一种并联主手的动力学建模与机构性能分析
摘要
力觉主手是遥操作和虚拟现实系统中的关键设备。本文结合国家863计划项目“遥操作辅助正骨医疗机器人系统研究”和“远程医疗机器人关键技术与系统研究”,建立一种6自由度缩放仪型并联主手的运动学和动力学数学模型,分析其工作空间和灵巧度性能,并对其进行机构参数优化。
针对6自由度缩放仪型并联机器人的串/并联混合构型的特点,在机构分析的基础上对其进行运动学建模。采用齐次坐标变换法建立位置逆解方程。采用Newton–Raphson法求解正动力学中的非线性方程组。采用矢量链方法建立输入速度与加速度分别和上平台的速度与加速度之间的映射关系方程。
针对该主手每条支链包含较多杆件所导致的用传统动力学方法建模计算量大的问题,在运动学模型的基础上,运用Kane方程建立该主手的逆动力学数学模型。并在动力学仿真软件Adams中建立其正动力学模型。对于主手上平台的两种运动实例,将由数学模型产生的驱动力矩变化规律输入到Adams模型中。再将Adams模型上平台的运动规律与最初设定的运动规律对比,来验证动力学数学模型的正确性。之后,建立只考虑上平台质量的简单动力学数学模型,通过对比分析来说明动力学建模的必要性。
在机构的工作空间研究中,采用能够直观描述物体姿态的一种新型Euler角,称为T&T角。引入球坐标扫描搜索法来获取完整的位置工作空间边界,并给出相应的工作空间体积的计算方法。之后,基于以上方法分析该主手的位置工作空间和姿态工作空间。并描述一种位姿耦合的新型工作空间,满足特定姿态能力要求的位置工作空间,来表征一种比较完整的工作空间能力。
为更有效地分析机构的灵巧度,针对机构的Jacobian矩阵中的元素单位不统一的问题,在现有单位归一化方法的基础上,提出一种新的通用的归一化方法。并且基于该方法分析主手的局部灵巧度分布。之后利用Matlab的遗传算法工具箱,以机构全局灵巧度为优化目标,将主手的工作空间要求和紧凑性要求作为双重约束条件,对主手的机构参数进行优化。然后通过对比优化前后机构的局部灵巧度分布情况,来验证优化的有效性。
Haptic devices are the key components in the teleoperation and virtual realitysystems. As part of the 863 projects of“Study of a Robot Assisted Orthopedic Tele-operation System”and“Study of the Key Technologies of Telemedical Robots and Sy-stems”, This thesis builds the kinematic and dynamic models of a 6-DOF 3-subchainhaptic device, analyzes its workspace and dexterity, and optimizes its kinematics pa-rameters.
Considering that the haptic device is a serial/parallel hybrid mechanism, the ki-nematic analysis is firstly performed after the mechanism analysis. The inverse ki-nematic models are built using the coordinate transformation method. The Newton-Raphson method is used to solve the nonlinear equations in the forward kinematics.And the velocity and acceleration mapping equations are derived through the vectorchain method.
Based on the kinematic model, the mathematic model of the inverse dynamicsis built by the Kane equation, since conventional dynamic modeling methods willresult in complex derivation and computation. And the forward dynamic model isbuilt using Adams. For two motion instances of the top plate, the mathematic model isused for obtaining the six input torques, which are then imported to the Adams modelto actuate the corresponding links. By comparing the motion of the Adams model withthe given motion, the mathematic model is validated. Then a simple dynamic modelsolely considering the mass of the top plate is built for comparison.
In the workspace analysis, a new set of Euler angles called T&T angles is utilizedto represent the orientation of the top plate. The spherical scanning search method isused to obtain complete position workspace. The position and orientation workspaceof the haptic device is analyzed based on those methods. Besides, a novel position-orientation-coupled workspace called position workspace with a specified orientationcapability requirement is presented as a more appropriate representation of workspacecapability.
