# Rapid Measurement and Identification Method for the Geometric Errors of CNC Machine Tools
## 4 Stepwise Identification of Geometric Errors
### Translation Error Identification  
The translation errors are decoupled by the integrated error model. By using the fitting polynomial to represent each PDGE, a solvable equation, constituted from Equations (4)-(6), which contains a finite number of unknown coefficients, can be obtained. Furthermore, under the theoretical guidance of the integrated error model, those equations can easily be solved by planning the measurement path reasonably. To put it more concretely, the points on the paraboloid surface are measured when the micro-angle sensor moves along the X-axis, as shown in Figure 3. As discussed in Section 4.2, by subtracting each measured value from the nominal value, the two-dimensional integrated error (\(d_{xi}\), \(d_{yi}\)) of each point can be obtained. Then, inputting them into Equations (4)-(6), we find  
\[dx_{i}=\delta_{x}(x_{i})+\delta_{x}(y)+\delta_{x}(z)-y\cdot(\alpha_{xy}+ \varepsilon_{x}(x_{i}))+z\cdot(\alpha_{xx}+\varepsilon_{y}(y)+\varepsilon_{y} (x_{i})) \tag{31}\]  
\[dy_{i}=\delta_{y}(x_{i})+\delta_{y}(y)+\delta_{y}(z)-z\cdot(\alpha_{zy}+ \varepsilon_{x}(y_{i})+\varepsilon_{x}(x)) \tag{32}\]  
The Y and Z axes do not move, so the individual errors relative to them will not change at each sampling point. Subtracting the integrated error of any two measurement points to reduce the number of unknown coefficients, we find  
\[dx_{i}-dx_{i-1}=t_{x11}(x_{i}-x_{i-1})+t_{x22}(x_{i}^{2}-x_{i-1}{}^{2})+t_{x3 3}(x_{i}^{3}-x_{i-1}{}^{3})-y\cdot(\varepsilon_{z}(x_{i})-\varepsilon_{z}(x_{ i-1}))+z\cdot(\varepsilon_{y}(x_{i})-\varepsilon_{y}(x_{i-1})) \tag{33}\]  
\[dy_{i}-dy_{i-1}=t_{y21}(x_{i}-x_{i-1})+t_{y22}(x_{i}^{2}-x_{i-1}{}^{2})+t_{y23 }(x_{i}{}^{3}-x_{i-1}{}^{3})-z\cdot(\varepsilon_{x}(x_{i})-\varepsilon_{x}(x_ {i-1})) \tag{34}\]  
where \(t_{xx1}\), \(t_{xx2}\), and \(t_{x3}\) are the coefficients of the positioning error \(\delta_{x}(x)\) polynomial; \(t_{yx1}\), \(t_{yx2}\), and \(t_{yx3}\) are the coefficients of the straightness error \(\delta_{y}(x)\) polynomial; and \(x_{i}\), \(y\), and \(z\) are the nominal motion. What is more, the angle error, such as \(\varepsilon_{y}(x)\), has been identified in Section 4.1. Theoretically, the coefficients of \(\delta_{x}(x)\) and \(\delta_{y}(x)\) of the X-axis can be solved by using the displacement information at 4 sampling points. Then, the fitting polynomial of them can also be obtained:  
\[\delta_{x}(x)=t_{xx1}x+t_{x22}x^{2}+t_{x33}x^{3} \tag{35}\]  
\[\delta_{y}(x)=t_{yx1}x+t_{y22}x^{2}+t_{y33}x^{3} \tag{36}\]  
Therefore, the positioning error \(\delta_{x}(x)\) and straightness error \(\delta_{y}(x)\) can be identified. Similarly, when the Y-axis moves, the positioning error \(\delta_{y}(y)\) and straightness error \(\delta_{x}(y)\) can be obtained.  
As shown in Figure 8a, the X-axis drives the micro-angle sensor to scan each paraboloid surface, so the integrated error of the X-axis in the z direction can be measured, and the polynomial of the straightness error \(\delta_{z}(x)\) can be calculated. Then, the Z-axis drives the micro-angle sensor to scan each paraboloid surface above the other arm, so the integrated error of the Z-axis in the x, z direction can be measured, and the polynomial of \(\delta_{y}(z)\) and \(\delta_{z}(z)\) can be calculated.  
At last, as shown in Figure 8b, the Y-axis drives the micro-angle sensor to scan each paraboloid surface, so the integrated error of the Y-axis in the y direction can be measured, and the polynomial of \(\delta_{z}(y)\) can be calculated. Then, the Z-axis drives the micro-angle sensor to scan each paraboloid surface above the other arm, so the integrated error of the Z-axis in the y direction can be measured, and the polynomial of \(\delta_{y}(z)\) can be calculated.  
Therefore, all 21 geometric errors can be identified with the stepwise identification method through the three placements of the L-shape standard piece.