Discrete second-order Euler-Poincar\'e equations. Applications to optimal control

TL;DR

This paper develops geometric numerical integrators for second-order Lagrangian systems on Lie groups, deriving discrete Euler-Poincaré equations and applying them to optimal control problems like rigid body dynamics and Cosserat rods.

Contribution

It introduces a discrete variational calculus approach for second-order systems on Lie groups, leading to new geometric integrators for optimal control.

Findings

Derived discrete second-order Euler-Lagrange equations.

Applied to optimal control of rigid bodies and Cosserat rods.

Demonstrated effectiveness of geometric integrators.

Abstract

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equations. Interesting applications as, for instance, a discrete derivation of the Euler-Poincar\'e equations for second-order Lagrangians and its application to optimal control of a rigid body, and of a Cosserat rod are shown at the end of the paper.