Nonlinear optimal control for robotic yoyo playing

Abstract

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.

Keywords

robotic yoyo / return map / reference trajectory / optimal control

References

[1] NelsonN., HanakT., LoewkeK., MillerD. N.. Modeling and implementation of McKibben actuators for a hopping robot [C]. Proceedings of the 12th International Conference on Advanced Robotics, 2005, Seattle, IEEE: 833-840

[2] CherouvimN., PapadopoulosE.. Single actuator control analysis of a planar 3DOF hopping robot [C]. Robotics: Science and Systems I, 2005, Cambridge, MIT Press: 145-152

[3] CherouvimN., PapadopoulosE.. Energy saving passive-dynamic gait for a one-legged hopping robot [J]. Robotica, 2006, 24: 491-498

[4] OliveiraV. M., LagesW. F.. Comparison between two actuation schemes for underactuated brachiation robots [C]. Proceedings of the International Conference on Advanced Intelligent Mechatronics, 2007, Zurich, IEEE/ASME: 1-6

[5] GomesM. W., RuinaA. L.. A five-link 2D brachiating ape model with life-like zero-energy-cost motions [J]. Journal of Theoretical Biology, 2005, 237: 265-278

[6] OliveiraV. M., LagesW. F.. MPC applied to motion control of an underactuated brachiation robot [C]. Proceedings of the Conference on Emerging Technologies & Factory Automation, 2006, Prague, IEEE: 985-988

[7] OliveiraV. M., LagesW. F.. Linear predictive control of a brachiation robot [C]. Proceedings of the Conference on Electrical and Computer Engineering, 2006, Ottawa, IEEE: 1518-1521

[8] HiraiH., MiyazakiF.. Dynamic coordination between robots: self-organized timing selection in a juggling-like ball-passing task [J]. IEEE Transactions on Systems, Man, and Cybernetics, 2006, 36(4): 738-754

[9] NakashimaA., SugiyamaY., HayakawaY.. Paddle juggling of one ball by robot manipulator with visual servo [C]. Proceedings of the 9th International Conference on Control, Automation, Robotics and Vision, 2006, Singapore, IEEE: 1-6

[10] BrogliatoaB., MabroukbM., RiocA. Z.. On the controllability of linear juggling mechanical systems [J]. Systems & Control Letters, 2006, 55: 350-367

[11] HashimotoK., NoritsuguT.. Modeling and control of robotic yoyo with visual feedback [C]. Proceedings of the International Conference on Robotics and Automation, 1996, Minneapolis, IEEE: 2650-2655

[12] JinH. L., ZacksenhouseM.. Robotic yoyo playing with visual feedback [J]. IEEE Transactions on Robotics, 2004, 20(4): 736-744

[13] JinH. L., ZacksenhouseM.. Oscillator-based yoyo control: implementation and comparison with model-based control [C]. Proceedings of the 12th International Conference on Advanced Robotics, 2005, Seattle, IEEE: 153-158

[14] JinH. L., YeQ., ZacksenhouseM.. Return maps, parameterization, and cycle-wise planning of yo-yo playing [J]. IEEE Transactions on Robotics, 2009, 25(2): 438-445

[15] JinH. L., ZacksenhouseM.. Yoyo dynamics: sequence of collisions captured by a restitution effect [J]. Journal of Dynamic Systems, Measurement and Control, 2002, 124(3): 390-397

[16] LynchK. M., BlackC. K.. Recurrence, controllability, and stabilization of juggling [J]. IEEE Transactions on Robotics and Automation, 2001, 17(2): 109-123

[17] BrysonA. E., HoY. C.Applied optimal control: optimization, estimation, and control [M], 1975, Washington, Hemisphere

[18] NaiduD. S.Optimal control systems [M], 2003, Cambridge, Cambridge University Press

[19] MilamM. B., MushambiK., MurrayR. M.. A new computational approach to real-time trajectory generation for constrained mechanical systems [C]. Proceedings of the Conference on Decision and Control, 2000, Sydney, IEEE: 845-851

[20] PetitN., MilamM. B., MurrayR. M.. Inversion based constrained trajectory optimization [C]. Proceedings of the 5th Symposium on Nonlinear Control Systems, 2001, Petersburg, IFAC: 1211-1216

[21] MisovecK., InancT., WohletzJ., MurrayR. M.. Low-observable nonlinear trajectory generation for unmanned air vehicles [C]. Proceedings of the 42nd Conference on Decision and Control, 2003, Hawaii, IEEE: 3103-3110

[22] de BoorC.A practical guide to splines [M], 1978, New York, Spring-Verlag

[23] PressW. H.Numerical recipes in C++, 20073rd edition [M]New York, Cambridge University Press