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An Angular Momentum and Energy Conserving Lie-Group Integration Scheme for Rigid Body Rotational Dynamics Originating from Störmer-Verlet Algorithm

The paper presents two novel 2nd order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. Firstly proposed algorithm is a fully explicit scheme that exactly conserves spatial angular momentum of a free spinning body. The method is inspired by the Störmer-Verlet integration algorithm for solving ordinary differential equations (ODEs), which is also momentum conservative when dealing with ODEs in linear spaces but loses its conservative properties in a non-linear regime, such as non-linear SO(3) rotational group. Then, we proposed an algorithm that is an implicit integration scheme with a direct update in SO(3). The method is algorithmically designed to conserve exactly both of the two ‘main’ motion integrals of a rotational rigid body, i.e. spatial angular momentum of a torque-free body as well as its kinetic energy. As it is shown in the paper, both methods also preserve Lagrangian top integrals of motion in a very good manner, and generally better than some of the most successful conservative schemes to whom the proposed methods were compared within the presented numerical examples. The proposed schemes can be easily applied within the integration algorithms of the dynamics of general rigid body motion.

Störmer-Verlet integration scheme; Geometric Integration Algorithm; Lie-groups; Multibody Systems Dynamics

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