## 4 Measurement Experiments
### Experimental Results  
The inner profile of the high aspect ratio aspheric workpiece was evaluated according to the measurement procedure described in Section 2 using the lateral displacement gauge and the axial displacement gauge. Figure 11 shows a schematic of a flowchart of the entire measurement procedure. As the first step, the nonlinearity of the lateral gauge was evaluated and calibrated according to the approach shown in Figure 5. As shown in Figure 12, through nonlinear fitting by the polynomial, the nonlinearity of the horizontal measurement before and after compensation was about \(\pm\)293 and \(\pm\)84 nm, respectively. It should be noted that only a third-order polynomial fit method is used to represent the real trend of the nonlinearity of the lateral gauge to prevent overfitting. In addition, with  
Figure 10: Photograph of the experimental setup. compensation, the crosstalk output of the lateral gauge is about \(\pm\)215 nm, corresponding to the Z-axial movement of 0.2 mm. Then, with the assistance of the reference sphere, tilt adjustments of the two gauges were carried out to reduce the cosine error. The results are presented in Figure 13 and the pitch angle and the roll angle of the gauges were reduced from 3\({}^{\circ}\) to about 0.2\({}^{\circ}\). The repeatability of the tilt adjustment method was about 0.1\({}^{\circ}\). The repeatability was caused by the motion errors of the slides, the roundness of the reference sphere, the errors of the gauges, and the roundness of the probing spheres. The roundness of 10 \(\upmu\)m of the probing sphere, according to Figure 4, led to a deviation of about 0.08\({}^{\circ}\) for the tilt angle and had little effect on the accuracy of the tilt adjustment method. After that, the roundness of the probing spheres of the lateral gauge and the axial gauge were evaluated according to the method shown in Figure 7. Figure 14a was the evaluated result of the left part of the probing sphere of the lateral gauge, and Figure 14b was that of the bottom part of the probing sphere of the axial gauge. The evaluating angle was about \(\pm\)60\({}^{\circ}\), which can fully meet the requirement of the measurement, and the repeatability of the evaluation method of the roundness of the probing spheres was about 80 nm. As the next step, motion errors of the \(Y\)-slide and \(Z\)-slide of the motion platform were evaluated according to the approach shown in Figure 9. Figure 15 shows the identification results of the slide motion errors. The straightness motion error of the \(Z\)-slide and the \(Y\)-slide were evaluated to be 2.489 and 0.871 m, respectively. The evaluated results are used to compensate for the measurement results. It should also be noted that the repeatability of the \(Z\)-slide and the \(Y\)-slide was 0.0835 and 0.1920 m, respectively. The repeatability was caused by the flatness of the standard gauge block and the errors of the gauges. Figure 11: Schematic of the measurement procedure for inner profile surface of the high aspect ratio aspheric workpiece. Figure 14: Experimental results of probing sphere roundness calibration for the lateral and axial displacement gauges. (**a**) Sphere roundness calibration of the lateral displacement gauge (probe A); (**b**) sphere roundness calibration of the axial displacement gauge (probe B). Figure 12: Experimental results of calibration of the lateral displacement gauge (probe A). (**a**) Horizontal calibration of the lateral displacement gauge (probe A); (**b**) vertical calibration of the lateral displacement gauge (probe A). Figure 13: Experimental results of the tilt alignments of the lateral and axial displacement gauges. (**a**) Tilt alignment of the lateral displacement gauge (probe A); (**b**) tilt alignment of the axial displacement gauge (probe B). As the final step of error calibration, the tilt of the high aspect ratio aspheric workpiece was adjusted. Figure 16 shows the results of the adjustment procedure.