, Volume 18, Issue 2, pp 181-214
Date: 18 Apr 2012

An intrinsic formulation of the problem on rolling manifolds

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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.

The first three authors are supported by the grant of the Norwegian Research Council No. 204726/V30 and by the grant of the European Science Foundation Networking Programme HCAA. The fourth author is partially supported by FCT under project PTDC/EEA-ACR/67020/2006.