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Abstract
A robust five-degree-of-freedom (5-DOF) measurement system is proposed in this paper. The compact optical configuration with high resolution is designed based on lens combination and multiple reflections. Beam drift and dual-beam parallelism are monitored and compensated by autocollimator units and a polarizer unit respectively. In addition, a protection method is proposed to reduce the intensity of air turbulence by reducing the Reynolds number of the beam path. The performance of the system is verified by experiments. The experimental results show that the self-compensation methods and air turbulence protection can effectively improve the accuracy and stability of the system under the long-term interference of external environments. The proposed system has high precision, desirable robustness, and convenient pre-calibration, which can be used for error measurement of precision machines.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of the manufacturing industry, higher requirements are put forward for the error measurement accuracy and resolution of precision machines, such as machine tools, coordinate measuring machines, and wafer stages. Any motion axis of precision machines, either linear or rotation, inherently possesses 6-DOF errors, including positioning, two straightness, pitch, yaw, and roll errors [1]. At present, there are some commercial instruments for geometric error measurement such as laser interferometers [2,3]. However, it can only measure a single error with a specified optical accessory each time. Therefore, the error measurement and compensation cannot be completed in real-time. Moreover, it is expensive and difficult to integrate with precision machinery.
The multi-degree-of-freedom measurement (MDFM) system, which detects the geometric errors of precision machines quickly and accurately, has been studied for many years. Fan [4,5] developed a system for simultaneously measuring 6-DOF motion errors, which combines multi-laser Doppler scales with quadrant photodetectors. Kuang [6] proposed a 4-DOF measurement system using a single-mode fiber-coupled laser to improve the stability of the light source. Kuang [7] developed a 5-DOF measurement system combining laser interference and collimation. Fan [8] proposed a compact MDFM system to detect the roll by relative position shift of two parallel laser beams. Zhai [9] proposed a method for 4-DOF error motions measurement based on a non-diffracting beam. Zheng [10] proposed a method for simultaneously measuring 6-DOF errors of the linear axis and the rotary axis. Huang [11] developed a 5-DOF sensor system that can be embedded in each axis of motion as additional feedback sensors of the machine tool. In the above MDFM measurement systems, the collimated dual-beam is the easiest to integrate and the most practical method for roll measurement [12–16].
In summary, the MDFM system based on collimation has been widely used in related fields due to the advantages of a large measurement range, low cost, and high integration. However, it is a major challenge to maintain the high precision and stability of MDFM systems under the long-term interference of external environments. The heat generated by the MDFM system and machine tool leads to thermal deformation and misalignment of optical components, and the glues and bolts will deteriorate after a long time of operation, which will eventually lead to the drift of the measuring beam [17]. Beam drift will also change the parallelism of dual-beam in MDFM systems based on the measurement of roll with the collimated dual-beam. Besides, air turbulence affects the stability of the measurement beam in the propagation path [18]. These conditions will lead to serious measurement errors when the MDFM system is used for a long time in the actual industrial environment. For the problem of beam drift, the previous solution is usually focused on compensating for the beam drift caused by the laser [19], but cannot solve the non-parallelism of dual-beam. Although the non-parallelism could be corrected by different methods [20,21], they are offline processes. Lou [22] compensates for the non-parallel error by a differential method, but it could not compensate for the initial installation error. For the influence of air turbulence on measurement, it is a common method to install shielding in the laser transmission path [23]. However, the relationship between the shape and size of the shield and the performance of air turbulence resistance has rarely been addressed and resolved before.
To solve the influence of beam drift, dual-beam parallelism change, and air turbulence, and improve the accuracy and stability of the MDFM under the long-term interference of external environments, a robust 5-DOF measurement system with beam drift compensation and air turbulence protection is proposed. A compact optical configuration with high resolution is designed based on lens combination and multiple reflections. The beam drift is compensated by collimation units, and a novel compensation method for dual-beam parallelism based on polarization is proposed. In addition, the principle of air turbulence protection is analyzed based on air turbulence phase screens and hydrodynamics, and an optimized air turbulence shield (ATS) is developed. Finally, the robustness and accuracy of the proposed system are verified by experiments.
