Nonholonomic mechanics and virtual constraints on Riemannian homogeneous spaces

TL;DR

This paper introduces a geometric framework for virtual nonholonomic constraints on Riemannian homogeneous spaces, establishing control laws that preserve invariance and analyzing the resulting dynamics with applications in robotics.

Contribution

It generalizes the concept of virtual constraints to Riemannian homogeneous spaces and proves the existence and uniqueness of control laws that maintain invariance.

Findings

Existence and uniqueness of control laws preserving invariant distributions.

Characterization of closed-loop dynamics via affine connections.

New robotics-inspired examples of nonholonomic control systems.

Abstract

Nonholonomic systems are, so to speak, mechanical systems with a prescribed restriction on the velocities. A virtual nonholonomic constraint is a controlled invariant distribution associated with an affine connection mechanical control system. A Riemannian homogeneous space is, a Riemannian manifold that looks the same everywhere, as you move through it by the action of a Lie group. These Riemannian manifolds are not necessarily Lie groups themselves, but nonetheless possess certain symmetries and invariances that allow for similar results to be obtained. In this work, we introduce the notion of virtual constraint on Riemannian homogeneous spaces in a geometric framework which is a generalization of the classical controlled invariant distribution setting and we show the existence and uniqueness of a control law preserving the invariant distribution. Moreover we characterize the closed-loop dynamics obtained using the unique control law in terms of an affine connection. We illustrate the theory with new examples of nonholonomic control systems inspired by robotics applications.