Time integration of rigid bodies modelled with three rotation parameters

Abstract

Three rotation parameters are commonly used in multibody dynamics or in spacecraft attitude determination to represent large spatial rotations. It is well known, however, that the direct time integration of kinematic equations with three rotation parameters is not possible in singular points. In standard formulations based on three rotation parameters, singular points are avoided, for example, by applying reparametrization strategies during the time integration of the kinematic equations. As an alternative, Euler parameters are commonly used to avoid singular points. State-of-the-art approaches use Lie group methods, specifically integrators, to model large rigid body rotations. However, the former methods are based on additional information, e.g. the rotation matrix, which must be computed in each time step. Thus, the latter method is difficult to incorporate into existing codes that are based on three rotation parameters. In this contribution, a novel approach for solving rotational kinematics in terms of three rotation parameters is presented. The proposed approach is illustrated by the example of the rotation vector and the Euler angles. In the proposed approach, Lie group time integration methods are used to compute consistent updates for the rotation vector or the Euler angles in each time step and therefore singular points can be surmounted and the accuracy is higher as compared to the direct time integration of rotation parameters. The proposed update formulas can be easily integrated into existing codes that use either the rotation vector or Euler angles. The advantages of the proposed approach are demonstrated with two numerical examples.