Exponentially Stabilizing Continuous-Time Controllers for multi-domain hybrid systems with application to 3D bipdeal walking

This paper presents a systematic approach to exponentially stabilize the periodic orbits of multi-domain hybrid systems arising from 3D bipedal walking. Firstly, the method of Poincaré sections is extended to the hybrid systems with multiple domains. Then, based on the properties of the Poincaré maps, a continuous piecewise feedback control strategy is presented, and three methods are furthermore given to design the controller parameters based on the developed theorems. By those design methods, the controller parameters in each continuous phase can be designed independently, which allows the strategy to be applied to hybrid systems with multiple domains. Finally, the proposed strategy is illustrated by a simulation example. To show that the proposed strategy is not limited to bipedal robots with left-right symmetry property which is assumed in some previous works, an underactuated 3D bipedal robot with asymmetric walking gait is considered.