Stability of a Groucho-Style Bounding Run in the Sagittal Plane

This paper develops a three-degree-of-freedom sagittal-plane hybrid dynamical systems model of a Groucho-style bounding quadrupedal run. Simple within-stance controls using a modular architecture yield a closed-form expression for a family of hybrid limit cycles that represent bounding behavior over a range of user-selected fore-aft speeds as a function of the model's kinematic and dynamical parameters. Controls acting on the hybrid transitions are structured so as to achieve a cascade composition of in-place bounding driving the fore-aft degree of freedom, thereby decoupling the linearized dynamics of an approximation to the stride map. Careful selection of the feedback channels used to implement these controls affords infinitesimal deadbeat stability, which is relatively robust against parameter mismatch. Experiments with a physical quadruped reasonably closely match the bounding behavior predicted by the hybrid limit cycle and its stable linearized approximation.