Reduced-order framework for exponential stabilization of periodic orbits on parameterized hybrid zero dynamics manifolds: Application to bipedal locomotion

This paper shows how controlled-invariant manifolds in hybrid dynamical systems can be used to reduce the offline computational burden associated with locally exponentially stabilizing periodic orbits. We recently introduced a method to systematically select stabilizing feedback controllers for hybrid periodic orbits from a family of parameterized control laws by solving offline optimization problems. These problems search for controller parameters as well as a set of Lyapunov matrices for the full-order hybrid systems. When the method is applied to mechanical systems with high degrees of freedom (DOF), the number of entries in the Lyapunov matrices may render the numerical optimization problems prohibitively slow. To address this challenge, the paper considers a family of attractive and parameterized hybrid zero dynamics (HZD) manifolds in the state space. It then investigates the properties of the associated Poincaré map to translate the full-order optimization framework to a reduced-order one on the parameterized HZD manifolds with lower-dimensional Lyapunov matrices. In addition, the paper provides a systematic approach to numerically compute the Jacobian linearization of the parameterized Poincaré map on the HZD manifolds. The power of the proposed framework is demonstrated by designing a set of stabilizing input-output linearizing controllers for walking gaits of an underactuated 3 D bipedal robot with 13 DOFs and 6 actuators. It is shown that the number of decision variables in the reduced-order optimization problem can be reduced by 70% compared to the full-order one.