m3.4.4" xref="alg1.l4.m3.4.4.cmml">N</mi><mo id="alg1.l4.m3.8.8.3.2.2.4" rspace="0.055em" stretchy="false" xref="alg1.l4.m3.8.8.3.2.1.cmml">}</mo></mrow><mo id="alg1.l4.m3.8.8.3.1" rspace="0.222em" xref="alg1.l4.m3.8.8.3.1.cmml">⊗</mo><mrow id="alg1.l4.m3.8.8.3.3.2" xref="alg1.l4.m3.8.8.3.3.1.cmml"><mo id="alg1.l4.m3.8.8.3.3.2.1" stretchy="false" xref="alg1.l4.m3.8.8.3.3.1.cmml">{</mo><mn id="alg1.l4.m3.5.5" xref="alg1.l4.m3.5.5.cmml">1</mn><mo id="alg1.l4.m3.8.8.3.3.2.2" xref="alg1.l4.m3.8.8.3.3.1.cmml">,</mo><mi id="alg1.l4.m3.6.6" mathvariant="normal" xref="alg1.l4.m3.6.6.cmml">⋯</mi><mo id="alg1.l4.m3.8.8.3.3.2.3" xref="alg1.l4.m3.8.8.3.3.1.cmml">,</mo><mi id="alg1.l4.m3.7.7" xref="alg1.l4.m3.7.7.cmml">N</mi><mo id="alg1.l4.m3.8.8.3.3.2.4" stretchy="false" xref="alg1.l4.m3.8.8.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m3.8b"><apply id="alg1.l4.m3.8.8.cmml" xref="alg1.l4.m3.8.8"><eq id="alg1.l4.m3.8.8.2.cmml" xref="alg1.l4.m3.8.8.2"></eq><interval closure="open" id="alg1.l4.m3.8.8.1.2.cmml" xref="alg1.l4.m3.8.8.1.1"><ci id="alg1.l4.m3.1.1.cmml" xref="alg1.l4.m3.1.1">𝑖</ci><apply id="alg1.l4.m3.8.8.1.1.1.cmml" xref="alg1.l4.m3.8.8.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m3.8.8.1.1.1.1.cmml" xref="alg1.l4.m3.8.8.1.1.1">superscript</csymbol><ci id="alg1.l4.m3.8.8.1.1.1.2.cmml" xref="alg1.l4.m3.8.8.1.1.1.2">𝑖</ci><ci id="alg1.l4.m3.8.8.1.1.1.3.cmml" xref="alg1.l4.m3.8.8.1.1.1.3">′</ci></apply></interval><apply id="alg1.l4.m3.8.8.3.cmml" xref="alg1.l4.m3.8.8.3"><csymbol cd="latexml" id="alg1.l4.m3.8.8.3.1.cmml" xref="alg1.l4.m3.8.8.3.1">tensor-product</csymbol><set id="alg1.l4.m3.8.8.3.2.1.cmml" xref="alg1.l4.m3.8.8.3.2.2"><cn id="alg1.l4.m3.2.2.cmml" type="integer" xref="alg1.l4.m3.2.2">1</cn><ci id="alg1.l4.m3.3.3.cmml" xref="alg1.l4.m3.3.3">⋯</ci><ci id="alg1.l4.m3.4.4.cmml" xref="alg1.l4.m3.4.4">𝑁</ci></set><set id="alg1.l4.m3.8.8.3.3.1.cmml" xref="alg1.l4.m3.8.8.3.3.2"><cn id="alg1.l4.m3.5.5.cmml" type="integer" xref="alg1.l4.m3.5.5">1</cn><ci id="alg1.l4.m3.6.6.cmml" xref="alg1.l4.m3.6.6">⋯</ci><ci id="alg1.l4.m3.7.7.cmml" xref="alg1.l4.m3.7.7">𝑁</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m3.8c">(i,i^{\prime})=\{1,\cdots,N\}\otimes\{1,\cdots,N\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m3.8d">( italic_i , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { 1 , ⋯ , italic_N } ⊗ { 1 , ⋯ , italic_N }</annotation></semantics></math> \If<math alttext="i==i^{\prime}" class="ltx_math_unparsed" display="inline" id="alg1.l4.m4.1"><semantics id="alg1.l4.m4.1a"><mrow id="alg1.l4.m4.1b"><mi id="alg1.l4.m4.1.1">i</mi><mo id="alg1.l4.m4.1.2" rspace="0em">=</mo><mo id="alg1.l4.m4.1.3" lspace="0em">=</mo><msup id="alg1.l4.m4.1.4"><mi id="alg1.l4.m4.1.4.2">i</mi><mo id="alg1.l4.m4.1.4.3">′</mo></msup></mrow><annotation encoding="application/x-tex" id="alg1.l4.m4.1c">i==i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m4.1d">italic_i = = italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> continue \EndIf\LComment(Optional) parallelize the following \For<math alttext="n=1,\dots,N_{end,M^{i}}^{i}" class="ltx_Math" display="inline" id="alg1.l4.m5.5"><semantics id="alg1.l4.m5.5a"><mrow id="alg1.l4.m5.5.5" xref="alg1.l4.m5.5.5.cmml"><mi id="alg1.l4.m5.5.5.3" xref="alg1.l4.m5.5.5.3.cmml">n</mi><mo id="alg1.l4.m5.5.5.2" xref="alg1.l4.m5.5.5.2.cmml">=</mo><mrow id="alg1.l4.m5.5.5.1.1" xref="alg1.l4.m5.5.5.1.2.cmml"><mn id="alg1.l4.m5.3.3" xref="alg1.l4.m5.3.3.cmml">1</mn><mo id="alg1.l4.m5.5.5.1.1.2" xref="alg1.l4.m5.5.5.1.2.cmml">,</mo><mi id="alg1.l4.m5.4.4" mathvariant="normal" xref="alg1.l4.m5.4.4.cmml">…</mi><mo id="alg1.l4.m5.5.5.1.1.3" xref="alg1.l4.m5.5.5.1.2.cmml">,</mo><msubsup id="alg1.l4.m5.5.5.1.1.1" xref="alg1.l4.m5.5.5.1.1.1.cmml"><mi id="alg1.l4.m5.5.5.1.1.1.2.2" xref="alg1.l4.m5.5.5.1.1.1.2.2.cmml">N</mi><mrow id="alg1.l4.m5.2.2.2.2" xref="alg1.l4.m5.2.2.2.3.cmml"><mrow id="alg1.l4.m5.1.1.1.1.1" xref="alg1.l4.m5.1.1.1.1.1.cmml"><mi id="alg1.l4.m5.1.1.1.1.1.2" xref="alg1.l4.m5.1.1.1.1.1.2.cmml">e</mi><mo id="alg1.l4.m5.1.1.1.1.1.1" xref="alg1.l4.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m5.1.1.1.1.1.3" xref="alg1.l4.m5.1.1.1.1.1.3.cmml">n</mi><mo id="alg1.l4.m5.1.1.1.1.1.1a" xref="alg1.l4.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l4.m5.1.1.1.1.1.4" xref="alg1.l4.m5.1.1.1.1.1.4.cmml">d</mi></mrow><mo id="alg1.l4.m5.2.2.2.2.3" xref="alg1.l4.m5.2.2.2.3.cmml">,</mo><msup id="alg1.l4.m5.2.2.2.2.2" xref="alg1.l4.m5.2.2.2.2.2.cmml"><mi id="alg1.l4.m5.2.2.2.2.2.2" xref="alg1.l4.m5.2.2.2.2.2.2.cmml">M</mi><mi id="alg1.l4.m5.2.2.2.2.2.3" xref="alg1.l4.m5.2.2.2.2.2.3.cmml">i</mi></msup></mrow><mi id="alg1.l4.m5.5.5.1.1.1.3" xref="alg1.l4.m5.5.5.1.1.1.3.cmml">i</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m5.5b"><apply id="alg1.l4.m5.5.5.cmml" xref="alg1.l4.m5.5.5"><eq id="alg1.l4.m5.5.5.2.cmml" xref="alg1.l4.m5.5.5.2"></eq><ci id="alg1.l4.m5.5.5.3.cmml" xref="alg1.l4.m5.5.5.3">𝑛</ci><list id="alg1.l4.m5.5.5.1.2.cmml" xref="alg1.l4.m5.5.5.1.1"><cn id="alg1.l4.m5.3.3.cmml" type="integer" xref="alg1.l4.m5.3.3">1</cn><ci id="alg1.l4.m5.4.4.cmml" xref="alg1.l4.m5.4.4">…</ci><apply id="alg1.l4.m5.5.5.1.1.1.cmml" xref="alg1.l4.m5.5.5.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m5.5.5.1.1.1.1.cmml" xref="alg1.l4.m5.5.5.1.1.1">superscript</csymbol><apply id="alg1.l4.m5.5.5.1.1.1.2.cmml" xref="alg1.l4.m5.5.5.1.1.1"><csymbol cd="ambiguous" id="alg1.l4.m5.5.5.1.1.1.2.1.cmml" xref="alg1.l4.m5.5.5.1.1.1">subscript</csymbol><ci id="alg1.l4.m5.5.5.1.1.1.2.2.cmml" xref="alg1.l4.m5.5.5.1.1.1.2.2">𝑁</ci><list id="alg1.l4.m5.2.2.2.3.cmml" xref="alg1.l4.m5.2.2.2.2"><apply id="alg1.l4.m5.1.1.1.1.1.cmml" xref="alg1.l4.m5.1.1.1.1.1"><times id="alg1.l4.m5.1.1.1.1.1.1.cmml" xref="alg1.l4.m5.1.1.1.1.1.1"></times><ci id="alg1.l4.m5.1.1.1.1.1.2.cmml" xref="alg1.l4.m5.1.1.1.1.1.2">𝑒</ci><ci id="alg1.l4.m5.1.1.1.1.1.3.cmml" xref="alg1.l4.m5.1.1.1.1.1.3">𝑛</ci><ci id="alg1.l4.m5.1.1.1.1.1.4.cmml" xref="alg1.l4.m5.1.1.1.1.1.4">𝑑</ci></apply><apply id="alg1.l4.m5.2.2.2.2.2.cmml" xref="alg1.l4.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l4.m5.2.2.2.2.2.1.cmml" xref="alg1.l4.m5.2.2.2.2.2">superscript</csymbol><ci id="alg1.l4.m5.2.2.2.2.2.2.cmml" xref="alg1.l4.m5.2.2.2.2.2.2">𝑀</ci><ci id="alg1.l4.m5.2.2.2.2.2.3.cmml" xref="alg1.l4.m5.2.2.2.2.2.3">𝑖</ci></apply></list></apply><ci id="alg1.l4.m5.5.5.1.1.1.3.cmml" xref="alg1.l4.m5.5.5.1.1.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m5.5c">n=1,\dots,N_{end,M^{i}}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m5.5d">italic_n = 1 , … , italic_N start_POSTSUBSCRIPT italic_e italic_n italic_d , italic_M start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l5"> 5:Update robot <math alttext="i" class="ltx_Math" display="inline" id="alg1.l5.m1.1"><semantics id="alg1.l5.m1.1a"><mi id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><ci id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.1d">italic_i</annotation></semantics></math> kinematic if needed \For<math alttext="n^{\prime}=1,\dots,N_{end,M^{i^{\prime}}}^{i^{\prime}}" class="ltx_Math" display="inline" id="alg1.l5.m2.5"><semantics id="alg1.l5.m2.5a"><mrow id="alg1.l5.m2.5.5" xref="alg1.l5.m2.5.5.cmml"><msup id="alg1.l5.m2.5.5.3" xref="alg1.l5.m2.5.5.3.cmml"><mi id="alg1.l5.m2.5.5.3.2" xref="alg1.l5.m2.5.5.3.2.cmml">n</mi><mo id="alg1.l5.m2.5.5.3.3" xref="alg1.l5.m2.5.5.3.3.cmml">′</mo></msup><mo id="alg1.l5.m2.5.5.2" xref="alg1.l5.m2.5.5.2.cmml">=</mo><mrow id="alg1.l5.m2.5.5.1.1" xref="alg1.l5.m2.5.5.1.2.cmml"><mn id="alg1.l5.m2.3.3" xref="alg1.l5.m2.3.3.cmml">1</mn><mo id="alg1.l5.m2.5.5.1.1.2" xref="alg1.l5.m2.5.5.1.2.cmml">,</mo><mi id="alg1.l5.m2.4.4" mathvariant="normal" xref="alg1.l5.m2.4.4.cmml">…</mi><mo id="alg1.l5.m2.5.5.1.1.3" xref="alg1.l5.m2.5.5.1.2.cmml">,</mo><msubsup id="alg1.l5.m2.5.5.1.1.1" xref="alg1.l5.m2.5.5.1.1.1.cmml"><mi id="alg1.l5.m2.5.5.1.1.1.2.2" xref="alg1.l5.m2.5.5.1.1.1.2.2.cmml">N</mi><mrow id="alg1.l5.m2.2.2.2.2" xref="alg1.l5.m2.2.2.2.3.cmml"><mrow id="alg1.l5.m2.1.1.1.1.1" xref="alg1.l5.m2.1.1.1.1.1.cmml"><mi id="alg1.l5.m2.1.1.1.1.1.2" xref="alg1.l5.m2.1.1.1.1.1.2.cmml">e</mi><mo id="alg1.l5.m2.1.1.1.1.1.1" xref="alg1.l5.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l5.m2.1.1.1.1.1.3" xref="alg1.l5.m2.1.1.1.1.1.3.cmml">n</mi><mo id="alg1.l5.m2.1.1.1.1.1.1a" xref="alg1.l5.m2.1.1.1.1.1.1.cmml">⁢</mo><mi id="alg1.l5.m2.1.1.1.1.1.4" xref="alg1.l5.m2.1.1.1.1.1.4.cmml">d</mi></mrow><mo id="alg1.l5.m2.2.2.2.2.3" xref="alg1.l5.m2.2.2.2.3.cmml">,</mo><msup id="alg1.l5.m2.2.2.2.2.2" xref="alg1.l5.m2.2.2.2.2.2.cmml"><mi id="alg1.l5.m2.2.2.2.2.2.2" xref="alg1.l5.m2.2.2.2.2.2.2.cmml">M</mi><msup id="alg1.l5.m2.2.2.2.2.2.3" xref="alg1.l5.m2.2.2.2.2.2.3.cmml"><mi id="alg1.l5.m2.2.2.2.2.2.3.2" xref="alg1.l5.m2.2.2.2.2.2.3.2.cmml">i</mi><mo id="alg1.l5.m2.2.2.2.2.2.3.3" xref="alg1.l5.m2.2.2.2.2.2.3.3.cmml">′</mo></msup></msup></mrow><msup id="alg1.l5.m2.5.5.1.1.1.3" xref="alg1.l5.m2.5.5.1.1.1.3.cmml"><mi id="alg1.l5.m2.5.5.1.1.1.3.2" xref="alg1.l5.m2.5.5.1.1.1.3.2.cmml">i</mi><mo id="alg1.l5.m2.5.5.1.1.1.3.3" xref="alg1.l5.m2.5.5.1.1.1.3.3.cmml">′</mo></msup></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m2.5b"><apply id="alg1.l5.m2.5.5.cmml" xref="alg1.l5.m2.5.5"><eq id="alg1.l5.m2.5.5.2.cmml" xref="alg1.l5.m2.5.5.2"></eq><apply id="alg1.l5.m2.5.5.3.cmml" xref="alg1.l5.m2.5.5.3"><csymbol cd="ambiguous" id="alg1.l5.m2.5.5.3.1.cmml" xref="alg1.l5.m2.5.5.3">superscript</csymbol><ci id="alg1.l5.m2.5.5.3.2.cmml" xref="alg1.l5.m2.5.5.3.2">𝑛</ci><ci id="alg1.l5.m2.5.5.3.3.cmml" xref="alg1.l5.m2.5.5.3.3">′</ci></apply><list id="alg1.l5.m2.5.5.1.2.cmml" xref="alg1.l5.m2.5.5.1.1"><cn id="alg1.l5.m2.3.3.cmml" type="integer" xref="alg1.l5.m2.3.3">1</cn><ci id="alg1.l5.m2.4.4.cmml" xref="alg1.l5.m2.4.4">…</ci><apply id="alg1.l5.m2.5.5.1.1.1.cmml" xref="alg1.l5.m2.5.5.1.1.1"><csymbol cd="ambiguous" id="alg1.l5.m2.5.5.1.1.1.1.cmml" xref="alg1.l5.m2.5.5.1.1.1">superscript</csymbol><apply id="alg1.l5.m2.5.5.1.1.1.2.cmml" xref="alg1.l5.m2.5.5.1.1.1"><csymbol cd="ambiguous" id="alg1.l5.m2.5.5.1.1.1.2.1.cmml" xref="alg1.l5.m2.5.5.1.1.1">subscript</csymbol><ci id="alg1.l5.m2.5.5.1.1.1.2.2.cmml" xref="alg1.l5.m2.5.5.1.1.1.2.2">𝑁</ci><list id="alg1.l5.m2.2.2.2.3.cmml" xref="alg1.l5.m2.2.2.2.2"><apply id="alg1.l5.m2.1.1.1.1.1.cmml" xref="alg1.l5.m2.1.1.1.1.1"><times id="alg1.l5.m2.1.1.1.1.1.1.cmml" xref="alg1.l5.m2.1.1.1.1.1.1"></times><ci id="alg1.l5.m2.1.1.1.1.1.2.cmml" xref="alg1.l5.m2.1.1.1.1.1.2">𝑒</ci><ci id="alg1.l5.m2.1.1.1.1.1.3.cmml" xref="alg1.l5.m2.1.1.1.1.1.3">𝑛</ci><ci id="alg1.l5.m2.1.1.1.1.1.4.cmml" xref="alg1.l5.m2.1.1.1.1.1.4">𝑑</ci></apply><apply id="alg1.l5.m2.2.2.2.2.2.cmml" xref="alg1.l5.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l5.m2.2.2.2.2.2.1.cmml" xref="alg1.l5.m2.2.2.2.2.2">superscript</csymbol><ci id="alg1.l5.m2.2.2.2.2.2.2.cmml" xref="alg1.l5.m2.2.2.2.2.2.2">𝑀</ci><apply id="alg1.l5.m2.2.2.2.2.2.3.cmml" xref="alg1.l5.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="alg1.l5.m2.2.2.2.2.2.3.1.cmml" xref="alg1.l5.m2.2.2.2.2.2.3">superscript</csymbol><ci id="alg1.l5.m2.2.2.2.2.2.3.2.cmml" xref="alg1.l5.m2.2.2.2.2.2.3.2">𝑖</ci><ci id="alg1.l5.m2.2.2.2.2.2.3.3.cmml" xref="alg1.l5.m2.2.2.2.2.2.3.3">′</ci></apply></apply></list></apply><apply id="alg1.l5.m2.5.5.1.1.1.3.cmml" xref="alg1.l5.m2.5.5.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l5.m2.5.5.1.1.1.3.1.cmml" xref="alg1.l5.m2.5.5.1.1.1.3">superscript</csymbol><ci id="alg1.l5.m2.5.5.1.1.1.3.2.cmml" xref="alg1.l5.m2.5.5.1.1.1.3.2">𝑖</ci><ci id="alg1.l5.m2.5.5.1.1.1.3.3.cmml" xref="alg1.l5.m2.5.5.1.1.1.3.3">′</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m2.5c">n^{\prime}=1,\dots,N_{end,M^{i^{\prime}}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m2.5d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 1 , … , italic_N start_POSTSUBSCRIPT italic_e italic_n italic_d , italic_M start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l6"> 6:Update robot <math alttext="i^{\prime}" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><msup id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><mi id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml">i</mi><mo id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1">superscript</csymbol><ci id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2">𝑖</ci><ci id="alg1.l6.m1.1.1.3.cmml" xref="alg1.l6.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> kinematic if needed \If<math alttext="t_{n^{\prime}}^{i^{\prime}}\geq t_{n}^{i}" class="ltx_Math" display="inline" id="alg1.l6.m2.1"><semantics id="alg1.l6.m2.1a"><mrow id="alg1.l6.m2.1.1" xref="alg1.l6.m2.1.1.cmml"><msubsup id="alg1.l6.m2.1.1.2" xref="alg1.l6.m2.1.1.2.cmml"><mi id="alg1.l6.m2.1.1.2.2.2" xref="alg1.l6.m2.1.1.2.2.2.cmml">t</mi><msup id="alg1.l6.m2.1.1.2.2.3" xref="alg1.l6.m2.1.1.2.2.3.cmml"><mi id="alg1.l6.m2.1.1.2.2.3.2" xref="alg1.l6.m2.1.1.2.2.3.2.cmml">n</mi><mo id="alg1.l6.m2.1.1.2.2.3.3" xref="alg1.l6.m2.1.1.2.2.3.3.cmml">′</mo></msup><msup id="alg1.l6.m2.1.1.2.3" xref="alg1.l6.m2.1.1.2.3.cmml"><mi id="alg1.l6.m2.1.1.2.3.2" xref="alg1.l6.m2.1.1.2.3.2.cmml">i</mi><mo id="alg1.l6.m2.1.1.2.3.3" xref="alg1.l6.m2.1.1.2.3.3.cmml">′</mo></msup></msubsup><mo id="alg1.l6.m2.1.1.1" xref="alg1.l6.m2.1.1.1.cmml">≥</mo><msubsup id="alg1.l6.m2.1.1.3" xref="alg1.l6.m2.1.1.3.cmml"><mi id="alg1.l6.m2.1.1.3.2.2" xref="alg1.l6.m2.1.1.3.2.2.cmml">t</mi><mi id="alg1.l6.m2.1.1.3.2.3" xref="alg1.l6.m2.1.1.3.2.3.cmml">n</mi><mi id="alg1.l6.m2.1.1.3.3" xref="alg1.l6.m2.1.1.3.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m2.1b"><apply id="alg1.l6.m2.1.1.cmml" xref="alg1.l6.m2.1.1"><geq id="alg1.l6.m2.1.1.1.cmml" xref="alg1.l6.m2.1.1.1"></geq><apply id="alg1.l6.m2.1.1.2.cmml" xref="alg1.l6.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.2.1.cmml" xref="alg1.l6.m2.1.1.2">superscript</csymbol><apply id="alg1.l6.m2.1.1.2.2.cmml" xref="alg1.l6.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.2.2.1.cmml" xref="alg1.l6.m2.1.1.2">subscript</csymbol><ci id="alg1.l6.m2.1.1.2.2.2.cmml" xref="alg1.l6.m2.1.1.2.2.2">𝑡</ci><apply id="alg1.l6.m2.1.1.2.2.3.cmml" xref="alg1.l6.m2.1.1.2.2.3"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.2.2.3.1.cmml" xref="alg1.l6.m2.1.1.2.2.3">superscript</csymbol><ci id="alg1.l6.m2.1.1.2.2.3.2.cmml" xref="alg1.l6.m2.1.1.2.2.3.2">𝑛</ci><ci id="alg1.l6.m2.1.1.2.2.3.3.cmml" xref="alg1.l6.m2.1.1.2.2.3.3">′</ci></apply></apply><apply id="alg1.l6.m2.1.1.2.3.cmml" xref="alg1.l6.m2.1.1.2.3"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.2.3.1.cmml" xref="alg1.l6.m2.1.1.2.3">superscript</csymbol><ci id="alg1.l6.m2.1.1.2.3.2.cmml" xref="alg1.l6.m2.1.1.2.3.2">𝑖</ci><ci id="alg1.l6.m2.1.1.2.3.3.cmml" xref="alg1.l6.m2.1.1.2.3.3">′</ci></apply></apply><apply id="alg1.l6.m2.1.1.3.cmml" xref="alg1.l6.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.3.1.cmml" xref="alg1.l6.m2.1.1.3">superscript</csymbol><apply id="alg1.l6.m2.1.1.3.2.cmml" xref="alg1.l6.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l6.m2.1.1.3.2.1.cmml" xref="alg1.l6.m2.1.1.3">subscript</csymbol><ci id="alg1.l6.m2.1.1.3.2.2.cmml" xref="alg1.l6.m2.1.1.3.2.2">𝑡</ci><ci id="alg1.l6.m2.1.1.3.2.3.cmml" xref="alg1.l6.m2.1.1.3.2.3">𝑛</ci></apply><ci id="alg1.l6.m2.1.1.3.3.cmml" xref="alg1.l6.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m2.1c">t_{n^{\prime}}^{i^{\prime}}\geq t_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m2.1d">italic_t start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ≥ italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> or <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="alg1.l6.m3.1"><semantics id="alg1.l6.m3.1a"><msubsup id="alg1.l6.m3.1.1" xref="alg1.l6.m3.1.1.cmml"><mi id="alg1.l6.m3.1.1.2.2" xref="alg1.l6.m3.1.1.2.2.cmml">v</mi><mi id="alg1.l6.m3.1.1.2.3" xref="alg1.l6.m3.1.1.2.3.cmml">n</mi><mi id="alg1.l6.m3.1.1.3" xref="alg1.l6.m3.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l6.m3.1b"><apply id="alg1.l6.m3.1.1.cmml" xref="alg1.l6.m3.1.1"><csymbol cd="ambiguous" id="alg1.l6.m3.1.1.1.cmml" xref="alg1.l6.m3.1.1">superscript</csymbol><apply id="alg1.l6.m3.1.1.2.cmml" xref="alg1.l6.m3.1.1"><csymbol cd="ambiguous" id="alg1.l6.m3.1.1.2.1.cmml" xref="alg1.l6.m3.1.1">subscript</csymbol><ci id="alg1.l6.m3.1.1.2.2.cmml" xref="alg1.l6.m3.1.1.2.2">𝑣</ci><ci id="alg1.l6.m3.1.1.2.3.cmml" xref="alg1.l6.m3.1.1.2.3">𝑛</ci></apply><ci id="alg1.l6.m3.1.1.3.cmml" xref="alg1.l6.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m3.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m3.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> depends on <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="alg1.l6.m4.1"><semantics id="alg1.l6.m4.1a"><msubsup id="alg1.l6.m4.1.1" xref="alg1.l6.m4.1.1.cmml"><mi id="alg1.l6.m4.1.1.2.2" xref="alg1.l6.m4.1.1.2.2.cmml">v</mi><msup id="alg1.l6.m4.1.1.2.3" xref="alg1.l6.m4.1.1.2.3.cmml"><mi id="alg1.l6.m4.1.1.2.3.2" xref="alg1.l6.m4.1.1.2.3.2.cmml">n</mi><mo id="alg1.l6.m4.1.1.2.3.3" xref="alg1.l6.m4.1.1.2.3.3.cmml">′</mo></msup><msup id="alg1.l6.m4.1.1.3" xref="alg1.l6.m4.1.1.3.cmml"><mi id="alg1.l6.m4.1.1.3.2" xref="alg1.l6.m4.1.1.3.2.cmml">i</mi><mo id="alg1.l6.m4.1.1.3.3" xref="alg1.l6.m4.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l6.m4.1b"><apply id="alg1.l6.m4.1.1.cmml" xref="alg1.l6.m4.1.1"><csymbol cd="ambiguous" id="alg1.l6.m4.1.1.1.cmml" xref="alg1.l6.m4.1.1">superscript</csymbol><apply id="alg1.l6.m4.1.1.2.cmml" xref="alg1.l6.m4.1.1"><csymbol cd="ambiguous" id="alg1.l6.m4.1.1.2.1.cmml" xref="alg1.l6.m4.1.1">subscript</csymbol><ci id="alg1.l6.m4.1.1.2.2.cmml" xref="alg1.l6.m4.1.1.2.2">𝑣</ci><apply id="alg1.l6.m4.1.1.2.3.cmml" xref="alg1.l6.m4.1.1.2.3"><csymbol cd="ambiguous" id="alg1.l6.m4.1.1.2.3.1.cmml" xref="alg1.l6.m4.1.1.2.3">superscript</csymbol><ci id="alg1.l6.m4.1.1.2.3.2.cmml" xref="alg1.l6.m4.1.1.2.3.2">𝑛</ci><ci id="alg1.l6.m4.1.1.2.3.3.cmml" xref="alg1.l6.m4.1.1.2.3.3">′</ci></apply></apply><apply id="alg1.l6.m4.1.1.3.cmml" xref="alg1.l6.m4.1.1.3"><csymbol cd="ambiguous" id="alg1.l6.m4.1.1.3.1.cmml" xref="alg1.l6.m4.1.1.3">superscript</csymbol><ci id="alg1.l6.m4.1.1.3.2.cmml" xref="alg1.l6.m4.1.1.3.2">𝑖</ci><ci id="alg1.l6.m4.1.1.3.3.cmml" xref="alg1.l6.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m4.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m4.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l7"> 7:continue \EndIf\If<math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="alg1.l7.m1.1"><semantics id="alg1.l7.m1.1a"><msubsup id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml"><mi id="alg1.l7.m1.1.1.2.2" xref="alg1.l7.m1.1.1.2.2.cmml">v</mi><msup id="alg1.l7.m1.1.1.2.3" xref="alg1.l7.m1.1.1.2.3.cmml"><mi id="alg1.l7.m1.1.1.2.3.2" xref="alg1.l7.m1.1.1.2.3.2.cmml">n</mi><mo id="alg1.l7.m1.1.1.2.3.3" xref="alg1.l7.m1.1.1.2.3.3.cmml">′</mo></msup><msup id="alg1.l7.m1.1.1.3" xref="alg1.l7.m1.1.1.3.cmml"><mi id="alg1.l7.m1.1.1.3.2" xref="alg1.l7.m1.1.1.3.2.cmml">i</mi><mo id="alg1.l7.m1.1.1.3.3" xref="alg1.l7.m1.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.1b"><apply id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.cmml" xref="alg1.l7.m1.1.1">superscript</csymbol><apply id="alg1.l7.m1.1.1.2.cmml" xref="alg1.l7.m1.1.1"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.2.1.cmml" xref="alg1.l7.m1.1.1">subscript</csymbol><ci id="alg1.l7.m1.1.1.2.2.cmml" xref="alg1.l7.m1.1.1.2.2">𝑣</ci><apply id="alg1.l7.m1.1.1.2.3.cmml" xref="alg1.l7.m1.1.1.2.3"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.2.3.1.cmml" xref="alg1.l7.m1.1.1.2.3">superscript</csymbol><ci id="alg1.l7.m1.1.1.2.3.2.cmml" xref="alg1.l7.m1.1.1.2.3.2">𝑛</ci><ci id="alg1.l7.m1.1.1.2.3.3.cmml" xref="alg1.l7.m1.1.1.2.3.3">′</ci></apply></apply><apply id="alg1.l7.m1.1.1.3.cmml" xref="alg1.l7.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.3.1.cmml" xref="alg1.l7.m1.1.1.3">superscript</csymbol><ci id="alg1.l7.m1.1.1.3.2.cmml" xref="alg1.l7.m1.1.1.3.2">𝑖</ci><ci id="alg1.l7.m1.1.1.3.3.cmml" xref="alg1.l7.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> collides with <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="alg1.l7.m2.1"><semantics id="alg1.l7.m2.1a"><msubsup id="alg1.l7.m2.1.1" xref="alg1.l7.m2.1.1.cmml"><mi id="alg1.l7.m2.1.1.2.2" xref="alg1.l7.m2.1.1.2.2.cmml">v</mi><mi id="alg1.l7.m2.1.1.2.3" xref="alg1.l7.m2.1.1.2.3.cmml">n</mi><mi id="alg1.l7.m2.1.1.3" xref="alg1.l7.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l7.m2.1b"><apply id="alg1.l7.m2.1.1.cmml" xref="alg1.l7.m2.1.1"><csymbol cd="ambiguous" id="alg1.l7.m2.1.1.1.cmml" xref="alg1.l7.m2.1.1">superscript</csymbol><apply id="alg1.l7.m2.1.1.2.cmml" xref="alg1.l7.m2.1.1"><csymbol cd="ambiguous" id="alg1.l7.m2.1.1.2.1.cmml" xref="alg1.l7.m2.1.1">subscript</csymbol><ci id="alg1.l7.m2.1.1.2.2.cmml" xref="alg1.l7.m2.1.1.2.2">𝑣</ci><ci id="alg1.l7.m2.1.1.2.3.cmml" xref="alg1.l7.m2.1.1.2.3">𝑛</ci></apply><ci id="alg1.l7.m2.1.1.3.cmml" xref="alg1.l7.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m2.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m2.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l8"> 8:Add type-2 edge from <math alttext="v_{n^{\prime}+1}^{i^{\prime}}" class="ltx_Math" display="inline" id="alg1.l8.m1.1"><semantics id="alg1.l8.m1.1a"><msubsup id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml"><mi id="alg1.l8.m1.1.1.2.2" xref="alg1.l8.m1.1.1.2.2.cmml">v</mi><mrow id="alg1.l8.m1.1.1.2.3" xref="alg1.l8.m1.1.1.2.3.cmml"><msup id="alg1.l8.m1.1.1.2.3.2" xref="alg1.l8.m1.1.1.2.3.2.cmml"><mi id="alg1.l8.m1.1.1.2.3.2.2" xref="alg1.l8.m1.1.1.2.3.2.2.cmml">n</mi><mo id="alg1.l8.m1.1.1.2.3.2.3" xref="alg1.l8.m1.1.1.2.3.2.3.cmml">′</mo></msup><mo id="alg1.l8.m1.1.1.2.3.1" xref="alg1.l8.m1.1.1.2.3.1.cmml">+</mo><mn id="alg1.l8.m1.1.1.2.3.3" xref="alg1.l8.m1.1.1.2.3.3.cmml">1</mn></mrow><msup id="alg1.l8.m1.1.1.3" xref="alg1.l8.m1.1.1.3.cmml"><mi id="alg1.l8.m1.1.1.3.2" xref="alg1.l8.m1.1.1.3.2.cmml">i</mi><mo id="alg1.l8.m1.1.1.3.3" xref="alg1.l8.m1.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.1b"><apply id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.1.cmml" xref="alg1.l8.m1.1.1">superscript</csymbol><apply id="alg1.l8.m1.1.1.2.cmml" xref="alg1.l8.m1.1.1"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.2.1.cmml" xref="alg1.l8.m1.1.1">subscript</csymbol><ci id="alg1.l8.m1.1.1.2.2.cmml" xref="alg1.l8.m1.1.1.2.2">𝑣</ci><apply id="alg1.l8.m1.1.1.2.3.cmml" xref="alg1.l8.m1.1.1.2.3"><plus id="alg1.l8.m1.1.1.2.3.1.cmml" xref="alg1.l8.m1.1.1.2.3.1"></plus><apply id="alg1.l8.m1.1.1.2.3.2.cmml" xref="alg1.l8.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.2.3.2.1.cmml" xref="alg1.l8.m1.1.1.2.3.2">superscript</csymbol><ci id="alg1.l8.m1.1.1.2.3.2.2.cmml" xref="alg1.l8.m1.1.1.2.3.2.2">𝑛</ci><ci id="alg1.l8.m1.1.1.2.3.2.3.cmml" xref="alg1.l8.m1.1.1.2.3.2.3">′</ci></apply><cn id="alg1.l8.m1.1.1.2.3.3.cmml" type="integer" xref="alg1.l8.m1.1.1.2.3.3">1</cn></apply></apply><apply id="alg1.l8.m1.1.1.3.cmml" xref="alg1.l8.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.3.1.cmml" xref="alg1.l8.m1.1.1.3">superscript</csymbol><ci id="alg1.l8.m1.1.1.3.2.cmml" xref="alg1.l8.m1.1.1.3.2">𝑖</ci><ci id="alg1.l8.m1.1.1.3.3.cmml" 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xref="alg1.l8.m2.1.1">superscript</csymbol><apply id="alg1.l8.m2.1.1.2.cmml" xref="alg1.l8.m2.1.1"><csymbol cd="ambiguous" id="alg1.l8.m2.1.1.2.1.cmml" xref="alg1.l8.m2.1.1">subscript</csymbol><ci id="alg1.l8.m2.1.1.2.2.cmml" xref="alg1.l8.m2.1.1.2.2">𝑣</ci><ci id="alg1.l8.m2.1.1.2.3.cmml" xref="alg1.l8.m2.1.1.2.3">𝑛</ci></apply><ci id="alg1.l8.m2.1.1.3.cmml" xref="alg1.l8.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m2.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m2.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> \EndIf\EndFor\EndFor\EndFor </div> <div class="ltx_listingline" id="alg1.l9"> 9:Simplify TPG with transitive reduction. </div> </div> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_4">\LComment</div> <div class="ltx_flex_cell ltx_flex_size_4">\For</div> <div class="ltx_flex_cell ltx_flex_size_4">\For</div> <div class="ltx_flex_cell ltx_flex_size_4">\If</div> </div> </figure> <div class="ltx_para" id="S4.SS3.p3"> Building a multi-modal TPG The process of constructing a TPG from a multi-robot motion plan can be interpreted as converting from the sequential robot trajectories to temporally dependent robot schedules. On a high level, inter-robot task dependencies are copied as type-2 edges for TPG, and type-2 edges for motion dependencies are constructed by scanning for collisions between all pair of TPG nodes. A key benefit of this partial-order representation is that TPG does not specify a fixed time between two consecutive nodes, and thus allows execution delays. We outline the detailed procedure below. </div> <div class="ltx_para" id="S4.SS3.p4"> The construction process begins by creating the nodes and type-1 edges. For each transit task <math alttext="\mathcal{T}_{m}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.1.m1.1"><semantics id="S4.SS3.p4.1.m1.1a"><msubsup id="S4.SS3.p4.1.m1.1.1" xref="S4.SS3.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p4.1.m1.1.1.2.2" xref="S4.SS3.p4.1.m1.1.1.2.2.cmml">𝒯</mi><mi id="S4.SS3.p4.1.m1.1.1.2.3" xref="S4.SS3.p4.1.m1.1.1.2.3.cmml">m</mi><mi id="S4.SS3.p4.1.m1.1.1.3" xref="S4.SS3.p4.1.m1.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.1.m1.1b"><apply id="S4.SS3.p4.1.m1.1.1.cmml" xref="S4.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.1.m1.1.1.1.cmml" xref="S4.SS3.p4.1.m1.1.1">superscript</csymbol><apply id="S4.SS3.p4.1.m1.1.1.2.cmml" xref="S4.SS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.1.m1.1.1.2.1.cmml" xref="S4.SS3.p4.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.p4.1.m1.1.1.2.2.cmml" xref="S4.SS3.p4.1.m1.1.1.2.2">𝒯</ci><ci id="S4.SS3.p4.1.m1.1.1.2.3.cmml" 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cd="ambiguous" id="S4.SS3.p4.2.m2.1.1.1.cmml" xref="S4.SS3.p4.2.m2.1.1">superscript</csymbol><apply id="S4.SS3.p4.2.m2.1.1.2.cmml" xref="S4.SS3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.2.m2.1.1.2.1.cmml" xref="S4.SS3.p4.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.p4.2.m2.1.1.2.2.cmml" xref="S4.SS3.p4.2.m2.1.1.2.2">𝑣</ci><ci id="S4.SS3.p4.2.m2.1.1.2.3.cmml" xref="S4.SS3.p4.2.m2.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p4.2.m2.1.1.3.cmml" xref="S4.SS3.p4.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.2.m2.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.2.m2.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is constructed for each configuration <math alttext="C_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.3.m3.1"><semantics id="S4.SS3.p4.3.m3.1a"><msubsup id="S4.SS3.p4.3.m3.1.1" 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xref="S4.SS3.p4.4.m4.1.1">subscript</csymbol><ci id="S4.SS3.p4.4.m4.1.1.2.2.cmml" xref="S4.SS3.p4.4.m4.1.1.2.2">𝜏</ci><ci id="S4.SS3.p4.4.m4.1.1.2.3.cmml" xref="S4.SS3.p4.4.m4.1.1.2.3">𝑚</ci></apply><ci id="S4.SS3.p4.4.m4.1.1.3.cmml" xref="S4.SS3.p4.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.4.m4.1c">\tau_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.4.m4.1d">italic_τ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. A skill node <math alttext="\tilde{v}_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.5.m5.1"><semantics id="S4.SS3.p4.5.m5.1a"><msubsup id="S4.SS3.p4.5.m5.1.1" xref="S4.SS3.p4.5.m5.1.1.cmml"><mover accent="true" id="S4.SS3.p4.5.m5.1.1.2.2" xref="S4.SS3.p4.5.m5.1.1.2.2.cmml"><mi id="S4.SS3.p4.5.m5.1.1.2.2.2" xref="S4.SS3.p4.5.m5.1.1.2.2.2.cmml">v</mi><mo id="S4.SS3.p4.5.m5.1.1.2.2.1" xref="S4.SS3.p4.5.m5.1.1.2.2.1.cmml">~</mo></mover><mi id="S4.SS3.p4.5.m5.1.1.2.3" xref="S4.SS3.p4.5.m5.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p4.5.m5.1.1.3" xref="S4.SS3.p4.5.m5.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.5.m5.1b"><apply id="S4.SS3.p4.5.m5.1.1.cmml" xref="S4.SS3.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.5.m5.1.1.1.cmml" xref="S4.SS3.p4.5.m5.1.1">superscript</csymbol><apply id="S4.SS3.p4.5.m5.1.1.2.cmml" xref="S4.SS3.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.5.m5.1.1.2.1.cmml" xref="S4.SS3.p4.5.m5.1.1">subscript</csymbol><apply id="S4.SS3.p4.5.m5.1.1.2.2.cmml" xref="S4.SS3.p4.5.m5.1.1.2.2"><ci id="S4.SS3.p4.5.m5.1.1.2.2.1.cmml" xref="S4.SS3.p4.5.m5.1.1.2.2.1">~</ci><ci id="S4.SS3.p4.5.m5.1.1.2.2.2.cmml" xref="S4.SS3.p4.5.m5.1.1.2.2.2">𝑣</ci></apply><ci id="S4.SS3.p4.5.m5.1.1.2.3.cmml" xref="S4.SS3.p4.5.m5.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p4.5.m5.1.1.3.cmml" xref="S4.SS3.p4.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.5.m5.1c">\tilde{v}_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.5.m5.1d">over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is created for every manipulation task. The timestamp in the input trajectory <math alttext="t_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.6.m6.1"><semantics id="S4.SS3.p4.6.m6.1a"><msubsup id="S4.SS3.p4.6.m6.1.1" xref="S4.SS3.p4.6.m6.1.1.cmml"><mi id="S4.SS3.p4.6.m6.1.1.2.2" xref="S4.SS3.p4.6.m6.1.1.2.2.cmml">t</mi><mi id="S4.SS3.p4.6.m6.1.1.2.3" xref="S4.SS3.p4.6.m6.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p4.6.m6.1.1.3" xref="S4.SS3.p4.6.m6.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.6.m6.1b"><apply id="S4.SS3.p4.6.m6.1.1.cmml" xref="S4.SS3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.6.m6.1.1.1.cmml" xref="S4.SS3.p4.6.m6.1.1">superscript</csymbol><apply id="S4.SS3.p4.6.m6.1.1.2.cmml" xref="S4.SS3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.6.m6.1.1.2.1.cmml" xref="S4.SS3.p4.6.m6.1.1">subscript</csymbol><ci id="S4.SS3.p4.6.m6.1.1.2.2.cmml" xref="S4.SS3.p4.6.m6.1.1.2.2">𝑡</ci><ci id="S4.SS3.p4.6.m6.1.1.2.3.cmml" xref="S4.SS3.p4.6.m6.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p4.6.m6.1.1.3.cmml" xref="S4.SS3.p4.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.6.m6.1c">t_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.6.m6.1d">italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> for each node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.7.m7.1"><semantics id="S4.SS3.p4.7.m7.1a"><msubsup id="S4.SS3.p4.7.m7.1.1" xref="S4.SS3.p4.7.m7.1.1.cmml"><mi id="S4.SS3.p4.7.m7.1.1.2.2" xref="S4.SS3.p4.7.m7.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p4.7.m7.1.1.2.3" xref="S4.SS3.p4.7.m7.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p4.7.m7.1.1.3" xref="S4.SS3.p4.7.m7.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.7.m7.1b"><apply id="S4.SS3.p4.7.m7.1.1.cmml" xref="S4.SS3.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.7.m7.1.1.1.cmml" xref="S4.SS3.p4.7.m7.1.1">superscript</csymbol><apply id="S4.SS3.p4.7.m7.1.1.2.cmml" xref="S4.SS3.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.7.m7.1.1.2.1.cmml" xref="S4.SS3.p4.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p4.7.m7.1.1.2.2.cmml" xref="S4.SS3.p4.7.m7.1.1.2.2">𝑣</ci><ci id="S4.SS3.p4.7.m7.1.1.2.3.cmml" xref="S4.SS3.p4.7.m7.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p4.7.m7.1.1.3.cmml" xref="S4.SS3.p4.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.7.m7.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.7.m7.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is also recorded. Each node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.8.m8.1"><semantics id="S4.SS3.p4.8.m8.1a"><msubsup id="S4.SS3.p4.8.m8.1.1" xref="S4.SS3.p4.8.m8.1.1.cmml"><mi id="S4.SS3.p4.8.m8.1.1.2.2" xref="S4.SS3.p4.8.m8.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p4.8.m8.1.1.2.3" xref="S4.SS3.p4.8.m8.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p4.8.m8.1.1.3" xref="S4.SS3.p4.8.m8.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.8.m8.1b"><apply id="S4.SS3.p4.8.m8.1.1.cmml" xref="S4.SS3.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.8.m8.1.1.1.cmml" xref="S4.SS3.p4.8.m8.1.1">superscript</csymbol><apply id="S4.SS3.p4.8.m8.1.1.2.cmml" xref="S4.SS3.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.8.m8.1.1.2.1.cmml" xref="S4.SS3.p4.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.p4.8.m8.1.1.2.2.cmml" xref="S4.SS3.p4.8.m8.1.1.2.2">𝑣</ci><ci id="S4.SS3.p4.8.m8.1.1.2.3.cmml" xref="S4.SS3.p4.8.m8.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p4.8.m8.1.1.3.cmml" xref="S4.SS3.p4.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.8.m8.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.8.m8.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is then connected to its successor <math alttext="v_{n+1}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.9.m9.1"><semantics id="S4.SS3.p4.9.m9.1a"><msubsup id="S4.SS3.p4.9.m9.1.1" xref="S4.SS3.p4.9.m9.1.1.cmml"><mi id="S4.SS3.p4.9.m9.1.1.2.2" xref="S4.SS3.p4.9.m9.1.1.2.2.cmml">v</mi><mrow id="S4.SS3.p4.9.m9.1.1.2.3" xref="S4.SS3.p4.9.m9.1.1.2.3.cmml"><mi id="S4.SS3.p4.9.m9.1.1.2.3.2" xref="S4.SS3.p4.9.m9.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p4.9.m9.1.1.2.3.1" xref="S4.SS3.p4.9.m9.1.1.2.3.1.cmml">+</mo><mn id="S4.SS3.p4.9.m9.1.1.2.3.3" xref="S4.SS3.p4.9.m9.1.1.2.3.3.cmml">1</mn></mrow><mi id="S4.SS3.p4.9.m9.1.1.3" xref="S4.SS3.p4.9.m9.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.9.m9.1b"><apply id="S4.SS3.p4.9.m9.1.1.cmml" xref="S4.SS3.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.9.m9.1.1.1.cmml" xref="S4.SS3.p4.9.m9.1.1">superscript</csymbol><apply id="S4.SS3.p4.9.m9.1.1.2.cmml" xref="S4.SS3.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.9.m9.1.1.2.1.cmml" xref="S4.SS3.p4.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p4.9.m9.1.1.2.2.cmml" xref="S4.SS3.p4.9.m9.1.1.2.2">𝑣</ci><apply id="S4.SS3.p4.9.m9.1.1.2.3.cmml" xref="S4.SS3.p4.9.m9.1.1.2.3"><plus id="S4.SS3.p4.9.m9.1.1.2.3.1.cmml" xref="S4.SS3.p4.9.m9.1.1.2.3.1"></plus><ci id="S4.SS3.p4.9.m9.1.1.2.3.2.cmml" xref="S4.SS3.p4.9.m9.1.1.2.3.2">𝑛</ci><cn id="S4.SS3.p4.9.m9.1.1.2.3.3.cmml" type="integer" xref="S4.SS3.p4.9.m9.1.1.2.3.3">1</cn></apply></apply><ci id="S4.SS3.p4.9.m9.1.1.3.cmml" xref="S4.SS3.p4.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.9.m9.1c">v_{n+1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.9.m9.1d">italic_v start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> by a type-1 edge and the end node of a task <math alttext="v_{end,m}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.10.m10.2"><semantics id="S4.SS3.p4.10.m10.2a"><msubsup id="S4.SS3.p4.10.m10.2.3" xref="S4.SS3.p4.10.m10.2.3.cmml"><mi id="S4.SS3.p4.10.m10.2.3.2.2" xref="S4.SS3.p4.10.m10.2.3.2.2.cmml">v</mi><mrow id="S4.SS3.p4.10.m10.2.2.2.2" xref="S4.SS3.p4.10.m10.2.2.2.3.cmml"><mrow id="S4.SS3.p4.10.m10.2.2.2.2.1" xref="S4.SS3.p4.10.m10.2.2.2.2.1.cmml"><mi id="S4.SS3.p4.10.m10.2.2.2.2.1.2" xref="S4.SS3.p4.10.m10.2.2.2.2.1.2.cmml">e</mi><mo id="S4.SS3.p4.10.m10.2.2.2.2.1.1" xref="S4.SS3.p4.10.m10.