## 3 Simultaneous Three-Axis Control Movements and Mathematical Model
### Mathematical Model
\begin{array}{cccc}1&0&\beta_{\rm CZ}&\delta x_{\rm CZ}\\ 0&1&-\alpha_{\rm CZ}&\delta y_{\rm CZ}\\ -\beta_{\rm CZ}&\alpha_{\rm CZ}&1&0\\ 0&0&0&1\end{array} \right] \tag{8}\]  
The rotation of C axis around Z axis is defined by the transformation matrix \(E_{\rm C}\) when the rotation angle \(\theta\) of the C axis is used. \[E_{\rm C}=\left[ \begin{array}{cccc}\cos\theta&-\sin\theta&0&0\\ \sin\theta&\cos\theta&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{9}\]  
Therefore, the center coordinate \(W\) of the W-side ball viewed from the machine coordinate system is denoted as following. \[W=M_{\rm CZ}E_{\rm C}W_{\rm C} \tag{10}\]  
3.2.3 Determination of the Initial Position \(W_{\rm C}\) of Center Coordinate of the W-Side Ball  
The W-side ball is positioned based on the center coordinate of the T-side ball, so it is necessary to add the influence of geometric deviations to the initial position of the W-side ball. The mounting position \(W^{\prime}_{\rm C}\) of the W-side ball in the machine coordinate system is same with the center coordinate \(T\) of the T-side ball in the machine coordinate system. Further, the initial position of the W-side ball is decided only by the command values of the translational axes without the B axis rotation. Therefore, it can be calculated by removing \(E_{\rm B}\) from the Equation (7), expressed as the following Equations (  
When W-side ball is set for the C axis measurement,  
\[ \begin{array}{c}\includegraphics[width=142.26378 pt]{Fig1}\end{array}  \tag{12}\]  
The transformation matrix from the machine coordinate system to the C axis coordinate system is performed by the inverse transformation of Equation (8). Moreover, when the W-side ball is mounted, the command value of the C axis is zero, so there is no need to consider the C axis rotation in the Equation (10). Therefore, the conversion from \(W_{\text{C}}^{\prime}\) to \(W_{\text{C}}\) is expressed by Equation (13). \[W_{\text{C}}=\left[ \begin{array}{cccc}1&0&-\beta_{\text{CZ}}&-\delta x_{ \text{CZ}}\\ 0&1&\alpha_{\text{CZ}}&-\delta y_{\text{CZ}}\\ \beta_{\text{CZ}}&-\alpha_{\text{CZ}}&1&0\\ 0&0&0&1\end{array} \right]W_{\text{C}}^{\prime} \tag{13}\]  
2.4 Calculation of the Difference \(\Delta L\) between Reference Length and Measured Length of Ball Bar  
From the above equations, the relative distance \(L\) between the T-side ball and the W-side ball can be calculated by the center coordinate \(T\left(x_{t},y_{t},z_{t}\right)\) of the T-side ball and the center coordinate \(W\) (\(x_{w}\), \(y_{w}\), \(z_{w}\) ) of the W-side ball when a command value is given to each axis. The relative distance \(L\) is different from the reference length \(L_{B}\) of the ball bar because of the existence of geometric deviations in the multi-tasking machine tools. Therefore, the ball bar length change amount \(\Delta L\) can be obtained from the relative distance \(L\) by subtracting the ball bar reference length \(L_{B}\) as written in Equation (14). \[\Delta L=\sqrt{\left(x_{t}-x_{w}\right)^{2}+\left(y_{t}-y_{w}\right)^{2}+\left( z_{t}-z_{w}\right)^{2}}-L_{B} \tag{14}\]