Optimal Robust Control for Constrained Nonlinear Hybrid Systems with Application to Bipedal Locomotion

Recent work on control Lyapunov functions and control Barrier functions has enabled addressing stability of nonlinear and underactuated hybrid systems while simultaneously enforcing input /state constraints and safety-critical constraints. However, under model uncertainty, these controllers break down and violate constraints. This paper presents a novel method of optimal robust control through quadratic programs that can handle stability, input /state dependent constraints, as well as safety-critical constraints, in the presence of high level of model uncertainty. Under the assumption of bounded uncertainty, the proposed controller strictly guarantees constraints without violating them. We evaluate our proposed control design for achieving dynamic bipedal locomotion that involves orbital stability of an underactuated nonlinear hybrid system subject to (a) torque saturation constraints (input constraints), (b) contact force constraints (state constraints), and (c) precise footstep placements (safety-critical constraints). We present numerical results on RABBIT, a five-link planar bipedal robot, subject to a large unknown load on its torso. Our proposed controller is able to demonstrate walking while strictly enforcing the above constraints with an unknown load of up to 15 Kg (47% of the robot mass.)