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Zaragoza, Spain; 2019. pp. 1587-90."><meta data-n-head="ssr" name="citation_journal_title" content="Intelligence & Robotics"><meta data-n-head="ssr" name="citation_publisher" content="OAE Publishing Inc."><meta data-n-head="ssr" name="citation_title" content="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping"><meta data-n-head="ssr" name="citation_publication_date" content="2022/10/12"><meta data-n-head="ssr" name="citation_online_date" content="2022/10/12"><meta data-n-head="ssr" name="citation_doi" content="10.20517/ir.2022.21"><meta data-n-head="ssr" name="citation_volume" content="2"><meta data-n-head="ssr" name="citation_issue" content="4"><meta data-n-head="ssr" name="citation_firstpage" content="333"><meta data-n-head="ssr" name="citation_lastpage" content="354"><meta data-n-head="ssr" name="citation_author" content="Timothy Sellers"><meta data-n-head="ssr" name="citation_author" content="Tingjun Lei"><meta data-n-head="ssr" name="citation_author" content="Chaomin Luo"><meta data-n-head="ssr" name="citation_author" content="Gene Eu Jan"><meta data-n-head="ssr" name="citation_author" content="Junfeng Ma"><meta data-n-head="ssr" name="prism.issn" content="ISSN 2770-3541 (Online)"><meta data-n-head="ssr" name="prism.publicationName" content="OAE Publishing Inc."><meta data-n-head="ssr" name="prism.publicationDate" content="2022-10-12"><meta data-n-head="ssr" name="prism.volume" content="2"><meta data-n-head="ssr" name="prism.section" content="Research Article"><meta data-n-head="ssr" name="prism.startingPag" content="333"><meta data-n-head="ssr" name="prism.url" content="https://www.oaepublish.com/articles/ir.2022.21"><meta data-n-head="ssr" name="prism.doi" content="doi:10.20517/ir.2022.21"><meta data-n-head="ssr" name="citation_journal_abbrev" content="ir"><meta data-n-head="ssr" name="citation_article_type" content="Research Article"><meta data-n-head="ssr" name="citation_language" content="en"><meta data-n-head="ssr" name="citation_doi" content="10.20517/ir.2022.21"><meta data-n-head="ssr" name="citation_id" content="ir.2022.21"><meta data-n-head="ssr" name="citation_issn" content="ISSN 2770-3541 (Online)"><meta data-n-head="ssr" name="citation_publication_date" content="2022-10-12"><meta data-n-head="ssr" name="citation_author_institution" content="Correspondence to: Prof. Chaomin Luo, Department of Electrical and Computer Engineering, Mississippi State University, 406 Hardy Road, Mississippi State, MS 39762, USA. E-mail: Chaomin.Luo@ece.msstate.edu; ORCID: 0000-0002-7578-3631"><meta data-n-head="ssr" name="citation_pdf_url" content="https://f.oaes.cc/xmlpdf/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170.pdf"><meta data-n-head="ssr" name="citation_fulltext_html_url" content="https://www.oaepublish.com/articles/ir.2022.21"><meta data-n-head="ssr" name="fulltext_pdf" content="https://f.oaes.cc/xmlpdf/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170.pdf"><meta data-n-head="ssr" name="twitter:type" content="article"><meta data-n-head="ssr" name="twitter:title" content="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping"><meta data-n-head="ssr" name="twitter:description" content="The paper proposes an ANS algorithm to find a node in the graph to connect waypoints that are utilized in the safety-aware multi-waypoint navigation and mapping by an improved PSO and GVD model."><meta data-n-head="ssr" name="og:url" content="https://www.oaepublish.com/articles/ir.2022.21"><meta 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data-v-6dffe839><h1 id="art_title" class="art_title2" data-v-6dffe839><span data-v-6dffe839>A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping</span></h1> <div class="art_seltte" data-v-6dffe839><i class="iconfont icon-yuyan" data-v-6dffe839></i> <div class="el-select selete_language" data-v-6dffe839><div class="el-input el-input--suffix"><input type="text" readonly="readonly" autocomplete="off" placeholder="" class="el-input__inner"><span class="el-input__suffix"><span class="el-input__suffix-inner"><i class="el-select__caret el-input__icon el-icon-arrow-up"></i></span></span></div><div class="el-select-dropdown el-popper" style="min-width:;display:none;"><div class="el-scrollbar" style="display:none;"><div class="el-select-dropdown__wrap el-scrollbar__wrap el-scrollbar__wrap--hidden-default"><ul class="el-scrollbar__view el-select-dropdown__list"><li class="el-select-dropdown__item" data-v-6dffe839><span>English</span></li><li 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" class="Crossref" data-v-6dffe839> <a href="/articles//citation/" target="_blank" style="color:#4475e1;margin-left:1px;" data-v-6dffe839>10</a> </span></div> <div id=" authorString" class="article-authors" data-v-6dffe839><span class="authors_item" data-v-6dffe839><div affNumList="" data-v-dc220f24 data-v-6dffe839><span class="pos_re" data-v-dc220f24><div role="tooltip" id="el-popover-350" aria-hidden="true" class="el-popover el-popper" style="width:300px;display:none;"><h3 class="font16 no_sup" style="color:#333;margin-bottom:20px;" data-v-dc220f24>Timothy Sellers<sup>1</sup></h3> <div class="Aff_current font14 no_sup" data-v-dc220f24><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>1</sup></label><addr-line>Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>2</sup></label><addr-line>Department of Electrical Engineering, National Taipei University and Tainan National University of the Arts, Taipei 72045, Taiwan.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>3</sup></label><addr-line>Department of Industrial and Systems Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div></div> <i class="close_btn el-icon-close" data-v-dc220f24></i> <a href="https://scholar.google.com/scholar?q=Timothy Sellers" target="_blank" data-v-dc220f24><button type="button" class="el-button el-button--primary el-button--mini" data-v-dc220f24><span>Google Scholar</span></button></a></div><span class="el-popover__reference-wrapper"><span class="author_name" data-v-dc220f24>Timothy Sellers<sup>1</sup></span></span></span></div>     <i data-v-6dffe839> , </i></span><span class="authors_item" data-v-6dffe839><div affNumList="" data-v-dc220f24 data-v-6dffe839><span class="pos_re" data-v-dc220f24><div role="tooltip" id="el-popover-1233" aria-hidden="true" class="el-popover el-popper" style="width:300px;display:none;"><h3 class="font16 no_sup" style="color:#333;margin-bottom:20px;" data-v-dc220f24>Tingjun Lei<sup>1</sup></h3> <div class="Aff_current font14 no_sup" data-v-dc220f24><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>1</sup></label><addr-line>Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>2</sup></label><addr-line>Department of Electrical Engineering, National Taipei University and Tainan National University of the Arts, Taipei 72045, Taiwan.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>3</sup></label><addr-line>Department of Industrial and Systems Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div></div> <i class="close_btn el-icon-close" data-v-dc220f24></i> <a href="https://scholar.google.com/scholar?q=Tingjun Lei" target="_blank" data-v-dc220f24><button type="button" class="el-button el-button--primary el-button--mini" data-v-dc220f24><span>Google Scholar</span></button></a></div><span class="el-popover__reference-wrapper"><span class="author_name" data-v-dc220f24>Tingjun Lei<sup>1</sup></span></span></span></div>     <i data-v-6dffe839> , ... </i></span><span class="authors_item" data-v-6dffe839></span><span class="authors_item" data-v-6dffe839></span><span class="authors_item" data-v-6dffe839><div affNumList="" data-v-dc220f24 data-v-6dffe839><span class="pos_re" data-v-dc220f24><div role="tooltip" id="el-popover-1237" aria-hidden="true" class="el-popover el-popper" style="width:300px;display:none;"><h3 class="font16 no_sup" style="color:#333;margin-bottom:20px;" data-v-dc220f24>Junfeng Ma<sup>3</sup></h3> <div class="Aff_current font14 no_sup" data-v-dc220f24><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>1</sup></label><addr-line>Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>2</sup></label><addr-line>Department of Electrical Engineering, National Taipei University and Tainan National University of the Arts, Taipei 72045, Taiwan.</addr-line></div></div><div data-v-dc220f24><div class="author_cont" data-v-dc220f24><label><sup>3</sup></label><addr-line>Department of Industrial and Systems Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div></div> <i class="close_btn el-icon-close" data-v-dc220f24></i> <a href="https://scholar.google.com/scholar?q=Junfeng Ma" target="_blank" data-v-dc220f24><button type="button" class="el-button el-button--primary el-button--mini" data-v-dc220f24><span>Google Scholar</span></button></a></div><span class="el-popover__reference-wrapper"><span class="author_name" data-v-dc220f24>Junfeng Ma<sup>3</sup></span></span></span></div>     </span> <button type="button" class="el-button el-button--primary el-button--mini" data-v-6dffe839><i class="el-icon-plus"></i><span>Show Authors</span></button></div> <div class="article-header-info" data-v-6dffe839><div data-v-6dffe839> <i>Intell Robot</i> 2022;2(4):333-54.</div> <div class="mgt5" data-v-6dffe839><a href="https://doi.org/10.20517/ir.2022.21" target="_blank" data-v-6dffe839>10.20517/ir.2022.21</a> | <span class="btn_link" data-v-6dffe839>© The Author(s) 2022</span></div></div> <div class="top_btn_box" data-v-6dffe839><div class="btn_item" data-v-6dffe839><i class="el-icon-caret-right" data-v-6dffe839></i><span data-v-6dffe839>Author Information</span></div> <div class="btn_item" data-v-6dffe839><i class="el-icon-caret-right" data-v-6dffe839></i><span data-v-6dffe839>Article Notes</span></div> <div class="btn_item" data-v-6dffe839><i class="el-icon-caret-right" data-v-6dffe839></i><span data-v-6dffe839>Cite This Article</span></div></div> <div class="author_box" style="display:none;" data-v-6dffe839><div data-v-6dffe839><div data-v-6dffe839><label><sup>1</sup></label><addr-line>Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div><div data-v-6dffe839><div data-v-6dffe839><label><sup>2</sup></label><addr-line>Department of Electrical Engineering, National Taipei University and Tainan National University of the Arts, Taipei 72045, Taiwan.</addr-line></div></div><div data-v-6dffe839><div data-v-6dffe839><label><sup>3</sup></label><addr-line>Department of Industrial and Systems Engineering, Mississippi State University, Mississippi State, MS 39759, USA.</addr-line></div></div> <div class="CorrsPlus" data-v-6dffe839><div data-v-6dffe839><span id="cirrsMail" data-v-6dffe839>Correspondence to: Prof. Chaomin Luo, Department of Electrical and Computer Engineering, Mississippi State University, 406 Hardy Road, Mississippi State, MS 39762, USA. E-mail: <email>Chaomin.Luo@ece.msstate.edu</email>; ORCID: 0000-0002-7578-3631</span></div></div></div> <div class="notes_box" style="display:none;" data-v-6dffe839><div class="articleDate mag_top10" data-v-6dffe839><span><b>Received:</b> 20 Jul 2022 | </span><span><b>First Decision:</b> 12 Aug 2022 | </span><span><b>Revised:</b> 18 Aug 2022 | </span><span><b>Accepted:</b> 25 Aug 2022 | </span><span><b>Published:</b> 12 Oct 2022</span></div> <div class="articleDate" data-v-6dffe839><span><b>Academic Editor:</b> Simon X. Yang | </span><span><b>Copy Editor:</b> Jia-Xin Zhang | </span><span><b>Production Editor:</b> Jia-Xin Zhang</span></div></div> <div class="article_bg" data-v-6dffe839><h2 id="art_Abstract" data-v-6dffe839>Abstract</h2> <div id="seo_des" class="article_Abstract mag_btn10" data-v-6dffe839><p>Autonomous robot multi-waypoint navigation and mapping have been demanded in many real-world applications found in search and rescue (SAR), environmental exploration, and disaster response. Many solutions to this issue have been discovered via graph-based methods in need of switching the robotos trajectory between the nodes and edges within the graph to create a trajectory for waypoint-to-waypoint navigation. However, studies of how waypoints are locally bridged to nodes or edges on the graphs have not been adequately undertaken. In this paper, an adjacent node selection (ANS) algorithm is developed to implement such a protocol to build up regional path from waypoints to nearest nodes or edges on the graph. We propose this node selection algorithm along with the generalized Voronoi diagram (GVD) and Improved Particle Swarm Optimization (IPSO) algorithm as well as a local navigator to solve the safety-aware concurrent graph-based multi-waypoint navigation and mapping problem. Firstly, GVD is used to form a Voronoi diagram in an obstacle populated environment to construct safety-aware routes. Secondly, the sequence of multiple waypoints is created by the IPSO algorithm to minimize the total travelling cost. Thirdly, while the robot attempts to visit multiple waypoints, it traverses along the edges of the GVD to plan a collision-free trajectory. The regional path from waypoints to the nearest nodes or edges needs to be created to join the trajectory by the proposed ANS algorithm. Finally, a sensor-based histogram local reactive navigator is adopted for moving obstacle avoidance while local maps are constructed as the robot moves. An improved <i>B</i>-spline curve-based smooth scheme is adopted that further refines the trajectory and enables the robot to be navigated smoothly. Simulation and comparison studies validate the effectiveness and robustness of the proposed model.</p></div>  <h2 id="art_Keywords" data-v-6dffe839>Keywords</h2> <div class="article_Abstract" data-v-6dffe839><span data-v-6dffe839><span data-v-6dffe839>Adjacent node selection (ANS) algorithm</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>safety-aware roads</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>path planning</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>multiple-waypoint optimization</span><i data-v-6dffe839>, </i></span><span data-v-6dffe839><span data-v-6dffe839>navigation and mapping</span></span></div></div> <div class="MoComment" data-v-6dffe839><div class="top_banner" data-v-6dffe839><div class="oae_header" data-v-6dffe839>Author's Talk</div> <div class="line" data-v-6dffe839></div> <div class="img_box" data-v-6dffe839><img src="https://i.oaes.cc/uploads/20240205/b263c1d7fa6a447f89d0b42385dcee47.jpg" alt="" data-itemid="5170" data-itemhref="https://v1.oaepublish.com/files/talkvideo/5170.mp4" data-itemimg="https://i.oaes.cc/uploads/20240205/b263c1d7fa6a447f89d0b42385dcee47.jpg" data-v-6dffe839> <i data-itemid="5170" data-itemhref="https://v1.oaepublish.com/files/talkvideo/5170.mp4" data-itemimg="https://i.oaes.cc/uploads/20240205/b263c1d7fa6a447f89d0b42385dcee47.jpg" class="bo_icon" data-v-6dffe839></i></div></div>  <div class="article_link" data-v-6dffe839><span data-v-6dffe839><a href="https://f.oaes.cc/xmlpdf/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170_down.pdf?v=98" data-v-6dffe839><b data-v-6dffe839><i class="icon-download icon_right4" data-v-6dffe839></i> Download PDF</b></a></span> <span data-v-6dffe839><i class="comment-l icon-commentl iconfont icon_right4" data-v-6dffe839></i> <b data-v-6dffe839>0</b></span> <span data-v-6dffe839><span data-v-6dffe839><div role="tooltip" id="el-popover-8184" aria-hidden="true" class="el-popover el-popper" style="width:170px;display:none;"><div class="icon_share" style="text-align:right;margin:0;" data-v-6dffe839><a href="http://pinterest.com/pin/create/button/?url=&media=&description=https://www.oaepublish.com/articles/" target="_blank" class="pinterest-sign" data-v-6dffe839><i class="iconfont icon-pinterest" data-v-6dffe839></i></a> <a href="https://www.facebook.com/sharer/sharer.php?u=https://www.oaepublish.com/articles/" target="_blank" class="facebook-sign" data-v-6dffe839><i aria-hidden="true" class="iconfont icon-facebook" data-v-6dffe839></i></a> <a href="https://twitter.com/intent/tweet?url=https://www.oaepublish.com/articles/" target="_blank" class="twitter-sign" data-v-6dffe839><i class="iconfont icon-tuite1" data-v-6dffe839></i></a> <a href="https://www.linkedin.com/shareArticle?url=https://www.oaepublish.com/articles/" target="_blank" class="linkedin-sign" data-v-6dffe839><i class="iconfont icon-linkedin" data-v-6dffe839></i></a></div> </div><span class="el-popover__reference-wrapper"><button type="button" class="el-button colorddd el-button--text el-button--mini" data-v-6dffe839><span><i class="icon-fenxiang iconfont icon_right4" data-v-6dffe839></i> <b data-v-6dffe839>0</b></span></button></span></span></span> <span data-v-6dffe839><span class="no_zan" data-v-6dffe839><i class="icon-like-line icon_right4" data-v-6dffe839></i> <i class="num_n" data-v-6dffe839><b data-v-6dffe839>4</b></i></span></span></div></div> <div id="artDivBox" class="art_cont" data-v-6dffe839><div id="sec11" class="article-Section"><h2 >1. INTRODUCTION</h2><p class="">Robotics system has been applied to numerous fields, such as transportation <sup>[<a href="#b1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b1">1</a>]</sup>, healthcare service <sup>[<a href="#b2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b2">2</a>, <a href="#b3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b3">3</a>]</sup>, agriculture <sup>[<a href="#b4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b4">4</a>]</sup>, manufacturing <sup>[<a href="#b5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b5">5</a>]</sup>, <i>etc.</i>, in recent years. Robot navigation is one of the fundamental components in robotic systems, which includes multi-waypoint navigation system <sup>[<a href="#b6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b6">6</a>-<a href="#b8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b8">8</a>]</sup>. As an increasing demand and limited onboard resources for autonomous robots, it requires the ability to visit several targets in one mission to optimize multiple objectives, including time, robot travel distance minimization, and spatial optimization <sup>[<a href="#b9" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b9">9</a>-<a href="#b15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b15">15</a>]</sup>. For example, due to a global pandemic, the world struggled to sanitize heavily populated areas, such as airports, hospitals, and educational buildings. Autonomous robots with multi-waypoint navigation systems can effectively sanitize all targeted areas without endangering the workers <sup>[<a href="#b14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b14">14</a>, <a href="#b16" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b16">16</a>]</sup>. As well as in agriculture management, multi-waypoint strategies allow the robotic system to navigate and survey multiple areas to assist in production and collection.</p><p class="">In order to employ robotic systems in real-world scenarios, one critical factor is to develop autonomous robot multi-waypoint navigation and mapping system <sup>[<a href="#b17" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b17">17</a>]</sup>. In order to solve the autonomous robot navigation problem, countless algorithms have been developed, such as graph-based<sup>[<a href="#b18" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b18">18</a>, <a href="#b19" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b19">19</a>]</sup>, ant colony optimization (ACO) <sup>[<a href="#b20" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b20">20</a>-<a href="#b22" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b22">22</a>]</sup>, bat-pigeon algorithm (BPA) <sup>[<a href="#b23" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b23">23</a>]</sup>, neural networks <sup>[<a href="#b24" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b24">24</a>-<a href="#b26" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b26">26</a>]</sup>, fuzzy logic <sup>[<a href="#b27" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b27">27</a>]</sup>, artificial potential field (APF) <sup>[<a href="#b28" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b28">28</a>]</sup>, sampling-based strategy <sup>[<a href="#b14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b14">14</a>, <a href="#b29" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b29">29</a>]</sup>, hybrid algorithms<sup>[<a href="#b30" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b30">30</a>]</sup>, task planning algorithm <sup>[<a href="#b31" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b31">31</a>]</sup>, <i>etc</i>. Chen <i>et al.</i> produced a hybrid graph-based reinforcement learning architecture to develop a method for robot navigation in crowds <sup>[<a href="#b18" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b18">18</a>]</sup>. Luo <i>et al.</i> proposed an improved vehicle navigation method, which utilizes a heading-enabled ACO algorithm to improve trajectory towards the target<sup>[<a href="#b20" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b20">20</a>]</sup>. Lei <i>et al.</i> developed a Bat-Pigeon algorithm with the ability to adjust the speed navigation of autonomous vehicles <sup>[<a href="#b23" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b23">23</a>]</sup>. Luo <i>et al.</i> developed the model for multiple robots complete coverage navigation while using a bio-inspired neural network to dynamically avoid obstacles<sup>[<a href="#b32" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b32">32</a>]</sup>. Na and Oh established a hybrid control system for autonomous mobile robot navigation that utilizes a neural network for environment classification and behavior-based control method to mimic the human steering commands<sup>[<a href="#b25" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b25">25</a>]</sup>.Lazreg and Benamrane <sup>[<a href="#b27" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b27">27</a>]</sup> developed a neuro-fuzzy inference system associated with a Particle Swarm Optimization (PSO) method for robot path planning using a variety of sensors to control the speed and position of a robot.Jensen-Nau <i>et al.</i><sup>[<a href="#b28" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b28">28</a>]</sup> integrated a Voronoi-based path generation algorithm and an artificial potential field path planning method, in which the latter is capable of establishing a path in an unknown environment in real-time for robot path planning and obstacle avoidance.Penicka and Scaramuzza <sup>[<a href="#b14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b14">14</a>]</sup> developed a sampling-based multi-waypoint minimum-time path planning model that allows obstacle avoidance in cluttered environments.Ortiz and Yu <sup>[<a href="#b30" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b30">30</a>]</sup> proposed a sliding control method in combination with simultaneous localization and mapping (SLAM) method to overcome the bounded uncertainties problem, which utilizes a genetic algorithm to improve path planning capabilities. Bernardo <i>et al.</i> proposed a task planning method for home environment ontology to translate tasks given by other robots or humans into feasible tasks for another robot agent<sup>[<a href="#b31" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b31">31</a>]</sup>.</p><p class="">In robotic path planning, one of the special topics is autonomous robot multi-waypoint navigation which has been studied for many years. For instance, Shair <i>et al.</i> proposed a model for real-world waypoint navigation using a variety of sensors for accurate environmental analysis<sup>[<a href="#b33" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b33">33</a>]</sup>. The system is designed utilizing the wide area augmentation system (WAAS) and the European geostationary navigation overlay service (EGNOS) for GPS in combination with aerial images to provide valuable positioning data to the system.Yang <sup>[<a href="#b34" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b34">34</a>]</sup> brought about a multi-waypoint navigation system based on terrestrial signals of opportunity (SOPs) transmitters, which has the ability to operate in environments that are not available to global navigation satellite systems (GNSS) for unmanned aerial vehicles (UAV).Janoš <i>et al.</i> proposed a sampling-based multi-waypoint path planner, space-filling forest method to solve the problem of finding collision-free trajectories while the sequence of waypoints is formed by multiple trees<sup>[<a href="#b29" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b29">29</a>]</sup>.However, the aforementioned approaches have not taken into account the safety of the robot during navigation. In real-world applications, an autonomous vehicle has odometry errors during operation. The safety-aware road model is developed utilizing the Generalized Voronoi diagram (GVD) approach. Once the safety-aware roads are defined, a particle swarm optimization (PSO) algorithm-based multi-waypoint path planning algorithm is proposed to visit each waypoint in an explicated sequence while simultaneously avoiding obstacles. In this paper, the Adjacent Node Selection (ANS) algorithm is developed to select the closest nodes on the safety-aware roads to generate the final collision-free trajectories with minimal distance. Furthermore, in our hybrid algorithm, we utilize a histogram-based reactive local navigator to avoid dynamic and unknown obstacles within the workspace. Through all of these methods, an effective and efficient safety-aware multiple waypoint navigation model was established, which has been validated by both simulations and comparison studies.</p><p class="">This paper proposes an Adjacent Node Selection (ANS) algorithm for obtaining an optimal access node into graph-based maps. To the best of our knowledge, there are no known similar algorithms that improve the paths created from the waypoint to the graph. The ANS algorithm can be applied to any graph-based mapping environment, which improves the various graph-based models used for autonomous robotic path planning systems. In finding an access point into the graph utilizing one of the nodes in the system, the ANS algorithm conducts point-to-point selection in dense obstacle field of environments to obtain a node to gain access to the graph that forms a resumable path from a waypoint to the graph. The algorithm's overall goal is to find a node in a graph-based map and shorten the overall path length from each waypoint to waypoint, and the waypoint to the graph.</p><p class="">One can see in <a href="#Figure1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure1">Figure 1</a> the overall framework of our proposed model. Initially, the model is provided with a global map of its environment and the location of each target waypoint. The GVD utilizes the global map to construct the safety-aware roads, which are used to guide the robot throughout the environment. The targeted waypoint locations are used as input to the improved PSO (IPSO) algorithm that act as our waypoint sequencing module, which is utilized to find a near-optimal sequence to visit each waypoint. The ANS algorithm utilizes the output from the previous stages. The ANS algorithm finds the best node for each waypoint to use as an access point to the graph. If there is no direct path from one waypoint to a node in the graph, the algorithm conduces point-to-point navigation with nodes within a specified range, which will be explained in later sections of the paper. Once each waypoint has found its access point, the safety-aware path is constructed, which is then applied to our path smoothing algorithm. Finally, we utilize a reactive local navigation system to detect obstacles autonomously and simultaneously build a map of the environment along the generated path.</p><div class="Figure-block" id="Figure1"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure1" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-1.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 1. Illustration of the proposed model framework.</p></div></div><p class="">The main contributions of this paper with this framework of concurrent multi-waypoint navigation and mapping with collision avoidance as are summarized as follows: (1) An adjacent node selection (ANS) algorithm is proposed to build up regional bridges from waypoints to nodes or edges on the graph in multi-waypoint navigation and mapping; (2) a concurrent multi-waypoint navigation and mapping framework of an autonomous robot is developed by the generalized Voronoi diagram (GVD) and IPSO algorithm as well as a local navigator; (3) a GVD method is employed to plan safety-aware trajectories in an obstacle populated environment, while the sequence of multiple waypoints is created by the IPSO algorithm to minimize the total distance cost; (4) a sensor-based histogram robot reactive navigator coupled with map building is adopted for moving obstacle avoidance and local mapping. An improved <inline-formula><tex-math id="M2">$$\mathcal{B}$$</tex-math></inline-formula>-spline curve-based speed modulation module is adopted for smoother navigation.</p><p class="">The structure of this paper is as follows. Section II presents the safety-aware model that constructs the safety-conscious roads for robot traversal. Section III describes the proposed adjacent node selection algorithm to find an access node into the graph-based map. Section IV, the improved PSO-based (IPSO) multi-waypoint navigation model for waypoint sequencing is explained. Section V illustrates the reactive local navigator for real-time workspace building and obstacle avoidance. Section VI depicts simulation results and comparative analyses. Several important properties of the proposed framework are summarized in Section VII.</p></div><div id="sec12" class="article-Section"><h2 >2. SAFETY-AWARE MODEL</h2><p class="">Computational geometry has exceedingly studied the Voronoi diagrams (VD) model, which is an elemental data structure used as a minimizing diagram of a finite set of continuous functions <sup>[<a href="#b28" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b28">28</a>, <a href="#b35" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b35">35</a>]</sup>. The minimized function defines the distantance to an object within the workspace. The VD model decomposes the workspace into several regions, each consisting of the points near a given object than the others. Let <inline-formula><tex-math id="M3">$$ \mathcal{G} = \{g_1\cdots, g_n\} $$</tex-math></inline-formula> be a set of points of <inline-formula><tex-math id="M4">$$ \mathbb{R}^d $$</tex-math></inline-formula> to each <inline-formula><tex-math id="M5">$$ g_i $$</tex-math></inline-formula> associated with a specific Voronoi region <inline-formula><tex-math id="M6">$$ V(g_i) $$</tex-math></inline-formula>. This can be expressed by the following equation<sup>[<a href="#b36" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b36">36</a>, <a href="#b37" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b37">37</a>]</sup>:</p><p class=""><div class="disp-formula"><label>(1)</label><tex-math id="E1"> $$ \begin{equation} V(g_i) = \{x \in \mathbb{R}^d: \| x-g_i \| \le \| x-g_j \|, \forall j \le n \} \end{equation} $$ </tex-math></div></p><p class="">The intersection of <inline-formula><tex-math id="M7">$$ n - 1 $$</tex-math></inline-formula> half spaces can be denoted by the region <inline-formula><tex-math id="M8">$$ V(g_i) $$</tex-math></inline-formula>. Each half space holds a point <inline-formula><tex-math id="M9">$$ g_i $$</tex-math></inline-formula> along with another point of <inline-formula><tex-math id="M10">$$ \mathcal{G} $$</tex-math></inline-formula>. The regions <inline-formula><tex-math id="M11">$$ V(g_i) $$</tex-math></inline-formula> are convex polyhedrons due to the bisectors acting as hyperplanes between each region. The generalized Voronoi diagram (GVD) is a modified version of the VD model defined as the set of points Euclidean distance from two obstacles<sup>[<a href="#b37" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b37">37</a>]</sup>. The workspace is represented as a graph by the GVD model consisting of nodes, edges, and vertices <sup>[<a href="#b28" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b28">28</a>]</sup>.</p><p class="">The proposed ANS algorithm utilizes nodes and edges obtained from the GVD illustrated in <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2A</a>. The nodes in the GVD act as junctions to connect waypoints, whereas the edges are used to determine the search range in view of waypoints. Solely the nodes in the search range are calculated, other than the entire workspace, thus reducing the computational expense. <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2B</a> reveals that the range of some edges fails to evaluate how spare of the workspace is configurated. These edges with short lengths are distractors for evaluating and computing the search range, thus need to be eliminated shown in <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2B</a>. In our ANS algorithm, those edges with the shortest 10% are eliminated, while the rest of the edges are averaged to compute the radius of the search range. One circumstance is exhibited in <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2C</a>, in which some nodes fail to fall into the search range once the excessively short edges are excluded. The original search space in solid circles and modified search space in dashed circles based on the ANS algorithm are shown in <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2C</a>. As may be seen, after excessively short edges are effectively eliminated, an appropriate search range is achieved.</p><div class="Figure-block" id="Figure2"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure2" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-2.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 2. Illustration of the ANS algorithm and the safety-aware model by creation of the GVD. (A) Depicts the equidistance property of the nodes and edges in the GVD model. (B) illustrates how we remove extraneous edge distances, which are small edges within the graph that reduce the search space or range in the ANS algorithm. (C) illustrates how the removal of those extraneous edges improves the range and how an access node to the graph can be found.</p></div></div><p class="">GVD nodes form the Euclidean distance between two or more obstacles, while the edges are the junction of two nodes that depict the distance between each neighboring node to another. Vertices are the connection points between three or more nodes. Using these features from the GVD model, an obstacle-free path with our safety-aware model is effectively created. The safety-aware model is constructed by inputting an image and extracting all significant features from the image, thus allowing the model to construct a map from the input image. The safety-aware roads are the clearest path between obstacles that occupy the available space in the map, which can be seen in <a href="#Figure2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure2">Figure 2</a>.</p><p class="">In this paper, in order to explain our proposed ANS algorithm, how the nodes are determined and constructed with the GVD will be presented in some detail. We will introduce some definitions and notations. Lee and Drysdale <sup>[<a href="#b36" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b36">36</a>]</sup> derive four basic definitions for determining the optimal placement of edges and nodes within the GVD graph.</p><p class="">DEFINITION 1. A closed line segment <inline-formula><tex-math id="M12">$$ M $$</tex-math></inline-formula> consists of two endpoints <inline-formula><tex-math id="M13">$$ \alpha $$</tex-math></inline-formula> and <inline-formula><tex-math id="M14">$$ \gamma $$</tex-math></inline-formula>. A straight line is denoted by (<inline-formula><tex-math id="M15">$$ \alpha $$</tex-math></inline-formula>, <inline-formula><tex-math id="M16">$$ \gamma $$</tex-math></inline-formula>), which is also known as an open segment. Elements within the derivation are referred to as points or segments. The straight line containing <inline-formula><tex-math id="M17">$$ M $$</tex-math></inline-formula> is denoted by <inline-formula><tex-math id="M18">$$ \overleftrightarrow{M} $$</tex-math></inline-formula>. The same line directed from <inline-formula><tex-math id="M19">$$ \alpha $$</tex-math></inline-formula> to <inline-formula><tex-math id="M20">$$ \gamma $$</tex-math></inline-formula> is denoted by <inline-formula><tex-math id="M21">$$ \vec M $$</tex-math></inline-formula>.