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Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage
Precision Engineering 30 (2006) 96–103
Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage Wei Gao a,∗ , Yoshikazu Arai a , Atsushi Shibuya a , Satoshi Kiyono a , Chun Hong Park b a
Department of Mechatronics and Precision Engineering, Tohoku University, Sendai 980-8579, Japan b Korea Institute of Machinery and Materials, Deajeon, South Korea Received 26 November 2004; received in revised form 10 May 2005; accepted 6 June 2005 Available online 10 October 2005
Abstract This paper describes the measurement of straightness error motions (vertical straightness and horizontal straightness) and rotational error motions (pitch, yaw and roll) of a commercial precision linear air-bearing stage actuated by a linear motor. Each of the error motions was measured by two different methods for assurance of reliability. The stage was placed in the XY-plane and moved along the X-direction. The pitch error and yaw error, which were measured by an autocollimator and the angle measurement kit of a laser interferometer, were about 8.7 and 1.6 arc-s, respectively, over a travel of 150 mm with a moving speed of 10 mm/s. The roll error was measured by the autocollimator through scanning a flat mirror along the X-direction. The second method for roll error measurement was to scan two capacitance-type displacement probes along the flat surface placed in the XZ-plane. The two probes with their sensing axes in the Y-direction were aligned with a certain spacing along the Z-axis. The roll error can be obtained by dividing the difference of the outputs of the two probes by the spacing between the two probes. The roll error was measured to be approximately 11.8 arc-s over the 150 mm travel. The horizontal straightness error and the vertical straightness error (Y- and Z-straightness errors) were measured by using the straightness measurement kit of the laser interferometer. The second method for straightness measurement was to scan the flat surface with a capacitance-type displacement probe. The horizontal and vertical straightness errors of the stage over the 150 mm travel were measured to be approximately 207 and 660 nm, respectively. © 2005 Elsevier Inc. All rights reserved. Keywords: Measurement; Error motion; Linear stage; Air-bearing; Linear motor; Straightness; Pitch; Yaw; Roll
1. Introduction Aerostatic bearings (air-bearings) are widely used in precision linear stages. The averaging effect of the air film on local surface errors allows the air-bearing to have higher precision and better repeatability of motion compared to slide bearings or roller bearings with mechanical contact between elements [1]. Air-bearings are especially suited for high-speed use because of their non-contact characteristics. The low viscosity of air also enables air-bearings to outperform hydrostatic bearings in terms of thermal performance. Thus, linear air-bearing stages have become the best choice for most high-accuracy, high-speed positioning applications ∗
Corresponding author. Tel.: +81 22 795 6951; fax: +81 22 795 6951. E-mail address: [email protected] (W. Gao).
0141-6359/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2005.06.003
in semiconductor manufacturing equipment, ultra-precision machine tools and scanning-type measuring instruments [2–8]. On the other hand, since error motions of a linear airbearing stage directly influence the performance of the precision positioning system in which the stage is used, measurement of the error motions is important for performance evaluation and/or error compensation of the positioning system. As can be seen in Fig. 1, there are six error motions for a linear stage, three translational errors (the positioning error, horizontal straightness error and vertical straightness error) and three rotational errors (the pitch error, roll error and yaw error) [9]. Among the six error motions, the positioning error, which is the difference between the command position and the actual position along the direction of motion, is the best measured and compensated error
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Fig. 1. Multi-degree-of-freedom error motions of a linear stage.
motion for most precision air-bearing stages through closedloop servo-control based on the position measured by a laser interferometer or a linear encoder [10–14]. The others are not well measured and compensated error motions, however, they are also important factors in precision positioning systems [15–23]. The only data that can be obtained from manufacturers of linear air-bearing stages are simple out-ofstraightness values (typically on the order of 100 nm over a 100 mm travel) or brief straightness error curves, which are not detailed enough for error-compensation purposes. Very little data is available on rotational error motions, which may dominate the error motion behavior of a precision linear airbearing stage at the nanometer scale. This paper provides measurements of multi-degree-offreedom (MDOF) error motions (except the well-reported positioning error) of a commercial linear air-bearing stage actuated by a linear motor. For assurance of reliability of the measurement data, each error motion was measured by two different kinds of instruments.
2. Measurement of rotational error motions Fig. 2 shows experimental setups for the pitch and yaw error measurements. An autocollimator [24] and a laser interferometer with an angle measurement kit [25] were used. The angular reflector of the interferometer and the target mirror of the autocollimator were mounted on the moving element of the linear stage so that the pitch and yaw errors of the stage could be measured by both the interferometer and the autocollimator. The autocollimator had a measurement range of ±600 arc-s, a resolution of 0.01 arc-s and an accuracy of 0.5 arc-s. The measurement range, resolution and accuracy of the laser interferometer were ±10◦ , 0.05 arcs and ±0.2% of the measured value, respectively. Because both the interferometer and the autocollimator are capable of
Fig. 2. Experimental setups for pitch and yaw error measurement: (a) pitch and (b) yaw error measurement.
