Six Degrees of Freedom Displacement Measurement System for Wafer Stage Composed of Hall Sensors
Abstract
:1. Introduction
2. Description of the Measurement System
3. Model Derivation
4. Matrixes Solution
4.1. Known Conditions
- XLaser, VHall can be obtained through calibration fixtures. Assuming that the number of calibration data points is N, the known point sequences are then {XLaser1,⋯XLaserk,⋯XLaserN} and {VHall1,⋯VHallk,⋯VHallN}.
- According to the mechanical structure of the device, the corresponding initial coordinates of Hall sensor probes are as follows (unit: mm):
4.2. Calibration Process
5. Experimental Results and Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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VHall (V) | a | b | c | d | e | MSE (×10−5) |
---|---|---|---|---|---|---|
−0.9000 | −4.8730 | 17.2287 | 133.4082 | 3.7653 | 0.4869 | 1.5882 |
−0.8300 | −4.9842 | 18.3401 | 135.3778 | 3.7967 | 0.4841 | 2.2378 |
−0.7600 | −4.5271 | 14.9682 | 139.7482 | 3.8022 | 0.4760 | 1.9036 |
−0.6900 | −4.4135 | 14.3637 | 137.7598 | 3.8248 | 0.4749 | 2.4322 |
−0.6200 | −3.7553 | 10.2714 | 139.4580 | 3.8035 | 0.4903 | 1.5774 |
−0.5500 | −3.5330 | 9.2171 | 145.8678 | 3.8191 | 0.4897 | 2.4767 |
−0.4800 | −3.1521 | 7.3729 | 147.0197 | 3.8213 | 0.5007 | 2.8644 |
−0.4100 | −2.7517 | 5.7210 | 151.4560 | 3.8091 | 0.5302 | 1.9099 |
−0.3400 | −2.1845 | 3.6350 | 151.6554 | 3.7735 | 0.5727 | 1.5192 |
−0.2700 | −1.7248 | 2.2788 | 151.2365 | 3.7611 | 0.5942 | 1.0114 |
−0.2000 | −1.2779 | 1.2677 | 150.9600 | 3.7432 | 0.6263 | 0.6854 |
−0.1300 | −0.7836 | 0.4641 | 151.4696 | 3.7438 | 0.5976 | 0.7116 |
−0.0600 | −0.3627 | 0.1004 | 153.1795 | 3.7372 | 0.6069 | 0.5898 |
0.0700 | 0.5670 | 0.2498 | 169.8377 | 3.8551 | 0.5516 | 2.6307 |
0.1400 | 0.9881 | 0.7461 | 156.1893 | 3.8146 | 0.5524 | 2.3675 |
0.2100 | 1.3435 | 1.3703 | 150.0127 | 3.7724 | 0.5773 | 1.5292 |
0.2800 | 1.9259 | 2.8212 | 151.7349 | 3.8086 | 0.5489 | 2.2555 |
0.3500 | 2.2771 | 3.9673 | 148.0845 | 3.7739 | 0.5801 | 1.6254 |
0.4200 | 2.6714 | 5.4854 | 145.5625 | 3.7626 | 0.5844 | 0.9534 |
0.4900 | 3.2424 | 7.9646 | 149.3453 | 3.8018 | 0.5338 | 1.9244 |
0.5600 | 3.4764 | 9.0506 | 141.6614 | 3.7855 | 0.5306 | 1.6327 |
0.6300 | 3.7473 | 10.5033 | 139.3210 | 3.7654 | 0.5412 | 1.2954 |
0.7000 | 4.1856 | 13.1069 | 139.8891 | 3.7720 | 0.5298 | 1.4367 |
0.7700 | 4.5792 | 15.3600 | 136.9817 | 3.8022 | 0.4724 | 2.2432 |
0.8400 | 5.2717 | 20.8156 | 142.4728 | 3.7998 | 0.4976 | 1.8012 |
0.9100 | 5.6072 | 22.8467 | 136.7349 | 3.8317 | 0.4365 | 1.4231 |
Aij (V/mm) | Aj1 | Aj2 | Aj3 | Aj4 | Aj5 | Aj6 |
---|---|---|---|---|---|---|
A1j | −1.5397 | −0.1037 | −0.3315 | −1.4055 | −1.3080 | −6.24751 |
A2j | 0.5659 | 2.2981 | 0 | 1.0508 | 0.5097 | 11.1114 |
A3j | 0.5148 | 0 | 3.6293 | 1.8000 | 1.3307 | 14.0375 |
A4j | 0.3224 | 0.9862 | −0.7829 | −2.5066 | 0 | 0 |
A5j | 0.2864 | 0.022993 | −0.5654 | 0 | 1.4780 | 0 |
A6j | 6.9354 | 5.1245 | −12.6979 | 0 | 0 | 14.7979 |
Bij (mm) | Bj1 | Bj2 | Bj3 | Bj4 | Bj5 | Bj6 |
---|---|---|---|---|---|---|
B1j | 20.3677 | −36.1377 | −43.1450 | 3.9887 | 4.4567 | −38.7739 |
X (mm) | Y (mm) | Z (mm) | θx | θy | θz | |
---|---|---|---|---|---|---|
(1) | 0.04 | 0.14 | 0.12 | 0.025° | 0.025° | 0.04° |
(2) | 0.04 | 0.13 | 0.16 | 0.04° | 0.03° | 0.05° |
(3) | 0.06 | 0.15 | 0.23 | 0.03° | 0.06° | 0.07° |
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Zhao, B.; Shi, W.; Zhang, J.; Zhang, M.; Qi, X.; Li, J.; Li, F.; Tan, J. Six Degrees of Freedom Displacement Measurement System for Wafer Stage Composed of Hall Sensors. Sensors 2018, 18, 2030. https://doi.org/10.3390/s18072030
Zhao B, Shi W, Zhang J, Zhang M, Qi X, Li J, Li F, Tan J. Six Degrees of Freedom Displacement Measurement System for Wafer Stage Composed of Hall Sensors. Sensors. 2018; 18(7):2030. https://doi.org/10.3390/s18072030
Chicago/Turabian StyleZhao, Bo, Weijia Shi, Jiawei Zhang, Ming Zhang, Xue Qi, Jiaxin Li, Feng Li, and Jiubin Tan. 2018. "Six Degrees of Freedom Displacement Measurement System for Wafer Stage Composed of Hall Sensors" Sensors 18, no. 7: 2030. https://doi.org/10.3390/s18072030
APA StyleZhao, B., Shi, W., Zhang, J., Zhang, M., Qi, X., Li, J., Li, F., & Tan, J. (2018). Six Degrees of Freedom Displacement Measurement System for Wafer Stage Composed of Hall Sensors. Sensors, 18(7), 2030. https://doi.org/10.3390/s18072030