## 2 Laser Transmitter Measurement System
### Single Transmitter Station Angle Measurement Model  
As shown in Figure 2, the coordinate measurement model of the wMPS can be simplified into two rotary-laser fan-shaped planes 1 and 2, where the rotation axis of the two scanning laser planes is given as the \(Z\) axis, the intersection of plane 1 and the \(Z\) axis is defined as the origin \(O\), the \(X\) axis initially lies in laser plane 1 and is perpendicular to the \(Z\) axis. The direction of the \(Y\) axis then follows from the standard right-hand rule of a Cartesian coordinate system.  
The internal parameters of the transmitter (i.e., the initial plane parameters) are calibrated after the instrument is assembled, where the initial normal vectors of plane 1 and plane 2 are given as \(\textbf{n}_{i1}=[a_{i1}\)\(b_{i1}\)\(c_{i1}]^{\text{T}}\) and \(\textbf{n}_{i2}=[a_{i2}\)\(b_{i2}\)\(c_{i2}]^{\text{T}}\), respectively, where \(i\) (\(i=1\), \(2\), \(\dots\), \(N\)) is the transmitter number, and represents the position of the head at the first pulse signal, and \(d_{i2}\) is the deviation between the two laser planes along the \(Z\) axis (Figure 2), which is caused by laser module assembly errors. Here,  
Figure 1: Image illustrating the design of a standard laser transmitter station.  
Figure 2: Schematic illustrating the scanning angle measurement of a laser transmitter station, where plane 1 and plane 2 are the two fan-shaped laser planes that rotate about the \(Z\) axis at a fixed rotational frequency.  
\({b_{i1}}^{2}+{c_{i1}}^{2}={a_{i2}}^{2}+{b_{i2}}^{2}+{c_{i2}}^{2}=1\), such that \({\bf n}_{i1}\) and \({\bf n}_{i2}\) are both unit vectors. Accordingly, the equations of plane 1 and plane 2 can be expressed as follows.  
\[\left\{\begin{array}{l}{a_{i1}}X+{b_{i1}}Y+{c_{i1}}Z+{d_{i1}}=0\\ {a_{i2}}X+{b_{i2}}Y+{c_{i2}}Z+{d_{i2}}=0\end{array}\right. \tag{1}\]  
When any laser plane sweeps across the receiver at position \(M(X,Y,Z)\) over time, the relationship between the current plane parameters \(\left({a^{\prime}_{ij}},{b^{\prime}_{ij}},{c^{\prime}_{ij}},{d^{\prime}_{ij}}\right)\), where \(j\) is an integer ranging from 1 to 2, and represents the times when the receiver at \(M(X,Y,Z)\) receives signals from planes 1 and 2, respectively, and the initial plane parameters can be expressed as follows.  
\[\left[\begin{array}{l}{a^{\prime}_{ij}}\\ {b^{\prime}_{ij}}\\ {c^{\prime}_{ij}}\\ {d^{\prime}_{ij}}\end{array}\right]=\left[\begin{array}{cc}{\bf R}\!\left( \theta_{j}\right)&0\\ 0&1\end{array}\right]\left[\begin{array}{l}{a_{ij}}\\ {b_{ij}}\\ {c_{ij}}\\ {d_{ij}}\end{array}\right] \tag{2}\]  
\[{\bf R}\!\left(\theta_{j}\right)=\left[\begin{array}{cc}\cos\!\left(\theta_ {j}\right)&-\sin\!\left(\theta_{j}\right)&0\\ \sin\!\left(\theta_{j}\right)&\cos\!\left(\theta_{j}\right)&0\\ 0&0&1\end{array}\right] \tag{3}\]  
here, \(\theta_{j}\) is the angle of rotation of the plane from its initial position to the receiver, which is calculated as follows. When the head returns to its initial position, the receiver receives the pulse signal at a time recorded as \(t_{0}\). When the head rotates to \(\theta_{1}\), the receiver receives the signal from laser plane 1 and records the time as \(t_{1}\). When the head rotates to \(\theta_{2}\), the receiver receives the light signal from laser plane 2 and records the time as \(t_{2}\). Finally, when the head again rotates to its initial position, the time is recorded as \(t^{\prime}_{0}\). As a result, \(T=t^{\prime}_{0}-t_{0}\) is the rotational period of the transmitter, which has been calibrated at the time of manufacture. Assuming that the head rotates with an angular velocity \(\omega\), \(\theta_{j}\) will be obtained as follows:  
\[\theta_{j}=\omega(t_{j}-t_{0})=\frac{2\pi(t_{j}-t_{0})}{(t^{\prime}_{0}-t_{0})}\]  
here, \(t_{1j}=t_{j}-t_{0}\) is the minimum time interval from the initial position to the receiver. Therefore, \(\theta_{j}\) can be expressed according to \(t_{1j}\) and \(T\) as follows:  
\[\theta_{j}=\frac{t_{1j}}