具有非驱动关节的机器人控制研究
|
摘要
具有非驱动关节机器人顾名思义，它不是所有关节都有驱动装置，也就是说，它有些关节是被动的，也称自由的。其特点是输入空间(即控制空间)维度小于构造空间维度，同时，自由关节的约束方程又是不可积的，具有非完全约束特性，这就增大了其在控制上的难度。这种机器人的优点是显而易见的，由于驱动的减少，使机器人的重量减轻了，成本也降低了，同时也就减少了能源的消耗，诸多优点使这种机器人很有研究价值，是近些年机器人研究领域的热点。
本文以该类机器人为对象，进行了位置控制的研究，主要完成以下几方面工作：
建立了具有非驱动关节机器人的动力学数学模型，对模型中的约束条件，即非驱动臂的动态方程的可积性进行了分析，给出可积性的判别方法，同时证明了该模型的约束条件为非完全约束。
通过相平面分析的方法分析了当驱动臂施加周期扰动对非驱动臂运动的影响，其结论为通过调整驱动臂周期扰动的幅度及极性，可以使非驱动臂沿着围绕平衡点±π／2的极限环轨迹运动。在此基础上，提出衰减因子的概念，通过衰减因子调整驱动臂扰动的幅度和周期，使非驱动臂能够稳定在平衡点±π／2。
以非驱动臂能够稳定在平衡点±π／2为基础，提出了一种对于具有非驱动关节机器人的基于神经网络的位置开环控制方法。利用前向BP网络能够对任意函数以任意精度逼近的特点，对开环控制策略中难以确定的函数进行学习，实现两关节的任意位置控制。
将驱动臂的运动作为非驱动臂的扰动，建立了非驱动臂的动态模型，利用动态方程有效的近似解法—平均法，对非驱动臂的动态模型进行简化，得到近似的平均系统，通过仿真计算验证了二者具有相同的动力学特性。
针对平均系统，利用李亚普诺夫稳定性方法设计了非驱动关节的位置闭环控制，使非驱动臂的运动从初始位置所在的极限环到达目标位置所在的极限环，为使非驱动臂能够稳定在目标位置，根据稳定目标位置速度为零的特点，推导出驱动臂扰动幅度在目标位置小邻域内的反馈规律，实现了非驱动臂的任意位置的臂环控制。
仿真结果表明本文设计的基于神经网络的位置开环控制方法及针对简化
哈尔滨工程大学博士学位论文
后的平均系统设计的基于李亚普诺夫函数方法的非驱动臂任意位置臂环控制
是非常有效的。
Underactuated manimulator is a kind of manipulator with one or more joint is underactuated which can be called passive joint or free joint. The major character is that the dimension of input space is less than dimension of output space.We can see that the constraint of the system is un-integrable.It is called nonholonomic constraint.All of these feature make the control prolem has a number of additional difficulties.But there are many merit of this kind of manipulator,for example,underactuated manipulator maybe useful in particular instances to reduce weight,cost or energy consumption,while still maintaining an adequate degree of dexterity.So It is benefit to research the method of control problem,and it becomes one of the major topics of the field of robotics recently.
The problem of position control is studied in this paper,and the results of my work are as follow:
In first the dynamic model of underactuated manipulator is established,the non-holonomic characters of the constraint is analysed,and then the judgement method that if constraint is integrable is gived. So it is proved that the constraint of the manipulator in the paper is un-integrable,and the dynamic system is nonholonomic.
Then the behavior of the free joint when we bring the periodic perturbation through controlling the actuated joint is analysed,We can find in the result that the behavior of the free joint in the phase plane follows an ordered closed trajectory with the center equilibrium point +2 in the case where the amplitude of the input is small,We can define a weakening factor to make the position of the free joint reach the center equilibrium point +2 .
Based on the result above,the open loop control strategy is bringed forward in the paper. We use the function approximation property of neural network to obtain the function through studying using neural network method.And then realize the position control.
