## 2 Basic Working Principle of wMPS  
The wMPS is connected via transmitting stations, receivers, and a computer (Figure 1). The transmitting station is composed of an upper rotating head and a bottom base (Figure 1). The two laser devices on the rotating head are spaced 90\({}^{\circ}\) apart in the horizontal direction of rotation. The laser can transmit a sector-shaped beam, and the two laser sectors and the horizontal plane are at 60\({}^{\circ}\) angles. A pulsed laser is installed on the base. When the system is in operation, the rotating head rotates around the fixed axis in a counterclockwise direction. The two laser devices continuously scan the space. When the rotor arrives at the initial position, the synchronous impulse transmitter on the base emits one round of synchronized optical signals and the timing begins accordingly. When any sector-shaped laser plane sweeps over the receiver, the rotation angle of the plane from the initial position to the receiver position can be obtained by counting pulses. When all the laser planes in the space scan the same receiver, a constraint equation based on multi-plane constraints can be established using the same method; thus, the coordinates of the receiver can be obtained. The digital model of the transmitting stations of the wMPS can be simplified to two sector half-planes rotating around an axis. The axis is defined as the \(Z\)-axis, and the intersection between plane 1 and the \(Z\)-axis is the origin of the measurement coordinates (point \(O\)). The initial position of the optical axis of plane 1 is the \(X\)-axis. On this basis, a right-handed coordinate system is established (Figure 2). The coordinate system is the transmitting station measurement coordinate system. Internal parameters (initial plane parameters) of the transmitting station \(i\) are calibrated after the instrument is assembled. The normal vectors of the initial plane \(n_{i1}=(a_{i1},b_{i1},c_{i1})\), \(n_{i2}=(a_{i2},b_{i2},c_{i2})\), and \(\Delta d\) are unit normal vectors of planes 1 and 2, while \(\Delta d\) is the deviation between the two laser planes along the \(Z\)-axis. In Figure 2, \((a_{i1},b_{i1},c_{i1},d_{i1})\) and \((a_{i2},b_{i2},c_{i2},d_{i2})\) are known variables. When any laser plane scans over the receiver \(P_{m}(x_{m},y_{m},z_{m})\), the relationship between the light-plane parameters \(\left(a_{ij}^{\prime},b_{ij}^{\prime},c_{ij}^{\prime},d_{ij}^{\prime}\right)\) and the initial plane parameters can be expressed as follows:  
\[\left[ \begin{array}{c}a_{ij}^{\prime}\\ b_{ij}^{\prime}\\ c_{ij}^{\prime}\\ d_{ij}^{\prime}\end{array} \right]=\left[ \begin{array}{cc}R\left(\theta_{j} \right)&0\\ 0&1\end{array} \right]\left[ \begin{array}{c}a_{ij}\\ b_{ij}\\ c_{ij}\\ d_{ij}\end{array} \right],i=1,2\cdots,j=1,2 \tag{1}\]  
Figure 1: Structure of the workspace measuring and positioning system (wMPS). Figure 2: Measurement coordinates. \[R\left(\theta_{j}\right)=\left[ \begin{array}{ccc}\cos\left(\theta_{j}\right)&- \sin\left(\theta_{j}\right)&0\\ \sin\left(\theta_{j}\right)&\cos\left(\theta_{j}\right)&0\\ 0&0&1\end{array} \right],j=1,2, \tag{2}\]  
where \(\theta_{j}(j=1,2)\) is the rotation angle of the plane from the initial position to the receiver. Based on the recording time of synchronous pulse transmitters and the rotation angle of the transmitting stations, the relationship can be expressed as  
\[\theta_{j}=\frac{t_{j}}{T}\times 2\pi,j=1,2, \tag{3}\]  
where \(T\) is the rotation period of the rotor. Suppose the regional independent coordinate system is a global one. The rotation and translation matrices of the measuring coordinate system of any transmitting station \(i\) to the global coordinate system are denoted \(R_{i}\) and \(T_{i}\), respectively. When the laser plane sweeps over the receiver, the light plane factor can be expressed as  
\[\left[ \begin{array}{c}a_{ij}^{{}^{\prime\prime}}\\ b_{ij}^{{}^{\prime\prime}}\\ c_{ij}^{{}^{\prime\prime}}\\ d_{ij}^{{}^{\prime\prime}}\end{array} \right]=\left[\left[ \begin{array}{cc }R_{i}&T_{i}\\ 0&1\end{array}