On Optimal Control Laws for a Class of Constrained Dynamical Systems (With Application to Control of Bipedal Locomotion)

Abstract

The physical and dynamical constraints of a constrained dynamical system are related to system controllability and stability. Proper formulation of these inequality constraints and treatment of the active ones leads to stabilizing controls with relatively smooth control efforts—in all cases, control laws. These approaches were useful in a study of the biomechanics of climbing and descending gaits by mathematical synthesis techniques, necessitated by the increased importance of terrain and lower extremity kinematics and incomplete specification of the tasks. The general criteria entail no uniqueness requirements on system motions and controls, although for the most common (and probably most desirable) condition of the constraints (fewer active constraints than system degrees of freedom) an “optimal” control law can be derived. Two examples are presented, and some general discussion is given relating mainly to the control of biped locomotion.