A Lie Algebra Approach to Lie Group Time Integration of Constrained Systems

Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 565)


Lie group integrators preserve by construction the Lie group structure of a nonlinear configuration space. In multibody dynamics, they support a representation of (large) rotations in a Lie group setting that is free of singularities. The resulting equations of motion are differential equations on a manifold with tangent spaces being parametrized by the corresponding Lie algebra. In the present paper, we discuss the time discretization of these equations of motion by a generalized-\(\alpha \) Lie group integrator for constrained systems and show how to exploit in this context the linear structure of the Lie algebra. This linear structure allows a very natural definition of the generalized-\(\alpha \) Lie group integrator, an efficient practical implementation and a very detailed error analysis. Furthermore, the Lie algebra approach may be combined with analytical transformations that help to avoid an undesired order reduction phenomenon in generalized-\(\alpha \) time integration. After a tutorial-like step-by-step introduction to the generalized-\(\alpha \) Lie group integrator, we investigate its convergence behaviour and develop a novel initialization scheme to achieve second-order accuracy in the application to constrained systems. The theoretical results are illustrated by a comprehensive set of numerical tests for two Lie group formulations of a rotating heavy top.



The work of this author was supported by the German Minister of Education and Research, BMBF grant 05M13NHB: “Modelling and structure preserving discretization of inelastic components in system simulation”.


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Copyright information

© CISM International Centre for Mechanical Sciences 2016

Authors and Affiliations

  • Martin Arnold
    • 1
  • Alberto Cardona
    • 2
  • Olivier Brüls
    • 3
  1. 1.Martin Luther University Halle-WittenbergHalle (Saale)Germany
  2. 2.Universidad Nacional Litoral - CONICETSanta FeArgentina
  3. 3.University of LiègeLiègeBelgium

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