We have simplified the model through obtaining the average system of the
underactuated robot using averaging method which is efficient to deal with nonlinear system. The simulation results show that the real system and the average system have the same trajectory in the phase plane under the disturbance of periodic input of the actuated joint of the robot.
Finally,the closed loop control strategy through nonlinear states feedback is designed by a Lyapunov function for the averaged system and realize the position control of the non-holonomic robot.
The simulation results show that the open-loop control method based on neural network and the feedback control method of the position are effective.
本文以该类机器人为对象，进行了位置控制的研究，主要完成以下几方面工作：
建立了具有非驱动关节机器人的动力学数学模型，对模型中的约束条件，即非驱动臂的动态方程的可积性进行了分析，给出可积性的判别方法，同时证明了该模型的约束条件为非完全约束。
通过相平面分析的方法分析了当驱动臂施加周期扰动对非驱动臂运动的影响，其结论为通过调整驱动臂周期扰动的幅度及极性，可以使非驱动臂沿着围绕平衡点±π／2的极限环轨迹运动。在此基础上，提出衰减因子的概念，通过衰减因子调整驱动臂扰动的幅度和周期，使非驱动臂能够稳定在平衡点±π／2。
以非驱动臂能够稳定在平衡点±π／2为基础，提出了一种对于具有非驱动关节机器人的基于神经网络的位置开环控制方法。利用前向BP网络能够对任意函数以任意精度逼近的特点，对开环控制策略中难以确定的函数进行学习，实现两关节的任意位置控制。
将驱动臂的运动作为非驱动臂的扰动，建立了非驱动臂的动态模型，利用动态方程有效的近似解法—平均法，对非驱动臂的动态模型进行简化，得到近似的平均系统，通过仿真计算验证了二者具有相同的动力学特性。
针对平均系统，利用李亚普诺夫稳定性方法设计了非驱动关节的位置闭环控制，使非驱动臂的运动从初始位置所在的极限环到达目标位置所在的极限环，为使非驱动臂能够稳定在目标位置，根据稳定目标位置速度为零的特点，推导出驱动臂扰动幅度在目标位置小邻域内的反馈规律，实现了非驱动臂的任意位置的臂环控制。
仿真结果表明本文设计的基于神经网络的位置开环控制方法及针对简化
哈尔滨工程大学博士学位论文
后的平均系统设计的基于李亚普诺夫函数方法的非驱动臂任意位置臂环控制
是非常有效的。
Underactuated manimulator is a kind of manipulator with one or more joint is underactuated which can be called passive joint or free joint. The major character is that the dimension of input space is less than dimension of output space.We can see that the constraint of the system is un-integrable.It is called nonholonomic constraint.All of these feature make the control prolem has a number of additional difficulties.But there are many merit of this kind of manipulator,for example,underactuated manipulator maybe useful in particular instances to reduce weight,cost or energy consumption,while still maintaining an adequate degree of dexterity.So It is benefit to research the method of control problem,and it becomes one of the major topics of the field of robotics recently.
The problem of position control is studied in this paper,and the results of my work are as follow:
In first the dynamic model of underactuated manipulator is established,the non-holonomic characters of the constraint is analysed,and then the judgement method that if constraint is integrable is gived. So it is proved that the constraint of the manipulator in the paper is un-integrable,and the dynamic system is nonholonomic.
Then the behavior of the free joint when we bring the periodic perturbation through controlling the actuated joint is analysed,We can find in the result that the behavior of the free joint in the phase plane follows an ordered closed trajectory with the center equilibrium point +2 in the case where the amplitude of the input is small,We can define a weakening factor to make the position of the free joint reach the center equilibrium point +2 .
Based on the result above,the open loop control strategy is bringed forward in the paper. We use the function approximation property of neural network to obtain the function through studying using neural network method.And then realize the position control.
