Discrete Variational Optimal Control

TL;DR

This paper introduces a numerical framework for optimal control of mechanical systems using discrete variational principles, ensuring structure-preserving, robust, and versatile algorithms applicable to complex systems.

Contribution

It extends discrete mechanics to solve optimal control problems via variational integrators, applicable to a wide range of mechanical systems with symmetries and constraints.

Findings

Developed structure-preserving numerical methods for optimal control.

Applicable to systems on Lie groups, nonholonomic, and underactuated systems.

Validated through theoretical analysis and practical examples like underwater vehicle control.

Abstract

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical and a practical examples, e.g. the control of an underwater vehicle, will illustrate the application of the proposed approach.