## 4 Measurement Experiments
### Data Processing and Analysis  
First of all, using the interpolation algorithm, the motion errors of the \(Z\)-slide and the \(Y\)-slide were applied to compensate for the measurement results. As described in Section 3.1, the measurement of the lateral gauge could not represent the inner profile because of the influence of the shape of the probing sphere of the gauge probe. The trajectory of the center of the probing sphere of the lateral gauge could be obtained by the following equations:  
\[{y_{ball}}^{i}={y_{slide}}^{i}+{S_{lateral}}^{i} \tag{34}\]  
\[{z_{ball}}^{i}={z_{slide}}^{i}+{h_{lateral}}^{i} \tag{35}\]  
\[{h_{lateral}}^{i}=f({S_{lateral}}^{i}) \tag{36}\]  
where \({y_{ball}}^{i}\) and \({z_{ball}}^{i}\) are the \(y\) position and the \(z\) position of the center of the lateral gauge, respectively, \({y_{slide}}^{i}\) and \({z_{slide}}^{i}\) are the position of the \(Y\)-slide and the position of the \(Z\)-slide, respectively, \({S_{lateral}}^{i}\) is the output of the lateral gauge and \({h_{lateral}}^{i}\) is the corresponding vertical displacement of the probing sphere obtained according to Figure 5b. Then, an algorithm was applied to evaluate the real sidewall of the inner profile of the workpiece by removing the influence of the probing sphere shape. The evaluation results are shown in Figure 17a. According to the geometric relationship, we could obtain the coordinates of the sidewall profile of the inner part of the workpiece from the trajectory of the center of the probing sphere:  
\[{y_{Ai}}^{i}={y_{Ai}}-{r_{Ai}}\cdot\cos({\theta_{Ai}}) \tag{37}\]  
\[{z_{Ai}}^{i}={z_{Ai}}-{r_{Ai}}\cdot\sin({\theta_{Ai}}) \tag{38}\]  
\[{\theta_{Ai}}=\arctan({k_{Ai}}) \tag{39}\]  
where \({y_{Ai}}^{i}\) and \({z_{Ai}}^{i}\) are the \(y\) and \(z\) coordinates of the side of the inner profile, respectively, \({y_{Ai}}\) and \({z_{Ai}}\) are the \(y\) and \(z\) coordinates of the center of the probing sphere of the lateral gauge, respectively, \({r_{Ai}}\) is the radius of the probing sphere of the gauge, \({k_{Ai}}\) is the slope of the trajectory of the center of the probing sphere at the measuring point, and \({\theta_{Ai}}\) is the inclination of the slope of the trajectory of the probing sphere. The results of the compensation algorithm of the sidewall of the inner profile are presented in Figure 17b. Similarly, the trajectory of the probing sphere of the axial gauge could be obtained from the coordinates of the slides and the output of the axial gauge. Then, the underside part of the inner profile was calculated using a similar compensation algorithm based on the trajectory of the probing sphere of the axial gauge and the roundness error obtained from Figure 14b. The results of the compensation algorithm of the underside of the inner profile are presented in Figure 17d. Finally, we applied a point cloud registration algorithm [24] to stitch the sidewall measured by the lateral gauge and the bottom part measured by the axial gauge to form an entire inner profile of the high aspect ratio aspheric workpiece, as shown in Figure 18. The overlap of the profiles measured by the lateral gauge and the axial gauge was determined through the moving range of the \(Y\)-slide, as shown in Figure 18a. Then the algorithm was applied to find the closest point pairs between the two overlap areas, and a transformation matrix was calculated based on the coordinates of the closest point pairs. The overlap area measured by the lateral gauge was then transformed through the calculated transformation matrix, and the average distance of the transformed closest point pairs was obtained. The procedure was repeated several times until the average distance was within a threshold of 100 nm, and the final transformation matrix was marked. Finally, the profile measured by the lateral gauge was transformed through the marked transformation matrix, and the entire inner profile was stitched.