Multibody System Dynamics

, Volume 14, Issue 3–4, pp 225–250 | Cite as

Multi-Criteria Optimization of a Hexapod Machine

  • Lars KÜbler
  • Christoph Henninger
  • Peter Eberhard


Alternative designs of a hexapod machine are proposed and investigated with the aims to reduce flexibility and to eliminate singular kinematic configurations that appear in the workspace for the current design of the machine. The hexapod is modeled as a rigid multibody system. Articular coordinates associated with desired tool trajectories are computed by inverse kinematics. Hence, dynamic forces and torques are not considered and, as there is no closed-loop control realized in the model, the actual rotational and translational position of the tool deviates from the desired position due to machining loads. These deviations serve as objective functions during a multi-criteria optimization in order to determine the best design regarding stiffness/flexibility of the machine. Further, a general approach for evaluating flexibility behavior of the machine in the complete workspace is introduced and the results from the optimization are verified. Besides flexibility, a crucial point for machining tools is the size of the feasible workspace. Therefore, the influence of the design modification on the workspace is also taken into account.


dynamic system design multi-criteria optimization parallel kinematics hexapod machine singular configurations flexibility analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bestle, D., Analyse und Optimierung von Mehrkörpersystemen. Berlin: Springer, 1994.Google Scholar
  2. 2.
    Bestle, D. and Eberhard, P., NEWOPT/AIMS 2.2. Ein Programmsystem zur Analyse und Optimierung von mechanischen Systemen. Manual AN–35. Institute B of Mechanics, University of Stuttgart, 1994.Google Scholar
  3. 3.
    Bestle, D. and Eberhard, P., ‘Dynamic system design via multicriteria optimization’, in Multiple Criteria Decision Making. {Proceedings of the Int. Conf. on MCDM, Hagen, 1995, Fandel, G. and Gal, T. (eds.), pp. 467–478, Berlin: Springer}, 1997.Google Scholar
  4. 4.
    Bestle, D. and Eberhard, P., ‘Analyzing and optimizing multibody systems’, Mechanics of Structures and Machines 20, 1989, 67–92.Google Scholar
  5. 5.
    Bischof, C., Carle, A., Khademi, P. and Mauer, A., ‘ADIFOR 2.0: Automatic Differentiation of Fortran 77 programs’, IEEE Computational Science & Engineering 3, 1996, 18–32.CrossRefGoogle Scholar
  6. 6.
    Boër, C., Molinari–Tosatti, L. and Smith, K. (eds.), Parallel Kinematic Machines: Theoretical Aspects and Industrial Requirements (Advanced Manufacturing). London: Springer, 1999.Google Scholar
  7. 7.
    Dasgupta, B. and Mruthyunjaya, T., ‘Force redundancy in parallel manipulators: Theoretical and practical issues., Mechanisms and Machine Theory 33(6), 1998, 727–742.MathSciNetMATHGoogle Scholar
  8. 8.
    Dignath, F., Zur Optimierung mechatronischer Systeme mit nichtdifferenzierbaren Kriterien. VDI–Fortschritt-Berichte, Series 8, No. 1031, Düsseldorf: VDI-Verlag, 2004.Google Scholar
  9. 9.
    Dignath, F. and Hempelmann, D., Grundlagenuntersuchungen zum thermischen Einfluss – Bericht 2002. Report ZB–131. Institute B of Mechanics, University of Stuttgart, 2002.Google Scholar
  10. 10.
    Eberhard, P., Zur Mehrkriterienoptimierung von Mehrkörpersystemen. VDI–Fortschritt-Berichte, Series 11, No. 227, Düsseldorf: VDI-Verlag, 1996.Google Scholar
  11. 11.
    Eberhard, P. and Bestle, D., ‘Integrated modeling, simulation and optimization of multibody systems’, in Integrated Systems Engineering. Johannsen, G. (ed.), pp. 35–40, Oxford: Pergamon, 1994.Google Scholar
  12. 12.
