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Research Papers

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

[+] Author and Article Information
Zdravko Terze

Department of Aeronautical Engineering,
Faculty of Mechanical Engineering
and Naval Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10002, Croatia
e-mail: zdravko.terze@fsb.hr

Andreas Müller

Institute for Robotics,
JKU Johannes Keppler University,
Altenbergerstraße 69,
Linz A-4040, Austria
e-mail: andreas.mueller@ieee.org

Dario Zlatar

Department of Aeronautical Engineering,
Faculty of Mechanical Engineering
and Naval Architecture,
University of Zagreb,
Ivana Lučića 5,
Zagreb 10002, Croatia
e-mail: dario.zlatar@fsb.hr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 22, 2014; final manuscript received September 24, 2014; published online April 2, 2015. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 10(5), 051005 (Apr 02, 2015) (11 pages) Paper No: CND-14-1053; doi: 10.1115/1.4028671 History: Received February 22, 2014

The paper presents two novel second order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. First 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 nonlinear regime, such as nonlinear 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 which 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 systems.

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References

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Figures

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Fig. 8

Convergence in the norm of the error in the spatial angular momentum of a free body rotational motion

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Fig. 9

Kinetic energy of a free body rotational motion

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Fig. 1

Error in the spatial angular momentum projection on the axis of the top

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Fig. 2

Error in the spatial angular momentum projection on the gravity vector (Lagrangian top)

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Fig. 3

Error in the spatial angular momentum projection on the gravity vector-zoom (Lagrangian top)

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Fig. 4

Convergence in the norm of the error in the rotation matrix of the top

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Fig. 5

Convergence in the norm of the error in the body-frame angular momentum of the top

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Fig. 6

Convergence in the norm of the error in the spatial angular momentum of the top

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Fig. 7

Hamiltonian function of Lagrangian top

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Fig. 10

Spatial angular momentum of free body rotational motion-deviation from the analytical solution (component y2)

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Fig. 11

Error in the spatial angular momentum projection on the gravity vector (Lagrangian top)

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Fig. 12

Error in the spatial angular momentum projection on the axis of the top

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