{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,20]],"date-time":"2026-07-20T23:59:18Z","timestamp":1784591958406,"version":"3.55.0"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T00:00:00Z","timestamp":1727395200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ecole Nationale de l\u2019Aviation Civile"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we address the problem of estimating the position of a mobile such as a drone from noisy position measurements using the framework of Lie groups. To model the motion of a rigid body, the relevant Lie group happens to be the Special Euclidean group SE(n), with n=2 or 3. Our work was carried out using a previously used parametric framework which derived equations for geodesic regression and polynomial regression on Riemannian manifolds. Based on this approach, our goal was to implement this technique in the Lie group SE(3) context. Given a set of noisy points in SE(3) representing measurements on the trajectory of a mobile, one wants to find the geodesic that best fits those points in a Riemannian least squares sense. Finally, applications to simulated data are proposed to illustrate this work. The limitations of such a method and future perspectives are discussed.<\/jats:p>","DOI":"10.3390\/e26100825","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T11:19:33Z","timestamp":1727435973000},"page":"825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Polynomial Regression on Lie Groups and Application to SE(3)"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-0887-9386","authenticated-orcid":false,"given":"Johan","family":"Aubray","sequence":"first","affiliation":[{"name":"Ecole Nationale de l\u2019Aviation Civile, Universit\u00e9 de Toulouse, 7, Avenue Edouard Belin, 31400 Toulouse, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2746-3828","authenticated-orcid":false,"given":"Florence","family":"Nicol","sequence":"additional","affiliation":[{"name":"Ecole Nationale de l\u2019Aviation Civile, Universit\u00e9 de Toulouse, 7, Avenue Edouard Belin, 31400 Toulouse, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1117\/12.280797","article-title":"New extension of the Kalman filter to nonlinear systems","volume":"Volume 3068","author":"Kadar","year":"1997","journal-title":"Proceedings of the Signal Processing, Sensor Fusion, and Target Recognition VI"},{"key":"ref_2","unstructured":"Bourmaud, G., M\u00e9gret, R., Giremus, A., and Berthoumieu, Y. (2013, January 9\u201313). Discrete Extended Kalman Filter on Lie groups. Proceedings of the 21st European Signal Processing Conference (EUSIPCO 2013), Marrakech, Morocco."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108812","DOI":"10.1016\/j.automatica.2020.108812","article-title":"Invariant extended Kalman filter on matrix Lie groups","volume":"114","author":"Phogat","year":"2020","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bonnabel, S. (2007, January 12\u201314). Left-invariant Extended Kalman Filter and attitude estimation. Proceedings of the 2007 46th IEEE Conference on Decision and Control, New Orleans, LA, USA.","DOI":"10.1109\/CDC.2007.4434662"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bonnabel, S., Martin, P., and Sala\u00fcn, E. (2009, January 15\u201318). Invariant Extended Kalman Filter: Theory and application to a velocity-aided attitude estimation problem. Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly with 2009 28th Chinese Control Conference, Shanghai, China.","DOI":"10.1109\/CDC.2009.5400372"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Fang, K., Cai, T., and Wang, B. (2024). The Kinematic Models of the SINS and Its Errors on the SE(3) Group in the Earth-Centered Inertial Coordinate System. Sensors, 24.","DOI":"10.3390\/s24123864"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jeong, D.B., Lee, B., and Ko, N.Y. (2024). Three-Dimensional Dead-Reckoning Based on Lie Theory for Overcoming Approximation Errors. Appl. Sci., 14.","DOI":"10.3390\/app14125343"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sun, J., Chen, Y., and Cui, B. (2024). An Improved Initial Alignment Method Based on SE2(3)\/EKF for SINS\/GNSS Integrated Navigation System with Large Misalignment Angles. Sensors, 24.","DOI":"10.3390\/s24092945"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1177\/027836499501400606","article-title":"A Lie Group Formulation of Robot Dynamics","volume":"14","author":"Park","year":"1995","journal-title":"Int. J. Robot. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"37850","DOI":"10.1109\/ACCESS.2024.3371159","article-title":"SE(3) Based LTV-MPC Algorithm for Multi-Obstacle Trajectory Tracking of Fully Driven Spacecraft","volume":"12","author":"Wang","year":"2024","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.3390\/e17041850","article-title":"Computing Bi-Invariant Pseudo-Metrics on Lie Groups for Consistent Statistics","volume":"17","author":"Miolane","year":"2015","journal-title":"Entropy"},{"key":"ref_12","unstructured":"Boisvert, J., Pennec, X., Ayache, N., Labelle, H., and Cheriet, K. (2006, January 6\u20139). 3D anatomical variability assessment of the scoliotic spine using statistics on Lie groups. Proceedings of the 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro, Arlington, VA, USA."