two-axis measurement, the pitch and yaw errors were measured simultaneously. Fig. 3 shows results of the stability test, in which the stage was kept stationary. The test duration was 10 min, and the sampling interval was 0.1 s. As can be seen in Fig. 3(a), the output of the autocollimator varied within a range of approximately 0.5 arc-s in both the pitch and yaw measurements. Fig. 3(b) shows that stabilities of the interferometer output in the pitch and yaw error measurements were 0.4 and 0.6 arc-s, respectively. Fig. 4 shows measurement results of pitch and yaw errors over a stage travel of 150 mm. The moving speed of the stage was 10 mm/s. The movement of the stage was servo-controlled by a PID controller based on the measurement result of a linear encoder. The stage travel range and moving speed were set to be the same for all experiments. The data of 10 repeated travels are shown in the figures. As can be seen in Fig. 4(a), the maximum pitch error of the stage over the 150 mm travel was measured to be 8.59 arc-s with a standard deviation of 0.15 and 8.77 arc-s with a standard deviation of 0.05 arc-s by the autocollimator and the laser interferometer, respectively. The maximum yaw error of the stage (Fig. 4(b)) was measured to be 1.72 arc-s with a standard deviation of 0.20 and 1.45 arc-s with a standard deviation of 0.07 arc-s by the autocollimator and the laser interferometer, respectively. Both the pitch error and yaw error varied linearly with the movement position. The pitch error was approximately five times larger than the yaw error. Considering the stability and accuracy levels of the two instruments, measurements by the autocollimator and laser interferometer showed good agreement. Fig. 5 shows the experimental setup for roll error measurement. The roll error was measured with the autocollimator
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Fig. 3. Results of the stability test for pitch and yaw error measurement: (a) autocollimator and (b) laser interferometer.
used for pitch and yaw measurements. Because roll measurement is not a standard function of a commercial laser interferometer, a displacement probe-assembly (Fig. 5) was employed as the second measurement instrument. As can be seen in Fig. 5, a Zerodur straightedge was mounted on the stage. The surface of the straightedge, which was coated with aluminum, was scanned by the displacement probe-assembly and the autocollimator. The probe-assembly consisted of two capacitance-type displacement probes, A and B [26]. The two probes were mounted along the Z-direction with a certain spacing L. Denoting the outputs of probes A and B by mA (x) and mB (x), the roll error eR (x) can be calculated via: eR (x) =
mA (x) − mB (x) L
Fig. 4. Measurement results of pitch and yaw errors: (a) pitch and (b) yaw error.
capacitance probe was 10 nm. Thus, the resolution for roll measurement was calculated to be 0.1 arc-s. The out-ofstraightness of Lines 1 and 2 on the straightedge surface, measured to be approximately 60 nm by the error-separation
(1)
The resolution for roll measurement is determined by that of the capacitance probe and the probe spacing L. In our experiment, the probe spacing L was set to 31 mm based on the dimension of the straightedge. The resolution of the
Fig. 5. Experimental setup for roll error measurement.
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3. Measurement of straightness error motions
Fig. 6. Results of stability test for roll error measurement.
technique based on the multi-probe method [27–30], was compensated for beforehand. The measurement range of the capacitance probe was ±50 m with an accuracy of 0.1% of the measurement range. Fig. 6 shows the results of the stability test. The test duration was 10 min, and the sampling interval was 0.1 s. As can be seen in Fig. 6, the autocollimator had a stability of 0.4 arcs, which was almost the same as those shown in Fig. 3(a). The stability of the capacitance probe-unit was approximately 0.3 arc-s. Fig. 7 shows the roll error measurement results. The results of 10 repeated travels are shown in the figure. The maximum roll error was measured to be 11.95 arc-s with a standard deviation of 0.14 arc-s by the autocollimator, and 11.64 arc-s with a standard deviation of 0.10 arc-s by the capacitance probe-unit. These results showed good agreement considering the stability and accuracy levels of the two methods.
Fig. 7. Roll error measurement results.
Fig. 8(a) shows the experimental setup for the horizontal straightness error measurement. The first measurement instrument was the laser interferometer with a straightness measurement kit. The reflector of the interferometer was mounted on the moving element, while the prism/receiver assembly was mounted outside the moving element. This arrangement had the advantage of a short optical path. The second instrument scanned a displacement probe over a straightedge. The straightedge was mounted on the moving element and the capacitance probe was kept stationary. Again, the experiment employed the capacitance probe and the straightedge used in roll error measurement. The out-of-straightness of the straightedge was compensated for beforehand. After the horizontal straightness measurement, the straightness optics of the interferometer and the capacitance probe were rotated 90◦ for measurement of the vertical straightness error motion (Fig. 8(b)). Because the resolution of the straightness measurement kit of the interferometer was on the order of 350 nm, which was not high enough for measurement of the high-precision linear stage, an averaging process was employed for resolution improvement so that a resolution of 10 nm could be obtained [25]. Fig. 9 shows the results of the stability test of the capacitance probe. The measurement term was 10 min, and the sampling interval was 0.1 s. The instability of the moving element in the horizontal direction was approximately 30 nm, which was mainly low-frequency component caused by thermal drift of the mechanical structure of the stage and long-term instability of the air pressure. In addition to the
Fig. 8. Experimental setup for straightness error measurement: (a) horizontal and (b) vertical straightness error measurement.
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Fig. 9. Results of stability test for straightness error measurement (airpressure fluctuation: 0.05 MPa).
Fig. 11. Results of stability test for straightness error measurement (airpressure fluctuation: 0.01 MPa).
low-frequency component, periodic component with a peakto-valley (PV) value of 60 nm and a period of 1 min was observed in the vertical direction. The period was found to correspond exactly to the fluctuation in compressed air supply pressure to the air-bearing of the stage. The 60 nm PV value corresponded to a pressure variation of 0.05 MPa (the nominal air pressure for the stage was 0.5 MPa). This was due to the structure of the specific linear stage used in the experiment. As can be seen in Fig. 10, a decrease/increase in the air supply pressure to the air-bearing of the stage caused a decrease/increase in pressure in both the vertical (PV ) and horizontal direction (PH ). In the horizontal direction, because the pressure was applied against the two sides of the guide way, the moving element was stable. In the vertical direction, however, because the pressure was applied against the guide way from only one side, a decrease/increase in pressure caused the moving element to move in the −Z/+Z direction. This means that a fluctuation in air pressure would cause a larger instability of the moving element in the vertical direction than in the horizontal direction for this particular kind of
Fig. 10. Influence of pressure fluctuations in the compressed air supply to the air-bearing of the stage.
