Development of a steel ball center alignment device based on Michelson interference concept Hau-Wei Lee and Chien-Hung Liu Citation: Review of Scientific Instruments 85, 095115 (2014); doi: 10.1063/1.4895669 View online: http://dx.doi.org/10.1063/1.4895669 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental surface plasmon resonance modulated radially sheared interference imaging using a birefringent lens Appl. Phys. Lett. 104, 251104 (2014); 10.1063/1.4884815 Geometric explanation of conic-section interference fringes in a Michelson interferometer Am. J. Phys. 81, 670 (2013); 10.1119/1.4811780 Simple method for thickness measurement in opaque samples with a MichelsonSagnac interferometer. AIP Conf. Proc. 992, 793 (2008); 10.1063/1.2926972 Observation of the interferences between the emitted beams in a 4Pi microscope by partial coherence interferometry Appl. Phys. Lett. 87, 181103 (2005); 10.1063/1.2120908 Three-dimensional display of light interference patterns Am. J. Phys. 67, 453 (1999); 10.1119/1.19288
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 095115 (2014)
Development of a steel ball center alignment device based on Michelson interference concept Hau-Wei Lee1,a) and Chien-Hung Liu2 1 2
Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu 300, Taiwan Department of Mechanical Engineering, National Chung Hsing University, Taichung 402, Taiwan
(Received 3 July 2014; accepted 1 September 2014; published online 23 September 2014) This study presents a ball center alignment method based on the Michelson interferometer where one of the reflecting mirrors is replaced by a lens and steel ball. By locating the ball away from the focal length of the lens, the beam is reflected as a spherical wave. The interference ring formed by the planar and spherical waves can be clearly observed using a camera without a lens. The distance of the offset of the ball center can be enhanced by more than 140% using this method. A fast ring profile fitting method can reduce circle fitting time to around a third of that needed for Hough transformation. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895669] I. INTRODUCTION
Balls are used in many different kinds of inspection, such as the measurement of axis rotational error using a ball bar with sensors as described in ISO 230-7:2006. Theoretically, when a spherical distance capacitance probe is used, the probe is placed near the center of the ball and this allows the center to be determined using the shortest distance method. However, angular deviations in sensor installation may result in cosine errors. Geometrical optical methods are also used for spindle rotational error measurement instead of attempts to reduce inherent ball center alignment errors.1, 2 Ball gauges, step ball gauges, and ball plates are normally used for the calibration of coordinate measuring machines (CMM),3 and alternative methods for calibrating CMMs are described in ISO 10360-4:2000, ISO 10360-5:2010, and ISO 103609:2013. Although the ball center can be located by CMM using multi-point measurement, this suffers from uncertainties that result from inaccurate assembly of the CMM platform as well as stylus tolerance. An optical calibration system for five-axis machine tools has been designed by Jywe et al.4, 5 Similar uncertainty may exist when the laser beam is not perfectly aligned with the ball lens center line. Thus, this measurement system requires careful calibration before it can be applied. In this application the reference plane of the planar reflective mirror is replaced by a steel ball to reduce the uncertainty of the Laser Tracer.6 In spite of reduced measurement uncertainty, the steel ball reflective surface may lead to a translational measurement error if the laser beam fails to reflect perpendicular to the sphere. A small shift of the ball through a distance of δ m , would lead to reflective beam deviation of k × δ m and k which is larger than one. This makes measurement uncertainty worse due to the optical path difference. Since balls are used in many different measurement systems, a means of checking that the beam is exactly aligned
a) Author to whom the correspondence should be addressed. Electronic mail:
[email protected]
0034-6748/2014/85(9)/095115/6/$30.00
to the ball center line is needed to reduce measurement uncertainty. In order to ensure the measurement stylus is orthogonal to the sphere surface and passing through the center, Yague et al. proposed a self-centering contact stylus method using three balls.7, 8 Based on the same concept, a Laser Tracker of smaller measurement range was proposed by Osawa et al.9 Although, these methods ensure the stylus and the beam are orthogonal to the sphere surface and coincide with the center line, the problem of contact wear would still influence measurement accuracy after long term use. Burke et al. used a lens, and convex and concave mirrors as well as geometric optics for ball center alignment.10 Yang et al.11 used a Fizeau interferometric microscope for 3D profile measurement. This microscope can construct a 3D profile for a spherical surface, but is limited to micro workpieces (e.g., micro lenses) and the image was more difficult to process due to the complex nature of the interference fringes. These two non-contact measurement methods do not work as well as the ball plate for rapid ball center alignment or Laser Tracer assembly, although they can produce a 3D spherical profile.
