## 3 Measurement Uncertainty Analysis of the Linear Scan Method  
### Mathematical Modeling  
The uncertainty of roundness and diameter measurements of a small cylinder with a diameter of 1.5 mm and length of 7.8 mm by the stitching linear scan method is carried out at first. Mathematical modelling was conducted to calculate the measurement uncertainty of the diameter and roundness. The arc coordinate (x\({}_{i}\), z\({}_{i}\)) in the rectangular coordinate system can be obtained by a profilometer with its stylus. The mathematical model used for radius calculation and the uncertainty factors deriving from the measurements are shown in Figure 3a. The radius of each arc can be fitted by the least square method. Radius R\({}_{i}\) of an arbitrary measuring point can be expressed by Equation (1), where (x\({}_{i}\), z\({}_{i}\)) is the arc coordinate. \[R_{i}=\sqrt{{x_{i}}^{2}+{z_{i}}^{2}} \tag{1}\]  
As derived from Equation (1), the combined standard uncertainty \(u({\rm R_{i}})\) of the measured arc radius can be expressed as follows:  
\[u^{2}(R_{i})=\left(\frac{\partial R_{i}}{\partial x_{i}}\right)u^{2}(x_{i})+ \left(\frac{\partial R_{i}}{\partial z_{i}}\right)u^{2}(z_{i})=\frac{{x_{i}}^{ 2}}{{z_{i}}^{2}+{x_{i}}^{2}}u^{2}(x_{i})+\frac{{z_{i}}^{2}}{{z_{i}}^{2}+{x_{i }}^{2}}u^{2}(z_{i}) \tag{2}\]  
Figure 2: The stitching procedure: (**a**) profile 1; (**b**) profile 2; (**c**) the stitching of profiles 1 and 2; (**d**) the stitching of the rest of the arc profiles. The uncertainties in the figure are as follows. 1. Uncertainty of output \(\mathrm{z_{i}}\) in the Z-axis direction. \(u(e_{calibration\_Z})\): Uncertainty of stylus calibration in the Z-axis direction. \(u(e_{resolution})\): Uncertainty due to stylus resolution. \(u(e_{repeat})\): Uncertainty of repeatability. 2. Uncertainty of output \(\mathrm{x_{i}}\) in the X-axis direction \(u(e_{calibration\_X})\): Uncertainty of stylus calibration in the X-axis direction. \(u(e_{alignment\_Z})\): Uncertainty due to position error around the Z-axis of the workpiece. \(u(e_{alignment\_X})\): Uncertainty due to position error around the X-axis of the workpiece. In summary, the combined standard uncertainty \(\mathrm{z_{i}}\) of the Z-axis output and the combined standard uncertainty of the X-axis coordinate \(x_{i}\) are expressed by Equations (3) and (4), respectively. \[u^{2}(z_{i})=u^{2}\big{(}e_{calibration\_z}\big{)}+u^{2}(e_{resolution})+u^{2} \big{(}e_{repeat}\big{)} \tag{3}\]  
\[u^{2}(x_{i})=u^{2}(e_{calibration\_X})+u^{2}\big{(}e_{alignment\_Z}\big{)}+u^{2} \big{(}e_{alignment\_X}\big{)} \tag{4}\]  
For the calculation of roundness, it is assumed that there is a deviation from the approximate center of the circle, rather than the radian of a perfect circle, and the approximate center of the circle is set as the origin. The output of \(\mathrm{z_{i}}\) can be obtained by Equation (5), where \(\mathrm{R_{i}}\) is the radius of the arbitrary point of the measured arc and \(\mathrm{R_{0}}\) is the peak radius. \[z_{i}=R_{0}-\sqrt{{R_{i}}^{2}-{x_{i}}^{2}} \tag{5}\]  
The radius \(\mathrm{R_{i}}\) after the stitching process is calculated according to Equation (6). \[u(\Delta zq)=u(\Delta r(\theta)_{max})+u(\Delta r(\theta)_{min}) \tag{6}\]  
\(\Delta r_{i}\) is the radial deviation at any measured point and \(\Delta r_{0}\) is the radial deviation at the arc apex. \[R_{i}=\overline{R}+\Delta r_{i}(\overline{R}\gg\Delta r_{i}) \tag{7}\]  
\[R_{0}=\overline{R}+\Delta r_{0}(\overline{R}\gg\Delta r_{0}) \tag{8}\]  
Figure 3: (**a**) Schematic of the radius measurement and the uncertainty components; (**b**) schematic of the roundness measurement and the uncertainty components. We square both sides of Equation (5), substitute this into Equations (7) and (8), and rearrange.