基于方向场理论的X-Y平台速度规划与轮廓控制研究
|
摘要
本文以国家自然科学基金资助项目“基于速度场的直接驱动XY平台任意轨迹规划与精密轮廓控制”(51175349)为背景，以永磁直线同步电机直接驱动X-Y运动平台伺服系统为研究对象，针对永磁直线同步电动机直接驱动的特点及其对伺服系统鲁棒性及轮廓性能的双重要求，提出一种基于方向场理论的X-Y平台速度规划与轮廓控制方法。
首先，采用几何法构建了基于隐函式的圆形轨迹的速度场。由于圆形轨迹能够以隐函式表示，因此，采用几何法对平面内任意一点到期望轨迹的最短距离向量与切线向量分别进行计算，构建期望轨迹的速度场，为避免传统轮廓-跟踪控制方法对于时间的限制问题，消除“轨径缩减”的现象提供理论依据。分别对速度场的构建、轮廓误差的计算进行仿真分析。
然后，针对NURBS参数曲线的任意性及其速度场难于构建的特殊性，提出采用方向场理论对直接驱动X-Y平台任意轨迹的初始速度进行间接规划。当初始轨迹为任意自由轨迹时，最短距离向量难以准确计算，因此，根据方向场理论，以划分网格的方式间接、离线规划期望轨迹的速度，将双轴协调控制问题转化为单轴速度控制问题。并且针对以方向场理论进行速度规划方法的正确性及有效性进行分析。在速度规划的框架之上，分别在理想控制条件下及扰动存在的两种情况下，对X-Y平台伺服系统进行传统PI控制仿真研究。
最后，针对由方向场理论进行速度规划方法的特殊性及X-Y平台伺服系统易受非线性、周期性扰动及参数摄动等不确定性因素的影响，提出一种基于Lyapunov函数的改进PI扰动补偿自适应轮廓控制方法。根据Lyapunov稳定性理论，以各单轴速度误差为控制目标，设计基于Lyapunov函数的速度控制器，直接提高系统的轮廓精度。针对系统存在的负载扰动及参数摄动，采用改进PI扰动补偿自适应控制方法，实时调整扰动补偿器的控制增益，实现扰动补偿，提高系统的鲁棒性。并分别针对上述轮廓控制方法进行仿真研究。
本文搭建了永磁直线电机直接驱动X-Y平台轮廓控制系统，在以方向场理论进行速度规划的基础之上，分别对传统PI控制，基于Lyapunov函数的速度控制方法(LVFC)，基于Lyapunov函数的传统PI扰动补偿方法(LVFC/PDCC)及基于Lyapunov函数的改进PI扰动补偿自适应控制方法(LVFC/IPDCAC)，在速度闭环情况下进行对比实验研究，验证了本文所提出的速度规划方法与控制策略的理论研究的有效性。
The paper is supported by National Natural Science Foundation of China “TrajectoryPlanning and Accurate Contour Control based Velocity field for Direct Drive X-Y Table”(NO.51175349). In this paper, permanent magnet linear synchronous motor (PMLSM)direct drive X-Y motion table servo system is the main study object. For the characteristicof direct drive system and requirements of robustness and contour performance for theX-Y table servo system, a method that combines velocity planning and contour controlmethod based on direction field theory was proposed.
Firstly, a geometric method was used to construct the velocity field for the circlecontour. For constructing the velocity field, calculate the distance vector and tangent vectorrespectively. To enhance the contour performance, avoid time restraint problem for thetraditional contour-tracking contour control method and eliminate the “radial reduction”phenomenon, a velocity planning method was adopted to transform contour-trackingbiaxial coordination control into velocity control of the single axis. Velocity fieldconstruction and contour error calculation was studied by simulation.
Secondly, in order to extend velocity field to contours represented in NURBS form,the direction field method was adopted for the direct drive X-Y table velocity planning.For the line or circle contour following tasks, it is easy to find the corresponding distancevector, whereas it is not the case for the free form contour following task. For planningdesired velocity of the system and transforming biaxial coordination control into thevelocity control of single axis, a novel meshing method based on direction field was usedto construct NURBS curve velocity field. Considering the ideal control condition and theexistence of load disturbances, the traditional PI control for X-Y table contour controlsystem was studied by simulation.
Last, for the influences of nonlinear, periodic disturbances and parameter changes, animproved PI disturbance compensation adaptive contour control method based onLyapunov function the direction field was proposed. Firstly, based on Lyapunov stability theorem, a velocity controller was designed based on Lyapunov function to reduce thevelocity error of the single axis. Secondly, in order to further enhance the robustperformance and improve the contour performance, an improved PI disturbancecompensation adaptive controller was used to compensation the disturbances force causedby the uncertainties of parameter changes and load disturbances of system. The abovecontrol methods were studied by simulation.
Contour control system for direct drive X-Y experimental table was built. Based onthe velocity planning principle, traditional PI control method based on direction field, aLyapunov function velocity field control (LVFC), PI disturbance compensation controlbased on Lyapunov function velocity field control (LVFC/PDCC) and Improved PIdisturbance compensation adaptive control based on Lyapunov function velocity fieldcontrol (LVFC/IPDCAC) were compared in the experiments under the circumstances ofthe loop. The comparative experimental results verified the validity of velocity planningmethod and control strategies proposed in theoretical analysis and simulation.
