機械手臂的動態往往包含各種不確定的元素，如未知的系統參數，或系統內部動態的不確定性，及外部干擾等，這些因素使得系統的數學模型無法明確地被建立，並且有可能使系統的輸出響應變差，或造成不穩定的現象。我們都知道 控制理論具有將系統的外部輸入對受控系統的受控輸出的影響降到最低的優點；因此本文提出了 -滑動綜合控制器，來針對具有不確定項及外部干擾的機械手臂，來設計控制器，我們利用滑動模式來控制系統，將滑動面的控制結構設定為各個機械手臂關節的誤差函數的微分項、比例項和積分項的總和；再利用回授線性化過後所得的線性誤差狀態方程式，化為標準的 控制問題；利用 控制理論來求解系統最佳的控制增益參數；如果閉迴路系統中未完全消除的不確定性參數，滿足本文所提出Lyapunov不等式，則系統為漸進穩定。
最後，本文針對非線性的機械手臂系統，做為模擬控制的對象，並用不同的質量負載來驗證所設計的 -滑動控制器的可行性。經由模擬的結果顯示，本文所設計的控制器在系統具有不確定性及外擾的存在下仍然保有良好的追蹤性能。

The dynamics of robot manipulators often contain various uncertainties, such as the unknown system parameters, the un-modeled dynamics and the external disturbances, which may make the proposed controller hard to achieve the desired performance. In this paper, the composite controller of the H∞-control and the sliding-mode control laws is proposed. The nonlinear controller of robots is formulated by utilizing the feedback linearization technique while the sliding surfaces in the control structure are set to be the summation of the derivation, the proportion, and the integration of the tracking errors in each generalized robot coordinate with respective adjustable controller gain parameters. Then these controller gain parameters can be obtained by the H∞-control methodology. If the remaining closed-loop un-modeled uncertainties satisfy the inequality derived from the Lyapunov stability theorem proposed in this paper, the proposed H∞-sliding controller will be able to offer the demanded performance for uncertain robot systems.
A three-degree of freedom robot manipulator is given as an example in this research to demonstrate the design methodology of the proposed H∞-sliding mode controller. The simulation results show that the tracking errors in every uncertain robot coordinate can be minimized by benefiting from the optimal control gain parameters obtained from H∞-sliding mode control methodology.

參考文獻
[1] Ayca Gokhan Ak, “Three Link Control with Fuzzy Sliding Mode Controller Based on RBF Neural Network”, Proceedings of the 2006 IEEE International Symposium on Intelligent Control Munich, Germany, October 4-6, 2006.

[2] C. H. AN, C. G. Atkeson, J. D. Griffiths, and J. M. Hollerbach,’’Experimental Evaluation of Feed forward and Computed Torque Control”, IEEE Transactions on Robotic and Automation,Vol.5, No.3, June 1989.

[3] C.N. Hwang, “Tracking of controllers for robot manipulators”, Master Dissertation, Michigan state University, 1986.

[4] C.N. Hwang, “Controller Synthesis for Input Output Tracking in Lure Type Uncertain Nonlinear Systems”, Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 1990.

[5] C.N. Hwang, “Design of Robust Controllers for Manipulators”, Journal of the National Cheng-Kung University, vol.26, Sci. Eng.&Med. Section, pp. 213-234,1991.

[6] C.N. Hwang, “Synthesis Procedure for Nonlinear Systems”, Proc. Natl. Sci. Counc. ROC(A) vol. 17 ,no. 4, pp. 279-294, 1993.

[7] C.N. Hwang, “Formulation of H2 and H Optimal Control Problems – A Variational Approach”, Journal of the Chinese Institute of Engineering’s, Vol. 16, No. 6, pp. 853-866, 1993.

[8] C.N. Hwang, “A Variational Approach to H2 and H Control Problems for Linear Nonautonomous System”, Proc. Natl. Sci. Counc. ROC(A) Vol. 19, No.5, 1995, pp. 408-422.

[9] Chieh-Li Chen and Chieh-Jer Lin, “A Sliding Mode Control Approach to Robotic Tracking Problem”, 2002 IFAC 15th Triennial World Congress, Barcelona, Spain.

[10] G-W. Lee and F.-T. Cheng, “Robust Control of Manipulators Using the Computed Torque Plus H Compensation Method”, IEE, 1996.

