We propose a new family of controllers for multi-jointed planar monoped runners, based on approximate but accurate models of the stance phase dynamics of a two degree of freedom "SLIP" leg. Unlike previous approaches, the new scheme gives control over all parameters of the system including the hopping height, forward speed and duty cycle. The control laws are "deadbeat" in nature, derived by computing the inverse of an approximate return map and corrected by integral compensation. We use the expressions obtained in this way to control the original SLIP leg as well as radically different, more realistic four degree of freedom legs. In each case, the performance of the deadbeat scheme in controlling forward running velocity is compared to a modified Raibert control strategy, whose experimental stability properties have been analyzed carefully in the low degree of freedom setting.

We characterize equilibrium gaits of a small knee monoped in terms of manifest parameters by recourse to approximate closed form expressions. We first eliminate gravity during stance and choose a very special model of potential energy storage in the knee. Next, we introduce simple closed form approximations, motivated by the mean value theorem, to the elliptic integrals arising in the more general case. In so doing, we derive a conjectured generalization applicable to small knee monopeds with an arbitrary knee potential. Finally, we introduce a new closed form perturbation intended to adjust the approximate coordinate transformations to the presence of gravity. Simulation data is offered as evidence for the efficacy (to within roughly 5-10% accuracy) of both the proposed generalization across knee potentials and the proposed perturbation for the presence of gravity during stance.

A simplified lossless model of the Raibert planar hopper is introduced for the purpose of analytically studying the control of forward velocity. A closed-form return map describing the robot's state at the next hop as a function of that at the current hop is derived. The Raibert forward velocity controller is introduced and the fixed points of the closed loop system are characterized as well as the stability of these points. A new control law inspired by this analysis is introduced and compared with the Raibert control law.