We have simplified the model through obtaining the average system of the
underactuated robot using averaging method which is efficient to deal with nonlinear system. The simulation results show that the real system and the average system have the same trajectory in the phase plane under the disturbance of periodic input of the actuated joint of the robot.
Finally,the closed loop control strategy through nonlinear states feedback is designed by a Lyapunov function for the averaged system and realize the position control of the non-holonomic robot.
The simulation results show that the open-loop control method based on neural network and the feedback control method of the position are effective.
引文
[1] 华中工学院数学教研室、软件教研室编．工程数学算法语言·计算方法．高等教育出版社出版社，1995：189-254页
[2] 浙江大学．概率论与数理统计(第二版)．高等教育出版社，1989
[3] 何广平，陆震，王凤翔编著．被动冗余度空间机器人运动学综合．机器人．2000，11(6)
[4] 熊光楞著．控制系统数字仿真．清华大学出版社，1994：37-62页，82-94页
[5] 冯登泰编．应用非线性振动力学．中国铁道出版社，1982：125-205页
[6] 熊有伦编．机器人技术基础．华中理工大学出版社，1999：15-141页
[7] 徐缤昌，阙志宏等．机器人控制工程．西北工业大学出版社，1991
[8] 周远清，张再兴等．智能机器人系统．清华大学出版社，1989
[9] 吴广玉，姜复兴等．机器人工程导论．哈尔滨工业大学出版社，1988
[10] 范印越。机器人技术。电子工业出版社，1988
[11] 申铁龙。机器人鲁棒控制基础。清华大学出版社，2000
[12] 蔡自兴。机器人学。清华大学出版社，2000
[13] 白井良明。机器人工程。科学出版社，2001
[14] 熊有伦，丁汉．机器人动力学性能指标及其优化．机械工程学报，1989，25(2)：9-15页
[15] 刘式适，刘式达．物理学中的非线性方程．北京大学出版社，1999：20-80页，127-163页
[16] 刘式适，刘式达．大气中的非线性椭圆余弦波和孤立波．中国科学，1982．4
[17] 冯纯伯，费树岷．非线性控制系统分析与设计(第2版)．电子工业出版社．
[18] 史忠科，郑永安等．控制系统的分析与设计．西北工业大学出版社，1995
[19] 夏德铃．反馈控制理论．哈尔滨工业大学出版社．1984
[20] 孙洪洲，韩其智等．李代数李超代数及其在物理忠的应用．北京大学出版社，1999
[21] 褚亦清，李翠英等．非线性振动分析．北京理工大学出版社，1996
[22] 曹建福，韩崇昭，方洋旺。非线性系统理论及应用。西安交通大学出版社，2001
[23]苏步青，刘鼎元。初等微分几何。上海科学技术出版社，1985
[24]秦元勋，王联，王慕秋．运动稳定性理论与应用，科学出版社，1980
[25]王朝文，霍伟。用滑动模型实现一类非完整动力学系统的指数镇定。自动化学报，2000，Vol．26，No．2，254-257页
[26]董文杰，霍伟。一类不确定非完整动力学系统的时变镇定。自动化学报，1999，Vol．25，No．3，402-405页
[27]董文杰，霍伟。不确定非完整动力学系统的指数镇定。控制与决策，1999，Vol．14，No．4，319-323页
[28]王朝文，霍伟。一类不确定非完整动力学系统的鲁棒镇定。机器人，1998，Vol．20，No．6，442-448页
[29]董文杰，霍伟。受非完整约束移动机器人的跟踪控制。自动化学报，2000，Vol．26，No．1，1-6页
[30]王朝文，霍伟，董文杰。不确定非完整动力学系统的变结构指数镇定。北京航空航天大学学报，1999，Vol．25，No．5，442-448页
[31]董文杰，霍伟。动态非完整控制系统的指数镇定及其在一类车式机器人中的应用。机器人，1998，Vol．20，No．2，88-92页
[32]Dong Wenjie,Huo Wei. Time-Varying Adaptive Control of Uncertain Dynamic Nonholonomic Systems. 控制理论及应用，1999，Vol．16，No．3，325-328页
[33]Kevin M.Lynch,Naoji Shiroma and Hirohiko Arai, "The Roles of Shape and Motion in Dynamic Manipulation:The Butterfly Example," IEEE International Conference on Robotics and Automation, vol.4,pp.203-214, 1989
[34]Kevin M.Lynch and Naoji Shiroma, "Control for A Sample Dynamic Underactuated Nonprehensile Manipulatio," IEEE International Conference on Robotics and Automation, vol.4,pp.203-214, 1989
