Motion planning of rolling surfaces

In this paper, we address the issues of controllability and motion planning for the control system S_R that results from the rolling without slipping nor spinning of a two dimensional Riemmanian manifold M₁ onto another one M₂. In the first part of the paper, we describe precisely the control system under consideration together with its Lie algebraic structure. This leads to a recovery of the result of Agrachev and Sachkov (1999) who provided a necessary and sufficient condition on the manifolds for complete controllability of S_R. In the main part of the paper, we present two procedures to tackle the motion planning problem when M₁ is a plane and M₂ a convex surface. The first approach is based on differential algebra. We show that S_R is a Liouvillian system and if M₂ has a symmetry of revolution, we compute a maximal linearizing output. The second technic consists of the use of a continuation method to attack the motion planning problem. 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