并联机构瞬轴面研究进展

doi:10.3901/JME.2023.21.131

摘要/Abstract

摘要： 沿动定瞬轴面连续地螺旋对滚，是并联机构固有的运动特性，也是并联机构转动和移动强耦合输出的几何表现形式。这种复杂空间刚体运动，极大地增加了并联机构的运动学标定和控制的难度。近年来，有学者利用瞬轴面来定性地描述这类机构的复杂运动特性，也有学者利用瞬轴面来优化设计机构。最近，有学者提出了动定瞬轴面螺旋对滚方程，来定量地计算这种复杂的空间运动。但是，瞬轴面在并联机构学中的研究和应用，并未受到国内外学者的广泛关注。为此，首先分析了并联机构的运动特性，综述了并联机构转轴和瞬轴面的分布规律，回顾了并联机构瞬轴面的计算方法和动定瞬轴面螺旋对滚方程的推导过程，并给出了该方程的几何应用。最后，展望了瞬轴面在并联机构学研究中的发展方向。

关键词: 瞬轴面, 并联机构, 本征基旋量, 拟圆柱面, 节距双曲面

Abstract: Screw rolling along the moving and fixed axodes is the intrinsic motion characteristic of PMs (parallel mechanisms). It is also the geometric presentation of the close coupling of rotation and translation of PMs. This close coupling motion greatly increases the difficulty of kinematic calibration and control of PMs. So, in recent years, some scholars have qualitatively described the complex motion characteristics of such PMs and optimized the mechanism design by the axode. One of them has developed the screw rolling equation between fixed and moving axodes, which can quantificationally compute the rigid motion produced by the screw rolling along the moving and fixed axodes. However, their research and application in the theory of PMs have never got much attention from domestic and foreign scholars. So, the motion characteristics of PMs are analyzed. The distribution regularity of the rotation axis and axodes of PMs, the computing method of axodes, and the derivation procedure of the screw rolling equation are reviewed. Then its application in the derivation of cycloid parameter equation is presented. Finally, the development tendency of axodes research in the theory of PMs has been prospected..

Key words: axode, parallel mechanism, proper basis screw, cylindroid, pitch-hyperboloid

中图分类号:

TH112

张雷雷, 赵延治, 赵铁石. 并联机构瞬轴面研究进展[J]. 机械工程学报, 2023, 59(21): 131-146.

ZHANG Leilei, ZHAO Yanzhi, ZHAO Tieshi. State of the Art of Axodes Traced by Parallel Mechanism[J]. Journal of Mechanical Engineering, 2023, 59(21): 131-146.

参考文献

[1] EDGE W L. The theory of ruled surfaces[M]. Cambridge:Cambridge University Press, 2011.
[2] POTTMANN H, WALLNER J. Computational line geometry[M]. Berlin:Springer Science & Business Media, 2009.
[3] 陈子明,黄真,曹文熬. 空间少自由度并联机构瞬时螺旋运动轴线曲面的几何表示方法[J]. 燕山大学学报, 2012, 36(2):113-118. CHEN Ziming, HUANG Zhen, CAO Wenao. Geometric expression of axodes of spatial lower mobility parallel mechanism[J]. Journal of Yanshan University, 2012, 36(2):113-118.
[4] 陈子明. 新型对称三自由度并联机构的研究[D]. 秦皇岛:燕山大学, 2012. CHEN Ziming. Study on novel symmetrical parallel mechanisms with three degrees of freedom[D]. Qinhuangdao:Yanshan University, 2012.
[5] YU J J, JIN Z, KONG X W. Identification and comparison for continuous motion characteristics of three two-degree-of-freedom pointing mechanisms[J]. Journal of Mechanisms and Robotics, 2017, 9(5):1-13.
[6] 于靖军,刘凯,孔宪文. 多模式机构研究进展[J]. 机械工程学报, 2020, 56(19):14-27. YU Jingjun, LIU Kai, KONG Xianwen. State of the art of multi-mode mechanisms[J]. Journal of Mechanical Engineering, 2020, 56(19):14-27.
