自由浮动空间柔性冗余机器人系统动力学建模与控制研究
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摘要
随着空间技术的发展,空间柔性冗余机器人的研究受到了广泛的关注。由于它结构的特殊性,柔性臂在运动中会发生变形和振动,这给空间柔性冗余机器人的定位和跟踪控制带来了很多的问题。因此,针对自由浮动空间柔性冗余机器人的动力学建模与振动控制及关节运动优化控制问题,本文进行了深入的研究,并进行了仿真实验验证。
首先,本文基于假设模态法、拉格朗日方程和动量守恒,推导了自由浮动空间柔性冗余机器人系统的动力学模型,并对负载轨迹进行跟踪控制,同时,还与刚性机器人系统比较分析,仿真结果说明了柔性臂存在弹性形变以及其影响着本体和负载的状态。
其次,针对柔性臂的弹性形变和振动问题,本文采用了基于滤波器的滑模振动控制方法,并与PD控制方法进行了比较,实验仿真结果说明了前一种控制方法的优越性。
再次,本文分析了自由浮动空间柔性冗余机器人弹性动力响应的特性,在结构参数不变的情况下,利用柔性冗余度机器人关节运动的冗余特征,构造出以冗余关节角为状态变量的控制结构,并由此得出基于关节运动参数的柔性冗余机器人的冗余关节控制方法。
最后,从自由浮动空间柔性冗余机器人的自运动出发,讨论自运动变量的选取,利用雅可比矩阵零空间矢量与自运动变量的关系,推导了自运动变量解析表达式,然后构造出以自运动变量为状态变量的优化控制系统,并由此得出基于自运动变量的自由浮动空间柔性冗余机器人优化控制方法。
With the development of space technology, the research on the flexible redundant space robot has attracted much attention. The structural flexibility inevitably causes the elastic deflection and vibration. As a consequence, the positioning and tracking performance of the flexible redundant robot will tend to be degraded. In this paper, the dynamic modeling、vibration suppression controll and joint motion optimization control are dealt with, and plenty of simulation tests have conducted to verify the effectiveness of the proposed methods.
Firstly, on the basis of the assumed modes method, Lagrange principle and momentum conservation, the dynamic modeling of a Free-floating Flexible Redundant Space Robot holding a rigid payload in a workspace is derived,and the trajectory tracking has been performed. At the same time, compared with the rigid robot system,simulation tests make out that the flexible structure presences flexibility vibration, which affects the state of the base and payload.
Secondly, for the elastic deflection and vibration of a Free-floating Flexible Redundant Space Robot, a sliding control approach based on the filter is developed,compared with the PD control method,simulation tests verify the advantage of the first approach.
Thirdly,the character for flexible dynamic response of the flexible redundant space robot is studied. In the situation of invariable structure parameter,the control structure for the redundant joint angle variables is constructed by making use of the redundant character of the flexible redundant joint motion. Then,an approach for redundant joint motion of the flexible redundant robot is presented.
Finally,based on the self-motion of the flexible redundant space robot,the self-motion variable is discussed. Making use of the relation between the Jacobi matrix null space vector and the self-motion variable, the self-motion expression of kinematics capability is presented.Then, an optimization control system is constructed, from which, a control algorithm of the flexible redundant space robot based on self-motion variable is presented.
首先,本文基于假设模态法、拉格朗日方程和动量守恒,推导了自由浮动空间柔性冗余机器人系统的动力学模型,并对负载轨迹进行跟踪控制,同时,还与刚性机器人系统比较分析,仿真结果说明了柔性臂存在弹性形变以及其影响着本体和负载的状态。
其次,针对柔性臂的弹性形变和振动问题,本文采用了基于滤波器的滑模振动控制方法,并与PD控制方法进行了比较,实验仿真结果说明了前一种控制方法的优越性。
再次,本文分析了自由浮动空间柔性冗余机器人弹性动力响应的特性,在结构参数不变的情况下,利用柔性冗余度机器人关节运动的冗余特征,构造出以冗余关节角为状态变量的控制结构,并由此得出基于关节运动参数的柔性冗余机器人的冗余关节控制方法。
最后,从自由浮动空间柔性冗余机器人的自运动出发,讨论自运动变量的选取,利用雅可比矩阵零空间矢量与自运动变量的关系,推导了自运动变量解析表达式,然后构造出以自运动变量为状态变量的优化控制系统,并由此得出基于自运动变量的自由浮动空间柔性冗余机器人优化控制方法。
With the development of space technology, the research on the flexible redundant space robot has attracted much attention. The structural flexibility inevitably causes the elastic deflection and vibration. As a consequence, the positioning and tracking performance of the flexible redundant robot will tend to be degraded. In this paper, the dynamic modeling、vibration suppression controll and joint motion optimization control are dealt with, and plenty of simulation tests have conducted to verify the effectiveness of the proposed methods.