II. CONCEPTUAL OUTLINE
This study proposes a non-contact ball center aligning method using the Michelson interference concept and geometrical optics in a layout shown in Figure 1(a). A polished steel ball replaces one of the reflective mirrors of the Michelson interferometer. The plane mirror serves to reflect the reference beam. The planar wave of the He-Ne laser remains planar when the reference beam is sent directly to the camera detector, through the BS spectroscope via the plane mirror, without a camera lens in the beam. When the steel ball surface is at the lens focus, the beam reflected by the sphere passes through the lens as a collimated beam which in the form of a planar wave and the interference fringes are difficult to observe at the camera detector. Therefore, we locate the ball away from the focal point so that the reflected beam is in the form of a spherical wave. The planar and spherical
85, 095115-1
© 2014 AIP Publishing LLC
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095115-2
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
FIG. 1. Measuring the ball center with the Michelson interferometer setup (with a steel ball of 25 mm diameter and a lens of 25 mm focal length). (a) A concentric interference fringe appears when beam reflection is coincident with the center of the ball and (b) an interference fringe from a ball offset from the center.
waves can be expressed as A (z) = Beikz ,
(1)
A (r) = Be±ikr ,
(2)
where k = 2π /λ; z is the beam translation direction of the planar wave; r is the radius of the spherical wave; constant B is the amplitude of the beam; and the plus and minus signs indicate the direction (i.e., divergence or convergence) of the spherical wave. The superimposition of the spherical and planar waves has no angular component, and thus the resulting waves can be expressed as A(φSL ) = B + Be±iφSL ,
(3)
where φ SL represents the phase shift. As shown in Figure 2(a) we know that δ 2 − 2rδ + x 2 = 0.
(4)
Since δ 2 1, the optical path difference between the planar and spherical waves can be expressed as δ = x 2 /2r.
(5)
To consider the optical path difference and ball center deviation, Eq. (5) can be revised to δ = δL + (x − xm )2 /2r,
(6)
where xm is the offset displacement of the sphere center. The phase shift between the planar and spherical waves can be derived from φSL = k[δL + (x − xm )2 /2r].
(7)
Thus, the interference beam intensity in 2D space can be expressed as I (x, y) = 4I0 (x, y) cos2 {k[δL + (x − xm )2 /2r + (y − ym )2 /2r]}.
(8)
Superimposing the two beams as shown in Figure 3(a) would generate interference images as shown in Figure 1(a). In a case where the reflective beam deviates from the reference beam’s axis as shown in Figure 3(b) another interference image would appear as shown in Figure 1(b). Bruke used the same idea for parabolic mirror alignment12 and also used a ball for calibration.13, 14 The angle and position of a nanorod can also be determined from beam waveform deformation.15, 16
FIG. 2. Relation of optical path difference between the spherical and planar waves. (a) No optical path difference and (b) some optical path difference.
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095115-3
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
FIG. 3. Interference image of spherical and planar waves. The figure shows both x-y plots of intensity (2D) and 3D simulations of x, y and intensity at the bottom of the figure. (a) Interference image of coincident beams and (b) interference image of deviated beams.
From Figure 4 and according to paraxial optics when din is very small:
III. GEOMETRICAL OPTICS MODEL A. Spherical reflective mirror
A model as shown in Figure 4 can be created using a steel ball with a reflective surface. Where Lin is the incident beam, Lout is the reflective beam, α is the angle of the incident and reflected beam to the normal line (i.e., the incident and reflect angle), din is the offset of the steel ball, R is the radius of the steel ball, θ d is the angle between the horizontal axis and the normal line, θ in is the angle between the incident beam and the horizontal line, and θ out is the angle between the reflective beam and the horizontal line. From Figure 4 we can derive α = θin − sin−1 (din /R),
(9)
din = R sin θd .
