Phase-based Nonlinear Model Predictive Control for Humanoid Walking Stabilization with Single and Double Support Time Adjustments

TL;DR

The paper tackles robust humanoid walking balance under disturbances across SSP and DSP by proposing a phase-based nonlinear model predictive controller (NMPC) that jointly optimizes ZMP modulation, step location, SSP duration, and DSP duration.It builds on nonlinear DCM–ZMP dynamics and introduces a phase-consistent representation, enforcing ZMP continuity across phase transitions and disabling footstep updates during DSP, enabling coherent balancing actions in both SSP and DSP.Key contributions include a unified NMPC framework with phase-duration adjustments, a nonlinear DCM error dynamics formulation, and a SQP-based solution with a QP-based whole-body controller, validated through extensive simulations and real TOCABI experiments showing improved robustness to pushes and terrain variations.The results demonstrate that coordinated use of ZMP modulation, step adjustment, and phase timing yields superior disturbance rejection and terrain handling, advancing practical humanoid locomotion in real-world settings.

Abstract

The contact sequence of humanoid walking consists of single and double support phases (SSP and DSP), and their coordination through proper duration and dynamic transition based on the robot's state is crucial for maintaining walking stability. Numerous studies have investigated phase duration optimization as an effective means of improving walking stability. This paper presents a phase-based Nonlinear Model Predictive Control (NMPC) framework that jointly optimizes Zero Moment Point (ZMP) modulation, step location, SSP duration (step timing), and DSP duration within a single formulation. Specifically, the proposed framework reformulates the nonlinear DCM (Divergent Component of Motion) error dynamics into a phase-consistent representation and incorporates them as dynamic constraints within the NMPC. The proposed framework also guarantees ZMP input continuity during contact-phase transitions and disables footstep updates during the DSP, thereby enabling dynamically reliable balancing control regardless of whether the robot is in SSP or DSP. The effectiveness of the proposed method is validated through extensive simulation and hardware experiments, demonstrating improved balance performance under external disturbances.

Figures (9)

• Figure 1: Graphical illustration of the reference DCM trajectory across gait phases: (a) SSP, where the ZMP is fixed at the center of the support foot and the reference DCM diverges toward the upcoming landing foot; (b) DSP, where the ZMP trajectory transitions linearly between both feet, enabling a smooth and continuous evolution of the reference DCM.
• Figure 2: Overall control framework for humanoid balance control utilizing the proposed phase-based Nonlinear Model Predictive Control.
• Figure 3: Comparative ablation results of the proposed NMPC under different combinations of balance strategies.
• Figure 4: Comparison of maximum recoverable impulses from different push directions for the proposed method, the LMPC method egle2023step, the heuristic method kim2023foot, and NMPC-w/o-DSP method choe2023seamless.
• Figure 5: Comparative simulation of robustness to external impulse timing variations between the proposed method, the LMPC method egle2023step, the heuristic method kim2023foot, and the NMPC-w/o-DSP method choe2023seamless. The graph represents the maximum impulse magnitude resistance in the forward direction.