Firstly, on the basis of the assumed modes method, Lagrange principle and momentum conservation, the dynamic modeling of a Free-floating Flexible Redundant Space Robot holding a rigid payload in a workspace is derived,and the trajectory tracking has been performed. At the same time, compared with the rigid robot system,simulation tests make out that the flexible structure presences flexibility vibration, which affects the state of the base and payload.
Secondly, for the elastic deflection and vibration of a Free-floating Flexible Redundant Space Robot, a sliding control approach based on the filter is developed,compared with the PD control method,simulation tests verify the advantage of the first approach.
Thirdly,the character for flexible dynamic response of the flexible redundant space robot is studied. In the situation of invariable structure parameter,the control structure for the redundant joint angle variables is constructed by making use of the redundant character of the flexible redundant joint motion. Then,an approach for redundant joint motion of the flexible redundant robot is presented.
Finally,based on the self-motion of the flexible redundant space robot,the self-motion variable is discussed. Making use of the relation between the Jacobi matrix null space vector and the self-motion variable, the self-motion expression of kinematics capability is presented.Then, an optimization control system is constructed, from which, a control algorithm of the flexible redundant space robot based on self-motion variable is presented.
引文
[1]洪炳镕,柳长安,刘宏,自由飞行空间机器人运动控制及仿真,北京,国防工业出版社,2005.
[2] Yoshida,Kazuya,Engineering test satellite VII flight experiments for space robot dynamics and control:Theories on laboratory test beds ten years ago,now in orbit,International Journal of Robotics Research,2003,22(5):p321~335.
[3]柳长安,李国栋,吴克河等,自由飞行空间机器人研究综述,机器人,2002,24(4):380~384.
[4] B.Liang,Y.Xu,M.Bergerman,Mapping a Space Manipulator to a Dynamically Equivalent Manipulator,ASME Journal of Dynamic Systems Measurement and Control,1998,120(1):1~7.
[5]王树国,丁希仑,蔡鹤皋,机器人柔性臂动力学有限元建模方法的研究,哈尔滨工业大学学报,1997,29(1):114~116.
[6]陆金伟,除卫良,达庆利,柔性机器人操作臂的动力学建模,东南大学学报,1997,27(6):12~18.
[7]边宇枢,陆震,柔性机器人动力学建模的一种方法,北京航空航天大学学报,1999,25(4): 486~490.
[8]王大龙,陆佑方,郭九大,单连杆柔性机械臂动力学模型分析,吉林工业大学学报,1998,28(2):51~56.
[9]苏文敬,吴立成,孙富春,等,空间柔性双臂机器人系统建模、控制与仿真研究,系统仿真学报,2003,15(8):1098~1100.
[10]张成新,余跃庆,柔性机器人协调操作的动力学建模与分析,机械工程学报,2003,39(5): 49~54.
[11] E.Papadopoulos,S.Dubowsky,Dynamic singularities in free-floating space manipulators ,Journal of Dynamic Systems,Measurement and Control,1993,115:44~52.
[12] Mitsuhiro Yamano,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[13]黄文虎,邵成勋等,多柔体系统动力学,北京,科学出版社,1996:132~142.
[14] Wenwei Xu,Xiaoping Zhang,Satish Nair,Dynamic Modeling of a Two-Link Flexible Manipulator Incorporating Experimental Data , Proceedings of the American Control Conference Arlington,Arlington,2001:4508~4513.
[15] Mitsuhiro Yamano,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[16]杨唐文,柔性太空机械手振动抑制和误差补偿控制—传感、控制和实验,[东南大学博士学位论文],南京,东南大学,2001年.