首先，采用几何法构建了基于隐函式的圆形轨迹的速度场。由于圆形轨迹能够以隐函式表示，因此，采用几何法对平面内任意一点到期望轨迹的最短距离向量与切线向量分别进行计算，构建期望轨迹的速度场，为避免传统轮廓-跟踪控制方法对于时间的限制问题，消除“轨径缩减”的现象提供理论依据。分别对速度场的构建、轮廓误差的计算进行仿真分析。
然后，针对NURBS参数曲线的任意性及其速度场难于构建的特殊性，提出采用方向场理论对直接驱动X-Y平台任意轨迹的初始速度进行间接规划。当初始轨迹为任意自由轨迹时，最短距离向量难以准确计算，因此，根据方向场理论，以划分网格的方式间接、离线规划期望轨迹的速度，将双轴协调控制问题转化为单轴速度控制问题。并且针对以方向场理论进行速度规划方法的正确性及有效性进行分析。在速度规划的框架之上，分别在理想控制条件下及扰动存在的两种情况下，对X-Y平台伺服系统进行传统PI控制仿真研究。
最后，针对由方向场理论进行速度规划方法的特殊性及X-Y平台伺服系统易受非线性、周期性扰动及参数摄动等不确定性因素的影响，提出一种基于Lyapunov函数的改进PI扰动补偿自适应轮廓控制方法。根据Lyapunov稳定性理论，以各单轴速度误差为控制目标，设计基于Lyapunov函数的速度控制器，直接提高系统的轮廓精度。针对系统存在的负载扰动及参数摄动，采用改进PI扰动补偿自适应控制方法，实时调整扰动补偿器的控制增益，实现扰动补偿，提高系统的鲁棒性。并分别针对上述轮廓控制方法进行仿真研究。
本文搭建了永磁直线电机直接驱动X-Y平台轮廓控制系统，在以方向场理论进行速度规划的基础之上，分别对传统PI控制，基于Lyapunov函数的速度控制方法(LVFC)，基于Lyapunov函数的传统PI扰动补偿方法(LVFC/PDCC)及基于Lyapunov函数的改进PI扰动补偿自适应控制方法(LVFC/IPDCAC)，在速度闭环情况下进行对比实验研究，验证了本文所提出的速度规划方法与控制策略的理论研究的有效性。
The paper is supported by National Natural Science Foundation of China “TrajectoryPlanning and Accurate Contour Control based Velocity field for Direct Drive X-Y Table”(NO.51175349). In this paper, permanent magnet linear synchronous motor (PMLSM)direct drive X-Y motion table servo system is the main study object. For the characteristicof direct drive system and requirements of robustness and contour performance for theX-Y table servo system, a method that combines velocity planning and contour controlmethod based on direction field theory was proposed.
Firstly, a geometric method was used to construct the velocity field for the circlecontour. For constructing the velocity field, calculate the distance vector and tangent vectorrespectively. To enhance the contour performance, avoid time restraint problem for thetraditional contour-tracking contour control method and eliminate the “radial reduction”phenomenon, a velocity planning method was adopted to transform contour-trackingbiaxial coordination control into velocity control of the single axis. Velocity fieldconstruction and contour error calculation was studied by simulation.
Secondly, in order to extend velocity field to contours represented in NURBS form,the direction field method was adopted for the direct drive X-Y table velocity planning.For the line or circle contour following tasks, it is easy to find the corresponding distancevector, whereas it is not the case for the free form contour following task. For planningdesired velocity of the system and transforming biaxial coordination control into thevelocity control of single axis, a novel meshing method based on direction field was usedto construct NURBS curve velocity field. Considering the ideal control condition and theexistence of load disturbances, the traditional PI control for X-Y table contour controlsystem was studied by simulation.
Last, for the influences of nonlinear, periodic disturbances and parameter changes, animproved PI disturbance compensation adaptive contour control method based onLyapunov function the direction field was proposed. Firstly, based on Lyapunov stability theorem, a velocity controller was designed based on Lyapunov function to reduce thevelocity error of the single axis. Secondly, in order to further enhance the robustperformance and improve the contour performance, an improved PI disturbancecompensation adaptive controller was used to compensation the disturbances force causedby the uncertainties of parameter changes and load disturbances of system. The abovecontrol methods were studied by simulation.
Contour control system for direct drive X-Y experimental table was built. Based onthe velocity planning principle, traditional PI control method based on direction field, aLyapunov function velocity field control (LVFC), PI disturbance compensation controlbased on Lyapunov function velocity field control (LVFC/PDCC) and Improved PIdisturbance compensation adaptive control based on Lyapunov function velocity fieldcontrol (LVFC/IPDCAC) were compared in the experiments under the circumstances ofthe loop. The comparative experimental results verified the validity of velocity planningmethod and control strategies proposed in theoretical analysis and simulation.
