## 5 Identification Procedures and Validity
### Formulae Considering Squareness of Translational Axes to Calculate Geometric Deviations  
To determine 12 variables from eight expressions obtained from above four sensitive direction measurements, measurement conditions should be devised as following.  
Firstly, as only one geometric deviation affects the X axis component of eccentricity in the B axis X-direction and Y-direction measurements, two deviations \(\gamma_{\mathrm{XB}}\) and \(\delta z_{\mathrm{BT}}\) are directly identified based on the corresponding eccentricities measured from the B axis X-direction and Y-direction, expressed as Equations (17) and (18).  
\[\gamma_{\mathrm{XB}}=-\frac{ex_{\mathrm{BY}}}{R_{\mathrm{B}}} \tag{17}\]  
\[\delta z_{\mathrm{BT}}=ex_{\mathrm{BX}} \tag{18}\]  
Secondly, when the distance between the center of the W-side ball and the workbench surface of the C axis is changed from \(Z_{\mathrm{C}}\) to \(Z_{\mathrm{C}}^{\prime}\), new eccentricities \(ex_{\mathrm{CR}}^{\prime}\) and \(ey_{\mathrm{CR}}^{\prime}\) are obtained by measuring the radial direction of the C axis again. Then, another six geometric deviations \(\delta x_{\mathrm{CZ}}\), \(\beta_{\mathrm{CZ}}\), \(\beta_{\mathrm{YZ}}\), \(\alpha_{\mathrm{CZ}}\), \(\alpha_{\mathrm{YZ}}\) and \(\alpha_{\mathrm{XB}}\) are calculated by the following Equations (19)-(24).  
\[\delta x_{\mathrm{CZ}}=ex_{\mathrm{CR}}+ex_{\mathrm{BX}}-\frac{Z_{\mathrm{C}} \Big{(}ex_{\mathrm{CR}}^{\prime}-ex_{\mathrm{CR}}\Big{)}}{Z_{\mathrm{C}}^{ \prime}-Z_{\mathrm{C}}} \tag{19}\]  
\[\beta_{\mathrm{CZ}}=\frac{ex_{\mathrm{CR}}^{\prime}-ex_{\mathrm{CR}}}{Z_{ \mathrm{C}}^{\prime}-Z_{\mathrm{C}}} \tag{20}\]  
\[\beta_{\mathrm{YZ}}=\frac{ex_{\mathrm{CR}}^{\prime}-ex_{\mathrm{CR}}}{Z_{ \mathrm{C}}^{\prime}-Z_{\mathrm{C}}}-\frac{ex_{\mathrm{CA}}}{R_{\mathrm{C}}} \tag{21}\]  
\[\alpha_{\mathrm{CZ}}=\frac{ey_{\mathrm{CR}}-ey_{\mathrm{CR}}^{\prime}}{Z_{ \mathrm{C}}^{\prime}-Z_{\mathrm{C}}} \tag{22}\]  
\[\alpha_{\mathrm{YZ}}=\frac{ey_{\mathrm{CR}}-ey_{\mathrm{CR}}^{\prime}}{Z_{ \mathrm{C}}^{\prime}-Z_{\mathrm{C}}}+\frac{ey_{\mathrm{CA}}}{R_{\mathrm{C}}} \tag{23}\]  
\[\alpha_{\mathrm{XB}}=\frac{ez_{\mathrm{BY}}}{R_{\mathrm{B}}}-\frac{ey_{ \mathrm{CA}}}{R_{\mathrm{C}}}-\frac{ey_{\mathrm{CR}}-ey_{\mathrm{CR}}^{\prime} }{Z_{\mathrm{C}}^{\prime}-Z_{\mathrm{C}}} \tag{24}\]  
Thirdly, when the radius value is changed from \(R_{\mathrm{B}}\) to \(R_{\mathrm{B}}^{\prime\prime}\), new eccentricities \(ex_{\mathrm{CR}}^{\prime\prime}\) and \(ey_{\mathrm{CR}}^{\prime\prime}\) are obtained from the C axis radial measurement. The remaining deviations \(\delta x_{\mathrm{BT}}\), \(\beta_{\mathrm{XB}}\), \(\alpha_{\mathrm{BT}}\) and \(\delta y_{\mathrm{CZ}}\) are calculated by the following Equations (25)-(28).  
\[\delta x_{\mathrm{BT}}=ez_{\mathrm{BX}}+\frac{R_{\mathrm{B}}\Big{(}ex_{ \mathrm{CR}}^{\prime\prime}-ex_{\mathrm{CR}}^{\prime}\Big{)}}{R_{\mathrm{B}}^{ \prime\prime}-R_{\mathrm{B}}}. \tag{25}\]  
\[\beta_{\mathrm{XB}}=\frac{ex_{\mathrm{CR}}^{\prime}-ex_{\mathrm{CR}}^{\prime \prime}}{R_{\mathrm{B}}-R_{\mathrm{B}}^{\prime\prime}}-\frac{ex_{\mathrm{CR} }^{\prime}-ex_{\mathrm{CR}}}{Z_{\mathrm{C}}^{\prime}-Z_{\mathrm{C}}}+\frac{ ex_{\mathrm{CA}}}{R_{\mathrm{C}}}. \tag{26}\]  
\[\alpha_{\mathrm{BT}}=\frac{ey_{\mathrm{CR}}^{\prime\prime}-ey_{\mathrm{CR}}^{ \prime}}{R_{\mathrm{B}}-R_{\mathrm{B}}^{\prime\prime}}-\frac{ez_{\mathrm{BY}}}{R _{\mathrm{B}}} \tag{27}\]\[\delta y_{\text{CZ}}=\frac{R_{\text{B}}\Big{(}ey^{\prime}_{\text{CR}}-ey^{\prime \prime}_{\text{CR}}\Big{)}}{R^{{}^{\prime\prime}}_{\text{B}}-R_{\text{B}}}+ \frac{Z^{\prime}_{\text{C}}ey_{\text{CR}}-Z_{\text{C}}ey^{\prime}_{\text{CR}}}{Z ^{\prime}_{\text{C}}-Z_{\text{C}}} \tag{28}\]