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Abstract
In order to improve the accuracy of linear stages, a compact, portable and easy installation of a six-degree-of-freedom (6DOF) geometric error measurement system, in which two mirrors with special position and orientation are innovatively regarded as the sensitive elements of the roll error, is proposed. A set of combined focus lenses is integrated into the 6DOF measurement system to improve the resolution of the roll error. The accuracy of a linear stage is evaluated by the positional errors at the functional point, which is located at the working volume of a linear stage. An error transformation model based on the Abbe principle and the Bryan principle is established to estimate the positional errors at the functional point according to those at the measurement point. A series of experiments are carried out to verify the capable of the designed system and the effectiveness of the established model.
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1. Introduction
The accuracy of a linear stage is evaluated by the positional accuracy at the functional point, which is located at the working volume of a linear stage [1]. According to the rigid body kinematics, the angular errors (pitch error, yaw error and roll error) of the rigid body are position-independent, while the positional errors (horizontal straightness error, vertical straightness error and positioning error) of the rigid body are position-dependent. The positional errors at the functional point, thus, will be different from those at the measurement point, which is located at the center of the error measurement system. In order to obtain the positional errors at the functional point, it is essential to measure the positional errors at the measurement point and establish an error transformation model to show the relationship of the positional errors between the functional point and the measurement point.
Due to the Abbe principle and the Bryan principle, the angular errors will affect the positional errors. Thus, it is necessary to measure not only the positional errors at the measurement point but also the angular errors of the linear stage. Many commercial instruments have been designed for simultaneously measuring six-degree-of-freedom (6DOF) geometric errors of the linear stage, such as XD6 (API, USA) and XM-60 (Renishaw, UK). However, those commercial instruments are too huge to mount on the machine tool for on-machine measurement. High-cost is the other disadvantage. To cope with such a problem, a wide variety of compact measurement systems have been proposed to on-machine detect the 6DOF geometric errors of linear stages [2–15]. Among the 6DOF geometric errors, the roll error is usually detected based on the principle of interference [10,11], polarization variation [12,13], parallel beams [14,15]. However, those roll error measurement methods possess either a complex signal processing or a complex optical system, which let the measurement system complex when those methods are instigated into it.
As above-mentioned, it is essential to establish an error transformation model to estimate the positional errors at the functional point from those at the measurement point. Homogeneous transformation matrix (HTM) [16–18], Denavit-Hartenberg (D-H) transformation matrix [19,20], screw theory [21–23] and product of exponential [24,25] are mostly adopted to establish the error model. However, the measurement trajectories for positional errors in above-motioned error model methods are not clearly specified.
Therefore, in order to evaluate the accuracy of linear stages, a compact, portable, wireless and easy-operation 6DOF measurement system was firstly designed for on-machine measuring the angular errors of the linear stage and the positional errors at the measurement point. An error transformation model based on the Abbe principle and the Bryan principle was then proposed to estimate the positional errors at the functional point from those at the measurement point. The accuracy of a linear stage was firstly taken as a test on which the modeling, measurement and compensation of positional errors at the functional point were conducted. The capability of the designed system and the effectiveness of the established model were verified by a series of experiments.
2. Measurement principle of 6DOF geometric error
A six-degree-of-freedom (6DOF) measurement system is designed for on-machine measuring the geometric errors of linear stages. It should be noted that the geometric errors here refer to the positional errors of the linear stages (horizontal straightness error, vertical straightness error and positioning error) at the measurement point (center of the sensitive elements of the straightness errors and the positioning error) and the angular errors (pitch error, yaw error and roll error) of the linear stages.
The schematic of this system, which consists of a sensor head and a detector, is shown in Fig. 1. A positional set (PS), a straightness set (SS) and two angular sets (AS1 and AS2) of the sensor head are employed to measure the positioning error (Δd), the straightness errors (δx and δy), and the angular errors (α, β and γ), respectively.
