Virtual Nonholonomic Constraints: A Geometric Approach

TL;DR

This paper introduces a geometric framework for virtual nonholonomic constraints in control systems, establishing their existence, uniqueness, and how they influence system trajectories and dynamics.

Contribution

It presents a novel geometric approach to virtual nonholonomic constraints, including existence, uniqueness, and their role in shaping system behavior.

Findings

Existence and uniqueness of control laws for virtual nonholonomic constraints

Characterization of trajectories as solutions of constrained mechanical systems

Conditions under which virtual constraints replicate nonholonomic dynamics

Abstract

Virtual constraints are invariant relations imposed on a control system via feedback as opposed to real physical constraints acting on the system. Nonholonomic systems are mechanical systems with non-integrable constraints on the velocities. In this work, we introduce the notion of virtual nonholonomic constraints in a geometric framework. More precisely, it is a controlled invariant distribution associated with an affine connection mechanical control system. We demonstrate the existence and uniqueness of a control law defining a virtual nonholonomic constraint and we characterize the trajectories of the closed-loop system as solutions of a mechanical system associated with an induced constrained connection. Moreover, we characterize the dynamics for nonholonomic systems in terms of virtual nonholonomic constraints, i.e., we characterize when can we obtain nonholonomic dynamics from virtual nonholonomic constraints.