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Abstract
In this paper, a self-compensation method for improving the accuracy of roll angle measurement of a linear stage caused by the non-parallelism of dual-beam due to time-dependent mechanical deformation of the support is proposed and integrated into a 5-DOF sensor to verify the feasibility. The non-parallelism between two laser beams is online real-time monitored by a pair of small autocollimator units. Through the ray-tracing analysis, the method to separate the roll angle of the moving stage and non-parallelism induced roll error is determined. A series of experiments under different supporting forces and ambient conditions have been carried out. The compensated P-V values of the roll angles are all within ±4 arc-sec, no matter how bad the originally measured value of the linear stage is. The average improvement of about 95% is significant. The effectiveness and robustness of the proposed measurement system in the changing environment are verified.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Precision measurement is essential for machine tools, measuring machines, and many other precision machines. There are six position-dependent geometric errors (PDGEs) inherited in the moving table supported by two linear guides, including one positioning error, two straightness errors (in horizontal and vertical directions), and three angular errors called the yaw, pitch, and roll around three axes, respectively. Currently, there are some commercial instruments for measuring PDGEs in one-by-one mode, which is deemed time-consuming and expensive, such as laser interferometer, autocollimator, precision level, etc. In recent years, various multi-degree-of-freedom measurement (MDFM) systems have been developed for fast and simultaneous measurement of PDGEs [1–4].
Recently, a variety of roll measurement methods based on different principles have been proposed. The roll angle measurement based on the autocollimator method [5–6] has high resolution and accuracy. As to the laser interferometer method [7–9] with differential configurations, it also has a high resolution in roll measurement, which makes use of complex processing system. The roll angle measurement based on the vision method [10,11] and the laser polarization state method [12–17] can achieve a larger measurement range.
However, there are some disadvantages in the methods mentioned above. The measuring distance of the autocollimator-based roll angle measurement is limited by the size of aperture of focus lens. The structure of the laser interferometer method is very complex. The vision-based method is difficult to achieve a high resolution and high real-time performance, and the laser polarization state method requires high-quality of laser, optics and signal processing.
Considering the feasibility of high resolution and long working distance, the parallel dual-beam method [18–22] is still the most practical industrial environment. Kuang [21] proposed a common path compensation principle for long-distance measurement, reducing environmental interference and improving measurement uncertainty. However, the parallelism of two measured laser beams is difficult to be kept unchanged due to the variation of ambient conditions and mechanical support when the time elapses. Although the non-parallelism could be corrected by different methods [23,24], they are all off-line processes. Besides, the thermal and structural deformations of the mount, the release of springs in the adjustment mechanism, and the deterioration of glues and bolts all have significant influences on the non-parallelism. These conditions will result in measurement errors.
To improve the long-term accuracy of dual beam roll angular measurement principle, an online real-time self-compensation method for non-parallelism error is proposed in this report. In Section 2, such a roll measurement module is embedded in a five-degree-of-freedom (5-DOF) geometric error measurement system and the error model of non-parallelism is derived. In Section 3, the real-time compensation effect and robustness of the developed measurement system are verified under different experimental conditions.
2. Methodology
The optical configuration of the proposed 5-DOF sensor is shown in Fig. 1. The laser part is fixed to a stationary base and the sensor part is mounted onto a moving stage. The reference beam from a diode laser is output and collimated by a single-mode fiber (SMF) and its collimating lens. A beam splitter (BS1) is used to separate the reference beam into two parts to generate the beam 2, which is initially adjusted by a bending mirror (M2) to ensure parallel to the reference beam 1. The sensor part is composed of a pair of 4-DOF sensor units, of which each unit contains a position sensitive detector (PSD) to detect two-directional straightness errors and an autocollimator unit (AC) to detect pitch and yaw errors of the moving stage [2]. Taking beam 1, PSD1 and AC1 as the main 4-DOF sensor system, the difference of y-directional movement of two PSDs can detect the roll angle of the stage. By comparing the difference of angular readings of AC1 and AC2, the non-parallelism between beam 1 and beam 2 in the y-direction can be automatically separated from the difference in pitch errors of the two ACs. Then, this extracted non-parallelism can be used to correct the measured roll angle in online and real-time mode. This is the innovative highlight of the proposed self-compensation method in roll error measurement.
As the measurement systems of 4-DOF and 5-DOF have already been reported in the author’s earlier papers [2,4], this report focuses only on the measurement principle of self-compensated roll angle.
