Robust Event-based Stabilization of Periodic Orbits for Hybrid Systems: Application to an Underactuated 3D Bipedal Robot

The first return map or Poincare map can be viewed as a discrete-time dynamical system evolving on a hyper surface that is transversal to a periodic orbit; the hyper surface is called a Poincare section. The Poincare map is a standard tool for assessing the stability of periodic orbits in non-hybrid as well as hybrid systems. In addition, it can be used for stabilization of periodic orbits if the underlying dynamics of the system depends on a set of parameters that can be updated by a feedback law when trajectories cross the Poincare section. This paper addresses an important practical obstacle that arises when designing feedback laws on the basis of the Jacobian linearization of the Poincare map. In almost all practical cases, the Jacobians must be estimated numerically, and when the underlying dynamics presents a wide range of time scales, the numerical approximations of the first partial derivatives are sufficiently inaccurate that controller tuning is very difficult. Here, a robust control formalism is proposed whereby a convex set of approximations to the Jacobian linearization is systematically generated and a stabilizing controller is designed through two appropriate sets of linear matrix inequalities (LMIs). The result is illustrated on a walking gait of a 3D underactuated bipedal robot.