2. Methodology
The optical configuration of the proposed 5-DOF measurement system is shown in Fig. 1, which consists of a stationary part and a moving part. The stationary part is fixed to a stationary base, which consists of a polarization-maintaining fiber-coupled laser (PFL), a quarter-wave plate (QWP), the first beam splitter (BS1), the second beam splitter (BS2), the fourth beam splitter (BS4), the first mirror (M1), the second mirror (M2), the third mirror (M3), the first polarization beam splitter (PBS1), the second polarization beam splitter (PBS2), the third lens (L3), the fourth lens (L4), the fifth lens (L5), the sixth lens (L6), the second position sensitive detector (PSD2), the third position sensitive detector (PSD3), the first quadrant detector (QD1) and the second quadrant detector (QD2). The moving part is mounted onto a moving stage, which consists of the third beam splitter (BS3), the third polarization beam splitter (PBS3), the first corner cube retroreflector (RR1), the second corner cube retroreflector (RR2), the first position sensitive detector (PSD1), the first photodiode (PD1), the second photodiode (PD2).
The measurement beam from a laser diode is output and collimated by PFL and collimating lens. The collimated beam is linearly polarized in the vertical direction and is converted into a circularly polarized beam by QWP. The fast axis of the QWP beam is 45° from the polarization direction of the emergent beam. As shown in Fig. 1, the green beam represents the circularly polarized beam. Then, the emergent beam from QWP is divided into the transmitted beam and the reflected beam by the BS1. After being reflected by the mirror (M1), the transmitted beam is divided into two beams, a p-polarized beam (beam 1) represented by red, and an s-polarized beam (beam 2) represented by blue, by a polarization beam splitter (PBS1). Beam 1 passes through BS2, BS3, RR1, M2, M3, RR2, BS4, and PBS2 in turn and is received by QD1 for measuring straightness error. The reflected beam of beam 1 at BS3 is received by PSD1 for measuring pitch and yaw error. The reflected beam of beam 1 at BS2 is received by PSD3 for measuring beam 1 drift. Beam 2 passes through PBS2, BS4, RR2, M3, M2, RR1, BS3, BS2, and PBS1 in turn and is received by QD2 for measuring the straightness error. The reflected beam of beam 2 at BS4 is received by PSD2 for measuring beam 2 drift. The roll error is measured by comparing the vertical straightness of QD1 and QD2. The reflected beam of BS1 is divided into two beams by PBS3, a p-polarized beam (beam 3) represented by red and an s-polarized beam (beam 4) represented by blue, which are received by PD1 and PD2 for compensating the parallelism of beam 1 and beam 2.
2.1 Principle of measurement system
As shown in Fig. 2, BS3 is used as a sensitive element in the moving part to measure pitch and yaw. Convex lens L1 and concave lens L2 are combined to form a combined lens. The focal length of the combined lens is larger than the mechanical size, so it can achieve higher angular measurement accuracy compared with a single lens under the same installation space. The focal length fPSD1 of the combined lens can be expressed as
The position of the light spot changes on the PSD1 when the moving part undergoes pitch εx and yaw εy movement. The reflected beam angle changes by 2εy and 2εx. The pitch and yaw are expressed as
A schematic diagram of the straightness measurement based on laser collimation is shown in Fig. 3. According to the characteristics of corner cube retroreflector, when the straightness δx and δy are generated by RR1 and RR2, the reflected beam position changes to 2δx and 2δy. The two-dimensional straightness can be expressed as
Similarly, two-dimensional straightness can also be measured by QD2, which can be expressed as
The roll is measured by the parallel dual-beam method. As shown in Fig. 4, two laser beams are reflected to QD1 and QD2 by two angular prism RR1 and RR2, respectively. Take beam 1 as an example, A is the incident point of the measuring beam, and B is the exit point on RR1. C is the incident point of the measuring beam on RR2, and D is the exit point on RR2. When there is no roll, the positions of B and C are the same in the Y direction. When the moving part produces a roll, the position of beam spots will change on RR1 and RR2 except for incident point A. The exit point of RR1 changes from B to B’, the incident point of RR2 changes from C to C’, and the exit point of RR2 changes from D to D’. Similarly, beam 2 is finally received by the QD2. The roll is expressed as
The beam of the similar 5-DOF measurement systems is usually directly received by the photoelectric sensor or reflected once by the corner cube reflector, and then received by the photoelectric sensor. To improve the straightness and roll measurement resolution, the distance l between two parallel beams is usually increased. However, each beam passes through the corner cube retroreflector twice in the proposed system, which can improve the measurement resolution of roll and two-dimensional straightness. Therefore, the distance l can be reduced to make the system structure more compact and easy to install on precision machinery such as machine tools.
