Graceful transitions between periodic motions for nonlinear and hybrid systems
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The objective of this dissertation is to provide a set of methods by which a graceful transition is synthesised for a large class of nonlinear and hybrid systems. A special focus of this thesis is on transitioning between periodic orbits. The primary motivation for this is in the application to legged locomotion. The Gluskabi Raccordation provides a general framework to accomplish this. In this thesis, we utilize the Gluskabi raccordation as a general framework for encapsulating the abstract notion of gracefulness. We extend the kernel method to a certain class of hybrid systems. We show how to construct a carefully formulated optimization problem, the solution of which yields graceful transitions. This is illustrated on hopping systems on elastic and granular terrain. The image method, which is dual to the kernel method, is also used as an alternative method to realize graceful transitions. This involves the careful formulation of a parameterized optimal control problem, the solution of which yields parameterized periodic orbits. A dynamically feasible trajectory is then constructed staying close to this orbit family, which yields a different notion of gracefulness. The method is illustrated on fully actuated and underactuated planar bipedal robots. Finally, energy efficient locomotion is also considered in the context of bipedal robots. The partial hybrid zero dynamics framework is employed to generate stable energy efficient periodic walking gaits. An optimal control problem is solved which generates energy efficient transitions between these stable periodic walking gaits.