Advertisement

Finite Motion Analysis of Parallel Mechanisms with Parasitic Motions Based on Conformal Geometric Algebra

Article
  • 45 Downloads

Abstract

Finite motion analysis of parallel mechanisms (PMs) denotes formulating the map between finite motion of end-effector and those of its component limbs. By employing conformal geometric algebra (CGA), this paper presents an analytical and accurate method to analyze the finite motions of PMs with parasitic motions. Herein, parasitic motions are defined as the dependent motions in the constraint Degrees-of-Freedom (DoFs) of PMs. Firstly, description of rigid body transformations based on CGA is reviewed. Then, the intersection algorithm of finite motions is introduced by exploiting the algebraic properties of CGA. Based on this, a method to formulate the finite motions of PMs with parasitic motions is proposed. Finally, Z3 mechanism is sketched as example by utilizing the approach. This method facilitates the invention of new mechanisms and can also be applied in the finite motion analysis of other kinds of PMs.

Keywords

Parallel mechanism Conformal geometric algebra Finite motion analysis Parasitic motion 

References

  1. 1.
    Carretero, J.A., Podhorodeski, R.P., Nahon, M.A., Gosselin, C.M.: Kinematic analysis and optimization of a new three degree-of-freedom spatial parallel manipulator. ASME J. Mech. Des. 122, 17–24 (2000)CrossRefGoogle Scholar
  2. 2.
    Chai, X.X., Li, Q.C.: Analytical mobility analysis of bennett linkage using Geometric Algebra. Adv. Appl. Clifford Algebras 1–13 (2017)Google Scholar
  3. 3.
    Clifford, W.K.: On the classification of geometric algebras. In: R. Tucker (Ed.) Mathematical Papers, pp. 397–401. Macmillian Publishers, London (1882)Google Scholar
  4. 4.
    Dai, J.S.: Finite displacement screw operators with embedded Chasles’ motion. ASME J. Mech. Robot. 4, 041002 (2012)CrossRefGoogle Scholar
  5. 5.
    Dai, J.S.: Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections. Mech. Mach. Theory. 92, 144–152 (2015)CrossRefGoogle Scholar
  6. 6.
    Dai, J.S., Huang, Z., Lipkin, H.: Mobility of overconstrained parallel mechanisms. ASME J. Mech. Des. 128, 220–229 (2006)CrossRefGoogle Scholar
  7. 7.
    Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science (Revised edition): an Object-oriented approach to geometry. Morgan Kaufmann 80–83 (2009)Google Scholar
  8. 8.
    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
  9. 9.
    Gao, F., Yang, J., Ge, Q.J.: Type synthesis of parallel mechanisms having the second class GF sets and two dimensional rotations. ASME J. Mech. Robot. 3, 011003 (8 pages) (2011)Google Scholar
  10. 10.
    Hennes, N.: Ecospeed: an innovative machining concept for high performance 5-axis-machining of large structural component in aircraft engineering, pp. 763–774. Proc. of the Third Chemnitz Parallel kinematics, Seminar (2002)Google Scholar
  11. 11.
    Hestenes, D.: New Foundations for Classical Mechanics, 2nd edn. Springer, Netherlands (1999)MATHGoogle Scholar
  12. 12.
    Huang, Z., Ge, Q.J.: A simple method for mobility analysis using reciprocal screws. Proc. ASME Des. Eng. Tech. Conf. Com. Inform. Eng. Conf. pp. 1229–1237, United States, Philadelphia (2006)Google Scholar
  13. 13.
    Hunt, K.H.: Structural kinematics of in-parallel-actuated robot arms. ASME, J. Mech. Trans. Auto. Des 105, 705–712 (1983)CrossRefGoogle Scholar
  14. 14.
    Jin, Q., Yang, T.L.: Theory for Topology Synthesis of Parallel Manipulators and Its Application to Three-Dimension-Translation Parallel Manipulators. ASME J. Mech. Des. 126, 625–639 (2004)CrossRefGoogle Scholar
  15. 15.
    Li, Q.C., Chen, Z., Chen, Q.H., Wu, C.Y., Hu, X.D.: Parasitic motion comparison of 3-PRS parallel mechanism with different limb arrangements. Robot. Com Int. Manuf. 27, 389–396 (2011)CrossRefGoogle Scholar
  16. 16.
    Li, Q.C., Hervé, J.M.: 1T2R Parallel Mechanisms without Parasitic Motion. IEEE Trans. Robot. 26, 401–410 (2010)CrossRefGoogle Scholar
  17. 17.
