Nonlinear control of multibody systems with symmetries via shape change.

Abstract

This dissertation makes contributions in three areas: local equilibrium controllability analysis for multibody systems controlled via shape change, motion planning for multibody systems via shape change, and control of specific multibody systems using shape change. Equations of motion are formulated using the variational principle of Hamilton for two classes of multibody systems with symmetries: multibody systems with base body symmetry and multibody systems dependent on an advected parameter. Equilibrium manifolds and conserved quantities are investigated. Due to the presence of conserved quantities, the complete state of the multibody systems may not be controllable. Local configuration controllability and local equilibrium controllability are studied for these two classes of multibody systems. A general formula for symmetric products evolving on a configuration manifold suitable for multibody systems is derived. Important properties are obtained for symmetric products only involving shape control vector fields and for symmetric products involving a potential vector field dependent on an advected parameter. These properties reflect the multibody symmetries. The symmetric product properties are applied to controllability analysis. Simplified sufficient conditions for local configuration accessibility and local equilibrium controllability are obtained. Motion planning algorithms are developed for general mechanical systems based on a series expansion, using a regular perturbation technique. These algorithms are applied to rest-to-rest maneuvers for the two classes of multibody systems via shape change. Three specific multibody control systems are studied. First, the Air Spindle Testbed controlled via two proof mass actuators is studied. Experiments are described that implement the motion planning approach. Experimental results demonstrate the effectiveness of the algorithms. Second, a free floating multibody spacecraft controlled via prismatic actuators is studied. Local fiber controllability analysis is carried out and global controllability properties are investigated. A motion planning algorithm is designed to achieve desired base body translational and rotational maneuvers. Third, the Triaxial Attitude Control Testbed controlled using shape change actuators is studied. Controllability conditions are obtained for several experimental setups; motion planning algorithms are constructed for rest-to-rest maneuvers.