基于Bursa模型的七参数空间三维坐标转换
一、Bursa模型简介
模型简介百度即可，这里不做介绍，因为不是自己整理的。
二、Bursa模型的推导
2.1 Bursa坐标转换模型
[ X Y Z ] C = [ 1 0 0 0 − Z D Y D X D 0 1 0 Z D 0 − X D Y D 0 0 1 − Y D X D 0 Z D ] [ T X T Y T Z ε X ε Y ε Z m ] + [ X Y Z ] D (1) {\left[ \begin{matrix} X\\ Y\\ Z \end{matrix} \right]}_C ={\left[ \begin{matrix} 1 & 0 & 0 & 0 & -Z_D & Y_D & X_D\\ 0 & 1 & 0 & Z_D & 0 & -X_D & Y_D\\ 0 & 0 & 1 & -Y_D & X_D & 0 & Z_D \end{matrix} \right]} {\left[ \begin{matrix} T_X\\ T_Y\\ T_Z\\ \varepsilon_X\\ \varepsilon_Y\\ \varepsilon_Z\\ m \end{matrix} \right]} + {\left[ \begin{matrix} X\\ Y\\ Z \end{matrix} \right]}_D\tag{1} ⎣⎡XYZ⎦⎤C=⎣⎡1000100010ZD−YD−ZD0XDYD−XD0XDYDZD⎦⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎡TXTYTZεXεYεZm⎦⎥⎥⎥⎥⎥⎥⎥⎥⎤+⎣⎡XYZ⎦⎤D(1)
2.3 建立间接平差模型
设有 N N N个重合点，七个参数看作必要观测数 t = 7 t=7 t=7，其中总观测数为 n = 3 N n=3N n=3N，多余观测数 r = n − t r=n-t r=n−t。由此关系可知，至少需要3个点才能完成结算。根据间接平差模型，列出误差方程为：
[ V X 1 V Y 1 V Z 1 ⋮ V X N V Y N V Z N ] C = [ 1 0 0 0 − Z D 1 Y D 1 X D 1 0 1 0 Z D 1 0 − X D 1 Y D 1 0 0 1 − Y D 1 X D 1 0 Z D 1 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 1 0 0 0 − Z D N Y D N X D N 0 1 0 Z D N 0 − X D N Y D N 0 0 1 − Y D N X D N 0 Z D N ] [ T X T Y T Z ε X ε Y ε Z m ] − [ X 1 Y 1 Z 1 ⋮ X N Y N Z N ] C + [ X 1 Y 1 Z 1 ⋮ X N Y N Z N ] D (2) {\left[ \begin{matrix} V_{X_1}\\ V_{Y_1}\\ V_{Z_1}\\ \vdots \\ V_{X_N}\\ V_{Y_N}\\ V_{Z_N} \end{matrix} \right]}_C ={\left[ \begin{matrix} 1 & 0 & 0 & 0 & -Z_{D_1} & Y_{D_1} & X_{D_1}\\ 0 & 1 & 0 & Z_{D_1} & 0 & -X_{D_1} & Y_{D_1}\\ 0 & 0 & 1 & -Y_{D_1} & X_{D_1} & {0} & Z_{D_1}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \\ 1 & 0 & 0 & 0 & -Z_{D_N} & Y_{D_N} & X_{D_N}\\ 0 & 1 & 0 & Z_{D_N} & 0 & -X_{D_N} & Y_{D_N}\\ 0 & 0 & 1 & -Y_{D_N} & X_{D_N} & {0} & Z_{D_N} \end{matrix} \right]} {\left[ \begin{matrix} T_X\\ T_Y\\ T_Z\\ \varepsilon_X\\ \varepsilon_Y\\ \varepsilon_Z\\ m \end{matrix} \right]}- {\left[ \begin{matrix} {X_1}\\ {Y_1}\\ {Z_1}\\ \vdots \\ {X_N}\\ {Y_N}\\ {Z_N} \end{matrix} \right]}_C + {\left[ \begin{matrix} {X_1}\\ {Y_1}\\ {Z_1}\\ \vdots \\ {X_N}\\ {Y_N}\\ {Z_N} \end{matrix} \right]}_D\tag{2} ⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡VX1VY1VZ1⋮VXNVYNVZN⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤C=⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡100⋮100010⋮010001⋮0010ZD1−YD1⋮0ZDN−YDN−ZD10XD1⋮−ZDN0XDNYD1−XD10⋮YDN−XDN0XD1YD1ZD1⋮XDNYDNZDN⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎡TXTYTZεXεYεZm⎦⎥⎥⎥⎥⎥⎥⎥⎥⎤−⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡X1Y1Z1⋮XNYNZN⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤C+⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡X1Y1Z1⋮XNYNZN⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤D(2)
