摘要
在伺服运动控制系统中,由于非线性摩擦环节的存在,系统的动静态性能受到很大影响。随着科技的不断发展,对伺服系统性能要求越来越高,而摩擦环节严重制约了伺服系统的发展,因此摩擦补偿技术已成为高性能伺服控制系统的关键技术,对该技术的研究具有重要的理论意义和广泛的应用前景。本文针对伺服运动控制系统中的摩擦环节,采用理论建模、仿真分析和实验研究相结合的方法,从LuGre摩擦模型的辨识、修正和控制策略的设计方面进行了深入研究。
首先,建立了开放式伺服系统各组成部分的数学模型,在此基础上建立了基于Simulink伺服系统仿真平台。基于该仿真平台对LuGre摩擦模型的动态特性进行了深入研究,分析了Stribeck模型与LuGre模型的区别与内在联系。此外,还通过该仿真平台完成了对摩擦参数的辨识方法和摩擦补偿方法的可行性和有效性验证。
其次,完成了LuGre摩擦模型的参数辨识与模型的修正。对伺服系统进行了转矩纹波补偿,消除了转矩纹波对辨识摩擦参数产生的不利影响;基于稳态误差反推摩擦力矩的方法,通过多组匀速运动的实验提取速度与摩擦力矩的对应关系,实现了对LuGre模型静态摩擦参数的精确辨识;提出了一种基于简化摩擦模型的转动惯量的辨识方法,完成对转动惯量的精确辨识,进而为LuGre摩擦模型动态参数的辨识提供条件;利用遗传算法实现了对LuGre摩擦模型动态参数的精确辨识;提出了一种LuGre摩擦模型的修正方法,解决了粘性摩擦系数在实际应用中存在的问题,修正后的摩擦模型能更完全、精确地反映伺服系统的摩擦特性,具有更高的实用价值。
最后,对摩擦补偿的控制策略进行了研究。根据LuGre修正摩擦模型的参数辨识结果,提出了采用转矩纹波补偿+速度加速度前馈补偿+LuGre修正模型的摩擦前馈补偿的控制方法,改善了伺服系统的跟踪性能。考虑到摩擦模型参数会随外界环境变化而改变,本文还提出了一种基于Backstepping设计的自适应摩擦补偿方法,保证了闭环系统的渐近稳定性,并能有效地克服非线性摩擦的影响,提高了伺服控制系统的跟踪精度。
Static and dynamic performance was greatly influenced by nonlinear friction in servo motion control system. With the continuous development of technology, higher performance of servo system was required, but friction was seriously restricting the development of servo system. Therefore, friction compensation has become the key technology of high-performance servo control system, which has important theoretical significance and wide application prospect. Aiming at friction of servo motion control system, the further research was made on identification and modification of LuGre model, and design of control strategy, through the methods of integrating theoretical modeling, simulation analysis and experimental research.
Firstly, mathematical models were established according to components of open servo system, thus a servo simulation platform was set up. The dynamic characteristics of LuGre model was studied based on the simulation platform, and the distinctions and relations between Stribeck model and LuGre model were analyzed. In addition, the feasibility and validity of the methods was verified, which included the friction parameter identification method and the friction compensation method.
Secondly, the parameter identification and modification of LuGre model were achieved. In order to eliminate the negative impact of torque ripple on the friction parameters identification, torque ripple compensation was carried out; Based on the method of steady-state error reverseing friction torque, the static friction parameters of LuGre model were identified accurately by the corresponding relation between speed and friction torque from multi-groups uniform motion experiments; an precise identification method of inertia was proposed base on simplified friction model, thus it provided conditions to the dynamic friction parameters identification of LuGre model;then dynamic parameters of the model were identified accurately using the genetic algorithm; Considering the problem of viscous friction coefficient in the practical application, a modified friction model based on the LuGre model was presented, furthermore, the modified friction model could completely and accurately reflect the friction characteristics of servo system, which had a higher practical value.
Finally, the control strategies for friction compensation were investigated. According to the friction parameters identification results of the modified LuGre model, a control method, including torque ripple compensation, speed and acceleration feedforward compensation, and friction feedforward compensation based on modified LuGre model was proposed, and it improved tracking performance of servo system. Considering the parameters of friction model could be changed with the variability of external environment, a method of adaptive friction compensation based on Backstepping design was presented in this paper, which could ensure the asymptotic stability of the close-loop system, overcome the influence of nonlinear friction, and enhance the tracking accuracy of servo control system.
