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Symbolic Processing of Multiloop Mechanism Dynamics Using Closed-Form Kinematics Solutions

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Abstract

This paper describes a method for the automated symbolic generation of the equations of motion of multibody systems using closed-form solutions of the kinematics at the position, velocity and acceleration levels where possible. The basic idea of the method is to employ the set of smallestindependent loops of the system as building blocks for the overall kinematics of the system. Using a geometric-algebraic approach, closed-form solutions are detected and generated for each loop where this is possible. These local solutions are then assembled at the global level, yielding a block diagram from which closed-form solutions for the overall system areproduced where possible. The equations of motion of minimal order are then generated by sums and products involving matrices from the kinematicprocessing. The resulting expressions are fully symbolic and do not contain redundant computations. The method was implemented in Mathematica and was applied to several mechanisms of practical relevance. A comparison of closed-form solutions with iterative solutions shows that the closed-formsolutions are more efficient by a factor of up to 8.

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References

  1. Andrews, G.C. and Kesavan, H.K., ‘The vector network model: A new approach to vector dynamics’, Mechanism and Machine Theory 10, 1975, 509–519.

    Google Scholar 

  2. Carré, B., Graphs and Networks, Oxford University Press, Oxford, 1979.

    Google Scholar 

  3. Gondran, M. and Minoux, M., Graphs and Algorithms, John Wiley & Sons, New York, 1984.

    Google Scholar 

  4. Hiller, M. and Kecskeméthy, A., ‘Equations of motion of complex multibody systems using kinematical differentials’, Transactions of the Canadian Society of Mechanical Engineers 13(4), 1989, 113–121.

    Google Scholar 

  5. Horton, J.D., ‘A polynomial-time algorithm to find the shortest cycle basis of a graph’, SIAM Journal of Computing 16(2), 1987, 358–366.

    Google Scholar 

  6. Kecskeméthy, A., ‘On closed form solutions of multiple-loop mechanisms’, in Computational Kinematics, J. Angeles, G. Hommel and P. Kovacs (eds), Kluwer Academic Publishers, Dordrecht, 1993, 263–274.

    Google Scholar 

  7. Kecskeméthy, A., ‘Kinematics of robots and mechanisms’, in International Summer School on Modelling and Control of Mechanisms and Robots, A.T.C. Melchiorri (ed.), World Scientific Publishing, Singapore, 1996, 39–79.

    Google Scholar 

  8. Kecskeméthy, A. and Hiller, M., ‘Automatic closed-form kinematics-solutions for recursive single-loop chains’, in Flexible Mechanisms, Dynamics, and Analysis, Proc. of the 22nd Biennal ASME-Mechanisms Conference Scottsdale (U.S.A.), Kinzel, Reinholtz, Tsai, Pennock, Cipra and Thompson (eds), 1992, 387–393.

  9. Krupp, T. and Kecskeméthy, A., ‘Application of symbolical kinematics to real-time vehicle dynamics’, Technical Report Contract No. N68171-9lC-9091, European Research Office of the U.S. Army, August 1995.

  10. Nikravesh, P., Computer-Aided Analysis of Mechanical Systems, Prentice Hall, London, 1988.

    Google Scholar 

  11. Orin, D.E. and Schrader, W.W., ‘Efficient Jacobian determination for robot manipulators’, in Proceedings of the Sixth World Congress on Theory of Machines and Mechanisms, 1983, 994–997.

  12. Raghavan, M. and Roth, B., ‘Inverse kinematics of the general 6R manipulator and related linkages’, Transactions of the ASME, Journal of Mechanical Design 115(3), 1993, 502–508.

    Google Scholar 

  13. Renaud, M., ‘Geometric and kinematic models of a robot manipulator: calculation of the Jacobian’, in 11th ISM, Tokyo, Japan, 1981, 757–763.

  14. Ribble, E.A., ‘Synthesis of human skeletal motion and the design of a special-purpose processor for real-time animation of human and animal figure motion’, Master's Thesis, The Ohio State University, 1982.

  15. Townsend, M., Discrete Mathematics: Applied Combinatorics and Graph Theory, The Benjamin/ Cummings Publishing Company, Menlo Park, CA, 1987.

    Google Scholar 

  16. Wittenburg, J., Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart, 1977.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mechanics, Technical University of Graz, Kopernikusgasse 24/III, A-8010, Graz, Austria

    A. Kecskeméthy

  2. Institute of Mechatronics, Gerhard-Mercator-Universitäat -- GH Duisburg, Lotharstrasse 1, 47057, Duisburg, Germany

    T. Krupp & M. Hiller

Authors
  1. A. Kecskeméthy
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  2. T. Krupp
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  3. M. Hiller
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Kecskeméthy, A., Krupp, T. & Hiller, M. Symbolic Processing of Multiloop Mechanism Dynamics Using Closed-Form Kinematics Solutions . Multibody System Dynamics 1, 23–45 (1997). https://doi.org/10.1023/A:1009743909765

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