Discrete Mechanics and Optimal Control: an Analysis

TL;DR

This paper introduces a discretization approach for optimal control of mechanical systems that preserves structural properties and demonstrates convergence, with applications in vehicle dynamics, space missions, and robotics.

Contribution

It proposes a variational discretization method (DMOC) that maintains system symmetries and integrals, and proves its equivalence to a symplectic Runge-Kutta scheme with convergence guarantees.

Findings

DMOC preserves continuous system properties in discrete solutions

The method is equivalent to a symplectic finite difference scheme

Numerical experiments show competitive performance

Abstract

The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.