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摘要
针对工作台运动误差，提出了一种基于计算全息的无衍射光莫尔条纹三自由度测量方法。通过液晶空间光调制器(SLM)生成无衍射光，利用两束无衍射光干涉生成莫尔条纹。设计了无衍射光莫尔条纹三自由度测量光路，建立了三自由度运动误差数学模型，并用几何分析法将三种运动误差(偏摆角、滚转角和俯仰角)进行分离。利用旋转台模拟不同大小的三自由度运动误差，带有误差信息的无衍射光和莫尔条纹图案分别由CCD1和CCD2接收。实验结果表明，通过光斑中心偏移量计算出的实际运动误差值接近理论值，测量误差不超过0.0104°，验证了无衍射光莫尔条纹三自由度测量系统的可行性与正确性。
Abstract
Aiming at the motion errors of the linear stage, a measurement method for the determination of three-degree-of-freedom (3-DOF) error motions is proposed based on non-diffracting Moiré fringes produced by computer-generated holograms (CGHs). A liquid crystal spatial light modulator (SLM) is used to generate non-diffracting beams, and two non-diffracting beams form Moiré fringes. A 3-DOF measuring optical path of non-diffracting Moiré fringes is designed. Meanwhile, a 3-DOF mathematical model of motion errors is established, and three kinds of motion errors (yaw angle, roll angle and pitch angle) are separated by geometric analysis method. A rotary table is used to simulate the 3-DOF motion errors on different conditions. The NDB and non-diffracting Moiré fringe patterns are obtained by CCD1 and CCD2 respectively. Experimental results show that the motion errors calculated by the positions of the central points agree well with the theoretical value with the error less than 0.0104°, which can verify the feasibility and correctness of the 3-DOF measurement system for non-diffracting Moiré fringes.
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Key words:
- 3-DOF measurement /
- non-diffracting beam /
- Moiré fringes /
- SLM
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Overview
Overview: Error motions of a linear stage directly influence the performance of the precision positioning system in which the stage is used. Therefore, it is a critical task to measure the error motions. A measurement method for the determination of three-degree-of-freedom (3-DOF) error motions based on non-diffracting Moiré fringes is proposed.
A semi-transparent mirror (STM), a beam splitter and a mirror are adopted as the measurement head, which is fixed on the moving stage in order to sense 3-DOF angular errors. Two CCDs are used to capture the non-diffracting beams patterns that are carrying the errors. Computer generated holograms (CGHs) are loaded into a liquid crystal spatial light modulator (SLM) to produce non-diffracting beams. A beam splitter prism (BS1), placed after the SLM, splits the non-diffracting beam into two beams, the transmitted beam and the reflection beam. The reflected non-diffracting beam, after reflection by mirror 2, traveling through BS3, is reflected by the STM, and reaches CCD1. The image of the non-diffracting beams is captured by CCD1. The other beam transmits from BS3 to CCD2 after travelling through STM and BS2. The transmitted non-diffracting beam exiting BS1 passes through an attenuator, and then is reflected by mirror 1, from where it meets the beam from the moving unit. These two non-diffracting beams generate non-diffracting Moiré fringes, which are captured by CCD2.
When the stage moves, the position of the central points of the non-diffracting beams (as received by CCD 1) and the forms of Moiré fringes (obtained by CCD 2) will change in relation to different errors. It can measure the 3-DOF errors, which are yaw, pitch, and roll. By analyzing the geometric position of these center points, mathematical models for 3-DOF motion errors are established. Obviously, one of central points in CCD2 will be unchanged while the other one will changes according to different motion errors of stage.
A rotary table is used to simulate the 3-DOF motion errors and demonstrate the theoretical analysis. The measurement head is fixed on the table, which rotates different small angles (0°, 0.125°, 0.25°, 0.375°, 0.5°). The on-diffracting beam and non-diffracting Moiré fringe patterns are obtained by CCD1 and CCD2, respectively. The actual angular displacement is calculated by the offset of the center of the spot with the mathematical models. Compared with the theoretical value, the angular displacement error is less than 0.0104°, which verifies the feasibility and correctness of the 3-DOF measurement system for non-diffracting Moiré fringes.
