为了解决三自由度平面并联机构的正运动学分析建模和求解时，需要建立坐标系和消元的问题，基于共形几何代数（Conformal geometric algebra，CGA）提出一种脱离坐标系的几何建模和免消元计算方法。在共形几何代数框架下通过基本几何体的相交、分离和对偶运算，表示出动平台上的两个铰链位置；根据动平台三角形面积的有向性，并经过一系列的几何代数运算和化简推导出该问题的特征多项式方程；通过半角正切变换、欧拉变换或不需要任何变换可直接获得任意构型的平面并联机构正运动学分析的一元高次方程。特征多项式方程的推导脱离了坐标系，不需要经过消元，且不需要任何前提条件。数字实例求解表明提出的方法对于特殊构型和一般构型的平面并联机构都是适用的，验证了算法的正确性，结果表明算法数值鲁棒性好，为平面并联机构运动学求解理论提供了一种新思路。
In order to cope with the requirements of the coordinate establishment and elimination process, in the process of the modelling and computing for the forward kinematic analysis of general planar parallel manipulators, a geometric modeling and free-elimination computing method for the forward kinematics of planar parallel manipulators is proposed using conformal geometric algebra (CGA). Under the frame of CGA, two of the three coordinates in the moving platform are formulated by the intersecting, dissecting and dual of the basic geometric entities; in terms with the area sign of the moving triangular platform, the characteristic polynomial equation is derived by a sequence of geometric algebra operation and simplification; a high-degree polynomial equation for planar parallel manipulators of any link parameters is deduced by tangent-half-angle substitution, Euler-angle substitution or no substitution. The derivation of the characteristic polynomial is free of coordinate and no elimination process and no assumption are required. Numerical examples are given to validate the correctness of the procedure and that the proposed algorithm is feasible to all cases of planar parallel manipulators including the special and general structures. At last, the results show that the proposed algorithm has a readily numerical robustness and provides a new sight for the theoretical solution to the kinematics of planar parallel manipulators.
[1] HUNT K H. Structural kinematics of in-parallel actuated robot arms[J]. ASME Journal of Mechanisms,Transmissions and Automation in Design,1983,105(4):705-712.
[2] MERLET J P. Parallel robots[M]. 2nd ed. Netherlands:Springer,2006.
[3] GOSSELIN C,SEFRIOUI J,RICHARD M J. Polynomial solutions to the direct kinematic problem of planar three degree-of-freedom parallel manipulators[J]. Mechanism and Machine Theory,1992,27:107-119.
[4] ROJAS N,THOMAS F. The forward kinematics of 3-RPR planar robots:A review and a distance-based formulation[J]. IEEE Transactions on Robotics,2013,27(1):143-150.
[5] 刘惠林,张同庄,丁洪生.3-RPR平面并联机构正解的吴方法[J]. 北京理工大学学报,2000,20(5):565-569. LIU Huilin,ZHANG Tongzhuang,DING Hongsheng. Forward solutions of the 3-RPR planar parallel mechanism with Wu's method[J]. Transaction of Beijing Institute of Technology,2000,20(5):565-569.
[6] 倪振松,廖启征,魏世民,等.基于共形几何代数的一种平面并联机构位置正解[J].北京邮电大学学报,2010,32(2):7-10-32. NI Zhensong,LIAO Qizheng,WEI Shimin,et al. Forward displacement of a planar parallel mechanisms position based on conformal geometric algebra[J]. Journal of Beijing University of Posts and Telecommunications,2010,32(2):7-10-32.
[7] 张忠海,李端玲.一种平面并联机构位置正解分析的共形几何代数及Sylvester结式方法[J].北京理工大学学报,2014,34(3):241-244. ZHANG Zhonghai,LI Duanling.Conformal geometric algebra and Sylvester resultant method for forward displacement analysis of a planar parallel mechanism[J]. Transaction of Beijing Institute of Technology,2014,34(3):241-244.
[8] GOSSELIN C M,MERLET J P. On the direct kinematics of planar parallel manipulators:Special architectures and number of solutions[J]. Mechanism and Machine Theory,1994,29(8):1083-1097.
[9] KONG X,GOSSELIN C M. Forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators[J]. Mechanism and Machine Theory,2001,36(9):1009-1018.
[10] WENGER P,CHABLAT D,ZEIN M. Degeneracy study of the forward kinematics of planar 3-RPR parallel manipulators[J]. ASME Journal of Mechanical Design,2007,129(12):1265-1268.
[11] 李洪波.共形几何代数-几何代数的新理论和计算框架[J].计算机辅助设计与图形学学报,2005,17(11):2383-2393. LI Hongbo.Conformal geometric algebra-A new framework for computational geometry[J].Journal of Computer-Aided Design and Computer Graphics,2005,17(11):2383-2393.
[12] ROSENHAHN B,SOMMER G. Pose estimation in conformal geometric algebra,part I:The stratification of mathematical spaces[J]. Journal of Mathematical Imaging and Vision,2005,22(1):27-48.
[13] HILDENBRAND D,FONTIJNE D,PERWASS C,et al. Geometric algebra and its application to computer graphics[C]//25th Annual Conference of the European Association for Computer Graphics,August 30-September 3,2004,Grenoble,France. France:INRIA and the Eurographics Association,2004:1-49.
[14] KLEPPE A L,EGELAND O. Inverse kinematics for industrial robots using conformal geometric algebra[J]. Modeling Identification and Control,2016,37(1):63-75.
[15] LI Q,XIANG J,CHAI X,et al. Singularity analysis of a 3-rps parallel manipulator using geometric algebra[J]. Chinese Journal of Mechanical Engineering,2015,28(6):1204-1212.
[16] ZHANG Y,LIAO Q,WEI S,et al. A novel geometric modeling and solution method for forward displacement analysis of 6-3 Stewart platforms[C]//Proceedings of ASIAN MMS 2016 and CCMMS 2016,December 15-17,2016,Guangzhou,China. Singapore:Springer,2016:911-924.
[17] ZHANG Y,LIAO Q,WEI S,et al. Forward kinematics of 3-RPS parallel mechanisms using conformal geometric algebra[C]//Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference,August 6-9,2017,Cleveland,Ohio. New York:ASME,2017:V05BT08A070.
