机械臂协调操作柔性负载系统建模与控制
|
摘要
机械臂已经被广泛应用于各种场合,到目前为止大多数研究对象都是刚性的,然而现实中有很多柔性负载,对它们的研究有很大意义。
本文针对机械臂协调操作柔性负载,首先给出了虚拟连杆的概念作为柔性体的基准,然后以这个基准把柔性梁分为4个单元。分析了由于坐标选择的原因,坐标原点两侧柔性坐标定义时出现的特殊情况,应用有限元法与Lagrange方程推导了柔性物体的刚性动力学方程与振动方程。为了使系统有一个统一坐标形式,给出了关节角坐标,末端执行器坐标与物体坐标之间的转换关系,在物体坐标建立了协调系统的刚性动力学方程与振动方程,分析了系统模型的一些特性。研究了James等人提出的刚性控制方法,在他们方法的基础上提出了针对本文控制对象的控制方法,并针对方法进行了仿真研究,结果证明了方法的有效性。然后利用力反馈作用设计了混合控制器,分析了力的有界性并通过仿真验证了这个结果。
Recently, robot manipulator has been used in many fields. There are many scholars who have done much work in manipulating rigid object, while in many fields, such as shipbuilding, car and air craft, flexible material has been used largely. So far, there are few scholars who have done such work. The system composed of robot manipulators and flexible payload has characteristics as following: exceeding nonlinearity, strong coupling and infinite degree of freedom. In the process of handling flexible payload two actions will happen that are rigid motion and vibration. Considering the current conditions domestic and abroad, the study of manipulating flexible payload is underway and the methods of control aren’t perfect. There are many questions needed to be solved.
In this paper the study of two manipulators handling a flexible payload has been done. The contributions of this paper are the modeling and the controller that is based on the other’s work. Firstly, the background and significance are introduced. The system of manipulator manipulating flexible payload is divided into two species single and cooperative and the conditions are introduced singly. The characteristics of current methods are analyzed.
In term of this paper, the structures that will appear in cooperative manipulation are analyzed and the structure we need in this paper is chose. Based on this structure the concept of virtual link is present to be the benchmark of flexible body. The object coordinate is located on this benchmark. The FEM is a new analytical method which depends on computer. The FEM can divide a complicated structure into finite elements and the structure is combined by the elements in the node. The displacement of the element is expressed by the interpolation function of node. The FEM has the same procedure from the choice of displacement to the modeling of dynamic equations, so it’s suited to be computed by computer. The physical meaning of flexible coordinates is clear because the generalized coordinates are the displacement of node not the mode. Because of these advantages we will use the FEM to model the dynamic equation of the flexible payload. The flexible payload is divided into four elements in the object coordinate. There is a special condition appearing when we define the flexible coordinates both sides of the object coordinate because of the choice of coordinates and the explanation is given. On the last basis the rigid dynamic equation and vibration equation are deduced. In this paper three kinds of coordinates will appear that are joint coordinates, end-effector coordinates and object coordinate. In order to have a uniform expression the relations of three coordinates are listed and the rigid dynamic equation and vibration equation of the system are derived in the object coordinate. The characteristics of the system are introduced.
So far, there are few scholars who have studied the whole system and some scholars have only studied certain problem because of the complexity of the system. The control of vibration is very difficult because the state vibration is unmeasurable. The method in this paper based on the‘rigid control’that it needs only the measurable coordinates in the controller which was brought out by James. The stability of the system is proved by the theory of stability and the results of the simulation validate the method of modeling and the controller. The hybrid controller is designed using the feedback signal of force. The boundary of the force is analyzed and simulation is done.
Some prepared work was done for two manipulators handling a flexible payload. Because it’s a very complex system many problems will appear in dealing with the system. This paper mostly solved the problem of model and designed a controller to validate the veracity but there are many works remained that needed to be solved.
