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1. 中国科学院 测量与地球物理研究所 大地测量与地球动力学国家重点实验室，湖北 武汉 430077；
2. 中国科学院大学，北京 100049；
3. 信息工程大学 地理空间信息学院，河南 郑州 450052

The Functional Gradient Description Method of Space Coordinate Transformation
DUAN Pengshuo1,2, LIU Genyou1 , GONG Youliang3, HAO Xiaoguang1, WANG Nazi1,2
1. State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China;
3. Institute of Geospatial Information, Information Engineering University, Zhengzhou 450052, China
First author: DUAN Pengshuo(1986—), male, PhD candidate, majors in space geodesy and time-series analysis. E-mail： duanpengshuo12@163.com
Abstract: The conception of coordinate transformation gradient field is proposed in this study, which can realize the space coordinate transformation from small angle to arbitrary angle and from static to dynamical. Based on the equivalent of the unit quaternion rotation matrix and the Rodrigues matrix, the mathematical relationship between the spatial coordinate transformation and the functional gradient is revealed and an arbitrary coordinate transformation formula expressed by functional gradient in space is derived. The results indicate that the essence of spatial coordinate transformation is potential field in mathematic means and we can unify all the space coordinate transformations by using the conception of field, which is the theoretical foundation for the further study of time continuous space coordinate transformation and this study also gives a new solution for the attitude determination of motion carriers.
Key words: spatial coordinate transformation     functional gradient field     complete equivalence     field
1 引 言

2 基于单位四元数的旋转矩阵与罗德里格矩阵

x2+y2+z2+w2=1

3 基于单位四元数的旋转矩阵和罗德里格矩阵的完全等价条件

x2+y2+z2+w2=1

,

0≤x2+y2+z2≤1

u为函数f(x,y,z)的梯度。由以上证明过程可知，罗德里格矩阵与基于单位四元数构造的旋转矩阵完全等价，针对的不是函数f(x,y,z)在某点处的具体的梯度向量，而是它的所有点的梯度向量，是所有点处的梯度向量所组成的区域或者空间。因此，罗德里格矩阵与基于单位四元数的旋转矩阵完全等价所要求的条件是，函数f(x,y,z)在其定义域内的所有点处的梯度向量所组成的向量空间。

F(M)=P(M)i+Q(M)j+R(M)k

4 空间坐标系变换与函数梯度场

4.1 空间坐标系变换的梯度场描述

x2+y2+z2+w2=1

(1) 如果旋转角θ=0，说明OXYZ没有发生旋转，则由式(10)可得XA2+YA2+ZA2=C (C为任意大于零的实数)，因此向量n=[XA YA ZA]是任意向量，其对应除原点(0，0，0)之外的整个三维矢量空间。

(2) 如果旋转角θ≠0，说明OXYZ发生了旋转，则由式(7)可得XA2+YA2+ZA2=1，因此，向量n=[XA YA ZA]是单位向量，其几何意义是单位球面。因此，单位四元数的归一化条件将空间任意角度的坐标系变换情况压缩到了单位球面上。