引文
[1]王先逵，陈定积，吴丹.机床进给系统用直线电动机综述.制造技术与机床，2001(8)：18-20.
[2]张威，刘春艳.先进制造与再制造技术的结合与发展.科学传播，2012(5)：18-20.
[3]胡楚雄.基于全局任务坐标系的精密轮廓控制研究(博士学位论文).浙江大学，2010.
[4] Wang G J, Lee T J. Neural network cross-coupled control system with application on circulartracking of linear motor X-Y table. IEEE Trans. On Automatic Control,2007,99(6):2194-2199.
[5]武志涛.直接驱动X-Y数控平台轮廓跟踪控制策略研究(博士学位论文).沈阳工业大学，2012.
[6]明强，张伟.基于X-Y数控平台的边条控制系统正弦运动研究.航空制造技术，2010(13):19-22.
[7]赵希梅，郭庆鼎，陈冬兰.通过直线伺服鲁棒跟踪控制方法提高轮廓加工精度.机械工程学报，2006，42(6)：166-169.
[8]王丽梅，武志涛，孙宜标.直接驱动XY平台轮廓误差分析及法向交叉耦合控制.电机与控制学报，2010,14(9)：63-68.
[9] Chen C L, Lin K C. Observer-based contouring controller design of a biaxial stage system subject tofriction. IEEE Trans. on Control System Technology,2008,16(2):322-329.
[10]孙宜标，闫峰，刘春芳.基于μ理论的永磁直线同步电机鲁棒重复控制.中国电机工程学报，2009,29(30):52-57.
[11]王丽梅，金抚颖，孙宜标.基于等效误差法的直线电机XY平台二阶滑模控制.中国电机工程学报，2009，30(6):88-92.
[12]王良勇，柴天佑.笛卡尔空间机械手间接轮廓控制.系统仿真学报，2008，20(22)：6218-6221.
[13] Li X, Yao B. Adaptive Robust Precision Motion Control of Linear Motors Negligible ElectricalDynamics. Theory and Experiments Proceedings of the American Control Conference,2000:2583-2587.
[14] Chen S L, Wu K C. Contouring control of smooth paths for multiaxis motion systems based onequivalent error. IEEE Trans on Control system technology,2007,15(6):1151-1158.
[15]孙建仁，胡赤兵，王宝民.CNC多轴运动控制系统轮廓误差分析.兰州理工大学学报，2010,36(4)：37-40.
[16]丛爽，刘宜.多轴协调运动中的交叉耦合控制.机械设计与制造，2006,10:166-168.
[17]赵希梅，郭庆鼎.提高轮廓加工精度的零相位自适应鲁棒交叉耦合控制.电工技术学报，2007，22(12)：170-174.
[18] Su K H, Cheng M Y. Contouring accuracy improvement using cross-coupled control and positionerror compensator. International Journal of Machine Tools&Manufacture,2008,48:1444-1453.
[19] Cheng M Y, Su K H, Wang S F. Contour error reduction for free-form contour following tasks ofbiaxial motion control systems. Robotics and computer Integrated Manufacturing,2009,25:323-333.
[20] Yang Jiangzhao, Li Zexiang. A novel contour error estimation for position loop-basedcross-coupled control. IEEE Trans on Mechatronics,2010,16(4):643-655.
[21] Su K H, Cheng M Y. Contouring accuracy improvement using a Tangential contouring with afuzzy logic based federate regulator. International Journal of Advanced ManufacturingTechnology,2009,41(2):75-85.
[22]孙宜标，宋杨，郭庆鼎.XY平台直线伺服系统的IP变增益交叉耦合控制.组合机床与自动化加工技术，2007，(11)：40-43.
[23] Barton K L, Alleyne A G. A cross-coupled iterative learning control design for precision motioncontrol. IEEE Trans on Control systems Technology,2008,16(6):1218-1231.
[24]王丽梅，金抚颖，孙宜标，刘春芳.基于等效误差法的直线电机X-Y平台轮廓控制.组合机床与自动化加工技术，2009，(1)：56-59.
[25]曲永印，赵希梅，郭庆鼎.基于零相位误差跟踪控制器的轮廓误差交叉耦合控制.中国机械工程，2006,17(11):1135-1137.
[26]孙兴伟，董蔚，王可.专用数控机床上全闭环控制方法的研究.机床与液压，2010，(2)：20-23.
[27]叶云岳.直线电机原理与应用.北京：机械工业出版社，2000.
[28]裴红星，刘建华，杨雷，刘晓.激光微加工系统中运动控制的实现.组合机床与自动化加工技术，2004，(3)：53-57.
[29]唐振宇，李锻能.高速加工与直线电机在数控机床的应用.机电工程技术，2005，34(11)：72-69.
[30]叶云岳，陆凯元．直线电机的PID控制与模糊控制．电工技术学报，2001，16(3)：12-15．
[31]蓝益鹏，孙文义，王雷．数控机床直线伺服系统非线性自适应鲁棒控制的研究．制造技术与机床，2009，(2)：146-149．
[32]张春良，陈子辰等.直线驱动新技术及其在加工装备上的应用.电工技术学报，2002，17(5)：45-49.
