Constraint Control of Nonholonomic Mechanical Systems

TL;DR

This paper develops an optimal control approach for nonholonomic mechanical systems by using the constraints themselves as control inputs, specifically applying it to Suslov's problem involving rigid body rotation.

Contribution

It introduces a novel control formulation for nonholonomic systems leveraging constraints as controls, with detailed derivations for arbitrary groups and $SO(3)$, and demonstrates complex trajectory tracking.

Findings

Control of Suslov's problem via constraint manipulation is feasible.

Numerical examples show tracking of complex trajectories like spirals.

The approach extends to arbitrary nonholonomic systems.

Abstract

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov's problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction $\boldsymbol{\xi}$. We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov's problem for the rotation group $SO(3)$. We show that it is possible to control the system using the constraint $\boldsymbol{\xi}(t)$ and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral.