## 4 Measurement Uncertainty Analysis of the Stitching Linear Scan Method  
Theoretical investigation of the measurement uncertainty of the stitching linear-scan method was carried out. In the method, several arc-profiles were first measured in the Cartesian coordinate system (\(x_{i}\),\(z_{i}\)). After that, the obtained data of the arc-profiles were converted into the polar coordinate system (\(r_{i}\),\(\theta_{i}\)) to reconstruct the workpiece profile through the stitching process. Finally, the roundness of the workpiece was evaluated by the reconstructed profile. Since the above procedure contained a fitting process based on the least-squares method, numerical calculations based on the Monte Carlo method were carried out for the estimation of the measurement uncertainty. Since the stitching linear-scan method was developed for roundness measurement of a small cylinder having a radius of a few millimeters, following the previous work by the authors [10], a series of arc-profiles obtained by measuring a needle roller with a diameter of 1.5 mm was employed for the following numerical calculations. First, the uncertainty of the radius of each sampling point was estimated. On the assumption that the workpiece had a perfect cylindrical profile, the radius \(r_{i}\) at the \(i\)th sampled point (\(x_{i}\),\(z_{i}\)) could be expressed by the following equation:  
\[r_{i}=\sqrt{x_{i}^{2}+z_{i}^{2}} \tag{20}\]  
The standard uncertainty of \(r_{i}\) (\(u_{ri}\)) can thus be obtained by the following equation:  
\[u_{ri}=\sqrt{\left(\frac{\partial R_{i}}{\partial x_{i}}\right)^{2}u_{xi}{}^{2 }+\left(\frac{\partial R_{i}}{\partial z_{i}}\right)^{2}u_{zi}{}^{2}}=\sqrt{ \frac{x_{i}^{2}}{x_{i}^{2}+z_{i}^{2}}u_{xi}{}^{2}+\frac{z_{i}^{2}}{x_{i}^{2} +z_{i}^{2}}u_{zi}{}^{2}} \tag{21}\]  
For the evaluation of \(u_{ri}\), the uncertainties of \(z_{i}\) and \(x_{i}\) need to be obtained. Table 3 summarizes the sources of uncertainty contributing to the uncertainty of \(z_{i}\). The stylus profilometer employed in the series of experiments was calibrated by using a master sphere artifact in advance of the roundness measurements. According to the specification sheet of the stylus profilometer [11], the uncertainty of the calibration is within \(\pm\)200 nm. It should be noted that the contribution due to the drift between calibrations is included in this value. Assuming the rectangular probability distribution with a divisor of \(\sqrt{3}\), the standard uncertainty of the calibration (\(u_{\mathrm{Cal}}\)) was evaluated to be 115.47 nm. It should be noted that \(u_{\mathrm{Cal}}\) contains the influences of the straightness of the linear scan axis, the form errors of the master sphere artifact, and the stylus tip. Also, the uncertainty associated with the resolution of the \(Z\)-reading (\(u_{\mathrm{res\_z}}\)) was evaluated to be 0.92 nm, with regard to the value (3.2 nm) shown in the specification sheet of the stylus profiler. Furthermore, the contribution of repeatability of the \(Z\)-reading (\(u_{\mathrm{rep\_z}}\)) was estimated from a standard deviation in ten repetitive arc-profile measurements (67.66 nm). A standard uncertainty of \(z_{i}\) (\(u_{zi}\)) was thus evaluated to be 133.84 nm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Angular Misalignment of the Pin Gage \(\theta_{z}\)** & **0.004\({}^{\circ}\)** & **0.951\({}^{\circ}\)** \\ \hline Diameter & Mean value & 2.99963 mm & 2.99898 mm \\  & Standard deviation & 0.10 μm & 0.13 μm \\ \hline Roundness & Mean value & 0.21 μm & 0.19 μm \\  & Standard deviation & 0.03 μm & 0.01 μm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Radius and roundness. The uncertainty of \(x_{i}\) was also estimated. Table 4 summarizes the sources of uncertainty contributing to the uncertainty of \(x_{i}\). According to the manufacturer of the stylus profilometer, the standard uncertainty of the reading of \(X\)-position (\(u_{\text{Read\_x}}\)) is 150 nm. It should be noted that this value contains the contributions from the repeatability, the drift, and so on.