Rigid Body Attitude Control

Abstract

Geometric numerical integration schemes have been studied for rigid body attitudecontrol problems. We consider symplectic integrators and present a Lie group varia-tional integrator and a discrete gradient method. The two methods preserve some ofthe geometric properties of rigid body dynamics, such as symplecticity, conservationof first integrals and the Lie group structure of the rotation matrices. We presentsome numerical experiments for an uncontrolled 3D pendulum where we investigatethe geometric properties of the two methods.The main part of this thesis concerns rigid body attitude control. We discuss theconcept of passivity and present a feedback control that rotates the rigid body froman initial state to a desired state. Passivity is a property that can be related to en-ergy conservation, and the feedback system presented in this paper is input-outputpassive. This means that the feedback system does not generate energy. Throughseveral numerical experiments, we investigate to which extent the Lie group varia-tional integrator and the discrete gradient method preserve the passivity condition.Our experiments indicate that the discrete gradient method preserves a discrete ver-sion of the passivity condition and for the Lie group variational integrator, the errorin the preservation of passivity is small. We compare our results with a Lie groupRunge-Kutta method and we observe that the method fails to preserve the discretepassivity condition for large step sizes.Lastly, we look at optimal attitude control. We present a time optimal attitudecontrol problem for a rigid body where the objective is to minimize the maneuvertime. The optimal control problem is formulated as a discrete-time optimizationproblem using a Lie group variational integrator. We present some numerical exper-iments to visualize the time optimal maneuver for a rigid body, where the controlinput is bounded.