(10)
θout ∼ = 2din /R − θin .
(11)
Thus, the ray transfer matrix of the spherical reflective mirror is dout 1 0 din = . (12) θout 2/R −1 θin Here dout = din as the location of the incident and reflective beams on the spherical mirror remains unchanged. B. Full model
The full model for steel ball center measurement is illustrated in Figure 5. A known ABCD lens matrix is expressed
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095115-4
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
The position and incident angle after the beam been reflected by the spherical surface and enters the lens are ∼d din3 = out1 + f θout1 − f θout2 . (16) θin3 = θout2 And the ABCD matrix of the beam after it passes through the lens is expressed as 1 −t 1 0 dout3 · = (n − 1) /R1 n θout3 0 1 FIG. 4. The mathematical model of the steel ball and the incident and reflected beams.
·
1
0
(1 − n) /nR2
1/n
·
din3 θin3
.
(17)
as
dout1 θout1
=
1
0
(n − 1) /R2
n
·
·
1
0
(1 − n) /nR1
1/n
1
t
C. Model simplification
0 1 ·
din1 θin1
,
(13)
where n is the refractive index, R1 and R2 are the spherical radii of both sides of the lens, t is the lens thickness, din1 is the offset displacement of the incident beam from the optic axis of the lens, θ in1 is the angle between the laser beam and normal line of the lens surface, which equals zero in this study as parallel beams are used, dout1 is the offset displacement of the emerging laser beam with respect to the optical axis of the lens, and θ out1 is the angle at which the laser beam leaves the lens. Thus, the position and angle when the beam is projected onto the spherical surface can be determined by the following equation: din2 = δm − dout1 − f θout1 . (14) θin2 = θout1 Since f is the distance between the mirror and steel ball surface. The reflective angle of the laser beam from the sphere can be derived from Eq. (12): din2 1 0 dout2 = . (15) θout2 θin2 2/R3 −1
FIG. 5. The full model of the steel ball center measurement.
Assuming the incident beam passes horizontally through the lens center (i.e., din1 = 0 and θ in1 = 0◦ ), position and angle when the beam projected on the spherical surface are din2 = δ m and θ in2 = 0◦ . In this study, a plano-convex lens was used (i.e., R2 = −∞) and because the incident beam is moving horizontally, θ in2 being zero, the ball center measuring equation will be dout3 = −2δm (nf + t)/nR3 . (18) θout3 = 2δm (nf − n2 f + nR1 − nt + t)/nR1 R3 This suggests a proportional relationship between dout3 , θ out3 , and δ m , therefore, Eq. (18) can be simplified to δ m = k1 p, in which p is the center of the interference fringe observed by the camera in units of one pixel, and k1 is a proportional constant derived from the equation below: k1 = δcamera [−2(nf + t)/nR3 − 2dcamera (nf − n2 f + nR1 − nt + t)/nR1 R3 ]−1 .
(19)
Here dcamera is the distance from the lens to the camera and δ camera is the pixel-displacement conversion constant in units of μm/pixel. In an actual application k1 can be determined by calibration. D. A method for finding the fringe center
The center of a circle can usually be found by Hough circle fitting. It is a time consuming method as a 1.3 mega-pixel image may take more than 100 ms to complete. Since the interference fringes are composed of black and white patterns they can be searched for in a way similar to straight line edge fitting.17 Using this method a line edge can easily be found in under 10 ms and so we used this method to find the edges of the interference fringes. As seen in Figure 6(a), the fringes are annular and separated into many ROIs (ranges of interest). The edge points can be found using line edge fitting17 as shown in Figure 6(b). After the edge points are found, the circle can be found by least squares circle fitting and the fringe center can then be determined.
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095115-5
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014) TABLE I. Calibrated values of k1 . Ball diameter (mm)
FIG. 6. Steps for finding the interference fringe center. Step (a): Split the interference fringe into multiple ROIs. Step (b): Find the correct edge points (the green crosses indicate correct and red the error edge points). The fringe center can then be easily found by least square circle fitting.
FIG. 7. Experimental setup.
