Skip to main content
SpringerLink
Log in

Simplified inverse dynamic models of parallel robots based on a Lagrangian approach

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

This paper proposes and analyzes some simplified dynamic modeling methods for parallel robots. The proposed methods are based on a Lagrangian approach where the key concept consists in simplifying the expression of the energy for robots, especially for the distal links. First, the slender link method is discussed, where the link energy is calculated from the endpoints’ velocities. Then, the commonly used equivalent point mass (EPM) method is analyzed. Since the EPM method uses point masses to replace links, the main problem is how to assign the values of point masses. According to the energy equivalence principle, some methods are proposed to solve the value-assigning problem. The derivations of the proposed methods are given, and the errors caused by each method are analyzed. Based on the error analyses, some simple approaches are introduced to improve the accuracy of the EPM methods. Inverse dynamic models of a spatial robot are established based on the proposed methods. Then, detailed comparisons and analyses of the different dynamic models are given to show that the proposed methods are simple and have relatively high accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31

Similar content being viewed by others

References

  1. Dasgupta B, Mruthyunjaya T (1998) A Newton–Euler formulation for the inverse dynamics of the Stewart platform manipulator. Mech Mach Theory 33(8):1135–1152. https://doi.org/10.1016/S0094-114X(97)00118-3

    Article  MathSciNet  Google Scholar 

  2. Dasgupta B, Mruthyunjaya T (1998) Closed-form dynamic equations of the general Stewart platform through the Newton–Euler approach. Mech Mach Theory 33(7):993–1012. https://doi.org/10.1016/S0094-114X(97)00087-6

    Article  MathSciNet  Google Scholar 

  3. Gosselin C (1996) Parallel computational algorithms for the kinematics and dynamics of planar and spatial parallel manipulators. ASME J Dyn Syst Meas Control 118(1):22–28. https://doi.org/10.1115/1.2801147

    Article  Google Scholar 

  4. Codourey A (1998) Dynamic modeling of parallel robots for computed-torque control implementation. Int J Robot Res 17(12):1325–1336. https://doi.org/10.1177/027836499801701205

    Article  Google Scholar 

  5. Wang J, Gosselin C (1998) A new approach for the dynamic analysis of parallel manipulators. Multibody Syst Dyn 2(3):317–334. https://doi.org/10.1023/A:1009740326195

    Article  MathSciNet  Google Scholar 

  6. Tsai LW (2000) Solving the inverse dynamics of a Stewart–Gough manipulator by the principle of virtual work. ASME J Mech Des 122(1):3–9. https://doi.org/10.1115/1.533540

    Article  Google Scholar 

  7. Shao P, Wang Z, Yang S et al (2019) Dynamic modeling of a two-DoF rotational parallel robot with changeable rotational axes. Mech Mach Theory 131:318–335. https://doi.org/10.1016/j.mechmachtheory.2018.08.020

    Article  Google Scholar 

  8. Gallardo-Alvarado J, Aguilar-Nájera CR, Casique-Rosas L et al (2008) Kinematics and dynamics of 2(3-RPS) manipulators by means of screw theory and the principle of virtual work. Mech Mach Theory 43(10):1281–1294. https://doi.org/10.1016/j.mechmachtheory.2007.10.009

    Article  Google Scholar 

  9. Abed Azad F, Ansari Rad S, Hairi Yazdi MR et al (2022) Dynamics analysis, offline–online tuning and identification of base inertia parameters for the 3-DOF Delta parallel robot under insufficient excitations. Meccanica 57(2):473–506. https://doi.org/10.1007/s11012-021-01464-7

    Article  MathSciNet  Google Scholar 

  10. Sun T, Yang S (2019) An approach to formulate the Hessian matrix for dynamic control of parallel robots. IEEE/ASME Trans Mechatron 24(1):271–281. https://doi.org/10.1109/TMECH.2019.2891297

    Article  Google Scholar 

  11. Begey J, Cuvillon L, Lesellier M et al (2018) Dynamic control of parallel robots driven by flexible cables and actuated by position-controlled winches. IEEE Trans Rob 35(1):286–293. https://doi.org/10.1109/TRO.2018.2875415

    Article  Google Scholar 

  12. Berti A, Gouttefarde M, Carricato M (2018) Dynamic recovery of cable-suspended parallel robots after a cable failure. In: Advances in robot kinematics 2016. Springer, pp 331–339. https://doi.org/10.1007/978-3-319-56802-7_35

  13. Jiang X, Barnett E, Gosselin C (2018) Dynamic point-to-point trajectory planning beyond the static workspace for six-DoF cable-suspended parallel robots. IEEE Trans Rob 34(3):781–793. https://doi.org/10.1109/TRO.2018.2794549

    Article  Google Scholar 

  14. Geng Z, Haynes LS, Lee JD et al (1992) On the dynamic model and kinematic analysis of a class of Stewart platforms. Robot Auton Syst 9(4):237–254. https://doi.org/10.1016/0921-8890(92)90041-V

    Article  Google Scholar 

  15. Lebret G, Liu K, Lewis FL (1993) Dynamic analysis and control of a Stewart platform manipulator. J Robot Syst 10(5):629–655. https://doi.org/10.1002/rob.4620100506

    Article  Google Scholar 

  16. Di Gregorio R, Parenti-Castelli V (2004) Dynamics of a class of parallel wrists. ASME J Mech Des 126(3):436–441. https://doi.org/10.1115/1.1737382

    Article  Google Scholar 

  17. Abdellatif H, Heimann B (2009) Computational efficient inverse dynamics of 6-DoF fully parallel manipulators by using the Lagrangian formalism. Mech Mach Theory 44(1):192–207. https://doi.org/10.1016/j.mechmachtheory.2008.02.003

