Analysis and Control of a Dissipative Spring-Mass Hopper with Torque Actuation

It has long been established that simple springmass models can accurately represent the dynamics of legged locomotion. Existing work in this domain, however, almost exclusively focuses on the idealized Spring-Loaded Inverted Pendulum (SLIP) model and neglects passive dissipative effects unavoidable in any physical robot or animal. In this paper, we extend on a recently proposed analytic approximation to the stance trajectories of a dissipative SLIP model to analyze stability properties of a planar hopper with a single rotary actuator at the hip. We first describe how a suitably chosen torque controller can compensate for damping losses, maintaining the same energy level across strides and hence reducing the return map to a single dimension. We then identify and characterize equilibrium points for this return map under a fixed leg placement policy and show that “uncontrolled” asymptotic stability is feasible for this energy-regulated system. Subsequent presentation of simulation evidence establishes that the predictions of this approximate model are consistent with the exact plant model. The paper concludes with the application of our energy-regulation scheme to the design of a task-level gait controller that uses explicit leg placement commands in conjunction with the hip torque.