引文
[1]舒志兵,周玮,李运华等,交流伺服运动控制系统,清华大学出版社,2006,1~3
[2]韩彦春,交流永磁同步电机伺服控制系统的研究:[硕士学位论文],辽宁;辽宁工程技术大学,2008
[3]崔宪莉,高速高精芯片封装平台的摩擦分析及补偿研究:[硕士学位论文],武汉;华中科技大学,2005
[4] http://baike.baidu.com/view/267111.htm
[5] http://www.gkong.com/zt/zidonghua/page_13.html
[6] http://baike.baidu.com/view/132346.htm
[7]吴琳,谭营,唐建,运动控制技术发展与展望,机床与液压,2007,35(7):231~234
[8]唐庆功,永磁同步电机伺服控制系统研究:[硕士学位论文],哈尔滨;哈尔滨工业大学,2009
[9]陈先锋,舒志兵,先进运动控制策略及运动控制新技术,电气时代,2005,2:129~131
[10]王军平,王安,敬忠良等,高性能运动控制在数控系统中的应用综述,信息与控制,2003,32(3):245~250
[11]韩彦春,交流永磁同步电机伺服控制系统的研究:[硕士学位论文],辽宁;辽宁工程技术大学,2008
[12]黄萌,全数字化高性能交流伺服控制系统的研究:[硕士学位论文],陕西;西北工业大学,2005
[13]盖廓,新型控制策略在交流伺服系统中的应用研究:[硕士学位论文],天津;天津大学,2006
[14]刘强,尔联洁,刘金琨,摩擦非线性环节的特性、建模与控制补偿综述,系统工程与电子技术,2002,24(11):45~52
[15]刘金琨,滑模变结构控制MATLAB仿真,清华大学出版社,2005,1~15
[16]董宁,自适应控制,北京理工大学出版社,2009,1~10
[17]钱积新,赵均,徐祖华,预测控制,化学工业出版社,2007,1~8
[18]易继锴,侯媛彬,智能控制技术,北京工业大学出版社,2004,1~15
[19]辛小南,贺莉,王宏洲,永磁同步电动机交流伺服系统控制策略综述,微特电机,2010(2),67~70
[20]窦汝振,高性能永磁交流伺服系统及其新型控制策略的研究:[硕士学位论文],天津;天津大学,2002
[21]Kyeong K., In-Cheol B., Myung Y., Simple and Robust Digital Current Control Scheme of Permanent Magnet Synchronous Motor Using Time Delay Control Approach, Elec.Letters, 1999, 35(12):1027~1028.
[22]Rahman A., Hoque A., On-line Adaptive Artificial Neural Network Based Vector Control of Permanent Magnet Synchronous Motors, Energy Conversion, 1998, 13(4):311~318
[23]Bimal K., Expert System, Fuzzy Logic and Neural Network Applications in Power Electronics and Motion Control, IEEE Proc., 1994, 82(8):1303~1323
[24]Baik C., Kim H.,Youn J., Robust Nonlinear Speed Control of PM Synchronous Motor Using Adaptive and Sliding Mode Control Techniques, IEEE Proc. Elec. Power Appl., 1998, 145(4) : 369~376
[25]金能强,何洪涛,稀土永磁同步电机的应用和发展状况,电工电能新技术,1995(3):17~21
[26]李骥,基于DSP的永磁同步电机矢量控制系统的研究.:[硕士学位论文],湖北;武汉理工大学工学,2008
[27]薛培军,刘冬敏,浅谈运动控制技术的发展现状及应用前景,中州大学学报,2007,24(1):125~126
[28]刘相扬,管雪梅,运动控制的实现技术与未来发展趋势,民营科技,2008,6:32~33
[29]郗志刚,周宏甫,运动控制器的发展与现状,电气传动自动化,2007,25(3):10~14
[30]杨占玺,韩秋实,智能数控系统发展现状及其关键技术,制造技术与机床,2008,12:63~66
[31]王霞,基于X-Y平台的摩擦补偿理论研究:[硕士学位论文],河北;河北大学,2005
[32]刘国平,机械系统中的摩擦模型及仿真:[硕士学位论文],陕西;西安理工大学,2007
[33]王玉兵,直线电机伺服控制系统的摩擦分析与补偿:[硕士学位论文],安徽;合肥工业大学,2009
[34]袭著燕,基于信号特征的数控交流伺服进给系统摩擦建模与补偿研究:[博士学位论文],山东;山东大学,2006
[35]王波,董申,赵万生,超精密定位中的几项关键技术,航空精密制造技术,1998,34(3):13~16
[36]徐圣林,陈耿,精密超精密定位技术及其应用,中国机械工程,1997,8(4):73-75
[37]王英,面向芯片封装的直线伺服系统的高速高精运动控制:[博士学位论文],上海;上海交通大学,2006
[38]魏立新,X-Y数控平台运动摩擦补偿及边缘跟踪力控制研究:[博士学位论文],河北;燕山大学,2006
[39]郭克尖,滑动导向系统摩擦建模及动力学研究:[博士学位论文],上海;上海交通大学,2006
[40]刘金琨,先进PID控制MATLAB仿真,电子工业出版社,2004,378~410
