Dynamic Walking 2010. Ian Manchester. Regions of Attraction to Limit Cycles of Nonlinear Hybrid Systems
This talk will present a method for computing estimates of regions of attraction to limit cycles of quite a general class of nonlinear hybrid systems. Such limit cycles typically represent the "nominal" motion of a walking robot, with alternating phases of continuous motion and distinct impacts upon footfall. The problems of stabilizing such motions and characterizing the regions of stability are central in dynamic walking, and this talk will present contributions on both problems. It is well-known that limit cycles cannot be asymptotically stable in the standard sense, since perturbations in phase are persistent. The more relevant concept is orbital stability, which can be analyzed via a lower-dimensional coordinate system transversal to the target orbit. A new analytical construction is given which is applicable to general nonlinear systems with impacts. This coordinate system is then used for both control via transverse linearization, and computing estimates of regions of attraction via Lyapunov's direct method and the sum-of-squares relaxation of polynomial boundedness. Both impacts (switching surfaces) and the need for well-posedness of dynamics present interesting issues in the selection of transversal coordinate system, which will be discussed. The method will be illustrated with examples including the van der Pol oscillator, the rimless wheel, and the compass-gait walker.