Document Zbl 1401.93204

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T.</a>; <a href="/authors/ishihara.joao-yoshiyuki" title="Author Profile">Ishihara, João Y.</a></div> <h2 class="title"> <strong>Unscented and square-root unscented Kalman filters for quaternionic systems.</strong> <i>(English)</i> <a class="label nowrap" href="/1401.93204">Zbl 1401.93204</a> </h2> <div class="source"> <a href="/serials/1790" title="Journal Profile">Int. J. Robust Nonlinear Control</a> <a href="/?q=in%3A415967" title="Articles in this Issue">28, No. 15, 4500-4527 (2018)</a>. </div> <div class="abstract">Summary: Proper construction of an Unscented Kalman Filter (UKF) for unit quaternionic systems is not straightforward due to the incompatibility between the algebraic properties of the unit quaternions and the common real vector space operations (additions and scalar multiplications) needed in the steps of a filter algorithm. This work studies, in detail, all UKFs and square-root UKFs for quaternionic systems proposed in the literature. First, we classify the algorithms according to the preservation of the unity norm of the quaternion variables. Second, we propose two new algorithms: the Quaternionic Additive Unscented Kalman Filter (QuAdUKF) and a square-root variant of it. The QuAdUKF encompasses all known UKFs for quaternionic systems of the literature preserving, in all steps, the norm of the unit quaternion variables. Besides, it can also yield new UKFs with this norm preservation property. The QuAdUKF’s square-root variant has better properties in comparison with all the square-root UKFs for quaternionic systems of the literature. Numerical experiments for a spacecraft attitude estimation problem illustrate the theoretical results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6979961">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E11" title="MSC2020">93E11</a> </td> <td class="space"> Filtering in stochastic control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93E10" title="MSC2020">93E10</a> </td> <td class="space"> Estimation and detection in stochastic control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A16H05" title="MSC2020">16H05</a> </td> <td class="space"> Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aattitude+estimation">attitude estimation</a>; <a href="/?q=ut%3Asquare-root+unscented+Kalman+filter+%28SRUKF%29">square-root unscented Kalman filter (SRUKF)</a>; <a href="/?q=ut%3Aunit+quaternion">unit quaternion</a>; <a href="/?q=ut%3Aunscented+Kalman+filter+%28UKF%29">unscented Kalman filter (UKF)</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1401.93204" data-ciurl="/ci/06979961" data-biburl="/bibtex/06979961.bib" data-amsurl="/amsrefs/06979961.bib" data-xmlurl="/xml/06979961.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06979961.pdf" title="Zbl 1401.93204 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1002/rnc.4249" aria-label="DOI for “Unscented and square-root unscented Kalman filters for quaternionic systems”" title="10.1002/rnc.4249">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Spong, MW; Hutchinson, S.; Vidyasagar, M., Robot Modeling and Control, 2006, Hoboken, NJ: John Wiley & Sons</td> </tr><tr> <td>[2]</td> <td class="space">Siciliano, B.; Sciavicco, L.; Villani, L.; Oriolo, G., Kinematics, Robotics. Advanced Textbooks in Control and Signal Processing, 2009, London, UK: Springer London</td> </tr><tr> <td>[3]</td> <td class="space">Selig, JM, Geometric Fundamentals of Robotics. 