# A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array
## 3 Calibration Method
### Verification Experiments  
Based on the characteristics of the composite module array that can be used to measure the four degrees of freedom error of the machine tool, we used the uncalibrated and calibrated composite module array to measure the CMM spindle's angle error and position error to verify the correctness of the calibration results. The CMM spindle drove the angle sensor to move along the X direction to scan the composite module array on the workbench, in which each composite module selected four measurement positions. At the same time, the laser interferometer was used to monitor the movement of the spindle. Comparing the two measurement results of the angle sensor and the laser interferometer, the error curve can be obtained, as shown in Figures 11 and 12.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Composite Module** & **Position in the X Direction /mm** & **Position in the Y Direction /mm** \\ \hline
1 & 0 & 0 \\
2 & 50.0972 & \(-\)0.0421 \\
3 & 100.0294 & 0.0484 \\
4 & 149.9785 & 0.0374 \\
5 & 199.9603 & 0.0261 \\
6 & 250.0125 & 0.0424 \\
7 & 299.9843 & \(-\)0.0393 \\
8 & 349.9831 & 0.008 \\
9 & 399.9664 & \(-\)0.0281 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Calibration results of the relative position.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Composite Module** & **Standard Deviation of the** & **Standard Deviation of the** \\  & **Position in the X Direction /\(\upmu\)m** & **Position in the Y Direction /\(\upmu\)m** \\ \hline
1 & 0 & 0 \\
2 & 0.28 & 0.49 \\
3 & 0.29 & 0.61 \\
4 & 0.40 & 0.59 \\
5 & 0.31 & 0.48 \\
6 & 0.35 & 0.58 \\
7 & 0.30 & 0.52 \\
8 & 0.36 & 0.52 \\
9 & 0.35 & 0.54 \\ \hline \hline \end{tabular}
\end{table}
Table 15: The standard deviation of the relative position.  
Root mean square (RMS) error was used to evaluate measurement accuracy. Compared with the uncalibrated array, the RMS error of the angle measurement around the X-axis of the calibrated array was reduced from 45.25\({}^{\prime\prime}\) to 0.43\({}^{\prime\prime}\), the RMS error of the angle measurement around the Y-axis was reduced from 26.22\({}^{\prime\prime}\) to 0.33\({}^{\prime\prime}\), and the total RMS error was reduced from 52.30\({}^{\prime\prime}\) to 0.54\({}^{\prime\prime}\). Meanwhile, the RMS error of the position measurement in the X direction was reduced from 93.95 to 0.34 \(\upmu\)m, the RMS error of the position measurement in the Y direction was reduced from 36.05 to 0.55 \(\upmu\)m, and the total RMS error was reduced from 100.63 to 0.65 \(\upmu\)m.  
Verification experiments satisfied the accuracy requirements of machine tool error measurement and eliminated the errors caused by the uncalibrated array. Therefore, it can be considered that the calibration data were reliable and guaranteed the accuracy of the composite reference array application. The measurement error of the calibrated composite module array mainly consisted of two aspects:  
1. Since the composite reference piece was made of metal, there was an oxidation problem with an extended time of use, which affected the surface shape of the composite reference piece.
2. The measurement with higher accuracy put forward higher requirements on the quality of the light source and other optical components of the angle sensor.  
Figure 11: Verification experiments with the uncalibrated composite module array: (**a**) comparison experiments for the angle; (**b**) comparison experiments for the position.  
Figure 12: Verification experiments with calibrated composite module array: (**a**) comparison experiments for the angle; (**b**) comparison experiments for the position.