[7] LIU K, KONG X, YU J. Operation mode analysis of lower-mobility parallel mechanisms based on dual quaternions[J]. Mechanism and Machine Theory, 2019, 142:1-6.
[8] LIU K, YU J, LIU X. Operation mode comparison of three 1T2R parallel mechanisms based on geometric constraints[C]//201916th International Conference on Ubiquitous Robots (UR), 2019:137-142.
[9] OPHASWONGSE C, AGRAWAL S K. Optimal design of a novel 3-DOF orientational parallel mechanism for pelvic assistance on a wheelchair:An approach based on kinematic geometry and screw theory[J]. IEEE Robotics and Automation Letters, 2020, 5(2):3315-3322.
[10] D'ALESSIO J, RUSSELL K, SODHI R S. On transfemoral prosthetic knee design using RRSS motion and axode generation[C]//Advances in Mechanism Design II, 2017:29-35.
[11] HUNT K H. Kinematic geometry of mechanisms[M]. Oxford, UK:Oxford University Press, 1978.
[12] 王德伦,汪伟. 机构运动微分几何学分析与综合[M]. 北京:机械工业出版社, 2014. WANG Delun, WANG Wei. The kinematic differential geometry and saddle synthesis of linkages[M]. Beijing:China Machine Press, 2014.
[13] WANG D L, WANG W. Kinematic differential geometry and saddle synthesis of linkages[M]. Hoboken, USA:John Wiley & Sons Inc, 2015.
[14] ONDER M, UGURLU H H, CALISKAN A. The euler-savary analogue equations of a point trajectory in lorentzian spatial motion[C]//Proceedings of the National Academy of Sciences India Section a-Physical Sciences, Springer-Verlag, 2013:119-127.
[15] BOKELBERG E H, HUNT K H, RIDLEY P R. Spatial motion-I:Points of inflection and the differential geometry of screws[J]. Mechanism and Machine Theory, 1992, 27(1):1-15.
[16] DOONER D B, GRIFFIS M W. On spatial euler-savary equations for envelopes[J]. Journal of Mechanical Design, 2006, 129(8):865-875.
[17] YANG A T, KIRSON Y, ROTH B. On a kinematic curvature theory for ruled surfaces[C]//Proc. 4th World Congress on Theory of Machines and Mechanisms, Newcastle-upon-Tyne, 1975:737-742.
[18] WANG D L, LIU J, XIAO D Z. Kinematic differential geometry of a rigid body in spatial motion-I. a new adjoint approach and instantaneous properties of a point trajectory in spatial kinematics[J]. Mechanism and Machine Theory, 1997, 32(4):419-432.
[19] WANG D L, LIU J, XIAO D Z. Kinematic differential geometry of a rigid body in spatial motion-II. a new adjoint approach and instantaneous properties of a line trajectory in spatial kinematics[J]. Mechanism and Machine Theory, 1997, 32(4):433-444.
[20] WANG D L, LIU J, XIAO D Z. Kinematic differential geometry of a rigid body in spatial motion-III. distribution of characteristic lines in the moving body in spatial motion[J]. Mechanism and Machine Theory, 1997, 32(4):445-457.
[21] 汪伟. 机构离散运动几何学研究[D]. 大连:大连理工大学, 2016. WANG Wei. Discrete kinematic geometry of mechanism[D]. Dalian:Dalian University of Technology, 2016.
[22] WANG P, ZHANG Y D, WAN M. Third order curvature analysis of conjugate surfaces[J]. Mechanism and Machine Theory, 2017, 107:87-104.
[23] VIA A J, PETUYA V. A proposal for a formula of absolute pole velocities between relative poles[J]. Mechanism and Machine Theory, 2017, 114:74-84.
