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The hybrid zero dynamics control concept for bipedal walking is extended toinclude a non-instantaneous double support phase. A symmetric robot thatconsists of five rigid body segments which are connected by four actuatedrevolute joints is considered. Periodic walking gaits with a constant averagewalking speed consists of alternating single (SSP) and double support phases(DSP). Hybrid zero dynamics control designs usually assume an instantaneousDSP, which is a severe limitation. The proposed controllers use continuous SSPsand 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 touchesdown at the end of the SSP. Due to the fact that the model during the DSP hasmore actuators (4) than degrees of freedom (3), the system is overactuated. Inorder to combine it with the underactuated SSP model and then formulate aperiodic walking gait, we suggest three controller designs for differentapplications. One with the underactuated DSP, one with the fully actuated DSP,and one with the overactuated DSP. A numerical optimization is used to generateenergy efficient gaits in an offline process. According to the optimizationresults, artificially creating an underactuated controller for the DSP resultsin the most efficient gaits. Adding control tasks utilizing the full actuationor overactuation during the DSP significantly improves the gait stability.

Hybrid Zero Dynamics Control for Bipedal Walking with a Non-Instantaneous Double Support Phase