## 2 Proposed Metrology System
### Correction Calculation for Workpiece Eccentricity  
Because the platform and workpiece are not on the same axis, eccentricity errors may occur during rotation. In the prototype system, the ring gauge coordinate center was obtained by using the ring gauge, and then the correction of the eccentricity value and the measured value were calculated by the C# program. On the rapid measuring machine, using the outer diameter of the ring gauge or the maximum outer diameter of the tooth profile in the workpiece, a circle center was obtained through regression analysis. In this study, the ring gauge diameter is 192.68 mm. As shown in Figure 4a, the rotating platform and workpiece (or ring gauge) are not on the same axis, and thus, the measurement result X varies due to eccentricity oscillation. The green zone is the LS-7030 measurement zone, C is the rotary table center, r is the sample or the ring gauge radius, r' is the eccentricity distance (\(\overline{\text{CP}}\)) from the ring gauge center to the rotation center, and \(\theta\) is the angle between the eccentricity distance (\(\overline{\text{CP}}\)) and the \(X\)-axis. In quadrant I, the angle is +\(\theta\). The measurement system moves in a clockwise rotation in the top view, and thus, the progressive rotation angle is \(-\theta\). Because the machine rotates in the clockwise direction, the maximum value of X is at 0\({}^{\circ}\); therefore, the measured values of X at the beginning of quadrants I and II gradually increase to the maximum value (\(\theta=0^{\circ}\)) as the platform rotates, after which they gradually decrease. The minimum value of X is at 180\({}^{\circ}\); therefore, the measured values of X at the beginning of quadrants III and IV gradually decrease to the minimum value (\(\theta=180^{\circ}\)) as the platform rotates, after which they gradually increase. As shown in Figure 4b, all X values can be simplified to a slider-crank mechanism, and the distance X can be obtained by using the slider-crank mechanism, as expressed in Equation (8). \[\text{X}=\text{r'}\cos\theta+\sqrt{\text{r}^{2}-(\text{r'}\sin\theta)^{2}} \tag{8}\]  
Equation (8) can be used to calculate the eccentricity sine curve where the input parameter of \(\theta\) was derived from the cosine theorem in Equation (9) and finally can be obtained by using Equation (10). When the prototype is used for ring gauge measurement, the eccentricity value (r') is calculated directly from the measured values; the value is obtained by dividing the difference between the maximum value of X and the minimum value of X by 2, as shown in Equation (11). The ring gauge radius (r) is 96.34 mm. On the rapid measuring machine, using the outer diameter of the ring gauge or the maximum outer diameter of the tooth profile in the workpiece, a circle profile is obtained through regression analysis. Figure 4: Eccentric correction calculation. (**a**) Eccentricity between rotary table center and ring gauge center. (**b**) Simplification to slider-crank mechanism. Then, the circle is used to calculate the center P. Therefore, the eccentricity value r', the addendum circle radius r, and the X distance can be obtained simultaneously. \[\rm{r}^{2}={\rm{r^{\prime}}}^{2}+X^{2}-2r^{\prime}X\cos\theta \tag{9}\]  
\[\rm{\theta}=\cos^{-1}\!\left(\frac{{\rm{r^{\prime}}}^{2}+X^{2}-r^{2}}{2 rrX}\right) \tag{10}\]  
\[\rm{r^{\prime}}=\frac{(Xmax-Xmin)}{2} \tag{11}\]  
To determine the error value of the measurement system, we take the result of the first measurement as the reference at origin 0 and then set the 1 st value to zero. The method is to deduct the first value of all measured results, and then the origin is translated by subtracting the starting X\({}_{0}\) value from the value of the first measured results. Figure 5 shows the error graphs obtained from different starting eccentricity angles, when the eccentricity is assumed to be 0.1 mm (r\({}^{\prime}\) = 0.1).