Rolling Manifolds of Different Dimensions

Acta Applicandae Mathematicae - Tập 139 - Trang 105-131 - 2014
Amina Mortada1,2, Petri Kokkonen3, Yacine Chitour4
1CNRS and STITS, Université Paris-Sud 11, Gif-sur-Yvette, France
2EDST, Université Libanaise, Hadath, Liban
3Helsinki, Finland
4CNRS and Supélec, Université Paris-Sud 11, Gif-sur-Yvette, France

Tóm tắt

If (M,g) and $(\hat{M},\hat{g})$ are two smooth connected complete oriented Riemannian manifolds of dimensions n and $\hat{n}$ respectively, we model the rolling of (M,g) onto $(\hat{M},\hat{g})$ as a driftless control affine systems describing two possible constraints of motion: the first rolling motion (Σ) NS captures the no-spinning condition only and the second rolling motion (Σ) R corresponds to rolling without spinning nor slipping. Two distributions of dimensions $(n + \hat{n})$ and n are then associated to the rolling motions (Σ) NS and (Σ) R respectively. This generalizes the rolling problems considered in Chitour and Kokkonen (Rolling manifolds and controllability: the 3D case, 2012) where both manifolds had the same dimension. The controllability issue is then addressed for both (Σ) NS and (Σ) R and completely solved for (Σ) NS . As regards to (Σ) R , basic properties for the reachable sets are provided as well as the complete study of the case $(n,\hat{n})=(3,2)$ and some sufficient conditions for non-controllability.

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