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arXiv · 2002.09744

Controllability results for the rolling of $2$-dim. against $3$-dim. Riemannian Manifolds

Amina Mortada·Yacine Chitour·Petri Kokkonen·Ali Wehbe
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Abstract

In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of \cite{MortadaKokkonenChitour}, which modelled the general setting of the rolling of two Riemannian connected manifolds with different dimensions as a driftless control affine system on a fibered space $Q$, with an emphasis on understanding the local structure of the rolling orbits, i.e., the reachable sets in $Q$. In this paper, the state space $Q$ has dimension eight and we show that the possible dimensions of non open rolling orbits belong to the set $\{2,5,6,7\}$. We describe the structures of orbits of dimension $2$, the possible local structures of rolling orbits of dimension $5$ and some of dimension $7$.

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Amina Mortada, Yacine Chitour, Petri Kokkonen, Ali Wehbe. 2020-02-22. Controllability results for the rolling of $2$-dim. against $3$-dim. Riemannian Manifolds. https://arxiv.org/abs/2002.09744

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