Journal of Dynamical and Control Systems

, Volume 18, Issue 2, pp 181–214

An intrinsic formulation of the problem on rolling manifolds


    • Department of MathematicsUniversity of Bergen
  • E. Grong
    • Department of MathematicsUniversity of Bergen
  • I. Markina
    • Department of MathematicsUniversity of Bergen
  • F. Silva Leite
    • Department of Mathematics and Institute of Systems and RoboticsUniversity of Coimbra

DOI: 10.1007/s10883-012-9139-2

Cite this article as:
Godoy Molina, M., Grong, E., Markina, I. et al. J Dyn Control Syst (2012) 18: 181. doi:10.1007/s10883-012-9139-2


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.

Key words and phrases

Rolling mapsmoving framesnonholonomic constraints

2000 Mathematics Subject Classification


Copyright information

© Springer Science+Business Media, LLC 2012