## 2 Measurement of Complete Geometric Errors
### Measurement of Coupled Geometric Errors  
The translation rails of a multi-axis machine tool are generally planar, and their shape and position errors are coupled by transverse and longitudinal geometric displacement and angle errors, as shown in Figure 5.  
As is shown in Figure 5, in the XOY plane, coupling geometric error items of \(\delta_{\mathrm{xz}}\) along X and Y direction cause shape errors. Coupling geometric error items of \(\varepsilon_{\mathrm{xx}}\) and \(\varepsilon_{\mathrm{xy}}\) causes position errors; they can be decoupled in the XOZ and YOZ plane, as shown in Figure 6. The decoupling steps are as follows:  
1. Set m and n points along the X and Y direction and measure shape errors of these m x n points. 2. Set the cross-sectional plane along each transverse and longitudinal line. 3. Fit shape errors along each transverse and longitudinal line. 4. Decouple error items of \(\varepsilon_{\mathrm{xx}}\) and \(\delta_{\mathrm{xz}}\) in the YOZ cross-sectional plane, and \(\varepsilon_{\mathrm{xy}}\) and \(\delta_{\mathrm{xz}}\) in the XOZ plane. 5. The tilt angles of the fitting line are angle error items of \(\varepsilon_{\mathrm{xx}}\) and \(\varepsilon_{\mathrm{xy}}\). The difference between the shape errors and fitting values along each transverse and longitudinal line is the displacement errors item of \(\varepsilon_{\mathrm{xx}}\), as shown in Figure 7.  
Figure 5: Geometric error coupling of the X-axis. Figure 6: Geometric error decoupling of the X-axis. Figure 7: Geometric error measuring and decoupling on the XOZ plane of the X-axis. According to the figure above,  
\[\left\{ \begin{array}{l}p_{i_{\cdot}}=\sum\limits_{j=1}^{n}p_{ij}/n(i=1,2, \ldots,m)\\ p_{\cdot j}=\sum\limits_{i=1}^{m}p_{ij}/n(j=1,2,\ldots,n)\end{array} \right., \tag{3}\]  
\[\left\{ \begin{array}{l}k_{i_{\cdot}}=\tan\varepsilon_{xx}^{i}\\ k_{\cdot j}=\tan\varepsilon_{xy}^{j}\end{array} \right., \tag{4}\]  
where \(k_{i_{\cdot}}\) and \(k_{\cdot j}\) are the slopes of the fitting lines in the transverse and longitudinal cross-sectional planes, respectively. In XOZ plane, the decoupling displacement and angle errors are obtained as Equations (5)-(7):  
\[\delta_{xz}^{i}=\sum\limits_{j=1}^{n}\left(p_{ij}-k_{i}y_{j}\right)/n(i=1,2, \ldots,m), \tag{5}\]  
\[\varepsilon_{xx}^{i}=\text{arctanh}k_{i_{\cdot}}(i=1,2,\ldots,m), \tag{6}\]  
\[\varepsilon_{xy}^{i}=\sum\limits_{j=1}^{n}\left(\text{arctanh}k_{j}\right)/n( i=1,2,\ldots,m). \tag{7}\]  
In YOZ plane, the decoupling displacement and angle errors are obtained as Equations (8)-(10):  
\[\delta_{xy}^{i}=\sum\limits_{j=1}^{n}\left(p_{ij}-k_{\cdot j}x_{i}\right)/n(i =1,2,\ldots,m), \tag{8}\]  
\[\varepsilon_{xx}^{i}=\text{arctanh}k_{i_{\cdot}}(i=1,2,\ldots,m), \tag{9}\]  
\[\varepsilon_{xz}^{i}=\sum\limits_{j=1}^{n}\Bigl{(}\text{arctanh}\cdot_{j} \Bigr{)}/n\left(i=1,2,\ldots,m\right). \tag{10}\]  
Geometric error item of \(\varepsilon_{xx}^{i}\) (Equation (11)) is derived from Equation (6) and Equation (9):  
\[\varepsilon_{xx}^{i}=(\varepsilon_{xx(XOY)}^{i}+\varepsilon_{xx(XOZ)}^{i})/2. \tag{11}\]  
Finally, complete geometric errors of the X-axis are all obtained (Equations (1), (5), (7), (8), (10), and (11)). Similarly, complete geometric errors of the Y-axis and Z-axis can be measured by the same method and steps. For the rotation axes: A, B and C-axis, considering that the final result is still expressed in the form of three-dimensional coordinates, we convert the cylindrical surfaces into partial planes to avoid the complex calculation process of transforming from 3D coordinates to cylindrical coordinates and then from cylindrical coordinates to 3D coordinates, as shown in Figure 8. By converting the cylindrical surfaces into partial planes, the geometrical shape defects of the surface will be transferred to the plane. Therefore, the transformed plane geometric defects described by measuring points are approximate to the geometrical shape defects of the surface.