Kinetic energy shaping for gait regulation of underactuated bipeds

This paper presents an underactuated control law for regulating the speed of bipedal walking robots. This control achieves speed regulation while requiring 96% less actuator energy per step than previously reported, fully-actuated controls that achieve speed regulation by velocity scaling. Our control arises from shaping the kinetic energy of a bipedal robot with actuators at the ankles only. While any kinetic energy shape may be imposed on a fully actuated robot, we show that underactuation restricts the achievable forms of the kinetic energy to those that solve a particular nonlinear partial differential equation (PDE). The general method of Auckly et al. for solving this PDE is reviewed and specialized to the case of mechanical systems. We show that the original nonlinear PDE may be reduced to one algebraic equation and two linear PDEs that may be solved simultaneously to find the set of achievable closed-loop kinetic energy shapes. We apply this approach to the compass-gait bipedal robot in the case of actuation at the ankle alone and present a particular kinetic energy form that yields energy-efficient speed control.