[41]Armstrong B., Dupont P., Canudas C., A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction, Automatica, 1994, 30 (7) : 1083~1138
[42] Amlstrong B., Stick Slip and Control in Low-Speed Motion. IEEE Trans. on AC, 1993, 38(10): 1483~1496
[43]郭涛,工作台误差测量与补偿系统的研究:[硕士学位论文],山东;山东大学,2002
[44]黄进,含摩擦环节伺服系统的分析及控制补偿研究:[博士学位论文],陕西;西安电子科技大学,1998
[45]Bowden F., Tabor D., Friction-An Introduction to Tribology, New York, Anchor Press/Doubleday, 1973
[46]Israelachvili J., Intermolecular and surface forces, New York, Academic, 1995
[47]Rabinowicz E., Friction and wear of materials, New York, Wiley, 1995
[48]Stribeck R., Die wesentlichen Eigenschaften der Gleit-und Rollenlager-The key qualities of sliding and roller bearings, Zeitschrift des Vereins Deutscher Ingenieure, 1902, 46(38-39) : 1342~1348, 1432~1437
[49]刘慧慧,基于速度相关和LuGre摩擦模型的滑动稳定性分析:[硕士学位论文],陕西;西安理工大学,2008
[50]Karnopp D., Computer simulation of stick-Slipfriction in mechanical dynamic systems, ASME J. Dyn. Sys. Meas Contr. 1985, 107(1): 100~103
[51]Dupont P., Hayward V., Armstrong B., Single state elasto-plastic friction models, IEEE Trans. Automat. Contr., 2002, 47(5): 787~792,.
[52]Dahl P., A solid friction model, The Aerospace Corp., El Segundo, CA, Tech. Rep. TOR-0158(3107-18)-1, 1968
[53]Hassig D., Friedland B. On the Modeling and simulation of friction, ASME Journal of Dynamic systems, Measurement and control, 1991, 113(3) : 354~362
[54]Canudas C., A New Model for Contml of Systems with Friction, IEEE Tran. on Automatic Control, 1995, 40(3): 419~425
[55]Armstrong B., Control of Machines with Friction, Norwell, MA, Kluwer, 1991
[56]Swevers J., Al-Bender F., Ganseman E., An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Trans. Automat. Contr., 2000, 45(4):675~686
[57]Lampaert V., Swevers J., Al-Bender F., Modification of the Leuven integrated friction model structure, IEEE Trans. Automat. Contr., 2002, 47(4):683~687
[58]Fun M., Hagan M., Modular Neural Networks for Friction Modeling and Compensation, Proceedings of the 1996 IEEE International Conference on Control Applications, Dearborn, 1996, 814~819
[59]覃媛媛,王道波,王志胜,CMAC神经网络在电动伺服摩擦补偿中的应用,兵工自动化,2004 ,23(1):41~43
[60]黄进,叶尚辉,基于CMAC网络的摩擦补偿研究,中国机械工程,1999,10(3):269~272
[61]Garcia C., Comparison of friction models applied to a control valve, Control Engineering Practice, 2008, 16(10):1231~1243
[62]Choudhury S., Shook D., Automatic detection and quantification of stiction in control valves, Control Engineering Practice, 2006, 14 (12):1395~1412.