In order to effectively analyze the dexterity, addressing the unit–inhomogeneityissue in the Jacobian matrix, a new general normalization method is proposed. Andthis method is used to analyze the local dexterity of the haptic device. Then the genetic algorithm tool in Matlab is used for the optimization of the kinematic parameters, withworkspace and compactness as the optimization constraints, and global dexterity asthe optimization objective. The result is validated by comparing the dexterity of themechanism before and after the optimization.
针对6自由度缩放仪型并联机器人的串/并联混合构型的特点,在机构分析的基础上对其进行运动学建模。采用齐次坐标变换法建立位置逆解方程。采用Newton–Raphson法求解正动力学中的非线性方程组。采用矢量链方法建立输入速度与加速度分别和上平台的速度与加速度之间的映射关系方程。
针对该主手每条支链包含较多杆件所导致的用传统动力学方法建模计算量大的问题,在运动学模型的基础上,运用Kane方程建立该主手的逆动力学数学模型。并在动力学仿真软件Adams中建立其正动力学模型。对于主手上平台的两种运动实例,将由数学模型产生的驱动力矩变化规律输入到Adams模型中。再将Adams模型上平台的运动规律与最初设定的运动规律对比,来验证动力学数学模型的正确性。之后,建立只考虑上平台质量的简单动力学数学模型,通过对比分析来说明动力学建模的必要性。
在机构的工作空间研究中,采用能够直观描述物体姿态的一种新型Euler角,称为T&T角。引入球坐标扫描搜索法来获取完整的位置工作空间边界,并给出相应的工作空间体积的计算方法。之后,基于以上方法分析该主手的位置工作空间和姿态工作空间。并描述一种位姿耦合的新型工作空间,满足特定姿态能力要求的位置工作空间,来表征一种比较完整的工作空间能力。
为更有效地分析机构的灵巧度,针对机构的Jacobian矩阵中的元素单位不统一的问题,在现有单位归一化方法的基础上,提出一种新的通用的归一化方法。并且基于该方法分析主手的局部灵巧度分布。之后利用Matlab的遗传算法工具箱,以机构全局灵巧度为优化目标,将主手的工作空间要求和紧凑性要求作为双重约束条件,对主手的机构参数进行优化。然后通过对比优化前后机构的局部灵巧度分布情况,来验证优化的有效性。
Haptic devices are the key components in the teleoperation and virtual realitysystems. As part of the 863 projects of“Study of a Robot Assisted Orthopedic Tele-operation System”and“Study of the Key Technologies of Telemedical Robots and Sy-stems”, This thesis builds the kinematic and dynamic models of a 6-DOF 3-subchainhaptic device, analyzes its workspace and dexterity, and optimizes its kinematics pa-rameters.
Considering that the haptic device is a serial/parallel hybrid mechanism, the ki-nematic analysis is firstly performed after the mechanism analysis. The inverse ki-nematic models are built using the coordinate transformation method. The Newton-Raphson method is used to solve the nonlinear equations in the forward kinematics.And the velocity and acceleration mapping equations are derived through the vectorchain method.
Based on the kinematic model, the mathematic model of the inverse dynamicsis built by the Kane equation, since conventional dynamic modeling methods willresult in complex derivation and computation. And the forward dynamic model isbuilt using Adams. For two motion instances of the top plate, the mathematic model isused for obtaining the six input torques, which are then imported to the Adams modelto actuate the corresponding links. By comparing the motion of the Adams model withthe given motion, the mathematic model is validated. Then a simple dynamic modelsolely considering the mass of the top plate is built for comparison.
In the workspace analysis, a new set of Euler angles called T&T angles is utilizedto represent the orientation of the top plate. The spherical scanning search method isused to obtain complete position workspace. The position and orientation workspaceof the haptic device is analyzed based on those methods. Besides, a novel position-orientation-coupled workspace called position workspace with a specified orientationcapability requirement is presented as a more appropriate representation of workspacecapability.
In order to effectively analyze the dexterity, addressing the unit–inhomogeneityissue in the Jacobian matrix, a new general normalization method is proposed. Andthis method is used to analyze the local dexterity of the haptic device. Then the genetic algorithm tool in Matlab is used for the optimization of the kinematic parameters, withworkspace and compactness as the optimization constraints, and global dexterity asthe optimization objective. The result is validated by comparing the dexterity of themechanism before and after the optimization.
