Feedback design for multi-contact push recovery via LMI approximation of the Piecewise-Affine Quadratic Regulator

To recover from large perturbations, a legged robot must make and break contact with its environment at various locations. These contact switches make it natural to model the robot as a hybrid system. If we apply Model Predictive Control to the feedback design of this hybrid system, the on/off behavior of contacts can be directly encoded using binary variables in a Mixed Integer Programming problem, which scales badly with the number of time steps and is too slow for online computation. We propose novel techniques for the design of stabilizing controllers for such hybrid systems. We approximate the dynamics of the system as a discrete-time Piecewise Affine (PWA) system, and compute the state feedback controllers across the hybrid modes offline via Lyapunov theory. The Lyapunov stability conditions are translated into Linear Matrix Inequalities. A Piecewise Quadratic Lyapunov function together with a Piecewise Linear (PL) feedback controller can be obtained by Semidefinite Programming (SDP). We show that we can embed a quadratic objective in the SDP, designing a controller approximating the Piecewise-Affine Quadratic Regulator. Moreover, we observe that our formulation restricted to the linear system case appears to always produce exactly the unique stabilizing solution to the Discrete Algebraic Riccati Equation. In addition, we extend the search from the PL controller to the PWA controller via Bilinear Matrix Inequalities. Finally, we demonstrate and evaluate our methods on a few PWA systems, including a simplified humanoid robot model.