Kinematic Optimal Design of a 2-DoF Parallel Positioning Mechanism Employing Geometric Algebra



This paper proposes a geometric algebra (GA) based approach to carry out inverse kinematics and design parameters of a 2-degree-of-freedom parallel mechanism with its topology structure 3-RSR&SS for the first time. Here, R and S denote respectively revolute and spherical joints. The inverse solutions are obtained easily by utilizing special geometric relations of 3-RSR&SS parallel positioning mechanism, which are proven by calculating relations among point, line and plane in virtue of operation rules. Three global indices of kinematic optimization are defined to evaluate kinematic performance of 3-RSR&SS parallel positioning mechanism in the light of shuffle and outer products. Finally, the kinematic optimal design of 3-RSR&SS parallel positioning mechanism is carried out by means of NSGA-II and then a set of optimal dimensional parameters is proposed. Comparing with traditional kinematic analysis and optimal design method, the approach employing GA has following merits, (1) kinematic analysis and optimal design would be carried out in concise and visual way by taking full advantage of the geometric conditions of the mechanism. (2) this approach is beneficial to kinematic analysis and optimal design of parallel mechanisms in automatic and visual manner using computer programming languages. This paper may lay a solid theoretical and technical foundation for prototype design and manufacture of 3-RSR&SS parallel positioning mechanism.


Parallel positioning mechanism Kinematic analysis Optimal design Geometric algebra 



This research work was supported by Tianjin Research Program of Application Foundation and Advanced Technology under Grant No. 14JCYBJC19500.