</p><p class="">DEFINITION 2. The projection <inline-formula><tex-math id="M22">$$ \sigma(\epsilon, M) $$</tex-math></inline-formula> of a point <inline-formula><tex-math id="M23">$$ q $$</tex-math></inline-formula> onto a closed segment <inline-formula><tex-math id="M24">$$ M $$</tex-math></inline-formula>, is the intersection of <inline-formula><tex-math id="M25">$$ \overleftrightarrow{M} $$</tex-math></inline-formula> and is perpendicular to <inline-formula><tex-math id="M26">$$ M $$</tex-math></inline-formula> and passing through <inline-formula><tex-math id="M27">$$ \epsilon $$</tex-math></inline-formula>.</p><p class="">DEFINITION 3. The distance between <inline-formula><tex-math id="M28">$$ \omega (\epsilon, M) $$</tex-math></inline-formula> is a point <inline-formula><tex-math id="M29">$$ \epsilon $$</tex-math></inline-formula> and a closed segment <inline-formula><tex-math id="M30">$$ M $$</tex-math></inline-formula> in the Euclidean metric is defined as the distance <inline-formula><tex-math id="M31">$$ \omega(\epsilon, \sigma(\epsilon, M)) $$</tex-math></inline-formula> between the point <inline-formula><tex-math id="M32">$$ \epsilon $$</tex-math></inline-formula> and its projection onto <inline-formula><tex-math id="M33">$$ M $$</tex-math></inline-formula> if <inline-formula><tex-math id="M34">$$ \sigma(\epsilon, M)) $$</tex-math></inline-formula> belongs to <inline-formula><tex-math id="M35">$$ M $$</tex-math></inline-formula> and is <inline-formula><tex-math id="M36">$$ \min $$</tex-math></inline-formula><inline-formula><tex-math id="M37">$$ (\omega(\epsilon, \alpha), \omega(\epsilon, \gamma)) $$</tex-math></inline-formula> otherwise. In other words, <inline-formula><tex-math id="M38">$$ \omega(\epsilon, M) = $$</tex-math></inline-formula><inline-formula><tex-math id="M39">$$ \min_{u \in M} (\epsilon, u ) $$</tex-math></inline-formula>. The point of <inline-formula><tex-math id="M40">$$ M $$</tex-math></inline-formula>, which is closest to <inline-formula><tex-math id="M41">$$ \epsilon $$</tex-math></inline-formula>, is called the image <inline-formula><tex-math id="M42">$$ I(\epsilon, M) $$</tex-math></inline-formula> of <inline-formula><tex-math id="M43">$$ \epsilon $$</tex-math></inline-formula> on <inline-formula><tex-math id="M44">$$ M $$</tex-math></inline-formula>.</p><p class="">DEFINITION 4. The bisector <inline-formula><tex-math id="M45">$$ \beta(c_{i}, c_{j}) $$</tex-math></inline-formula> of two elements {<inline-formula><tex-math id="M46">$$ c_{i} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M47">$$ c_{j} $$</tex-math></inline-formula>} is the locus of points equidistant from <inline-formula><tex-math id="M48">$$ c_{i} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M49">$$ c_{j} $$</tex-math></inline-formula>. The bisector <inline-formula><tex-math id="M50">$$ \beta(Z, Q) $$</tex-math></inline-formula> of two sets of elements <inline-formula><tex-math id="M51">$$ Z $$</tex-math></inline-formula> and <inline-formula><tex-math id="M52">$$ Q $$</tex-math></inline-formula> is defined to be the locus of points equidistant from <inline-formula><tex-math id="M53">$$ Z $$</tex-math></inline-formula> and <inline-formula><tex-math id="M54">$$ Q $$</tex-math></inline-formula>, where the distance <inline-formula><tex-math id="M55">$$ \omega(\epsilon, Z) $$</tex-math></inline-formula> between a point <inline-formula><tex-math id="M56">$$ \epsilon $$</tex-math></inline-formula> and a set of elements <inline-formula><tex-math id="M57">$$ Z $$</tex-math></inline-formula> is defined to be <inline-formula><tex-math id="M58">$$ \min_{c \in Z}\omega(\epsilon, c ) $$</tex-math></inline-formula>. The bisector <inline-formula><tex-math id="M59">$$ \beta(c_{i}, c_{j}) $$</tex-math></inline-formula> is said to be oriented if a direction is imposed upon it so that elements <inline-formula><tex-math id="M60">$$ c_{i} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M61">$$ c_{j} $$</tex-math></inline-formula> lie to the left and the right of it, respectively. An oriented bisector <inline-formula><tex-math id="M62">$$ \beta(Z, Q) $$</tex-math></inline-formula> is defined similarly.</p><p class="">Utilizing these four definitions, we can easily establish {edges} {among} spaces or obstacles, which uses the equidistant to create an optimal edge that lies directly between the two spaces. As seen in the above {definitions, } we can also create edges between irregular-shaped spaces {and} obstacles.</p><p class="">To characterize the GVD, we expect the robot to be operated at a point within the workspace, a <inline-formula><tex-math id="M63">$$ W $$</tex-math></inline-formula>, which is populated by convex obstacles <inline-formula><tex-math id="M64">$$ C_{1}, . . . , C_{n} $$</tex-math></inline-formula>. Non-convex obstacles are displayed as the union of convex shapes. The distance between a point and an obstacle is the minimal distance between the point and all points of the obstacle. The distance function, and its "gradient" are represented as:</p><p class=""><div class="disp-formula"><label>(2)</label><tex-math id="E2"> $$ \begin{equation} d_i(x) = min_{c_{0} \in C_{i}} \| x-c_0 \| \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(3)</label><tex-math id="E3"> $$ \begin{equation} \nabla d_i(x) = \frac{x - c_{0}} {\| x-c_0 \|} \end{equation} $$ </tex-math></div></p><p class="">where in Equation (2) <inline-formula><tex-math id="M65">$$ d_i $$</tex-math></inline-formula> is the distance to obstacle <inline-formula><tex-math id="M66">$$ C_i $$</tex-math></inline-formula> from a point <inline-formula><tex-math id="M67">$$ x $$</tex-math></inline-formula>, and in Equation (3) <inline-formula><tex-math id="M68">$$ \nabla d_i(x) $$</tex-math></inline-formula> is the unit vector in the direction from <inline-formula><tex-math id="M69">$$ x $$</tex-math></inline-formula> to <inline-formula><tex-math id="M70">$$ c_0 $$</tex-math></inline-formula>, where <inline-formula><tex-math id="M71">$$ c_0 $$</tex-math></inline-formula> is the closest point to <inline-formula><tex-math id="M72">$$ x $$</tex-math></inline-formula> in <inline-formula><tex-math id="M73">$$ C_i $$</tex-math></inline-formula>. The essential structure block of the GVD is the arrangement of points equidistant to two sets <inline-formula><tex-math id="M74">$$ C_i $$</tex-math></inline-formula> and <inline-formula><tex-math id="M75">$$ C_j $$</tex-math></inline-formula>, with the end goal that each set in this set is the minimal distance to the obstacles <inline-formula><tex-math id="M76">$$ C_i $$</tex-math></inline-formula> and <inline-formula><tex-math id="M77">$$ C_j $$</tex-math></inline-formula> than other obstacles. This type of structure is known as the two-equidistant face,</p><p class=""><div class="disp-formula"><label>(4)</label><tex-math id="E4"> $$ \begin{equation} f(i, j)=\left\{\begin{array}{c} x \in \mathbb{R}^{m}: 0 \leq d_{i}(x)=d_{h}(x)\\ \forall d_{i} \neq i, j \\ \nabla d_{i}(x) \neq \nabla d_{j}(x) \end{array}\right. \end{equation} $$ </tex-math></div></p><p class="">Each face has a co-dimension in the ambient space, which causes the two-equidistant faces to be seen as one-dimensional. The intersection of both faces forms the GVD and is denoted by the following equation:</p><p class=""><div class="disp-formula"><label>(5)</label><tex-math id="E5"> $$ \begin{equation} GVD=\bigcup\limits_{i=1}^{n-1} \bigcup\limits_{j=i+1}^{n} f(i, j) \end{equation} $$ </tex-math></div></p></div><div id="sec13" class="article-Section"><h2 >3. ADJACENT NODE SELECTION ALGORITHM</h2><p class="">The details of the adjacent node selection algorithm are shown in <a href="#Figure3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure3">Figure 3</a>. Within the search range obtained, we applyIPSO-based path planning algorithm to generate the connection path from the waypoint to all the potential nodes in the range. The connection path is planned in the grid-based map as shown in <a href="#Figure3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure3">Figure 3A</a>. The search range in the larger map is shown in <a href="#Figure3" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure3">Figure 3B</a>, which is also a part of <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6</a>. Finally, with the formation of a collision-free path to the adjacent nodes from the two waypoints, we obtain the optimal path selection by calculating the overall path length <inline-formula><tex-math id="M78">$$ \mathcal{L}_{c} + \mathcal{L}_{e} $$</tex-math></inline-formula>.</p><div class="Figure-block" id="Figure3"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure3" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-3.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 3. Details of the Adjacent Node Selection (ANS) algorithm. (A) The IPSO-based connection path planning in the search range. (B) The search range determination and node selection inside. (C) The final generated path with the minimum path length.</p></div></div><p class="">To show the necessity of IPSO-based path planning from waypoints to adjacent nodes, a more specific scenario is shown in <a href="#Figure4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure4">Figure 4</a>. In <a href="#Figure4" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure4">Figure 4A</a>, all straight lines connecting nodes and waypoints are separated by obstacles. Among them, points <inline-formula><tex-math id="M79">$$ i $$</tex-math></inline-formula> and point <inline-formula><tex-math id="M80">$$ j $$</tex-math></inline-formula> are located in our search range. If we only rely on the Euclidean distance between node and waypoint, it can be found that the point <inline-formula><tex-math id="M81">$$ i $$</tex-math></inline-formula> is closer to the waypoint. However, after considering the path with obstacle avoidance, the path for point <inline-formula><tex-math id="M82">$$ i $$</tex-math></inline-formula> to the waypoint is longer. To sum up, the obtained search range with ANS algorithm and the IPSO-based path planning algorithm can reduce the computational cost while ensuring a short and safe trajectory. The details of the procedure are described in Algorithm 2.</p><div class="Figure-block" id="Figure4"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure4" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-4.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 4. Illustration of ANS method within a more specific sense, where an obstacle obstructing the connection path. (A) The multiple connection paths have been obstructed by the obstacles. (B) It selects the nodes in the defined range. (C) It conducts IPSO point-to-point algorithm to achieve the optimal path to the selected node.</p></div></div><p class="">We carry out further discussions on our proposed ANS algorithm. <a href="#Figure5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure5">Figure 5A</a> and <a href="#Figure5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure5">Figure 5B</a> are parts of <a href="#Figure11" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure11">Figure 11</a> and <a href="#Figure13" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure13">Figure 13</a> (enclosed in pink dashed boxes), respectively. As shown in <a href="#Figure5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure5">Figure 5A</a>, the solid circles represent the search radius of the waypoints in the simulation and the red solid dots depict the nodes in the workspace. In <a href="#Figure11" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure11">Figure 11</a>, we end up with the trajectory that follows the generated safe-aware road. However, if the space in the map is more sparse, our search radius <inline-formula><tex-math id="M83">$$ \mathcal{R} $$</tex-math></inline-formula> may increase to <inline-formula><tex-math id="M84">$$ \mathcal{R}' $$</tex-math></inline-formula>, which may achieve the waypoints in their respective search spaces. Thus, instead of following the safe-awareness road, a new connection path is obtained through the improved PSO algorithm directly, as shown in dashed lines in <a href="#Figure5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure5">Figure 5A</a>. Moreover, the sparseness of the overall workspace may not represent the complexity of local obstacles. Therefore, the choice of the radius of the search space may require more mathematical proof and analysis.</p><div class="Figure-block" id="Figure5"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure5" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-5.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 5. Illustration of the ANS algorithm analysis. (A) From <a href="#Figure11" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure11">Figure 11</a> it is enclosed by a pink dashed box. (B) From <a href="#Figure13" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure13">Figure 13</a> it is enclosed by a pink dashed box.</p></div></div><div class="Figure-block" id="Figure6"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure6" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-6.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 6. Illustration of the Adjacent Node Selection (ANS) algorithm. (A) The workspace with nodes, edges and waypoints. (B) The node selection in the search range. (C) The final generated path.</p></div></div><div class="Figure-block" id="Figure7"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure7" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-7.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 7. Illustration of the B-spline function. (A) The improved segmented B-spline curve. (B) The same path smoothed by the fundamental B-spline and the improved B-spline function.</p></div></div><div class="Figure-block" id="Figure8"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure8" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-8.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 8. Illustration of how the VHF uses a probability along with histogram-based grid to detect and build a map simultaneously.</p></div></div><div class="Figure-block" id="Figure9"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure9" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-9.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 9. Robot sensor configuration for multi-waypoint navigation and mapping.</p></div></div><div class="Figure-block" id="Figure10"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure10" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-10.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 10. Illustration of the path created from the other models<sup>[<a href="#b43" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b43">43</a>]</sup>. (A) It depicts the path created by Zhang <i>et al</i>.'s model by the green lines (redrawn by Zhang <i>et al</i>., 2021<sup>[<a href="#b43" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b43">43</a>]</sup>). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles.</p></div></div><div class="Figure-block" id="Figure11"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure11" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-11.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 11. Illustration of the path created from the other models<sup>[<a href="#b44" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b44">44</a>]</sup>. (A) It depicts the path created by Asl and Taghirad model by the green lines (redrawn by Asl and Taghirad, 2019<sup>[<a href="#b44" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b44">44</a>]</sup>). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles.</p></div></div><div class="Figure-block" id="Figure12"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure12" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-12.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 12. Illustration of the path created from the compared models<sup>[<a href="#b45" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b45">45</a>]</sup>. (A) It depicts the path created by Zhuang <i>et al</i>.'s model by the green lines (redrawn from Zhuang <i>et al</i>., 2021<sup>[<a href="#b45" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b45">45</a>]</sup>). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles.</p></div></div><div class="Figure-block" id="Figure13"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure13" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-13.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 13. Illustration of the path created from the compared models<sup>[<a href="#b46" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b46">46</a>]</sup>. (A) It depicts the path created by Vonásek's model shown by the green lines (redrawn from Vonásek and Penicka, 2019<sup>[<a href="#b46" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b46">46</a>]</sup>). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles. (C) It depicts how the B-spline curve is applied to the known path, which smooths and reduces the path for a local navigator to traverse.</p></div></div><p class="">With the increasing radius of the potential search range, more nodes are applicable for selection, such as the nodes connected with dashed lines in <a href="#Figure5" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure5">Figure 5B</a>. Enlarging the search space may avoid some unnecessary detours and give a shorter path. Therefore, the trade-off between path length and safety of the autonomous robot still requires more consideration.</p></div><div id="sec14" class="article-Section"><h2 >4. IMPROVED PSO-BASED MULTI-WAYPOINT NAVIGATION</h2><p class="">The particle swarm optimization (PSO) algorithm is a swarm-based bio-inspired algorithm based on the behavioral observation of birds. It uses an iterative methodology to optimize randomly initialized particles to define a path from the initial position to the goal <sup>[<a href="#b27" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b27">27</a>]</sup>. In this section an improved PSO (IPSO) algorithm by introduction of a weighted particles is addressed to resolve the multi-waypoint sequence issue.</p><div id="sec22" class="article-Section"><h3 >4.1. Multi-waypoint visiting sequence</h3><p class="">In real-world scenarios, one important factor is that the GPS coordinates provide portions for the multiple waypoints. Traveling from one waypoint to another, the distance between them determines their associated cost. The primary purpose of traveling from one waypoint to another is to simultaneously find the minimal cost of all generated trajectories. Using the coordinates of each waypoint and the PSO algorithm, the minimal-distance path can be found within the environment. The PSO algorithm finds the best waypoint visiting sequence by initializing randomized particles. The algorithm denotes the local best position as {<inline-formula><tex-math id="M85">$$ x^b $$</tex-math></inline-formula>} and the global best position as {<inline-formula><tex-math id="M86">$$ x^g $$</tex-math></inline-formula>}. Then by taking advantage of a fitness function, the algorithm guides each particle towards the local and global best positions. The particle velocities are updated as follows:</p><p class=""><div class="disp-formula"><label>(6)</label><tex-math id="E6"> $$ \begin{equation} \begin{array}{l} v_{p}(t+1)=v_{p}(t)+ \alpha_{1}\omega_{1}[ {x^b_{p}(t)}-x_{p}(t)]+ \alpha_{2}\omega_{2}[ {x^g_{p}(t)}-x_{p}(t)] \end{array} \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(7)</label><tex-math id="E7"> $$ \begin{equation} \begin{array}{l} {x_{p}^{(t+1)}=x_{p}^{(t)}+ v_{p}^{(t+1)}} \end{array} \end{equation} $$ </tex-math></div></p><p class="">where, <inline-formula><tex-math id="M87">$$ v_p(t) $$</tex-math></inline-formula> represents the velocity of particle <inline-formula><tex-math id="M88">$$ p $$</tex-math></inline-formula> at instant <inline-formula><tex-math id="M89">$$ t $$</tex-math></inline-formula>, {<inline-formula><tex-math id="M90">$$ x_p(t) $$</tex-math></inline-formula>} is the position of particle <inline-formula><tex-math id="M91">$$ p $$</tex-math></inline-formula> at instant <inline-formula><tex-math id="M92">$$ t $$</tex-math></inline-formula>, <inline-formula><tex-math id="M93">$$ \alpha_{1} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M94">$$ \alpha_{2} $$</tex-math></inline-formula> are the positive acceleration constants used to scale the contribution of cognitive and social components. <inline-formula><tex-math id="M95">$$ \omega_{1} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M96">$$ \omega_{2} $$</tex-math></inline-formula> are the uniform random number between 0 and 1. {<inline-formula><tex-math id="M97">$$ x^b_p(t) $$</tex-math></inline-formula>} is the best position the particle <inline-formula><tex-math id="M98">$$ p $$</tex-math></inline-formula> achieved up to instant <inline-formula><tex-math id="M99">$$ t $$</tex-math></inline-formula> at current iteration, {<inline-formula><tex-math id="M100">$$ x^g_p(t) $$</tex-math></inline-formula>} is the {global} position that any of <inline-formula><tex-math id="M101">$$ p $$</tex-math></inline-formula>'s neighbors has reached up to instant <inline-formula><tex-math id="M102">$$ t $$</tex-math></inline-formula>. However, if a particle <inline-formula><tex-math id="M103">$$ p_i $$</tex-math></inline-formula> lies close to the <inline-formula><tex-math id="M104">$$ x^b_p(t) $$</tex-math></inline-formula> and <inline-formula><tex-math id="M105">$$ x^g_p(t) $$</tex-math></inline-formula>, only one term guides the <inline-formula><tex-math id="M106">$$ p_i $$</tex-math></inline-formula> to search the potential solution.The optimization process in our navigation issue is more likely trapped in local minima. Thus, an improved PSO algorithm is utilized to provide a more promising search direction for all particles during the optimization process.</p><p class=""><div class="disp-formula"><label>(8)</label><tex-math id="E8"> $$ \begin{equation} x^{w}=\sum\limits_{i=1}^{P} \bar{\alpha}_{i}^{w} {x^b_i(t)} \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(9)</label><tex-math id="E9"> $$ \begin{equation} \bar{\alpha}_{i}^{W}=\frac{\hat{\alpha}_{i}^{W}}{\sum_{j=1}^{P} \hat{\alpha}_{i}^{W}} \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(10)</label><tex-math id="E10"> $$ \begin{equation} \begin{aligned} &\hat{\alpha}_{i}^{W}=\frac{\max _{1 \leq k \leq M}\left(\mathcal{F}\left( {x^b_k(t)}\right)\right)-\mathcal{F}\left( {x^b_i(t)}\right)+\varepsilon}{\max _{1 \leq k \leq M}\left(\mathcal{F}\left( {x^b_k(t)}\right)\right)-\min _{1 \leq k \leq M}\left(\mathcal{F}\left( {x^b_k(t)}\right)\right)+\varepsilon}, \quad i=1, 2, \ldots, M, \end{aligned} \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M107">$$ \varepsilon $$</tex-math></inline-formula> is a positive constant, <inline-formula><tex-math id="M108">$$ \hat{\alpha} $$</tex-math></inline-formula> is the weighted constant of each particle. <inline-formula><tex-math id="M109">$$ \mathcal{F}(\cdot) $$</tex-math></inline-formula> is the fitness function. The worst and the best fitness values of all personal best particles are represented by <inline-formula><tex-math id="M110">$$ \max _{1 \leq k \leq M}(\mathcal{F}\left(x_{k}^{P}\right)) $$</tex-math></inline-formula> and <inline-formula><tex-math id="M111">$$ \min _{1 \leq k \leq M}(\mathcal{F}\left(x_{k}^{P}\right)) $$</tex-math></inline-formula>, respectively.</p><p class="">The order is optimized through this method, in which each waypoint is visited. A sequence of particles are initialized to compose a population in the original PSO algorithm. A possible optimal solution to an optimization issue in our multi-waypoint sequence is discovered by one particle in the PSO. This particle indicates a possible optimal solution to the multi-waypoint navigation issue and moves to explore an optimal solution in a certain search space. In this paper, a weighted particle is introduced into a swarm to suggest a more reasonable search direction for all the particles. As a result, the best position of particle and neighbor guides the particle to move along the corrected direction for better covergence.</p><p class="">The multi-waypoint visiting sequence problem can be used to solve the transportation planning problem and Covid-19 disinfection robot path planning in hospitals, in which agents (vehicles) need to be delivered as well as the overall cost and time need to be minimized <sup>[<a href="#b14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b14">14</a>]</sup>. The algorithm of the improved PSO to finding multi-waypoint visiting sequence is explained in Algorithm 1. The objective of the algorithm is to minimize the total trajectory length of Cartesian coordinates <inline-formula><tex-math id="M112">$$ (X_n, Y_n) $$</tex-math></inline-formula> of waypoints given.</p><p class="">In the IPSO model the <inline-formula><tex-math id="M113">$$ \mathcal{P}_A $$</tex-math></inline-formula> represents the particle agent, which is denoted by the <inline-formula><tex-math id="M114">$$ \mathcal{P}_A $$</tex-math></inline-formula> : {<inline-formula><tex-math id="M115">$$ \mathcal{P}_b $$</tex-math></inline-formula>, <inline-formula><tex-math id="M116">$$ \mathcal{P}_g $$</tex-math></inline-formula>, <inline-formula><tex-math id="M117">$$ \mathcal{N} $$</tex-math></inline-formula>}. <inline-formula><tex-math id="M118">$$ \mathcal{N} $$</tex-math></inline-formula> holds a group of particle agents which are predetermined as neighbors of <inline-formula><tex-math id="M119">$$ \mathcal{P}_A $$</tex-math></inline-formula>. The <inline-formula><tex-math id="M120">$$ \mathcal{P}_A $$</tex-math></inline-formula> are defined by the following parameters.</p><p class="">(1) Each <inline-formula><tex-math id="M121">$$ \mathcal{P}_A $$</tex-math></inline-formula> requests each neighbor's current personal best location.</p><p class="">(2) Each <inline-formula><tex-math id="M122">$$ \mathcal{P}_A $$</tex-math></inline-formula> returns its current personal best to neighbors.</p><p class="">(3) Obtains the center location of each surrounding cluster.</p><p class="">(4) Determines if the current position has been visited.</p><p class="">(5) Determines if current position is optimal if not record current position.</p><p class="">Every <inline-formula><tex-math id="M123">$$ \mathcal{P}_A $$</tex-math></inline-formula> obtains a set of neighbors of positions during the initial setup. Utilizing a variety of topologies one can create numerous properties of neighboring particle agents to obtain better performance. Within the proposed model, we assume that each <inline-formula><tex-math id="M124">$$ \mathcal{P}_A $$</tex-math></inline-formula> has a static set of neighbors. Each <inline-formula><tex-math id="M125">$$ \mathcal{P}_A $$</tex-math></inline-formula> keeps track of its local best solution, <inline-formula><tex-math id="M126">$$ \mathcal{P}_b $$</tex-math></inline-formula>, which is where a solution closest to the optimal solution is found in the problem space, while the global best solution is recorded in the parameter <inline-formula><tex-math id="M127">$$ \mathcal{P}_g $$</tex-math></inline-formula>. During each iteration the <inline-formula><tex-math id="M128">$$ \mathcal{P}_A $$</tex-math></inline-formula> evaluates its current position and determines if it needs to perform a fitness evaluation, while simultaneous checking if the termination criteria has been met. If the termination criteria have not been met then it updates its <inline-formula><tex-math id="M129">$$ \mathcal{P}_b $$</tex-math></inline-formula>, obtains the neighbor's <inline-formula><tex-math id="M130">$$ \mathcal{P}_b $$</tex-math></inline-formula> and calculates the <inline-formula><tex-math id="M131">$$ \mathcal{P}_g $$</tex-math></inline-formula>, and marks the current position as visited.</p><p class=""><table-wrap><table><tbody><tr><td style="class:table_top_border2" align="left"><b>Algorithm 1:</b> Improved PSO (IPSO) algorithm for waypoint sequencing</td></tr><tr><td style="class:table_top_border2" align="left"><b>Initialize a population of particles</b></td></tr><tr><td align="left">Set the size of the swarm to <inline-formula><tex-math id="M132">$$ S_{p} $$</tex-math></inline-formula>, the maximum number of iteration <inline-formula><tex-math id="M133">$$ T_{max} $$</tex-math></inline-formula>.</td></tr><tr><td style="class:table_bottom_border"><inline-formula id="5170-M1"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" class="inline-graphic" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-M1.jpg" /></inline-formula></td></tr></tbody></table></table-wrap></p></div><div id="sec23" class="article-Section"><h3 >4.2. Safety-aware IPSO multi-waypoint path planning</h3><p class="">To ensure the robot safely reaches the waypoints via the planned visiting sequence, the safety-aware road is selected to guide the autonomous robot. Nevertheless, there is a problem with the connection path from the location of the waypoints to safety-aware roads. When we integrate the position information of a waypoint in the workspace, it may be necessary to obtain the collision-free connection path length from the waypoint to all nodes, which is computationally expensive. As shown in <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6A</a>, with <inline-formula><tex-math id="M134">$$ \mathcal{N} $$</tex-math></inline-formula> nodes obtained in the workspace, there are <inline-formula><tex-math id="M135">$$ \mathcal{N} $$</tex-math></inline-formula> possible connection paths from the waypoints to the nodes in the workspace. With the initial <inline-formula><tex-math id="M136">$$ \mathcal{N} \times \mathcal{N} $$</tex-math></inline-formula> adjacent distance matrix obtained from GVD graph and the increasing <inline-formula><tex-math id="M137">$$ \mathcal{M} $$</tex-math></inline-formula> waypoints in the workspace, the size of new distance matrix is expanded to <inline-formula><tex-math id="M138">$$ (\mathcal{N}+\mathcal{M}) \times (\mathcal{N}+\mathcal{M}) $$</tex-math></inline-formula>. Since most of the connection path computations are unnecessary, a new adjacent node selection (ANS) algorithm is proposed to reduce the computational effort by restricting the search space in local regions rather than the entire working region.</p><p class="">The local regions are shown in <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6B</a>, and the dark blue nodes, such as <inline-formula><tex-math id="M139">$$ \alpha $$</tex-math></inline-formula> and <inline-formula><tex-math id="M140">$$ \beta $$</tex-math></inline-formula> nodes in the regions, are potential adjacent nodes to connect. The radius <inline-formula><tex-math id="M141">$$ \mathcal{R} $$</tex-math></inline-formula> of the local area shall be determined by the potential workspace, which determines the number of nodes in the search range. For instance, in an area with clustered obstacles, there are many nodes in the environment; thus, the search radius may be small. However, In an area with sparse obstacles, there are fewer nodes in the environment; thus, the search radius needs to be larger to include all potential nodes.</p><p class="">Since the entire workspace is projected through the GVD graph, we can interpret the sparseness of the entire workspace through the distance of the edge list <inline-formula><tex-math id="M142">$$ \mathcal{E} $$</tex-math></inline-formula>. Nevertheless, some edge distances cannot represent the sparseness of the entire workspace, such as the edges enclosed by the red dotted line in <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6B</a>.</p><p class="">Therefore, to exclude these extraneous edges of distance, we sort the entire edge list before removing the lowest <inline-formula><tex-math id="M143">$$ 10\% $$</tex-math></inline-formula>. The radius <inline-formula><tex-math id="M144">$$ \mathcal{R} $$</tex-math></inline-formula> of the search range is defined as the average of the remaining edge distance. After obtaining the nodes in the search environment, we plan a collision-free trajectory from the current waypoint to reach each node through the IPSO algorithm.</p><p class="">The IPSO navigation algorithm is initiated in the search range to obtain the optimal path. The local search region is interpreted into grid-based map, where the obstacle areas are inaccessible grids in the workspace. Dijkstra's algorithm is utilized as a local search algorithm and the cost function within the model. For graph-based maps, they must be a method for traveling from one node to another utilizing the edges within the graph. One method of this is Dijkstra's algorithm, which utilizes a weighted graph to determine the shortest path from a source node to a target node. The algorithm also keeps track of the known shortest distances from each node while simultaneously updating their weights to improve the overall shortest path from each node. By recursively establishing a path with random solutions generated in the workspace, the IPSO algorithm can construct the collision-free path with the least fitness value, which also represents the path with minimum length. Therefore, the length of the connecting path <inline-formula><tex-math id="M145">$$ \mathcal{L}_{c} $$</tex-math></inline-formula> can be obtained, and by combining the length of the GVD path <inline-formula><tex-math id="M146">$$ \mathcal{L}_{e} $$</tex-math></inline-formula>, the optimal safety-aware trajectory is obtained as shown in <a href="#Figure6" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure6">Figure 6C</a>.</p><p class="">To effectively reduce and smooth the overall path from each waypoint to the waypoint, various methods have been developed to achieve this goal, for example, the <inline-formula><tex-math id="M147">$$ \mathcal{B} $$</tex-math></inline-formula>-spline curve, which works by taking a set of points that are used to curve sharp close turns around obstacles and is very effective and thus widely used in smooth polylines, due to its closed-form expression of the position coordinates. The original methodology of the <inline-formula><tex-math id="M148">$$ \mathcal{B} $$</tex-math></inline-formula>-spline curve method, in some cases, changes the trajectory of the original path created by the global navigation system. The problem can be mediated by implementing the piecewise <inline-formula><tex-math id="M149">$$ \mathcal{B} $$</tex-math></inline-formula>-spline method, which only smooths the path around each obstacle. Lei <i>et al.</i> evaluated the effectiveness of the improved <inline-formula><tex-math id="M150">$$ \mathcal{B} $$</tex-math></inline-formula>-spline method, and ultimately found that the hybrid method was able to reduce the overall all path in point-to-point navigation <sup>[<a href="#b38" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b38">38</a>]</sup>. The <inline-formula><tex-math id="M151">$$ \mathcal{B} $$</tex-math></inline-formula>-spline curve can be defined by a cardinal functions <inline-formula><tex-math id="M152">$$ L_{j, r}(q) $$</tex-math></inline-formula>, control points <inline-formula><tex-math id="M153">$$ B_{j} $$</tex-math></inline-formula> and degree <inline-formula><tex-math id="M154">$$ (r - 1), $$</tex-math></inline-formula> which is given by the following equations.