Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forces

TL;DR

This paper introduces an affine connection framework to realize nonholonomic constraints via viscous friction, providing a coordinate-free description and recursive approximation methods for the slow manifold and slip velocities.

Contribution

It presents a novel affine connection approach with invariance conditions and recursive procedures for approximating slow manifolds in nonholonomic systems with friction.

Findings

Coordinate-free representation of the slow manifold.

Recursive method for approximating slip velocities.

First and second order approximations of the dynamics.

Abstract

In this paper, we study an affine connection approach to realizing nonholonomic mechanical systems mediated by viscous friction forces with large coefficients, viewed as a singular perturbation of the nonholonomic system. We show that the associated slow manifold is represented coordinate-free as the image of a section over the nonholonomic distribution. We propose a novel invariance condition based on covariant derivatives and prove that this condition is equivalent to the classical invariance condition based on time derivatives. Accordingly, we propose a novel recursive procedure to approximate the slow manifold based on the covariant derivatives of a formal power series expansion of the section. Using this recurrence relation, we derive, up to second order, approximations of the slip velocities residing in the slow manifold, as well as the associated approximated dynamics up to first order. Lastly, we illustrate our approach with a case study of a vertical rolling disk.