On the motion planning of rolling surfaces

In this paper, we address the issue of motion planning for the control system S R that results from the rolling without slipping nor spinning of a two dimensional Riemannian mani- fold M1 onto another one M2. We present two procedures to tackle the motion planning problem when M1 is a plane and M2 a convex surface. The first approach rests on the Liou- villian character of S R. More precisely, if just one of the manifolds has a symmetry of revolu- tion, then S 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 S R admits nontrivial abnormal extremals, we are still able to successfully apply the continuation method if M2 admits a stable periodic geodesic.

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