B1394
Title: Regression on lie groups: Application to estimation of positions of a mobile
Authors: Johan Aubray - ENAC (France) [presenting]
Florence Nicol - ENAC (France)
Stephane Puechmorel - Ecole nationale de l'aviation civile (France)
Abstract: The problem of estimating the position of a mobile, such as a drone, from noisy position measurements is addressed. To model the motion of a rigid body, rather than considering trajectories in the state space as is usually done in functional data analysis, the framework of differential geometry is used. More precisely, the trajectory of the mobile is modelled as a Lie group-valued curve. The relevant Lie group for poses of a rigid object happens to be the special Euclidean group SE(n), with $n = 2$ or 3. A parametric framework is placed which extends linear regression in an Euclidean space to geodesic regression in a Riemannian manifold. This method was later extended to higher-order polynomials on Riemannian manifolds and explicitly written in SO(3). Based on this approach, the goal is to implement this technique in the Liegroup 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. A more general mathematical formulation is established by using differential forms. Finally, applications to simulated data are shown. The limitations of such a method and future perspectives are discussed.
