The hybrid zero dynamics control concept for bipedal walking is extended to include a non-instantaneous double support phase. A symmetric robot that consists of five rigid body segments which are connected by four actuated revolute joints is considered. Periodic walking gaits with a constant average walking speed consists of alternating single (SSP) and double support phases (DSP). Hybrid zero dynamics control designs usually assume an instantaneous DSP, which is a severe limitation. The proposed controllers use continuous SSPs and DSPs. Transitions between both phases are modeled as instantaneous events, when the rear leg lifts off at the end of the DSP and the swing leg touches down at the end of the SSP. Due to the fact that the model during the DSP has more actuators (4) than degrees of freedom (3), the system is overactuated. In order to combine it with the underactuated SSP model and then formulate a periodic walking gait, we suggest three controller designs for different applications. One with the underactuated DSP, one with the fully actuated DSP, and one with the overactuated DSP. A numerical optimization is used to generate energy efficient gaits in an offline process. According to the optimization results, artificially creating an underactuated controller for the DSP results in the most efficient gaits. Adding control tasks utilizing the full actuation or overactuation during the DSP significantly improves the gait stability.

翻译：双足行走的混合零动态控制概念被扩展，以包含非瞬时双支撑阶段。考虑一个由四个驱动旋转关节连接的五个刚性体段组成的对称机器人。以恒定平均步行速度行走的周期性步态由交替的单支撑阶段和双支撑阶段构成。混合零动态控制设计通常假设双支撑阶段为瞬时事件，这是一个严重的限制。所提出的控制器采用连续的单支撑阶段和双支撑阶段。两个阶段之间的过渡被建模为瞬时事件，即后腿在双支撑阶段结束时离地，以及摆动腿在单支撑阶段结束时触地。由于双支撑阶段模型拥有比自由度（3）更多的驱动器（4），系统处于过驱动状态。为了将其与欠驱动单支撑阶段模型结合并制定周期性步行步态，我们针对不同应用提出了三种控制器设计方案：一种采用欠驱动双支撑阶段，一种采用全驱动双支撑阶段，以及一种采用过驱动双支撑阶段。通过数值优化在离线过程中生成节能步态。根据优化结果，人为为双支撑阶段创建欠驱动控制器可获得最高效的步态。而在双支撑阶段利用全驱动或过驱动来添加控制任务，则能显著提升步态稳定性。