Iterative Robust Stabilization Algorithm for Periodic Orbits of Hybrid Dynamical Systems: Application to Bipedal Running ?

This paper presents a systematic numerical algorithm to design optimal H∞ continuous-time controllers to robustly stabilize periodic orbits for hybrid dynamical systems in the presence of discrete-time uncertainties. A parameterized set of closed-loop hybrid systems is assumed for which there exists a common periodic orbit. The algorithm is created based on an iterative sequence of optimization problems involving Bilinear and Linear Matrix Inequalities (BMIs and LMIs). At each iteration, the optimal H∞ problem is translated into a BMI optimization problem which can be easily solved using available software packages. Some sufficient conditions for the convergence of the iterative algorithm are presented. The power of the algorithm is then demonstrated in designing robust stabilizing virtual constraints for running of a highly underactuated bipedal robot with 7 degrees of underactuation in the presence of impact model uncertainties.