[33]林春.永磁同步直线电机伺服控制系统研究(博士学位论文).浙江大学，2005.
[34]李义强，周惠兴.精密伺服用无铁心永磁同步直线电动机研究综述.微电机，2008，41(5)：71-76.
[35]黄东兆周会成李斌唐小琦.面向轮廓精度控制的误差补偿方法.华中科技大学学报.36(2):13-16.
[36] Tomizuka M. Zero phase error tracking algorithm for digital control. ASME Journal of DynamicSystems,1987,109(1):65-68.
[37] Park H S, Chang P H, Lee D Y. continuous zero phase error tracking controller with gain errorcompensation. American Control Conference,1999,5:3554-3558.
[38]宋宝，周云飞.多轴运动轴协同控制的研究.机床与液压，2004，(10):141-143.
[39]徐月同.高速精密永磁直线同步电机进给系统及控制技术研究：(博士学位论文).浙江：浙江大学，2004.
[40]宋亦旭，王春洪，尹文生，贾培发.永磁直线同步电动机的自适应学习控制.中国电机工程学报，2005，25(20)：151-156.
[41]王丽梅，武志涛，刘春芳.永磁直线伺服系统最优参数负载扰动补偿方法.电工技术学报，2012,27(3):133-138.
[42] Ming-Yang Cheng, Cheng-Chien Lee. Motion controller design for contour-following tasks basedon real-time contour error estimation. IEEE Trans. on Industrial Electronics,2007,54(3):1686-1695.
[43]孙宜标，杨雪，夏加宽．采用鲁棒微分器的永磁直线同步电机二阶滑模控制．中国电机工程学报，2007，27(33)：6-10．
[44] George T C Chin, Tomizuka M. Contouring control of machine toot feed drive system: a taskcoordinate frame approach. IEEE Trans on Control System Technology,2001,9(1):130-139.
[45]王丽梅，陈吉超，郑浩.直接驱动X-Y平台全局快速Terminal滑模控制.沈阳工业大学学报，2012，34(1)：1-5.
[46]丛爽,刘宜.多轴协调运动中的交叉耦合控制[J].机械设计与制造，2006，(10)：166-168.
[47]李宏胜.数控机床进给运动的零相位跟踪鲁棒控制.制造业自动化，2005，27(4)：22-26.
[48]王波，梁迎春，董申．超精密机床模糊轮廓控制技术研究．哈尔滨工业大学学报，2005，37(6):830-832．
[49]陈静，刘强，齐畅．基于自适应和模糊控制的新型XY平台同步控制研究．系统仿真学报，2008，20(12):3212-3215．
[50]赵希梅,郭庆鼎.数控机床多轴联动伺服电机的零相位自适应鲁棒交叉祸合控制．中国电机工程学报，2005，25(12)：129-133．
[51]朱年军，王文，季国顺.数控伺服系统跟踪及轮廓误差分析.机床与液压，2006，(10):20-23.
[52]肖本贤，郭福权，邓红辉，王群京.基于干扰观测器的轮廓误差耦合控制研究.系统仿真学报，2004,16(4)：787-793.
[54]孟祥忠,刘作宗. QFT在高速线性电机直接驱动平面运动定位控制系统中的应用.控制理论与应用，2006,23(2)：225-228.
[55]李静，何永义，方明伦，沈南燕等.曲轴非圆磨削中基于差分进化算法的变参数交叉耦合轮廓控制.机械工程学报，2011，47(9)：139-145.
[56]王广炎.数控机床误差控制理论及开放化数控系统的研究(博士学位论文).西安：西安交通大学，2000.
[57] Li P Y, Horowitz R. Passive velocity field control of mechanical manipulators. IEEE Trans. onRobotics and Automation,1999,15(4)：751-763.
[58] Yamakita M, Asano F, Furuta K. Passive velocity field control of biped walking robot. Proceedingsof IEEE Conference on Robotics and Automation,2000:3057-3062.
[59] Saitoh Y, Luo Z, Watanalbe K. Environmental adaptive PVFC for a robot manipulator.Proceedings of the41th SICE Annual Conference,2002:228-229.
[60] Cervantes I, Kelly R, Moreno J, et al. A robust velocity field control. IEEE Trans. on ControlSystems Technology,2002,10(6):888-894.
[61] Moreno-Valenzuela J. On passive velocity field control of robot arms. Proceeding of the45th IEEEConference on Decision and Control,2006:2955-2960.
[62] Lee D, Li P Y. Passive bilateral control and tool dynamics redering for nonlinear mechanicalteleoperators. IEEE Trans. on Robotics and Automation,2003,21(5):936-951.
[63] Yamakita M, Asano F, Furuta K. Passive velocity field control of biped walking robot. Proceedingof IEEE Conference on Robotics and Automation,2005:3057-3062.
[64] Kim T, Sarma S E. optimal sweeping paths on a2-manifold a new class of optimization problemsdefined by path structures. IEEE Trans on Robotics and Automation,19(4):613-636.
