## 4 Measurement Experiments
### Data Processing and Analysis
The stitched inner profile is presented in Figure 18e, and the mean stitching error was evaluated to be 87 nm according to the final average distance of the closest point pairs, meeting the requirement of micrometer measurement accuracy. For a clearer view of the measured inner profile, a three-dimensional inner profile error is represented by rotating the stitching profile around the central axis of the high aspect ratio aspheric workpiece, as shown in Figure 18f. The results of the compensation algorithm of the underside of the inner profile are presented in Figure 18g. The results of the compensation algorithm of the underside of the inner profile are presented in Figure 18g. Finally, the compensation algorithm of the underside of the inner profile is presented in Figure 18g. Figure 19a,b. The machining error has an RMS (root-mean-square) value of 25.126 \(\upmu\)m. The error distribution is mainly considered to be caused by the manufacturing error of the machine tool. The machining error of the upper part is relatively small while it becomes larger when the machine tool processes the bottom part, which might be influenced by the stiffness of the machine tool. The radius of the bottom sphere is evaluated as 29.5923 mm, and the roundness is evaluated as 0.4808 mm. The steepness of the sidewall is evaluated as 84.9671\({}^{\circ}\), and the straightness is evaluated as 0.2794 mm. It should be noted that the above results are calculated based on the overall inner profile after stitching. Furthermore, the results obtained by the measurement system were verified by comparing with the measurement results obtained by using a coordinate measurement machine (CMM) (Hexagon Bridge-explorer 060806), which has a qualified accuracy of 2.9 \(\upmu\)m with a standard test. Since the employed CMM could not measure the bottom part of the inner profile of the workpiece due to its relatively large probe, only the result of the upper part of the inner profile was compared. The difference between the profile measured by the proposed measurement system and that of CMM is presented in Figure 20b. The peak-to-valley value of the difference was evaluated to be about \(\pm\)1.350 \(\upmu\)m. The comparison result confirms the reliability of the proposed method for the evaluation of the inner profile of the high aspect ratio aspheric workpiece. Figure 17: Schematic and experimental results of evaluating the inner profile of the high aspect ratio aspheric workpiece. (**a**) Schematic of evaluating the inner sidewall profile; (**b**) experimental results of evaluating the inner sidewall profile; (**c**) schematic of evaluating the inner underside profile; (**d**) experimental results of evaluating the inner underside profile. Figure 19: Three-dimensional machining error of the inner profile of the high aspect ratio aspheric workpiece. (**a**) Three-dimensional machining error of the high aspect ratio aspheric workpiece compared with the theory profile; (**b**) top view of three-dimensional machining error of the high aspect ratio aspheric workpiece. Figure 18: Schematic and experimental results of the point cloud registration algorithm for inner profile stitching. (**a**) Schematic of obtaining an overlap measurement profile of probe A and probe B, (**b**) schematic of finding the closest point pairs of the point cloud registration algorithm; (**c**) schematic of coordinate transformation of the point cloud registration algorithm; (**d**) schematic of coordinate transformation for Measurement profile A; (**e**) experimental results of the point cloud registration algorithm for inner profile stitching.