Nonlinear optimal control for robotic yoyo playing

A general approach for controlling of periodical dynamic systems was presented by taking robotic yoyo as an example. The height of the robot arm when the yoyo arrives at the bottom was chosen as virtual control. The initial amplitude of yoyo could be mapped to the desired final amplitude by adjusting the virtual control. First, the yoyo motion was formulated into a nonlinear optimal control problem which contained the virtual control. The reference trajectory of robot could be obtained by solving the optimal problem with analytic method or more general numerical approach. Then, both PI and deadbeat control methods were used to control the yoyo system. The simulation results show that the analytic solution of the reference trajectory is identical to the numerical solution, which mutually validates the correctness of the two solution methods. In simulation, the initial amplitude of yoyo is set to be 0.22 m which is 10% higher than the desired final amplitude of 0.2 m. It can be seen that the amplitude achieves the desired value asymptotically in about five periods when using PI control, while it needs only one period with deadbeat control. The reference trajectory of robot is generated by optimizing a certain performance index; therefore, it is globally optimal. This is essentially different from those traditional control methods, in which the reference trajectories are empirically imposed on robot. What’s more, by choosing the height of the robot arm when the yoyo arrives at the bottom as the virtual control, the motion of the robot arm may not be out of its stroke limitation. The proposed approach may also be used in the control of other similar periodical dynamic systems.