Focal length (mm)
k1 (μm/pixel)
15
25 50
−1.8 −1.4
25
25 50
−2.2 −1.6
IV. EXPERIMENTAL RESULTS
The experimental setup is shown in Figure 7. The HeNe laser is a 5 mW, 633 nm device and a circular variable ND (neutral density) filter was placed in the beam path to regulate intensity and make it easier to observe the interference fringes in the signal from the camera. The camera used in this study was a Basler acA 1300 μm at resolution 1280 × 960 with an element size of 3.75 μm2 . The environmental temperature was 20 ◦ C ± 0.2 ◦ C. The camera was used without a lens to avoid image distortion. Steel balls of 15 mm and 25 mm were used and the plano-convex lenses used were of 25 mm and 50 mm focal length. The steel balls were moved in 25 μm horizontal steps along the y-axis (as seen in Figure 7). Figures 8(a) and 8(b) are the offset variance charts of the interference fringe centers in the horizontal and vertical directions observed from the camera. Table I shows the pixel-displacement conversion constant k1 after calibration. Figure 8(c) shows the residual error after k1 was calibrated.
FIG. 8. Experimental results of steel ball offset 100 μm in the negative y-direction. Changes of the interference fringe center: (a) in the horizontal direction at the camera; (b) in the vertical direction at the camera; and (c) residual error after calibration.
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095115-6
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
FIG. 9. Illustration of the proposed method applies to assembling of a laser tracer. The module is fixed with the bearing on the right hand side and the laser beam passes through the center of bearing. Using the ball center alignment module, the bearing center can be adjusted to approach the ball center.
V. CONCLUSIONS
A steel ball center determination method based on the Michelson interferometer was proposed in this study. The experiment suggests that a camera without a lens would magnify the ball offset more than 140%. The actual magnification varies with the diameter of the ball, the focal length of the lens, and the distance between the camera and plano-convex lens. Changing the distance between the camera and the plano-convex lens changes the value of the pixeldisplacement conversion constant. The larger the radius of steel ball the lower the measurement sensitivity. Ideally, the pixel-displacement conversion constant can be doubled by using a 2X telecentric lens with the camera and if lens distortion is ignored. According to this idea, we designed a ball center alignment device for assembly for laser tracer, as seen in Figure 9. In the future, we suggest that this device can be applied on the CMM stylus in a step ball gauge, for ball/hole plate measurement and also on roundness measuring machines after measurement precision and accuracy has been improved. ACKNOWLEDGMENTS
The work was supported by Ministry of Science and Technology, Taiwan (MOST 102-2218-E-005-012).
1 C.-H.
Liu, W.-Y. Jywe, and H.-W. Lee, Meas. Sci. Technol. 15(9), 1733 (2004). 2 C.-H. Liu, W.-Y. Jywe, S.-C. Tzeng, and Y.-S. Lin, Opt. Eng. 44(9), 097003–097006 (2005). 3 S. Osawa, T. Takatsuji, H. Noguchi, and T. Kurosawa, Precis. Eng. 26(2), 214–221 (2002). 4 W. Jywe, T.-H. Hsu, and C. H. Liu, Int. J. Mach. Tools Manuf. 59, 16–23 (2012). 5 H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, CIRP Ann.-Manuf. Technol. 57(2), 660–675 (2008). 6 E. B. Hughes, A. Wilson, and G. N. Peggs, CIRP Ann.-Manuf. Technol. 49(1), 391–394 (2000). 7 J. Santolaria, J.-J. Aguilar, J.-A. Yagüe, and J. Pastor, Precis. Eng. 32(4), 251–268 (2008). 8 E. Trapet, J.-J. Aguilar Martín, J.-A. Yagüe, H. Spaan, and V. Zelený, Precis. Eng. 30(2), 165–179 (2006). 9 S. Osawa, T. Takatsuji, J. Hong, H. Noguchi, and T. Kurosawa, Proc. SPIE 4401, 127–135 (2001). 10 J. Burke, K. Green, W. Stuart, E. Puhanic, A. Leistner, and B. Oreb, Proc. SPIE 7064, 70640E (2008). 11 S.-W. Yang, C.-S. Lin, and S.-K. Lin, Opt. Lasers Eng. 51(4), 348–357 (2013). 12 J. Burke, Opt. Express 17(5), 3242–3254 (2009). 13 J. Burke and D. S. Wu, Appl. Opt. 49(31), 6014–6023 (2010). 14 U. Griesmann, Q. Wang, J. Soons, and R. Carakos, Proc. SPIE 5869, 58690S (2005). 15 M. Grießhammer and A. Rohrbach, Opt. Express 22(5), 6114–6132 (2014). 16 P. B. Bareil and Y. Sheng, Opt. Express 18(25), 26388–26398 (2010). 17 H.-W. L. Lee, C. H. Liu, and J.-S. Chen, Adv. Mech. Eng. 2014, 904061 (2014).