本文针对机械臂协调操作柔性负载,首先给出了虚拟连杆的概念作为柔性体的基准,然后以这个基准把柔性梁分为4个单元。分析了由于坐标选择的原因,坐标原点两侧柔性坐标定义时出现的特殊情况,应用有限元法与Lagrange方程推导了柔性物体的刚性动力学方程与振动方程。为了使系统有一个统一坐标形式,给出了关节角坐标,末端执行器坐标与物体坐标之间的转换关系,在物体坐标建立了协调系统的刚性动力学方程与振动方程,分析了系统模型的一些特性。研究了James等人提出的刚性控制方法,在他们方法的基础上提出了针对本文控制对象的控制方法,并针对方法进行了仿真研究,结果证明了方法的有效性。然后利用力反馈作用设计了混合控制器,分析了力的有界性并通过仿真验证了这个结果。
Recently, robot manipulator has been used in many fields. There are many scholars who have done much work in manipulating rigid object, while in many fields, such as shipbuilding, car and air craft, flexible material has been used largely. So far, there are few scholars who have done such work. The system composed of robot manipulators and flexible payload has characteristics as following: exceeding nonlinearity, strong coupling and infinite degree of freedom. In the process of handling flexible payload two actions will happen that are rigid motion and vibration. Considering the current conditions domestic and abroad, the study of manipulating flexible payload is underway and the methods of control aren’t perfect. There are many questions needed to be solved.
In this paper the study of two manipulators handling a flexible payload has been done. The contributions of this paper are the modeling and the controller that is based on the other’s work. Firstly, the background and significance are introduced. The system of manipulator manipulating flexible payload is divided into two species single and cooperative and the conditions are introduced singly. The characteristics of current methods are analyzed.
In term of this paper, the structures that will appear in cooperative manipulation are analyzed and the structure we need in this paper is chose. Based on this structure the concept of virtual link is present to be the benchmark of flexible body. The object coordinate is located on this benchmark. The FEM is a new analytical method which depends on computer. The FEM can divide a complicated structure into finite elements and the structure is combined by the elements in the node. The displacement of the element is expressed by the interpolation function of node. The FEM has the same procedure from the choice of displacement to the modeling of dynamic equations, so it’s suited to be computed by computer. The physical meaning of flexible coordinates is clear because the generalized coordinates are the displacement of node not the mode. Because of these advantages we will use the FEM to model the dynamic equation of the flexible payload. The flexible payload is divided into four elements in the object coordinate. There is a special condition appearing when we define the flexible coordinates both sides of the object coordinate because of the choice of coordinates and the explanation is given. On the last basis the rigid dynamic equation and vibration equation are deduced. In this paper three kinds of coordinates will appear that are joint coordinates, end-effector coordinates and object coordinate. In order to have a uniform expression the relations of three coordinates are listed and the rigid dynamic equation and vibration equation of the system are derived in the object coordinate. The characteristics of the system are introduced.
So far, there are few scholars who have studied the whole system and some scholars have only studied certain problem because of the complexity of the system. The control of vibration is very difficult because the state vibration is unmeasurable. The method in this paper based on the‘rigid control’that it needs only the measurable coordinates in the controller which was brought out by James. The stability of the system is proved by the theory of stability and the results of the simulation validate the method of modeling and the controller. The hybrid controller is designed using the feedback signal of force. The boundary of the force is analyzed and simulation is done.
Some prepared work was done for two manipulators handling a flexible payload. Because it’s a very complex system many problems will appear in dealing with the system. This paper mostly solved the problem of model and designed a controller to validate the veracity but there are many works remained that needed to be solved.
