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.

翻译：人形机器人行走的接触序列由单支撑期（SSP）与双支撑期（DSP）构成，根据机器人状态通过恰当的持续时间和动态转换来协调这两个阶段，对于维持行走稳定性至关重要。大量研究已探讨了将相位持续时间优化作为提升行走稳定性的有效手段。本文提出一种基于相位的非线性模型预测控制（NMPC）框架，该框架在单一公式中联合优化零力矩点（ZMP）调节、落脚点位置、SSP持续时间（步态时序）以及DSP持续时间。具体而言，所提框架将非线性发散运动分量（DCM）误差动力学重构为相位一致表示，并将其作为动态约束纳入NMPC中。该框架还保证了接触相位转换期间ZMP输入的连续性，并在DSP期间禁用落脚点更新，从而使得无论机器人处于SSP还是DSP，都能实现动态可靠的平衡控制。通过大量仿真与硬件实验验证了所提方法的有效性，结果表明其在外部扰动下具有更优的平衡性能。