## 3 Identification of Position Error of Rotary Axis  
### Identification of Average Position Error of A-Axis  
Six position-independent geometric errors of the A-axis were established in this paper, which were the average errors of the A-axis deviating from the ideal space position. An on-machine identification method was developed for the calibration of average position error by measuring spherical center coordinates before and after rotation. Finally, five equations were obtained for calculating average position error. As shown in Figure 2, \(W_{IA}\) is the established the A-axis theoretical coordinate system, \(W_{rA}\) is the actual coordinate system of the A-axis in the presence of errors, and its coordinate origin is located at the theoretical intersection of the A- and C-axes. As shown in Figure 2, precision ball columns 1 and 2 were mounted on rotary table. These two ball columns did not need very high positioning accuracy and they could be quickly installed and positioned. It was necessary to ensure uniform dispersion of the positions of the two ball columns to avoid ball column interference after the rotation of the A-axis. Detection plane, consisting of four detection points for detecting the center of sphere, was ideally a horizontal plane passing through the center of sphere and probing needle was detected along the X and Y directions. By assuming a point \(P_{1}=\left[ \begin{array}{cccc}x_{p}&y_{p}&z_{p}&1\end{array} \right]^{T}\) in the actual coordinate system \(W_{rA}\) in the presence of A-axis error, after the A-axis was rotated by \(90^{\circ}\), its theoretical position was changed from \(S_{1}=\left[ \begin{array}{cccc}x_{1}&y_{1}&z_{1}&1\end{array} \right]^{T}\) to \(S_{1}^{\prime}=\left[ \begin{array}{cccc}x_{1}^{\prime}&y_{1}^{\prime}&z_{1}^ {\prime}&1\end{array} \right]^{T}\). \(S_{1}\) and \(S_{1}^{\prime}\) were two points in \(W_{IA}\), obtained by the center coordinate transformer of measured ball through matrix \(T_{AC}\) between MCS and \(W_{IA}\). The mathematical relationship between \(P_{1}\) and \(S_{1}\), \(S_{1}^{\prime}\) was as follows:  
\[\left\{ \begin{array}{l}S_{1}=T_{AC}\cdot T_{EA}\cdot R_{EA}\cdot P_{1}\\ S_{1}^{\prime}=T_{AC}\cdot T_{EA}\cdot R_{EA}\cdot R(a)\cdot P_{1}\end{array} \right. \tag{14}\]  
where \(T_{AC}=\left[ \begin{array}{cccc}1&0&0&m_{x}\\ 0&1&0&m_{y}\\ 0&0&1&m_{z}\\ 0&0&0&1\end{array} \right]d=90^{\circ}\), was further simplified to obtain position error solution Equation (14):  
\[R(-a)\cdot R_{EA}^{-1}\cdot T_{EA}^{-1}\cdot T_{AC}^{-1}\cdot S_{1}^{\prime}- R_{EA}^{-1}\cdot T_{EA}^{-1}\cdot T_{AC}^{-1}\cdot S_{1}=0 \tag{15}\]  
This equation described the relationship between the spatial position coordinates of measuring points before and after 90-degree rotation of the A-axis and geometric error elements at each position. The first set of Equation (15) used to solve the A-axis average position error was obtained by matrix operation from Equation (14) as follows:  
\[\left\{ \begin{array}{l}\Delta x_{1}+\Delta y_{1}\cdot\varepsilon_{\gamma A} -\Delta z_{1}\cdot\varepsilon_{\beta A}=0\\ \varepsilon_{\gamma A}\cdot(z_{1}-m_{z}+y_{1}^{\prime}-m_{y})=\delta_{\psi A}- \delta_{zA}+C_{1}\\ \varepsilon_{\alpha A}\cdot(z_{1}^{\prime}-m_{z}-y_{1}+m_{y})=\delta_{\psi A}+ \delta_{zA}+C_{2}\end{array} \right. \tag{16}\]  
Figure 2: A-axis position error detection method. (**a**) Measuring positions of two precision spheres at A = \(0^{\circ}\); (**b**) Measuring positions of two precision spheres at A = \(90^{\circ}\). where \(\Delta x_{1}=x_{1}^{\prime}-x_{1}\), \(\Delta y_{1}=y_{1}^{\prime}-y_{1}\), and \(\Delta z_{1}=z_{1}^{\prime}-z_{1}\). The second set of spherical coordinates \(S_{2}\) and \(S_{2}^{\prime}\) of ball column 2 were introduced by the same method as expressed in Equation (16)  
\[\left\{