引文
[1] T. Yukawa, M. Uchiyama and H. Inooka, Handling of a Constrained Flexible Object by a Robot, IEEE International Conference on Robotics and Automation, 1995,v1, pp324-329
[2] Jianqing Wu, Zhiwei Luo and Koji Ito, Gain Scheduled Control for Robot Manipulator's Contact Tasks on Flexible Environments, International Workshop on Advanced Motion Control, 1996, v2, pp512-517
[3] Jianqing Wu, Zhiwei Luo, Masaki Yamakita and Koji Ito, Gain Scheduled Control of Robot Manipulators for Contact Tasks on Uncertain Flexible Objects, IEEE International Conference on Systems, Man and Cybernetics, 1996,v1, pp41-46
[4] Ru Lai and Fujio Ohkawa, Digital Control of A Manipulator Handling Flexible Objects, The World Congress on Intelligent Control and Automation, 2004, v5, pp4595-459
[5] Yuan F. Zhang, Run Pei and Chichyang Chen, Strategies For Automatic Assembly of Deformable Objects, IEEE International Conference on robotics and automation, 1991, v3, pp2598-2603
[6] Fumihito ARAI, Lili RONG and Toshio FUKUDA, Trajectory Control of Flexible Plate Using Neural Network, 1993, v1, pp155-159
[7] Nakagaki, Hirofumi, Kitagaki, Kosei, Tsukune, Hideo, Study of Insertion Task of a Flexible Beam into a Hole, IEEE International Conference on Robotics and Automation, 1995, v1, pp330-335
[8] Werner Kraus Jr. and Brenan J. McCarragher, Force Fields in the Manipulation of Flexible Materials, IEEE International Conference on Robotics and Automation, 1996, v3, pp2352-2357
[9] Hirofumi Nakagaki, Kosei Kitagaki, Tsukasa Ogasawara and Hideo Tsukune, Study of Deformation and Insertion Tasks of a Flexible Wire,IEEE International Conference on Robotics and Automation Albuquerque, 1997, v3, pp2397-2402
[10] Ming Z. Chen and Yuan F. Zheng, Vibration-free movement of deformable beams by robot manipulator, IEEE International Conference on Robotics and Automation, 1993, v2, pp456-461
[11] Jain Sandeep and Khorrami Farshad, Vibration Suppression of Unknown Flexible Payloads Using a Wrist Mounted Force/Torque Sensor for Remote Manipulator Systems, IEEE International Conference on Robotics and Automation,1994, pp2119-2124
[12] Sandeep Jain and Farshad Khorrami, Positioning of unknown flexible payloads for robotic arms using a wrist-mounted force/torque sensor, IEEE Transactions on Control Systems Technology, 3(2), 1995, pp189-20
[13] T. Zhou, A. A. Goldenberg and J. W. Zu, Modal Force Based Input Shaper for Vibration Suppression of Flexible Payloads, IEEE International Conference on Robotics and Automation , 2002, v3, pp2430-2435
[14] T. Zhou, J. W. Zu and A. A. Goldenberg, Vibration Controllability of Flexible Robot-Payload Systems, IEEE International Conference on Robotics and Automation, 2000, v2, pp 1484-1489
[15] N. Fahantidis, K. Paraschidis, V. Petridis, Z. Doulgeri, L. Petrou, G. Hasapis, Robot Handling of Flat Textile Materials, IEEE Robotics and Automation Magazine, 1997, 4(1), pp34-41
[16] Mills James K. and Jerry G.-L. Ing, Robotic Fixtureless Assembly of Sheet Metal Parts Using Dynamic Finite Element Models: Modelling and Simulation, IEEE International Conference on Robotics and Automation, 1995, v3, pp2350-2357
[17] Nguyen Wai and Mills James K., Multi-Robot Control For Flexible Fixtureless Assembly of Flexible Sheet Metal Auto Body Parts, IEEE International Conference on Robotics and Automation, 1996, v3, pp2340-2345
[18] Dong Sun and Yunhui Liu, Modeling and Impedance Control of aTwo-Manipulator System Handling a Flexible Beam, IEEE International Conference on Robotics and Automation, 1997, v2, pp1787-1792
