Lie Group and Lie Algebra Modeling for Numerical Calculation of Rigid Body Dynamics

BAI Long; DONG Zhi-feng; GE Xin-sheng

doi:10.3879/j.issn.1000-0887.2015.08.005

Volume 36 Issue 8

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2015 > 36(8): 833-843

BAI Long, DONG Zhi-feng, GE Xin-sheng. Lie Group and Lie Algebra Modeling for Numerical Calculation of Rigid Body Dynamics[J]. Applied Mathematics and Mechanics, 2015, 36(8): 833-843. doi: 10.3879/j.issn.1000-0887.2015.08.005

Citation:

PDF( 1749 KB)

Lie Group and Lie Algebra Modeling for Numerical Calculation of Rigid Body Dynamics

doi: 10.3879/j.issn.1000-0887.2015.08.005

¹School of Mechanical Electronic & Information Engineering, China University of Mining and Technology(Beijing), Beijing 100083, P.R.China;²Mechanical & Electrical Engineering School, Beijing Information Science & Technology University, Beijing 100192, P.R.China

Funds: The National Natural Science Foundation of China（11472058）

Received Date: 2015-01-30
Rev Recd Date: 2015-06-20
Publish Date: 2015-08-15

Abstract

Abstract

The Lie group dynamics equation for rigid bodies was derived based on the exponent mapping equivalence relationship between the Lie group and Lie algebra. The discrete Lie group variational integrator was derived according to the discrete variation theory. The momentum conservation of the 2 Lie group equations was demonstrated. The Lie group dynamics equation was processed so that every part has the same dimension and the equation can be solved with the RungeKutta method directly. The RungeKutta method to directly solve the Lie group dynamics equation with different dimensions was also built. The Lie group variational integrator was solved with the Lie algebraic transform, the Cayley transform and Newton iteration, respectively. The computation results of the 3 algorithms are highly identical to each other, the structure conservation and momentum conservation both have high precisions.
- Lie group,
- Lie algebra,
- Runge-Kutta,
- discrete variational integrator,
- free rigid body

FullText(HTML)

References(16)

References

[1]	Chaturvedi N A, Lee T, Leok M, McClamroch N H. Nonlinear dynamics of the 3D pendulum[J].Nonlinear Science,2011,21(1): 3-32.
[2]	边宇枢, 高志慧. 6自由度水下机器人动力学分析与运动控制[J]. 机械工程学报, 2007,43(7): 87-92.（BIAN Yu-shu, GAO Zhi-hui. Dynamic analysis and motion control of 6-dof underwater robot [J].Chinese Journal of Mechanical Engineering,2007,43(7):87-92.(in Chinese)）
[3]	徐正武, 唐国元. 四元数在水下航行体运动建模中的应用[J]. 中国舰船研究, 2014,9(2): 12-29.（XU Zheng-wu, TANG Guo-yuan. Applying the four-parameter approach to establish the motion model of an AUV[J].Chinese Journal of Ship Research,2014,9(2): 12-29.(in Chinese)）
[4]	LIN Xi-chuan, GUO Shu-xiang. A simplified dynamics modeling of a spherical underwater vehicle[C]// Proceeding of the 2008 IEEE International Conference on Robotics and Blomimetics,2009: 1140-1145.
[5]	Lee T. Computational geometric mechanics and control of rigid bodies[D]. Ph D Thesis. Michigan: University of Michigan, 2008.
[6]	Lee T, McClamroch H N, Leok M. A Lie group variational integrator for the attitude dynamics of a rigid body with application to the 3D pendulum[C]//Proceedings of the IEEE International Conference on Control Applications,2005: 962-967.
[7]	Nordkvist N, Sanyal A K. A Lie group variational integrator for rigid body motion in SE(3) with applications to underwater vehicle dynamics[C]//49th IEEE Conference on Decision and Control,2010: 5414-5419.
[8]	Jiménez F, Kobilarov M, de Diego D M. Discrete variational optimal control[J].Journal of Nonlinear Science,2013,23(3): 393-426.
[9]	丁希仑, 刘颖. 用李群李代数分析具有空间柔性变形杆件的机器人动力学[J]. 机械工程学报, 2007,43(12):184-189.（DING Xi-lun, LIU Ying. Dynamics analysis of robot with spatial compliant links using Lie group and Lie algebra[J].Chinese Journal of Mechanical Engineering,2007,43(12): 184-189.(in Chinese)）
[10]	张继锋, 邓子辰, 张凯. 结构动力方程求解的改进精细Runge-Kutta方法[J]. 应用数学和力学, 2015,36(4): 378-385.(ZHANG Ji-feng, DENG Zi-chen, ZHANG Kai. An improved precise Runge-Kutta method for structural dynamic equations[J].Applied Mathematics and Mechanics,2015,36(4): 378-385.(in Chinese))
[11]	李庆军, 叶学华, 王博, 王艳. 辛Runge-Kutta方法在卫星交会对接中的非线性动力学应用研究[J]. 应用数学和力学, 2014,〖STHZ〗35(12): 1299-1370.(LI Qing-jun, YE Xue-hua, WANG Bo, WANG Yan. Nonlinear dynamic behavior of the satellite rendezvous and docking based on the symplectic Runge-Kutta method[J].Applied Mathematics and Mechanics,2014,35(12): 1299-1370.(in Chinese))
[12]	白龙, 戈新生. 基于李群离散变分积分子3D摆姿态动力学研究[J]. 北京信息科技大学学报, 2013,28（3）: 14-18.（BAI Long, GE Xin-sheng. Attitude dynamics of 3D pendulum based on the Lie group variational integrator[J].Journal of Beijing Information Science and Technology University,2013,28（3）: 14-18.(in Chinese)）
[13]	白龙, 戈新生. 基于球摆模型的离散变分积分子算法研究[J]. 动力学与控制学报, 2013,11(4): 295-300.（BAI Long, GE Xin-sheng. The discrete variational integrators method of the spherical pendulum[J].Journal of Dynamics and Control, 2013,11(4): 295-300.(in Chinese)）
[14]	刘延柱. 高等动力学[M]. 北京: 高等教育出版社, 2001: 93-105.（LIU Yan-zhu.Advanced Dynamics [M]. Beijing: Higher Education Press, 2001: 93-105.(in Chinese)）
[15]	丁丽娟, 程杞元. 数值计算方法[M]. 北京: 高等教育出版社, 2011: 300-305.（DING Li-juan, CHENG Qi-yuan.Numerical Computation Method [M]. Beijing: Higher Education Press, 2011: 300-305.(in Chinese)）
[16]	Onishchik A L.Lie Groups and Lie Algebras: Foundations of Lie Theory, Lie Transformation Groups [M]. Beijing: Science Press, 2008: 44-52.