  1. 1.
    Baumann, R., Maeder, W., Glauser, D., Clavel, R.: The PantoScope: a spherical remote-center-of-motion parallel manipulator for force reflection. In: Proc. IEEE Inter. Conf. Robot. Auto., pp. 718-723, Albuquerque (1997)Google Scholar
  2. 2.
    Bayro-Corrochano, E., Falcon, L.E.: Geometric algebra of points, lines, planes and spheres for computer vision and robotics. Robotica 23, 755–770 (2005)CrossRefGoogle Scholar
  3. 3.
    Carricato, M., Parenti-Castelli, V.: A novel fully decoupled two-degrees-of-freedom parallel wrist. ASME J. Mech. Robot. 23, 661–667 (2004)Google Scholar
  4. 4.
    Chai, X.X., Li, Q.C.: Analytical mobility analysis of bennett linkage using geometric algebra. Adv. Appl. Clifford Algebras 27, 2083–2095 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Clifford, W.K.: On the classification of geometric algebras. In: Tucker, R. (ed.) Mathematical Papers, pp. 397–401. Macmillian Publishers, London (1882)Google Scholar
  6. 6.
    Dong, X., Yu, J.J., Chen, B., Zong, G.H.: Geometric approach for kinematic analysis of a class of 2-DOF rotational parallel manipulators. Chin. J. Mech. Eng. 25, 24–247 (2012)CrossRefGoogle Scholar
  7. 7.
    Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science (revised edition): an object-oriented approach to geometry, pp. 80–83. Morgan Kaufmann, Massachusettes (2009)Google Scholar
  8. 8.
    Dunlop, G.R., Jones, T.P.: Position analysis of a two DOF parallel mechanism–the Canterbury tracker. Mech. Mach. Theory 34, 599–614 (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Fu, Z.T., Yang, W.Y., Yang, Z.: Solution of inverse kinematics for 6R robot manipulators with offset wrist based on geometric algebra. ASME J. Mech. Robot. 5, 031010 (2013)CrossRefGoogle Scholar
  10. 10.
    Hestenes, D.: New foundations for classical mechanics, 2nd edn. Springer, The Netherlands (1999)MATHGoogle Scholar
  11. 11.
    Hestenes, D.: New tools for computational geometry and rejuvenation of screw theory. In: Bayro-Corrochano EJ, Scheuermann G (eds) Geometric algebra computing in engineering and computer science, pp. 3–35. Springer, Berlin (2010)Google Scholar
  12. 12.
    Hildenbrand, D., Zamora, J., Bayro-Corrochano, E.: Inverse kinematics computation in computer graphics and robotics using conformal geometric algebra. Adv. Appl. Clifford Algebras 18, 699–713 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hitzer, E.M.S.: Euclidean geometric objects in the clifford geometric algebra of origin, 3space, infinity. Bull. Belg. Math. Soc. Simon Stevin 11, 653–662 (2005)MATHGoogle Scholar
  14. 14.
    Hu, B., Yu, J.J., Lu, Y., Sui, C.P., Han, J.D., Yu, J.J.: Statics and stiffness model of serial-parallel manipulator formed by $k$ parallel manipulators connected in series. ASME J. Mech. Robot. 4, 021012 (2012)CrossRefGoogle Scholar
  15. 15.
    Huang, Z., Liu, J.F., Li, Q.C.: A unified methodology for mobility analysis based on screw theory. Springer, Berlin (2008)CrossRefGoogle Scholar
  16. 16.
    Huo, X.M., Sun, T., Song, Y.M.: An analytical approach to determine motions/constraints of serial kinematic chains based on Clifford algebra. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 231, 1324–1338 (2017)CrossRefGoogle Scholar
  17. 17.
    Jiang, A., Zhao, H.: Tracking mode and tacking precision analysis of navigation satellite. Modern Radar 33, 78–82 (2011)Google Scholar
  18. 18.
    Kim, J.S., Jin, H.J., Park, J.H.: Inverse kinematics and geometric singularity analysis of a 3-SPS/S redundant motion mechanism using conformal geometric algebra. Mech. Mach. Theory 90, 23–36 (2015)CrossRefGoogle Scholar
  19. 19.
    Li, H.B.: Conformal geometric algebra—a new framework for computational geometry. J. Comput. Aided Design Comput. Graph. 17, 2383–2393 (2005)Google Scholar
  20. 20.
    Li, J.R.: Elementary Analysis on Vertex Tracking of X-Y Type Antenna Pedestal, Modern Electronics Technique, pp. 29–31 (2010)Google Scholar
  21. 21.
    Li, C.: Cartesian stiffness evaluation of a novel 2 DoF parallel wrist under redundant and antagonistic actuation. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 959–964 (2014)Google Scholar
  22. 22.
    Li, J.J., Jia, Y.H.: Configuration Design of X-Y Style Antenna Pedestal. Electro-Mechanical Engineering, pp. 41–44 (2014)Google Scholar
  23. 23.
    Li, Q.C., Chai, X.X., Xiang, J.N.: Mobility analysis of limited-DOF parallel mechanisms in the framework of geometric algebra. ASME J. Mech. Robot. 8, 041005 (2014)CrossRefGoogle Scholar
  24. 24.
    Li, W.Z., Sun, T., Huo, X.M., Song, Y.M.: CGA approach to kinematic analysis of a 2-DOF parallel positioning mechanism. In: Proc. ASME 2016 Inter. Design Eng. Tech. Conf. and Com. Inform. Eng. Conf., Charlotte, North Carolina, 21–24 August (2016)Google Scholar
  25. 25.
    Lian, B.B., Sun, T., Song, Y.M., Jin, Y., Price, M.: Stiffness analysis and experiment of a novel 5-DoF parallel kinematic machine considering gravitational effects. Int. J. Mach. Tools Manuf. 95, 82–96 (2015)CrossRefGoogle Scholar
  26. 26.
