Geometric structure-preserving optimal control of the rigid body

TL;DR

This paper develops a geometric, structure-preserving optimal control method for rigid bodies using discrete variational principles, ensuring accurate motion on Lie groups and providing a numerical example.

Contribution

It introduces a discrete variational optimal control framework for rigid bodies that preserves geometric structures using Lie group methods, improving accuracy over traditional discretizations.

Findings

The method accurately computes optimal control trajectories on SO(3).

Discrete equations derived from variational principles better approximate true dynamics.

Numerical example demonstrates effectiveness in rigid body maneuvers.

Abstract

In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange--d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.