</p><p class=""><div class="disp-formula"><label>(11)</label><tex-math id="E11"> $$ \begin{equation} \begin{array}{l} K(q)= \sum L_{j, r}(q) B_{j} \end{array} \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M155">$$ B_{j} = [B_{jx}, B_{jy}] $$</tex-math></inline-formula> are the <inline-formula><tex-math id="M156">$$ (n + 1) $$</tex-math></inline-formula> control points and a knot vector <inline-formula><tex-math id="M157">$$ u $$</tex-math></inline-formula>. <inline-formula><tex-math id="M158">$$ N_{i, k}(u) $$</tex-math></inline-formula> are the basic functions, which are defined recursively as follows:</p><p class=""><div class="disp-formula"><label>(12)</label><tex-math id="E12"> $$ \begin{equation} \begin{array}{l} L_{j, r}(q) = \frac{(q-x_{j})}{x_{j+r-1}-x_{j}} N_{j, r-1}(q) + \frac{(x_{j+r}-q)}{x_{j+r}-x_{j+1}} N_{j+1, r-1}(q) \end{array} \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(13)</label><tex-math id="E13"> $$ \begin{equation} L_{j, r}(q) = \begin{cases} 1, \; \; \; \; q_{j} q_{j}\le q \le q_{j+1}\\ 0, \; \; \; \; \; \;\; \; otherwise \end{cases} ; \; q \in [0, 1] \end{equation} $$ </tex-math></div></p><p class="">Geometric continuity <inline-formula><tex-math id="M159">$$ G^2 $$</tex-math></inline-formula> is the metric used to evaluate smoothing methods, which is defined by the tangent unit and curvature vector at the intersection of two continuous segments <sup>[<a href="#b38" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b38">38</a>]</sup> [<a href="#Figure7" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure7">Figure 7A</a>]. To achieve <inline-formula><tex-math id="M160">$$ G^2 $$</tex-math></inline-formula> continuity, the control points <inline-formula><tex-math id="M161">$$ B_{j} $$</tex-math></inline-formula> of <inline-formula><tex-math id="M162">$$ \mathcal{B} $$</tex-math></inline-formula>-spline curve to the path point, <inline-formula><tex-math id="M163">$$ Y_{j} $$</tex-math></inline-formula>, is defined as</p><p class=""><div class="disp-formula"><label>(14)</label><tex-math id="E14"> $$ \begin{equation} \begin{array}{l} B_{1} = Y_{j} - (1 +v)s_{2}q_{j-1}\\ B_{2} = -s_{2}q_{j-1}\\ B_{3} = Y_{j}\\ B_{4} = Y_{j}+s_{2}q_{j}\\ B_{5} = Y_{j} +(1 +v)s_{2}q_{j-1} \end{array} \end{equation} $$ </tex-math></div></p><p class="">where <inline-formula><tex-math id="M164">$$ c $$</tex-math></inline-formula> is smoothing length ratio <inline-formula><tex-math id="M165">$$ v= s_{1}/s_{2} $$</tex-math></inline-formula>, and <inline-formula><tex-math id="M166">$$ q_{j-1} $$</tex-math></inline-formula> defines the unit vector of <inline-formula><tex-math id="M167">$$ Y_{j-1}Y_{j} $$</tex-math></inline-formula>. <inline-formula><tex-math id="M168">$$ q_{j} $$</tex-math></inline-formula> represents the unit vector of the line <inline-formula><tex-math id="M169">$$ Y_{j}Y_{j+1} $$</tex-math></inline-formula>. The combined sum of <inline-formula><tex-math id="M170">$$ s_{1} $$</tex-math></inline-formula> and <inline-formula><tex-math id="M171">$$ s_{2} $$</tex-math></inline-formula> is the smoothed length. The half of the corner angle is denoted as: <inline-formula><tex-math id="M172">$$ \Gamma = \beta / 2 $$</tex-math></inline-formula>. Using a knot vector of [0, 0, 0, 0, 0.5, 1, 1, 1, 1], the smoothing error distance <inline-formula><tex-math id="M173">$$ \epsilon $$</tex-math></inline-formula> and maximum curvature <inline-formula><tex-math id="M174">$$ K_{max} $$</tex-math></inline-formula> within the smooth path can be expressed as:</p><p class=""><div class="disp-formula"><label>(15)</label><tex-math id="E15"> $$ \begin{equation} \begin{array}{l} \epsilon = \frac{s_{2}\sin{\Gamma}}{2} \end{array} \end{equation} $$ </tex-math></div></p><p class=""><div class="disp-formula"><label>(16)</label><tex-math id="E16"> $$ \begin{equation} \begin{array}{l} R_{max} = \frac{4\sin{\Gamma}}{3s_{2}\cos^2{\Gamma}} \end{array} \end{equation} $$ </tex-math></div></p><p class="">Using the previous equations the smoothing error distance <inline-formula><tex-math id="M175">$$ \epsilon $$</tex-math></inline-formula> can be defined by the existing maximum curvature <inline-formula><tex-math id="M176">$$ R_{max} $$</tex-math></inline-formula> given by the robot:</p><p class=""><div class="disp-formula"><label>(17)</label><tex-math id="E17"> $$ \begin{equation} \begin{array}{l} \epsilon = \frac{2\tan^2{\Gamma}}{3R_{max}} \end{array} \end{equation} $$ </tex-math></div></p><p class="">The improved <inline-formula><tex-math id="M177">$$\mathcal{B}$$</tex-math></inline-formula>-spline model has specific advantages over the basic <inline-formula><tex-math id="M178">$$\mathcal{B}$$</tex-math></inline-formula>-spline model, one of which is its ability to smooth many different trajectories with various angles, as seen in <a href="#Figure7" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure7">Figure 7B</a>. The curve produced by the improved <inline-formula><tex-math id="M179">$$\mathcal{B}$$</tex-math></inline-formula>-spline model is significantly closer to the original path than the original model. When considering the constraints of the robot, the improved <inline-formula><tex-math id="M180">$$\mathcal{B}$$</tex-math></inline-formula>-spline mode performs {better} in various degrees of angles. The overall advantages of the improved <inline-formula><tex-math id="M181">$$\mathcal{B}$$</tex-math></inline-formula>-spline model are as follows:</p><p class="">(1) The path generated is tangent and curvature continuity, so that the robot can have a smooth steering command, which can correct any discontinuity of normal acceleration and establish a safer path for the robot to follow.</p><p class="">(2) The improved model generates a better curve by solely affecting the two lines within the corner of the original trajectory. Each curve generated affects others within the lines.</p><p class="">(3) The improved model easily adjusts to the smoothed path based on the environment constraints or the robot.</p><p class=""><table-wrap><table><tbody><tr><td style="class:table_top_border2" align="left"><b>Algorithm 2:</b> Pseudocode for the adjacent node selection (ANS) algorithm</td></tr><tr><td style="class:table_top_border2" align="left"><b>Input:</b> Edge list <inline-formula><tex-math id="M182">$$ \mathcal{E} $$</tex-math></inline-formula>, <inline-formula><tex-math id="M183">$$ \mathcal{N} $$</tex-math></inline-formula> nodes coordinates (<inline-formula><tex-math id="M184">$$ \mathbb{N}_x, \mathbb{N}_y $$</tex-math></inline-formula>), <inline-formula><tex-math id="M185">$$ \mathcal{N} \times \mathcal{N} $$</tex-math></inline-formula> distance matrix <inline-formula><tex-math id="M186">$$ \mathcal{D} $$</tex-math></inline-formula> and the location of the waypoint <inline-formula><tex-math id="M187">$$ \alpha $$</tex-math></inline-formula>, (<inline-formula><tex-math id="M188">$$ \alpha_x, \alpha_y $$</tex-math></inline-formula>) and the waypoint <inline-formula><tex-math id="M189">$$ \beta $$</tex-math></inline-formula>, (<inline-formula><tex-math id="M190">$$ \beta_x, \beta_y $$</tex-math></inline-formula>).</td></tr><tr><td align="left"><b>Output:</b> The path length of the trajectory <inline-formula><tex-math id="M191">$$ \mathcal{L}_{t} $$</tex-math></inline-formula></td></tr><tr><td align="left"><inline-formula><tex-math id="M192">$$ N_e = size(\mathcal{E}) $$</tex-math></inline-formula>; // Number of the edges in the workspace</td></tr><tr><td align="left"><inline-formula><tex-math id="M193">$$ [\mathcal{E}_s, sortInd]=sort(\mathcal{E}) $$</tex-math></inline-formula>; // Sort the edge list from low to high</td></tr><tr><td align="left"><inline-formula><tex-math id="M194">$$ N_s = \lceil \frac{N_e}{10} \rceil $$</tex-math></inline-formula>; // Exclude 10% of extraneous edge distance</td></tr><tr><td style="class:table_bottom_border" align="left"><inline-formula id="5170-M2"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" class="inline-graphic" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-M2.jpg" /></inline-formula></td></tr></tbody></table></table-wrap></p></div></div><div id="sec15" class="article-Section"><h2 >5. REACTIVE LOCAL NAVIGATION</h2><p class="">A crucial aspect in developing a multi-waypoint model is accounting for moving and unknown obstacles <sup>[<a href="#b39" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b39">39</a>, <a href="#b40" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b40">40</a>]</sup>. In a real-world setting, not all objects are static and known. To develop a more efficient model, we propose the use of a local navigator to remedy this issue.</p><p class="">In order to avoid dynamic and unknown obstacles, the proposed model employs the Vector Field Histogram (VFH) model as a reactive local navigator. An autonomous robot uses a velocity command to and from each waypoint, provided by the local navigator <sup>[<a href="#b41" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b41">41</a>]</sup>. By applying the VFH to the overall global trajectory with a sequence of markers, the path can be broken down into various segments to improve the efficiency in obstacle-populated workspaces. The local navigator builds a map depicting the free space and obstacles in the map by utilizing a 2D histogram grid with equally sized cells <sup>[<a href="#b42" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b42">42</a>]</sup>. As the robot follows the generated trajectory within the workspace, the map is simultaneously built, shown in <a href="#Figure8" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure8">Figure 8</a>. In developing an autonomous obstacle avoidance model, concurrent map building and navigation are crucial. The robot pose <inline-formula><tex-math id="M196">$$ (X, Y, Yaw) $$</tex-math></inline-formula> is used to determine the map building. Thus, the precise registration of the built local map as a part of the global map can</p><p class="">be carried out. This map building aims to construct an occupancy-cell-based map. The values for each cell in the map vary over the range [-127, 128] <sup>[<a href="#b42" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b42">42</a>]</sup>. The initial value is zero, which indicates that the cell is neither occupied nor unoccupied. The value is 128 if one cell is occupied with certainty and -127 if one cell is unoccupied with certainty. The values falling into (-127, 128) express contain a level of certainty in the range. When the VFH model is employed in conjunction with the GVD and IPSO algorithm, the robot can be successfully navigated through our built map with obstacle avoidance. In combination with our local navigator, a sensor configuration can be developed for the local navigator to perform it. In <a href="#Figure9" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure9">Figure 9</a> one can see the overview of our sensor configuration. The proposed configuration utilizes a 270-degree SICK LMS LiDAR sensor to detect obstacles within a range of 20 m @ 0.25-degree resolution. The LiDAR sensor scans at a rate of 25Hz. Then, it needs a method of finding our current position and the waypoints within the map. A Novatel's ProPak-LB Plus DGPS sensor is utilized to obtain our current position and how it correlates to the coordinates of each waypoint. Next, a PNI TCM6 digital compass is employed to establish our heading with an accuracy of 0.5 degrees. The sensor updates at 20Hz, which allows the robot to operate efficiently. Lastly, the configuration utilizes an AVT Stingray F-080C 1/3" CCD camera, which enables our robot to sense obstacles of various heights, shapes and sizes. The stingray camera is perfect for robot vision because it uses the IIDC IEEE 1394B protocol to transfer images. The system needs a computer system to house our operating system, sensor data, and programs for the robot. In this portion of the sensor configuration, a MackBook Pro i s equipped to suit our needs. The last step in the process is to establish a method of communication from the sensors to the computer systems. A sort of UART to USB hub is utilized for this purpose and fuses the sensor data together without losing any sensor information. The type of sensor confusion can be used on most ground-based robot systems for indoor and outdoor use.</p></div><div id="sec16" class="article-Section"><h2 >6. SIMULATION AND COMPARISON STUDIES</h2><p class="">In this section, simulation and comparison studies are performed to illustrate the value and vitality of the proposed model. In the first experiment, simulations are conducted using a well-known Traveling Salesman Problem (TSP) based data set, and results are compared with other heuristic-based algorithms. The proposed model is thoroughly evaluated in the second experiment through a comparison study using a similar model proven to work effectively for multi-waypoint navigation.</p><div id="sec22" class="article-Section"><h3 >6.1. Comparison studies with benchmark datasets</h3><p class="">To show the effectiveness of our IPSO waypoint sequencing model, a comparison study was conducted with well-known TSP data sets and various heuristic-based algorithms. The employed datasets and algorithms are as follows: (a) 561-city problem by Kleinschmidt (pa561); (b) 299-city problem by Patberg/Rinaldi (pr299); (c) 200-city problem A, by Krolik/Felts/Nelson (kroA200); and (d) 150-city problem by Chur Ritz (ch150). The selected datasets have been verified and widely used to prove the validity of multi-waypoint sequencing models. The Simulated Annealing (SA) algorithm, Grey Wolf Optimization (GWO) algorithm, Ant Colony Optimization (ACO) algorithm, Genetic Algorithms (GA), Imperialist Competitive Algorithm (ICA), and Self-Organizing Maps (SOM) were chosen as the heuristic-based algorithms used in the comparison studies.</p><p class="">The ICA algorithm is a biologically inspired algorithm by the human, which simulates the social-political process of imperialism and imperialistic competition. The SOM algorithm is similar to a typical artificial neural network algorithm, except it utilizes a competitive learning process instead of backpropagation that utilizes gradient descent.</p><p class="">Heuristic-based algorithms have similar attributes; due to this feature, the same parameters can be used to construct a stable comparison study for our proposed IPSO algorithm. The conducted comparison studies focus on six key attributes such as: min length (<inline-formula><tex-math id="M197">$$ m $$</tex-math></inline-formula>), average length (<inline-formula><tex-math id="M198">$$ m $$</tex-math></inline-formula>), length standard deviation (<inline-formula><tex-math id="M199">$$ m $$</tex-math></inline-formula>), min time (<inline-formula><tex-math id="M200">$$ s $$</tex-math></inline-formula>), average time (<inline-formula><tex-math id="M201">$$ s $$</tex-math></inline-formula>), and time standard deviation (<inline-formula><tex-math id="M202">$$ s $$</tex-math></inline-formula>). The variance between each algorithm can be seen by assessing each parameter. The above analyses show how effective the IPSO model can generate the minimum overall global trajectory in <a href="#Table1" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table1">Table 1</a>. The global trajectory generated by the compared algorithms is notably larger than the IPSO model. However, regarding the time aspect, the IPSO model was unable to achieve the shortest time. The significance of the proposed model can be seen in the STD evaluation parameter. The results of the comparison studies more than show the validity and performance of the proposed model to discover the optimal waypoint visiting sequence.</p><div id="Table1" class="Figure-block"><div class="table-note"><span class="">Table 1</span><p class="">Comparison of minimum path length, average path length, STD of path length, minimum time, average time and STD of time with other models. The parameter for the test of each model was: 100 initialized particles, 10 runs per data set, and a maximum of 10 minutes per run</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td style="class:table_top_border" align="center"><b>Datasets</b></td><td style="class:table_top_border" align="center"><b>Model</b></td><td style="class:table_top_border" align="center"><b>Min length (<i>m</i>)</b></td><td style="class:table_top_border" align="center"><b>Average length (<i>m</i>)</b></td><td style="class:table_top_border" align="center"><b>Length STD (<i>m</i>)</b></td><td style="class:table_top_border" align="center"><b>Min time (<i>s</i>)</b></td><td style="class:table_top_border" align="center"><b>Average time (<i>s</i>)</b></td><td style="class:table_top_border" align="center"><b>Time STD (<i>s</i>)</b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="left" rowspan="7">Ch150</td><td style="class:table_top_border2" align="center">Proposed model</td><td style="class:table_top_border2" align="center">1.67E+04</td><td style="class:table_top_border2" align="center">1.77E+04</td><td style="class:table_top_border2" align="center">5.99E+02</td><td style="class:table_top_border2" align="center">1.75E+04</td><td style="class:table_top_border2" align="center">1.31E+01</td><td style="class:table_top_border2" align="center">6.09E-02</td></tr><tr><td align="center">ACO</td><td align="center">1.84E+04</td><td align="center">2.23E+04</td><td align="center">4.40E+03</td><td align="center">1.67E+04</td><td align="center">1.76E+02</td><td align="center">1.32E+00</td></tr><tr><td align="center">GA</td><td align="center">4.22E+04</td><td align="center">5.04E+04</td><td align="center">5.38E+03</td><td align="center">1.79E+04</td><td align="center">1.39E-02</td><td align="center">1.25E-03</td></tr><tr><td align="center">SA</td><td align="center">2.47E+04</td><td align="center">3.05E+04</td><td align="center">4.11E+03</td><td align="center">1.75E+04</td><td align="center">6.51E+00</td><td align="center">7.25E-01</td></tr><tr><td align="center">GWO</td><td align="center">3.23E+04</td><td align="center">3.66E+04</td><td align="center">2.64E+03</td><td align="center">5.78E-01</td><td align="center">9.39E-01</td><td align="center">1.73E-01</td></tr><tr><td align="center">SOM</td><td align="center">4.26E+04</td><td align="center">4.70E+04</td><td align="center">2.932E+03</td><td align="center">1.84E+03</td><td align="center">2.26E+03</td><td align="center">1.77E+02</td></tr><tr><td align="center">ICA</td><td align="center">3.29E+04</td><td align="center">3.50E+04</td><td align="center">9.34E+02</td><td align="center">1.55E+03</td><td align="center">2.04E+03</td><td align="center">1.79E+02</td></tr><tr><td align="left" rowspan="7">KroA200</td><td align="center">Proposed model</td><td align="center">1.09E+05</td><td align="center">1.17E+05</td><td align="center">5.50E+03</td><td align="center">2.15E+01</td><td align="center">2.15E+01</td><td align="center">9.73E-02</td></tr><tr><td align="center">ACO</td><td align="center">1.30E+05</td><td align="center">2.81E+05</td><td align="center">4.09E+05</td><td align="center">2.44E+02</td><td align="center">5.17E+02</td><td align="center">8.36E+02</td></tr><tr><td align="center">GA</td><td align="center">2.39E+05</td><td align="center">2.57E+05</td><td align="center">1.15E+04</td><td align="center">1.21E-02</td><td align="center">1.54E-02</td><td align="center">4.45E-03</td></tr><tr><td align="center">SA</td><td align="center">1.96E+05</td><td align="center">2.13E+05</td><td align="center">1.18E+04</td><td align="center">6.27E+00</td><td align="center">6.62E+00</td><td align="center">8.70E-01</td></tr><tr><td align="center">GWO</td><td align="center">2.17E+05</td><td align="center">2.40E+05</td><td align="center">1.24E+04</td><td align="center">1.12E+00</td><td align="center">1.45E+00</td><td align="center">3.68E-01</td></tr><tr><td align="center">SOM</td><td align="center">2.13E+05</td><td align="center">2.60E+05</td><td align="center">2.192E+04</td><td align="center">5.49E+03</td><td align="center">2.60E+05</td><td align="center">2.31E+04</td></tr><tr><td align="center">ICA</td><td align="center">2.12E+05</td><td align="center">2.60E+05</td><td align="center">6.671E+03</td><td align="center">3.22E+02</td><td align="center">6.81E+02</td><td align="center">1.38E+02</td></tr><tr><td align="left" rowspan="7">PR299</td><td align="center">Proposed model</td><td align="center">2.77E+05</td><td align="center">2.89E+05</td><td align="center">5.92E+03</td><td align="center">4.28E+01</td><td align="center">4.29E+01</td><td align="center">6.06E-02</td></tr><tr><td align="center">ACO</td><td align="center">3.37E+05</td><td align="center">4.33E+05</td><td align="center">4.82E+04</td><td align="center">2.09E+02</td><td align="center">2.37E+02</td><td align="center">1.41E+01</td></tr><tr><td align="center">GA</td><td align="center">3.19E+05</td><td align="center">3.44E+05</td><td align="center">1.17E+04</td><td align="center">3.24E-02</td><td align="center">3.38E-02</td><td align="center">8.43E-04</td></tr><tr><td align="center">SA</td><td align="center">5.26E+05</td><td align="center">5.59E+05</td><td align="center">1.94E+04</td><td align="center">6.27E+00</td><td align="center">6.52E+00</td><td align="center">6.68E-01</td></tr><tr><td align="center">GWO</td><td align="center">2.90E+05</td><td align="center">3.90E+05</td><td align="center">8.15E+04</td><td align="center">6.71E+01</td><td align="center">8.00E+01</td><td align="center">7.51E+00</td></tr><tr><td align="center">SOM</td><td align="center">2.72E+05</td><td align="center">3.15E+05</td><td align="center">6.34E+04</td><td align="center">4.25E+03</td><td align="center">4.75E+03</td><td align="center">3.45E+02</td></tr><tr><td align="center">ICA</td><td align="center">4.81E+05</td><td align="center">4.96E+05</td><td align="center">1.34E+04</td><td align="center">2.70E+03</td><td align="center">2.81E+03</td><td align="center">7.60E+01</td></tr><tr><td style="class:table_bottom_border" align="left" rowspan="7">PA561</td><td align="center">Proposed model</td><td align="center">1.11E+05</td><td align="center">1.14E+05</td><td align="center">1.60E+03</td><td align="center">1.41E+02</td><td align="center">1.42E+02</td><td align="center">3.65E-01</td></tr><tr><td align="center">ACO</td><td align="center">—</td><td align="center">—</td><td align="center">—</td><td align="center">—</td><td align="center">—</td><td align="center">—</td></tr><tr><td align="center">GA</td><td align="center">1.37E+05</td><td align="center">1.93E+05</td><td align="center">6.30E+04</td><td align="center">9.36E-02</td><td align="center">9.57E-02</td><td align="center">2.19E-03</td></tr><tr><td align="center">SA</td><td align="center">1.88E+05</td><td align="center">1.92E+05</td><td align="center">2.07E+03</td><td align="center">6.27E+00</td><td align="center">6.42E+00</td><td align="center">4.29E-01</td></tr><tr><td align="center">GWO</td><td align="center">3.02E+05</td><td align="center">4.21E+05</td><td align="center">5.94E+04</td><td align="center">7.64E+01</td><td align="center">8.26E+01</td><td align="center">2.21E+00</td></tr><tr><td align="center">SOM</td><td align="center">1.01E+05</td><td align="center">1.21E+05</td><td align="center">1.84E+04</td><td align="center">1.1E+01</td><td align="center">8.19E+02</td><td align="center">3.94E+02</td></tr><tr><td style="class:table_bottom_border" align="center">ICA</td><td style="class:table_bottom_border" align="center">1.48E+05</td><td style="class:table_bottom_border" align="center">1.51E+05</td><td style="class:table_bottom_border" align="center">2.43E+03</td><td style="class:table_bottom_border" align="center">1.22E+03</td><td style="class:table_bottom_border" align="center">2.04E+03</td><td style="class:table_bottom_border" align="center">1.79E+02</td></tr></tbody></table></div><div class="table_footer"></div></div></div><div id="sec23" class="article-Section"><h3 >6.2. Model comparison studies</h3><p class="">The compared models were developed to address the issues of multi-waypoint navigation and mapping in various applications. Each model uses some variation of a global navigation system in combination with an obstacle avoidance technique. The models were selected based on their map configuration and overall efficiency in solving the multi-waypoint navigation problem. Our comparison studies analyze the number of nodes, the trajectories produced, and the total time to fulfill the fastest route.</p><p class="">It is clear that the waypoint order and paths obtained by each model are created in an obstacle-free environment, as illustrated in <a href="#Figure10" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure10">Figure 10A</a>. The length created by the Zhang's model was 240.84 <i>m</i>, while the proposed model produced a shorter trajectory of 219.99 <i>m</i>. This is due to the founded waypoint orders in the environment. In Zhang's comparison study, the proposed model establishes more nodes, and the overall path is expanded by 1.09%, but the proposed model generates a solution 6.1% faster than the compared model. Zhang's proposed model has to utilize a node selection algorithm to establish its shortest path, while the proposed model does not. Due to this feature, the compared model was evaluated before this crucial step and discovered that the nodes established were vastly greater than the proposed model, as seen in <a href="#Table2" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Table2">Table 2</a>. Considering this factor, the proposed model can surpass and outperform Zhang's model. Asl and Taghirad aimed to solve the multi-goal navigation problem by developing a traveling salesman problem in the belief space <sup>[<a href="#b44" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b44">44</a>]</sup>.</p><div id="Table2" class="Figure-block"><div class="table-note"><span class="">Table 2</span><p class="">An illustration of the number of nodes, distance, and time spent traversing with the map to each waypoint</p></div><div class="table-responsive article-table"><table class="a-table"><thead><tr><td style="class:table_top_border" align="left"><b>Model</b></td><td style="class:table_top_border" align="center"><b>Nodes</b></td><td style="class:table_top_border" align="center"><b>Distance</b></td><td style="class:table_top_border" align="center"><b>Time spent <i>s</i></b></td></tr></thead><tbody><tr><td style="class:table_top_border2" align="left">Zhang's model before node reduction</td><td style="class:table_top_border2" align="center">242</td><td style="class:table_top_border2" align="center">271.1</td><td style="class:table_top_border2" align="center">2.25</td></tr><tr><td align="left">Zhang's model after node reduction</td><td align="center">24</td><td align="center">253.4</td><td align="center">0.66</td></tr><tr><td style="class:table_bottom_border" align="left">Proposed model</td><td style="class:table_bottom_border" align="center">38</td><td style="class:table_bottom_border" align="center">277.7</td><td style="class:table_bottom_border" align="center">0.40</td></tr></tbody></table></div><div class="table_footer"></div></div><p class="">From <a href="#Figure11" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure11">Figure 11</a>, one can see the established path using both Asl's method as well as the method proposed in this paper. The method proposed by Asl and Taghirad has the advantage of creating a shorter path but requires a greater number of nodes than the proposed method. <a href="#Figure12" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure12">Figure 12</a> depicts the model comparison between Zhuang <i>et al.</i>'s model and the proposed method<sup>[<a href="#b45" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b45">45</a>]</sup>. The simulation studies reveal that the proposed model had an increased length of approximately 0.05% over Zhuang <i>et al.</i>'s model. Once the compared model requires a greater number of nodes to complete its multi-waypoint navigation, another key point from this comparison is the path created from Zhuang <i>et al.</i>'s model and the proximity to the obstacles in the map. In a real-world environment, the robot could obtain server damage or cause an accident if it is too close to the surrounding obstacles<sup>[<a href="#b45" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b45">45</a>]</sup>. The proposed method established an effective path without risking the robots well being. Von{á}sek and P{e}ni{c}ka <sup>[<a href="#b46" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b46">46</a>]</sup> models had similar results to the previous models, with an increased length of approximately 0.05%, as seen in <a href="#Figure13" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure13">Figure 13</a>. The two compared models have the same problem as Zhuang <i>et al.</i>' model<sup>[<a href="#b45" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b45">45</a>]</sup>. The paths are excessively close to obstacles in the map and thus are not efficient for real-world implementation.</p><p class="">In such an environment, it is very important to consider the robot safety because of the narrow paths created being tightly packed with triangular shaped obstacles. Although most of our model comparison results showed that the path constructed with the proposed model increased from the compared models, we achieved our goal of constructing safety-aware roads for robot safety and establishing an obstacle-free path.</p><p class="">From <a href="#Figure14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure14">Figure 14</a>, it is observed how the local navigator establishes a map through vision sensors such as LiDAR. In <a href="#Figure14" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure14">Figure 14</a>, it is obvious that the map in various stages is shown as the robot traverses along the generated trajectory found in the Vonásek's simulation [<a href="#Figure13" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure13">Figure 13</a>]<sup>[<a href="#b46" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="b46">46</a>]</sup>.</p><div class="Figure-block" id="Figure14"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure14" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-14.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 14. Illustration of the scenario in <a href="#Figure13" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure13">Figure 13</a> navigation and mapping simulation. (A) It depicts the robot traversing a majority of the map, while avoiding obstacles in the environment. (B) It illustrates a polar histogram and how the obstacles are viewed while the robot is in motion, as well as points of high impact. The obstacles are viewed as lines since the LiDAR sensor can only see the part of the object that faces the LiDAR sensor. The picked direction portion of part (B) depicts the probability of colliding with obstacles while also selecting the best direction to move the robot. (C) It demonstrates the map being simultaneously built as the the robot traverses the established trajectory.</p></div></div><p class="">The robot is able to reconstruct the outer boundary of the obstacles through the LiDAR scan. These are depicted as the poly-shaped figures with a rough background and a white center.</p><p class="">In <a href="#Figure15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure15">Figure 15A</a>, it is clear to see the original starting position as well as the planned trajectory, which was found utilizing our proposed IPSO model. From the figures, one could observe fully and partly detected obstacles as well as the outer boundary being detected. The map depicted in <a href="#Figure15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure15">Figure 15</a> has a height and width of 60 <inline-formula><tex-math id="M203">$$ m $$</tex-math></inline-formula>. The dimensions of the robot are approximately 0.82 <inline-formula><tex-math id="M204">$$ m $$</tex-math></inline-formula> long and 0.68 <inline-formula><tex-math id="M205">$$ m $$</tex-math></inline-formula> wide. In <a href="#Figure15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure15">Figure 15A</a>, the robot has successfully traversed one third of the map, while simultaneously avoiding the obstacles. In <a href="#Figure15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure15">Figure 15B</a>, it illustrates the robot's planned trajectory and the map being simultaneously constructed along the path. The portions of the figure depicted in a yellow field are the built map sensed by the onboard sensors. Finally, in <a href="#Figure15" class="Link_style" data-jats-ref-type="bibr" data-jats-rid="Figure15">Figure 15C</a>, the robot has successfully visited each waypoint and reached its final destination, and it shows the complete depiction of the map along the projected path.</p><div class="Figure-block" id="Figure15"><div xmlns="http://www.w3.org/1999/xhtml" class="article-figure-image"><a href="/articles/ir.2022.21/image/Figure15" class="Article-img" alt="" target="_blank"><img alt="A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping" src="https://image.oaes.cc/b6bde10c-1a17-43d0-9758-88dabfd18e62/5170-15.jpg" class="" title="" alt="" /></a></div><div class="article-figure-note"><p class="figure-note"></p><p class="figure-note">Figure 15. Illustration of the navigation and mapping ability of the proposed model. (A) The robot follows the generated trajectory and detects obstacle boundaries by the LiDAR sensor. (B) The simultaneous map building and navigation capabilities of the proposed model. (C) The fully generated trajectory and established map.</p></div></div></div></div><div id="sec17" class="article-Section"><h2 >7. CONCLUSION</h2><p class="">We proposed an adjacent node selection (ANS) algorithm to find a node in the graph to connect waypoints. This algorithm is utilized in the safety-aware multi-waypoint navigation and mapping by an improved PSO and GVD model.An IPSO-based multi-waypoint algorithm has been developed to define an order for waypoint navigation. Through our proposed ANS algorithm, connections among the waypoints and the safety-aware routes to reach multi-objective optimization can be created. The feasibility and effectiveness of our model by conducting a benchmark test and model comparison studies and analyses have been demonstrated.</p></div><div id="sec18" class="article-Section"><h2 >DECLARATIONS</h2><div id="sec21" class="article-Section"><h3 >Acknowledgments</h3><p class="">The authors would like to thank the editor-in-chief, the associate editor, and the anonymous reviewers for their valuable comments.</p></div><div id="sec22" class="article-Section"><h3 >Authors' contributions</h3><p class="">Made substantial contributions to the research, idea generation, algorithm design, simulation, wrote and edited the original draft: Sellers T, Lei T, Luo C</p><p class="">Performed critical review, commentary and revision, as well as provided administrative, technical, and material support: Jan G, Ma J</p></div><div id="sec23" class="article-Section"><h3 >Financial support and sponsorship</h3><p class="">None.</p></div><div id="sec24" class="article-Section"><h3 >Availability of data and materials</h3><p class="">Not applicable.</p></div><div id="sec25" class="article-Section"><h3 >Conflicts of interest</h3><p class="">All authors declared that there are no conflicts of interest.</p></div><div id="sec26" class="article-Section"><h3 >Ethical approval and consent to participate</h3><p class="">Not applicable.</p></div><div id="sec27" class="article-Section"><h3 >Consent for publication</h3><p class="">Not applicable.</p></div><div id="sec28" class="article-Section"><h3 >Copyright</h3><p class="">© The Author(s) 2022.</p></div></div></div>  <div class="art_list" data-v-6dffe839></div> <div class="article_references" data-v-6dffe839><div class="ReferencesBox" data-v-6dffe839><h2 id="References" class="bg_d" data-v-6dffe839><span data-v-6dffe839><a href="/articles//reference" data-v-6dffe839>REFERENCES</a></span> <span class="icon" data-v-6dffe839><i class="el-icon-arrow-down" data-v-6dffe839></i> <i class="el-icon-arrow-up hidden" data-v-6dffe839></i></span></h2> <div class="references_list heightHide" data-v-6dffe839><div id="b1" class="references_item" data-v-6dffe839><p data-v-6dffe839><span data-v-6dffe839>1. </span> <span data-v-6dffe839>Chu Z, Wang F, Lei T, Luo C. 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Zaragoza, Spain; 2019. pp. 1587-90.</span></p> <div class="refrences" data-v-6dffe839><a href="https://dx.doi.org/10.1109/ETFA. 2019.8869521" target="_blank" data-v-6dffe839><button type="button" class="el-button el-button--text el-button--mini" data-v-6dffe839><span>DOI</span></button></a>  </div></div></div> <div class="line" data-v-6dffe839></div></div> <div class="article_cite cite_layout" data-v-6dffe839><div id="cite" data-v-6dffe839></div> <div class="el-row" style="margin-left:-10px;margin-right:-10px;" data-v-6dffe839><div class="el-col el-col-24 el-col-xs-24 el-col-sm-16" style="padding-left:10px;padding-right:10px;" data-v-6dffe839><div class="left_box" data-v-6dffe839><div data-v-6dffe839><h2 style="margin-top:0!important;padding-top:0;" data-v-6dffe839>Cite This Article</h2> <div class="cite_article" data-v-6dffe839><div class="cite_article_sec" data-v-6dffe839>Research Article</div> <div class="cite_article_open" style="color:#aa0c2f;" data-v-6dffe839><img src="https://g.oaes.cc/oae/nuxt/img/open_icon.bff5dde.png" alt="" style="width:10px;" data-v-6dffe839> Open Access</div> <div class="cite_article_tit" data-v-6dffe839><span data-v-6dffe839>A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping</span></div> <div class="cite_article_editor" data-v-6dffe839><span data-v-6dffe839>Timothy Sellers, ... Junfeng Ma</span></div></div></div> <div class="color_000" data-v-6dffe839><h2 data-v-6dffe839>How to Cite</h2> <p data-v-6dffe839>Sellers, T.; Lei, T.; Luo, C.; Eu Jan, G.; Ma, J. A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping. <i>Intell. Robot.