Evaluation and implementation of Lie group integration methods for rigid multibody systems

As commonly known, standard time integration of the kinematic equations of rigid bodies modeled 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 the ${\mathbb{R}}^3\times SO(3)$ $\mathbb{R}^{3}\times SO(3)$ Lie group formulation and several typical rigid multibody systems. It is shown that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turns out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the Lie group integration method is computationally the more efficient one. 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.