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.10.m10.2.2.2.2.1.3" xref="S4.SS3.p4.10.m10.2.2.2.2.1.3.cmml">n</mi><mo id="S4.SS3.p4.10.m10.2.2.2.2.1.1a" xref="S4.SS3.p4.10.m10.2.2.2.2.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.10.m10.2.2.2.2.1.4" xref="S4.SS3.p4.10.m10.2.2.2.2.1.4.cmml">d</mi></mrow><mo id="S4.SS3.p4.10.m10.2.2.2.2.2" xref="S4.SS3.p4.10.m10.2.2.2.3.cmml">,</mo><mi id="S4.SS3.p4.10.m10.1.1.1.1" xref="S4.SS3.p4.10.m10.1.1.1.1.cmml">m</mi></mrow><mi id="S4.SS3.p4.10.m10.2.3.3" xref="S4.SS3.p4.10.m10.2.3.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.10.m10.2b"><apply id="S4.SS3.p4.10.m10.2.3.cmml" xref="S4.SS3.p4.10.m10.2.3"><csymbol cd="ambiguous" id="S4.SS3.p4.10.m10.2.3.1.cmml" xref="S4.SS3.p4.10.m10.2.3">superscript</csymbol><apply id="S4.SS3.p4.10.m10.2.3.2.cmml" xref="S4.SS3.p4.10.m10.2.3"><csymbol cd="ambiguous" id="S4.SS3.p4.10.m10.2.3.2.1.cmml" xref="S4.SS3.p4.10.m10.2.3">subscript</csymbol><ci id="S4.SS3.p4.10.m10.2.3.2.2.cmml" xref="S4.SS3.p4.10.m10.2.3.2.2">𝑣</ci><list id="S4.SS3.p4.10.m10.2.2.2.3.cmml" xref="S4.SS3.p4.10.m10.2.2.2.2"><apply id="S4.SS3.p4.10.m10.2.2.2.2.1.cmml" xref="S4.SS3.p4.10.m10.2.2.2.2.1"><times id="S4.SS3.p4.10.m10.2.2.2.2.1.1.cmml" xref="S4.SS3.p4.10.m10.2.2.2.2.1.1"></times><ci id="S4.SS3.p4.10.m10.2.2.2.2.1.2.cmml" xref="S4.SS3.p4.10.m10.2.2.2.2.1.2">𝑒</ci><ci id="S4.SS3.p4.10.m10.2.2.2.2.1.3.cmml" xref="S4.SS3.p4.10.m10.2.2.2.2.1.3">𝑛</ci><ci id="S4.SS3.p4.10.m10.2.2.2.2.1.4.cmml" xref="S4.SS3.p4.10.m10.2.2.2.2.1.4">𝑑</ci></apply><ci id="S4.SS3.p4.10.m10.1.1.1.1.cmml" xref="S4.SS3.p4.10.m10.1.1.1.1">𝑚</ci></list></apply><ci id="S4.SS3.p4.10.m10.2.3.3.cmml" xref="S4.SS3.p4.10.m10.2.3.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.10.m10.2c">v_{end,m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.10.m10.2d">italic_v start_POSTSUBSCRIPT italic_e italic_n italic_d , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is connected to the start node of the next task <math alttext="v_{start,m+1}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.11.m11.2"><semantics id="S4.SS3.p4.11.m11.2a"><msubsup id="S4.SS3.p4.11.m11.2.3" xref="S4.SS3.p4.11.m11.2.3.cmml"><mi id="S4.SS3.p4.11.m11.2.3.2.2" xref="S4.SS3.p4.11.m11.2.3.2.2.cmml">v</mi><mrow id="S4.SS3.p4.11.m11.2.2.2.2" xref="S4.SS3.p4.11.m11.2.2.2.3.cmml"><mrow id="S4.SS3.p4.11.m11.1.1.1.1.1" xref="S4.SS3.p4.11.m11.1.1.1.1.1.cmml"><mi id="S4.SS3.p4.11.m11.1.1.1.1.1.2" xref="S4.SS3.p4.11.m11.1.1.1.1.1.2.cmml">s</mi><mo id="S4.SS3.p4.11.m11.1.1.1.1.1.1" xref="S4.SS3.p4.11.m11.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.11.m11.1.1.1.1.1.3" xref="S4.SS3.p4.11.m11.1.1.1.1.1.3.cmml">t</mi><mo id="S4.SS3.p4.11.m11.1.1.1.1.1.1a" xref="S4.SS3.p4.11.m11.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.11.m11.1.1.1.1.1.4" xref="S4.SS3.p4.11.m11.1.1.1.1.1.4.cmml">a</mi><mo id="S4.SS3.p4.11.m11.1.1.1.1.1.1b" xref="S4.SS3.p4.11.m11.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.11.m11.1.1.1.1.1.5" xref="S4.SS3.p4.11.m11.1.1.1.1.1.5.cmml">r</mi><mo id="S4.SS3.p4.11.m11.1.1.1.1.1.1c" xref="S4.SS3.p4.11.m11.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p4.11.m11.1.1.1.1.1.6" xref="S4.SS3.p4.11.m11.1.1.1.1.1.6.cmml">t</mi></mrow><mo id="S4.SS3.p4.11.m11.2.2.2.2.3" xref="S4.SS3.p4.11.m11.2.2.2.3.cmml">,</mo><mrow id="S4.SS3.p4.11.m11.2.2.2.2.2" xref="S4.SS3.p4.11.m11.2.2.2.2.2.cmml"><mi id="S4.SS3.p4.11.m11.2.2.2.2.2.2" xref="S4.SS3.p4.11.m11.2.2.2.2.2.2.cmml">m</mi><mo id="S4.SS3.p4.11.m11.2.2.2.2.2.1" xref="S4.SS3.p4.11.m11.2.2.2.2.2.1.cmml">+</mo><mn id="S4.SS3.p4.11.m11.2.2.2.2.2.3" xref="S4.SS3.p4.11.m11.2.2.2.2.2.3.cmml">1</mn></mrow></mrow><mi id="S4.SS3.p4.11.m11.2.3.3" xref="S4.SS3.p4.11.m11.2.3.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.11.m11.2b"><apply id="S4.SS3.p4.11.m11.2.3.cmml" xref="S4.SS3.p4.11.m11.2.3"><csymbol cd="ambiguous" id="S4.SS3.p4.11.m11.2.3.1.cmml" xref="S4.SS3.p4.11.m11.2.3">superscript</csymbol><apply id="S4.SS3.p4.11.m11.2.3.2.cmml" xref="S4.SS3.p4.11.m11.2.3"><csymbol cd="ambiguous" id="S4.SS3.p4.11.m11.2.3.2.1.cmml" xref="S4.SS3.p4.11.m11.2.3">subscript</csymbol><ci id="S4.SS3.p4.11.m11.2.3.2.2.cmml" xref="S4.SS3.p4.11.m11.2.3.2.2">𝑣</ci><list id="S4.SS3.p4.11.m11.2.2.2.3.cmml" xref="S4.SS3.p4.11.m11.2.2.2.2"><apply id="S4.SS3.p4.11.m11.1.1.1.1.1.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1"><times id="S4.SS3.p4.11.m11.1.1.1.1.1.1.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1.1"></times><ci id="S4.SS3.p4.11.m11.1.1.1.1.1.2.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1.2">𝑠</ci><ci id="S4.SS3.p4.11.m11.1.1.1.1.1.3.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1.3">𝑡</ci><ci id="S4.SS3.p4.11.m11.1.1.1.1.1.4.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1.4">𝑎</ci><ci id="S4.SS3.p4.11.m11.1.1.1.1.1.5.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1.5">𝑟</ci><ci id="S4.SS3.p4.11.m11.1.1.1.1.1.6.cmml" xref="S4.SS3.p4.11.m11.1.1.1.1.1.6">𝑡</ci></apply><apply id="S4.SS3.p4.11.m11.2.2.2.2.2.cmml" xref="S4.SS3.p4.11.m11.2.2.2.2.2"><plus id="S4.SS3.p4.11.m11.2.2.2.2.2.1.cmml" xref="S4.SS3.p4.11.m11.2.2.2.2.2.1"></plus><ci id="S4.SS3.p4.11.m11.2.2.2.2.2.2.cmml" xref="S4.SS3.p4.11.m11.2.2.2.2.2.2">𝑚</ci><cn id="S4.SS3.p4.11.m11.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS3.p4.11.m11.2.2.2.2.2.3">1</cn></apply></list></apply><ci id="S4.SS3.p4.11.m11.2.3.3.cmml" xref="S4.SS3.p4.11.m11.2.3.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.11.m11.2c">v_{start,m+1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.11.m11.2d">italic_v start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. When a robot waits at a node <math alttext="v_{m}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.12.m12.1"><semantics id="S4.SS3.p4.12.m12.1a"><msubsup id="S4.SS3.p4.12.m12.1.1" xref="S4.SS3.p4.12.m12.1.1.cmml"><mi id="S4.SS3.p4.12.m12.1.1.2.2" xref="S4.SS3.p4.12.m12.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p4.12.m12.1.1.2.3" xref="S4.SS3.p4.12.m12.1.1.2.3.cmml">m</mi><mi id="S4.SS3.p4.12.m12.1.1.3" xref="S4.SS3.p4.12.m12.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.12.m12.1b"><apply id="S4.SS3.p4.12.m12.1.1.cmml" xref="S4.SS3.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.12.m12.1.1.1.cmml" xref="S4.SS3.p4.12.m12.1.1">superscript</csymbol><apply id="S4.SS3.p4.12.m12.1.1.2.cmml" xref="S4.SS3.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.12.m12.1.1.2.1.cmml" xref="S4.SS3.p4.12.m12.1.1">subscript</csymbol><ci id="S4.SS3.p4.12.m12.1.1.2.2.cmml" xref="S4.SS3.p4.12.m12.1.1.2.2">𝑣</ci><ci id="S4.SS3.p4.12.m12.1.1.2.3.cmml" xref="S4.SS3.p4.12.m12.1.1.2.3">𝑚</ci></apply><ci id="S4.SS3.p4.12.m12.1.1.3.cmml" xref="S4.SS3.p4.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.12.m12.1c">v_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.12.m12.1d">italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, that is <math alttext="C_{m}^{i}=C_{m-1}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.13.m13.1"><semantics id="S4.SS3.p4.13.m13.1a"><mrow id="S4.SS3.p4.13.m13.1.1" xref="S4.SS3.p4.13.m13.1.1.cmml"><msubsup id="S4.SS3.p4.13.m13.1.1.2" xref="S4.SS3.p4.13.m13.1.1.2.cmml"><mi id="S4.SS3.p4.13.m13.1.1.2.2.2" xref="S4.SS3.p4.13.m13.1.1.2.2.2.cmml">C</mi><mi id="S4.SS3.p4.13.m13.1.1.2.2.3" xref="S4.SS3.p4.13.m13.1.1.2.2.3.cmml">m</mi><mi id="S4.SS3.p4.13.m13.1.1.2.3" xref="S4.SS3.p4.13.m13.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p4.13.m13.1.1.1" xref="S4.SS3.p4.13.m13.1.1.1.cmml">=</mo><msubsup id="S4.SS3.p4.13.m13.1.1.3" xref="S4.SS3.p4.13.m13.1.1.3.cmml"><mi id="S4.SS3.p4.13.m13.1.1.3.2.2" xref="S4.SS3.p4.13.m13.1.1.3.2.2.cmml">C</mi><mrow id="S4.SS3.p4.13.m13.1.1.3.2.3" xref="S4.SS3.p4.13.m13.1.1.3.2.3.cmml"><mi id="S4.SS3.p4.13.m13.1.1.3.2.3.2" xref="S4.SS3.p4.13.m13.1.1.3.2.3.2.cmml">m</mi><mo id="S4.SS3.p4.13.m13.1.1.3.2.3.1" xref="S4.SS3.p4.13.m13.1.1.3.2.3.1.cmml">−</mo><mn id="S4.SS3.p4.13.m13.1.1.3.2.3.3" xref="S4.SS3.p4.13.m13.1.1.3.2.3.3.cmml">1</mn></mrow><mi id="S4.SS3.p4.13.m13.1.1.3.3" xref="S4.SS3.p4.13.m13.1.1.3.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.13.m13.1b"><apply id="S4.SS3.p4.13.m13.1.1.cmml" xref="S4.SS3.p4.13.m13.1.1"><eq id="S4.SS3.p4.13.m13.1.1.1.cmml" xref="S4.SS3.p4.13.m13.1.1.1"></eq><apply id="S4.SS3.p4.13.m13.1.1.2.cmml" xref="S4.SS3.p4.13.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p4.13.m13.1.1.2.1.cmml" xref="S4.SS3.p4.13.m13.1.1.2">superscript</csymbol><apply id="S4.SS3.p4.13.m13.1.1.2.2.cmml" xref="S4.SS3.p4.13.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p4.13.m13.1.1.2.2.1.cmml" xref="S4.SS3.p4.13.m13.1.1.2">subscript</csymbol><ci id="S4.SS3.p4.13.m13.1.1.2.2.2.cmml" xref="S4.SS3.p4.13.m13.1.1.2.2.2">𝐶</ci><ci id="S4.SS3.p4.13.m13.1.1.2.2.3.cmml" xref="S4.SS3.p4.13.m13.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS3.p4.13.m13.1.1.2.3.cmml" xref="S4.SS3.p4.13.m13.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p4.13.m13.1.1.3.cmml" xref="S4.SS3.p4.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p4.13.m13.1.1.3.1.cmml" xref="S4.SS3.p4.13.m13.1.1.3">superscript</csymbol><apply id="S4.SS3.p4.13.m13.1.1.3.2.cmml" xref="S4.SS3.p4.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p4.13.m13.1.1.3.2.1.cmml" xref="S4.SS3.p4.13.m13.1.1.3">subscript</csymbol><ci id="S4.SS3.p4.13.m13.1.1.3.2.2.cmml" xref="S4.SS3.p4.13.m13.1.1.3.2.2">𝐶</ci><apply id="S4.SS3.p4.13.m13.1.1.3.2.3.cmml" xref="S4.SS3.p4.13.m13.1.1.3.2.3"><minus id="S4.SS3.p4.13.m13.1.1.3.2.3.1.cmml" xref="S4.SS3.p4.13.m13.1.1.3.2.3.1"></minus><ci id="S4.SS3.p4.13.m13.1.1.3.2.3.2.cmml" xref="S4.SS3.p4.13.m13.1.1.3.2.3.2">𝑚</ci><cn id="S4.SS3.p4.13.m13.1.1.3.2.3.3.cmml" type="integer" xref="S4.SS3.p4.13.m13.1.1.3.2.3.3">1</cn></apply></apply><ci id="S4.SS3.p4.13.m13.1.1.3.3.cmml" xref="S4.SS3.p4.13.m13.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.13.m13.1c">C_{m}^{i}=C_{m-1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.13.m13.1d">italic_C start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_C start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, the node <math alttext="v_{m}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p4.14.m14.1"><semantics id="S4.SS3.p4.14.m14.1a"><msubsup id="S4.SS3.p4.14.m14.1.1" xref="S4.SS3.p4.14.m14.1.1.cmml"><mi id="S4.SS3.p4.14.m14.1.1.2.2" xref="S4.SS3.p4.14.m14.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p4.14.m14.1.1.2.3" xref="S4.SS3.p4.14.m14.1.1.2.3.cmml">m</mi><mi id="S4.SS3.p4.14.m14.1.1.3" xref="S4.SS3.p4.14.m14.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p4.14.m14.1b"><apply id="S4.SS3.p4.14.m14.1.1.cmml" xref="S4.SS3.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.14.m14.1.1.1.cmml" xref="S4.SS3.p4.14.m14.1.1">superscript</csymbol><apply id="S4.SS3.p4.14.m14.1.1.2.cmml" xref="S4.SS3.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p4.14.m14.1.1.2.1.cmml" xref="S4.SS3.p4.14.m14.1.1">subscript</csymbol><ci id="S4.SS3.p4.14.m14.1.1.2.2.cmml" xref="S4.SS3.p4.14.m14.1.1.2.2">𝑣</ci><ci id="S4.SS3.p4.14.m14.1.1.2.3.cmml" xref="S4.SS3.p4.14.m14.1.1.2.3">𝑚</ci></apply><ci id="S4.SS3.p4.14.m14.1.1.3.cmml" xref="S4.SS3.p4.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p4.14.m14.1c">v_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p4.14.m14.1d">italic_v start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is removed because removing wait action does not change the robot path or any temporal dependency in the TPG. </div> <div class="ltx_para" id="S4.SS3.p5"> Next, all the type-2 edges are identified the TPG. First, all inter-robot task dependencies <math alttext="(\mathcal{T}_{m}^{i}\rightarrow\mathcal{T}_{m^{\prime}}^{i^{\prime}})\in% \mathcal{E}" class="ltx_Math" display="inline" id="S4.SS3.p5.1.m1.1"><semantics id="S4.SS3.p5.1.m1.1a"><mrow id="S4.SS3.p5.1.m1.1.1" xref="S4.SS3.p5.1.m1.1.1.cmml"><mrow id="S4.SS3.p5.1.m1.1.1.1.1" xref="S4.SS3.p5.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS3.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S4.SS3.p5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p5.1.m1.1.1.1.1.1" xref="S4.SS3.p5.1.m1.1.1.1.1.1.cmml"><msubsup id="S4.SS3.p5.1.m1.1.1.1.1.1.2" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.2" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.2.cmml">𝒯</mi><mi id="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.3.cmml">m</mi><mi id="S4.SS3.p5.1.m1.1.1.1.1.1.2.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p5.1.m1.1.1.1.1.1.1" stretchy="false" xref="S4.SS3.p5.1.m1.1.1.1.1.1.1.cmml">→</mo><msubsup id="S4.SS3.p5.1.m1.1.1.1.1.1.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.2" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.2.cmml">𝒯</mi><msup id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.cmml"><mi id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.2" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.2.cmml">m</mi><mo id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.cmml"><mi id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.2" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.2.cmml">i</mi><mo id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.3" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.3.cmml">′</mo></msup></msubsup></mrow><mo id="S4.SS3.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S4.SS3.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS3.p5.1.m1.1.1.2" xref="S4.SS3.p5.1.m1.1.1.2.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.1.m1.1.1.3" xref="S4.SS3.p5.1.m1.1.1.3.cmml">ℰ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.1.m1.1b"><apply id="S4.SS3.p5.1.m1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1"><in id="S4.SS3.p5.1.m1.1.1.2.cmml" xref="S4.SS3.p5.1.m1.1.1.2"></in><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1"><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.1">→</ci><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.1.1.2.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2">superscript</csymbol><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.2">𝒯</ci><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.2.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.1.1.3.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3">superscript</csymbol><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.2">𝒯</ci><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3">superscript</csymbol><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.2">𝑚</ci><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.1.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3">superscript</csymbol><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.2.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.2">𝑖</ci><ci id="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.3.cmml" xref="S4.SS3.p5.1.m1.1.1.1.1.1.3.3.3">′</ci></apply></apply></apply><ci id="S4.SS3.p5.1.m1.1.1.3.cmml" xref="S4.SS3.p5.1.m1.1.1.3">ℰ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.1.m1.1c">(\mathcal{T}_{m}^{i}\rightarrow\mathcal{T}_{m^{\prime}}^{i^{\prime}})\in% \mathcal{E}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.1.m1.1d">( caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → caligraphic_T start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) ∈ caligraphic_E</annotation></semantics></math> from the task graph <math alttext="\mathcal{G}" class="ltx_Math" display="inline" id="S4.SS3.p5.2.m2.1"><semantics id="S4.SS3.p5.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.2.m2.1.1" xref="S4.SS3.p5.2.m2.1.1.cmml">𝒢</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.2.m2.1b"><ci id="S4.SS3.p5.2.m2.1.1.cmml" xref="S4.SS3.p5.2.m2.1.1">𝒢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.2.m2.1c">\mathcal{G}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.2.m2.1d">caligraphic_G</annotation></semantics></math> are added as type-2 edges to the TPG. Specifically, for every edge in the task graph <math alttext="\mathcal{T}_{m}^{i}\rightarrow\mathcal{T}_{m^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p5.3.m3.1"><semantics id="S4.SS3.p5.3.m3.1a"><mrow id="S4.SS3.p5.3.m3.1.1" xref="S4.SS3.p5.3.m3.1.1.cmml"><msubsup id="S4.SS3.p5.3.m3.1.1.2" xref="S4.SS3.p5.3.m3.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.3.m3.1.1.2.2.2" xref="S4.SS3.p5.3.m3.1.1.2.2.2.cmml">𝒯</mi><mi id="S4.SS3.p5.3.m3.1.1.2.2.3" xref="S4.SS3.p5.3.m3.1.1.2.2.3.cmml">m</mi><mi id="S4.SS3.p5.3.m3.1.1.2.3" xref="S4.SS3.p5.3.m3.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p5.3.m3.1.1.1" stretchy="false" xref="S4.SS3.p5.3.m3.1.1.1.cmml">→</mo><msubsup id="S4.SS3.p5.3.m3.1.1.3" xref="S4.SS3.p5.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.3.m3.1.1.3.2.2" xref="S4.SS3.p5.3.m3.1.1.3.2.2.cmml">𝒯</mi><msup id="S4.SS3.p5.3.m3.1.1.3.2.3" xref="S4.SS3.p5.3.m3.1.1.3.2.3.cmml"><mi id="S4.SS3.p5.3.m3.1.1.3.2.3.2" xref="S4.SS3.p5.3.m3.1.1.3.2.3.2.cmml">m</mi><mo id="S4.SS3.p5.3.m3.1.1.3.2.3.3" xref="S4.SS3.p5.3.m3.1.1.3.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p5.3.m3.1.1.3.3" xref="S4.SS3.p5.3.m3.1.1.3.3.cmml"><mi id="S4.SS3.p5.3.m3.1.1.3.3.2" xref="S4.SS3.p5.3.m3.1.1.3.3.2.cmml">i</mi><mo id="S4.SS3.p5.3.m3.1.1.3.3.3" xref="S4.SS3.p5.3.m3.1.1.3.3.3.cmml">′</mo></msup></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.3.m3.1b"><apply id="S4.SS3.p5.3.m3.1.1.cmml" xref="S4.SS3.p5.3.m3.1.1"><ci id="S4.SS3.p5.3.m3.1.1.1.cmml" xref="S4.SS3.p5.3.m3.1.1.1">→</ci><apply id="S4.SS3.p5.3.m3.1.1.2.cmml" xref="S4.SS3.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.1.1.2.1.cmml" xref="S4.SS3.p5.3.m3.1.1.2">superscript</csymbol><apply id="S4.SS3.p5.3.m3.1.1.2.2.cmml" xref="S4.SS3.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.1.1.2.2.1.cmml" xref="S4.SS3.p5.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS3.p5.3.m3.1.1.2.2.2.cmml" xref="S4.SS3.p5.3.m3.1.1.2.2.2">𝒯</ci><ci id="S4.SS3.p5.3.m3.1.1.2.2.3.cmml" xref="S4.SS3.p5.3.m3.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS3.p5.3.m3.1.1.2.3.cmml" xref="S4.SS3.p5.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p5.3.m3.1.1.3.cmml" xref="S4.SS3.p5.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.1.1.3.1.cmml" xref="S4.SS3.p5.3.m3.1.1.3">superscript</csymbol><apply id="S4.SS3.p5.3.m3.1.1.3.2.cmml" xref="S4.SS3.p5.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.1.1.3.2.1.cmml" xref="S4.SS3.p5.3.m3.1.1.3">subscript</csymbol><ci id="S4.SS3.p5.3.m3.1.1.3.2.2.cmml" xref="S4.SS3.p5.3.m3.1.1.3.2.2">𝒯</ci><apply id="S4.SS3.p5.3.m3.1.1.3.2.3.cmml" xref="S4.SS3.p5.3.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.1.1.3.2.3.1.cmml" xref="S4.SS3.p5.3.m3.1.1.3.2.3">superscript</csymbol><ci id="S4.SS3.p5.3.m3.1.1.3.2.3.2.cmml" xref="S4.SS3.p5.3.m3.1.1.3.2.3.2">𝑚</ci><ci id="S4.SS3.p5.3.m3.1.1.3.2.3.3.cmml" xref="S4.SS3.p5.3.m3.1.1.3.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p5.3.m3.1.1.3.3.cmml" xref="S4.SS3.p5.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p5.3.m3.1.1.3.3.1.cmml" xref="S4.SS3.p5.3.m3.1.1.3.3">superscript</csymbol><ci id="S4.SS3.p5.3.m3.1.1.3.3.2.cmml" xref="S4.SS3.p5.3.m3.1.1.3.3.2">𝑖</ci><ci id="S4.SS3.p5.3.m3.1.1.3.3.3.cmml" xref="S4.SS3.p5.3.m3.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.3.m3.1c">\mathcal{T}_{m}^{i}\rightarrow\mathcal{T}_{m^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.3.m3.1d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → caligraphic_T start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, a type-2 edge is added from the beginning of task <math alttext="\mathcal{T}_{m+1}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p5.4.m4.1"><semantics id="S4.SS3.p5.4.m4.1a"><msubsup id="S4.SS3.p5.4.m4.1.1" xref="S4.SS3.p5.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.4.m4.1.1.2.2" xref="S4.SS3.p5.4.m4.1.1.2.2.cmml">𝒯</mi><mrow id="S4.SS3.p5.4.m4.1.1.2.3" xref="S4.SS3.p5.4.m4.1.1.2.3.cmml"><mi id="S4.SS3.p5.4.m4.1.1.2.3.2" xref="S4.SS3.p5.4.m4.1.1.2.3.2.cmml">m</mi><mo id="S4.SS3.p5.4.m4.1.1.2.3.1" xref="S4.SS3.p5.4.m4.1.1.2.3.1.cmml">+</mo><mn id="S4.SS3.p5.4.m4.1.1.2.3.3" xref="S4.SS3.p5.4.m4.1.1.2.3.3.cmml">1</mn></mrow><mi id="S4.SS3.p5.4.m4.1.1.3" xref="S4.SS3.p5.4.m4.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.4.m4.1b"><apply id="S4.SS3.p5.4.m4.1.1.cmml" xref="S4.SS3.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.4.m4.1.1.1.cmml" xref="S4.SS3.p5.4.m4.1.1">superscript</csymbol><apply id="S4.SS3.p5.4.m4.1.1.2.cmml" xref="S4.SS3.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.4.m4.1.1.2.1.cmml" xref="S4.SS3.p5.4.m4.1.1">subscript</csymbol><ci id="S4.SS3.p5.4.m4.1.1.2.2.cmml" xref="S4.SS3.p5.4.m4.1.1.2.2">𝒯</ci><apply id="S4.SS3.p5.4.m4.1.1.2.3.cmml" xref="S4.SS3.p5.4.m4.1.1.2.3"><plus id="S4.SS3.p5.4.m4.1.1.2.3.1.cmml" xref="S4.SS3.p5.4.m4.1.1.2.3.1"></plus><ci id="S4.SS3.p5.4.m4.1.1.2.3.2.cmml" xref="S4.SS3.p5.4.m4.1.1.2.3.2">𝑚</ci><cn id="S4.SS3.p5.4.m4.1.1.2.3.3.cmml" type="integer" xref="S4.SS3.p5.4.m4.1.1.2.3.3">1</cn></apply></apply><ci id="S4.SS3.p5.4.m4.1.1.3.cmml" xref="S4.SS3.p5.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.4.m4.1c">\mathcal{T}_{m+1}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.4.m4.1d">caligraphic_T start_POSTSUBSCRIPT italic_m + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="v_{{start,m+1}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p5.5.m5.2"><semantics id="S4.SS3.p5.5.m5.2a"><msubsup id="S4.SS3.p5.5.m5.2.3" xref="S4.SS3.p5.5.m5.2.3.cmml"><mi id="S4.SS3.p5.5.m5.2.3.2.2" xref="S4.SS3.p5.5.m5.2.3.2.2.cmml">v</mi><mrow id="S4.SS3.p5.5.m5.2.2.2.2" xref="S4.SS3.p5.5.m5.2.2.2.3.cmml"><mrow id="S4.SS3.p5.5.m5.1.1.1.1.1" xref="S4.SS3.p5.5.m5.1.1.1.1.1.cmml"><mi id="S4.SS3.p5.5.m5.1.1.1.1.1.2" xref="S4.SS3.p5.5.m5.1.1.1.1.1.2.cmml">s</mi><mo id="S4.SS3.p5.5.m5.1.1.1.1.1.1" xref="S4.SS3.p5.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.5.m5.1.1.1.1.1.3" xref="S4.SS3.p5.5.m5.1.1.1.1.1.3.cmml">t</mi><mo id="S4.SS3.p5.5.m5.1.1.1.1.1.1a" xref="S4.SS3.p5.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.5.m5.1.1.1.1.1.4" xref="S4.SS3.p5.5.m5.1.1.1.1.1.4.cmml">a</mi><mo id="S4.SS3.p5.5.m5.1.1.1.1.1.1b" xref="S4.SS3.p5.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.5.m5.1.1.1.1.1.5" xref="S4.SS3.p5.5.m5.1.1.1.1.1.5.cmml">r</mi><mo id="S4.SS3.p5.5.m5.1.1.1.1.1.1c" xref="S4.SS3.p5.5.m5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.5.m5.1.1.1.1.1.6" xref="S4.SS3.p5.5.m5.1.1.1.1.1.6.cmml">t</mi></mrow><mo id="S4.SS3.p5.5.m5.2.2.2.2.3" xref="S4.SS3.p5.5.m5.2.2.2.3.cmml">,</mo><mrow id="S4.SS3.p5.5.m5.2.2.2.2.2" xref="S4.SS3.p5.5.m5.2.2.2.2.2.cmml"><mi id="S4.SS3.p5.5.m5.2.2.2.2.2.2" xref="S4.SS3.p5.5.m5.2.2.2.2.2.2.cmml">m</mi><mo id="S4.SS3.p5.5.m5.2.2.2.2.2.1" xref="S4.SS3.p5.5.m5.2.2.2.2.2.1.cmml">+</mo><mn id="S4.SS3.p5.5.m5.2.2.2.2.2.3" xref="S4.SS3.p5.5.m5.2.2.2.2.2.3.cmml">1</mn></mrow></mrow><mi id="S4.SS3.p5.5.m5.2.3.3" xref="S4.SS3.p5.5.m5.2.3.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.5.m5.2b"><apply id="S4.SS3.p5.5.m5.2.3.cmml" xref="S4.SS3.p5.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.5.m5.2.3.1.cmml" xref="S4.SS3.p5.5.m5.2.3">superscript</csymbol><apply id="S4.SS3.p5.5.m5.2.3.2.cmml" xref="S4.SS3.p5.5.m5.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.5.m5.2.3.2.1.cmml" xref="S4.SS3.p5.5.m5.2.3">subscript</csymbol><ci id="S4.SS3.p5.5.m5.2.3.2.2.cmml" xref="S4.SS3.p5.5.m5.2.3.2.2">𝑣</ci><list id="S4.SS3.p5.5.m5.2.2.2.3.cmml" xref="S4.SS3.p5.5.m5.2.2.2.2"><apply id="S4.SS3.p5.5.m5.1.1.1.1.1.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1"><times id="S4.SS3.p5.5.m5.1.1.1.1.1.1.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1.1"></times><ci id="S4.SS3.p5.5.m5.1.1.1.1.1.2.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1.2">𝑠</ci><ci id="S4.SS3.p5.5.m5.1.1.1.1.1.3.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1.3">𝑡</ci><ci id="S4.SS3.p5.5.m5.1.1.1.1.1.4.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1.4">𝑎</ci><ci id="S4.SS3.p5.5.m5.1.1.1.1.1.5.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1.5">𝑟</ci><ci id="S4.SS3.p5.5.m5.1.1.1.1.1.6.cmml" xref="S4.SS3.p5.5.m5.1.1.1.1.1.6">𝑡</ci></apply><apply id="S4.SS3.p5.5.m5.2.2.2.2.2.cmml" xref="S4.SS3.p5.5.m5.2.2.2.2.2"><plus id="S4.SS3.p5.5.m5.2.2.2.2.2.1.cmml" xref="S4.SS3.p5.5.m5.2.2.2.2.2.1"></plus><ci id="S4.SS3.p5.5.m5.2.2.2.2.2.2.cmml" xref="S4.SS3.p5.5.m5.2.2.2.2.2.2">𝑚</ci><cn id="S4.SS3.p5.5.m5.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS3.p5.5.m5.2.2.2.2.2.3">1</cn></apply></list></apply><ci id="S4.SS3.p5.5.m5.2.3.3.cmml" xref="S4.SS3.p5.5.m5.2.3.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.5.m5.2c">v_{{start,m+1}}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.5.m5.2d">italic_v start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, to the beginning of task <math alttext="\mathcal{T}_{m^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p5.6.m6.1"><semantics id="S4.SS3.p5.6.m6.1a"><msubsup id="S4.SS3.p5.6.m6.1.1" xref="S4.SS3.p5.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.6.m6.1.1.2.2" xref="S4.SS3.p5.6.m6.1.1.2.2.cmml">𝒯</mi><msup id="S4.SS3.p5.6.m6.1.1.2.3" xref="S4.SS3.p5.6.m6.1.1.2.3.cmml"><mi id="S4.SS3.p5.6.m6.1.1.2.3.2" xref="S4.SS3.p5.6.m6.1.1.2.3.2.cmml">m</mi><mo id="S4.SS3.p5.6.m6.1.1.2.3.3" xref="S4.SS3.p5.6.m6.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p5.6.m6.1.1.3" xref="S4.SS3.p5.6.m6.1.1.3.cmml"><mi id="S4.SS3.p5.6.m6.1.1.3.2" xref="S4.SS3.p5.6.m6.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p5.6.m6.1.1.3.3" xref="S4.SS3.p5.6.m6.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.6.m6.1b"><apply id="S4.SS3.p5.6.m6.1.1.cmml" xref="S4.SS3.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.6.m6.1.1.1.cmml" xref="S4.SS3.p5.6.m6.1.1">superscript</csymbol><apply id="S4.SS3.p5.6.m6.1.1.2.cmml" xref="S4.SS3.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.6.m6.1.1.2.1.cmml" xref="S4.SS3.p5.6.m6.1.1">subscript</csymbol><ci id="S4.SS3.p5.6.m6.1.1.2.2.cmml" xref="S4.SS3.p5.6.m6.1.1.2.2">𝒯</ci><apply id="S4.SS3.p5.6.m6.1.1.2.3.cmml" xref="S4.SS3.p5.6.m6.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.6.m6.1.1.2.3.1.cmml" xref="S4.SS3.p5.6.m6.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p5.6.m6.1.1.2.3.2.cmml" xref="S4.SS3.p5.6.m6.1.1.2.3.2">𝑚</ci><ci id="S4.SS3.p5.6.m6.1.1.2.3.3.cmml" xref="S4.SS3.p5.6.m6.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p5.6.m6.1.1.3.cmml" xref="S4.SS3.p5.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.6.m6.1.1.3.1.cmml" xref="S4.SS3.p5.6.m6.1.1.3">superscript</csymbol><ci id="S4.SS3.p5.6.m6.1.1.3.2.cmml" xref="S4.SS3.p5.6.m6.1.1.3.2">𝑖</ci><ci id="S4.SS3.p5.6.m6.1.1.3.3.cmml" xref="S4.SS3.p5.6.m6.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.6.m6.1c">\mathcal{T}_{m^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.6.m6.1d">caligraphic_T start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="v_{{start,{m^{\prime}}}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p5.7.m7.2"><semantics id="S4.SS3.p5.7.m7.2a"><msubsup id="S4.SS3.p5.7.m7.2.3" xref="S4.SS3.p5.7.m7.2.3.cmml"><mi id="S4.SS3.p5.7.m7.2.3.2.2" xref="S4.SS3.p5.7.m7.2.3.2.2.cmml">v</mi><mrow id="S4.SS3.p5.7.m7.2.2.2.2" xref="S4.SS3.p5.7.m7.2.2.2.3.cmml"><mrow id="S4.SS3.p5.7.m7.1.1.1.1.1" xref="S4.SS3.p5.7.m7.1.1.1.1.1.cmml"><mi id="S4.SS3.p5.7.m7.1.1.1.1.1.2" xref="S4.SS3.p5.7.m7.1.1.1.1.1.2.cmml">s</mi><mo id="S4.SS3.p5.7.m7.1.1.1.1.1.1" xref="S4.SS3.p5.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.7.m7.1.1.1.1.1.3" xref="S4.SS3.p5.7.m7.1.1.1.1.1.3.cmml">t</mi><mo id="S4.SS3.p5.7.m7.1.1.1.1.1.1a" xref="S4.SS3.p5.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.7.m7.1.1.1.1.1.4" xref="S4.SS3.p5.7.m7.1.1.1.1.1.4.cmml">a</mi><mo id="S4.SS3.p5.7.m7.1.1.1.1.1.1b" xref="S4.SS3.p5.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.7.m7.1.1.1.1.1.5" xref="S4.SS3.p5.7.m7.1.1.1.1.1.5.cmml">r</mi><mo id="S4.SS3.p5.7.m7.1.1.1.1.1.1c" xref="S4.SS3.p5.7.m7.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p5.7.m7.1.1.1.1.1.6" xref="S4.SS3.p5.7.m7.1.1.1.1.1.6.cmml">t</mi></mrow><mo id="S4.SS3.p5.7.m7.2.2.2.2.3" xref="S4.SS3.p5.7.m7.2.2.2.3.cmml">,</mo><msup id="S4.SS3.p5.7.m7.2.2.2.2.2" xref="S4.SS3.p5.7.m7.2.2.2.2.2.cmml"><mi id="S4.SS3.p5.7.m7.2.2.2.2.2.2" xref="S4.SS3.p5.7.m7.2.2.2.2.2.2.cmml">m</mi><mo id="S4.SS3.p5.7.m7.2.2.2.2.2.3" xref="S4.SS3.p5.7.m7.2.2.2.2.2.3.cmml">′</mo></msup></mrow><msup id="S4.SS3.p5.7.m7.2.3.3" xref="S4.SS3.p5.7.m7.2.3.3.cmml"><mi id="S4.SS3.p5.7.m7.2.3.3.2" xref="S4.SS3.p5.7.m7.2.3.3.2.cmml">i</mi><mo id="S4.SS3.p5.7.m7.2.3.3.3" xref="S4.SS3.p5.7.m7.2.3.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.7.m7.2b"><apply id="S4.SS3.p5.7.m7.2.3.cmml" xref="S4.SS3.p5.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.7.m7.2.3.1.cmml" xref="S4.SS3.p5.7.m7.2.3">superscript</csymbol><apply id="S4.SS3.p5.7.m7.2.3.2.cmml" xref="S4.SS3.p5.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.7.m7.2.3.2.1.cmml" xref="S4.SS3.p5.7.m7.2.3">subscript</csymbol><ci id="S4.SS3.p5.7.m7.2.3.2.2.cmml" xref="S4.SS3.p5.7.m7.2.3.2.2">𝑣</ci><list id="S4.SS3.p5.7.m7.2.2.2.3.cmml" xref="S4.SS3.p5.7.m7.2.2.2.2"><apply id="S4.SS3.p5.7.m7.1.1.1.1.1.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1"><times id="S4.SS3.p5.7.m7.1.1.1.1.1.1.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1.1"></times><ci id="S4.SS3.p5.7.m7.1.1.1.1.1.2.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1.2">𝑠</ci><ci id="S4.SS3.p5.7.m7.1.1.1.1.1.3.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1.3">𝑡</ci><ci id="S4.SS3.p5.7.m7.1.1.1.1.1.4.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1.4">𝑎</ci><ci id="S4.SS3.p5.7.m7.1.1.1.1.1.5.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1.5">𝑟</ci><ci id="S4.SS3.p5.7.m7.1.1.1.1.1.6.cmml" xref="S4.SS3.p5.7.m7.1.1.1.1.1.6">𝑡</ci></apply><apply id="S4.SS3.p5.7.m7.2.2.2.2.2.cmml" xref="S4.SS3.p5.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p5.7.m7.2.2.2.2.2.1.cmml" xref="S4.SS3.p5.7.m7.2.2.2.2.2">superscript</csymbol><ci id="S4.SS3.p5.7.m7.2.2.2.2.2.2.cmml" xref="S4.SS3.p5.7.m7.2.2.2.2.2.2">𝑚</ci><ci id="S4.SS3.p5.7.m7.2.2.2.2.2.3.cmml" xref="S4.SS3.p5.7.m7.2.2.2.2.2.3">′</ci></apply></list></apply><apply id="S4.SS3.p5.7.m7.2.3.3.cmml" xref="S4.SS3.p5.7.m7.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p5.7.m7.2.3.3.1.cmml" xref="S4.SS3.p5.7.m7.2.3.3">superscript</csymbol><ci id="S4.SS3.p5.7.m7.2.3.3.2.cmml" xref="S4.SS3.p5.7.m7.2.3.3.2">𝑖</ci><ci id="S4.SS3.p5.7.m7.2.3.3.3.cmml" xref="S4.SS3.p5.7.m7.2.3.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.7.m7.2c">v_{{start,{m^{\prime}}}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.7.m7.2d">italic_v start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t , italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. This type-2 edge ensures that robot <math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p5.8.m8.1"><semantics id="S4.SS3.p5.8.m8.1a"><mi id="S4.SS3.p5.8.m8.1.1" xref="S4.SS3.p5.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.8.m8.1b"><ci id="S4.SS3.p5.8.m8.1.1.cmml" xref="S4.SS3.p5.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.8.m8.1d">italic_i</annotation></semantics></math> only starts <math alttext="\mathcal{T}_{m^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p5.9.m9.1"><semantics id="S4.SS3.p5.9.m9.1a"><msubsup id="S4.SS3.p5.9.m9.1.1" xref="S4.SS3.p5.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.9.m9.1.1.2.2" xref="S4.SS3.p5.9.m9.1.1.2.2.cmml">𝒯</mi><msup id="S4.SS3.p5.9.m9.1.1.2.3" xref="S4.SS3.p5.9.m9.1.1.2.3.cmml"><mi id="S4.SS3.p5.9.m9.1.1.2.3.2" xref="S4.SS3.p5.9.m9.1.1.2.3.2.cmml">m</mi><mo id="S4.SS3.p5.9.m9.1.1.2.3.3" xref="S4.SS3.p5.9.m9.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p5.9.m9.1.1.3" xref="S4.SS3.p5.9.m9.1.1.3.cmml"><mi id="S4.SS3.p5.9.m9.1.1.3.2" xref="S4.SS3.p5.9.m9.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p5.9.m9.1.1.3.3" xref="S4.SS3.p5.9.m9.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.9.m9.1b"><apply id="S4.SS3.p5.9.m9.1.1.cmml" xref="S4.SS3.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.9.m9.1.1.1.cmml" xref="S4.SS3.p5.9.m9.1.1">superscript</csymbol><apply id="S4.SS3.p5.9.m9.1.1.2.cmml" xref="S4.SS3.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.9.m9.1.1.2.1.cmml" xref="S4.SS3.p5.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p5.9.m9.1.1.2.2.cmml" xref="S4.SS3.p5.9.m9.1.1.2.2">𝒯</ci><apply id="S4.SS3.p5.9.m9.1.1.2.3.cmml" xref="S4.SS3.p5.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p5.9.m9.1.1.2.3.1.cmml" xref="S4.SS3.p5.9.m9.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p5.9.m9.1.1.2.3.2.cmml" xref="S4.SS3.p5.9.m9.1.1.2.3.2">𝑚</ci><ci id="S4.SS3.p5.9.m9.1.1.2.3.3.cmml" xref="S4.SS3.p5.9.m9.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p5.9.m9.1.1.3.cmml" xref="S4.SS3.p5.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p5.9.m9.1.1.3.1.cmml" xref="S4.SS3.p5.9.m9.1.1.3">superscript</csymbol><ci id="S4.SS3.p5.9.m9.1.1.3.2.cmml" xref="S4.SS3.p5.9.m9.1.1.3.2">𝑖</ci><ci id="S4.SS3.p5.9.m9.1.1.3.3.cmml" xref="S4.SS3.p5.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.9.m9.1c">\mathcal{T}_{m^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.9.m9.1d">caligraphic_T start_POSTSUBSCRIPT italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> after robot <math alttext="i^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p5.10.m10.1"><semantics id="S4.SS3.p5.10.m10.1a"><msup id="S4.SS3.p5.10.m10.1.1" xref="S4.SS3.p5.10.m10.1.1.cmml"><mi id="S4.SS3.p5.10.m10.1.1.2" xref="S4.SS3.p5.10.m10.1.1.2.cmml">i</mi><mo id="S4.SS3.p5.10.m10.1.1.3" xref="S4.SS3.p5.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.10.m10.1b"><apply id="S4.SS3.p5.10.m10.1.1.cmml" xref="S4.SS3.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.10.m10.1.1.1.cmml" xref="S4.SS3.p5.10.m10.1.1">superscript</csymbol><ci id="S4.SS3.p5.10.m10.1.1.2.cmml" xref="S4.SS3.p5.10.m10.1.1.2">𝑖</ci><ci id="S4.SS3.p5.10.m10.1.1.3.cmml" xref="S4.SS3.p5.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.10.m10.1c">i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.10.m10.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> finishes task <math alttext="\mathcal{T}_{m}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p5.11.m11.1"><semantics id="S4.SS3.p5.11.m11.1a"><msubsup id="S4.SS3.p5.11.m11.1.1" xref="S4.SS3.p5.