Open Access
Add to BibSonomy
Abstract
A novel measurement system for a high-precision roll angle measurement of long working distance on the basis of two parallel beams in association with two detectors is presented. The measurement system consists of a light source part and a detecting part. The light source part uses transmission grating and a plane mirror to produce a pair of high-precision parallel beams. The nonparallelism of the dual beam caused by the installation error can be compressed to ensure the measurement system achieves high-precision measurement and long working distance. The effectiveness of the measurement system and proposed methods are demonstrated by a series of experiments. The resolution of 0.5′′ and measurement accuracy of 1.1′′ can be obtained by the set-up measurement system.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Precision measurement plays an important role in precision machines, precision manufacturing and precision assembly. There are six geometric errors need to be measured with high accuracy and resolution in these fields, namely one positioning error, two sraightness errors and three angular errors (pitch, yaw and roll). Currently, based on conventional optical methods, positioning error and two sraightness errors are usually measured using interferometer, pitch and yaw are usually measured using autocollimator. However, the roll is the most challenging geometric error to be measured using optical methods.
Currently, some methods based on the different principles and structures are present for the roll measurement. Monocular vision method [1,2], interferometric method [3,4], polarization variation method [5–10], autocollimation method [11,12] and parallel beams method [13–16] are the typical representative among all the methods. Monocular vision method can achieve large measurement range but low accuracy. Interferometric method can ahieve high resolution with complex signal processing system, however, it is not suitable for long working distance measurement because of the non-parallelism of the two measuring beams. Polarization variation method has large measurement range but low resolution, it is not suitable for high-precision measurement occasions. Autocollimation method is simple and can ahieve high resolution and accuracy, but the mothod is difficult to be applied in industrial measurement because the working distance is too short. Compared to the measurement method mentioned above, the parallel beams method fully takes into account the resolution, accuracy, complexity and working distance, the method is a more balanced measurement method and suitable for industrial fields. However, the measuring beams cannot be parallel due to machining error and installation error. Based on the measurement principle, the non-parallelism of the measuring beams will severely limit the measurement accuracy and measurement range. Improving the parallelism of the measuring beams is a direct and effective method to improve the measurement accuracy and working distance. Hence, this paper proposed a novel parallel beam generation method and structure which can greatly improve the parallelism of the two measuring beams under the condition that the machining error and installation error remain unchanged. At the same time, to reduce parallelism error, a method of parallelism error measurement is proposed to guide system installation.
2. Principle and analysis
2.1 Measurement principle
Typical roll measurement method based on a pair of parallel beams is shown in Fig. 1. Two collimation beams are incident perpendicular to the PSDs (position sensitive detector) fixed on the target to be detected. When the target rotates around the optical axis with γ, the positions of the light spots on the PSDs will change accordingly. Assuming the displacement of light spot in vertical direction on two PSDs is Δy1 and Δy2, respectively. Assuming the distance between two measuring beams is d. Because the roll γ is small angle, it can be expressed as
Based on Eq. (1), the measurement accuracy of roll angle is mainly determained by the measurement accuracy of the displacement of the light spot. To obtain higher precision measurement results, accurate measurement of distance d is also necessary. Considering to achieve the long working distance, the angle between the two measuring beams in both horizontal and vertical direction should be decrease as samll as possible.2.2 Error analysis
The non-parallelism of the measuring beams is the main source of the roll angle measurement error. The non-parallelism of the the measuring beams can be represented by parallelism error, namely the angle between the two measuring beams in horizontal and vertical direction. Assuming the angle between the two measuring beams is θh and θv in horizontal and vertical direction, respectively. Assuming the working distance is L, the actual value of roll angle γa can be expressed as
Compare Eq. (1) and Eq. (2), it can be known that the measurement accuracy of the roll angle depends on the value of θh, θv and L. The value of L is determined by actual work requirements and is not the source of measurement error. Hence, reducing the value of θh and θv is the key to improving measurement accuracy. At the same time, reducing the value of θh and θv can also prevent the measuring beam from overflowing the PSD at long working distances, which will cause the measuring system to fail as shown in Fig. 2. Figure 2(a) and Fig. 2(b) shows the schematic diagram of the measurement system when parallelism error exists in horizontal and vertical direction, respectively. At Position 1, two measuring beams can be detected by position sensitive detectors (PSD1 and PSD2). With increasing working distance, at Position 2, one measuring beam will overflow the PSD in horizontal or vertical direction and cause measurement failure. Therefore, reducing the parallelism error both in horizontal and vertical direction is the key to increasing the working distance.
