Rolling Manifolds: Intrinsic Formulation and Controllability

Yacine Chitour (1) , Petri Kokkonen (1)

1 L2S - Laboratoire des signaux et systèmes

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for the rolling problem $(R)$, there is no relative spin and also no relative slip. Since the manifolds are not assumed to be embedded into an Euclidean space, we provide an intrinsic description of the two constraints ''without spinning'' and ''without slipping'' in terms of the Levi-Civita connections $\nabla^{g}$ and $\nabla^{\hg}$. For that purpose, we recast the two rolling problems within the framework of geometric control and associate to each of them a distribution and a control system. We then investigate the relationships between the two control systems and we address for both of them the issue of complete controllability. For the rolling $(NS)$, the reachable set (from any point) can be described exactly in terms of the holonomy groups of $(M,g)$ and $(\hM,\hg)$ respectively, and thus we achieve a complete understanding of the controllability properties of the corresponding control system. As for the rolling $(R)$, the problem turns out to be more delicate. We first provide general properties for the reachable set and determine the associated Lie bracket structure. Regarding the controllability issue, we only have partial results, for instance dealing with the situation where one of the manifold is a space form. Finally, we extend the two types of rolling to the case where the manifolds have different dimensions.

Domaines: Géométrie différentielle [math.DG]; Optimisation et contrôle [math.OC]

Yacine Chitour, Petri Kokkonen. Rolling Manifolds: Intrinsic Formulation and Controllability. 2010. ⟨hal-00535711v1⟩