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS3.p5.11.m11.1.1.2.2" xref="S4.SS3.p5.11.m11.1.1.2.2.cmml">𝒯</mi><mi id="S4.SS3.p5.11.m11.1.1.2.3" xref="S4.SS3.p5.11.m11.1.1.2.3.cmml">m</mi><mi id="S4.SS3.p5.11.m11.1.1.3" xref="S4.SS3.p5.11.m11.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p5.11.m11.1b"><apply id="S4.SS3.p5.11.m11.1.1.cmml" xref="S4.SS3.p5.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.11.m11.1.1.1.cmml" xref="S4.SS3.p5.11.m11.1.1">superscript</csymbol><apply id="S4.SS3.p5.11.m11.1.1.2.cmml" xref="S4.SS3.p5.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p5.11.m11.1.1.2.1.cmml" xref="S4.SS3.p5.11.m11.1.1">subscript</csymbol><ci id="S4.SS3.p5.11.m11.1.1.2.2.cmml" xref="S4.SS3.p5.11.m11.1.1.2.2">𝒯</ci><ci id="S4.SS3.p5.11.m11.1.1.2.3.cmml" xref="S4.SS3.p5.11.m11.1.1.2.3">𝑚</ci></apply><ci id="S4.SS3.p5.11.m11.1.1.3.cmml" xref="S4.SS3.p5.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p5.11.m11.1c">\mathcal{T}_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p5.11.m11.1d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. </div> <div class="ltx_para" id="S4.SS3.p6"> Then, motion-level dependencies are identified by iterating over all pairs of nodes in the TPG with a double loop iterating over <math alttext="(i,i^{\prime})" class="ltx_Math" display="inline" id="S4.SS3.p6.1.m1.2"><semantics id="S4.SS3.p6.1.m1.2a"><mrow id="S4.SS3.p6.1.m1.2.2.1" xref="S4.SS3.p6.1.m1.2.2.2.cmml"><mo id="S4.SS3.p6.1.m1.2.2.1.2" stretchy="false" xref="S4.SS3.p6.1.m1.2.2.2.cmml">(</mo><mi id="S4.SS3.p6.1.m1.1.1" xref="S4.SS3.p6.1.m1.1.1.cmml">i</mi><mo id="S4.SS3.p6.1.m1.2.2.1.3" xref="S4.SS3.p6.1.m1.2.2.2.cmml">,</mo><msup id="S4.SS3.p6.1.m1.2.2.1.1" xref="S4.SS3.p6.1.m1.2.2.1.1.cmml"><mi id="S4.SS3.p6.1.m1.2.2.1.1.2" xref="S4.SS3.p6.1.m1.2.2.1.1.2.cmml">i</mi><mo id="S4.SS3.p6.1.m1.2.2.1.1.3" xref="S4.SS3.p6.1.m1.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.SS3.p6.1.m1.2.2.1.4" stretchy="false" xref="S4.SS3.p6.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.1.m1.2b"><interval closure="open" id="S4.SS3.p6.1.m1.2.2.2.cmml" xref="S4.SS3.p6.1.m1.2.2.1"><ci id="S4.SS3.p6.1.m1.1.1.cmml" xref="S4.SS3.p6.1.m1.1.1">𝑖</ci><apply id="S4.SS3.p6.1.m1.2.2.1.1.cmml" xref="S4.SS3.p6.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.1.m1.2.2.1.1.1.cmml" xref="S4.SS3.p6.1.m1.2.2.1.1">superscript</csymbol><ci id="S4.SS3.p6.1.m1.2.2.1.1.2.cmml" xref="S4.SS3.p6.1.m1.2.2.1.1.2">𝑖</ci><ci id="S4.SS3.p6.1.m1.2.2.1.1.3.cmml" xref="S4.SS3.p6.1.m1.2.2.1.1.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.1.m1.2c">(i,i^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.1.m1.2d">( italic_i , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Robot kinematics can be incrementally changed if node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.2.m2.1"><semantics id="S4.SS3.p6.2.m2.1a"><msubsup id="S4.SS3.p6.2.m2.1.1" xref="S4.SS3.p6.2.m2.1.1.cmml"><mi id="S4.SS3.p6.2.m2.1.1.2.2" xref="S4.SS3.p6.2.m2.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.2.m2.1.1.2.3" xref="S4.SS3.p6.2.m2.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.2.m2.1.1.3" xref="S4.SS3.p6.2.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.2.m2.1b"><apply id="S4.SS3.p6.2.m2.1.1.cmml" xref="S4.SS3.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.2.m2.1.1.1.cmml" xref="S4.SS3.p6.2.m2.1.1">superscript</csymbol><apply id="S4.SS3.p6.2.m2.1.1.2.cmml" xref="S4.SS3.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.2.m2.1.1.2.1.cmml" xref="S4.SS3.p6.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.p6.2.m2.1.1.2.2.cmml" xref="S4.SS3.p6.2.m2.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.2.m2.1.1.2.3.cmml" xref="S4.SS3.p6.2.m2.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.2.m2.1.1.3.cmml" xref="S4.SS3.p6.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.2.m2.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.2.m2.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> is the start of a new task that attaches or detaches an object. For each node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.3.m3.1"><semantics id="S4.SS3.p6.3.m3.1a"><msubsup id="S4.SS3.p6.3.m3.1.1" xref="S4.SS3.p6.3.m3.1.1.cmml"><mi id="S4.SS3.p6.3.m3.1.1.2.2" xref="S4.SS3.p6.3.m3.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.3.m3.1.1.2.3" xref="S4.SS3.p6.3.m3.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.3.m3.1.1.3" xref="S4.SS3.p6.3.m3.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.3.m3.1b"><apply id="S4.SS3.p6.3.m3.1.1.cmml" xref="S4.SS3.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.3.m3.1.1.1.cmml" xref="S4.SS3.p6.3.m3.1.1">superscript</csymbol><apply id="S4.SS3.p6.3.m3.1.1.2.cmml" xref="S4.SS3.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.3.m3.1.1.2.1.cmml" xref="S4.SS3.p6.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.p6.3.m3.1.1.2.2.cmml" xref="S4.SS3.p6.3.m3.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.3.m3.1.1.2.3.cmml" xref="S4.SS3.p6.3.m3.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.3.m3.1.1.3.cmml" xref="S4.SS3.p6.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.3.m3.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.3.m3.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, it is checked against all other nodes <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.4.m4.1"><semantics id="S4.SS3.p6.4.m4.1a"><msubsup id="S4.SS3.p6.4.m4.1.1" xref="S4.SS3.p6.4.m4.1.1.cmml"><mi id="S4.SS3.p6.4.m4.1.1.2.2" xref="S4.SS3.p6.4.m4.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.4.m4.1.1.2.3" xref="S4.SS3.p6.4.m4.1.1.2.3.cmml"><mi id="S4.SS3.p6.4.m4.1.1.2.3.2" xref="S4.SS3.p6.4.m4.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.4.m4.1.1.2.3.3" xref="S4.SS3.p6.4.m4.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.4.m4.1.1.3" xref="S4.SS3.p6.4.m4.1.1.3.cmml"><mi id="S4.SS3.p6.4.m4.1.1.3.2" xref="S4.SS3.p6.4.m4.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.4.m4.1.1.3.3" xref="S4.SS3.p6.4.m4.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.4.m4.1b"><apply id="S4.SS3.p6.4.m4.1.1.cmml" xref="S4.SS3.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.4.m4.1.1.1.cmml" xref="S4.SS3.p6.4.m4.1.1">superscript</csymbol><apply id="S4.SS3.p6.4.m4.1.1.2.cmml" xref="S4.SS3.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.4.m4.1.1.2.1.cmml" xref="S4.SS3.p6.4.m4.1.1">subscript</csymbol><ci id="S4.SS3.p6.4.m4.1.1.2.2.cmml" xref="S4.SS3.p6.4.m4.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.4.m4.1.1.2.3.cmml" xref="S4.SS3.p6.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.4.m4.1.1.2.3.1.cmml" xref="S4.SS3.p6.4.m4.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.4.m4.1.1.2.3.2.cmml" xref="S4.SS3.p6.4.m4.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.4.m4.1.1.2.3.3.cmml" xref="S4.SS3.p6.4.m4.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.4.m4.1.1.3.cmml" xref="S4.SS3.p6.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.4.m4.1.1.3.1.cmml" xref="S4.SS3.p6.4.m4.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.4.m4.1.1.3.2.cmml" xref="S4.SS3.p6.4.m4.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.4.m4.1.1.3.3.cmml" xref="S4.SS3.p6.4.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.4.m4.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.4.m4.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> from a different robot (<math alttext="i\neq i^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p6.5.m5.1"><semantics id="S4.SS3.p6.5.m5.1a"><mrow id="S4.SS3.p6.5.m5.1.1" xref="S4.SS3.p6.5.m5.1.1.cmml"><mi id="S4.SS3.p6.5.m5.1.1.2" xref="S4.SS3.p6.5.m5.1.1.2.cmml">i</mi><mo id="S4.SS3.p6.5.m5.1.1.1" xref="S4.SS3.p6.5.m5.1.1.1.cmml">≠</mo><msup id="S4.SS3.p6.5.m5.1.1.3" xref="S4.SS3.p6.5.m5.1.1.3.cmml"><mi id="S4.SS3.p6.5.m5.1.1.3.2" xref="S4.SS3.p6.5.m5.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.5.m5.1.1.3.3" xref="S4.SS3.p6.5.m5.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.5.m5.1b"><apply id="S4.SS3.p6.5.m5.1.1.cmml" xref="S4.SS3.p6.5.m5.1.1"><neq id="S4.SS3.p6.5.m5.1.1.1.cmml" xref="S4.SS3.p6.5.m5.1.1.1"></neq><ci id="S4.SS3.p6.5.m5.1.1.2.cmml" xref="S4.SS3.p6.5.m5.1.1.2">𝑖</ci><apply id="S4.SS3.p6.5.m5.1.1.3.cmml" xref="S4.SS3.p6.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.5.m5.1.1.3.1.cmml" xref="S4.SS3.p6.5.m5.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.5.m5.1.1.3.2.cmml" xref="S4.SS3.p6.5.m5.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.5.m5.1.1.3.3.cmml" xref="S4.SS3.p6.5.m5.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.5.m5.1c">i\neq i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.5.m5.1d">italic_i ≠ italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>) and has an earlier timestamp <math alttext="t^{i^{\prime}}_{n^{\prime}}\leq t_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.6.m6.1"><semantics id="S4.SS3.p6.6.m6.1a"><mrow id="S4.SS3.p6.6.m6.1.1" xref="S4.SS3.p6.6.m6.1.1.cmml"><msubsup id="S4.SS3.p6.6.m6.1.1.2" xref="S4.SS3.p6.6.m6.1.1.2.cmml"><mi id="S4.SS3.p6.6.m6.1.1.2.2.2" xref="S4.SS3.p6.6.m6.1.1.2.2.2.cmml">t</mi><msup id="S4.SS3.p6.6.m6.1.1.2.3" xref="S4.SS3.p6.6.m6.1.1.2.3.cmml"><mi id="S4.SS3.p6.6.m6.1.1.2.3.2" xref="S4.SS3.p6.6.m6.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.6.m6.1.1.2.3.3" xref="S4.SS3.p6.6.m6.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.6.m6.1.1.2.2.3" xref="S4.SS3.p6.6.m6.1.1.2.2.3.cmml"><mi id="S4.SS3.p6.6.m6.1.1.2.2.3.2" xref="S4.SS3.p6.6.m6.1.1.2.2.3.2.cmml">i</mi><mo id="S4.SS3.p6.6.m6.1.1.2.2.3.3" xref="S4.SS3.p6.6.m6.1.1.2.2.3.3.cmml">′</mo></msup></msubsup><mo id="S4.SS3.p6.6.m6.1.1.1" xref="S4.SS3.p6.6.m6.1.1.1.cmml">≤</mo><msubsup id="S4.SS3.p6.6.m6.1.1.3" xref="S4.SS3.p6.6.m6.1.1.3.cmml"><mi id="S4.SS3.p6.6.m6.1.1.3.2.2" xref="S4.SS3.p6.6.m6.1.1.3.2.2.cmml">t</mi><mi id="S4.SS3.p6.6.m6.1.1.3.2.3" xref="S4.SS3.p6.6.m6.1.1.3.2.3.cmml">n</mi><mi id="S4.SS3.p6.6.m6.1.1.3.3" xref="S4.SS3.p6.6.m6.1.1.3.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.6.m6.1b"><apply id="S4.SS3.p6.6.m6.1.1.cmml" xref="S4.SS3.p6.6.m6.1.1"><leq id="S4.SS3.p6.6.m6.1.1.1.cmml" xref="S4.SS3.p6.6.m6.1.1.1"></leq><apply id="S4.SS3.p6.6.m6.1.1.2.cmml" xref="S4.SS3.p6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p6.6.m6.1.1.2.1.cmml" xref="S4.SS3.p6.6.m6.1.1.2">subscript</csymbol><apply id="S4.SS3.p6.6.m6.1.1.2.2.cmml" xref="S4.SS3.p6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p6.6.m6.1.1.2.2.1.cmml" xref="S4.SS3.p6.6.m6.1.1.2">superscript</csymbol><ci id="S4.SS3.p6.6.m6.1.1.2.2.2.cmml" xref="S4.SS3.p6.6.m6.1.1.2.2.2">𝑡</ci><apply id="S4.SS3.p6.6.m6.1.1.2.2.3.cmml" xref="S4.SS3.p6.6.m6.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.6.m6.1.1.2.2.3.1.cmml" xref="S4.SS3.p6.6.m6.1.1.2.2.3">superscript</csymbol><ci id="S4.SS3.p6.6.m6.1.1.2.2.3.2.cmml" xref="S4.SS3.p6.6.m6.1.1.2.2.3.2">𝑖</ci><ci id="S4.SS3.p6.6.m6.1.1.2.2.3.3.cmml" xref="S4.SS3.p6.6.m6.1.1.2.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.6.m6.1.1.2.3.cmml" xref="S4.SS3.p6.6.m6.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.6.m6.1.1.2.3.1.cmml" xref="S4.SS3.p6.6.m6.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.6.m6.1.1.2.3.2.cmml" xref="S4.SS3.p6.6.m6.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.6.m6.1.1.2.3.3.cmml" xref="S4.SS3.p6.6.m6.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.6.m6.1.1.3.cmml" xref="S4.SS3.p6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.6.m6.1.1.3.1.cmml" xref="S4.SS3.p6.6.m6.1.1.3">superscript</csymbol><apply id="S4.SS3.p6.6.m6.1.1.3.2.cmml" xref="S4.SS3.p6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.6.m6.1.1.3.2.1.cmml" xref="S4.SS3.p6.6.m6.1.1.3">subscript</csymbol><ci id="S4.SS3.p6.6.m6.1.1.3.2.2.cmml" xref="S4.SS3.p6.6.m6.1.1.3.2.2">𝑡</ci><ci id="S4.SS3.p6.6.m6.1.1.3.2.3.cmml" xref="S4.SS3.p6.6.m6.1.1.3.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.6.m6.1.1.3.3.cmml" xref="S4.SS3.p6.6.m6.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.6.m6.1c">t^{i^{\prime}}_{n^{\prime}}\leq t_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.6.m6.1d">italic_t start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≤ italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. Collision checking for pose nodes means that the robot links and attached objects at the corresponding poses <math alttext="C_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.7.m7.1"><semantics id="S4.SS3.p6.7.m7.1a"><msubsup id="S4.SS3.p6.7.m7.1.1" xref="S4.SS3.p6.7.m7.1.1.cmml"><mi id="S4.SS3.p6.7.m7.1.1.2.2" xref="S4.SS3.p6.7.m7.1.1.2.2.cmml">C</mi><mi id="S4.SS3.p6.7.m7.1.1.2.3" xref="S4.SS3.p6.7.m7.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.7.m7.1.1.3" xref="S4.SS3.p6.7.m7.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.7.m7.1b"><apply id="S4.SS3.p6.7.m7.1.1.cmml" xref="S4.SS3.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.7.m7.1.1.1.cmml" xref="S4.SS3.p6.7.m7.1.1">superscript</csymbol><apply id="S4.SS3.p6.7.m7.1.1.2.cmml" xref="S4.SS3.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.7.m7.1.1.2.1.cmml" xref="S4.SS3.p6.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p6.7.m7.1.1.2.2.cmml" xref="S4.SS3.p6.7.m7.1.1.2.2">𝐶</ci><ci id="S4.SS3.p6.7.m7.1.1.2.3.cmml" xref="S4.SS3.p6.7.m7.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.7.m7.1.1.3.cmml" xref="S4.SS3.p6.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.7.m7.1c">C_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.7.m7.1d">italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="C_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.8.m8.1"><semantics id="S4.SS3.p6.8.m8.1a"><msubsup id="S4.SS3.p6.8.m8.1.1" xref="S4.SS3.p6.8.m8.1.1.cmml"><mi id="S4.SS3.p6.8.m8.1.1.2.2" xref="S4.SS3.p6.8.m8.1.1.2.2.cmml">C</mi><msup id="S4.SS3.p6.8.m8.1.1.2.3" xref="S4.SS3.p6.8.m8.1.1.2.3.cmml"><mi id="S4.SS3.p6.8.m8.1.1.2.3.2" xref="S4.SS3.p6.8.m8.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.8.m8.1.1.2.3.3" xref="S4.SS3.p6.8.m8.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.8.m8.1.1.3" xref="S4.SS3.p6.8.m8.1.1.3.cmml"><mi id="S4.SS3.p6.8.m8.1.1.3.2" xref="S4.SS3.p6.8.m8.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.8.m8.1.1.3.3" xref="S4.SS3.p6.8.m8.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.8.m8.1b"><apply id="S4.SS3.p6.8.m8.1.1.cmml" xref="S4.SS3.p6.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.8.m8.1.1.1.cmml" xref="S4.SS3.p6.8.m8.1.1">superscript</csymbol><apply id="S4.SS3.p6.8.m8.1.1.2.cmml" xref="S4.SS3.p6.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.8.m8.1.1.2.1.cmml" xref="S4.SS3.p6.8.m8.1.1">subscript</csymbol><ci id="S4.SS3.p6.8.m8.1.1.2.2.cmml" xref="S4.SS3.p6.8.m8.1.1.2.2">𝐶</ci><apply id="S4.SS3.p6.8.m8.1.1.2.3.cmml" xref="S4.SS3.p6.8.m8.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.8.m8.1.1.2.3.1.cmml" xref="S4.SS3.p6.8.m8.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.8.m8.1.1.2.3.2.cmml" xref="S4.SS3.p6.8.m8.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.8.m8.1.1.2.3.3.cmml" xref="S4.SS3.p6.8.m8.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.8.m8.1.1.3.cmml" xref="S4.SS3.p6.8.m8.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.8.m8.1.1.3.1.cmml" xref="S4.SS3.p6.8.m8.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.8.m8.1.1.3.2.cmml" xref="S4.SS3.p6.8.m8.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.8.m8.1.1.3.3.cmml" xref="S4.SS3.p6.8.m8.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.8.m8.1c">C_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.8.m8.1d">italic_C start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> are checked. For skill nodes, all robot poses on the reference trajectory have to be collision-free simultaneously. If there is any collision between <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.9.m9.1"><semantics id="S4.SS3.p6.9.m9.1a"><msubsup id="S4.SS3.p6.9.m9.1.1" xref="S4.SS3.p6.9.m9.1.1.cmml"><mi id="S4.SS3.p6.9.m9.1.1.2.2" xref="S4.SS3.p6.9.m9.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.9.m9.1.1.2.3" xref="S4.SS3.p6.9.m9.1.1.2.3.cmml"><mi id="S4.SS3.p6.9.m9.1.1.2.3.2" xref="S4.SS3.p6.9.m9.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.9.m9.1.1.2.3.3" xref="S4.SS3.p6.9.m9.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.9.m9.1.1.3" xref="S4.SS3.p6.9.m9.1.1.3.cmml"><mi id="S4.SS3.p6.9.m9.1.1.3.2" xref="S4.SS3.p6.9.m9.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.9.m9.1.1.3.3" xref="S4.SS3.p6.9.m9.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.9.m9.1b"><apply id="S4.SS3.p6.9.m9.1.1.cmml" xref="S4.SS3.p6.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.9.m9.1.1.1.cmml" xref="S4.SS3.p6.9.m9.1.1">superscript</csymbol><apply id="S4.SS3.p6.9.m9.1.1.2.cmml" xref="S4.SS3.p6.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.9.m9.1.1.2.1.cmml" xref="S4.SS3.p6.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p6.9.m9.1.1.2.2.cmml" xref="S4.SS3.p6.9.m9.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.9.m9.1.1.2.3.cmml" xref="S4.SS3.p6.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.9.m9.1.1.2.3.1.cmml" xref="S4.SS3.p6.9.m9.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.9.m9.1.1.2.3.2.cmml" xref="S4.SS3.p6.9.m9.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.9.m9.1.1.2.3.3.cmml" xref="S4.SS3.p6.9.m9.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.9.m9.1.1.3.cmml" xref="S4.SS3.p6.9.m9.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.9.m9.1.1.3.1.cmml" xref="S4.SS3.p6.9.m9.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.9.m9.1.1.3.2.cmml" xref="S4.SS3.p6.9.m9.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.9.m9.1.1.3.3.cmml" xref="S4.SS3.p6.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.9.m9.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.9.m9.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.10.m10.1"><semantics id="S4.SS3.p6.10.m10.1a"><msubsup id="S4.SS3.p6.10.m10.1.1" xref="S4.SS3.p6.10.m10.1.1.cmml"><mi id="S4.SS3.p6.10.m10.1.1.2.2" xref="S4.SS3.p6.10.m10.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.10.m10.1.1.2.3" xref="S4.SS3.p6.10.m10.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.10.m10.1.1.3" xref="S4.SS3.p6.10.m10.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.10.m10.1b"><apply id="S4.SS3.p6.10.m10.1.1.cmml" xref="S4.SS3.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.10.m10.1.1.1.cmml" xref="S4.SS3.p6.10.m10.1.1">superscript</csymbol><apply id="S4.SS3.p6.10.m10.1.1.2.cmml" xref="S4.SS3.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.10.m10.1.1.2.1.cmml" xref="S4.SS3.p6.10.m10.1.1">subscript</csymbol><ci id="S4.SS3.p6.10.m10.1.1.2.2.cmml" xref="S4.SS3.p6.10.m10.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.10.m10.1.1.2.3.cmml" xref="S4.SS3.p6.10.m10.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.10.m10.1.1.3.cmml" xref="S4.SS3.p6.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.10.m10.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.10.m10.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, a type-2 edge is added from the earlier node’s successor, <math alttext="v_{n^{\prime}+1}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.11.m11.1"><semantics id="S4.SS3.p6.11.m11.1a"><msubsup id="S4.SS3.p6.11.m11.1.1" xref="S4.SS3.p6.11.m11.1.1.cmml"><mi id="S4.SS3.p6.11.m11.1.1.2.2" xref="S4.SS3.p6.11.m11.1.1.2.2.cmml">v</mi><mrow id="S4.SS3.p6.11.m11.1.1.2.3" xref="S4.SS3.p6.11.m11.1.1.2.3.cmml"><msup id="S4.SS3.p6.11.m11.1.1.2.3.2" xref="S4.SS3.p6.11.m11.1.1.2.3.2.cmml"><mi id="S4.SS3.p6.11.m11.1.1.2.3.2.2" xref="S4.SS3.p6.11.m11.1.1.2.3.2.2.cmml">n</mi><mo id="S4.SS3.p6.11.m11.1.1.2.3.2.3" xref="S4.SS3.p6.11.m11.1.1.2.3.2.3.cmml">′</mo></msup><mo id="S4.SS3.p6.11.m11.1.1.2.3.1" xref="S4.SS3.p6.11.m11.1.1.2.3.1.cmml">+</mo><mn id="S4.SS3.p6.11.m11.1.1.2.3.3" xref="S4.SS3.p6.11.m11.1.1.2.3.3.cmml">1</mn></mrow><msup id="S4.SS3.p6.11.m11.1.1.3" xref="S4.SS3.p6.11.m11.1.1.3.cmml"><mi id="S4.SS3.p6.11.m11.1.1.3.2" xref="S4.SS3.p6.11.m11.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.11.m11.1.1.3.3" xref="S4.SS3.p6.11.m11.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.11.m11.1b"><apply id="S4.SS3.p6.11.m11.1.1.cmml" xref="S4.SS3.p6.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.11.m11.1.1.1.cmml" xref="S4.SS3.p6.11.m11.1.1">superscript</csymbol><apply id="S4.SS3.p6.11.m11.1.1.2.cmml" xref="S4.SS3.p6.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.11.m11.1.1.2.1.cmml" xref="S4.SS3.p6.11.m11.1.1">subscript</csymbol><ci id="S4.SS3.p6.11.m11.1.1.2.2.cmml" xref="S4.SS3.p6.11.m11.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.11.m11.1.1.2.3.cmml" xref="S4.SS3.p6.11.m11.1.1.2.3"><plus id="S4.SS3.p6.11.m11.1.1.2.3.1.cmml" xref="S4.SS3.p6.11.m11.1.1.2.3.1"></plus><apply id="S4.SS3.p6.11.m11.1.1.2.3.2.cmml" xref="S4.SS3.p6.11.m11.1.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p6.11.m11.1.1.2.3.2.1.cmml" xref="S4.SS3.p6.11.m11.1.1.2.3.2">superscript</csymbol><ci id="S4.SS3.p6.11.m11.1.1.2.3.2.2.cmml" xref="S4.SS3.p6.11.m11.1.1.2.3.2.2">𝑛</ci><ci id="S4.SS3.p6.11.m11.1.1.2.3.2.3.cmml" xref="S4.SS3.p6.11.m11.1.1.2.3.2.3">′</ci></apply><cn id="S4.SS3.p6.11.m11.1.1.2.3.3.cmml" type="integer" xref="S4.SS3.p6.11.m11.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS3.p6.11.m11.1.1.3.cmml" xref="S4.SS3.p6.11.m11.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.11.m11.1.1.3.1.cmml" xref="S4.SS3.p6.11.m11.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.11.m11.1.1.3.2.cmml" xref="S4.SS3.p6.11.m11.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.11.m11.1.1.3.3.cmml" xref="S4.SS3.p6.11.m11.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.11.m11.1c">v_{n^{\prime}+1}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.11.m11.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, to the later node, <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.12.m12.1"><semantics id="S4.SS3.p6.12.m12.1a"><msubsup id="S4.SS3.p6.12.m12.1.1" xref="S4.SS3.p6.12.m12.1.1.cmml"><mi id="S4.SS3.p6.12.m12.1.1.2.2" xref="S4.SS3.p6.12.m12.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.12.m12.1.1.2.3" xref="S4.SS3.p6.12.m12.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.12.m12.1.1.3" xref="S4.SS3.p6.12.m12.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.12.m12.1b"><apply id="S4.SS3.p6.12.m12.1.1.cmml" xref="S4.SS3.p6.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.12.m12.1.1.1.cmml" xref="S4.SS3.p6.12.m12.1.1">superscript</csymbol><apply id="S4.SS3.p6.12.m12.1.1.2.cmml" xref="S4.SS3.p6.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.12.m12.1.1.2.1.cmml" xref="S4.SS3.p6.12.m12.1.1">subscript</csymbol><ci id="S4.SS3.p6.12.m12.1.1.2.2.cmml" xref="S4.SS3.p6.12.m12.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.12.m12.1.1.2.3.cmml" xref="S4.SS3.p6.12.m12.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.12.m12.1.1.3.cmml" xref="S4.SS3.p6.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.12.m12.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.12.m12.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. This way, if <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.13.m13.1"><semantics id="S4.SS3.p6.13.m13.1a"><msubsup id="S4.SS3.p6.13.m13.1.1" xref="S4.SS3.p6.13.m13.1.1.cmml"><mi id="S4.SS3.p6.13.m13.1.1.2.2" xref="S4.SS3.p6.13.m13.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.13.m13.1.1.2.3" xref="S4.SS3.p6.13.m13.1.1.2.3.cmml"><mi id="S4.SS3.p6.13.m13.1.1.2.3.2" xref="S4.SS3.p6.13.m13.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.13.m13.1.1.2.3.3" xref="S4.SS3.p6.13.m13.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.13.m13.1.1.3" xref="S4.SS3.p6.13.m13.1.1.3.cmml"><mi id="S4.SS3.p6.13.m13.1.1.3.2" xref="S4.SS3.p6.13.m13.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.13.m13.1.1.3.3" xref="S4.SS3.p6.13.m13.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.13.m13.1b"><apply id="S4.SS3.p6.13.m13.1.1.cmml" xref="S4.SS3.p6.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.13.m13.1.1.1.cmml" xref="S4.SS3.p6.13.m13.1.1">superscript</csymbol><apply id="S4.SS3.p6.13.m13.1.1.2.cmml" xref="S4.SS3.p6.13.m13.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.13.m13.1.1.2.1.cmml" xref="S4.SS3.p6.13.m13.1.1">subscript</csymbol><ci id="S4.SS3.p6.13.m13.1.1.2.2.cmml" xref="S4.SS3.p6.13.m13.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.13.m13.1.1.2.3.cmml" xref="S4.SS3.p6.13.m13.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.13.m13.1.1.2.3.1.cmml" xref="S4.SS3.p6.13.m13.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.13.m13.1.1.2.3.2.cmml" xref="S4.SS3.p6.13.m13.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.13.m13.1.1.2.3.3.cmml" xref="S4.SS3.p6.13.m13.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.13.m13.1.1.3.cmml" xref="S4.SS3.p6.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.13.m13.1.1.3.1.cmml" xref="S4.SS3.p6.13.m13.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.13.m13.1.1.3.2.cmml" xref="S4.SS3.p6.13.m13.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.13.m13.1.1.3.3.cmml" xref="S4.SS3.p6.13.m13.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.13.m13.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.13.m13.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> has a smaller timestamp than <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.14.m14.1"><semantics id="S4.SS3.p6.14.m14.1a"><msubsup id="S4.SS3.p6.14.m14.1.1" xref="S4.SS3.p6.14.m14.1.1.cmml"><mi id="S4.SS3.p6.14.m14.1.1.2.2" xref="S4.SS3.p6.14.m14.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.14.m14.1.1.2.3" xref="S4.SS3.p6.14.m14.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.14.m14.1.1.3" xref="S4.SS3.p6.14.m14.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.14.m14.1b"><apply id="S4.SS3.p6.14.m14.1.1.cmml" xref="S4.SS3.p6.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.14.m14.1.1.1.cmml" xref="S4.SS3.p6.14.m14.1.1">superscript</csymbol><apply id="S4.SS3.p6.14.m14.1.1.2.cmml" xref="S4.SS3.p6.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.14.m14.1.1.2.1.cmml" xref="S4.SS3.p6.14.m14.1.1">subscript</csymbol><ci id="S4.SS3.p6.14.m14.1.1.2.2.cmml" xref="S4.SS3.p6.14.m14.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.14.m14.1.1.2.3.cmml" xref="S4.SS3.p6.14.m14.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.14.m14.1.1.3.cmml" xref="S4.SS3.p6.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.14.m14.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.14.m14.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> in the input sequential plan, i.e., <math alttext="t_{n^{\prime}}^{i^{\prime}}<t_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.15.m15.1"><semantics id="S4.SS3.p6.15.m15.1a"><mrow id="S4.SS3.p6.15.m15.1.1" xref="S4.SS3.p6.15.m15.1.1.cmml"><msubsup id="S4.SS3.p6.15.m15.1.1.2" xref="S4.SS3.p6.15.m15.1.1.2.cmml"><mi id="S4.SS3.p6.15.m15.1.1.2.2.2" xref="S4.SS3.p6.15.m15.1.1.2.2.2.cmml">t</mi><msup id="S4.SS3.p6.15.m15.1.1.2.2.3" xref="S4.SS3.p6.15.m15.1.1.2.2.3.cmml"><mi id="S4.SS3.p6.15.m15.1.1.2.2.3.2" xref="S4.SS3.p6.15.m15.1.1.2.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.15.m15.1.1.2.2.3.3" xref="S4.SS3.p6.15.m15.1.1.2.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.15.m15.1.1.2.3" xref="S4.SS3.p6.15.m15.1.1.2.3.cmml"><mi id="S4.SS3.p6.15.m15.1.1.2.3.2" xref="S4.SS3.p6.15.m15.1.1.2.3.2.cmml">i</mi><mo id="S4.SS3.p6.15.m15.1.1.2.3.3" xref="S4.SS3.p6.15.m15.1.1.2.3.3.cmml">′</mo></msup></msubsup><mo id="S4.SS3.p6.15.m15.1.1.1" xref="S4.SS3.p6.15.m15.1.1.1.cmml"><</mo><msubsup id="S4.SS3.p6.15.m15.1.1.3" xref="S4.SS3.p6.15.m15.1.1.3.cmml"><mi id="S4.SS3.p6.15.m15.1.1.3.2.2" xref="S4.SS3.p6.15.m15.1.1.3.2.2.cmml">t</mi><mi id="S4.SS3.p6.15.m15.1.1.3.2.3" xref="S4.SS3.p6.15.m15.1.1.3.2.3.cmml">n</mi><mi id="S4.SS3.p6.15.m15.1.1.3.3" xref="S4.SS3.p6.15.m15.1.1.3.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.15.m15.1b"><apply id="S4.SS3.p6.15.m15.1.1.cmml" xref="S4.SS3.p6.15.m15.1.1"><lt id="S4.SS3.p6.15.m15.1.1.1.cmml" xref="S4.SS3.p6.15.m15.1.1.1"></lt><apply id="S4.SS3.p6.15.m15.1.1.2.cmml" xref="S4.SS3.p6.15.m15.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p6.15.m15.1.1.2.1.cmml" xref="S4.SS3.p6.15.m15.1.1.2">superscript</csymbol><apply id="S4.SS3.p6.15.m15.1.1.2.2.cmml" xref="S4.SS3.p6.15.m15.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p6.15.m15.1.1.2.2.1.cmml" xref="S4.SS3.p6.15.m15.1.1.2">subscript</csymbol><ci id="S4.SS3.p6.15.m15.1.1.2.2.2.cmml" xref="S4.SS3.p6.15.m15.1.1.2.2.2">𝑡</ci><apply id="S4.SS3.p6.15.m15.1.1.2.2.3.cmml" xref="S4.SS3.p6.15.m15.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.15.m15.1.1.2.2.3.1.cmml" xref="S4.SS3.p6.15.m15.1.1.2.2.3">superscript</csymbol><ci id="S4.SS3.p6.15.m15.1.1.2.2.3.2.cmml" xref="S4.SS3.p6.15.m15.1.1.2.2.3.2">𝑛</ci><ci id="S4.SS3.p6.15.m15.1.1.2.2.3.3.cmml" xref="S4.SS3.p6.15.m15.1.1.2.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.15.m15.1.1.2.3.cmml" xref="S4.SS3.p6.15.m15.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.15.m15.1.1.2.3.1.cmml" xref="S4.SS3.p6.15.m15.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.15.m15.1.1.2.3.2.cmml" xref="S4.SS3.p6.15.m15.1.1.2.3.2">𝑖</ci><ci id="S4.SS3.p6.15.m15.1.1.2.3.3.cmml" xref="S4.SS3.p6.15.m15.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.15.m15.1.1.3.cmml" xref="S4.SS3.p6.15.m15.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.15.m15.1.1.3.1.cmml" xref="S4.SS3.p6.15.m15.1.1.3">superscript</csymbol><apply id="S4.SS3.p6.15.m15.1.1.3.2.cmml" xref="S4.SS3.p6.15.m15.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.15.m15.1.1.3.2.1.cmml" xref="S4.SS3.p6.15.m15.1.1.3">subscript</csymbol><ci id="S4.SS3.p6.15.m15.1.1.3.2.2.cmml" xref="S4.SS3.p6.15.m15.1.1.3.2.2">𝑡</ci><ci id="S4.SS3.p6.15.m15.1.1.3.2.3.cmml" xref="S4.SS3.p6.15.m15.1.1.3.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.15.m15.1.1.3.3.cmml" xref="S4.SS3.p6.15.m15.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.15.m15.1c">t_{n^{\prime}}^{i^{\prime}}<t_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.15.m15.1d">italic_t start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT < italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.16.m16.1"><semantics id="S4.SS3.p6.16.m16.1a"><msubsup id="S4.SS3.p6.16.m16.1.1" xref="S4.SS3.p6.16.m16.1.1.cmml"><mi id="S4.SS3.p6.16.m16.1.1.2.2" xref="S4.SS3.p6.16.m16.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.16.m16.1.1.2.3" xref="S4.SS3.p6.16.m16.1.1.2.3.cmml"><mi id="S4.SS3.p6.16.m16.1.1.2.3.2" xref="S4.SS3.p6.16.m16.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.16.m16.1.1.2.3.3" xref="S4.SS3.p6.16.m16.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.16.m16.1.1.3" xref="S4.SS3.p6.16.m16.1.1.3.cmml"><mi id="S4.SS3.p6.16.m16.1.1.3.2" xref="S4.SS3.p6.16.m16.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.16.m16.1.1.3.3" xref="S4.SS3.p6.16.m16.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.16.m16.1b"><apply id="S4.SS3.p6.16.m16.1.1.cmml" xref="S4.SS3.p6.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.16.m16.1.1.1.cmml" xref="S4.SS3.p6.16.m16.1.1">superscript</csymbol><apply id="S4.SS3.p6.16.m16.1.1.2.cmml" xref="S4.SS3.p6.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.16.m16.1.1.2.1.cmml" xref="S4.SS3.p6.16.m16.1.1">subscript</csymbol><ci id="S4.SS3.p6.16.m16.1.1.2.2.cmml" xref="S4.SS3.p6.16.m16.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.16.m16.1.1.2.3.cmml" xref="S4.SS3.p6.16.m16.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.16.m16.1.1.2.3.1.cmml" xref="S4.SS3.p6.16.m16.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.16.m16.1.1.2.3.2.cmml" xref="S4.SS3.p6.16.m16.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.16.m16.1.1.2.3.3.cmml" xref="S4.SS3.p6.16.m16.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.16.m16.1.1.3.cmml" xref="S4.SS3.p6.16.m16.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.16.m16.1.1.3.1.cmml" xref="S4.SS3.p6.16.m16.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.16.m16.1.1.3.2.cmml" xref="S4.SS3.p6.16.m16.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.16.m16.1.1.3.3.cmml" xref="S4.SS3.p6.16.m16.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.16.m16.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.16.m16.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> must still be executed before <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.17.m17.1"><semantics id="S4.SS3.p6.17.m17.1a"><msubsup id="S4.SS3.p6.17.m17.1.1" xref="S4.SS3.p6.17.m17.1.1.cmml"><mi id="S4.SS3.p6.17.m17.1.1.2.2" xref="S4.SS3.p6.17.m17.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.17.m17.1.1.2.3" xref="S4.SS3.p6.17.m17.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.17.m17.1.1.3" xref="S4.SS3.p6.17.m17.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.17.m17.1b"><apply id="S4.SS3.p6.17.m17.1.1.cmml" xref="S4.SS3.p6.17.m17.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.17.m17.1.1.1.cmml" xref="S4.SS3.p6.17.m17.1.1">superscript</csymbol><apply id="S4.SS3.p6.17.m17.1.1.2.cmml" xref="S4.SS3.p6.17.m17.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.17.m17.1.1.2.1.cmml" xref="S4.SS3.p6.17.m17.1.1">subscript</csymbol><ci id="S4.SS3.p6.17.m17.1.1.2.2.cmml" xref="S4.SS3.p6.17.m17.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.17.m17.1.1.2.3.cmml" xref="S4.SS3.p6.17.m17.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.17.m17.1.1.3.cmml" xref="S4.SS3.p6.17.m17.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.17.m17.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.17.m17.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> can start, avoiding any potential collisions. When iterating <math alttext="n^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p6.18.m18.1"><semantics id="S4.SS3.p6.18.m18.1a"><msup id="S4.SS3.p6.18.m18.1.1" xref="S4.SS3.p6.18.m18.1.1.cmml"><mi id="S4.SS3.p6.18.m18.1.1.2" xref="S4.SS3.p6.18.m18.1.1.2.cmml">n</mi><mo id="S4.SS3.p6.18.m18.1.1.3" xref="S4.SS3.p6.18.m18.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.18.m18.1b"><apply id="S4.SS3.p6.18.m18.1.1.cmml" xref="S4.SS3.p6.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.18.m18.1.1.1.cmml" xref="S4.SS3.p6.18.m18.1.1">superscript</csymbol><ci id="S4.SS3.p6.18.m18.1.1.2.cmml" xref="S4.SS3.p6.18.m18.1.1.2">𝑛</ci><ci id="S4.SS3.p6.18.m18.1.1.3.cmml" xref="S4.SS3.p6.18.m18.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.18.m18.1c">n^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.18.m18.1d">italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, it is unnecessary to check collisions for any node <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.19.m19.1"><semantics id="S4.SS3.p6.19.m19.1a"><msubsup id="S4.SS3.p6.19.m19.1.1" xref="S4.SS3.p6.19.m19.1.1.cmml"><mi id="S4.SS3.p6.19.m19.1.1.2.2" xref="S4.SS3.p6.19.m19.