Skip to main content
SpringerLink
Log in

Recursive Modeling of Serial Flexible Manipulators

  • Original Article
  • Published:
The Journal of the Astronautical Sciences Aims and scope Submit manuscript

Abstract

This paper discusses the modeling of serial manipulators with flexible links and joints. The model of a flexible link includes bending in two perpendicular directions and torsion around the longitudinal axis. Second-order strain-displacement relationships, coupled with curvatures as generalized coordinates, are used to represent the foreshortening effect. Then, the dynamic equations are exact to first order in terms of the generalized coordinates associated with the flexible links. These generalized coordinates and their first time-derivatives are assumed to be small (of first order). The model developed captures all the important phenomena, such as stiffening due to the angular speed or buckling due to large payloads. The dynamic equations are developed recursively using Jourdain’s principle to allow an efficient symbolic implementation. The equations associated with the flexible links are reformulated to enable off-line symbolic integration. The gyroscopic effects of motors with reducers are included.

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

Similar content being viewed by others

References

  1. ERDMAN, A. (editor) Modern Kinematics: Developments in the Last Forty Years, Wiley Interscience, New York, 1993, pp. 333–368.

    Google Scholar 

  2. SHABANA, A.A. Dynamics of Multibody Systems, John Wiley & Sons, New York, 1989, pp. 220–252.

    MATH  Google Scholar 

  3. AMIROUCHE, F. M. L. Computational Methods in Multibody Dynamics, Prentice Hall, New Jersey, 1992, pp. 322–423.

    MATH  Google Scholar 

  4. TABARROK, B., and RIMROTT, F. P. J. Variational Methods and Complementary Formulations in Dynamics, Kluwer Academic Publishers, Dordrecht, 1994, pp. 67–117.

    Book  Google Scholar 

  5. BREMER, H. “Über eine Zentralgleichung in der Dynamik,” ZAMM Z. angew. Math. Mech., Vol. 68, No. 7, 1988, pp. 307–311.

    MATH  Google Scholar 

  6. GARCÍA DE JALÓN, J., and BAYO, E. Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge, Springer-Verlag, New York, 1994, pp. 271–324.

    Book  Google Scholar 

  7. HAUG, E. J., and DEYO, R. C. (editors) Real-Time Integration Methods for Mechanical System Simulation, NATO ASI Series F: Computer and Systems Sciences, Springer-Verlag, Berlin, 1991.

    Book  Google Scholar 

  8. JOURDAIN, P.E.B. “Note on an Analogue of Gauss’ Principle of Least Constraint,” The Quarterly Journal of Pure and Applied Mathematics, Vol. 40, 1909, pp. 153–157.

    MATH  Google Scholar 

  9. PIEDBŒUF, J.-C. “A Comparison Between Kane’s Equations and Jourdain’s Principle,” Technical Report 931201, Mechanical Engineering Department, Royal Military College of Canada, Kingston, Ontario, Canada, December 1993.

    Google Scholar 

  10. KANE, T.R. “Dynamics of Nonholonomic Systems,” Transactions of the ASME Journal of Applied Mechanics, 1961, pp. 574–578.

  11. ROBERSON, R.E., and SCWERTASSEK, R. Dynamics of Multibody Systems, Springer-Verlag, 1988.

  12. SIMO, J.C., and VU-QUOC, L. “The Role of Non-Linear Theories in Transient Dynamic Analysis of Flexible Structures,” Journal of Sound and Vibration, Vol. 119, No. 3, 1987, pp. 487–508.

    Article  MathSciNet  Google Scholar 

  13. WALLRAPP, O., and SCHWERTASSEK, R. “Representation of Geometric Stiffening in Multibody System Simulation,” International Journal of Numerical Methods in Engineering, Vol. 32, 1991, pp. 1833–1850.

    Article  Google Scholar 

  14. PADILLA, C.E., and VON FLOWTOW, “Nonlinear Strain-Displacement Relations and Flexible Multibody Dynamics,” Journal of Guidance, Control, and Dynamics, Vol. 15, 1992, pp. 128–136.

    Article  Google Scholar 

  15. PIEDBŒUF, J.-C., and HURTEAU, R. “Modelling and Analysis of a Two-Degree-of-Freedom Robot with a Flexible Forearm,” Canadian Journal of Electrical and Computer Engineering, Vol. 16, No. 4, 1991, pp. 127–134.

    Article  Google Scholar 

  16. RYU, J., KIM, S.-S., and KIM, S.-S. “A General Approach to Stress Stiffening Effects on Flexible Multibody Dynamic Systems,” Mechanism of Structures and Machines, Vol. 22, No. 2, 1994, pp. 157–180.

    Article  Google Scholar 

  17. SHARF, I. “Geometric Stiffening in Multibody Dynamics Formulations,” Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia, Canada, Personal communication, 1993.

    Google Scholar 

  18. DAMAREN, C., and SHARF, I. “Simulation of Flexible-Link Manipulators with Inertial and Geometric Nonlinearities,” Transactions of the ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 117, 1995, pp. 74–87.

    Article  Google Scholar 

  19. KAZA, K. R. V., and KVATERNIK, R. G. “Nonlinear Flag-Lag-Axial Equations of a Rotating Beam,” AIAA Journal, Vol. 15, 1977, pp. 871–874.

    Article  Google Scholar 

  20. HODGES, D. H., and DOWELL, E. H. “Nonlinear Equations of Motion for the Elastic Bending and Torsion of Twisted Nonuniform Rotor Blades,” Technical Report TN D-7818, NASA Ames Research Center, Moffet Field, California, December 1974.

