## 2 Measurement Principles  
Due to the requirement for high measurement accuracy, the principles of multi-station and time-sharing measurements developed by [PERSON] et al. were adopted and improved in this paper to obtain and identify the 6-DOF geometric errors of rotary tables [33; 39]. In addition, to obtain the spatial coordinate position of the LaserTRACER and the reflectors, the multilateration method was also adopted in this paper [30; 31]. The multilateration method was proposed by [PERSON] et al. by solving a set of four equations [40; 41]. Accordingly, the positions of the LaserTRACER and the reflectors were identified through the measurement principle and are depicted in this section.  
Since the detailed mathematical derivation of the multilateration method can be found in our previous publications [30; 31], this section only reviews it briefly and presents how it is different here. Based on laser interferometry techniques, the operation principle of the LaserTRACER only involves the length measurement. When the LaserTRACER was placed at each base station, the distances between the LaserTRACER and each reflector position were obtained and recorded sequentially. As a result, the position of each LaserTRACER base station can be calculated and obtained through Equation (1):  
\[l_{ij}+l_{oj}+r_{ij}=s_{j}\sqrt{\left(x_{i}-x_{0j}\right)^{2}+\left(y_{i}-y_{ 0j}\right)^{2}+\left(z_{i}-z_{0j}\right)^{2}}, \tag{1}\]  
where (\(x_{i}\), \(y_{i}\), \(z_{i}\)) represents the coordinate position of the \(i\)-th measurement point; (\(x_{0j}\), \(y_{0j}\), \(z_{0j}\)) represents the coordinate position of the \(j\)-th LaserTRACER; when the reflector is located at the \(i\)-th measurement point, \(l_{ij}\) is the measured length; \(l_{0j}\) is the initial distance between the reflector and the \(j\)-th LaserTRACER; and \(s_{j}\) is the scale factor for the \(j\)-th LaserTRACER; finally, \(r_{ij}\) is the residual error.  
As shown in Figure 1, the reflectors with the reflector standard fixture were mounted on the rotary table, and the LaserTRACER was installed at different base stations externally to the measured rotary table for the 6-DOF geometric error measurements. Moreover, Pi (\(i\) = 1-6) in Figure 1 represents the six different LaserTRACER base stations, even though, according to the literature, using four different reflector positions and LaserTRACER base stations are enough to calibrate the 6-DOF geometric errors of a rotary table. However, for a much better convergence of the measurement equations, the original principle of multi-station and time-sharing measurements was improved and utilized in this paper. Consequently, six different reflector positions and six different LaserTRACER base stations were used in this paper to identify the position and orientation of the rotary table.  
First, the rotary table (A-axis) was driven to the home position (zero position). When the LaserTRACER was placed at the first base station (P1), the distances between P1 and each reflector were measured. By using Equation (1) and the Levenberg-Marquardt method, the position of the LaserTRACER at the first base station (P1) can be identified. Then, the positions of LaserTRACER base stations P1 to P6 were also obtained sequentially. Second, the rotary table was driven to each measurement position. Afterwards, the distances between each reflector at each measurement position and the LaserTRACER base station were measured. Likewise, the positions of each reflector at each measurement position were also calculated by using Equation (1) and the Levenberg-Marquardt method.