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.19.m19.1.1.2.3" xref="S4.SS3.p6.19.m19.1.1.2.3.cmml"><mi id="S4.SS3.p6.19.m19.1.1.2.3.2" xref="S4.SS3.p6.19.m19.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.19.m19.1.1.2.3.3" xref="S4.SS3.p6.19.m19.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.19.m19.1.1.3" xref="S4.SS3.p6.19.m19.1.1.3.cmml"><mi id="S4.SS3.p6.19.m19.1.1.3.2" xref="S4.SS3.p6.19.m19.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.19.m19.1.1.3.3" xref="S4.SS3.p6.19.m19.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.19.m19.1b"><apply id="S4.SS3.p6.19.m19.1.1.cmml" xref="S4.SS3.p6.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.19.m19.1.1.1.cmml" xref="S4.SS3.p6.19.m19.1.1">superscript</csymbol><apply id="S4.SS3.p6.19.m19.1.1.2.cmml" xref="S4.SS3.p6.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.19.m19.1.1.2.1.cmml" xref="S4.SS3.p6.19.m19.1.1">subscript</csymbol><ci id="S4.SS3.p6.19.m19.1.1.2.2.cmml" xref="S4.SS3.p6.19.m19.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.19.m19.1.1.2.3.cmml" xref="S4.SS3.p6.19.m19.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.19.m19.1.1.2.3.1.cmml" xref="S4.SS3.p6.19.m19.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.19.m19.1.1.2.3.2.cmml" xref="S4.SS3.p6.19.m19.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.19.m19.1.1.2.3.3.cmml" xref="S4.SS3.p6.19.m19.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.19.m19.1.1.3.cmml" xref="S4.SS3.p6.19.m19.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.19.m19.1.1.3.1.cmml" xref="S4.SS3.p6.19.m19.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.19.m19.1.1.3.2.cmml" xref="S4.SS3.p6.19.m19.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.19.m19.1.1.3.3.cmml" xref="S4.SS3.p6.19.m19.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.19.m19.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.19.m19.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> that has a larger timestamp than <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.20.m20.1"><semantics id="S4.SS3.p6.20.m20.1a"><msubsup id="S4.SS3.p6.20.m20.1.1" xref="S4.SS3.p6.20.m20.1.1.cmml"><mi id="S4.SS3.p6.20.m20.1.1.2.2" xref="S4.SS3.p6.20.m20.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.20.m20.1.1.2.3" xref="S4.SS3.p6.20.m20.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.20.m20.1.1.3" xref="S4.SS3.p6.20.m20.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.20.m20.1b"><apply id="S4.SS3.p6.20.m20.1.1.cmml" xref="S4.SS3.p6.20.m20.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.20.m20.1.1.1.cmml" xref="S4.SS3.p6.20.m20.1.1">superscript</csymbol><apply id="S4.SS3.p6.20.m20.1.1.2.cmml" xref="S4.SS3.p6.20.m20.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.20.m20.1.1.2.1.cmml" xref="S4.SS3.p6.20.m20.1.1">subscript</csymbol><ci id="S4.SS3.p6.20.m20.1.1.2.2.cmml" xref="S4.SS3.p6.20.m20.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.20.m20.1.1.2.3.cmml" xref="S4.SS3.p6.20.m20.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.20.m20.1.1.3.cmml" xref="S4.SS3.p6.20.m20.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.20.m20.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.20.m20.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, i.e., <math alttext="t_{n^{\prime}}^{i^{\prime}}\geq t_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.21.m21.1"><semantics id="S4.SS3.p6.21.m21.1a"><mrow id="S4.SS3.p6.21.m21.1.1" xref="S4.SS3.p6.21.m21.1.1.cmml"><msubsup id="S4.SS3.p6.21.m21.1.1.2" xref="S4.SS3.p6.21.m21.1.1.2.cmml"><mi id="S4.SS3.p6.21.m21.1.1.2.2.2" xref="S4.SS3.p6.21.m21.1.1.2.2.2.cmml">t</mi><msup id="S4.SS3.p6.21.m21.1.1.2.2.3" xref="S4.SS3.p6.21.m21.1.1.2.2.3.cmml"><mi id="S4.SS3.p6.21.m21.1.1.2.2.3.2" xref="S4.SS3.p6.21.m21.1.1.2.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.21.m21.1.1.2.2.3.3" xref="S4.SS3.p6.21.m21.1.1.2.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.21.m21.1.1.2.3" xref="S4.SS3.p6.21.m21.1.1.2.3.cmml"><mi id="S4.SS3.p6.21.m21.1.1.2.3.2" xref="S4.SS3.p6.21.m21.1.1.2.3.2.cmml">i</mi><mo id="S4.SS3.p6.21.m21.1.1.2.3.3" xref="S4.SS3.p6.21.m21.1.1.2.3.3.cmml">′</mo></msup></msubsup><mo id="S4.SS3.p6.21.m21.1.1.1" xref="S4.SS3.p6.21.m21.1.1.1.cmml">≥</mo><msubsup id="S4.SS3.p6.21.m21.1.1.3" xref="S4.SS3.p6.21.m21.1.1.3.cmml"><mi id="S4.SS3.p6.21.m21.1.1.3.2.2" xref="S4.SS3.p6.21.m21.1.1.3.2.2.cmml">t</mi><mi id="S4.SS3.p6.21.m21.1.1.3.2.3" xref="S4.SS3.p6.21.m21.1.1.3.2.3.cmml">n</mi><mi id="S4.SS3.p6.21.m21.1.1.3.3" xref="S4.SS3.p6.21.m21.1.1.3.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.21.m21.1b"><apply id="S4.SS3.p6.21.m21.1.1.cmml" xref="S4.SS3.p6.21.m21.1.1"><geq id="S4.SS3.p6.21.m21.1.1.1.cmml" xref="S4.SS3.p6.21.m21.1.1.1"></geq><apply id="S4.SS3.p6.21.m21.1.1.2.cmml" xref="S4.SS3.p6.21.m21.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p6.21.m21.1.1.2.1.cmml" xref="S4.SS3.p6.21.m21.1.1.2">superscript</csymbol><apply id="S4.SS3.p6.21.m21.1.1.2.2.cmml" xref="S4.SS3.p6.21.m21.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p6.21.m21.1.1.2.2.1.cmml" xref="S4.SS3.p6.21.m21.1.1.2">subscript</csymbol><ci id="S4.SS3.p6.21.m21.1.1.2.2.2.cmml" xref="S4.SS3.p6.21.m21.1.1.2.2.2">𝑡</ci><apply id="S4.SS3.p6.21.m21.1.1.2.2.3.cmml" xref="S4.SS3.p6.21.m21.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.21.m21.1.1.2.2.3.1.cmml" xref="S4.SS3.p6.21.m21.1.1.2.2.3">superscript</csymbol><ci id="S4.SS3.p6.21.m21.1.1.2.2.3.2.cmml" xref="S4.SS3.p6.21.m21.1.1.2.2.3.2">𝑛</ci><ci id="S4.SS3.p6.21.m21.1.1.2.2.3.3.cmml" xref="S4.SS3.p6.21.m21.1.1.2.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.21.m21.1.1.2.3.cmml" xref="S4.SS3.p6.21.m21.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.21.m21.1.1.2.3.1.cmml" xref="S4.SS3.p6.21.m21.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.21.m21.1.1.2.3.2.cmml" xref="S4.SS3.p6.21.m21.1.1.2.3.2">𝑖</ci><ci id="S4.SS3.p6.21.m21.1.1.2.3.3.cmml" xref="S4.SS3.p6.21.m21.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.21.m21.1.1.3.cmml" xref="S4.SS3.p6.21.m21.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.21.m21.1.1.3.1.cmml" xref="S4.SS3.p6.21.m21.1.1.3">superscript</csymbol><apply id="S4.SS3.p6.21.m21.1.1.3.2.cmml" xref="S4.SS3.p6.21.m21.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.21.m21.1.1.3.2.1.cmml" xref="S4.SS3.p6.21.m21.1.1.3">subscript</csymbol><ci id="S4.SS3.p6.21.m21.1.1.3.2.2.cmml" xref="S4.SS3.p6.21.m21.1.1.3.2.2">𝑡</ci><ci id="S4.SS3.p6.21.m21.1.1.3.2.3.cmml" xref="S4.SS3.p6.21.m21.1.1.3.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.21.m21.1.1.3.3.cmml" xref="S4.SS3.p6.21.m21.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.21.m21.1c">t_{n^{\prime}}^{i^{\prime}}\geq t_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.21.m21.1d">italic_t start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ≥ italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, because those would be checked when the iterated robot pair (<math alttext="i" class="ltx_Math" display="inline" id="S4.SS3.p6.22.m22.1"><semantics id="S4.SS3.p6.22.m22.1a"><mi id="S4.SS3.p6.22.m22.1.1" xref="S4.SS3.p6.22.m22.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.22.m22.1b"><ci id="S4.SS3.p6.22.m22.1.1.cmml" xref="S4.SS3.p6.22.m22.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.22.m22.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.22.m22.1d">italic_i</annotation></semantics></math>, <math alttext="i^{\prime}" class="ltx_Math" display="inline" id="S4.SS3.p6.23.m23.1"><semantics id="S4.SS3.p6.23.m23.1a"><msup id="S4.SS3.p6.23.m23.1.1" xref="S4.SS3.p6.23.m23.1.1.cmml"><mi id="S4.SS3.p6.23.m23.1.1.2" xref="S4.SS3.p6.23.m23.1.1.2.cmml">i</mi><mo id="S4.SS3.p6.23.m23.1.1.3" xref="S4.SS3.p6.23.m23.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.23.m23.1b"><apply id="S4.SS3.p6.23.m23.1.1.cmml" xref="S4.SS3.p6.23.m23.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.23.m23.1.1.1.cmml" xref="S4.SS3.p6.23.m23.1.1">superscript</csymbol><ci id="S4.SS3.p6.23.m23.1.1.2.cmml" xref="S4.SS3.p6.23.m23.1.1.2">𝑖</ci><ci id="S4.SS3.p6.23.m23.1.1.3.cmml" xref="S4.SS3.p6.23.m23.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.23.m23.1c">i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.23.m23.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>) are swapped. Also, if the current <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.24.m24.1"><semantics id="S4.SS3.p6.24.m24.1a"><msubsup id="S4.SS3.p6.24.m24.1.1" xref="S4.SS3.p6.24.m24.1.1.cmml"><mi id="S4.SS3.p6.24.m24.1.1.2.2" xref="S4.SS3.p6.24.m24.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.24.m24.1.1.2.3" xref="S4.SS3.p6.24.m24.1.1.2.3.cmml"><mi id="S4.SS3.p6.24.m24.1.1.2.3.2" xref="S4.SS3.p6.24.m24.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.24.m24.1.1.2.3.3" xref="S4.SS3.p6.24.m24.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.24.m24.1.1.3" xref="S4.SS3.p6.24.m24.1.1.3.cmml"><mi id="S4.SS3.p6.24.m24.1.1.3.2" xref="S4.SS3.p6.24.m24.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.24.m24.1.1.3.3" xref="S4.SS3.p6.24.m24.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.24.m24.1b"><apply id="S4.SS3.p6.24.m24.1.1.cmml" xref="S4.SS3.p6.24.m24.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.24.m24.1.1.1.cmml" xref="S4.SS3.p6.24.m24.1.1">superscript</csymbol><apply id="S4.SS3.p6.24.m24.1.1.2.cmml" xref="S4.SS3.p6.24.m24.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.24.m24.1.1.2.1.cmml" xref="S4.SS3.p6.24.m24.1.1">subscript</csymbol><ci id="S4.SS3.p6.24.m24.1.1.2.2.cmml" xref="S4.SS3.p6.24.m24.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.24.m24.1.1.2.3.cmml" xref="S4.SS3.p6.24.m24.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.24.m24.1.1.2.3.1.cmml" xref="S4.SS3.p6.24.m24.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.24.m24.1.1.2.3.2.cmml" xref="S4.SS3.p6.24.m24.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.24.m24.1.1.2.3.3.cmml" xref="S4.SS3.p6.24.m24.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.24.m24.1.1.3.cmml" xref="S4.SS3.p6.24.m24.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.24.m24.1.1.3.1.cmml" xref="S4.SS3.p6.24.m24.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.24.m24.1.1.3.2.cmml" xref="S4.SS3.p6.24.m24.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.24.m24.1.1.3.3.cmml" xref="S4.SS3.p6.24.m24.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.24.m24.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.24.m24.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is a predecessor of <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.25.m25.1"><semantics id="S4.SS3.p6.25.m25.1a"><msubsup id="S4.SS3.p6.25.m25.1.1" xref="S4.SS3.p6.25.m25.1.1.cmml"><mi id="S4.SS3.p6.25.m25.1.1.2.2" xref="S4.SS3.p6.25.m25.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.25.m25.1.1.2.3" xref="S4.SS3.p6.25.m25.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.25.m25.1.1.3" xref="S4.SS3.p6.25.m25.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.25.m25.1b"><apply id="S4.SS3.p6.25.m25.1.1.cmml" xref="S4.SS3.p6.25.m25.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.25.m25.1.1.1.cmml" xref="S4.SS3.p6.25.m25.1.1">superscript</csymbol><apply id="S4.SS3.p6.25.m25.1.1.2.cmml" xref="S4.SS3.p6.25.m25.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.25.m25.1.1.2.1.cmml" xref="S4.SS3.p6.25.m25.1.1">subscript</csymbol><ci id="S4.SS3.p6.25.m25.1.1.2.2.cmml" xref="S4.SS3.p6.25.m25.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.25.m25.1.1.2.3.cmml" xref="S4.SS3.p6.25.m25.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.25.m25.1.1.3.cmml" xref="S4.SS3.p6.25.m25.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.25.m25.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.25.m25.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> in the graph, <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p6.26.m26.1"><semantics id="S4.SS3.p6.26.m26.1a"><msubsup id="S4.SS3.p6.26.m26.1.1" xref="S4.SS3.p6.26.m26.1.1.cmml"><mi id="S4.SS3.p6.26.m26.1.1.2.2" xref="S4.SS3.p6.26.m26.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p6.26.m26.1.1.2.3" xref="S4.SS3.p6.26.m26.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p6.26.m26.1.1.3" xref="S4.SS3.p6.26.m26.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.26.m26.1b"><apply id="S4.SS3.p6.26.m26.1.1.cmml" xref="S4.SS3.p6.26.m26.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.26.m26.1.1.1.cmml" xref="S4.SS3.p6.26.m26.1.1">superscript</csymbol><apply id="S4.SS3.p6.26.m26.1.1.2.cmml" xref="S4.SS3.p6.26.m26.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.26.m26.1.1.2.1.cmml" xref="S4.SS3.p6.26.m26.1.1">subscript</csymbol><ci id="S4.SS3.p6.26.m26.1.1.2.2.cmml" xref="S4.SS3.p6.26.m26.1.1.2.2">𝑣</ci><ci id="S4.SS3.p6.26.m26.1.1.2.3.cmml" xref="S4.SS3.p6.26.m26.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p6.26.m26.1.1.3.cmml" xref="S4.SS3.p6.26.m26.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.26.m26.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.26.m26.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> already waits for <math alttext="v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p6.27.m27.1"><semantics id="S4.SS3.p6.27.m27.1a"><msubsup id="S4.SS3.p6.27.m27.1.1" xref="S4.SS3.p6.27.m27.1.1.cmml"><mi id="S4.SS3.p6.27.m27.1.1.2.2" xref="S4.SS3.p6.27.m27.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p6.27.m27.1.1.2.3" xref="S4.SS3.p6.27.m27.1.1.2.3.cmml"><mi id="S4.SS3.p6.27.m27.1.1.2.3.2" xref="S4.SS3.p6.27.m27.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p6.27.m27.1.1.2.3.3" xref="S4.SS3.p6.27.m27.1.1.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p6.27.m27.1.1.3" xref="S4.SS3.p6.27.m27.1.1.3.cmml"><mi id="S4.SS3.p6.27.m27.1.1.3.2" xref="S4.SS3.p6.27.m27.1.1.3.2.cmml">i</mi><mo id="S4.SS3.p6.27.m27.1.1.3.3" xref="S4.SS3.p6.27.m27.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p6.27.m27.1b"><apply id="S4.SS3.p6.27.m27.1.1.cmml" xref="S4.SS3.p6.27.m27.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.27.m27.1.1.1.cmml" xref="S4.SS3.p6.27.m27.1.1">superscript</csymbol><apply id="S4.SS3.p6.27.m27.1.1.2.cmml" xref="S4.SS3.p6.27.m27.1.1"><csymbol cd="ambiguous" id="S4.SS3.p6.27.m27.1.1.2.1.cmml" xref="S4.SS3.p6.27.m27.1.1">subscript</csymbol><ci id="S4.SS3.p6.27.m27.1.1.2.2.cmml" xref="S4.SS3.p6.27.m27.1.1.2.2">𝑣</ci><apply id="S4.SS3.p6.27.m27.1.1.2.3.cmml" xref="S4.SS3.p6.27.m27.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p6.27.m27.1.1.2.3.1.cmml" xref="S4.SS3.p6.27.m27.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p6.27.m27.1.1.2.3.2.cmml" xref="S4.SS3.p6.27.m27.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p6.27.m27.1.1.2.3.3.cmml" xref="S4.SS3.p6.27.m27.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p6.27.m27.1.1.3.cmml" xref="S4.SS3.p6.27.m27.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p6.27.m27.1.1.3.1.cmml" xref="S4.SS3.p6.27.m27.1.1.3">superscript</csymbol><ci id="S4.SS3.p6.27.m27.1.1.3.2.cmml" xref="S4.SS3.p6.27.m27.1.1.3.2">𝑖</ci><ci id="S4.SS3.p6.27.m27.1.1.3.3.cmml" xref="S4.SS3.p6.27.m27.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p6.27.m27.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p6.27.m27.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and avoid collisions, so it becomes unnecessary to check them again. </div> <div class="ltx_para" id="S4.SS3.p7"> The total number of collision checks needed depends on the number of robots, discretized steps, and type-2 edges added from the task graph. Once every node has been checked against potentially colliding nodes, a transition reduction algorithm is applied, similar to <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib33" title="">33</a>]</cite> to simplify the TPG. A type-2 edge <math alttext="v_{n}^{i}\rightarrow v_{n^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S4.SS3.p7.1.m1.1"><semantics id="S4.SS3.p7.1.m1.1a"><mrow id="S4.SS3.p7.1.m1.1.1" xref="S4.SS3.p7.1.m1.1.1.cmml"><msubsup id="S4.SS3.p7.1.m1.1.1.2" xref="S4.SS3.p7.1.m1.1.1.2.cmml"><mi id="S4.SS3.p7.1.m1.1.1.2.2.2" xref="S4.SS3.p7.1.m1.1.1.2.2.2.cmml">v</mi><mi id="S4.SS3.p7.1.m1.1.1.2.2.3" xref="S4.SS3.p7.1.m1.1.1.2.2.3.cmml">n</mi><mi id="S4.SS3.p7.1.m1.1.1.2.3" xref="S4.SS3.p7.1.m1.1.1.2.3.cmml">i</mi></msubsup><mo id="S4.SS3.p7.1.m1.1.1.1" stretchy="false" xref="S4.SS3.p7.1.m1.1.1.1.cmml">→</mo><msubsup id="S4.SS3.p7.1.m1.1.1.3" xref="S4.SS3.p7.1.m1.1.1.3.cmml"><mi id="S4.SS3.p7.1.m1.1.1.3.2.2" xref="S4.SS3.p7.1.m1.1.1.3.2.2.cmml">v</mi><msup id="S4.SS3.p7.1.m1.1.1.3.2.3" xref="S4.SS3.p7.1.m1.1.1.3.2.3.cmml"><mi id="S4.SS3.p7.1.m1.1.1.3.2.3.2" xref="S4.SS3.p7.1.m1.1.1.3.2.3.2.cmml">n</mi><mo id="S4.SS3.p7.1.m1.1.1.3.2.3.3" xref="S4.SS3.p7.1.m1.1.1.3.2.3.3.cmml">′</mo></msup><msup id="S4.SS3.p7.1.m1.1.1.3.3" xref="S4.SS3.p7.1.m1.1.1.3.3.cmml"><mi id="S4.SS3.p7.1.m1.1.1.3.3.2" xref="S4.SS3.p7.1.m1.1.1.3.3.2.cmml">i</mi><mo id="S4.SS3.p7.1.m1.1.1.3.3.3" xref="S4.SS3.p7.1.m1.1.1.3.3.3.cmml">′</mo></msup></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p7.1.m1.1b"><apply id="S4.SS3.p7.1.m1.1.1.cmml" xref="S4.SS3.p7.1.m1.1.1"><ci id="S4.SS3.p7.1.m1.1.1.1.cmml" xref="S4.SS3.p7.1.m1.1.1.1">→</ci><apply id="S4.SS3.p7.1.m1.1.1.2.cmml" xref="S4.SS3.p7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p7.1.m1.1.1.2.1.cmml" xref="S4.SS3.p7.1.m1.1.1.2">superscript</csymbol><apply id="S4.SS3.p7.1.m1.1.1.2.2.cmml" xref="S4.SS3.p7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p7.1.m1.1.1.2.2.1.cmml" xref="S4.SS3.p7.1.m1.1.1.2">subscript</csymbol><ci id="S4.SS3.p7.1.m1.1.1.2.2.2.cmml" xref="S4.SS3.p7.1.m1.1.1.2.2.2">𝑣</ci><ci id="S4.SS3.p7.1.m1.1.1.2.2.3.cmml" xref="S4.SS3.p7.1.m1.1.1.2.2.3">𝑛</ci></apply><ci id="S4.SS3.p7.1.m1.1.1.2.3.cmml" xref="S4.SS3.p7.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p7.1.m1.1.1.3.cmml" xref="S4.SS3.p7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p7.1.m1.1.1.3.1.cmml" xref="S4.SS3.p7.1.m1.1.1.3">superscript</csymbol><apply id="S4.SS3.p7.1.m1.1.1.3.2.cmml" xref="S4.SS3.p7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p7.1.m1.1.1.3.2.1.cmml" xref="S4.SS3.p7.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS3.p7.1.m1.1.1.3.2.2.cmml" xref="S4.SS3.p7.1.m1.1.1.3.2.2">𝑣</ci><apply id="S4.SS3.p7.1.m1.1.1.3.2.3.cmml" xref="S4.SS3.p7.1.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS3.p7.1.m1.1.1.3.2.3.1.cmml" xref="S4.SS3.p7.1.m1.1.1.3.2.3">superscript</csymbol><ci id="S4.SS3.p7.1.m1.1.1.3.2.3.2.cmml" xref="S4.SS3.p7.1.m1.1.1.3.2.3.2">𝑛</ci><ci id="S4.SS3.p7.1.m1.1.1.3.2.3.3.cmml" xref="S4.SS3.p7.1.m1.1.1.3.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p7.1.m1.1.1.3.3.cmml" xref="S4.SS3.p7.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS3.p7.1.m1.1.1.3.3.1.cmml" xref="S4.SS3.p7.1.m1.1.1.3.3">superscript</csymbol><ci id="S4.SS3.p7.1.m1.1.1.3.3.2.cmml" xref="S4.SS3.p7.1.m1.1.1.3.3.2">𝑖</ci><ci id="S4.SS3.p7.1.m1.1.1.3.3.3.cmml" xref="S4.SS3.p7.1.m1.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.1.m1.1c">v_{n}^{i}\rightarrow v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.1.m1.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is redundant if node <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p7.2.m2.1"><semantics id="S4.SS3.p7.2.m2.1a"><msubsup id="S4.SS3.p7.2.m2.1.1" 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id="S4.SS3.p7.3.m3.1b"><apply id="S4.SS3.p7.3.m3.1.1.cmml" xref="S4.SS3.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p7.3.m3.1.1.1.cmml" xref="S4.SS3.p7.3.m3.1.1">superscript</csymbol><apply id="S4.SS3.p7.3.m3.1.1.2.cmml" xref="S4.SS3.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p7.3.m3.1.1.2.1.cmml" xref="S4.SS3.p7.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.p7.3.m3.1.1.2.2.cmml" xref="S4.SS3.p7.3.m3.1.1.2.2">𝑣</ci><apply id="S4.SS3.p7.3.m3.1.1.2.3.cmml" xref="S4.SS3.p7.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p7.3.m3.1.1.2.3.1.cmml" xref="S4.SS3.p7.3.m3.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p7.3.m3.1.1.2.3.2.cmml" xref="S4.SS3.p7.3.m3.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p7.3.m3.1.1.2.3.3.cmml" xref="S4.SS3.p7.3.m3.1.1.2.3.3">′</ci></apply></apply><apply id="S4.SS3.p7.3.m3.1.1.3.cmml" xref="S4.SS3.p7.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p7.3.m3.1.1.3.1.cmml" xref="S4.SS3.p7.3.m3.1.1.3">superscript</csymbol><ci id="S4.SS3.p7.3.m3.1.1.3.2.cmml" xref="S4.SS3.p7.3.m3.1.1.3.2">𝑖</ci><ci id="S4.SS3.p7.3.m3.1.1.3.3.cmml" xref="S4.SS3.p7.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p7.3.m3.1c">v_{n^{\prime}}^{i^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p7.3.m3.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> after the edge is removed. We remove all such redundant edges to reduce the total number of scheduling constraints and communication overheads during execution. </div> <div class="ltx_para" id="S4.SS3.p8"> The primary bottleneck of the TPG construction process is the number of collision checks, which scales quadratically with respect to the number of robots and the number of nodes. To alleviate this, collision checking can be parallelized across many CPU threads, reducing its runtime. A pseudocode of the entire construction process is provided in Algo. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#alg1" title="Algorithm 1 ‣ IV-C Asynchronous Execution ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">1</a>. </div> <figure class="ltx_figure" id="S4.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="414" id="S4.F4.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 4: Illustration of the execution setup. TPG maintains and controls the execution schedule of all robots on a central server, gradually sending new trajectory segments that can be safely executed. Each robot maintains a controller-sensing loop independently while updating its state with the central server.</figcaption> </figure> <div class="ltx_para" id="S4.SS3.p9"> Further Optimization An optional step to further reduce the execution makespan and smooth the trajectory is to skip the intermediate transition to HOME after every assembly step. We use the following shortcutting algorithm, similar to the strategy implemented in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib41" title="">41</a>]</cite>, to achieve that while maintaining a collision- and deadlock-free plan. </div> <div class="ltx_para" id="S4.SS3.p10"> The anytime algorithm works by randomly sampling two nodes of the TPG (<math alttext="v_{n}^{i},v_{n^{\prime}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.1.m1.2"><semantics id="S4.SS3.p10.1.m1.2a"><mrow id="S4.SS3.p10.1.m1.2.2.2" xref="S4.SS3.p10.1.m1.2.2.3.cmml"><msubsup id="S4.SS3.p10.1.m1.1.1.1.1" xref="S4.SS3.p10.1.m1.1.1.1.1.cmml"><mi id="S4.SS3.p10.1.m1.1.1.1.1.2.2" xref="S4.SS3.p10.1.m1.1.1.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p10.1.m1.1.1.1.1.2.3" xref="S4.SS3.p10.1.m1.1.1.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p10.1.m1.1.1.1.1.3" xref="S4.SS3.p10.1.m1.1.1.1.1.3.cmml">i</mi></msubsup><mo id="S4.SS3.p10.1.m1.2.2.2.3" xref="S4.SS3.p10.1.m1.2.2.3.cmml">,</mo><msubsup id="S4.SS3.p10.1.m1.2.2.2.2" xref="S4.SS3.p10.1.m1.2.2.2.2.cmml"><mi id="S4.SS3.p10.1.m1.2.2.2.2.2.2" xref="S4.SS3.p10.1.m1.2.2.2.2.2.2.cmml">v</mi><msup id="S4.SS3.p10.1.m1.2.2.2.2.2.3" xref="S4.SS3.p10.1.m1.2.2.2.2.2.3.cmml"><mi id="S4.SS3.p10.1.m1.2.2.2.2.2.3.2" xref="S4.SS3.p10.1.m1.2.2.2.2.2.3.2.cmml">n</mi><mo id="S4.SS3.p10.1.m1.2.2.2.2.2.3.3" xref="S4.SS3.p10.1.m1.2.2.2.2.2.3.3.cmml">′</mo></msup><mi id="S4.SS3.p10.1.m1.2.2.2.2.3" xref="S4.SS3.p10.1.m1.2.2.2.2.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.1.m1.2b"><list id="S4.SS3.p10.1.m1.2.2.3.cmml" xref="S4.SS3.p10.1.m1.2.2.2"><apply id="S4.SS3.p10.1.m1.1.1.1.1.cmml" xref="S4.SS3.p10.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.1.m1.1.1.1.1.1.cmml" xref="S4.SS3.p10.1.m1.1.1.1.1">superscript</csymbol><apply id="S4.SS3.p10.1.m1.1.1.1.1.2.cmml" xref="S4.SS3.p10.1.m1.1.1.1.1"><csymbol cd="ambiguous" 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encoding="application/x-tex" id="S4.SS3.p10.2.m2.1c">\mathcal{T}_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.2.m2.1d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>, checking whether connecting them in a shortcut is feasible. Consecutive transit tasks passing through the robot’s HOME pose are merged as a single task. This allows the HOME pose to be skipped. A shortcut path directly interpolates between <math alttext="C_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.3.m3.1"><semantics id="S4.SS3.p10.3.m3.1a"><msubsup id="S4.SS3.p10.3.m3.1.1" xref="S4.SS3.p10.3.m3.1.1.cmml"><mi id="S4.SS3.p10.3.m3.1.1.2.2" xref="S4.SS3.p10.3.m3.1.1.2.2.cmml">C</mi><mi id="S4.SS3.p10.3.m3.1.1.2.3" xref="S4.SS3.p10.3.m3.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p10.3.m3.1.1.3" xref="S4.SS3.p10.3.m3.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.3.m3.1b"><apply id="S4.SS3.p10.3.m3.1.1.cmml" xref="S4.SS3.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.3.m3.1.1.1.cmml" xref="S4.SS3.p10.3.m3.1.1">superscript</csymbol><apply id="S4.SS3.p10.3.m3.1.1.2.cmml" xref="S4.SS3.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.3.m3.1.1.2.1.cmml" xref="S4.SS3.p10.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.p10.3.m3.1.1.2.2.cmml" xref="S4.SS3.p10.3.m3.1.1.2.2">𝐶</ci><ci id="S4.SS3.p10.3.m3.1.1.2.3.cmml" xref="S4.SS3.p10.3.m3.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p10.3.m3.1.1.3.cmml" xref="S4.SS3.p10.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.3.m3.1c">C_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.3.m3.1d">italic_C start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="C_{n^{\prime}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.4.m4.1"><semantics id="S4.SS3.p10.4.m4.1a"><msubsup id="S4.SS3.p10.4.m4.1.1" xref="S4.SS3.p10.4.m4.1.1.cmml"><mi id="S4.SS3.p10.4.m4.1.1.2.2" xref="S4.SS3.p10.4.m4.1.1.2.2.cmml">C</mi><msup id="S4.SS3.p10.4.m4.1.1.2.3" xref="S4.SS3.p10.4.m4.1.1.2.3.cmml"><mi id="S4.SS3.p10.4.m4.1.1.2.3.2" xref="S4.SS3.p10.4.m4.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p10.4.m4.1.1.2.3.3" xref="S4.SS3.p10.4.m4.1.1.2.3.3.cmml">′</mo></msup><mi id="S4.SS3.p10.4.m4.1.1.3" xref="S4.SS3.p10.4.m4.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.4.m4.1b"><apply id="S4.SS3.p10.4.m4.1.1.cmml" xref="S4.SS3.p10.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.4.m4.1.1.1.cmml" xref="S4.SS3.p10.4.m4.1.1">superscript</csymbol><apply id="S4.SS3.p10.4.m4.1.1.2.cmml" xref="S4.SS3.p10.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.4.m4.1.1.2.1.cmml" xref="S4.SS3.p10.4.m4.1.1">subscript</csymbol><ci id="S4.SS3.p10.4.m4.1.1.2.2.cmml" xref="S4.SS3.p10.4.m4.1.1.2.2">𝐶</ci><apply id="S4.SS3.p10.4.m4.1.1.2.3.cmml" xref="S4.SS3.p10.4.m4.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p10.4.m4.1.1.2.3.1.cmml" xref="S4.SS3.p10.4.m4.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p10.4.m4.1.1.2.3.2.cmml" xref="S4.SS3.p10.4.m4.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p10.4.m4.1.1.2.3.3.cmml" xref="S4.SS3.p10.4.m4.1.1.2.3.3">′</ci></apply></apply><ci id="S4.SS3.p10.4.m4.1.1.3.cmml" xref="S4.SS3.p10.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.4.m4.1c">C_{n^{\prime}}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.4.m4.1d">italic_C start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> to generate a sequence of poses with the same step size <math alttext="\Delta t" class="ltx_Math" display="inline" id="S4.SS3.p10.5.m5.1"><semantics id="S4.SS3.p10.5.m5.1a"><mrow id="S4.SS3.p10.5.m5.1.1" xref="S4.SS3.p10.5.m5.1.1.cmml"><mi id="S4.SS3.p10.5.m5.1.1.2" mathvariant="normal" xref="S4.SS3.p10.5.m5.1.1.2.cmml">Δ</mi><mo id="S4.SS3.p10.5.m5.1.1.1" xref="S4.SS3.p10.5.m5.1.1.1.cmml">⁢</mo><mi id="S4.SS3.p10.5.m5.1.1.3" xref="S4.SS3.p10.5.m5.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.5.m5.1b"><apply id="S4.SS3.p10.5.m5.1.1.cmml" xref="S4.SS3.p10.5.m5.1.1"><times id="S4.SS3.p10.5.m5.1.1.1.cmml" xref="S4.SS3.p10.5.m5.1.1.1"></times><ci id="S4.SS3.p10.5.m5.1.1.2.cmml" xref="S4.SS3.p10.5.m5.1.1.2">Δ</ci><ci id="S4.SS3.p10.5.m5.1.1.3.cmml" xref="S4.SS3.p10.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.5.m5.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.5.m5.1d">roman_Δ italic_t</annotation></semantics></math>. The shortcut must be collision-checked against any independent nodes (i.e., nodes that are not predecessors of <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.6.m6.1"><semantics id="S4.SS3.p10.6.m6.1a"><msubsup id="S4.SS3.p10.6.m6.1.1" xref="S4.SS3.p10.6.m6.1.1.cmml"><mi id="S4.SS3.p10.6.m6.1.1.2.2" xref="S4.SS3.p10.6.m6.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p10.6.m6.1.1.2.3" xref="S4.SS3.p10.6.m6.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p10.6.m6.1.1.3" xref="S4.SS3.p10.6.m6.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.6.m6.1b"><apply id="S4.SS3.p10.6.m6.1.1.cmml" xref="S4.SS3.p10.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.6.m6.1.1.1.cmml" xref="S4.SS3.p10.6.m6.1.1">superscript</csymbol><apply id="S4.SS3.p10.6.m6.1.1.2.cmml" xref="S4.SS3.p10.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.6.m6.1.1.2.1.cmml" xref="S4.SS3.p10.6.m6.1.1">subscript</csymbol><ci id="S4.SS3.p10.6.m6.1.1.2.2.cmml" xref="S4.SS3.p10.6.m6.1.1.2.2">𝑣</ci><ci id="S4.SS3.p10.6.m6.1.1.2.3.cmml" xref="S4.SS3.p10.6.m6.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p10.6.m6.1.1.3.cmml" xref="S4.SS3.p10.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.6.m6.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.6.m6.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> or successors to <math alttext="v_{n^{\prime}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.7.m7.1"><semantics id="S4.SS3.p10.7.m7.1a"><msubsup id="S4.SS3.p10.7.m7.1.1" xref="S4.SS3.p10.7.m7.1.1.cmml"><mi id="S4.SS3.p10.7.m7.1.1.2.2" xref="S4.SS3.p10.7.m7.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p10.7.m7.1.1.2.3" xref="S4.SS3.p10.7.m7.1.1.2.3.cmml"><mi id="S4.SS3.p10.7.m7.1.1.2.3.2" xref="S4.SS3.p10.7.m7.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p10.7.m7.1.1.2.3.3" xref="S4.SS3.p10.7.m7.1.1.2.3.3.cmml">′</mo></msup><mi id="S4.SS3.p10.7.m7.1.1.3" xref="S4.SS3.p10.7.m7.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.7.m7.1b"><apply id="S4.SS3.p10.7.m7.1.1.cmml" xref="S4.SS3.p10.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.7.m7.1.1.1.cmml" xref="S4.SS3.p10.7.m7.1.1">superscript</csymbol><apply id="S4.SS3.p10.7.m7.1.1.2.cmml" xref="S4.SS3.p10.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.7.m7.1.1.2.1.cmml" xref="S4.SS3.p10.7.m7.1.1">subscript</csymbol><ci id="S4.SS3.p10.7.m7.1.1.2.2.cmml" xref="S4.SS3.p10.7.m7.1.1.2.2">𝑣</ci><apply id="S4.SS3.p10.7.m7.1.1.2.3.cmml" xref="S4.SS3.p10.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p10.7.m7.1.1.2.3.1.cmml" xref="S4.SS3.p10.7.m7.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p10.7.m7.1.1.2.3.2.cmml" xref="S4.SS3.p10.7.m7.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p10.7.m7.1.1.2.3.3.cmml" xref="S4.SS3.p10.7.m7.1.1.2.3.3">′</ci></apply></apply><ci id="S4.SS3.p10.7.m7.1.1.3.cmml" xref="S4.SS3.p10.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.7.m7.1c">v_{n^{\prime}}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.7.m7.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> in the TPG). On a multi-modal TPG, the collision checking must include any attached objects to the robot, as well as any independent object nodes on a task graph (i.e., object nodes that are not predecessors or successors of the current task). Once a valid shortcut is found, the original nodes between <math alttext="v_{n}^{i},v_{n^{\prime}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.8.m8.2"><semantics id="S4.SS3.p10.8.m8.2a"><mrow id="S4.SS3.p10.8.m8.2.2.2" xref="S4.SS3.p10.8.m8.2.2.3.cmml"><msubsup id="S4.SS3.p10.8.m8.1.1.1.1" xref="S4.SS3.p10.8.m8.1.1.1.1.cmml"><mi id="S4.SS3.p10.8.m8.1.1.1.1.2.2" xref="S4.SS3.p10.8.m8.1.1.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p10.8.m8.1.1.1.1.2.3" xref="S4.SS3.p10.8.m8.1.1.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p10.8.m8.1.1.1.1.3" xref="S4.SS3.p10.8.m8.1.1.1.1.3.cmml">i</mi></msubsup><mo id="S4.SS3.p10.8.m8.2.2.2.3" xref="S4.SS3.p10.8.m8.2.2.3.cmml">,</mo><msubsup id="S4.SS3.p10.8.m8.2.2.2.2" xref="S4.SS3.p10.8.m8.2.2.2.2.cmml"><mi id="S4.SS3.p10.8.m8.2.2.2.2.2.2" xref="S4.SS3.p10.8.m8.2.2.2.2.2.2.cmml">v</mi><msup id="S4.SS3.p10.8.m8.2.2.2.2.2.3" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3.cmml"><mi id="S4.SS3.p10.8.m8.2.2.2.2.2.3.2" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3.2.cmml">n</mi><mo id="S4.SS3.p10.8.m8.2.2.2.2.2.3.3" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3.3.cmml">′</mo></msup><mi id="S4.SS3.p10.8.m8.2.2.2.2.3" xref="S4.SS3.p10.8.m8.2.2.2.2.3.cmml">i</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.8.m8.2b"><list id="S4.SS3.p10.8.m8.2.2.3.cmml" xref="S4.SS3.p10.8.m8.2.2.2"><apply id="S4.SS3.p10.8.m8.1.1.1.1.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.8.m8.1.1.1.1.1.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1">superscript</csymbol><apply id="S4.SS3.p10.8.m8.1.1.1.1.2.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.8.m8.1.1.1.1.2.1.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1">subscript</csymbol><ci id="S4.SS3.p10.8.m8.1.1.1.1.2.2.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1.2.2">𝑣</ci><ci id="S4.SS3.p10.8.m8.1.1.1.1.2.3.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p10.8.m8.1.1.1.1.3.cmml" xref="S4.SS3.p10.8.m8.1.1.1.1.3">𝑖</ci></apply><apply id="S4.SS3.p10.8.m8.2.2.2.2.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p10.8.m8.2.2.2.2.1.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2">superscript</csymbol><apply id="S4.SS3.p10.8.m8.2.2.2.2.2.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS3.p10.8.m8.2.2.2.2.2.1.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2">subscript</csymbol><ci id="S4.SS3.p10.8.m8.2.2.2.2.2.2.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2.2.2">𝑣</ci><apply id="S4.SS3.p10.8.m8.2.2.2.2.2.3.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.SS3.p10.8.m8.2.2.2.2.2.3.1.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3">superscript</csymbol><ci id="S4.SS3.p10.8.m8.2.2.2.2.2.3.2.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3.2">𝑛</ci><ci id="S4.SS3.p10.8.m8.2.2.2.2.2.3.3.