Fig. 12. Results of straightness error measurement: (a) horizontal and (b) vertical straightness error.
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Fig. 13. Influence of the yaw error on straightness error measurement by the interferometer with a moving reflector.
stage structure. A stable air supply pressure is thus necessary for stabilizing the moving element of the stage. Fig. 11 shows the results obtained with the air-pressure fluctuation kept to within 0.01 MPa. It can be seen that the instability of the stage in the vertical direction was reduced to approximately 20 nm, which was almost the same as that in the horizontal direction. The stability test was not carried out with the interferometer because of the limitation in the data acquisition function of the straightness measurement kit of the interferometer. Fig. 12 shows the results of straightness error motion measurement. The results of 10 repeated travels are shown in the figure. As can be seen in Fig. 12(a), the maximum horizontal straightness error over a travel of 150 mm was measured to be 209 nm with a standard deviation of 16 nm by
Fig. 14. Errors in straightness measurement caused by the pitch and yaw in Fig. 4.
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Fig. 15. Improved setup for straightness error measurement: (a) horizontal and (b) vertical straightness error measurement.
the capacitance probe, and 339 nm with a standard deviation of 38 nm by the interferometer, respectively. The maximum vertical straightness error (Fig. 12(b)) was measured to be 639 nm with a standard deviation of 28 nm by the capacitance probe, and 1188 nm with a standard deviation of 86 nm by the interferometer. Clearly, the capacitance probe and interferometer results differed substantially, especially in the vertical direction. An analysis revealed that this difference was caused by stage rotational errors (pitch error and/or yaw error) in the setup of the straightness measurement kit of the interferometer shown in Fig. 8. Fig. 13 shows the case for horizontal straightness measurement, which is affected by the yaw error. If the reflector of the interferometer is mounted on the moving element of the stage, the reflector will rotate by exactly the same angle as the yaw error eYAW , causing an optical path change eOP in Beam 1 and −eOP in Beam 2, respectively. For simplicity, let us assume that the center of rotation is the same as that of the reflector. Let the distance between the center of the prism (O1 ) and the center of the reflector (O2 ) at the starting position of the moving element of the stage (x = 0) be D0 , and the angle between Beams 1 and 2 be φ, eOP at position x can then be expressed as: ϕ eOP (x) = (D0 + x) sin tan(eYAW (x)) (2) 2 When the yaw error eYAW is small enough, eOP becomes ϕ eOP (x) = (D0 + x) sin eYAW (x) (3) 2 The error in the horizontal straightness error measurement then becomes: ϕ eS−H (x) = eOPD (x)/ sin (4) = 2(D0 + x)eYAW (x) 2
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where eOPD (x) = 2eOP (x) is the optical path difference between Beams 1 and 2. Similarly, the error in the vertical straightness measurement, which is caused by the pitch error of the moving element of the stage (eP (x)), can be expressed by: eS−V (x) = 2(D0 + x)eP (x)
Fig. 16. Influence of the yaw error on straightness measurement by the interferometer with a moving prism/receiver assembly.
(5)
Fig. 14 shows the calculated eS−H (x) and eS−V (x) based on the measured pitch error and yaw error shown in Fig. 4. It can be seen that differences between the capacitance probe and laser interferometer measurements (Fig. 12) were mainly caused by eS−H (x) and eS−V (x). To avoid the error component due to the rotational error, an improved setup (Fig. 15) was used. Instead of the movingreflector arrangement, the prism/receiver assembly of the interferometer was mounted on the moving element of the stage and the reflector was kept stationary. As can be seen in Fig. 16, a rotation of the prism/receiver assembly causes an equal optical path change in both Beams 1 and 2, which does not affect the measurements of the straightness error motion. Fig. 17 shows the measured straightness errors obtained using the improved setup shown in Fig. 15. The maximum horizontal and vertical straightness errors were measured to be 205 nm (standard deviation: 35 nm) and 680 nm (standard deviation: 57 nm), respectively, by the interferometer. The capacitance probe is also shown in the figure for comparison. The two methods are seen to produce close results owing to the improved setup.
4. Conclusion
Fig. 17. Results of straightness error measurement with the improved experimental setup: (a) horizontal and (b) vertical straightness error.
Multi-degree-of-freedom error motions (except the wellreported positioning error) of a linear air-bearing stage driven by a linear motor were measured. For comparison, each error motion was measured by two different methods, both of which are based on commercially available measuring instruments and sensors, and can easily be constructed. An autocollimator and a laser interferometer with an angle measurement kit were used for pitch and yaw error measurements. The roll error was measured by the autocollimator as well as a displacement probe-unit consisting of two capacitance probes. The rotational error motions of the stage were measured to be approximately 8.7 arc-s (pitch error), 1.6 arc-s (yaw error) and 11.8 arc-s (roll error) over a travel of 150 mm. The straightness errors were measured by a laser interferometer with a straightness measurement kit, as well as by a capacitance probe that was used to scan a straightedge. It has been confirmed that the influence of air-pressure fluctuations on the stability of the stage varies with the stage structure. Applying air pressure from opposite directions against the guide way of the stage results in higher stage stability. It has also been verified the moving-reflector arrangement of the straightness measurement kit suffers from the rotational error in the stage. By contrast, the arrangement of moving-
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prism/receiver assembly enables higher measurement accuracy. The horizontal and vertical straightness errors were measured to be approximately 207 and 660 nm over a travel of 150 mm. Each error motion was successfully measured by two different methods showing good agreement within the range of stability and accuracy of the methods, it should be pointed out that we have not analyzed the uncertainty in either of these measurement methods. This and the investigation of MDOF motion errors of the stage under different conditions, such as movement speed and load, is left for future work.