  21. PARK, K. C., DOWNER, J. D., CHIOU, J. C., and FARHAT, C. “A Modular Multibody Analysis Capability for High-Precision, Active Control and Real-Time Applications,” International Journal of Numerical Methods in Engineering, Vol. 32, 1991, pp. 1767–1798.

    Article  Google Scholar 

  22. GÉRADIN, M. “General Methods for Flexible Multibody Systems,” Proceedings of the 2nd Conference on Mechatronics and Robotics, Duisburg, Germany, September 1993, pp. 1–16.

  23. HAERING, W. J., RYAN, R.R., and SCOTT, R.A. “New Formulation for General Spatial Motion of Flexible Beams,” Journal of Guidance, Control, and Dynamics, Vol. 18, No. 1, 1995, pp. 82–86.

    Article  Google Scholar 

  24. PIEDBŒUF, J.-C. “The Jacobian Matrix for a Flexible Manipulator,” Journal of Robotic Systems, Vol. 12, No. 11, 1995, pp. 709–726.

    Article  Google Scholar 

  25. BREMER, H., and PFEIFFER, F. Elastiche Mehrkörpersystem, B. G. Teubner Stuttgart, 1992, pp. 42–87.

  26. JUNKINS, J. L., and KIM, Y. Introduction to Dynamics and Control of Flexible Structures, AIAA Education Series, Washington, 1993, pp. 139–234.

    Book  Google Scholar 

  27. BAGLEY, R.L., and TORVIK, P.J. “Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures,” AIAA Journal, Vol. 23, 1985, pp. 918–925.

    Article  Google Scholar 

  28. SKAAR, S.B., MICHEL, A.N., and MILLER, R.K. “Stability of Viscoelastic Control Systems,” IEEE Transactions on Automatic Control, Vol. 33, 1988, pp. 348–357.

    Article  MathSciNet  Google Scholar 

  29. MAKROGLOU, S. S., MILLER, R. K., and SKAAR, S. “Computational Results for a Feedback Control for a Rotating Viscoelastic Beam,” Journal of Guidance, Control, and Dynamics, Vol. 17, No. 1, 1994, pp. 84–90.

    Article  Google Scholar 

  30. POTVIN, M.-J., PIEDBŒUF, J.-C., and NEMES, J. A. “Solution d’équations Dynamiques Non-Linéaires avec loi de Comportement Viscoélastique d’ordre Fractionnaire,” Proceedings of the 16th Canadian Congress of Applied Mechanics (CANCAM97), Université Laval, Québec, Canada, June 1997, pp. 75–76.

    Google Scholar 

  31. POTVIN, M.-J., JESWIET, J., and PIEDBŒUF, J.-C. “Réponse Temporelle d’une Poutre en Polymère Modélisée par un Amortissement Fractionnaire de Voigt-Kelvin,” Transactions of the CSME, Vol. 19, No. 1, 1995, pp. 13–24.

    Google Scholar 

  32. PILKEY, W.D., and WUNDERLICH, W. Mechanics of Structures: Variational and Computational Methods, CRC Press, 1994, pp. 669–710.

  33. HODGES, D.H., ORMISTON, R.A., and PETERS, D.A. “On the Nonlinear Deformation Geometry of Euler-Bernoulli Beams,” Technical Report, TP 1566, NASA Ames Research Center, Moffet Field, California, 1980.

  34. JOHANNI, R. “Automatisches Aufstellen der Bewegungsgleichungen von Baumstrukturierten Mehrkörpersystemen mit Elastischen Bauteilen,” Technical Report 8501, Institute B for Mechanics, Technical University of Munich, 1985.

  35. SADIGH, M. J., and MISRA, A.K. “A Control-Oriented Symbolic Multibody Formulation with Application to Flexible Space Manipulator,” Technical Report, Mechanical Engineering Department, McGill University, 1994.

    Google Scholar 

  36. SAAD, M., PIEDBCEUF, J.-C., AKHRIF, O., and SAYDY, L. “A Comparison of Different Shape Function in Assumed-Mode Model of a Flexible Slewing Beam,” The International Journal of Robotics Research, Submitted for publication, May 1997.

  37. PIEDBŒUF, J.-C. “Kane’s Equation or Jourdain’s Principle,” 36th Midwest Symposium on Circuits and Systems, Detroit, Michigan, August 1993.

  38. CHUN, H.M., TURNER, J.D., and FRISCH, H. P. “Recursive Multibody Formulations for Robotics Applications with Harmonic Drives,” International Conference on Dynamics of Flexible Structures in Space, Cranfield, United Kingdom, May 1990.

  39. PIEDBŒUF, J.-C. “SYMOFROS: Symbolic Modelling of Flexible Robots and Simulation,” Proceedings of the AAS/AIAA Astrodynamics Conference, Halifax, Nova Scotia, Canada, August 1995, pp. 949–968.

Download references

Author information

Authors and Affiliations

  1. Space Technologies, Canadian Space Agency, 6767 Route de l’Aéroport, St-Hubert, J3Y 8Y9, Québec, Canada

    Jean-Claude Pidbœuf

Authors
  1. Jean-Claude Pidbœuf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Claude Pidbœuf.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pidbœuf, JC. Recursive Modeling of Serial Flexible Manipulators. J of Astronaut Sci 46, 1–24 (1998). https://doi.org/10.1007/BF03546190

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03546190

Search

Navigation