Reachability-based Push Recovery for Humanoid Robots with Variable-Height Inverted Pendulum

This paper studies push recovery for humanoid robots based on a variable-height inverted pendulum (VHIP) model. We first develop an approach for treating zero-step capturability of the VHIP with a novel methodology based on Hamilton-Jacobi (HJ) reachability analysis. Such an approach uses the sub-zero level set of a value function to encode capturability of the VHIP, where the value function is obtained by numerically solving a HJ variational inequality offline. Based on this analysis, a simple and effective method for adjusting foothold locations is then devised for cases where the VHIP state is not zero-step capturable. In addition, the HJ reachability analysis naturally induces an optimal control law that allows for rapid planning with the VHIP during push recovery online. To enable use of the strategy with a position-controlled humanoid robot, an associated differential inverse kinematics based tracking controller is employed. The effectiveness of the overall framework is demonstrated with the UBTECH Walker robot in the MuJoCo simulator. Simulation validations show a significant improvement in push robustness as compared to the methods based on the classical linear inverted pendulum model.