[17] V. Feliu,K.S. Rattan,H. Brown,Modeling and Control of Single-Link Flexible Arms with Lumped Masses,Journal of Dynamic Systems,Measurement and Control,1992,114:59~69
[18] Mitsuhiro Yamano ,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[19] Bayo E,Paden B. On trajectory generation for flexible robots[J] Journal of Robotic Systems,1987,4(2):229~235.
[20] Singer N C Seering W P. Preshaping command inputs to reduce system vibration ASME Transactions,Journal of Dynamic Systems,Measurements,and control,1990,112:76-82.
[21]张志谊,傅志方,饶柱石,模态的可观可控度与作动器、传感器的布置,振动与冲击,1998,17(2):5~8.
[22]王波,王荣秀,结构振动的独立模态和耦合模态的组合控制,振动与冲击,2004,23(4):28~32.
[23]关玉明,阎绍泽,张铁民,等,柔性机械臂振动控制研究,河北工业大学学报,1999,28(4):17~20.
[24]吕月娥,颜世科,单连杆柔性臂加速度反馈抑振的研究,青岛大学学报,2002,17(4):16~19.
[25] S. K. Tso,Xu W L,Sun Z Q,T. W. Yang ,Vibration control for a flexible-link robot arm with deflection feedback,International Journal of Non-Linear Mechanics,2003,38(1):51~62.
[26]窦建武,余跃庆,柔性机器人协调操作闭链负载机构的一般模型,机器人,2001,23(1): 6~10.
[27]窦建武,余跃庆,基于目标运动规划的柔性机器人协调操作闭链刚性负载的动力学模型,机械科学与技术,2001,20(2):161~163.
[28]窦建武,余跃庆,两柔性机器人协调操作的动力学模型及其逆动力学分析,机器人,2000,22(1):39~46.
[29]刘迎春,余跃庆,陈新荣,冗余度柔性机器人协调操作刚性负载的动力学模型及仿真,机械科学与技术,2004,23(2):178~181.
[30]张成新,余跃庆,柔性机器人协调操作的动力学建模与分析,机械工程学报,2003,39(5):49~54.
[31] Fumitoshi Matsuno,Michinori Hatayama,Quasi-Static Cooperative Control of Two Two-Link Flexible Manipulators,IEEE International Conference on Robotics and Automation,1995:615~620.
[32] Mitsuhiro Yamano,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[33] Fumitoshi Matsuno,Michinori Hatayama,Quasi-Static Cooperative Control of Two Two-Link Flexible Manipulators,IEEE International Conference on Robotics and Automation,1995:615~620.
[34] Chantalakhana C,Stability Improvement of Modal Controllers for Suppressing Beam Vibration using a Hybrid Control Scheme,Industrial Technology,IEEE ICIT '02.2002 IEEE International Conference,2002:549~554.
[35]杨辉,洪嘉振,余征跃,一类刚-柔耦合系统的模态特性与实验研究,宇航学报,2002,23(2):67~72.
[36]宋轶民,余跃庆,张策,邵宏宇,柔性冗余度机器人残余振动主动控制,振动工程学报,2002,15(2):189~193.
[37]宋轶民,张永新,张策,余跃庆,冗余度柔性机器人复模态振动主动控制,中国机械工程学报,2004,15(6):546~560.
[38]王建军,宋轶民,张策,余跃庆,柔性冗余度机器人振动主动控制系统特性分析,机械设计,2005,22(5):7~9.
[39]刘才山,刘又午,王建明,柔性机械臂的动力学模型及滑模变结构控制,振动与冲击,1998,17(1):24~29.
[40]熊有伦,机器人技术基础,武汉,华中科技大学出版社,2003:55~75.
[41]石宗坤,自由浮动空间双臂机器人的动力学控制研究,[南京航空航天大学硕士学位论文],南京,南京航空航天大学,2005.3.
[42]孙富春,朱纪洪,刘国栋,等译,机器人学导论-分析、系统及应用,北京,电子工业出版社,2005:107~125.
[43]邱志成,柔性机械臂的变结构振动控制研究,动力学与控制学报,2007,05(1):62~67.
[44]张志涌,精通MATLAB7.0,北京,北京航空航天大学出版社,2003:210~230.