[63]丛爽,运动控制中先进控制策略的研究综述,微特电机,1998,1:2~8
[64]Friedlalld B., Park J., On Adaptive Friction Compensation, IEEE Trans on Automatic Control, 1992, 37(10):1609~1612
[65]袭著燕,张涛,路长厚,数控伺服进给系统中摩擦补偿控制研究进展,现代制造工程,2006,1:21~25
[66]Phillips S M., Ballou K R., Friction Modeling and Compensation for an Industrial Robot, Journal of Robotic Systems , 1993 , 10 (7) : 947~971
[67]Weiping L., Cheng X., Adaptive High-Precision Control of Positioning Tables-Theory and Experiments, IEEE Trans. on Control Systems Technology, 1992, 2: 265~270
[68]Canudas C., Robust Control for Servo-Mechanisms under Inexact Friction Compensation, Automatica, 1993, 29 (30) : 757~761
[69]张伟英,张友安,非线性观测器用于高精度甚低速系统的动态补偿,控制与决策, 1989,1:35~37
[70]张从鹏,刘强,直线电机定位平台的摩擦建模与补偿,北京航空航天大学学报,2008,34(1):47~50
[71]曾鸣,王忠山,王学智,基于非线性摩擦模型参数观测器的自适应摩擦补偿方法的研究,航空精密制造技术, 2005,41(3):17~22
[72]李书训,姚郁,王子才,一种自适应摩擦补偿方法的研究,电机与控制学报,1999,3(3):129~134
[73]Feemster M., Vedagarbha P., Haste D., Adaptive Control Techniques for Friction Compensation, Mechatronics, 1999, 9: 125~145
[74]Cheok K. C., Hu H., Loh N K., Modeling and Identification of a Class of Servomechanism Systems with Stick2Slip Friction, ASME Journal of Dynamic Systems, Measurement and Control, 1988, 110: 324~328
[75]Lee S W., Kim J H., Friction Identification Using Evolution Strategies and Robust Control of Positioning Tables, ASME Journal of Dynamic Systems, Measurement and Control, 1999, 121: 619~624
[76]Canudas De Wit C., Lischinsky P., Adaptive Friction Compensation with Partially Known Dynamic Friction Model, International Journal of Adaptive Control and Signal Processing, 1997, 11:65~80
[77]Lischinsky P., Canudas de Wit C., Morel G. Friction Compensation for an Industrial Hydraulic Robot, IEEE Control Systems, 1999, 2: 25~32
[78]Truong Ngoc Minh, Kiyoshi Ohishi, Masasuke Takata, et al., Adaptive Friction Compensation Design for Sub-micrometer Positioning of High Precision Stage, Proceedings of International Conference on Mechatronics, 2007, 8(10): 1~6
[79]刘强,扈宏杰,刘金琨,基于遗传算法的伺服系统摩擦参数辨识研究,系统工程与电子技术,2003,25(1): 77~79
[80]Wenjing Z., Parameter Identification of LuGre Friciton Model in Servo System Based on Improved Particle Swarm Optimmization Algorithm, Proceeding of the 26th Chinese Control Conference, 2007, 135~139
[81]Lin J., Chen H. C., A Novel Fuzzy Friction Compensation Approach for Tracking of a Linear Motion Stage, Proceedings of the 2006 Americm Control Conference, 2006, 3188~3198
[82]Yaolong T., Jie C., Hualin T., Adaptive Backstepping Control and Friction ComDensation for AC Servo wim Inertia and Load Uncenainties, IEEE Transactions on Industrial Electronics, 2003, 50(5):944~952
[83]BaSilio B., Marina I., Nicola S., Rapid Prototyping of a Model-Based Control with Friction Compensation for a Direct-Drive Robot, IEEE/ASME Transactions on on Mechatronics, 2006, 11(5):576~584
[84]Emmanuel C., Dean L., Vicente P. V., Visual Servoing for Constrained Planar Robots Subject to Complex Friction, IEEE Transactions on Mechatronics, 2006, 11(4):389~400
[85]Nitin P., Christopher E., Sarah K S., A sliding Mode Observer for Tyre Friction Estimation during Braking, Proceedings of the 2006 American Control Confcrence, 2006:5867~5872.