[24] CERA M, PENNESTR E. Generalized burmester points computation by means of bottema's instantaneous invariants and intrinsic geometry[J]. Mechanism and Machine Theory, 2018, 129:316-335.
[25] KAHVECI D, YAYL Y. Geometric kinematics of persistent rigid motions in three-dimensional minkowski space[J]. Mechanism and Machine Theory, 2022, 167:1-3.
[26] CARRETERO J A, PODHORODESKI R P, NAHON M A, et al. Kinematic analysis and optimization of a new three degree-of-freedom spatial parallel manipulator[J]. Journal of Mechanical Design, 1999, 122(1):17-24.
[27] 黄真,赵永生,赵铁石. 高等空间机构学[M]. 2版. 北京:高等教育出版社, 2014. HUANG Zhen, ZHAO Yongsheng, ZHAO Tieshi. Advanced spatial mechanism[M]. 2nd ed. Beijing:Higher Education Press, 2014.
[28] LI Q C, HERV J M. 1T2R parallel mechanisms without parasitic motion[J]. IEEE Transactions on Robotics, 2010, 26(3):401-410.
[29] CHEN Z M, DING H F, CAO W A, et al. Axodes analysis of the multi-dof parallel mechanisms and parasitic motion[C]//Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Portland, Oregon, USA:The American Society of Mechanical Engineers, 2013:1-9.
[30] ZHANG L L, ZHAO Y Z, REN J K. Screw rolling between moving and fixed axodes traced by lower-mobility parallel mechanism[J]. Mechanism and Machine Theory, 2021, 163(104354):1-35.
[31] ZHANG L L, ZHAO Y Z, ZHAO T S. Fundamental equation of mechanism kinematic geometry:Mapping curve in se(3) to counterpart in SE(3)[J]. Mechanism and Machine Theory, 2020, 146(103732):1-23.
[32] 赵铁石,祁晓园,鹿玲,等. 被约束刚体连续运动的充分必要条件[J]. 燕山大学学报, 2000, 24(3):224-227. ZHAO Tieshi, QI Xiaoyuan, LU Lin, et al. Sufficient and necessary conditions of realizing continuous motion for a restricted rigid body[J]. Journal of Yanshan University, 2000, 24(3):224-227.
[33] 赵铁石,赵永生,黄真. 欠秩并联机器人能连续转动转轴存在的物理条件和数学判据[J]. 机器人, 1999, 21(5):347-351. ZHAO Tieshi, ZHAO Yongsheng, HUANG Zhen. Physical and mathematical conditions of existence of axes about which platform of deficient-rank parallel robots can rotate continuously[J]. Robot, 1999, 21(5):347-351.
[34] 李秦川,柴馨雪,陈巧红,等. 2-UPR-SPR并联机构转轴分析[J]. 机械工程学报, 2013, 49(21):62-69. LI Qinchuan, CHAI Xinxue, CHEN Qiaohong, et al. Analysis of rotational axes of 2-UPR-SPR parallel mechanism[J]. Journal of Mechanical Engineering, 2013, 49(21):62-69.
[35] HUANG Z, TAO W S, FANG Y F. Study on the kinematic characteristics of 3 DOF in-parallel actuated platform mechanisms[J]. Mechanism and Machine Theory, 1996, 31(8):999-1007.
[36] XU Y D, ZHAO Y, YUE Y, et al. Type synthesis of overconstrained 2R1T parallel mechanisms with the fewest kinematic joints based on the ultimate constraint wrenches[J]. Mechanism and Machine Theory, 2020, 147:1-13.
[37] 许允斗,胡建华,张东胜,等. 一种所有转轴均连续的五自由度混联机器人机构[J]. 机械工程学报, 2018, 54(21):19-24. XU Yundou, HU Jianhua, ZHANG Dongsheng, et al. A kind of five degrees of freedom hybrid robot with all rotating shafts continuous[J]. Journal of Mechanical Engineering, 2018, 54(21):19-24.