引文
1 R. C. Goertz. Fundamentals of General-purpose Remote Manipulators. Nucleo-nics. 1952, 10:36–42
2 K. Vlachos, E. Papadopoulos, D. N. Mitropoulos. Design and Implementationof a Haptic Device for Training in Urological Operations. IEEE Trans. Robot.Automat. 2003, 19(5):801–809
3 K. Y. Woo, B. D. Jin, D.-S. Kwon. A 6 DOF Force-re?ecting Hand ControllerUsing the Fivebar Parallel Mechanism. Proc. IEEE Int. Conf. Robot. Automat.Leuven, Belgium, 1998:1–6
4 H. Iwata. Artificial Reality with Force Feadback: Development of DesktopVirtual Space with Compact Master Manipulator. Computer Graphics. 1990,24(4):165–170
5 Y.Tsumaki, H.Naruse, D.N.Nenchev,等. Design of a Compact 6-DOF HapticInterface. Proc. IEEE Int. Conf. Robot. Automat. Leuven, Belgium, 1998:2580–2585
6 B. Dasgupta, T. S. Mruthyunjaya. The Stewart Platform Manipulator: A Review.Mech. Mach. Theory. 2000, 35:15–40
7 J.-P. Merlet. Parallel Robots. Kluwer Academic Publishers, 2000
8黄真,赵永生,赵铁石.空间机构学.宇航出版社, 2007
9 E. F. Fichter. A Stewart Platform-based Manipulator: General Theory and Prac-tical Constructions. Int. J. Robot. Res. 1986, 5(2):157–182
10 B. Dasgupta, T. S. Mruthyunjaya. A Newton–euler Formulation for the InverseDynamics of the Stewart Platform Manipulator. Mech. Mach. Theory. 1998,33(8):1135–1152
11 B. Dasgupta, T. S. Mruthyunjaya. A Newton–euler Formulation for the InverseDynamics of the Stewart Platform Manipulator. Mech. Mach. Theory. 1998,33(7):993–1012
12 L.-W. Tsai. Solving the Inverse Dynamics of a Stewart–gough Manipulator bythe Principle of Virtual Work. J. Mech. Des. 2000, 122:3–9
13 J. Wang, C. M. Gosselin. A New Approach for the Dynamic Analysis of ParallelManipulators. Multibody System Dynamics. 1998, 2:317–334
14 Z. Geng, L. S. Haynes, J. D. Lee, et al. Parallel Manipulator. Robot. and Auto-nomous Sys. 1992, 9:237–254
15 K. Liu, F. Lewis, G. Lebret, et al. The Singularities and Dynamics of a StewartPlatform Manipulator. J. Intel. Robot. Sys. 1993, 8:287–308
16 T. Geike, J. McPhee. Inverse Dynamic Analysis of Parallel Manipulators withFull Mobility. Mech. Mach. Theory. 2003, 38:549–562
17 B. Monsarrat, C. M. Gosselin. Workspace Analysis and Optimal Design of a3-leg 6-DOF Parallel Platform Mechanism. IEEE Trans. Robot. Automat. 2003,19(6):954–966
18 J.-P. Merlet. Designing a Parallel Manipulator for a Specific Workspace. Int. J.Robot. Res. 1997, 16(4):3726–3731
19 J. K. Salisbury, J. J. Craig. Articulated Hands: Force Control and KinematicIssues. Int. J. Robot. Res. 1982, 1(1):4–17
20 H. Lipkin, J. Duffy. Hybrid Twist and Wrench Control for a Robotic Manipulator.ASME J. Mech. Trans. Automat. Des. 1988, 110(6):138–144
21 C. M. Gosselin. Dexterity Indices for Planar and Spatial Robotic Manipulators.Proc. IEEE Int. Conf. Robot. Automat. Cincinnati, OH, 1990:650–655
22 S.-G. Kim, J. Ryu. New Dimensionally Homogeneous Jacobian Matrix Formu-lation by Three End-effector Points for Optimal Design of Parallel Manipulators.IEEE Trans. Robot. Automat. 2003, 19(4):731–737