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 095115 (2014)
Development of a steel ball center alignment device based on Michelson interference concept Hau-Wei Lee1,a) and Chien-Hung Liu2 1 2
Center for Measurement Standards, Industrial Technology Research Institute, Hsinchu 300, Taiwan Department of Mechanical Engineering, National Chung Hsing University, Taichung 402, Taiwan
(Received 3 July 2014; accepted 1 September 2014; published online 23 September 2014) This study presents a ball center alignment method based on the Michelson interferometer where one of the reflecting mirrors is replaced by a lens and steel ball. By locating the ball away from the focal length of the lens, the beam is reflected as a spherical wave. The interference ring formed by the planar and spherical waves can be clearly observed using a camera without a lens. The distance of the offset of the ball center can be enhanced by more than 140% using this method. A fast ring profile fitting method can reduce circle fitting time to around a third of that needed for Hough transformation. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895669] I. INTRODUCTION
Balls are used in many different kinds of inspection, such as the measurement of axis rotational error using a ball bar with sensors as described in ISO 230-7:2006. Theoretically, when a spherical distance capacitance probe is used, the probe is placed near the center of the ball and this allows the center to be determined using the shortest distance method. However, angular deviations in sensor installation may result in cosine errors. Geometrical optical methods are also used for spindle rotational error measurement instead of attempts to reduce inherent ball center alignment errors.1, 2 Ball gauges, step ball gauges, and ball plates are normally used for the calibration of coordinate measuring machines (CMM),3 and alternative methods for calibrating CMMs are described in ISO 10360-4:2000, ISO 10360-5:2010, and ISO 103609:2013. Although the ball center can be located by CMM using multi-point measurement, this suffers from uncertainties that result from inaccurate assembly of the CMM platform as well as stylus tolerance. An optical calibration system for five-axis machine tools has been designed by Jywe et al.4, 5 Similar uncertainty may exist when the laser beam is not perfectly aligned with the ball lens center line. Thus, this measurement system requires careful calibration before it can be applied. In this application the reference plane of the planar reflective mirror is replaced by a steel ball to reduce the uncertainty of the Laser Tracer.6 In spite of reduced measurement uncertainty, the steel ball reflective surface may lead to a translational measurement error if the laser beam fails to reflect perpendicular to the sphere. A small shift of the ball through a distance of δ m , would lead to reflective beam deviation of k × δ m and k which is larger than one. This makes measurement uncertainty worse due to the optical path difference. Since balls are used in many different measurement systems, a means of checking that the beam is exactly aligned
a) Author to whom the correspondence should be addressed. Electronic mail:
[email protected]
0034-6748/2014/85(9)/095115/6/$30.00
to the ball center line is needed to reduce measurement uncertainty. In order to ensure the measurement stylus is orthogonal to the sphere surface and passing through the center, Yague et al. proposed a self-centering contact stylus method using three balls.7, 8 Based on the same concept, a Laser Tracker of smaller measurement range was proposed by Osawa et al.9 Although, these methods ensure the stylus and the beam are orthogonal to the sphere surface and coincide with the center line, the problem of contact wear would still influence measurement accuracy after long term use. Burke et al. used a lens, and convex and concave mirrors as well as geometric optics for ball center alignment.10 Yang et al.11 used a Fizeau interferometric microscope for 3D profile measurement. This microscope can construct a 3D profile for a spherical surface, but is limited to micro workpieces (e.g., micro lenses) and the image was more difficult to process due to the complex nature of the interference fringes. These two non-contact measurement methods do not work as well as the ball plate for rapid ball center alignment or Laser Tracer assembly, although they can produce a 3D spherical profile.
