## 4 Simulation Verification
### Geometric Error and Set-Up Error Identification Experiment  
As shown in Figure 3, the laboratory-built prototype of the reflector standard fixture with reflectors was designed and presented. To obtain and identify the 6-DOF geometric errors of the rotary table more precisely and robustly, it was important to note that the reflector standard fixture was designed for the placement of six non-coplanar holders. In addition, obtaining the relative spatial coordinate positions of the reflectors was important for identifying the LaserTRACER positions. As a result, the relative spatial coordinate positions of the six holders in the reflector standard fixture were measured with the Leitz Ultra coordinate measuring machine (MPE: 0.4 + 1.176 \(\times 10^{-7}L\) um). Table 3 shows the measured data of the six holder positions. In this regard, evaluating the uncertainty associated with the measured data was a must. Since a detailed mathematical derivation can be found in our previous publication and ISO/IEC Guide 98-3:2008, we only briefly review them here and present the results of the measurement uncertainty [7; 19; 42]. According to ISO/IEC Guide 98-3:2008, the influential factors such as the temperature variation in the reflector standard fixture, repeatability, and the maximum permissible error (MPE) of the coordinate measuring machine (CMM) were considered to estimate the measurement uncertainty. Based on the expression of measurement uncertainty described in [7; 42], the combined standard uncertainty \(u_{c}(\Delta L)\) was defined as follows:  
\[u_{c}(\Delta L)=\sqrt{\left(\left|\frac{\partial\Delta L}{\partial\delta_{ \text{thermal}}}\right|u(\delta_{\text{thermal}})\right)^{2}+\left(\left|\frac {\partial\Delta L}{\partial\delta_{\text{repeatability}}}\right|u(\delta_{ \text{repeatability}})\right)^{2}+\left(\left|\frac{\partial\Delta L}{ \partial\delta_{\text{mpe}}}\right|u(\delta_{\text{mpe}})\right)^{2}}, \tag{10}\]  
where \(u(\delta_{\text{thermal}})\), \(u(\delta_{\text{repeatability}})\), and \(u(\delta_{\text{mpe}})\) represent the standard uncertainties of the temperature variation, repeatability, and MPE of the CMM, respectively. Considering the temperature variation in the reflector standard fixture, repeatability and MPE of the CMM during measurement, the standard uncertainties \(u(\delta_{\text{thermal}})\), \(u(\delta_{\text{repeatability}})\), and \(u(\delta_{\text{mpe}})\) were calculated as \(3.12\times 10^{-7}L\) um, 0.0289 um, and 0.4 + 1.176 \(\times 10^{-7}L\) um, respectively. The combined standard uncertainty \(u_{c}(\Delta L)\) can be expressed with Equation (11). Furthermore, based on a 95% confidence level, the coverage factor \(k\) = 2, and the expanded measurement uncertainty equation \(U=k\times u_{c}\), the expanded measurement  
Figure 2: Experimental setup. uncertainty can be expressed as Equation (12). Consequently, Table 3 shows the measured data of the six holder positions with the expanded measurement uncertainty. \[u_{c}(\Delta L)=\sqrt{0.401^{2}+\left(3.32\times 10^{-7}L\right)^{2}}. \tag{11}\]  
\[U=2\times\sqrt{0.401^{2}+\left(3.32\times 10^{-7}L\right)^{2}} \tag{12}\]  
The complete flowchart of geometric error and set-up error identification experiments is shown in Figure 4. Following the flowchart depicted in Figure 4, the reflector standard fixture with reflectors was firstly put on the rotary table, and then the LaserTRACER was placed in front of the rotary table, as shown in Figure 2. As described in Section 2, the positions of the LaserTRACER at base stations P1 to P6 and the reflector at each measurement position were obtained sequentially during the measurement experiment. Furthermore, based on the equations mentioned in Section 3 and the measured data obtained in this section, six geometric errors from the rotary table and four set-up errors were obtained. Table 4 shows the measured results of the set-up errors of the reflector standard fixture. As shown, the maximum linear offset error and angular error were 192.6 \(\upmu\)m and 108.70 arcsec.