Decentralized Feedback Controllers for Robust Stabilization of Periodic Orbits of Hybrid Systems: Application to Bipedal Walking

This paper presents a systematic algorithm to design time-invariant decentralized feedback controllers to exponentially and robustly stabilize periodic orbits for hybrid dynamical systems against possible uncertainties in discrete-time phases. The algorithm assumes a family of parameterized and decentralized nonlinear controllers to co-ordinate interconnected hybrid subsystems based on a common phasing variable. The exponential and $\mathcal {H}_{2}$ robust stabilization problems of periodic orbits are translated into an iterative sequence of optimization problems involving bilinear and linear matrix inequalities. By investigating the properties of the Poincaré map, some sufficient conditions for the convergence of the iterative algorithm are presented. The power of the algorithm is finally demonstrated through designing a set of robust stabilizing local nonlinear controllers for walking of an underactuated 3-D autonomous bipedal robot with nine degrees of freedom, impact model uncertainties, and a decentralization scheme motivated by amputee locomotion with a transpelvic prosthetic leg.