## 4 Measurement Uncertainty Analysis of the Stitching Linear Scan Method
By combining these contributions based on Equation (21), \(u_{ri}\) was evaluated to be 178.75 nm, as summarized in Table 5. It should be noted that the sensitivity coefficients \((\partial r_{i}/\partial x_{i})\) and \((\partial r_{i}/\partial z_{i})\) were treated to have maximum values \((\partial r_{i}/\partial x_{i})\ =\ \sin(\varphi/2)\) = 0.67559 and \((\partial r_{i}/\partial z_{i})\) = 1, respectively, to avoid underestimation. A standard uncertainty of \(\theta_{i}\left(u_{\theta i}\right)\) was also evaluated. Since \(\theta_{i}\) can be calculated as \(\theta_{i}\) = \(\arctan(z_{i}/x_{i})\), \(u_{\theta i}\) can be calculated by the following equation:  
\[u_{\theta i}=\sqrt{\left(\frac{\partial\theta_{i}}{\partial x_{i}}\right)^{2} u_{xi}^{2}+\left(\frac{\partial\theta_{i}}{\partial z_{i}}\right)^{2}}u_{zi} ^{2}=\sqrt{\left(\frac{z_{i}}{{x_{i}}^{2}+{z_{i}}^{2}}\right)^{2}}u_{xi}^{2}+ \left(\frac{x_{i}}{{x_{i}}^{2}+{z_{i}}^{2}}\right)^{2}}u_{zi}^{2} \tag{25}\]  
The sensitivity coefficients \((\partial\theta_{i}/\partial x_{i})\) and \((\partial\theta_{i}/\partial z_{i})\) were calculated as \((\partial\theta_{i}/\partial x_{i})\) = \(1/R\) and \((\partial\theta_{i}/\partial z_{i})\ =(1/R)\sin(\varphi/2)\), respectively, to avoid underestimation. As a result, \(u_{\theta i}\) was evaluated to be \(u_{\theta i}\) = \(1.1\times 10^{-4}\) rad (0.0065 degree), as summarized in Table 6.  
By using the obtained \(u_{ri}\) and \(u_{\theta i}\), numerical calculations were carried out based on the Monte Carlo method. The procedure of the numerical calculations was as follows:  
1. Prepare the data of the series of arc profiles in the polar coordinate system by using the circumferential profile of a small cylinder obtained in experiments after filtering (50 URP). 2. Apply a random value in a Gaussian distribution with a standard deviation of \(u_{\theta}\) to the \(\theta\)-coordinate of each point. 3. Apply a random value in a Gaussian distribution with a standard deviation of \(u_{R}\) to the \(R\)-coordinate of each point. 4. Convert the arc-profile data into the Cartesian coordinate system. 5. Carry out the stitching process in the same manner as the experiments. Figure 19 shows the results of numerical calculations. A trial number of \(1\times 10^{5}\) was used for sufficient numerical stability of the output parameters. Expanded uncertainties of diameter and roundness were thus evaluated to be 0.032 \(\upmu\)m and 0.024 \(\upmu\)m (\(k\) = 2, 95% confidence), respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Uncertainty Sources** & **Symbol** & **Value** & \
\begin{tabular}{c} **Sensitivity** \\ **Coefficient** \\ \end{tabular} &
\begin{tabular}{c} **Standard** \\ **Uncertainty** \\ \end{tabular} \\ \hline Standard uncertainty of \(z_{i}\) & \(u_{zi}\) & 133.84 nm & 666.67 rad/mm & 0.0000892 rad \\ Standard uncertainty of \(x_{i}\) & \(u_{xi}\) & 154.28 nm & 450.39 rad/mm & 0.0000695 rad \\ \hline Standard uncertainty of \(\theta_{i}\) & \(u_{\theta i}\) & 0.00011 rad (0.0065 deg.) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Uncertainty of the angular position of each sampled point. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sources of Uncertainty** & **Symbol** & **Value** & \
\begin{tabular}{c} **Sensitivity** \\ **Coefficient** \\ \end{tabular} &
\begin{tabular}{c} **Standard** \\ **Uncertainty** \\ \end{tabular} \\ \hline Standard uncertainty of \(z\) & \(u_{zi}\) & 133.84 nm & 0.67559 & 90.42 nm \\ Standard uncertainty of \(x\) & \(u_{xi}\) & 154.28 nm & 1 & 154.28 nm \\ \hline Standard uncertainty of \(r_{i}\) & & \(u_{ri}\) & 178.75 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Uncertainty of the radius of each sampled point.