## 3 Measurement Uncertainty Analysis of the Linear Scan Method
### Measurement Uncertainty Evaluation
\[h=R\times(1-\cos\frac{\varphi}{2}) \tag{16}\]  
\[u\big{(}e_{alignment\_Z}\big{)}=\frac{x_{edge}}{\cos\frac{\theta_{Z}}{\sqrt{3} }}-x_{edge}=R\sin\frac{\varphi}{2}(\frac{1}{\cos\frac{\theta_{Z}}{\sqrt{3}}}-1) \tag{17}\]In the actual measurement, the alignment of the rotating stage and the scanning of the stylus were repeated to converge the obtained arc length to the minimum, and the measurement was performed with the error angle \(\theta_{Z}\) being as small as possible. Assuming that the error angle at this time is \(\pm 1^{\circ}\), the uncertainty due to the attitude error around the Z-axis can be obtained as follows, by substituting the radius after stitching into Equation (18). \[u(e_{alignment\_Z})=0.74940\times\sin\frac{85^{\circ}}{2}\left(\frac{1}{\cos \frac{1^{\circ}}{\sqrt{3}}}-1\right)=0.00002568\;\mathrm{mm}=25.68\;\mathrm{ nm} \tag{18}\]  
(3) Uncertainty due to position error around the X-axis of the workpiece \(u(e_{alignment\_Z})\). The influence of the position error around the X-axis of the measured cylinder is shown in Figure 5b, in which D is the workpiece diameter, R is the radius, the coordinates (x\({}_{\mathrm{edge}}\), z\({}_{\mathrm{edge}}\)) are the end point of the obtained arc, and \(\theta_{X}\) is the angle of position error, which causes measurements of geometric circles to geometric cylinders to appear as ellipses, with the major axis along the Z-axis. Since the arc with height h from the vertex is extracted in the same way as when there is a posture error around the Z-axis, an error occurs in the X-axis coordinate. Assuming that the error angle is \(\theta_{X}\), which forms a rectangular distribution, the uncertainty due to the position error around the X-axis is expressed by the following equation. \[u(e_{alignment\_X})=x_{edge}-x_{edge}\mathrm{cos}\;\frac{\theta_{X}}{\sqrt{3}} =R\sin\frac{\varphi}{2}(1-\cos\frac{\theta_{X}}{\sqrt{3}}) \tag{19}\]  
In the actual measurement, the manual stage was moved in the Y-axis direction while the stylus was in contact with the flat surface of the mounting table, and alignment was performed using the tilting stage so that the change in displacement output was minimized. As a result, the error angle \(\theta_{X}\) was measured as small as possible. Assuming that the error angle at this time is \(\pm 0.1^{\circ}\), the uncertainty due to the attitude error around the X-axis is obtained as follows, by substituting the radius after stitching processing into Equation (20). Figure 5: (**a**) Influence of attitude error of roll workpiece around Z-axis; (**b**) influence of attitude error of roll workpiece around X-axis. Figure 4:0  
Tables 1 and 2 show a summary of the calculation results for each uncertainty. From these results, the combined standard uncertainty of the Z-axis direction output z\({}_{i}\) and the combined standard uncertainty of the X-axis coordinate x\({}_{i}\) are expressed by Equation (21). \[ \begin{split} u(z_{i})&=\sqrt{u^{2}\Big{(}e_{ calibration\_Z}\Big{)}+u^{2}(e_{resolution})+u^{2}\big{(}e_{repeat}\big{)}}\\ &=\sqrt{33.7^{2}+0.92^{2}+18.95^{2}}=38.45\;\mathrm{nm}\end{split}  \tag{21}\]  
\[ \begin{split} u(x_{i})&=\sqrt{u^{2}(e_{calibration\_X})+u^{2}\Big{(}e_{ alignment\_Z}\Big{)}+u^{2}\Big{(}e_{alignment\_X}\Big{)}}\\ &=\sqrt{33.7^{2}+25.68^{2}+0.26^{2}}=42.37\;\mathrm{nm}\end{split}  \tag{22}\]  
Using the calculated uncertainties, the combined standard uncertainty \(u(\mathrm{R_{i}})\) of the radius of the measured arc can be obtained from Equation (23). \(u(\mathrm{R_{i}})\) can be obtained by Equation (24). \[u(\Delta r(\theta))=u(m(\theta))=26.92\;\mathrm{nm} \tag{23}\]