23 G. Pond, J. A. Carretero. Formulating Jacobian Matrices for the Dexterity Ana-lysis of Parallel Manipulators. Mech. Mach. Theory. 2006, 41(12):1505–1519
24 G. Pond, J. A. Carretero. Quantitative Dexterous Workspace Com-parison of Parallel Manipulators. Mech. Mach. Theory (2006),doi:10.1016/j.mechmachtheory.2006.10.004
25 M. Tandirci, J. Angeles, F. Ranjbaran. The Characteristic Point and CharacteristicLength of Robotic Manipulators. Proc. ASME 22nd Biennial Conf. Robotics,Spatial Mechanics, Mechanical Systems. Scottsdale, AZ, 1992:203–208
26 O. Ma, J. Angeles. Optimum Architecture Design of Platform Manipulators.Proc. IEEE Int. Conf. Adv. Robot. Scottsdale, AZ, 1991:1130–1135
27 J. Angeles. Kinematic Isotropy in Humans and Machines. Proc. IFTOMM 9thWorld Congr. Theory of Mach. Mech. Milan, Italy, 1995:XLII–XLIX
28 J. Angeles, F. Ranjbaran, R. V. Ratel. On the Design of the Kinematic Structureof Seven-axes Redundant Manipulators for Maximum Conditioning. Proc. IEEEInt. Conf. Robot. Automat. Nice, France, 1992:494–499
29 G. Nawratil07. New Performance Indices for 6R Robots. Mech. Mach. Theory(2007), doi:10.1016/j.mechmachtheory.2006.12.007
30 L. J. Stocco, S. E. Salcudean, F. Sassani. Matrix Normalization for OptimalKinematic Design. Proc. IEEE Int. Conf. Robot. Automat. Leuven, Belgium,1998:1–6
31 L. J. Stocco, S. E. Salcudean, F. Sassani. On the Use of Scaling Matrices forTask-specific Robot Design. IEEE Trans. Robot. Automat. 1999, 15(5):958–96532 L. J. Stocco, S. E. Salcudean, F. Sassani. Optimal Kinematic Design of a HapticPen. IEEE/ASME Trans. Mechatronics. 2001, 6(3):210–220
33 A. Bowling, O. Khatib. The Dynamic Capability Equations: A New Tool for Ana-lyzing Robotic Manipulator Performance. IEEE Trans. Robot. 2005, 21(1):115–123
34 O. Altuzarra, O. Salgado, V. Petuya, et al. Point-based Jacobian Formulationfor Computational Kinematics of Manipulators. Mech. Mach. Theory. 2006,41:1407–1423
35 R. Boudreau, C. M. Gosselin. The Synthesis of Planar Parallel Manipulators witha Genetic Algorithm. ASME J. Mech. Des. 1999, 121(4):533–537
36 J. A. Cabrera, A. Simon, M. Prado. Optimal Synthesis of Mechanisms withGenetic Algorithms. Mech. Mach. Theory. 2002, 37:1165–1177
37 P. S. Shiakolas, D. Koladiya, J. Kebrle. On the Optimum Synthesis Six-bar Lin-kage Using Differential Evolution and the Geometric Centroid of Precision Po-sitions Technique. Mech. Mach. Theory. 2005, 40:319–335
38 R. Storn, K. Price. Differential Evolution: A Simple and Efficient Heuristic Sche-me for Global Optimization Over Continuous Spaces. J. Global Optimization.1997, 11:341–359
39 J. A. Cabrera, F. Nadal, J. P. Munoz, et al. Multiobjective Constrained OptimalSynthesis of Planar Mechanisms Using a New Evolutionary Algorithm. Mech.Mach. Theory. 2007, 42:791–806
40 A. B. K. Rao, P. V. M. Rao, S. K. Saha. Dimensional Design of Hexaslides forOptimal Workspace and Dexterity. IEEE Trans. Robot. 2005, 21(3):444–449
