This paper studies the formulation of the dynamics of multibody systems with large rotation variables and kinematic constraints as differential-algebraic equations on a matrix Lie group. Those equations can then be solved using a Lie group time integration method proposed in a previous work. The general structure of the equations of motion are derived from Hamilton principle in a general and unifying framework. Then, in the case of rigid body dynamics, two particular formulations are developed and compared from the viewpoint of the structure of the equations of motion, of the accuracy of the numerical solution obtained by time integration, and of the computational cost of the iteration matrix involved in the Newton iterations at each time step. In the first formulation, the equations of motion are described on a Lie group defined as the Cartesian product of the group of translations R3 (the Euclidean space) and the group of rotations SO(3) (the special group of 3 by 3 proper orthogonal transformations). In the second formulation, the equations of motion are described on the group of Euclidean transformations SE(3) (the group of 4 by 4 homogeneous transformations). Both formulations lead to a second-order accurate numerical solution. For an academic example, we show that the formulation on SE(3) offers the advantage of an almost constant iteration matrix.
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ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
August 28–31, 2011
Washington, DC, USA
Conference Sponsors:
- Design Engineering Division and Computers and Information in Engineering Division
ISBN:
978-0-7918-5481-5
PROCEEDINGS PAPER
Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations
Olivier Bru¨ls,
Olivier Bru¨ls
University of Lie`ge, Lie`ge, Belgium
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Martin Arnold,
Martin Arnold
Martin Luther University Halle-Wittenberg, Halle (Saale), Germany
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Alberto Cardona
Alberto Cardona
Universidad Nacional del Litoral - Conicet, Santa Fe, Argentina
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Olivier Bru¨ls
University of Lie`ge, Lie`ge, Belgium
Martin Arnold
Martin Luther University Halle-Wittenberg, Halle (Saale), Germany
Alberto Cardona
Universidad Nacional del Litoral - Conicet, Santa Fe, Argentina
Paper No:
DETC2011-48132, pp. 85-94; 10 pages
Published Online:
June 12, 2012
Citation
Bru¨ls, O, Arnold, M, & Cardona, A. "Two Lie Group Formulations for Dynamic Multibody Systems With Large Rotations." Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A and B. Washington, DC, USA. August 28–31, 2011. pp. 85-94. ASME. https://doi.org/10.1115/DETC2011-48132
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