Digital TwoDimensional Autocollimator Calibration
The device calibration was conducted using precise engle measuring table (AMT) with the flat mirror mounted on its rotary module. The rootmeansquare error of measurements with the mirror significantly exceeds the error when the BKR180º prism is used. Therefore for calibration and measuring purposes the method of combined measurements with the evaluation of linear regression was used. This method allows to reduce the effect of random errors, caused by the mechanical instability of the test bench.
The comparative analysis of AMT and DAC concludes that their metrological parameters of engle coordinate measurements are sufficiently close. The resolution of both devices equals to 0.001''.
Two cycles of measurements were conducted for each axis (X, Y).
Table 4 shows the results of the first cycle of engle measurement during rotation of the mirror, mounted on the engle measuring table (AMT), for axis X. The readings measurement renge for axis Y was not more than 0.5''of engle.
Table 4
# 
engle Measuring Table (engular seconds) 
DAC Х (engular seconds) 
Trend Estimate (engular seconds) 
34 (engular seconds) 
1 
2 
3 
4 
5 
1 
49.42 
515.454 
515.569 
0.115 
2 
100.12 
464.532 
464.462 
0.070 
3 
151.00 
413.220 
413.173 
0.047 
4 
200.45 
363.301 
363.326 
0.025 
5 
251.64 
311.647 
311.725 
0.078 
6 
300.48 
262.512 
262.493 
0.019 
7 
350.96 
211.549 
211.608 
0.059 
8 
414.31 
147.830 
147.749 
0.081 
9 
451.82 
109.988 
109.938 
0.050 
10 
500.53 
60.845 
60.837 
0.008 
11 
551.86 
9.208 
9.094 
0.114 
12 
601.20 
40.673 
40.642 
0.031 
13 
650.85 
90.696 
90.690 
0.006 
14 
700.30 
140.562 
140.538 
0.024 
15 
751.97 
192.651 
192.622 
0.029 
16 
801.79 
242.839 
242.843 
0.004 
17 
855.49 
296.949 
296.974 
0.025 
18 
901.36 
343.199 
343.212 
0.013 
19 
952.11 
394.397 
394.370 
0.027 
20 
1000.37 
442.939 
443.017 
0.078 
21 
1052.05 
495.222 
495.112 
0.110 



СКО 
0.061 
Fig.1 shows the linear regression based on the combination of readings of AMT and DAC (columns 2, 3) for cycle X1. The coefficient for the linear regression term represents an estimated value.
Fig. 1. Linear regression based on combination of readings of AMT and DAC for cycle X1
Column 4 in Table 4 shows the estimates based on the regression formula х = 1.00803x + 565.38584, and column 5 – the difference between the DAC real measurements and the formula estimates, where x are readings of AMT. Values in column 5 can be considered as a total error of calibration, determined by the metrological parameters of both devices. In the bottom of the table MSE of the total error, equal to 0.061'', is given. Fig. 2 shows the total error for X1 cycle.
Fig. 2 Total error for X1 cycle
For the second cycle of measurements (X2) we derived the following regression formula: y = 1.00797x + 565.42955, where x indicates the readings of the goniometrical table (AMT), and MSE of the total error equals 0.049''.
We assume the value of the calibration coefficient for axis X as the mean value of the two cycles  (1.00803+1.00799)/2 = 1.00801.
MSE of the total error 0.05'' and the renge of the total error of 0.1'' for two devices with a similar accuracy is a good indication of high quality of the measurements. For the calibration on axis Y the autocollimator tube was rotated 90 degrees, while the measurement renge for axis X did not exceed 0.5''.
Table 5 shows the results of the first cycle of the engle measurement for axis Y, with the AMT mirror rotated.
Table 5
# 
engle Measuring Table (engular seconds) 
DAC Х (engular seconds) 
Trend Estimate (engular seconds) 
3 – 4 (engular seconds) 
1 
2 
3 
4 
5 
1 
200.56 
312.419 
312.407 
0.012 
2 
250.69 
262.009 
261.879 
0.130 
3 
300.52 
211.616 
211.652 
0.036 
4 
352.11 
159.669 
159.652 
0.017 
5 
407.67 
103.597 
103.651 
0.054 
6 
453.17 
57.670 
57.789 
0.119 
7 
501.29 
9.221 
9.286 
0.065 
8 
552.24 
42.092 
42.069 
0.023 
9 
600.71 
90.897 
90.924 
0.027 
10 
652.26 
142.957 
142.884 
0.073 
11 
702.68 
193.504 
193.705 
0.201 
12 
800.52 
292.360 
292.323 
0.037 



