# Rapid Measurement and Identification Method for the Geometric Errors of CNC Machine Tools
## 2 Integrated Error Model of 3-Axis MTs
### Integrated Error Model of FXYZ 3-Axis MTs  
In MBS, the relationship between each adjacent body is transformed into the relationship between adjacent coordinate systems. According to the right-hand rule [21], the worktable coordinate system 0\({}_{0}\)-X\({}_{0}\)Y\({}_{0}\)Z\({}_{0}\) is established as a reference coordinate system. Then, the coordinate system of the workpiece, X-axis, Y-axis, spindle/Z-axis and measurement system are established as 0\({}_{1}\)-X\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\), 0\({}_{2}\)-X\({}_{2}\)Y\({}_{2}\)Z\({}_{2}\), 0\({}_{3}\)-X\({}_{3}\)Y\({}_{3}\)Z\({}_{3}\), 0\({}_{4}\)-X\({}_{4}\)Y\({}_{4}\)Z\({}_{4}\), and 0\({}_{5}\)-X\({}_{5}\)Y\({}_{5}\)Z\({}_{5}\), respectively.  
To analyze the geometric error of 3-axis MTs, the coordinate system 0\({}_{k}\)-X\({}_{k}\)Y\({}_{k}\)Z\({}_{k}\) (\(k=1\), 2, 3, 4, 5) is made to coincide with the reference coordinate system in the initial time. The relationship between the coordinate systems can be described by the 4 \(\times\) 4 homogeneous Denavit-Hartenberg (D-H) matrix. The feature matrix \(T_{\bar{i}}\) between the body \(j\) and the body \(i\) can be determined by the nominal motion matrix and error matrix [22]. It is assumed that the initial homogeneous coordinates of the forming point is \(Pt\) = [\(x_{p}\)\(y_{p}\)\(z_{p}\)\(1\)]\({}^{T}\). When there is nominal motion of the \(X\), \(Y\), and \(Z\) axes of the MT, then its actual forming point P and ideal forming point \(P_{w}\) can be obtained as follows:  
\[P = \left[\begin{array}{cccc}1&-\varepsilon_{z}(x)&\varepsilon_{y}( x)&x+\delta_{x}(x)\\ \varepsilon_{z}(x)&1&-\varepsilon_{x}(x)&\delta_{y}(x)\\ -\varepsilon_{y}(x)&\varepsilon_{x}(x)&1&\delta_{z}(x)\\ 0&0&0&1\\ \end{array}\right]\left[\begin{array}{cccc}1&-\alpha_{xy}&0&0\\ \alpha_{xy}&1&0&0\\ 0&0&1\\ \end{array}\right]\left[\begin{array}{cccc}1&-\varepsilon_{z}(y)&\varepsilon _{y}(y)&\delta_{x}(y)\\ \varepsilon_{z}(y)&1&-\varepsilon_{x}(y)&y+\delta_{y}(y)\\ 0&0&1\\ \end{array}\right] \tag{1}\] \[\left[\begin{array}{cccc}1&0&\alpha_{xz}\\ 0&1&-\alpha_{yz}&0\\ -\alpha_{xz}&\alpha_{yz}&1&0\\ 0&0&0&1\\ \end{array}\right]\left[\begin{array}{cccc}1&-\alpha_{yz}&0&0\\ \alpha_{xy}&1&0&0\\ 0&0&1\\ \end{array}\right]\left[\begin{array}{cccc}1&-\varepsilon_{z}(y)&\varepsilon _{y}(y)&\delta_{x}(y)\\ \varepsilon_{z}(y)&1&-\varepsilon_{x}(y)&y+\delta_{y}(y)\\ 0&0&1\\ \end{array}\right]\]  
\[P_{w}=\left[\begin{array}{cccc}x+x_{p}&y+y_{p}&z+z_{p}&1\\ \end{array}\right]^{T} \tag{2}\]  
Then, the integrated error model of FXYZ MT is  
\[E=P-P_{w} \tag{3}\]  
The geometric errors of MTs are very small, so the second and higher order errors can be neglected. Moreover, \(x_{p}=y_{p}=z_{p}=0\) is made to simplify the calculation. Then, Equation (3) is expanded, with only the first-order error remaining, and the integrated error model, represented by 21 error terms, can be expressed as  
\[dx=\delta_{x}(x)+\delta_{x}(y)+\delta_{x}(z)-y\times(\alpha_{xy}+\varepsilon _{z}(x))+z\times(\alpha_{xz}+\varepsilon_{y}(y)+\varepsilon_{y}(x)) \tag{4}\]  
\[dy=\delta_{y}(x)+\delta_{y}(y)+\delta_{y}(z)-z\times(\alpha_{yz}+\varepsilon_ {y}(y)+\varepsilon_{z}(x)) \tag{5}\]  
\[dz=\delta_{z}(x)+\delta_{z}(y)+\delta_{z}(z)+y\times\varepsilon_{x}(x). \tag{6}\]  
The PDGEs, including translation errors and angle errors, are functions of the motion axis [23]. For example, the angle error \(\varepsilon_{x}(x)\) can be expressed as  
\[\varepsilon_{x}(x)=f(x)=\sum_{i=1}^{n}\left(r_{xxi}\times x^{i}\right)=r_{xx1 }\times x+r_{xx2}\times x^{2}+r_{x33}\times x^{3}+\cdots+r_{xxn}\times x^{n} \tag{7}\]  
where \(r_{xxi}\) (\(i=1\), 2,..., \(n\)) are the coefficients of the fitting polynomial.