Acknowledgements This work was financially supported by grants from the New Energy and Industrial Technology Development Organization (NEDO) and the Ministry of Science and Technology of Korea. The authors thank Dr. Y. Uda of Nikon Cooperation for providing the straightedge.
References [1] Slocum AH. Precision machine design. Prentice Hall; 1992 [Chapter 9]. [2] Slocum AH, Basaran M, Cortesi R, Anastasios JH. Linear motion carriage with aerostatic bearings preloaded by inclined iron core linear electric motor. Precision Eng 2003;27:382–94. [3] Eom TB, Han JW. A precision length measuring system for a variety of linear artifacts. Meas Sci Technol 2001;12:698–701. [4] Mekid S, Bonis M. Conceptual design and study of high precision translational stages: application to an optical delay line. Precision Eng 1997;21:29–35. [5] Israel W, Tiemann I, Metz G, Yamaryo Y, Maeda E, Simomura T. An international length comparison at an industrial level using a photoelectric incremental encoder as transfer standard. Precision Eng 2003;27:151–6. [6] Sawabe M, Maeda F, Yamaryo Y, Simonura T, Saruki Y, Kudo T, et al. A new vacuum interferometric comparator for calibrating the fine linear encoders and scales. Precision Eng 2004;28:320–8. [7] Mekid S. High precision linear slide. Part I: design and construction. Int J Mach Tools Manuf 2000;40:1039–50. [8] Lee C-W, Kim S-W. An ultraprecision stage for alignment of wafers in advanced microlithography. Precision Eng 1997;21(2–3):113–22. [9] Tyler EW. Calibration and use of optical straightedges in the metrology of precision machines. Opt Eng 1985;24(3):372–9.
103
[10] Kunzmann H, Pfeifer T, Flugge J. Scales vs. laser interferometers, performance and comparison of two measuring systems. Ann CIRP 1993;42(2):753–67. [11] Steinmetz RC. Sub-micron position measurement and control on precision machine tools with laser interferometry. Precision Eng 1990;12(1):12–24. [12] Teimel A. Technology and applications of grating interferometers in high-precision measurement. Precision Eng 1992;14(3):147–54. [13] Onda H. Development of a unique, high precision linear motor integrated air slide table, and its application to laser beam writers. Opt Rev 1999;6(1):88–92. [14] Mao J, Tachikawa H, Shimokohbe A. Precision positioning of a DC-motor-driven aerostatic slide system. Precision Eng 2003;27(1): 32–41. [15] Huang PS, Li Y. Laser measurement instrument for fast calibration of machine tools. In: Proceedings of the American Society for Precision Engineering, 1996 Annual Meeting. 1996. p. 644–7. [16] Fan K-C, Chen M-J. A 6-degree-of-freedom measurement system for the accuracy of X–Y stages. Precision Eng 2000;24(1):15–23. [17] Gao W, Dejima S, Shimizu Y, Kiyono S. Precision measurement of two-axis positions and tilt motions using a surface encoder. Ann CIRP 2003;52(1):435–8. [18] Gao W, Araki T, Kiyono S, Okazaki Y, Yamanaka M. Precision nanofabrication and evaluation of a large area sinusoidal grid surface for a surface encoder. Precision Eng 2003;27(3):289–98. [19] Lee QS, Gweon GD. A new 3-DOF Z-tilts micropositioning system using electromagnetic actuators and air bearings. Precision Eng 2000;24(1):24–31. [20] Tomita Y, Koyanagawa Y, Satoh F. A surface motor-driven precision positioning system. Precision Eng 1994;16(3):184–91. [21] Gao W, Dejima S, Yanai H, Katakura K, Kiyono S, Tomita Y. A surface motor-driven planar motion stage integrated with an XYθ Z surface encoder for precision positioning. Precision Eng 2004;28(3): 329–37. [22] Watanabe Y, Gao W, Shimizu H, Kiyono S. Analysis of a surface encoder in wave optics. Key Eng Mater 2003;257–258:219–24. [23] Gao W, Dejima S, Kiyono S. A dual-mode surface encoder for position measurement. Sens Actuators A 2004;111(2–3):216–21. [24] Autocollimator catalogue, Tokyo, Japan: Nikon Corporation. [25] Laser interferometer catalogue, Palo Alto, CA, USA: Agilent Technologies Inc. [26] Capacitance gage catalogue, Westwood, MA, USA: ADE Corporation. [27] Kiyono S, Gao W. Profile measurement of machined surface with combined method. Precision Eng 1994;16(3):212–8. [28] Gao W, Kiyono S. High accuracy profile measurement of machined surface by the combined method. Measurement 1996;19(1):55–64. [29] Gao W, Yokoyama J, Kojima H, Kiyono S. Precision measurement of cylinder straightness using a scanning multi-probe system. Precision Eng 2002;26(3):279–88. [30] Arai Y, Gao W, Kiyono S, Kuriyagawa T. Measurement of the straightness of a leadscrew-driven precision stage. Key Eng Mater 2005;295–296:259–64.
Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage Wei Gao a,∗ , Yoshikazu Arai a , Atsushi Shibuya a , Satoshi Kiyono a , Chun Hong Park b a
Department of Mechatronics and Precision Engineering, Tohoku University, Sendai 980-8579, Japan b Korea Institute of Machinery and Materials, Deajeon, South Korea Received 26 November 2004; received in revised form 10 May 2005; accepted 6 June 2005 Available online 10 October 2005
Abstract This paper describes the measurement of straightness error motions (vertical straightness and horizontal straightness) and rotational error motions (pitch, yaw and roll) of a commercial precision linear air-bearing stage actuated by a linear motor. Each of the error motions was measured by two different methods for assurance of reliability. The stage was placed in the XY-plane and moved along the X-direction. The pitch error and yaw error, which were measured by an autocollimator and the angle measurement kit of a laser interferometer, were about 8.7 and 1.6 arc-s, respectively, over a travel of 150 mm with a moving speed of 10 mm/s. The roll error was measured by the autocollimator through scanning a flat mirror along the X-direction. The second method for roll error measurement was to scan two capacitance-type displacement probes along the flat surface placed in the XZ-plane. The two probes with their sensing axes in the Y-direction were aligned with a certain spacing along the Z-axis. The roll error can be obtained by dividing the difference of the outputs of the two probes by the spacing between the two probes. The roll error was measured to be approximately 11.8 arc-s over the 150 mm travel. The horizontal straightness error and the vertical straightness error (Y- and Z-straightness errors) were measured by using the straightness measurement kit of the laser interferometer. The second method for straightness measurement was to scan the flat surface with a capacitance-type displacement probe. The horizontal and vertical straightness errors of the stage over the 150 mm travel were measured to be approximately 207 and 660 nm, respectively. © 2005 Elsevier Inc. All rights reserved. Keywords: Measurement; Error motion; Linear stage; Air-bearing; Linear motor; Straightness; Pitch; Yaw; Roll
1. Introduction Aerostatic bearings (air-bearings) are widely used in precision linear stages. The averaging effect of the air film on local surface errors allows the air-bearing to have higher precision and better repeatability of motion compared to slide bearings or roller bearings with mechanical contact between elements [1]. Air-bearings are especially suited for high-speed use because of their non-contact characteristics. The low viscosity of air also enables air-bearings to outperform hydrostatic bearings in terms of thermal performance. Thus, linear air-bearing stages have become the best choice for most high-accuracy, high-speed positioning applications ∗
Corresponding author. Tel.: +81 22 795 6951; fax: +81 22 795 6951. E-mail address: [email protected] (W. Gao).
0141-6359/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2005.06.003
in semiconductor manufacturing equipment, ultra-precision machine tools and scanning-type measuring instruments [2–8]. On the other hand, since error motions of a linear airbearing stage directly influence the performance of the precision positioning system in which the stage is used, measurement of the error motions is important for performance evaluation and/or error compensation of the positioning system. As can be seen in Fig. 1, there are six error motions for a linear stage, three translational errors (the positioning error, horizontal straightness error and vertical straightness error) and three rotational errors (the pitch error, roll error and yaw error) [9]. Among the six error motions, the positioning error, which is the difference between the command position and the actual position along the direction of motion, is the best measured and compensated error
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97
Fig. 1. Multi-degree-of-freedom error motions of a linear stage.
motion for most precision air-bearing stages through closedloop servo-control based on the position measured by a laser interferometer or a linear encoder [10–14]. The others are not well measured and compensated error motions, however, they are also important factors in precision positioning systems [15–23]. The only data that can be obtained from manufacturers of linear air-bearing stages are simple out-ofstraightness values (typically on the order of 100 nm over a 100 mm travel) or brief straightness error curves, which are not detailed enough for error-compensation purposes. Very little data is available on rotational error motions, which may dominate the error motion behavior of a precision linear airbearing stage at the nanometer scale. This paper provides measurements of multi-degree-offreedom (MDOF) error motions (except the well-reported positioning error) of a commercial linear air-bearing stage actuated by a linear motor. For assurance of reliability of the measurement data, each error motion was measured by two different kinds of instruments.
2. Measurement of rotational error motions Fig. 2 shows experimental setups for the pitch and yaw error measurements. An autocollimator [24] and a laser interferometer with an angle measurement kit [25] were used. The angular reflector of the interferometer and the target mirror of the autocollimator were mounted on the moving element of the linear stage so that the pitch and yaw errors of the stage could be measured by both the interferometer and the autocollimator. The autocollimator had a measurement range of ±600 arc-s, a resolution of 0.01 arc-s and an accuracy of 0.5 arc-s. The measurement range, resolution and accuracy of the laser interferometer were ±10◦ , 0.05 arcs and ±0.2% of the measured value, respectively. Because both the interferometer and the autocollimator are capable of
Fig. 2. Experimental setups for pitch and yaw error measurement: (a) pitch and (b) yaw error measurement.
