Conservative integration of rigid body motion by quaternion parameters with implicit constraints

Martin Bjerre Nielsen, Steen Krenk

Research output: Contribution to journalJournal articleResearchpeer-review


An angular momentum and energy‐conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four‐component conjugate momentum vector via Hamilton's equations. The introduction of an extended mass matrix leads to a symmetric set of eight state‐space equations of motion. The extra inertial parameter serves as a multiplier on the kinematic constraint, and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments. External loads enter these equations via the set of momentum equations. Initially, the normalization of the quaternion array is introduced via a Lagrange multiplier. However, this Lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential, and elimination of the Lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient. An algorithm is developed by forming a finite increment of the Hamiltonian. This procedure identifies the proper selection of increments and mean values, and leads to an algorithm with conservation of momentum and energy. Implementation, conservation properties, and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top. Copyright © 2012 John Wiley & Sons, Ltd.
Original languageEnglish
JournalInternational Journal for Numerical Methods in Engineering
Issue number8
Pages (from-to)734-752
Publication statusPublished - 2012


  • Rigid body dynamics
  • Quaternion parameters
  • Conservative integration
  • Energy conservation
  • Time integration


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