2.1 Measurement principle of positioning error
The measurement principle of positioning error is based on the principle of Michelson interferometer. A laser beam emitted from a laser diode (LD) is split into two beams by a polarized beam splitter (P1), namely a transmitted measurement beam and a reflected reference beam. The transmitted measurement beam passes through a quarter waveplate (Q1) and enters into the P1 again after being reflected by a corner cube retroreflector (C1). The reflected reference beam passes through a Q2 and enters into the P1 again after being reflected by a C2. The reflected measurement and reference beams interfere with each other at the P1. The interference signals enter into the PS, which is composed of three polarized beamsplitters (P2, P3 and P4), a non-polarized beamsplitter (B3) and four photodetectors (D1 to D4). The C1 is moved with the worktable of the linear stage. The motion of the C1 with a distance of d will induce the variation of the count of the interference fringes and further causes the output of the detectors. The positioning error (Δd) of the linear stage can be obtained by
where, λ is the wavelength of the LD, n is the refractive index of air and N is the count of interference fringes. d is the commanded position of the linear stage.2.2 Measurement principle of straightness errors
The SS consists of a quadrant photodetector (QD1). The transmitted measurement beam from the P1 is reflected by the C1 and split into two beams by the B2, as shown in Fig. 1. The reflected beam from the B2 is projected onto the SS for measuring δx and δy.
The reflected beam from the B2 should be projected onto the center of the QD1 when the worktable is at the starting point. Otherwise, there will have initial shifts (Δx1 and Δy1) along the X- and Y-directions, respectively, of the QD1. δx and δy can be calculated by
2.3 Measurement principle of pitch and yaw errors
The measurement of α and β are based on the principle of the 2D laser auto-collimation. The transmitted measurement beam from the P1 is divided into two beams by the B1, as shown in Fig. 1. The reflected beam from the B1 projects onto the AS1 after being reflected by a semi-reflective-film (SRF), which is pasted on the B4. The AS1 is composed of a focus lens (F1) and a QD2. When the worktable has α or β, the X- and Y-directional outputs of the QD2 will be varied to Δx2 or Δy2, respectively. α and β, thus, can be evaluated by
2.4 Measurement principle of the roll error
As shown in Fig. 1 that the reflected beam from the B1 projects onto the AS2, which is composed of a F4 and a QD3, after being continuously bent by a set of mirrors (M1, M4, M5, M2) and passing through a F2 and a F3.
The measurement principle of γ is shown in Fig. 2. Here only the M4 and the M5, which are the sensitive elements of the roll error, is shown in Fig. 2 for the sake of clarity. A coordinate system O-XYZ is established by taking one vertex O of the B4 as the origin. The coordinate system O1-X1Y1Z1 is then established by rotating the O-XYZ 45° counterclockwise around the X-axis and moving the origin of the O-XYZ (Point O) to the center of the M4 (Point O1). Similarly, the coordinate system O2-X2Y2Z2 is established by rotating the O-XYZ 45°+θx clockwise and θy clockwise around the X- and Y-axes, respectively. The center of the M5 (Point O2) is set to be the origin of the O2-X2Y2Z2. The direction vector of reflection planes of the M4 and the M5 can be expressed by
In order to simplify the calculation, cos(θy+π/4), -sin(θy+π/4), cosθx and -sinθx are replaced by E, F, G and H, respectively. N5, thus, can be simplified to
When the worktable has angular errors of α, β and γ around the X-, Y- and Z-axes, N4 and N5 will be revised to
For this case, when an incident beam with a direction vector I1 of ${\left( {\begin{array}{{ccc}} 0&0&1 \end{array}} \right)^T}$ projects onto the M4, the direction vector I3 of the reflected beam from the M5 can be deduced to
Therefore, according to Eq. (8), the X- and Y-directional outputs of the QD3 can be expressed by
Equation (9) is a general case revealing the influence of α, β and γ on the reflected beam direction of I3 if the M5 is rotated with both 45°+θx and θy. In practice, θx is set to be zero. Therefore, γ can be obtained by
It can be seen from Eq. (10) that the resolution of γ significantly depends on θy. A series of simulations are carried out to investigate the relationship between the resolution of γ and θy. The results are shown in Fig. 3(a). As seen that the resolution of γ increases with the increase of θy. The resolution of γ is evaluated to be 5.15 arcsec when θy is set to be 45°. However, θy is limited by the size of the F4, as shown in Fig. 3(b). The range of θy can be evaluated by
where, s represents the distance between the sensor head and the detector at the starting point. l represents the diameter of the F4. According to the dimension of the F4, θy is preferably less than 15°, corresponding resolution of γ is evaluated to be 10.32 arcsec, which is much larger than the target resolution.