II. CONCEPTUAL OUTLINE
This study proposes a non-contact ball center aligning method using the Michelson interference concept and geometrical optics in a layout shown in Figure 1(a). A polished steel ball replaces one of the reflective mirrors of the Michelson interferometer. The plane mirror serves to reflect the reference beam. The planar wave of the He-Ne laser remains planar when the reference beam is sent directly to the camera detector, through the BS spectroscope via the plane mirror, without a camera lens in the beam. When the steel ball surface is at the lens focus, the beam reflected by the sphere passes through the lens as a collimated beam which in the form of a planar wave and the interference fringes are difficult to observe at the camera detector. Therefore, we locate the ball away from the focal point so that the reflected beam is in the form of a spherical wave. The planar and spherical
85, 095115-1
© 2014 AIP Publishing LLC
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095115-2
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
FIG. 1. Measuring the ball center with the Michelson interferometer setup (with a steel ball of 25 mm diameter and a lens of 25 mm focal length). (a) A concentric interference fringe appears when beam reflection is coincident with the center of the ball and (b) an interference fringe from a ball offset from the center.
waves can be expressed as A (z) = Beikz ,
(1)
A (r) = Be±ikr ,
(2)
where k = 2π /λ; z is the beam translation direction of the planar wave; r is the radius of the spherical wave; constant B is the amplitude of the beam; and the plus and minus signs indicate the direction (i.e., divergence or convergence) of the spherical wave. The superimposition of the spherical and planar waves has no angular component, and thus the resulting waves can be expressed as A(φSL ) = B + Be±iφSL ,
(3)
where φ SL represents the phase shift. As shown in Figure 2(a) we know that δ 2 − 2rδ + x 2 = 0.
(4)
Since δ 2 1, the optical path difference between the planar and spherical waves can be expressed as δ = x 2 /2r.
(5)
To consider the optical path difference and ball center deviation, Eq. (5) can be revised to δ = δL + (x − xm )2 /2r,
(6)
where xm is the offset displacement of the sphere center. The phase shift between the planar and spherical waves can be derived from φSL = k[δL + (x − xm )2 /2r].
(7)
Thus, the interference beam intensity in 2D space can be expressed as I (x, y) = 4I0 (x, y) cos2 {k[δL + (x − xm )2 /2r + (y − ym )2 /2r]}.
(8)
Superimposing the two beams as shown in Figure 3(a) would generate interference images as shown in Figure 1(a). In a case where the reflective beam deviates from the reference beam’s axis as shown in Figure 3(b) another interference image would appear as shown in Figure 1(b). Bruke used the same idea for parabolic mirror alignment12 and also used a ball for calibration.13, 14 The angle and position of a nanorod can also be determined from beam waveform deformation.15, 16
FIG. 2. Relation of optical path difference between the spherical and planar waves. (a) No optical path difference and (b) some optical path difference.
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095115-3
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
FIG. 3. Interference image of spherical and planar waves. The figure shows both x-y plots of intensity (2D) and 3D simulations of x, y and intensity at the bottom of the figure. (a) Interference image of coincident beams and (b) interference image of deviated beams.
From Figure 4 and according to paraxial optics when din is very small:
III. GEOMETRICAL OPTICS MODEL A. Spherical reflective mirror
A model as shown in Figure 4 can be created using a steel ball with a reflective surface. Where Lin is the incident beam, Lout is the reflective beam, α is the angle of the incident and reflected beam to the normal line (i.e., the incident and reflect angle), din is the offset of the steel ball, R is the radius of the steel ball, θ d is the angle between the horizontal axis and the normal line, θ in is the angle between the incident beam and the horizontal line, and θ out is the angle between the reflective beam and the horizontal line. From Figure 4 we can derive α = θin − sin−1 (din /R),
(9)
din = R sin θd .
(10)
θout ∼ = 2din /R − θin .