</i> <b>2022</b>, <i>2</i>, 333-54. http://dx.doi.org/10.20517/ir.2022.21</p></div> <div data-v-6dffe839><h2 data-v-6dffe839>Download Citation</h2> <div class="font_12" data-v-6dffe839>If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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E-mail: \u003Cemail\u003EChaomin.Luo@ece.msstate.edu\u003C\u002Femail\u003E; ORCID: 0000-0002-7578-3631",editor:[],editor_time:"\u003Cspan\u003E\u003Cb\u003EReceived:\u003C\u002Fb\u003E 20 Jul 2022 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EFirst Decision:\u003C\u002Fb\u003E 12 Aug 2022 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003ERevised:\u003C\u002Fb\u003E 18 Aug 2022 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EAccepted:\u003C\u002Fb\u003E 25 Aug 2022 | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EPublished:\u003C\u002Fb\u003E 12 Oct 2022\u003C\u002Fspan\u003E",cop_link:"https:\u002F\u002Fcreativecommons.org\u002Flicenses\u002Fby\u002F4.0\u002F",cop_info:"© The Author(s) 2022. \u003Cb\u003EOpen Access\u003C\u002Fb\u003E This article is licensed under a Creative Commons Attribution 4.0 International License (\u003Ca target=\"_blank\"href=\"https:\u002F\u002Fcreativecommons.org\u002Flicenses\u002Fby\u002F4.0\u002F\" xmlns:xlink=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink\"\u003Ehttps:\u002F\u002Fcreativecommons.org\u002Flicenses\u002Fby\u002F4.0\u002F\u003C\u002Fa\u003E), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.",keywords:["Adjacent node selection (ANS) algorithm","safety-aware roads","path planning","multiple-waypoint optimization","navigation and mapping"],issue:f,image:t,tag:" \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E 2022;2(4):333-54.",authors:"Timothy Sellers, ... Junfeng Ma",picurl:b,expicurl:b,picabstract:b,interview_pic:a,interview_url:a,review:a,cop_statement:"© The Author(s) 2022",seo:{title:ay,keywords:bH,description:cA},video_img:cB,lpage:354,author:[{base:"Timothy Sellers\u003Csup\u003E1\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Tingjun Lei\u003Csup\u003E1\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Chaomin Luo\u003Csup\u003E1\u003C\u002Fsup\u003E",email:R,orcid:"http:\u002F\u002Forcid.org\u002F0000-0002-7578-3631"},{base:"Gene Eu Jan\u003Csup\u003E2\u003C\u002Fsup\u003E",email:a,orcid:a},{base:"Junfeng Ma\u003Csup\u003E3\u003C\u002Fsup\u003E",email:a,orcid:a}],specialissue:a,specialinfo:a,date_published_stamp:cC,year1:az,CitedImage:"https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Fjournals\u002FCrossref.png",article_editor:[],editoruser:"\u003Cspan\u003E\u003Cb\u003EAcademic Editor:\u003C\u002Fb\u003E Simon X. Yang | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003ECopy Editor:\u003C\u002Fb\u003E Jia-Xin Zhang | \u003C\u002Fspan\u003E\u003Cspan\u003E\u003Cb\u003EProduction Editor:\u003C\u002Fb\u003E Jia-Xin Zhang\u003C\u002Fspan\u003E",commentsNums:g,oaestyle:cD,amastyle:cE,ctstyle:cF,acstyle:cG,copyImage:"https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Fjournals\u002Fccb_4.png",affiliation:[{id:68469,article_id:d,Content:"\u003Clabel\u003E\u003Csup\u003E1\u003C\u002Fsup\u003E\u003C\u002Flabel\u003E\u003Caddr-line\u003EDepartment of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39759, USA.\u003C\u002Faddr-line\u003E"},{id:68470,article_id:d,Content:"\u003Clabel\u003E\u003Csup\u003E2\u003C\u002Fsup\u003E\u003C\u002Flabel\u003E\u003Caddr-line\u003EDepartment of Electrical Engineering, National Taipei University and Tainan National University of the Arts, Taipei 72045, Taiwan.\u003C\u002Faddr-line\u003E"},{id:68471,article_id:d,Content:"\u003Clabel\u003E\u003Csup\u003E3\u003C\u002Fsup\u003E\u003C\u002Flabel\u003E\u003Caddr-line\u003EDepartment of Industrial and Systems Engineering, Mississippi State University, Mississippi State, MS 39759, USA.\u003C\u002Faddr-line\u003E"}],related:[{article_id:aA,journal_id:n,section_id:r,path:q,journal:o,ar_title:aB,date_published:cH,doi:aC,author:[{first_name:cI,middle_name:a,last_name:cJ,ans:e,email:a,bio:a,photoUrl:a},{first_name:cK,middle_name:a,last_name:cL,ans:e,email:cM,bio:a,photoUrl:a},{first_name:cN,middle_name:a,last_name:cO,ans:e,email:a,bio:a,photoUrl:a},{first_name:cP,middle_name:a,last_name:aD,ans:e,email:a,bio:a,photoUrl:a}]},{article_id:bI,journal_id:n,section_id:r,path:q,journal:o,ar_title:bJ,date_published:cQ,doi:bK,author:[{first_name:cR,middle_name:a,last_name:cS,ans:e,email:a,bio:a,photoUrl:a},{first_name:cT,middle_name:a,last_name:aE,ans:e,email:cU,bio:a,photoUrl:a},{first_name:cV,middle_name:a,last_name:aF,ans:e,email:a,bio:a,photoUrl:a},{first_name:cW,middle_name:a,last_name:aD,ans:e,email:a,bio:a,photoUrl:a}]},{article_id:bL,journal_id:n,section_id:r,path:q,journal:o,ar_title:bM,date_published:cX,doi:bN,author:[{first_name:cY,middle_name:a,last_name:cZ,ans:e,email:c_,bio:a,photoUrl:a},{first_name:c$,middle_name:a,last_name:da,ans:e,email:a,bio:a,photoUrl:a},{first_name:db,middle_name:a,last_name:A,ans:e,email:a,bio:a,photoUrl:a}]},{article_id:bO,journal_id:n,section_id:r,path:q,journal:o,ar_title:bP,date_published:dc,doi:bQ,author:[{first_name:dd,middle_name:a,last_name:de,ans:aG,email:a,bio:a,photoUrl:a},{first_name:aH,middle_name:a,last_name:aI,ans:aG,email:a,bio:a,photoUrl:a},{first_name:aJ,middle_name:a,last_name:aK,ans:H,email:R,bio:a,photoUrl:a},{first_name:df,middle_name:a,last_name:dg,ans:dh,email:a,bio:a,photoUrl:a},{first_name:di,middle_name:a,last_name:dj,ans:dk,email:a,bio:a,photoUrl:a}]},{article_id:bR,journal_id:n,section_id:r,path:q,journal:o,ar_title:bS,date_published:bG,doi:bT,author:[{first_name:aH,middle_name:a,last_name:aI,ans:H,email:a,bio:a,photoUrl:a},{first_name:dl,middle_name:a,last_name:A,ans:dm,email:a,bio:a,photoUrl:a},{first_name:aJ,middle_name:a,last_name:aK,ans:H,email:R,bio:a,photoUrl:a},{first_name:A,middle_name:a,last_name:dn,ans:do0,email:a,bio:a,photoUrl:a},{first_name:dp,middle_name:a,last_name:aF,ans:dq,email:a,bio:a,photoUrl:a},{first_name:dr,middle_name:a,last_name:ds,ans:dt,email:a,bio:a,photoUrl:a}]},{article_id:bU,journal_id:n,section_id:I,path:q,journal:o,ar_title:bV,date_published:du,doi:bW,author:[{first_name:aL,middle_name:a,last_name:J,ans:e,email:a,bio:a,photoUrl:a},{first_name:aM,middle_name:a,last_name:aN,ans:e,email:a,bio:a,photoUrl:a},{first_name:aO,middle_name:a,last_name:aP,ans:e,email:dv,bio:a,photoUrl:a}]},{article_id:bX,journal_id:n,section_id:I,path:q,journal:o,ar_title:bY,date_published:dw,doi:bZ,author:[{first_name:dx,middle_name:a,last_name:A,ans:e,email:a,bio:a,photoUrl:a},{first_name:dy,middle_name:a,last_name:dz,ans:e,email:a,bio:a,photoUrl:a},{first_name:aL,middle_name:a,last_name:J,ans:e,email:a,bio:a,photoUrl:a},{first_name:dA,middle_name:a,last_name:aE,ans:e,email:a,bio:a,photoUrl:a},{first_name:aM,middle_name:a,last_name:aN,ans:e,email:a,bio:a,photoUrl:a},{first_name:J,middle_name:a,last_name:dB,ans:e,email:a,bio:a,photoUrl:a},{first_name:aO,middle_name:a,last_name:aP,ans:e,email:a,bio:a,photoUrl:a}]}],down:"https:\u002F\u002Ff.oaes.cc\u002Fris\u002F5170.ris",xml:{id:3242,article_id:d,xml_down:m,cite_click:S,export_click:T},zan:g,cited_type:"cited",subarray:[],issn:b_,uuid:a,abstractUuid:a,apiurl:a,api_abstract_url:a,journal_id:aQ,journal_path:q},loadingAbs:void 0,loading:U,ArtDataC:{content:"\u003Cdiv id=\"sec11\" class=\"article-Section\"\u003E\u003Ch2 \u003E1. INTRODUCTION\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003ERobotics system has been applied to numerous fields, such as transportation \u003Csup\u003E[\u003Ca href=\"#b1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b1\"\u003E1\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, healthcare service \u003Csup\u003E[\u003Ca href=\"#b2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b2\"\u003E2\u003C\u002Fa\u003E, \u003Ca href=\"#b3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b3\"\u003E3\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, agriculture \u003Csup\u003E[\u003Ca href=\"#b4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b4\"\u003E4\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, manufacturing \u003Csup\u003E[\u003Ca href=\"#b5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b5\"\u003E5\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, \u003Ci\u003Eetc.\u003C\u002Fi\u003E, in recent years. Robot navigation is one of the fundamental components in robotic systems, which includes multi-waypoint navigation system \u003Csup\u003E[\u003Ca href=\"#b6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b6\"\u003E6\u003C\u002Fa\u003E-\u003Ca href=\"#b8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b8\"\u003E8\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. As an increasing demand and limited onboard resources for autonomous robots, it requires the ability to visit several targets in one mission to optimize multiple objectives, including time, robot travel distance minimization, and spatial optimization \u003Csup\u003E[\u003Ca href=\"#b9\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b9\"\u003E9\u003C\u002Fa\u003E-\u003Ca href=\"#b15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b15\"\u003E15\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. For example, due to a global pandemic, the world struggled to sanitize heavily populated areas, such as airports, hospitals, and educational buildings. Autonomous robots with multi-waypoint navigation systems can effectively sanitize all targeted areas without endangering the workers \u003Csup\u003E[\u003Ca href=\"#b14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b14\"\u003E14\u003C\u002Fa\u003E, \u003Ca href=\"#b16\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b16\"\u003E16\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. As well as in agriculture management, multi-waypoint strategies allow the robotic system to navigate and survey multiple areas to assist in production and collection.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn order to employ robotic systems in real-world scenarios, one critical factor is to develop autonomous robot multi-waypoint navigation and mapping system \u003Csup\u003E[\u003Ca href=\"#b17\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b17\"\u003E17\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. In order to solve the autonomous robot navigation problem, countless algorithms have been developed, such as graph-based\u003Csup\u003E[\u003Ca href=\"#b18\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b18\"\u003E18\u003C\u002Fa\u003E, \u003Ca href=\"#b19\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b19\"\u003E19\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, ant colony optimization (ACO) \u003Csup\u003E[\u003Ca href=\"#b20\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b20\"\u003E20\u003C\u002Fa\u003E-\u003Ca href=\"#b22\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b22\"\u003E22\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, bat-pigeon algorithm (BPA) \u003Csup\u003E[\u003Ca href=\"#b23\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b23\"\u003E23\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, neural networks \u003Csup\u003E[\u003Ca href=\"#b24\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b24\"\u003E24\u003C\u002Fa\u003E-\u003Ca href=\"#b26\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b26\"\u003E26\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, fuzzy logic \u003Csup\u003E[\u003Ca href=\"#b27\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b27\"\u003E27\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, artificial potential field (APF) \u003Csup\u003E[\u003Ca href=\"#b28\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b28\"\u003E28\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, sampling-based strategy \u003Csup\u003E[\u003Ca href=\"#b14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b14\"\u003E14\u003C\u002Fa\u003E, \u003Ca href=\"#b29\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b29\"\u003E29\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, hybrid algorithms\u003Csup\u003E[\u003Ca href=\"#b30\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b30\"\u003E30\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, task planning algorithm \u003Csup\u003E[\u003Ca href=\"#b31\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b31\"\u003E31\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E, \u003Ci\u003Eetc\u003C\u002Fi\u003E. Chen \u003Ci\u003Eet al.\u003C\u002Fi\u003E produced a hybrid graph-based reinforcement learning architecture to develop a method for robot navigation in crowds \u003Csup\u003E[\u003Ca href=\"#b18\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b18\"\u003E18\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Luo \u003Ci\u003Eet al.\u003C\u002Fi\u003E proposed an improved vehicle navigation method, which utilizes a heading-enabled ACO algorithm to improve trajectory towards the target\u003Csup\u003E[\u003Ca href=\"#b20\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b20\"\u003E20\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Lei \u003Ci\u003Eet al.\u003C\u002Fi\u003E developed a Bat-Pigeon algorithm with the ability to adjust the speed navigation of autonomous vehicles \u003Csup\u003E[\u003Ca href=\"#b23\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b23\"\u003E23\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Luo \u003Ci\u003Eet al.\u003C\u002Fi\u003E developed the model for multiple robots complete coverage navigation while using a bio-inspired neural network to dynamically avoid obstacles\u003Csup\u003E[\u003Ca href=\"#b32\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b32\"\u003E32\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. Na and Oh established a hybrid control system for autonomous mobile robot navigation that utilizes a neural network for environment classification and behavior-based control method to mimic the human steering commands\u003Csup\u003E[\u003Ca href=\"#b25\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b25\"\u003E25\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.Lazreg and Benamrane \u003Csup\u003E[\u003Ca href=\"#b27\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b27\"\u003E27\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E developed a neuro-fuzzy inference system associated with a Particle Swarm Optimization (PSO) method for robot path planning using a variety of sensors to control the speed and position of a robot.Jensen-Nau \u003Ci\u003Eet al.\u003C\u002Fi\u003E\u003Csup\u003E[\u003Ca href=\"#b28\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b28\"\u003E28\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E integrated a Voronoi-based path generation algorithm and an artificial potential field path planning method, in which the latter is capable of establishing a path in an unknown environment in real-time for robot path planning and obstacle avoidance.Penicka and Scaramuzza \u003Csup\u003E[\u003Ca href=\"#b14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b14\"\u003E14\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E developed a sampling-based multi-waypoint minimum-time path planning model that allows obstacle avoidance in cluttered environments.Ortiz and Yu \u003Csup\u003E[\u003Ca href=\"#b30\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b30\"\u003E30\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E proposed a sliding control method in combination with simultaneous localization and mapping (SLAM) method to overcome the bounded uncertainties problem, which utilizes a genetic algorithm to improve path planning capabilities. Bernardo \u003Ci\u003Eet al.\u003C\u002Fi\u003E proposed a task planning method for home environment ontology to translate tasks given by other robots or humans into feasible tasks for another robot agent\u003Csup\u003E[\u003Ca href=\"#b31\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b31\"\u003E31\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn robotic path planning, one of the special topics is autonomous robot multi-waypoint navigation which has been studied for many years. For instance, Shair \u003Ci\u003Eet al.\u003C\u002Fi\u003E proposed a model for real-world waypoint navigation using a variety of sensors for accurate environmental analysis\u003Csup\u003E[\u003Ca href=\"#b33\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b33\"\u003E33\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The system is designed utilizing the wide area augmentation system (WAAS) and the European geostationary navigation overlay service (EGNOS) for GPS in combination with aerial images to provide valuable positioning data to the system.Yang \u003Csup\u003E[\u003Ca href=\"#b34\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b34\"\u003E34\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E brought about a multi-waypoint navigation system based on terrestrial signals of opportunity (SOPs) transmitters, which has the ability to operate in environments that are not available to global navigation satellite systems (GNSS) for unmanned aerial vehicles (UAV).Janoš \u003Ci\u003Eet al.\u003C\u002Fi\u003E proposed a sampling-based multi-waypoint path planner, space-filling forest method to solve the problem of finding collision-free trajectories while the sequence of waypoints is formed by multiple trees\u003Csup\u003E[\u003Ca href=\"#b29\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b29\"\u003E29\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.However, the aforementioned approaches have not taken into account the safety of the robot during navigation. In real-world applications, an autonomous vehicle has odometry errors during operation. The safety-aware road model is developed utilizing the Generalized Voronoi diagram (GVD) approach. Once the safety-aware roads are defined, a particle swarm optimization (PSO) algorithm-based multi-waypoint path planning algorithm is proposed to visit each waypoint in an explicated sequence while simultaneously avoiding obstacles. In this paper, the Adjacent Node Selection (ANS) algorithm is developed to select the closest nodes on the safety-aware roads to generate the final collision-free trajectories with minimal distance. Furthermore, in our hybrid algorithm, we utilize a histogram-based reactive local navigator to avoid dynamic and unknown obstacles within the workspace. Through all of these methods, an effective and efficient safety-aware multiple waypoint navigation model was established, which has been validated by both simulations and comparison studies.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThis paper proposes an Adjacent Node Selection (ANS) algorithm for obtaining an optimal access node into graph-based maps. To the best of our knowledge, there are no known similar algorithms that improve the paths created from the waypoint to the graph. The ANS algorithm can be applied to any graph-based mapping environment, which improves the various graph-based models used for autonomous robotic path planning systems. In finding an access point into the graph utilizing one of the nodes in the system, the ANS algorithm conducts point-to-point selection in dense obstacle field of environments to obtain a node to gain access to the graph that forms a resumable path from a waypoint to the graph. The algorithm's overall goal is to find a node in a graph-based map and shorten the overall path length from each waypoint to waypoint, and the waypoint to the graph.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EOne can see in \u003Ca href=\"#Figure1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure1\"\u003EFigure 1\u003C\u002Fa\u003E the overall framework of our proposed model. Initially, the model is provided with a global map of its environment and the location of each target waypoint. The GVD utilizes the global map to construct the safety-aware roads, which are used to guide the robot throughout the environment. The targeted waypoint locations are used as input to the improved PSO (IPSO) algorithm that act as our waypoint sequencing module, which is utilized to find a near-optimal sequence to visit each waypoint. The ANS algorithm utilizes the output from the previous stages. The ANS algorithm finds the best node for each waypoint to use as an access point to the graph. If there is no direct path from one waypoint to a node in the graph, the algorithm conduces point-to-point navigation with nodes within a specified range, which will be explained in later sections of the paper. Once each waypoint has found its access point, the safety-aware path is constructed, which is then applied to our path smoothing algorithm. Finally, we utilize a reactive local navigation system to detect obstacles autonomously and simultaneously build a map of the environment along the generated path.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure1\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure1\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-1.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 1. Illustration of the proposed model framework.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EThe main contributions of this paper with this framework of concurrent multi-waypoint navigation and mapping with collision avoidance as are summarized as follows: (1) An adjacent node selection (ANS) algorithm is proposed to build up regional bridges from waypoints to nodes or edges on the graph in multi-waypoint navigation and mapping; (2) a concurrent multi-waypoint navigation and mapping framework of an autonomous robot is developed by the generalized Voronoi diagram (GVD) and IPSO algorithm as well as a local navigator; (3) a GVD method is employed to plan safety-aware trajectories in an obstacle populated environment, while the sequence of multiple waypoints is created by the IPSO algorithm to minimize the total distance cost; (4) a sensor-based histogram robot reactive navigator coupled with map building is adopted for moving obstacle avoidance and local mapping. An improved \u003Cinline-formula\u003E\u003Ctex-math id=\"M2\"\u003E$$\\mathcal{B}$$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline curve-based speed modulation module is adopted for smoother navigation.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe structure of this paper is as follows. Section II presents the safety-aware model that constructs the safety-conscious roads for robot traversal. Section III describes the proposed adjacent node selection algorithm to find an access node into the graph-based map. Section IV, the improved PSO-based (IPSO) multi-waypoint navigation model for waypoint sequencing is explained. Section V illustrates the reactive local navigator for real-time workspace building and obstacle avoidance. Section VI depicts simulation results and comparative analyses. Several important properties of the proposed framework are summarized in Section VII.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec12\" class=\"article-Section\"\u003E\u003Ch2 \u003E2. SAFETY-AWARE MODEL\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EComputational geometry has exceedingly studied the Voronoi diagrams (VD) model, which is an elemental data structure used as a minimizing diagram of a finite set of continuous functions \u003Csup\u003E[\u003Ca href=\"#b28\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b28\"\u003E28\u003C\u002Fa\u003E, \u003Ca href=\"#b35\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b35\"\u003E35\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The minimized function defines the distantance to an object within the workspace. The VD model decomposes the workspace into several regions, each consisting of the points near a given object than the others. Let \u003Cinline-formula\u003E\u003Ctex-math id=\"M3\"\u003E$$ \\mathcal{G} = \\{g_1\\cdots, g_n\\} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E be a set of points of \u003Cinline-formula\u003E\u003Ctex-math id=\"M4\"\u003E$$ \\mathbb{R}^d $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E to each \u003Cinline-formula\u003E\u003Ctex-math id=\"M5\"\u003E$$ g_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E associated with a specific Voronoi region \u003Cinline-formula\u003E\u003Ctex-math id=\"M6\"\u003E$$ V(g_i) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. This can be expressed by the following equation\u003Csup\u003E[\u003Ca href=\"#b36\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b36\"\u003E36\u003C\u002Fa\u003E, \u003Ca href=\"#b37\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b37\"\u003E37\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(1)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E1\"\u003E $$ \\begin{equation} V(g_i) = \\{x \\in \\mathbb{R}^d: \\| x-g_i \\| \\le \\| x-g_j \\|, \\forall j \\le n \\} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe intersection of \u003Cinline-formula\u003E\u003Ctex-math id=\"M7\"\u003E$$ n - 1 $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E half spaces can be denoted by the region \u003Cinline-formula\u003E\u003Ctex-math id=\"M8\"\u003E$$ V(g_i) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Each half space holds a point \u003Cinline-formula\u003E\u003Ctex-math id=\"M9\"\u003E$$ g_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E along with another point of \u003Cinline-formula\u003E\u003Ctex-math id=\"M10\"\u003E$$ \\mathcal{G} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The regions \u003Cinline-formula\u003E\u003Ctex-math id=\"M11\"\u003E$$ V(g_i) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are convex polyhedrons due to the bisectors acting as hyperplanes between each region. The generalized Voronoi diagram (GVD) is a modified version of the VD model defined as the set of points Euclidean distance from two obstacles\u003Csup\u003E[\u003Ca href=\"#b37\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b37\"\u003E37\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The workspace is represented as a graph by the GVD model consisting of nodes, edges, and vertices \u003Csup\u003E[\u003Ca href=\"#b28\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b28\"\u003E28\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe proposed ANS algorithm utilizes nodes and edges obtained from the GVD illustrated in \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2A\u003C\u002Fa\u003E. The nodes in the GVD act as junctions to connect waypoints, whereas the edges are used to determine the search range in view of waypoints. Solely the nodes in the search range are calculated, other than the entire workspace, thus reducing the computational expense. \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2B\u003C\u002Fa\u003E reveals that the range of some edges fails to evaluate how spare of the workspace is configurated. These edges with short lengths are distractors for evaluating and computing the search range, thus need to be eliminated shown in \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2B\u003C\u002Fa\u003E. In our ANS algorithm, those edges with the shortest 10% are eliminated, while the rest of the edges are averaged to compute the radius of the search range. One circumstance is exhibited in \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2C\u003C\u002Fa\u003E, in which some nodes fail to fall into the search range once the excessively short edges are excluded. The original search space in solid circles and modified search space in dashed circles based on the ANS algorithm are shown in \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2C\u003C\u002Fa\u003E. As may be seen, after excessively short edges are effectively eliminated, an appropriate search range is achieved.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure2\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure2\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-2.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 2. Illustration of the ANS algorithm and the safety-aware model by creation of the GVD. (A) Depicts the equidistance property of the nodes and edges in the GVD model. (B) illustrates how we remove extraneous edge distances, which are small edges within the graph that reduce the search space or range in the ANS algorithm. (C) illustrates how the removal of those extraneous edges improves the range and how an access node to the graph can be found.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EGVD nodes form the Euclidean distance between two or more obstacles, while the edges are the junction of two nodes that depict the distance between each neighboring node to another. Vertices are the connection points between three or more nodes. Using these features from the GVD model, an obstacle-free path with our safety-aware model is effectively created. The safety-aware model is constructed by inputting an image and extracting all significant features from the image, thus allowing the model to construct a map from the input image. The safety-aware roads are the clearest path between obstacles that occupy the available space in the map, which can be seen in \u003Ca href=\"#Figure2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure2\"\u003EFigure 2\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn this paper, in order to explain our proposed ANS algorithm, how the nodes are determined and constructed with the GVD will be presented in some detail. We will introduce some definitions and notations. Lee and Drysdale \u003Csup\u003E[\u003Ca href=\"#b36\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b36\"\u003E36\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E derive four basic definitions for determining the optimal placement of edges and nodes within the GVD graph.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EDEFINITION 1. A closed line segment \u003Cinline-formula\u003E\u003Ctex-math id=\"M12\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E consists of two endpoints \u003Cinline-formula\u003E\u003Ctex-math id=\"M13\"\u003E$$ \\alpha $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M14\"\u003E$$ \\gamma $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. A straight line is denoted by (\u003Cinline-formula\u003E\u003Ctex-math id=\"M15\"\u003E$$ \\alpha $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M16\"\u003E$$ \\gamma $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), which is also known as an open segment. Elements within the derivation are referred to as points or segments. The straight line containing \u003Cinline-formula\u003E\u003Ctex-math id=\"M17\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is denoted by \u003Cinline-formula\u003E\u003Ctex-math id=\"M18\"\u003E$$ \\overleftrightarrow{M} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The same line directed from \u003Cinline-formula\u003E\u003Ctex-math id=\"M19\"\u003E$$ \\alpha $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E to \u003Cinline-formula\u003E\u003Ctex-math id=\"M20\"\u003E$$ \\gamma $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is denoted by \u003Cinline-formula\u003E\u003Ctex-math id=\"M21\"\u003E$$ \\vec M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EDEFINITION 2. The projection \u003Cinline-formula\u003E\u003Ctex-math id=\"M22\"\u003E$$ \\sigma(\\epsilon, M) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of a point \u003Cinline-formula\u003E\u003Ctex-math id=\"M23\"\u003E$$ q $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E onto a closed segment \u003Cinline-formula\u003E\u003Ctex-math id=\"M24\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, is the intersection of \u003Cinline-formula\u003E\u003Ctex-math id=\"M25\"\u003E$$ \\overleftrightarrow{M} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and is perpendicular to \u003Cinline-formula\u003E\u003Ctex-math id=\"M26\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and passing through \u003Cinline-formula\u003E\u003Ctex-math id=\"M27\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EDEFINITION 3. The distance between \u003Cinline-formula\u003E\u003Ctex-math id=\"M28\"\u003E$$ \\omega (\\epsilon, M) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is a point \u003Cinline-formula\u003E\u003Ctex-math id=\"M29\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and a closed segment \u003Cinline-formula\u003E\u003Ctex-math id=\"M30\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E in the Euclidean metric is defined as the distance \u003Cinline-formula\u003E\u003Ctex-math id=\"M31\"\u003E$$ \\omega(\\epsilon, \\sigma(\\epsilon, M)) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E between the point \u003Cinline-formula\u003E\u003Ctex-math id=\"M32\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and its projection onto \u003Cinline-formula\u003E\u003Ctex-math id=\"M33\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E if \u003Cinline-formula\u003E\u003Ctex-math id=\"M34\"\u003E$$ \\sigma(\\epsilon, M)) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E belongs to \u003Cinline-formula\u003E\u003Ctex-math id=\"M35\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and is \u003Cinline-formula\u003E\u003Ctex-math id=\"M36\"\u003E$$ \\min $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M37\"\u003E$$ (\\omega(\\epsilon, \\alpha), \\omega(\\epsilon, \\gamma)) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E otherwise. In other words, \u003Cinline-formula\u003E\u003Ctex-math id=\"M38\"\u003E$$ \\omega(\\epsilon, M) = $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M39\"\u003E$$ \\min_{u \\in M} (\\epsilon, u ) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The point of \u003Cinline-formula\u003E\u003Ctex-math id=\"M40\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, which is closest to \u003Cinline-formula\u003E\u003Ctex-math id=\"M41\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, is called the image \u003Cinline-formula\u003E\u003Ctex-math id=\"M42\"\u003E$$ I(\\epsilon, M) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of \u003Cinline-formula\u003E\u003Ctex-math id=\"M43\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E on \u003Cinline-formula\u003E\u003Ctex-math id=\"M44\"\u003E$$ M $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EDEFINITION 4. The bisector \u003Cinline-formula\u003E\u003Ctex-math id=\"M45\"\u003E$$ \\beta(c_{i}, c_{j}) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of two elements {\u003Cinline-formula\u003E\u003Ctex-math id=\"M46\"\u003E$$ c_{i} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M47\"\u003E$$ c_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E} is the locus of points equidistant from \u003Cinline-formula\u003E\u003Ctex-math id=\"M48\"\u003E$$ c_{i} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M49\"\u003E$$ c_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The bisector \u003Cinline-formula\u003E\u003Ctex-math id=\"M50\"\u003E$$ \\beta(Z, Q) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of two sets of elements \u003Cinline-formula\u003E\u003Ctex-math id=\"M51\"\u003E$$ Z $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M52\"\u003E$$ Q $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is defined to be the locus of points equidistant from \u003Cinline-formula\u003E\u003Ctex-math id=\"M53\"\u003E$$ Z $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M54\"\u003E$$ Q $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, where the distance \u003Cinline-formula\u003E\u003Ctex-math id=\"M55\"\u003E$$ \\omega(\\epsilon, Z) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E between a point \u003Cinline-formula\u003E\u003Ctex-math id=\"M56\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and a set of elements \u003Cinline-formula\u003E\u003Ctex-math id=\"M57\"\u003E$$ Z $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is defined to be \u003Cinline-formula\u003E\u003Ctex-math id=\"M58\"\u003E$$ \\min_{c \\in Z}\\omega(\\epsilon, c ) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The bisector \u003Cinline-formula\u003E\u003Ctex-math id=\"M59\"\u003E$$ \\beta(c_{i}, c_{j}) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is said to be oriented if a direction is imposed upon it so that elements \u003Cinline-formula\u003E\u003Ctex-math id=\"M60\"\u003E$$ c_{i} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M61\"\u003E$$ c_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E lie to the left and the right of it, respectively. An oriented bisector \u003Cinline-formula\u003E\u003Ctex-math id=\"M62\"\u003E$$ \\beta(Z, Q) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is defined similarly.