2.2 Principle of beam drift compensation
The measurement beam will drift due to the instability of the light source, thermal deformation, and mechanical stress release of the optical components, which severely limits the application of the MDFM system. The stationary part and the moving part of the system are made of the same material. The displacement drift has no effect on the measurement of pitch and yaw, and both the stationary part and the moving part have part of the displacement drift caused by thermal deformation, which can cancel each other. According to the measurement principle, the influence of angle drift on two-dimensional straightness and roll measurement increases with distance. Therefore, only the laser angle drift is monitored and compensated by the autocollimator units, which are composed of L3, L4, L5, L6, PSD2, and PSD3.
The compensation principle is shown in Fig. 5, the spot position on PSD2 changes to (ΔyPSD2, ΔzPSD2) when the beam 1 angle drifts εx1 and εy1. Similarly, the spot position on PSD3 changes to (ΔyPSD3, ΔzPSD3) when the beam 2 angle drifts εx2 and εy2. The 5-DOF measurement compensation model is expressed as
2.3 Principle of dual-beam parallelism compensation
Compared with other roll error measurement principles mentioned in the introduction, the dual-beam method has the advantages of better resolution and accuracy. However, due to initial installation error, environmental variations, and mechanical deformation, the parallelism between beam 1 and beam 2 is impossible to be kept stable, which will cause serious roll measurement errors, as shown in Fig. 6(a). The compensation in Section 2.2 can only improve the drift of the laser and cannot fully compensate for the parallelism of dual-beam. In addition, beam drift compensation cannot compensate for the initial installation error, resulting in similar systems requiring to pay particular efforts to pre-calibrate the parallelism of dual-beam. Even so, it still has a serious impact over a long distance. The non-parallelism in Y-direction between parallel dual-beam will lead to a measurement error proportional to the length of the working distance d. As shown in Fig. 6(b), if beam 2 is drifted relative to beam 1 with an angle εdrift or there is an initial installation error, it will generate a straightness error Δydrift in the Y-direction. A similar error in the X-direction has no effect on the measurement of the roll angle, as long as the spot does not leave the quadrant detector in the measurement range by pre-calibration. In this paper, the compensation method of dual-beam parallelism in the Y direction is designed based on polarization.
Fig. 6. The roll measurement error caused by non-parallelism: (a) the angle between beam 1 and beam 2 is εdrift (b) spot position measured by QD1 and QD2.
Since at each time of measurement the roll error is reset at the starting position d0, the Δydrift at the current position d1 will be
The straightness error will cause the following roll measurement error as below.
The roll measurement error εzd is a time-dependent random error. It seriously affects the accuracy of roll measurement and cannot be calibrated in advance. However, the roll measurement based on polarization is not sensitive to the spot position and only depends on the beam intensity, which can be used to compensate for the parallelism of dual-beam. As shown in Fig. 7, a dual-beam parallelism self-compensation method is proposed to solve the non-parallelism of dual-beam and initial installation errors. The polarizer unit is composed of PBS3, PD1, and PD2. Even if the beam of the stationary part drifts due to external environmental interference, it will not affect the beam intensity on PD1 and PD2. The beam is split by BS1 after exiting the PFL and QWP, where the reflected beam is circularly polarized. Then, the polarized light reflected by BS1 is split by PBS3, where p-polarized beam 3 is received by PD1 and s-polarized beam 4 is received by PD2. The intensity of beam 3 and beam 4 changes when the sensitive element PBS3 appears a roll angle εpz. Therefore, εpz can be calculated by the voltage of PD1 and PD2.
