基于Lie群的刚体动力学建模及数值计算方法研究

白龙; 董志峰; 戈新生

doi:10.3879/j.issn.1000-0887.2015.08.005

基于Lie群的刚体动力学建模及数值计算方法研究

doi: 10.3879/j.issn.1000-0887.2015.08.005

¹中国矿业大学(北京) 机电与信息工程学院, 北京 100083;²北京信息科技大学机电工程学院, 北京 100192

基金项目: 国家自然科学基金（11472058）

详细信息

作者简介:
白龙（1988—），男，山东人，博士生(通讯作者. E-mail: bailong0316jn@126.com).

中图分类号: TH123;O302
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- 文章访问数: 1711
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- 被引次数: 0
出版历程
- 收稿日期: 2015-01-30
- 修回日期: 2015-06-20
- 刊出日期: 2015-08-15

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

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）

摘要

摘要: 基于Lie群和Lie代数之间的指数映射等价关系，推导了基于Lie群的自由刚体连续动力学方程.结合离散变分原理，推导了其Lie群离散变分积分子.通过证明可知连续和离散动力学系统都具有动量守恒性.对连续动力学方程进行同维化处理，使其变为常规非线性方程组的形式，利用Runge-Kutta法进行求解；基于Runge-Kutta基本理论，推导了直接用于Lie群的Runge-Kutta法，从而使Runge-Kutta法可用于求解变维非线性方程组；通过Lie代数变换，利用Kelly变换和Newton迭代对Lie群离散变分积分子进行求解.仿真对比结果表明，3种算法下的计算结果高度吻合，且能高精度地保持系统的结构守恒和动量守恒性.
- Lie群 /
- Lie代数 /
- Runge-Kutta法 /
- 离散变分积分子 /
- 自由刚体
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

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参考文献(16)

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