Finite-Step Capturability and Recursive Feasibility for Bipedal Walking in Constrained Regions

This paper presents a Model Predictive Control (MPC) formulation for bipedal footstep planning based on the Linear Inverted Pendulum (LIP) model, ensuring recursive feasibility when navigating restricted regions. The proposed approach incorporates capturability and introduces a new constraint that forces the Divergent Component of Motion (DCM) into a finite-step capture region, adjusted between consecutive MPC calls. This constraint enables the MPC to anticipate beyond its prediction horizon, preventing collisions with the walking surface boundaries. We validate the approach through high-fidelity simulations with the bipedal robot Digit, demonstrating recursively feasible MPC footstep planning in restricted regions. Future efforts will extend the approach to general polytopic constraints, thereby facilitating footstep planning in cluttered environments while preserving the MPC's recursive feasibility.