Robust Push Recovery on Bipedal Robots: Leveraging Multi-Domain Hybrid Systems with Reduced-Order Model Predictive Control

TL;DR

This work addresses robust push recovery for bipedal robots by unifying hybrid locomotion dynamics with a reduced-order MPC grounded in an augmented LIP model with ZMP (the ZLIP model). By jointly optimizing foot placement, step timing, and ankle/ ZMP-related commands within a hybrid-domain framework, the approach expands the disturbance rejection envelope for both flat-footed and multi-domain gaits. The method achieves real-time feasibility (8 ms NLP solves) and demonstrates substantial stability improvements in high-fidelity Cassie simulations, including extreme sagittal and lateral pushes. The framework holds practical significance for robust, agile walking in unstructured environments and provides a path toward extending to full-body humanoids with centroidal momentum regulation.

Abstract

In this paper, we present a novel control framework to achieve robust push recovery on bipedal robots while locomoting. The key contribution is the unification of hybrid system models of locomotion with a reduced-order model predictive controller determining: foot placement, step timing, and ankle control. The proposed reduced-order model is an augmented Linear Inverted Pendulum model with zero moment point coordinates; this is integrated within a model predictive control framework for robust stabilization under external disturbances. By explicitly leveraging the hybrid dynamics of locomotion, our approach significantly improves stability and robustness across varying walking heights, speeds, step durations, and is effective for both flat-footed and more complex multi-domain heel-to-toe walking patterns. The framework is validated with high-fidelity simulation on Cassie, a 3D underactuated robot, showcasing real-time feasibility and substantially improved stability. The results demonstrate the robustness of the proposed method in dynamic environments.

Figures (10)

• Figure 1: Cassie modifying its step time, foot placement, and using ankle torque to recover from unknown disturbances of -100N for 0.5s.
• Figure 2: A directed graph showing the hybrid system model used to describe flat-footed and multi-domain bipedal walking.
• Figure 3: Visualization of the ZLIP model when applied for heel-to-toe walking, highlighting the key phases of the gait cycle. The swing foot location at foot touchdown is denoted as $u_\text{sw}$. During the OA phase, the ZMP (yellow dots) shifts from the back toe to the front heel, while during the FA phase, it moves from heel to toe. The traveled distance in FA phase, $l$, corresponds to the foot curve length $\rho$ for the depicted heel-to-toe walking.
• Figure 4: Top-down view of the ZMP constraints as shaded red areas for (a) flat-footed and (b) multi-domain walking for OA (blue), FA (yellow), and UA (green) phases, with foot placement vectors shown in grey.
• Figure 5: Schematic of Cassie's physical structure and the output definitions used in the control framework.