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2.2.3.3">′</ci></apply></apply><ci id="S4.SS3.p10.8.m8.2.2.2.2.3.cmml" xref="S4.SS3.p10.8.m8.2.2.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.8.m8.2c">v_{n}^{i},v_{n^{\prime}}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.8.m8.2d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> are replaced with new nodes corresponding to the shortcut. If adding a valid shortcut from <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.9.m9.1"><semantics id="S4.SS3.p10.9.m9.1a"><msubsup id="S4.SS3.p10.9.m9.1.1" xref="S4.SS3.p10.9.m9.1.1.cmml"><mi id="S4.SS3.p10.9.m9.1.1.2.2" xref="S4.SS3.p10.9.m9.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p10.9.m9.1.1.2.3" xref="S4.SS3.p10.9.m9.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p10.9.m9.1.1.3" xref="S4.SS3.p10.9.m9.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.9.m9.1b"><apply id="S4.SS3.p10.9.m9.1.1.cmml" xref="S4.SS3.p10.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.9.m9.1.1.1.cmml" xref="S4.SS3.p10.9.m9.1.1">superscript</csymbol><apply id="S4.SS3.p10.9.m9.1.1.2.cmml" xref="S4.SS3.p10.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.9.m9.1.1.2.1.cmml" xref="S4.SS3.p10.9.m9.1.1">subscript</csymbol><ci id="S4.SS3.p10.9.m9.1.1.2.2.cmml" xref="S4.SS3.p10.9.m9.1.1.2.2">𝑣</ci><ci id="S4.SS3.p10.9.m9.1.1.2.3.cmml" xref="S4.SS3.p10.9.m9.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p10.9.m9.1.1.3.cmml" xref="S4.SS3.p10.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.9.m9.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.9.m9.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="v_{n^{\prime}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.10.m10.1"><semantics id="S4.SS3.p10.10.m10.1a"><msubsup id="S4.SS3.p10.10.m10.1.1" xref="S4.SS3.p10.10.m10.1.1.cmml"><mi id="S4.SS3.p10.10.m10.1.1.2.2" xref="S4.SS3.p10.10.m10.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p10.10.m10.1.1.2.3" xref="S4.SS3.p10.10.m10.1.1.2.3.cmml"><mi id="S4.SS3.p10.10.m10.1.1.2.3.2" xref="S4.SS3.p10.10.m10.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p10.10.m10.1.1.2.3.3" xref="S4.SS3.p10.10.m10.1.1.2.3.3.cmml">′</mo></msup><mi id="S4.SS3.p10.10.m10.1.1.3" xref="S4.SS3.p10.10.m10.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.10.m10.1b"><apply id="S4.SS3.p10.10.m10.1.1.cmml" xref="S4.SS3.p10.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.10.m10.1.1.1.cmml" xref="S4.SS3.p10.10.m10.1.1">superscript</csymbol><apply id="S4.SS3.p10.10.m10.1.1.2.cmml" xref="S4.SS3.p10.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.10.m10.1.1.2.1.cmml" xref="S4.SS3.p10.10.m10.1.1">subscript</csymbol><ci id="S4.SS3.p10.10.m10.1.1.2.2.cmml" xref="S4.SS3.p10.10.m10.1.1.2.2">𝑣</ci><apply id="S4.SS3.p10.10.m10.1.1.2.3.cmml" xref="S4.SS3.p10.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p10.10.m10.1.1.2.3.1.cmml" xref="S4.SS3.p10.10.m10.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p10.10.m10.1.1.2.3.2.cmml" xref="S4.SS3.p10.10.m10.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p10.10.m10.1.1.2.3.3.cmml" xref="S4.SS3.p10.10.m10.1.1.2.3.3">′</ci></apply></apply><ci id="S4.SS3.p10.10.m10.1.1.3.cmml" xref="S4.SS3.p10.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.10.m10.1c">v_{n^{\prime}}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.10.m10.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math> skips any outgoing edges between these two nodes, the start node of these outgoing edges is moved to <math alttext="v_{n}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.11.m11.1"><semantics id="S4.SS3.p10.11.m11.1a"><msubsup id="S4.SS3.p10.11.m11.1.1" xref="S4.SS3.p10.11.m11.1.1.cmml"><mi id="S4.SS3.p10.11.m11.1.1.2.2" xref="S4.SS3.p10.11.m11.1.1.2.2.cmml">v</mi><mi id="S4.SS3.p10.11.m11.1.1.2.3" xref="S4.SS3.p10.11.m11.1.1.2.3.cmml">n</mi><mi id="S4.SS3.p10.11.m11.1.1.3" xref="S4.SS3.p10.11.m11.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.11.m11.1b"><apply id="S4.SS3.p10.11.m11.1.1.cmml" xref="S4.SS3.p10.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.11.m11.1.1.1.cmml" xref="S4.SS3.p10.11.m11.1.1">superscript</csymbol><apply id="S4.SS3.p10.11.m11.1.1.2.cmml" xref="S4.SS3.p10.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.11.m11.1.1.2.1.cmml" xref="S4.SS3.p10.11.m11.1.1">subscript</csymbol><ci id="S4.SS3.p10.11.m11.1.1.2.2.cmml" xref="S4.SS3.p10.11.m11.1.1.2.2">𝑣</ci><ci id="S4.SS3.p10.11.m11.1.1.2.3.cmml" xref="S4.SS3.p10.11.m11.1.1.2.3">𝑛</ci></apply><ci id="S4.SS3.p10.11.m11.1.1.3.cmml" xref="S4.SS3.p10.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.11.m11.1c">v_{n}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.11.m11.1d">italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. If any incoming edges are skipped, then the end node of these incoming edges is moved to <math alttext="v_{n^{\prime}}^{i}" class="ltx_Math" display="inline" id="S4.SS3.p10.12.m12.1"><semantics id="S4.SS3.p10.12.m12.1a"><msubsup id="S4.SS3.p10.12.m12.1.1" xref="S4.SS3.p10.12.m12.1.1.cmml"><mi id="S4.SS3.p10.12.m12.1.1.2.2" xref="S4.SS3.p10.12.m12.1.1.2.2.cmml">v</mi><msup id="S4.SS3.p10.12.m12.1.1.2.3" xref="S4.SS3.p10.12.m12.1.1.2.3.cmml"><mi id="S4.SS3.p10.12.m12.1.1.2.3.2" xref="S4.SS3.p10.12.m12.1.1.2.3.2.cmml">n</mi><mo id="S4.SS3.p10.12.m12.1.1.2.3.3" xref="S4.SS3.p10.12.m12.1.1.2.3.3.cmml">′</mo></msup><mi id="S4.SS3.p10.12.m12.1.1.3" xref="S4.SS3.p10.12.m12.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS3.p10.12.m12.1b"><apply id="S4.SS3.p10.12.m12.1.1.cmml" xref="S4.SS3.p10.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.12.m12.1.1.1.cmml" xref="S4.SS3.p10.12.m12.1.1">superscript</csymbol><apply id="S4.SS3.p10.12.m12.1.1.2.cmml" xref="S4.SS3.p10.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS3.p10.12.m12.1.1.2.1.cmml" xref="S4.SS3.p10.12.m12.1.1">subscript</csymbol><ci id="S4.SS3.p10.12.m12.1.1.2.2.cmml" xref="S4.SS3.p10.12.m12.1.1.2.2">𝑣</ci><apply id="S4.SS3.p10.12.m12.1.1.2.3.cmml" xref="S4.SS3.p10.12.m12.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.p10.12.m12.1.1.2.3.1.cmml" xref="S4.SS3.p10.12.m12.1.1.2.3">superscript</csymbol><ci id="S4.SS3.p10.12.m12.1.1.2.3.2.cmml" xref="S4.SS3.p10.12.m12.1.1.2.3.2">𝑛</ci><ci id="S4.SS3.p10.12.m12.1.1.2.3.3.cmml" xref="S4.SS3.p10.12.m12.1.1.2.3.3">′</ci></apply></apply><ci id="S4.SS3.p10.12.m12.1.1.3.cmml" xref="S4.SS3.p10.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p10.12.m12.1c">v_{n^{\prime}}^{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p10.12.m12.1d">italic_v start_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. These two steps ensure that any dependencies before adding a shortcut still exist after, and the TPG remains collision-free. Since each shortcut is collision-free, no new dependencies in the TPG are introduced and TPG remains deadlock-free. The shortcutting algorithm keeps identifying valid shortcuts until a user-defined time limit is reached, removing redundant transitions to HOME poses in the process. </div> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> IV-D TPG Execution </h3> <div class="ltx_para" id="S4.SS4.p1"> Executing a motion plan on robot arms often requires a position controller for movement and other specific controllers for manipulation skills. These controllers may have delays or uncertainties that affect the real-robot execution time. The TPG formulation provides an easy way to execute a multi-robot plan. Here, we present one centralized mechanism to coordinate multiple robot arms. </div> <div class="ltx_para" id="S4.SS4.p2"> As shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.F4" title="In IV-C Asynchronous Execution ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 4</a>, the TPG is hosted on a central server that communicates with each robot’s execution thread. Each pose node in the TPG corresponds to an action that moves the robot’s position to the node’s configuration. Each skill node corresponds to an action that executes the predefined robot skill. An action can be safely executed if there are no incoming edges from any nodes that are not executed. </div> <div class="ltx_para" id="S4.SS4.p3"> If an action is safe to execute based on the TPG, the central server sends it to the robot’s action queue. Each robot maintains its own controller-sensing loop and actuates the robot according to upcoming commands and its state estimation. The state estimation is also shared with the TPG, which then updates the TPG when a node is being executed or completed. Newly completed nodes may also enqueue new nodes if their outgoing edges were previously preventing unsafe actions. During execution, TPG can be interpreted as a control law that maintains the safe scheduling of individual robot actions. </div> <figure class="ltx_figure" id="S4.F7.sf1"> </figure> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> V Results </h2> <div class="ltx_para" id="S5.p1"> To evaluate the performance, we apply the proposed APEX-MR to bimanual LEGO assembly tasks. Given a customized LEGO design as shown in LABEL:fig:eval_lego, APEX-MR coordinates the robots to construct the desired structure as shown in LABEL:fig:lego_assembly_structures using available LEGO bricks. We deliberate the inputs (introduced in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S3" title="III Preliminaries ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. III</a>) to APEX-MR below. </div> <div class="ltx_para" id="S5.p2"> Environment Setup LABEL:fig:sim_env_setup and LABEL:fig:env_setup illustrate the simulation environment and the real setup, which includes two Yaskawa GP4 robots. Following the task convention in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib30" title="">30</a>]</cite>, we consider building LEGO structures on a baseplate, which is calibrated111We calibrate the transformation from the robots to the baseplate by teleoperating the robot to touch the plate. Note that we only measure the translation (<math alttext="X" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><mi id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><ci id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">X</annotation><annotation encoding="application/x-llamapun" id="footnote1.m1.1e">italic_X</annotation></semantics></math>, <math alttext="Y" class="ltx_Math" display="inline" id="footnote1.m2.1"><semantics id="footnote1.m2.1b"><mi id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="footnote1.m2.1c"><ci id="footnote1.m2.1.1.cmml" xref="footnote1.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.1d">Y</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.1e">italic_Y</annotation></semantics></math>, <math alttext="Z" class="ltx_Math" display="inline" id="footnote1.m3.1"><semantics id="footnote1.m3.1b"><mi id="footnote1.m3.1.1" xref="footnote1.m3.1.1.cmml">Z</mi><annotation-xml encoding="MathML-Content" id="footnote1.m3.1c"><ci id="footnote1.m3.1.1.cmml" xref="footnote1.m3.1.1">𝑍</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m3.1d">Z</annotation><annotation encoding="application/x-llamapun" id="footnote1.m3.1e">italic_Z</annotation></semantics></math>) and yaw angle while assuming no roll and pitch offsets. and placed in between the two robots, using commercial standard LEGO bricks initially stored on the baseplate. Each robot is equipped with an ATI Gamma force-torque sensor (FTS) and the end-of-arm tool (EOAT) is mounted on the FTS. The simulation consists of the entire workspace, which includes the robots, FTS, EOAT, LEGOs, nearby workstations, etc. The complete digital environment provides rich and accurate information for APEX-MR to safely coordinate the robot collaboration. </div> <figure class="ltx_figure" id="S5.F7.sf2"> </figure> <div class="ltx_para" id="S5.p3"> Assembly Plan Given a LEGO structure, we employ the physics-aware assembly planning in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib28" title="">28</a>]</cite> with customized LEGO physics reasoning <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib29" title="">29</a>]</cite> to generate a physically valid assembly sequence. Specifically, a physically valid assembly sequence enforces that for each step after assembling a brick, the structure is stable and does not collapse. Note that the definition of a physically valid assembly sequence can be different for other cooperative assembly tasks. For other applications, the assembly sequence can be obtained via planners, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib58" title="">58</a>]</cite>, and the proposed APEX-MR is also applicable downstream. </div> <div class="ltx_para" id="S5.p4"> Manipulation Skills Manipulating LEGO bricks is a non-trivial contact-rich manipulation problem beyond simple pick and stack. A robot EOAT and manipulation policy (i.e., insert-and-twist) was presented in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib30" title="">30</a>]</cite>, which enables a robot to manipulate commercial standard LEGO bricks, i.e., pick, and place-down in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>. However, a robot can only use it to manipulate a LEGO brick from its top, which limits the system from constructing complex structures. To enhance the system capability, we present a new LEGO tool (LT-V2), as shown in LABEL:fig:eoat. In particular, LT-V2 has LEGO studs added to the side of the tooltip. The new design enables the robot to manipulate a brick from its bottom as shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>, i.e., handover and place-up. With LT-V2, we define the manipulation skills as shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>, including 1) goal reaching with force feedback (i.e., support-bottom and support-top), and 2) learned force policy (i.e., pick, place-down, place-up, handover). For each skill, we generate a maximum of one feasible LEGO grasp pose and one support pose if necessary to use in task planning. </div> <figure class="ltx_figure" id="S4.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="255" id="S4.F7.g1" src="x5.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 7: Normalized makespan and wait time of APEX-MR (Sequential) versus the synchronized planner across example evaluation environment. The results are normalized by the makespan of the sequential motion plan before TPG and averaged over 4 random seeds. The unnormalized makespan and wait time are labeled for each entry and the dashed horizontal line corresponds to the makespan of the sequential motion plan before TPG. </figcaption> </figure> <figure class="ltx_figure" id="S5.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="207" id="S5.F8.g1" src="x6.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 8: Breakdown of wall clock time of APEX-MR (Sequential) and the synchronized planner baseline across evaluation environments. The results are normalized by the total of APEX-MR planning time for each task (excluding shortcutting) and averaged over 4 random seeds. The running sum of unnormalized wall clock time is labeled for each component. </figcaption> </figure> <div class="ltx_para" id="S5.p5"> Implementation We implement the TPG algorithm and manipulation skills in C++ with ROS-Noetic and MoveIt <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib7" title="">7</a>]</cite>. The ILP in task planning is solved with the pulp Python package. The RRT-Connect motion planning uses MoveIt’s OMPL <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib57" title="">57</a>]</cite> plug-in. Trajectories are discretized using <math alttext="\Delta t=0.05" class="ltx_Math" display="inline" id="S5.p5.1.m1.1"><semantics id="S5.p5.1.m1.1a"><mrow id="S5.p5.1.m1.1.1" xref="S5.p5.1.m1.1.1.cmml"><mrow id="S5.p5.1.m1.1.1.2" xref="S5.p5.1.m1.1.1.2.cmml"><mi id="S5.p5.1.m1.1.1.2.2" mathvariant="normal" xref="S5.p5.1.m1.1.1.2.2.cmml">Δ</mi><mo id="S5.p5.1.m1.1.1.2.1" xref="S5.p5.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S5.p5.1.m1.1.1.2.3" xref="S5.p5.1.m1.1.1.2.3.cmml">t</mi></mrow><mo id="S5.p5.1.m1.1.1.1" xref="S5.p5.1.m1.1.1.1.cmml">=</mo><mn id="S5.p5.1.m1.1.1.3" xref="S5.p5.1.m1.1.1.3.cmml">0.05</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.1.m1.1b"><apply id="S5.p5.1.m1.1.1.cmml" xref="S5.p5.1.m1.1.1"><eq id="S5.p5.1.m1.1.1.1.cmml" xref="S5.p5.1.m1.1.1.1"></eq><apply id="S5.p5.1.m1.1.1.2.cmml" xref="S5.p5.1.m1.1.1.2"><times id="S5.p5.1.m1.1.1.2.1.cmml" xref="S5.p5.1.m1.1.1.2.1"></times><ci id="S5.p5.1.m1.1.1.2.2.cmml" xref="S5.p5.1.m1.1.1.2.2">Δ</ci><ci id="S5.p5.1.m1.1.1.2.3.cmml" xref="S5.p5.1.m1.1.1.2.3">𝑡</ci></apply><cn id="S5.p5.1.m1.1.1.3.cmml" type="float" xref="S5.p5.1.m1.1.1.3">0.05</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.1.m1.1c">\Delta t=0.05</annotation><annotation encoding="application/x-llamapun" id="S5.p5.1.m1.1d">roman_Δ italic_t = 0.05</annotation></semantics></math> when the maximum <math alttext="L_{1}" class="ltx_Math" display="inline" id="S5.p5.2.m2.1"><semantics id="S5.p5.2.m2.1a"><msub id="S5.p5.2.m2.1.1" xref="S5.p5.2.m2.1.1.cmml"><mi id="S5.p5.2.m2.1.1.2" xref="S5.p5.2.m2.1.1.2.cmml">L</mi><mn id="S5.p5.2.m2.1.1.3" xref="S5.p5.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p5.2.m2.1b"><apply id="S5.p5.2.m2.1.1.cmml" xref="S5.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p5.2.m2.1.1.1.cmml" xref="S5.p5.2.m2.1.1">subscript</csymbol><ci id="S5.p5.2.m2.1.1.2.cmml" xref="S5.p5.2.m2.1.1.2">𝐿</ci><cn id="S5.p5.2.m2.1.1.3.cmml" type="integer" xref="S5.p5.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.2.m2.1c">L_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p5.2.m2.1d">italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> joint velocity is 1 rad/s. <math alttext="\Delta t" class="ltx_Math" display="inline" id="S5.p5.3.m3.1"><semantics id="S5.p5.3.m3.1a"><mrow id="S5.p5.3.m3.1.1" xref="S5.p5.3.m3.1.1.cmml"><mi id="S5.p5.3.m3.1.1.2" mathvariant="normal" xref="S5.p5.3.m3.1.1.2.cmml">Δ</mi><mo id="S5.p5.3.m3.1.1.1" xref="S5.p5.3.m3.1.1.1.cmml">⁢</mo><mi id="S5.p5.3.m3.1.1.3" xref="S5.p5.3.m3.1.1.3.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.3.m3.1b"><apply id="S5.p5.3.m3.1.1.cmml" xref="S5.p5.3.m3.1.1"><times id="S5.p5.3.m3.1.1.1.cmml" xref="S5.p5.3.m3.1.1.1"></times><ci id="S5.p5.3.m3.1.1.2.cmml" xref="S5.p5.3.m3.1.1.2">Δ</ci><ci id="S5.p5.3.m3.1.1.3.cmml" xref="S5.p5.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.3.m3.1c">\Delta t</annotation><annotation encoding="application/x-llamapun" id="S5.p5.3.m3.1d">roman_Δ italic_t</annotation></semantics></math> is adjusted linearly based on the max velocity to ensure the same density. All simulation experiments are conducted on an AMD 7840HS laptop. Algo. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#alg1" title="Algorithm 1 ‣ IV-C Asynchronous Execution ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">1</a> parallelizes collision checking with 16 threads. </div> <div class="ltx_para" id="S5.p6"> Experiment Objective While APEX-MR itself is a full pipeline for multi-robot tasks and motion planning, the key innovation that enables asynchronous innovation is the TPG execution framework. Thus, we are interested in the following questions when evaluating APEX-MR: <ul class="ltx_itemize" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> • <div class="ltx_para" id="S5.I1.i1.p1"> (Q1) How significant is the benefit of asynchronous execution, enabled by the TPG execution framework? </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> • <div class="ltx_para" id="S5.I1.i2.p1"> (Q2) How is the quality of plans produced by APEX-MR and what are the computational costs? </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> • <div class="ltx_para" id="S5.I1.i3.p1"> (Q3) How well does APEX-MR perform in physical LEGO assembly and can it safely execute planned trajectories despite uncertainties? </div> </li> </ul> </div> <div class="ltx_para" id="S5.p7"> Q1 and Q2 will be examined closely in simulation, whereas Q3 will be the focus of our real robot experiments. </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> V-A Simulation Performance </h3> <div class="ltx_para" id="S5.SS1.p1"> We first conduct experiments in simulation. To our knowledge, no existing simulator can reliably simulate the connections between LEGO bricks. Thus, all robot skills are reduced to deterministic operations when evaluated in simulation, and any variations are due to the stochasticity of planning in APEX-MR. </div> <div class="ltx_para" id="S5.SS1.p2"> Dataset We evaluate the performance of APEX-MR on a suite of nine LEGO assembly tasks as shown in LABEL:fig:eval_lego. The complexity of these tasks varies significantly in terms of the number of objects in the assembly plan, stability, orientation, and manipulation skills required for physical assembly. The Chair shown in LABEL:fig:chair_render has 258 objects, but the structure is solid and stable, and thus, no collaborative skills are needed. On the other hand, many of the bricks along the span of the Bridge (LABEL:fig:bridge_render) require a robot to support them when assembling from the top, whereas building the Cliff (LABEL:fig:cliff_render) and Faucet (LABEL:fig:faucet_render) requires an object reorientation and collaborative assembly from bottom. </div> <div class="ltx_para" id="S5.SS1.p3"> Metrics We use the execution makespan and wait time as our evaluation metrics for plan quality. Since our output is a TPG, we first roll out the asynchronous trajectory from the TPG, assuming no controller delay. The rollout trajectory converts each pose node back to a configuration. Actions in the skill nodes are executed based on the reference trajectory, and force feedback is switched off in the simulation. The timestamp for each configuration in the rollout trajectory is the earliest possible time to reach this node based on incoming type-2 edges. 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Wait time is defined as the total amount of time any robot spends waiting in the rollout asynchronous trajectory from TPG, or the original sequential or synchronous trajectory. In particular, we are interested in whether TPG processing can successfully reduce wait time when initialized with a sequential task and motion plan. </div> <div class="ltx_para" id="S5.SS1.p4"> Baseline We also design a baseline for a synchronized task and motion planning as a comparison to APEX-MR. Although APEX-MR uses a sequential planner for simplicity and efficiency, our TPG can also improve synchronized motion plans which is a common approach in MR-TAMP. Thus, we evaluate the performance improvement of TPG on synchronized plan as well. In this synchronous planner, robots can execute tasks in parallel but must wait for all robots to finish their current task before proceeding to the next set of tasks. We use an algorithm to convert the sequential task plan from APEX-MR to a synchronous task plan, as shown by <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.F15" title="In -B Details of Lego Manipulation Policy ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 15</a> in the Appendix. The main idea is to execute the sequential task plan in parallel if executing the next task does not violate inter-robot task dependencies or block tasks scheduled at an earlier time. This process is similar to building a TPG on tasks instead of motions for parallelization. For every robot <math alttext="i" class="ltx_Math" display="inline" id="S5.SS1.p4.1.m1.1"><semantics id="S5.SS1.p4.1.m1.1a"><mi id="S5.SS1.p4.1.m1.1.1" xref="S5.SS1.p4.1.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.1.m1.1b"><ci id="S5.SS1.p4.1.m1.1.1.cmml" xref="S5.SS1.p4.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.1.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.1.m1.1d">italic_i</annotation></semantics></math> and its task <math alttext="\mathcal{T}_{m}^{i}" class="ltx_Math" display="inline" id="S5.SS1.p4.2.m2.1"><semantics id="S5.SS1.p4.2.m2.1a"><msubsup id="S5.SS1.p4.2.m2.1.1" xref="S5.SS1.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p4.2.m2.1.1.2.2" xref="S5.SS1.p4.2.m2.1.1.2.2.cmml">𝒯</mi><mi id="S5.SS1.p4.2.m2.1.1.2.3" xref="S5.SS1.p4.2.m2.1.1.2.3.cmml">m</mi><mi id="S5.SS1.p4.2.m2.1.1.3" xref="S5.SS1.p4.2.m2.1.1.3.cmml">i</mi></msubsup><annotation-xml 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end_POSTSUPERSCRIPT</annotation></semantics></math> that satisfies <math alttext="m^{\prime}<m" class="ltx_Math" display="inline" id="S5.SS1.p4.6.m6.1"><semantics id="S5.SS1.p4.6.m6.1a"><mrow id="S5.SS1.p4.6.m6.1.1" xref="S5.SS1.p4.6.m6.1.1.cmml"><msup id="S5.SS1.p4.6.m6.1.1.2" xref="S5.SS1.p4.6.m6.1.1.2.cmml"><mi id="S5.SS1.p4.6.m6.1.1.2.2" xref="S5.SS1.p4.6.m6.1.1.2.2.cmml">m</mi><mo id="S5.SS1.p4.6.m6.1.1.2.3" xref="S5.SS1.p4.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S5.SS1.p4.6.m6.1.1.1" xref="S5.SS1.p4.6.m6.1.1.1.cmml"><</mo><mi id="S5.SS1.p4.6.m6.1.1.3" xref="S5.SS1.p4.6.m6.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.6.m6.1b"><apply id="S5.SS1.p4.6.m6.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1"><lt id="S5.SS1.p4.6.m6.1.1.1.cmml" xref="S5.SS1.p4.6.m6.1.1.1"></lt><apply id="S5.SS1.p4.6.m6.1.1.2.cmml" xref="S5.SS1.p4.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.SS1.p4.6.m6.1.1.2.1.cmml" xref="S5.SS1.p4.6.m6.1.1.2">superscript</csymbol><ci id="S5.SS1.p4.6.m6.1.1.2.2.cmml" xref="S5.SS1.p4.6.m6.1.1.2.2">𝑚</ci><ci id="S5.SS1.p4.6.m6.1.1.2.3.cmml" xref="S5.SS1.p4.6.m6.1.1.2.3">′</ci></apply><ci id="S5.SS1.p4.6.m6.1.1.3.cmml" xref="S5.SS1.p4.6.m6.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.6.m6.1c">m^{\prime}<m</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.6.m6.1d">italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT < italic_m</annotation></semantics></math>. If there exists a collision, then robot <math alttext="i" class="ltx_Math" display="inline" id="S5.SS1.p4.7.m7.1"><semantics id="S5.SS1.p4.7.m7.1a"><mi id="S5.SS1.p4.7.m7.1.1" xref="S5.SS1.p4.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.7.m7.1b"><ci id="S5.SS1.p4.7.m7.1.1.cmml" xref="S5.SS1.p4.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.7.m7.1d">italic_i</annotation></semantics></math> must wait for robot <math alttext="i^{\prime}" class="ltx_Math" display="inline" id="S5.SS1.p4.8.m8.1"><semantics id="S5.SS1.p4.8.m8.1a"><msup id="S5.SS1.p4.8.m8.1.1" xref="S5.SS1.p4.8.m8.1.1.cmml"><mi id="S5.SS1.p4.8.m8.1.1.2" xref="S5.SS1.p4.8.m8.1.1.2.cmml">i</mi><mo id="S5.SS1.p4.8.m8.1.1.3" xref="S5.SS1.p4.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.8.m8.1b"><apply id="S5.SS1.p4.8.m8.1.1.cmml" xref="S5.SS1.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.8.m8.1.1.1.cmml" xref="S5.SS1.p4.8.m8.1.1">superscript</csymbol><ci id="S5.SS1.p4.8.m8.1.1.2.cmml" xref="S5.SS1.p4.8.m8.1.1.2">𝑖</ci><ci id="S5.SS1.p4.8.m8.1.1.3.cmml" xref="S5.SS1.p4.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.8.m8.1c">i^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.8.m8.1d">italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to complete task <math alttext="\mathcal{T}_{m^{\prime}}^{i^{\prime}}" class="ltx_Math" display="inline" id="S5.SS1.p4.9.m9.1"><semantics id="S5.SS1.p4.9.m9.1a"><msubsup id="S5.SS1.p4.9.m9.1.1" xref="S5.SS1.p4.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p4.9.m9.1.1.2.2" xref="S5.SS1.p4.9.m9.1.1.2.2.cmml">𝒯</mi><msup id="S5.SS1.p4.9.m9.1.1.2.3" xref="S5.SS1.p4.9.m9.1.1.2.3.cmml"><mi id="S5.SS1.p4.9.m9.1.1.2.3.2" xref="S5.SS1.p4.9.m9.1.1.2.3.2.cmml">m</mi><mo id="S5.SS1.p4.9.m9.1.1.2.3.3" xref="S5.SS1.p4.9.m9.1.1.2.3.3.cmml">′</mo></msup><msup id="S5.SS1.p4.9.m9.1.1.3" xref="S5.SS1.p4.9.m9.1.1.3.cmml"><mi id="S5.SS1.p4.9.m9.1.1.3.2" xref="S5.SS1.p4.9.m9.1.1.3.2.cmml">i</mi><mo id="S5.SS1.p4.9.m9.1.1.3.3" xref="S5.SS1.p4.9.m9.1.1.3.3.cmml">′</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.9.m9.1b"><apply id="S5.SS1.p4.9.m9.1.1.cmml" xref="S5.SS1.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.1.1.1.cmml" xref="S5.SS1.p4.9.m9.1.1">superscript</csymbol><apply id="S5.SS1.p4.9.m9.1.1.2.cmml" xref="S5.SS1.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.1.1.2.1.cmml" xref="S5.SS1.p4.9.m9.1.1">subscript</csymbol><ci id="S5.SS1.p4.9.m9.1.1.2.2.cmml" xref="S5.SS1.p4.9.m9.1.1.2.2">𝒯</ci><apply id="S5.SS1.p4.9.m9.1.1.2.3.cmml" xref="S5.SS1.p4.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S5.SS1.p4.9.m9.1.1.2.3.1.cmml" 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starting task <math alttext="\mathcal{T}_{m}^{i}" class="ltx_Math" display="inline" id="S5.SS1.p4.10.m10.1"><semantics id="S5.SS1.p4.10.m10.1a"><msubsup id="S5.SS1.p4.10.m10.1.1" xref="S5.SS1.p4.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.SS1.p4.10.m10.1.1.2.2" xref="S5.SS1.p4.10.m10.1.1.2.2.cmml">𝒯</mi><mi id="S5.SS1.p4.10.m10.1.1.2.3" xref="S5.SS1.p4.10.m10.1.1.2.3.cmml">m</mi><mi id="S5.SS1.p4.10.m10.1.1.3" xref="S5.SS1.p4.10.m10.1.1.3.cmml">i</mi></msubsup><annotation-xml encoding="MathML-Content" id="S5.SS1.p4.10.m10.1b"><apply id="S5.SS1.p4.10.m10.1.1.cmml" xref="S5.SS1.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.10.m10.1.1.1.cmml" xref="S5.SS1.p4.10.m10.1.1">superscript</csymbol><apply id="S5.SS1.p4.10.m10.1.1.2.cmml" xref="S5.SS1.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S5.SS1.p4.10.m10.1.1.2.1.cmml" xref="S5.SS1.p4.10.m10.1.1">subscript</csymbol><ci id="S5.SS1.p4.10.m10.1.1.2.2.cmml" xref="S5.SS1.p4.10.m10.1.1.2.2">𝒯</ci><ci id="S5.SS1.p4.10.m10.1.1.2.3.cmml" xref="S5.SS1.p4.10.m10.1.1.2.3">𝑚</ci></apply><ci id="S5.SS1.p4.10.m10.1.1.3.cmml" xref="S5.SS1.p4.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.p4.10.m10.1c">\mathcal{T}_{m}^{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.p4.10.m10.1d">caligraphic_T start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT</annotation></semantics></math>. A synchronous task graph can then be generated by combining these calculated dependencies, with existing inter-robot dependencies. Then, composite RRT-Connect is used as the multi-robot motion planner. Synchronous trajectories for tasks executing in parallel are generated by planning all degrees of freedom as a single robot. </div> <figure class="ltx_figure" id="S5.F14.sf1"> </figure> <div class="ltx_para" id="S5.SS1.p5"> Performance <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.F7" title="In V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 7</a> shows the quality of the solution of APEX-MR on a variety of tasks. First, the TPG post-processing and applying and shortcut, as described in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S4.SS3" title="IV-C Asynchronous Execution ‣ IV APEX-MR: Asynchronous Planning and Execution for Multi-Robot System ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Sec. IV-C</a>, significantly reduces makespan by 48% and wait time by 85% on average, compared to the initial sequential motion plan at the horizontal dashed line. Compared to the synchronized motion plan, our asynchronous plans from APEX-MR are consistently shorter and have less wait time. When applied to the synchronous plan, TPG also significantly reduces the makespan by 36% and wait time by 77% on average. Note that the post-processed sequential plan from APEX-MR still slightly outperforms the post-processed synchronized motion plan by 3% in terms of makespan. This is due to the path produced by a multi-robot motion plan being suboptimal compared to sequential motion planning. Still, the wait time for the synchronized plan after TPG post-processing is minimal. </div> <div class="ltx_para" id="S5.SS1.p6"> Runtime <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5.F8" title="In V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 8</a> and Table <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.T1" title="Table. I ‣ -B Details of Lego Manipulation Policy ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">I</a> in the appendix shows the wall clock time for APEX-MR. On average, the TPG construction time is always lower than the task motion planning time except for the Chair, which has a very long assembly sequence. On the other hand, running a synchronized planner can be much more expensive than the simple sequential planner used in APEX-MR, which requires more careful coordination in task planning, and multi-robot motion planning with more degrees of freedom. By combining a simple sequential task and motion planner with TPG post-processing, APEX-MR produces higher-quality multi-robot plans with 26% lower computational overhead on average than a synchronized multi-robot task and motion planner alone. </div> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> V-B System Deployment </h3> <figure class="ltx_figure" id="S5.F14"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="415" id="S5.F14.g1" src="x7.png" width="830"/> <figcaption class="ltx_caption ltx_centering">Figure 14: Distribution of executing various FTS-feedback-controlled LEGO manipulation policy on real robot. The execution time is collected from assembling the Cliff structure in LABEL:fig:cliff repeatedly. This variation adds to the uncertainty in real-time execution.</figcaption> </figure> <div class="ltx_para" id="S5.SS2.p1"> We deploy the proposed APEX-MR to a real bimanual setup for cooperative LEGO assembly. LABEL:fig:env_setup illustrates the environment setup of the dual-arm system. Note that despite the environment being pre-calibrated, errors (<math alttext="\sim 1" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><mrow id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml"><mi id="S5.SS2.p1.1.m1.1.1.2" xref="S5.SS2.p1.1.m1.1.1.2.cmml"></mi><mo id="S5.SS2.p1.1.m1.1.1.1" xref="S5.SS2.p1.1.m1.1.1.1.cmml">∼</mo><mn id="S5.SS2.p1.1.m1.1.1.3" xref="S5.SS2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><apply id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1"><csymbol cd="latexml" id="S5.SS2.p1.1.m1.1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1.1">similar-to</csymbol><csymbol cd="latexml" id="S5.SS2.p1.1.m1.1.1.2.cmml" xref="S5.SS2.p1.1.m1.1.1.2">absent</csymbol><cn id="S5.