2nd ed, 2005, New York, NY: Springer · <a href="/1062.93002" class="nowrap">Zbl 1062.93002</a></td> </tr><tr> <td>[4]</td> <td class="space">Altmann, SL, Rotations, Quaternions, and Double Groups, 1986, Mineola, NY: Dover Publications, Inc. · <a href="/0683.20037" class="nowrap">Zbl 0683.20037</a></td> </tr><tr> <td>[5]</td> <td class="space">Cheon, Y‐J; Kim, J‐H, Unscented filtering in a unit quaternion space for spacecraft attitude estimation, 2007, Vigo, Spain</td> </tr><tr> <td>[6]</td> <td class="space">Crassidis, JL; Markley, FL, Unscented filtering for spacecraft attitude estimation, J Guid Control Dyn, 26, 4, 536-542, 2003</td> </tr><tr> <td>[7]</td> <td class="space">Tang, X.; Yan, J.; Zhong, D., Square‐root sigma‐point Kalman filtering for spacecraft relative navigation, Acta Astronaut, 66, 5‐6, 704-713, 2010</td> </tr><tr> <td>[8]</td> <td class="space">Challa, MS; Moore, JG; Rogers, DJ, A simple attitude unscented Kalman filter: theory and evaluation in a magnetometer‐only spacecraft scenario, IEEE Access, 4, 1845-1858, 2016</td> </tr><tr> <td>[9]</td> <td class="space">Khoder, W.; Fassinut‐Mombot, B.; Benjelloun, M., Inertial navigation attitude velocity and position algorithms using quaternion scaled unscented Kalman filtering, 2008, Orlando, FL</td> </tr><tr> <td>[10]</td> <td class="space">Du, G.; Shao, H.; Chen, Y.; Zhang, P.; Liu, X., An online method for serial robot self‐calibration with CMAC and UKF, Robot Comput‐Integr Manuf, 42, 39-48, 2016</td> </tr><tr> <td>[11]</td> <td class="space">Enayati, N.; Momi, ED; Ferrigno, G., A quaternion‐based unscented Kalman filter for robust optical/inertial motion tracking in computer‐assisted surgery, IEEE Trans Instrum Meas, 64, 8, 2291-2301, 2015</td> </tr><tr> <td>[12]</td> <td class="space">Hauberg, S.; Lauze, F.; Pedersen, KS, Unscented Kalman filtering on Riemannian manifolds, J Math Imaging Vis, 46, 1, 103-120, 2013 · <a href="/1316.94028" class="nowrap">Zbl 1316.94028</a></td> </tr><tr> <td>[13]</td> <td class="space">Xuedong, W.; Xinhua, J.; Rongjin, Z.; Jiashan, H., An application of unscented Kalman filter for pose and motion estimation based on monocular vision, 2006, Montreal, Canada</td> </tr><tr> <td>[14]</td> <td class="space">Vartiainen, P.; Bragge, T.; Arokoski, JP; Karjalainen, PA, A nonlinear state‐space modeling of human motion using 2‐D marker observations, IEEE Trans Biomed Eng, 61, 7, 2167-2178, 2014</td> </tr><tr> <td>[15]</td> <td class="space">Julier, SJ; Uhlmann, JK, Unscented filtering and nonlinear estimation, Proc IEEE, 92, 3, 401-422, 2004</td> </tr><tr> <td>[16]</td> <td class="space">Menegaz, HMT; Ishihara, JY; Borges, GA; Vargas, AN, A systematization of the unscented Kalman filter theory, IEEE Trans Autom Control, 60, 10, 2583-2598, 2015 · <a href="/1360.93705" class="nowrap">Zbl 1360.93705</a></td> </tr><tr> <td>[17]</td> <td class="space">Kraft, E., A quaternion‐based unscented Kalman filter for orientation tracking, 2003, Cairns, Australia · <a href="/1040.68137" class="nowrap">Zbl 1040.68137</a></td> </tr><tr> <td>[18]</td> <td class="space">Larsen, JA; Vinther, K., Inexpensive CubeSat attitude estimation using COTS components and