起重機是工廠、港口或倉儲貨櫃場中，不可或缺的運輸工具。控制起重機在運送貨物過程中不搖晃，是相當不容易的，考慮安全性的問題，降低其運動造成的搖擺更是相當重要。
本研究對起重機系統的防擺和輸入成型控制的控制方法進行了研究，考慮一種非線性的起重機模型，並利用尤拉-拉格朗日方程式(Euler-Lagrange equation)推導了該系統的動力學模型，在控制理論部分，輸入成型控制是一種用於減少殘餘震動量的開迴路控制技術，其理論是由線性系統理論開發出來的，其參數設計需先得知系統的自然頻率以及阻尼比，而這對於非線性系統來說是不可行的，根據多目標最佳化近年來的發展，本研究將利用一種多目標最佳化演算法來確定控制器參數，目標是找到一組精確的輸入成型控制器參數，其實驗結果的特性是優於線性系統理論的結果。
硬體與實驗部分，本研究使用導螺桿移動平台搭配伺服旋轉馬達驅動移動滑車，並在滑車上設置一單擺系統，以此裝置模擬起重機系統的運動情形，採用RS232做為電腦與控制器UTC400之間的通訊架構，並將Matlab數學軟體執行多目標最佳化的控制結果透過UTC Panel人機介面輸入於控制器中，再由單擺上的旋轉編碼器讀取滑車移動後的單擺擺動狀況。

An industrial crane is very common and crucial in factories, ports of call, and storage warehouses of a container yard. For the issue of safety, it is necessary to reduce the sway caused by the movement of crane.
This study investigates the development of anti-swaying control scheme for a crane system by using the input shaping control. A nonlinear overhead crane system is considered, and the dynamic model of the system is derived using the Euler-Lagrange equations. The input shaping control (ISC) is one of open-loop control techniques for reducing residual vibration, and the ISC was developed based on linear system theory. Conventional designs of the ISC require a priori of natural frequencies and damping ratios of the systems, which may not be available for nonlinear systems. Motivated by recent advances of multi-objective optimization, this study utilizes multi-objective optimization to determine the parameters of the ISC, and the results show that the parameters provide better experiment performances than those obtained from linear system theory.
In the hardware and experimental parts, this study establishes a moving platform consisting of a cart and a pendulum, where the motion of the platform is driven by a servo motor through a ball screw. The port RS232 is used as the communication framework between a personal computer and the controller UTC400. Before performing the experiments, the software Matlab is used to calculate the controller parameters through multi-objective optimization, and then the ISC design is implemented in the controller through the UTC Panel interface. The residual vibrations of the pendulum are recorded by using a rotary encoder mounted on the pendulum.

[1] O. J. M. Smith, “Posicast control of damped oscillatory systems.”Proceeding of the IRE, pp.1249-1255, 1957.
[2] L. Y. Pao, M. A. Lau, “Robust input shaper control design for paremeter variations in flexible structures,” ASME Journal of Dynamic system, Measurement and Control, 122(1), pp. 63-70, 2000.
[3] T. C. Lin, “Design an input shaper to reduce operation-induced vibration,” IEEE American Control Conference, June, San Francisco, pp. 2502-2505, 1993.
[4] N. C. Singer, W. P. Seering, “Design and comparison of command shaping methods for controlling residual vibration,” IEEE International Conference on Robotics and Automation, IEEE Computer Society, pp. 888-893, 1989.
[5] N. C. Singer, W. P. Seering, “Preshaping Command Inputs to Reduce System Vibration,” Journal of Dynamic Systems, Measurement, and Control, pp. 76-82, 1990.
[6] W. Singhose, “Command Generation for Flexible Systems,” Doctoral dissertation, Messachusetts Institude of Technology, pp. 285, 1997.
[7] D. O. Popa, B. H. Kang, H. E. Stephanou, “Dynamic modeling and input shaping of thermal bimorph MEMS actuators,” IEEE International Conference on Robotics and Automation, September 14-19, Taipei, Taiwan, vol. 1, no. 14-19, pp. 1470-1475, 2003.
[8] T. Singh, “Optimal Reference Shaping for Dynamical Systems,” CRC Press, pp. 24-46, 2009.
[9] W. Singhose, D. Blackburn, J. Kitchen, V. Patrangenaru, A. Taura, “Command shaping for nonlinear crane dynamics,” Journal of Vibration and Control, vol. 4, no. 16, pp. 477-501, 2010.
[10] M. J. Maghsoudi, Z. Mohamed, A. R. Husain, H. I. Jaafar, “Improved input shaping technique for a nonlinear system,” IEEE International Conference on Control System, Computing and Engineering, November 28-30, Penang, Malaysia, pp. 261-266, 2014.
[11] A. M. Abdullahi, A. Mohamed, M. S. Z. Abidin, “Adaptive input shaping for sway control of 3D crane using a pole-zero cancellation method,” IEEE Student Conference on Research and Development, pp. 1-6, 2015.
