Polynomial Regression on Lie Groups and Application to SE(3)

被引:0
作者
Aubray, Johan [1 ]
Nicol, Florence [1 ]
机构
[1] Univ Toulouse, Ecole Natl Aviat Civile, 7, Ave Edouard Belin, F-31400 Toulouse, France
关键词
Lie groups; SE(n); regression; Riemannian manifolds; affine connection; linear connection; air traffic management; KALMAN FILTER;
D O I
10.3390/e26100825
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
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.
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页数:21
相关论文
共 38 条
[1]  
Amari Shun-ichi, 2000, Translations of Mathematical Monographs, V191
[2]  
Boisvert J, 2006, I S BIOMED IMAGING, P750
[3]   Geometric variability of the scoliotic spine using statistics on articulated shape models [J].
Boisvert, Jonathan ;
Cheriet, Farida ;
Pennec, Xavier ;
Labelle, Hubert ;
Ayache, Nicholas .
IEEE TRANSACTIONS ON MEDICAL IMAGING, 2008, 27 (04) :557-568
[4]  
Bonnabel S, 2007, IEEE DECIS CONTR P, P4007
[5]   Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem [J].
Bonnabel, Silvere ;
Martin, Philippe ;
Salauen, Erwan .
PROCEEDINGS OF THE 48TH IEEE CONFERENCE ON DECISION AND CONTROL, 2009 HELD JOINTLY WITH THE 2009 28TH CHINESE CONTROL CONFERENCE (CDC/CCC 2009), 2009, :1297-1304
[6]  
Boumal N., 2023, An Introduction to Optimization on Smooth Manifolds, DOI DOI 10.1017/9781009166164
[7]  
Bourmaud G, 2013, 2013 PROCEEDINGS OF THE 21ST EUROPEAN SIGNAL PROCESSING CONFERENCE (EUSIPCO)
[8]   On the geometry of Riemannian cubic polynomials [J].
Camarinha, M ;
Leite, FS ;
Crouch, P .
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 2001, 15 (02) :107-135
[9]   High-Order Splines on Riemannian Manifolds [J].
Camarinha, Margarida ;
Silva Leite, Fatima ;
Crouch, Peter E. .
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS, 2023, 321 (01) :158-178
[10]   A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion [J].
Duan, Xiaomin ;
Sun, Huafei ;
Zhao, Xinyu .
ENTROPY, 2019, 21 (05)