Decentralized feedback controllers for exponential stabilization of hybrid periodic orbits: Application to robotic walking

This paper presents a systematic algorithm to design time-invariant decentralized feedback controllers to exponentially stabilize periodic orbits for a class of hybrid dynamical systems arising from bipedal walking. The algorithm assumes a class of parameterized and nonlinear decentralized feedback controllers which coordinate lower-dimensional hybrid subsystems based on a common phasing variable. The exponential stabilization problem is translated into an iterative sequence of optimization problems involving bilinear and linear matrix inequalities, which can be easily solved with available software packages. A set of sufficient conditions for the convergence of the iterative algorithm to a stabilizing decentralized feedback control solution is presented. The power of the algorithm is demonstrated by designing a set of local nonlinear controllers that cooperatively produce stable walking for a 3D autonomous biped with 9 degrees of freedom, 3 degrees of underactuation, and a decentralization scheme motivated by amputee locomotion with a transpelvic prosthetic leg.