## 6 Conclusions  
In this paper, the influence of gravity, acceleration, inertia, and other factors existing in the operation of the A-axis on the average position error of the C-axis were considered, and 10 key average position error elements for the two rotating axes were determined. According to the error transfer chain relationship between machine tool axes and the principle of homogeneous coordinate transformation, the mathematical model of ideal and actual positions of the tool tip and tool posture of a five-axis CNC machine tool TOPNC VMC-C50 were determined. The main conclusions of this paper were as follows:  
1. A model was developed for solving the average position errors of rotation A and C axes. Further, 10 average position errors of two rotating axes were obtained by bringing the coordinates of the center of spheres before and after 90\({}^{\circ}\) of rotation into the model.
2. An on-machine detection method was developed for average position errors of rotating axes based on the precision spherical column and a multi-degree of freedom, step-by-step contact scheme. The measurement accuracy obtained by this method was about 91.8% of traditional measurement equipment, such as laser interferometers and double-ball-bar measuring systems. However, the cost of the proposed method was about one tenth of the traditional methods and each test took about 20-30 min.
3. The spatial geometric error field of the five-axis CNC machine tool was determined based on the position errors of two identified rotating axes. By analyzing comprehensive error fields obtained by simulations, the following conclusions were drawn: (a) the comprehensive error of rotating spaces of A and C presented an increasing trend from inside to outside, (b) comprehensive error increased with the increase of the rotation angle of the A-axis, and therefore the large rotation angles of the A-axis had to be avoided, and (c) there was a machining space with less error in work center. By changing the A-axis angle of the table, the axis change of the optimal machining space had a certain offset law. These laws could provide guidance for error compensation, machining path, and parameter optimization, etc.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Position Error (\(\mu\)m)** & \(\delta_{yA}\) & \(\delta_{xA}\) & \(\epsilon_{aA}\) & \(\epsilon_{BA}\) & \(\epsilon_{YA}\) & \(\delta_{xC}\) & \(\delta_{yC}\) & \(\epsilon_{aC}\) & \(\epsilon_{pC}\) & \(\epsilon_{YC}\) \\ \hline Precision balls method & 10.6 & \(-\)19.4 & 29.8 & 25.7 & \(-\)15.1 & 13.7 & 23.9 & \(-\)18.5 & \(-\)21.3 & 28.2 \\ [PERSON]’s method & 12.7 & \(-\)20.1 & 27.6 & 24.0 & \(-\)12.3 & 15.6 & 22.4 & \(-\)18.4 & \(-\)19.8 & 27.2 \\ A and C axis Accuracy & & & & 89.5\% & & & & 94.1\% & \\ Comparison & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of measurement accuracy.  
**Author Contributions:** Data curation, S.W.; Funding acquisition, X.L.; Investigation, R.L.; Methodology, H.S.; Writing--original draft, C.L. and H.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** The authors express their gratitude to the National Natural Science Foundation of China (NSFC) (Nos. 51720105009) for it support.  
**Conflicts of Interest:** The authors declare no conflict of interest.