In the actual measurement system, it is difficult to generate two ideally parallel laser beams due to the existence machining error and installation error. The non-parallelism of two measuring beams caused by machining error and installation error in typical parallel beams acquisition system described in [13–16] as shown in Fig. 3. In Fig. 3(a), the parallel beams are obtained by PBS and mirror, the installation error of mirror is θix1 and θiy1 in horizontal and vertical direction, respectively. Based on reflection principle, the angle between the two measuring beams is θh1 and θv1 in horizontal and vertical direction, respectively. They can be expressed as
In Fig. 3(b), the parallel beams are obtained through splitting pentagonal prism (SPP) and pentagonal prism (PP). Based on the characteristics of the pentaprism prism, the measurement error caused by the installation error can be eliminated, but the measurement error caused by the machining error of the pentaprism is inevitable in horizontal direction. Assuming the machining error of the SPP and PP is θmx1 and θmx2, respectively. The angle between the two measuring beams in horizontal direction can be given
In contrast, the installation error in the vertical direction cannot be ignored, assuming the installation error of the PP is θiy2, the angle between the two measuring beams in vertical direction can be given
As shown in Eqs. (3), (4) and (5), the parallelism error is twice the installation error and machining error. It means that the parallel beam generation method mentioned above will amplify machining errors and installation errors which will produce a greater measurement error. Hence, proposing a novel parallel beam generation method which can reduce the parallelism error while the machining error and installation error remain unchanged is the key to improving measurement accuracy.2.3 Principle of the roll angle measurement system based on the novel parallel beam generation method
As shown in Fig. 4, a roll angle measurement system based on a novel parallel beams acquisition system is designed for high accuracy and long working distance roll angle measurement by reducing parallelism error. The system is composed of two parts: light source part and detecting part. The light source part is used to generate high parallelism parallel measuring beams. The detecting part is mounted on the object to be measured for detecting roll angle. As shown in Fig. 4, beam I0 from laser diode (LD) is projected onto a transmission grating after being reflected by a rereflective mirror (RM). After being diffracted by the transmission grating, the ±1-order diffraction beams I1 and I-1 are projected onto a mirror whose reflective surface is parallel to the transmission grating. Then the two reflected beams Id1 and Id-1 from mirror are projected onto TG, the -1-order diffraction beam Im1 of incident beam Id1 and the +1-order diffraction beam Im-1 of incident beam Id-1 are selected as the measuring beams. The two measuring beams are projected onto the detecting part, perpendicularly. The detecting part is composed of two PSDs for detecting the position displacement of light spots formed by measuring beams.
2.4 Principle of the parallel beam generation method
As mentioned above, the installation error can not be avoided, but it can be compressed by the parallel beams acquisition system. As shown in Fig. 5(a), when there is no installation error in the system, the angle between I1 and Z-axis in horizontal direction is θ1 or θ2, the angle between Id1 and Z-axis in horizontal direction is θ3 or θ4. Based on diffraction principle, +1-order diffraction angle can be given as
where λ is wavelength of LD, p is grating constant of transmission grating.
Fig. 5. Schematic of parallel beams acquisition system: (a) without installation error; (b) installation error in horizontal direction; (c) installation error in vertical direction.
Based on reflection principle, θ4 can be given as
Assume the between Im1 and Z-axis in horizontal direction is θm1, the diffraction principle is shown as Eq. (8) where θm is the diffraction angle of m order outgoing beam, θi is the incident angle, m is diffraction order.As shown in Fig. 5(a), beam Id1 is the incident beam, the incident angle is -θ4, using Eq. (7), Eq. (8) can be simplified to
It means that Im1 is parallel to Z-axis. Similarly, Im-1 is also parallel to Z-axis. Hence, Im1 is parallel to Im-1.