[12] T. Zhou, H. Guo, J. Xu, C. Lin, “Adaptive robust control with input shaping technology for solar array drive system,” Acta Astronautica, vol. 140, pp. 264-272, 2017.
[13] M. J. Maghsoudi, Z. Mohamed, “An improved input shaping design for an effcient sway control of a nonlinear 3D overhead crane with friction,” Mechanical Systems and Signal Processing, vol. 92, pp. 364-378, 2017.
[14] D. Newman, J. Vaughan, “Command shaping of a boom crane subject to nonzero initial conditions,” IEEE Conference on Control Technology and Applications, August 27-30, Hawaii, USA, pp. 1189-1194, 2017.
[15] S. Grazioso, G. Gironimo, W. Singhose, B. Siciliano, “Input predictive shaping for vibration control of flexible systems,” IEEE Conference on Control Technology and Applications, August 27-30, Hawaii, USA, pp. 305-310, 2017.
[16] S. M. Fasih, Z. Mohamed, A. R. Husain, “Payload swing control of a tower crane using a neural network-based input shaper,” Measurement and Control, vol. 53, pp. 1171-1182, 2020.
[17] H. D. Tho, N. Uchiyama, K. Terashima, “Model reference input shaping control of a nonlinear rotary crane with time-varying rope length,” IEEE International Conference on Control, Decision and Information Technologies, June 29- July 2, Prague, Czech Republic, pp. 166-171, 2020.
[18] T. Singh, G. R. Heppler, “Shaped input for a multimode system,” IEEE International Conference on Robotics and Automation, pp. 484-489, 1993.
[19] J. Y. Smith, K. Kozak, W. Singhose, “Input shaping for a simple nonlinear system,” Proceeding of the American Control Conference, May 8-10, Anchorage, AK, vol. 1, pp. 821-826, 2002.
[20] W. Singhose, “A vector diagram approach to shaping inputs for vibration reduction,” MIT Artificial Intelligence Lab Memo, no. 1223, 1990.
[21] W. Singhose, W. Seering, N. Singer, “Residual vibration reduction using vector diagrams to generate shaped inputs,” Journal of Mechanical Design, vol. 116, no. 2 pp.654-659, 1994.
[22] W. Singhose, L. Porter, N. Singer, “Vibration reduction using multi-hump extra-insensitive input shapers,” Proceedings of the American Control Conference, American Automatic Control Council, vol. 5, pp.3830-3834, 1995.
[23] W. Singhose, W. Seering, N. Singer, “Input shaping for vibration reduction with specified insensitivity to modeling errors,” Japan-USA Symposium On Flexible Automation, pp. 307-313, 1996.
[24] J. Vaughan, A. Yano, W. Singhose, “Comparison od robust input shapers,” Journal of Sound and Vibration, vol. 315, no. 4, pp. 797-815, 2008.
[25] D. Fujioka, M. Shah, W. Singhose, “Robustness analysis of input-shaped model reference control on a double-pendulum crane,” IEEE American Control Conference, July 1-3, Chicago, USA, pp. 2561-2566, 2015.
[26] C. W. Ha, D. Lee, K. H. Rew, K. S. Kim, “An impulse-time perturbation approach for a symmetric extra-insensitive input shaper,” International Journal of Control, Automation and Systems, vol. 16, no. 3, pp. 1239-1246, 2018.
[27] J. Xu, H. Fang, T. Zhou, Y. H. Chen, “Optimal robust position control with input shaping for flexible solar array drive system: a fuzzy-set theoretic approach,” IEEE Transactions on fuzzy systems, vol. 27, no. 9, pp. 1807-1817, 2019.
[28] A. Alshaya, A. Alshayji, “Robust multi-steps input command for liquid sloshing control,” Journal of Vibration and Control, vol. 0, no. 0, pp. 1-18, 2021.
[29] M. S. Alam, M. O. Tokhi, “Design of a command shaper for vibration control of flexible system: a genetic algorithm optimization approach,” Journal of low frequency noise, vibration and active control, vol. 26, no. 4, pp. 295-310, 2007.
[30] M. S. Alam, M. O. Tokhi, “Designing feedforward command shapers with multi-objective genetic optimisation for vibration control of a single-link flexible manipulator,” Engineering Applications of Artificial Intelligence, vol. 21, no. 2, pp. 229-246, 2008.
[31] H. I. Jaafar, Z. Mohamed, A. F. Z. Abidin, Z. A. Ghani, “PSO-tuned PID controller for a nonlinear gantry crane system,” IEEE International Conference on Control System, Control System, Computing and Engineering, November 23-25, Penang, Malaysia, pp. 515-519, 2012.
[32] M. Hamdy, R. Shalaby, M. Sallam, “Experimental verification of a hybrid control scheme with chaotic whale optimization algorithm for nonlinear gantry crane: A comparative study,” ISA Transactions, vol. 98, pp.418-433, 2020.
[33] Mathworks, “Optimization Toolbox R2020a Documentation,” Available: https://www.mathworks.com/products/optimization.html
[34] J. H. Holland, “Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intellgence,” MIT press, 1992.