[38] 陈子明,张扬,黄坤,等. 一种无伴随运动的对称两转一移并联机构[J]. 机械工程学报, 2016, 52(3):9-17. CHEN Ziming, ZHANG Yang, HUANG Kun, et al. Symmetrical 2R1T parallel mechanism without parasitic motion[J]. Journal of Mechanical Engineering, 2016, 52(3):9-17.
[39] 李博. 两种三自由度并联机构运动等效机构特性研究[D]. 秦皇岛:燕山大学, 2017. LI Bo. Characteristic analysis of two kinds of 3-DOF kinematically identical paralllel manipulators[D]. Qinhuangdao:Yanshan University, 2017.
[40] CHEN Z, CHEN X, GAO M, et al. Motion characteristics analysis of a novel spherical two-degree-of-freedom parallel mechanism[J]. Chinese Journal of Mechanical Engineering, 2022, 35(1):29.
[41] ZHAO C, SONG J, CHEN X, et al. Optimum seeking of redundant actuators for M-RCM 3-UPU parallel mechanism[J]. Chinese Journal of Mechanical Engineering, 2021, 34(1):121.
[42] 李仕华,田志立,王子义,等. 具有连续转轴的对称2R1T三自由度并联机构型综合[J]. 机械工程学报, 2017, 53(23):74-82. LI Shihua, TIAN Zhili, WANG Ziyi, et al. Type synthesis of 2R1T symmetrical parallel mechanism with continuous rotation axes[J]. Journal of Mechanical Engineering, 2017, 53(23):74-82.
[43] 田志立. 基于线几何与螺旋理论的几种并联机构连续转轴的研究[D]. 秦皇岛:燕山大学, 2017. TIAN Zhili. Continuous ratation axes study of some parallel mechanisms based on linegeometry and screw theory[D]. Qinhuangdao:Yanshan University, 2017.
[44] 陈淼,张氢,陈文韬,等. 基于空间几何的无伴随运动并联机构分析与综合[J]. 机械工程学报, 2021, 57(1):77-85. CHEN Miao, ZHANG Qing, CHEN Wentao, et al. Analysis and type synthesis of parallel mechanisms without parasitic motion based on spatial geometry[J].Journal of Mechanical Engineering, 2021, 57(1):77-85.
[45] 曹毅,居勇健,黄河,等. 具有两条连续确定转轴的PRR型并联机构型综合[J]. 华中科技大学学报, 2022, 50(1):125-131. CAO Yi, JU Yongjian, HUANG He, et al. Type synthesis of PRR parallel mechanisms with two continuous and determined rotational axes[J]. Journal Huazhong University of Science And Technology, 2022, 50(1):125-131.
[46] 王靖朝. 无伴随运动的含耦合链的混联机构构型综合研究[D]. 秦皇岛:燕山大学, 2020. WANG Jingchao. Type synthesis of spatial multi-loop coupling mechanisms without parasitic motion[D]. Qinhuangdao:Yanshan University, 2020.
[47] LI Q C, CHEN Q H, WU C Y, et al. Geometrical distribution of rotational axes of 3-[P] [S] parallel mechanisms[J]. Mechanism and Machine Theory, 2013, 65:46-57.
[48] HUANG C, SUGIMOTO K, PARKIN I. The correspondence between finite screw systems and projective spaces[J]. Mechanism and Machine Theory, 2008, 43(1):50-56.
[49] ZLATANOV D. The representation of the general three-system of screws by a sphere[J]. Mechanism and Machine Theory, 2012, 49:315-331.
[50] HATHOUT F, BEKAR M, YAYLI Y. Ruled surfaces and tangent bundle of unit 2-sphere[J]. International Journal of Geometric Methods in Modern Physics, 2017, 14(10):1-14.
[51] SELIG J M. 机器人学的几何基础[M]. 2版. 杨向东, 译. 北京:清华大学出版社, 2008. SELIG J M. Geometric fundamentals of robotics[M]. 2nd ed. Translatef by Xiangdong YANG. Beijing:Tsinghua University Press, 2008.