    Liang, D., Song, Y.M., Sun, T., Dong, G.: Optimum design of a novel redundantly actuated parallel manipulator with multiple actuation modes for high kinematic and dynamic performance. Nonlinear Dyn. 83, 631–658 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lin, R., Guo, W., Gao, F.: Type synthesis of a family of novel four, five, and six degrees-of-freedom sea lion ball mechanisms with three limbs. ASME J. Mech. Robot. 8, 021023 (12 pages) (2016)Google Scholar
  19. 19.
    Liu, X.J., Wu, C., Wang, J.S., Bonev, I.: Attitude description method of [PP]S type parallel robotic mechanisms. Chin. J. Mech. Eng. 44, 19–23 (2008)CrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Perez-Gracia, A., McCarthy, J.M.: Kinematic synthesis of spatial serial chains using Clifford algebra exponentials. Proc. IMechE Part C: J. Mech. Eng. Sci 220, 953–968 (2006)CrossRefGoogle Scholar
  22. 22.
    Perwass, C., Forstner, W.: Uncertain geometry with circles, spheres and conics. In: Klette R, Kozera R, Noakes L, et al. (Eds.) Geometric Properties from Incomplete Data, Computational Imaging and Vision, pp. 23-41. Springer, Netherlands 31 (2006)Google Scholar
  23. 23.
    Qi, Y., Sun, T., Song, Y.M.: Type synthesis of parallel tracking mechanism with varied axes by modeling its finite motions algebraically. ASME J. Mech. Robot. 9(5), 054504-1–054504-6 (2017)CrossRefGoogle Scholar
  24. 24.
    Qi, Y., Sun, T., Song, Y.M., Jin, Y.: Topology synthesis of three-legged spherical parallel manipulators employing Lie group theory. Proc. Ins. Mech. Eng., Part C J. Mech. Eng. Sci. 229, 1873–1886 (2015)CrossRefGoogle Scholar
  25. 25.
    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. Com. Int. Manuf. 30, 508–516 (2014)CrossRefGoogle Scholar
  26. 26.
    Sun, T., Lian, B.B.: Stiffness and mass optimization of parallel kinematic machine. Mech. Mach. Theory 120, 73–88 (2018)CrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    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. Chin. Tech. Sci. 53, 168–174 (2010)CrossRefMATHGoogle Scholar
  29. 29.
    Sun, T., Wang, P.F., Lian, B.B., Zhai, Y.P., Liu, S.: Geometric accuracy design and error compensation of a 1-translational and 3-rotational parallel mechanism with articulated travelling plate. Proc. Ins. Mech. Eng., Part B. J. Mech. Eng. Sci. Manuf. (2017).  https://doi.org/10.1177/09544054
  30. 30.
    Sun, T., Wu, H., Lian, B.B., Qi, Y., Wang, P.F.: Stiffness modeling, analysis and evaluation of a 5 degree of freedom hybrid manipulator for friction stir welding. Proc. Ins. Mech. Eng. Part C: J. Mech. Eng. Sci. 231, 4441–4456 (2017).  https://doi.org/10.1177/0954406216668911 CrossRefGoogle Scholar
  31. 31.
    Sun, T., Yang, S.F., Huang, T., Dai, J.S.: A way of relating instantaneous and finite screws based on the screw triangle product. Mech. Mach. Theory. 108, 75–82 (2017)CrossRefGoogle Scholar
  32. 32.
    Ting, K.L., Zhang, Y.: Rigid body motion characteristics and unified instantaneous motion representation of points, lines, and planes. J. Mech. Des. 126, 593–601 (2004)CrossRefGoogle Scholar
  33. 33.
    Tsai, L.W., Kim, H.S.: Kinematic synthesis of a spatial 3-RPS parallel manipulator. ASME J. Mech. Des. 125, 92–97 (2003)CrossRefGoogle Scholar
  34. 34.
    Tsai, M.S., Shiau, T.N., Tsai, Y.J., Chang, T.H.: Direct kinematic analysis of a 3-PRS parallel mechanism. Mech. Mach. Theory 38, 71–83 (2003)CrossRefMATHGoogle Scholar
  35. 35.
    Wahl, J.: Articulated tool head. WIPO Patent WO/2000/025976 (2000)Google Scholar
  36. 36.
    Xie, F.G., Liu, X.J., Wang, J.S.: A 3-DOF parallel manufacturing module and its kinematic optimization. Robot. Com. Int. Manuf. 28, 334–343 (2012)CrossRefGoogle Scholar
  37. 37.
    Yang, S.F., Sun, T., Huang, T.: Type synthesis of parallel mechanisms having 3T1R motion with variable rotational axis. Mech. Mach. Theory 109, 220–230 (2017)CrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    Zamora-Esquivel, J., Bayro-Corrochano, E.: Robot object manipulation using stereoscopic vision and conformal geometric algebra. Appl. Bionics Biomech. 8, 411–428 (2011)CrossRefGoogle 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