## 5 Identification Procedures and Validity  
### Mathematical Expressions between Eccentricities and Geometric Deviations  
Mathematical expressions between eccentricities of circular trajectories and geometric deviations are summarized in Table 5 according to the simulation results. _ex_, _ey_ and _ez_ represent the components of eccentricities in X-, Y- and Z-direction, respectively. The subscripts indicate the type of measurements, for example, CA and BY represent measurements of the C axis axial direction and B axis Y- direction, respectively. Among the eccentricity, the positional deviation appears as eccentricity is, while the angular deviation, multiplied by coefficient \(Z_{\mathrm{C}}\), \(R_{\mathrm{B}}\) or \(R_{\mathrm{C}}\), appears in eccentricity. The deviation is positive if the direction of the deviation is identical with the direction of the eccentricity. On the contrary, the deviation is negative.  
It is found that the expressions of the C axis tangential measurement have only opposite arithmetic signs comparing to those of the C axis radial measurement, and the expressions of the B axis X-direction measurement is similar with those of the B axis Z-direction measurement. Furthermore, it has been  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **ex** & **ey** & **ez** \\ \hline B: X-direction & \(\delta_{\mathrm{ZBT}}\) & — & \(\delta_{\mathrm{XBT}}-R_{\mathrm{B}}\beta_{\mathrm{XB}}-R_{\mathrm{B}}\beta_{ \mathrm{YZ}}\) \\ \hline B: Y-direction & \(-R_{\mathrm{BYXB}}\) & — & \(R_{\mathrm{B}}\alpha_{\mathrm{XB}}+R_{\mathrm{B}}\alpha_{\mathrm{YZ}}\) \\ \hline B: Z-direction & \(R_{\mathrm{B}}\beta_{\mathrm{XB}}-\delta_{\mathrm{XBT}}\) & & \(\delta_{\mathrm{ZBT}}\) \\ \hline C: Radial & \(-\delta_{\mathrm{XBT}}+R_{\mathrm{B}}\beta_{\mathrm{XB}}+\delta_{\mathrm{XCZ}}+ &-R_{\mathrm{B}}\alpha_{\mathrm{XB}}-R_{\mathrm{B}}\alpha_{\mathrm{BT}}+ \delta_{\mathrm{YCZ}}-\) & — \\  & \(Z_{\mathrm{C}}\beta_{\mathrm{CZ}}+R_{\mathrm{B}}\beta_{\mathrm{YZ}}\) & \(Z_{\mathrm{C}}\alpha_{\mathrm{CZ}}-R_{\mathrm{B}}\alpha_{\mathrm{YZ}}\) & — \\ \hline C: Tangential & \(R_{\mathrm{B}}\alpha_{\mathrm{XB}}+R_{\mathrm{B}}\alpha_{\mathrm{BGT}}-\delta_{ \mathrm{YCZ}}+\) & \(-\delta_{\mathrm{XBT}}+R_{\mathrm{B}}\beta_{\mathrm{XB}}+\delta_{\mathrm{XCZ}}+\) & \\  & \(Z_{\mathrm{C}}\alpha_{\mathrm{CZ}}+R_{\mathrm{B}}\alpha_{\mathrm{YZ}}\) & \(Z_{\mathrm{C}}\beta_{\mathrm{CZ}}+R_{\mathrm{B}}\beta_{\mathrm{YZ}}\) & \\ \hline C: Axial & \(R_{\mathrm{C}}\beta_{\mathrm{CZ}}-R_{\mathrm{C}}\beta_{\mathrm{YZ}}\) & \(R_{\mathrm{C}}\alpha_{\mathrm{YZ}}-R_{\mathrm{C}}\alpha_{\mathrm{CZ}}\) & — \\ \hline \hline \end{tabular}
\end{table}
Table 5: Mathematical expressions between eccentricities and geometric deviations.  
known that a pitch error of the worm gear affects the eccentricity of circular trajectory measured in the tangential direction. Thus, the C axis tangential measurement and B axis Z-direction measurement are not discussed to identify geometric deviations in this study.  
In summary, considering the squareness of translational axes, twelve geometric deviations for multi-tasking machine tools can be identified by measuring the eccentricities of circular trajectories of the B axis X-direction, B axis Y-direction, C axis radial direction and C axis axial direction.