[86]Adres G., Modelling End-of-roll Dynamics in Position Servos, Control Engineering Practice, 2004, 12(2):217~224
[87]Armstrong B., Stick Slip and Control in Low-Speed Motion, IEEE Transaction on Automatic Contrrol, 1993, 38(10):1483-1496
[88] Armstrong B., Amin B., PID Control in the Presence of Static Friction: A Comparison of Algebraic and Describing Function Analysis, Automatic, 1996,32(5):676~692
[89]黄进,叶尚辉,陈其昌,含摩擦环节的伺服系统的低速爬行研究,机械设计,1998,10:39~41
[90]黄进,叶尚辉,陈其昌,含有摩擦环节伺服系统的Volterra泛函级数分析,西安电子科技大学学报,1998,25(4):515～520
[91]Armstrong B., Neevel D., Kusik T, New Results in NPID Control: Tracking, Integral Control, Friction Compensation and Experimental Results, IEEE Transantion on Control Systems Technology, 2001,9(2):399-405
[92]Horowitz L., Oidak S., Shapiro A., Extensions of Dithered Feedback Systems, International Journal of Control, 1991, 54 (1):83~109
[93]Yang S., Tomizuka M., Adaptive Pulse Width Control for Precise Positioning under Influence of Sticktion and Coulomb Friciton, ASME Journal of Dynamic Systems, Measurement, and Control, 1988, 110 (3): 221~227.
[94]张丹,含摩擦环节伺服系统的补偿控制:[硕士学位论文],陕西;西安电子科技大学,2008
[95] Morel G.., Iagnemma K., The Precise control of Manipulators with HighJoint Friction Using Base Force/Torque Sensing, Automatica, 2000, 36:931~941
[96]李泉,X-Y工作台的摩擦建模与仿真研究:[硕士学位论文],山东;山东大学,2005
[97] Ohaishi K., A New Sevro Mehtod inMeehatronies, Transactions of Japanaese Soeiety of Electrical Engineers, 1987, 107-D.83~86
[98]Seong Lee H., Robust Digital Traeking Controllers for High-speed, Hligh-Accuracy PositioningSystems, Bekreley, USA:University of Caliofnria, 1994
[99]梅志千,机电伺服系统中的补偿技术研究:[博士学位论文],上海;上海交通大学,2003
[100]熊田忠,交流伺服系统中的干扰观测器的设计研究:[硕士学位论文],南京;河海大学,2006
[101]于爽,付庄,闫维新等,基于干扰观测器的惯性平台摩擦补偿方法,哈尔滨工业大学学报,2008,40(11):1830~1833
[102]Southward S.C., Radcliffe C. J., Mac C. R., Robust Nonlinear Stick-Slip Friction Compensation, ASME Journal of Dynamic Systems, Measurement, and Control, 1991, 113: 639~645
[103]Altintas Y., Erkokmaz K., Sliding Mode Controller Design for High Feed Drives, CIRP Annals - Manufacturing Technology, 2000, 49(1):265~270
[104]Tzes A., Peng P Y., Guthy J., Genetic-Based Fuzzy Clustering for DC-Motor Friction Identification and Compensation, IEEE Trans. on Control Systems Technology, 1998, 6 (1): 462~472.
[105]黄进,叶尚辉,陈其昌,基于自调整量化因子模糊控制器的摩擦补偿研究,机械设计与研究,1999 (1) :27~29
[106]Selmic R., Lewis F L., Neural Network Approximation of Piecewise Continuous Functions: Application to Friction Compensation, In Proceedings of the 12th IEEE International Symposium on Intelligent Control, 1997, 227~232.
[107]Rastko R S., Lewis F L., Deadzone Compensation in Motion Control Systems Using Neural Networks, IEEE Trans. on Automatic Control, 2000, 45(4):602~613.
