## 3 Measurement Uncertainty Analysis of the Linear Scan Method
### Variation of Uncertainty Due to Change in Workpiece Diameter  
For the purpose of examining the cylindrical work diameter suitable for roundness measurement using the profilometer, which is the stitching linear scan method, a simulation is performed to calculate the measurement uncertainty when the work diameter is changed. Of the uncertainty factors in (1) to (3) of 1 and (1) to (3) of 2, shown in the previous section, the ones that change significantly depending on the workpiece diameter are the uncertainty \(u(e_{alignment\_Z})\) due to the workpiece orientation error around the Z-axis and the uncertainty \(u(e_{alignment\_X})\) due to the workpiece orientation error around the X-axis. Figure 6a,b show the graphs of the uncertainty \(u(\text{x}_{\text{i}})\) of the X-axis coordinate when the workpiece diameter is varied from 0.01 mm to 50.00 mm using Equations (13) and (14). From Figure 6a, as the diameter increases, \(u(e_{alignment\_Z})\) increases and becomes almost the same value as \(u(\text{x}_{\text{i}})\), but the amount of change in \(u(e_{alignment\_X})\) is small. In other words, \(u(e_{alignment\_Z})\) is the dominant factor for changes in \(u(\text{x}_{\text{i}})\). This is because the attitude error angles \(\theta_{\text{Z}}=\pm 1^{\circ}\) and \(\theta_{\text{X}}=\pm 0.1^{\circ}\) are set in consideration of the alignment method and stylus resolution, and \(\theta_{\text{Z}}\) is estimated to be larger than \(\theta_{\text{X}}\). Figure 7a shows a graph showing changes in diameter expanded uncertainty U(D) and roundness expanded uncertainty U(\(\Delta\text{zq}\)) created by substituting \(u(\text{x}_{\text{i}})\), which varies depending on the workpiece diameter, and other standard uncertainties. It can be seen that both U(D) and U(\(\Delta\text{zq}\)) increase as the workpiece diameter increases due to the influence of \(u(\text{x}_{\text{i}})\). Figure 7b shows an enlarged graph of the work diameter range from 0.01 mm to 10 mm. From this graph, when the attitude error angle \(\theta_{\text{Z}}=\pm 1^{\circ}\) and \(\theta_{\text{X}}=\pm 0.1^{\circ}\), it can be read that the conditions for the workpiece diameter that can theoretically achieve the target measurement uncertainty within \(\pm\)0.1 \(\upmu\)m are a U(D) of 5.58 mm or less and a U(\(\Delta\text{zq}\)) of 2.11 mm or less. In the previous simulation, the position error angles were set to \(\theta_{Z}=\pm 1^{\circ}\) and \(\theta_{X}=\pm 0.1^{\circ}\), but improving the alignment increases the upper limit of the workpiece diameter that satisfies the target measurement uncertainty. There is room for improvement in the position error angle \(\theta_{Z}\) around the \(Z\)-axis. Therefore, Figure 8a,b show the changes in the expanded uncertainty of diameter U(D) and the expanded uncertainty of roundness U(\(\Delta\)zq) when the attitude angle error occurs in a rectangular distribution of \(\theta_{Z}=\pm 0.1^{\circ}\), \(\pm 0.3^{\circ}\), \(\pm 0.5^{\circ}\), \(\pm 0.6^{\circ}\), \(\pm 0.7^{\circ}\), \(\pm 0.8^{\circ}\), \(\pm 0.9^{\circ}\), \(\pm 1^{\circ}\), \(\pm 1.5^{\circ}\) and \(\pm 2^{\circ}\). Table 4 summarizes the upper limit of the workpiece diameter that can achieve the target uncertainty of diameter and roundness within \(\pm 0.1~{}\upmu\)m at each attitude error angle \(\theta_{Z}\). From Table 4, we see that it is desirable to converge \(\theta_{Z}\) as small as possible by more accurate alignment when measuring cylindrical workpieces with a large diameter. It can be confirmed that the proposed method can cover all diameters of less than \(\phi\) 3 mm, which are difficult to measure using the rotational scanning measurement method, as long as \(\theta_{Z}\) is kept smaller than \(\pm 0.8^{\circ}\). Furthermore, if \(\theta_{Z}\) can be made smaller, the proposed method can be used for workpieces with a larger diameter. If alignment is possible up to \(\theta_{Z}=\theta_{X}=\pm 0.1^{\circ}\), there is no limitation on the workpiece diameter theoretically.