[52] CHEN G L, WANG H, LIN Z Q, et al. Identification of canonical basis of screw systems using general-special decomposition[J]. Journal of Mechanisms and Robotics, 2018, 10(3):1-8.
[53] BALL R S. A Treatise on the theory of screws[M]. Cambridge, UK:Cambridge University Press, 1998.
[54] PHILLIPS J. Freedom in machinery[M]. Cambridge, UK:Cambridge University Press, 2007.
[55] ZHANG L, ZHAO Y, WEI X. Distribution characteristics of screw axes belonging to general three-system of screws[J]. Machines, 2022, 10(3):172.
[56] HUANG Z, LI Q C, DING H F. Theory of parallel mechanisms[M]. Dordrecht, NED:Springer Science & Business Media, 2013.
[57] TSAI M J, LEE H W. On the special bases of two- and three-screw systems[J]. Journal of Mechanical Design, 1993, 115(3):540-546.
[58] PARKIN J A. Co-ordinate transformations of screws with applications to screw systems and finite twists[J]. Mechanism and Machine Theory, 1990, 25(6):689-699.
[59] PARKIN I A. Finding the principal axes of screw systems[C]//ASME 1997 Design Engineering Technical Conferences, 1997:1-9.
[60] RICO J M, DUFFY J. A general method for the computation of the canonical form of three-systems of infinitesimal screws[J]. Robotica, 1998, 16(1):37-45.
[61] BANDYOPADHYAY S, GHOSAL A. Analytical determination of principal twists in serial, parallel and hybrid manipulators using dual vectors and matrices[J]. Mechanism and Machine Theory, 2004, 39(12):1289-1305.
[62] BANDYOPADHYAY S, GHOSAL A. An eigenproblem approach to classical screw theory[J]. Mechanism and Machine Theory, 2009, 44(6):1256-1269.
[63] ZHAO J S, ZHOU H X, FENG Z J, et al. An algebraic methodology to identify the principal screws and pitches of screw systems[C]//Proceedings of the Institution of Mechanical Engineers, Part C:Journal of Mechanical Engineering Science, SAGE Journals, 2009:1931-1941.
[64] ZHANG W X, XU Z C. Algebraic construction of the three-system of screws[J]. Mechanism and Machine Theory, 1998, 33(7):925-930.
[65] FANG Y F, HUANG Z. Analytical identification of the principal screws of the third order screw system[J]. Mechanism and Machine Theory, 1998, 33(7):987-992.
[66] WALDRON K J. The constraint analysis of mechanisms[J]. Journal of Mechanisms, 1966, 1(2):101-114.
[67] HUANG Z, LIU J F, ZENG D X. A general methodology for mobility analysis of mechanisms based on constraint screw theory[J]. Science in China Series E:Technological Sciences, 2009, 52(5):1337-1347.
[68] 黄真. 空间机构学[M]. 北京:机械工业出版社, 1991. HUANG Zhen. Advanced spatial mechanism[M]. Beijing:China Machine Press, 1991.
[69] 黄真,孔令富,方跃法. 并联机器人机构学理论及控制[M]. 北京:机械工业出版社, 1997. HUANG Zhen, KONG Lingfu, FANG Yuefa. Mechanism theory and control of parallel manipulators[M]. Beijing:China Machine Press, 1997.
[70] DAI J S, HUANG Z, LIPKIN H. Mobility of overconstrained parallel mechanisms[J]. Journal of Mechanical Design, 2004, 128(1):220-229.
[71] YUAN M S C, FREUDENSTEIN F, WOO L S. Kinematic analysis of spatial mechanisms by means of screw coordinates:Part 2-analysis of spatial mechanisms[J]. Journal of Engineering for Industry, 1971, 93(1):67-73.
[72] SUTHERLAND G, ROTH B. A transmission index for spatial mechanisms[J]. Journal of Engineering for Industry, 1973, 95(2):589-597.