[65] Aguiar A P, Hespanha J P. Trajectory-tracking and path-following of underactuated autonomousvehicles with parametric modeling uncertainty. IEEE Trans on Automation Control,2007,52(8):1686-1695.
[66] Moreno J,Kelly R. Herarchical velocity field control for robot manipulators. IEEE IternationalConference on Rbotics and Atomation,2003:4374-4379.
[67] Chen Chaoyun, Cheng Mingyang, Wang Yinghui. Velocity field control for free-form contourfollowing tasks by a two link robot. International Workshop on Robotic and Snsors Evironments,2008:64-69.
[68] Li P Y. Adaptive passive velocity field control.Proceedings of American Control Conference,1999:774-779.
[69] Changsu Ha, Choi F B, Dongjun Lee. Preliminary results on passive velocity field control ofquadrotors. International Conference on Ubiquitous Robots and Ambient Intelligence,2012:378-379.
[70] Erdogan A, Satici A C, Patoglu V. Passive velocity field control of a forearm-wrist rehabilitationrobot. IEEE International Conference on Rehabilitation Robotics,2011:1-8.
[71] Mcintyre M L, Dixon W E, Dawson D M, er al. Adaptive tracking control of on-line pathplanners:velocity fields and navigation functions. Proceedings of American Control Conference,2005:3168-3173.
[72]田艳丰.永磁直线同步电动机鲁棒控制策略研究(博士学位论文).沈阳：沈阳工业大学，2010.
[73]郭庆鼎，孙宜标，王丽梅.现代永磁电动机交流伺服系统.中国电力出版社，2006.
[74]叶云岳.新型直线驱动装置与系统.北京：冶金工业出版社，2000.
[75]王秀和.永磁电机.中国电力出版社，2007.
[76] Zhao Ximei, Guo Qingding, Sun Yibiao. Zero phase two-degrees of freedom H∞robust trackingcontrol for permanent magnet linear synchronous motor. Proceedings of IEEE InternationalConference on Mechatronics,2005:230-234.
[77]赵镜红，张俊洪.永磁直线同步电机扰动观测器仿真研究.武汉理工大学学报，2007，31(2)：224-227.
[78] Jee Sungchul and Yoram Koren. Adaptive fuzzy logic controller for feed drives of a CNC machinetool.Mechatronics,2004,14:299-326.
[79] Jiangzhao Yang, Dongjun Zhang, Zexiang Li. Position loop-based cross-coupled control forhigh-speed machining. Proceedings of the7th World Congress on Intelligent Control andAutomation,2008:4285-4290.
[80] Hu C,Yao B,Wang Q.Coordinated adaptive robust contouring controller design for an industrialbiaxial precision gantry. IEEE Trans on Mechatronics,2010,15(5):728-735.
[81]张春良，陈子辰等.直线驱动新技术及其在加工装备上的应用.电工技术学报，2002，17(5)：45-49.
[82]林春.永磁同步直线电机伺服控制系统研究(博士学位论文).杭州：浙江大学，2005.
[83]李义强，周惠兴.精密伺服用无铁心永磁同步直线电动机研究综述.微电机，2008，41(5)：71-76.
[84]解旭辉.超精密机床数控系统与伺服控制技术研究(博士学位论文).长沙：国防科技大学，1998.
[85]郭庆鼎，王成元，周美文等.直线交流伺服系统的精密控制技术.机械工业出版社，2000.
[86]夏加宽，王成元，李翱东等.高精度数控机床用直线电动机端部效应分析及神经网络补偿技术研究.中国电机工程学报.2003，23(8):100-104.
[87] Nagle R K, Saff E B. Fundamentals of differential equations and boundary value problems.Addison-Wesley,1993.
[88]施法中.计算机辅助几何设计与非均匀有理B样条.北京航空航天大学出版社,1994.
[89]王兴波,李圣怡.平面NURBS曲线的椭圆弧自适应逼近.国防科技大学学报,2000,22(4)：24-26.
[90]孙海洋.NURBS曲线刀具路径实时插补技术研究(博士学位论文).北京：国防科学技术大学，2008.
[91]赵罡，穆国正，王拉柱.非均匀有理B样条.第二版.北京：清华大学出版社，2010.
[92] Isacson E, Keller H B. Analysis of numberical methods, Wiley,1996.
[93]丁同仁，李承治.常微分方程教程.北京：高等教育出版社，1991.
[94]秦元勋.微分方程所定义的积分曲线.北京：科学出版社，1959.
[95]许淞庆.常微分方程稳定性理论.上海：上海科学技术出版社，1962.
[96] Demyanov V F, Malozemov V N. Introduction to minimax, John Wiley&Son,Inc,1994.
[97] Bartle R G. The Elements of real analysis, John Wiley&Son,1976.
[98]叶彦谦.极限环论.上海：上海科学技术出版社，1964.
[99]张国柱，陈杰，李志平.基于复合自适应律的直线电机自适应鲁棒控制.控制理论与应用，2009,26(8):833-837.
