﻿ 基于实四元数的大旋转角三维坐标转换的改进谱修正迭代解法
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 大地测量与地球动力学  2019, Vol. 39 Issue (11): 1182-1187  DOI: 10.14075/j.jgg.2019.11.016

引用本文

JIANG Pan, YOU Wei. The Improved Iteration Method by Correcting Characteristic Value for Transformation of Three-Dimensional Coordinates Based on Large Rotation Angle and Quaternions[J]. Journal of Geodesy and Geodynamics, 2019, 39(11): 1182-1187.

Foundation support

National Natural Science Foundation of China, No. 41574018, 41604068, 41404018.

Corresponding author

YOU Wei, PhD, associate professor, majors in satellite gravimetry and measurement data processing, E-mail:youwei1985@foxmail.com.

About the first author

JIANG Pan, postgraduate, majors in geodetic and measurement data processing, E-mail: macfreestream@foxmail.com.

文章历史

1. 西南交通大学地球科学与环境工程学院，成都市犀安路999号，611756;
2. 高速铁路运营安全空间信息技术国家地方联合工程实验室，成都市二环路北一段111号，610031

1 单位实四元数与三维坐标转换

 $\left[ {\begin{array}{*{20}{c}} {{{X'}_i}}\\ {{{Y'}_i}}\\ {{{Z'}_i}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{X_0}}\\ {{Y_0}}\\ {{Z_0}} \end{array}} \right] + (1 + \mu )\mathit{\boldsymbol{R}}\left[ {\begin{array}{*{20}{c}} {{X_i}}\\ {{Y_i}}\\ {{Z_i}} \end{array}} \right]$ (1)

 $\begin{array}{c} \mathit{\boldsymbol{R}} = {\mathit{\boldsymbol{R}}_X}(\alpha ){\mathit{\boldsymbol{R}}_Y}(\beta ){\mathit{\boldsymbol{R}}_Z}(\gamma ) = \\ \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&{\cos \alpha }&{\sin \alpha }\\ 0&{ - \sin \alpha }&{\cos \alpha } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos \beta }&0&{ - \sin \beta }\\ 0&1&0\\ {\sin \beta }&0&{\cos \beta } \end{array}} \right]\\ \left[ {\begin{array}{*{20}{c}} {\cos \gamma }&{\sin \gamma }&0\\ { - \sin \gamma }&{\cos \gamma }&0\\ 0&0&1 \end{array}} \right] \end{array}$ (2)

 $\mathit{\boldsymbol{R}} = \left[ {\begin{array}{*{20}{c}} {q_0^2 + q_1^2 - q_2^2 - q_3^2}&{2({q_1}{q_2} - {q_0}{q_3})}&{2({q_1}{q_3} + {q_0}{q_2})}\\ {2({q_1}{q_2} + {q_0}{q_3})}&{q_0^2 - q_1^2 + q_2^2 - q_3^2}&{2({q_2}{q_3} - {q_0}{q_1})}\\ {2({q_1}{q_3} - {q_0}{q_2})}&{2({q_2}{q_3} + {q_0}{q_1})}&{q_0^2 - q_1^2 - q_2^2 + q_3^2} \end{array}} \right]$ (3)

 $q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}=1$ (4)

 ${\mathit{\boldsymbol{V}}_i} = {\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{\hat x}} - {\mathit{\boldsymbol{L}}_i}, i = 1, 2, \cdots , n$ (5)

 $\mathit{\boldsymbol{B\hat x}} + \mathit{\boldsymbol{W}} = 0$ (6)

N=ATPA，权阵P一般设为单位阵，利用拉格朗日乘数法按附有限制条件的间接平差解算式(5)和式(6)，得到法方程：

 $\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{N}}&{{\mathit{\boldsymbol{B}}^T}}\\ \mathit{\boldsymbol{B}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat x}}}\\ \mathit{\boldsymbol{K}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}}}\\ { - \mathit{\boldsymbol{W}}} \end{array}} \right]$ (7)

2 病态问题的岭估计求解

 $\mathit{\boldsymbol{L}} = \mathit{\boldsymbol{AX}} + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }}$ (8)

 ${\mathit{\boldsymbol{\hat x}}_{LS}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PA}})^{ - 1}}({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}})$ (9)

 $\mathit{\boldsymbol{\hat x}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PA}} + \alpha \mathit{\boldsymbol{I}})^{ - 1}}({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}})$ (10)

3 基于岭参数的谱修正迭代法 3.1 谱修正迭代法

 ${\mathit{\boldsymbol{\hat X}}^{(n)}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PA}} + \mathit{\boldsymbol{I}})^{ - 1}}({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}} + {\mathit{\boldsymbol{\hat X}}^{(n - 1)}})$ (11)

3.2 改进谱修正迭代法

 ${\mathit{\boldsymbol{\hat X}}^{(n)}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PA}} + (\alpha + 1)\mathit{\boldsymbol{I}})^{ - 1}}({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}} + {\mathit{\boldsymbol{\hat X}}^{(n - 1)}})$ (12)