[108]Opart G., Compensation and Estimation of Friction by Using Extended Kalman Filter, SICE-ICASE International Joint Conference, 2006, 5032~5035
[109]李萌,沈炯,基于自适应遗传算法的过热汽温PID参数优化控制仿真研究,中国电机工程学报, 2002, 22 (8):145~149
[110]LIN C. L., LEE G.Y., Hierarchical Fuzzy Control for C-Axis of CNCTurning Centers Using Genetic Algorithms, Journal of Intelligent and Robotic Systems.Journal of Intelligent and Robotic Systems, 1999, 25(3):255~275
[111]Lib C. L., LIN J. Y., Fuzzy-enhanced Adaptive Control for Flexible Drive System with Friction Using Genetic Algorithms, 1998, 23(2-4):379~405
[112]LIAW C. D., HUANQ T.J., Contact Friction Compensation for Robots Using Genetic Learning Algorithms, Journal ofIntelligem and Robotic Systems, 1998, 23(2-4):331~349
[113]Tung D. E., Anwar G., Tomizuka M., Low Velocity Friction Compensation and Feedforward Solution Based on Repetitive Control, Journal of Dynamic Systems,Measurement, 1993, l15(7):279~284
[114]陈永生,重复控制方法在伺服系统中的应用研究:[硕士学位论文],北京;北京航空航天大学,1999
[115]贾亮,具有重复控制的高精度系统跟踪控制系统的研究和应用:[硕士学位论文],北京航空航天大学,1997
[116]丁汉,吴建华,王英,高加速度系统的快速高精度定位控制,自然科学进展,2008,18(10):1143~1150
[117]Teeter T. J., Chow M., Briddey J. J., A Novel Fuzzy Friction Compensation Approach to Improve the Performance of a DC Motor Control System, IEEE Tractions on Industrial Electronics, 1996, 43(1):1 13~120
[118] Popovie R. M., Gorinevsky D. D., High-precision of a Mechanism with Nonlinesr Friction Using Fuzzy Logic Pulse Controller.IEEE Transaction on Control Systems Technology, 2000, 8(1):151~158
[119]丛爽,机电系统中模糊与模糊神经元网络控制策略的研究,中国电机工程学报,1999,19(7):30～32
[120]Krim J., Friction at the atomic scale, Scientific Amer, 1996, 275(4): 74~80
[121]Kstic M., Kanellakopoulos I., Kokotovic P., Nonlmear and Adaptive Control Design, Wiley, 1995
[122]Deng H., Krstic M., Stochastic nonlinear stabilization-Part I: A backstepping design, Systems and Control Letters, 1997, 1(32):143-150
[123]Ikhouane F., Krstic M., Adaptive backstepping with parameter projection: robustness and asymptotic performance, Automatica, 1998, 34:429~435
[124]Liu W., Krstic M., Backstepping boundary control of Burgers’equation with actuator dynamics, Systems&Control Letters, 2000, 41:291~303
[125] Boskovic D., Krstic M., Backstepping control of chemical tubular reactors,Computers and Chemical Engineering, 2002, 26:1077~1085
[126]Boskovic D., Balogh A., Krstic M., Backstepping in infinite dimension for a classof parabolic distributed parameter systems, Mathematics of Control, Signals, and Systems, 2003, 16:44~75
[127]Balogh A., Krstic M., Stability of partial difference equations governing controlgains in infinite dimensional backstepping, Systems and Control Letters, 2004, 51:151~164
[128]王莉,Backstepping设计方法及应用,自动化博览,2004,6:57~61
[129]胡寿松,自动控制原理(第四版),北京:科学出版社,2001.501~510
[130]王家军,交流伺服系统的反推式控制策略研究:[博士后研究工作报告],浙江;浙江大学,2005
[131]牛志刚,张建民,基于Turbo PMAC的数控系统自定义伺服算法的嵌入和实现,新技术新工艺,2005,7:11~13
[132]王清华,韩秋实,孙志永等,基于Turbo PMAC数控系统的PID在线调节,微计算机信息,2007,23(6):226~229
[133]刘宇,刘杰,戴丽等,基于神经网络和PID算法的数控机床并行混合控制模型,信息与控制,2006,35(1):30~42
[134]解本铭鞠红超,基于PMAC打磨机进给伺服系统的研究,制造技术与机床,2009,4:42~44
[135]Σ-Ⅱ系列SGM□H/SGDM用户手册,安川电机株式会社,2004
[136]吴大正,信号与线性系统分析,北京:高等教育出版社,2002.120~122
[137]Depeng L., Genetic Algorithms Based Parameter Identification for Nonlinear Mechanical Servo Systems, 2006 1st IEEE Conference on Industrial Electronics and Applications, 2006
[138]Erkorkmaz K., High Bandwidth Control of Ball screw Drives, Manufacturing Technology, 2006, 55(1):393~398
[139]Courtney-Pratt J., Eisner E., The Effect of a Tangential Force on the Contact of Metallic Bodies, Proc. Royal Society, 1957, A238:529~550
[140]PMAC用户手册,北京元茂兴控制设备技术有限责任公司,2002