[19] Yunhui Liu and Dong Sun, Stabilizing a Flexible Beam Handled by Two Manipulators Via PD Feedback, IEEE Transactions on Automatic Control, 2000, 45(11), pp2159-2164
[20] Dong Sun, James K. Mills and Yunhui Liu, Hybrid Position and Force Control of Two Industrial Robots Manipulating a Flexible Sheet: Theory and Experiment, IEEE International Conference on Robotics & Automation, 1998, v2, pp1835-1840
[21] Dong Sun and Yunhui Liu, Position and Force Tracking of a Two-Manipulator System Manipulating a Flexible Beam Payload, IEEE International Conference on Robotics & Automation, 2001, v4, pp3483-3488
[22] Dong Sun and Yunhui Liu, Position and Force Tracking of a Two-Manipulator System Manipulating a Flexible Beam, Journal of Robotic Systems, 2001, 18(4), pp197-212
[23] Dong Sun, Xiao-lun Shi and Yun-hui Liu, Modeling and Cooperation of Two-Arm Robotic System Manipulating a Deformable Object, IEEE International Conference on Robotics and Automation, 1996, v3, pp2346-2351
[24] Dong Sun James K. Mills Yunhui Liu Position Control of Multiple Robots Manipulating a Flexible Payload, Proceedings of the American Control Conference Phtladelphia, Pennsylvania - June 1998 pp456-460
[25] Dong Sun, James K. Mills and Yunhui Liu, Position Control of Robot Manipulators Manipulating a Flexible Payload, The International Journal of Robotics Research, 1999, 18(3), pp. 319-332
[26] Yuan F. Zheng and Ming Z. Chen, Trajectory planning for two manipulators to deform flexible beams, IEEE International Conference on Robotics and Automation, 1993, v1, pp1019-1024
[27] Ming Z. Chen and Yuan F. Zheng, Deform flexible beams by two 55manipulators through neural network learning, IEEE International Conference on Robotics and Automation, 1994, v4, pp3180-3185
[28] Omar Al-Jarrah, Yuan F. Zheng and Keon-Young Yi, Efficient trajectory planning for two manipulators to deform flexible materials with experiments, IEEE International Conference on Robotics and Automation, 1995, v1, pp312-317
[29] Al-Jarrah Omar, Yuan F. Zheng, Yi, Keon-Young, Trajectory Planning for Two Manipulators to Deform Flexible Materials Using Compliant Motion, IEEE International Conference on Robotics and Automation, 1995, v2, pp1517-1522
[30] Takayuki MATSUNO, Daichi TAMAKI, Fumihito ARAI and Toshio FUKUDA, Manipulation of Deformable Linear Objects with Knot Invariant to Classify Condition, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2005, v2, pp893-898
[31] Takayuki Matsuno, Toshio Fukuda and Fumihito Arai, Flexible Rope Manipulation by Dual Manipulator System Using Vision Sensor, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2001, v2, pp677-682
[32] Toshio Fukuda, Takayuki Matsuno and Fumihito Arai, Flexible Object Manipulation by Dual Manipulator System, IEEE International Conference on Robotics & Automation, 2000, v2, pp1955-1960
[33] T. Yukava, M. Uchiyama, D. N. Nenchev and H. Inooka, Stability of Control System in Handling of a Flexible Object by Rigid Arm Robots, IEEE International Conference on Robotics and Automation ,1996, v3, pp2332-2339
[34] M. M. Svinin, and M. Uchiyama, Coordination Strategy for a System of Manipulators Coupled Via a Flexible Object, Industrial Electronics Conference, 1994, v2, pp685-690
[35] T. Yukawa, M. Uchiyama and H. Inooka, Cooperative Control of a Vibrating Flexible Object by a Rigid Dual-Arm Robot, IEEE InternationalConference on Robotics and Automation, 1995, v2, pp1820-1826