[100]富月，柴天佑.一类非线性最小相位系统的直接自适应控制.控制理论与应用，2006,23（6）：886-890.
[2]张威，刘春艳.先进制造与再制造技术的结合与发展.科学传播，2012(5)：18-20.
[3]胡楚雄.基于全局任务坐标系的精密轮廓控制研究(博士学位论文).浙江大学，2010.
[4] Wang G J, Lee T J. Neural network cross-coupled control system with application on circulartracking of linear motor X-Y table. IEEE Trans. On Automatic Control,2007,99(6):2194-2199.
[5]武志涛.直接驱动X-Y数控平台轮廓跟踪控制策略研究(博士学位论文).沈阳工业大学，2012.
[6]明强，张伟.基于X-Y数控平台的边条控制系统正弦运动研究.航空制造技术，2010(13):19-22.
[7]赵希梅，郭庆鼎，陈冬兰.通过直线伺服鲁棒跟踪控制方法提高轮廓加工精度.机械工程学报，2006，42(6)：166-169.
[8]王丽梅，武志涛，孙宜标.直接驱动XY平台轮廓误差分析及法向交叉耦合控制.电机与控制学报，2010,14(9)：63-68.
[9] Chen C L, Lin K C. Observer-based contouring controller design of a biaxial stage system subject tofriction. IEEE Trans. on Control System Technology,2008,16(2):322-329.
[10]孙宜标，闫峰，刘春芳.基于μ理论的永磁直线同步电机鲁棒重复控制.中国电机工程学报，2009,29(30):52-57.
[11]王丽梅，金抚颖，孙宜标.基于等效误差法的直线电机XY平台二阶滑模控制.中国电机工程学报，2009，30(6):88-92.
[12]王良勇，柴天佑.笛卡尔空间机械手间接轮廓控制.系统仿真学报，2008，20(22)：6218-6221.
[13] Li X, Yao B. Adaptive Robust Precision Motion Control of Linear Motors Negligible ElectricalDynamics. Theory and Experiments Proceedings of the American Control Conference,2000:2583-2587.
[14] Chen S L, Wu K C. Contouring control of smooth paths for multiaxis motion systems based onequivalent error. IEEE Trans on Control system technology,2007,15(6):1151-1158.
[15]孙建仁，胡赤兵，王宝民.CNC多轴运动控制系统轮廓误差分析.兰州理工大学学报，2010,36(4)：37-40.
[16]丛爽，刘宜.多轴协调运动中的交叉耦合控制.机械设计与制造，2006,10:166-168.
[17]赵希梅，郭庆鼎.提高轮廓加工精度的零相位自适应鲁棒交叉耦合控制.电工技术学报，2007，22(12)：170-174.
[18] Su K H, Cheng M Y. Contouring accuracy improvement using cross-coupled control and positionerror compensator. International Journal of Machine Tools&Manufacture,2008,48:1444-1453.
[19] Cheng M Y, Su K H, Wang S F. Contour error reduction for free-form contour following tasks ofbiaxial motion control systems. Robotics and computer Integrated Manufacturing,2009,25:323-333.
[20] Yang Jiangzhao, Li Zexiang. A novel contour error estimation for position loop-basedcross-coupled control. IEEE Trans on Mechatronics,2010,16(4):643-655.
[21] Su K H, Cheng M Y. Contouring accuracy improvement using a Tangential contouring with afuzzy logic based federate regulator. International Journal of Advanced ManufacturingTechnology,2009,41(2):75-85.
[22]孙宜标，宋杨，郭庆鼎.XY平台直线伺服系统的IP变增益交叉耦合控制.组合机床与自动化加工技术，2007，(11)：40-43.
[23] Barton K L, Alleyne A G. A cross-coupled iterative learning control design for precision motioncontrol. IEEE Trans on Control systems Technology,2008,16(6):1218-1231.
[24]王丽梅，金抚颖，孙宜标，刘春芳.基于等效误差法的直线电机X-Y平台轮廓控制.组合机床与自动化加工技术，2009，(1)：56-59.
[25]曲永印，赵希梅，郭庆鼎.基于零相位误差跟踪控制器的轮廓误差交叉耦合控制.中国机械工程，2006,17(11):1135-1137.
[26]孙兴伟，董蔚，王可.专用数控机床上全闭环控制方法的研究.机床与液压，2010，(2)：20-23.
[27]叶云岳.直线电机原理与应用.北京：机械工业出版社，2000.
[28]裴红星，刘建华，杨雷，刘晓.激光微加工系统中运动控制的实现.组合机床与自动化加工技术，2004，(3)：53-57.
[29]唐振宇，李锻能.高速加工与直线电机在数控机床的应用.机电工程技术，2005，34(11)：72-69.
[30]叶云岳，陆凯元．直线电机的PID控制与模糊控制．电工技术学报，2001，16(3)：12-15．
[31]蓝益鹏，孙文义，王雷．数控机床直线伺服系统非线性自适应鲁棒控制的研究．制造技术与机床，2009，(2)：146-149．
[32]张春良，陈子辰等.直线驱动新技术及其在加工装备上的应用.电工技术学报，2002，17(5)：45-49.
