An intrinsic formulation of the problem on rolling manifolds

Abstract

We present an intrinsic formulation of the kinematic problem of two n-dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an n(n?+?3)/2-dimensional manifold. The conditions of no-twisting and no-slipping are encoded by means of a distribution of rank n. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the n-dimensional sphere over $ {\mathbb{R}^n} $ is controllable. In contrast with this, we show that in the case of SE(3) rolling over $ \mathfrak{s}\mathfrak{e}(3) $ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.