[45]张立科,MATLAB7.0从入门到精髓,北京,人民邮电出版社,2006:126~213.
[46]王海宁,张晓江,基于滤波器的机器人滑模控制,2008,34(4):64~66.
[47]解永春,吴宏鑫,黄金分割在自适应控制器设计中的应用,自动化学报,1992,18(2):177~185.
[48]张奇志,孙增圻,孙富春,基于奇异摄动的柔性机械手预测控制,机械科学与技术,2001,20(2):172~173.
[49] Mohamed,Martins,Tokhi,Vibration control of a very flexible manipulator system,Control Engineering Practice,2005,13(3):267~277.
[50] Ei-Sinawi,M.N.Hamdan,optimal vibration estimation of a non-linear flexible beam mounted on a rotating compliant hub,Journal of Sound and Vibration,2003,259(4):857~872.
[51] Joono Cheong,Wan Kyun Chung,Youngil Youm,Fast Suppression of Vibration for Multi-Link Flexible Robots Using Parameter Adaptive Control,International Conference on Intelligent aRobots and Systems,Hawaii,USA,2001:913~918.
[52]高志慧,员超,提高柔性冗余度机器人动态性能的冗余关节运动规划方法,2004,15(13):1132~1136.
[53]马保离,霍伟,基座姿态可控的空间机器人系统关节角轨迹的自适应算法,控制与决策,1995,10(6):514~518.
[54]高志慧,员超,柔性冗余度机器人减振运动规划研究,机械科学与技术学报,2004,23(12):1515~1518.
[55] Liegeois A. Automatic supervisory control of the configuration and behavior of multibody mechanisms. IEEE trans.System,Man,Cybenetics,1977,7(12):868~871.
[56] Chan T F,Dubey R V.A Weighted least-norm solution based scheme for aviding joint limits for redundant manipulators. Proc. of IEEE Int. Conf. On R.& A,1993:395~402.
[57]陈伟海,冗余自由度机器人优化控制研究:[博士学位论文].北京:北京航空航天大学,1996.
[58] Nedungadi A,Kazweounian K. A Local Solution with Global Characteristics for the Joint Torque Optimization of Redundant Manipulators. Journal of Robotics System,1989,6(5):631~654.
[59]边宇枢,陆震,柔性冗余度机器人力矩优化的研究,1998,20(5):382~388.
[60]唐世明,张启先,冗余度机器人优化的自运动控制方法,2000,36(1):41~44.
[61]叶平,基于自运动控制的冗余度机器人运动学优化,2004,40(12):128~132.
[62] Hollerbach,J.M,Such K,C.Redundancy resolution of Manipulators though torgue optimization. In Proc. of IEEE. Inter. Conf. On R.& A.1985:308~316.
[63]赵占芳,机器人冗余特性和力控制研究[D].北京:北京航空航天大学研究生院,1991.
[64] Lee and Heow Pueh. Motions with Minimal Joint Torques for Redundant Manipulators [J].ASME Journal of Mechanical design,1993,115:599~603.
[2] Yoshida,Kazuya,Engineering test satellite VII flight experiments for space robot dynamics and control:Theories on laboratory test beds ten years ago,now in orbit,International Journal of Robotics Research,2003,22(5):p321~335.
[3]柳长安,李国栋,吴克河等,自由飞行空间机器人研究综述,机器人,2002,24(4):380~384.
[4] B.Liang,Y.Xu,M.Bergerman,Mapping a Space Manipulator to a Dynamically Equivalent Manipulator,ASME Journal of Dynamic Systems Measurement and Control,1998,120(1):1~7.
[5]王树国,丁希仑,蔡鹤皋,机器人柔性臂动力学有限元建模方法的研究,哈尔滨工业大学学报,1997,29(1):114~116.
[6]陆金伟,除卫良,达庆利,柔性机器人操作臂的动力学建模,东南大学学报,1997,27(6):12~18.
[7]边宇枢,陆震,柔性机器人动力学建模的一种方法,北京航空航天大学学报,1999,25(4): 486~490.
[8]王大龙,陆佑方,郭九大,单连杆柔性机械臂动力学模型分析,吉林工业大学学报,1998,28(2):51~56.
