## 3 Identification of Position Error of Rotary Axis
### Identification of Average Position Error of C-Axis  
The identification method of C-axis position error was almost similar to that of A-axis and the only difference lied in the need to solve various position errors of C-axis under different A-axis rotation angles Ai, and divide Ai (I = 1\(\sim\)3) into three angles. : A1 = 0\({}^{\circ}\), A2 = 45\({}^{\circ}\), A3 = 90\({}^{\circ}\). In previous section, the six average position errors of the A-axis were identified. Since the C-axis was connected to the A-axis, the position error of A-axis needed to be added when identifying the position error of C-axis. Figure 3 shows the installation arrangement of precision ball columns 1 and 2 and on-line measurement processes before and after rotation where \(W_{iC}\) is the theoretical coordinate system of C axis coinciding with \(W_{iA}\), The actual coordinate system in the presence of C-axis error was assumed to be \(W_{iC}\). The spherical point \(O=\left[ \begin{array}{cccccc}x&y&z&1\end{array} \right]^{T}\)\(O^{\prime}=\left[ \begin{array}{cccccc}x^{\prime}&y^{\prime}&z^{\prime}&1 \end{array} \right]^{T}\) was obtained by measuring the probe in MCS. The coordinate of the spherical point coordinate O was S in coordinate \(W_{iC}\), and the coordinate transformation relationship between them was:\(O=T_{AC}^{-1}\cdot S\) where \(T_{AC}\) is the transform matrix between the two coordinate systems. Equation (19) was obtained based on the coordinate transformation relationship between actual spherical point coordinate \(P_{1}\) in \(W_{rC}\) and two sets of spherical points \(S_{1}\), \(S_{1}^{\prime}\), \(S_{2}\), \(S_{2}^{\prime}\) in \(W_{iC}\) as:  
\[\left\{ \begin{array}{l}S_{1}=T_{EA}\cdot R_{EA}\cdot R(a)\cdot T_{EC}\cdot R_{ EC}\cdot P_{1}\\ S_{1}^{\prime}=T_{EA}\cdot R_{EA}\cdot R(a)\cdot T_{EC}\cdot R_{EC}\cdot R(c) \cdot P_{1}\end{array} \right. \tag{20}\]  
Equation (20) was obtained to solve C-axis position error after the subtraction of the two equations in Equation (19),  
\[R(-c)\cdot R_{EC}^{-1}\cdot T_{EC}^{-1}\cdot R(-a)\cdot R_{EA}^{-1}\cdot T_{EA }^{-1}\cdot T_{AC}^{-1}\cdot S_{1}^{\prime}-R_{EC}^{-1}\cdot T_{EC}^{-1}\cdot R (-a)\cdot R_{EA}^{-1}\cdot T_{EA}^{-1}\cdot T_{AC}^{-1}\cdot S_{1}=0 \tag{21}\]  
After matrix operations, the two tilt error sums \(\varepsilon_{aC}\) and \(\varepsilon_{BC}\) of C-axis were obtained as stated in Equations (21) and (22), in which the position error of A-axis was taken as a known constant. \[\varepsilon_{aC}=-\varepsilon_{aA}+(\Delta x_{1}\Delta z_{2}-\Delta z_{1} \Delta x_{2})/(\Delta x_{1}\Delta y_{2}-\Delta y_{1}\Delta x_{2}) \tag{22}\]  
\[\varepsilon_{BC}=-\varepsilon_{BA}+(\Delta y_{1}\Delta z_{2}-\Delta z_{1} \Delta y_{2})/(\Delta x_{1}\Delta y_{2}-\Delta y_{1}\Delta x_{2}) \tag{23}\]  
\[\varepsilon_{YC}= \begin{array}{l}\left[\left(z_{1}+z_{1}^{\prime}-z_{2}-z_{ 2}^{\prime}\right)\cdot\varepsilon_{BC}+\left(\Delta z_{1}-\Delta z_{2}\right) \cdot\varepsilon_{aC}+C_{5}-C_{6}\right]\\ \left/\left(y_{1}+y_{1}^{\prime}+\Delta x_{1}-y_{2}-y_{2}^{\prime}-\Delta x_{ 2}\right)\right.\end{array}  \tag{24}\]  
It was seen from Equation (23) that the tilt error of C-axis about Z-axis was related to the other two tilt errors of C-axis. The two key linear errors of C-axis \(\delta_{xC}\) and \(\delta_{YC}\) were obtained by Equations (24) and (25), respectively, as:  
\[\delta_{xC}=\left[\left(y_{1}+y_{1}^{\prime}+\Delta x_{1}\right)\cdot \varepsilon_{YC}-\left(z_{1}+z_{1}^{\prime}\right)\cdot\varepsilon_{BC}-\Delta z _{1}\cdot\varepsilon_{aC}-C_{5}\right]/2 \tag{25}\]  
\[\delta_{YC}=\left[\left(y_{1}^{\prime}-y_{1}-x_{1}^{\prime}-x_{1}\right)\cdot \varepsilon_{YC}-\Delta z_{1}\cdot\varepsilon_{BC}+\Delta z_{1}\cdot \varepsilon_{aC}+C_{7}\right]/2 \tag{26}\]  
A series of constant terms were represented by \(C_{5}\), \(C_{6}\) and \(C_{7}\), in which were defined by Equation (26) as:  
\[\left\{