On the motion planning of rolling surfaces

In this paper, we address the issue of motion planning for the control system L R that results from the rolling without slipping nor spinning of a two dimensional Riemannian manifold M 1 onto another one M 2 . We present two procedures to tackle the motion planning problem when M 1 is a plane and M 2 a convex surface. The first approach rests on the Liouvillian character of L R . More precisely, if just one of the manifolds has a symmetry of revolution, then L R is shown to be a Liouvillian system. If, in addition, that manifold is convex and the other one is a plane, then a maximal linearizing output is explicitely computed. The second approach consists of the use of a continuation method. Even though L R admits nontrivial abnormal extremals, we are still able to successfully apply the continuation method if M 2 admits a stable periodic geodesic