# Rapid Measurement and Identification Method for the Geometric Errors of CNC Machine Tools
## 4 Stepwise Identification of Geometric Errors
### Angle Error Identification  
As shown in Figure 3, the two arms of the standard piece are placed along the X, Y-axis of the three-axis MT, and the micro-angle sensor is placed along the Z-axis. The X-axis drives the micro-angle sensor to scan each plane surface to measure the roll error \(\varepsilon_{x}(x_{i})\) and pitch error \(\varepsilon_{y}(x_{i})\), fitting the mean curve for each discrete datum. Then, the roll and pitch error in the whole measurement range can be obtained from the following polynomial:  
\[\varepsilon_{x}(x)=r_{xx1}\times x+r_{xx2}\times x^{2}+r_{xx3}\times x^{3} \tag{25}\]  
\[\varepsilon_{y}(x)=r_{yx1}\times x+r_{yx2}\times x^{2}+r_{yx3}\times x^{3} \tag{26}\]  
where \(r_{xx1}\), \(r_{xx2}\), and \(r_{xx3}\) are coefficients of the roll error \(\varepsilon_{x}(x)\) polynomial; and \(r_{yx1}\), \(r_{yx2}\), and \(r_{yx3}\) are the coefficients of the pitch error \(\varepsilon_{y}(x)\) polynomial.  
Next, the Y-axis drives the micro-angle sensor to scan each plane surface above the other arm of the L-shape standard piece. Considering the structure of the CMM in Figure 2a, this will cause a pitch and yaw between the micro-angle sensor and the composite module when there is a pitch error \(\varepsilon_{x}(y)\) and roll error \(\varepsilon_{y}(y)\) of the Y-axis, respectively. Thus, \(\varepsilon_{x}(y)\) and \(\varepsilon_{y}(y)\) can be measured and their polynomial can be fitted:  
\[\varepsilon_{x}(y)=r_{xy1}\times y+r_{xy2}\times y^{2}+r_{xy3}\times y^{3} \tag{27}\]  
\[\varepsilon_{y}(y)=r_{yy1}\times y+r_{yy2}\times y^{2}+r_{yy3}\times y^{3} \tag{28}\]  
where \(r_{xy1}\), \(r_{xy2}\), and \(r_{xy3}\) are the coefficients of the pitch error \(\varepsilon_{x}(y)\) polynomial; and \(r_{yy1}\), \(r_{yy2}\), and \(r_{yy3}\) are the coefficients of the roll error \(\varepsilon_{y}(y)\) polynomial.  
As shown in Figure 8a, the two arms of the standard piece are placed along the X, Z-axis of the MT, and the micro-angle sensor is placed along the Y-axis. The X-axis drives the micro-angle sensor to scan each plane surface. This will cause a yaw between the micro-angle sensor and the composite module, when there is a yaw error \(\varepsilon_{z}(x)\) of the X-axis. Thus, \(\varepsilon_{z}(x)\) can be measured, and its polynomial can be fitted. Then, the Z axis drives the micro-angle sensor to scan each plane surface above the other arm. This will cause a pitch and yaw between the micro-angle sensor and the composite module, when there is a pitch error \(\varepsilon_{x}(z)\) and roll error \(\varepsilon_{z}(z)\) of Z-axis, respectively. Thus, \(\varepsilon_{x}(z)\) and \(\varepsilon_{z}(z)\) can be measured, and their polynomials can be fitted.  
At last, as shown in Figure 8b, the two arms of the standard piece are placed along the Y, Z-axis of the MT, and the micro-angle sensor is placed along the \(X\) axis. The Y-axis drives the micro-angle sensor to scan each plane surface. This will cause a yaw between the micro-angle sensor and the composite module, when there is a yaw error \(\varepsilon_{z}(y)\) of Y-axis. Thus, the \(\varepsilon_{z}(y)\) can be measured, and its polynomial can be fitted. Then, the Z-axis drives the micro-angle sensor to scan each plane surface above the other arm. This will cause a yaw between the micro-angle sensor and the composite module, when there is a yaw error \(\varepsilon_{y}(z)\) of the Z-axis. Thus, \(\varepsilon_{y}(z)\) can be measured, and its polynomial can be fitted.  
To sum up, through the three placements of the L-shape standard piece, the measurement and identification of all 9 angle errors can be implemented.