Continuous-time controllers for stabilizing periodic orbits of hybrid systems: Application to an underactuated 3D bipedal robot

This paper presents a systematic approach to exponentially stabilize periodic orbits in nonlinear systems with impulse effects, a special class of hybrid systems. Stabilization is achieved with a time invariant continuous-time controller. The presented method assumes a parametrized family of continuous-time controllers has been designed so that (1) a periodic orbit is induced, and (2) the orbit itself is invariant under the choice of parameters in the controllers. By investigating the properties of the Poincaré return map, a sensitivity analysis is presented that translates the stabilization problem into a set of Bilinear Matrix Inequalities (BMIs). A BMI optimization problem is set up to select the parameters of the continuous-time controller to achieve exponential stability. We illustrate the power of the approach by finding new stabilizing solutions for periodic orbits of an underactuated 3D bipedal robot.