## 3 Calibration of the Measurement System
### Calibration of the Measuring Probes
According to Figure 6a, in the right triangle of \(O_{An}T_{A}P_{A}\), we have the following geometric relationships:  
\[O_{An}P_{A}=S_{An}-S_{A0} \tag{7}\]  
\[O_{An}T_{A}=O_{An}P_{A}\cdot\cos\alpha_{A}=(S_{An}-S_{A0})\cdot\cos\alpha_{A} \tag{8}\]  
\[P_{A}T_{A}=O_{An}P_{A}\cdot\sin\alpha_{A}=(S_{An}-S_{A0})\cdot\sin\alpha_{A} \tag{9}\]  
In the right triangle of \(O_{A}Q_{A}O_{An}\), we have the following geometric relationships:  
\[O_{A}O_{An}=R+r_{A} \tag{10}\]  
Figure 4:0  
Figure 4:1  
Figure 4:2  
Figure 4:3  
Then, according to Equation (14), the roll angle \(\alpha_{A}\) of the lateral displacement gauge in the \(YZ\)-plane can be evaluated. Similarly, the tilt alignment of the axial displacement gauge with a roll angle in the \(YZ\)-plane of \(\alpha_{B}\) is carried out as shown in Figure 6b, and we can formulate a similar right triangle of \(O_{B}Q_{B}O_{Bn}\), which follows the relationships:  
Figure 4:4  
Figure 4:5  
Figure 4:6  
Figure 4:7  
Figure 4:8  
Figure 4:9 Schematic of tilt alignment of the measuring probes. (**a**) Schematic of the lateral probe (probe A) tilt alignment; (**b**) schematic of the axial probe (probe B) tilt alignment. According to Equation (19), the roll angle \(\alpha_{B}\) of the axial displacement gauge in the \(Y\)Z-plane can be evaluated. Similarly, the pitch angle of the measuring probe can be evaluated by changing the scanning direction to the \(X\)-axis. The roll and pitch angles can thus be corrected by adjusting the corresponding manual tilt stage, which is mounted behind the probe fixture. In addition, since the inner profile is the envelope of the trajectory of the center of the probing sphere, the inner profile can be calculated based on the trajectory of the center of the probing sphere and the sphere radius. The probing sphere roundness error of the two gauges would affect the calculation accuracy and reduce the accuracy of the measurement profile. It is necessary to evaluate and compensate for the roundness error of the probing sphere of the gauge probes. The aligned measuring probes are shown in Figure 7, where the lateral gauge measurement is mounted vertically to measure the exact horizontal displacement, and the axial gauge is mounted vertically to measure the exact vertical displacement. To evaluate the probing sphere roundness of the lateral gauge, the lateral gauge is moved along the \(Z\) direction, as shown in Figure 7a, which is similar to that of Figure 6a. \(Z_{A0}\) is the start position of the lateral gauge, and \(Z_{Ai}\) is an intermediate calibration position. At the position of \(Z_{Ai}\), in the right triangle of \(O_{A}Q_{Ai}O_{Ai}\), we have the following geometric relationships:  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]0  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]1  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]2  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]3  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]4  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]5  
\[e_{i}=(S_{i+1}-S_{i})-(Y_{i+1}-Y_{i}) \tag{5}\]6 Schematic of sphere roundness calibration of the measuring probes. (**a**) Schematic of measuring sphere roundness calibration for lateral probe (probe A); (**b**) schematic of measuring sphere roundness calibration for axial probe (probe B). According to Equations (24) and (25), the exact radius \(r_{A0}\) could be measured by a coordinate measurement machine, and the radius \(r_{Ai}\) of the probing sphere of the lateral gauge at an angular range \(\theta_{Ai}\) of \(-60^{\circ}\leq\theta_{Ai}\leq-60^{\circ}\) due to the limitation of its measuring range can be evaluated.