two-axis measurement, the pitch and yaw errors were measured simultaneously. Fig. 3 shows results of the stability test, in which the stage was kept stationary. The test duration was 10 min, and the sampling interval was 0.1 s. As can be seen in Fig. 3(a), the output of the autocollimator varied within a range of approximately 0.5 arc-s in both the pitch and yaw measurements. Fig. 3(b) shows that stabilities of the interferometer output in the pitch and yaw error measurements were 0.4 and 0.6 arc-s, respectively. Fig. 4 shows measurement results of pitch and yaw errors over a stage travel of 150 mm. The moving speed of the stage was 10 mm/s. The movement of the stage was servo-controlled by a PID controller based on the measurement result of a linear encoder. The stage travel range and moving speed were set to be the same for all experiments. The data of 10 repeated travels are shown in the figures. As can be seen in Fig. 4(a), the maximum pitch error of the stage over the 150 mm travel was measured to be 8.59 arc-s with a standard deviation of 0.15 and 8.77 arc-s with a standard deviation of 0.05 arc-s by the autocollimator and the laser interferometer, respectively. The maximum yaw error of the stage (Fig. 4(b)) was measured to be 1.72 arc-s with a standard deviation of 0.20 and 1.45 arc-s with a standard deviation of 0.07 arc-s by the autocollimator and the laser interferometer, respectively. Both the pitch error and yaw error varied linearly with the movement position. The pitch error was approximately five times larger than the yaw error. Considering the stability and accuracy levels of the two instruments, measurements by the autocollimator and laser interferometer showed good agreement. Fig. 5 shows the experimental setup for roll error measurement. The roll error was measured with the autocollimator
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Fig. 3. Results of the stability test for pitch and yaw error measurement: (a) autocollimator and (b) laser interferometer.
used for pitch and yaw measurements. Because roll measurement is not a standard function of a commercial laser interferometer, a displacement probe-assembly (Fig. 5) was employed as the second measurement instrument. As can be seen in Fig. 5, a Zerodur straightedge was mounted on the stage. The surface of the straightedge, which was coated with aluminum, was scanned by the displacement probe-assembly and the autocollimator. The probe-assembly consisted of two capacitance-type displacement probes, A and B [26]. The two probes were mounted along the Z-direction with a certain spacing L. Denoting the outputs of probes A and B by mA (x) and mB (x), the roll error eR (x) can be calculated via: eR (x) =
mA (x) − mB (x) L
Fig. 4. Measurement results of pitch and yaw errors: (a) pitch and (b) yaw error.
capacitance probe was 10 nm. Thus, the resolution for roll measurement was calculated to be 0.1 arc-s. The out-ofstraightness of Lines 1 and 2 on the straightedge surface, measured to be approximately 60 nm by the error-separation
(1)
The resolution for roll measurement is determined by that of the capacitance probe and the probe spacing L. In our experiment, the probe spacing L was set to 31 mm based on the dimension of the straightedge. The resolution of the
Fig. 5. Experimental setup for roll error measurement.
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3. Measurement of straightness error motions
Fig. 6. Results of stability test for roll error measurement.
technique based on the multi-probe method [27–30], was compensated for beforehand. The measurement range of the capacitance probe was ±50 m with an accuracy of 0.1% of the measurement range. Fig. 6 shows the results of the stability test. The test duration was 10 min, and the sampling interval was 0.1 s. As can be seen in Fig. 6, the autocollimator had a stability of 0.4 arcs, which was almost the same as those shown in Fig. 3(a). The stability of the capacitance probe-unit was approximately 0.3 arc-s. Fig. 7 shows the roll error measurement results. The results of 10 repeated travels are shown in the figure. The maximum roll error was measured to be 11.95 arc-s with a standard deviation of 0.14 arc-s by the autocollimator, and 11.64 arc-s with a standard deviation of 0.10 arc-s by the capacitance probe-unit. These results showed good agreement considering the stability and accuracy levels of the two methods.
Fig. 7. Roll error measurement results.
Fig. 8(a) shows the experimental setup for the horizontal straightness error measurement. The first measurement instrument was the laser interferometer with a straightness measurement kit. The reflector of the interferometer was mounted on the moving element, while the prism/receiver assembly was mounted outside the moving element. This arrangement had the advantage of a short optical path. The second instrument scanned a displacement probe over a straightedge. The straightedge was mounted on the moving element and the capacitance probe was kept stationary. Again, the experiment employed the capacitance probe and the straightedge used in roll error measurement. The out-of-straightness of the straightedge was compensated for beforehand. After the horizontal straightness measurement, the straightness optics of the interferometer and the capacitance probe were rotated 90◦ for measurement of the vertical straightness error motion (Fig. 8(b)). Because the resolution of the straightness measurement kit of the interferometer was on the order of 350 nm, which was not high enough for measurement of the high-precision linear stage, an averaging process was employed for resolution improvement so that a resolution of 10 nm could be obtained [25]. Fig. 9 shows the results of the stability test of the capacitance probe. The measurement term was 10 min, and the sampling interval was 0.1 s. The instability of the moving element in the horizontal direction was approximately 30 nm, which was mainly low-frequency component caused by thermal drift of the mechanical structure of the stage and long-term instability of the air pressure. In addition to the
Fig. 8. Experimental setup for straightness error measurement: (a) horizontal and (b) vertical straightness error measurement.
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Fig. 9. Results of stability test for straightness error measurement (airpressure fluctuation: 0.05 MPa).
Fig. 11. Results of stability test for straightness error measurement (airpressure fluctuation: 0.01 MPa).
low-frequency component, periodic component with a peakto-valley (PV) value of 60 nm and a period of 1 min was observed in the vertical direction. The period was found to correspond exactly to the fluctuation in compressed air supply pressure to the air-bearing of the stage. The 60 nm PV value corresponded to a pressure variation of 0.05 MPa (the nominal air pressure for the stage was 0.5 MPa). This was due to the structure of the specific linear stage used in the experiment. As can be seen in Fig. 10, a decrease/increase in the air supply pressure to the air-bearing of the stage caused a decrease/increase in pressure in both the vertical (PV ) and horizontal direction (PH ). In the horizontal direction, because the pressure was applied against the two sides of the guide way, the moving element was stable. In the vertical direction, however, because the pressure was applied against the guide way from only one side, a decrease/increase in pressure caused the moving element to move in the −Z/+Z direction. This means that a fluctuation in air pressure would cause a larger instability of the moving element in the vertical direction than in the horizontal direction for this particular kind of
Fig. 10. Influence of pressure fluctuations in the compressed air supply to the air-bearing of the stage.