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EUtilizing these four definitions, we can easily establish {edges} {among} spaces or obstacles, which uses the equidistant to create an optimal edge that lies directly between the two spaces. As seen in the above {definitions, } we can also create edges between irregular-shaped spaces {and} obstacles.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ETo characterize the GVD, we expect the robot to be operated at a point within the workspace, a \u003Cinline-formula\u003E\u003Ctex-math id=\"M63\"\u003E$$ W $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, which is populated by convex obstacles \u003Cinline-formula\u003E\u003Ctex-math id=\"M64\"\u003E$$ C_{1}, . . . , C_{n} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Non-convex obstacles are displayed as the union of convex shapes. The distance between a point and an obstacle is the minimal distance between the point and all points of the obstacle. The distance function, and its \"gradient\" are represented as:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(2)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E2\"\u003E $$ \\begin{equation} d_i(x) = min_{c_{0} \\in C_{i}} \\| x-c_0 \\| \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(3)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E3\"\u003E $$ \\begin{equation} \\nabla d_i(x) = \\frac{x - c_{0}} {\\| x-c_0 \\|} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere in Equation (2) \u003Cinline-formula\u003E\u003Ctex-math id=\"M65\"\u003E$$ d_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the distance to obstacle \u003Cinline-formula\u003E\u003Ctex-math id=\"M66\"\u003E$$ C_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E from a point \u003Cinline-formula\u003E\u003Ctex-math id=\"M67\"\u003E$$ x $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and in Equation (3) \u003Cinline-formula\u003E\u003Ctex-math id=\"M68\"\u003E$$ \\nabla d_i(x) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the unit vector in the direction from \u003Cinline-formula\u003E\u003Ctex-math id=\"M69\"\u003E$$ x $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E to \u003Cinline-formula\u003E\u003Ctex-math id=\"M70\"\u003E$$ c_0 $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, where \u003Cinline-formula\u003E\u003Ctex-math id=\"M71\"\u003E$$ c_0 $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the closest point to \u003Cinline-formula\u003E\u003Ctex-math id=\"M72\"\u003E$$ x $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E in \u003Cinline-formula\u003E\u003Ctex-math id=\"M73\"\u003E$$ C_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The essential structure block of the GVD is the arrangement of points equidistant to two sets \u003Cinline-formula\u003E\u003Ctex-math id=\"M74\"\u003E$$ C_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M75\"\u003E$$ C_j $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, with the end goal that each set in this set is the minimal distance to the obstacles \u003Cinline-formula\u003E\u003Ctex-math id=\"M76\"\u003E$$ C_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M77\"\u003E$$ C_j $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E than other obstacles. This type of structure is known as the two-equidistant face,\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(4)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E4\"\u003E $$ \\begin{equation} f(i, j)=\\left\\{\\begin{array}{c} \t\tx \\in \\mathbb{R}^{m}: 0 \\leq d_{i}(x)=d_{h}(x)\\\\ \t\t\\forall d_{i} \\neq i, j \\\\ \t\t\\nabla d_{i}(x) \\neq \\nabla d_{j}(x) \t\\end{array}\\right. \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EEach face has a co-dimension in the ambient space, which causes the two-equidistant faces to be seen as one-dimensional. The intersection of both faces forms the GVD and is denoted by the following equation:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(5)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E5\"\u003E $$ \\begin{equation} GVD=\\bigcup\\limits_{i=1}^{n-1} \\bigcup\\limits_{j=i+1}^{n} f(i, j) \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec13\" class=\"article-Section\"\u003E\u003Ch2 \u003E3. ADJACENT NODE SELECTION ALGORITHM\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EThe details of the adjacent node selection algorithm are shown in \u003Ca href=\"#Figure3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure3\"\u003EFigure 3\u003C\u002Fa\u003E. Within the search range obtained, we applyIPSO-based path planning algorithm to generate the connection path from the waypoint to all the potential nodes in the range. The connection path is planned in the grid-based map as shown in \u003Ca href=\"#Figure3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure3\"\u003EFigure 3A\u003C\u002Fa\u003E. The search range in the larger map is shown in \u003Ca href=\"#Figure3\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure3\"\u003EFigure 3B\u003C\u002Fa\u003E, which is also a part of \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6\u003C\u002Fa\u003E. Finally, with the formation of a collision-free path to the adjacent nodes from the two waypoints, we obtain the optimal path selection by calculating the overall path length \u003Cinline-formula\u003E\u003Ctex-math id=\"M78\"\u003E$$ \\mathcal{L}_{c} + \\mathcal{L}_{e} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure3\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure3\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-3.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 3. Details of the Adjacent Node Selection (ANS) algorithm. (A) The IPSO-based connection path planning in the search range. (B) The search range determination and node selection inside. (C) The final generated path with the minimum path length.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003ETo show the necessity of IPSO-based path planning from waypoints to adjacent nodes, a more specific scenario is shown in \u003Ca href=\"#Figure4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure4\"\u003EFigure 4\u003C\u002Fa\u003E. In \u003Ca href=\"#Figure4\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure4\"\u003EFigure 4A\u003C\u002Fa\u003E, all straight lines connecting nodes and waypoints are separated by obstacles. Among them, points \u003Cinline-formula\u003E\u003Ctex-math id=\"M79\"\u003E$$ i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and point \u003Cinline-formula\u003E\u003Ctex-math id=\"M80\"\u003E$$ j $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are located in our search range. If we only rely on the Euclidean distance between node and waypoint, it can be found that the point \u003Cinline-formula\u003E\u003Ctex-math id=\"M81\"\u003E$$ i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is closer to the waypoint. However, after considering the path with obstacle avoidance, the path for point \u003Cinline-formula\u003E\u003Ctex-math id=\"M82\"\u003E$$ i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E to the waypoint is longer. To sum up, the obtained search range with ANS algorithm and the IPSO-based path planning algorithm can reduce the computational cost while ensuring a short and safe trajectory. The details of the procedure are described in Algorithm 2.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure4\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure4\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-4.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 4. Illustration of ANS method within a more specific sense, where an obstacle obstructing the connection path. (A) The multiple connection paths have been obstructed by the obstacles. (B) It selects the nodes in the defined range. (C) It conducts IPSO point-to-point algorithm to achieve the optimal path to the selected node.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EWe carry out further discussions on our proposed ANS algorithm. \u003Ca href=\"#Figure5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure5\"\u003EFigure 5A\u003C\u002Fa\u003E and \u003Ca href=\"#Figure5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure5\"\u003EFigure 5B\u003C\u002Fa\u003E are parts of \u003Ca href=\"#Figure11\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure11\"\u003EFigure 11\u003C\u002Fa\u003E and \u003Ca href=\"#Figure13\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure13\"\u003EFigure 13\u003C\u002Fa\u003E (enclosed in pink dashed boxes), respectively. As shown in \u003Ca href=\"#Figure5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure5\"\u003EFigure 5A\u003C\u002Fa\u003E, the solid circles represent the search radius of the waypoints in the simulation and the red solid dots depict the nodes in the workspace. In \u003Ca href=\"#Figure11\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure11\"\u003EFigure 11\u003C\u002Fa\u003E, we end up with the trajectory that follows the generated safe-aware road. However, if the space in the map is more sparse, our search radius \u003Cinline-formula\u003E\u003Ctex-math id=\"M83\"\u003E$$ \\mathcal{R} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E may increase to \u003Cinline-formula\u003E\u003Ctex-math id=\"M84\"\u003E$$ \\mathcal{R}' $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, which may achieve the waypoints in their respective search spaces. Thus, instead of following the safe-awareness road, a new connection path is obtained through the improved PSO algorithm directly, as shown in dashed lines in \u003Ca href=\"#Figure5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure5\"\u003EFigure 5A\u003C\u002Fa\u003E. Moreover, the sparseness of the overall workspace may not represent the complexity of local obstacles. Therefore, the choice of the radius of the search space may require more mathematical proof and analysis.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure5\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure5\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-5.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 5. Illustration of the ANS algorithm analysis. (A) From \u003Ca href=\"#Figure11\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure11\"\u003EFigure 11\u003C\u002Fa\u003E it is enclosed by a pink dashed box. (B) From \u003Ca href=\"#Figure13\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure13\"\u003EFigure 13\u003C\u002Fa\u003E it is enclosed by a pink dashed box.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure6\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure6\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-6.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 6. Illustration of the Adjacent Node Selection (ANS) algorithm. (A) The workspace with nodes, edges and waypoints. (B) The node selection in the search range. (C) The final generated path.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure7\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure7\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-7.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 7. Illustration of the B-spline function. (A) The improved segmented B-spline curve. (B) The same path smoothed by the fundamental B-spline and the improved B-spline function.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure8\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure8\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-8.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 8. Illustration of how the VHF uses a probability along with histogram-based grid to detect and build a map simultaneously.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure9\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure9\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-9.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 9. Robot sensor configuration for multi-waypoint navigation and mapping.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure10\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure10\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-10.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 10. Illustration of the path created from the other models\u003Csup\u003E[\u003Ca href=\"#b43\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b43\"\u003E43\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. (A) It depicts the path created by Zhang \u003Ci\u003Eet al\u003C\u002Fi\u003E.'s model by the green lines (redrawn by Zhang \u003Ci\u003Eet al\u003C\u002Fi\u003E., 2021\u003Csup\u003E[\u003Ca href=\"#b43\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b43\"\u003E43\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure11\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure11\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-11.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 11. Illustration of the path created from the other models\u003Csup\u003E[\u003Ca href=\"#b44\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b44\"\u003E44\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. (A) It depicts the path created by Asl and Taghirad model by the green lines (redrawn by Asl and Taghirad, 2019\u003Csup\u003E[\u003Ca href=\"#b44\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b44\"\u003E44\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure12\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure12\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-12.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 12. Illustration of the path created from the compared models\u003Csup\u003E[\u003Ca href=\"#b45\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b45\"\u003E45\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. (A) It depicts the path created by Zhuang \u003Ci\u003Eet al\u003C\u002Fi\u003E.'s model by the green lines (redrawn from Zhuang \u003Ci\u003Eet al\u003C\u002Fi\u003E., 2021\u003Csup\u003E[\u003Ca href=\"#b45\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b45\"\u003E45\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure13\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure13\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-13.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 13. Illustration of the path created from the compared models\u003Csup\u003E[\u003Ca href=\"#b46\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b46\"\u003E46\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. (A) It depicts the path created by Vonásek's model shown by the green lines (redrawn from Vonásek and Penicka, 2019\u003Csup\u003E[\u003Ca href=\"#b46\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b46\"\u003E46\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E). (B) It represents the proposed method point order and traversed path. The point order is illustrated by the violet arrows, while the orange path represents the robot path. Safety-aware roads are depicted by the blue dashed lines. The waypoints are illustrated by the violet circles. (C) It depicts how the B-spline curve is applied to the known path, which smooths and reduces the path for a local navigator to traverse.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EWith the increasing radius of the potential search range, more nodes are applicable for selection, such as the nodes connected with dashed lines in \u003Ca href=\"#Figure5\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure5\"\u003EFigure 5B\u003C\u002Fa\u003E. Enlarging the search space may avoid some unnecessary detours and give a shorter path. Therefore, the trade-off between path length and safety of the autonomous robot still requires more consideration.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec14\" class=\"article-Section\"\u003E\u003Ch2 \u003E4. IMPROVED PSO-BASED MULTI-WAYPOINT NAVIGATION\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EThe particle swarm optimization (PSO) algorithm is a swarm-based bio-inspired algorithm based on the behavioral observation of birds. It uses an iterative methodology to optimize randomly initialized particles to define a path from the initial position to the goal \u003Csup\u003E[\u003Ca href=\"#b27\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b27\"\u003E27\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. In this section an improved PSO (IPSO) algorithm by introduction of a weighted particles is addressed to resolve the multi-waypoint sequence issue.\u003C\u002Fp\u003E\u003Cdiv id=\"sec22\" class=\"article-Section\"\u003E\u003Ch3 \u003E4.1. Multi-waypoint visiting sequence\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EIn real-world scenarios, one important factor is that the GPS coordinates provide portions for the multiple waypoints. Traveling from one waypoint to another, the distance between them determines their associated cost. The primary purpose of traveling from one waypoint to another is to simultaneously find the minimal cost of all generated trajectories. Using the coordinates of each waypoint and the PSO algorithm, the minimal-distance path can be found within the environment. The PSO algorithm finds the best waypoint visiting sequence by initializing randomized particles. The algorithm denotes the local best position as {\u003Cinline-formula\u003E\u003Ctex-math id=\"M85\"\u003E$$ x^b $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E} and the global best position as {\u003Cinline-formula\u003E\u003Ctex-math id=\"M86\"\u003E$$ x^g $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E}. Then by taking advantage of a fitness function, the algorithm guides each particle towards the local and global best positions. The particle velocities are updated as follows:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(6)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E6\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\tv_{p}(t+1)=v_{p}(t)+ \\alpha_{1}\\omega_{1}[ {x^b_{p}(t)}-x_{p}(t)]+ \\alpha_{2}\\omega_{2}[ {x^g_{p}(t)}-x_{p}(t)] \t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(7)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E7\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t {x_{p}^{(t+1)}=x_{p}^{(t)}+ v_{p}^{(t+1)}} \t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere, \u003Cinline-formula\u003E\u003Ctex-math id=\"M87\"\u003E$$ v_p(t) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the velocity of particle \u003Cinline-formula\u003E\u003Ctex-math id=\"M88\"\u003E$$ p $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E at instant \u003Cinline-formula\u003E\u003Ctex-math id=\"M89\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, {\u003Cinline-formula\u003E\u003Ctex-math id=\"M90\"\u003E$$ x_p(t) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E} is the position of particle \u003Cinline-formula\u003E\u003Ctex-math id=\"M91\"\u003E$$ p $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E at instant \u003Cinline-formula\u003E\u003Ctex-math id=\"M92\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M93\"\u003E$$ \\alpha_{1} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M94\"\u003E$$ \\alpha_{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are the positive acceleration constants used to scale the contribution of cognitive and social components. \u003Cinline-formula\u003E\u003Ctex-math id=\"M95\"\u003E$$ \\omega_{1} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M96\"\u003E$$ \\omega_{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are the uniform random number between 0 and 1. {\u003Cinline-formula\u003E\u003Ctex-math id=\"M97\"\u003E$$ x^b_p(t) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E} is the best position the particle \u003Cinline-formula\u003E\u003Ctex-math id=\"M98\"\u003E$$ p $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E achieved up to instant \u003Cinline-formula\u003E\u003Ctex-math id=\"M99\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E at current iteration, {\u003Cinline-formula\u003E\u003Ctex-math id=\"M100\"\u003E$$ x^g_p(t) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E} is the {global} position that any of \u003Cinline-formula\u003E\u003Ctex-math id=\"M101\"\u003E$$ p $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E's neighbors has reached up to instant \u003Cinline-formula\u003E\u003Ctex-math id=\"M102\"\u003E$$ t $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. However, if a particle \u003Cinline-formula\u003E\u003Ctex-math id=\"M103\"\u003E$$ p_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E lies close to the \u003Cinline-formula\u003E\u003Ctex-math id=\"M104\"\u003E$$ x^b_p(t) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M105\"\u003E$$ x^g_p(t) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, only one term guides the \u003Cinline-formula\u003E\u003Ctex-math id=\"M106\"\u003E$$ p_i $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E to search the potential solution.The optimization process in our navigation issue is more likely trapped in local minima. Thus, an improved PSO algorithm is utilized to provide a more promising search direction for all particles during the optimization process.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(8)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E8\"\u003E $$ \\begin{equation} x^{w}=\\sum\\limits_{i=1}^{P} \\bar{\\alpha}_{i}^{w} {x^b_i(t)} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(9)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E9\"\u003E $$ \\begin{equation} \\bar{\\alpha}_{i}^{W}=\\frac{\\hat{\\alpha}_{i}^{W}}{\\sum_{j=1}^{P} \\hat{\\alpha}_{i}^{W}} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(10)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E10\"\u003E $$ \\begin{equation} \\begin{aligned} \t\t&\\hat{\\alpha}_{i}^{W}=\\frac{\\max _{1 \\leq k \\leq M}\\left(\\mathcal{F}\\left( {x^b_k(t)}\\right)\\right)-\\mathcal{F}\\left( {x^b_i(t)}\\right)+\\varepsilon}{\\max _{1 \\leq k \\leq M}\\left(\\mathcal{F}\\left( {x^b_k(t)}\\right)\\right)-\\min _{1 \\leq k \\leq M}\\left(\\mathcal{F}\\left( {x^b_k(t)}\\right)\\right)+\\varepsilon}, \\quad i=1, 2, \\ldots, M, \\end{aligned} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M107\"\u003E$$ \\varepsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is a positive constant, \u003Cinline-formula\u003E\u003Ctex-math id=\"M108\"\u003E$$ \\hat{\\alpha} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the weighted constant of each particle. \u003Cinline-formula\u003E\u003Ctex-math id=\"M109\"\u003E$$ \\mathcal{F}(\\cdot) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the fitness function. The worst and the best fitness values of all personal best particles are represented by \u003Cinline-formula\u003E\u003Ctex-math id=\"M110\"\u003E$$ \\max _{1 \\leq k \\leq M}(\\mathcal{F}\\left(x_{k}^{P}\\right)) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M111\"\u003E$$ \\min _{1 \\leq k \\leq M}(\\mathcal{F}\\left(x_{k}^{P}\\right)) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, respectively.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe order is optimized through this method, in which each waypoint is visited. A sequence of particles are initialized to compose a population in the original PSO algorithm. A possible optimal solution to an optimization issue in our multi-waypoint sequence is discovered by one particle in the PSO. This particle indicates a possible optimal solution to the multi-waypoint navigation issue and moves to explore an optimal solution in a certain search space. In this paper, a weighted particle is introduced into a swarm to suggest a more reasonable search direction for all the particles. As a result, the best position of particle and neighbor guides the particle to move along the corrected direction for better covergence.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe multi-waypoint visiting sequence problem can be used to solve the transportation planning problem and Covid-19 disinfection robot path planning in hospitals, in which agents (vehicles) need to be delivered as well as the overall cost and time need to be minimized \u003Csup\u003E[\u003Ca href=\"#b14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b14\"\u003E14\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The algorithm of the improved PSO to finding multi-waypoint visiting sequence is explained in Algorithm 1. The objective of the algorithm is to minimize the total trajectory length of Cartesian coordinates \u003Cinline-formula\u003E\u003Ctex-math id=\"M112\"\u003E$$ (X_n, Y_n) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of waypoints given.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn the IPSO model the \u003Cinline-formula\u003E\u003Ctex-math id=\"M113\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the particle agent, which is denoted by the \u003Cinline-formula\u003E\u003Ctex-math id=\"M114\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E : {\u003Cinline-formula\u003E\u003Ctex-math id=\"M115\"\u003E$$ \\mathcal{P}_b $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M116\"\u003E$$ \\mathcal{P}_g $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M117\"\u003E$$ \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E}. \u003Cinline-formula\u003E\u003Ctex-math id=\"M118\"\u003E$$ \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E holds a group of particle agents which are predetermined as neighbors of \u003Cinline-formula\u003E\u003Ctex-math id=\"M119\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The \u003Cinline-formula\u003E\u003Ctex-math id=\"M120\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are defined by the following parameters.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(1) Each \u003Cinline-formula\u003E\u003Ctex-math id=\"M121\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E requests each neighbor's current personal best location.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(2) Each \u003Cinline-formula\u003E\u003Ctex-math id=\"M122\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E returns its current personal best to neighbors.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(3) Obtains the center location of each surrounding cluster.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(4) Determines if the current position has been visited.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(5) Determines if current position is optimal if not record current position.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EEvery \u003Cinline-formula\u003E\u003Ctex-math id=\"M123\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E obtains a set of neighbors of positions during the initial setup. Utilizing a variety of topologies one can create numerous properties of neighboring particle agents to obtain better performance. Within the proposed model, we assume that each \u003Cinline-formula\u003E\u003Ctex-math id=\"M124\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E has a static set of neighbors. Each \u003Cinline-formula\u003E\u003Ctex-math id=\"M125\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E keeps track of its local best solution, \u003Cinline-formula\u003E\u003Ctex-math id=\"M126\"\u003E$$ \\mathcal{P}_b $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, which is where a solution closest to the optimal solution is found in the problem space, while the global best solution is recorded in the parameter \u003Cinline-formula\u003E\u003Ctex-math id=\"M127\"\u003E$$ \\mathcal{P}_g $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. During each iteration the \u003Cinline-formula\u003E\u003Ctex-math id=\"M128\"\u003E$$ \\mathcal{P}_A $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E evaluates its current position and determines if it needs to perform a fitness evaluation, while simultaneous checking if the termination criteria has been met. If the termination criteria have not been met then it updates its \u003Cinline-formula\u003E\u003Ctex-math id=\"M129\"\u003E$$ \\mathcal{P}_b $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, obtains the neighbor's \u003Cinline-formula\u003E\u003Ctex-math id=\"M130\"\u003E$$ \\mathcal{P}_b $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and calculates the \u003Cinline-formula\u003E\u003Ctex-math id=\"M131\"\u003E$$ \\mathcal{P}_g $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and marks the current position as visited.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Ctable-wrap\u003E\u003Ctable\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"left\"\u003E\u003Cb\u003EAlgorithm 1:\u003C\u002Fb\u003E Improved PSO (IPSO) algorithm for waypoint sequencing\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"left\"\u003E\u003Cb\u003EInitialize a population of particles\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\"\u003ESet the size of the swarm to \u003Cinline-formula\u003E\u003Ctex-math id=\"M132\"\u003E$$ S_{p} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, the maximum number of iteration \u003Cinline-formula\u003E\u003Ctex-math id=\"M133\"\u003E$$ T_{max} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E.\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\"\u003E\u003Cinline-formula id=\"5170-M1\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" class=\"inline-graphic\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-M1.jpg\" \u002F\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Ftable-wrap\u003E\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec23\" class=\"article-Section\"\u003E\u003Ch3 \u003E4.2. Safety-aware IPSO multi-waypoint path planning\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ETo ensure the robot safely reaches the waypoints via the planned visiting sequence, the safety-aware road is selected to guide the autonomous robot. Nevertheless, there is a problem with the connection path from the location of the waypoints to safety-aware roads. When we integrate the position information of a waypoint in the workspace, it may be necessary to obtain the collision-free connection path length from the waypoint to all nodes, which is computationally expensive. As shown in \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6A\u003C\u002Fa\u003E, with \u003Cinline-formula\u003E\u003Ctex-math id=\"M134\"\u003E$$ \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E nodes obtained in the workspace, there are \u003Cinline-formula\u003E\u003Ctex-math id=\"M135\"\u003E$$ \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E possible connection paths from the waypoints to the nodes in the workspace. With the initial \u003Cinline-formula\u003E\u003Ctex-math id=\"M136\"\u003E$$ \\mathcal{N} \\times \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E adjacent distance matrix obtained from GVD graph and the increasing \u003Cinline-formula\u003E\u003Ctex-math id=\"M137\"\u003E$$ \\mathcal{M} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E waypoints in the workspace, the size of new distance matrix is expanded to \u003Cinline-formula\u003E\u003Ctex-math id=\"M138\"\u003E$$ (\\mathcal{N}+\\mathcal{M}) \\times (\\mathcal{N}+\\mathcal{M}) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Since most of the connection path computations are unnecessary, a new adjacent node selection (ANS) algorithm is proposed to reduce the computational effort by restricting the search space in local regions rather than the entire working region.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe local regions are shown in \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6B\u003C\u002Fa\u003E, and the dark blue nodes, such as \u003Cinline-formula\u003E\u003Ctex-math id=\"M139\"\u003E$$ \\alpha $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M140\"\u003E$$ \\beta $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E nodes in the regions, are potential adjacent nodes to connect. The radius \u003Cinline-formula\u003E\u003Ctex-math id=\"M141\"\u003E$$ \\mathcal{R} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of the local area shall be determined by the potential workspace, which determines the number of nodes in the search range. For instance, in an area with clustered obstacles, there are many nodes in the environment; thus, the search radius may be small. However, In an area with sparse obstacles, there are fewer nodes in the environment; thus, the search radius needs to be larger to include all potential nodes.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ESince the entire workspace is projected through the GVD graph, we can interpret the sparseness of the entire workspace through the distance of the edge list \u003Cinline-formula\u003E\u003Ctex-math id=\"M142\"\u003E$$ \\mathcal{E} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Nevertheless, some edge distances cannot represent the sparseness of the entire workspace, such as the edges enclosed by the red dotted line in \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6B\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ETherefore, to exclude these extraneous edges of distance, we sort the entire edge list before removing the lowest \u003Cinline-formula\u003E\u003Ctex-math id=\"M143\"\u003E$$ 10\\% $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The radius \u003Cinline-formula\u003E\u003Ctex-math id=\"M144\"\u003E$$ \\mathcal{R} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of the search range is defined as the average of the remaining edge distance. After obtaining the nodes in the search environment, we plan a collision-free trajectory from the current waypoint to reach each node through the IPSO algorithm.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe IPSO navigation algorithm is initiated in the search range to obtain the optimal path. The local search region is interpreted into grid-based map, where the obstacle areas are inaccessible grids in the workspace. Dijkstra's algorithm is utilized as a local search algorithm and the cost function within the model. For graph-based maps, they must be a method for traveling from one node to another utilizing the edges within the graph. One method of this is Dijkstra's algorithm, which utilizes a weighted graph to determine the shortest path from a source node to a target node. The algorithm also keeps track of the known shortest distances from each node while simultaneously updating their weights to improve the overall shortest path from each node. By recursively establishing a path with random solutions generated in the workspace, the IPSO algorithm can construct the collision-free path with the least fitness value, which also represents the path with minimum length. Therefore, the length of the connecting path \u003Cinline-formula\u003E\u003Ctex-math id=\"M145\"\u003E$$ \\mathcal{L}_{c} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E can be obtained, and by combining the length of the GVD path \u003Cinline-formula\u003E\u003Ctex-math id=\"M146\"\u003E$$ \\mathcal{L}_{e} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, the optimal safety-aware trajectory is obtained as shown in \u003Ca href=\"#Figure6\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure6\"\u003EFigure 6C\u003C\u002Fa\u003E.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003ETo effectively reduce and smooth the overall path from each waypoint to the waypoint, various methods have been developed to achieve this goal, for example, the \u003Cinline-formula\u003E\u003Ctex-math id=\"M147\"\u003E$$ \\mathcal{B} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline curve, which works by taking a set of points that are used to curve sharp close turns around obstacles and is very effective and thus widely used in smooth polylines, due to its closed-form expression of the position coordinates. The original methodology of the \u003Cinline-formula\u003E\u003Ctex-math id=\"M148\"\u003E$$ \\mathcal{B} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline curve method, in some cases, changes the trajectory of the original path created by the global navigation system. The problem can be mediated by implementing the piecewise \u003Cinline-formula\u003E\u003Ctex-math id=\"M149\"\u003E$$ \\mathcal{B} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline method, which only smooths the path around each obstacle. Lei \u003Ci\u003Eet al.