Capturability as Controlled-Invariant Sets: Recursive Feasibility for Variable-Stepping Time S2S NMPC

Capturability characterizes a safe region of states for humanoid walking and is most commonly constructed by analyzing the one-dimensional divergent component of motion (DCM) of the center of mass. In this work, by exploiting the mathematical structure of the step-to-step (S2S) dynamics, we characterize capturability directly at the S2S level using the notion of a controlled positive invariant set (CPIS), which is computed via backward reachable sets (BRS). Although the inclusion of step time as a control input makes the S2S dynamics bilinear and renders these BRS difficult to compute directly, we show that, thanks to the specific mathematical structure of the S2S model, the capturable CPIS of the original nonlinear system can be exactly represented by the controlled invariant set of a linear system obtained at the minimum step time Tmin. Building on this set-valued characterization, we first propose a variable-step-time ALIP-based NMPC that employs the CPIS as a terminal set to guarantee recursive feasibility. We validated our approach through numerical simulations of the ALIP model, showing that an NMPC formulation equipped with a CPIS terminal set increases the feasibility (success) rate for challenging initial conditions near the boundary of the capturable set. We further demonstrated the effectiveness of the proposed method via both simulation and real-world hardware experiments on the Bruce humanoid robot, as well as simulation experiments on the Unitree G1.