Fig. 12. Results of straightness error measurement: (a) horizontal and (b) vertical straightness error.
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Fig. 13. Influence of the yaw error on straightness error measurement by the interferometer with a moving reflector.
stage structure. A stable air supply pressure is thus necessary for stabilizing the moving element of the stage. Fig. 11 shows the results obtained with the air-pressure fluctuation kept to within 0.01 MPa. It can be seen that the instability of the stage in the vertical direction was reduced to approximately 20 nm, which was almost the same as that in the horizontal direction. The stability test was not carried out with the interferometer because of the limitation in the data acquisition function of the straightness measurement kit of the interferometer. Fig. 12 shows the results of straightness error motion measurement. The results of 10 repeated travels are shown in the figure. As can be seen in Fig. 12(a), the maximum horizontal straightness error over a travel of 150 mm was measured to be 209 nm with a standard deviation of 16 nm by
Fig. 14. Errors in straightness measurement caused by the pitch and yaw in Fig. 4.
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Fig. 15. Improved setup for straightness error measurement: (a) horizontal and (b) vertical straightness error measurement.
the capacitance probe, and 339 nm with a standard deviation of 38 nm by the interferometer, respectively. The maximum vertical straightness error (Fig. 12(b)) was measured to be 639 nm with a standard deviation of 28 nm by the capacitance probe, and 1188 nm with a standard deviation of 86 nm by the interferometer. Clearly, the capacitance probe and interferometer results differed substantially, especially in the vertical direction. An analysis revealed that this difference was caused by stage rotational errors (pitch error and/or yaw error) in the setup of the straightness measurement kit of the interferometer shown in Fig. 8. Fig. 13 shows the case for horizontal straightness measurement, which is affected by the yaw error. If the reflector of the interferometer is mounted on the moving element of the stage, the reflector will rotate by exactly the same angle as the yaw error eYAW , causing an optical path change eOP in Beam 1 and −eOP in Beam 2, respectively. For simplicity, let us assume that the center of rotation is the same as that of the reflector. Let the distance between the center of the prism (O1 ) and the center of the reflector (O2 ) at the starting position of the moving element of the stage (x = 0) be D0 , and the angle between Beams 1 and 2 be φ, eOP at position x can then be expressed as: ϕ eOP (x) = (D0 + x) sin tan(eYAW (x)) (2) 2 When the yaw error eYAW is small enough, eOP becomes ϕ eOP (x) = (D0 + x) sin eYAW (x) (3) 2 The error in the horizontal straightness error measurement then becomes: ϕ eS−H (x) = eOPD (x)/ sin (4) = 2(D0 + x)eYAW (x) 2
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where eOPD (x) = 2eOP (x) is the optical path difference between Beams 1 and 2. Similarly, the error in the vertical straightness measurement, which is caused by the pitch error of the moving element of the stage (eP (x)), can be expressed by: eS−V (x) = 2(D0 + x)eP (x)
Fig. 16. Influence of the yaw error on straightness measurement by the interferometer with a moving prism/receiver assembly.
(5)
Fig. 14 shows the calculated eS−H (x) and eS−V (x) based on the measured pitch error and yaw error shown in Fig. 4. It can be seen that differences between the capacitance probe and laser interferometer measurements (Fig. 12) were mainly caused by eS−H (x) and eS−V (x). To avoid the error component due to the rotational error, an improved setup (Fig. 15) was used. Instead of the movingreflector arrangement, the prism/receiver assembly of the interferometer was mounted on the moving element of the stage and the reflector was kept stationary. As can be seen in Fig. 16, a rotation of the prism/receiver assembly causes an equal optical path change in both Beams 1 and 2, which does not affect the measurements of the straightness error motion. Fig. 17 shows the measured straightness errors obtained using the improved setup shown in Fig. 15. The maximum horizontal and vertical straightness errors were measured to be 205 nm (standard deviation: 35 nm) and 680 nm (standard deviation: 57 nm), respectively, by the interferometer. The capacitance probe is also shown in the figure for comparison. The two methods are seen to produce close results owing to the improved setup.
4. Conclusion
Fig. 17. Results of straightness error measurement with the improved experimental setup: (a) horizontal and (b) vertical straightness error.
Multi-degree-of-freedom error motions (except the wellreported positioning error) of a linear air-bearing stage driven by a linear motor were measured. For comparison, each error motion was measured by two different methods, both of which are based on commercially available measuring instruments and sensors, and can easily be constructed. An autocollimator and a laser interferometer with an angle measurement kit were used for pitch and yaw error measurements. The roll error was measured by the autocollimator as well as a displacement probe-unit consisting of two capacitance probes. The rotational error motions of the stage were measured to be approximately 8.7 arc-s (pitch error), 1.6 arc-s (yaw error) and 11.8 arc-s (roll error) over a travel of 150 mm. The straightness errors were measured by a laser interferometer with a straightness measurement kit, as well as by a capacitance probe that was used to scan a straightedge. It has been confirmed that the influence of air-pressure fluctuations on the stability of the stage varies with the stage structure. Applying air pressure from opposite directions against the guide way of the stage results in higher stage stability. It has also been verified the moving-reflector arrangement of the straightness measurement kit suffers from the rotational error in the stage. By contrast, the arrangement of moving-
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prism/receiver assembly enables higher measurement accuracy. The horizontal and vertical straightness errors were measured to be approximately 207 and 660 nm over a travel of 150 mm. Each error motion was successfully measured by two different methods showing good agreement within the range of stability and accuracy of the methods, it should be pointed out that we have not analyzed the uncertainty in either of these measurement methods. This and the investigation of MDOF motion errors of the stage under different conditions, such as movement speed and load, is left for future work.