41 X.-J. Liu, J. Wang, G. Pritschow. Performance Atlases and Optimum Designof Planar 5R Symmetrical Parallel Mechanisms. Mech. Mach. Theory. 2006,41:119–144
42 X.-J. Liu, J. Wang. A New Methodology for Optimal Kinema-tic Design of Parallel Mechanisms. Mech. Mach. Theory (2006),doi:10.1016/j.mechmachtheory.2006.08.002
43 I. A. Bonev, J. Ryu. Orientation Workspace Analysis of 6-DOF Parallel Ma-nipulators. Proc. ASME Des. Eng. Tech. Conf. Las Vegas, NV, 1999:PaperDETC99/DAC–8646
44 I. A. Bonev, J. Ryu. A New Approach to Orientation Workspace Analysis of6-DOF Parallel Manipulators. Mech. Mach. Theory. 2001, 36:15–18
45 T. Huang, J. Wang, D. J. Whitehouse. Closed Form Solution to Workspace ofHexapod-based Virtual Axis Machine Tools. J. Mech. Des. 1999, 121(1):26–31
46 Y. Wang. Workspace Analysis of a Novel Closed-chain Manipulator. Masterthesis, Case Western Reserve Univ. 1999
47 N. R. Crawford, G. T. Yamaguchi, C. A. Dickman. A New Technique forDetermining 3-d Joint Angles: The Tilt/twist Method. Clin. Biomech. 1999,14(3):153–165
48 J. U. Korein. A Geometric Investigation of Reach. Cambridge, MA, 1984
49 O. Masory, J. Wang. Workspace Evaluation of Stewart Platforms. Proc. ASMEDesign Engineering Tech. Conf. Scottsdale, AZ, 1992:337–346
50 J. R. Conti, C. M. Clinton, G. Zhang, et al. Dynamic Variation of the Workspaceof an Octahedral Hexapod Machine Tool During Machining. Inst. Syst. Res.,Univ. Maryland, College Park, 1997
51 J. H. Holland. Genetic Algorithms and the Optimal Allocations of Trials. SIAMJ. Computing. 1973, 2(2):88–105
52 D. E. Goldberg. Genetic Algorithms in Search, Optimization and Machine Lear-ning. Addison-Wesley Professional, 1989
2 K. Vlachos, E. Papadopoulos, D. N. Mitropoulos. Design and Implementationof a Haptic Device for Training in Urological Operations. IEEE Trans. Robot.Automat. 2003, 19(5):801–809
3 K. Y. Woo, B. D. Jin, D.-S. Kwon. A 6 DOF Force-re?ecting Hand ControllerUsing the Fivebar Parallel Mechanism. Proc. IEEE Int. Conf. Robot. Automat.Leuven, Belgium, 1998:1–6
4 H. Iwata. Artificial Reality with Force Feadback: Development of DesktopVirtual Space with Compact Master Manipulator. Computer Graphics. 1990,24(4):165–170
5 Y.Tsumaki, H.Naruse, D.N.Nenchev,等. Design of a Compact 6-DOF HapticInterface. Proc. IEEE Int. Conf. Robot. Automat. Leuven, Belgium, 1998:2580–2585
6 B. Dasgupta, T. S. Mruthyunjaya. The Stewart Platform Manipulator: A Review.Mech. Mach. Theory. 2000, 35:15–40
7 J.-P. Merlet. Parallel Robots. Kluwer Academic Publishers, 2000
8黄真,赵永生,赵铁石.空间机构学.宇航出版社, 2007
9 E. F. Fichter. A Stewart Platform-based Manipulator: General Theory and Prac-tical Constructions. Int. J. Robot. Res. 1986, 5(2):157–182
10 B. Dasgupta, T. S. Mruthyunjaya. A Newton–euler Formulation for the InverseDynamics of the Stewart Platform Manipulator. Mech. Mach. Theory. 1998,33(8):1135–1152
11 B. Dasgupta, T. S. Mruthyunjaya. A Newton–euler Formulation for the InverseDynamics of the Stewart Platform Manipulator. Mech. Mach. Theory. 1998,33(7):993–1012