[9]苏文敬,吴立成,孙富春,等,空间柔性双臂机器人系统建模、控制与仿真研究,系统仿真学报,2003,15(8):1098~1100.
[10]张成新,余跃庆,柔性机器人协调操作的动力学建模与分析,机械工程学报,2003,39(5): 49~54.
[11] E.Papadopoulos,S.Dubowsky,Dynamic singularities in free-floating space manipulators ,Journal of Dynamic Systems,Measurement and Control,1993,115:44~52.
[12] Mitsuhiro Yamano,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[13]黄文虎,邵成勋等,多柔体系统动力学,北京,科学出版社,1996:132~142.
[14] Wenwei Xu,Xiaoping Zhang,Satish Nair,Dynamic Modeling of a Two-Link Flexible Manipulator Incorporating Experimental Data , Proceedings of the American Control Conference Arlington,Arlington,2001:4508~4513.
[15] Mitsuhiro Yamano,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[16]杨唐文,柔性太空机械手振动抑制和误差补偿控制—传感、控制和实验,[东南大学博士学位论文],南京,东南大学,2001年.
[17] V. Feliu,K.S. Rattan,H. Brown,Modeling and Control of Single-Link Flexible Arms with Lumped Masses,Journal of Dynamic Systems,Measurement and Control,1992,114:59~69
[18] Mitsuhiro Yamano ,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[19] Bayo E,Paden B. On trajectory generation for flexible robots[J] Journal of Robotic Systems,1987,4(2):229~235.
[20] Singer N C Seering W P. Preshaping command inputs to reduce system vibration ASME Transactions,Journal of Dynamic Systems,Measurements,and control,1990,112:76-82.
[21]张志谊,傅志方,饶柱石,模态的可观可控度与作动器、传感器的布置,振动与冲击,1998,17(2):5~8.
[22]王波,王荣秀,结构振动的独立模态和耦合模态的组合控制,振动与冲击,2004,23(4):28~32.
[23]关玉明,阎绍泽,张铁民,等,柔性机械臂振动控制研究,河北工业大学学报,1999,28(4):17~20.
[24]吕月娥,颜世科,单连杆柔性臂加速度反馈抑振的研究,青岛大学学报,2002,17(4):16~19.
[25] S. K. Tso,Xu W L,Sun Z Q,T. W. Yang ,Vibration control for a flexible-link robot arm with deflection feedback,International Journal of Non-Linear Mechanics,2003,38(1):51~62.
[26]窦建武,余跃庆,柔性机器人协调操作闭链负载机构的一般模型,机器人,2001,23(1): 6~10.
[27]窦建武,余跃庆,基于目标运动规划的柔性机器人协调操作闭链刚性负载的动力学模型,机械科学与技术,2001,20(2):161~163.
[28]窦建武,余跃庆,两柔性机器人协调操作的动力学模型及其逆动力学分析,机器人,2000,22(1):39~46.
[29]刘迎春,余跃庆,陈新荣,冗余度柔性机器人协调操作刚性负载的动力学模型及仿真,机械科学与技术,2004,23(2):178~181.
[30]张成新,余跃庆,柔性机器人协调操作的动力学建模与分析,机械工程学报,2003,39(5):49~54.
[31] Fumitoshi Matsuno,Michinori Hatayama,Quasi-Static Cooperative Control of Two Two-Link Flexible Manipulators,IEEE International Conference on Robotics and Automation,1995:615~620.
[32] Mitsuhiro Yamano,Jin-soo Kim,cooperative control of 3D dual-flexible-arm robot,Journal of Intelligent and Robotics Systems,2004,39:1~15.
[33] Fumitoshi Matsuno,Michinori Hatayama,Quasi-Static Cooperative Control of Two Two-Link Flexible Manipulators,IEEE International Conference on Robotics and Automation,1995:615~620.
[34] Chantalakhana C,Stability Improvement of Modal Controllers for Suppressing Beam Vibration using a Hybrid Control Scheme,Industrial Technology,IEEE ICIT '02.2002 IEEE International Conference,2002:549~554.
[35]杨辉,洪嘉振,余征跃,一类刚-柔耦合系统的模态特性与实验研究,宇航学报,2002,23(2):67~72.