[36] M. M. Svinin and M. Uchiyama, Cartesian-Level Control Strategy for a System of Manipulators Coupled Through a Flexible Object, IEEE International Conference on Intelligent Robots and Systems, 1994, v1, pp687-694
[37] Kosuge Kazuhiro, Hashimoto Satoshi, Yoshida Hidehiro, Human-Robots Collaboration System for Flexible Object Handling, IEEE International Conference on Robotics, 1998, v2, pp1841-1846
[38] W. F. Dellinger and J. N. Anderson, Interactive Force Dynamics of Two Robotic Manipulators Grasping a Non-rigid Object, IEEE International Conference on Robotics and Automation, 1992, v3, pp2205-2210
[39] Ka-Ming Yuen and Gary M. Bone, Robotic Assembly of Flexible Sheet Metal Parts, IEEE International Conference on Robotics and Automation, 1996, v2, pp1511-1516
[40] Werner Kraus Jr. and Brenan J. McCarragher, Case studies in the manipulation of flexible parts using a hybrid position/force approach, IEEE International Conference on Robotics and Automation, 1997, v1, pp202-207
[41] Herbert G. Tanner and Kostas J. Kyriakopoulos, Analysis of Deformable Object Handling, IEEE International Conference on Robotics & Automation,1999, v4, pp2674-2679
[42] Makoto Kudo, Yasuo Nasu, Kazuhisa Mitobe and Branislav Borovac, Multi-arm robot control system for manipulation of Flexible materials in sewing operation Mechatronics, 2000, 10(3), pp371-402
[43] Young-Chun Ji and Youn-sik Park, Optimal input design for a cooperating robot to reduce vibration when carrying flexible objects, Robotica, 2001, 19(2), pp 209–215
[44] V.Yu. Rutkovsky, V.M. Sukhanov, V.M. Glumov, S.D. Zemlyakov and S.D. Dodds, Computer simulation of an adaptive control system for a free-flying space robotic module with flexible payload’s transportation,Mathematics and Computers in Simulation, 2002, 58 (4-6), pp407–421
[45] 刘迎春,余跃庆, 冗余度柔性机器人协调操作刚性和柔性负载的仿真分析, 机械科学与技术, 2005, 24(6)
[46] 刘迎春, 余跃庆, 窦建武 柔性机器人协调操作柔性负载的动力学模型, 机械科学与技术 2002, 21(4)
[47] 刘迎春,余跃庆 考虑负载变形的柔性机器人协调操作动力学, 中国机械工程 2002,13(14)
[48] 杨荣柏, 机械结构分析的有限元法. 华中理工大学出版社.1989
[49] Y.C. Li, B.J. Tang, Z.X. Shi, et al. Experimental study for trajectory tracking of a two-link flexible manipulator, International Journal of Systems Science, 2000, 31(1), pp3-9.
[50] Y.C. Li, F. Matsuno, Trajectory control two-degree-of freedom flexible manipulators, The Annual Conference of the Robotics Society of Japan, 1995, v1, pp81-82.
[51] S.S. Ge, T.H. Lee, G. Zhu, Nonlinear feedback controller for a single-link flexible manipulator based on a finite element model, Journal of Robotic Systems, 1997, 14(3), pp165-178.
[52] H. Du, M.K. Lim, K.M. Liew, A nonlinear finite element model for dynamics of flexible manipulators, Mechanism and Machine Theory, 1996, 31(8), pp1109-1119.
[53] J.M. Martins, Z. Mohamed, M.O. Tokhi, et al. Approaches for dynamic modelling of flexible manipulator systems, IEE Proceedings: Control Theory and Applications, 2003, 150(4), pp401-411
[54] J.G. Han, W. X. Ren, Y. Huang, A wavelet-based stochastic finite element method of thin plate bending, Applied Mathematical Modelling, 2007, 31, pp181-193
[55] S. S. Ge, T. H. Lee, and G. Zhu, Comparison studies between assumed modes method and finite element method in modeling a single-link flexible manipulator, Proceedings - IEEE International Conference on intelligent control and instrument, 1995, pp376-381.
[56] R.J. Theodore, A. Ghosal, Comparison of the assumed modes and finite element models for flexible multilink manipulators, International Journal of Robotics Research, 1995, 14(2), pp91-111.
