Stiffness Modeling and Analysis of a 3-DOF Parallel Kinematic Machine

  • Yanqin Zhao
  • Chensheng Wang
  • Wentie Niu
  • Zhaobo Mei
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)


To evaluate rigidity of a 3-degree-of-freedom parallel kinematic machine which forms a novel hybrid robot, a stiffness model is established by combining the matrix structure method with the virtual joint method. Compliance of joints and limbs is considered in the presented model by simplifying limbs into spatial limbs and treating joints as virtual springs with equivalent stiffness. Static equilibrium equations of active/passive limbs and the platform are derived respectively, after which the governing static equilibrium equation of the system can be obtained by introducing deformation compatibility conditions. To verify the presented model, finite element analysis is conducted. Based on the presented model, contribution rates of structural parameters to the system’s rigidity are evaluated to provide useful information on optimal design.


Parallel kinematic machine stiffness matrix structure method virtual joint method 


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This work was supported by the National Key Research Project under Grant 2017ZX04013001.


  1. 1.
    Neumann, K.: System and method for controlling a robot. U.S. Patent 6301525, October 9, 2010.Google Scholar
  2. 2.
    Neumann, K.: Tricept application. In: Proceedings of the 3rd Chemnitz Parallel Kinematics Seminar, pp. 547–551. Zwickau (2002).Google Scholar
  3. 3.
    Wang, Y. Y., Liu, H. T., Huang, T., Chetwynd, D. G.: Stiffness Modeling of the Tricept Robot Using the Overall Jacobian Matrix. ASME Journal of Mechanisms Robotics 1(2), 021002–021002-8 (2009).CrossRefGoogle Scholar
  4. 4.
    Karthick S.: Design and dynamic analysis of Tricept parallel manipulator. Journal for Research 3(2), 73–78 (2017).Google Scholar
  5. 5.
    Dong, C. L., Liu, H. T, Yue, W., Huang, T.: Stiffness modeling and analysis of a novel 5-DOF hybrid robot. Mechanism and Machine Theory 125(2018), 80–93 (2018).CrossRefGoogle Scholar
  6. 6.
    Wu, L., Wang, G. F., Liu, H. T., Huang, T.: An approach for elastodynamic modeling of hybrid robots based on substructure synthesis technique. Mechanism and Machine Theory 123(2018), 124–136 (2018).CrossRefGoogle Scholar
  7. 7.
    Bouzgarrou, B. C., Fauroux, J. C., Gogu, G., Heerah Y.: Rigidity of T3R1 parallel robot with uncoupled kinematics. In: Proceedings of the 35th International Symposium on Robotics (ISR), pp. 1–6. Paris (2004).Google Scholar
  8. 8.
    Nigus, H.: Semi-analytical approach for stiffness estimation of 3-DOF PKM. Modern Mechanical Engineering 4(2), 108–118 (2014).CrossRefGoogle Scholar
  9. 9.
    Wang, Y. Y., Huang, T., Zhao, X. M., Mei, J. P., Chetwynd, D. G.: A semi-analytical approach for stiffness modeling of PKM by considering compliance of machine frame with complex geometry. Chinese Science Bulletin 53(16), 2565–2574 (2008).CrossRefGoogle Scholar
  10. 10.
    Zhang, D., Wang, L. H.: Conceptual development of an enhanced tripod mechanism for machine tool. Robotics and Computer-Integrated Manufacturing 21(4–5), 318–327 (2005).CrossRefGoogle Scholar
  11. 11.
    Majou, F., Gosselin, C., Wenger, P., Chablat D.: Parametric stiffness analysis of the orthoglide. Mechanism and Machine Theory 42(3), 296–311 (2007).CrossRefGoogle Scholar
  12. 12.
    Darvekar, S. K., Rao, A. B. K., Praveen, J. V. S.: Stiffness Estimation of a 2-DoF Parallel Kinematic Machine. In: International Conference on Computing Communication Control and Automation IEEE, pp. 23–28. Pune (2015).Google Scholar
  13. 13.
    Huang, T., Zhao, X. Y., Whitehouse, D. J.: Stiffness estimation of a tripod-based parallel kinematic machine. IEEE Trans on Robotics and Automation 18(1), 50–58 (2002).Google Scholar
  14. 14.
    Ceccarelli, M., Carbone, G.: A stiffness analysis for CaPaMan (Cassino Parallel Manipulator). Mechanism and Machine Theory 37(5), 427–439 (2002).CrossRefGoogle Scholar
  15. 15.
    Zhang, J., Zhao, Y. Q., Jin, Y.: Kinetostatic-model-based stiffness analysis of Exechon PKM. Robotics and Computer-Integrated Manufacturing 37(2016), 208–220 (2016).CrossRefGoogle Scholar
  16. 16.
    Liu, Q., Huang, T.: Inverse kinematics of a 5-axis hybrid robot with non-sigular tool path generation. Robotics and Computer-Integrated Manufacturing 56(2019), 140–148 (2019).CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yanqin Zhao
    • 1
  • Chensheng Wang
    • 1
  • Wentie Niu
    • 1
  • Zhaobo Mei
    • 2
  1. 1.Key Laboratory of Mechanism Theory and Equipment Design of Ministry of EducationTianjin UniversityTianjinChina
  2. 2.School of PhysicsBUAA, Beihang UniversityBeijingChina

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