\u003C\u002Fi\u003E evaluated the effectiveness of the improved \u003Cinline-formula\u003E\u003Ctex-math id=\"M150\"\u003E$$ \\mathcal{B} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline method, and ultimately found that the hybrid method was able to reduce the overall all path in point-to-point navigation \u003Csup\u003E[\u003Ca href=\"#b38\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b38\"\u003E38\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The \u003Cinline-formula\u003E\u003Ctex-math id=\"M151\"\u003E$$ \\mathcal{B} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline curve can be defined by a cardinal functions \u003Cinline-formula\u003E\u003Ctex-math id=\"M152\"\u003E$$ L_{j, r}(q) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, control points \u003Cinline-formula\u003E\u003Ctex-math id=\"M153\"\u003E$$ B_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and degree \u003Cinline-formula\u003E\u003Ctex-math id=\"M154\"\u003E$$ (r - 1), $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E which is given by the following equations.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(11)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E11\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t\tK(q)= \\sum L_{j, r}(q) B_{j} \t\t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M155\"\u003E$$ B_{j} = [B_{jx}, B_{jy}] $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are the \u003Cinline-formula\u003E\u003Ctex-math id=\"M156\"\u003E$$ (n + 1) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E control points and a knot vector \u003Cinline-formula\u003E\u003Ctex-math id=\"M157\"\u003E$$ u $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. \u003Cinline-formula\u003E\u003Ctex-math id=\"M158\"\u003E$$ N_{i, k}(u) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E are the basic functions, which are defined recursively as follows:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(12)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E12\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t\tL_{j, r}(q) = \\frac{(q-x_{j})}{x_{j+r-1}-x_{j}}\tN_{j, r-1}(q) + \\frac{(x_{j+r}-q)}{x_{j+r}-x_{j+1}} N_{j+1, r-1}(q) \t\t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(13)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E13\"\u003E $$ \\begin{equation} L_{j, r}(q) = \t\t\\begin{cases} \t\t\t1, \\; \\; \\; \\; q_{j} q_{j}\\le q \\le q_{j+1}\\\\ \t\t\t0, \\; \\; \\; \\; \\; \\;\\; \\; otherwise \t\t\\end{cases} \t\t; \\; q \\in [0, 1] \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EGeometric continuity \u003Cinline-formula\u003E\u003Ctex-math id=\"M159\"\u003E$$ G^2 $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the metric used to evaluate smoothing methods, which is defined by the tangent unit and curvature vector at the intersection of two continuous segments \u003Csup\u003E[\u003Ca href=\"#b38\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b38\"\u003E38\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E [\u003Ca href=\"#Figure7\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure7\"\u003EFigure 7A\u003C\u002Fa\u003E]. To achieve \u003Cinline-formula\u003E\u003Ctex-math id=\"M160\"\u003E$$ G^2 $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E continuity, the control points \u003Cinline-formula\u003E\u003Ctex-math id=\"M161\"\u003E$$ B_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E of \u003Cinline-formula\u003E\u003Ctex-math id=\"M162\"\u003E$$ \\mathcal{B} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline curve to the path point, \u003Cinline-formula\u003E\u003Ctex-math id=\"M163\"\u003E$$ Y_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, is defined as\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(14)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E14\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t\tB_{1} = Y_{j} - (1 +v)s_{2}q_{j-1}\\\\ \t\t\tB_{2} = -s_{2}q_{j-1}\\\\ \t\t\tB_{3} = Y_{j}\\\\ \t\t\tB_{4} = Y_{j}+s_{2}q_{j}\\\\ \t\t\tB_{5} = Y_{j} +(1 +v)s_{2}q_{j-1} \t\t\t \t\t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ewhere \u003Cinline-formula\u003E\u003Ctex-math id=\"M164\"\u003E$$ c $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is smoothing length ratio \u003Cinline-formula\u003E\u003Ctex-math id=\"M165\"\u003E$$ v= s_{1}\u002Fs_{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, and \u003Cinline-formula\u003E\u003Ctex-math id=\"M166\"\u003E$$ q_{j-1} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E defines the unit vector of \u003Cinline-formula\u003E\u003Ctex-math id=\"M167\"\u003E$$ Y_{j-1}Y_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. \u003Cinline-formula\u003E\u003Ctex-math id=\"M168\"\u003E$$ q_{j} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E represents the unit vector of the line \u003Cinline-formula\u003E\u003Ctex-math id=\"M169\"\u003E$$ Y_{j}Y_{j+1} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The combined sum of \u003Cinline-formula\u003E\u003Ctex-math id=\"M170\"\u003E$$ s_{1} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and \u003Cinline-formula\u003E\u003Ctex-math id=\"M171\"\u003E$$ s_{2} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is the smoothed length. The half of the corner angle is denoted as: \u003Cinline-formula\u003E\u003Ctex-math id=\"M172\"\u003E$$ \\Gamma = \\beta \u002F 2 $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. Using a knot vector of [0, 0, 0, 0, 0.5, 1, 1, 1, 1], the smoothing error distance \u003Cinline-formula\u003E\u003Ctex-math id=\"M173\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and maximum curvature \u003Cinline-formula\u003E\u003Ctex-math id=\"M174\"\u003E$$ K_{max} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E within the smooth path can be expressed as:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(15)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E15\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t\t \t\t\t \\epsilon = \\frac{s_{2}\\sin{\\Gamma}}{2} \t\t\t \t\t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(16)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E16\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t\t R_{max} = \\frac{4\\sin{\\Gamma}}{3s_{2}\\cos^2{\\Gamma}} \t\t\t \t\t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EUsing the previous equations the smoothing error distance \u003Cinline-formula\u003E\u003Ctex-math id=\"M175\"\u003E$$ \\epsilon $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E can be defined by the existing maximum curvature \u003Cinline-formula\u003E\u003Ctex-math id=\"M176\"\u003E$$ R_{max} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E given by the robot:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Cdiv class=\"disp-formula\"\u003E\u003Clabel\u003E(17)\u003C\u002Flabel\u003E\u003Ctex-math id=\"E17\"\u003E $$ \\begin{equation} \\begin{array}{l} \t\t\t \\epsilon = \\frac{2\\tan^2{\\Gamma}}{3R_{max}} \t\t\t \t\t\\end{array} \\end{equation} $$ \u003C\u002Ftex-math\u003E\u003C\u002Fdiv\u003E\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe improved \u003Cinline-formula\u003E\u003Ctex-math id=\"M177\"\u003E$$\\mathcal{B}$$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline model has specific advantages over the basic \u003Cinline-formula\u003E\u003Ctex-math id=\"M178\"\u003E$$\\mathcal{B}$$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline model, one of which is its ability to smooth many different trajectories with various angles, as seen in \u003Ca href=\"#Figure7\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure7\"\u003EFigure 7B\u003C\u002Fa\u003E. The curve produced by the improved \u003Cinline-formula\u003E\u003Ctex-math id=\"M179\"\u003E$$\\mathcal{B}$$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline model is significantly closer to the original path than the original model. When considering the constraints of the robot, the improved \u003Cinline-formula\u003E\u003Ctex-math id=\"M180\"\u003E$$\\mathcal{B}$$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline mode performs {better} in various degrees of angles. The overall advantages of the improved \u003Cinline-formula\u003E\u003Ctex-math id=\"M181\"\u003E$$\\mathcal{B}$$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E-spline model are as follows:\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(1) The path generated is tangent and curvature continuity, so that the robot can have a smooth steering command, which can correct any discontinuity of normal acceleration and establish a safer path for the robot to follow.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(2) The improved model generates a better curve by solely affecting the two lines within the corner of the original trajectory. Each curve generated affects others within the lines.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E(3) The improved model easily adjusts to the smoothed path based on the environment constraints or the robot.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003E\u003Ctable-wrap\u003E\u003Ctable\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"left\"\u003E\u003Cb\u003EAlgorithm 2:\u003C\u002Fb\u003E Pseudocode for the adjacent node selection (ANS) algorithm\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"left\"\u003E\u003Cb\u003EInput:\u003C\u002Fb\u003E Edge list \u003Cinline-formula\u003E\u003Ctex-math id=\"M182\"\u003E$$ \\mathcal{E} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, \u003Cinline-formula\u003E\u003Ctex-math id=\"M183\"\u003E$$ \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E nodes coordinates (\u003Cinline-formula\u003E\u003Ctex-math id=\"M184\"\u003E$$ \\mathbb{N}_x, \\mathbb{N}_y $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), \u003Cinline-formula\u003E\u003Ctex-math id=\"M185\"\u003E$$ \\mathcal{N} \\times \\mathcal{N} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E distance matrix \u003Cinline-formula\u003E\u003Ctex-math id=\"M186\"\u003E$$ \\mathcal{D} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E and the location of the waypoint \u003Cinline-formula\u003E\u003Ctex-math id=\"M187\"\u003E$$ \\alpha $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, (\u003Cinline-formula\u003E\u003Ctex-math id=\"M188\"\u003E$$ \\alpha_x, \\alpha_y $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E) and the waypoint \u003Cinline-formula\u003E\u003Ctex-math id=\"M189\"\u003E$$ \\beta $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E, (\u003Cinline-formula\u003E\u003Ctex-math id=\"M190\"\u003E$$ \\beta_x, \\beta_y $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E).\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\"\u003E\u003Cb\u003EOutput:\u003C\u002Fb\u003E The path length of the trajectory \u003Cinline-formula\u003E\u003Ctex-math id=\"M191\"\u003E$$ \\mathcal{L}_{t} $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M192\"\u003E$$ N_e = size(\\mathcal{E}) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E; \u002F\u002F Number of the edges in the workspace\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M193\"\u003E$$ [\\mathcal{E}_s, sortInd]=sort(\\mathcal{E}) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E; \u002F\u002F Sort the edge list from low to high\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\"\u003E\u003Cinline-formula\u003E\u003Ctex-math id=\"M194\"\u003E$$ N_s = \\lceil \\frac{N_e}{10} \\rceil $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E; \u002F\u002F Exclude 10% of extraneous edge distance\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"left\"\u003E\u003Cinline-formula id=\"5170-M2\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" class=\"inline-graphic\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-M2.jpg\" \u002F\u003E\u003C\u002Finline-formula\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Ftable-wrap\u003E\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec15\" class=\"article-Section\"\u003E\u003Ch2 \u003E5. REACTIVE LOCAL NAVIGATION\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EA crucial aspect in developing a multi-waypoint model is accounting for moving and unknown obstacles \u003Csup\u003E[\u003Ca href=\"#b39\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b39\"\u003E39\u003C\u002Fa\u003E, \u003Ca href=\"#b40\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b40\"\u003E40\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. In a real-world setting, not all objects are static and known. To develop a more efficient model, we propose the use of a local navigator to remedy this issue.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn order to avoid dynamic and unknown obstacles, the proposed model employs the Vector Field Histogram (VFH) model as a reactive local navigator. An autonomous robot uses a velocity command to and from each waypoint, provided by the local navigator \u003Csup\u003E[\u003Ca href=\"#b41\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b41\"\u003E41\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. By applying the VFH to the overall global trajectory with a sequence of markers, the path can be broken down into various segments to improve the efficiency in obstacle-populated workspaces. The local navigator builds a map depicting the free space and obstacles in the map by utilizing a 2D histogram grid with equally sized cells \u003Csup\u003E[\u003Ca href=\"#b42\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b42\"\u003E42\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. As the robot follows the generated trajectory within the workspace, the map is simultaneously built, shown in \u003Ca href=\"#Figure8\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure8\"\u003EFigure 8\u003C\u002Fa\u003E. In developing an autonomous obstacle avoidance model, concurrent map building and navigation are crucial. The robot pose \u003Cinline-formula\u003E\u003Ctex-math id=\"M196\"\u003E$$ (X, Y, Yaw) $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E is used to determine the map building. Thus, the precise registration of the built local map as a part of the global map can\u003C\u002Fp\u003E\u003Cp class=\"\"\u003Ebe carried out. This map building aims to construct an occupancy-cell-based map. The values for each cell in the map vary over the range [-127, 128] \u003Csup\u003E[\u003Ca href=\"#b42\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b42\"\u003E42\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The initial value is zero, which indicates that the cell is neither occupied nor unoccupied. The value is 128 if one cell is occupied with certainty and -127 if one cell is unoccupied with certainty. The values falling into (-127, 128) express contain a level of certainty in the range. When the VFH model is employed in conjunction with the GVD and IPSO algorithm, the robot can be successfully navigated through our built map with obstacle avoidance. In combination with our local navigator, a sensor configuration can be developed for the local navigator to perform it. In \u003Ca href=\"#Figure9\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure9\"\u003EFigure 9\u003C\u002Fa\u003E one can see the overview of our sensor configuration. The proposed configuration utilizes a 270-degree SICK LMS LiDAR sensor to detect obstacles within a range of 20 m @ 0.25-degree resolution. The LiDAR sensor scans at a rate of 25Hz. Then, it needs a method of finding our current position and the waypoints within the map. A Novatel's ProPak-LB Plus DGPS sensor is utilized to obtain our current position and how it correlates to the coordinates of each waypoint. Next, a PNI TCM6 digital compass is employed to establish our heading with an accuracy of 0.5 degrees. The sensor updates at 20Hz, which allows the robot to operate efficiently. Lastly, the configuration utilizes an AVT Stingray F-080C 1\u002F3\" CCD camera, which enables our robot to sense obstacles of various heights, shapes and sizes. The stingray camera is perfect for robot vision because it uses the IIDC IEEE 1394B protocol to transfer images. The system needs a computer system to house our operating system, sensor data, and programs for the robot. In this portion of the sensor configuration, a MackBook Pro i s equipped to suit our needs. The last step in the process is to establish a method of communication from the sensors to the computer systems. A sort of UART to USB hub is utilized for this purpose and fuses the sensor data together without losing any sensor information. The type of sensor confusion can be used on most ground-based robot systems for indoor and outdoor use.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec16\" class=\"article-Section\"\u003E\u003Ch2 \u003E6. SIMULATION AND COMPARISON STUDIES\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EIn this section, simulation and comparison studies are performed to illustrate the value and vitality of the proposed model. In the first experiment, simulations are conducted using a well-known Traveling Salesman Problem (TSP) based data set, and results are compared with other heuristic-based algorithms. The proposed model is thoroughly evaluated in the second experiment through a comparison study using a similar model proven to work effectively for multi-waypoint navigation.\u003C\u002Fp\u003E\u003Cdiv id=\"sec22\" class=\"article-Section\"\u003E\u003Ch3 \u003E6.1. Comparison studies with benchmark datasets\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ETo show the effectiveness of our IPSO waypoint sequencing model, a comparison study was conducted with well-known TSP data sets and various heuristic-based algorithms. The employed datasets and algorithms are as follows: (a) 561-city problem by Kleinschmidt (pa561); (b) 299-city problem by Patberg\u002FRinaldi (pr299); (c) 200-city problem A, by Krolik\u002FFelts\u002FNelson (kroA200); and (d) 150-city problem by Chur Ritz (ch150). The selected datasets have been verified and widely used to prove the validity of multi-waypoint sequencing models. The Simulated Annealing (SA) algorithm, Grey Wolf Optimization (GWO) algorithm, Ant Colony Optimization (ACO) algorithm, Genetic Algorithms (GA), Imperialist Competitive Algorithm (ICA), and Self-Organizing Maps (SOM) were chosen as the heuristic-based algorithms used in the comparison studies.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EThe ICA algorithm is a biologically inspired algorithm by the human, which simulates the social-political process of imperialism and imperialistic competition. The SOM algorithm is similar to a typical artificial neural network algorithm, except it utilizes a competitive learning process instead of backpropagation that utilizes gradient descent.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EHeuristic-based algorithms have similar attributes; due to this feature, the same parameters can be used to construct a stable comparison study for our proposed IPSO algorithm. The conducted comparison studies focus on six key attributes such as: min length (\u003Cinline-formula\u003E\u003Ctex-math id=\"M197\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), average length (\u003Cinline-formula\u003E\u003Ctex-math id=\"M198\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), length standard deviation (\u003Cinline-formula\u003E\u003Ctex-math id=\"M199\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), min time (\u003Cinline-formula\u003E\u003Ctex-math id=\"M200\"\u003E$$ s $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), average time (\u003Cinline-formula\u003E\u003Ctex-math id=\"M201\"\u003E$$ s $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E), and time standard deviation (\u003Cinline-formula\u003E\u003Ctex-math id=\"M202\"\u003E$$ s $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E). The variance between each algorithm can be seen by assessing each parameter. The above analyses show how effective the IPSO model can generate the minimum overall global trajectory in \u003Ca href=\"#Table1\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table1\"\u003ETable 1\u003C\u002Fa\u003E. The global trajectory generated by the compared algorithms is notably larger than the IPSO model. However, regarding the time aspect, the IPSO model was unable to achieve the shortest time. The significance of the proposed model can be seen in the STD evaluation parameter. The results of the comparison studies more than show the validity and performance of the proposed model to discover the optimal waypoint visiting sequence.\u003C\u002Fp\u003E\u003Cdiv id=\"Table1\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 1\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EComparison of minimum path length, average path length, STD of path length, minimum time, average time and STD of time with other models. The parameter for the test of each model was: 100 initialized particles, 10 runs per data set, and a maximum of 10 minutes per run\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EDatasets\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EModel\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EMin length (\u003Ci\u003Em\u003C\u002Fi\u003E)\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EAverage length (\u003Ci\u003Em\u003C\u002Fi\u003E)\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003ELength STD (\u003Ci\u003Em\u003C\u002Fi\u003E)\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EMin time (\u003Ci\u003Es\u003C\u002Fi\u003E)\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EAverage time (\u003Ci\u003Es\u003C\u002Fi\u003E)\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003ETime STD (\u003Ci\u003Es\u003C\u002Fi\u003E)\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"left\" rowspan=\"7\"\u003ECh150\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E1.67E+04\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E1.77E+04\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E5.99E+02\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E1.75E+04\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E1.31E+01\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E6.09E-02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EACO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.84E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.23E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.40E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.67E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.76E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.32E+00\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.22E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.04E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.38E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.79E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.39E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.25E-03\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.47E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.05E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.11E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.75E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.51E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E7.25E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGWO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.23E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.66E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.64E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.78E-01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E9.39E-01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.73E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESOM\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.26E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.70E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.932E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.84E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.26E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.77E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EICA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.29E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.50E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E9.34E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.55E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.04E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.79E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\" rowspan=\"7\"\u003EKroA200\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.09E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.17E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.50E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.15E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.15E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E9.73E-02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EACO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.30E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.81E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.09E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.44E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.17E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.36E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.39E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.57E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.15E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.21E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.54E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.45E-03\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.96E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.13E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.18E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.27E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.62E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.70E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGWO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.17E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.40E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.24E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.12E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.45E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.68E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESOM\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.13E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.60E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.192E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.49E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.60E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.31E+04\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EICA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.12E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.60E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.671E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.22E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.81E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.38E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\" rowspan=\"7\"\u003EPR299\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.77E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.89E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.92E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.28E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.29E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.06E-02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EACO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.37E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.33E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.82E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.09E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.37E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.41E+01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.19E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.44E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.17E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.24E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.38E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.43E-04\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.26E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.59E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.94E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.27E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.52E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.68E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGWO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.90E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.90E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.15E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.71E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.00E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E7.51E+00\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESOM\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.72E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.15E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.34E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.25E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.75E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.45E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EICA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.81E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.96E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.34E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.70E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.81E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E7.60E+01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"left\" rowspan=\"7\"\u003EPA561\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.11E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.14E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.60E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.41E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.42E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.65E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EACO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E—\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E—\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E—\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E—\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E—\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E—\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.37E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.93E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.30E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E9.36E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E9.57E-02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.19E-03\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESA\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.88E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.92E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.07E+03\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.27E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E6.42E+00\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.29E-01\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003EGWO\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.02E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E4.21E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E5.94E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E7.64E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.26E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E2.21E+00\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"center\"\u003ESOM\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.01E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.21E+05\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.84E+04\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E1.1E+01\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E8.19E+02\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E3.94E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003EICA\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E1.48E+05\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E1.51E+05\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E2.43E+03\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E1.22E+03\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E2.04E+03\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E1.79E+02\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec23\" class=\"article-Section\"\u003E\u003Ch3 \u003E6.2. Model comparison studies\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EThe compared models were developed to address the issues of multi-waypoint navigation and mapping in various applications. Each model uses some variation of a global navigation system in combination with an obstacle avoidance technique. The models were selected based on their map configuration and overall efficiency in solving the multi-waypoint navigation problem. Our comparison studies analyze the number of nodes, the trajectories produced, and the total time to fulfill the fastest route.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIt is clear that the waypoint order and paths obtained by each model are created in an obstacle-free environment, as illustrated in \u003Ca href=\"#Figure10\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure10\"\u003EFigure 10A\u003C\u002Fa\u003E. The length created by the Zhang's model was 240.84 \u003Ci\u003Em\u003C\u002Fi\u003E, while the proposed model produced a shorter trajectory of 219.99 \u003Ci\u003Em\u003C\u002Fi\u003E. This is due to the founded waypoint orders in the environment. In Zhang's comparison study, the proposed model establishes more nodes, and the overall path is expanded by 1.09%, but the proposed model generates a solution 6.1% faster than the compared model. Zhang's proposed model has to utilize a node selection algorithm to establish its shortest path, while the proposed model does not. Due to this feature, the compared model was evaluated before this crucial step and discovered that the nodes established were vastly greater than the proposed model, as seen in \u003Ca href=\"#Table2\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Table2\"\u003ETable 2\u003C\u002Fa\u003E. Considering this factor, the proposed model can surpass and outperform Zhang's model. Asl and Taghirad aimed to solve the multi-goal navigation problem by developing a traveling salesman problem in the belief space \u003Csup\u003E[\u003Ca href=\"#b44\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b44\"\u003E44\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.\u003C\u002Fp\u003E\u003Cdiv id=\"Table2\" class=\"Figure-block\"\u003E\u003Cdiv class=\"table-note\"\u003E\u003Cspan class=\"\"\u003ETable 2\u003C\u002Fspan\u003E\u003Cp class=\"\"\u003EAn illustration of the number of nodes, distance, and time spent traversing with the map to each waypoint\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table-responsive article-table\"\u003E\u003Ctable class=\"a-table\"\u003E\u003Cthead\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border\" align=\"left\"\u003E\u003Cb\u003EModel\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003ENodes\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003EDistance\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border\" align=\"center\"\u003E\u003Cb\u003ETime spent \u003Ci\u003Es\u003C\u002Fi\u003E\u003C\u002Fb\u003E\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Fthead\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_top_border2\" align=\"left\"\u003EZhang's model before node reduction\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E242\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E271.1\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_top_border2\" align=\"center\"\u003E2.25\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd align=\"left\"\u003EZhang's model after node reduction\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E24\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E253.4\u003C\u002Ftd\u003E\u003Ctd align=\"center\"\u003E0.66\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003Ctr\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"left\"\u003EProposed model\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E38\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E277.7\u003C\u002Ftd\u003E\u003Ctd style=\"class:table_bottom_border\" align=\"center\"\u003E0.40\u003C\u002Ftd\u003E\u003C\u002Ftr\u003E\u003C\u002Ftbody\u003E\u003C\u002Ftable\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"table_footer\"\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EFrom \u003Ca href=\"#Figure11\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure11\"\u003EFigure 11\u003C\u002Fa\u003E, one can see the established path using both Asl's method as well as the method proposed in this paper. The method proposed by Asl and Taghirad has the advantage of creating a shorter path but requires a greater number of nodes than the proposed method. \u003Ca href=\"#Figure12\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure12\"\u003EFigure 12\u003C\u002Fa\u003E depicts the model comparison between Zhuang \u003Ci\u003Eet al.