将(2)上式简化为新的矩阵形式为：
V = B X ^ − L (3) V = B \hat X -L \tag{3} V=BX^−L(3)
然后利用最小二乘法来求解坐标转换参数，这种方法利用了所有的公共点，可以得到较好的结果，但是由于将每个点的坐标精度都视为精度相同的观测值，所以得到是一种近似的结果。因此，各点的坐标视为同精度独立观测值，P为单位矩阵，则可由(3)式得到：
X ^ = ( B T B ) − 1 ( B T L ) (4) \hat X = (B^T B)^{-1}(B^T L)\tag{4} X^=(BTB)−1(BTL)(4)
精度评定，其单位权中误差为：
σ 0 = V T P V n − t (5) \sigma _0 = \sqrt{\frac{V^T PV}{n-t}}\tag{5} σ0=n−tVTPV(5)
注意
需要将旋转参数 ε X \varepsilon _X εX、 ε Y \varepsilon _Y εY和 ε Z \varepsilon _Z εZ的单位为弧度，要将其转换到秒；尺度参数m单位换位“ppm”，ppmpart per million 百万分之……。
具体计算公式为：将旋转角乘以206265即可换为“s”，将尺度参数乘以1000000单位即为 “ppm”。
1 弧 度 ( r a d ) = 206264.8062471 秒 1弧度(rad) = 206264.8062471秒 1弧度(rad)=206264.8062471秒
| 七参数 | 单位 |
|---|---|
| T X T_X TX | m m m |
| T Y T_Y TY | m m m |
| T Z T_Z TZ | m m m |
| ε X \varepsilon _X εX | s s s |
| ε Y \varepsilon _Y εY | s s s |
| ε Z \varepsilon _Z εZ | s s s |
| m m m | p p m ppm ppm |
2.4 非重合点的坐标转换
利用求得的七参数，将数据代入即可解得转换后的坐标。
精度评定：无法像重合点那样可以利用原有的坐标与转换的坐标来计算残差。此时可利用配置法，将重合点的转换值改正数作为已知值，然后对非公共点进行配置，具体的方法为：
①计算重合点转换值得改正数，其重合点的坐标采用已知值。
V = 已 知 值 − 转 换 值 (6) V = 已知值-转换值\tag{6} V=已知值−转换值(6)
②采用配置可计算出非公共点转换值的改正数。
V ′ = ∑ 1 n P i V i ∑ 1 n P i (7) V' = \frac{\sum_{1}^n{P_i V_i}}{\sum_{1}^n{P_i}}\tag{7} V′=∑1nPi∑1nPiVi(7)
n n n为选择重合点的个数，根据非重合点与重合点的距离来定权，其权为：
P i = 1 S i 2 (8) P_i = \frac {1} {S_i^2}\tag 8 Pi=Si21(8)
三、程序的实现
# 忽略烦人的红色提示
import warnings
warnings.filterwarnings("ignore")
# 导入Python的数据处理库pandas，相当于python里的excel
import pandas as pd
# 导入python绘图matplotlib
import matplotlib.pyplot as plt
# 使用ipython的魔法方法，将绘制出的图像直接嵌入在notebook单元格中
%matplotlib inline
# 设置绘图大小
plt.style.use({'figure.figsize':(25,20)})
plt.rcParams['font.sans-serif']=['SimHei'] # 用来正常显示中文标签
plt.rcParams['axes.unicode_minus']=False # 用来正常显示负号
3.1 坐标的导入
# 读取CGCS2000坐标系下的坐标
cc = pd.read_csv('new.csv')
# 查看坐标
cc
| 点号 | X | Y | Z | |
|---|---|---|---|---|
| 0 | GPS04 | -1964642.836 | 4484908.586 | 4075486.898 |
| 1 | GPS30 | -1967082.716 | 4490541.646 | 4068048.151 |
| 2 | GPS18 | -1958106.370 | 4482074.179 | 4082054.321 |
| 3 | GPS22 | -1958396.995 | 4485396.445 | 4077966.297 |
| 4 | GPS27 | -1953364.459 | 4481502.655 | 4084942.265 |
