## 4 Measurement Experiments  
### Experimental Setup  
A photograph of the experimental setup is shown in Figure 10. A lateral displacement gauge (Keyence AT2-52) and an axial displacement gauge (Heidenhain CT6001) were applied as Probe A and Probe B, respectively. Both gauges are contact type, and the measurement principle is based on photoelectric scanning. The two probes were mounted on the \(Z\)-slide of the four-axis motion platform fixed by a designed gauge fixture. The lateral displacement gauge has a resolution of 73 nm and linearity of about 0.5% over a measurement range of 1.2 mm, according to its specification sheet. The probing sphere of the gauge has a nominal radius of 1.6 mm. The axial displacement gauge has a resolution of 100 nm over a measurement range of 60 mm. The probing sphere of the gauge has a nominal radius of 4.5 mm. The measuring points obtained from the gauges were collected by using the NI PCIe-6353 capture card with a developed computer program based on the Qt platform. To adjust the tilt of both the displacement gauges, an \(\mathit{XZ}\)-tilt adjustment stage, which has a resolution of 0.01\({}^{\circ}\) with an adjustment range of \(\pm\)10\({}^{\circ}\), was fixed between the measuring gauge and the \(Z\)-slide of the motion platform. To calibrate the tilt errors of the measuring gauges, a reference sphere (Zeiss, A-1034-0002) with a nominal radius of 12.5 mm and a nominal roundness of 25 nm was applied. The spindle of the scanning stage on which the workpiece was mounted has a resolution of 0.5 arc-second and a maximum rotation speed of 200 rpm. The spindle was directly driven by using the PC connected to the scanning stage. The air bearing technique is integrated with the scanning stage to reduce the influence of the vibration. As the measurement target, a high aspect ratio aspheric workpiece, which is a cone-shaped structure with a thickness of 10 mm, was employed. The sidewall of the inner profile of the workpiece is part of a conical surface with a maximum radius of 40 mm and steepness of 84.89\({}^{\circ}\). The bottom part of the workpiece  
Figure 9: Schematic of slide motion error calibration. (**a**) Preparation for \(Z\)-slide motion error calibration; (**b**) \(Z\)-slide motion error calibration; (**c**) preparation for \(Y\)-slide motion error calibration; (**d**) \(Y\)-slide motion error calibration.  
is part of a spherical surface with a radius of 30 mm. The total depth of the inner profile of the workpiece is 140 mm. Since the machining process of the workpiece is rotationally symmetrical, one side of a cross-section of the workpiece could represent the geometric shape of the workpiece.  
The experiment target is to measure one side of a cross-section of the workpiece to verify this measurement method. Taking into consideration the machining accuracy is desired to be about 10 \(\upmu\)m, the expected measurement accuracy is 1 \(\upmu\)m. It should be noted that the minimum measurable radius of the workpiece is 33.3 mm when the depth of the workpiece is larger than 83 mm, due to the limitation of the body size of the axial gauge of Heidenhain CT6001, and is 4.5 mm when the depth of the workpiece is less than 83 mm. An _XY_-tilt stage and an _XY_-linear stage were fixed under the workpiece fixture for tilt adjustment of the workpiece. To identify the motion errors of the _Y_-slide and the _Z_-slide, a standard gauge block with a flatness of 30 mm was applied. The travel range of the _XYZ_-slides is 200 mm along with the three directions with a resolution of 20 nm on each axis.