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Geometric Error Modeling of Parallel Manipulators Based on Conformal Geometric Algebra

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Abstract

An approach for geometric error modeling of parallel manipulators (PMs) based on the visual representation and direct calculation of conformal geometric algebra is introduced in this paper. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then the error sources are separated in terms of joint perturbations and geometric errors. Next, motions and constraints of PMs are analyzed visually by their reciprocal property. Finally geometric error model of PMs are formulated considering the actuations and constraints. The merits of this new approach are twofold: (1) complete and continuous geometric error modeling can be achieved since finite motions are considered, (2) visual and analytical computation of motions and constraints are applied for transferring geometric errors from the open-loop chain to the PM. A 2-DoF rotational PM is applied to demonstrate the geometric error modeling process. Comparisons between simulation and analytical models show that this approach is highly effective.

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References

  1. Alessandro, V., Gianni, C., Antonio, S.: Axis geometric errors analysis through a performance test to evaluate kinematic error in a five axis tilting-rotary table machine tool. Precis. Eng. 39, 224–233 (2015)

    Article  Google Scholar 

  2. Carbonari, L., Callegari, M., Palmieri, C., Palpacelli, M.C.: Analysis of kinematics and reconfigurability of a spherical parallel manipulator. IEEE Trans. Robot. 30(6), 1541–1547 (2014)

    Article  Google Scholar 

  3. Charker, A., Mlika, A., Laribi, M.A., Romdhane, L., Zeghloul, S.: Accuracy analysis of non-overconstrianed spherical parallel manipulators. Eur. J. Mech. A Solid 47, 362–372 (2014)

    Article  ADS  Google Scholar 

  4. Chen, I.M., Yang, G.L., Tan, C.T., Yeo, S.H.: Local POE model for robot kinematic calibration. Mech. Mach. Theory. 36, 1215–1239 (2001)

    Article  MATH  Google Scholar 

  5. Chen, J.X., Lin, S.W., Zhou, Z.L.: A comprehensive error analysis method for the geometric error of multi-axis machine tool. Int. J. Mach. Tools Manuf. 106, 56–66 (2016)

    Article  Google Scholar 

  6. Dong, G., Sun, T., Song, Y.M., Gao, H., Lian, B.B.: Mobility analysis and kinematic synthesis of a novel 4-DoF parallel manipulator. Robotica 34(5), 1010–1025 (2016)

    Article  Google Scholar 

  7. Frisoli, A., Solazzi, M., Pellegrinetti, D., Bergamasco, M.: A screw theory method for the estimation of position accuracy in spatial parallel manipulators with revolute joint clearances. Mech. Mach. Theory 46, 1929–1949 (2011)

    Article  Google Scholar 

  8. Fu, J.X., Gao, F., Chen, W.X., Pan, Y., Lin, R.F.: Kinematic accuracy research of a novel six-degree-of-freedom parallel robot with three legs. Mech. Mach. Theory 102, 86–102 (2016)

    Article  Google Scholar 

  9. He, B., Zhang, P.C., Hou, S.C.: Accuracy analysis of a spherical 3-DOF parallel underactuated robot wrist. Int. Adv. Manuf. Technol. 79, 395–404 (2015)

    Article  Google Scholar 

  10. Hestenes, D.: New tools for computational geometry and rejuvenation of screw theory. In: Bayro-Corrochano, E.J., Scheuermann, G. (eds.) Geometric Algebra Computing Engineering and Computer Science, pp. 3–35. Springer, Berlin (2010)

    Chapter  Google Scholar 

  11. Huang, Z., Li, Q.C., Ding, H.: Theory of parallel mechanisms. Mech. Mach. Sci. 6, 47–69 (2013)

    Article  Google Scholar 

  12. Huang, P., Wang, J., Wang, L.P.: Kinematical calibration of a hybrid machine tool with regularization method. Int. J. Mach. Tools Manuf. 17(8), 210–220 (2011)