2.1 Measurement principle of the roll error
Figure 2 indicates that the measurement principle of roll error is based on the traditional dual-straightness reference method specified in ISO230-1 [25]. The roll error of the tested linear stage is proportional to the difference of vertical readings of PSD1 and PSD2.
Fig. 2. The measurement principle of roll error; (a) optical configuration, (b) relationship between the roll error and detected spot positions of PSD1 and PSD2.
Compared with other roll error measurement principles as mentioned in the introduction section, the dual-beam method has the advantage of a better resolution and accuracy. However, the shortage of unable to keep parallelism of two beams at all times due to unavoidable problems, like the creep of the mechanical support, the release of spring force, and thermal deformation of the stage, etc., this measurement method would limit its field applications. Therefore, the technique of error separation and compensation becomes very important in practice.
2.2 Non-parallelism error of roll measurement
Due to the variation of ambient conditions and mechanical support, the parallelism between two reference laser beams is impossible to be kept stable, which will cause roll angle measurement errors, as shown in Fig. 3(a). The non-parallelism in y-direction between two reference beams will lead to a measurement error proportional to the length of the optical path. As shown in Fig. 3(b), if the beam 2 is drifted relative to beam 1 with an angle ${\varepsilon _{py}}$, it will generate a straightness error in y-direction. Since at each time of measurement the roll error is reset at the starting position d0, the induced straightness error in y-direction at the current position d1 will be
This generated straightness error will then induce the roll measurement error as below.
Fig. 3. The roll error measurement caused by non-parallelism; (a) optical configuration, (b) detected spot relative position between PSD1 and PSD2.
The occurrence of this induced measurement error is obviously serious to the dual-beam roll measurement system. For example, let the offset L be 70 mm. Just 1 arc-sec non-parallelism of two beams would cause 11.43 arc-sec error in roll measurement at an optical length of 800 mm. This kind of non-parallelism induced roll measurement error is a time-dependent random error, which cannot be calibrated in advance and has to be removed in real-time by a special technique as proposed below.
Conventionally, the non-parallelism is calibrated by a precision level. However, the pre-calibrated non-parallelism would be changed, due to the variations of non-parallelism caused by the manufacturing errors, structural deformation and the release of spring force.
2.3 Self-compensation principle of non-parallelism between reference beams
To solve the problem mentioned above, a self-compensation method is proposed and embedded into a 5-DOF error measurement sensor, as shown in Fig. 4(a). A series of coordinate frames are designated. The reference frame R (xr, yr, zr) is set on the fixed laser part, frame S (xs, ys, zs) is set on the base plate of the moving sensor part, frame AC1 (xac1, yac1, zac1) set on the AC1 of the sensor part, and frame AC2 (xac2-yac2-zac2) is set on the AC2. Let frame AC1 be parallel to frame S and frame AC2 is rotated180 degrees around the Y-axis relative to frame AC1. Figure 4(b) shows the error parameters in these three coordinate frames.
Fig. 4. The configuration of the 5-DOF measurement system with self-compensation of roll measurement; (a)definitions of the coordinate system, (b) definitions of error parameters
The ray-tracing analysis needs to be performed to calculate the vector of each beam in the optical path based on the law of reflection. The directional vector of the emitted light from the laser unit is ${}^{r}{\vec{n}_{LD}} = {\left[ {\begin{array}{ccc} { - 1}&0&0 \end{array}} \right]^T}$. The unit normal vector of the reflecting surface i, representing M and BS in general, can be expressed by the vector form of
The transformation matrix of the reflecting surface i can be expressed by
The ideal directional vector of beam 1 and beam 2 are expressed by
In reference frame R, the ideal normalized vectors and the ideal reflecting transformation matrices of BS2 and BS3 are listed in Table 1.
Table 1. The normal vector and transformation matrix of beam splitters
Then, the directional vectors of incident beams of AC1 and AC2 in the ideal condition can be expressed as:
However, in actual conditions, the non-parallelism between beam1 and beam 2 will change the directional vector of beam 2. Both the geometric errors of the measured linear stage and the initial installation error of AC2 relative to AC1 have direct influences on readings of QPD1 in AC1 and QPD2 in AC2. In order to obtain εpy for roll error compensation, ray-tracing analysis in the actual condition is necessary. In this study, three error factors in actual measurement conditions are considered and listed in Table 2, including the non-parallelism error, geometric errors, and installation errors. Details are expressed as below.