 ${\mathit{\boldsymbol{\hat X}}^{(n)}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PA}} + \mathit{\boldsymbol{K}})^{ - 1}}({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}} + \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{\hat X}}^{(n - 1)}})$ (13)

 ${\mathit{\boldsymbol{\hat X}}^{(n)}} = {({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PA}} + \alpha \mathit{\boldsymbol{T}})^{ - 1}}({\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}} + \alpha \mathit{\boldsymbol{T}}{\mathit{\boldsymbol{\hat X}}^{(n - 1)}})$ (14)

3.3 三维坐标转换模型求解

 $\left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{N}}&{{\mathit{\boldsymbol{B}}^T}}\\ \mathit{\boldsymbol{B}}&0 \end{array}} \right] = \mathit{\boldsymbol{M}}, \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat x}}}\\ \mathit{\boldsymbol{K}} \end{array}} \right] = \mathit{\boldsymbol{Y}}, \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{A}}^T}\mathit{\boldsymbol{PL}}}\\ { - \mathit{\boldsymbol{W}}} \end{array}} \right] = \mathit{\boldsymbol{U}}$ (15)

 $\mathit{\boldsymbol{MY}} = \mathit{\boldsymbol{U}}$ (16)

 $(\boldsymbol{M}+\alpha \boldsymbol{T}) \boldsymbol{Y}=\boldsymbol{U}+\alpha \boldsymbol{T} \boldsymbol{Y}$ (17)

 $\hat{\boldsymbol{Y}}^{(k)}=(\boldsymbol{M}+\alpha \boldsymbol{T})^{-1}\left(\boldsymbol{U}+\alpha \boldsymbol{T} \hat{\boldsymbol{Y}}^{(k-1)}\right)$ (18)

4 算例与分析 4.1 算例1

1) 对未知参数的初始值采用以下3种不同的取值方案。

2) 采用本文提出的算法和文献[4]算法(算法二)，使表 1中的公共点全部参与解算，得到的结果见表 3

3) 初值取为1)中3种方案时，本文算法和文献[4]算法(算法二)的机器耗时分别如表 4(单位s)所示，结果均利用MATLAB编程计算得到。

4.2 算例2

 图 1 内符合精度 Fig. 1 Inner precision

 图 2 外符合精度 Fig. 2 External precision

4.3 算例3

 图 3 坐标较差 Fig. 3 Coordinate difference

5 结语

1) 本文将三维旋转变换矩阵用四元数进行表示，构造了基于单位实四元数的三维坐标变换模型，并引入岭参数和泛函矩阵，得到改进的谱修正迭代算法。相对于用方向余弦、欧拉角和罗德里格矩阵构造的三维旋转变换矩阵，实四元数几何意义更加直观，且采用单位实四元数构造旋转矩阵无需进行复杂的三角函数求导，易于线性化，系数矩阵更为简洁。

2) 在进行三维坐标转换时，由于选取的公共点都是非均匀分布且都在局部区域，转换模型法方程矩阵的条件数较大，即法方程矩阵存在严重的病态性。考虑到模型法方程矩阵的病态性，引入岭参数和泛函矩阵改进谱修正迭代法求解转换参数，可有效降低法方程病态问题带来的不利影响，使方程求解达到稳定，同时方程迭代求解时解的估计值接近真值的程度较谱修正迭代法高。

3) 通过模拟数据和实测数据对算法进行验证，并与其他算法进行比较表明，本文算法具有收敛速度快、不依赖转换参数初值、全局收敛、解是无偏的、便于程序实现等特点。

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The Improved Iteration Method by Correcting Characteristic Value for Transformation of Three-Dimensional Coordinates Based on Large Rotation Angle and Quaternions
JIANG Pan1     YOU Wei1,2
1. Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, 999 Xi'an Road, Chengdu 611756, China;
2. State-Province Joint Engineering Laboratory of Spatial Information Technology for High-Speed Railway Safety, 111 First Section of North-Erhuan Road, Chengdu 610031, China
Abstract: We propose a new method for ill-posed problems in the large rotation angle three-dimensional coordinate transformation based on the unit quaternion. This method constructs a rotation matrix by unit quaternion, which avoids complex trigonometric function derivations, is easy to linearize, and has a more concise coefficient matrix. Considering the ill-posedness of the normal equation matrix, the ridge parameter and the functional matrix are introduced, which reduces the adverse effects caused by the ill-posedness of the equation, and generates a stable equation solution. At the same time, the estimated value of the solution when the equation is iteratively solved is closer to the true value than the traditional iteration method by correcting characteristic value. The algorithm is verified by simulation and measured data, showing that the algorithm has the characteristics of fast convergence, no initial value of conversion parameters, global convergence, unbiased solution and easy program implementation. It provides a new way for general coordinate transformation.
Key words: quaternion; ill-posed problem; the iteration method by correcting characteristic value; ridge parameter