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">\sim 1</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">∼ 1</annotation></semantics></math>mm) still exist since the calibration is imperfect and the structure could be tilted due to the passive connection nature. Thus, we integrate real-time force feedback using the FTS to improve the manipulation robustness. For operation skills (i.e., pick, place-down, place-up, handover), we use the force feedback to detect successful insertion and update the manipulation policy accordingly. For supporting skills (i.e., support-bottom and support-top), we use force feedback to sense a slight touch with the structure to avoid either over or under-supporting. LABEL:fig:lego_assembly_structures showcases example LEGO structures accomplished by the bimanual system with APEX-MR. The robots can safely collaborate and efficiently build customized and complex LEGO objects, including fragile overhanging structures. </div> <div class="ltx_para" id="S5.SS2.p2"> Note that the key difference between assembling in real and in simulation is with and without force feedback. Integrating force feedback into the manipulation skills improves the system robustness, but also brings uncertainty with respect to execution time. In particular, we are interested in whether the TPG execution framework ensures safe execution and avoids collisions despite uncertainties from manipulation skills. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S5.F14" title="In V-B System Deployment ‣ V Results ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 14</a> depicts the distribution of execution times of six manipulation skills, i.e., pick, place-down, place-up, handover, support-bottom, and support-top (see <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a> for illustrations). All of these skills are designed to execute with force feedback to ensure proper contact between the EOAT and the object being manipulated and minimize the effect of imperfect calibration or tilted structure. As a consequence of these mm-level adjustments with force feedback, the execution time can vary as much as 2 seconds, or as much as 23% from the median. Nevertheless, our TPG execution framework can reliably adjust the robot schedule if any delay could cause a collision or require another robot to wait longer until a skill is completed. </div> <div class="ltx_para" id="S5.SS2.p3"> In practice, APEX-MR also allows the two robot arms to operate in close proximity asynchronously thanks to the use of TPG. For example, if one robot is stopped due to a controller issue or because the user presses emergency stop, the other robot would automatically stop if it is unsafe to continue its execution. Although each robot’s feedback controller operates independently, only those actions that are deemed safe are passed to the robot’s action queue. Thus, APEX-MR can significantly reduce the risk of unsafe action in real multi-robot execution. </div> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> VI Limitations and Future Work </h2> <div class="ltx_para" id="S6.p1"> Although the proposed APEX-MR pipeline enables efficient and safe execution for multiple robot arms, it still has several algorithmic limitations which we discuss below. </div> <div class="ltx_para" id="S6.p2"> Offline Computation Currently, both the TPG processing and motion planning in APEX-MR are performed offline before real execution. This can be a drawback in real assembly tasks, where new tasks are continuously assigned once the robots finish existing assembly on an assembly line. Another limitation of offline computation is that the APEX-MR cannot adapt to changes in the collision environment or assembly steps easily. A principled framework to address these lifelong planning techniques is to use a windowed multi-robot planner and only convert the first <math alttext="n" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mi id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><ci id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">italic_n</annotation></semantics></math> part of the robot plan to a TPG, similar to how <cite class="ltx_cite ltx_citemacro_citet">Varambally et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib59" title="">59</a>]</cite> address automated warehousing. Taking a reduced-horizon approach will significantly reduce planning time and allow plans to be continuously updated concurrently with execution. </div> <div class="ltx_para" id="S6.p3"> Planning for Robot Dynamics While APEX-MR are reliable and safe on real robots, APEX-MR requires a good position or force controller for the robot because the generated plan does not consider robot dynamics, such as acceleration and jerk constraints. This is challenging because the planner must generate continuous velocity and acceleration profiles while also avoiding inter-robot collisions at all times. We plan to incorporate dynamics as part of the TPG post-processing step, such as solving a linear program on top of TPG-imposed constraints and velocity constraints, as suggested in <cite class="ltx_cite ltx_citemacro_citet">Hönig et al. [<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib20" title="">20</a>]</cite>. Another interesting problem with robot arms is the speed constraints may be imposed on the task space, due to the attached object at the end-effector or even closed kinematic chains formed by concurrent manipulation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib60" title="">60</a>]</cite>. </div> <div class="ltx_para" id="S6.p4"> Manipulation Policy While the proposed APEX-MR enables efficient and safe dual-arm cooperative LEGO assembly, the current system assembles each object based on pre-defined skills. With the additional force feedback, each single operation can be performed robustly. However, due to the passive connection nature of LEGO structures, i.e., established connections could be gradually loosened and the structure can be tilted or even collapse due to subsequent operations, the long-horizon assembly could still fail. Therefore, the dual-arm system, at its current stage, is not robust enough to construct large-scale LEGO structures that have multiple fragile overhanging geometries. To further improve the robustness from a system perspective, we aim to investigate methods for failure detection <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib4" title="">4</a>]</cite> and recovery, e.g., reinforce the connections that are loosened due to later operations. Failure detection and recovery can also be integrated as part of an online replanning framework that dynamically reschedules the robot’s tasks if a failure occurs and intervention becomes necessary. </div> <div class="ltx_para" id="S6.p5"> Other Cooperative Assemblies Building on the APEX-MR pipeline, we plan to extend its application to a broader range of cooperative tasks, such as the NIST Box Assembly <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib40" title="">40</a>]</cite>, and other industrial assembly scenarios. By doing so, we aim to address the unique challenges posed by real-world manufacturing environments, gaining deeper insights into how cooperative systems can be optimized for complex, large-scale production tasks. While the discussion in this paper is based on LEGO assembly, the components in APEX-MR, especially TPG post-processing, can be applied to other multi-robot assembly tasks. A concrete step would be to investigate how to integrate manipulation policies that are more complex than those used for LEGO assembly, e.g., diffusion policy <cite class="ltx_cite ltx_citemacro_citep">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib6" title="">6</a>]</cite>, to the TPG framework. </div> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> VII Conclusion </h2> <div class="ltx_para" id="S7.p1"> For many robotic manipulation tasks, a team of cooperative robot arms is often necessary and beneficial because cooperation can improve dexterity, flexibility, and versatility. A reliable framework for coordinating robot arms should possess several key qualities: efficiency to maximize throughput, scalability to long-horizon and complex tasks, and safety during real execution. </div> <div class="ltx_para" id="S7.p2"> With these criteria in mind, we have proposed APEX-MR, a pipeline for multi-robot asynchronous planning and execution. Our proposed pipeline combines a sequential task and motion planner with a TPG to post-process the plan for asynchronous execution. Specifically, TPG post-processing can significantly speed up the execution of otherwise sequential and synchronous multi-robot task plans by 48% and 36% on our simulated assembly tasks. Because coordination is easier, sequential motion planning is far more efficient than planning for synchronous execution. </div> <div class="ltx_para" id="S7.p3"> We demonstrated that our proposed algorithm can be successfully deployed and integrated for a real bimanual cooperative task. LEGO structures, as an example, presented a challenging manipulation task due to the need for high precision and the non-rigid nature of their connections. We presented a set of manipulation skills for complex cooperative assembly, including supported placement and object handover, based on the end-effector design LT-V2. The dual-arm system successfully performs customized LEGO assembly and is the first robotic system to do so with commercial LEGO bricks. Additionally, we showed that the integration with the TPG execution framework is robust in the presence of uncertain execution time. In the end, we hope that this framework can advance and bring closer to more real use of multi-robot arm collaboration algorithms. </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_font_smallcaps ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> This work is in part supported by the Manufacturing Futures Institute, Carnegie Mellon University, through a grant from the Richard King Mellon Foundation. The authors would also like to thank Yifan Sun for helping design LT-V2. </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> Berndt et al. [2020] Alexander Berndt, Niels Van Duijkeren, Luigi Palmieri, and Tamas Keviczky. 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Eng., PP(99):1–20, 2024. </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_pagination ltx_role_newpage"></div> <section class="ltx_subsection" id="A0.SS1"> <h3 class="ltx_title ltx_title_subsection"> -A Details of Task Planning Formulation </h3> <div class="ltx_para" id="A0.SS1.p1"> Each assembly step <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p1.1.m1.1"><semantics id="A0.SS1.p1.1.m1.1a"><msub id="A0.SS1.p1.1.m1.1.1" xref="A0.SS1.p1.1.m1.1.1.cmml"><mi id="A0.SS1.p1.1.m1.1.1.2" xref="A0.SS1.p1.1.m1.1.1.2.cmml">a</mi><mi id="A0.SS1.p1.1.m1.1.1.3" xref="A0.SS1.p1.1.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.1.m1.1b"><apply id="A0.SS1.p1.1.m1.1.1.cmml" xref="A0.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.1.m1.1.1.1.cmml" xref="A0.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="A0.SS1.p1.1.m1.1.1.2.cmml" xref="A0.SS1.p1.1.m1.1.1.2">𝑎</ci><ci id="A0.SS1.p1.1.m1.1.1.3.cmml" xref="A0.SS1.p1.1.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.1.m1.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> requires a specific lego type, and each brick in environment has a specific type. Typically, there are many bricks of the same type for in the environment. Mathematically, we represent whether a brick <math alttext="b_{k}" class="ltx_Math" display="inline" id="A0.SS1.p1.2.m2.1"><semantics id="A0.SS1.p1.2.m2.1a"><msub id="A0.SS1.p1.2.m2.1.1" xref="A0.SS1.p1.2.m2.1.1.cmml"><mi id="A0.SS1.p1.2.m2.1.1.2" xref="A0.SS1.p1.2.m2.1.1.2.cmml">b</mi><mi id="A0.SS1.p1.2.m2.1.1.3" xref="A0.SS1.p1.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.2.m2.1b"><apply id="A0.SS1.p1.2.m2.1.1.cmml" xref="A0.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.2.m2.1.1.1.cmml" xref="A0.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="A0.SS1.p1.2.m2.1.1.2.cmml" xref="A0.SS1.p1.2.m2.1.1.2">𝑏</ci><ci id="A0.SS1.p1.2.m2.1.1.3.cmml" xref="A0.SS1.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.2.m2.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.2.m2.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is the correct type for the assembly step <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p1.3.m3.1"><semantics id="A0.SS1.p1.3.m3.1a"><msub id="A0.SS1.p1.3.m3.1.1" xref="A0.SS1.p1.3.m3.1.1.cmml"><mi id="A0.SS1.p1.3.m3.1.1.2" xref="A0.SS1.p1.3.m3.1.1.2.cmml">a</mi><mi id="A0.SS1.p1.3.m3.1.1.3" xref="A0.SS1.p1.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.3.m3.1b"><apply id="A0.SS1.p1.3.m3.1.1.cmml" xref="A0.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.3.m3.1.1.1.cmml" xref="A0.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="A0.SS1.p1.3.m3.1.1.2.cmml" xref="A0.SS1.p1.3.m3.1.1.2">𝑎</ci><ci id="A0.SS1.p1.3.m3.1.1.3.cmml" xref="A0.SS1.p1.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.3.m3.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> using a binary variable <math alttext="\delta^{t}_{j,k}" class="ltx_Math" display="inline" id="A0.SS1.p1.4.m4.2"><semantics id="A0.SS1.p1.4.m4.2a"><msubsup id="A0.SS1.p1.4.m4.2.3" xref="A0.SS1.p1.4.m4.2.3.cmml"><mi id="A0.SS1.p1.4.m4.2.3.2.2" xref="A0.SS1.p1.4.m4.2.3.2.2.cmml">δ</mi><mrow id="A0.SS1.p1.4.m4.2.2.2.4" xref="A0.SS1.p1.4.m4.2.2.2.3.cmml"><mi id="A0.SS1.p1.4.m4.1.1.1.1" xref="A0.SS1.p1.4.m4.1.1.1.1.cmml">j</mi><mo id="A0.SS1.p1.4.m4.2.2.2.4.1" xref="A0.SS1.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="A0.SS1.p1.4.m4.2.2.2.2" xref="A0.SS1.p1.4.m4.2.2.2.2.cmml">k</mi></mrow><mi id="A0.SS1.p1.4.m4.2.3.2.3" xref="A0.SS1.p1.4.m4.2.3.2.3.cmml">t</mi></msubsup><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.4.m4.2b"><apply id="A0.SS1.p1.4.m4.2.3.cmml" xref="A0.SS1.p1.4.m4.2.3"><csymbol cd="ambiguous" id="A0.SS1.p1.4.m4.2.3.1.cmml" xref="A0.SS1.p1.4.m4.2.3">subscript</csymbol><apply id="A0.SS1.p1.4.m4.2.3.2.cmml" xref="A0.SS1.p1.4.m4.2.3"><csymbol cd="ambiguous" id="A0.SS1.p1.4.m4.2.3.2.1.cmml" xref="A0.SS1.p1.4.m4.2.3">superscript</csymbol><ci id="A0.SS1.p1.4.m4.2.3.2.2.cmml" xref="A0.SS1.p1.4.m4.2.3.2.2">𝛿</ci><ci id="A0.SS1.p1.4.m4.2.3.2.3.cmml" xref="A0.SS1.p1.4.m4.2.3.2.3">𝑡</ci></apply><list id="A0.SS1.p1.4.m4.2.2.2.3.cmml" xref="A0.SS1.p1.4.m4.2.2.2.4"><ci id="A0.SS1.p1.4.m4.1.1.1.1.cmml" xref="A0.SS1.p1.4.m4.1.1.1.1">𝑗</ci><ci id="A0.SS1.p1.4.m4.2.2.2.2.cmml" xref="A0.SS1.p1.4.m4.2.2.2.2">𝑘</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.4.m4.2c">\delta^{t}_{j,k}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.4.m4.2d">italic_δ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\delta^{t}_{j,k}=1" class="ltx_Math" display="inline" id="A0.SS1.p1.5.m5.2"><semantics id="A0.SS1.p1.5.m5.2a"><mrow id="A0.SS1.p1.5.m5.2.3" xref="A0.SS1.p1.5.m5.2.3.cmml"><msubsup id="A0.SS1.p1.5.m5.2.3.2" xref="A0.SS1.p1.5.m5.2.3.2.cmml"><mi id="A0.SS1.p1.5.m5.2.3.2.2.2" xref="A0.SS1.p1.5.m5.2.3.2.2.2.cmml">δ</mi><mrow id="A0.SS1.p1.5.m5.2.2.2.4" xref="A0.SS1.p1.5.m5.2.2.2.3.cmml"><mi id="A0.SS1.p1.5.m5.1.1.1.1" xref="A0.SS1.p1.5.m5.1.1.1.1.cmml">j</mi><mo id="A0.SS1.p1.5.m5.2.2.2.4.1" xref="A0.SS1.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="A0.SS1.p1.5.m5.2.2.2.2" xref="A0.SS1.p1.5.m5.2.2.2.2.cmml">k</mi></mrow><mi id="A0.SS1.p1.5.m5.2.3.2.2.3" xref="A0.SS1.p1.5.m5.2.3.2.2.3.cmml">t</mi></msubsup><mo id="A0.SS1.p1.5.m5.2.3.1" xref="A0.SS1.p1.5.m5.2.3.1.cmml">=</mo><mn id="A0.SS1.p1.5.m5.2.3.3" xref="A0.SS1.p1.5.m5.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.5.m5.2b"><apply id="A0.SS1.p1.5.m5.2.3.cmml" xref="A0.SS1.p1.5.m5.2.3"><eq id="A0.SS1.p1.5.m5.2.3.1.cmml" xref="A0.SS1.p1.5.m5.2.3.1"></eq><apply id="A0.SS1.p1.5.m5.2.3.2.cmml" xref="A0.SS1.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="A0.SS1.p1.5.m5.2.3.2.1.cmml" xref="A0.SS1.p1.5.m5.2.3.2">subscript</csymbol><apply id="A0.SS1.p1.5.m5.2.3.2.2.cmml" xref="A0.SS1.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="A0.SS1.p1.5.m5.2.3.2.2.1.cmml" xref="A0.SS1.p1.5.m5.2.3.2">superscript</csymbol><ci id="A0.SS1.p1.5.m5.2.3.2.2.2.cmml" xref="A0.SS1.p1.5.m5.2.3.2.2.2">𝛿</ci><ci id="A0.SS1.p1.5.m5.2.3.2.2.3.cmml" xref="A0.SS1.p1.5.m5.2.3.2.2.3">𝑡</ci></apply><list id="A0.SS1.p1.5.m5.2.2.2.3.cmml" xref="A0.SS1.p1.5.m5.2.2.2.4"><ci id="A0.SS1.p1.5.m5.1.1.1.1.cmml" xref="A0.SS1.p1.5.m5.1.1.1.1">𝑗</ci><ci id="A0.SS1.p1.5.m5.2.2.2.2.cmml" xref="A0.SS1.p1.5.m5.2.2.2.2">𝑘</ci></list></apply><cn id="A0.SS1.p1.5.m5.2.3.3.cmml" type="integer" xref="A0.SS1.p1.5.m5.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.5.m5.2c">\delta^{t}_{j,k}=1</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.5.m5.2d">italic_δ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j , italic_k 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end_POSTSUBSCRIPT</annotation></semantics></math> is suitable and <math alttext="\delta^{t}_{j,k}=0" class="ltx_Math" display="inline" id="A0.SS1.p1.7.m7.2"><semantics id="A0.SS1.p1.7.m7.2a"><mrow id="A0.SS1.p1.7.m7.2.3" xref="A0.SS1.p1.7.m7.2.3.cmml"><msubsup id="A0.SS1.p1.7.m7.2.3.2" xref="A0.SS1.p1.7.m7.2.3.2.cmml"><mi id="A0.SS1.p1.7.m7.2.3.2.2.2" xref="A0.SS1.p1.7.m7.2.3.2.2.2.cmml">δ</mi><mrow id="A0.SS1.p1.7.m7.2.2.2.4" xref="A0.SS1.p1.7.m7.2.2.2.3.cmml"><mi id="A0.SS1.p1.7.m7.1.1.1.1" xref="A0.SS1.p1.7.m7.1.1.1.1.cmml">j</mi><mo id="A0.SS1.p1.7.m7.2.2.2.4.1" xref="A0.SS1.p1.7.m7.2.2.2.3.cmml">,</mo><mi id="A0.SS1.p1.7.m7.2.2.2.2" xref="A0.SS1.p1.7.m7.2.2.2.2.cmml">k</mi></mrow><mi id="A0.SS1.p1.7.m7.2.3.2.2.3" xref="A0.SS1.p1.7.m7.2.3.2.2.3.cmml">t</mi></msubsup><mo id="A0.SS1.p1.7.m7.2.3.1" xref="A0.SS1.p1.7.m7.2.3.1.cmml">=</mo><mn id="A0.SS1.p1.7.m7.2.3.3" xref="A0.SS1.p1.7.m7.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.7.m7.2b"><apply id="A0.SS1.p1.7.m7.2.3.cmml" xref="A0.SS1.p1.7.m7.2.3"><eq id="A0.SS1.p1.7.m7.2.3.1.cmml" xref="A0.SS1.p1.7.m7.2.3.1"></eq><apply id="A0.SS1.p1.7.m7.2.3.2.cmml" xref="A0.SS1.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="A0.SS1.p1.7.m7.2.3.2.1.cmml" xref="A0.SS1.p1.7.m7.2.3.2">subscript</csymbol><apply id="A0.SS1.p1.7.m7.2.3.2.2.cmml" xref="A0.SS1.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="A0.SS1.p1.7.m7.2.3.2.2.1.cmml" xref="A0.SS1.p1.7.m7.2.3.2">superscript</csymbol><ci id="A0.SS1.p1.7.m7.2.3.2.2.2.cmml" xref="A0.SS1.p1.7.m7.2.3.2.2.2">𝛿</ci><ci id="A0.SS1.p1.7.m7.2.3.2.2.3.cmml" xref="A0.SS1.p1.7.m7.2.3.2.2.3">𝑡</ci></apply><list id="A0.SS1.p1.7.m7.2.2.2.3.cmml" xref="A0.SS1.p1.7.m7.2.2.2.4"><ci id="A0.SS1.p1.7.m7.1.1.1.1.cmml" xref="A0.SS1.p1.7.m7.1.1.1.1">𝑗</ci><ci id="A0.SS1.p1.7.m7.2.2.2.2.cmml" xref="A0.SS1.p1.7.m7.2.2.2.2">𝑘</ci></list></apply><cn id="A0.SS1.p1.7.m7.2.3.3.cmml" type="integer" xref="A0.SS1.p1.7.m7.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.7.m7.2c">\delta^{t}_{j,k}=0</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.7.m7.2d">italic_δ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j , italic_k end_POSTSUBSCRIPT = 0</annotation></semantics></math> otherwise. The assembly sequence also specifies whether each step <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p1.8.m8.1"><semantics id="A0.SS1.p1.8.m8.1a"><msub id="A0.SS1.p1.8.m8.1.1" xref="A0.SS1.p1.8.m8.1.1.cmml"><mi id="A0.SS1.p1.8.m8.1.1.2" xref="A0.SS1.p1.8.m8.1.1.2.cmml">a</mi><mi id="A0.SS1.p1.8.m8.1.1.3" xref="A0.SS1.p1.8.m8.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.8.m8.1b"><apply id="A0.SS1.p1.8.m8.1.1.cmml" xref="A0.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.8.m8.1.1.1.cmml" xref="A0.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="A0.SS1.p1.8.m8.1.1.2.cmml" xref="A0.SS1.p1.8.m8.1.1.2">𝑎</ci><ci id="A0.SS1.p1.8.m8.1.1.3.cmml" xref="A0.SS1.p1.8.m8.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.8.m8.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.8.m8.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> requires multiple robots for cooperative assembly, denoted by another binary variable <math alttext="\delta^{s}_{j}" class="ltx_Math" display="inline" id="A0.SS1.p1.9.m9.1"><semantics id="A0.SS1.p1.9.m9.1a"><msubsup id="A0.SS1.p1.9.m9.1.1" xref="A0.SS1.p1.9.m9.1.1.cmml"><mi id="A0.SS1.p1.9.m9.1.1.2.2" xref="A0.SS1.p1.9.m9.1.1.2.2.cmml">δ</mi><mi id="A0.SS1.p1.9.m9.1.1.3" xref="A0.SS1.p1.9.m9.1.1.3.cmml">j</mi><mi id="A0.SS1.p1.9.m9.1.1.2.3" xref="A0.SS1.p1.9.m9.1.1.2.3.cmml">s</mi></msubsup><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.9.m9.1b"><apply id="A0.SS1.p1.9.m9.1.1.cmml" xref="A0.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.9.m9.1.1.1.cmml" xref="A0.SS1.p1.9.m9.1.1">subscript</csymbol><apply id="A0.SS1.p1.9.m9.1.1.2.cmml" xref="A0.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.9.m9.1.1.2.1.cmml" xref="A0.SS1.p1.9.m9.1.1">superscript</csymbol><ci id="A0.SS1.p1.9.m9.1.1.2.2.cmml" xref="A0.SS1.p1.9.m9.1.1.2.2">𝛿</ci><ci id="A0.SS1.p1.9.m9.1.1.2.3.cmml" xref="A0.SS1.p1.9.m9.1.1.2.3">𝑠</ci></apply><ci id="A0.SS1.p1.9.m9.1.1.3.cmml" xref="A0.SS1.p1.9.m9.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.9.m9.1c">\delta^{s}_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.9.m9.1d">italic_δ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\delta^{s}_{j}=1" class="ltx_Math" display="inline" id="A0.SS1.p1.10.m10.1"><semantics id="A0.SS1.p1.10.m10.1a"><mrow id="A0.SS1.p1.10.m10.1.1" xref="A0.SS1.p1.10.m10.1.1.cmml"><msubsup id="A0.SS1.p1.10.m10.1.1.2" xref="A0.SS1.p1.10.m10.1.1.2.cmml"><mi id="A0.SS1.p1.10.m10.1.1.2.2.2" xref="A0.SS1.p1.10.m10.1.1.2.2.2.cmml">δ</mi><mi id="A0.SS1.p1.10.m10.1.1.2.3" xref="A0.SS1.p1.10.m10.1.1.2.3.cmml">j</mi><mi id="A0.SS1.p1.10.m10.1.1.2.2.3" xref="A0.SS1.p1.10.m10.1.1.2.2.3.cmml">s</mi></msubsup><mo id="A0.SS1.p1.10.m10.1.1.1" xref="A0.SS1.p1.10.m10.1.1.1.cmml">=</mo><mn id="A0.SS1.p1.10.m10.1.1.3" xref="A0.SS1.p1.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.10.m10.1b"><apply id="A0.SS1.p1.10.m10.1.1.cmml" xref="A0.SS1.p1.10.m10.1.1"><eq id="A0.SS1.p1.10.m10.1.1.1.cmml" xref="A0.SS1.p1.10.m10.1.1.1"></eq><apply id="A0.SS1.p1.10.m10.1.1.2.cmml" xref="A0.SS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="A0.SS1.p1.10.m10.1.1.2.1.cmml" xref="A0.SS1.p1.10.m10.1.1.2">subscript</csymbol><apply id="A0.SS1.p1.10.m10.1.1.2.2.cmml" xref="A0.SS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="A0.SS1.p1.10.m10.1.1.2.2.1.cmml" xref="A0.SS1.p1.10.m10.1.1.2">superscript</csymbol><ci id="A0.SS1.p1.10.m10.1.1.2.2.2.cmml" xref="A0.SS1.p1.10.m10.1.1.2.2.2">𝛿</ci><ci id="A0.SS1.p1.10.m10.1.1.2.2.3.cmml" xref="A0.SS1.p1.10.m10.1.1.2.2.3">𝑠</ci></apply><ci id="A0.SS1.p1.10.m10.1.1.2.3.cmml" xref="A0.SS1.p1.10.m10.1.1.2.3">𝑗</ci></apply><cn id="A0.SS1.p1.10.m10.1.1.3.cmml" type="integer" xref="A0.SS1.p1.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.10.m10.1c">\delta^{s}_{j}=1</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.10.m10.1d">italic_δ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> if two robots are needed for <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p1.11.m11.1"><semantics id="A0.SS1.p1.11.m11.1a"><msub id="A0.SS1.p1.11.m11.1.1" xref="A0.SS1.p1.11.m11.1.1.cmml"><mi id="A0.SS1.p1.11.m11.1.1.2" xref="A0.SS1.p1.11.m11.1.1.2.cmml">a</mi><mi id="A0.SS1.p1.11.m11.1.1.3" xref="A0.SS1.p1.11.m11.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p1.11.m11.1b"><apply id="A0.SS1.p1.11.m11.1.1.cmml" xref="A0.SS1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A0.SS1.p1.11.m11.1.1.1.cmml" xref="A0.SS1.p1.11.m11.1.1">subscript</csymbol><ci id="A0.SS1.p1.11.m11.1.1.2.cmml" xref="A0.SS1.p1.11.m11.1.1.2">𝑎</ci><ci id="A0.SS1.p1.11.m11.1.1.3.cmml" xref="A0.SS1.p1.11.m11.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p1.11.m11.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p1.11.m11.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_para" id="A0.SS1.p2"> Let <math alttext="X_{ijkg}" class="ltx_Math" display="inline" id="A0.SS1.p2.1.m1.1"><semantics id="A0.SS1.p2.1.m1.1a"><msub id="A0.SS1.p2.1.m1.1.1" xref="A0.SS1.p2.1.m1.1.1.cmml"><mi id="A0.SS1.p2.1.m1.1.1.2" xref="A0.SS1.p2.1.m1.1.1.2.cmml">X</mi><mrow id="A0.SS1.p2.1.m1.1.1.3" xref="A0.SS1.p2.1.m1.1.1.3.cmml"><mi id="A0.SS1.p2.1.m1.1.1.3.2" xref="A0.SS1.p2.1.m1.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p2.1.m1.1.1.3.1" xref="A0.SS1.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.1.m1.1.1.3.3" xref="A0.SS1.p2.1.m1.1.1.3.3.cmml">j</mi><mo id="A0.SS1.p2.1.m1.1.1.3.1a" xref="A0.SS1.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.1.m1.1.1.3.4" xref="A0.SS1.p2.1.m1.1.1.3.4.cmml">k</mi><mo id="A0.SS1.p2.1.m1.1.1.3.1b" xref="A0.SS1.p2.1.m1.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.1.m1.1.1.3.5" xref="A0.SS1.p2.1.m1.1.1.3.5.cmml">g</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.1.m1.1b"><apply id="A0.SS1.p2.1.m1.1.1.cmml" xref="A0.SS1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.1.m1.1.1.1.cmml" xref="A0.SS1.p2.1.m1.1.1">subscript</csymbol><ci id="A0.SS1.p2.1.m1.1.1.2.cmml" xref="A0.SS1.p2.1.m1.1.1.2">𝑋</ci><apply id="A0.SS1.p2.1.m1.1.1.3.cmml" xref="A0.SS1.p2.1.m1.1.1.3"><times id="A0.SS1.p2.1.m1.1.1.3.1.cmml" xref="A0.SS1.p2.1.m1.1.1.3.1"></times><ci id="A0.SS1.p2.1.m1.1.1.3.2.cmml" xref="A0.SS1.p2.1.m1.1.1.3.2">𝑖</ci><ci id="A0.SS1.p2.1.m1.1.1.3.3.cmml" xref="A0.SS1.p2.1.m1.1.1.3.3">𝑗</ci><ci id="A0.SS1.p2.1.m1.1.1.3.4.cmml" xref="A0.SS1.p2.1.m1.1.1.3.4">𝑘</ci><ci id="A0.SS1.p2.1.m1.1.1.3.5.cmml" xref="A0.SS1.p2.1.m1.1.1.3.5">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.1.m1.1c">X_{ijkg}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.1.m1.1d">italic_X start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Y_{ijg}" class="ltx_Math" display="inline" id="A0.SS1.p2.2.m2.1"><semantics id="A0.SS1.p2.2.m2.1a"><msub id="A0.SS1.p2.2.m2.1.1" xref="A0.SS1.p2.2.m2.1.1.cmml"><mi id="A0.SS1.p2.2.m2.1.1.2" xref="A0.SS1.p2.2.m2.1.1.2.cmml">Y</mi><mrow id="A0.SS1.p2.2.m2.1.1.3" xref="A0.SS1.p2.2.m2.1.1.3.cmml"><mi id="A0.SS1.p2.2.m2.1.1.3.2" xref="A0.SS1.p2.2.m2.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p2.2.m2.1.1.3.1" xref="A0.SS1.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.2.m2.1.1.3.3" xref="A0.SS1.p2.2.m2.1.1.3.3.cmml">j</mi><mo id="A0.SS1.p2.2.m2.1.1.3.1a" xref="A0.SS1.p2.2.m2.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.2.m2.1.1.3.4" xref="A0.SS1.p2.2.m2.1.1.3.4.cmml">g</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.2.m2.1b"><apply id="A0.SS1.p2.2.m2.1.1.cmml" xref="A0.SS1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.2.m2.1.1.1.cmml" xref="A0.SS1.p2.2.m2.1.1">subscript</csymbol><ci id="A0.SS1.p2.2.m2.1.1.2.cmml" xref="A0.SS1.p2.2.m2.1.1.2">𝑌</ci><apply id="A0.SS1.p2.2.m2.1.1.3.cmml" xref="A0.SS1.p2.2.m2.1.1.3"><times id="A0.SS1.p2.2.m2.1.1.3.1.cmml" xref="A0.SS1.p2.2.m2.1.1.3.1"></times><ci id="A0.SS1.p2.2.m2.1.1.3.2.cmml" xref="A0.SS1.p2.2.m2.1.1.3.2">𝑖</ci><ci id="A0.SS1.p2.2.m2.1.1.3.3.cmml" xref="A0.SS1.p2.2.m2.1.1.3.3">𝑗</ci><ci id="A0.SS1.p2.2.m2.1.1.3.4.cmml" xref="A0.SS1.p2.2.m2.1.1.3.4">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.2.m2.1c">Y_{ijg}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.2.m2.1d">italic_Y start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT</annotation></semantics></math> be the binary decision variables. <math alttext="X_{ijkg=1}" class="ltx_Math" display="inline" id="A0.SS1.p2.3.m3.1"><semantics id="A0.SS1.p2.3.m3.1a"><msub id="A0.SS1.p2.3.m3.1.1" xref="A0.SS1.p2.3.m3.1.1.cmml"><mi id="A0.SS1.p2.3.m3.1.1.2" xref="A0.SS1.p2.3.m3.1.1.2.cmml">X</mi><mrow id="A0.SS1.p2.3.m3.1.1.3" xref="A0.SS1.p2.3.m3.1.1.3.cmml"><mrow id="A0.SS1.p2.3.m3.1.1.3.2" xref="A0.SS1.p2.3.m3.1.1.3.2.cmml"><mi id="A0.SS1.p2.3.m3.1.1.3.2.2" xref="A0.SS1.p2.3.m3.1.1.3.2.2.cmml">i</mi><mo id="A0.SS1.p2.3.m3.1.1.3.2.1" xref="A0.SS1.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="A0.SS1.p2.3.m3.1.1.3.2.3" xref="A0.SS1.p2.3.m3.1.1.3.2.3.cmml">j</mi><mo id="A0.SS1.p2.3.m3.1.1.3.2.1a" xref="A0.SS1.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="A0.SS1.p2.3.m3.1.1.3.2.4" xref="A0.SS1.p2.3.m3.1.1.3.2.4.cmml">k</mi><mo id="A0.SS1.p2.3.m3.1.1.3.2.1b" xref="A0.SS1.p2.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="A0.SS1.p2.3.m3.1.1.3.2.5" xref="A0.SS1.p2.3.m3.1.1.3.2.5.cmml">g</mi></mrow><mo id="A0.SS1.p2.3.m3.1.1.3.1" xref="A0.SS1.p2.3.m3.1.1.3.1.cmml">=</mo><mn id="A0.SS1.p2.3.m3.1.1.3.3" xref="A0.SS1.p2.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.3.m3.1b"><apply id="A0.SS1.p2.3.m3.1.1.cmml" xref="A0.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.3.m3.1.1.1.cmml" xref="A0.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="A0.SS1.p2.3.m3.1.1.2.cmml" xref="A0.SS1.p2.3.m3.1.1.2">𝑋</ci><apply id="A0.SS1.p2.3.m3.1.1.3.cmml" xref="A0.SS1.p2.3.m3.1.1.3"><eq id="A0.SS1.p2.3.m3.1.1.3.1.cmml" xref="A0.SS1.p2.3.m3.1.1.3.1"></eq><apply id="A0.SS1.p2.3.m3.1.1.3.2.cmml" xref="A0.SS1.p2.3.m3.1.1.3.2"><times id="A0.SS1.p2.3.m3.1.1.3.2.1.cmml" xref="A0.SS1.p2.3.m3.1.1.3.2.1"></times><ci id="A0.SS1.p2.3.m3.1.1.3.2.2.cmml" xref="A0.SS1.p2.3.m3.1.1.3.2.2">𝑖</ci><ci id="A0.SS1.p2.3.m3.1.1.3.2.3.cmml" xref="A0.SS1.p2.3.m3.1.1.3.2.3">𝑗</ci><ci id="A0.SS1.p2.3.m3.1.1.3.2.4.cmml" xref="A0.SS1.p2.3.m3.1.1.3.2.4">𝑘</ci><ci id="A0.SS1.p2.3.m3.1.1.3.2.5.cmml" xref="A0.SS1.p2.3.m3.1.1.3.2.5">𝑔</ci></apply><cn id="A0.SS1.p2.3.m3.1.1.3.3.cmml" type="integer" xref="A0.SS1.p2.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.3.m3.1c">X_{ijkg=1}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g = 1 end_POSTSUBSCRIPT</annotation></semantics></math> means that robot <math alttext="i" class="ltx_Math" display="inline" id="A0.SS1.p2.4.m4.1"><semantics id="A0.SS1.p2.4.m4.1a"><mi id="A0.SS1.p2.4.m4.1.1" xref="A0.SS1.p2.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.4.m4.1b"><ci id="A0.SS1.p2.4.m4.1.1.cmml" xref="A0.SS1.p2.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.4.m4.1d">italic_i</annotation></semantics></math> is assigned to assembly step <math alttext="j" class="ltx_Math" display="inline" id="A0.SS1.p2.5.m5.1"><semantics id="A0.SS1.p2.5.m5.1a"><mi id="A0.SS1.p2.5.m5.1.1" xref="A0.SS1.p2.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.5.m5.1b"><ci id="A0.SS1.p2.5.m5.1.1.cmml" xref="A0.SS1.p2.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.5.m5.1d">italic_j</annotation></semantics></math>, using object <math alttext="b_{k}" class="ltx_Math" display="inline" id="A0.SS1.p2.6.m6.1"><semantics id="A0.SS1.p2.6.m6.1a"><msub id="A0.SS1.p2.6.m6.1.1" xref="A0.SS1.p2.6.m6.1.1.cmml"><mi id="A0.SS1.p2.6.m6.1.1.2" xref="A0.SS1.p2.6.m6.1.1.2.cmml">b</mi><mi id="A0.SS1.p2.6.m6.1.1.3" xref="A0.SS1.p2.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.6.m6.1b"><apply id="A0.SS1.p2.6.m6.1.1.cmml" xref="A0.SS1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.6.m6.1.1.1.cmml" xref="A0.SS1.p2.6.m6.1.1">subscript</csymbol><ci id="A0.SS1.p2.6.m6.1.1.2.cmml" xref="A0.SS1.p2.6.m6.1.1.2">𝑏</ci><ci id="A0.SS1.p2.6.m6.1.1.3.cmml" xref="A0.SS1.p2.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.6.m6.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.6.m6.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and the <math alttext="g^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.7.m7.1"><semantics id="A0.SS1.p2.7.m7.1a"><msup id="A0.SS1.p2.7.m7.1.1" xref="A0.SS1.p2.7.m7.1.1.cmml"><mi id="A0.SS1.p2.7.m7.1.1.2" xref="A0.SS1.p2.7.m7.1.1.2.cmml">g</mi><mrow id="A0.SS1.p2.7.m7.1.1.3" xref="A0.SS1.p2.7.m7.1.1.3.cmml"><mi id="A0.SS1.p2.7.m7.1.1.3.2" xref="A0.SS1.p2.7.m7.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.7.m7.1.1.3.1" xref="A0.SS1.p2.7.m7.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.7.m7.1.1.3.3" xref="A0.SS1.p2.7.m7.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.7.m7.1b"><apply id="A0.SS1.p2.7.m7.1.1.cmml" xref="A0.SS1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.7.m7.1.1.1.cmml" xref="A0.SS1.p2.7.m7.1.1">superscript</csymbol><ci id="A0.SS1.p2.7.m7.1.1.2.cmml" xref="A0.SS1.p2.7.m7.1.1.2">𝑔</ci><apply id="A0.SS1.p2.7.m7.1.1.3.cmml" xref="A0.SS1.p2.7.m7.1.1.3"><times id="A0.SS1.p2.7.m7.1.1.3.1.cmml" xref="A0.SS1.p2.7.m7.1.1.3.1"></times><ci id="A0.SS1.p2.7.m7.1.1.3.2.cmml" xref="A0.SS1.p2.7.m7.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.7.m7.1.1.3.3.cmml" xref="A0.SS1.p2.7.m7.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.7.m7.1c">g^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.7.m7.1d">italic_g start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> grasp pose. <math alttext="Y_{ijg}=1" class="ltx_Math" display="inline" id="A0.SS1.p2.8.m8.1"><semantics id="A0.SS1.p2.8.m8.1a"><mrow id="A0.SS1.p2.8.m8.1.1" xref="A0.SS1.p2.8.m8.1.1.cmml"><msub id="A0.SS1.p2.8.m8.1.1.2" xref="A0.SS1.p2.8.m8.1.1.2.cmml"><mi id="A0.SS1.p2.8.m8.1.1.2.2" xref="A0.SS1.p2.8.m8.1.1.2.2.cmml">Y</mi><mrow id="A0.SS1.p2.8.m8.1.1.2.3" xref="A0.SS1.p2.8.m8.1.1.2.3.cmml"><mi id="A0.SS1.p2.8.m8.1.1.2.3.2" xref="A0.SS1.p2.8.m8.1.1.2.3.2.cmml">i</mi><mo id="A0.SS1.p2.8.m8.1.1.2.3.1" xref="A0.SS1.p2.8.m8.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.8.m8.1.1.2.3.3" xref="A0.SS1.p2.8.m8.1.1.2.3.3.cmml">j</mi><mo id="A0.SS1.p2.8.m8.1.1.2.3.1a" xref="A0.SS1.p2.8.m8.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.8.m8.1.1.2.3.4" xref="A0.SS1.p2.8.m8.1.1.2.3.4.cmml">g</mi></mrow></msub><mo id="A0.SS1.p2.8.m8.1.1.1" xref="A0.SS1.p2.8.m8.1.1.1.cmml">=</mo><mn id="A0.SS1.p2.8.m8.1.1.3" xref="A0.SS1.p2.8.m8.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.8.m8.1b"><apply id="A0.SS1.p2.8.m8.1.1.cmml" xref="A0.SS1.p2.8.m8.1.1"><eq id="A0.SS1.p2.8.m8.1.1.1.cmml" xref="A0.SS1.p2.8.m8.1.1.1"></eq><apply id="A0.SS1.p2.8.m8.1.1.2.cmml" xref="A0.SS1.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="A0.SS1.p2.8.m8.1.1.2.1.cmml" xref="A0.SS1.p2.8.m8.1.1.2">subscript</csymbol><ci id="A0.SS1.p2.8.m8.1.1.2.2.cmml" xref="A0.SS1.p2.8.m8.1.1.2.2">𝑌</ci><apply id="A0.SS1.p2.8.m8.1.1.2.3.cmml" xref="A0.SS1.p2.8.m8.1.1.2.3"><times id="A0.SS1.p2.8.m8.1.1.2.3.1.cmml" xref="A0.SS1.p2.8.m8.1.1.2.3.1"></times><ci id="A0.SS1.p2.8.m8.1.1.2.3.2.cmml" xref="A0.SS1.p2.8.m8.1.1.2.3.2">𝑖</ci><ci id="A0.SS1.p2.8.m8.1.1.2.3.3.cmml" xref="A0.SS1.p2.8.m8.1.1.2.3.3">𝑗</ci><ci id="A0.SS1.p2.8.m8.1.1.2.3.4.cmml" xref="A0.SS1.p2.8.m8.1.1.2.3.4">𝑔</ci></apply></apply><cn id="A0.SS1.p2.8.m8.1.1.3.cmml" type="integer" xref="A0.SS1.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.8.m8.1c">Y_{ijg}=1</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.8.m8.1d">italic_Y start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT = 1</annotation></semantics></math> means that the <math alttext="i^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.9.m9.1"><semantics id="A0.SS1.p2.9.m9.1a"><msup id="A0.SS1.p2.9.m9.1.1" xref="A0.SS1.p2.9.m9.1.1.cmml"><mi id="A0.SS1.p2.9.m9.1.1.2" xref="A0.SS1.p2.9.m9.1.1.2.cmml">i</mi><mrow id="A0.SS1.p2.9.m9.1.1.3" xref="A0.SS1.p2.9.m9.1.1.3.cmml"><mi id="A0.SS1.p2.9.m9.1.1.3.2" xref="A0.SS1.p2.9.m9.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.9.m9.1.1.3.1" xref="A0.SS1.p2.9.m9.