- (1) Firstly, the non-parallelism errors of two beams are considered. It could be due to many factors, such as the thermal and structural deformations of the mounts of the laser part and sensor part, the release of springs in the adjustment mechanisms for M1 and M2, and the deterioration of gluing force of optics and tightening force of bolts, etc. Taking beam 1 as the reference beam, the non-parallelism angle of the actual beam 2 relatives to beam 1 is denoted by εpx in x-direction and εpy in y-direction.
Then, the directional vector of incident light of AC2 under the influence of the non-parallelism can be expressed by:
- (2) Secondly, due to the manufacturing imperfections and the assembling errors of the linear stage, the sensor part mounted on the moving linear stage inherits rotational geometric errors of εyaw and εpitch that can be obtained from the readings of AC1.
Therefore, the vectors of the reflected beams of the BS2 and BS3 are:
$${}^r\overrightarrow n _{_{_{\textrm{BS2}}}}^{\prime} = {}_s^r{R^e}{}^r{\overrightarrow n _{_{\textrm{BS2}}}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{ccc} {1 - {\varepsilon_{yaw}}}\\ {{\varepsilon_{roll}} + {\varepsilon_{pitch}}}\\ { - 1 - {\varepsilon_{yaw}}} \end{array}} \right]$$$${}^r\overrightarrow n _{_{_{\textrm{BS3}}}}^{\prime} = {}_s^r{R^e}{}^r{\overrightarrow n _{_{\textrm{BS3}}}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{ccc} { - 1 - {\varepsilon_{yaw}}}\\ { - {\varepsilon_{roll}} + {\varepsilon_{pitch}}}\\ { - 1 + {\varepsilon_{yaw}}} \end{array}} \right]$$The actual transformation matrices ${}^rR_{BS2}^{\prime}$ and ${}^rR_{BS3}^{\prime}$ can be obtained by substituting Eqs. (10, 11) into Eq. (5). Therefore, the actual directional vector of the incident beam of AC1 can be calculated as:
$${}^s\vec{n}_{AC1}^{\prime\prime} = {}^{ac1}\vec{n}_{AC1}^{\prime\prime} = {({}_s^r{R^e})^{ - 1}}({}^rR_{BS2}^{\prime}{}^r{\overrightarrow n _{_{beam1}}}) \approx {\left[ {\begin{array}{ccc} 1&{{\varepsilon_{pitch}}}&{\textrm{ - }{\varepsilon_{yaw}}} \end{array}} \right]^T}\textrm{ = }{\left[ {\begin{array}{ccc} 1&{{\varepsilon_{y\_Q1}}}&{{\varepsilon_{x\_Q1}}} \end{array}} \right]^T}$$where, εy_Q1 and εx_Q1 are readings of QPD1 in the Y-direction and X-direction, respectively. Likewise, the actual directional vector of the incident beam of AC2 can be expressed by the following form. - (3) Thirdly, due to the installation error of the sensor components, the initial installation error of AC2 relative to AC1 can be expressed as a transformation matrix ${}_{ac1}^{ac2}R^e$, as listed in Table 2. It is noted that the coordinate frame of AC2 is rotated by 180 degrees around Y-axis relative to AC1. Hence, the actual transformation matrix of AC2 relative to AC1 is calculated as:$$\begin{aligned}{}_{ac2}^{ac1}{R_{AC}} &= {}_{ac2}^{ac1}{R^e}{}_{ac2}^{ac1}{R_{ideal}}\; = \left[ {\begin{array}{ccc} 1&{ - {\varepsilon_{pitch\_ac}}}&{{\varepsilon_{yaw\_ac}}}\\ {{\varepsilon_{pitch\_ac}}}&1&{ - {\varepsilon_{roll\_ac}}}\\ { - {\varepsilon_{yaw\_ac}}}&{{\varepsilon_{roll\_ac}}}&1 \end{array}} \right]\left[ {\begin{array}{ccc} { - 1}&0&0\\ 0&1&0\\ 0&0&{ - 1} \end{array}} \right] \\ &= \left[ {\begin{array}{ccc} { - 1}&{ - {\varepsilon_{pitch\_ac}}}&{ - {\varepsilon_{yaw\_ac}}}\\ { - {\varepsilon_{pitch\_ac}}}&1&{{\varepsilon_{roll\_ac}}}\\ {{\varepsilon_{yaw\_ac}}}&{{\varepsilon_{roll\_ac}}}&{ - 1} \end{array}} \right] \end{aligned}$$
Table 2. Error parameters in actual measurement conditions
Because the frame S and the frame AC1 are in the same direction, in frame AC2 the directional vector of the incident beam of AC2 can be expressed as:
The non-parallelism between 2 reference beams in Y-direction can be expressed as:
Therefore, according to Eq. (3) and Eq. (17), the real-time compensation roll error ${\varepsilon _{roll}}$ can be expressed as
2.4 Structure and software design of the self-compensated 5-DOF sensor
The design of the new 5-DOF sensor part with self-compensated roll measurement is shown in Fig. 5(a). It is a symmetrical structure to ensure the thermal balance during long-term or long-distance operation. For the purpose of lightweight and compact size, the overall dimension is only 90×45×28 mm3 and the beam separation is 70 mm. The process flowchart of the self-compensated measurement system is given in Fig. 5(b).