    Article  Google Scholar 

  18. Meng G, Zhao X, Li B (2010) Inverse dynamic modeling for a 3-RRRT parallel manipulator. In: 2010 IEEE international conference on robotics and biomimetics (ROBIO), Tianjin, China, pp 495–500. https://doi.org/10.1109/ROBIO.2010.5723376

  19. Sherwood A, Hockey B (1969) The optimisation of mass distribution in mechanisms using dynamically similar systems. J Mech 4(3):243–260. https://doi.org/10.1016/0022-2569(69)90005-6

    Article  Google Scholar 

  20. Gherman B, Pisla D, Vaida C et al (2012) Development of inverse dynamic model for a surgical hybrid parallel robot with equivalent lumped masses. Robot Comput Integr Manuf 28(3):402–415. https://doi.org/10.1016/j.rcim.2011.11.003

    Article  Google Scholar 

  21. Ardestani MA, Asgari M (2012) Modeling and analysis of a novel 3-DoF spatial parallel robot. In: 2012 19th International conference on mechatronics and machine vision in practice (M2VIP), Auckland, New Zealand. IEEE, pp 162–167

  22. Wen K, Nguyen TS, Harton D et al (2020) A backdrivable kinematically redundant (6 + 3)-degree-of-freedom hybrid parallel robot for intuitive sensorless physical human–robot interaction. IEEE Trans Rob 37(4):1222–1238. https://doi.org/10.1109/TRO.2020.3043723

    Article  Google Scholar 

  23. Li Y, Xu Q (2007) Design and development of a medical parallel robot for cardiopulmonary resuscitation. IEEE/ASME Trans Mechatron 12(3):265–273. https://doi.org/10.1109/TMECH.2007.897257

    Article  MathSciNet  Google Scholar 

  24. Xiao C, Jiang H, Zhang G et al (2019) Decoupling control and simulation of a 3-RSR spheroid parallel wrist. In: 2019 IEEE international conference on robotics and biomimetics (ROBIO), Dali, China, pp 989–994. https://doi.org/10.1109/ROBIO49542.2019.8961596

  25. Codourey A (1996) Dynamic modelling and mass matrix evaluation of the DELTA parallel robot for axes decoupling control. In: Proceedings of IEEE/RSJ international conference on intelligent robots and systems (IROS), Osaka, Japan, pp 1211–1218. https://doi.org/10.1109/IROS.1996.568973

  26. Hao J, Xie X, Bian G et al (2015) Dynamic modeling and control simulation of a modified DELTA manipulator. In: 2015 IEEE international conference on information and automation, Lijiang, China, pp 1573–1578. https://doi.org/10.1109/ICInfA.2015.7279537

  27. Li Y, Xu Q (2009) Dynamic modeling and robust control of a 3-PRC translational parallel kinematic machine. Robot Comput Integr Manuf 25(3):630–640. https://doi.org/10.1016/j.rcim.2008.05.006

    Article  Google Scholar 

  28. Nakamura Y, Ghodoussi M (1989) Dynamics computation of closed-link robot mechanisms with nonredundant and redundant actuators. IEEE Trans Robot Autom 5(3):294–302. https://doi.org/10.1109/70.34765

    Article  Google Scholar 

  29. Cheng H, Yiu YK, Li Z (2003) Dynamics and control of redundantly actuated parallel manipulators. IEEE/ASME Trans Mechatron 8(4):483–491. https://doi.org/10.1109/TMECH.2003.820006

    Article  Google Scholar 

  30. Lee SH, Song JB, Choi WC et al (2003) Position control of a Stewart platform using inverse dynamics control with approximate dynamics. Mechatronics 13(6):605–619. https://doi.org/10.1016/S0957-4158(02)00033-8

    Article  Google Scholar 

  31. Nguyen TS, Harton D, Campeau-Lecours A et al (2021) Motion control algorithms based on the dynamic modelling of kinematically redundant hybrid parallel robots. Mechatronics 76:102555. https://doi.org/10.1016/j.mechatronics.2021.102555

    Article  Google Scholar 

  32. Liu XJ, Wu C, Wang J (2012) A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators. ASME J Mech Robot. https://doi.org/10.1115/1.4007004

    Article  Google Scholar 

  33. Stigger T, Siegele J, Scharler DF et al (2019) Analysis of a 3-RUU parallel manipulator using algebraic constraints. Mech Mach Theory 136:256–268. https://doi.org/10.1016/j.mechmachtheory.2019.03.011

    Article  Google Scholar 

  34. Gallardo-Alvarado J (2023) Unified infinitesimal kinematics of a 3-RRR/PRR six-degree-of-freedom parallel-serial manipulator. Meccanica 58(4):795–811. https://doi.org/10.1007/s11012-023-01648-3

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would also like to thank Arda Yiğit for the discussion on the use of the Lagrangian equations.

Author information

Authors and Affiliations

  1. Département de Génie Mécanique, Université Laval, 1065 Avenue de la Médecine, Québec, G1V 0A6, QC, Canada

    Zhou Zhou & Clément Gosselin

Authors
  1. Zhou Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clément Gosselin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clément Gosselin.

Ethics declarations

Conflict of interest

The authors declare that there are no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, Z., Gosselin, C. Simplified inverse dynamic models of parallel robots based on a Lagrangian approach. Meccanica 59, 657–680 (2024). https://doi.org/10.1007/s11012-024-01782-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-024-01782-6

Keywords

Search

Navigation