[35]Yoshihiko Nakamura and Ranjan Mukherjee,"Nonholonomic Path Planning of Space Robots via a Bidirectional Approach."IEEE Transactions on Roboics and Automation ,vol.7,NO.4,1991
[36]R.M.Murray and S.S.Sastry," Grasping and manipulation using multifingered robot hands",Mem.m. UCB/ERLM90/24,Unite of California at Berkeley, 1990
[37]H.Arai and S.Tachi, "Dynamic control of a manipulator with passive joints experments by a prototype manipulator",in Proc. 1990 IEEE International Workshop on Intelligent Robots and Systems,pp.935-940,1990.
[38] A lessandro De Luca, Raffaella Mattone and Giuseppe Oriolo, "Control of Underactuated Mechanica Systems:Application to the Planar 2R Robot," Proceeding of the 35th Conference on Decision and Control Kobe Janpan December 1996
[39] Naoji Shiroma, Hirohiko Arai and Kazuo Tanie, "Nonlinear Control of a Planar Free Link under a Nonholonmic Constraint," Proceeding of 8th International Conference on Advanced Robotics(ICAR'97),pp.103-109, Monterey, California,July 1997.
[40] Ablowitz M J,Clarkson P A.Solitons, "Nonlinear evolutions and inverse scattering transform," Cambridge University Press, 1991
[41] J.P. Laumond, "Controllability of a multibody mobile robot," in Proc.5th International Conference on Advanced Robotics(ICAR'91)(Pisa,Italy), pp. 1033-1038, 1991
[42] G. Oriolo and Y. Nakamura, "Underactuated manipulators:motion control under second-order nonholonomic constraints",Technical Report,University of California at Santa Barbara, 1991
[43] Ames W F. "Nonlinear partial differential equations in engineering," Vol 1.Academic Press, 1965
[44] Mahmut Reyhanoglu, Arian van der Schaft, N.Harris McClamroch. "Nonlinear Control of a Class of Underactuated Systems." Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
[45] Y.Umetani and K. Yoshida, "Resolved motion rate control of space manipulators with generalized jacobian matrix" IEEE Trans on Robotics and Automation,vol.5,pp.303-314, 1989
[46] A.De Luca and G. Oriolo, "Nonholonomy in redundant robot under kinematic inversion," 4th IFAC Symp. On Robot Control,Capri,I,pp. 179-184,1994
[47] K.M.Lynch and M.T. Mason,"Controllability of pushing,"1995 IEEE Int. Conf. On Robotics and Automation, Nagoya, J, pp. 112-119,1995
[48] A.M.Bloch,M.Reyhanoglu,and Reyhanoglu,and N.H.McClamroch,"Control and stabilization of nonholonmic dynamic systems" IEEE Trans. On Automatic Control,vol.37,pp. 1746-1757,1992.
[49] H. Arai, "Controllability of a 3-DOF manipulator with a passive joint under a nonholonomic constraint,"1996 IEEE Int. Conf. on Robotics and Automation, Minneapolis, MN, pp. 3707-3713, 1996
[50] G. Oriolo and Y. Nakamura,"Free-Joint Manipulators:Motion Control under Second-Order Nonholonomic Constraints,"Proc.of IROS'91, pp.1248-1253,1991.