    Article  Google Scholar 

  13. Huo, X.M., Sun, T., Song, Y.M., Qi, Y., Wang, P.F.: An analytical approach to determine motions/constraints of serial kinematic chains based on Clifford algebra. Proc. Inst. Mech. Eng. C J. Mech. 231(7), 1324–1338 (2017)

    Article  Google Scholar 

  14. Huo, X.M., Sun, T., Song, Y.M.: A geometric algebra approach to determine motion/constraint, mobility and singularity of parallel mechanism. Mech. Mach. Theory 116, 273–293 (2017)

    Article  Google Scholar 

  15. Jin, Y., Chen, I.M.: Effects of constraint errors on parallel manipulators with decoupled motion. Mech. Mach. Theory 41, 912–928 (2006)

    Article  MATH  Google Scholar 

  16. Kumaraswamy, U., Shunmugam, M.S., Sujatha, S.: A unified framework for tolerance analysis of planar and spatial mechanisms using screw theory. Mech. Mach. Theory 69, 168–184 (2013)

    Article  Google Scholar 

  17. Li, J., Xie, F.G., Liu, X.J.: Geometric error modeling and sensitivity analysis of a five-axis machine tool. Int. Adv. Manuf. Technol. 82, 2037–2051 (2016)

    Article  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. Liang, D., Song, Y.M., Sun, T., Jin, X.Y.: Rigid-flexible coupling dynamic modeling and investigation of a redundantly actuated parallel manipulator with multiple actuation modes. J. Sound Vib. 403, 129–151 (2017)

    Article  ADS  Google Scholar 

  20. Liu, H.T., Huang, T., Derek, G.C.: A general approach for geometric error modeling of lower mobility parallel manipulators. J. Mech. Robot. 3, 021013 (2011)

    Article  Google Scholar 

  21. Nategh, M.J., Agheli, M.M.: A total solution to kinematic calibration of hexapod machine tools with a minimum number of measurement configurations and superior accuracies. Int. J. Mach. Tools Manuf. 49, 1155–1164 (2009)

    Article  Google Scholar 

  22. Okamura, K., Park, F.C.: Kinematic calibration and product of exponential formula. Robotica 14, 415–421 (1996)

    Article  Google Scholar 

  23. Park, F.C., Okamura, K.: Kinematic Calibration and Product of Exponential Formula. Advances in Robot Kinematics and Computational Geometry, pp. 119–128. MIT Press, Cambridge (1994)

    Book  Google Scholar 

  24. Qi, Y., Sun, T., Song, Y.M., Jin, Y.: Topology synthesis of three-legged spherical parallel manipulators employing Lie group theory. Proc. Inst. Mech. Eng. C. J. Mech. 229(10), 1873–1886 (2015)

    Article  Google Scholar 

  25. Qi, Y., Sun, T., Song, Y.M.: Type synthesis of parallel tracking mechanism with varied axes by modeling its finite motions algebraically. J. Mech. Robot. 9(5), 054504-1–054504-6 (2017)

    Article  Google Scholar 

  26. Sommers, G., Computergrafik Datenverarbeitung, G.: Geometric Computing with Clifford Algebras. Springer, Berlin (2001)

    Book  Google Scholar 

  27. Song, Y.M., Qi, Y., Dong, G., Sun, T.: Type synthesis of 2-DoF rotational parallel mechanisms actuating the inter-satellite link antenna. Chin. J. Aeronaut. 29(6), 1795–1805 (2016)

    Article  Google Scholar 

  28. Song, Y.M., Zhang, J.T., Lian, B.B., Sun, T.: Kinematic calibration of a 5-DoF parallel kinematic machine. Precis. Eng. 45, 242–261 (2016)

    Article  Google Scholar 

  29. Stone, H.W.: Kinematic Modeling, Identification, and Control of Robotic Manipulators. Kluwer Academic Publishers, Dordrecht (1987)

    Book  Google Scholar 

  30. Sun, T., Song, Y.M., Li, Y.G., Zhang, J.: Workspace decomposition based dimensional synthesis of a novel hybrid reconfigurable robot. J. Mech. Robot. 2(3), 31009 (2010)

