Composing limit cycles for motion planning of 3D bipedal walkers

This paper presents a framework for navigation of 3D dynamically walking bipeds. The framework is based on extracting gait primitives in the form of limit-cycle locomotion behaviors, which are then composed by a higher-level planning algorithm with the purpose of navigating the biped to a goal location while avoiding obstacles. By formulating motion planning as a discrete-time switched system with multiple equilibria - each corresponding to a gait primitive - we provide analytical conditions that constrain the frequency of the switching signal so that the biped is guaranteed to stably execute a suggested plan. Effectively, these conditions distill the stability limitations of the system dynamics in a form that can be readily incorporated to the planning algorithm. We demonstrate the feasibility of the method in the context of a 3D bipedal model, walking dynamically under the influence of a Hybrid Zero Dynamics (HZD) controller. It is shown that the dimensional reduction afforded by HZD greatly facilitates the application of the method by allowing certificates of stability for gait primitives using sums-of-squares programming.

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