| 5 | GPS26 | -1957928.755 | 4492765.305 | 4070011.563 |
import numpy as np
np.set_printoptions(suppress=True)
C_XYZ = np.empty((0,3))
for index,row in cc.iterrows():
pd.set_option('precision', 4)
C_x = row['X']
C_y = row['Y']
C_z = row['Z']
C_xyz = np.array([[C_x,C_y,C_z]])
C_XYZ = np.append(C_XYZ,C_xyz,axis=0)
print(C_XYZ)
[[-1964642.836 4484908.586 4075486.898]
[-1967082.716 4490541.646 4068048.151]
[-1958106.37 4482074.179 4082054.321]
[-1958396.995 4485396.445 4077966.297]
[-1953364.459 4481502.655 4084942.265]
[-1957928.755 4492765.305 4070011.563]]
# 读取地方坐标系下的坐标
dd = pd.read_csv('old.csv')
# 查看信息
print(dd.shape)
dd
(6, 4)
| 点号 | X | Y | Z | |
|---|---|---|---|---|
| 0 | GPS04 | -1.9647e+06 | 4.4848e+06 | 4.0754e+06 |
| 1 | GPS30 | -1.9672e+06 | 4.4904e+06 | 4.0679e+06 |
| 2 | GPS18 | -1.9582e+06 | 4.4819e+06 | 4.0820e+06 |
| 3 | GPS22 | -1.9585e+06 | 4.4853e+06 | 4.0779e+06 |
| 4 | GPS27 | -1.9535e+06 | 4.4814e+06 | 4.0848e+06 |
| 5 | GPS26 | -1.9580e+06 | 4.4926e+06 | 4.0699e+06 |
import numpy as np
D_XYZ = np.empty((0,3))
for index,row in dd.iterrows():
pd.set_option('precision', 4)
D_x = row['X']
D_y = row['Y']
D_z = row['Z']
D_xyz = np.array([[D_x,D_y,D_z]])
D_XYZ = np.append(D_XYZ,D_xyz,axis=0)
print(D_XYZ)
[[-1964734.964 4484768.547 4075386.77 ]
[-1967174.802 4490401.508 4067948.166]
[-1958198.61 4481934.193 4081954.089]
[-1958489.229 4485256.399 4077866.134]
[-1953456.789 4481362.679 4084841.963]
[-1958020.995 4492625.151 4069911.537]]
3.2 解算七参数
X ^ = ( B T B ) − 1 ( B T L ) (4) \hat X = (B^T B)^{-1}(B^T L)\tag{4} X^=(BTB)−1(BTL)(4)
L = []
B = np.empty((0,7))
for i in range(D_XYZ.shape[0]):
# 提取元素
X_C = C_XYZ[i][0]
Y_C = C_XYZ[i][1]
Z_C = C_XYZ[i][2]
X_D = D_XYZ[i][0]
Y_D = D_XYZ[i][1]
Z_D = D_XYZ[i][2]
# 构建L矩阵
L.extend((X_C - X_D,Y_C - Y_D,Z_C - Z_D))
#L = np.append(L,LL,axis=1)
# 构建B矩阵
b1 = np.array([1,0,0,0,-Z_D,Y_D,X_D])
b2 = np.array([0,1,0,Z_D,0,-X_D,Y_D])
b3 = np.array([0,0,1,-Y_D,X_D,0,Z_D])
BB = np.row_stack((b1,b2,b3))
B = np.append(B,BB,axis=0)
B = B
L = np.array([L]).T
# print("L矩阵为：\n",L)
# print("B矩阵为：\n",B)
a = np.linalg.inv(np.dot(B.T,B))
b = np.dot(B.T,L)
# 求取伪七参数
x = np.dot(a,b)
print("解算的七参数为：")
print("X平移参数：{} m".format(x[0][0]))
print("Y平移参数：{} m".format(x[1][0]))
print("Z平移参数：{} m".format(x[2][0]))
print("X旋转参数：{} s".format(x[3][0]* 206265))
print("Y旋转参数：{} s".format(x[4][0]* 206265))
