## 2 Establishment of Measurement Control Network Model
### Length-Angle Mixed Intersection Adjustment Method  
Based on the error characteristics analysis of the laser tracker and total station described above, an adjustment method based on length-angle mixed intersection is established. As shown in Figure 7, the track orientation is defined as the x-axis direction, with the direction perpendicular to the track defined as the y-axis direction. Target points are placed equidistantly along the x-axis on one side of the track, consisting of fixed base and target ball. The target ball is a spherical reflector. To establish a linkage between the total station (Level III) and the laser tracker (Level II), reference points are deployed around the track, which consist of movable base and target ball, and the target ball is a spherical reflector. The laser tracker performs coordinate measurements of both target and reference points within its measuring range, while the total station conducts the coordinate measurements of reference points within its range. For the construction of a measuring network for target and reference points depicted in Figure 7, we assume that \(m\) measuring stations are set up inside the track with the laser tracker and total station measurement systems. These systems observe distances and angles of \(n\) measuring points. The length and angle observation from the \(i\)th measuring station to the \(j\)th point is denoted as \(\left(D_{ij},\alpha_{ij},\beta_{ij}\right)\), with the spatial position parameters of the station's center being \(\left(t_{xi},t_{yi},t_{zi}\right)\) (\(i=1,\,2,\,\ldots,\,m\)). The coordinates of the \(j\)th measuring point in the coordinate system (local coordinate system) of the \(i\)th measuring station are represented as \(\left(X_{ij},Y_{ij},Z_{ij}\right)\) (\(j=1,\,2,\,\ldots,\,n\)), and in the global coordinate system, as \(\left(X_{j},Y_{j},Z_{j}\right)\). Defining the coordinate system of the total station as the global coordinate system, the rotation matrix \(\mathbf{R}_{i}\) and translation vector \(\mathbf{T}_{i}\) relative to the global coordinate system for the other measuring  
Figure 7: Distribution of target points, reference points, and stations on a long track. stations' coordinate systems are obtained through a common point transformation model. The relationship between \(\left(X_{j},Y_{j},Z_{j}\right)\) and \(\left(X_{ij},Y_{ij},Z_{ij}\right)\) is expressed as follows:  
\[ \begin{pmatrix}X_{j}\\ Y_{j}\\ Z_{j}\end{pmatrix} =\mathbf{R}_{i}\times \begin{pmatrix}X_{ij}\\ Y_{ij}\\ Z_{ij}\end{pmatrix} +\mathbf{T}_{i} \tag{2}\]  
In the formula, \(\mathbf{T}_{i}=\left(t_{xi},t_{yi},t_{zi}\right)^{T}\), \(\mathbf{R}_{i}=\mathbf{R}y_{i}\times\mathbf{R}x_{i}\times\mathbf{R}z_{i}\) (\(\mathbf{R}x_{i}\), \(\mathbf{R}y_{i}\), \(\mathbf{R}z_{i}\) are the rotation matrices for the \(x\), \(y\), and \(z\) axes, respectively). As can be seen from Formula (1), the functional relationship between the coordinates of the measuring points at the \(i\)th station and the horizontal and zenith angles are as follows:  
\[\alpha_{ij}= \begin{cases}\arctan\frac{Y_{ij}}{X_{ij}},&X_{ij}>0\\ \pi+arctan\frac{Y_{ij}}{X_{ij}},&X_{ij}<0,Y_{ij}>0\\ -\pi+arctan\frac{Y_{ij}}{X_{ij}},&X_{ij}<0,Y_{ij}<0\end{cases}  \tag{3}\]  
\[\beta_{ij}=\frac{\pi}{2}-arctan\frac{Z_{ij}}{\sqrt{X_{ij}^{2}+Y_{ij}^{2}}} \tag{4}\]  
The coordinates of the measuring points in the above two equations are expressed in the local coordinate system.