Aubray Johan, Nicol Florence
Ecole Nationale de l'Aviation Civile, Université de Toulouse, 7, Avenue Edouard Belin, 31400 Toulouse, France.
Entropy (Basel). 2024 Sep 27;26(10):825. doi: 10.3390/e26100825.
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.
在本文中，我们利用李群框架解决了从有噪声的位置测量中估计无人机等移动设备位置的问题。为了对刚体的运动进行建模，相关的李群恰好是特殊欧几里得群SE(n)，其中n = 2或3。我们的工作是在先前使用的参数框架下进行的，该框架推导了黎曼流形上测地线回归和多项式回归的方程。基于这种方法，我们的目标是在李群SE(3)的背景下实现该技术。给定一组在SE(3)中表示移动设备轨迹测量的噪声点，人们希望找到在黎曼最小二乘意义下最适合这些点的测地线。最后，提出了对模拟数据的应用以说明这项工作。讨论了这种方法的局限性和未来展望。