[57] J. E. Slotine, W.P. Li, Applied Nonlinear Control, Prentice-Hall, USA, 1991, 461
[2] Jianqing Wu, Zhiwei Luo and Koji Ito, Gain Scheduled Control for Robot Manipulator's Contact Tasks on Flexible Environments, International Workshop on Advanced Motion Control, 1996, v2, pp512-517
[3] Jianqing Wu, Zhiwei Luo, Masaki Yamakita and Koji Ito, Gain Scheduled Control of Robot Manipulators for Contact Tasks on Uncertain Flexible Objects, IEEE International Conference on Systems, Man and Cybernetics, 1996,v1, pp41-46
[4] Ru Lai and Fujio Ohkawa, Digital Control of A Manipulator Handling Flexible Objects, The World Congress on Intelligent Control and Automation, 2004, v5, pp4595-459
[5] Yuan F. Zhang, Run Pei and Chichyang Chen, Strategies For Automatic Assembly of Deformable Objects, IEEE International Conference on robotics and automation, 1991, v3, pp2598-2603
[6] Fumihito ARAI, Lili RONG and Toshio FUKUDA, Trajectory Control of Flexible Plate Using Neural Network, 1993, v1, pp155-159
[7] Nakagaki, Hirofumi, Kitagaki, Kosei, Tsukune, Hideo, Study of Insertion Task of a Flexible Beam into a Hole, IEEE International Conference on Robotics and Automation, 1995, v1, pp330-335
[8] Werner Kraus Jr. and Brenan J. McCarragher, Force Fields in the Manipulation of Flexible Materials, IEEE International Conference on Robotics and Automation, 1996, v3, pp2352-2357
[9] Hirofumi Nakagaki, Kosei Kitagaki, Tsukasa Ogasawara and Hideo Tsukune, Study of Deformation and Insertion Tasks of a Flexible Wire,IEEE International Conference on Robotics and Automation Albuquerque, 1997, v3, pp2397-2402
[10] Ming Z. Chen and Yuan F. Zheng, Vibration-free movement of deformable beams by robot manipulator, IEEE International Conference on Robotics and Automation, 1993, v2, pp456-461
[11] Jain Sandeep and Khorrami Farshad, Vibration Suppression of Unknown Flexible Payloads Using a Wrist Mounted Force/Torque Sensor for Remote Manipulator Systems, IEEE International Conference on Robotics and Automation,1994, pp2119-2124
[12] Sandeep Jain and Farshad Khorrami, Positioning of unknown flexible payloads for robotic arms using a wrist-mounted force/torque sensor, IEEE Transactions on Control Systems Technology, 3(2), 1995, pp189-20
[13] T. Zhou, A. A. Goldenberg and J. W. Zu, Modal Force Based Input Shaper for Vibration Suppression of Flexible Payloads, IEEE International Conference on Robotics and Automation , 2002, v3, pp2430-2435
[14] T. Zhou, J. W. Zu and A. A. Goldenberg, Vibration Controllability of Flexible Robot-Payload Systems, IEEE International Conference on Robotics and Automation, 2000, v2, pp 1484-1489
[15] N. Fahantidis, K. Paraschidis, V. Petridis, Z. Doulgeri, L. Petrou, G. Hasapis, Robot Handling of Flat Textile Materials, IEEE Robotics and Automation Magazine, 1997, 4(1), pp34-41
[16] Mills James K. and Jerry G.-L. Ing, Robotic Fixtureless Assembly of Sheet Metal Parts Using Dynamic Finite Element Models: Modelling and Simulation, IEEE International Conference on Robotics and Automation, 1995, v3, pp2350-2357
[17] Nguyen Wai and Mills James K., Multi-Robot Control For Flexible Fixtureless Assembly of Flexible Sheet Metal Auto Body Parts, IEEE International Conference on Robotics and Automation, 1996, v3, pp2340-2345
[18] Dong Sun and Yunhui Liu, Modeling and Impedance Control of aTwo-Manipulator System Handling a Flexible Beam, IEEE International Conference on Robotics and Automation, 1997, v2, pp1787-1792