Fig. 3. Relationship between θy and γ, (a) Resolution of γ (b) Range of θy (c) Combined focus lens set.
Therefore, a combined focus lens set, which consists of a F2 and a F3, as shown in Fig. 3(c), is particularly designed to improve the resolution of γ. The relationship between the incident angle (θ1) of the F2 and the emergent angle (θ2) of the F3 can be expressed by
where, f2 and f3 are the focal lengths of the F2 and the F3, respectively. The resolution of γ can be improved by f2/f3 times by using the combined focus lens set. f2 and f3 are 100 mm and 12.5 mm in the designed 6DOF measurement system. The resolution of γ, thus, is evaluated to be 1.29 arcsec, which is acceptable for roll measurement.3. Error transformation model of the linear stage
The positional relationship among the measurement point, functional point and the working volume of a linear stage is shown in Fig. 4. The functional point could be moved to any commanded point within the working volume of the linear stage. The measurement point is located at the center of the C1 of the 6DOF measurement system. According to the Abbe principle and the Bryan principle, when the measurement axes of the positioning and the straightness error are not in line with the moving axis of the functional point, the angular errors of the worktable will induce the Abbe error and the Bryan error at the functional point. The X-, Y- and Z-directional shifts from the measurement axes of the positioning and the straightness error to the functional point are called Abbe offsets and Bryan offsets, respectively. Since the sensitive elements of the straightness errors and the positioning error are all the center of the C1, the Abbe offsets and the Bryan offsets are the same, which are label as in Lxb, Lyb and Lzb Fig. 4.
Fig. 4. Relationship among the measurement point, functional point, feedback point, and the working volume of a linear stage.
According to the principle of rigid body kinematics, the Abbe and the Bryan principles, the positional errors at the functional point, including two straightness errors (δxf, δyf) and a positioning error (Δdm), can be estimated by the following error transformation model.
The accuracy of the linear stage can be improved by compensating the positioning error at the functional point caused by the Abbe errors using the following equation.
4. Test of 6DOF measurement system
The geometric errors of a linear stage (KSA100-11-X, Zolix, China) were detected, modeled and compensated to verify the effectiveness of the designed system and the established model. The experimental setup is shown in Fig. 5.
Firstly, the stability of the designed 6DOF measurement system was tested. The tested results are shown in Fig. 6. A median filter and a wavelet filter were employed to reduce the effects caused by the noise. The sampling frequency was set to be 100 Hz. It can be seen that the stability of the signals for measuring the straightness errors and the angler errors were within ±0.5 µm and ±0.5 arcsec, respectively, during the sampling time of 40 minutes. It can be seen that the designed 6DOF measurement system satisfied stability for geometric error measurements with sub-micron and sub-arcsecond precision.
The measurement accuracy of the designed 6DOF measurement system was, then, investigated by comparing with standard instruments. Figure 7 shows the measured geometric errors with three time. The horizontal axis of the figure is the geometric errors measured by the standard instruments. The left-vertical axis of the figure is the geometric errors measured by the designed 6DOF measurement system and the right-vertical axis of the figure is the residual. Figures 7(a) and 7(b) show the measured straightness errors by using the designed 6DOF measurement system and a commercial digital indicator (P12DHR, Sylvac, Switzerland) with a measurement accuracy of 0.22 µm. The measurement accuracy of the straightness errors was evaluated to be ±1.5 µm within the measurement range of ±60 µm. A laser autocollimator (5000U3050, AutoMat, China) and an electronic level (WL/AL11, Qianshao, China) were employed to confirm the measurement accuracy of the angular errors. Results are shown in Fig. 7(c) to Fig. 7(e). As seen that the measurement accuracy of the pitch error, the yaw error and the roll error were evaluated to be ±1.0 arcsec, ±1.0 arcsec and ±1.8 arcsec, respectively, within the measurement range of ±60 arcsec. The displacement measurement accuracy of the 6DOF measurement system was evaluated to be ±500 nm by comparing with a laser interferometer (MCV500, Optodyne, USA). It is verified that the accuracy of the designed 6DOF measurement system was satisfied for measuring the geometric errors of the linear stage.