unscented Kalman filtering, 2011, Istanbul, Turkey</td> </tr><tr> <td>[19]</td> <td class="space">LaViola, JJ, A comparison of unscented and extended Kalman filtering for estimating quaternion motion, 2003, Denver, CO</td> </tr><tr> <td>[20]</td> <td class="space">Leeghim, H.; Choi, Y.; Jaroux, BA, Uncorrelated unscented filtering for spacecraft attitude determination, Acta Astron, 67, 1‐2, 135-144, 2009</td> </tr><tr> <td>[21]</td> <td class="space">Romanovas, M.; Klingbeil, L.; Trachtler, M.; Manoli, Y., Efficient orientation estimation algorithm for low cost inertial and magnetic sensor systems, 2009, Cardiff, UK</td> </tr><tr> <td>[22]</td> <td class="space">Sekhavat, P.; Gong, Q.; Ross, IM, NPSAT1 parameter estimation using unscented Kalman filtering, 2007, New York, NY</td> </tr><tr> <td>[23]</td> <td class="space">Sipos, BJ, Application of the manifold‐constrained unscented Kalman filter, 2008, Monterey, CA</td> </tr><tr> <td>[24]</td> <td class="space">Soken, HE; Sakai, S‐I, UKF based on‐orbit gyro and magnetometer bias estimation as a part of the attitude determination procedure for a small satellite, 2011, Gyeonggi‐do, South Korea</td> </tr><tr> <td>[25]</td> <td class="space">Teixeira, BOS; Chandrasekar, J.; Tôrres, LAB; Aguirre, LA; Bernstein, DS, State estimation for linear and non‐linear equality‐constrained systems, Int J Control, 82, 5, 918-936, 2009 · <a href="/1165.93033" class="nowrap">Zbl 1165.93033</a></td> </tr><tr> <td>[26]</td> <td class="space">Xing, Y.; Cao, X.; Zhang, S., Modified UKF for integrated orbit and attitude determination based on gyro and magnetometer, 2008, Shenzhen, China</td> </tr><tr> <td>[27]</td> <td class="space">Yang, Y.; Zhou, J.; Loffeld, O., Quaternion‐based Kalman filtering on INS/GPS, 2012, Singapore</td> </tr><tr> <td>[28]</td> <td class="space">Zhang, Z‐Q; Ji, L‐Y; Huang, Z‐P; Wu, J‐K, Adaptive information fusion for human upper limb movement estimation, IEEE Trans Syst Man Cybern A, 42, 5, 1100-1108, 2012</td> </tr><tr> <td>[29]</td> <td class="space">Zhao, L.; Nie, Q.; Guo, Q., Unscented Kalman filtering for SINS attitude estimation, 2007, Guangzhou, China</td> </tr><tr> <td>[30]</td> <td class="space">Amirhosseini, SF; Romanovas, M.; Schwarze, T.; Schwaab, M.; Traechtler, M.; Manoli, Y., Stochastic cloning unscented Kalman filtering for pedestrian localization applications, 2013, Montbeliard‐Belfort, France</td> </tr><tr> <td>[31]</td> <td class="space">Chang, L.; Hu, B.; Chang, G., Modified unscented quaternion estimator based on quaternion averaging, J Guid Control Dyn, 37, 1, 305-308, 2014</td> </tr><tr> <td>[32]</td> <td class="space">Julier, SJ; LaViola, JJ, On Kalman filtering with nonlinear equality constraints, IEEE Trans Signal Process, 55, 6, 2774-2784, 2007 · <a href="/1391.94261" class="nowrap">Zbl 1391.94261</a></td> </tr><tr> <td>[33]</td> <td class="space">Kondo, A.; Doki, H.; Hirose, K., An estimation method of 3D posture using quaternion‐based unscented Kalman filter, 2013, Nagoya, Japan</td> </tr><tr> <td>[34]</td> <td class="space">Le, HX; Matunaga, S., A residual based adaptive unscented Kalman filter for fault recovery in attitude determination system of microsatellites, Acta Astronaut, 105, 1, 30-39, 2014</td> </tr><tr> <td>[35]</td> <td class="space">da Silva, NBF; Wilson, DB; Branco, KRLJ, Performance evaluation of