2.5 Parallelism error analysis in horizontal direction
Assuming that the TG is fixed and used as a reference for other device installation errors. Figure 5(b) shows the change of beam propagation direction when there is installation error. The installation error of RM is θix1 and the installation error of mirror is α. The incident angle of beam I0 relative to TG is θi, it can be expressed as
Based on diffraction principle, +1-order diffraction angle θd1 and -1-order diffraction angle θd-1 can be expressed as
The diffraction beams I1 and I-1 are then reflected by mirror with installation error α, the incident angles of the second-time diffraction are θr1 and θr-1 and they can be given as Beam Id1 and Id-1 are then diffracted by TG again, the -1-order diffraction angle θm1 of beam Id1 and +1-order diffraction angle θm-1 of beam Id-1 are θm1 and θm-1, they can be given as Because θm1 and θm-1 are small angles, using Eqs. (11), (12), (13), (14), (15) and (16), the parallelism error θh3 of beam Im1 and Im-1 can be givenIn actual installation, the installation error can be limited within ±1800′′. Hence, the installation error α and θix1 are both less than 1800′′. The λ and p in the measurement system are 0.632 µm and 4 µm, respectively. The simulation results of Eq. (17) are shown in Fig. 6. The maximum and minimum of parallelism error θh3 are 30.04′′ and -10.18′′, respectively. When the installation error is the same, based on Eq. (3), the maximum parallelism error θh1 as shown in Fig. 3 is 3600′′. Hence, the parallelism error has been compressed about 120 times in the proposed method compared with the method in Fig. 3(a) by comparing the maximum of θh3 and θh1. It means that under the same conditions, the working distance has increased by 120 times.
By querying the parameters of commercial PP and SPP, the machining error θix1 and θix2 is ±60′′. Based on Eq. (4), the parallelism error θh2 can be ±240′′. The parallelism error θh2 caused by machining error is inevitable and unadjustable. Relatively, the parallelism error of the proposed system can be measured and adjusted, it will be introduced below.
2.6 Parallelism error analysis in vertical direction
The parallelism error in vertical direction may caused by the installation error of RM and mirror in vertical direction as shown in Fig. 5(c). Because the TG has no diffraction in the vertical direction, the measuring beams are only reflected by RM and mirror. Based on reflection principle, the change in the propagation direction of the two measuring beams are consistent. So, there is no parallelism error in vertical direction caused by installation error.
Another factor that may cause parallelism errors are the angular errors of the RM and mirror around Z-axis. When the RM has an angular error around Z-axis, there will be angular errors in horizontal and vertical directions of beam I0. The angular errors in horizontal and vertical directions as shown in Fig. 5(b) and Fig. 5(c) have been analysed. When the mirror has an angular error around Z-axis, there will be no angular errors in horizontal or vertical directions because of the characteristics of the mirror.
Hence, as analysed above, the parallelism error in horizontal direction can be compressed about 120 times and parallelism error in vertical direction can be completely eliminated in the proposed system. So, Eq. (2) can be simplified to
where θhc is the parallelism error in horizontal direction after being compressed about 120 times.Compare Eq. (18) and Eq. (2), the vertical parallelism error θv as the main factor affecting measurement accuracy is completely eliminated. As the secondary factor affecting measurement accuracy, the horizontal parallelism error θh is greatly compressed. Hence, the measurement accuracy of roll angle can be greatly improved and the working distance can be greatly increased.