    Eberhard, P., Dignath, F. and Kübler, L., ‘Parallel evolutionary optimization of multibody systems with application to railway dynamics’, Multibody System Dynamics 9, 2003, 143–164.CrossRefMATHGoogle Scholar
  13. 13.
    Fletcher, R., Practical Methods of Optimization. Chichester: John Wiley & Sons, 1987.MATHGoogle Scholar
  14. 14.
    Gosselin, C. and Angeles, J., ‘Singularity analysis of closed–loop kinematic chains’, IEEE Transactions on Robotics and Automation 6(3), 1990, 281–290.Google Scholar
  15. 15.
    Heisel, U., ‘Precision requirements of hexapod-machines and investigation results’, in Proceedings of the First European–American Forum on Parallel Kinematic Machines. Mailand, 1998.Google Scholar
  16. 16.
    Heisel, U., Maier, V. and Lunz, E. ‘Auslegung von maschinenkonstruktionen mit Gelenkstab-Kinematik-Grundaufbau, Tools, Komponentenauswahl, Methoden und Erfahrungen’, wt–Werkstatttechnik 88 4, 1998, 75–78.Google Scholar
  17. 17.
    Kirchner, J., Mehrkriterielle Optimierung von Parallelkinematiken. Berichte aus dem IWU, No. 12, Chemnitz: Verlag Wissenschaftliche Scripten, 2000.Google Scholar
  18. 18.
    Kreuzer, E. and Leister, G., Programmsystem NEWEUL'90. Manual AN–24. Institute B of Mechanics, University of Stuttgart, 1991.Google Scholar
  19. 19.
    Liu, J., Kinematics and Dynamics of Spatial Hexapod Motions with Thermal Disturbances, Report IB–38. Institute B of Mechanics, University of Stuttgart, 2001.Google Scholar
  20. 20.
    Merlet, J.-P., Parallel Robots. Dordrecht: Kluwer, 2000.MATHGoogle Scholar
  21. 21.
    Merlet, J.-P., ‘Singular configurations of parallel manipulators and grassmann geometry’, Journal of Robotics Research 8(5), 1989, 45–56.Google Scholar
  22. 22.
    O'Brien, J. and Wen, J., ‘Redundant actuation for improving kinematic manipulability’, in Proceedings of the 1999 IEEE International Conference on Robotics and Automation, Detroit, Michigan, Vol. 2, pp. 1520–1525, 1999.Google Scholar
  23. 23.
    Riebe, S. and Ulbrich, H., ‘Modelling and online computation of the dynamics of a parallel Kinematic with Six Degrees-of-Freedom’, Archive of Applied Mechanics 72, 2003, 817–829.Google Scholar
  24. 24.
    Schiehlen, W., Technische Dynamik. Stuttgart: B.G. Teubner, 1986.MATHGoogle Scholar
  25. 25.
    Takeda, Y. and Funabashi, H., ‘Kinematic and static characteristics of in–parallel actuated manipulators at Singular Points and in their Neighborhood’, JSME International Journal, Series C, 39(1), 1996, 85–93.Google Scholar
  26. 26.
    Takeda, Y. and Funabashi, H., ‘Kinematic Synthesis of in–parallel actuated mechanisms based on the global isotropy index’, Journal of Robotics and Mechatronics 11(5), 1999, 404–410.Google Scholar
  27. 27.
    Tsai, L.-W. Robot Analysis: The Mechanics of Serial and Parallel Manipulators, New York: John Wiley, 1999.Google Scholar
  28. 28.
    Valášek, M., Šika, Z., Bauma, V. and Vampola, T., ‘Design methodology for redundant parallel robots’, in Proc. of AED 2001, 2nd Int. Conf. on Advanced Engineering Design, Glasgow, pp. 243–248, 2001.Google Scholar
  29. 29.
    Zhengyi, X., Fengfeng, X. and Mechefske, C., ‘Kinetostatic analysis and optimization of a Tripod Attachment’, in Proc. of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators. October 3–4, 2002, Quebec City, Canada, Gosselin, C. and Ebert-Uphoff, I. (eds.), pp. 294–303, 2002.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Lars KÜbler
    • 1
  • Christoph Henninger
    • 2
  • Peter Eberhard
    • 2
  1. 1.Institute of Applied MechanicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Institute B of MechanicsUniversity of StuttgartStuttgartGermany

Personalised recommendations