This paper develops a three degree-of-freedom sagittal-plane hybrid dynamical systems model of a bounding quadruped. Simple within-stance controls 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.