According to Eq. (18), the only parallelism error is θhc, hence, the theoretical working distance is depend on θhc and the effective area of PSD. Assume the size of the PSD is D×D mm, the theoretical working distance Lt can be given
According Eq. (18) and Fig. 1, the maximum value of Δy1 and Δy2 is D, the values of Δy1 and Δy2 are always opposite, hence, the theoretical roll angle measurement range γt can be given3. Principle of parallelism error measurement
To compensate the parallelism error and verify the effectiveness of the proposed parallel beam generation method, a parallelism error measurement method is proposed as shown in Fig. 7. The measuring beams Im1 and Im-1 from light soure part are detected by CCD (charge coupled device) after being focus by focus lens. Using autocollimation principle, θm1 and θm-1 can be given as
where x1 and x2 are horizontal positions of light spots of beam Im1 and Im-1, f is the focal length of focus lens.The parallelism error of beam Im1 and Im-1 in horizontal direction can be given as
Similarly, the parallelism error of beam Im1 and Im-1 in vertical direction can be given as where y1 and y2 are vertical positions of light spots of beam Im1 and Im-1.The measurement result of the parallelism error can be used to guide the installation of the mirror until the installation error meets the requirements.
4. Experiment
A prototype of the roll angle measurement system and parallelism error measurement system based on Fig. 4 and Fig. 7 are constructed as shown in Fig. 8. The parallelism error is measured by parallelism error measurement system. The PSDs are fixed on an angular displacement stage (ADS). The resolution and accuracy of the roll angle measurement system are tested.
Fig. 8. Prototype of the roll angle measurement system: (a) roll angle measurement system; (b) parallelism error measurement system.
4.1 Measurement of the parallelism error
A prototype of parallelism error measurement system is constructed as shown in Fig. 8(b). The displacement resolution and effective area of the CCD is 200 nm and 10×10 mm. The focal length f is 400 mm. Hence, the angular displacement resolution and measurement range of the parallelism error measurement system can be 0.1′′ and 5000′′.
The parallelism errors are measured ten times in 100s, the measurement results are shown in Table 1. The average value of x1 and x2 are 153.2 µm and 159.5 µm. The average value of y1 and y2 are 55.3 µm and 55.7 µm, respectively. Hence, based on Eqs. (23) and (24), the parallelism error in horizontal and vertical direction are 3.15′′ and 0.2′′, respectively. The parallelism error in vertical direction may caused by air disturbance and measurement error of CCD.
Table 1. The results of parallelism errors measurement
4.2 Resolution test
To test the resolution of the roll measurement system, the two PSDs are fixed on an angular displacement stage. The minimum angular displacement and angular movement range og the stage are 0.01′′ and 70′′, respectively. The distance d between Im1 and Im-1 is 40 mm, it can be adjusted by changing the distance between the mirror and the grating. The displacement resolution of PSD is 0.1 µm, therefore the resolution of the roll angle measurement is 0.5′′, theoretically. In the experiment, the step size of the moving stage is 0.5′′. The corresponding time interval for each step was 10 s. The results of resolution test are shown in Fig. 9. The results show that the output of the roll angle γ can reveal each step clearly, which indicates that the resolution of the system is less than 0.5′′.
4.3 Calibration experiment
A schematic diagram of calibration experiment setup is shown in Fig. 10. The propoed system is calibrated by a commercial autollimator, the resolution and the accuracy of the autocollimator is 0.01′′ and 0.25′′, respectively. The two PSDs and a mirror are fixed on the angular displacement stage (ADS), perpendicularly. The mirror is the angle sensor of the commercial autollimator, the beam from the autocollimator is incident perpendicular to the mirror. The pitch angle detected by the autocollimator is the same as the roll angle detected by the proposed system and they can be read out simultaneously. The calibration result is shown in Fig. 11, it can be seen that the residuals are from −1.0′′ to 1.1′′ within the range of 70′′ and the standard deviation in roll is 0.7′′.
5. Conclusion
To achieve long working distance and high-precision roll angle measurement, a novel parallel beam generation method for roll angle measurement is presented. The method uses the characteristics of grating and mirror to generate a pair of parallel beams. The parallelism error of the parallel beams in the vertical direction can be completely eliminated as the main factor affecting measurement accuracy. The parallelism error of the parallel beams in the horizontal direction can be effectively compressed. In order to further reduce the parallelism error, a method of parallelism error measurement is proposed as a guide for system installation. A series of experiments have been carried out to verify the feasibility of this method. The resolution and accuracy of the proposed system can be 0.5′′ and 1.1′′.