[51] H. Arail and S. Tachi," Position Control of a Manipulator with Passive Joints UsingDynamicCoupling," IEEE Trans.on Robotic and Automation, Vol.7,No.4, pp.528-534,1991
[52] Jun-ichi Imura,Keigo Kobayashi,"Nonholonomic control of 3 Link Planar Manipulator", Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
[53] K.M.Lynch,N.Shiroma,"Motion Planning for a 3-DOF Robot with a Passive Joint", 1998 IEEE International Conference on Robotics and Automation
[54] Muruhan Rathinam,Richard M.Murray,"Configuration Flatness of Lagrangian Systems Underactuated by One Control", Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
[55] Giuseppe Oriolo,Yoshihiro Nakamura,"Control of Mechanical Systems with Second-Order Nonholonomic Constraints:Underactuated Manipulators", Proceedings of the 30th Conference on Decision and Control Brighton, England, December 1991
[56] Takahiro Suzuki,Masabumi Koinuma,"Chaos and Nonlinear Control of a Nonholonomic Free-Joint Manipulator", Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, Minnesota-April 1996
[57] Rafael Fierro,Frank L.Lewis,"Hybrid Control for a Class of Underactuated Mechanical Systems",IEEE Transaction on Systems,Man,and Cybernrtics-Part A:Systems and Human,Vol.29,No.6,November, 1999
[58] Hirohiko Arai,Kazuo Tanie,"Time-scaling Control of an Underactuated Manipulator", Proceedings of 1998 Conference on Robotics and Automation(ICRA'98), Leuven,Bdgium,May 1998
[59] S.Suzuki and H.Inooka,"A new golf-swing robot model emulating golfer's skill",Sports Engineering,(1999) 2
[60] Bergerman,M.;Xu,Y. "Planning collision-free motions for underactuated manipulators in constrained configration space." IEEE International Conference on Robitics and Automation,Albuquerque,NM,EUA,April 1997
[61] Kevin M. Lynch,Naoji Shiroma. "Experiments in Dynamic Rolling Manipulation"
[62] Kevin M. Lynch,Naoji Shiroma. "Experiments in Motion Planning for an Underactuated robot"
[63] 中村仁彦。非第1回 非何?。日本学会誌。Vol．11 No．4，pp．521-528，1993．
[64] 中村仁彦。非第2回 几何学的非拘束下运动计画。日本学会誌。Vol．11 No．5，pp．655-662，1993．
[65] 中村仁彦。非第3回 几何学的非拘束下运动计画。日本学会誌。Vol．11 No．6，pp．837-844，1993．
[66] 中村仁彦。非第4回 动力学的非拘束下运动。日本学会誌。Vol．11 No．7，pp．999-1005，1993．
[67] 中村仁彦。非第5回 动力学的非拘束下运动制御。日本学会誌。Vol．12 No．2，pp．231-239，1994．
[68] 中村仁彦。非。日本机械学会誌。Vol．97，No．908，pp．42-45，1994．
[69] 中村仁彦。项非机械制御。日本学会誌。Vol．13 No．6，pp．830-837，1995．
[70] 鈴木 聡一郎。剛性低碗有动特性。日本机械学会论文集，Vol．61，No．586，pp2508-2512，1995．
[71] 荒井裕彦。2階非系制御。计测制御，Vol．36，No．6，pp．404-411，1997．
[72] 鈴木 聡一郎。非驱动关关节低剛性有动特性。日本机械学会论文集，Vol．63，No．611，pp2351-2359，1997．
[73] 小林啟吾。自由关节含3自由度非制御。计测自动制御学会论文集。Vol．33，No．8，pp．799-804，1997．
