# Rapid Measurement and Identification Method for the Geometric Errors of CNC Machine Tools
## 3 Measurement Principles
### Measurement Principle of the Micro-Angle and Displacement
#### 3.1.1 Measurement Principle of the Micro-Angle  
The XOZ section of the measurement system is taken as an example to illustrate the measurement process of the pitch error \(\varepsilon_{y}(x)\), as shown in Figure 4.  
At the initial time, the angle between the beam and the axis of the composite module in the x direction is \(\alpha_{x0}\). The incident beam is emitted to the plane surface, then the reflected beam forms a spot \(A_{0}\) on the CCD through the convergence of the imaging lenses. As shown in Figure 4a, the position of the image spot does not change when there is a displacement in the x direction between the micro-angle sensor and the composite module. However, when there is a pitch \(\varepsilon_{y}(x)\), as shown in Figure 4b, the angle between the beam and the axis of the composite module in the x direction changes  
Figure 4: XOZ section of the measurement system, when detected at the plane surface: (**a**) The micro-angle sensor has a displacement in the X direction; (**b**) The micro-angle sensor has a pitch error \(\varepsilon_{y}(x)\).  
Figure 3: Composition of the measurement system.  
to \(\alpha_{x1}\). Moreover, the imaging point of the reflective beam on the CCD moves from point \(A_{0}\) to \(B_{0}\). Equation (8) can easily be derived:  
\[\alpha_{x0}=\alpha_{x1}+\varepsilon_{y}(x). \tag{8}\]  
Equation (9) can be derived from the principle of laser collimation [24]:  
\[\tan(2\alpha_{x1})-\tan(2\alpha_{x0})=\frac{S_{1x}-S_{0x}}{f} \tag{9}\]  
where \(f\) is the focal length of the imaging lens, and its value is 100 mm. \(S_{0x}\), \(S_{1x}\) is the distance between the center of the spot \(A_{0}\), \(B_{0}\) and the optical axis in the \(x\) direction. From Equations (8) and (9), the tangent value is replaced by the angle value due to the measurement of the micro-angle, then the pitch error \(\varepsilon_{y}(x)\) can be expressed as  
\[\varepsilon_{y}(x)=(s_{1x}-s_{0x})/2f \tag{10}\]  
The X-axis drives the micro-angle sensor to scan the plane surface, so that the pitch error \(\varepsilon_{y}(x_{i})\) of each sampling point, relative to the initial one, can be collected. The least-squares method is used to fit the mean curve of each discrete datum, and the first three terms are selected for the identification study. Then, the pitch error at each point of the composite module surface can be obtained from the following polynomial:  
\[\varepsilon_{y}(x)=r_{yx1}\times x+r_{yx2}\times x^{2}+r_{yx3}\times x^{3} \lim_{x\to\infty} \tag{11}\]  
where \(r_{yx1}\), \(r_{yx2}\), and \(r_{yx3}\) are the coefficients of the pitch error polynomial, and x is the nominal motion of the X-axis, which can be obtained from the feedback value of the MT. Considering the structure of the CMM in Figure 2a, a roll error of the X-axis will cause a yaw between the micro-angle sensor and the composite module. Therefore, the roll error \(\varepsilon_{x}(x)\) can be obtained in a similar way by analyzing the Y0Z section of the measurement system. Thus, the pitch and roll error of the X-axis can be separated and measured.  
Since it is difficult to know the intersection \(\mathcal{O}\) of the optical axis and CCD, in the actual measurement process, the spot position on the CCD of the first measurement point is taken as the reference point. By detecting the projection distance between the imaging spots of other measurement points and the reference spot in the \(x\) and \(y\) direction, respectively, the two-dimensional micro-angle variation of each sampling point, relative to the initial measurement point, can be obtained through Equation (10).