Time Optimal Attitude Control for a Rigid Body

TL;DR

This paper develops a geometric approach to solve the time optimal attitude control problem for rigid bodies, avoiding singularities and ensuring computational efficiency through Lie group variational integrators.

Contribution

It introduces a novel geometric optimality condition directly on the special orthogonal group and a discrete control method using Lie group integrators for efficient computation.

Findings

Successfully applied to a large-angle maneuver of an elliptic cylinder

Demonstrates the method's geometric exactness and numerical efficiency

Provides a singularity-free solution on the rotation group

Abstract

A time optimal attitude control problem is studied for the dynamics of a rigid body. The objective is to minimize the time to rotate the rigid body to a desired attitude and angular velocity while subject to constraints on the control input. Necessary conditions for optimality are developed directly on the special orthogonal group using rotation matrices. They completely avoid singularities associated with local parameterizations such as Euler angles, and they are expressed as compact vector equations. In addition, a discrete control method based on a geometric numerical integrator, referred to as a Lie group variational integrator, is proposed to compute the optimal control input. The computational approach is geometrically exact and numerically efficient. The proposed method is demonstrated by a large-angle maneuver for an elliptic cylinder rigid body.