12 L.-W. Tsai. Solving the Inverse Dynamics of a Stewart–gough Manipulator bythe Principle of Virtual Work. J. Mech. Des. 2000, 122:3–9
13 J. Wang, C. M. Gosselin. A New Approach for the Dynamic Analysis of ParallelManipulators. Multibody System Dynamics. 1998, 2:317–334
14 Z. Geng, L. S. Haynes, J. D. Lee, et al. Parallel Manipulator. Robot. and Auto-nomous Sys. 1992, 9:237–254
15 K. Liu, F. Lewis, G. Lebret, et al. The Singularities and Dynamics of a StewartPlatform Manipulator. J. Intel. Robot. Sys. 1993, 8:287–308
16 T. Geike, J. McPhee. Inverse Dynamic Analysis of Parallel Manipulators withFull Mobility. Mech. Mach. Theory. 2003, 38:549–562
17 B. Monsarrat, C. M. Gosselin. Workspace Analysis and Optimal Design of a3-leg 6-DOF Parallel Platform Mechanism. IEEE Trans. Robot. Automat. 2003,19(6):954–966
18 J.-P. Merlet. Designing a Parallel Manipulator for a Specific Workspace. Int. J.Robot. Res. 1997, 16(4):3726–3731
19 J. K. Salisbury, J. J. Craig. Articulated Hands: Force Control and KinematicIssues. Int. J. Robot. Res. 1982, 1(1):4–17
20 H. Lipkin, J. Duffy. Hybrid Twist and Wrench Control for a Robotic Manipulator.ASME J. Mech. Trans. Automat. Des. 1988, 110(6):138–144
21 C. M. Gosselin. Dexterity Indices for Planar and Spatial Robotic Manipulators.Proc. IEEE Int. Conf. Robot. Automat. Cincinnati, OH, 1990:650–655
22 S.-G. Kim, J. Ryu. New Dimensionally Homogeneous Jacobian Matrix Formu-lation by Three End-effector Points for Optimal Design of Parallel Manipulators.IEEE Trans. Robot. Automat. 2003, 19(4):731–737
23 G. Pond, J. A. Carretero. Formulating Jacobian Matrices for the Dexterity Ana-lysis of Parallel Manipulators. Mech. Mach. Theory. 2006, 41(12):1505–1519
24 G. Pond, J. A. Carretero. Quantitative Dexterous Workspace Com-parison of Parallel Manipulators. Mech. Mach. Theory (2006),doi:10.1016/j.mechmachtheory.2006.10.004
25 M. Tandirci, J. Angeles, F. Ranjbaran. The Characteristic Point and CharacteristicLength of Robotic Manipulators. Proc. ASME 22nd Biennial Conf. Robotics,Spatial Mechanics, Mechanical Systems. Scottsdale, AZ, 1992:203–208
26 O. Ma, J. Angeles. Optimum Architecture Design of Platform Manipulators.Proc. IEEE Int. Conf. Adv. Robot. Scottsdale, AZ, 1991:1130–1135
27 J. Angeles. Kinematic Isotropy in Humans and Machines. Proc. IFTOMM 9thWorld Congr. Theory of Mach. Mech. Milan, Italy, 1995:XLII–XLIX
28 J. Angeles, F. Ranjbaran, R. V. Ratel. On the Design of the Kinematic Structureof Seven-axes Redundant Manipulators for Maximum Conditioning. Proc. IEEEInt. Conf. Robot. Automat. Nice, France, 1992:494–499
29 G. Nawratil07. New Performance Indices for 6R Robots. Mech. Mach. Theory(2007), doi:10.1016/j.mechmachtheory.2006.12.007
30 L. J. Stocco, S. E. Salcudean, F. Sassani. Matrix Normalization for OptimalKinematic Design. Proc. IEEE Int. Conf. Robot. Automat. Leuven, Belgium,1998:1–6
