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.

A novel formulation for the description of spatial rigid body motion using six non-redundant, homogeneous local velocity coordinates is presented. In contrast to common practice, the formulation proposed here does not distinguish between a translational and rotational motion in the sense that only translational velocity coordinates are used to describe the spatial motion of a rigid body. We obtain these new velocity coordinates by using the body-fixed translational velocity vectors of six properly selected points on the rigid body. These vectors are projected into six local directions and thus give six scalar velocities. Importantly, the equations of motion are derived without the aid of the rotation matrix or the angular velocity vector. The position coordinates and orientation of the body are obtained using the exponential map on the special Euclidean group SE(3). Furthermore, we introduce the appropriate inverse tangent operator on SE(3) in order to be able to solve the incremental motion vector differential equation. In addition, we present a modified version of a recently introduced a fourth-order Runge-Kutta Lie-group time integration scheme such that it can be used directly in our formulation. To demonstrate the applicability of our approach, we simulate the unstable rotation of a rigid body.

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The ability of a multibody dynamics model to accurately predict the behavior of a real system depends heavily on the correct choice of model parameters. The identification of unknown system parameters, which cannot be directly computed or measured is usually time consuming and costly. If experimental measurement data of the real system is available, the parameters in the mathematical model can be determined by minimizing the error between the model response and the measurement data. The latter task can be solved by means of optimization. While many optimization methods are available, optimization with a genetic algorithm is a promising approach for searching optimal solutions for complex engineering problems, as reported in a paper of one of the authors. So far, however, there is no general approach how to apply genetic optimization algorithms for complex multibody system dynamics models in order to obtain unknown parameters automatically — which is however of great importance when dealing with real flexible multibody systems. In the present paper we present a methodology to determine several unknown system parameters applied to a flexible rotor system which is excited with periodic impacts. Experiments were performed on the physical system to obtain measurement data which is used to identify the impact force as well as the support stiffnesses of the rotor system using genetic optimization.

As commonly known, standard time integration of the kinematic equations of rigid bodies modelled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using several typical rigid multibody systems. It is shown, that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turned out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the average number of Newton iterations required per time step is most often smaller for the Lie group integration method. It also turns out, that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly.