Controllability of rolling without twisting or slipping in higher dimensions

TL;DR

This paper investigates the controllability of a higher-dimensional rolling system without slipping or twisting, by analyzing the lifted nonholonomic system on frame bundles and deriving conditions based on curvature properties.

Contribution

It introduces a framework for analyzing controllability in higher dimensions using principal Ehresmann connections and curvature tensors, extending previous lower-dimensional results.

Findings

Derived sufficient conditions for local controllability based on curvature and sectional curvatures.

Provided specific results for locally symmetric and complete manifolds.

Connected geometric properties to controllability criteria.

Abstract

We describe how the dynamical system of rolling two $n$-dimensional connected, oriented Riemannian manifolds $M$ and $\hat M$ without twisting or slipping, can be lifted to a nonholonomic system of elements in the product of the oriented orthonormal frame bundles belonging to the manifolds. By considering the lifted problem and using properties of the elements in the respective principal Ehresmann connections, we obtain sufficient conditions for the local controllability of the system in terms of the curvature tensors and the sectional curvatures of the manifolds involved. We also give some results for the particular cases when $M$ and $\hat M$ are locally symmetric or complete.