Dynamic Output Controllers for Exponential Stabilization of Periodic Orbits for Multidomain Hybrid Models of Robotic Locomotion

The primary goal of this paper is to develop an analytical framework to systematically design dynamic output feedback controllers that exponentially stabilize multidomain periodic orbits for hybrid dynamical models of robotic locomotion. We present a class of parameterized dynamic output feedback controllers such that (1) a multidomain periodic orbit is induced for the closed-loop system and (2) the orbit is invariant under the change of the controller parameters. The properties of the Poincaré map are investigated to show that the Jacobian linearization of the Poincaré map around the fixed point takes a triangular form. This demonstrates the nonlinear separation principle for hybrid periodic orbits. We then employ an iterative algorithm based on a sequence of optimization problems involving bilinear matrix inequalities to tune the controller parameters. A set of sufficient conditions for the convergence of the algorithm to stabilizing parameters is presented. Full-state stability and stability modulo yaw under dynamic output feedback control are addressed. The power of the analytical approach is ultimately demonstrated through designing a nonlinear dynamic output feedback controller for walking of a three-dimensional (3D) humanoid robot with 18 state variables and 325 controller parameters.