    Article  Google Scholar 

  31. 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(1), 168–174 (2010)

    Article  MATH  Google Scholar 

  32. Sun, T., Song, Y.M., Li, Y.G., Xu, L.: Separation of comprehensive geometrical errors of a 3-dof parallel manipulator based on Jacobian matrix and its sensitivity analysis with Monte-Carlo method. Chin. J. Mech. Eng. Engl. 24(3), 406–413 (2011)

    Article  Google Scholar 

  33. Sun, T., Song, Y.M., Yan, K.: Kineto-static analysis of a novel high-speed parallel manipulator with rigid-flexible coupled links. J. Cent. South Univ. Technol. Engl. Ed. 18(3), 593–599 (2011)

    Article  Google Scholar 

  34. Sun, X.Y., Xie, Z.J., Shi, W.K.: Error analysis and calibration of 6-PSS parallel mechanism. Robot. Comput. Int. Manuf. 18(12), 2659–2666 (2012)

    Google Scholar 

  35. Sun, T., Song, Y.M., Gao, H., Qi, Y.: Topology synthesis of a 1T3R parallel manipulator with an articulated traveling plate. J. Mech. Robot. 7(3), 310151–310159 (2015)

    Article  Google Scholar 

  36. 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. Int. Manuf. 41, 78–91 (2016)

    Article  Google Scholar 

  37. 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)

    Article  Google Scholar 

  38. Wang, J., Masory, O.: On the accuracy of a Stewart platform-part I: the effect of manufacturing tolerances. In: Proceedings of IEEE International Conference on Robotics and Automation, Alanta, Georgia, USA, pp. 114–120 (1993)

  39. Whitney, D.E., Lozinski, C.A., Rourke, J.M.: Industrial robot forward calibration method and results. J. Dyn. Syst. 108, 1–8 (1986)

    Article  MATH  Google Scholar 

  40. Wu, Y., Klimchik, A., Caro, S., Furet, B., Pashkevich, A.: Geometric calibration of industrial robots using enhanced partial pose measurements and design of experiments. Robot. Comput. Int. Manuf. 35, 151–168 (2015)

    Article  Google Scholar 

  41. Xu, Q.S., Li, Y.M.: Error analysis and optimal design of a class of translational parallel kinematic machine using particle swarm optimization. Robotica 27, 67–78 (2009)

    Article  Google Scholar 

  42. Yang, S.F., Sun, T., Huang, T., Li, Q.C., Gu, D.B.: A finite screw approach to type synthesis of three-DoF transaltional parallel mechanisms. Mech. Mach. Theory 104, 405–419 (2016)

    Article  Google Scholar 

  43. Yang, S.F., Sun, T., Huang, T.: Type synthesis of parallel mechanisms having 3T1R motion with variable rotational axis. Mach. Theory 109, 220–230 (2017)

    Article  Google Scholar 

  44. Zhuang, H., Roth, Z.S., Hamano, F.: A complete and parametrically continuous kinematic model for robot manipulators. IEEE Trans. Rob. Autom. 8(4), 451–463 (1992)

    Article  Google Scholar 

  45. Zi, B., Ding, H.F., Wu, X., Kesckemethy, A.: Error modeling and sensitivity of a hybrid-driven based cable parallel manipulator. Percis. Eng. 38, 197–211 (2014)

    Article  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, Tianjin University, Tianjin, 300350, China

    Binbin Lian

  2. Department of Machine Design, School of Industrial Engineering and Management, KTH Royal Institute of Technology, Brinellvägen 83, 10044, Stockholm, Sweden

    Binbin Lian

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  1. Binbin Lian
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Correspondence to Binbin Lian.

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Communicated by Hongbo Li.

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Lian, B. Geometric Error Modeling of Parallel Manipulators Based on Conformal Geometric Algebra. Adv. Appl. Clifford Algebras 28, 30 (2018). https://doi.org/10.1007/s00006-018-0831-5

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  • DOI: https://doi.org/10.1007/s00006-018-0831-5

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