## 2 Establishment of Measurement Control Network Model
### Length-Angle Mixed Intersection Adjustment Method
Using Formula (2) to convert them to the global coordinate system, Formula (4) is expressed as follows:  
\[\beta_{ij}^{0}=\frac{\pi}{2}-arctan\frac{Z_{j}-t_{zi}}{\sqrt{\left(X_{j}-t_{ xi}\right)^{2}+\left(Y_{j}-t_{yi}\right)^{2}}} \tag{5}\]  
The relationship between the coordinates of points and distance observations in the global coordinate system is as follows:  
\[D_{ij}^{0}=\sqrt{\left(X_{j}-t_{xi}\right)^{2}+\left(Y_{j}-t_{yi}\right)^{2}+ \left(Z_{j}-t_{zi}\right)^{2}} \tag{6}\]  
Due to the rotation of the coordinate systems, there is an angle between the O-X axis of the global coordinate system \(O_{1}\)-\(X_{1}\)\(Y_{1}\)\(Z_{1}\) and the local coordinate system \(O_{i}\)-\(X_{i}\)\(Y_{i}\)\(Z_{i}\) (\(i\) = 2,..., \(m\)) as shown in Figure 8.  
Therefore, when calculating the horizontal angle using the transformed global coordinates of the measurement points, it is necessary to consider not only the angle and direction between the x axes but also which quadrant the projection of vector \(\mathbf{O_{1}^{\dagger}P}\) falls onto. The  
Figure 8: The schematic diagram for calculating the horizontal angle \(\alpha\) in the global coordinate system. calculation formula for the horizontal angle \(\alpha_{ij}^{0}\) in the global coordinate system should be divided into four cases:  
(1) when vector \(\vec{O_{1}P}\) projects onto the first quadrant,  
\[\alpha_{ij}^{0}=\left\{ \begin{array}{l}arctan\frac{\gamma_{j}-t_{yi}}{X_{j}-t _{xi}}-\alpha_{OX}^{1i},\alpha_{OX}^{1i}>\alpha_{0}-\pi\\ -2\pi+arctan\frac{\gamma_{j}-t_{yi}}{X_{j}-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX }^{1i}\in[-2\pi,-\frac{\pi}{2}]\cup\alpha_{OX}^{1i}<\alpha_{0}-\pi\end{array} \right. \tag{7}\]  
(2) when vector \(\vec{O_{1}P}\) projects onto the second quadrant,  
\[\alpha_{ij}^{0}=\left\{ \begin{array}{l}\pi+arctan\frac{\gamma_{j}-t_{yi}}{X_ {j}-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX}^{1i}\
otin(-2\pi,-\frac{\pi}{2})\cup \alpha_{OX}^{1i}>\alpha_{0}\\ -\pi+arctan\frac{\gamma_{j}-t_{yi}}{X_{j}-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX }^{1i}<\alpha_{0}\end{array} \right. \tag{8}\]  
(3) when vector \(\vec{O_{1}P}\) projects onto the third quadrant,  
\[\alpha_{ij}^{0}=\left\{ \begin{array}{l}\pi+arctan\frac{\gamma_{j}-t_{yi}}{X _{j}-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX}^{1i}\in[max(\frac{\pi}{2},\alpha_{ 0}),2\pi]\\ -\pi+arctan\frac{\gamma_{j}-t_{yi}}{X_{j}-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX }^{1i}\in(-\frac{\pi}{2},min(0,\alpha_{0})]\end{array} \right. \tag{9}\]  
(4) when vector \(\vec{O_{1}P}\) projects onto the fourth quadrant,  
\[\alpha_{ij}^{0}=\left\{ \begin{array}{l}arctan\frac{\gamma_{j}-t_{yi}}{X_{j }-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX}^{1i}<\alpha_{0}+\pi\\ 2\pi+arctan\frac{\gamma_{j}-t_{yi}}{X_{j}-t_{xi}}-\alpha_{OX}^{1i},\alpha_{OX }^{1i}>\alpha_{0}+\pi\end{array} \right. \tag{10}\]  
where \(\alpha_{0}=arctan\frac{\gamma_{j}-t_{yi}}{X_{j}-t_{xi}}\), \(\alpha_{OX}^{1i}\) represents the angle between the \(x\) axis of the global coordinate system and the local coordinate system. The general form of the equations for calculating distances and angles in the global coordinate system are as follows:  
\[\left\{ \begin{array}{l}D_{ij}^{0}=F_{ij}^{D}\big{(}t_{x1},t_{y1},t_{z1}, \ldots,t_{xi},t_{yi},t_{zi},X_{1},Y_{1},Z_{1},\ldots,X_{j},Y_{j},Z_{j}\big{)}\\ \alpha_{ij}^{0}=F_{ij}^{0}\big{(}t_{x1},t_{y1},t_{z1},\ldots,t_{xi},t_{yi},t_{zi },X_{1},Y_{1},Z_{1},\ldots,X_{j},Y_{j},Z_{j}\big{)}\\ \beta_{ij}^{0}=F_{ij}^{0}\big{(}t_{x1},t_{y1},t_{z1},\ldots,t_{xi},t_{yi},t_{zi },X_{1},Y_{1},Z_{1},\ldots,X_{j},Y_{j},Z_{j}\big{)}\end{array} \right. \tag{11}\]  
Since Formula (11) is a nonlinear function, it is necessary to linearize the observation equation. Linearization of the observation equation generally involves using the Taylor series expansion, taking the first-order term and omitting terms of second order and higher.