Abstract: This paper presented a general approach for planning trajectories for cyclic dynamic systems, and demonstrated its application for robotic yoyo playing. The approach is based on the return map parameterization using intermediate states of the system. The height of the robot arm when the yoyo arrives at the bottom is selected as the intermediate state. By solving the optimization problems during each phase of a cycle, a reference trajectory, a nominal control for the robot and the corresponding return map of the object are generated simultaneously. The method can also be applied to other similar periodic dynamic systems.