\u003C\u002Fi\u003E's model and the proposed method\u003Csup\u003E[\u003Ca href=\"#b45\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b45\"\u003E45\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The simulation studies reveal that the proposed model had an increased length of approximately 0.05% over Zhuang \u003Ci\u003Eet al.\u003C\u002Fi\u003E's model. Once the compared model requires a greater number of nodes to complete its multi-waypoint navigation, another key point from this comparison is the path created from Zhuang \u003Ci\u003Eet al.\u003C\u002Fi\u003E's model and the proximity to the obstacles in the map. In a real-world environment, the robot could obtain server damage or cause an accident if it is too close to the surrounding obstacles\u003Csup\u003E[\u003Ca href=\"#b45\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b45\"\u003E45\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The proposed method established an effective path without risking the robots well being. Von{á}sek and P{e}ni{c}ka \u003Csup\u003E[\u003Ca href=\"#b46\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b46\"\u003E46\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E models had similar results to the previous models, with an increased length of approximately 0.05%, as seen in \u003Ca href=\"#Figure13\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure13\"\u003EFigure 13\u003C\u002Fa\u003E. The two compared models have the same problem as Zhuang \u003Ci\u003Eet al.\u003C\u002Fi\u003E' model\u003Csup\u003E[\u003Ca href=\"#b45\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b45\"\u003E45\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E. The paths are excessively close to obstacles in the map and thus are not efficient for real-world implementation.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn such an environment, it is very important to consider the robot safety because of the narrow paths created being tightly packed with triangular shaped obstacles. Although most of our model comparison results showed that the path constructed with the proposed model increased from the compared models, we achieved our goal of constructing safety-aware roads for robot safety and establishing an obstacle-free path.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EFrom \u003Ca href=\"#Figure14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure14\"\u003EFigure 14\u003C\u002Fa\u003E, it is observed how the local navigator establishes a map through vision sensors such as LiDAR. In \u003Ca href=\"#Figure14\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure14\"\u003EFigure 14\u003C\u002Fa\u003E, it is obvious that the map in various stages is shown as the robot traverses along the generated trajectory found in the Vonásek's simulation [\u003Ca href=\"#Figure13\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure13\"\u003EFigure 13\u003C\u002Fa\u003E]\u003Csup\u003E[\u003Ca href=\"#b46\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"b46\"\u003E46\u003C\u002Fa\u003E]\u003C\u002Fsup\u003E.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure14\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure14\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-14.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 14. Illustration of the scenario in \u003Ca href=\"#Figure13\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure13\"\u003EFigure 13\u003C\u002Fa\u003E navigation and mapping simulation. (A) It depicts the robot traversing a majority of the map, while avoiding obstacles in the environment. (B) It illustrates a polar histogram and how the obstacles are viewed while the robot is in motion, as well as points of high impact. The obstacles are viewed as lines since the LiDAR sensor can only see the part of the object that faces the LiDAR sensor. The picked direction portion of part (B) depicts the probability of colliding with obstacles while also selecting the best direction to move the robot. (C) It demonstrates the map being simultaneously built as the the robot traverses the established trajectory.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cp class=\"\"\u003EThe robot is able to reconstruct the outer boundary of the obstacles through the LiDAR scan. These are depicted as the poly-shaped figures with a rough background and a white center.\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EIn \u003Ca href=\"#Figure15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure15\"\u003EFigure 15A\u003C\u002Fa\u003E, it is clear to see the original starting position as well as the planned trajectory, which was found utilizing our proposed IPSO model. From the figures, one could observe fully and partly detected obstacles as well as the outer boundary being detected. The map depicted in \u003Ca href=\"#Figure15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure15\"\u003EFigure 15\u003C\u002Fa\u003E has a height and width of 60 \u003Cinline-formula\u003E\u003Ctex-math id=\"M203\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E. The dimensions of the robot are approximately 0.82 \u003Cinline-formula\u003E\u003Ctex-math id=\"M204\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E long and 0.68 \u003Cinline-formula\u003E\u003Ctex-math id=\"M205\"\u003E$$ m $$\u003C\u002Ftex-math\u003E\u003C\u002Finline-formula\u003E wide. In \u003Ca href=\"#Figure15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure15\"\u003EFigure 15A\u003C\u002Fa\u003E, the robot has successfully traversed one third of the map, while simultaneously avoiding the obstacles. In \u003Ca href=\"#Figure15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure15\"\u003EFigure 15B\u003C\u002Fa\u003E, it illustrates the robot's planned trajectory and the map being simultaneously constructed along the path. The portions of the figure depicted in a yellow field are the built map sensed by the onboard sensors. Finally, in \u003Ca href=\"#Figure15\" class=\"Link_style\" data-jats-ref-type=\"bibr\" data-jats-rid=\"Figure15\"\u003EFigure 15C\u003C\u002Fa\u003E, the robot has successfully visited each waypoint and reached its final destination, and it shows the complete depiction of the map along the projected path.\u003C\u002Fp\u003E\u003Cdiv class=\"Figure-block\" id=\"Figure15\"\u003E\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" class=\"article-figure-image\"\u003E\u003Ca href=\"\u002Farticles\u002Fir.2022.21\u002Fimage\u002FFigure15\" class=\"Article-img\" alt=\"\" target=\"_blank\"\u003E\u003Cimg alt=\"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" src=\"https:\u002F\u002Fimage.oaes.cc\u002Fb6bde10c-1a17-43d0-9758-88dabfd18e62\u002F5170-15.jpg\" class=\"\" title=\"\" alt=\"\" \u002F\u003E\u003C\u002Fa\u003E\u003C\u002Fdiv\u003E\u003Cdiv class=\"article-figure-note\"\u003E\u003Cp class=\"figure-note\"\u003E\u003C\u002Fp\u003E\u003Cp class=\"figure-note\"\u003EFigure 15. Illustration of the navigation and mapping ability of the proposed model. (A) The robot follows the generated trajectory and detects obstacle boundaries by the LiDAR sensor. (B) The simultaneous map building and navigation capabilities of the proposed model.\r\n(C) The fully generated trajectory and established map.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec17\" class=\"article-Section\"\u003E\u003Ch2 \u003E7. CONCLUSION\u003C\u002Fh2\u003E\u003Cp class=\"\"\u003EWe proposed an adjacent node selection (ANS) algorithm to find a node in the graph to connect waypoints. This algorithm is utilized in the safety-aware multi-waypoint navigation and mapping by an improved PSO and GVD model.An IPSO-based multi-waypoint algorithm has been developed to define an order for waypoint navigation. Through our proposed ANS algorithm, connections among the waypoints and the safety-aware routes to reach multi-objective optimization can be created. The feasibility and effectiveness of our model by conducting a benchmark test and model comparison studies and analyses have been demonstrated.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec18\" class=\"article-Section\"\u003E\u003Ch2 \u003EDECLARATIONS\u003C\u002Fh2\u003E\u003Cdiv id=\"sec21\" class=\"article-Section\"\u003E\u003Ch3 \u003EAcknowledgments\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EThe authors would like to thank the editor-in-chief, the associate editor, and the anonymous reviewers for their valuable comments.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec22\" class=\"article-Section\"\u003E\u003Ch3 \u003EAuthors' contributions\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EMade substantial contributions to the research, idea generation, algorithm design, simulation, wrote and edited the original draft: Sellers T, Lei T, Luo C\u003C\u002Fp\u003E\u003Cp class=\"\"\u003EPerformed critical review, commentary and revision, as well as provided administrative, technical, and material support: Jan G, Ma J\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec23\" class=\"article-Section\"\u003E\u003Ch3 \u003EFinancial support and sponsorship\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ENone.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec24\" class=\"article-Section\"\u003E\u003Ch3 \u003EAvailability of data and materials\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ENot applicable.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec25\" class=\"article-Section\"\u003E\u003Ch3 \u003EConflicts of interest\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003EAll authors declared that there are no conflicts of interest.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec26\" class=\"article-Section\"\u003E\u003Ch3 \u003EEthical approval and consent to participate\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ENot applicable.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec27\" class=\"article-Section\"\u003E\u003Ch3 \u003EConsent for publication\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003ENot applicable.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003Cdiv id=\"sec28\" class=\"article-Section\"\u003E\u003Ch3 \u003ECopyright\u003C\u002Fh3\u003E\u003Cp class=\"\"\u003E© The Author(s) 2022.\u003C\u002Fp\u003E\u003C\u002Fdiv\u003E\u003C\u002Fdiv\u003E",translate:[{language:"en",new_title:ay,new_abstract:cy,new_keywords:bH,is_check:l},{language:"cn",new_title:"一个节点选择算法用于基于图的多路径点优化导航和地图绘制",new_abstract:"自主机器人多路径点导航和地图绘制在许多实际应用中被需求，如搜索和救援（SAR）、环境探索和灾难响应。许多解决这个问题的方法都是通过基于图的方法来发现的，需要在图中的节点和边之间切换机器人的轨迹，从而创建路径点导航的轨迹。然而，关于路径点如何与图中的节点或边进行本地连接的研究尚未被充分进行。本文提出了一种相邻节点选择（ANS）算法来实现这样一个协议，从路径点到图中最近的节点或边上建立区域路径。我们提出了这种节点选择算法以及广义Voronoi图（GVD）和改进粒子群优化（IPSO）算法以及一个本地导航器来解决基于图的多路径点导航和地图绘制问题。首先，在障碍物密集的环境中利用GVD构造Voronoi图来形成安全路线。其次，IPSO算法创建多路径点的顺序，以最小化总旅行成本。第三，当机器人尝试访问多个路径点时，沿着GVD的边缘遍历以规划无碰撞的轨迹。从路径点到最近节点或边的区域路径需要通过提出的ANS算法创建，以连接轨迹。最后，采用基于传感器直方图的本地响应导航器进行运动障碍物避免，同时机器人移动时构建本地地图。采用基于改进B样条曲线的平滑方案，进一步优化轨迹，并使机器人平稳导航。模拟和比较研究验证了所提出模型的有效性和鲁棒性。",new_keywords:"邻接节点选择（ANS）算法、安全感知道路、路径规划、多航点优化、导航与地图绘制",is_check:i},{language:"de",new_title:"Ein Knotenauswahlalgorithmus zur graphbasierten Mehrwegpunkt-Optimierungs-Navigation und Kartierung.",new_abstract:"Autonome Robotermultiwegpunktnavigation und Kartierung werden in vielen realen Anwendungen wie Such- und Rettungseinsätzen (SAR), Umwelterkundung und Katastrophenbewältigung gefordert. Viele Lösungen für dieses Problem wurden mittels graphenbasierter Methoden entdeckt, bei denen der Roboter seine Trajektorie zwischen den Knoten und Kanten innerhalb des Graphen wechseln muss, um eine Trajektorie für die Wegpunkt-zu-Wegpunkt-Navigation zu erstellen. Es wurden jedoch keine ausreichenden Studien darüber durchgeführt, wie Wegpunkte lokal mit Knoten oder Kanten auf den Graphen verbunden werden. In diesem Paper wurde ein Algorithmus für die Auswahl benachbarter Knoten (ANS) entwickelt, um ein entsprechendes Protokoll zur Erstellung eines regionalen Pfades von Wegpunkten zu den nächstgelegenen Knoten oder Kanten auf dem Graphen umzusetzen. Wir schlagen diesen Knotenauswahlalgorithmus zusammen mit dem verallgemeinerten Voronoi-Diagramm (GVD) und dem verbesserten Partikelschwarmoptimierung (IPSO) Algorithmus sowie einem lokalen Navigator vor, um das sicherheitsbewusste gleichzeitige graphenbasierte Multiwegpunkt-Navigations- und Kartierungsproblem zu lösen. Zunächst wird GVD verwendet, um ein Voronoi-Diagramm in einer von Hindernissen besiedelten Umgebung zu erstellen, um sicherheitsbewusste Routen zu konstruieren. Zweitens wird die Reihenfolge mehrerer Wegpunkte mithilfe des IPSO-Algorithmus erstellt, um die Gesamtreisekosten zu minimieren. Drittens, während der Roboter versucht, mehrere Wegpunkte zu besuchen, durchläuft er die Kanten des GVD, um eine kollisionsfreie Trajektorie zu planen. Der regionale Pfad von den Wegpunkten zu den nächstgelegenen Knoten oder Kanten muss erstellt werden, um die Trajektorie durch den vorgeschlagenen ANS-Algorithmus zu verbinden. Schließlich wird ein sensorgesteuerter Histogramm-Ortsreaktionsnavigator zur Vermeidung von bewegten Hindernissen verwendet, während vor Ort Karten erstellt werden, während der Roboter sich bewegt. Ein verbessertes \u003Ci\u003EB\u003C\u002Fi\u003E-Spline-Kurvenbasiertes Glättungsschema wird übernommen, um die Trajektorie weiter zu verfeinern und es dem Roboter zu ermöglichen, reibungslos zu navigieren. Simulationen und Vergleichsstudien bestätigen die Wirksamkeit und Robustheit des vorgeschlagenen Modells.",new_keywords:"Die Auswahl benachbarter Knoten (ANS) Algorithmus, sicherheitsbewusste Straßen, Pfadplanung, Mehrwegeoptimierung von Wegpunkten, Navigation und Kartierung",is_check:i},{language:"fa",new_title:"Un algorithme de sélection de nœuds pour la navigation et la cartographie multi-points basée sur un graphe",new_abstract:"La navigation et la cartographie multi-points des robots autonomes ont été demandées dans de nombreuses applications du monde réel telles que les opérations de recherche et de sauvetage, l'exploration environnementale et la réponse aux catastrophes. De nombreuses solutions à ce problème ont été découvertes grâce à des méthodes basées sur des graphes nécessitant de faire passer la trajectoire du robot entre les nœuds et les arêtes du graphe pour créer une trajectoire de navigation de point à point. Cependant, des études sur la manière dont les points de passage sont localement reliés aux nœuds ou aux arêtes des graphes n'ont pas été suffisamment entreprises. Dans cet article, un algorithme de sélection de nœud adjacent (ANS) est développé pour implémenter un tel protocole afin de construire un chemin régional des points de passage aux nœuds ou aux arêtes les plus proches du graphe. Nous proposons cet algorithme de sélection de nœud ainsi que le diagramme de Voronoi généralisé (GVD) et l'algorithme d'optimisation par essaim de particules amélioré (IPSO) ainsi qu'un navigateur local pour résoudre le problème de navigation et de cartographie multi-points basé sur des graphes concurrents conscients de la sécurité. Premièrement, le GVD est utilisé pour former un diagramme de Voronoi dans un environnement peuplé d'obstacles pour construire des itinéraires conscients de la sécurité. Deuxièmement, la séquence de plusieurs points de passage est créée par l'algorithme IPSO pour minimiser le coût total du déplacement. Troisièmement, tandis que le robot tente de visiter plusieurs points de passage, il traverse le long des arêtes du GVD pour planifier une trajectoire exempte de collisions. Le chemin régional des points de passage aux nœuds ou aux arêtes les plus proches doit être créé pour rejoindre la trajectoire par l'algorithme ANS proposé. Enfin, un navigateur local réactif basé sur un histogramme sensoriel est adopté pour l'évitement des obstacles en mouvement pendant que des cartes locales sont construites lorsque le robot se déplace. Un schéma lisse basé sur des courbes en B-spline amélioré est adopté pour affiner davantage la trajectoire et permettre au robot de naviguer en douceur. Des études de simulation et de comparaison valident l'efficacité et la robustesse du modèle proposé.",new_keywords:"Sélection du nœud adjacent (SNA) algorithme, routes conscientes de la sécurité, planification de trajet, optimisation de plusieurs points de passage, navigation et cartographie.",is_check:i},{language:"jp",new_title:"グラフベースのマルチウェイポイント最適化ナビゲーションおよびマッピングのためのノード選択アルゴリズム",new_abstract:"自律ロボットの複数ウェイポイントナビゲーションとマッピングは、救助活動（SAR）、環境探査、災害対応など、多くの現実世界のアプリケーションで求められています。この問題に対する多くの解決策は、グラフベースの方法を用いて発見されており、ロボットの軌道をグラフ内のノードとエッジ間で切り替えて、ウェイポイント間のナビゲーションのための軌道を作成しています。しかし、ウェイポイントが局所的にノードやエッジとどのようにつながるかについての研究は不十分です。本研究では、隣接ノード選択（ANS）アルゴリズムを開発し、提案されたプロトコルを実装してウェイポイントからグラフ上の最も近いノードやエッジにリージョナルパスを構築することを目指します。我々は、このノード選択アルゴリズムを一般化ボロノイ図（GVD）と改良粒子群最適化（IPSO）アルゴリズム、およびローカルナビゲーターと共に、安全意識を持った同時グラフベースの複数ウェイポイントナビゲーションとマッピング問題を解決することを提案します。最初に、GVDを使用して障害物の多い環境でボロノイ図を形成し、安全意識のある経路を構築します。次に、IPSOアルゴリズムによって複数のウェイポイントのシーケンスが作成され、総移動コストを最小化します。最後に、ロボットが複数のウェイポイントを訪れようとする際、GVDのエッジを沿って移動して衝突しない軌道を計画します。ウェイポイントから最も近いノードやエッジへのリージョナルパスが提案されたANSアルゴリズムによって作成され、軌道に結合されます。最後に、ロボットが移動する間に移動障害物を回避するためにセンサーベースのヒストグラムローカルリアクティブナビゲーターが採用され、ロボットがスムーズにナビゲートされるように改良されたBスプライン曲線ベースのスムース計画が採用されます。シミュレーションおよび比較研究は、提案モデルの効果的さと堅牢さを検証しています。",new_keywords:"隣接ノード選択（ANS）アルゴリズム、安全意識の高い道路、経路計画、複数ウェイポイントの最適化、ナビゲーションやマッピング",is_check:i},{language:"py",new_title:"Алгоритм выбора узла для оптимизации навигации и картографирования с использованием графа и нескольких путевых точек.",new_abstract:"Автономная навигация и создание карт роботов по множеству точек маршрута требуется во многих реальных приложениях, таких как поисково-спасательные операции, исследования окружающей среды и реагирование на чрезвычайные ситуации. Множество решений этой проблемы были обнаружены с использованием графовых методов, требующих переключения траектории роботов между узлами и рёбрами графа для создания маршрута навигации от точки к точке. Однако исследования о том, как точки маршрута локально соединяются с узлами или рёбрами на графах, не были должным образом проведены. В данной статье разработан алгоритм выбора смежного узла (ANS) для реализации такого протокола построения регионального пути от точек маршрута к ближайшим узлам или рёбрам на графе. Мы предлагаем этот алгоритм выбора узла вместе с обобщённой диаграммой Вороного (GVD) и улучшенным алгоритмом оптимизации роя частиц (IPSO), а также для решения проблемы навигации и создания карт на основе графов с учётом безопасности. Во-первых, GVD используется для построения диаграммы Вороного в среде, заполненной препятствиями, чтобы создать маршруты с учётом безопасности. Во-вторых, последовательность из нескольких точек маршрута создаётся алгоритмом IPSO для минимизации общих расходов на путешествие. В-третьих, когда робот пытается посетить несколько точек маршрута, он движется по рёбрам GVD, чтобы спланировать свободный от столкновений маршрут. Региональный путь от точек маршрута к ближайшим узлам или рёбрам должен быть создан для соединения траектории посредством предложенного алгоритма ANS. Наконец, для избегания движущихся препятствий используется реактивный навигатор на основе сенсоров, в то время как на лету создаются локальные карты. Применяется улучшенная схема сглаживания на основе кривых \u003Ci\u003EB\u003C\u002Fi\u003E-сплайн, которая дополнительно улучшает траекторию и позволяет роботу двигаться плавно. Исследования симуляции и сравнения подтверждают эффективность и надёжность предложенной модели.",new_keywords:"Выбор смежного узла (ANS) алгоритм, дороги, учитывающие безопасность, планирование пути, оптимизация маршрута с несколькими точками, навигация и картографирование.",is_check:i},{language:"sk",new_title:"그래프 기반 다중 웨이포인트 최적화 탐색 및 매핑을 위한 노드 선택 알고리즘",new_abstract:"자율 로봇 다중 웨이포인트 내비게이션 및 매핑은 다수의 실제 응용분야에서 요구되고 있으며, 구조 탐색 및 구조 구조 탐색, 환경 탐사, 재난 대응 등이 있다.이 문제에 대한 많은 해결책이 발견되어 왔고, 이는 그래프 기반 방법을 통해 로봇의 궤적을 그래프 내의 노드와 엣지 사이로 전환하여 웨이포인트 간 내비게이션을 만드는 데 필요하다.그러나 웨이포인트가 로컬리 노드나 그래프의 엣지들과 어떻게 연결되는지에 대한 연구는 충분히 이루어지지 않았다.본 논문에서는 인접 노드 선택 (ANS) 알고리즘을 개발하여 이러한 프로토콜을 구현하고 웨이포인트에서 그래프 상에서 가장 가까운 노드나 엣지로 지역 경로를 구축한다.우리는이 노드 선택 알고리즘과 일반화된 Voronoi 다이어그램 (GVD) 및 향상된 Particle Swarm Optimization (IPSO) 알고리즘 및 로컬 내비게이터를 제안하여 안전하게 공동으로 그래프 기반 다중 웨이포인트 내비게이션 및 매핑 문제를 해결한다.먼저, GVD를 사용하여 장애물이 많은 환경에서 Voronoi 다이어그램을 형성하여 안전한 경로를 구성한다.둘째, 여러 웨이포인트의 순서를 IPSO 알고리즘을 통해 최소 비용으로 생성하여 로봇이 여러 웨이포인트를 방문하려고 할 때 GVD의 엣지를 따라 충돌 없이 궤적을 계획한다.예약된 ANS 알고리즘에 의해 제안 된 궤적에 가입 함에는 가장 가까운 노드나 엣지로부터 웨이포인트까지의 국부 경로가 생성되어야 한다.마지막으로 로컬 지도가 로봇 이동시마다 생성될 때 로컬 이동 장애물 회피를 위해 센서 기반 히스토그램 로컬 반응 내비게이터가 채택되었다.개선 된 B-Spline 곡선을 사용하여 이러한 추세를 보완하고 로봇을 부드럽게 탐험 할 수 있도록하는 효과적인 경로가 채택되었다.시뮬레이션 및 비교 연구를 통해 제안된 모델의 효과성과 견고성이 검증되었다.",new_keywords:"인접 노드 선택 (ANS) 알고리즘, 안전 인식 도로, 경로 계획, 다중 경유지 최적화, 내비게이션 및 맵핑",is_check:i},{language:"it",new_title:"Un algoritmo di selezione dei nodi per la navigazione e la mappatura ottimizzata a più punti di passaggio basata su grafi",new_abstract:"La navigazione e la mappatura multi-waypoint di robot autonomi sono richieste in molte applicazioni del mondo reale, come soccorso in caso di catastrofi, esplorazione ambientale e risposta a disastri. Moltissime soluzioni a questo problema sono state scoperte attraverso metodi basati su grafi che richiedono di cambiare la traiettoria del robot tra i nodi e gli archi all'interno del grafo per creare una traiettoria per la navigazione da waypoint a waypoint. Tuttavia, non sono stati adeguatamente affrontati studi su come i waypoint sono collegati localmente a nodi o archi sui grafi. In questo articolo, è stato sviluppato un algoritmo di selezione di nodi adiacenti (ANS) per implementare un tale protocollo per creare un percorso regionale dai waypoint ai nodi o archi più vicini sul grafo. Proponiamo questo algoritmo di selezione dei nodi insieme al diagramma di Voronoi generalizzato (GVD) e all'algoritmo di ottimizzazione dello sciame di particelle migliorato (IPSO) insieme a un navigatore locale per risolvere il problema della navigazione e mappatura concorrente basata su grafi di waypoint multipli, consapevoli della sicurezza. In primo luogo, il GVD viene utilizzato per formare un diagramma di Voronoi in un ambiente popolato da ostacoli per costruire percorsi consapevoli della sicurezza. In secondo lu luogo, la sequenza di waypoint multipli è creata dall'algoritmo IPSO per minimizzare il costo totale del viaggio. In terzo luogo, mentre il robot cerca di visitare più waypoint, percorre lungo gli archi del GVD per pianificare una traiettoria senza collisioni. Il percorso regionale dai waypoint ai nodi o archi più vicini deve essere creato per unire la traiettoria tramite l'algoritmo ANS proposto. Infine, un navigatore reattivo locale basato su istogrammi sensoriali è adottato per evitare gli ostacoli in movimento mentre vengono create mappe locali mentre il robot si sposta. Viene adottato uno schema di smussatura basato su curve \u003Ci\u003EB\u003C\u002Fi\u003E-spline migliorato che raffina ulteriormente la traiettoria e consente al robot di essere navigato dolcemente. Studi di simulazione e confronto convalidano l'efficacia e la robustezza del modello proposto.",new_keywords:"Selezione del nodo adiacente (ANS) algoritmo, strade consapevoli della sicurezza, pianificazione del percorso, ottimizzazione multipunto, navigazione e mappatura.",is_check:i},{language:"fs",new_title:"Un algoritmo de selección de nodos para la navegación y cartografía de optimización de múltiples puntos de referencia basada en gráficos.",new_abstract:"La navegación y mapeo de varios puntos de destino de robots autónomos ha sido demandada en muchas aplicaciones del mundo real, como la búsqueda y rescate (SAR), la exploración ambiental y la respuesta a desastres. Muchas soluciones a este problema han sido descubiertas a través de métodos basados en grafos que requieren cambiar la trayectoria del robot entre los nodos y bordes dentro del grafo para crear una trayectoria de navegación de punto a punto. Sin embargo, los estudios de cómo los puntos de destino están conectados localmente a los nodos o bordes en los grafos no han sido adecuadamente abordados. En este trabajo, se ha desarrollado un algoritmo de selección de nodos adyacentes (ANS) para implementar un protocolo para construir una ruta regional desde los puntos de destino hasta los nodos o bordes más cercanos en el grafo. Proponemos este algoritmo de selección de nodos junto con el diagrama de Voronoi generalizado (GVD) y el algoritmo de optimización de enjambre de partículas mejorado (IPSO), así como un navegador local para resolver el problema de navegación y mapeo de varios puntos de destino basado en grafos concurrentes y seguros. Primero, se utiliza GVD para formar un diagrama de Voronoi en un entorno poblado de obstáculos para construir rutas seguras. En segundo lugar, la secuencia de múltiples puntos de destino es creada por el algoritmo IPSO para minimizar el costo total de viaje. En tercer lugar, mientras el robot intenta visitar múltiples puntos de destino, recorre a lo largo de los bordes del GVD para planificar una trayectoria sin colisiones. La ruta regional desde los puntos de destino a los nodos o bordes más cercanos debe ser creada para unir la trayectoria mediante el algoritmo ANS propuesto. Por último, se adopta un navegador reactivo local basado en histogramas sensoriales para evitar obstáculos en movimiento mientras se construyen mapas locales a medida que avanza el robot. Se adopta un esquema suave basado en curvas B-spline mejorado que refina aún más la trayectoria y permite que el robot navegue suavemente. Estudios de simulación y comparación validan la efectividad y robustez del modelo propuesto.",new_keywords:"Selección de nodos adyacentes (ANS) algoritmo, carreteras conscientes de la seguridad, planificación de rutas, optimización de múltiples puntos de paso, navegación y mapeo.",is_check:i},{language:"po",new_title:"Um algoritmo de seleção de nós para navegação e mapeamento baseados em grafo de otimização multi-ponto.",new_abstract:"A navegação e mapeamento multi-pontos por robô autônomo tem sido demandada em muitas aplicações do mundo real encontradas em busca e salvamento (SAR), exploração ambiental e resposta a desastres. Muitas soluções para este problema foram descobertas através de métodos baseados em gráficos que precisam alternar a trajetória do robô entre os nós e arestas dentro do gráfico para criar uma trajetória de navegação ponto a ponto. No entanto, estudos sobre como os pontos de passagem são localmente conectados aos nós ou arestas nos gráficos não foram adequadamente realizados. Neste artigo, um algoritmo de seleção de nós adjacentes (ANS) foi desenvolvido para implementar tal protocolo e construir um caminho regional a partir dos pontos de passagem para os nós ou arestas mais próximos no gráfico. Propomos este algoritmo de seleção de nós juntamente com o diagrama de Voronoi generalizado (GVD) e o algoritmo de Otimização de Enxame de Partículas Melhorado (IPSO), bem como um navegador local para resolver o problema de navegação e mapeamento multi-pontos baseado em gráfico simultaneamente atento à segurança. Em primeiro lugar, o GVD é usado para formar um diagrama de Voronoi em um ambiente populado por obstáculos para construir rotas conscientes da segurança. Em segundo lugar, a sequência de múltiplos pontos de passagem é criada pelo algoritmo IPSO para minimizar o custo total de viagem. Em terceiro lugar, enquanto o robô tenta visitar múltiplos pontos de passagem, ele percorre ao longo das arestas do GVD para planejar uma trajetória livre de colisões. O caminho regional dos pontos de passagem para os nós ou arestas mais próximos precisa ser criado para unir a trajetória pelo algoritmo ANS proposto. Por fim, um navegador reativo local baseado em histograma sensível a sensores é adotado para evitar obstáculos móveis enquanto mapas locais são construídos enquanto o robô se move. Um esquema suave baseado em curva de B-spline melhorado é adotado que refina ainda mais a trajetória e permite que o robô seja navegado suavemente. Estudos de simulação e comparação validam a eficácia e robustez do modelo proposto.",new_keywords:"Seleção de nó adjacente (ANS) algoritmo, estradas cientes de segurança, planejamento de caminhos, otimização de múltiplos pontos de passagem, navegação e mapeamento.",is_check:i}]},ArtDataF:[{id:1483952,article_id:d,reference_num:i,reference:"Chu Z, Wang F, Lei T, Luo C. Path planning based on deep reinforcement learning for autonomous underwater vehicles under ocean current disturbance. \u003Ci\u003EIEEE Trans Intell Veh\u003C\u002Fi\u003E 2022:1-1.",refdoi:"https:\u002F\u002Fdx.doi.org\u002F10.1109\u002FTIV.2022.3153352",pubmed:a,pmc:a},{id:1483953,article_id:d,reference_num:l,reference:"Zhao W, Lun R, Gordon C, et al. A privacy-aware Kinect-based system for healthcare professionals. In: IEEE International Conference on Electro Information Technology. 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Path planning, a critical aspect for achieving autonomous aerial navigation, has consistently been a focal point in UAV research. However, traditional ant colony algorithms need to be improved for the drawbacks of susceptibility to local optima and weak convergence capabilities. Consequently, a novel path planning methodology is proposed based on a dual-strategy ant colony algorithm. In detail, an improved state transition probability rule is introduced, redefining ant movement rules by integrating the state transition strategy of deterministic selection during the iterative process. Additionally, heuristic information on adjacent node distance and mountain height is added to further improve the search efficiency of the algorithm. Then, a new dynamically adjusted pheromone update strategy is proposed. The update strategy is continuously adjusted during the iteration process, which is beneficial to the algorithm’s global search in the early stage and accelerated convergence in the later stage, preventing the algorithm from falling into local optimality and improving its convergence. Based on the above improvements, a new variation of ant colony optimization (ACO) called dual-strategy ACO algorithm is formed. Experimental results prove that dual-strategy ACO has superior global search capabilities and convergence characteristics from four key aspects: path length, fitness values, iteration number, and running time.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Fe37f9659-d974-440e-8cb1-ff98344d3663\u002Fir3037.pdf",elocation_id:b,fpage:666,article_id:bI,viewed:g,downloaded:g,video_url:b,volume:k,year:cm,tag:"666-84",image:t,authors:dV,video_img:a,journal_path:x,lpage:684,author:dV,specialissue:{id:dW,name:dX},specialinfo:a,url_doi:bK},{date_published:1698422400,section:y,section_id:r,title:bM,doi:"10.20517\u002Fir.2023.30",abstract:"\u003Cp\u003EThis paper focuses on the problem of regional cooperative search using multiple unmanned aerial vehicles (UAVs) for targets that have the ability to perceive and evade. When UAVs search for moving targets in a mission area, the targets can perceive the positions and flight direction of UAVs within certain limits and take corresponding evasive actions, which makes the search more challenging than traditional search problems. To address this problem, we first define a detailed motion model for such targets and design various search information maps and their update methods to describe the environmental information based on the prediction of moving targets and the search results of UAVs. We then establish a multi-UAV search path planning optimization model based on the model predictive control, which includes various newly designed objective functions of search benefits and costs. We propose a priority-encoded improved genetic algorithm with a fine-adjustment mechanism to solve this model. The simulation results show that the proposed method can effectively improve the cooperative search efficiency, and more targets can be found at a much faster rate compared to traditional search methods.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Feae351ed-07a4-4292-b118-78b2879997eb\u002Fir3030.pdf",elocation_id:b,fpage:538,article_id:bL,viewed:F,downloaded:g,video_url:b,volume:k,year:cm,tag:"538-64",image:t,authors:dY,video_img:a,journal_path:x,lpage:564,author:dY,specialissue:{id:dW,name:dX},specialinfo:a,url_doi:bN},{date_published:1687881600,section:y,section_id:r,title:bP,doi:"10.20517\u002Fir.2023.14",abstract:"\u003Cp\u003EWith the growing applications of autonomous robots and vehicles in unknown environments, studies on image-based localization and navigation have attracted a great deal of attention. This study is significantly motivated by the observation that relatively little research has been published on the integration of cutting-edge path planning algorithms for robust, reliable, and effective image-based navigation. To address this gap, a \u003Ci\u003Ebiologically inspired\u003C\u002Fi\u003E Bat Algorithm (BA) is introduced and adopted for image-based path planning in this paper. The proposed algorithm utilizes visual features as the reference in generating a path for an autonomous vehicle, and these features are extracted from the obtained images by convolutional neural networks (CNNs). The paper proceeds as follows: first, the requirements for image-based localization and navigation are described. Second, the principles of the BA are explained in order to expound on the justifications for its successful incorporation in image-based navigation. Third, in the proposed image-based navigation system, the BA is developed and implemented as a path planning tool for global path planning. Finally, the performance of the BA is analyzed and verified through simulation and comparison studies to demonstrate its effectiveness.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F7973e866-76d1-4b79-98f7-a6115ae7018f\u002F5791.pdf",elocation_id:b,fpage:dZ,article_id:bO,viewed:640,downloaded:69,video_url:b,volume:k,year:cm,tag:"222-41",image:"https:\u002F\u002Foaepublishstorage.blob.core.windows.net\u002F7973e866-76d1-4b79-98f7-a6115ae7018f\u002F5791-coverimg.jpg",authors:d_,video_img:a,journal_path:x,lpage:241,author:d_,specialissue:a,specialinfo:a,url_doi:bQ,image_list:"https:\u002F\u002Fimage.oaes.cc\u002F7973e866-76d1-4b79-98f7-a6115ae7018f\u002F5791-coverimg.jpg"},{date_published:cC,section:y,section_id:r,title:bS,doi:"10.20517\u002Fir.2022.18",abstract:"\u003Cp\u003EMany real-world robot applications, as found in precision agriculture, poultry farms, disaster response, and environment monitoring, require search, locate, and removal (SLR) operations by autonomous mobile robots. In such application settings, the robots initially search and explore the entire workspace to find the targets, so that the subsequent robots conveniently move directly to the targets to fulfill the task. A multi-layer robot navigation system is necessary for SLR operations. The scenario of interest is the removal of broiler mortality by autonomous robots in poultry barns in this paper. Daily manual collection of broiler mortality is time- and labor-consuming, and an autonomous robotic system can solve this issue effectively. In this paper, a multi-layer navigation system is developed to detect and remove broiler mortality with two robots. One robot is assigned to search a large-scale workspace in a coverage mode and find and locate objects, whereas the second robot directly moves to the located targets to remove the objects. Directed coverage path planning (DCPP) fused with an informative planning protocol (IPP) is proposed to efficiently search the entire workspace. IPP is proposed for coverage directions in DCPP devoted to rapidly achieving spatial coverage with the least estimation uncertainty in the decomposed grids. The detection robot consists of a developed informative-based directed coverage path planner and a You Only Look Once (YOLO) V4-based dead bird detector. It refines and optimizes the coverage path based on historical data on broiler mortality distribution in a broiler barn. The removal robot collects dead broilers driven by a new hub-based multi-target path routing (HMTR) scheme, which is applicable to row-based environments. The proposed methods show great potential to navigate in broiler barns efficiently and safely, thus being a useful component for robotics. The effectiveness and robustness of the proposed methods are validated through simulation and comparison studies.