Modeling the Dynamics of Quadrupedal Running

Abstract

In this paper, a 2-DOF model for quadrupedal running-in-place is presented, consisting of a rigid body on springy legs. "Energy- pumping " feedback is included to excite the natural dynamics of the system. The model exhibits at least two periodic solutions, which correspond to the bound and pronk gaits of four-legged animals. Approximate return maps are constructed around both trajectories, and these are used to derive explicit expressions for the amplitude and stability of the gaits. The pronk is shown to produce significantly more height than the bound for the same amount of effort. However, the bound has more desirable stability characteristics: simulations and analysis demonstrate that the bound is unstable if a dimension less body inertia is greater than 1, which is an unlikely situation for either a robot or an animal. The pronk's stability, however, exhibits coupling between height and inertia: a given inertia is more likely to cause instability for large heights than for small heights. These results indicate that body inertia is a critical design parameter for running robots, and suggest passive mechanisms for balance and gait transition in legged robots and animals.