Exponentially Stabilizing and Time-Varying Virtual Constraint Controllers for Dynamic Quadrupedal Bounding *

This paper aims to develop time-varying virtual constraint controllers that allow stable and agile bounding gaits for full-order hybrid dynamical models of quadrupedal locomotion. As opposed to state-based nonlinear controllers, time-varying controllers can initiate locomotion from zero velocity. Motivated by this property, we investigate the stability guarantees that can be provided by the time-varying approach. In particular, we systematically establish necessary and sufficient conditions that guarantee exponential stability of periodic orbits for time-varying hybrid dynamical systems utilizing the Poincare return map. Leveraging the results of the presented proof, we develop time-varying virtual constraint controllers to stabilize bounding gaits of a 14 degree of freedom planar quadrupedal robot, Minitaur. A framework for choosing the parameters of virtual constraint controllers to achieve exponential stability is shown, and the feasibility of the analytical results is numerically validated in full-order simulation models of Minitaur.