[33]林春.永磁同步直线电机伺服控制系统研究(博士学位论文).浙江大学，2005.
[34]李义强，周惠兴.精密伺服用无铁心永磁同步直线电动机研究综述.微电机，2008，41(5)：71-76.
[35]黄东兆周会成李斌唐小琦.面向轮廓精度控制的误差补偿方法.华中科技大学学报.36(2):13-16.
[36] Tomizuka M. Zero phase error tracking algorithm for digital control. ASME Journal of DynamicSystems,1987,109(1):65-68.
[37] Park H S, Chang P H, Lee D Y. continuous zero phase error tracking controller with gain errorcompensation. American Control Conference,1999,5:3554-3558.
[38]宋宝，周云飞.多轴运动轴协同控制的研究.机床与液压，2004，(10):141-143.
[39]徐月同.高速精密永磁直线同步电机进给系统及控制技术研究：(博士学位论文).浙江：浙江大学，2004.
[40]宋亦旭，王春洪，尹文生，贾培发.永磁直线同步电动机的自适应学习控制.中国电机工程学报，2005，25(20)：151-156.
[41]王丽梅，武志涛，刘春芳.永磁直线伺服系统最优参数负载扰动补偿方法.电工技术学报，2012,27(3):133-138.
[42] Ming-Yang Cheng, Cheng-Chien Lee. Motion controller design for contour-following tasks basedon real-time contour error estimation. IEEE Trans. on Industrial Electronics,2007,54(3):1686-1695.
[43]孙宜标，杨雪，夏加宽．采用鲁棒微分器的永磁直线同步电机二阶滑模控制．中国电机工程学报，2007，27(33)：6-10．
[44] George T C Chin, Tomizuka M. Contouring control of machine toot feed drive system: a taskcoordinate frame approach. IEEE Trans on Control System Technology,2001,9(1):130-139.
[45]王丽梅，陈吉超，郑浩.直接驱动X-Y平台全局快速Terminal滑模控制.沈阳工业大学学报，2012，34(1)：1-5.
[46]丛爽,刘宜.多轴协调运动中的交叉耦合控制[J].机械设计与制造，2006，(10)：166-168.
[47]李宏胜.数控机床进给运动的零相位跟踪鲁棒控制.制造业自动化，2005，27(4)：22-26.
[48]王波，梁迎春，董申．超精密机床模糊轮廓控制技术研究．哈尔滨工业大学学报，2005，37(6):830-832．
[49]陈静，刘强，齐畅．基于自适应和模糊控制的新型XY平台同步控制研究．系统仿真学报，2008，20(12):3212-3215．
[50]赵希梅,郭庆鼎.数控机床多轴联动伺服电机的零相位自适应鲁棒交叉祸合控制．中国电机工程学报，2005，25(12)：129-133．
[51]朱年军，王文，季国顺.数控伺服系统跟踪及轮廓误差分析.机床与液压，2006，(10):20-23.
[52]肖本贤，郭福权，邓红辉，王群京.基于干扰观测器的轮廓误差耦合控制研究.系统仿真学报，2004,16(4)：787-793.
[54]孟祥忠,刘作宗. QFT在高速线性电机直接驱动平面运动定位控制系统中的应用.控制理论与应用，2006,23(2)：225-228.
[55]李静，何永义，方明伦，沈南燕等.曲轴非圆磨削中基于差分进化算法的变参数交叉耦合轮廓控制.机械工程学报，2011，47(9)：139-145.
[56]王广炎.数控机床误差控制理论及开放化数控系统的研究(博士学位论文).西安：西安交通大学，2000.
[57] Li P Y, Horowitz R. Passive velocity field control of mechanical manipulators. IEEE Trans. onRobotics and Automation,1999,15(4)：751-763.
[58] Yamakita M, Asano F, Furuta K. Passive velocity field control of biped walking robot. Proceedingsof IEEE Conference on Robotics and Automation,2000:3057-3062.
[59] Saitoh Y, Luo Z, Watanalbe K. Environmental adaptive PVFC for a robot manipulator.Proceedings of the41th SICE Annual Conference,2002:228-229.
[60] Cervantes I, Kelly R, Moreno J, et al. A robust velocity field control. IEEE Trans. on ControlSystems Technology,2002,10(6):888-894.
[61] Moreno-Valenzuela J. On passive velocity field control of robot arms. Proceeding of the45th IEEEConference on Decision and Control,2006:2955-2960.
[62] Lee D, Li P Y. Passive bilateral control and tool dynamics redering for nonlinear mechanicalteleoperators. IEEE Trans. on Robotics and Automation,2003,21(5):936-951.
[63] Yamakita M, Asano F, Furuta K. Passive velocity field control of biped walking robot. Proceedingof IEEE Conference on Robotics and Automation,2005:3057-3062.
[64] Kim T, Sarma S E. optimal sweeping paths on a2-manifold a new class of optimization problemsdefined by path structures. IEEE Trans on Robotics and Automation,19(4):613-636.