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Fd6f884ad-91fd-4b28-9214-c2afee46bb1f\u002F5169.pdf",elocation_id:b,fpage:313,article_id:bR,viewed:895,downloaded:505,video_url:"https:\u002F\u002Fv1.oaepublish.com\u002Ffiles\u002Ftalkvideo\u002F5169.mp4",volume:m,year:az,tag:"313-32",image:t,authors:d$,video_img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240205\u002Ff2ea9e6e2aaf41c9881a7e09531bb1be.jpg",journal_path:x,lpage:332,author:d$,specialissue:a,specialinfo:a,url_doi:bT},{date_published:1657209600,section:ea,section_id:I,title:bV,doi:"10.20517\u002Fir.2022.13",abstract:"\u003Cp\u003EThe motion planning and tracking control techniques of unmanned underwater vehicles (UUV) are fundamentally significant for efficient and robust UUV navigation, which is crucial for underwater rescue, facility maintenance, marine resource exploration, aquatic recreation, etc. Studies on UUV motion planning and tracking control have been growing rapidly worldwide, which are usually sorted into the following topics: task assignment of the multi-UUV system, UUV path planning, and UUV trajectory tracking. This paper provides a comprehensive review of conventional and intelligent technologies for motion planning and tracking control of UUVs. Analysis of the benefits and drawbacks of these various methodologies in literature is presented. In addition, the challenges and prospects of UUV motion planning and tracking control are provided as possible developments for future research.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002Fd2072f93-382c-47f8-b11d-89a475d1f391\u002F5032.pdf",elocation_id:b,fpage:200,article_id:bU,viewed:1174,downloaded:902,video_url:"https:\u002F\u002Fv1.oaepublish.com\u002Ffiles\u002Ftalkvideo\u002F5032.mp4",volume:m,year:az,tag:"200-22",image:t,authors:eb,video_img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240205\u002Fd754278c51784ba18a1c6e1b027cb762.jpg",journal_path:x,lpage:dZ,author:eb,specialissue:a,specialinfo:a,url_doi:bW},{date_published:1633968000,section:ea,section_id:I,title:bY,doi:"10.20517\u002Fir.2021.08",abstract:"\u003Cp\u003EIn the past decades, considerable attention has been paid to bio-inspired intelligence and its applications to robotics. This paper provides a comprehensive survey of bio-inspired intelligence, with a focus on neurodynamics approaches, to various robotic applications, particularly to path planning and control of autonomous robotic systems. Firstly, the bio-inspired shunting model and its variants (additive model and gated dipole model) are introduced, and their main characteristics are given in detail. Then, two main neurodynamics applications to real-time path planning and control of various robotic systems are reviewed. A bio-inspired neural network framework, in which neurons are characterized by the neurodynamics models, is discussed for mobile robots, cleaning robots, and underwater robots. The bio-inspired neural network has been widely used in real-time collision-free navigation and cooperation without any learning procedures, global cost functions, and prior knowledge of the dynamic environment. In addition, bio-inspired backstepping controllers for various robotic systems, which are able to eliminate the speed jump when a large initial tracking error occurs, are further discussed. Finally, the current challenges and future research directions are discussed in this paper.\u003C\u002Fp\u003E",pdfurl:"https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F446fa24e-e8c7-4d94-93c5-bbfff4f32e6b\u002F4353.pdf",elocation_id:b,fpage:cn,article_id:bX,viewed:2873,downloaded:905,video_url:"https:\u002F\u002Fv1.oaepublish.com\u002Ffiles\u002Ftalkvideo\u002F4353.mp4",volume:c,year:2021,tag:"58-83",image:t,authors:ec,video_img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240205\u002F29f1fb64c19a470fae5fe533b38418e7.jpg",journal_path:x,lpage:co,author:ec,specialissue:{id:1201,name:" Advances in Human-Assistive Technologies and Human-Robot Interactions"},specialinfo:a,url_doi:bZ},{date_published:dE,section:y,section_id:r,title:dF,doi:dG,abstract:dH,pdfurl:dI,elocation_id:b,fpage:dJ,article_id:dK,viewed:g,downloaded:g,video_url:b,volume:f,year:aV,tag:dL,image:t,authors:aW,video_img:a,journal_path:x,lpage:dM,author:aW,specialissue:a,specialinfo:a,url_doi:dN},{date_published:dO,section:y,section_id:r,title:aB,doi:dP,abstract:dQ,pdfurl:dR,elocation_id:b,fpage:dS,article_id:aA,viewed:g,downloaded:g,video_url:b,volume:f,year:aV,tag:dT,image:t,authors:aX,video_img:a,journal_path:x,lpage:dU,author:aX,specialissue:a,specialinfo:a,url_doi:aC}],ArtNum:{view:2079,click:g,down:882,read:g,like:f,comment:t,xml_down:k,cite_click:cp,export_click:n,cite_count:z,share_count:g,tran_click:G,mp3_click:g,sharenum:g,id:d},articleShow:U,ArtBase:{seo:{title:ay,keywords:bH,description:cA},picabstract:a,interview_pic:a,interview_url:a,review:a,video_url:cz,video_img:cB,oaestyle:cD,amastyle:cE,ctstyle:cF,acstyle:cG,related:[{article_id:aA,journal_id:n,section_id:r,path:q,journal:o,ar_title:aB,date_published:cH,doi:aC,author:[{first_name:cI,middle_name:a,last_name:cJ,ans:e,email:a,bio:a,photoUrl:a},{first_name:cK,middle_name:a,last_name:cL,ans:e,email:cM,bio:a,photoUrl:a},{first_name:cN,middle_name:a,last_name:cO,ans:e,email:a,bio:a,photoUrl:a},{first_name:cP,middle_name:a,last_name:aD,ans:e,email:a,bio:a,photoUrl:a}]},{article_id:bI,journal_id:n,section_id:r,path:q,journal:o,ar_title:bJ,date_published:cQ,doi:bK,author:[{first_name:cR,middle_name:a,last_name:cS,ans:e,email:a,bio:a,photoUrl:a},{first_name:cT,middle_name:a,last_name:aE,ans:e,email:cU,bio:a,photoUrl:a},{first_name:cV,middle_name:a,last_name:aF,ans:e,email:a,bio:a,photoUrl:a},{first_name:cW,middle_name:a,last_name:aD,ans:e,email:a,bio:a,photoUrl:a}]},{article_id:bL,journal_id:n,section_id:r,path:q,journal:o,ar_title:bM,date_published:cX,doi:bN,author:[{first_name:cY,middle_name:a,last_name:cZ,ans:e,email:c_,bio:a,photoUrl:a},{first_name:c$,middle_name:a,last_name:da,ans:e,email:a,bio:a,photoUrl:a},{first_name:db,middle_name:a,last_name:A,ans:e,email:a,bio:a,photoUrl:a}]},{article_id:bO,journal_id:n,section_id:r,path:q,journal:o,ar_title:bP,date_published:dc,doi:bQ,author:[{first_name:dd,middle_name:a,last_name:de,ans:aG,email:a,bio:a,photoUrl:a},{first_name:aH,middle_name:a,last_name:aI,ans:aG,email:a,bio:a,photoUrl:a},{first_name:aJ,middle_name:a,last_name:aK,ans:H,email:R,bio:a,photoUrl:a},{first_name:df,middle_name:a,last_name:dg,ans:dh,email:a,bio:a,photoUrl:a},{first_name:di,middle_name:a,last_name:dj,ans:dk,email:a,bio:a,photoUrl:a}]},{article_id:bR,journal_id:n,section_id:r,path:q,journal:o,ar_title:bS,date_published:bG,doi:bT,author:[{first_name:aH,middle_name:a,last_name:aI,ans:H,email:a,bio:a,photoUrl:a},{first_name:dl,middle_name:a,last_name:A,ans:dm,email:a,bio:a,photoUrl:a},{first_name:aJ,middle_name:a,last_name:aK,ans:H,email:R,bio:a,photoUrl:a},{first_name:A,middle_name:a,last_name:dn,ans:do0,email:a,bio:a,photoUrl:a},{first_name:dp,middle_name:a,last_name:aF,ans:dq,email:a,bio:a,photoUrl:a},{first_name:dr,middle_name:a,last_name:ds,ans:dt,email:a,bio:a,photoUrl:a}]},{article_id:bU,journal_id:n,section_id:I,path:q,journal:o,ar_title:bV,date_published:du,doi:bW,author:[{first_name:aL,middle_name:a,last_name:J,ans:e,email:a,bio:a,photoUrl:a},{first_name:aM,middle_name:a,last_name:aN,ans:e,email:a,bio:a,photoUrl:a},{first_name:aO,middle_name:a,last_name:aP,ans:e,email:dv,bio:a,photoUrl:a}]},{article_id:bX,journal_id:n,section_id:I,path:q,journal:o,ar_title:bY,date_published:dw,doi:bZ,author:[{first_name:dx,middle_name:a,last_name:A,ans:e,email:a,bio:a,photoUrl:a},{first_name:dy,middle_name:a,last_name:dz,ans:e,email:a,bio:a,photoUrl:a},{first_name:aL,middle_name:a,last_name:J,ans:e,email:a,bio:a,photoUrl:a},{first_name:dA,middle_name:a,last_name:aE,ans:e,email:a,bio:a,photoUrl:a},{first_name:aM,middle_name:a,last_name:aN,ans:e,email:a,bio:a,photoUrl:a},{first_name:J,middle_name:a,last_name:dB,ans:e,email:a,bio:a,photoUrl:a},{first_name:aO,middle_name:a,last_name:aP,ans:e,email:a,bio:a,photoUrl:a}]}],editor:[]}}],fetch:{"data-v-0baa1603:0":{qKname:q,component:U,screenwidth:a}},error:b,state:{token:a,index:{data:{data:{footer:{},info:{},middle:{},nav:{},top:{}}},oaeNav:[{name:ed,sort:i,children:[{name:"Company",sort:i,url:"\u002Fabout\u002Fwho_we_are"},{name:"Latest 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permanently:",url:"https:\u002F\u002Fwww.portico.org\u002Fpublishers\u002Foae\u002F",img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20230911\u002F67d78ebf8c55485db6ae5b5b4bcda421.jpg"},follow:[{title:"LinkedIn",url:"https:\u002F\u002Fwww.linkedin.com\u002Fin\u002Fintelligence-robotics\u002F",icon:"icon-linkedin"},{title:"Twitter",url:mb,icon:"icon-tuite1"}],wechat_img:a,twitter:{url:mb,img:"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20230824\u002F5249ddabb6d642558c9843fba9283219.png"}},top:{path:q,pid:n,journal_img:cs,journal_name:o,mpt:"40 days",issn:ma,indexing:{ESCI:"https:\u002F\u002Fwww.oaepublish.com\u002Fnews\u002Fir.852",Scopus:bn,"Google Scholar":"https:\u002F\u002Fscholar.google.com.hk\u002Fcitations?view_op=list_works&hl=zh-CN&hl=zh-CN&user=-Hx5OVYAAAAJ",Dimensions:"https:\u002F\u002Fapp.dimensions.ai\u002Fdiscover\u002Fpublication?and_facet_source_title=jour.1423782",Lens:"https:\u002F\u002Fwww.lens.org\u002Flens\u002Fsearch\u002Fscholar\u002Flist?p=0&n=10&s=date_published&d=%2B&f=false&e=false&l=en&authorField=author&dateFilterField=publishedYear&orderBy=%2Bdate_published&presentation=false&preview=true&stemmed=true&useAuthorId=false&publicationType.must=journal%20article&sourceTitle.must=Intelligence%20%26%20Robotics&publisher.must=OAE%20Publishing%20Inc."},editor:ct,journal_rank:a,journal_flyer:"https:\u002F\u002Ff.oaes.cc\u002Findex_ad\u002Fflyer\u002FIR-flyer.pdf",qksearch:["Intelligence","Robotics","Reinforcement Learning","Machine Learning","Unmanned Vehicles","UAV"],sitetag:"Intell Robot",ad:[],colour_tag:cr,score:a,mobile_top_img:a,impact_factor:[{factor:cw,url:bn},{factor:j,url:a}],rgba:cu,log_image:cv},webinfo:{},searchKey:a,loading:U,appid:a,videoPlay:{show:cx,href:a}},editer:{editList:{list:{}}},userdata:{showLogin:cx,logined:cx}},serverRendered:U,routePath:"\u002Farticles\u002Fir.2022.21",config:{_app:{basePath:mc,assetsPath:mc,cdnURL:"https:\u002F\u002Fg.oaes.cc\u002Foae\u002Fnuxt\u002F"}}}}("",null,1,5170,"[]",4,0,"2021","1","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240506\u002Fea3d9071c35b4bf3982ffe25f1083620.png",3,"2",2,40,"Intelligence & Robotics","3","ir",927,5,"0","4","5","6","IR","Research Article",10,"Li","https:\u002F\u002Fmjl.clarivate.com\u002Fsearch-results","2022","2017","33",7,15,"[\"1\"]",935,"Zhu",20,"Lens",34,6,35,"2020",32,"Chaomin.Luo@ece.msstate.edu",23,39,true,"39","Extracellular Vesicles and Circulating Nucleic Acids","Microbiome Research Reports","One Health & Implementation Research","Chemical Synthesis","Energy Materials","Journal of Materials Informatics","Microstructures","Minerals and Mineral Materials","Soft Science","Complex Engineering Systems","Disaster Prevention and Resilience","Green Manufacturing Open","Journal of Smart Environments and Green Computing","Journal of Surveillance, Security and Safety","Ageing and Neurodegenerative Diseases","Artificial Intelligence Surgery","Cancer Drug Resistance","Connected Health And Telemedicine","Hepatoma Research","Journal of Cancer Metastasis and Treatment","Journal of Translational Genetics and Genomics","Metabolism and Target Organ Damage","Mini-invasive Surgery","Plastic and Aesthetic Research","Rare Disease and Orphan Drugs Journal","The Journal of Cardiovascular Aging","Vessel Plus","Carbon Footprints","Journal of Environmental Exposure Assessment","Water Emerging Contaminants & Nanoplastics","A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping",2022,7352,"Improved DDPG algorithm-based path planning for unmanned surface vehicles","ir.2024.22","Chen","Dong","Liu","[\"1\",\"#\"]","Tingjun","Lei","Chaomin","Luo","Danjie","Tao","Yan","Simon X.","Yang","40","7","8","9","11",2024,"Shiqi Liu, ... Eugene Levin","Menglong Hua, ... Zihao Chen",65,22,14,25,30,46,8,24,9,"ESCI, CAS, Dimensions, Lens, CNKI",17,11,55,12,"2015","ESCI, CAS, Scopus, Wanfang Data, CNKI, Dimensions, Embase",13,"https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101199351",90,16,"Journal of Unexplored Medical Data",18,21,26,"ESCI, Scopus, CAS, Dimensions, Lens, CNKI","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240809\u002Fccc8ac23b789404781787655e4495af4.png",117,33,"Neuroimmunology and Neuroinflammation","2014",29,31,"ISSN XXXX-XXXX (Coming soon)","Soil Health","Space Mission Planning & Operations","Stomatological Disease and Science","2022-10-12 00:00:00","Adjacent node selection (ANS) algorithm, safety-aware roads, path planning, multiple-waypoint optimization, navigation and mapping",6379,"UAV path planning based on a dual-strategy ant colony optimization algorithm","ir.2023.37",5988,"Cooperative search for moving targets with the ability to perceive and evade using multiple UAVs","ir.2023.30",5791,"A bio-inspired algorithm in image-based path planning and localization using visual features and maps","ir.2023.14",5169,"An informative planning-based multi-layer robot navigation system as applied in a poultry barn","ir.2022.18",5032,"Motion planning and tracking control of unmanned underwater vehicles: technologies, challenges and prospects","ir.2022.13",4353,"Bio-inspired intelligence with applications to robotics: a survey","ir.2021.08","ISSN 2770-3541 (Online)","10","14","15","16","20","25","28","29","30","36","37","42","44",2023,58,83,27,"Contact Us","#0047bb","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002F59802903b17e4eebae240e004311d193.jpg","Simon X. Yang","rgb(0,71,187)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fir.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240813\u002F49390c7e86ab40a58ee862e8c1af65ba.png",false,"\u003Cp\u003EAutonomous robot multi-waypoint navigation and mapping have been demanded in many real-world applications found in search and rescue (SAR), environmental exploration, and disaster response. Many solutions to this issue have been discovered via graph-based methods in need of switching the robotos trajectory between the nodes and edges within the graph to create a trajectory for waypoint-to-waypoint navigation. However, studies of how waypoints are locally bridged to nodes or edges on the graphs have not been adequately undertaken. In this paper, an adjacent node selection (ANS) algorithm is developed to implement such a protocol to build up regional path from waypoints to nearest nodes or edges on the graph. We propose this node selection algorithm along with the generalized Voronoi diagram (GVD) and Improved Particle Swarm Optimization (IPSO) algorithm as well as a local navigator to solve the safety-aware concurrent graph-based multi-waypoint navigation and mapping problem. Firstly, GVD is used to form a Voronoi diagram in an obstacle populated environment to construct safety-aware routes. Secondly, the sequence of multiple waypoints is created by the IPSO algorithm to minimize the total travelling cost. Thirdly, while the robot attempts to visit multiple waypoints, it traverses along the edges of the GVD to plan a collision-free trajectory. The regional path from waypoints to the nearest nodes or edges needs to be created to join the trajectory by the proposed ANS algorithm. Finally, a sensor-based histogram local reactive navigator is adopted for moving obstacle avoidance while local maps are constructed as the robot moves. An improved \u003Ci\u003EB\u003C\u002Fi\u003E-spline curve-based smooth scheme is adopted that further refines the trajectory and enables the robot to be navigated smoothly. Simulation and comparison studies validate the effectiveness and robustness of the proposed model.\u003C\u002Fp\u003E","https:\u002F\u002Fv1.oaepublish.com\u002Ffiles\u002Ftalkvideo\u002F5170.mp4","The paper proposes an ANS algorithm to find a node in the graph to connect waypoints that are utilized in the safety-aware multi-waypoint navigation and mapping by an improved PSO and GVD model.","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240205\u002Fb263c1d7fa6a447f89d0b42385dcee47.jpg",1665504000,"Sellers T, Lei T, Luo C, Eu Jan G, Ma J. A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping. \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E 2022;2(4):333-54. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2022.21","Sellers T, Lei T, Luo C, Eu Jan G, Ma J. A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping. \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E. 2022;2:333-54. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2022.21","Sellers, Timothy, Gene Eu Jan, and Junfeng Ma. 2022. \"A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping\" \u003Ci\u003EIntell Robot\u003C\u002Fi\u003E. 2, no.4: 333-54. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2022.21","Sellers, T.; Lei, T.; Luo, C.; Eu Jan, G.; Ma, J. A node selection algorithm to graph-based multi-waypoint optimization navigation and mapping. \u003Ci\u003EIntell. Robot.\u003C\u002Fi\u003E \u003Cb\u003E2022\u003C\u002Fb\u003E, \u003Ci\u003E2\u003C\u002Fi\u003E, 333-54. http:\u002F\u002Fdx.doi.org\u002F10.20517\u002Fir.2022.21","2024-11-13 00:00:00","Menglong","Hua","Weixiang","Zhou","zhouwx@shmtu.edu.cn","Hongying","Cheng","Zihao","2023-12-21 00:00:00","Xiaoming","Mai","Na","dongna@tju.edu.cn","Shuai","Hao","2023-10-28 00:00:00","Ziyi","Wang","mariowang@njust.edu.cn","Wencheng","Zou","Sheng","2023-06-28 00:00:00","Daniel","Short","Daniel W.","Carruth","[\"2\"]","Zhuming","Bi","[\"3\"]","Guoming","[\"2\",\"3\"]","Zhang","[\"4\"]","Lantao","[\"5\"]","Richard","Stephen Gates","[\"2\",\"3\",\"6\"]","2022-07-08 00:00:00","syang@uoguelph.ca","2021-10-12 00:00:00","Junfei","Zhe","Xu","Kevin","Zeng","12","13",1731945600,"Path planning method for USVs based on improved DWA and COLREGs","10.20517\u002Fir.2024.23","\u003Cp\u003EIn the navigation of unmanned surface vehicles (USVs), various types of obstacles may be encountered, which can be categorized into real-time collision avoidance among multiple USVs and obstacle avoidance between USVs and other obstacles. Most existing autonomous obstacle avoidance algorithms do not account for the nonlinear motion characteristics of USVs, often resulting in non-compliance with the International Regulations for Preventing Collisions at Sea (COLREGs) and a tendency to fall into local optima. To address these issues, this paper proposes a path planning algorithm that integrates the dynamic window approach (DWA) considering nonlinear characteristics with COLREGs, making the USV's motion trajectory more applicable to practical engineering scenarios. A kinematic mathematical model is established based on the motion characteristics of USVs, and an evaluation function for the optimal path is constructed using DWA. The fully informed search algorithm (FISA) is employed to select the optimal set of velocities and steering angles from the velocity sampling set, based on different cost calculation methods. The USVs use a laser radar for local obstacle detection, enabling real-time dynamic obstacle avoidance. To address the real-time collision avoidance problem among multiple USVs in open waters, the algorithm filters out COLREGs-compliant avoidance maneuvers during path planning. The correctness and feasibility of the fusion algorithm were verified through comparative simulations. In the simulated environment model, the influence of ocean currents on the USV was introduced, and multiple sets of experiments under different conditions were conducted to compare the motion trajectories, average travel distances, and average travel times of the USV. The simulation results indicate that the USV can perform accurate obstacle avoidance when encountering various types of obstacles. Compared to the traditional DWA algorithm, the proposed approach demonstrates advantages in terms of travel distance and travel time, while still achieving effective obstacle avoidance.\u003C\u002Fp\u003E","https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F2e840c3e-2dc5-4bea-bb3c-0b62e2478ad6\u002Fir4023.pdf",385,7378,"385-405",405,"ir.2024.23",1731427200,"10.20517\u002Fir.2024.22","\u003Cp\u003EAs a promising mode of water transportation, unmanned surface vehicles (USVs) are used in various fields owing to their small size, high flexibility, favorable price, and other advantages. Traditional navigation algorithms are affected by various path planning issues. To address the limitations of the traditional deep deterministic policy gradient (DDPG) algorithm, namely slow convergence speed and sparse reward and punishment functions, we proposed an improved DDPG algorithm for USV path planning. First, the principle and workflow of the DDPG deep reinforcement learning (DRL) algorithm are described. Second, the improved method (based on the USVs kinematic model) is proposed, and a continuous state and action space is designed. The reward and punishment function are improved, and the principle of collision avoidance at sea is introduced. Dynamic region restriction is added, distant obstacles in the state space are ignored, and the nearby obstacles are observed to reduce the number of algorithm iterations and save computational resources. The introduction of a multi-intelligence approach combined with a prioritized experience replay mechanism accelerates algorithm convergence, thereby increasing the efficiency and robustness of training. Finally, through a combination of theory and simulation, the DDPG DRL is explored for USV obstacle avoidance and optimal path planning.\u003C\u002Fp\u003E","https:\u002F\u002Ff.oaes.cc\u002Fxmlpdf\u002F69bc157a-c03e-48d7-b08c-14cc87ec711f\u002Fir4022.pdf",363,"363-84",384,"Xiaoming Mai, ... Hao Chen",1517," Swarm Intelligence for Robotic Systems","Ziyi Wang\u003Ca href='https:\u002F\u002Forcid.org\u002F0009-0004-2527-7409' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Ziyi Wang'\u003E\u003C\u002Fa\u003E, ... Sheng Li\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0002-8897-6889' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Sheng Li'\u003E\u003C\u002Fa\u003E",222,"Daniel Short\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0002-6340-2918' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Daniel Short'\u003E\u003C\u002Fa\u003E, ... Zhuming Bi\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0002-8145-7883' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Zhuming Bi'\u003E\u003C\u002Fa\u003E","Tingjun Lei, ... Richard Stephen Gates","Review","Danjie Zhu, ... Simon X. Yang\u003Ca href='https:\u002F\u002Forcid.org\u002F0000-0002-6888-7993' target='_blank'\u003E\u003Cimg src='https:\u002F\u002Fi.oaes.cc\u002Fimages\u002Forcid.png' class='author_id' alt='Simon X. Yang'\u003E\u003C\u002Fa\u003E","Junfei Li, ... Simon X. Yang","About","Editorial Policies","Journals","Academic Talks","Biology & Life Science","Chemistry & Materials Science","Computer Science & Engineering","\u002Fir","Medicine & Public Health","Environmental Science","#837fbc","ISSN 2769-5301 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F6135b005a8674b878a7d1a5c91ac9869.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fand","Wei-Dong Le","CAS, Dimensions, Lens, CNKI","rgb(99,94,171)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fand.jpg","and","#030072","ISSN 2771-0408 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F14bc775310574f91b457ff09d6c6d950.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fais","Andrew A. Gumbs","ESCI, Scopus, Dimensions, Lens","rgb(3,0,114)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fais.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240813\u002Fb7358d0632aa413fa84630ef0a2fb7a0.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101189039","ais","#e8b475","ISSN 2578-532X (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231113\u002F9db4b73f2e954900807499f00094fcf0.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcdr","Godefridus J. (Frits) Peters","2018","337","ESCI, PMC, Scopus, CAS, CNKI, Dimensions, Lens, Embase","rgb(224,155,71)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fcdr.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240104\u002F502e35965fc448e0bc9f2d6c1b0bb3d7.png","https:\u002F\u002Fwww.ncbi.nlm.nih.gov\u002Fpmc\u002Fjournals\u002F4180\u002F","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240611\u002F5def3ab5829743ce8d55e664cabe0dcb.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101047803","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240620\u002Fed79fcbfce8e4cd38a36cc4d5f7d473d.png","https:\u002F\u002Fjcr.clarivate.com\u002Fjcr-jp\u002Fjournal-profile?journal=CANCER%20DRUG%20RESIST&year=2022","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240620\u002F86311aafb7a945a98a7ee95c3627173b.png","cdr",418,38,"#1d57a5","ISSN 2770-6249 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Foae\u002Fcover_comengsys.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcomengsys","Hamid Reza Karimi","Scopus, Dimensions, Lens","rgb(29,87,165)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fces.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240722\u002F4748814aab87441d809e5b9ddcd9d56e.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101189038","comengsys",71,"#75ac9d","ISSN 2993-2920 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F0b7a80c0f6e545818e143ea48fe9be90.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fchatmed","Yuan-Ting Zhang","rgb(117, 172, 157)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fchatmed.jpg","chatmed","#00a5b3","ISSN 2769-5247 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002F1598f33bf2284642830511e489eb31bf.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcs","Bao-Lian Su","67","Scopus, CAS, Dimensions, Lens, CNKI","rgb(0,165,179)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fcs.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240813\u002F6d91bea2ff474cb9b7ec10017b26ec2a.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101189036","cs",157,51,"#6cc24a","ISSN 2831-932X (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fb1a3c337f6a54bcb945e0d9ab6afe0ec.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fcf","Yong Geng","rgb(86,164,55)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fcf.jpg","cf","#01588b","ISSN 2832-4056 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe5808de306274ba287f6f24bf45eb910.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fdpr","Jie Li","rgb(1,88,139)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fdpr.jpg","dpr",41,"#008c15","ISSN 2770-5900 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe1df1779a10c4a7486e89c5f5c4994f3.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fenergymater","Yuping Wu","92","rgb(0,140,21)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fem.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240621\u002Fab41932db942489c841586b032349a31.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240625\u002F3f1358bb24954c8eae11907a77ab8b18.png","energymater",196,"#b15ee9","ISSN 2767-6641 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fb7fc9d6a51bf4adda4ff6dec5002690b.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fevcna","Yoke Peng Loh","68","ESCI, Scopus, CAS, CNKI, Dimensions, 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Cao","465","rgb(239,108,62)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fhr.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240313\u002Fee358186a06449d5b2c8ea15f712cbed.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101058282","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240625\u002Faf5ee9028a554c6aa6f0030bbcbfe55d.png","https:\u002F\u002Fjcr.clarivate.com\u002Fjcr-jp\u002Fjournal-profile?journal=HEPATOMA%20RES&year=2023","hr",543,"https:\u002F\u002Fwww.oaepublish.com\u002Fir","ESCI, Scopus, Google Scholar, Dimensions, Lens",60,"#44b762","ISSN 2454-2857 (Online)\u003Cbr\u003EISSN 2394-4722 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Pedrycz","Lens, Dimensions","rgb(0,162,168)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjsegc.jpg",80,"jsegc",43,19,"#3e9aff","ISSN 2694-1015 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Ffac7bed365f1465799f61e5b2212492b.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjsss","CNKI, Dimensions, Lens","rgb(62,154,255)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjsss.jpg",85,"jsss",49,47,"#00b2a9","ISSN 2771-5949 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231115\u002F1dd8d978556f47cda9d63c2e03bb7e29.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjeea","Stuart Harrad","Scopus, CAS, Dimensions, Lens","rgb(0,128,121)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjeea.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240617\u002F3a87e8e1c3f145e78ac49d8d775b0e7d.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101196042","jeea",68,52,"#2d68c4","ISSN 2831-2597 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Lens","rgb(123,164,219)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fmrr.jpg",101,"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240104\u002F206342e31750460bb7449112e6aafba6.png","https:\u002F\u002Fwww.ncbi.nlm.nih.gov\u002Fpmc\u002Fjournals\u002F4522\u002F","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240621\u002F1dfbaaeea68e40c98d7494b8c702b695.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101195678?origin=resultslist","https:\u002F\u002Fwww.oaepublish.com\u002Fnews\u002Fmrr.836","mrr","#6f7bd4","ISSN 2770-2995 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231113\u002F1738c68b2d0a41edae268adf4ea5c7e2.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fmicrostructures","Shujun Zhang","53","ESCI, Scopus, CAS, Dimensions, 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Dimensions","rgb(41,167,168)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fmis.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240208\u002F6020c4a092dd44578a776056b5057abc.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101115668","mis",395,"#33a7d9","ISSN 2349-6142 (Online)\u003Cbr\u003EISSN 2347-8659 (Print)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F026484b9f2b042fda9b7afffc0d9f62e.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fneurosciences","296","rgb(51,167,217)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fnn.jpg",104,"neurosciences",296,28,"#43b02a","ISSN 2769-6413 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Ff65f06a7eae24d1484bad38ff0a2bef2.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fohir","Dimensions","Jose M. Martin-Moreno","rgb(67,176,42)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fohir.jpg",105,"ohir","#c45284","ISSN 2349-6150 (Online)\u003Cbr\u003EISSN 2347-9264 (Print)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F8d72c3f669d5490ea3b9abc424d66cb6.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fpar","Wen-Guo Cui","520","rgb(196,82,132)","ESCI, Scopus, Dimensions, CNKI, Lens, Wanfang","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fpar.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240607\u002F9bbbcad710cf457e85d63b1811faefb1.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101111784","https:\u002F\u002Fclarivate.com\u002Fproducts\u002Fscientific-and-academic-research\u002Fresearch-discovery-and-workflow-solutions\u002Fwebofscience-platform\u002Fweb-of-science-core-collection\u002Femerging-sources-citation-index\u002F","par",620,"#d62598","ISSN 2771-2893 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231121\u002F6ee75e8ab583446b90c97865c2c68464.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Frdodj","Daniel Scherman","Dimensions, Lens","rgb(214,37,152)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Frdodj.jpg",106,"rdodj",78,"#1f4e79","ISSN 2769-5441 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002F98e2ed5023ee466c8890d749bfd97597.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fss","YongAn Huang","58","ESCI, Scopus, CAS, Lens, Dimensions, CNKI","rgb(31,78,121)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fss.jpg",107,"https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240606\u002Fcfa200290630438c8e3a3cae1536af37.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101178114","ss",54,"#a96318","https:\u002F\u002Fi.oaes.cc\u002Fimages_2018\u002Foae\u002Fcover_sh.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fsh","Manoj Shukla","rgb(169,99,24)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fsh.jpg",108,"sh","#4475e1","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F79ccda0bc909441587a0b70e0a8ef1cc.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fsmpo","Madjid Tavana","rgb(68,117,225)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fsmpo.jpg",109,"smpo","#53be9b","ISSN 2573-0002 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231218\u002F54ba11ec5c57448a84169c318bb94fd4.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fsds","Nikolaos G. Nikitakis","rgb(62,163,130)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fsds.jpg",110,"sds","#0038a0","ISSN 2768-5993 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231113\u002F695c0f0de82c43f189957ff572f0798c.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fjca","89","rgb(0,56,160)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fjca.png","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240611\u002Fd7d617bceee246428da2ed07a84efd5e.png","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101176723","https:\u002F\u002Fwww.oaepublish.com\u002Fnews\u002Fjca.838","jca",129,36,"#db6868","ISSN 2574-1209 (Online)","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20231110\u002Fe64051237c8a454085bb02abfdc6dd12.jpg","https:\u002F\u002Fwww.oaepublish.com\u002Fvp","290","CAS, Scopus, CNKI, Dimensions, Lens","rgb(219,104,104)","https:\u002F\u002Fi.oaes.cc\u002Fupload\u002Fjournal_logo\u002Fvp.jpg","https:\u002F\u002Fi.oaes.cc\u002Fuploads\u002F20240606\u002Fa5cb40102c2d4ec4b1a362fe1f27d324.jpg","https:\u002F\u002Fwww.scopus.com\u002Fsourceid\u002F21101064905","vp",351,"5555","\u002Fir\u002Fcontact_us","Video Abstract Guidelines","\u002Fir\u002Fvideo_abstract_guidelines","2770-3541 (Online)","https:\u002F\u002Ftwitter.com\u002FOAE_IR","\u002F"));</script><script src="https://g.oaes.cc/oae/nuxt/b06ddfb.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/d1923ac.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/0a3b980.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/3e8004d.js" defer></script><script src="https://g.oaes.cc/oae/nuxt/b19d7ea.js" defer></script> </body> </html> <div id="noIe" style="display: none;"> <style> #noIe { background: rgba(99, 125, 255, 1); width: 100%; height: 100vh; position: fixed; top: 0; left: 0; z-index: 999999; } #noIe .container 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padding-top: 100px; } #noIe .logo { height: 24px; } @media screen and (max-width: 820px) { #noIe .container { width: 432px; left: 50%; margin-left: -216px; margin-top: 48px; position: relative; top: 0; } #noIe .container ul { width: 290px; height: 352px; margin-left: -30px; margin-top: 40px; } #noIe .container li { margin-left: 30px; margin-top: 24px; } #noIe .logo-container { padding-top: 121px; padding-bottom: 20px; } } </style> <div class="container"> <p class="title">The current browser is not compatible</p> <p class="title2">The following browsers are recommended for the best use experience</p> <ul> <li> <a href="https://www.google.cn/chrome/" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i2/O1CN01Nn0IoE1cmXZ6gFiM3_!!6000000003643-2-tps-230-230.png" /> <p>Chrome</p> </a> </li> <li> <a href="http://www.firefox.com.cn/" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i3/O1CN01P8aqdX1HdHczGialK_!!6000000000780-2-tps-230-230.png" /> <p>Firefox</p> </a> </li> <li> <a href="https://www.apple.com/safari/" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i4/O1CN01vVxDF11chVD0nsbiZ_!!6000000003632-2-tps-230-230.png" /> <p>Safari</p> <p class="tip">Only supports Mac</p> </a> </li> <li> <a href="https://www.microsoft.com/zh-cn/edge" target="_blank"> <img src="https://gw.alicdn.com/imgextra/i4/O1CN01UW7hs31Xa6jfm2a2O_!!6000000002939-2-tps-230-230.png" /> <p>Edge</p> </a> </li> </ul> <div class="logo-container"> <img 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" 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