A geometric approach to the optimal control of nonholnomic mechanical systems

TL;DR

This paper develops a geometric framework using Lagrangian and Hamiltonian formalisms for optimal control of nonholonomic mechanical systems, enabling efficient motion planning and obstacle avoidance.

Contribution

It introduces a novel geometric approach to formulate and solve optimal control problems for nonholonomic systems using adapted vector fields and Riemannian metrics.

Findings

Effective control strategies for nonholonomic systems demonstrated

Application to obstacle avoidance and transmission problems shown

Framework unifies Lagrangian and Hamiltonian methods for constrained systems

Abstract

In this paper, we describe a constrained Lagrangian and Hamiltonian formalism for the optimal control of nonholonomic mechanical systems. In particular, we aim to minimize a cost functional, given initial and final conditions where the controlled dynamics is given by nonholonomic mechanical system. In our paper, the controlled equations are derived using a basis of vector fields adapted to the nonholonomic distribution and the Riemannian metric determined by the kinetic energy. Given a cost function, the optimal control problem is understood as a constrained problem or equivalently, under some mild regularity conditions, as a Hamiltonian problem on the cotangent bundle of the nonholonomic distribution. A suitable Lagrangian submanifold is also shown to lead to the correct dynamics. We demonstrate our techniques in several examples including a continuously variable transmission problem and motion planning for obstacle avoidance problems.