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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.

A method to improve the energy efficiency of a bipedal robot by coupling its thighs with compliant smart mechanisms is introduced. The walking gaits are driven by electric motors in its revolute joints, whose reference trajectories are generated via numerical optimization. The optimized nonlinear characteristic of the compliant mechanism modifies the free oscillation frequency of the system that matches the current double step frequency even under different conditions, which results in a very high energy efficiency.