一种避开大转动奇异点的角速度数值积分方法

张志刚; 侯俊剑; 齐朝晖

doi:10.21656/1000-0887.380222

一种避开大转动奇异点的角速度数值积分方法

doi: 10.21656/1000-0887.380222

¹郑州轻工业学院机电工程学院, 郑州 450003;²工业装备结构分析国家重点实验室(大连理工大学), 辽宁大连 116023

基金项目: 国家自然科学基金（11602228;51505433）

详细信息

作者简介:
张志刚（1984—），男，讲师，博士(通讯作者. E-mail: zhigangzhang@foxmail.com).

中图分类号: O302
计量
- 文章访问数: 1444
- HTML全文浏览量: 276
- PDF下载量: 714
- 被引次数: 0
出版历程
- 收稿日期: 2017-08-04
- 修回日期: 2017-10-06
- 刊出日期: 2018-04-15

A Numerical Integration Method for Angular Velocity Vectors to Avoid Singularity of Large Rotation

¹Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, P.R.China;²School of State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology ), Dalian, Liaoning 116023, P.R.China

Funds: The National Natural Science Foundation of China（11602228;51505433）

摘要

摘要: 采用三参数描述有限转动会不可避免的遇到奇异性问题，这给由角速度积分求解转动参数带来了数值困难.系统地研究了采用转动矢量描述空间大转动的奇异性问题，在此基础上提出了一种避开转动矢量奇异点的数值积分方法.利用方向相同、模相差2π的两个转动矢量对应同一有限转动这一性质，在数值积分过程中将靠近奇异点的转动矢量切换到与之对应但远离奇异点的数值稳定区，从而避开了转动矢量奇异性给角速度数值积分带来的困难.数值算例表明所提方法简单、稳定、有效.
- 有限转动 /
- 转动矢量 /
- 奇异性 /
- 角速度矢量 /
- 数值积分
Abstract: Using 3 parameters to describe finite rotations will inevitably have the singularity problem, which leads to numerical difficulties in solving the rotational parameters from the integration of the angular velocity. Based on systematical studies of the singularity of the rotation vector, a new numerical integration method, which can overcome singular points of the rotation vector, was proposed. With the property that the 2 rotation vectors with the same direction but different norms correspond to the same finite rotation, the rotation vector near the singular point was switched to its corresponding one far away from the singular point and in the numerical stability region, during the numerical integration. This method can avoid the difficulties in the numerical integration caused by the singularity of rotation vectors for the angular velocity vectors. Numerical examples show that the proposed method is simple, stable and effective.
- finite rotation /
- rotation vector /
- singularity /
- angular velocity vector /
- numerical integration

HTML全文

参考文献(18)

