# Rapid Measurement and Identification Method for the Geometric Errors of CNC Machine Tools
## 3 Measurement Principles
### Measurement Principle of the Micro-Angle and Displacement
#### 3.1.2 Measurement Principle of Micro-Displacement  
Based on the principle of angle measurement, the measurement principle of the displacement is illustrated by the _XOZ_ section of the proposed system, as shown in Figure 5.  
Figure 5: XOZ section of the measurement system, when detected at the rotating paraboloid surface.  
The incident beam is emitted to the paraboloid surface, then the reflected beam forms a spot \(C_{0}\) on the CCD. When the incident beam moves from point C to D, the imaging point of the reflective beam on the CCD moves from point \(C_{0}\) to \(D_{0}\). Equation (12) can be derived from the principle of laser collimation.  
\[\tan(2\gamma_{x1})-\tan(2\gamma_{x0})=\frac{d_{1x}-d_{0x}}{f} \tag{12}\]  
where \(d_{0x}\) is the distance between the center of the spot \(C_{0}\) and the optical axis in the \(x\) direction, and the distance between the center of the spot \(D_{0}\) and the optical axis in the \(x\) direction is \(d_{1x}\).  
The surface equation of the rotating paraboloid is  
\[\frac{x^{2}}{k^{2}}+\frac{y^{2}}{k^{2}}=2z \tag{13}\]  
where \(k^{2}\) is the characteristic parameter of the rotating paraboloid, and its value is 0.057; and \(x\), \(y\) and \(z\) represent the three-dimensional coordinates of the measurement point. Then, the tangent slope in the x direction of sampling point C and D can be obtained as follows:  
\[\tan(\beta_{x0})=\frac{\partial z}{\partial x}=\frac{x_{0}}{k^{2}} \tag{14}\]  
\[\tan(\beta_{x1})=\frac{\partial z}{\partial x}=\frac{x_{1}}{k^{2}} \tag{15}\]  
where \(x_{0}\) and \(x_{1}\) are the distance between the sampling point C and D and the center point of the rotating paraboloid. Then, the distance between the sampling points C and D is \(l_{x}\), which can be expressed as  
\[l_{x}=k^{2}\times(\tan(\beta_{x1})-\tan(\beta_{x0})). \tag{16}\]  
Equation (17) can be derived easily, in accordance with Figure 5:  
\[(\beta_{x1}-\beta_{x0})=\left(\gamma_{x1}-\varepsilon_{y}(x_{1})\right)-\left( \gamma_{x0}-\varepsilon_{y}(x_{0})\right)=\left(\gamma_{x1}-\gamma_{x0}\right) -\left(\varepsilon_{y}(x_{1})-\varepsilon_{y}(x_{0})\right) \tag{17}\]  
where \(\varepsilon_{y}(x_{1})\) and \(\varepsilon_{y}(x_{0})\) are the pitch error at the sampling point C and D of the micro-angle sensor, which can be obtained from Equation (11).  
From Equations (12), (16), and (17), and the tangent value can be replaced by the angle value for the micro-angle, then the displacement \(l_{x}\) can be expressed as  
\[l_{x}=k^{2}\times(\frac{d_{1x}-d_{0x}}{2f}-\left(\varepsilon_{y}(x_{1})- \varepsilon_{y}(x_{0})\right)) \tag{18}\]  
The displacement in the \(y\) direction can be obtained in a similar way by analyzing the Y0Z section of the measurement system. Similar to angle measurement, the two-dimensional displacement variation of each sampling point, relative to the initial one, can be obtained through Equation (18).  
In this paper, a CCD detector, with a pixel size of 2.2 \(\mu\)m \(\times\) 2.2 \(\mu\)m was selected, and 3 times the image segmentation was realized by the image segmentation algorithm. Thus, the resolution \(\delta_{m}\). of the position detection for the imaging spot is 0.73 \(\mu\)m. The convergent lens was used to shrink the laser beam. From the principle of laser collimation, the angle resolution of the micro-angle sensor can be calculated as  
\[\delta_{k}=\frac{\delta_{m}}{2f}=0.76^{\prime\prime} \tag{19}\]  
Then, the displacement resolution can be calculated as  
\[\delta_{s}=k^{2}\times\tan\frac{\delta_{m}}{2f}=0.19\ \mu \tag{20}\]The processing quality of the rotating paraboloid surface is detected by a 3D Surface Measuring system (LuphoScan), as shown in Figure 6. The result shows that the RMS is 0.021 \(\upmu\)m. What is more, the PV is 0.148, which indicates that the slope deviation of each point on the rotating paraboloid is within 0.148, compared with the design value. According to Equation (14), its contribution to the measurement accuracy is within 0.008 \(\upmu\)m, which can be ignored.