In this paper, stable hopping of a one-legged, articulated robot with a flat foot is investigated. The robot has a special feature that before taking off, it goes through an underactuated phase in which the foot rotates about the unactuated toe on the ground. By having the underactuated phase, the robot can perform stable human-like hops with longer hopping distances. To devise a systematic trajectory design methodology for the robot, its dynamics including the ground-foot impact and the hopping constraints are carefully studied. An optimization procedure is then proposed to plan the feasible actuated trajectories which not only meet specific performance requirements but also attain certain optimality with respect to actuation energy. The hopping strategy and the optimal trajectories are verified by simulations and hardware implementation. Experiments indicate that the robot not only can stably perform hops with different hopping distances on the level ground but also can successfully hop up/down staircases.
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