## 3 Results and Discussion
### Rapid Measurement System
#### 3.2.3 Virtual Measurement System Repeatability and Accuracy Test  
The results presented in the previous section indicate that the Butterworth filter can improve measurement accuracy and does not affect eccentric correction. In this section, we discuss how both  
Figure 10: Ring gauge measurement results translated to X\({}_{0}\) relative offset data. The blue line indicates the X\({}_{0}\) relative offset data of the original measurement result. The orange line indicates the data after being filtered using a Butterworth filter. (**a**) Measurement results translated to X\({}_{0}\) relative offset data. (**b**) Calculation result of ring gauge eccentricity curve. (**c**) Eccentric correction results. filters affect different parameters. The direct simple filter was used for the tooth-shape workpieces, and the Butterworth filter was used for the ring gauge process. First, the stability of the rapid system was tested. Table 9 lists the system stability results obtained via the unloading test; the measurement was repeated thrice. To easily compare the three measurement results, the first measurement result is subtracted from the second and third measurement results, to obtain the repeatability measurement error. The repeatability measurement result indicated a maximum difference of 0.007 mm, which demonstrates that the accuracy and repeatability of the radial measurement are highly satisfactory in our system. This result is slightly higher than that of the prototype system (0.004 mm). This implies that the Butterworth filter is effective and can achieve the desired goal. After comparing with the ring gauge diameter of 192.68 mm, we find that the diameter error value obtained from using this system is \(-0.02\) mm, and the tooth height error value is within \(-0.02\) mm, which is comparable to the optical projector result. Both comparison results show the same error for the radial measurement. When the radial results shown in Tables 4 and 9 are combined, the difference between the measurement system and the actual size is seen to be \(-0.02\) mm, which means that the system accuracy and repeatability of the diameter measurement are good. In the tooth-shape workpiece, the measurements were conducted thrice in an unloading test. The tooth height and tooth thickness of 16 teeth were measured, as shown in Tables 10 and 11. To easily compare the three measurement results, the first measurement result was subtracted from the second and third measurement results, to obtain the repeatability measurement error. As seen in Table 10, the maximum repeatability measurement error was only 0.023 mm for the tooth height, which indicates that the accuracy and repeatability of the radial measurement are highly satisfactory. Table 11 lists the repeatability measurement results for tooth thickness; the maximum repeatability error is 0.479 mm because the total number of data points is reduced to 31,817. As mentioned in Section 3.2.1 and indicated in Table 8, in the worst case, the sampling interval translates into an arc length of 0.019 mm (sampling blind spot size). The maximum sampling error for tooth thickness is twice the sampling interval, and thus, it is twice the arc length; therefore, the maximum sampling error is 0.038 mm (\(0.019\times 2\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**System Repeatability Study**} \\ \hline \multicolumn{1}{c}{**(Unit: mm)**} & \multicolumn{1}{c}{**Ring Gage Correction Deviation Value**} & \multicolumn{2}{c}{**Difference**} \\ \hline
**Results** & **Test 1** & **Test 2** & **Test 3** & **2 nd–1 st** & **3 rd–1 st** \\ \hline Diameter & 192.66 & 192.66 & 192.66 & 0.00 & 0.00 \\ Positive value & 0.009 & 0.002 & 0.015 & \(-0.007\) & 0.006 \\ Negative value & \(-0.028\) & \(-0.035\) & \(-0.028\) & \(-0.007\) & 0.000 \\ Total value & 0.036 & 0.037 & 0.043 & 0.000 & 0.006 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Measurement system repeatability test (measurements repeated thrice). Figure 11 shows the thickness error distribution.