## 3 Single Transmitter Station Calibration Method
### Positioning Optimization Equation  
The coordinates of receivers at any point within the \(O\)-\(XYZ\) frame are denoted as (\(X\), \(Y\), \(Z\)), the corresponding coordinates in the \(o\)-\(xyz\) frame are denoted as (\(x\), \(y\), \(z\)), and \(\mathbf{R}\) and \(\mathbf{T}\) are the respective rotation and translation matrices that relate (\(x\), \(y\), \(z\)) to (\(X\), \(Y\), \(Z\)) as follows.  
\[\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]=\mathbf{R}\cdot\left[\begin{array}{c}x\\ y\\ z\end{array}\right]+\mathbf{T} \tag{5}\]  
\[\mathbf{R}=\left[\begin{array}{ccc}r_{1}&r_{2}&r_{3}\\ r_{4}&r_{5}&r_{6}\\ r_{7}&r_{8}&r_{9}\end{array}\right] \tag{6}\]  
\[\mathbf{T}=\left[\begin{array}{c}t_{1}\\ t_{2}\\ t_{3}\end{array}\right] \tag{7}\]  
In addition, we note that \(\mathbf{R}\) in Equation (6) is an orthogonal matrix. Substituting Equation (5) into Equation (1) yields the following.  
\[\left[\begin{array}{cc}a_{ij}&b_{ij}&c\\ \end{array}\right]\cdot\left[\begin{array}{c}r_{1}x+r_{2}y+r_{3}z+t_{1}\\ r_{4}x+r_{5}y+r_{6}z+t_{2}\\ r_{7}x+r_{8}y+r_{9}z+t_{3}\end{array}\right]+d_{ij}=0 \tag{8}\]  
The scanning process of the two fan-shaped lasers on the head provide two plane Equations (8) for each control point. Therefore, \(N\) control points obtain \(2N\) plane equations given by Equation (8). Accordingly, this includes nine rotation parameters \(r_{l}\), \(l\) = 1, 2,..., 9, and three translation parameters \(t_{1}\), \(t_{2}\), and \(t_{3}\), resulting in a total of \(9\) + \(3\) = 12 unknown parameters. Obtaining these 12 parameters would yield the coordinates of the individual receivers mounted on the array in the \(O\)-\(XYZ\) frame. This solution process is configured as follows.  
Setting the distance between the control point and the two laser planes as \(e_{ij}\), yields the following.  
\[e_{ij}=\frac{a_{ij}X+b_{ij}Y+c_{ij}Z+d_{ij}}{\sqrt{a_{ij}{}^{2}+b_{ij}{}^{2}+c _{ij}{}^{2}}} \tag{9}\]  
This can be rewritten according to Equations (5) and (8) as follows.  
\[e_{ij}=\left[\begin{array}{ccc}a_{ij}&b_{ij}&c\\ \end{array}\right]\cdot\left[\begin{array}{c}r_{1}x+r_{2}y+r_{3}z+t_{1}\\ r_{4}x+r_{5}y+r_{6}z+t_{2}\\ r_{7}x+r_{8}y+r_{9}z+t_{3}\end{array}\right]+d_{ij} \tag{10}\]  
Then, Formula (10) can be expanded for the \(k\)-th receiver, \(k\) = 1, 2,..., \(N\), mounted on the receiver array as follows.  
\[\begin{array}{l}e_{ijk}=a_{ij}r_{1}x_{k}+a_{ij}r_{2}y_{k}+a_{ij}r_{3}z_{k}+a _{ij}t_{1}+b_{ij}r_{4}x_{k}+b_{ij}r_{5}y_{k}\\ +b_{ij}r_{6}z_{k}+b_{ij}t_{2}+c_{ij}r_{7}x_{k}+c_{ij}r_{8}y_{k}+c_{ij}r_{9}z_{k} +c_{ij}t_{3}+d_{ij}\end{array} \tag{11}\]However, in practical applications, determining \(\mathbf{R}\) by this method does not in general satisfy the orthogonal constraint condition. Therefore, we adopt a nonlinear method to determine \(\mathbf{R}\) and \(\mathbf{T}\). In this method, the elements in Equation (11) satisfy the following orthogonal constraint conditions.  
\[\left\{\begin{array}{l}f_{1}=r_{1}^{2}+r_{2}^{2}+r_{2}^{2}-1=0\\ f_{2}=r_{4}^{2}+r_{5}^{2}+r_{6}^{2}-1=0\\ f_{3}=r_{7}^{2}+r_{8}^{2}+r_{9}^{2}-1=0\\ f_{4}=r_{1}r_{4}+r_{2}r_{5}+r_{3}r_{6}=0\\ f_{5}=r_{1}r_{7}+r_{2}r_{8}+r_{3}r_{9}=0\\ f_{6}=r_{4}r_{7}+r_{5}r_{8}+r_{6}r_{9}=0\end{array}\right. \tag{12}\]  
Accordingly, we obtain the unknown elements in \(\mathbf{R}\) and \(\mathbf{T}\) by employing (12) as penalty constraints in the following nonlinear optimization algorithm:  
\[F=\min\left\{\sum\sum e_{ijk}^{2}+\mu\sum_{m=1}^{6}f_{m}^{2}\right\} \tag{13}\]  
where \(\mu\) is a penalty factor. The solution of Equation (13) yields \((X,Y,Z)\) from Equation (5) based on the known values of \((x,y,z)\).