Return Maps, Parameterization, and Cycle-Wise Planning of Yo-Yo Playing

This note presents a unified approach to nominal control generation and return map parameterization for yo-yo playing. The approach is based on choosing an intermediate state of the yo-yo as the parameter and formulating an optimization problem with appropriate boundary conditions that incorporate the corresponding parameter. By solving the optimization problems, both the return map and the nominal optimal control for the robot can be obtained. The return map is naturally parameterized with the intermediate state, which can be treated as a virtual control. The return map facilitates the application of standard analysis and design methods for discrete-time control systems. The approach is promising as it might be extended to control other cyclic dynamic systems.