Acknowledgements This work was financially supported by grants from the New Energy and Industrial Technology Development Organization (NEDO) and the Ministry of Science and Technology of Korea. The authors thank Dr. Y. Uda of Nikon Cooperation for providing the straightedge.
References [1] Slocum AH. Precision machine design. Prentice Hall; 1992 [Chapter 9]. [2] Slocum AH, Basaran M, Cortesi R, Anastasios JH. Linear motion carriage with aerostatic bearings preloaded by inclined iron core linear electric motor. Precision Eng 2003;27:382–94. [3] Eom TB, Han JW. A precision length measuring system for a variety of linear artifacts. Meas Sci Technol 2001;12:698–701. [4] Mekid S, Bonis M. Conceptual design and study of high precision translational stages: application to an optical delay line. Precision Eng 1997;21:29–35. [5] Israel W, Tiemann I, Metz G, Yamaryo Y, Maeda E, Simomura T. An international length comparison at an industrial level using a photoelectric incremental encoder as transfer standard. Precision Eng 2003;27:151–6. [6] Sawabe M, Maeda F, Yamaryo Y, Simonura T, Saruki Y, Kudo T, et al. A new vacuum interferometric comparator for calibrating the fine linear encoders and scales. Precision Eng 2004;28:320–8. [7] Mekid S. High precision linear slide. Part I: design and construction. Int J Mach Tools Manuf 2000;40:1039–50. [8] Lee C-W, Kim S-W. An ultraprecision stage for alignment of wafers in advanced microlithography. Precision Eng 1997;21(2–3):113–22. [9] Tyler EW. Calibration and use of optical straightedges in the metrology of precision machines. Opt Eng 1985;24(3):372–9.
103
[10] Kunzmann H, Pfeifer T, Flugge J. Scales vs. laser interferometers, performance and comparison of two measuring systems. Ann CIRP 1993;42(2):753–67. [11] Steinmetz RC. Sub-micron position measurement and control on precision machine tools with laser interferometry. Precision Eng 1990;12(1):12–24. [12] Teimel A. Technology and applications of grating interferometers in high-precision measurement. Precision Eng 1992;14(3):147–54. [13] Onda H. Development of a unique, high precision linear motor integrated air slide table, and its application to laser beam writers. Opt Rev 1999;6(1):88–92. [14] Mao J, Tachikawa H, Shimokohbe A. Precision positioning of a DC-motor-driven aerostatic slide system. Precision Eng 2003;27(1): 32–41. [15] Huang PS, Li Y. Laser measurement instrument for fast calibration of machine tools. In: Proceedings of the American Society for Precision Engineering, 1996 Annual Meeting. 1996. p. 644–7. [16] Fan K-C, Chen M-J. A 6-degree-of-freedom measurement system for the accuracy of X–Y stages. Precision Eng 2000;24(1):15–23. [17] Gao W, Dejima S, Shimizu Y, Kiyono S. Precision measurement of two-axis positions and tilt motions using a surface encoder. Ann CIRP 2003;52(1):435–8. [18] Gao W, Araki T, Kiyono S, Okazaki Y, Yamanaka M. Precision nanofabrication and evaluation of a large area sinusoidal grid surface for a surface encoder. Precision Eng 2003;27(3):289–98. [19] Lee QS, Gweon GD. A new 3-DOF Z-tilts micropositioning system using electromagnetic actuators and air bearings. Precision Eng 2000;24(1):24–31. [20] Tomita Y, Koyanagawa Y, Satoh F. A surface motor-driven precision positioning system. Precision Eng 1994;16(3):184–91. [21] Gao W, Dejima S, Yanai H, Katakura K, Kiyono S, Tomita Y. A surface motor-driven planar motion stage integrated with an XYθ Z surface encoder for precision positioning. Precision Eng 2004;28(3): 329–37. [22] Watanabe Y, Gao W, Shimizu H, Kiyono S. Analysis of a surface encoder in wave optics. Key Eng Mater 2003;257–258:219–24. [23] Gao W, Dejima S, Kiyono S. A dual-mode surface encoder for position measurement. Sens Actuators A 2004;111(2–3):216–21. [24] Autocollimator catalogue, Tokyo, Japan: Nikon Corporation. [25] Laser interferometer catalogue, Palo Alto, CA, USA: Agilent Technologies Inc. [26] Capacitance gage catalogue, Westwood, MA, USA: ADE Corporation. [27] Kiyono S, Gao W. Profile measurement of machined surface with combined method. Precision Eng 1994;16(3):212–8. [28] Gao W, Kiyono S. High accuracy profile measurement of machined surface by the combined method. Measurement 1996;19(1):55–64. [29] Gao W, Yokoyama J, Kojima H, Kiyono S. Precision measurement of cylinder straightness using a scanning multi-probe system. Precision Eng 2002;26(3):279–88. [30] Arai Y, Gao W, Kiyono S, Kuriyagawa T. Measurement of the straightness of a leadscrew-driven precision stage. Key Eng Mater 2005;295–296:259–64.