f(x,y,z)= 0≤x2+y2+z2≤1

4.2 布尔莎(Bursa)模型的梯度场描述

5 结 论

 [1] PAN Guorong, ZHAO Pengfei. 3D Datum Transformation Model Based on Space Vector[J]. Journal of Geodesy and Geodynamics, 2009, 29(6):79-82.(潘国荣, 赵鹏飞. 基于空间向量的三维基准转换模型[J]. 大地测量与地球动力学,2009, 29(6): 79-82.) [2] CHENG Yingyan, CHENG Pengfei, BEI Jinzhong, et al. A Study of Coordinate Transformation Methods from WGS-84 to 1980 Xi’an Coordinate System in Large Space Domain[J]. Bulletin of Surveying and Mapping, 2007(12):5-8.(成英燕,程鹏飞,秘金钟,等. 大尺度空间域下1980西安坐标系与WGS-84坐标系转换方法研究[J]. 测绘通报,2007(12):5-8.) [3] WANG Jiexian, WANG Jun, LU Caiping. Problem of Coordinate Transformation between WGS-84 and Beijing54[J]. Journal of Geodesy and Geodynamics, 2003,23(3):70-73.(王解先,王军,陆彩萍. WGS-84与北京54坐标的转换问题[J]. 大地测量与地球动力学,2003,23(3):70-73.) [4] PAN Guorong, ZHOU Yueyin. Comparison between Two Ways of Calculation of Coordinate Transfer[J]. Journal of Geodesy and Geodynamics, 2011,31(3):58-62. (潘国荣,周跃寅. 两种坐标系转换计算方法的比较[J]. 大地测量与地球动力学,2011,31(3):58-62.) [5] SHEN Yunzhong, HU Leiming, LI Bofeng. Ill-posed Problem in Determination of Coordinate Transformation Parameters with Small Area’s Data Based on Bursa Model[J]. Acta Geodaetica et Cartographica Sinica, 2006,35(2):95-98.(沈云中,胡雷鸣,李博峰. Bursa 模型用于局部坐标变换的病态问题及其解法[J]. 测绘学报,2006,35(2):95-98.) [6] LV Zhiping, ZHU Huatong. The Testing and the Unified Expression of Coordinate Transformation Models[J]. Acta Geodaetica et Cartographica Sinica, 1993,22(3):161-168.(吕志平,朱华统. 坐标转换模型的检验及统一表达[J]. 测绘学报,1993,22(3):161-168.) [7] YUAN Qing, LOU Lizhi, CHEN Weixian. The Application of the Weighted Total Least-squares to Tree Dimensional-Datum Transformation[J]. Acta Geodaetica et Cartographica Sinica, 2011,40(sup): 115-119.(袁庆,楼立志,陈玮娴.加权总体最小二乘在三维基准转换中的应用[J]. 测绘学报,2011,40(S0):115-119.) [8] ZENG Wenxian, TAO Benzao. Non-linear Adjustment Model of Three-dimensional Coordinate Transformation[J]. Geomatics and Information Science of Wuhan University,2003,28(5):566-568.(曾文宪,陶本藻. 三维坐标转换的非线性模型[J]. 武汉大学学报:信息科学版,2003,28(5):566-568.) [9] SHEN Yunzhong, WEI Gang. Improvement of Three Dimensional Coordinate Transformation Model by Use of Interim Coordinate System[J]. Acta Geodaetica et Cartographica Sinica, 1998,27(2):161-165.(沈云中,卫刚. 利用过渡坐标系改进3维坐标转换模型[J]. 测绘学报,1998,27(2):161-165.) [10] YAO Jili, HAN Baomin, YANG Yuanxi. Application of Lodrigues Matrix in 3D Coordinate Transformation[J]. Geomatics and Information Science of Wuhan University, 2006, 31(12): 1094-1096.(姚吉利, 韩保民, 杨元喜. 罗德里格矩阵在三维坐标系转换严密解算中的应用[J]. 武汉大学学报:信息科学版,2006,31(12): 1094-1096.) [11] YANG Shiping, FAN Dongming, LONG Yuchun. Three-dimensional Coordination Transformation Adapted to Arbitrary Rotation Angle Based on Total Least Squares Method[J]. Journal of Geodesy and Geodynamics, 2013,33(2):114-119.(杨仕平,范东明,龙玉春.基于整体最小二乘法的任意旋转角度三维坐标转换[J]. 大地测量与地球动力学,2013,33(2):114-119.) [12] CHEN Yi, SHEN Yunzhong, LIU Dajie. A Simplified Model of Three Dimensional-datum Transformation Adapted to Big Rotation Angle[J]. Geomatics and Information Science of Wuhan University, 2004,29(12):1101-1105.(陈义,沈云中,刘大杰.适用于大旋转角的三维基准转换的一种简便模型[J].武汉大学学报:信息科学版,2004,29(12):1101-1105.) [13] YOU Wei, FAN Dongming,HUANG Ruijin. A Method of 3D Rectangular Coordinate Transformation Adapted to Any Rotation Angle[J]. Science of Surveying and Mapping, 2009,34(5):154-155.(游为,范东明,黄瑞金. 适用于任意旋转角的三维直角坐标转换方法[J].测绘科学,2009,34(5):154-155.) [14] ZHAO Shuangming, GUO Qiuyan, LUO Yan, et al. Quaternion-based 3D Similarity Transformation Algorithm[J]. Geomatics and Information Science of Wuhan University, 2009, 34(10):1214-1217.(赵双明,郭秋燕,罗研,等. 基于四元数的三维空间相似变换解算[J].武汉大学学报:信息科学版,2009, 34(10): 1214-1217.) [15] JIANG Gangwu, JIANG Ting, WANG Yong, et al. Space Resection Independent of Initial Value Based on Unit Quaternions[J]. Acta Geodaetica et Cartographica Sinica, 2007, 36(2): 169-175. (江刚武,姜挺,王勇, 等. 基于单位四元数的无初值依赖空间后方交会[J]. 测绘学报,2007,36(2):169-175.) [16] LIU Jun, WANG Donghong, ZHANG Yongsheng. et al. Bundle Adjustment of Airborne Three Line Array Imagery Based on Unit Quaternion[J]. Acta Geodaetica et Cartographica Sinica, 2008,37(4):451-457.(刘军,王冬红,张永生,等.基于单位四元数的机载三线阵影像光束法平差[J].测绘学报,2008,37(4):451-457.) [17] GUAN Yunlan, CHENG Xiaojun, ZHOU Shijian, et al. A Solution to Space Research Based on Unit Quaternion[J]. Acta Geodaetica et Cartographica Sinica, 2008,37(1):30-35.(官云兰,程效军,周世健,等. 基于单位四元数的空间后方交会解算[J].测绘学报,2008,37(1):30-35.) [18] GONG Hui, JIANG Ting, JIANG Gangwu, et al. Solution of Exterior Orientation Parameters for High-resolution Imagery Based on Quaternion Differential Equation[J]. Acta Geodaetica et Cartographica Sinica, 2012,41(3):409-416.(龚辉,姜挺,江刚武,等. 四元数微分方程的高分辨率卫星遥感影像外方位元素求解[J]. 测绘学报,2012,41(3):409-416.) [19] ZHANG Senlin. Rodrigues Marix in the Use of Rigorous Solution of the Collinearity Equation[J]. Journal of Wuhan Technical University of Surveying and Mapping, 1987,12(1):81-91.(张森林.罗德里格矩阵在共线方程严密解法中的应用[J].武汉测绘科技大学学报,1987,12(1):81-91.) [20] YUAN Yulei, JIANG Lixing, LIU Lingjie. Applications of Rodrigues Matrix in Coordinates Transformation[J]. Science of Surveying and Mapping,2010,35(2): 178-179.(原玉磊,蒋理兴,刘灵杰. 罗德里格矩阵在坐标系转换中的应用[J].测绘科学,2010,35(2):178-179.)
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0145

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#### 文章信息

DUAN Pengshuo, LIU Genyou, GONG Youliang, et al.

The Functional Gradient Description Method of Space Coordinate Transformation

Acta Geodaeticaet Cartographica Sinica, 2014, 43(10): 1005-1012.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0145