Acta Mechanica

, Volume 227, Issue 12, pp 3381–3389 | Cite as

Improved quaternion-based integration scheme for rigid body motion

  • L. J. H. Seelen
  • J. T. Padding
  • J. A. M. Kuipers
Open Access
Original Paper

Abstract

Rotation quaternions are frequently used for describing the orientation of non-spherical rigid bodies. Their compact representation by four numbers and disappearance of numerical problems, such as gimbal lock, are reasons for using them. We describe an improvement of a predictor–corrector direct multiplication (PCDM) method for numerically integrating the rigid body equations of motion with rotation quaternions. The method only uses quaternions to describe the orientation, so no rotation matrices are needed in the implementation. A predictor–corrector approach is used to update the quaternions each time step, such that no renormalization is needed at the end of the time step. The PCDM method suggested by Zhao and Van Wachem is improved such that forces and torques are calculated at the correct time using position and orientation information at that same time. This is achieved by using a leapfrog approach in the improved version, in which the linear and angular velocities and rotation quaternions are defined at half time steps, while whole time step information of these quantities is calculated as part of the improved integration scheme. The improved PCDM scheme is compared with the original implementation for rotational kinetic energy conservation, accuracy of object orientation and angular velocity, and rate of convergence for different time steps. With the modifications that we propose, the improved method has a true second-order rate of convergence, without the need for explicit renormalization of the quaternions. Furthermore, the method is applicable to problems with position and velocity dependent torques, while still only a single force/torque evaluation is needed per time step. For objects experiencing torque, the improved PCDM method performs better than the original method, now showing a true second-order rate of convergence, and much smaller errors in the prediction of object orientation and angular velocity while still requiring only a single torque evaluation per time step.

References

  1. 1.
    Betsch, P., Siebert, R.: Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration. Int. J. Numer. Methods Eng. 79(4), 444–473 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Betsch, P., Steinmann, P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191(3–5), 467–488 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cleary, P.W.: DEM prediction of industrial and geophysical particle flows. Particuology 8(2), 106–118 (2010)CrossRefGoogle Scholar
  4. 4.
    Cundall, P.A., Strack, O.D.: A discrete numerical model for granular assemblies. Geotechnique 29(1), 47–65 (1979)CrossRefGoogle Scholar
  5. 5.
    Diebel, J.: Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors. Tech. rep., Stanford University, Stanford, California, pp. 94301–9010 (2006)Google Scholar
  6. 6.
    Dullweber, A., Leimkuhler, B., McLachlan, R.: Symplectic splitting methods for rigid body molecular dynamics. J. Chem. Phys. 107(15), 5840–5851 (1997)CrossRefGoogle Scholar
  7. 7.
    Evans, D.J., Murad, S.: Singularity free algorithm for molecular dynamics simulation of rigid polyatomics. Mol. Phys. 34(2), 327–331 (1977)CrossRefGoogle Scholar
  8. 8.
    Kleppmann, M.: Simulation of Colliding Constrained Rigid Bodies. Tech. Rep. UCAM-CL-TR-683, University of Cambridge, Cambridge (2007)Google Scholar
  9. 9.
    Kuipers, J.B.: Quaternions and Rotation Sequences, vol. 66. Princeton University Press, Princeton (1999)MATHGoogle Scholar
  10. 10.
    Martys, N.S., Mountain, R.D.: Velocity verlet algorithm for dissipative-particle-dynamics-based models of suspensions. Phys. Rev. E 59(3), 3733 (1999)CrossRefGoogle Scholar
  11. 11.
    Miller Iii, T., Eleftheriou, M., Pattnaik, P., Ndirango, A., Newns, D., Martyna, G.: Symplectic quaternion scheme for biophysical molecular dynamics. J. Chem. Phys. 116(20), 8649–8659 (2002)CrossRefGoogle Scholar
  12. 12.
    Nielsen, M.B., Krenk, S.: Conservative integration of rigid body motion by quaternion parameters with implicit constraints. Int. J. Numer. Methods Eng. 92(8), 734–752 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rapaport, D.: Molecular dynamics simulation using quaternions. J. Comput. Phys. 60(2), 306–314 (1985)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Shuster, M.D.: A survey of attitude representations. Navigation 8(9), 439–517 (1993)MathSciNetGoogle Scholar
  15. 15.
    Vattulainen, I., Karttunen, M., Besold, G., Polson, J.M.: Integration schemes for dissipative particle dynamics simulations: from softly interacting systems towards hybrid models. J. Chem. Phys. 116(10), 3967–3979 (2002)CrossRefGoogle Scholar
  16. 16.
    Wang, Y., Abe, S., Latham, S., Mora, P.: Implementation of particle-scale rotation in the 3-d lattice solid model. Pure Appl. Geophys. 163(9), 1769–1785 (2006)CrossRefGoogle Scholar
  17. 17.
    Zhao, F., van Wachem, B.: A novel quaternion integration approach for describing the behaviour of non-spherical particles. Acta Mech. 224(12), 3091–3109 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • L. J. H. Seelen
    • 1
  • J. T. Padding
    • 1
  • J. A. M. Kuipers
    • 1
  1. 1.Department of Chemical Engineering and ChemistryEindhoven University of TechnologyEindhovenThe Netherlands

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