[65] Aguiar A P, Hespanha J P. Trajectory-tracking and path-following of underactuated autonomousvehicles with parametric modeling uncertainty. IEEE Trans on Automation Control,2007,52(8):1686-1695.
[66] Moreno J,Kelly R. Herarchical velocity field control for robot manipulators. IEEE IternationalConference on Rbotics and Atomation,2003:4374-4379.
[67] Chen Chaoyun, Cheng Mingyang, Wang Yinghui. Velocity field control for free-form contourfollowing tasks by a two link robot. International Workshop on Robotic and Snsors Evironments,2008:64-69.
[68] Li P Y. Adaptive passive velocity field control.Proceedings of American Control Conference,1999:774-779.
[69] Changsu Ha, Choi F B, Dongjun Lee. Preliminary results on passive velocity field control ofquadrotors. International Conference on Ubiquitous Robots and Ambient Intelligence,2012:378-379.
[70] Erdogan A, Satici A C, Patoglu V. Passive velocity field control of a forearm-wrist rehabilitationrobot. IEEE International Conference on Rehabilitation Robotics,2011:1-8.
[71] Mcintyre M L, Dixon W E, Dawson D M, er al. Adaptive tracking control of on-line pathplanners:velocity fields and navigation functions. Proceedings of American Control Conference,2005:3168-3173.
[72]田艳丰.永磁直线同步电动机鲁棒控制策略研究(博士学位论文).沈阳：沈阳工业大学，2010.
[73]郭庆鼎，孙宜标，王丽梅.现代永磁电动机交流伺服系统.中国电力出版社，2006.
[74]叶云岳.新型直线驱动装置与系统.北京：冶金工业出版社，2000.
[75]王秀和.永磁电机.中国电力出版社，2007.
[76] Zhao Ximei, Guo Qingding, Sun Yibiao. Zero phase two-degrees of freedom H∞robust trackingcontrol for permanent magnet linear synchronous motor. Proceedings of IEEE InternationalConference on Mechatronics,2005:230-234.
[77]赵镜红，张俊洪.永磁直线同步电机扰动观测器仿真研究.武汉理工大学学报，2007，31(2)：224-227.
[78] Jee Sungchul and Yoram Koren. Adaptive fuzzy logic controller for feed drives of a CNC machinetool.Mechatronics,2004,14:299-326.
[79] Jiangzhao Yang, Dongjun Zhang, Zexiang Li. Position loop-based cross-coupled control forhigh-speed machining. Proceedings of the7th World Congress on Intelligent Control andAutomation,2008:4285-4290.
[80] Hu C,Yao B,Wang Q.Coordinated adaptive robust contouring controller design for an industrialbiaxial precision gantry. IEEE Trans on Mechatronics,2010,15(5):728-735.
[81]张春良，陈子辰等.直线驱动新技术及其在加工装备上的应用.电工技术学报，2002，17(5)：45-49.
[82]林春.永磁同步直线电机伺服控制系统研究(博士学位论文).杭州：浙江大学，2005.
[83]李义强，周惠兴.精密伺服用无铁心永磁同步直线电动机研究综述.微电机，2008，41(5)：71-76.
[84]解旭辉.超精密机床数控系统与伺服控制技术研究(博士学位论文).长沙：国防科技大学，1998.
[85]郭庆鼎，王成元，周美文等.直线交流伺服系统的精密控制技术.机械工业出版社，2000.
[86]夏加宽，王成元，李翱东等.高精度数控机床用直线电动机端部效应分析及神经网络补偿技术研究.中国电机工程学报.2003，23(8):100-104.
[87] Nagle R K, Saff E B. Fundamentals of differential equations and boundary value problems.Addison-Wesley,1993.
[88]施法中.计算机辅助几何设计与非均匀有理B样条.北京航空航天大学出版社,1994.
[89]王兴波,李圣怡.平面NURBS曲线的椭圆弧自适应逼近.国防科技大学学报,2000,22(4)：24-26.
[90]孙海洋.NURBS曲线刀具路径实时插补技术研究(博士学位论文).北京：国防科学技术大学，2008.
[91]赵罡，穆国正，王拉柱.非均匀有理B样条.第二版.北京：清华大学出版社，2010.
[92] Isacson E, Keller H B. Analysis of numberical methods, Wiley,1996.
[93]丁同仁，李承治.常微分方程教程.北京：高等教育出版社，1991.
[94]秦元勋.微分方程所定义的积分曲线.北京：科学出版社，1959.
[95]许淞庆.常微分方程稳定性理论.上海：上海科学技术出版社，1962.
[96] Demyanov V F, Malozemov V N. Introduction to minimax, John Wiley&Son,Inc,1994.
[97] Bartle R G. The Elements of real analysis, John Wiley&Son,1976.
[98]叶彦谦.极限环论.上海：上海科学技术出版社，1964.
[99]张国柱，陈杰，李志平.基于复合自适应律的直线电机自适应鲁棒控制.控制理论与应用，2009,26(8):833-837.
[100]富月，柴天佑.一类非线性最小相位系统的直接自适应控制.控制理论与应用，2006,23（6）：886-890.