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.9.m9.1.1.3.3" xref="A0.SS1.p2.9.m9.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.9.m9.1b"><apply id="A0.SS1.p2.9.m9.1.1.cmml" xref="A0.SS1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.9.m9.1.1.1.cmml" xref="A0.SS1.p2.9.m9.1.1">superscript</csymbol><ci id="A0.SS1.p2.9.m9.1.1.2.cmml" xref="A0.SS1.p2.9.m9.1.1.2">𝑖</ci><apply id="A0.SS1.p2.9.m9.1.1.3.cmml" xref="A0.SS1.p2.9.m9.1.1.3"><times id="A0.SS1.p2.9.m9.1.1.3.1.cmml" xref="A0.SS1.p2.9.m9.1.1.3.1"></times><ci id="A0.SS1.p2.9.m9.1.1.3.2.cmml" xref="A0.SS1.p2.9.m9.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.9.m9.1.1.3.3.cmml" xref="A0.SS1.p2.9.m9.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.9.m9.1c">i^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.9.m9.1d">italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> robot is assigned to support the assembly step <math alttext="j" class="ltx_Math" display="inline" id="A0.SS1.p2.10.m10.1"><semantics id="A0.SS1.p2.10.m10.1a"><mi id="A0.SS1.p2.10.m10.1.1" xref="A0.SS1.p2.10.m10.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.10.m10.1b"><ci id="A0.SS1.p2.10.m10.1.1.cmml" xref="A0.SS1.p2.10.m10.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.10.m10.1c">j</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.10.m10.1d">italic_j</annotation></semantics></math>, using the <math alttext="g^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.11.m11.1"><semantics id="A0.SS1.p2.11.m11.1a"><msup id="A0.SS1.p2.11.m11.1.1" xref="A0.SS1.p2.11.m11.1.1.cmml"><mi id="A0.SS1.p2.11.m11.1.1.2" xref="A0.SS1.p2.11.m11.1.1.2.cmml">g</mi><mrow id="A0.SS1.p2.11.m11.1.1.3" xref="A0.SS1.p2.11.m11.1.1.3.cmml"><mi id="A0.SS1.p2.11.m11.1.1.3.2" xref="A0.SS1.p2.11.m11.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.11.m11.1.1.3.1" xref="A0.SS1.p2.11.m11.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.11.m11.1.1.3.3" xref="A0.SS1.p2.11.m11.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.11.m11.1b"><apply id="A0.SS1.p2.11.m11.1.1.cmml" xref="A0.SS1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.11.m11.1.1.1.cmml" xref="A0.SS1.p2.11.m11.1.1">superscript</csymbol><ci id="A0.SS1.p2.11.m11.1.1.2.cmml" xref="A0.SS1.p2.11.m11.1.1.2">𝑔</ci><apply id="A0.SS1.p2.11.m11.1.1.3.cmml" xref="A0.SS1.p2.11.m11.1.1.3"><times id="A0.SS1.p2.11.m11.1.1.3.1.cmml" xref="A0.SS1.p2.11.m11.1.1.3.1"></times><ci id="A0.SS1.p2.11.m11.1.1.3.2.cmml" xref="A0.SS1.p2.11.m11.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.11.m11.1.1.3.3.cmml" xref="A0.SS1.p2.11.m11.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.11.m11.1c">g^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.11.m11.1d">italic_g start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> support pose. <math alttext="C_{ijkg}" class="ltx_Math" display="inline" id="A0.SS1.p2.12.m12.1"><semantics id="A0.SS1.p2.12.m12.1a"><msub id="A0.SS1.p2.12.m12.1.1" xref="A0.SS1.p2.12.m12.1.1.cmml"><mi id="A0.SS1.p2.12.m12.1.1.2" xref="A0.SS1.p2.12.m12.1.1.2.cmml">C</mi><mrow id="A0.SS1.p2.12.m12.1.1.3" xref="A0.SS1.p2.12.m12.1.1.3.cmml"><mi id="A0.SS1.p2.12.m12.1.1.3.2" xref="A0.SS1.p2.12.m12.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p2.12.m12.1.1.3.1" xref="A0.SS1.p2.12.m12.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.12.m12.1.1.3.3" xref="A0.SS1.p2.12.m12.1.1.3.3.cmml">j</mi><mo id="A0.SS1.p2.12.m12.1.1.3.1a" xref="A0.SS1.p2.12.m12.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.12.m12.1.1.3.4" xref="A0.SS1.p2.12.m12.1.1.3.4.cmml">k</mi><mo id="A0.SS1.p2.12.m12.1.1.3.1b" xref="A0.SS1.p2.12.m12.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.12.m12.1.1.3.5" xref="A0.SS1.p2.12.m12.1.1.3.5.cmml">g</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.12.m12.1b"><apply id="A0.SS1.p2.12.m12.1.1.cmml" xref="A0.SS1.p2.12.m12.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.12.m12.1.1.1.cmml" xref="A0.SS1.p2.12.m12.1.1">subscript</csymbol><ci id="A0.SS1.p2.12.m12.1.1.2.cmml" xref="A0.SS1.p2.12.m12.1.1.2">𝐶</ci><apply id="A0.SS1.p2.12.m12.1.1.3.cmml" xref="A0.SS1.p2.12.m12.1.1.3"><times id="A0.SS1.p2.12.m12.1.1.3.1.cmml" xref="A0.SS1.p2.12.m12.1.1.3.1"></times><ci id="A0.SS1.p2.12.m12.1.1.3.2.cmml" xref="A0.SS1.p2.12.m12.1.1.3.2">𝑖</ci><ci id="A0.SS1.p2.12.m12.1.1.3.3.cmml" xref="A0.SS1.p2.12.m12.1.1.3.3">𝑗</ci><ci id="A0.SS1.p2.12.m12.1.1.3.4.cmml" xref="A0.SS1.p2.12.m12.1.1.3.4">𝑘</ci><ci id="A0.SS1.p2.12.m12.1.1.3.5.cmml" xref="A0.SS1.p2.12.m12.1.1.3.5">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.12.m12.1c">C_{ijkg}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.12.m12.1d">italic_C start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="C^{s}_{ijg}" class="ltx_Math" display="inline" id="A0.SS1.p2.13.m13.1"><semantics id="A0.SS1.p2.13.m13.1a"><msubsup id="A0.SS1.p2.13.m13.1.1" xref="A0.SS1.p2.13.m13.1.1.cmml"><mi id="A0.SS1.p2.13.m13.1.1.2.2" xref="A0.SS1.p2.13.m13.1.1.2.2.cmml">C</mi><mrow id="A0.SS1.p2.13.m13.1.1.3" xref="A0.SS1.p2.13.m13.1.1.3.cmml"><mi id="A0.SS1.p2.13.m13.1.1.3.2" xref="A0.SS1.p2.13.m13.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p2.13.m13.1.1.3.1" xref="A0.SS1.p2.13.m13.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.13.m13.1.1.3.3" xref="A0.SS1.p2.13.m13.1.1.3.3.cmml">j</mi><mo id="A0.SS1.p2.13.m13.1.1.3.1a" xref="A0.SS1.p2.13.m13.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.13.m13.1.1.3.4" xref="A0.SS1.p2.13.m13.1.1.3.4.cmml">g</mi></mrow><mi id="A0.SS1.p2.13.m13.1.1.2.3" xref="A0.SS1.p2.13.m13.1.1.2.3.cmml">s</mi></msubsup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.13.m13.1b"><apply id="A0.SS1.p2.13.m13.1.1.cmml" xref="A0.SS1.p2.13.m13.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.13.m13.1.1.1.cmml" xref="A0.SS1.p2.13.m13.1.1">subscript</csymbol><apply id="A0.SS1.p2.13.m13.1.1.2.cmml" xref="A0.SS1.p2.13.m13.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.13.m13.1.1.2.1.cmml" xref="A0.SS1.p2.13.m13.1.1">superscript</csymbol><ci id="A0.SS1.p2.13.m13.1.1.2.2.cmml" xref="A0.SS1.p2.13.m13.1.1.2.2">𝐶</ci><ci id="A0.SS1.p2.13.m13.1.1.2.3.cmml" xref="A0.SS1.p2.13.m13.1.1.2.3">𝑠</ci></apply><apply id="A0.SS1.p2.13.m13.1.1.3.cmml" xref="A0.SS1.p2.13.m13.1.1.3"><times id="A0.SS1.p2.13.m13.1.1.3.1.cmml" xref="A0.SS1.p2.13.m13.1.1.3.1"></times><ci id="A0.SS1.p2.13.m13.1.1.3.2.cmml" xref="A0.SS1.p2.13.m13.1.1.3.2">𝑖</ci><ci id="A0.SS1.p2.13.m13.1.1.3.3.cmml" xref="A0.SS1.p2.13.m13.1.1.3.3">𝑗</ci><ci id="A0.SS1.p2.13.m13.1.1.3.4.cmml" xref="A0.SS1.p2.13.m13.1.1.3.4">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.13.m13.1c">C^{s}_{ijg}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.13.m13.1d">italic_C start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT</annotation></semantics></math> denote the estimated cost for assigning <math alttext="X_{ijkg}=1" class="ltx_Math" display="inline" id="A0.SS1.p2.14.m14.1"><semantics id="A0.SS1.p2.14.m14.1a"><mrow id="A0.SS1.p2.14.m14.1.1" xref="A0.SS1.p2.14.m14.1.1.cmml"><msub id="A0.SS1.p2.14.m14.1.1.2" xref="A0.SS1.p2.14.m14.1.1.2.cmml"><mi id="A0.SS1.p2.14.m14.1.1.2.2" xref="A0.SS1.p2.14.m14.1.1.2.2.cmml">X</mi><mrow id="A0.SS1.p2.14.m14.1.1.2.3" xref="A0.SS1.p2.14.m14.1.1.2.3.cmml"><mi id="A0.SS1.p2.14.m14.1.1.2.3.2" xref="A0.SS1.p2.14.m14.1.1.2.3.2.cmml">i</mi><mo id="A0.SS1.p2.14.m14.1.1.2.3.1" xref="A0.SS1.p2.14.m14.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.14.m14.1.1.2.3.3" xref="A0.SS1.p2.14.m14.1.1.2.3.3.cmml">j</mi><mo id="A0.SS1.p2.14.m14.1.1.2.3.1a" xref="A0.SS1.p2.14.m14.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.14.m14.1.1.2.3.4" xref="A0.SS1.p2.14.m14.1.1.2.3.4.cmml">k</mi><mo id="A0.SS1.p2.14.m14.1.1.2.3.1b" xref="A0.SS1.p2.14.m14.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.14.m14.1.1.2.3.5" xref="A0.SS1.p2.14.m14.1.1.2.3.5.cmml">g</mi></mrow></msub><mo id="A0.SS1.p2.14.m14.1.1.1" xref="A0.SS1.p2.14.m14.1.1.1.cmml">=</mo><mn id="A0.SS1.p2.14.m14.1.1.3" xref="A0.SS1.p2.14.m14.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.14.m14.1b"><apply id="A0.SS1.p2.14.m14.1.1.cmml" xref="A0.SS1.p2.14.m14.1.1"><eq id="A0.SS1.p2.14.m14.1.1.1.cmml" xref="A0.SS1.p2.14.m14.1.1.1"></eq><apply id="A0.SS1.p2.14.m14.1.1.2.cmml" xref="A0.SS1.p2.14.m14.1.1.2"><csymbol cd="ambiguous" id="A0.SS1.p2.14.m14.1.1.2.1.cmml" xref="A0.SS1.p2.14.m14.1.1.2">subscript</csymbol><ci id="A0.SS1.p2.14.m14.1.1.2.2.cmml" xref="A0.SS1.p2.14.m14.1.1.2.2">𝑋</ci><apply id="A0.SS1.p2.14.m14.1.1.2.3.cmml" xref="A0.SS1.p2.14.m14.1.1.2.3"><times id="A0.SS1.p2.14.m14.1.1.2.3.1.cmml" xref="A0.SS1.p2.14.m14.1.1.2.3.1"></times><ci id="A0.SS1.p2.14.m14.1.1.2.3.2.cmml" xref="A0.SS1.p2.14.m14.1.1.2.3.2">𝑖</ci><ci id="A0.SS1.p2.14.m14.1.1.2.3.3.cmml" xref="A0.SS1.p2.14.m14.1.1.2.3.3">𝑗</ci><ci id="A0.SS1.p2.14.m14.1.1.2.3.4.cmml" xref="A0.SS1.p2.14.m14.1.1.2.3.4">𝑘</ci><ci id="A0.SS1.p2.14.m14.1.1.2.3.5.cmml" xref="A0.SS1.p2.14.m14.1.1.2.3.5">𝑔</ci></apply></apply><cn id="A0.SS1.p2.14.m14.1.1.3.cmml" type="integer" xref="A0.SS1.p2.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.14.m14.1c">X_{ijkg}=1</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.14.m14.1d">italic_X start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="Y_{ijg}=1" class="ltx_Math" display="inline" id="A0.SS1.p2.15.m15.1"><semantics id="A0.SS1.p2.15.m15.1a"><mrow id="A0.SS1.p2.15.m15.1.1" xref="A0.SS1.p2.15.m15.1.1.cmml"><msub id="A0.SS1.p2.15.m15.1.1.2" xref="A0.SS1.p2.15.m15.1.1.2.cmml"><mi id="A0.SS1.p2.15.m15.1.1.2.2" xref="A0.SS1.p2.15.m15.1.1.2.2.cmml">Y</mi><mrow id="A0.SS1.p2.15.m15.1.1.2.3" xref="A0.SS1.p2.15.m15.1.1.2.3.cmml"><mi id="A0.SS1.p2.15.m15.1.1.2.3.2" xref="A0.SS1.p2.15.m15.1.1.2.3.2.cmml">i</mi><mo id="A0.SS1.p2.15.m15.1.1.2.3.1" xref="A0.SS1.p2.15.m15.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.15.m15.1.1.2.3.3" xref="A0.SS1.p2.15.m15.1.1.2.3.3.cmml">j</mi><mo id="A0.SS1.p2.15.m15.1.1.2.3.1a" xref="A0.SS1.p2.15.m15.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.15.m15.1.1.2.3.4" xref="A0.SS1.p2.15.m15.1.1.2.3.4.cmml">g</mi></mrow></msub><mo id="A0.SS1.p2.15.m15.1.1.1" xref="A0.SS1.p2.15.m15.1.1.1.cmml">=</mo><mn id="A0.SS1.p2.15.m15.1.1.3" xref="A0.SS1.p2.15.m15.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.15.m15.1b"><apply id="A0.SS1.p2.15.m15.1.1.cmml" xref="A0.SS1.p2.15.m15.1.1"><eq id="A0.SS1.p2.15.m15.1.1.1.cmml" xref="A0.SS1.p2.15.m15.1.1.1"></eq><apply id="A0.SS1.p2.15.m15.1.1.2.cmml" xref="A0.SS1.p2.15.m15.1.1.2"><csymbol cd="ambiguous" id="A0.SS1.p2.15.m15.1.1.2.1.cmml" xref="A0.SS1.p2.15.m15.1.1.2">subscript</csymbol><ci id="A0.SS1.p2.15.m15.1.1.2.2.cmml" xref="A0.SS1.p2.15.m15.1.1.2.2">𝑌</ci><apply id="A0.SS1.p2.15.m15.1.1.2.3.cmml" xref="A0.SS1.p2.15.m15.1.1.2.3"><times id="A0.SS1.p2.15.m15.1.1.2.3.1.cmml" xref="A0.SS1.p2.15.m15.1.1.2.3.1"></times><ci id="A0.SS1.p2.15.m15.1.1.2.3.2.cmml" xref="A0.SS1.p2.15.m15.1.1.2.3.2">𝑖</ci><ci id="A0.SS1.p2.15.m15.1.1.2.3.3.cmml" xref="A0.SS1.p2.15.m15.1.1.2.3.3">𝑗</ci><ci id="A0.SS1.p2.15.m15.1.1.2.3.4.cmml" xref="A0.SS1.p2.15.m15.1.1.2.3.4">𝑔</ci></apply></apply><cn id="A0.SS1.p2.15.m15.1.1.3.cmml" type="integer" xref="A0.SS1.p2.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.15.m15.1c">Y_{ijg}=1</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.15.m15.1d">italic_Y start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT = 1</annotation></semantics></math>, respectively. <math alttext="P" class="ltx_Math" display="inline" id="A0.SS1.p2.16.m16.1"><semantics id="A0.SS1.p2.16.m16.1a"><mi id="A0.SS1.p2.16.m16.1.1" xref="A0.SS1.p2.16.m16.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.16.m16.1b"><ci id="A0.SS1.p2.16.m16.1.1.cmml" xref="A0.SS1.p2.16.m16.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.16.m16.1c">P</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.16.m16.1d">italic_P</annotation></semantics></math> is the maximum number of feasible grasp poses and is set based on the objects and user preferences. <math alttext="X_{ijkg}=1" class="ltx_Math" display="inline" id="A0.SS1.p2.17.m17.1"><semantics id="A0.SS1.p2.17.m17.1a"><mrow id="A0.SS1.p2.17.m17.1.1" xref="A0.SS1.p2.17.m17.1.1.cmml"><msub id="A0.SS1.p2.17.m17.1.1.2" xref="A0.SS1.p2.17.m17.1.1.2.cmml"><mi id="A0.SS1.p2.17.m17.1.1.2.2" xref="A0.SS1.p2.17.m17.1.1.2.2.cmml">X</mi><mrow id="A0.SS1.p2.17.m17.1.1.2.3" xref="A0.SS1.p2.17.m17.1.1.2.3.cmml"><mi id="A0.SS1.p2.17.m17.1.1.2.3.2" xref="A0.SS1.p2.17.m17.1.1.2.3.2.cmml">i</mi><mo id="A0.SS1.p2.17.m17.1.1.2.3.1" xref="A0.SS1.p2.17.m17.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.17.m17.1.1.2.3.3" xref="A0.SS1.p2.17.m17.1.1.2.3.3.cmml">j</mi><mo id="A0.SS1.p2.17.m17.1.1.2.3.1a" xref="A0.SS1.p2.17.m17.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.17.m17.1.1.2.3.4" xref="A0.SS1.p2.17.m17.1.1.2.3.4.cmml">k</mi><mo id="A0.SS1.p2.17.m17.1.1.2.3.1b" xref="A0.SS1.p2.17.m17.1.1.2.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.17.m17.1.1.2.3.5" xref="A0.SS1.p2.17.m17.1.1.2.3.5.cmml">g</mi></mrow></msub><mo id="A0.SS1.p2.17.m17.1.1.1" xref="A0.SS1.p2.17.m17.1.1.1.cmml">=</mo><mn id="A0.SS1.p2.17.m17.1.1.3" xref="A0.SS1.p2.17.m17.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.17.m17.1b"><apply id="A0.SS1.p2.17.m17.1.1.cmml" xref="A0.SS1.p2.17.m17.1.1"><eq id="A0.SS1.p2.17.m17.1.1.1.cmml" xref="A0.SS1.p2.17.m17.1.1.1"></eq><apply id="A0.SS1.p2.17.m17.1.1.2.cmml" xref="A0.SS1.p2.17.m17.1.1.2"><csymbol cd="ambiguous" id="A0.SS1.p2.17.m17.1.1.2.1.cmml" xref="A0.SS1.p2.17.m17.1.1.2">subscript</csymbol><ci id="A0.SS1.p2.17.m17.1.1.2.2.cmml" xref="A0.SS1.p2.17.m17.1.1.2.2">𝑋</ci><apply id="A0.SS1.p2.17.m17.1.1.2.3.cmml" xref="A0.SS1.p2.17.m17.1.1.2.3"><times id="A0.SS1.p2.17.m17.1.1.2.3.1.cmml" xref="A0.SS1.p2.17.m17.1.1.2.3.1"></times><ci id="A0.SS1.p2.17.m17.1.1.2.3.2.cmml" xref="A0.SS1.p2.17.m17.1.1.2.3.2">𝑖</ci><ci id="A0.SS1.p2.17.m17.1.1.2.3.3.cmml" xref="A0.SS1.p2.17.m17.1.1.2.3.3">𝑗</ci><ci id="A0.SS1.p2.17.m17.1.1.2.3.4.cmml" xref="A0.SS1.p2.17.m17.1.1.2.3.4">𝑘</ci><ci id="A0.SS1.p2.17.m17.1.1.2.3.5.cmml" xref="A0.SS1.p2.17.m17.1.1.2.3.5">𝑔</ci></apply></apply><cn id="A0.SS1.p2.17.m17.1.1.3.cmml" type="integer" xref="A0.SS1.p2.17.m17.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.17.m17.1c">X_{ijkg}=1</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.17.m17.1d">italic_X start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT = 1</annotation></semantics></math> means the <math alttext="i^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.18.m18.1"><semantics id="A0.SS1.p2.18.m18.1a"><msup id="A0.SS1.p2.18.m18.1.1" xref="A0.SS1.p2.18.m18.1.1.cmml"><mi id="A0.SS1.p2.18.m18.1.1.2" xref="A0.SS1.p2.18.m18.1.1.2.cmml">i</mi><mrow id="A0.SS1.p2.18.m18.1.1.3" xref="A0.SS1.p2.18.m18.1.1.3.cmml"><mi id="A0.SS1.p2.18.m18.1.1.3.2" xref="A0.SS1.p2.18.m18.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.18.m18.1.1.3.1" xref="A0.SS1.p2.18.m18.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.18.m18.1.1.3.3" xref="A0.SS1.p2.18.m18.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.18.m18.1b"><apply id="A0.SS1.p2.18.m18.1.1.cmml" xref="A0.SS1.p2.18.m18.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.18.m18.1.1.1.cmml" xref="A0.SS1.p2.18.m18.1.1">superscript</csymbol><ci id="A0.SS1.p2.18.m18.1.1.2.cmml" xref="A0.SS1.p2.18.m18.1.1.2">𝑖</ci><apply id="A0.SS1.p2.18.m18.1.1.3.cmml" xref="A0.SS1.p2.18.m18.1.1.3"><times id="A0.SS1.p2.18.m18.1.1.3.1.cmml" xref="A0.SS1.p2.18.m18.1.1.3.1"></times><ci id="A0.SS1.p2.18.m18.1.1.3.2.cmml" xref="A0.SS1.p2.18.m18.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.18.m18.1.1.3.3.cmml" xref="A0.SS1.p2.18.m18.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.18.m18.1c">i^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.18.m18.1d">italic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> robot and object <math alttext="b_{k}" class="ltx_Math" display="inline" id="A0.SS1.p2.19.m19.1"><semantics id="A0.SS1.p2.19.m19.1a"><msub id="A0.SS1.p2.19.m19.1.1" xref="A0.SS1.p2.19.m19.1.1.cmml"><mi id="A0.SS1.p2.19.m19.1.1.2" xref="A0.SS1.p2.19.m19.1.1.2.cmml">b</mi><mi id="A0.SS1.p2.19.m19.1.1.3" xref="A0.SS1.p2.19.m19.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.19.m19.1b"><apply id="A0.SS1.p2.19.m19.1.1.cmml" xref="A0.SS1.p2.19.m19.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.19.m19.1.1.1.cmml" xref="A0.SS1.p2.19.m19.1.1">subscript</csymbol><ci id="A0.SS1.p2.19.m19.1.1.2.cmml" xref="A0.SS1.p2.19.m19.1.1.2">𝑏</ci><ci id="A0.SS1.p2.19.m19.1.1.3.cmml" xref="A0.SS1.p2.19.m19.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.19.m19.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.19.m19.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is assigned for the assembly step <math alttext="j" class="ltx_Math" display="inline" id="A0.SS1.p2.20.m20.1"><semantics id="A0.SS1.p2.20.m20.1a"><mi id="A0.SS1.p2.20.m20.1.1" xref="A0.SS1.p2.20.m20.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.20.m20.1b"><ci id="A0.SS1.p2.20.m20.1.1.cmml" xref="A0.SS1.p2.20.m20.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.20.m20.1c">j</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.20.m20.1d">italic_j</annotation></semantics></math>, using the <math alttext="g^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.21.m21.1"><semantics id="A0.SS1.p2.21.m21.1a"><msup id="A0.SS1.p2.21.m21.1.1" xref="A0.SS1.p2.21.m21.1.1.cmml"><mi id="A0.SS1.p2.21.m21.1.1.2" xref="A0.SS1.p2.21.m21.1.1.2.cmml">g</mi><mrow id="A0.SS1.p2.21.m21.1.1.3" xref="A0.SS1.p2.21.m21.1.1.3.cmml"><mi id="A0.SS1.p2.21.m21.1.1.3.2" xref="A0.SS1.p2.21.m21.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.21.m21.1.1.3.1" xref="A0.SS1.p2.21.m21.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.21.m21.1.1.3.3" xref="A0.SS1.p2.21.m21.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.21.m21.1b"><apply id="A0.SS1.p2.21.m21.1.1.cmml" xref="A0.SS1.p2.21.m21.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.21.m21.1.1.1.cmml" xref="A0.SS1.p2.21.m21.1.1">superscript</csymbol><ci id="A0.SS1.p2.21.m21.1.1.2.cmml" xref="A0.SS1.p2.21.m21.1.1.2">𝑔</ci><apply id="A0.SS1.p2.21.m21.1.1.3.cmml" xref="A0.SS1.p2.21.m21.1.1.3"><times id="A0.SS1.p2.21.m21.1.1.3.1.cmml" xref="A0.SS1.p2.21.m21.1.1.3.1"></times><ci id="A0.SS1.p2.21.m21.1.1.3.2.cmml" xref="A0.SS1.p2.21.m21.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.21.m21.1.1.3.3.cmml" xref="A0.SS1.p2.21.m21.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.21.m21.1c">g^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.21.m21.1d">italic_g start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> grasp pose. The cost <math alttext="C_{ijkg}" class="ltx_Math" display="inline" id="A0.SS1.p2.22.m22.1"><semantics id="A0.SS1.p2.22.m22.1a"><msub id="A0.SS1.p2.22.m22.1.1" xref="A0.SS1.p2.22.m22.1.1.cmml"><mi id="A0.SS1.p2.22.m22.1.1.2" xref="A0.SS1.p2.22.m22.1.1.2.cmml">C</mi><mrow id="A0.SS1.p2.22.m22.1.1.3" xref="A0.SS1.p2.22.m22.1.1.3.cmml"><mi id="A0.SS1.p2.22.m22.1.1.3.2" xref="A0.SS1.p2.22.m22.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p2.22.m22.1.1.3.1" xref="A0.SS1.p2.22.m22.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.22.m22.1.1.3.3" xref="A0.SS1.p2.22.m22.1.1.3.3.cmml">j</mi><mo id="A0.SS1.p2.22.m22.1.1.3.1a" xref="A0.SS1.p2.22.m22.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.22.m22.1.1.3.4" xref="A0.SS1.p2.22.m22.1.1.3.4.cmml">k</mi><mo id="A0.SS1.p2.22.m22.1.1.3.1b" xref="A0.SS1.p2.22.m22.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.22.m22.1.1.3.5" xref="A0.SS1.p2.22.m22.1.1.3.5.cmml">g</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.22.m22.1b"><apply id="A0.SS1.p2.22.m22.1.1.cmml" xref="A0.SS1.p2.22.m22.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.22.m22.1.1.1.cmml" xref="A0.SS1.p2.22.m22.1.1">subscript</csymbol><ci id="A0.SS1.p2.22.m22.1.1.2.cmml" xref="A0.SS1.p2.22.m22.1.1.2">𝐶</ci><apply id="A0.SS1.p2.22.m22.1.1.3.cmml" xref="A0.SS1.p2.22.m22.1.1.3"><times id="A0.SS1.p2.22.m22.1.1.3.1.cmml" xref="A0.SS1.p2.22.m22.1.1.3.1"></times><ci id="A0.SS1.p2.22.m22.1.1.3.2.cmml" xref="A0.SS1.p2.22.m22.1.1.3.2">𝑖</ci><ci id="A0.SS1.p2.22.m22.1.1.3.3.cmml" xref="A0.SS1.p2.22.m22.1.1.3.3">𝑗</ci><ci id="A0.SS1.p2.22.m22.1.1.3.4.cmml" xref="A0.SS1.p2.22.m22.1.1.3.4">𝑘</ci><ci id="A0.SS1.p2.22.m22.1.1.3.5.cmml" xref="A0.SS1.p2.22.m22.1.1.3.5">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.22.m22.1c">C_{ijkg}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.22.m22.1d">italic_C start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT</annotation></semantics></math> is computed as the sum of joint space distance from robot <math alttext="i" class="ltx_Math" display="inline" id="A0.SS1.p2.23.m23.1"><semantics id="A0.SS1.p2.23.m23.1a"><mi id="A0.SS1.p2.23.m23.1.1" xref="A0.SS1.p2.23.m23.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.23.m23.1b"><ci id="A0.SS1.p2.23.m23.1.1.cmml" xref="A0.SS1.p2.23.m23.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.23.m23.1c">i</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.23.m23.1d">italic_i</annotation></semantics></math>’s HOME pose to <math alttext="g^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.24.m24.1"><semantics id="A0.SS1.p2.24.m24.1a"><msup id="A0.SS1.p2.24.m24.1.1" xref="A0.SS1.p2.24.m24.1.1.cmml"><mi id="A0.SS1.p2.24.m24.1.1.2" xref="A0.SS1.p2.24.m24.1.1.2.cmml">g</mi><mrow id="A0.SS1.p2.24.m24.1.1.3" xref="A0.SS1.p2.24.m24.1.1.3.cmml"><mi id="A0.SS1.p2.24.m24.1.1.3.2" xref="A0.SS1.p2.24.m24.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.24.m24.1.1.3.1" xref="A0.SS1.p2.24.m24.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.24.m24.1.1.3.3" xref="A0.SS1.p2.24.m24.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.24.m24.1b"><apply id="A0.SS1.p2.24.m24.1.1.cmml" xref="A0.SS1.p2.24.m24.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.24.m24.1.1.1.cmml" xref="A0.SS1.p2.24.m24.1.1">superscript</csymbol><ci id="A0.SS1.p2.24.m24.1.1.2.cmml" xref="A0.SS1.p2.24.m24.1.1.2">𝑔</ci><apply id="A0.SS1.p2.24.m24.1.1.3.cmml" xref="A0.SS1.p2.24.m24.1.1.3"><times id="A0.SS1.p2.24.m24.1.1.3.1.cmml" xref="A0.SS1.p2.24.m24.1.1.3.1"></times><ci id="A0.SS1.p2.24.m24.1.1.3.2.cmml" xref="A0.SS1.p2.24.m24.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.24.m24.1.1.3.3.cmml" xref="A0.SS1.p2.24.m24.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.24.m24.1c">g^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.24.m24.1d">italic_g start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> robot pose for picking object <math alttext="b_{k}" class="ltx_Math" display="inline" id="A0.SS1.p2.25.m25.1"><semantics id="A0.SS1.p2.25.m25.1a"><msub id="A0.SS1.p2.25.m25.1.1" xref="A0.SS1.p2.25.m25.1.1.cmml"><mi id="A0.SS1.p2.25.m25.1.1.2" xref="A0.SS1.p2.25.m25.1.1.2.cmml">b</mi><mi id="A0.SS1.p2.25.m25.1.1.3" xref="A0.SS1.p2.25.m25.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.25.m25.1b"><apply id="A0.SS1.p2.25.m25.1.1.cmml" xref="A0.SS1.p2.25.m25.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.25.m25.1.1.1.cmml" xref="A0.SS1.p2.25.m25.1.1">subscript</csymbol><ci id="A0.SS1.p2.25.m25.1.1.2.cmml" xref="A0.SS1.p2.25.m25.1.1.2">𝑏</ci><ci id="A0.SS1.p2.25.m25.1.1.3.cmml" xref="A0.SS1.p2.25.m25.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.25.m25.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.25.m25.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> to the robot target pose for placing the object at assembly step <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p2.26.m26.1"><semantics id="A0.SS1.p2.26.m26.1a"><msub id="A0.SS1.p2.26.m26.1.1" xref="A0.SS1.p2.26.m26.1.1.cmml"><mi id="A0.SS1.p2.26.m26.1.1.2" xref="A0.SS1.p2.26.m26.1.1.2.cmml">a</mi><mi id="A0.SS1.p2.26.m26.1.1.3" xref="A0.SS1.p2.26.m26.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.26.m26.1b"><apply id="A0.SS1.p2.26.m26.1.1.cmml" xref="A0.SS1.p2.26.m26.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.26.m26.1.1.1.cmml" xref="A0.SS1.p2.26.m26.1.1">subscript</csymbol><ci id="A0.SS1.p2.26.m26.1.1.2.cmml" xref="A0.SS1.p2.26.m26.1.1.2">𝑎</ci><ci id="A0.SS1.p2.26.m26.1.1.3.cmml" xref="A0.SS1.p2.26.m26.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.26.m26.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.26.m26.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, <math alttext="C^{s}_{ijg}" class="ltx_Math" display="inline" id="A0.SS1.p2.27.m27.1"><semantics id="A0.SS1.p2.27.m27.1a"><msubsup id="A0.SS1.p2.27.m27.1.1" xref="A0.SS1.p2.27.m27.1.1.cmml"><mi id="A0.SS1.p2.27.m27.1.1.2.2" xref="A0.SS1.p2.27.m27.1.1.2.2.cmml">C</mi><mrow id="A0.SS1.p2.27.m27.1.1.3" xref="A0.SS1.p2.27.m27.1.1.3.cmml"><mi id="A0.SS1.p2.27.m27.1.1.3.2" xref="A0.SS1.p2.27.m27.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p2.27.m27.1.1.3.1" xref="A0.SS1.p2.27.m27.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.27.m27.1.1.3.3" xref="A0.SS1.p2.27.m27.1.1.3.3.cmml">j</mi><mo id="A0.SS1.p2.27.m27.1.1.3.1a" xref="A0.SS1.p2.27.m27.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.27.m27.1.1.3.4" xref="A0.SS1.p2.27.m27.1.1.3.4.cmml">g</mi></mrow><mi id="A0.SS1.p2.27.m27.1.1.2.3" xref="A0.SS1.p2.27.m27.1.1.2.3.cmml">s</mi></msubsup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.27.m27.1b"><apply id="A0.SS1.p2.27.m27.1.1.cmml" xref="A0.SS1.p2.27.m27.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.27.m27.1.1.1.cmml" xref="A0.SS1.p2.27.m27.1.1">subscript</csymbol><apply id="A0.SS1.p2.27.m27.1.1.2.cmml" xref="A0.SS1.p2.27.m27.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.27.m27.1.1.2.1.cmml" xref="A0.SS1.p2.27.m27.1.1">superscript</csymbol><ci id="A0.SS1.p2.27.m27.1.1.2.2.cmml" xref="A0.SS1.p2.27.m27.1.1.2.2">𝐶</ci><ci id="A0.SS1.p2.27.m27.1.1.2.3.cmml" xref="A0.SS1.p2.27.m27.1.1.2.3">𝑠</ci></apply><apply id="A0.SS1.p2.27.m27.1.1.3.cmml" xref="A0.SS1.p2.27.m27.1.1.3"><times id="A0.SS1.p2.27.m27.1.1.3.1.cmml" xref="A0.SS1.p2.27.m27.1.1.3.1"></times><ci id="A0.SS1.p2.27.m27.1.1.3.2.cmml" xref="A0.SS1.p2.27.m27.1.1.3.2">𝑖</ci><ci id="A0.SS1.p2.27.m27.1.1.3.3.cmml" xref="A0.SS1.p2.27.m27.1.1.3.3">𝑗</ci><ci id="A0.SS1.p2.27.m27.1.1.3.4.cmml" xref="A0.SS1.p2.27.m27.1.1.3.4">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.27.m27.1c">C^{s}_{ijg}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.27.m27.1d">italic_C start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT</annotation></semantics></math> is the distance from robot <math alttext="i" class="ltx_Math" display="inline" id="A0.SS1.p2.28.m28.1"><semantics id="A0.SS1.p2.28.m28.1a"><mi id="A0.SS1.p2.28.m28.1.1" xref="A0.SS1.p2.28.m28.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.28.m28.1b"><ci id="A0.SS1.p2.28.m28.1.1.cmml" xref="A0.SS1.p2.28.m28.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.28.m28.1c">i</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.28.m28.1d">italic_i</annotation></semantics></math>’s HOME pose, to the <math alttext="g^{th}" class="ltx_Math" display="inline" id="A0.SS1.p2.29.m29.1"><semantics id="A0.SS1.p2.29.m29.1a"><msup id="A0.SS1.p2.29.m29.1.1" xref="A0.SS1.p2.29.m29.1.1.cmml"><mi id="A0.SS1.p2.29.m29.1.1.2" xref="A0.SS1.p2.29.m29.1.1.2.cmml">g</mi><mrow id="A0.SS1.p2.29.m29.1.1.3" xref="A0.SS1.p2.29.m29.1.1.3.cmml"><mi id="A0.SS1.p2.29.m29.1.1.3.2" xref="A0.SS1.p2.29.m29.1.1.3.2.cmml">t</mi><mo id="A0.SS1.p2.29.m29.1.1.3.1" xref="A0.SS1.p2.29.m29.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p2.29.m29.1.1.3.3" xref="A0.SS1.p2.29.m29.1.1.3.3.cmml">h</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.29.m29.1b"><apply id="A0.SS1.p2.29.m29.1.1.cmml" xref="A0.SS1.p2.29.m29.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.29.m29.1.1.1.cmml" xref="A0.SS1.p2.29.m29.1.1">superscript</csymbol><ci id="A0.SS1.p2.29.m29.1.1.2.cmml" xref="A0.SS1.p2.29.m29.1.1.2">𝑔</ci><apply id="A0.SS1.p2.29.m29.1.1.3.cmml" xref="A0.SS1.p2.29.m29.1.1.3"><times id="A0.SS1.p2.29.m29.1.1.3.1.cmml" xref="A0.SS1.p2.29.m29.1.1.3.1"></times><ci id="A0.SS1.p2.29.m29.1.1.3.2.cmml" xref="A0.SS1.p2.29.m29.1.1.3.2">𝑡</ci><ci id="A0.SS1.p2.29.m29.1.1.3.3.cmml" xref="A0.SS1.p2.29.m29.1.1.3.3">ℎ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.29.m29.1c">g^{th}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.29.m29.1d">italic_g start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT</annotation></semantics></math> support pose for step <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p2.30.m30.1"><semantics id="A0.SS1.p2.30.m30.1a"><msub id="A0.SS1.p2.30.m30.1.1" xref="A0.SS1.p2.30.m30.1.1.cmml"><mi id="A0.SS1.p2.30.m30.1.1.2" xref="A0.SS1.p2.30.m30.1.1.2.cmml">a</mi><mi id="A0.SS1.p2.30.m30.1.1.3" xref="A0.SS1.p2.30.m30.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p2.30.m30.1b"><apply id="A0.SS1.p2.30.m30.1.1.cmml" xref="A0.SS1.p2.30.m30.1.1"><csymbol cd="ambiguous" id="A0.SS1.p2.30.m30.1.1.1.cmml" xref="A0.SS1.p2.30.m30.1.1">subscript</csymbol><ci id="A0.SS1.p2.30.m30.1.1.2.cmml" xref="A0.SS1.p2.30.m30.1.1.2">𝑎</ci><ci id="A0.SS1.p2.30.m30.1.1.3.cmml" xref="A0.SS1.p2.30.m30.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p2.30.m30.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p2.30.m30.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. The solution to the following ILP program then forms a complete sequential task plan. </div> <div class="ltx_para" id="A0.SS1.p3"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A0.EGx1"> <tbody id="A0.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{arg\,min}_{X,Y}" class="ltx_Math" display="inline" id="A0.E1.m1.2"><semantics id="A0.E1.m1.2a"><munder id="A0.E1.m1.2.3" xref="A0.E1.m1.2.3.cmml"><mrow id="A0.E1.m1.2.3.2" xref="A0.E1.m1.2.3.2.cmml"><mi id="A0.E1.m1.2.3.2.2" xref="A0.E1.m1.2.3.2.2.cmml">arg</mi><mo id="A0.E1.m1.2.3.2.1" lspace="0.170em" xref="A0.E1.m1.2.3.2.1.cmml">⁢</mo><mi id="A0.E1.m1.2.3.2.3" xref="A0.E1.m1.2.3.2.3.cmml">min</mi></mrow><mrow id="A0.E1.m1.2.2.2.4" xref="A0.E1.m1.2.2.2.3.cmml"><mi id="A0.E1.m1.1.1.1.1" xref="A0.E1.m1.1.1.1.1.cmml">X</mi><mo id="A0.E1.m1.2.2.2.4.1" xref="A0.E1.m1.2.2.2.3.cmml">,</mo><mi id="A0.E1.m1.2.2.2.2" xref="A0.E1.m1.2.2.2.2.cmml">Y</mi></mrow></munder><annotation-xml encoding="MathML-Content" id="A0.E1.m1.2b"><apply id="A0.E1.m1.2.3.cmml" xref="A0.E1.m1.2.3"><csymbol cd="ambiguous" id="A0.E1.m1.2.3.1.cmml" xref="A0.E1.m1.2.3">subscript</csymbol><apply id="A0.E1.m1.2.3.2.cmml" xref="A0.E1.m1.2.3.2"><times id="A0.E1.m1.2.3.2.1.cmml" xref="A0.E1.m1.2.3.2.1"></times><ci id="A0.E1.m1.2.3.2.2.cmml" xref="A0.E1.m1.2.3.2.2">arg</ci><ci id="A0.E1.m1.2.3.2.3.cmml" xref="A0.E1.m1.2.3.2.3">min</ci></apply><list id="A0.E1.m1.2.2.2.3.cmml" xref="A0.E1.m1.2.2.2.4"><ci id="A0.E1.m1.1.1.1.1.cmml" xref="A0.E1.m1.1.1.1.1">𝑋</ci><ci id="A0.E1.m1.2.2.2.2.cmml" xref="A0.E1.m1.2.2.2.2">𝑌</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.E1.m1.2c">\displaystyle\operatorname*{arg\,min}_{X,Y}</annotation><annotation encoding="application/x-llamapun" id="A0.E1.m1.2d">start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_X , italic_Y end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{ijkg}C_{ijkg}X_{ijkg}+\sum_{ijg}C^{s}_{ijg}Y_{ijg}+\lambda% \sum_{j}(Z^{M}_{j}-Z^{m}_{j})" class="ltx_Math" display="inline" id="A0.E1.m2.1"><semantics id="A0.E1.m2.1a"><mrow id="A0.E1.m2.1.1" xref="A0.E1.m2.1.1.cmml"><mrow id="A0.E1.m2.1.1.3" xref="A0.E1.m2.1.1.3.cmml"><mstyle displaystyle="true" id="A0.E1.m2.1.1.3.1" xref="A0.E1.m2.1.1.3.1.cmml"><munder id="A0.E1.m2.1.1.3.1a" xref="A0.E1.m2.1.1.3.1.cmml"><mo id="A0.E1.m2.1.1.3.1.2" movablelimits="false" xref="A0.E1.m2.1.1.3.1.2.cmml">∑</mo><mrow id="A0.E1.m2.1.1.3.1.3" xref="A0.E1.m2.1.1.3.1.3.cmml"><mi id="A0.E1.m2.1.1.3.1.3.2" xref="A0.E1.m2.1.1.3.1.3.2.cmml">i</mi><mo id="A0.E1.m2.1.1.3.1.3.1" xref="A0.E1.m2.1.1.3.1.3.1.cmml">⁢</mo><mi id="A0.E1.m2.1.1.3.1.3.3" xref="A0.E1.m2.1.1.3.1.3.3.cmml">j</mi><mo id="A0.E1.m2.1.1.3.1.3.1a" xref="A0.E1.m2.1.1.3.1.3.1.cmml">⁢</mo><mi id="A0.E1.m2.1.1.3.1.3.4" xref="A0.E1.m2.1.1.3.1.3.4.cmml">k</mi><mo id="A0.E1.m2.1.1.3.1.3.1b" xref="A0.E1.m2.1.1.3.1.3.1.cmml">⁢</mo><mi id="A0.E1.m2.1.1.3.1.3.5" xref="A0.E1.m2.1.1.3.1.3.5.cmml">g</mi></mrow></munder></mstyle><mrow id="A0.E1.m2.1.1.3.2" xref="A0.E1.m2.1.1.3.2.cmml"><msub id="A0.E1.m2.1.1.3.2.2" xref="A0.E1.m2.1.1.3.2.2.cmml"><mi id="A0.E1.m2.1.1.3.2.2.2" xref="A0.E1.m2.1.1.3.2.2.2.cmml">C</mi><mrow id="A0.E1.m2.1.1.3.2.2.3" xref="A0.E1.m2.1.1.3.2.2.3.cmml"><mi id="A0.E1.m2.1.1.3.2.2.3.2" xref="A0.E1.m2.1.1.3.2.2.3.2.cmml">i</mi><mo id="A0.E1.m2.1.1.3.2.2.3.1" xref="A0.E1.m2.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="A0.E1.m2.1.1.3.2.2.3.3" xref="A0.E1.m2.1.1.3.2.2.3.3.cmml">j</mi><mo id="A0.E1.m2.1.1.3.2.2.3.1a" xref="A0.E1.m2.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="A0.E1.m2.1.1.3.2.2.3.4" xref="A0.E1.m2.1.1.3.2.2.3.4.cmml">k</mi><mo id="A0.E1.m2.1.1.3.2.2.3.1b" xref="A0.E1.m2.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="A0.E1.m2.1.1.3.2.2.3.5" 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italic_X start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT italic_Y start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT + italic_λ ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_Z start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_Z start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(1)</td> </tr></tbody> <tbody id="A0.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell">subect to</td> <td class="ltx_td ltx_align_left ltx_eqn_cell"> task, robot, and object constraints</td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A0.SS1.p4"> The first two terms of optimization minimize the total cost of executing a sequential task plan. The last auxiliary cost term penalizes cases where a single robot is assigned for multiple consecutive tasks and prevents parallelization. The constraints include <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A0.EGx2"> <tbody id="A0.