An intrinsic formulation of the problem on rolling manifolds

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 en...

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Bibliographic Details
Published inJournal of dynamical and control systems Vol. 18; no. 2; pp. 181 - 214
Main Authors Godoy Molina, M., Grong, E., Markina, I., Silva Leite, F.
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.04.2012
Subjects
Calculus of Variations and Optimal Control; Optimization
Control
Dynamical Systems
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
Systems Theory
Vibration
53A55
37J60
53A17
nonholonomic constraints
moving frames
Rolling maps
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Summary: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 is controllable. In contrast with this, we show that in the case of SE(3) rolling over the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-012-9139-2