The geometric errors of the linear stage were measured by the 6DOF measurement system. The detected results for three measurements are shown in Fig. 8. The maximum straightness errors, the angular errors and the positioning error at the measurement point were detected to be 4.04 µm, −6.04 µm, −4.28 arcsec, 9.03 arcsec, 19.96 arcsec and 2.92 µm, respectively. The standard deviations of the measured δxm, δym, αm, βm, γm and Δdm were evaluated to be 0.15 µm, 0.17 µm, 0.10 arcsec, 0.05 arcsec, 0.2 arcsec and 0.18 µm, respectively.
Fig. 8. Geometric errors of the linear stage at the measurement point (a) δxm (b) δym (c) αm (d) βm (e) γm (f) Δdm.
A point in the working volume of a linear stage with X-, Y- and Z-directional shifts of 120 mm, 130 mm and 200 mm from the measurement point is set to be the functional ponit. The positional errors at the functional ponit were then estimated by substituting the measured results shown in Fig. 8 into the proposed error transformation model Eq. (13). In order to verify the correctness of the estimated results, the positional errors at the functional ponit were also measured by the 6DOF measurement system. The estimated and measured positional errors at the functional point were shown in Fig. 9. As seen that the estimated results and the measured results are in a good correspondence with each other, from which the effectiveness of the error transformation model was verified.
Fig. 9. Estimated and measured (a) δxf (b) δyf and (c) Δdf of the linear stage at the functional point
Finally, the positioning error at the functional point caused by the Abbe errors was compensated based on the proposed error compensation model. The positioning error with and without compensation were shown in Fig. 10. It can be seen that the maximum positioning error was improved to be 0.52 µm after compensation from 3.41 µm before compensation. The actual positioning error of the linear stage has been reduced by about 84% (from maximum 3.41 µm to 0.52 µm), from which the effectiveness of the error transformation model was demonstrated.
5. Conclusion
A method of on-machine measuring 6DOF geometric errors of linear stages was proposed. Differ from the roll error measurement method in conventional researches, a novel roll error measurement method, in which two mirrors with special position and orientation are as the sensitive elements of the roll error, was proposed. In addition, an error transformation model, in which the measurement trajectories for geometric errors of the linear stage were clearly specified, based on the Abbe principle and the Bryan principle was established to estimate the positional errors at the functional point. The proposed model can be employed for evaluate the positional errors at any point within the working volume of the linear stage. The proposed model can also be applied for compensating the positioning error induced by the geometric errors of one-dimensional linear stages during its working volume. A 6DOF geometric error measurement system was designed. It has been confirmed that the measurement accuracy of the designed measurement system for straightness errors, angular errors and positioning errors were evaluated to be ±1.5 µm, ±1.8 arcsec and ±500 nm, respectively. The positional errors at the functional point was simultaneously estimated and measured by the established error transformation model and the designed 6DOF measurement system. The estimated and measured results were in a good correspondence with each other, from which the effectiveness of the established model was verified. The positioning errors at the functional point was then compensated. It has been verified that the actual positioning error of the linear stage has been reduced by about 84% (from maximum 3.41 µm to 0.52 µm).
Funding
National Natural Science Foundation of China (51905078); National Key Research and Development Program of China (2018YFB2001400).
Acknowledgments
The authors would like to thank Qi Sang for his help in the preparation of the experimental setup.
Disclosures
The authors declare no conflict 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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