The Jones vector of the incident light after QWP can be express
When the moving part generates a roll εpz, the Jones matrix of the incident light converted from XOY to X′OY′ can be express
The Jones matrix of PBS3 is
The Jones matrix of the polarized beam after passing through PSB3 is
Therefore, the light intensity on PD1 and PD2 is expressed as
Then, the intensity difference ΔI between beam 3 and beam 4 is expressed as
Due to the roll is small, the εpz measured by the polarizer unit is expressed as
Therefore, the angle between the two beams in the Y direction εdrift can be expressed as
After compensating for the parallelism of dual-beam, the roll of the 5-DOF measurement system is expressed as
The self-compensation method can compensate for the beam drift caused by the change of environment and also can compensate for the initial installation error. Therefore, compared with other MDFM systems, the proposed 5-DOF system is more robust, accurate, and easy to pre-calibration.
2.4 Protection method for air turbulence
Air turbulence leads to the random change of atmospheric refractive index with time and space. As shown in Fig. 8(a), beam propagation through the air with an uneven refractive index is similar to passing through bubbles of different sizes, which results in uneven distribution of light intensity, spot deformation, and beam drift. The accuracy and stability of the 5-DOF measurement based on laser collimation are severely affected by air turbulence. The propagation process of the measurement beam is analyzed by the phase screen method. As shown in Fig. 8(b), phase screen theory refers to the equivalent of air turbulence as phase screens. The influence of turbulence is reflected in the distortion of the wavefront phase when the beam is transmitted in a turbulent medium. Therefore, multiple discrete phase screens can be superimposed on the wavefront phase to analyze the influence of air turbulence on the beam.
Fig. 8. (a) Beam propagates in air turbulence; (b) Diagram of air turbulence phase screens. (n1: refractive index outside the bubble; n2: refractive index inside the bubble; θ1: beam incidence angle; θ2: beam exit angle; Δz: distance between two adjacent phase screens.)
The whole process can be expressed by [23]
As shown in Fig. 9, An ATS with a movable and simple structure has been developed by the author’s group. The ATS is made of rubber with good thermal insulation performance to make the temperature field of the measurement environment more uniform. A more uniform temperature field has been proven to reduce the disturbance of the additional phase Φ(r,zj) of air turbulence [23]. According to Eq. (18), the intensity of air turbulence is also reduced. However, the actual beam path of the proposed 5-DOF system is 4 times the working distance, the influence of air turbulence on measurement is more serious. So the protection method for air turbulence needs to be further studied and optimized.
There are three types of airflow in the ATS: laminar, turbulent, and transitional flow. The motion in turbulent flow is chaotic, while the motion in laminar flow is relatively stable. The transition from laminar to turbulent flow is determined by the dimensionless Reynolds number Re, which is defined as
where V is the velocity of fluid movement, E is the equivalent diameter of the pipe, and U is the kinematic viscosity of the fluid.Rec is the critical Reynolds number for the transformation of laminar flow into turbulence. In a small-diameter pipe, the turbulence that is transient at low Re becomes sustained after a distinct Rec [24]. The critical point for transiting to sustained turbulence is decided when the local proliferation of puffs outweighs their decay. The experimental results show that a sharp intersection is formed at Rec = 2040 ± 10, which marks the transition from laminar flow to continuous turbulence in the pipeline fluid [25]. Generally, the fluid is a turbulent flow when Re is greater than 4000, is a transitional flow when Re is less than 4000 and greater than Rec and is a laminar flow when Re is less than Rec. According to Eq. (19), the Reynolds number of the system can be reduced by decreasing the equivalent diameter of the ATS. As shown in Fig. 10, the measurement beam of the 5 DOF measurement system propagates from the stationary part to the moving part. The beam is affected by air turbulence when it is not protected. The intensity of air turbulence decreases when protected by ATS. Subsequently, the protective effect of ATS is optimized by reducing the equivalent diameter. The turbulence intensity along the optical path is reduced, and even the turbulence is transformed into the laminar flow. Thus, the ability of the 5-DOF measurement system to resist air turbulence is improved.