[73] TSAI M J, LEE H W. Generalized evaluation for the transmission performance of mechanisms[J]. Mechanism and Machine Theory, 1994, 29(4):607-618.
[74] LIU X J, WU C, WANG J S. A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators[J]. Journal of Mechanisms and Robotics, 2012, 4(4):1-10.
[75] MCCARTHY J M. An introduction to theoretical kinematics[M]. Cambridge, Massachusetts, London:The MIT Press, 1990.
[76] 戴建生. 机构学与机器人学的几何基础与旋量代数[M]. 北京:高等教育出版社, 2014. DAI Jiansheng. Geometrical foundations and screw algebra for mechanisms and robotics[M]. Beijing:Higher Education Press, 2014.
[77] CHEN C, ANGELES J. Generalized transmission index and transmission quality for spatial linkages[J]. Mechanism and Machine Theory, 2007, 42(9):1225-1237.
[78] HUANG Z, LI Q C. General methodology for type synthesis of symmetrical lower-mobility parallel manipulators and several novel manipulators[J]. The International Journal of Robotics Research, 2002, 21(2):131-145.
[79] HUANG Z, LI S H, ZUO R G. Feasible instantaneous motions and kinematic characteristics of a special 3-DOF 3-UPU parallel manipulator[J]. Mechanism and Machine Theory, 2004, 39(9):957-970.
[80] WANG J S, WU C, LIU X J. Performance evaluation of parallel manipulators:motion/force transmissibility and its index[J]. Mechanism and Machine Theory, 2010, 45(10):1462-1476.
[81] SKREINER M. A study of the geometry and the kinematics of instantaneous spatial motion[J]. Journal of Mechanisms, 1966, 1(2):115-143.
[82] YUAN M S C, FREUDENSTEIN F. Kinematic analysis of spatial mechanisms by means of screw coordinates:part 1-screw coordinates[J]. Journal of Engineering for Industry, 1971, 93(1):61-66.
[83] SLAVUTIN M, SHAI O, SHEFFER A, et al. A novel criterion for singularity analysis of parallel mechanisms[J]. Mechanism and Machine Theory, 2019, 137:459-475.
[84] FIGLIOLINI G, REA P, ANGELES J. The synthesis of the axodes of RCCC linkages[J]. Journal of Mechanisms and Robotics, 2015, 8(2):0210111- 0210119.10.1115/1.4031950.
[85] BROCKETT R W, STOKES A, PARK F. A geometrical formulation of the dynamical equations describing kinematic chains[C]//Proceedings of IEEE International Conference on Robotics and Automation, Atlanta, GA, USA:IEEE, 1993:637-641.
[86] SOMMER H J. Jerk analysis and axode geometry of spatial linkages[J]. Journal of Mechanical Design, 2008, 130(4):1-6.
[87] NURAHMI L, SCHADLBAUER J, CARO S, et al. Kinematic analysis of the 3-RPS cube parallel manipulator[J]. Journal of Mechanisms and Robotics, 2015, 7(1):1-11.10.1115/1.4029305.
[88] NURAHMI L, SCHADLBAUER J, HUSTY M, et al. Motion capability of the 3-RPS cube parallel manipulator[C]//Advanced in Robot Kinematics, Cham:Springer, 2014:527-535.
[89] NURAHMI L, GAN D M. Reconfiguration of a 3-(rR)PS metamorphic parallel mechanism based on complete workspace and operation mode analysis[J]. Journal of Mechanisms and Robotics, 2019, 12(1):1-15.
[90] PFURNER M, SCHADLBAUER J. The instantaneous screw axis of motions in the kinematic image space[C]//Computational Kinematics Mechanisms and Machine Science, Cham:Springer, 2018:544-551.
[91] SUGIMOTO K, DUFFY J. Application of linear algebra to screw systems[J]. Mechanism and Machine Theory, 1982, 17(1):73-83.
[92] YU J J, YU J Z, WU K, et al. Design of constant-velocity transmission devices using parallel kinematics principle[C]//International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2014:1-9.
