Galerkin-based time integration approaches to rigid body dynamics in terms of unit quaternions

Galerkin-based time integration approaches are investigated for discrete mechanical systems subject to holonomic constraints. In this connection, rigid body rotations are described by using unit quaternions, resulting in a configuration-dependent mass matrix. Two different Galerkin methods are developed: (i) a Petrov-Galerkin approach leading to the common time-stepping format of numerical integrators for initial value problems and (ii) a Bubnov-Galerkin method leading to a global solution procedure. The numerical performance of the alternative schemes is compared, including their algorithmic conservation properties, accuracy, and satisfaction of the constraints on configuration and velocity level. The global Galerkin-based approach is deemed to be advantageous for the solution of inverse dynamics and optimal control problems, in which a flow of information backwards in time is required.
Numerical examples involving large rotations are investigated to assess the numerical performance of the alternative Galerkin approaches. In addition to that, a comparison of the quaternion-based rigid body formulation with a director-based formulation is included.