Fig. 5. Design and process flowchart of the new 5-DOF sensor part with self-compensated roll measurement.
3. Performance tests
As shown in Fig. 6(b), a prototype of the new 5-DOF sensors was designed and manufactured to verify the performance of the proposed measurement system. The experiments were carried out under a laboratory condition that can control the ambient temperature and vibration. The reference precision level, which was developed by the author’s group [26], was adopted as the reference of roll measurement. It has been calibrated by a commercial autocollimator (model 5000U, AutoMat Co., China) and compared with the commercial level (WL/AL11, Qianshao, China) to prove its performance that can meet the requirement of precision machine error measurement. The sensor part and reference precision level were both fixed on the moving stage, as schematically shown in Fig. 6(a), so that the comparison test of roll error of the tested linear stage measured by the proposed 5-DOF sensor and reference precision level can be made possible.
Fig. 6. Prototype of proposed 5-DOF sensor and the experiment condition, (a)schematic of the performance test, (b) photo of the experimental set-up
3.1 Calibration of the initial installation error
The parallelism of the two beams has to be adjusted by M2 during the installation. In real practice, it is difficult to reach perfectly parallel due to the assembly error of related optics. Therefore, the initial installation error (${\varepsilon _{pitch\_ac}}$) has to be recorded for error compensation. The measurement of the initial installation error takes the dual-axis precision level as a reference. The standard operation procedure (SOP) of parallel beams is as follows.
- Step 1: Reset both the 5-DOF sensor reading and the level reading of roll angle at the starting point of the linear stage.
- Step 2: Move the linear stage to its maximum stroke and record the roll angle readings of the 5-DOF sensor and level.
- Step 3: Adjust M2 until the reading of 5-DOF's roll error is the same as the level.
- Step 4: Move the linear stage back to the origin, and save the readings of AC1 and AC2. The difference between the readings would be the initial installation error, which is needed in the compensation software.
3.2 Comparison tests and stability of roll measurement
In order to test the accuracy of the roll measurement, the linear stage was driven by the stepping motor to 800 mm travel with a 100 mm step size. The comparison experiment between the measured roll error of 5-DOF sensor and the precision level has been carried out five times. It is clearly seen from Fig. 7(a) and Fig. 7(b) that the measurement repeatability of each instrument is very good. Figure 7(c) shows that the measured roll angles by the dual-beam method are very consistent with the readings of the precision level. The residuals are less than one arc-sec.
Fig. 7. Comparison results of the measured roll errors of the linear stage between the 5-DOF measurement system and precision level. (a) 5DOF, (b)Precision level, (c) average error and residual.
The long-term stability of the roll measurement was also tested by fixing the sensor part at a distance of 400 mm from the laser part. The sampling rate was set to 2 Hz. Recorded data for forty minutes is shown in Fig. 8. The measured stability variation of only 0.41 arc-sec of the peak to valley value certainly meets the measurement requirement.
3.3 Performance of the self-compensated roll error measurement
A series of verification experiments were carried out to prove the effectiveness of the self-compensation method. At first, the screw on M2 was intentionally loosened or tightened to create the vertical drift angle of beam 2 in experiment 1, and the experimental results prove that the compensation method can adapt to the different non-parallelism conditions. Figure 9 shows the measured roll angles with and without self-compensation with reference to the precision level. It can be seen that the influence of non-parallelism on the roll angle is very significant. The larger the non-parallelism error the bigger the measured roll error, as shown in Fig. 9(a). However, if the self-compensation technique is activated, the induced error due to non-parallelism could be effectively detected and immediately compensated, as shown in Fig. 9(b). The compensated P-V values of the roll angles are all within ±3 arc-sec, no matter how bad the originally measured value of the linear stage is. The average improvement of about 95% is significant.
