In this paper we propose the fastest converging control policy (also known as deadbeat control) for walking with the bipedal spring-mass model, which serves as an abstraction of a robot on compliant legs. To fully leverage the passive dynamics of the system, the touchdown angle of the swing-leg is assigned as the only control input of the system. We show that two steps (or one stride) are necessary and sufficient to converge to target walking gaits. We first analyze the dynamics of the system to identify the limit cycles as well as the limitations of the control authority within the definition of walking. Then, we present the two-step deadbeat control policy that guarantees stability with the fastest possible convergence rate for the system. For each equilibrium gait, the basin of attraction in which this two-step control exists is a measure of the robustness of the system. The simulation results show that human-like walking gaits (double hump ground reaction force profile) have relatively large basins of attraction. Finally, we extend the policy to various energy levels to accommodate walking on uneven ground that has height changes. We show in simulation that the system indeed rejects various disturbances and converges to the desired equilibrium gait in two steps.
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