[74] 荒井裕彦。非驱动关节有时间轴伸缩制御。日本学会誌。Vol．16 No．4，pp．561-568，1998．
[75] 鈴木 聡一郎。特性适应关考察。日本机械学会论文集，Vol．64，No．623，pp．2341-2346，1998．
[76] 鈴木 聡一郎。特性适应。实验的检讨。日本机械学会论文集，Vol．65，No．637，pp．216-222，1999．
[2] 浙江大学．概率论与数理统计(第二版)．高等教育出版社，1989
[3] 何广平，陆震，王凤翔编著．被动冗余度空间机器人运动学综合．机器人．2000，11(6)
[4] 熊光楞著．控制系统数字仿真．清华大学出版社，1994：37-62页，82-94页
[5] 冯登泰编．应用非线性振动力学．中国铁道出版社，1982：125-205页
[6] 熊有伦编．机器人技术基础．华中理工大学出版社，1999：15-141页
[7] 徐缤昌，阙志宏等．机器人控制工程．西北工业大学出版社，1991
[8] 周远清，张再兴等．智能机器人系统．清华大学出版社，1989
[9] 吴广玉，姜复兴等．机器人工程导论．哈尔滨工业大学出版社，1988
[10] 范印越。机器人技术。电子工业出版社，1988
[11] 申铁龙。机器人鲁棒控制基础。清华大学出版社，2000
[12] 蔡自兴。机器人学。清华大学出版社，2000
[13] 白井良明。机器人工程。科学出版社，2001
[14] 熊有伦，丁汉．机器人动力学性能指标及其优化．机械工程学报，1989，25(2)：9-15页
[15] 刘式适，刘式达．物理学中的非线性方程．北京大学出版社，1999：20-80页，127-163页
[16] 刘式适，刘式达．大气中的非线性椭圆余弦波和孤立波．中国科学，1982．4
[17] 冯纯伯，费树岷．非线性控制系统分析与设计(第2版)．电子工业出版社．
[18] 史忠科，郑永安等．控制系统的分析与设计．西北工业大学出版社，1995
[19] 夏德铃．反馈控制理论．哈尔滨工业大学出版社．1984
[20] 孙洪洲，韩其智等．李代数李超代数及其在物理忠的应用．北京大学出版社，1999
[21] 褚亦清，李翠英等．非线性振动分析．北京理工大学出版社，1996
[22] 曹建福，韩崇昭，方洋旺。非线性系统理论及应用。西安交通大学出版社，2001
[23]苏步青，刘鼎元。初等微分几何。上海科学技术出版社，1985
[24]秦元勋，王联，王慕秋．运动稳定性理论与应用，科学出版社，1980
[25]王朝文，霍伟。用滑动模型实现一类非完整动力学系统的指数镇定。自动化学报，2000，Vol．26，No．2，254-257页
[26]董文杰，霍伟。一类不确定非完整动力学系统的时变镇定。自动化学报，1999，Vol．25，No．3，402-405页
[27]董文杰，霍伟。不确定非完整动力学系统的指数镇定。控制与决策，1999，Vol．14，No．4，319-323页
[28]王朝文，霍伟。一类不确定非完整动力学系统的鲁棒镇定。机器人，1998，Vol．20，No．6，442-448页
[29]董文杰，霍伟。受非完整约束移动机器人的跟踪控制。自动化学报，2000，Vol．26，No．1，1-6页
[30]王朝文，霍伟，董文杰。不确定非完整动力学系统的变结构指数镇定。北京航空航天大学学报，1999，Vol．25，No．5，442-448页
[31]董文杰，霍伟。动态非完整控制系统的指数镇定及其在一类车式机器人中的应用。机器人，1998，Vol．20，No．2，88-92页
[32]Dong Wenjie,Huo Wei. Time-Varying Adaptive Control of Uncertain Dynamic Nonholonomic Systems. 控制理论及应用，1999，Vol．16，No．3，325-328页
[33]Kevin M.Lynch,Naoji Shiroma and Hirohiko Arai, "The Roles of Shape and Motion in Dynamic Manipulation:The Butterfly Example," IEEE International Conference on Robotics and Automation, vol.4,pp.203-214, 1989
[34]Kevin M.Lynch and Naoji Shiroma, "Control for A Sample Dynamic Underactuated Nonprehensile Manipulatio," IEEE International Conference on Robotics and Automation, vol.4,pp.203-214, 1989
[35]Yoshihiko Nakamura and Ranjan Mukherjee,"Nonholonomic Path Planning of Space Robots via a Bidirectional Approach."IEEE Transactions on Roboics and Automation ,vol.7,NO.4,1991
[36]R.M.Murray and S.S.Sastry," Grasping and manipulation using multifingered robot hands",Mem.m. UCB/ERLM90/24,Unite of California at Berkeley, 1990
[37]H.Arai and S.Tachi, "Dynamic control of a manipulator with passive joints experments by a prototype manipulator",in Proc. 1990 IEEE International Workshop on Intelligent Robots and Systems,pp.935-940,1990.