the extended Kalman filter and unscented Kalman filter, 2015, Denver, CO</td> </tr><tr> <td>[36]</td> <td class="space">Menegaz, HMT, Unscented Kalman Filtering on Euclidean and Riemannian Manifolds, 2016, Brasília, Brazil: Universidade de Brasília</td> </tr><tr> <td>[37]</td> <td class="space">Cohn, PM, Further Algebra and Applications, 452, 2003, London, UK: Springer · <a href="/1006.00001" class="nowrap">Zbl 1006.00001</a></td> </tr><tr> <td>[38]</td> <td class="space">Markley, FL, Attitude error representations for Kalman filtering, J Guid Control Dyn, 26, 2, 311-317, 2003</td> </tr><tr> <td>[39]</td> <td class="space">Schaub, H.; Junkins, JL, Stereographic orientation parameters for attitude dynamics: a generalization of the Rodrigues parameters, J Astronaut Sci, 44, 1, 1-19, 1996</td> </tr><tr> <td>[40]</td> <td class="space">Julier, SJ; Uhlmann, JK; Durrant‐Whyte, HF, A new approach for filtering nonlinear systems, 1995, Seatle, WA</td> </tr><tr> <td>[41]</td> <td class="space">Vaccarella, A.; Momi, ED; Enquobahrie, A.; Ferrigno, G., Unscented Kalman filter based sensor fusion for robust optical and electromagnetic tracking in surgical navigation, IEEE Trans Instrum Meas, 62, 7, 2067-2081, 2013</td> </tr><tr> <td>[42]</td> <td class="space">Soken, HE; Hajiyev, C., Adaptive unscented Kalman filter with multiple fading factors for pico satellite attitude estimation, 2009, Istanbul, Turkey</td> </tr><tr> <td>[43]</td> <td class="space">Wan, EA; Van Der Merwe, R., The unscented Kalman filter for nonlinear estimation, 2000, Lake Louise, Canada</td> </tr><tr> <td>[44]</td> <td class="space">Julier, SJ, The spherical simplex unscented transformation, 2003, Denver, CO</td> </tr><tr> <td>[45]</td> <td class="space">Pennec, X., Computing the mean of geometric features application to the mean rotation, 1998</td> </tr><tr> <td>[46]</td> <td class="space">Markley, FL; Cheng, Y.; Crassidis, JL; Oshman, Y., Averaging quaternions, J Guid Control Dyn, 30, 4, 1675-1682, 2007</td> </tr><tr> <td>[47]</td> <td class="space">Lerner, UN, Hybrid Bayesian Networks for Reasoning About Complex Systems, 2002, Stanford, CA: Stanford University</td> </tr><tr> <td>[48]</td> <td class="space">Van Der Merwe, R.; Wan, EA, The square‐root unscented Kalman filter for state and parameter‐estimation, 2001, Salt Lake City, UT</td> </tr><tr> <td>[49]</td> <td class="space">Björck, A., Numerical Methods for Least Squares Problems, 408, 1996, Philadelphia, PA: SIAM · <a href="/0847.65023" class="nowrap">Zbl 0847.65023</a></td> </tr><tr> <td>[50]</td> <td class="space">Arasaratnam, I.; Haykin, S., Cubature Kalman filters, IEEE Trans Autom Control, 54, 6, 1254-1269, 2009 · <a href="/1367.93637" class="nowrap">Zbl 1367.93637</a></td> </tr><tr> <td>[51]</td> <td class="space">Crassidis, JL; Junkins, JL, Optimal Estimation of Dynamic Systems. 2nd ed., 2012, Boca Raton, FL: CRC Press</td> </tr><tr> <td>[52]</td> <td class="space">Menegaz, HM; Ishihara, JY; Borges, GA, A new smallest sigma set for the unscented transform and its applications on SLAM, 2011, Orlando, FL</td> </tr><tr> <td>[53]</td> <td class="space">Grewal, MS; Andrews, AP, Kalman Filtering: Theory and Practice: Using Matlab, 2001, New York, NY: John Wiley & Sons</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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