Computational Geometric Optimal Control of Rigid Bodies

TL;DR

This paper develops a geometric framework for solving optimal control problems of rigid bodies using Lie group variational integrators, ensuring preservation of geometric properties during numerical computation.

Contribution

It introduces a geometric formulation of rigid body optimal control problems and computational methods based on Lie group variational integrators that preserve geometric features.

Findings

Preserves geometric structure in numerical solutions.

Applicable to single and multiple rigid body problems.

Demonstrates computational efficiency and accuracy.

Abstract

This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each rigid body is viewed as evolving on a configuration manifold that is a Lie group. Discrete-time dynamics of each rigid body are developed that evolve on the configuration manifold according to a discrete version of Hamilton's principle so that the computations preserve geometric features of the dynamics and guarantee evolution on the configuration manifold; these discrete-time dynamics are referred to as Lie group variational integrators. Rigid body optimal control problems are formulated as discrete-time optimization problems for discrete Lagrangian/Hamiltonian dynamics, to which standard numerical optimization algorithms can be applied. This general approach is illustrated by presenting results for several different optimal control problems for a single rigid body and for multiple interacting rigid bodies. The computational advantages of the approach, that arise from correctly modeling the geometry, are discussed.