## 1 Introduction
### Influences of the Angular Misalignments of a Roll Workpiece  
In the stitching linear-scan method, the angular misalignment \(\theta_{X}\) of a small cylinder about the \(X\)-axis could affect the roundness measurement. Figure 4 shows a schematic of the influence of \(\theta_{X}\) on the measured arc profile of a small cylinder. As can be seen in the figure, the radius of the observed arc becomes larger than the real one. Denoting the diameter of an ideal small cylinder as \(D\), the \(X\)- and \(Z\)-coordinates \((x_{i},z_{i})\) of the \(i\)th sampled point satisfy the following equation:  
\[x_{i}^{2}+z_{i}^{2}\cos^{2}\theta_{X}=D^{2}/4 \tag{10}\]  
Denoting the \(i\)th sampled point in the polar coordinate system as \((r_{i}\theta_{i})\), the above equation can be modified as follows:  
\[r_{i}(\theta_{i})=\frac{D}{2}\Big{(}\cos^{2}\theta_{i}+\sin^{2}\theta_{i}\cos ^{2}\theta_{X}\Big{)}^{-\frac{1}{2}} \tag{11}\]  
The influence of the angular misalignment of the small cylinder about the \(X\)-axis \(e_{OX}\) can thus be obtained by the following equation:  
\[e_{OX}(\theta_{i})=\frac{D}{2}\bigg{[}1-\Big{(}\cos^{2}\theta_{i}+\sin^{2} \theta_{i}\cos^{2}\theta_{X}\Big{)}^{-\frac{1}{2}}\bigg{]} \tag{12}\]  
Figure 5a shows the variation of \(e_{OX}\) over an angular range from \(\theta=42.5^{\circ}\) to \(\theta=137.5^{\circ}\) under the condition of \(D=1\) mm and \(\theta_{X}=1^{\circ}\). The error becomes maximum at the top of the arc (namely, \(\theta=90^{\circ}\)). Numerical calculations are extended to further investigate the influence of \(\theta_{X}\). Figure 5b shows the variation of the maximum \(e_{OX}\) as the increase of \(\theta_{X}\) under the condition of \(\theta_{i}=42.5^{\circ}\). Calculations are carried out for the cases with different workpiece diameter \(D\) ranging from 1 mm to 6 mm. As can be seen in the figure, the influence of \(\theta_{X}\) becomes larger with the increase of \(D\). Meanwhile, the calculation results show that \(e_{OX}\) can be suppressed to be less than 0.01 \(\upmu\)m by reducing \(\theta_{X}\) to less than 0.3\({}^{\circ}\) for a small cylinder with a diameter \(D\) of less than 3 mm, for which it is difficult to carry out roundness measurement by the conventional rotary-scan method. Figure 4: A schematic of the influence of angular misalignment \(\theta_{X}\) on the arc-profile to be obtained in measurement: (**a**) definition of \(\theta_{X}\); (**b**) error caused by \(\theta_{X}\). The angular misalignment \(\theta_{Z}\) of a small cylinder about the \(Z\)-axis could also affect the roundness measurement. Figure 6 shows a schematic of the influence of \(\theta_{Z}\) on the measured profile of a small cylinder. As can be seen in the figure, the radius of the observed arc becomes larger than the real one. The influence of the angular misalignment of the small cylinder about the \(Z\)-axis \(e_{\theta Z}\) can be expressed by the following equation:  
\[e_{\theta z}(\theta_{i})=\frac{D}{2}\bigg{[}1-\left(\cos^{2}\theta_{i}\cos^{2 }\theta_{z}+\sin^{2}\theta_{i}\right)^{-\frac{1}{2}}\bigg{]} \tag{13}\]  
In the same manner as \(\theta_{X}\), the influence of \(\theta_{Z}\) is also investigated through numerical calculations. Figure 7a shows the variation of \(e_{\theta Z}\) under the condition of \(D=1\) mm and \(\theta_{Z}=1^{\circ}\). The error becomes maximum at the edges of the arc (namely, \(\theta_{i}=42.5^{\circ}\) and \(137.5^{\circ}\)). Figure 7b shows the variation of the maximum \(e_{\theta Z}\) as the increase of \(\theta_{Z}\) under the condition of \(\theta_{i}=42.5^{\circ}\). According to the calculation results, \(e_{\theta Z}\) can be suppressed to be less than \(0.01~{}\upmu\)m by reducing \(\theta_{Z}\) to less than \(0.2^{\circ}\) for a small cylinder with a diameter \(D\) of less than \(3\) mm. Figure 5: Radial error induced by the angular misalignment \(\theta_{X}\) of a cylindrical workpiece: (**a**) radial error at each \(\theta_{i}\); (**b**) variation of the maximum radial error due to the increase of \(\theta_{X}\).