Motion Control along Relative Equilibria

The subject of this thesis is control of mechanical systems as they evolve along the steady motions called relative equilibria. These trajectories are of interest in theory and applications and have the characterizing property that the system's body-fixed velocity is constant. For example, constant-speed rotation about a principal axis is a relative equilibrium of a rigid body in three dimensions. We focus our study on simple mechanical control systems on Lie groups, i.e., mechanical systems with the following properties: the configuration manifold is a matrix Lie group, the total energy is equal to the kinetic energy (i.e., no potential energy is present), and the kinetic energy and control forces both satisfy an invariance condition. The novel contributions of this thesis are twofold. First, we develop sufficient conditions, algebraic in nature, that ensure that a simple mechanical control system on a Lie group is locally controllable along a relative equilibrium. These conditions subsume the well-known local controllability conditions for equilibrium points. Second, for systems that have fewer controls than degrees of freedom, we present a novel algorithm to control simple mechanical control systems on Lie groups along relative equilibria. Under some assumptions, we design iterative small-amplitude control forces to accelerate along, decelerate along, and stabilize relative equilibria. The technical approach is based upon perturbation analysis and the design of inversion primitives and composition methods. We finally apply the algorithms to a planar rigid body and a satellite with two thrusters.