Geometry and Inverse Optimality in Global Attitude Stabilization

Abstract

Theproblemofgloballystabilizingtheattitudeofarigidbodyisconsidered.Topologicalandgeometricproperties of the space of rotations relevant to the stabilization problem are discussed. Chevalley' s exponential coordinates for a Lie group are used to represent points in this space. An appropriate attitude error is formulated and used for control design. A control Lyapunov function approach is used to design globally stabilizing feedback laws that have desirable optimality properties. Their performance is compared to the performance of previously developed proportional-derivative-typecontrol laws. Thenew control laws achievethe sameorgreaterstabilization ratewith lesscontroleffort. SpecialissuesintheLyapunov stabilityproofsrelated to thetopology ofthespaceof rotationsare identie ed and resolved. The simplerproblem of stabilization on thespace of planarrotationsis treated throughout the paper to provide insight. HE problem of controlling the orientation of a rigid body is central to the control of aircraft and spacecraft. Large-angle maneuvers long have been of interest for satellite control and more recently have become relevant for agile aircraft and missiles. We address the attitude control of a rigid body subject only to torques due to its own inertia and to control effectors, either thrusters or momentum wheels, but much of the discussion and results also are extendable to a rigid body subject to gravitational and aerodynamic torques. We assume that the controls can be continuously varied; the results are thus applicable to control by proportional thrusters, momentum wheels, or on off thrusters operated in a pulse-width, pulse-frequency modulation mode. The particular control problem that we consider is the global feedback stabilization of a specie ed inertialpointing direction. The stabilizationproblemprovides asuf- e ciently rich contextforthepointsthatwe wishtomake,though our discussion and results could be extended to the global asymptotic tracking problem. There aretwo featuresofthe globalattitudestabilization problem that motivate our study. First, the statespaceis amanifold that is not equivalent to a linear vector space. The dynamics of the orientation of a rigid body evolve on the tangent bundle to the rotation group SO.3/. SO.3/is a compactmanifold withoutboundary. This topol- ogy has profound implications for control design. We approach the formulation and solution of the global stabilization problem from a geometric perspectiveof SO.3/asboth aRiemannian manifold and a Lie group. Chevalley' s canonical coordinates for a Lie group are used to represent points in SO.3/. The coordinates themselves are notnew; they arethe elements ofwhat usually is called the principal rotation vector. 1 The geometric perspective is prominent in the ap- proaches to attitude control by Meyer, 2 Crouch, 3 Koditschek, 4 Wen and Kreutz-Delgado, 5 Paielli and Bach, 6 and Bullo and Murray. 7