[38] A lessandro De Luca, Raffaella Mattone and Giuseppe Oriolo, "Control of Underactuated Mechanica Systems:Application to the Planar 2R Robot," Proceeding of the 35th Conference on Decision and Control Kobe Janpan December 1996
[39] Naoji Shiroma, Hirohiko Arai and Kazuo Tanie, "Nonlinear Control of a Planar Free Link under a Nonholonmic Constraint," Proceeding of 8th International Conference on Advanced Robotics(ICAR'97),pp.103-109, Monterey, California,July 1997.
[40] Ablowitz M J,Clarkson P A.Solitons, "Nonlinear evolutions and inverse scattering transform," Cambridge University Press, 1991
[41] J.P. Laumond, "Controllability of a multibody mobile robot," in Proc.5th International Conference on Advanced Robotics(ICAR'91)(Pisa,Italy), pp. 1033-1038, 1991
[42] G. Oriolo and Y. Nakamura, "Underactuated manipulators:motion control under second-order nonholonomic constraints",Technical Report,University of California at Santa Barbara, 1991
[43] Ames W F. "Nonlinear partial differential equations in engineering," Vol 1.Academic Press, 1965
[44] Mahmut Reyhanoglu, Arian van der Schaft, N.Harris McClamroch. "Nonlinear Control of a Class of Underactuated Systems." Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
[45] Y.Umetani and K. Yoshida, "Resolved motion rate control of space manipulators with generalized jacobian matrix" IEEE Trans on Robotics and Automation,vol.5,pp.303-314, 1989
[46] A.De Luca and G. Oriolo, "Nonholonomy in redundant robot under kinematic inversion," 4th IFAC Symp. On Robot Control,Capri,I,pp. 179-184,1994
[47] K.M.Lynch and M.T. Mason,"Controllability of pushing,"1995 IEEE Int. Conf. On Robotics and Automation, Nagoya, J, pp. 112-119,1995
[48] A.M.Bloch,M.Reyhanoglu,and Reyhanoglu,and N.H.McClamroch,"Control and stabilization of nonholonmic dynamic systems" IEEE Trans. On Automatic Control,vol.37,pp. 1746-1757,1992.
[49] H. Arai, "Controllability of a 3-DOF manipulator with a passive joint under a nonholonomic constraint,"1996 IEEE Int. Conf. on Robotics and Automation, Minneapolis, MN, pp. 3707-3713, 1996
[50] G. Oriolo and Y. Nakamura,"Free-Joint Manipulators:Motion Control under Second-Order Nonholonomic Constraints,"Proc.of IROS'91, pp.1248-1253,1991.
