Skip to main content
SpringerLink
Log in

Spatial Compliance Modeling Using a Quaternion-Based Potential Function Method

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

This paper looks at the modeling of elastically coupled rigidbodies. The elastic deformation is assumed to be localized, which is aparticularly valid assumption for flexural joints. A generic, lumpedparameter, Euclidean geometric, potential function based approach ispresented using quaternion calculus. The potential functions are similarto the functions presented in the spatial compliance control literature.Rigid body displacements are represented using a combination ofCartesian coordinates and quaternions. To demonstrate the utility of theproposed methods for computer analysis, a nontrivial example isconsidered. The system consists of two rigid bodies coupled by anasymmetric flexure incorporating crossed leaf springs. While thecompliant constitutive equations are well defined for arbitrary rigidbody displacements, it is only claimed that the model is accurate forsmall displacements.

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. Blanchet, P. and Lipkin, H., 'New geometric properties for modelled planar vibration', in Proceedings ASME Design Engineering Technical Conference, Vol. DETC97/VIB-4176, CD-ROM, ASME, New York, 1997.

    Google Scholar 

  2. Brockett, R.W. and Lončarić, J., 'The geometry of compliance programming', 'in Theory and Applications of Nonlinear Control Systems, C.I. Byrnes and A. Lindquist (eds), North-Holland, Amsterdam, 1986, 35-42.

    Google Scholar 

  3. Caccavale, F., Natale, C., Siciliano, B. and Villani, L., 'Six-DOF impedance control based on angle/axis representations', IEEE Transactions on Robotics and Automation 15, 1999, 289-300.

    Google Scholar 

  4. Caccavale, F., Siciliano, B. and Villani, L., 'Quaternion-based robot impedance control with nondiagonal stiffness for robot manipulators', in Proceedings American Control Conference, Philadelphia, PA, 1998, 468-472.

  5. Ciblak, N., 'Analysis of Cartesian stiffness and compliance with applications', Ph.D. Thesis, Georgia Institute of Technology, 1998.

  6. Ciblak, N. and Lipkin, H., 'Asymmetric Cartesian stiffness for the modelling of compliant robotic systems', in Proceedings ASME Design Engineering Technical Conference, Vol. 72, ASME, New York, 1996, 197-204.

    Google Scholar 

  7. Ciblak, N. and Lipkin, H., 'Centers of stiffness, compliance, and elasticity in the modelling of robotic systems', in Proceedings ASME Design Engineering Technical Conference, Vol. 26, ASME, New York, 1996, 185-195.

    Google Scholar 

  8. Ciblak, N. and Lipkin, H., 'Remote center of compliance reconsidered', Proceedings ASME Design Engineering Technical Conference, Vol. 96-DETC-MECH-1167, CD-ROM, ASME, New York, 1996.

    Google Scholar 

  9. Dimentberg, F.M., The Screw Calculus and Its Applications in Mechanics, U.S. Department of Commerce, 1965, Translation No. AD680993.

  10. Drake, S.H., 'Using compliance in lieu of sensory feedback for automatic assembly', Ph.D. Thesis, MIT, Cambridge, MA, 1977.

    Google Scholar 

  11. Fasse, E.D., 'On the spatial compliance of robotic manipulators', ASME Journal of Dynamic Systems, Measurement and Control 119, 1997, 839-844.

    Google Scholar 

  12. Fasse, E.D. and Breedveld, P.C., 'Modelling of elastically coupled bodies: Part I: General theory and geometric potential function method', ASME Journal of Dynamic Systems, Measurement and Control 120, 1998, 496-500.

    Google Scholar 

  13. Fasse, E.D. and Breedveld, P.C., 'Modelling of elastically coupled bodies: Part II: Exponential and generalized coordinate methods', ASME Journal of Dynamic Systems, Measurement and Control 120, 1998, 501-506.

    Google Scholar 

  14. Fasse, E.D. and Broenink, J.F., 'A spatial impedance controller for robotic manipulation', IEEE Transactions on Robotics and Automation 13, 1997, 546-556.

    Google Scholar 

  15. Fasse, E.D. and Gosselin, C.M., 'Spatio-geometric impedance control of Gough-Stewart platforms', IEEE Transactions on Robotics and Automation 15, 1999, 281-288.

    Google Scholar 

  16. Fasse, E.D. and Zhang, S., 'Lumped-parameter modelling of spatial compliance using a twistbased potential function', in Proceedings of the ASME Dynamic Systems and Control Division, 1999, accepted for publication.

  17. Griffis,M. and Duffy, J., 'Kinestatic control: A novel theory for simultaneously regulating force and displacement', ASME Journal of Mechanical Design 113, 1991, 508-515.

