Optimal Control of Underactuated Mechanical Systems: A Geometric Approach

TL;DR

This paper develops a geometric framework for optimal control of underactuated mechanical systems, extending classical methods to higher-order constraints and establishing a symplectic structure for better numerical integration.

Contribution

It introduces a novel geometric formalism adapting Skinner and Rusk's approach to higher-order Lagrangian dynamics with constraints, enabling new integrators.

Findings

Established a symplectic framework for the problem

Proved uniqueness of the dynamics vector field

Proposed new geometric integrators based on discrete variational calculus

Abstract

In this paper, we consider a geometric formalism for optimal control of underactuated mechanical systems. Our techniques are an adaptation of the classical Skinner and Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics of the optimal control problem. These developments will allow us to develop a new class of geometric integrators based on discrete variational calculus.