Advertisement

O(n) Algorithm for Elastic Link/Joint Robots with End-Effector Contact

  • Hubert Gattringer
  • Andreas Müller
  • Florian Pucher
  • Alexander Reiter
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 33)

Abstract

This paper deals with the dynamical modeling of flexible multibody systems like elastic robots that go in contact with the environment. Models for elastic systems have a large degree of freedom leading to longer calculation times for solving the equations of motion (EOM). Conventionally, this includes the inversion of the mass matrix with a cubic run time complexity O(n3). By using a subsystem formulation and the Projection Equation an O(n) algorithm can be formulated that significantly reduces the simulation time. Additional contacts with the environment can be included in the equations of motion by the corresponding constraint Jacobian and the contact forces. For the explicit calculation of these forces, normally the inverse of the mass matrix is needed again. A novel algorithm to avoid this inversion is presented. Therein, the contact forces are calculated by additional runs of the same O(n) algorithm that is used without contact. The transition phase between different contact states is treated with the help of Newton’s impact law, again avoiding the inversion of the mass matrix. Simulation results for an elastic robot show the effectiveness of the proposed algorithms.

Keywords

Dynamical modeling Elastic multibody systems Ritz approximation O(n) algorithm Contact 

Notes

Acknowledgements

This work has been supported by the Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM).

References

  1. 1.
    Attia, H.A.: A recursive method for the dynamic analysis of mechanical systems in spatial motion. Acta Mech. 167, 41–55 (2004)CrossRefGoogle Scholar
  2. 2.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 490–501 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brandl, H., Johanni, R., Otter, M.: A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix. In: Proceedings of the IFAC International Symposium on Theory of Robots, Viennam, pp. 365–370 (1986)Google Scholar
  4. 4.
    Brandl, H., Johanni, R., Otter, M.: An algorithm for the simulation of multibody systems with kinematic loops. In: Proceedings of 7th IFToMM World Congress on the Theory of Machines and Mechanisms, Sevilla, pp. 407–411 (1987)Google Scholar
  5. 5.
    Bremer, H.: Elastic Multibody Dynamics: A Direct Ritz Approach. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Dwivedy, S.K., Eberhard, P.: Dynamic analysis of flexible manipulators, a literature review. Mech. Mach. Theory 41(7), 749–777 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Eigen C++ Library http://eigen.tuxfamily.org
  8. 8.
    Featherstone, R.: Rigid Body Dynamics Algorithms. Springer, New York (2008)CrossRefGoogle Scholar
  9. 9.
    Gattringer, H.: Starr-elastische Robotersysteme (Theorie und Anwendungen). Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Gattringer, H., Müller, A.: Dynamic formulations and computational algorithms. In: Goswami, A., Vadakkepat, P. (eds.) Humanoid Robotics: A Reference. Springer, Netherlands (2017)Google Scholar
  11. 11.
    Gattringer, H., Bremer, H., Kastner, M.: Efficient dynamic modeling for rigid multibody systems with contact and impact. Acta Mech. 219, 111–128 (2011)CrossRefGoogle Scholar
  12. 12.
    Gattringer, H., Müller, A., Springer, K., Jörgl, M.: An efficient method for the dynamical modeling of serial elastic link/joint robots. In: Moreno-Diaz, R., et al. (eds.) Computer Aided Systems Theory – EUROCAST 2015. Series Lecture Notes in Computer Science, vol. 9520, pp. 689–697. Springer, Heidelberg (2015)Google Scholar
  13. 13.
    Glocker, C.: Set-Valued Force Laws – Dynamics of Non-smooth Systems. Springer, Berlin/Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Höbarth, W., Gattringer, H., Bremer, H.: Modeling and control of an articulated robot with flexible links/joints. In: Proceedings of the 9th International Conference on Motion and Vibration Control, Garching (2008)Google Scholar
  15. 15.
    Khalil, W.: Dynamic modeling robots using recursive Newton-Euler techniques. In: Filipe, J., Cetto, J., Ferrier, J. (eds.) Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, vol. 2, pp. 19–31. SciTePress, Portugal (2010)Google Scholar
  16. 16.
    Luh, J.Y.S., Walker, M.W., Paul, R.P.C.: On-line computational scheme for mechanical manipulators. ASME J. Dyn. Syst. Meas. Control 102, 69–76 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mohan, A., Saha, S.K.: A recursive, numerically stable, and efficient simulation algorithm for serial robots. Multibody Syst. Dyn. 17, 291–319 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pfeiffer, F.: Mechanical System Dynamics. Springer, Berlin (2008)CrossRefGoogle Scholar
  19. 19.
    Siciliano, B., Khatib, O.: Springer Handbook of Robotics. Springer, Cham (2016)CrossRefGoogle Scholar
  20. 20.
    Staufer, P., Gattringer, H.: State estimation on flexible robots using accelerometers and angular rate sensors. Mechatronics 22, 1042–1049 (2012)CrossRefGoogle Scholar
  21. 21.
    Walker, M.W., Orin, D.E.: Efficient dynamic computer simulation of robotic mechanisms. ASME J. Dyn. Syst. Meas. Control 104, 205–211 (1982)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hubert Gattringer
    • 1
  • Andreas Müller
    • 1
  • Florian Pucher
    • 1
  • Alexander Reiter
    • 1
  1. 1.Institute of RoboticsJohannes Kepler University LinzLinzAustria

Personalised recommendations