## 5 Conclusions  
In this study, the laser tracer multi-station measurement system was used to calibrate the angular positioning deviation of the rotary table. The algorithm error of the high-precision rotary table method based on the laser tracer multi-station measurement system was mainly analyzed. A laser tracer multi-station measurement system was built. The rotary table was mounted on the working plane of CMM, and the laser tracer was placed on the rotary table. The rotary table rotated with the same angle intervals. With the redundancy equation of the laser tracer multi-station measurement system, the nonlinear least squares method was used to self-calibrate the coordinates of laser tracer stations. The coordinates of laser tracer stations were iterated with the Levenberg\(-\)Marquardt algorithm. The singular value decomposition (Singular Value Decomposition) of the covariance matrix was used to fit the m coordinates of laser tracer stations obtained from the calibration algorithm to obtain a plane. Then \(m\) coordinates of laser tracer stations were projected onto the fitting plane to obtain the optimized coordinates of laser tracer stations. The circle was fitted with the coordinates of optimized laser tracer stations, and then the coordinates of the center of the fitting circle were solved. Then, the geometric relationship between the coordinates of optimized laser tracer stations and the center of the obtained fitting circle was established, and the actual rotation angle of the rotary table was obtained. In this way, the calibration of the angular positioning deviation of the rotary table was realized.  
The experimental parameters were as follows. The spatial range of measured points given by CMM was \(450\times 600\times 450\) mm\({}^{3}\); the number of measured points was \(64\); the number of laser tracer  
Figure 8: Angular positioning deviation of the rotary table at each indexing angle.  
stations was 11; the angle range of rotary table was [\(-145^{\circ}\),\(155^{\circ}\)]; the rotation angle interval of the rotary table was \(30^{\circ}\). Under the above experimental conditions, the conclusions were drawn as follows: The angular positioning deviation of the rotary table was as low as \(\pm 0.9^{\circ}\) compared with other work. Besides, the simple structure made it possible to meet the requirements of fast and efficient calibration. The algorithm error of the laser tracer multi-station measurement system for calibrating angular positioning deviation of the rotary table was \(\pm 0.4^{\prime\prime}\). The experiment proved the feasibility of the proposed calibration method. In actual industrial measurement, the angular positioning deviation demand of high-precision rotary tables is low.  
Although there is a limitation for this case that the rotary table should not be linked with the CMM, it is especially suitable for calibration of angular positioning deviation of large high-precision rotary tables. It also makes a theoretical preparation for the calibration of four-axis machine tools.  
Data curation, H.C. and B.J.; Investigation, H.C.; B.J. and H.L.; Methodology, H.C. and B.J.; Resources, Z.S.; Writing--original draft, S.Z., H.S. and Y.S.; Writing--review & editing, S.Z., H.S. and Y.S.  
The research was funded by National Key Research and Development Program \"Making basic technology and key components\" special project under Grant No. 2018 YFB2001402, Natural Science Foundation of Beijing, China under Grant No.3182005, National Natural Science Foundation of China under Grant No. 51635001.  
The authors declare no conflict of interest.