## 4 Measurement Uncertainty Analysis of the Stitching Linear Scan Method
Also, the contribution of the resolution of the \(x\)-reading (\(u_{\text{Res\_x}}\)) was evaluated to be 36.08 nm, with regard to the value (125 nm) shown in the specification sheet of the stylus profiler and the rectangular probability distribution with a divisor of \(2\sqrt{3}\). Meanwhile, for the evaluation of \(u_{xi}\), influences of the angular misalignments of a workpiece about the \(Z\)- and \(X\)-axes need to be considered. Denoting the angular misalignment of a workpiece about the \(Z\)-axis as \(\theta_{Z}\), the resultant error in the \(X\)-coordinate of each sampled point \(\Delta x_{i\_0Z}\) can be expressed by the following equation from the geometric relationship:  
\[\Delta x_{i\_0Z}=R\sin(\varphi/2)\cdot[(1/\cos\theta_{Z})-1] \tag{22}\]  
The contribution of \(\theta_{Z}\) (\(u_{\theta Z}\)) can thus be evaluated by the following equation:  
\[u_{\theta Z}=\sqrt{\left(\frac{\partial\Delta x_{i\_0Z}}{\partial\theta_{Z}} \right)^{2}u^{2}(\theta_{Z})}=\sqrt{\left[R\sin(\frac{\varphi}{2})\sin\theta_ {Z}/\cos^{2}\theta_{Z}\right]^{2}\cdot u^{2}(\theta_{Z})} \tag{23}\]  
On the assumption that \(\theta_{Z}\) was within \(\pm 0.1^{\circ}\), by employing a mean radius \(\overline{R}\) of 1.5 mm after the stitching process, \(u_{\theta Z}\) was evaluated to be 0.89 nm. The contribution of the angular misalignment of a workpiece about the \(X\)-axis \(\theta_{X}\) was also estimated. According to the geometric relationship, the error in the \(X\)-coordinate due to \(\theta_{X}\) (\(\Delta x_{i\_0Z}\)) can be expressed by the following equation:  
\[\Delta x_{i\_0X}=r_{i}\sin(\varphi/2)\cdot[1-\cos\theta_{X}] \tag{24}\]  
On the assumption that \(\theta_{X}\) was within \(\pm 0.1^{\circ}\), by employing a mean radius \(\overline{R}\) of 1.5 mm after the stitching process, \(u_{\theta X}\) was evaluated to be 0.89 nm in the same manner as \(u_{\theta Z}\). By combining these contributions, the standard uncertainty of \(x_{i}\) (\(u_{xi}\)) was evaluated to be 154.28 nm. Compensations of the angular misalignment of a workpiece are effective in reducing the standard uncertainties of \(x_{i}\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sources of Uncertainty** & **Symbol** & **Type** & \
\begin{tabular}{c} **Probability** \\ **Distribution** \\ \end{tabular} &
\begin{tabular}{c} **Standard** \\ **Uncertainty** \\ \end{tabular} \\ \hline Reading of \(x\) (including the  contributions from the repeatability,  the drift and so on) & \(u_{\text{Read\_x}}\) & B & Gaussian & 150.00 nm \\ Resolution of \(x\)-reading & \(u_{\text{Res\_x}}\) & B & Rectangular & 36.08 nm \\ Angular misalignment \(\theta_{Z}\) & \(u_{\theta Z}\) & B & Rectangular & 0.89 nm \\ Angular misalignment \(\theta_{X}\) & \(u_{\theta X}\) & B & Rectangular & 0.89 nm \\ \hline Standard uncertainty of \(x_{i}\) & \(u_{xi}\) & & & 154.28 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Uncertainty of the \(X\)-coordinate of each sampled point. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sources of Uncertainty** & **Symbol** & **Type** & \
\begin{tabular}{c} **Probability** \\ **Distribution** \\ \end{tabular} &
\begin{tabular}{c} **Standard** \\ **Uncertainty** \\ \end{tabular} \\ \hline Calibration of the probe (including  the contribution due to the drift  between calibrations) & \(u_{\text{Cal}}\) & B & Rectangular & 115.47 nm \\ Resolution of probe & \(u_{\text{Res\_z}}\) & B & Rectangular & 0.92 nm \\ Repeatability & \(u_{\text{Rep}}\) & A & Gaussian & 67.66 nm \\ \hline Standard uncertainty of \(z_{i}\) & \(u_{zi}\) & & & 133.84 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Uncertainty of the \(Z\)-coordinate of each sampled point. \(z_{i}\), and the contributions from the specifications of the stylus profiler were found to be dominant in these uncertainties.