[36]宋轶民,余跃庆,张策,邵宏宇,柔性冗余度机器人残余振动主动控制,振动工程学报,2002,15(2):189~193.
[37]宋轶民,张永新,张策,余跃庆,冗余度柔性机器人复模态振动主动控制,中国机械工程学报,2004,15(6):546~560.
[38]王建军,宋轶民,张策,余跃庆,柔性冗余度机器人振动主动控制系统特性分析,机械设计,2005,22(5):7~9.
[39]刘才山,刘又午,王建明,柔性机械臂的动力学模型及滑模变结构控制,振动与冲击,1998,17(1):24~29.
[40]熊有伦,机器人技术基础,武汉,华中科技大学出版社,2003:55~75.
[41]石宗坤,自由浮动空间双臂机器人的动力学控制研究,[南京航空航天大学硕士学位论文],南京,南京航空航天大学,2005.3.
[42]孙富春,朱纪洪,刘国栋,等译,机器人学导论-分析、系统及应用,北京,电子工业出版社,2005:107~125.
[43]邱志成,柔性机械臂的变结构振动控制研究,动力学与控制学报,2007,05(1):62~67.
[44]张志涌,精通MATLAB7.0,北京,北京航空航天大学出版社,2003:210~230.
[45]张立科,MATLAB7.0从入门到精髓,北京,人民邮电出版社,2006:126~213.
[46]王海宁,张晓江,基于滤波器的机器人滑模控制,2008,34(4):64~66.
[47]解永春,吴宏鑫,黄金分割在自适应控制器设计中的应用,自动化学报,1992,18(2):177~185.
[48]张奇志,孙增圻,孙富春,基于奇异摄动的柔性机械手预测控制,机械科学与技术,2001,20(2):172~173.
[49] Mohamed,Martins,Tokhi,Vibration control of a very flexible manipulator system,Control Engineering Practice,2005,13(3):267~277.
[50] Ei-Sinawi,M.N.Hamdan,optimal vibration estimation of a non-linear flexible beam mounted on a rotating compliant hub,Journal of Sound and Vibration,2003,259(4):857~872.
[51] Joono Cheong,Wan Kyun Chung,Youngil Youm,Fast Suppression of Vibration for Multi-Link Flexible Robots Using Parameter Adaptive Control,International Conference on Intelligent aRobots and Systems,Hawaii,USA,2001:913~918.
[52]高志慧,员超,提高柔性冗余度机器人动态性能的冗余关节运动规划方法,2004,15(13):1132~1136.
[53]马保离,霍伟,基座姿态可控的空间机器人系统关节角轨迹的自适应算法,控制与决策,1995,10(6):514~518.
[54]高志慧,员超,柔性冗余度机器人减振运动规划研究,机械科学与技术学报,2004,23(12):1515~1518.
[55] Liegeois A. Automatic supervisory control of the configuration and behavior of multibody mechanisms. IEEE trans.System,Man,Cybenetics,1977,7(12):868~871.
[56] Chan T F,Dubey R V.A Weighted least-norm solution based scheme for aviding joint limits for redundant manipulators. Proc. of IEEE Int. Conf. On R.& A,1993:395~402.
[57]陈伟海,冗余自由度机器人优化控制研究:[博士学位论文].北京:北京航空航天大学,1996.
[58] Nedungadi A,Kazweounian K. A Local Solution with Global Characteristics for the Joint Torque Optimization of Redundant Manipulators. Journal of Robotics System,1989,6(5):631~654.
[59]边宇枢,陆震,柔性冗余度机器人力矩优化的研究,1998,20(5):382~388.
[60]唐世明,张启先,冗余度机器人优化的自运动控制方法,2000,36(1):41~44.
[61]叶平,基于自运动控制的冗余度机器人运动学优化,2004,40(12):128~132.
[62] Hollerbach,J.M,Such K,C.Redundancy resolution of Manipulators though torgue optimization. In Proc. of IEEE. Inter. Conf. On R.& A.1985:308~316.
[63]赵占芳,机器人冗余特性和力控制研究[D].北京:北京航空航天大学研究生院,1991.
[64] Lee and Heow Pueh. Motions with Minimal Joint Torques for Redundant Manipulators [J].ASME Journal of Mechanical design,1993,115:599~603.