    Lum, M.J.H., Jacob, R., Sinanan, M.N., Blake, H.: Kinematic optimization of a spherical mechanism for a minimally invasive surgical robot. IEEE Trans. Bio-Med. Eng. 53, 1440–1445 (2006)CrossRefGoogle Scholar
  27. 27.
    Ma, S., Shi, Z.P., Shao, Z.Z., Guan, Y., Li, L.M., Li, Y.D.: Higher-order logic formalization of conformal geometric algebra and its application in verifying a robotic manipulation algorithm. Adv. Appl. Clifford Algebras 26, 1–26 (2016)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Mauro, S., Battezzato, A., Biondi, G., Scarzella, C.: Design and test of a parallel kinematic solar tracker. Mech. Eng. 7, 1–16 (2015)Google Scholar
  29. 29.
    Nikulin, V.V., Sofka, J., Skormin, V.A.: Decentralized control of an Omni Wrist laser beam tracking system. In: Proceedings of SPIE on Free-Space Laser Communication Technologies XVI, pp. 194-203, San Jose (2004)Google Scholar
  30. 30.
    Porteous, I.: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  31. 31.
    Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Berlin (2009)MATHGoogle Scholar
  32. 32.
    Shi, M.X.: Introduction to the highly reliable antenna pedestal of an unmanned weather radar. Electro-Mech. Eng. 29, 22–26 (2013)Google Scholar
  33. 33.
    Selig, J.M.: Geometric Fundamentals of Robotics. Springer, London (2005)MATHGoogle Scholar
  34. 34.
    Sofka, J., Skormin, V.A., Nikulin, V.V.: New generation of gimbals systems for laser positioning applications. In: Proc. SPIE on Free-Space Laser Communication and Active Illumination III, pp. 182–191. San Diego (2004)Google Scholar
  35. 35.
    Sofka, J., Skormin, V.: Integrated approach to electromechanical design of a digitally controlled high precision actuator for aerospace applications. In: Proc. IEEE Inter. Conf. Control Appl. in Com. Aided Control Sys. Des., pp. 261–265, Munich (2006)Google Scholar
  36. 36.
    Song, Y.M., Gao, H., Sun, T., Dong, G., Lian, B.B., Qi, Y.: Kinematic analysis and optimal design of a novel 1T3R parallel manipulator with an articulated travelling plate. Robot. Comput. Integr. Manuf. 30, 508–516 (2014)CrossRefGoogle Scholar
  37. 37.
    Song, Y.M., Lian, B.B., Sun, T.: A novel 5-DoF parallel manipulator and its kinematic optimization. ASME J. Mech. Robot. 6, 500–508 (2014)Google Scholar
  38. 38.
    Song, Y.M., Dong, G., Sun, T., Lian, B.B.: Elasto-dynamic analysis of a novel 2-DoF rotational parallel mechanism with an articulated travelling platform. Meccanica 51, 1547–1557 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Sun, T., Song, Y.M., Li, Y.G., Zhang, J.: Workspace decomposition based dimensional synthesis of a novel hybrid reconfigurable robot. ASME J. Mech. Robot. 2, 310091–310098 (2010)CrossRefGoogle Scholar
  40. 40.
    Sun, T., Song, Y.M., Li, Y.G., Liu, L.S.: Dimensional synthesis of a 3-DOF parallel manipulator based on dimensionally homogeneous Jacobian matrix. Sci. China Technol. Sci. 53, 168–174 (2010)CrossRefMATHGoogle Scholar
  41. 41.
    Sun, T., Song, Y.M., Dong, G., Lian, B.B., Liu, J.P.: Optimal design of a parallel mechanism with three rotational degrees of freedom. Robot. Comput. Integr. Manuf. 28, 500–508 (2012)CrossRefGoogle Scholar
  42. 42.
    Sun, T., Song, Y.M., Gao, H., Qi, Y.: Topology synthesis of a 1T3R parallel manipulator with an articulated traveling plate. ASME J. Mech. Robot. 7, 310151–310159 (2015)CrossRefGoogle Scholar
  43. 43.
    Sun, T., Zhai, Y.P., Song, Y.M., Zhang, J.T.: Kinematic calibration of a 3-DoF rotational parallel manipulator using laser tracker. Robot. Comput. Integr. Manuf. 41, 78–91 (2016)CrossRefGoogle Scholar
  44. 44.
    Sun, T., Lian, B.B., Song, Y.M.: Stiffness analysis of a 2-DoF over-constrained RPM with an articulated traveling platform. Mech. Mach. Theory 96, 165–178 (2016)CrossRefGoogle Scholar
  45. 45.
    Wu, W.L., Shi, J.Z.: Research on relation of antenna pedestal type and high elevation tracking. Computer and Network, pp. 50–53 (2010)Google Scholar
  46. 46.
    Wu, C., Liu, X.J., Wang, L.P., Wang, J.S.: Optimal design of spherical 5R parallel manipulators considering the motion/force transmissibility. ASME J. Mech. Des. 132, 1–10 (2010)CrossRefGoogle Scholar
  47. 47.
    Wu, Y.Q., Lowe, H., Carricato, M., Li, Z.: Inversion symmetry of the Euclidean group: theory and application to robot kinematics. IEEE Trans. Robot. 32, 312–326 (2016)CrossRefGoogle Scholar
  48. 48.
    Yang, S.F., Sun, T., Huang, T., Li, Q.C., Gu, D.B.: A finite screw approach to type synthesis of three-DOF translational parallel mechanisms. Mech. Mach. Theory 104, 405–419 (2016)CrossRefGoogle Scholar
  49. 49.
    Zeng, D.X., Hou, Y.L., Huang, Z., Lu, W.J.: Type synthesis and characteristic analysis of a family of 2-DOF rotational decoupled parallel mechanisms. Chin. J. Mech. Eng. 22, 833–840 (2009)CrossRefGoogle Scholar
  50. 50.
    Zhuo, P.H.: The research on vertex tracking of the antenna. Electro-Mechanical Engineering, pp. 46–48 (2010)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Key Laboratory of Mechanism Theory and Equipment Design of Ministry of EducationTianjin UniversityTianjinChina

Personalised recommendations