## 2 Laser Tracer Multi-Station Measurement System  
The laser tracer multi-station measurement model is shown in Figure 1. The rotary table is mounted on the working plane of the CMM (coordinate measurement machine), and the laser tracer is placed on the rotary table. The multi-station measuring system of the laser tracer consists of CMM, rotary table, and laser tracer. The coordinate system of the rotary table is the same as the coordinate system of the CMM, as shown in Figure 2. The cat eye reflector of the laser tracer [9; 10; 11] is fixed on the probe of the CMM as the measured point. The cat eye reflector has the same trajectory as the probe of the CMM. When the CMM controls the probe to move in the measurement space, the cat eye reflector also follows the probe of the CMM. The laser beam from the laser tracer is incident on the cat eye reflector and reflected back to the tracking head of the laser tracer. After receiving the reflected beam from the cat eye reflector, the laser tracer measures the relative displacement between the cat eye reflector (i.e., the measured point) and the laser tracer. The CMM provides the coordinates of measured points. It is assumed that the laser tracer is used to measure \(n\) measured points at \(m\) stations. The coordinates of laser tracer stations are \(P_{j}(X_{j},Y_{j},Z_{j})\); the coordinates of measured points are \(A_{i}(x_{i},y_{i},z_{i})\); the distance from the laser tracer at each station \(P_{j}\) to the initial measured point \(A_{1}\) is \(d_{j}\); the high-precision interferometric length measured by the laser tracer is \(l_{ij}\). According to the distance formula of two points in three-dimensional (3D) space, the following relationship is established. The distance \(d_{ji}\) of the laser tracer from each station to the measured point is obtained,  
\[d_{ji}=\sqrt{\left(x_{i}-X_{j}\right)^{2}+\left(y_{i}-Y_{j}\right)^{2}+\left(z _{i}-Z_{j}\right)^{2}}=d_{j}+l_{ij} \tag{1}\]  
where \(i\) = 1,2,3... \(n\), \(j\) = 1,2,3,... \(m\). Since \(n\) measured points are measured at \(m\) stations, according to Equation (1), the number of unknowns is \(4m+3n\). In order to solve Equation (1), the following relationship should hold:  
\[m\times n\geq 4m+3n \tag{2}\]  
From Equation (2), we get \(m\geq 4\), \(n\geq 16\). With the high-precision interferometric length measured by the laser tracer as the constraint condition, Equation (1) can be rewritten as the error equation,  
Figure 1: Laser tracer multi-station measurement model. CMM: coordinate measurement machine. Figure 2: Schematic diagram of the rotary table coordinate system. \[v_{ij}=\sqrt{\left(x_{i}-X_{j}\right)^{2}+\left(y_{i}-Y_{j}\right)^{2}+\left(z_{i} -Z_{j}\right)^{2}}-d_{j}-l_{ij} \tag{3}\]  
Since Equation (3) is a nonlinear equation, it is difficult to directly solve it. According to a linearization method, the nonlinear function is converted into a linear function, which is then processed according to linear parameters. Let \(\sqrt{\left(x_{i}-X_{j}\right)^{2}+\left(y_{i}-Y_{j}\right)^{2}+\left(z_{i}-Z _{j}\right)^{2}}\) be recorded as \(L_{ij}\), and the first-order Taylor series expansion of Equation (3) is obtained as,  
\[v_{ij}=L_{ij}|_{0}+\frac{x_{i}|_{0}-X_{j}|_{0}}{L_{ij}|_{0}}\cdot\left(\Delta x _{i}-\Delta X_{j}\right)+\frac{y_{i}|_{0}-Y_{j}|_{0}}{L_{ij}|_{0}}\cdot\left( \Delta y_{i}-\Delta Y_{j}\right)+\frac{z_{i}|_{0}-Z_{j}|_{0}}{L_{ij}|_{0}} \cdot\left(\Delta z_{i}-\Delta Z_{j}\right)-d_{j}-l_{ij} \tag{4}\]  
where \(L_{ij}|_{0}=\sqrt{\left(x_{i}|_{0}-X_{j}|_{0}\right)^{2}+\left(y_{i}|_{0}-Y_{ j}|_{0}\right)^{2}+\left(z_{i}|_{0}-Z_{j}|_{0}\right)^{2}}\); the mark symbol \(|_{0}\) is the approximate of this value, and the approximate values \(x_{i}|_{0},y_{i}|_{0}\) and \(z_{i}|_{0}\) of the measured points in the actual calculation are replaced by the measured points in the coordinate system of CMM.