[11] J.C. Doyle , K. Glover , P.P. Khargonekar and B.A. Francis, “State-Space Solutions to Standard H2 and H Control Problems”, IEEE Transactions on Automatic Control, Vol. 34,No.8, pp.831-847, 1989.

[12] J.J. Slotine and S.S. Sastry, “Tracking Control of Nonlinear Systems Using Sliding Surfaces, with Application to Robot Manipulators”, Int.J. Control, vol. 38, no. 2, pp.465-492, 1983.

[13] J. J. Slotine, “Sliding controller design for non-linear systems,” Int. J. Contr., vol. 40, pp. 421 – 434, 1984.

[14] J.J. Slotine and W. Li, “Applied Nonlinear Control”, Prentice Hall, Englewood Cliffs, NJ, 1991.

[15] J.shi, H.Liu and N. Bajcinca, “Robust Control of Robotic Manipulators Based on Integral Sliding Mode”, International Journal of Control Vol.81, No. 10, October 2008, 1537-1548, 2008.

[16] Juhing-Perng Su and Chun-Chieh Wang, “Complement Sliding control of non-linear systems”, Int. J. Control, 2002, Vol. 75, No.5, 360-368.

[17] K. Ogata, “Modern Control Engineering”, Fourth Edition, Prentice Hall, 2001.

[18] M.onder Efe, Okyay Kaynak, Xinghuo Yu, “Sliding Mode Control of a Three Degrees of Freedom Anthropoid Robot by Driving the Controller Parameters to an Equivalent Regime”, Transactions of the ASME, Vol.122 December 2000.

[19] Mu Xiaojiang, Chen Yangzhou, “Decoupling Neural Sliding Mode Control for Multi-Link Robots”, Proceeding for the 17th World Congress the International Federation of Automatic Control Seoul, Korea, July 6-11, 2008.

[20] Meysar Zeinali, and Leila Notash, “Adaptive Sliding Mode Control with Uncertainty Estimator for Robot Manipulators” Elsevier Mechanism and Machine Theory 45(2010)80-90, 2010.

[21] Vadim I. Utkin “Sliding Modes in Control and Optimization”, Springer-Verlag. 1992.

[22] Vadim I. Utkin and Jingxin Shi, “Integral Sliding Mode in Systems Operating under Uncertainty Conditions”, Proceedings of the 35th Conference on Decision and Control Kobe, Japan. December 1996.

[23] Vadim I. Utkin, “Variable Structure Systems with Sliding Modes”, IEEE Transactions on Automatic Control, April 1977.

[24] Vadim I. Utkin, Jurgen Guldner and Jingxin Shi, “Sliding Mode Control in Electromechanical Systems”, Taylor &Francis 1999.

[25] Vadim I. Utkin and Hao-Chi Chang, “Sliding Mode Control on Electro-Mechanical Systems” Mathematical Problems in Engineering, 2002, Vol.8(4-5), pp. 451-473 Taylor &Francis Group.

[26] Y. Orlov, J. Alvarez, L. Acho and L. Aguilar, “Global Position Regulation of Friction Manipulators Via Switched Chattering Control”, INT.J.Control,2003, Vol.76, NO.14.

[27] Zames G., “On H∞ Optimal Sensitivity Theory for SISO Feedback Systems”, IEEE Transactions on Automatic Control, Vol.AC-29, NO.1,pp.9-16,1984.

[28] 楊憲東、葉芳柏，“線性與非線性H控制理論”，全華科技圖書股份公司，1997。

[29] 楊憲東，“H控制理論及應用講義”，國立成功大學航太系，1999。

[30] 楊憲東，“非線性控制講義”，國立成功大學航太系，2011。

[31] 謝秉澂、黃正能，“非線性多變數適應-H∞控制器之設計研究”，國立成功大學系統及船舶機電工程研究所碩士論文，2005。

[32] 蘇義閎、黃正能，“H-模糊滑動複合控制設計與其在水下載具上之應用”，國立成功大學系統及船舶機電工程研究所碩士論文，2010。

[33] 黃瀚生、黃正能， “機械臂之H-類神經滑動綜合控制器設計”，國立成功大學系統及船舶機電工程研究所碩士論文，2011。