[93] 王国彪,刘辛军. 初论现代数学在机构学研究中的作用与影响[J]. 机械工程学报, 2013, 49(3):1-9. WANG Guobiao, LIU Xinjun. Role and influence of modern mathematics in mechanisms[J]. Journal of Mechanical Engineering, 2013, 49(3):1-9.
[94] MURRAY R M, LI Z X, SASTRY S S. A mathematical introduction to robotics manipulation[M]. Boca Raton, USA:CRC Press, 1994.
[95] 于靖军,刘辛军,丁希仑,等. 机器人机构学的数学基础[M]. 北京:机械工业出版社, 2008. YU Jingjun, LIU Xinjun, DING Xilun, et al. Mathematic foundation of mechanisms and robotics[M]. Beijing:China Machine Press, 2008.
[96] SELIG J M. Geometric fundamentals of robotics[M]. 2nd ed. New York:Springer Science & Business Media, 2004.
[97] SELIG J M, CARRICATO M. Persistent rigid-body motions and study's "ribaucour" problem[J]. Journal of Geometry, 2017, 108(1):149-169.
[98] BOTTEMA O, ROTH B. Theoretical kinematics[M]. New York:Dover Publications, Inc., 1990.
[99] WU Y Q, CARRICATO M. Identification and geometric characterization of lie triple screw systems and their exponential images[J]. Mechanism and Machine Theory, 2017, 107:305-323.
[100] PARK F C. Computational aspects of the product-of-exponentials formula for robot kinematics[J]. IEEE Transactions on Automatic Control, 1994, 39(3):643-647.
[101] ŽEFRAN M, KUMAR V, CROKE C. Choice of riemannian metrics for rigid body kinematics[C]//International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Irvine, California, USA:American Society of Mechanical Engineers, 1996:1-11.
[102] MCCARTHY J M, ROTH B. The curvature theory of line trajectories in spatial kinematics[J]. Journal of Mechanical Design, 1981, 103(4):718-724.
[103] CUI L, DAI J S. Posture, workspace, and manipulability of the metamorphic multifingered hand with an articulated palm[J]. Journal of Mechanisms and Robotics, 2011, 3(2):021001-021007.
[104] LIU H L, LIU Y X, JUNG S D. Ruled invariants and associated ruled surfaces of a space curve[J]. Applied Mathematics and Computation, 2019, 348:479-486.
[105] OH Y, ABHISHESH P, RYUH B. Study on robot trajectory planning by robot end-effector using dual curvature theory of the ruled surface[J]. International Journal of Mechanical and Materials Engineering, 2017, 11(3):577-582.
[106] KIRSON Y D. Higher order curvature theory in space kinematics[D]. Berkeley:University of California, Berkeley, 1975.
[107] KIRSON Y D, YANG A T. Instantaneous invariants in three-dimensional kinematics[J]. Journal of Applied Mechanics, 1978, 45(2):409-414.
[108] MERLET J P. Jacobian, manipulability, condition number, and accuracy of parallel robots[J]. Journal of Mechanical Design, 2005, 128(1):199-206.
[109] LIU X J, WANG J S. Parallel kinematics:Type, kinematics, and optimal design[M]. Berlin, Heidelberg:Springer Science & Business, 2014.
[110] 陈祥,谢福贵,刘辛军. 并联机构中运动/力传递功率最大值的评价[J]. 机械工程学报, 2014, 50(3):1-9. CHEN Xiang, XIE Fugui, LIU Xinjun. Evaluation of the maximum value of motion/force transmission power in parallel manipulators[J]. Journal of Mechanical Engineering, 2014, 50(3):1-9.
[111] 数学辞海编辑委员会. 数学辞海(第一卷)[M]. 太原:山西教育出版社, 2002. Editorial Board of Mathematics Dictionary. Mathematics dictionary(volume 1)[M]. Taiyuan:Shanxi Education Press, 2002.