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图 2 三自由度运动误差测量系统示意图
Figure 2. Schematic diagram of measurement system of 3-DOF error motions
图 3 CCD1中偏摆角误差产生的位移量。
Figure 3. Displacement of the central point generated by the yaw α in CCD1. (a) Schematic diagram; (b) Measuring principle
图 4 CCD1中俯仰角误差产生的位移量。
Figure 4. Displacement of the central point generated by the pitch β in CCD1. (a) Schematic diagram; (b) Measuring principle
图 5 CCD2中偏摆角误差产生的位移量。(a)示意图；(b)简化原理图
Figure 5. Displacement of the central point generated by the yaw α in CCD2. (a) Schematic diagram; (b) Measuring principle
图 6 CCD2中俯仰角误差产生的位移量。(a)模型；(b)原理图
Figure 6. Displacement of the central point generated by the pitch β in CCD2. (a) Schematic diagram; (b) Measuring principle
图 7 CCD2中滚转角误差产生的位移量。(a)示意图；(b)简化原理图
Figure 7. Displacement of the central point generated by the roll γ in CCD2. (a) Schematic diagram; (b) Measuring princip
图 10 实验结果。(a) CCD1；(b) CCD2
Figure 10. Experimental results calculated by the images in (a) CCD1 and (b) CCD2
表 1 CCD1、CCD2中三自由度变化对应的光斑中心
Table 1. Central positions of the non-diffracting beams for 3-DOF in CCD1 and CCD2
0° 0.125° 0.25° 0.375° 0.5° CCD1 Yaw α (577, 500) (578, 521) (578, 540) (577, 560) (579, 579) Pitch β (606, 489) (587, 489) (568, 490) (548, 490) (528, 491) Roll γ (561, 551) (560, 551) (560, 550) (561, 550) (560, 551) CCD2 Yaw α (601, 502) (580, 502) (559, 503) (538, 503) (518, 501) Pitch β (600, 530) (599, 550) (598, 569) (599, 589) (602, 608) Roll γ (611, 532) (590, 531) (568, 532) (547, 533) (527, 533) 
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表 2 根据CCD1、CCD2中图像运动误差计算结果
Table 2. Motion errors calculated by the images in CCD1 and CCD2
Type of errors Variation error/(°) Pixels moving Pixels increment Calculated values/(°) Measuring error/(°) CCD1 Yaw α 0° 0 0 0° 0° 0.125° 21 21 0.1318° 0.0068° 0.25° 40 19 0.1203° -0.0047° 0.375° 60 20 0.1261° 0.0011° 0.5° 79 19 0.1203° -0.0047° Pitch β 0° 0 0 0° 0° 0.125° 19 19 0.1203° -0.0047° 0.25° 38 19 0.1203° -0.0047° 0.375° 58 20 0.1261° 0.0011° 0.5° 78 20 0.1261° 0.0011° CCD2 Yaw α 0° 0 0 0° 0° 0.125° 21 21 0.1318° 0.0068° 0.25° 42 21 0.1318° 0.0068° 0.375° 63 21 0.1318° 0.0068° 0.5° 83 20 0.1203° -0.0047° Pitch β 0° 0 0 0° 0° 0.125° 20 20 0.1203° -0.0047° 0.25° 39 19 0.1146° -0.0104° 0.375° 59 20 0.1203° -0.0047° 0.5° 78 19 0.1146° -0.0047° Roll γ 0° 0 0 0° 0° 0.125° 21 21 0.1318° 0.0068° 0.25° 43 22 0.1353° 0.0103° 0.375° 64 21 0.1318° 0.0068° 0.5° 84 20 0.1203° -0.0047°
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