print("Z旋转参数：{} s".format(x[5][0]* 206265))
print("m尺度参数：{} ppm".format(x[6][0]* 1000000))
解算的七参数为：
X平移参数：121.62369966506958 m
Y平移参数：55.88656985759735 m
Z平移参数：31.898368503898382 m
X旋转参数：0.18627551317509372 s
Y旋转参数：-0.06667437699100276 s
Z旋转参数：0.17122195111095806 s
m尺度参数：17.579272022061332 ppm
3.3 重合点精度评定
V = B X ^ − L (3) V = B \hat X -L \tag{3} V=BX^−L(3)
σ 0 = V T P V n − t (5) \sigma _0 = \sqrt{\frac{V^T PV}{n-t}}\tag{5} σ0=n−tVTPV(5)
V = np.dot(B,x)-L
print(V)
[[-0.00272118]
[-0.0020901 ]
[-0.00234788]
[-0.0013403 ]
[-0.00675914]
[ 0.00558849]
[-0.00004679]
[ 0.00158902]
[ 0.00954763]
[ 0.00228071]
[-0.00345956]
[ 0.00377805]
[-0.00622995]
[ 0.000214 ]
[-0.01070229]
[ 0.00805748]
[ 0.01050535]
[-0.00586396]]
import math
xx = math.sqrt(np.dot(V.T,V)/(3*D_XYZ.shape[0]-7))
print(xx)
0.007287575001997314
3.4 非重合点的转换
# 读取地方坐标系下的坐标
ddno = pd.read_csv('oldno.csv')
# 查看信息
print(ddno.shape)
ddno
(5, 4)
| 点号 | X | Y | Z | |
|---|---|---|---|---|
| 0 | GPS21 | -1.9545e+06 | 4.4903e+06 | 4.0742e+06 |
| 1 | GPS17 | -1.9520e+06 | 4.4853e+06 | 4.0811e+06 |
| 2 | GPS29 | -1.9569e+06 | 4.4974e+06 | 4.0652e+06 |
| 3 | GPS25 | -1.9519e+06 | 4.4949e+06 | 4.0704e+06 |
| 4 | GPS28 | -1.9535e+06 | 4.5010e+06 | 4.0629e+06 |
import numpy as np
DN_XYZ = np.empty((0,3))
for index,row in ddno.iterrows():
pd.set_option('precision', 4)
Dno_x = row['X']
Dno_y = row['Y']
Dno_z = row['Z']
DN_xyz = np.array([[Dno_x,Dno_y,Dno_z]])
DN_XYZ = np.append(DN_XYZ,DN_xyz,axis=0)
print(DN_XYZ.shape)
(5, 3)
[ X Y Z ] C = [ 1 0 0 0 − Z D Y D X D 0 1 0 Z D 0 − X D Y D 0 0 1 − Y D X D 0 Z D ] [ T X T Y T Z ε X ε Y ε Z m ] + [ X Y Z ] D (1) {\left[ \begin{matrix} X\\ Y\\ Z \end{matrix} \right]}_C ={\left[ \begin{matrix} 1 & 0 & 0 & 0 & -Z_D & Y_D & X_D\\ 0 & 1 & 0 & Z_D & 0 & -X_D & Y_D\\ 0 & 0 & 1 & -Y_D & X_D & 0 & Z_D \end{matrix} \right]} {\left[ \begin{matrix} T_X\\ T_Y\\ T_Z\\ \varepsilon_X\\ \varepsilon_Y\\ \varepsilon_Z\\ m \end{matrix} \right]} + {\left[ \begin{matrix} X\\ Y\\ Z \end{matrix} \right]}_D\tag{1} ⎣⎡XYZ⎦⎤C=⎣⎡1000100010ZD−YD−ZD0XDYD−XD0XDYDZD⎦⎤⎣⎢⎢⎢⎢⎢⎢⎢⎢⎡TXTYTZεXεYεZm⎦⎥⎥⎥⎥⎥⎥⎥⎥⎤+⎣⎡XYZ⎦⎤D(1)
for i in range(DN_XYZ.shape[0]):
# 提取元素
XN_D = DN_XYZ[i][0]
YN_D = DN_XYZ[i][1]
ZN_D = DN_XYZ[i][2]
LN = np.row_stack((XN_D,YN_D,ZN_D))
# 构建L矩阵
# 构建B矩阵
bn1 = np.array([1,0,0,0,-ZN_D,YN_D,XN_D])
bn2 = np.array([0,1,0,ZN_D,0,-XN_D,YN_D])
bn3 = np.array([0,0,1,-YN_D,XN_D,0,ZN_D])
BN = np.row_stack((b1,b2,b3))