Fig. 9. Comparison results of roll residuals in experiment 1, (a) original error, (b) with self-compensation technique.
Experiment 2 shows that the ambient temperature of the experiment condition was changed by the air conditioner. The corresponding residual errors in different temperatures of roll error measurement were obtained. The experiment data was shown in Fig. 10(a), it can be seen that the non-parallelism is significant influenced by the temperature. The measurement errors of roll data before compensation were enormously large and increased with the distance. On the contrary, the compensated roll errors were very consistent with the data of precision level.
Fig. 10. Comparison results of roll residuals in different conditions, (a) temperature change, (b) different days.
In Experiment 3, the drift of beam 2 with respect to time was investigated. The experimental setup was fixed on the tested linear stages for a month. And comparison tests with the precision level on the linear stage were carried out once a week. Figure 10(b) obviously shows that the accuracy of uncompensated roll error was significantly increased with time. It obviously implies that the parallelism of two beams could not be properly maintained after initial installation. After applying the proposed self-compensation method, a significant reduction in the residual values has been achieved. The performance of the proposed method in the accuracy improvement of roll measurement is satisfactory. The compensation effect evaluated by the P-V value is listed in detail in Table 3. It can be seen that after compensation, the P-V values are all within ±4 arc-sec, no matter how bad the originally measured value of the linear stage is. The average improvement of about 95% is significant.
Table 3. Accuracy improvement of the 5DOF system
3.4 Measurement uncertainty analysis
Based on the experimental data of Section 3.1 to 3.3, the measurement uncertainty of the roll angle can be analyzed. The combined standard uncertainty can be obtained by five sources:
These five sources of uncertainty and the combined standard uncertainty are listed in Table 4. The first influential source of uncertainty, uac_roll, is the standard uncertainty of the roll calibration experiments. The standard uncertainty of the calibrated data in the form of rectangular distribution is 0.159 arc-sec. The second source of uncertainty, ures_roll, is obtained from the resolution test in the short-term stability experiments. Taking the data of the first 60 seconds from Fig. 8, the standard uncertainty of the short-term stability data corresponding to the rectangular distribution is 0.032 arc-sec. The third source of uncertainty, urep_roll, is analyzed from the repeatability experiments. Experimental data are obtained from Fig. 7(a), the standard uncertainty of the measurement repeatability corresponding to the gaussian distribution is 0.676 arc-sec. The fourth source of uncertainty, usta_roll, is the standard uncertainty of the long-term stability experiments. From Fig. 8, the standard uncertainty of the long-term stability data corresponding to the gaussian distribution is 0.118 arc-sec. The last source of uncertainty, urep_roll, is related to time-dependent mechanical deformation with measured data collected from Fig. 10(b). Compared with the reference precision level for four weeks, the standard uncertainty of the residuals is found to be 5.745 arc-sec from original roll data and it is reduced to 0.283 arc-sec. An obvious improvement in the measurement uncertainty can be obtained after the use of self-compensation function. The overall combined standard uncertainty is reduced from 5.788 arc-sec of original data to 0.76 arc-sec of compensated data.
Table 4. Uncertainty of roll angle measurement of the 5DOF system
4. Conclusion
Measurement of roll error of the linear stage is a major challenge in the research of multi-degree-of-freedom measurement technology. The parallelism between two measured laser beams is difficult to keep unchanged due to the variation of ambient conditions and mechanical structure, which becomes the main error source in the dual-beams roll error measurement principle. In this paper, a method for self-compensation of non-parallelism between two beams has been proposed based on the ray-tracing analysis. A series of experiments under different conditions have been carried out to verify the feasibility of the proposed method. After compensation, not only the the accuracy of measured roll angle but also the measurement uncertainty can be significantly improved. The effectiveness and robustness of the proposed measurement system and analytical method are verified. One of the goals of this study is to evaluate the possibility of embedding the 5-DOF measurement system in a precision machine as a feedback sensor to monitor the real-time machine accuracy for a long-term. The results of this study confirm the applicability of the dual-beam roll angle measurement with the implementation of the proposed self-compensation method.
Funding
National Key Research and Development Program of China (2017YFF0204800); Department of Science and Technology of Liaoning Province (2020JH6/10500017).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available at this time but may be obtained from the authors upon reasonable request.
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