Rolling Manifolds of Different Dimensions

Amina Mortada Yacine Chitour 1 Petri Kokkonen
1 Division Systèmes - L2S
L2S - Laboratoire des signaux et systèmes : 1289
Abstract : 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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Submitted on : Tuesday, February 9, 2016 - 8:47:18 AM
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Amina Mortada, Yacine Chitour, Petri Kokkonen. Rolling Manifolds of Different Dimensions. Acta Applicandae Mathematicae, Springer Verlag, 2015, 139 (1), pp.105-131. ⟨10.1007/s10440-014-9972-2 ⟩. ⟨hal-01271291⟩



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