NCC = np.dot(BN,x) + LN
print('第{}个点的坐标为：{},{},{}'.format(i,NCC[0][0],NCC[1][0],NCC[2][0]))
第0个点的坐标为：-1954403.217942523,4490401.535505347,4074332.8511360423
第1个点的坐标为：-1951867.9859425232,4485404.294505347,4081154.3441360416
第2个点的坐标为：-1956813.4649425233,4497507.388505346,4065328.6971360417
第3个点的坐标为：-1951778.264942523,4495070.800505347,4070465.314136042
第4个点的坐标为：-1953427.302942523,4501107.370505347,4062974.232136042
3.5 非重合点的精度评定
3.5.1 定权
V ′ = ∑ 1 n P i V i ∑ 1 n P i (7) V' = \frac{\sum_{1}^n{P_i V_i}}{\sum_{1}^n{P_i}}\tag{7} V′=∑1nPi∑1nPiVi(7)
P i = 1 S i 2 (8) P_i = \frac {1} {S_i^2}\tag 8 Pi=Si21(8)
aa = list(range(0,V.shape[0],3))
bb = list(range(1,V.shape[0],3))
cc = list(range(2,V.shape[0],3))
# 已知点的x残差
v_x = V[aa,:]
# 已知点的y残差
v_y = V[bb,:]
# 已知点的z残差
v_z = V[cc,:]
v_x.shape
(6, 1)
V_xx,V_yy,V_zz = [],[],[]
for i in range(DN_XYZ.shape[0]):
# 提取元素
XN_D = DN_XYZ[i][0]
YN_D = DN_XYZ[i][1]
ZN_D = DN_XYZ[i][2]
LC = np.row_stack((XN_D,YN_D,ZN_D))
# 定权
PP = np.dot(D_XYZ,LC).T
ccc = []
for j in range(PP.shape[1]):
ccc.append(1/PP[0][j])
# 公式(7)
VV_S = sum(ccc)
print(VV_S)
V_x = np.dot(np.array([ccc]),v_x)[0][0] / VV_S
V_y = np.dot(np.array([ccc]),v_y)[0][0] / VV_S
V_z = np.dot(np.array([ccc]),v_z)[0][0] / VV_S
V_xx.append(V_x)
V_yy.append(V_y)
V_yy.append(V_z)
print("非重合点GPS{}的残差V_x{} V_y{} V_z{} ".format(i+1,V_x,V_y,V_z))
1.478477870203965e-13
非重合点GPS1的残差V_x2.621759300967811e-08 V_y-8.17546918711022e-08 V_z1.1506213912802006e-08
1.478462584622879e-13
非重合点GPS2的残差V_x3.02685943659054e-08 V_y-8.170440346251001e-08 V_z1.3431354650058274e-08
1.4784816011772054e-13
非重合点GPS3的残差V_x2.0702207439612064e-08 V_y-8.220030115808458e-08 V_z9.309236986800881e-09
1.4784765461761397e-13
非重合点GPS4的残差V_x2.3313321582638324e-08 V_y-8.337122325453742e-08 V_z1.190322739292841e-08
1.4784846884028183e-13
非重合点GPS5的残差V_x1.8683221941032832e-08 V_y-8.387564655386071e-08 V_z1.0205456918699615e-08
3.5.2 精度评定
# X的中误差
S = 0
for i in V_xx:
S += i*i
VXX = math.sqrt(S/(len(V_xx)-1))
print("空间直角坐标X残差中误差",VXX)
# Y的中误差
S = 0
for i in V_yy:
S += i*i
VYY = math.sqrt(S/(len(V_yy)-1))
print("空间直角坐标Y残差中误差",VYY)
# Z的中误差
S = 0
for i in V_zz:
S += i*i
VZZ = math.sqrt(S/(len(V_zz)-1))
print("空间直角坐标Z残差中误差",VZZ)
# 点位中误差
print("空间直角坐标点位中误差",math.sqrt(VXX**2 + VYY**2 + VZZ**2))
空间直角坐标X残差中误差 2.7040271477441625e-08
空间直角坐标Y残差中误差 6.21356115133806e-08
空间直角坐标Z残差中误差 -0.0
空间直角坐标点位中误差 6.776437485667154e-08