Is there an optimal choice of configuration space for Lie group integration schemes applied to constrained MBS?

TL;DR

This paper investigates whether there is an optimal configuration-space choice for Lie-group integration of constrained multibody systems by directly comparing $SE(3)$ and $SO(3)\times \mathbb{R}^{3}$ within the Munthe-Kaas framework. By formulating the constrained Boltzmann-Hamel dynamics on each group and applying the MK method to index-1 ODEs, the authors quantify constraint violations and energy drift across multiple benchmark MBS, including a heavy top and a spherical double pendulum. The key finding is that $SE(3)$ generally yields smaller constraint violations and energy drift when absolute motions belong to a motion subgroup (notably ground-connected constraints), while in other cases the two formulations are effectively equivalent, with $SO(3)\times \mathbb{R}^{3}$ offering lower numerical complexity. The results provide practical guidance for selecting configuration spaces in Lie-group based MBS simulations and motivate a hybrid approach that uses $SE(3)$ where it offers a clear benefit.

Abstract

Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: $SE\left( 3\right) $ and $SO\left( 3\right) \times \mathbb{R}^{3}$. Since the performance of the numerical integration scheme clearly depends on the underlying configuration space it is important to analyze the effect of using either variant. For constrained MBS a crucial aspect is the constraint satisfaction. In this paper the constraint violation observed for the two variants are investigated. It is concluded that the $SE\left( 3\right) $ formulation outperforms the $SO\left( 3\right) \times \mathbb{R}^{3}$ formulation if the absolute motions of the rigid bodies, as part of a constrained MBS, belong to a motion subgroup. In all other cases both formulations are equivalent. In the latter cases the $SO\left( 3\right) \times \mathbb{R}^{3}$ formulation should be used since the $SE\left( 3\right) $ formulation is numerically more complex, however.

Figures (15)

• Figure 1: Model of a heavy top.
• Figure 2: Drift of rotation center for integration on a) $SE(3)$, and b)$SO(3)\times \mathbb{R}^{3}$.
• Figure 3: Drift of kinetic energy for a) $SE(3)$ and b) $SO(3)\times \mathbb{R}^{3}$ formulation.
• Figure 4: Spherical double pendulum.
• Figure 5: Violation of geometric constraints of joint 1 when integrating the $SE(3)$ (a), and $SO(3)\times \mathbb{R}^{3}$ (b) formulation.