Contingency Model Predictive Control for Bipedal Locomotion on Moving Surfaces with a Linear Inverted Pendulum Model

TL;DR

The paper tackles bipedal locomotion on moving surfaces with uncertain motion by introducing Contingency Model Predictive Control (CMPC) built on a Linear Inverted Pendulum (LIP) model. CMPC anticipates worst-case surface trajectories by deriving bounded envelopes for the surface acceleration and optimizing two extreme ZMP sequences that share an initial control segment, enforcing stability through a derived constraint on the unstable component. The approach yields a robust control framework that maintains stability under both sinusoidal and random disturbances, outperforming a traditional, intrinsically stable MPC in simulations with a NAO-like model. The work advances safe locomotion on dynamically moving platforms and paves the way for integration with foot-placement strategies and whole-body control in future experimental validation.

Abstract

Gait control of legged robotic walkers on dynamically moving surfaces (e.g., ships and vehicles) is challenging due to the limited balance control actuation and unknown surface motion. We present a contingent model predictive control (CMPC) for bipedal walker locomotion on moving surfaces with a linear inverted pendulum (LIP) model. The CMPC is a robust design that is built on regular model predictive control (MPC) to incorporate the "worst case" predictive motion of the moving surface. Integrated with an LIP model and walking stability constraints, the CMPC framework generates a set of consistent control inputs considering to anticipated uncertainties of the surface motions. Simulation results and comparison with the regular MPC for bipedal walking are conducted and presented. The results confirm the feasibility and superior performance of the proposed CMPC design over the regular MPC under various motion profiles of moving surfaces.

Figures (6)

• Figure 1: Schematic of bipedal robotic walker on a moving surface frame $\mathcal{S}$ (left) and an LIP model (right).
• Figure 2: The schematic of acceleration $\ddot{x}_s$ calculation at $t_k$ within the MPC control horizon $T_c$.
• Figure 3: Schematic of step and ZMP configuration of bipedal walker.
• Figure 4: Two types of surface motion disturbances (red: $x$-direction and blue: in $y$-direction). (a) Sinusoidal disturbance. (b) Random disturbance.
• Figure 5: The $\mathcal{C}_1$ controls of two disturbance scenarios. ZMP trajectories w. r. t two extreme bounded disturbances, $\ddot{x}_{s}^{u}(\ddot{y}_{s}^{u})$ and $\ddot{x}_{s}^{l}(\ddot{y}_{s}^{l})$, respectively in one prediction horizon, $T_c=1$ s. Green boxes indicate ZMP geometry constraints. (a) x-t plot (b) y-t plot (c) x-y plot of ZMP under periodic sinusoidal disturbance and the prediction started at $t_k=1.02$ s. (d)-(f) Under random disturbance and the prediction started at $t_k=0.8$ s.

Theorems & Definitions (1)

• Remark 1