Funding
National Natural Science Foundation of China (52075131).
Disclosures
The authors declare no conflict of interest.
References
1. H. X. Dong, Q. Fu, X. Zhao, Q. Quan, and R. F. Zhang, “Practical rotation angle measurement method by monocular vision,” Appl. Opt. 54(3), 425–435 (2015). [CrossRef]
2. W. Li, J. Jin, X. Li, and B. Li, “Method of rotation angle measurement in machine vision based on calibration pattern with spot array,” Appl. Opt. 49(6), 1001–1006 (2010). [CrossRef]
3. K. Shi, J. Su, and W. Hou, “Roll angle measurement system based on differential plane mirror interferometer,” Opt. Express 26(16), 19826–19834 (2018). [CrossRef]
4. T. Jin, G. Xia, W. Hou, Y. Le, and S. Han, “High resolution and stability roll angle measurement method for precision linear displacement stages,” Rev. Sci. Instrum. 88(2), 023102 (2017). [CrossRef]
5. J. Qi, Z. Wang, J. Huang, Q. Wang, and J. Gao, “Heterodyne interferometer with two parallel-polarized input beams for high-resolution roll angle measurement,” Opt. Express 27(10), 13820–13830 (2019). [CrossRef]
6. P. Z. Zhang, Y. C. Wang, C. F. Kuang, S. Li, and X. Liu, “Measuring roll angle displacement based on ellipticity with high resolution and large range,” Opt. Laser. Technol. 65, 126–130 (2015). [CrossRef]
7. S. R. Gillmer, X. Yu, C. Wang, and J. D. Ellis, “Robust high-dynamic-range optical roll sensing,” Opt. Lett. 40(11), 2497–2500 (2015). [CrossRef]
8. S. Tang, Z. Wang, M. Li, J. Gao, Y. Jiang, and Y. Zhang, “Note: Optimal choice of the reflector by phaseanalysis for heterodyne interferometric roll angle measurement,” Rev. Sci. Instrum. 87(2), 026101 (2016). [CrossRef]
9. E. Plosker and S. Arnon, “Statistics of remote roll angle measurement,” Appl. Opt. 53(11), 2437–2440 (2014). [CrossRef]
10. S. Tang, Z. Wang, M. Li, W. Zhang, F. Yang, and X. Zhang, “A small roll angle measurement method with enhanced resolution based on a heterodyne interferometer,” Rev. Sci. Instrum. 86(9), 096104 (2015). [CrossRef]
11. W. Gao, Y. Saito, H. Muto, Y. Arai, and Y. Shimizu, “A three-axis autocollimator for detection of angular error motions of a precision stage,” CIRP Annals 60(1), 515–518 (2011). [CrossRef]
12. R. Li, M. Zhou, I. Konyakhin, K. Di, Y. Lu, J. Guo, and Y. Liu, “Cube-corner autocollimator with expanded measurement range,” Opt. Express 27(5), 6389–6403 (2019). [CrossRef]
13. Y. S. Zhai, Z. F. Zhang, Y. L. Su, X. J. Wang, and Q. B. Feng, “A high-precision roll angle measurement method,” Optik 126(24), 4837–4840 (2015). [CrossRef]
14. Y. Zhu, S. Liu, C. Kuang, S. Li, and X. Liu, “Roll angle measurement based on common path compensation principle,” Opt. Lasers Eng. 67, 66–73 (2015). [CrossRef]
15. T. Zhang, Q. Feng, C. Cui, and B. Zhang, “Research on error compensation method for dual-beam measurement of roll angle based on rhombic prism,” Chin. Opt. Lett. 12(7), 071201–71204 (2014). [CrossRef]
16. Y. D. Cai, B. H. Yang, and K. C. Fan, “Robust roll angular error measurement system for precision machines,” Opt. Express 27(6), 8027–8036 (2019). [CrossRef]