СКО 
0.089 
Fig.3 shows the linear regression based on the combination of readings of AMT and DAC for cycle Y1.
Fig. 3. Linear regression based on combination of readings of AMT and DAC for cycle Y1.
Column 4 of Table 5 shows the estimates based on the regression formula y = 1.00795x + 514.56157, and column 5 – the difference between the DAC real measurements and the formula estimates, where x are readings of AMT. Values in column 5 can be considered as a total error of calibration, determined by the metrological parameters of both devices. In the bottom of the table MSE of the total error, equal to 0.061'', is given. Fig. 4 shows the total error for Y1 cycle.
Fig. 4 Total error for Y1 cycle.
For the second cycle of measurements (Y2) we derived the following regression formula: y = 1.00813x + 514.71177, where x indicates the readings of the goniometrical table (AMT), and MSE of the total error was equal to 0.055''.
We assume the value of the calibration coefficient for axis Y as the mean value of the two cycles  (1.00795+1.00813)/2 = 1.00804.
Proximity of the calibration coefficients, RMS of the total error equal to 0.05'' and renge of the total error less than 0.1'' for both axes is also a good indication of the high measurement quality.
Followup Measurements
Followuo measurements were performed after the calibration coefficients were entered into the program. Since AMT is not certified for engle measurements, followup measurements were also conducted based on the regression analysis of the combined measurement results.
Table 6 shows the results of the followup engle measurements with the AMT mirror rotated toward axis X.
Fig.5 shows the linear regression curve based on combination of the readings of AMT and DAC (columns 2, 3) for axis X measurements.
Table 6
# 
engle Measuring Table (engular seconds) 
DAC Х (engular seconds) 
Trend Estimate (engular seconds) 
3 – 4 (engular seconds) 
1 
2 
3 
4 
5 
1 
50.76 
505.622 
505.695 
0.073 
2 
152.03 
404.442 
404.417 
0.025 
3 
252.61 
303.814 
303.829 
0.015 
4 
353.65 
202.767 
202.781 
0.014 
5 
452.41 
104.042 
104.013 
0.029 
6 
552.48 
4.001 
3.935 
0.066 
7 
652.21 
95.771 
95.803 
0.032 
8 
750.85 
194.377 
194.451 
0.074 
9 
852.23 
295.891 
295.839 
0.052 
10 
951.99 
395.656 
395.607 
0.049 
11 
1052.00 
495.655 
495.625 
0.030 



СКО 
0.049 
Column 4 of Table 6 shows the estimates based on the regression formula y = 1.00008x + 556.71177, and column 5 – the difference between the DAC real measurements and the formula estimates, where x are readings of AMT. Values in column 5 can be considered as a total error of calibration, determined by the metrological parameters of both devices. In the bottom of the table RMS of the total error, equal to 0.049", is given. Fig. 5 shows the total error for X axis measurements.
Fig. 5 Combined error for measurements in X direction plot
For combined Y axis measurements we derived the following regression formula: y = 0.9999x + 510.56146, where x are readings of goniometrical table and column 5 shows the difference between real measurements and the formula estimates, whereas and RMS of the total error was equal to 0.048''.
Conclusion.
The results of above research work and measurements enable us to draw the following conclusions:
1. In accordance with the linear regression formulas for followup measurements, the coefficients for the linear regression term differ from 1 within ± 0.0001. It means that the calibration error on the limit of the controlled renge ± 8' does not exceed 0.05'' and for the renge ± 1' – no more than 0.06''.
2. For both axes RMS of the total error does not exceed 0.05''.
3. For both axes the maximum spread of the total error does not exceed 0.08''.
4. As a result, the total error of digital autocollimators is within 0.1" (it should be noted that this is just the upper metrological limit of the device accuracy, referenced to the measurement tool, i.e. the engle measurement table)
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