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.

Robotic yoyo playing is a challenging, open-loop unstable game, which requires dynamic dexterity to stabilize. The switching control strategy proposed here stabilizes the yoyo by determining when to start its activation. A straightforward implementation is presented and demonstrated, using continuous state estimation provided by visual feedback and the previously developed dynamic model. The discrete return map associated with the original continuous system is analyzed and shown to possess a single physically relevant fixed point. Stability is guaranteed as long as the control parameter, which determines the magnitude of the acceleration, is above a certain level. The stabilizing power of the proposed control algorithm is successfully demonstrated on a real-time robotic yoyo playing system. Theoretical predictions regarding the fixed point of the return map are confirmed experimentally, providing further support for our modeling and analysis.

Different networks of coupled oscillators were developed for open-loop control of periodic motion. However, some tasks, like yo-yo playing, are open-loop unstable and require proper phase locking to stabilize. Given the phase-locking property of coupled oscillators, we investigate their application to closed-loop control of open-loop unstable systems, concentrating on the challenging task of yo-yo control. In particular, we focus on pulse-coupling, where the yo-yo sends a feedback upon reaching the bottom of the string and the onset of the oscillatory cycle is used to trigger the movement. Four networks involving either a stand-alone or a circuit level oscillator with either excitatory or inhibitory couplings are considered. Working curve analysis indicates that three of the networks cannot stabilize the yo-yo. The fourth network, which is based on a circuit-level oscillator, is analyzed using the return map and the region of stability is determined and verified by simulations. The resulting pulse-coupled oscillatory control provides a model-free control strategy that operates with an easy-to-measure low-rate feedback.