Modelling and Integration Concepts of Multibody Systems on Lie Groups

Abstract

Lie group integration schemes for multibody systems (MBS) are attractive as they provide a coordinate-free and thus singularity-free approach to MBS modeling. The Lie group setting also allows for developing integration schemes that preserve motion integrals and coadjoint orbits. Most of the recently proposed Lie group integration schemes are based on variants of generalized alpha Newmark schemes. In this chapter constrained MBS are modeled by a system of differential-algebraic equations (DAE) on a configuration space being a subvariety of the Lie group \(SE(3)^{n}\). This is transformed to an index 1 DAE system that is integrated with Munthe-Kaas (MK) integration scheme. The chapter further addresses geometric integration schemes that preserve integrals of motion. In this context, a non-canonical Lie-group Stormer-Verlet integration scheme with direct \(SO(3)\) rotational update is presented. The method is 2nd order accurate and it is angular momentum preserving for a free-spinning body. Moreover, although being fully explicit, the method achieves excellent conservation of the angular momentum of a free rotational body and the motion integrals of the Lagrangian top. A higher-order coadjoint-preserving integration scheme on \(SO(3)\) is also presented. This method exactly preserves spatial angular momentum of a free body and it is particularly numerically efficient.