## 3 Measurement Uncertainty Analysis of the Linear Scan Method
### Measurement Uncertainty Evaluation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source of Uncertainty** & **Symbol** & **Type** & **Coverage Factor** & **Standard Uncertainty** & **Sensitivity Coefficient** & \(1c_{i}1\times u(x_{i})\) \\ \hline Calibration of probe & \(u(e_{calibration\_Z})\) & A & — & 33.7 & 1 & 33.7 \\ \hline Resolution & \(u(e_{resolution})\) & B & \(\sqrt{3}\) & 0.92 & 1 & 0.92 \\ \hline Repeatability & \(u(e_{repeat})\) & A & — & 18.95 & 1 & 18.95 \\ \hline Combined standard uncertainty & \(u(z_{i})\) & & — & & 38.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Uncertainty budget of the output in the Z-axis direction. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Source of Uncertainty** & **Symbol** & **Type** & **Coverage Factor** & **Standard Uncertainty** & **Sensitivity Coefficient** & \(1c_{i}1\times u(x_{i})\) \\ \hline Calibration of probe & \(u(e_{calibration\_X})\) & A & — & 33.7 & 1 & 33.7 \\ \hline Attitude error around & \(u(e_{alignment\_Z})\) & B & \(\sqrt{3}\) & 25.68 & 1 & 25.68 \\ \hline Attitude error around & \(u(e_{alignment\_X})\) & B & \(\sqrt{3}\) & 0.26 & 1 & 0.26 \\ \hline Combined standard uncertainty & \(u(x_{i})\) & & — & & & 42.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Uncertainty budget of the X-axis coordinate. \[u(R_{i})=\sqrt{\frac{x_{i}^{2}}{z_{i}^{2}+x_{i}^{2}}}u^{2}(x_{i})+ \frac{z_{i}^{2}}{z_{i}^{2}+x_{i}^{2}}u^{2}(z_{i})\] \[=\sqrt{\frac{0.50575^{2}}{0.55280^{2}+0.50575^{2}}}\times 42.37^{2}+ \frac{0.55280^{2}}{0.55280^{2}+0.50575^{2}}\times 38.45^{2}} \tag{24}\] \[=40.28\;\mathrm{nm}\]  
The uncertainty of the radius after the stitching process can be obtained by substituting the values obtained in Equation (13) into Equation (11) to obtain the following equation. \[u\big{(}\overline{R}\big{)}=\frac{u(R_{i})}{\sqrt{8}}=\frac{40.28}{\sqrt{8}}=14.24\;\mathrm{nm} \tag{25}\]  
The uncertainty of the diameter measurement is twice the uncertainty of the radius. \[u(D)=2\times u(R_{i})=2\times 14.24=28.48\;\mathrm{nm} \tag{26}\]  
The expanded uncertainty can be calculated when k = 2. \[U(D)=ku(D)=2\times 28.48=56.96\;\mathrm{nm} \tag{27}\]  
As can be seen above, the uncertainty of the diameter measurement of the 1.5 mm small cylindrical workpiece using the linear scanning method was estimated to be \(\pm 47.24\;\mathrm{nm}\), achieving a target measurement uncertainty within \(\pm 0.1\;\mathrm{\SIUnitSymbolMicro m}\). Table 3 summarizes the results of the standard uncertainty. Using the calculated uncertainty, the combined standard uncertainty \(u(\Delta r_{i})\) of the radial deviation of an arbitrary measured point on the arc can be obtained from Equation (10). Substituting the combination of (x\({}_{i}\), m\({}_{i}\)), the coordinates of the endpoint of the arc (x\({}_{\mathrm{edge}}\), z\({}_{\mathrm{edge}}\)) = (0.50575, 0.55280) and the radius after the stitching process, for which u obtained from the measurement \(u(\Delta r_{i})\) is maximum, the following equation is obtained. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Source of Uncertainty** & **Symbol** & **Type** & **Coverage Factor** & **Standard Uncertainty** & **Sensitivity Coefficient** & \(1c_{i}\times u(x_{i})\) \\ \hline Output in & \(u(z_{i})\) & — & — & 38.45 & 1 & 38.45 \\ \hline Coordinate of & \(u(x_{i})\) & — & — & 42.37 & 1 & 42.37 \\ \hline Combined & \(u(\mathrm{R_{i}})\) & & — & & 40.28 \\ standard uncertainty & \(u(\mathrm{R})\) & & — & & 14.24 \\ \hline Radius after & \(u(R)\) & & — & & 28.48 \\ stitching process & \(u(D)\) & & — & & 56.96 \\ \hline Diameter & \(u(D)\) & & — & & 56.96 \\ \hline Expanded uncertainty & U(D) & & — & & 56.96 \\ \((k=2)\) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Uncertainty budget of diameter measurement by the proposed method. \[