Event-Based Stabilization of Periodic Orbits for Underactuated 3-D Bipedal Robots With Left-Right Symmetry

Models of robotic bipedal walking are hybrid, with a differential equation that describes the stance phase and a discrete map describing the impact event, that is, the nonstance leg contacting the walking surface. The feedback controllers for these systems can be hybrid as well, including both continuous and discrete (event-based) actions. This paper concentrates on the event-based portion of the feedback design problem for 3-D bipedal walking. The results are developed in the context of robustly stabilizing periodic orbits for a simulation model of ATRIAS 2.1, which is a highly underactuated 3-D bipedal robot with series-compliant actuators and point feet, against external disturbances as well as parametric and nonparametric uncertainty. It is shown that left-right symmetry of the model can be used to both simplify and improve the design of event-based controllers. Here, the event-based control is developed on the basis of the Poincaré map, linear matrix inequalities and robust optimal control. The results are illustrated by designing a controller that enhances the lateral stability of ATRIAS 2.1.