[19] Yunhui Liu and Dong Sun, Stabilizing a Flexible Beam Handled by Two Manipulators Via PD Feedback, IEEE Transactions on Automatic Control, 2000, 45(11), pp2159-2164
[20] Dong Sun, James K. Mills and Yunhui Liu, Hybrid Position and Force Control of Two Industrial Robots Manipulating a Flexible Sheet: Theory and Experiment, IEEE International Conference on Robotics & Automation, 1998, v2, pp1835-1840
[21] Dong Sun and Yunhui Liu, Position and Force Tracking of a Two-Manipulator System Manipulating a Flexible Beam Payload, IEEE International Conference on Robotics & Automation, 2001, v4, pp3483-3488
[22] Dong Sun and Yunhui Liu, Position and Force Tracking of a Two-Manipulator System Manipulating a Flexible Beam, Journal of Robotic Systems, 2001, 18(4), pp197-212
[23] Dong Sun, Xiao-lun Shi and Yun-hui Liu, Modeling and Cooperation of Two-Arm Robotic System Manipulating a Deformable Object, IEEE International Conference on Robotics and Automation, 1996, v3, pp2346-2351
[24] Dong Sun James K. Mills Yunhui Liu Position Control of Multiple Robots Manipulating a Flexible Payload, Proceedings of the American Control Conference Phtladelphia, Pennsylvania - June 1998 pp456-460
[25] Dong Sun, James K. Mills and Yunhui Liu, Position Control of Robot Manipulators Manipulating a Flexible Payload, The International Journal of Robotics Research, 1999, 18(3), pp. 319-332
[26] Yuan F. Zheng and Ming Z. Chen, Trajectory planning for two manipulators to deform flexible beams, IEEE International Conference on Robotics and Automation, 1993, v1, pp1019-1024
[27] Ming Z. Chen and Yuan F. Zheng, Deform flexible beams by two 55manipulators through neural network learning, IEEE International Conference on Robotics and Automation, 1994, v4, pp3180-3185
[28] Omar Al-Jarrah, Yuan F. Zheng and Keon-Young Yi, Efficient trajectory planning for two manipulators to deform flexible materials with experiments, IEEE International Conference on Robotics and Automation, 1995, v1, pp312-317
[29] Al-Jarrah Omar, Yuan F. Zheng, Yi, Keon-Young, Trajectory Planning for Two Manipulators to Deform Flexible Materials Using Compliant Motion, IEEE International Conference on Robotics and Automation, 1995, v2, pp1517-1522
[30] Takayuki MATSUNO, Daichi TAMAKI, Fumihito ARAI and Toshio FUKUDA, Manipulation of Deformable Linear Objects with Knot Invariant to Classify Condition, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2005, v2, pp893-898
[31] Takayuki Matsuno, Toshio Fukuda and Fumihito Arai, Flexible Rope Manipulation by Dual Manipulator System Using Vision Sensor, IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2001, v2, pp677-682
[32] Toshio Fukuda, Takayuki Matsuno and Fumihito Arai, Flexible Object Manipulation by Dual Manipulator System, IEEE International Conference on Robotics & Automation, 2000, v2, pp1955-1960
[33] T. Yukava, M. Uchiyama, D. N. Nenchev and H. Inooka, Stability of Control System in Handling of a Flexible Object by Rigid Arm Robots, IEEE International Conference on Robotics and Automation ,1996, v3, pp2332-2339
[34] M. M. Svinin, and M. Uchiyama, Coordination Strategy for a System of Manipulators Coupled Via a Flexible Object, Industrial Electronics Conference, 1994, v2, pp685-690
[35] T. Yukawa, M. Uchiyama and H. Inooka, Cooperative Control of a Vibrating Flexible Object by a Rigid Dual-Arm Robot, IEEE InternationalConference on Robotics and Automation, 1995, v2, pp1820-1826
