## 3 Measuring Process and Results
### Measuring the System Construction
Data fitting could obtain another extreme value, and the extreme Y-coordinate value was the Y-coordinate of the center of the standard sphere. The position of two intersections was the vertex of the standard sphere, marking the position of the probe shaft at this time. The standard ball was replaced by the DOE of the measured surface base, and the workpiece alignment was also performed by the inductor. The adjustment center coaxiality accuracy of the rotating workpiece on the air-floating rotary table was within 500 nm, which could ensure the center alignment accuracy of the DOE after installation and realize the calibration of the optical axis of the probe and the vertex of the workpiece. Figure 10: Error analysis of the measurement path. A standard ball with a radius of 12.7 mm was adsorbed on the vacuum chuck for adjusting the axis of rotation of the spindle, the optical axis of the probe, and the Z direction of the center vertex of the standard ball, which were coaxial. The standard sphere had a high enough geometric accuracy and was also used to verify the measurement accuracy of the probe. In order to keep the measurement stable during the alignment process, the speed of the machine tool was set to 0.1 mm/s and the sampling frequency was set to 1000 points per second. The fitting results of the two sets of data are shown in Figure 12a,b. Fitting results show that the position error of data center point was small, and the alignment accuracy could be better than 1 \(\upmu\)m after adjustment. The machining accuracy of DOE was on the order of micrometers. In order to achieve a complete and accurate measurement of the workpiece, it was also necessary to calibrate the systematic error of the measurement system. After the alignment of the optical axis of the probe with the vertex of the standard sphere, the probe measured the standard ball at a constant scanning speed along the standard ball profile to verify the measurement accuracy of the probe. Because the standard sphere profile was measured, the height directly measured by the probe was the measurement error of the system, so as to evaluate the measurement accuracy of the measurement system. The error data obtained after processing are shown in Figure 12c, and obvious gross errors have been eliminated. After removing them, the error was controlled within 100 nm, and the PV (Maximum peak-to-valley value of surface topography) value is 0.117 \(\upmu\)m, which showed that the measurement accuracy of the measurement system was guaranteed. The DOEs were replaced by the standard sphere, and then the measurement experiment of the curve-based DOE was carried out. The workpiece alignment was completed by the inductance meter of the machine tool. We adjusted the center coaxial accuracy of the measured DOE and the spindle to less than 500 nm, which would ensure the center alignment accuracy of the DOE adsorbed by the spindle. The planned measurement path was scanned at a constant speed along the DOE substrate contour through the DOE vertex coordinate O point, at an equal angle of 5\({}^{\circ}\). The probe acquisition frequency was 400 Hz, the scanning speed was 0.5 mm/s along the contour, and three sets of measured data were recorded. According to the data processing method introduced in Section 3, the probe did not need to read the machine axis coordinate from the machine control system in real time, it only needed to record the position coordinates of the start and end points of the measured motion path. After the measurement was completed, the data processing of one set of the measured data and the three-dimensional reconstruction of the DOE surface were carried out based on the proposed data processing algorithm. The two remaining sets of measured data were used as validation to verify the feasibility, accuracy, and repeatability of the proposed data processing algorithm.