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1109\/TMI.2007.911474","article-title":"Geometric Variability of the Scoliotic Spine Using Statistics on Articulated Shape Models","volume":"27","author":"Boisvert","year":"2008","journal-title":"IEEE Trans. Med Imaging"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1137\/21M1410373","article-title":"Bi-Invariant Dissimilarity Measures for Sample Distributions in Lie Groups","volume":"4","author":"Hanik","year":"2022","journal-title":"Siam J. Math. Data Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1080\/00207179.2020.1780474","article-title":"Minimal control effort and time Lie-group synchronisation design based on proportional-derivative control","volume":"95","author":"Fiori","year":"2022","journal-title":"Int. J. Control"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Duan, X., Sun, H., and Zhao, X. (2019). A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion. Entropy, 21.","DOI":"10.3390\/e21050531"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Fiori, S. (2022). Manifold Calculus in System Theory and Control\u2014Second Order Structures and Systems. Symmetry, 14.","DOI":"10.3390\/sym14061144"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1610","DOI":"10.1109\/TSP.2005.845428","article-title":"Covariance, subspace, and intrinsic Cramer-Rao bounds","volume":"53","author":"Smith","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Labsir, S., Renaux, A., Vil\u00e0-Valls, J., and Chaumette, \u00c9. (2023, January 4\u201310). Cram\u00e9r-Rao Bound on Lie Groups with Observations on Lie Groups: Application to SE(2). Proceedings of the ICASSP 2023\u20142023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Rhodes Island, Greece.","DOI":"10.1109\/ICASSP49357.2023.10096257"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109232","DOI":"10.1016\/j.sigpro.2023.109232","article-title":"An intrinsic Bayesian bound for estimators on the Lie groups SO(3) and SE(3)","volume":"214","author":"Labsir","year":"2024","journal-title":"Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2973","DOI":"10.1214\/22-AOS2218","article-title":"Nonparametric regression on Lie groups with measurement errors","volume":"50","author":"Jeon","year":"2022","journal-title":"Ann. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0926-2245(01)00054-7","article-title":"On the geometry of Riemannian cubic polynomials","volume":"15","author":"Camarinha","year":"2001","journal-title":"Differ. Geom. Its Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1134\/S0081543823020128","article-title":"High-Order Splines on Riemannian Manifolds","volume":"321","author":"Camarinha","year":"2023","journal-title":"Proc. Steklov Inst. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., and Schmid, C. (2012, January 7\u201313). Polynomial Regression on Riemannian Manifolds. Proceedings of the Computer Vision\u2014ECCV 2012, Florence, Italy.","DOI":"10.1007\/978-3-642-33709-3"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.jat.2007.03.002","article-title":"B\u00e9zier curves and C2 interpolation in Riemannian manifolds","volume":"148","author":"Popiel","year":"2007","journal-title":"J. Approx. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Martel, A.L., Abolmaesumi, P., Stoyanov, D., Mateus, D., Zuluaga, M.A., Zhou, S.K., Racoceanu, D., and Joskowicz, L. (2020, January 4\u20138). Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic B\u00e9zier Splines. Proceedings of the Medical Image Computing and Computer Assisted Intervention\u2014MICCAI 2020, Lima, Peru.","DOI":"10.1007\/978-3-030-59719-1"},{"key":"ref_27","unstructured":"Pennec, X., Joshi, S., and Nielsen, M. (2011, January 22). Geodesic Regression on Riemannian Manifolds. Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy\u2014Geometrical and Statistical Methods for Modelling Biological Shape Variability, Westin Harbour Castle, TO, Canada."},{"key":"ref_28","unstructured":"Nielsen, F., and Barbaresco, F. (2013, January 28\u201330). Bi-invariant Means on Lie Groups with Cartan-Schouten Connections. Proceedings of the Geometric Science of Information, Paris, France."},{"key":"ref_29","unstructured":"Amari, S.I., and Nagaoka, H. (2000). Methods of Information Geometry, American Mathematical Society and Oxford University Press. Translations of Mathematical Monographs 191."},{"key":"ref_30","unstructured":"Husem\u00f6ller, D. (2013). Fibre Bundles, Springer. Graduate Texts in Mathematics."},{"key":"ref_31","unstructured":"Willmore, T. (1996). Riemannian Geometry, Clarendon Press. Oxford Science Publications."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Agrachev, A.A., and Sachkov, Y.L. (2004). Control Theory from the Geometric Viewpoint, Springer. Encyclopaedia of Mathematical Sciences.","DOI":"10.1007\/978-3-662-06404-7"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Saunders, D.J. (1989). The Geometry of Jet Bundles, Cambridge University Press.","DOI":"10.1017\/CBO9780511526411"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Boumal, N. (2023). An Introduction to Optimization on Smooth Manifolds, Cambridge University Press.","DOI":"10.1017\/9781009166164"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Marsden, J., and Ratiu, T. (1999). Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Springer. Texts in Applied Mathematics.","DOI":"10.1007\/978-0-387-21792-5"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Niethammer, M., Huang, Y., and Vialard, F.X. (2011, January 18\u201322). Geodesic Regression on Image Time Series. Proceedings of the Medical Image Computing and Computer-Assisted Intervention: MICCAI\u2014International Conference on Medical Image Computing and Computer-Assisted Intervention, Toronto, ON, Canada.","DOI":"10.1007\/978-3-642-23629-7_80"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Duistermaat, J., and Kolk, J. (1999). Lie Groups, Springer. Universitext.","DOI":"10.1007\/978-3-642-56936-4"},{"key":"ref_38","first-page":"215","article-title":"Les \u00e9l\u00e9ments al\u00e9atoires de nature quelconque dans un espace distanci\u00e9","volume":"10","year":"1948","journal-title":"Ann. 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