[36] M. M. Svinin and M. Uchiyama, Cartesian-Level Control Strategy for a System of Manipulators Coupled Through a Flexible Object, IEEE International Conference on Intelligent Robots and Systems, 1994, v1, pp687-694
[37] Kosuge Kazuhiro, Hashimoto Satoshi, Yoshida Hidehiro, Human-Robots Collaboration System for Flexible Object Handling, IEEE International Conference on Robotics, 1998, v2, pp1841-1846
[38] W. F. Dellinger and J. N. Anderson, Interactive Force Dynamics of Two Robotic Manipulators Grasping a Non-rigid Object, IEEE International Conference on Robotics and Automation, 1992, v3, pp2205-2210
[39] Ka-Ming Yuen and Gary M. Bone, Robotic Assembly of Flexible Sheet Metal Parts, IEEE International Conference on Robotics and Automation, 1996, v2, pp1511-1516
[40] Werner Kraus Jr. and Brenan J. McCarragher, Case studies in the manipulation of flexible parts using a hybrid position/force approach, IEEE International Conference on Robotics and Automation, 1997, v1, pp202-207
[41] Herbert G. Tanner and Kostas J. Kyriakopoulos, Analysis of Deformable Object Handling, IEEE International Conference on Robotics & Automation,1999, v4, pp2674-2679
[42] Makoto Kudo, Yasuo Nasu, Kazuhisa Mitobe and Branislav Borovac, Multi-arm robot control system for manipulation of Flexible materials in sewing operation Mechatronics, 2000, 10(3), pp371-402
[43] Young-Chun Ji and Youn-sik Park, Optimal input design for a cooperating robot to reduce vibration when carrying flexible objects, Robotica, 2001, 19(2), pp 209–215
[44] V.Yu. Rutkovsky, V.M. Sukhanov, V.M. Glumov, S.D. Zemlyakov and S.D. Dodds, Computer simulation of an adaptive control system for a free-flying space robotic module with flexible payload’s transportation,Mathematics and Computers in Simulation, 2002, 58 (4-6), pp407–421
[45] 刘迎春,余跃庆, 冗余度柔性机器人协调操作刚性和柔性负载的仿真分析, 机械科学与技术, 2005, 24(6)
[46] 刘迎春, 余跃庆, 窦建武 柔性机器人协调操作柔性负载的动力学模型, 机械科学与技术 2002, 21(4)
[47] 刘迎春,余跃庆 考虑负载变形的柔性机器人协调操作动力学, 中国机械工程 2002,13(14)
[48] 杨荣柏, 机械结构分析的有限元法. 华中理工大学出版社.1989
[49] Y.C. Li, B.J. Tang, Z.X. Shi, et al. Experimental study for trajectory tracking of a two-link flexible manipulator, International Journal of Systems Science, 2000, 31(1), pp3-9.
[50] Y.C. Li, F. Matsuno, Trajectory control two-degree-of freedom flexible manipulators, The Annual Conference of the Robotics Society of Japan, 1995, v1, pp81-82.
[51] S.S. Ge, T.H. Lee, G. Zhu, Nonlinear feedback controller for a single-link flexible manipulator based on a finite element model, Journal of Robotic Systems, 1997, 14(3), pp165-178.
[52] H. Du, M.K. Lim, K.M. Liew, A nonlinear finite element model for dynamics of flexible manipulators, Mechanism and Machine Theory, 1996, 31(8), pp1109-1119.
[53] J.M. Martins, Z. Mohamed, M.O. Tokhi, et al. Approaches for dynamic modelling of flexible manipulator systems, IEE Proceedings: Control Theory and Applications, 2003, 150(4), pp401-411
[54] J.G. Han, W. X. Ren, Y. Huang, A wavelet-based stochastic finite element method of thin plate bending, Applied Mathematical Modelling, 2007, 31, pp181-193
[55] S. S. Ge, T. H. Lee, and G. Zhu, Comparison studies between assumed modes method and finite element method in modeling a single-link flexible manipulator, Proceedings - IEEE International Conference on intelligent control and instrument, 1995, pp376-381.
[56] R.J. Theodore, A. Ghosal, Comparison of the assumed modes and finite element models for flexible multilink manipulators, International Journal of Robotics Research, 1995, 14(2), pp91-111.
[57] J. E. Slotine, W.P. Li, Applied Nonlinear Control, Prentice-Hall, USA, 1991, 461