Based on nonlinear trajectory generation (NTG) software package,a general approach (i.e.numerical solution) to trajectory planning for yoyo motion is presented.For the real-time control of such periodical dynamic system,a critical problem is how to implement fast solving the optimal trajectory,so as to meet the real-time demand.However,traditional numerical solution methods are very time-consuming.In this paper,the optimization problem is solved by mapping the problem to a lower-dimension space.And combined with multithread programming technology,the computation time for solving the optimal trajectory is greatly reduced.Simulation results show that the numerical solution is identical to the analytic one,which demonstrates the correctness of the proposed method.The computation time of one cycle of yoyo simulation is about 10 ms,which shows that the proposed numerical method can be applied to the real-time control of 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.
