## 3 Simultaneous Three-Axis Control Movements and Mathematical Model
### Mathematical Model  
Since the angular deviations are generally less than \(1^{\circ}\), the small angle approximation (\(\sin\theta\approx\theta\)\(\cos\theta\approx 1\), if the angle \(\theta<0.244\) radians \((14^{\circ})\), the relative error does not exceed 1%) is assumed and second order errors are neglected. Moreover, for small rotation matrices, the order of rotation matrix could be interchanged and it is possible to add and subtract matrices. Therefore, according to this minute rotation approximation theory, homogeneous transformation matrix (HTM) can be simplified in the mathematical model to express the center coordinates of both the tool spindle side ball (T-side ball) and the work spindle side ball (W-side ball) viewed from the machine coordinate system. 2.1 Determination of Center Coordinate T of the T-Side Ball Viewed from the Machine Coordinate System  
When the distance from the rotational center of B axis to the center of the T-side ball is \(R_{\text{B}}\), the center coordinates \(T_{\text{T}}\) of the T-side ball in the tool spindle axis coordinate system are expressed by Equation (1). \[T_{\text{T}}=\left[ \begin{array}{ccc}0&0&-R_{\text{B}}&1\end{array} \right] ^{\text{T}} \tag{1}\]  
As there are angular deviation \(\alpha_{\text{BT}}\) and positional deviations \(\delta x_{\text{BT}}\), \(\delta z_{\text{BT}}\) between B axis and the tool spindle axis, the homogeneous transformation matrix \(M_{\text{BT}}\) from the tool spindle axis coordinate system to the B axis coordinate system is expressed by Equation (2) if the above simplified homogeneous coordinate transformation is used. \[M_{\text{BT}}=\left[ \begin{array}{ccc}1&0&0&\delta x_{\text{BT}}\\ 0&1&-\alpha_{\text{BT}}&0\\ 0&\alpha_{\text{BT}}&1&\delta z_{\text{BT}}\\ 0&0&0&1\end{array} \right] \tag{2}\]  
In the same way, the homogeneous transformation matrix \(M_{\text{XB}}\) between the X axis and B axis is defined as following. \[M_{\text{XB}}=\left[ \begin{array}{ccc}1&-\gamma_{\text{XB}}&\beta_{\text{XB }}&0\\ \gamma_{\text{XB}}&1&-\alpha_{\text{XB}}&0\\ -\beta_{\text{XB}}&\alpha_{\text{XB}}&1&0\\ 0&0&0&1\end{array} \right] \tag{3}\]  
The T-side ball circularly moves around the B axis with radius \(R_{\text{B}}\). The circular motion of the T-side ball is expressed by the transformation matrix \(E_{\text{B}}\) shown in Equation (4), using the rotation angle \(\varphi\) of the B axis. \[E_{\text{B}}=\left[ \begin{array}{ccc}\cos\varnothing&0&\sin\varnothing&0 \\ 0&1&0&0\\ -\sin\varnothing&0&\cos\varnothing&0\\ 0&0&0&1\end{array} \right] \tag{4}\]  
The squareness of the X, Y and Z translational axes can be expressed by the homogeneous transformation matrix \(M_{\text{YZ}}\) and \(M_{\text{XY}}\) based on the theory of form-shaping system, as followings. \[M_{\text{YZ}}=\left[ \begin{array}{ccc}1&0&\beta_{\text{YZ}}&0\\ 0&1&-\alpha_{\text{YZ}}&0\\ -\beta_{\text{YZ}}&\alpha_{\text{YZ}}&1&0\\ 0&0&0&1\end{array} \right] \tag{5}\]\[M_{\rm XY}=\left[ \begin{array}{cccc}1&-\gamma_{\rm XY}&0&0\\ \gamma_{\rm XY}&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{6}\]  
If the translational motion controlled by the command values of X, Y and Z axis is respectively expressed by \(E_{\rm X}\), \(E_{\rm Y}\) and \(E_{\rm Z}\), the center coordinate of the T-side ball viewed from the machine coordinate system is denoted as following. \[T=E_{\rm Z}M_{\rm YZ}E_{\rm Y}M_{\rm XY}E_{\rm X}M_{\rm XB}E_{\rm B}M_{\rm BT}T _{\rm T} \tag{7}\]  
2.2 Determination of Center Coordinate W of the W-Side Ball Viewed from the Machine Coordinate System  
If the initial position of the center coordinate of the W-side ball in the C axis coordinate system is \(W_{\rm C}\) (\(x_{\rm wC}\), \(y_{\rm wC}\), \(z_{\rm wC}\)), the center coordinates \(W\) of the W-side ball viewed from the machine coordinate system are calculated as following. The homogeneous transformation matrix \(M_{\rm CZ}\) between the C axis and Z axis is expressed by Equation (8). \[M_{\rm CZ}=\left[