Airflow velocity in the ATS is affected by the speed of the moving part. For example, when the average velocity of the airflow is 0.3 m/s. The kinematic viscosity U of air at 20°C and one standard atmospheric pressure is 1.48 × 10−5 m2/s. The cross-section of the unoptimized ATS is a rectangle with length a of 376 mm and width b of 76 mm. The optimized ATS cavity section is three circles with a diameter c of 20 mm. Regardless of the shape of the shield, it can be equivalent to a circular pipe, and the equivalent diameter of the pipe can be expressed as
According to Eqs. (19) and (20), the Re of the unoptimized ATS is 2563 so the internal airflow is a transition flow. The Re of the optimized ATS is 405 so the internal airflow is stable laminar flow. Therefore, the protection method can further reduce the disturbance of air turbulence and make the 5-DOF measurement system more accurate and stable. The proposed protection method can be applied to other similar micro displacement and angle measurements based on laser collimation.
3. Experiments and results
A series of experiments were carried out on a precision linear stage to evaluate the feasibility and effectiveness of the proposed system. The stationary part of the laser 5-DOF measurement system is fixed on the marble optical platform, the moving part moves with the slider. During the experiments, the sampling speed is 10 Hz, the fPSD1 is 400 mm, the fPSD2 and fPSD3 are 200 mm, and the l is 275 mm. The distance between the moving part and the stationary part was 800 mm. The laser interferometer (SJ6000, Chotest, angle measurement accuracy of ± (0.02%R + 0.1 + 0.02d) ″, R is the display value and d is the measured distance, angle measurement resolution of 0.01 ″, straightness measurement accuracy of ± 0.5 ppm, straightness measurement resolution of 1 nm) and electronic level (DEG-IL, Auleadson Electronic, accuracy of ± 1 µrad, resolution of 1 µrad) were used to calibrate and compare the 5-DOF measurement system. All experiments were conducted in a non-environmental controlled open-type laboratory.
3.1 Performance of beam drift compensation
To verify the effectiveness of the beam drift compensation, the stability experiments with and without the beam drift compensation were carried out under the same ambient condition. The measurement principle of yaw and pitch angle is the same. The measurement principle of vertical straightness and horizontal straightness is the same, while the roll is calculated by two-dimensional straightness. Therefore, pitch and horizontal straightness measurements were given as examples. The total experiment time lasted 1.5 hours and the experimental results are shown in Fig. 11. The peak-to-valley (P-V) values of the pitch are 0.681 arcsec and 0.442 arcsec with and without compensation, respectively. The P-V values of the horizontal straightness are 6.178 um and 1.053 um with and without compensation, respectively. The experimental results show that the proposed beam drift compensation method can effectively suppress the measurement error caused by laser drift. In addition, it should be noted that the measurement results after compensation still have drift, which may be caused by thermal deformation of the moving part, air turbulence disturbance on the beam propagation path, and changes in the laser intensity distribution.
3.2 Performance of dual-beam parallelism self-compensation
The parallelism of dual-beam in the Y direction has an initial installation error that cannot be eliminated, and the non-parallelism of dual-beam inevitably occurs with the change of the environment. The performance test of dual-beam parallelism compensation was carried out. The dual-beam parallelism of the 5-DOF system was pre-calibrated, which was placed in the laboratory for a month. As shown in Fig. 12, environmental disturbances still affect the parallelism of dual-beam over time. The non-parallelism of dual-beam has a serious influence on the measurement of roll, and the error increases with the increase of measurement distance. The standard deviation (SD) of residuals before compensation is 33.812 arcsec, and the SD of residuals after compensation is 0.991 arcsec. The roll measurement SD of residual is reduced by 96% using the dual-beam parallelism compensation method, which is significant. The experimental results show that as long as the measuring beam does not exceed the measuring range of QD1 and QD2, no matter how poor the parallelism change and initial parallelism are, the roll measurement error can be corrected, so the system is robust and convenient for pre-calibration.