    Google Scholar 

  18. Howard, W.S., Žefran, M. and Kumar, V., 'On the 6 x 6 stiffness matrix for three dimensional motions', in Proceedings 9th IFToMM, Milano, Italy, 1995, 1575-1579.

  19. Huang, S., 'The analysis and synthesis of spatial compliance', Ph.D. Thesis, Marquette University, Milwaukee, WI, 1998.

    Google Scholar 

  20. Huang, S. and Schimmels, J.M., 'The realizable space of spatial stiffnesses achieved with a parallel connection of simple springs', in Proceedings of the ASME Dynamic Systems and Control Division, ASME, New York, 1997, 519-525.

    Google Scholar 

  21. Huang, S. and Schimmels, J., 'Achieving an arbitrary spatial stiffness with springs connected in parallel', ASME Journal of Mechanical Design 120, 1998, 520-526.

    Google Scholar 

  22. Huang, S. and Schimmels, J., 'The bounds and realization of spatial stiffness achieved with simple springs connected in parallel', IEEE Transactions on Robotics and Automation 14, 1998, 466-475.

    Google Scholar 

  23. Huang, S. and Schimmels, J.M., 'A procedure for the synthesis of simple spring realizable spatial stiffnesses', in Proceedings of the ASME Dynamic Systems and Control Division, Vol. 62, ASME, New York, 1998, 637-642.

    Google Scholar 

  24. Koster, M.P., Constructieprincipes, Philips Electronics, Eindhoven, 1994 [in Dutch].

    Google Scholar 

  25. Lončarić, J., 'Geometrical analysis of compliant mechanisms in robotics', Ph.D. Thesis, Harvard University, Cambridge, MA, 1985.

    Google Scholar 

  26. Lončarić, J., 'Normal forms of stiffness and compliance matrices', IEEE Transactions on Robotics and Automation 3, 1987, 567-572.

    Google Scholar 

  27. Lončarić, J., 'On statics of elastic systems and networks of rigid bodies', Technical Report SCR TR 88-13, Systems Research Center, University of Maryland, 1988.

  28. Marsden, J.E. and Hughes, T.J.R., Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, NJ, 1983.

    Google Scholar 

  29. Nikravesh, P.E., Computer-Aided Analysis of Mechanical Systems, Prentice Hall, Englewood Cliffs, NJ, 1988.

    Google Scholar 

  30. Paros, J.M. and Weisbord, L., 'How to design flexure hinges', Machine Design, November 1965, 151-156.

  31. Patterson, T. and Lipkin, H., 'A classification of robot compliance', ASME Journal of Mechanical Design 115, 1993, 581-584.

    Google Scholar 

  32. Patterson, T. and Lipkin, H., 'Structure of robot compliance', ASME Journal of Mechanical Design 115, 1993, 576-580.

    Google Scholar 

  33. Shabana, A.A., 'Flexible multibody dynamics: Review of past and recent developments', Multibody System Dynamics 1 1998, 189-222.

    Google Scholar 

  34. Smith, S.T. and Chetwynd, D.G., Foundations of Ultraprecision Mechanism Design, Gordon and Breach Science Publishers, 1992.

  35. Timoshenko, S.P. and Goodier, J.N., Theory of Elasticity, third edition, McGraw-Hill, New York, 1970.

    Google Scholar 

  36. Žefran, M. and Kumar, V., 'Affine connections for the Cartesian stiffness matrix', in IEEE International Conference on Robotics and Automation, IEEE, New York, 1997, 1376-1381.

    Google Scholar 

  37. Žefran, M., Kumar, V. and Croke, C., 'Metrics and connections for rigid body kinematics', International Journal of Robotics Research 18, 1999, 243-258.

    Google Scholar 

  38. Whitney, D., 'Quasi-static assembly of compliantly supported rigid parts', ASME Journal of Dynamic Systems, Measurement and Control 104, 1982, 65-77.

    Google Scholar 

  39. Zhang, S., 'Lumped-parameter modelling of elastically coupled bodies: Derivation of constitutive equations and determination of stiffness matrices', Ph.D. Thesis, The University of Arizona, 1999.

  40. Zhang, S. and Fasse, E.D., 'A finite-element-based method to determine canonical stiffness parameters of elastic elements applied to notch hinges', ASME Journal of Mechanical Design, 1998, submitted.

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, College of Engineering and Mines, The University of Arizona, 1130 N. Mountain, P.O. Box 210019, Tucson, AZ, 85721-0119, U.S.A.

    Shilong Zhang

Authors
  1. Shilong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ernest D. Fasse
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, S., Fasse, E.D. Spatial Compliance Modeling Using a Quaternion-Based Potential Function Method. Multibody System Dynamics 4, 75–101 (2000). https://doi.org/10.1023/A:1009895915332

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009895915332

Search

Navigation