[1]	王德春, 芮健, 张杰. 捷联惯性导航系统姿态算法综述[J]. 战术导弹控制技术, 2009,31(2): 41-44.(WANG Dechun, RUI Jian, ZHANG Jie. A review of attitude algorithms of strapdown inertial navigation system[J]. Control Technology of Tactical Missile,2009,31(2): 41-44.(in Chinese))
[2]	刘延柱, 洪嘉振, 杨海兴. 多刚体系统动力学[M]. 北京: 高等教育出版社, 1989.(LIU Yanzhu, HONG Jiazhen, YANG Haixing. Rigid Multibody System Dynamics [M]. Beijing: Higher Education Press, 1989.(in Chinese))
[3]	洪嘉振. 计算多体系统动力学[M]. 北京: 高等教育出版社, 1999.(HONG Jiazhen. Computational Dynamics of Multibody Systems [M]. Beijing: Higher Education Press, 1999.(in Chinese))
[4]	SIMO J C. A finite strain beam formulation, the three-dimensional dynamic problem: part I[J]. Computer Methods in Applied Mechanics and Engineering,1985,49(1): 55-70.
[5]	SIMO J C, VU-QUOC L. A three-dimensional finite-strain rod model, part II: computational aspects[J]. Computer Methods in Applied Mechanics and Engineering,1986,58(1): 79-116.
[6]	ZUPAN D, SAJE M. Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures[J]. Computer Methods in Applied Mechanics and Engineering,2003,192(49/50): 5209-5248.
[7]	ZUPAN D, SAJE M. The three-dimensional beam theory: finite element formulation based on curvature[J]. Computers & Structures,2003,81(18/19): 1875-1888.
[8]	张志刚, 齐朝晖, 吴志刚. 基于曲率插值的大变形梁单元[J]. 应用数学和力学, 2013,34(6):620-629.(ZHANG Zhigang, QI Zhaohui, WU Zhigang. Large deformation beam element based on curvature interpolation[J]. Applied Mathematics and Mechanics,2013,34(6): 620-629.(in Chinese))
[9]	张志刚, 齐朝晖, 吴志刚. 一种基于应变插值大变形梁单元的刚-柔耦合动力学分析[J]. 振动工程学报, 2015,28(3): 337-344.(ZHANG Zhigang, QI Zhaohui, WU Zhigang. Rigid-flexible dynamics analysis of a large deformation beam element based on interpolation of strains[J]. Journal of Vibration Engineering,2015,28(3): 337-344.(in Chinese))
[10]	黄雪樵. 克服欧拉方程奇异性的双欧法[J]. 飞行力学, 1994,12(4): 28-37.(HUANG Xueqiao. The dual-Euler method for overcoming the singularity of Euler equation[J]. Flight Dynamics,1994,12(4): 28-37.(in Chinese))
[11]	周伟, 张晓今, 寇保华, 秦子增. 双欧法在克服伞-弹系统欧拉方程奇异性中的应用[J]. 航天返回与遥感, 2003,24(3): 4-8.(ZHOU Wei, ZHANG Xiaojin, KOU Baohua, et al. The application of the dual-Euler method for overcoming the singularity of Euler equation in parachute-missile system[J]. Spacecraft Recovery & Remote Sensing,2003,24(3): 4-8.(in Chinese))
[12]	孙丽, 秦永元. 捷联惯导系统姿态算法比较[J]. 中国惯性技术学报, 2006,14(3): 6-10.(SUN Li, QIN Yongyuan. Comparison of attitude algorithms of SINS[J]. Journal of Chinese Inertial Technology,2006,14(3): 6-10.(in Chinese))
[13]	李跃军, 阎超. 飞行器姿态角解算的全角度双欧法[J]. 北京航空航天大学学报, 2007,33(5): 505-508.(LI Yuejun, YAN Chao. Improvement of dual-Euler method for full scale Eulerian angles solution of aircraft[J]. Journal of Beijing University of Aeronautics and Astronautics,2007,33(5): 505-508.(in Chinese))
[14]	王勇军, 秦永元, 杨波. 四元数、Rodrigues参数在卫星姿态解算上的对比研究[J]. 中国空间科学技术, 2007,27(3): 18-23.(WANG Yongjun, QIN Yongyuan, YANG Bo. Comparison of quaternion and Rodrigues parameters on attitude algorithm of secondary planet[J]. Chinese Space Science and Technology,2007,27(3): 18-23.(in Chinese))
[15]	张荣辉, 贾宏光, 陈涛, 等. 基于四元数法的捷联式惯性导航系统的姿态解算[J]. 光学精密工程, 2008,16(10): 1963-1970.(ZHANG Yonghui, JIA Hongguang, CHEN Tao, et al. Attitude solution for strapdown inertial navigation system based on quaternion algorithm[J]. Optics and Precision Engineering,2008,16(10): 1963-1970.(in Chinese))
[16]	BAUCHAU O A, TRAINELLI L. The vectorial parameterization of rotation[J]. Nonlinear dynamics,2003,32(1): 71-92.
[17]	ZUPAN E, SAJE M. Integrating rotation from angular velocity[J]. Advances in Engineering Software,2011,42(9): 723-733.
[18]	齐朝晖. 多体系统动力学[M]. 北京: 科学出版社, 2008.(QI Zhaohui. Dynamics of Multibody Systems [M]. Beijing: Science Press, 2008.(in Chinese))