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{i=1}^{N}\sum_{k=1}^{N_{b}}\sum_{g=1}^{P}X_{ijkg}" class="ltx_Math" display="inline" id="A0.E2.m1.1"><semantics id="A0.E2.m1.1a"><mrow id="A0.E2.m1.1.1" xref="A0.E2.m1.1.1.cmml"><mstyle displaystyle="true" id="A0.E2.m1.1.1.1" xref="A0.E2.m1.1.1.1.cmml"><munderover id="A0.E2.m1.1.1.1a" xref="A0.E2.m1.1.1.1.cmml"><mo id="A0.E2.m1.1.1.1.2.2" movablelimits="false" xref="A0.E2.m1.1.1.1.2.2.cmml">∑</mo><mrow id="A0.E2.m1.1.1.1.2.3" xref="A0.E2.m1.1.1.1.2.3.cmml"><mi id="A0.E2.m1.1.1.1.2.3.2" xref="A0.E2.m1.1.1.1.2.3.2.cmml">i</mi><mo id="A0.E2.m1.1.1.1.2.3.1" xref="A0.E2.m1.1.1.1.2.3.1.cmml">=</mo><mn id="A0.E2.m1.1.1.1.2.3.3" xref="A0.E2.m1.1.1.1.2.3.3.cmml">1</mn></mrow><mi 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id="A0.E4.m1.1.1.3.2.3.1.cmml" xref="A0.E4.m1.1.1.3.2.3.1"></times><ci id="A0.E4.m1.1.1.3.2.3.2.cmml" xref="A0.E4.m1.1.1.3.2.3.2">𝑖</ci><ci id="A0.E4.m1.1.1.3.2.3.3.cmml" xref="A0.E4.m1.1.1.3.2.3.3">𝑗</ci><ci id="A0.E4.m1.1.1.3.2.3.4.cmml" xref="A0.E4.m1.1.1.3.2.3.4">𝑔</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.E4.m1.1c">\displaystyle\sum_{k=1}^{N_{b}}\sum_{g=1}^{P}X_{ijkg}+\sum_{g=1}^{P}Y_{ijg}</annotation><annotation encoding="application/x-llamapun" id="A0.E4.m1.1d">∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_g = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT italic_X start_POSTSUBSCRIPT italic_i italic_j italic_k italic_g end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_g = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT italic_Y start_POSTSUBSCRIPT italic_i italic_j italic_g end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq 1\quad\forall i\in[1,\dots,N],\;\forall j" class="ltx_Math" display="inline" id="A0.E4.m2.5"><semantics id="A0.E4.m2.5a"><mrow id="A0.E4.m2.5.5.2" xref="A0.E4.m2.5.5.3.cmml"><mrow id="A0.E4.m2.4.4.1.1" xref="A0.E4.m2.4.4.1.1.cmml"><mi id="A0.E4.m2.4.4.1.1.2" xref="A0.E4.m2.4.4.1.1.2.cmml"></mi><mo id="A0.E4.m2.4.4.1.1.1" xref="A0.E4.m2.4.4.1.1.1.cmml">≤</mo><mn id="A0.E4.m2.4.4.1.1.3" xref="A0.E4.m2.4.4.1.1.3.cmml">1</mn></mrow><mspace id="A0.E4.m2.5.5.2.3" width="1.167em" xref="A0.E4.m2.5.5.3a.cmml"></mspace><mrow id="A0.E4.m2.5.5.2.2" xref="A0.E4.m2.5.5.2.2.cmml"><mrow id="A0.E4.m2.5.5.2.2.4" xref="A0.E4.m2.5.5.2.2.4.cmml"><mo id="A0.E4.m2.5.5.2.2.4.1" rspace="0.167em" xref="A0.E4.m2.5.5.2.2.4.1.cmml">∀</mo><mi id="A0.E4.m2.5.5.2.2.4.2" xref="A0.E4.m2.5.5.2.2.4.2.cmml">i</mi></mrow><mo 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Planning and Execution for Cooperative Assembly">2</a> ensures that each task is assigned exactly one primary robot. Equ. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.E3" title="Equation 3 ‣ -A Details of Task Planning Formulation ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">3</a> assigns a second support robot for each assembly step that needs one. Eqn. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.E4" title="Equation 4 ‣ -A Details of Task Planning Formulation ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">4</a> prevents each robot from being the primary robot and the support robot at the same step. Eqn. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.E5" title="Equation 5 ‣ -A Details of Task Planning Formulation ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">5</a> matches an object of the correct type for each assembly step. Eqn. <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.E6" title="Equation 6 ‣ -A Details of Task Planning Formulation ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">6</a> ensures that each object is used at most once. 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encoding="application/x-tex" id="A0.SS1.p5.1.m1.3c">\forall i\in[1,\dots,N]</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p5.1.m1.3d">∀ italic_i ∈ [ 1 , … , italic_N ]</annotation></semantics></math> and <math alttext="\forall j=[1,\dots,N_{a}-N+1]" class="ltx_Math" display="inline" id="A0.SS1.p5.2.m2.3"><semantics id="A0.SS1.p5.2.m2.3a"><mrow id="A0.SS1.p5.2.m2.3.3" xref="A0.SS1.p5.2.m2.3.3.cmml"><mrow id="A0.SS1.p5.2.m2.3.3.3" xref="A0.SS1.p5.2.m2.3.3.3.cmml"><mo id="A0.SS1.p5.2.m2.3.3.3.1" rspace="0.167em" xref="A0.SS1.p5.2.m2.3.3.3.1.cmml">∀</mo><mi id="A0.SS1.p5.2.m2.3.3.3.2" xref="A0.SS1.p5.2.m2.3.3.3.2.cmml">j</mi></mrow><mo id="A0.SS1.p5.2.m2.3.3.2" xref="A0.SS1.p5.2.m2.3.3.2.cmml">=</mo><mrow id="A0.SS1.p5.2.m2.3.3.1.1" xref="A0.SS1.p5.2.m2.3.3.1.2.cmml"><mo id="A0.SS1.p5.2.m2.3.3.1.1.2" stretchy="false" xref="A0.SS1.p5.2.m2.3.3.1.2.cmml">[</mo><mn id="A0.SS1.p5.2.m2.1.1" xref="A0.SS1.p5.2.m2.1.1.cmml">1</mn><mo id="A0.SS1.p5.2.m2.3.3.1.1.3" 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]</annotation></semantics></math> </div> <div class="ltx_para" id="A0.SS1.p6"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A0.EGx3"> <tbody id="A0.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle z_{i,j}" class="ltx_Math" display="inline" id="A0.E7.m1.2"><semantics id="A0.E7.m1.2a"><msub id="A0.E7.m1.2.3" xref="A0.E7.m1.2.3.cmml"><mi id="A0.E7.m1.2.3.2" xref="A0.E7.m1.2.3.2.cmml">z</mi><mrow id="A0.E7.m1.2.2.2.4" xref="A0.E7.m1.2.2.2.3.cmml"><mi id="A0.E7.m1.1.1.1.1" xref="A0.E7.m1.1.1.1.1.cmml">i</mi><mo id="A0.E7.m1.2.2.2.4.1" xref="A0.E7.m1.2.2.2.3.cmml">,</mo><mi id="A0.E7.m1.2.2.2.2" xref="A0.E7.m1.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.E7.m1.2b"><apply id="A0.E7.m1.2.3.cmml" xref="A0.E7.m1.2.3"><csymbol cd="ambiguous" id="A0.E7.m1.2.3.1.cmml" 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xref="A0.E8.m2.6.6.2.2.2.2.3">𝑚</ci></apply><ci id="A0.E8.m2.6.6.2.2.2.3.cmml" xref="A0.E8.m2.6.6.2.2.2.3">𝑗</ci></apply><apply id="A0.E8.m2.6.6.2.2.3.cmml" xref="A0.E8.m2.6.6.2.2.3"><csymbol cd="ambiguous" id="A0.E8.m2.6.6.2.2.3.1.cmml" xref="A0.E8.m2.6.6.2.2.3">subscript</csymbol><ci id="A0.E8.m2.6.6.2.2.3.2.cmml" xref="A0.E8.m2.6.6.2.2.3.2">𝑧</ci><list id="A0.E8.m2.4.4.2.3.cmml" xref="A0.E8.m2.4.4.2.4"><ci id="A0.E8.m2.3.3.1.1.cmml" xref="A0.E8.m2.3.3.1.1">𝑖</ci><ci id="A0.E8.m2.4.4.2.2.cmml" xref="A0.E8.m2.4.4.2.2">𝑗</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.E8.m2.6c">\displaystyle\geq z_{i,j},\quad Z^{m}_{j}\leq z_{i,j}</annotation><annotation encoding="application/x-llamapun" id="A0.E8.m2.6d">≥ italic_z start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_Z start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≤ italic_z start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1">(8)</td> </tr></tbody> </table> </div> <div class="ltx_para" id="A0.SS1.p7"> <math alttext="z_{ij}" class="ltx_Math" display="inline" id="A0.SS1.p7.1.m1.1"><semantics id="A0.SS1.p7.1.m1.1a"><msub id="A0.SS1.p7.1.m1.1.1" xref="A0.SS1.p7.1.m1.1.1.cmml"><mi id="A0.SS1.p7.1.m1.1.1.2" xref="A0.SS1.p7.1.m1.1.1.2.cmml">z</mi><mrow id="A0.SS1.p7.1.m1.1.1.3" xref="A0.SS1.p7.1.m1.1.1.3.cmml"><mi id="A0.SS1.p7.1.m1.1.1.3.2" xref="A0.SS1.p7.1.m1.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p7.1.m1.1.1.3.1" xref="A0.SS1.p7.1.m1.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p7.1.m1.1.1.3.3" xref="A0.SS1.p7.1.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.1.m1.1b"><apply id="A0.SS1.p7.1.m1.1.1.cmml" xref="A0.SS1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.1.m1.1.1.1.cmml" 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id="A0.SS1.p7.2.m2.1.1.cmml" xref="A0.SS1.p7.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.2.m2.1d">italic_i</annotation></semantics></math> from the task window <math alttext="a_{j}" class="ltx_Math" display="inline" id="A0.SS1.p7.3.m3.1"><semantics id="A0.SS1.p7.3.m3.1a"><msub id="A0.SS1.p7.3.m3.1.1" xref="A0.SS1.p7.3.m3.1.1.cmml"><mi id="A0.SS1.p7.3.m3.1.1.2" xref="A0.SS1.p7.3.m3.1.1.2.cmml">a</mi><mi id="A0.SS1.p7.3.m3.1.1.3" xref="A0.SS1.p7.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.3.m3.1b"><apply id="A0.SS1.p7.3.m3.1.1.cmml" xref="A0.SS1.p7.3.m3.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.3.m3.1.1.1.cmml" xref="A0.SS1.p7.3.m3.1.1">subscript</csymbol><ci id="A0.SS1.p7.3.m3.1.1.2.cmml" xref="A0.SS1.p7.3.m3.1.1.2">𝑎</ci><ci id="A0.SS1.p7.3.m3.1.1.3.cmml" xref="A0.SS1.p7.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.3.m3.1c">a_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="a_{j+N-1}" class="ltx_Math" display="inline" id="A0.SS1.p7.4.m4.1"><semantics id="A0.SS1.p7.4.m4.1a"><msub id="A0.SS1.p7.4.m4.1.1" xref="A0.SS1.p7.4.m4.1.1.cmml"><mi id="A0.SS1.p7.4.m4.1.1.2" xref="A0.SS1.p7.4.m4.1.1.2.cmml">a</mi><mrow id="A0.SS1.p7.4.m4.1.1.3" xref="A0.SS1.p7.4.m4.1.1.3.cmml"><mrow id="A0.SS1.p7.4.m4.1.1.3.2" xref="A0.SS1.p7.4.m4.1.1.3.2.cmml"><mi id="A0.SS1.p7.4.m4.1.1.3.2.2" xref="A0.SS1.p7.4.m4.1.1.3.2.2.cmml">j</mi><mo id="A0.SS1.p7.4.m4.1.1.3.2.1" xref="A0.SS1.p7.4.m4.1.1.3.2.1.cmml">+</mo><mi id="A0.SS1.p7.4.m4.1.1.3.2.3" xref="A0.SS1.p7.4.m4.1.1.3.2.3.cmml">N</mi></mrow><mo id="A0.SS1.p7.4.m4.1.1.3.1" xref="A0.SS1.p7.4.m4.1.1.3.1.cmml">−</mo><mn id="A0.SS1.p7.4.m4.1.1.3.3" xref="A0.SS1.p7.4.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.4.m4.1b"><apply id="A0.SS1.p7.4.m4.1.1.cmml" xref="A0.SS1.p7.4.m4.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.4.m4.1.1.1.cmml" xref="A0.SS1.p7.4.m4.1.1">subscript</csymbol><ci id="A0.SS1.p7.4.m4.1.1.2.cmml" xref="A0.SS1.p7.4.m4.1.1.2">𝑎</ci><apply id="A0.SS1.p7.4.m4.1.1.3.cmml" xref="A0.SS1.p7.4.m4.1.1.3"><minus id="A0.SS1.p7.4.m4.1.1.3.1.cmml" xref="A0.SS1.p7.4.m4.1.1.3.1"></minus><apply id="A0.SS1.p7.4.m4.1.1.3.2.cmml" xref="A0.SS1.p7.4.m4.1.1.3.2"><plus id="A0.SS1.p7.4.m4.1.1.3.2.1.cmml" xref="A0.SS1.p7.4.m4.1.1.3.2.1"></plus><ci id="A0.SS1.p7.4.m4.1.1.3.2.2.cmml" xref="A0.SS1.p7.4.m4.1.1.3.2.2">𝑗</ci><ci id="A0.SS1.p7.4.m4.1.1.3.2.3.cmml" xref="A0.SS1.p7.4.m4.1.1.3.2.3">𝑁</ci></apply><cn id="A0.SS1.p7.4.m4.1.1.3.3.cmml" type="integer" xref="A0.SS1.p7.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.4.m4.1c">a_{j+N-1}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_j + italic_N - 1 end_POSTSUBSCRIPT</annotation></semantics></math>, whereas <math alttext="Z^{M}_{j}" class="ltx_Math" display="inline" id="A0.SS1.p7.5.m5.1"><semantics id="A0.SS1.p7.5.m5.1a"><msubsup id="A0.SS1.p7.5.m5.1.1" xref="A0.SS1.p7.5.m5.1.1.cmml"><mi id="A0.SS1.p7.5.m5.1.1.2.2" xref="A0.SS1.p7.5.m5.1.1.2.2.cmml">Z</mi><mi id="A0.SS1.p7.5.m5.1.1.3" xref="A0.SS1.p7.5.m5.1.1.3.cmml">j</mi><mi id="A0.SS1.p7.5.m5.1.1.2.3" xref="A0.SS1.p7.5.m5.1.1.2.3.cmml">M</mi></msubsup><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.5.m5.1b"><apply id="A0.SS1.p7.5.m5.1.1.cmml" xref="A0.SS1.p7.5.m5.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.5.m5.1.1.1.cmml" xref="A0.SS1.p7.5.m5.1.1">subscript</csymbol><apply id="A0.SS1.p7.5.m5.1.1.2.cmml" xref="A0.SS1.p7.5.m5.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.5.m5.1.1.2.1.cmml" xref="A0.SS1.p7.5.m5.1.1">superscript</csymbol><ci id="A0.SS1.p7.5.m5.1.1.2.2.cmml" xref="A0.SS1.p7.5.m5.1.1.2.2">𝑍</ci><ci id="A0.SS1.p7.5.m5.1.1.2.3.cmml" xref="A0.SS1.p7.5.m5.1.1.2.3">𝑀</ci></apply><ci id="A0.SS1.p7.5.m5.1.1.3.cmml" xref="A0.SS1.p7.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.5.m5.1c">Z^{M}_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.5.m5.1d">italic_Z start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Z^{m}_{j}" class="ltx_Math" display="inline" id="A0.SS1.p7.6.m6.1"><semantics id="A0.SS1.p7.6.m6.1a"><msubsup id="A0.SS1.p7.6.m6.1.1" xref="A0.SS1.p7.6.m6.1.1.cmml"><mi id="A0.SS1.p7.6.m6.1.1.2.2" xref="A0.SS1.p7.6.m6.1.1.2.2.cmml">Z</mi><mi id="A0.SS1.p7.6.m6.1.1.3" xref="A0.SS1.p7.6.m6.1.1.3.cmml">j</mi><mi id="A0.SS1.p7.6.m6.1.1.2.3" xref="A0.SS1.p7.6.m6.1.1.2.3.cmml">m</mi></msubsup><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.6.m6.1b"><apply id="A0.SS1.p7.6.m6.1.1.cmml" xref="A0.SS1.p7.6.m6.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.6.m6.1.1.1.cmml" xref="A0.SS1.p7.6.m6.1.1">subscript</csymbol><apply id="A0.SS1.p7.6.m6.1.1.2.cmml" xref="A0.SS1.p7.6.m6.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.6.m6.1.1.2.1.cmml" xref="A0.SS1.p7.6.m6.1.1">superscript</csymbol><ci id="A0.SS1.p7.6.m6.1.1.2.2.cmml" xref="A0.SS1.p7.6.m6.1.1.2.2">𝑍</ci><ci id="A0.SS1.p7.6.m6.1.1.2.3.cmml" xref="A0.SS1.p7.6.m6.1.1.2.3">𝑚</ci></apply><ci id="A0.SS1.p7.6.m6.1.1.3.cmml" xref="A0.SS1.p7.6.m6.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.6.m6.1c">Z^{m}_{j}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.6.m6.1d">italic_Z start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are the maximum and minimum <math alttext="z_{ij}" class="ltx_Math" display="inline" id="A0.SS1.p7.7.m7.1"><semantics id="A0.SS1.p7.7.m7.1a"><msub id="A0.SS1.p7.7.m7.1.1" xref="A0.SS1.p7.7.m7.1.1.cmml"><mi id="A0.SS1.p7.7.m7.1.1.2" xref="A0.SS1.p7.7.m7.1.1.2.cmml">z</mi><mrow id="A0.SS1.p7.7.m7.1.1.3" xref="A0.SS1.p7.7.m7.1.1.3.cmml"><mi id="A0.SS1.p7.7.m7.1.1.3.2" xref="A0.SS1.p7.7.m7.1.1.3.2.cmml">i</mi><mo id="A0.SS1.p7.7.m7.1.1.3.1" xref="A0.SS1.p7.7.m7.1.1.3.1.cmml">⁢</mo><mi id="A0.SS1.p7.7.m7.1.1.3.3" xref="A0.SS1.p7.7.m7.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.7.m7.1b"><apply id="A0.SS1.p7.7.m7.1.1.cmml" xref="A0.SS1.p7.7.m7.1.1"><csymbol cd="ambiguous" id="A0.SS1.p7.7.m7.1.1.1.cmml" xref="A0.SS1.p7.7.m7.1.1">subscript</csymbol><ci id="A0.SS1.p7.7.m7.1.1.2.cmml" xref="A0.SS1.p7.7.m7.1.1.2">𝑧</ci><apply id="A0.SS1.p7.7.m7.1.1.3.cmml" xref="A0.SS1.p7.7.m7.1.1.3"><times id="A0.SS1.p7.7.m7.1.1.3.1.cmml" xref="A0.SS1.p7.7.m7.1.1.3.1"></times><ci id="A0.SS1.p7.7.m7.1.1.3.2.cmml" xref="A0.SS1.p7.7.m7.1.1.3.2">𝑖</ci><ci id="A0.SS1.p7.7.m7.1.1.3.3.cmml" xref="A0.SS1.p7.7.m7.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.7.m7.1c">z_{ij}</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.7.m7.1d">italic_z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> at each window for all robot <math alttext="i" class="ltx_Math" display="inline" id="A0.SS1.p7.8.m8.1"><semantics id="A0.SS1.p7.8.m8.1a"><mi id="A0.SS1.p7.8.m8.1.1" xref="A0.SS1.p7.8.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="A0.SS1.p7.8.m8.1b"><ci id="A0.SS1.p7.8.m8.1.1.cmml" xref="A0.SS1.p7.8.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A0.SS1.p7.8.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="A0.SS1.p7.8.m8.1d">italic_i</annotation></semantics></math>. </div> </section> <section class="ltx_subsection" id="A0.SS2"> <h3 class="ltx_title ltx_title_subsection"> -B Details of Lego Manipulation Policy </h3> <div class="ltx_para" id="A0.SS2.p1"> Our LEGO manipulation policy is composed of 6 manipulation skills as shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#S1.F1" title="In I Introduction ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">Fig. 1</a>. For the learned force policy (i.e., pick, place-down, handover, place-up), the robot follows a two-step motion sequence - attach, twist to manipulate the LEGO brick. The twist angle and axis of the EOAT are learned with a safe-learning framework following <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#bib.bib30" title="">30</a>]</cite> to ensure the EOAT can successfully attach or release the LEGO brick. For goal-reaching policy (i.e., support-up, support-down), the robot moves to its target pose in a one-step motion. </div> <div class="ltx_para" id="A0.SS2.p2"> APEX-MR requires a reference trajectory for each manipulation policy to check collisions in the TPG construction and shortcutting process. Thus, for each learned force policy, we generate a sequence of subgoals (i.e., robot target pose) that corresponds to each of the two steps, attach and twist. Algorithmically, the EOAT pose in attach subgoal is calculated based on the known LEGO pose in the assembly plate, the specific press side, and the press pose offset for the LEGO brick. The twist is set by rotating the attach subgoal with the learned twist parameters. Then, the two joint space subgoals are computed with inverse kinematics for each robot to determine their feasibility. The target pose of the previous transit task is computed by adding a small z-offset (e.g., 5mm) to the attach subgoal, where the z-axis is orthogonal to the assembly plate. </div> <div class="ltx_para" id="A0.SS2.p3"> For support-up, the EOAT is set to support from below the LEGO brick in the current substructure that would sit below the current brick that is being placed down. Similarly for support-down pose, the EOAT is set to support from the top of the LEGO brick in the current substructure that sits above the current brick that is being placed-up. Then, the feasibility of these goal-reaching policies can be determined from the inverse kinematic for each robot. Also, the goal pose of the previous transit task is computed by adding a small z-offset to the support pose. </div> <div class="ltx_para" id="A0.SS2.p4"> Since the contact models of LEGO bricks are not directly simulated, these manipulation skills are treated as a (sequence of) motion primitive(s) and directed executed similar to transit tasks in simulation. In the real bi-manual manipulation setup, force feedback determines contact location and successful attach positions. Specifically, for the learned force policy (i.e., pick, place down, handover, place up), the robot EOAT moves toward the direction of the attach pose in the z-axis until a predetermined force threshold (e.g., 15N). Once this force threshold is reached, the stopping point determines the new twist axis, and a new twist pose is calculated for the robot. For the goal-reaching policy (i.e., support-up, support-down), the robot EOAT moves in the direction of the support pose until a small predetermined force threshold is reached (e.g., 0.1N), and the stopping point becomes the actual support pose. The robot moves at low speed during the support skill to ensure that our robot can stop in time (<math alttext="\leq 30ms" class="ltx_Math" display="inline" id="A0.SS2.p4.1.m1.1"><semantics id="A0.SS2.p4.1.m1.1a"><mrow id="A0.SS2.p4.1.m1.1.1" xref="A0.SS2.p4.1.m1.1.1.cmml"><mi id="A0.SS2.p4.1.m1.1.1.2" xref="A0.SS2.p4.1.m1.1.1.2.cmml"></mi><mo id="A0.SS2.p4.1.m1.1.1.1" xref="A0.SS2.p4.1.m1.1.1.1.cmml">≤</mo><mrow id="A0.SS2.p4.1.m1.1.1.3" xref="A0.SS2.p4.1.m1.1.1.3.cmml"><mn id="A0.SS2.p4.1.m1.1.1.3.2" xref="A0.SS2.p4.1.m1.1.1.3.2.cmml">30</mn><mo id="A0.SS2.p4.1.m1.1.1.3.1" xref="A0.SS2.p4.1.m1.1.1.3.1.cmml">⁢</mo><mi id="A0.SS2.p4.1.m1.1.1.3.3" xref="A0.SS2.p4.1.m1.1.1.3.3.cmml">m</mi><mo id="A0.SS2.p4.1.m1.1.1.3.1a" xref="A0.SS2.p4.1.m1.1.1.3.1.cmml">⁢</mo><mi id="A0.SS2.p4.1.m1.1.1.3.4" xref="A0.SS2.p4.1.m1.1.1.3.4.cmml">s</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A0.SS2.p4.1.m1.1b"><apply id="A0.SS2.p4.1.m1.1.1.cmml" xref="A0.SS2.p4.1.m1.1.1"><leq id="A0.SS2.p4.1.m1.1.1.1.cmml" xref="A0.SS2.p4.1.m1.1.1.1"></leq><csymbol cd="latexml" id="A0.SS2.p4.1.m1.1.1.2.cmml" xref="A0.SS2.p4.1.m1.1.1.2">absent</csymbol><apply id="A0.SS2.p4.1.m1.1.1.3.cmml" xref="A0.SS2.p4.1.m1.1.1.3"><times id="A0.SS2.p4.1.m1.1.1.3.1.cmml" xref="A0.SS2.p4.1.m1.1.1.3.1"></times><cn id="A0.SS2.p4.1.m1.1.1.3.2.cmml" type="integer" xref="A0.SS2.p4.1.m1.1.1.3.2">30</cn><ci id="A0.SS2.p4.1.m1.1.1.3.3.cmml" xref="A0.SS2.p4.1.m1.1.1.3.3">𝑚</ci><ci id="A0.SS2.p4.1.m1.1.1.3.4.cmml" xref="A0.SS2.p4.1.m1.1.1.3.4">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.SS2.p4.1.m1.1c">\leq 30ms</annotation><annotation encoding="application/x-llamapun" id="A0.SS2.p4.1.m1.1d">≤ 30 italic_m italic_s</annotation></semantics></math>) to avoid damaging the stability of the existing structure. </div> <figure class="ltx_figure" id="A0.F15"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="252" id="A0.F15.g1" src="extracted/6295254/figs/sequential_v_sync.png" width="598"/> <figcaption class="ltx_caption">Figure 15: Converting a sequential task plan to a synchronous task plan. Sequential task plan produces turn-based, sequential motion plan, whereas in synchronous task plan, each robot waits for other robots to complete the current task before starting the next task as the same time. If it is unsafe to complete two tasks simulateously due to collision, tasks scheduled later in the sequential plan would wait until earlier robot completes its task and it becomes safe to proceed simulateously. </figcaption> </figure> <figure class="ltx_table" id="A0.T1"> <figcaption class="ltx_caption ltx_centering">TABLE I: Wall clock time for planning, TPG construction, and shortcutting across environments (mean ± std)</figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="A0.T1.4"> <tr class="ltx_tr" id="A0.T1.4.1"> <td class="ltx_td ltx_border_tt" id="A0.T1.4.1.1" style="padding-left:4.0pt;padding-right:4.0pt;"></td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.2" style="padding-left:4.0pt;padding-right:4.0pt;">Version</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.3" style="padding-left:4.0pt;padding-right:4.0pt;">Cliff</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.4" style="padding-left:4.0pt;padding-right:4.0pt;">Branched Stairs</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.5" style="padding-left:4.0pt;padding-right:4.0pt;">Faucet</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.6" style="padding-left:4.0pt;padding-right:4.0pt;">Bridge</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.7" style="padding-left:4.0pt;padding-right:4.0pt;">Fish</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.8" style="padding-left:4.0pt;padding-right:4.0pt;">Chair</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.9" style="padding-left:4.0pt;padding-right:4.0pt;">Vessel</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="A0.T1.4.1.10" style="padding-left:4.0pt;padding-right:4.0pt;">Guitar</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_tt" id="A0.T1.4.1.11" style="padding-left:4.0pt;padding-right:4.0pt;">RSS</td> </tr> <tr class="ltx_tr" id="A0.T1.4.2"> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.1" style="padding-left:4.0pt;padding-right:4.0pt;"># Objects</td> <td class="ltx_td" id="A0.T1.4.2.2" style="padding-left:4.0pt;padding-right:4.0pt;"></td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.3" style="padding-left:4.0pt;padding-right:4.0pt;">11</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.4" style="padding-left:4.0pt;padding-right:4.0pt;">14</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.5" style="padding-left:4.0pt;padding-right:4.0pt;">14</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.6" style="padding-left:4.0pt;padding-right:4.0pt;">38</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.7" style="padding-left:4.0pt;padding-right:4.0pt;">29</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.8" style="padding-left:4.0pt;padding-right:4.0pt;">258</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.9" style="padding-left:4.0pt;padding-right:4.0pt;">36</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.2.10" style="padding-left:4.0pt;padding-right:4.0pt;">24</td> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A0.T1.4.2.11" style="padding-left:4.0pt;padding-right:4.0pt;">47</td> </tr> <tr class="ltx_tr" id="A0.T1.4.3"> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.1" rowspan="2" style="padding-left:4.0pt;padding-right:4.0pt;"> Task Planning </td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.2" style="padding-left:4.0pt;padding-right:4.0pt;">APEX-MR</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.3" style="padding-left:4.0pt;padding-right:4.0pt;">0.6 ± 0.0</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.4" style="padding-left:4.0pt;padding-right:4.0pt;">1.1 ± 0.0</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.5" style="padding-left:4.0pt;padding-right:4.0pt;">1.1 ± 0.0</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.6" style="padding-left:4.0pt;padding-right:4.0pt;">9.7 ± 0.5</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.7" style="padding-left:4.0pt;padding-right:4.0pt;">6.5 ± 0.2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.8" style="padding-left:4.0pt;padding-right:4.0pt;">13.0 ± 0.2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.9" style="padding-left:4.0pt;padding-right:4.0pt;">4.9 ± 0.6</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.3.10" style="padding-left:4.0pt;padding-right:4.0pt;">2.3 ± 0.1</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="A0.T1.4.3.11" style="padding-left:4.0pt;padding-right:4.0pt;">14.5 ± 0.5</td> </tr> <tr class="ltx_tr" id="A0.T1.4.4"> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.1" style="padding-left:4.0pt;padding-right:4.0pt;">Sync</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.2" style="padding-left:4.0pt;padding-right:4.0pt;">1.5 ± 0.0</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.3" style="padding-left:4.0pt;padding-right:4.0pt;">1.9 ± 0.0</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.4" style="padding-left:4.0pt;padding-right:4.0pt;">2.3 ± 0.0</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.5" style="padding-left:4.0pt;padding-right:4.0pt;">10.8 ± 0.5</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.6" style="padding-left:4.0pt;padding-right:4.0pt;">8.8 ± 0.2</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.7" style="padding-left:4.0pt;padding-right:4.0pt;">27.1 ± 0.3</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.8" style="padding-left:4.0pt;padding-right:4.0pt;">6.7 ± 0.5</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.4.9" style="padding-left:4.0pt;padding-right:4.0pt;">3.9 ± 0.0</td> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A0.T1.4.4.10" style="padding-left:4.0pt;padding-right:4.0pt;">18.1 ± 0.5</td> </tr> <tr class="ltx_tr" id="A0.T1.4.5"> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.1" rowspan="2" style="padding-left:4.0pt;padding-right:4.0pt;"> Motion Planning </td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.2" style="padding-left:4.0pt;padding-right:4.0pt;">APEX-MR</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.3" style="padding-left:4.0pt;padding-right:4.0pt;">11.9 ± 0.3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.4" style="padding-left:4.0pt;padding-right:4.0pt;">14.7 ± 0.3</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.5" style="padding-left:4.0pt;padding-right:4.0pt;">14.3 ± 0.2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.6" style="padding-left:4.0pt;padding-right:4.0pt;">52.9 ± 0.8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.7" style="padding-left:4.0pt;padding-right:4.0pt;">37.8 ± 0.6</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.8" style="padding-left:4.0pt;padding-right:4.0pt;">220.3 ± 3.5</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.9" style="padding-left:4.0pt;padding-right:4.0pt;">30.5 ± 0.7</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.5.10" style="padding-left:4.0pt;padding-right:4.0pt;">20.2 ± 0.1</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="A0.T1.4.5.11" style="padding-left:4.0pt;padding-right:4.0pt;">72.5 ± 4.6</td> </tr> <tr class="ltx_tr" id="A0.T1.4.6"> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.1" style="padding-left:4.0pt;padding-right:4.0pt;">Sync</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.2" style="padding-left:4.0pt;padding-right:4.0pt;">26.7 ± 0.5</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.3" style="padding-left:4.0pt;padding-right:4.0pt;">36.4 ± 1.2</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.4" style="padding-left:4.0pt;padding-right:4.0pt;">37.2 ± 0.5</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.5" style="padding-left:4.0pt;padding-right:4.0pt;">124.8 ± 2.8</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.6" style="padding-left:4.0pt;padding-right:4.0pt;">96.7 ± 1.0</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.7" style="padding-left:4.0pt;padding-right:4.0pt;">501.9 ± 5.7</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.8" style="padding-left:4.0pt;padding-right:4.0pt;">77.4 ± 0.8</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.6.9" style="padding-left:4.0pt;padding-right:4.0pt;">46.8 ± 0.3</td> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A0.T1.4.6.10" style="padding-left:4.0pt;padding-right:4.0pt;">180.9 ± 7.6</td> </tr> <tr class="ltx_tr" id="A0.T1.4.7"> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.1" rowspan="2" style="padding-left:4.0pt;padding-right:4.0pt;"> TPG Construction </td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.2" style="padding-left:4.0pt;padding-right:4.0pt;">APEX-MR</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.3" style="padding-left:4.0pt;padding-right:4.0pt;">9.1 ± 0.0</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.4" style="padding-left:4.0pt;padding-right:4.0pt;">9.6 ± 0.1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.5" style="padding-left:4.0pt;padding-right:4.0pt;">10.3 ± 0.1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.6" style="padding-left:4.0pt;padding-right:4.0pt;">35.7 ± 0.6</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.7" style="padding-left:4.0pt;padding-right:4.0pt;">29.2 ± 0.1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.8" style="padding-left:4.0pt;padding-right:4.0pt;">341.9 ± 1.4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.9" style="padding-left:4.0pt;padding-right:4.0pt;">16.2 ± 0.8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.7.10" style="padding-left:4.0pt;padding-right:4.0pt;">16.7 ± 0.1</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="A0.T1.4.7.11" style="padding-left:4.0pt;padding-right:4.0pt;">52.2 ± 0.4</td> </tr> <tr class="ltx_tr" id="A0.T1.4.8"> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.1" style="padding-left:4.0pt;padding-right:4.0pt;">Sync</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.2" style="padding-left:4.0pt;padding-right:4.0pt;">9.4 ± 0.2</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.3" style="padding-left:4.0pt;padding-right:4.0pt;">9.7 ± 0.0</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.4" style="padding-left:4.0pt;padding-right:4.0pt;">10.7 ± 0.3</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.5" style="padding-left:4.0pt;padding-right:4.0pt;">30.0 ± 0.6</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.6" style="padding-left:4.0pt;padding-right:4.0pt;">27.0 ± 0.2</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.7" style="padding-left:4.0pt;padding-right:4.0pt;">345.5 ± 4.3</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.8" style="padding-left:4.0pt;padding-right:4.0pt;">16.8 ± 0.2</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.8.9" style="padding-left:4.0pt;padding-right:4.0pt;">16.8 ± 0.0</td> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A0.T1.4.8.10" style="padding-left:4.0pt;padding-right:4.0pt;">53.8 ± 1.0</td> </tr> <tr class="ltx_tr" id="A0.T1.4.9"> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.1" rowspan="2" style="padding-left:4.0pt;padding-right:4.0pt;"> Total Planning </td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.2" style="padding-left:4.0pt;padding-right:4.0pt;">APEX-MR</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.3" style="padding-left:4.0pt;padding-right:4.0pt;">21.6 ± 0.4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.4" style="padding-left:4.0pt;padding-right:4.0pt;">25.3 ± 0.4</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.5" style="padding-left:4.0pt;padding-right:4.0pt;">25.7 ± 0.2</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.6" style="padding-left:4.0pt;padding-right:4.0pt;">98.2 ± 1.8</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.7" style="padding-left:4.0pt;padding-right:4.0pt;">73.5 ± 0.9</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.8" style="padding-left:4.0pt;padding-right:4.0pt;">575.2 ± 5.1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.9" style="padding-left:4.0pt;padding-right:4.0pt;">51.6 ± 2.1</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A0.T1.4.9.10" style="padding-left:4.0pt;padding-right:4.0pt;">9.2 ± 0.2</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="A0.T1.4.9.11" style="padding-left:4.0pt;padding-right:4.0pt;">139.3 ± 5.5</td> </tr> <tr class="ltx_tr" id="A0.T1.4.10"> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.1" style="padding-left:4.0pt;padding-right:4.0pt;">Sync</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.2" style="padding-left:4.0pt;padding-right:4.0pt;">37.5 ± 0.8</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.3" style="padding-left:4.0pt;padding-right:4.0pt;">48.0 ± 1.2</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.4" style="padding-left:4.0pt;padding-right:4.0pt;">50.1 ± 0.8</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.5" style="padding-left:4.0pt;padding-right:4.0pt;">165.6 ± 3.8</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.6" style="padding-left:4.0pt;padding-right:4.0pt;">132.5 ± 1.4</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.7" style="padding-left:4.0pt;padding-right:4.0pt;">874.6 ± 10.3</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.8" style="padding-left:4.0pt;padding-right:4.0pt;">100.9 ± 1.5</td> <td class="ltx_td ltx_align_center" id="A0.T1.4.10.9" style="padding-left:4.0pt;padding-right:4.0pt;">67.5 ± 0.3</td> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A0.T1.4.10.10" style="padding-left:4.0pt;padding-right:4.0pt;">252.8 ± 9.1</td> </tr> <tr class="ltx_tr" id="A0.T1.4.11"> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.1" style="padding-left:4.0pt;padding-right:4.0pt;">TPG Shortcut</td> <td class="ltx_td ltx_border_bb ltx_border_t" id="A0.T1.4.11.2" style="padding-left:4.0pt;padding-right:4.0pt;"></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.3" style="padding-left:4.0pt;padding-right:4.0pt;">20</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.4" style="padding-left:4.0pt;padding-right:4.0pt;">20</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.5" style="padding-left:4.0pt;padding-right:4.0pt;">20</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.6" style="padding-left:4.0pt;padding-right:4.0pt;">20</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.7" style="padding-left:4.0pt;padding-right:4.0pt;">20</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.8" style="padding-left:4.0pt;padding-right:4.0pt;">60</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.9" style="padding-left:4.0pt;padding-right:4.0pt;">60</td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.10" style="padding-left:4.0pt;padding-right:4.0pt;">60</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_bb ltx_border_t" id="A0.T1.4.11.11" style="padding-left:4.0pt;padding-right:4.0pt;">60</td> </tr> </table> </figure> </section> <section class="ltx_subsection" id="A0.SS3"> <h3 class="ltx_title ltx_title_subsection"> -C Numerical Results </h3> <div class="ltx_para" id="A0.SS3.p1"> Table <a class="ltx_ref" href="https://arxiv.org/html/2503.15836v1#A0.T1" title="Table. I ‣ -B Details of Lego Manipulation Policy ‣ APEX-MR: Multi-Robot Asynchronous Planning and Execution for Cooperative Assembly">I</a> shows the wall clock time for planning, TPG construction, and shortcutting across environments averaged over 4 random seeds. </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Mar 20 04:18:20 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/">LaTeXML<img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>