## 2 Principle
Additionally, the static error matrix between adjacent bodies on a machine tool is a unit matrix. Therefore, the error transformation homogeneous matrix of the C-axis can be represented by Equation (2). \[T_{e}^{C}= \begin{bmatrix}1&-\varepsilon_{zC}&\varepsilon_{yC}+S_{XC}&\delta_ {xC}\\ \varepsilon_{zC}&1&-\varepsilon_{xC}-S_{YC}&\delta_{yC}\\ -\varepsilon_{yC}-S_{XC}&\varepsilon_{xC}+S_{YC}&1&\delta_{zC}\\ 0&0&0&1\end{bmatrix}  \tag{2}\]  
Figure 2: (**a**) Change caused by angular motion error. (**b**) Error caused by probe deviating from the ideal direction. Similarly, the error homogeneous transformation matrix of the A-axis can be represented by Equation (3). \[T_{e}^{A}= \begin{bmatrix}1&-\varepsilon_{ZA}-S_{YA}&\varepsilon_{yA}+S_{ZA}& \delta_{xA}\\ \varepsilon_{ZA}+\varepsilon_{YA}&1&-\varepsilon_{xA}&\delta_{yA}\\ -\varepsilon_{yA}-S_{ZA}&\varepsilon_{xA}&1&\delta_{zA}\\ 0&0&0&1\end{bmatrix}  \tag{3}\]  
On the Y-axis, the Z-axis motion distances are Y and Z. After the A rotary table and C rotary table are rotated by \(\theta_{\rm A}\) and \(\theta_{\rm C}\) angles, respectively, the sensor's signal is L. At this time, the ideal error-free and actual conditions between the probe and the workpiece are measured. The comprehensive error transformation matrix is shown in Equation (4). \[\Delta E=T_{actual}-T_{ideal}=\left(T_{i}^{\rm C}\cdot T_{e}^{\rm C}\right)^{- 1}\cdot T_{i}^{\rm Y}\cdot T_{e}^{\rm Y}\cdot T_{i}^{\rm Z}\cdot T_{e}^{\rm Z }\cdot T_{i}^{A}\cdot T_{e}^{A}-\left(T_{i}^{\rm C}\right)^{-1}\cdot T_{i}^{ \rm Y}\cdot T_{i}^{\rm Z}\cdot T_{i}^{A} \tag{4}\]  
where \(T_{actual}\) is the actual transformation matrix between the probe and the measured workpiece;\(T_{ideal}\) is the ideal transformation matrix between the probe and measured workpiece; \(T_{i}^{j}\) is the ideal pose transformation matrix between adjacent axes; and \(T_{e}^{j}\) is the actual pose transformation matrix between adjacent axes, where \(j\) = C, Y, Z, A. Overall, based on the assumption of a small error and excluding the small amount of second-order and above errors, the components \(E_{x}\), \(E_{y}\), and \(E_{z}\) of the comprehensive error model of the measuring machine in the X-, Y-, and Z-directions is obtained as shown in Equations (5)-(7). \[ \begin{array}{l}E_{x}=\cos\!\theta_{\rm C}\cdot\delta_{xA}-\cos\!\theta_{\rm C }\cdot\delta_{xC}+\cos\!\theta_{\rm C}\cdot\delta_{xY}+\cos\!\theta_{\rm C} \cdot\delta_{xZ}+\sin\!\theta_{\rm C}\cdot\delta_{yA}-\sin\!\theta_{\rm C} \cdot\delta_{yC}\\ \quad+\sin\!\theta_{\rm C}\cdot\delta_{yY}+\sin\!\theta_{\rm C}\cdot\delta_{yZ }+\mathrm{Y}\cdot\cos\!\theta_{\rm C}\cdot\varepsilon_{zC}-\mathrm{Z}\cdot \cos\!\theta_{\rm C}\cdot\mathrm{S}_{YZ}-\mathrm{Z}\cdot\cos\!\theta_{\rm C} \\ \quad\cdot S_{CX}-\mathrm{Z}\cdot\cos\!\theta_{\rm C}\cdot\varepsilon_{yC}+ \mathrm{Z}\cdot\sin\!\theta_{\rm C}\cdot\varepsilon_{xC}+\mathrm{Z}\cdot\sin \!\theta_{\rm C}\cdot S_{YC}+\mathrm{Y}\cdot\cos\!\theta_{\rm C}\cdot\varepsilon _{zC}\\ \quad-\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\cos\!\theta_{\rm C}\cdot \varepsilon_{yC}-\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\cos\!\theta_{\rm C }\cdot\mathrm{S}_{CX}-\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\sin\!\theta_{\rm C }\\ \quad\cdot\varepsilon_{xA}+\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\sin\!\theta_{ \rm C}\cdot\varepsilon_{xC}+\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\sin\! \theta_{\rm C}\cdot S_{YC}-\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\sin\! \theta_{\rm C}\\ \quad\cdot S_{YZ}+\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\cos\!\theta_{\rm C }\cdot\varepsilon_{yA}+\mathrm{L}\cdot\cos\!\theta_{\rm A}\cdot\cos\!\theta_{ \rm C}\cdot\mathrm{S}_{ZA}-\mathrm{L}\cdot\sin\!\theta_{\rm A}\\ \quad\cdot\cos\!\theta_{\rm C}\cdot\varepsilon_{zC}+\mathrm{L}\cdot\sin\!\theta_{ \rm A}\cdot\cos\!\theta_{\rm C}\cdot\varepsilon_{zA}+\mathrm{L}\cdot\sin\! \theta_{\rm A}\cdot\cos\!\theta_{\rm C}\cdot S_{YA}\end{array}  \tag{5}\]  
Figure 3: (**a**) Six TDGEs in the rotary stage. (**b**) Four PIGEs in the rotary stage. \[