31 L. J. Stocco, S. E. Salcudean, F. Sassani. On the Use of Scaling Matrices forTask-specific Robot Design. IEEE Trans. Robot. Automat. 1999, 15(5):958–96532 L. J. Stocco, S. E. Salcudean, F. Sassani. Optimal Kinematic Design of a HapticPen. IEEE/ASME Trans. Mechatronics. 2001, 6(3):210–220
33 A. Bowling, O. Khatib. The Dynamic Capability Equations: A New Tool for Ana-lyzing Robotic Manipulator Performance. IEEE Trans. Robot. 2005, 21(1):115–123
34 O. Altuzarra, O. Salgado, V. Petuya, et al. Point-based Jacobian Formulationfor Computational Kinematics of Manipulators. Mech. Mach. Theory. 2006,41:1407–1423
35 R. Boudreau, C. M. Gosselin. The Synthesis of Planar Parallel Manipulators witha Genetic Algorithm. ASME J. Mech. Des. 1999, 121(4):533–537
36 J. A. Cabrera, A. Simon, M. Prado. Optimal Synthesis of Mechanisms withGenetic Algorithms. Mech. Mach. Theory. 2002, 37:1165–1177
37 P. S. Shiakolas, D. Koladiya, J. Kebrle. On the Optimum Synthesis Six-bar Lin-kage Using Differential Evolution and the Geometric Centroid of Precision Po-sitions Technique. Mech. Mach. Theory. 2005, 40:319–335
38 R. Storn, K. Price. Differential Evolution: A Simple and Efficient Heuristic Sche-me for Global Optimization Over Continuous Spaces. J. Global Optimization.1997, 11:341–359
39 J. A. Cabrera, F. Nadal, J. P. Munoz, et al. Multiobjective Constrained OptimalSynthesis of Planar Mechanisms Using a New Evolutionary Algorithm. Mech.Mach. Theory. 2007, 42:791–806
40 A. B. K. Rao, P. V. M. Rao, S. K. Saha. Dimensional Design of Hexaslides forOptimal Workspace and Dexterity. IEEE Trans. Robot. 2005, 21(3):444–449
41 X.-J. Liu, J. Wang, G. Pritschow. Performance Atlases and Optimum Designof Planar 5R Symmetrical Parallel Mechanisms. Mech. Mach. Theory. 2006,41:119–144
42 X.-J. Liu, J. Wang. A New Methodology for Optimal Kinema-tic Design of Parallel Mechanisms. Mech. Mach. Theory (2006),doi:10.1016/j.mechmachtheory.2006.08.002
43 I. A. Bonev, J. Ryu. Orientation Workspace Analysis of 6-DOF Parallel Ma-nipulators. Proc. ASME Des. Eng. Tech. Conf. Las Vegas, NV, 1999:PaperDETC99/DAC–8646
44 I. A. Bonev, J. Ryu. A New Approach to Orientation Workspace Analysis of6-DOF Parallel Manipulators. Mech. Mach. Theory. 2001, 36:15–18
45 T. Huang, J. Wang, D. J. Whitehouse. Closed Form Solution to Workspace ofHexapod-based Virtual Axis Machine Tools. J. Mech. Des. 1999, 121(1):26–31
46 Y. Wang. Workspace Analysis of a Novel Closed-chain Manipulator. Masterthesis, Case Western Reserve Univ. 1999
47 N. R. Crawford, G. T. Yamaguchi, C. A. Dickman. A New Technique forDetermining 3-d Joint Angles: The Tilt/twist Method. Clin. Biomech. 1999,14(3):153–165
48 J. U. Korein. A Geometric Investigation of Reach. Cambridge, MA, 1984
49 O. Masory, J. Wang. Workspace Evaluation of Stewart Platforms. Proc. ASMEDesign Engineering Tech. Conf. Scottsdale, AZ, 1992:337–346
50 J. R. Conti, C. M. Clinton, G. Zhang, et al. Dynamic Variation of the Workspaceof an Octahedral Hexapod Machine Tool During Machining. Inst. Syst. Res.,Univ. Maryland, College Park, 1997
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