3.3 Performance of air turbulence protection
The effectiveness of the air disturbance protection method was tested by experiment. As shown in Fig. 13, the stability of pitch and horizontal straightness were tested under three conditions: without ATS, with unoptimized ATS, and with optimized ATS. The optimized ATS reduces the Reynolds number by reducing the equivalent diameter, thus transforming the internal air turbulence into a more stable laminar flow.
The experimental results are shown in Fig. 14. Compared with the system without ATS, the SD of pitch error noise of the system with ATS decreases by about 66% and the SD of pitch error noise of the system with optimized ATS decreases by about 75%. At the same time, the SD of the horizontal straightness error noise of the system with ATS decreases by about 83%, and the SD of the horizontal straightness error noise of the system with optimized ATS decreases by about 88%. The air turbulence protection method effectively improves the stability and accuracy of the 5-DOF measurement system. This method provides a theoretical basis for the design of shields for other similar error measurement systems, which is meaningful. The beam path of straightness measurement travels four times as far as the angle measurement, so the effect of air turbulence protection in the straightness measurement is more obvious. With the increase in measurement distance, the error measurement based on the laser collimation principle needs more reasonable and feasible optical path protection designs.
3.4 Comparison tests of the 5-DOF measurement system
Repeatability and comparison experiments were performed using the proposed system and above commercial standard instruments. Air turbulence protection, angle drift compensation, and dual-beam parallelism compensation were carried out. The experiments were repeated five times under identical conditions. The measured distance was 800 mm, and the measurement interval was 80 mm. The proposed system and commercial standard instruments are shown in Fig. 15, the laser interferometer was used for the comparison of pitch, yaw, and two straightness, and the electronic level was used for the comparison of the roll.
The experimental results are shown in Fig. 16, the SD of the average deviation error of pitch and yaw is 0.27 arcsec and 0.33 arcsec, respectively. The SD of the average deviation of horizontal and vertical straightness is 0.51 um and 0.71 um, respectively. The SD of the average deviation error of the roll is 0.53 arcsec. The experimental results show that the 5-DOF measurement system has high accuracy and desirable repeatability on the premise of maintaining high resolution, and can effectively resist the interference of the environment on the measurement.
The measurement errors mainly include random errors, manufacturing and installation deviations, and error crosstalk. Due to the limitation of length, error analysis and models will be presented in future work.
4. Conclusions
It is a major challenge to maintain the high precision and stability of MDFM systems under the long-term interference of external environments. To solve this problem, a robust five-degree-of-freedom measurement system with beam drift compensation and air turbulence protection was proposed in this paper. Not only the resolution of the system was improved, but also the compactness of the system was improved based on lens combination and multiple reflections. The beam drift was compensated by collimation units, and a novel compensation method for dual-beam parallelism based on polarization was proposed. In addition, the principle of air turbulence protection was analyzed based on air turbulence phase screens and hydrodynamics, and an optimized ATS was developed. A series of experiments have been carried out in a non-environmental controlled open-type laboratory to verify the feasibility and robustness of the proposed system. The experimental results show that the proposed beam drift compensation methods effectively compensate for the measurement error and improve the accuracy. The optimized ATS successfully reduces the influence of air turbulence on measurement accuracy and improves the resistance to air turbulence of the system. It has been verified that the proposed system has the advantages of high precision, desirable robustness, and convenient pre-calibration. In addition, compared with traditional commercial instruments, the proposed system has low cost, compact structure, and high measurement efficiency, which is of significance for the error measurement of precision machines. Future works will focus on system optimization to maintain high accuracy over longer measurement distances.
Funding
National key research and development plan project (2020YFB2010800); National Natural Science Foundations of China under Grant (61905175); the Fok Ying Tung education foundation (171055), Young Elite Scientists Sponsorship Program by CAST (2021QNRC001); Guangdong Province key research and development plan project (2020B0404030001); National Defense Science and Technology Key Laboratory Fund (6142212210304).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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