## 3 Measurement Uncertainty Analysis of the Linear Scan Method
### Measurement Uncertainty Evaluation  
The standard uncertainty of each uncertainty factor can be analyzed and obtained according to GUM. 1. Uncertainty coefficient of the Z-axis coordinate \(z_{i}\). 2. Uncertainty of stylus calibration in Z-axis direction \(u(e_{calibration\_Z})\). The calibration of the employed Form Talysurf PGI420 is conducted with the dedicated master sphere shown in Figure 4a. The Pt value, which is the minimum and maximum width after linear expansion of the arc measured at the known radius values shown in Figure 4b, includes the shape error of the master ball, the shape error of the stylus tip diameter, the electrical noise of the system, and mechanical and environmental disturbances. The Pt value = 0.0337 \(\upmu\)m exists in the radial direction, but assuming that it exists in the Z-axis direction and the X-axis direction, respectively, the uncertainty of the probe calibration in the Z-axis direction can be expressed by the following equation. \[u(e_{calibration\_Z})=0.0337\ \upmu\text{m}=33.7\ \text{nm} \tag{12}\]  
Figure 4: (**a**) Standard ball; (**b**) calibration result. (2) Uncertainty due to stylus resolution \(u(e_{resolution})\). The stylus of the Form Talysurf PGI420 has a vertical resolution of 3.2 nm. Since the uncertainty due to the resolution of the probe is considered to be a square distribution of \(\pm 1.6\) nm, it is obtained by the following equation using B-type evaluation. \[u(e_{resolution})=\frac{3.2/2}{\sqrt{3}}=0.92\;\mathrm{nm} \tag{13}\]  
(3) Uncertainty of repeated measurements \(u(e_{repeat})\). The uncertainty due to the measurement error can be evaluated by the results of the repeated measurements of the small cylinder with a diameter 1.5 mm. Since it has a repeatability of 59.93 nm from the results of 10 repeated measurements, the uncertainty of repeated measurements can be obtained by the following formula using A type evaluation. \[u\big{(}e_{repeat}\big{)}=\frac{59.93}{\sqrt{10}}=18.95\;\mathrm{nm} \tag{14}\]  
2. Uncertainty coefficient of the X-axis coordinate \(\mathrm{x_{i}}\). (1) Uncertainty of stylus calibration in the X-axis direction \(u(e_{calibration\_X})\). When picking up the output \(z_{i}\) in the Z-axis direction as the uncertainty coefficient, it is assumed that Pt values exist in the Z-axis and X-axis directions, respectively, so the uncertainty of the stylus calibration in the X-axis direction is the same as that in the X-axis direction, expressed by the following equation. \[u(e_{calibration\_X})=33.7\;\mathrm{nm} \tag{15}\]  
(2) Uncertainty due to position error around the Z-axis of the workpiece \(u(e_{alignment\_Z})\). The uncertainty due to the workpiece position error can be obtained according to B type evaluation by using the value of the calculated measurement error, which can be obtained from the mathematical modeling. The influence of the position error around the Z-axis of the measured cylinder is shown in Figure 5a, in which D is the diameter of the workpiece, R is the radius, (\(\mathrm{x_{edge}}\), \(\mathrm{z_{edge}}\)) is the end point coordinate of the obtained arc, and \(\theta_{Z}\) is the angle of the position error around the Z-axis. The position error around the Z-axis causes the measurement result of the geometric circle of the geometric cylinder to appear as an ellipse, with its main axis in the X-axis direction. Since the arc of \(\varphi=85^{\circ}\), that is, the arc of height h from the vertex shown in Equation (17), is extracted regardless of the size of the workpiece diameter, an error occurs in the X-axis coordinate. If the error angle is \(\pm \theta_{2}\)\({}^{\circ}\) and this error has a rectangular distribution, the uncertainty due to the position error around the Z-axis can be represented by the following equation.