[51] H. Arail and S. Tachi," Position Control of a Manipulator with Passive Joints UsingDynamicCoupling," IEEE Trans.on Robotic and Automation, Vol.7,No.4, pp.528-534,1991
[52] Jun-ichi Imura,Keigo Kobayashi,"Nonholonomic control of 3 Link Planar Manipulator", Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
[53] K.M.Lynch,N.Shiroma,"Motion Planning for a 3-DOF Robot with a Passive Joint", 1998 IEEE International Conference on Robotics and Automation
[54] Muruhan Rathinam,Richard M.Murray,"Configuration Flatness of Lagrangian Systems Underactuated by One Control", Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996
[55] Giuseppe Oriolo,Yoshihiro Nakamura,"Control of Mechanical Systems with Second-Order Nonholonomic Constraints:Underactuated Manipulators", Proceedings of the 30th Conference on Decision and Control Brighton, England, December 1991
[56] Takahiro Suzuki,Masabumi Koinuma,"Chaos and Nonlinear Control of a Nonholonomic Free-Joint Manipulator", Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, Minnesota-April 1996
[57] Rafael Fierro,Frank L.Lewis,"Hybrid Control for a Class of Underactuated Mechanical Systems",IEEE Transaction on Systems,Man,and Cybernrtics-Part A:Systems and Human,Vol.29,No.6,November, 1999
[58] Hirohiko Arai,Kazuo Tanie,"Time-scaling Control of an Underactuated Manipulator", Proceedings of 1998 Conference on Robotics and Automation(ICRA'98), Leuven,Bdgium,May 1998
[59] S.Suzuki and H.Inooka,"A new golf-swing robot model emulating golfer's skill",Sports Engineering,(1999) 2
[60] Bergerman,M.;Xu,Y. "Planning collision-free motions for underactuated manipulators in constrained configration space." IEEE International Conference on Robitics and Automation,Albuquerque,NM,EUA,April 1997
[61] Kevin M. Lynch,Naoji Shiroma. "Experiments in Dynamic Rolling Manipulation"
[62] Kevin M. Lynch,Naoji Shiroma. "Experiments in Motion Planning for an Underactuated robot"
[63] 中村仁彦。非第1回 非何?。日本学会誌。Vol．11 No．4，pp．521-528，1993．
[64] 中村仁彦。非第2回 几何学的非拘束下运动计画。日本学会誌。Vol．11 No．5，pp．655-662，1993．
[65] 中村仁彦。非第3回 几何学的非拘束下运动计画。日本学会誌。Vol．11 No．6，pp．837-844，1993．
[66] 中村仁彦。非第4回 动力学的非拘束下运动。日本学会誌。Vol．11 No．7，pp．999-1005，1993．
[67] 中村仁彦。非第5回 动力学的非拘束下运动制御。日本学会誌。Vol．12 No．2，pp．231-239，1994．
[68] 中村仁彦。非。日本机械学会誌。Vol．97，No．908，pp．42-45，1994．
[69] 中村仁彦。项非机械制御。日本学会誌。Vol．13 No．6，pp．830-837，1995．
[70] 鈴木 聡一郎。剛性低碗有动特性。日本机械学会论文集，Vol．61，No．586，pp2508-2512，1995．
[71] 荒井裕彦。2階非系制御。计测制御，Vol．36，No．6，pp．404-411，1997．
[72] 鈴木 聡一郎。非驱动关关节低剛性有动特性。日本机械学会论文集，Vol．63，No．611，pp2351-2359，1997．
[73] 小林啟吾。自由关节含3自由度非制御。计测自动制御学会论文集。Vol．33，No．8，pp．799-804，1997．
[74] 荒井裕彦。非驱动关节有时间轴伸缩制御。日本学会誌。Vol．16 No．4，pp．561-568，1998．
[75] 鈴木 聡一郎。特性适应关考察。日本机械学会论文集，Vol．64，No．623，pp．2341-2346，1998．
[76] 鈴木 聡一郎。特性适应。实验的检讨。日本机械学会论文集，Vol．65，No．637，pp．216-222，1999．