(11)
Thus, the ray transfer matrix of the spherical reflective mirror is dout 1 0 din = . (12) θout 2/R −1 θin Here dout = din as the location of the incident and reflective beams on the spherical mirror remains unchanged. B. Full model
The full model for steel ball center measurement is illustrated in Figure 5. A known ABCD lens matrix is expressed
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095115-4
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
The position and incident angle after the beam been reflected by the spherical surface and enters the lens are ∼d din3 = out1 + f θout1 − f θout2 . (16) θin3 = θout2 And the ABCD matrix of the beam after it passes through the lens is expressed as 1 −t 1 0 dout3 · = (n − 1) /R1 n θout3 0 1 FIG. 4. The mathematical model of the steel ball and the incident and reflected beams.
·
1
0
(1 − n) /nR2
1/n
·
din3 θin3
.
(17)
as
dout1 θout1
=
1
0
(n − 1) /R2
n
·
·
1
0
(1 − n) /nR1
1/n
1
t
C. Model simplification
0 1 ·
din1 θin1
,
(13)
where n is the refractive index, R1 and R2 are the spherical radii of both sides of the lens, t is the lens thickness, din1 is the offset displacement of the incident beam from the optic axis of the lens, θ in1 is the angle between the laser beam and normal line of the lens surface, which equals zero in this study as parallel beams are used, dout1 is the offset displacement of the emerging laser beam with respect to the optical axis of the lens, and θ out1 is the angle at which the laser beam leaves the lens. Thus, the position and angle when the beam is projected onto the spherical surface can be determined by the following equation: din2 = δm − dout1 − f θout1 . (14) θin2 = θout1 Since f is the distance between the mirror and steel ball surface. The reflective angle of the laser beam from the sphere can be derived from Eq. (12): din2 1 0 dout2 = . (15) θout2 θin2 2/R3 −1
FIG. 5. The full model of the steel ball center measurement.
Assuming the incident beam passes horizontally through the lens center (i.e., din1 = 0 and θ in1 = 0◦ ), position and angle when the beam projected on the spherical surface are din2 = δ m and θ in2 = 0◦ . In this study, a plano-convex lens was used (i.e., R2 = −∞) and because the incident beam is moving horizontally, θ in2 being zero, the ball center measuring equation will be dout3 = −2δm (nf + t)/nR3 . (18) θout3 = 2δm (nf − n2 f + nR1 − nt + t)/nR1 R3 This suggests a proportional relationship between dout3 , θ out3 , and δ m , therefore, Eq. (18) can be simplified to δ m = k1 p, in which p is the center of the interference fringe observed by the camera in units of one pixel, and k1 is a proportional constant derived from the equation below: k1 = δcamera [−2(nf + t)/nR3 − 2dcamera (nf − n2 f + nR1 − nt + t)/nR1 R3 ]−1 .
(19)
Here dcamera is the distance from the lens to the camera and δ camera is the pixel-displacement conversion constant in units of μm/pixel. In an actual application k1 can be determined by calibration. D. A method for finding the fringe center
The center of a circle can usually be found by Hough circle fitting. It is a time consuming method as a 1.3 mega-pixel image may take more than 100 ms to complete. Since the interference fringes are composed of black and white patterns they can be searched for in a way similar to straight line edge fitting.17 Using this method a line edge can easily be found in under 10 ms and so we used this method to find the edges of the interference fringes. As seen in Figure 6(a), the fringes are annular and separated into many ROIs (ranges of interest). The edge points can be found using line edge fitting17 as shown in Figure 6(b). After the edge points are found, the circle can be found by least squares circle fitting and the fringe center can then be determined.
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095115-5
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014) TABLE I. Calibrated values of k1 . Ball diameter (mm)
FIG. 6. Steps for finding the interference fringe center. Step (a): Split the interference fringe into multiple ROIs. Step (b): Find the correct edge points (the green crosses indicate correct and red the error edge points). The fringe center can then be easily found by least square circle fitting.
FIG. 7. Experimental setup.
Focal length (mm)
k1 (μm/pixel)
15
25 50
−1.8 −1.4
25
25 50
−2.2 −1.6
IV. EXPERIMENTAL RESULTS
The experimental setup is shown in Figure 7. The HeNe laser is a 5 mW, 633 nm device and a circular variable ND (neutral density) filter was placed in the beam path to regulate intensity and make it easier to observe the interference fringes in the signal from the camera. The camera used in this study was a Basler acA 1300 μm at resolution 1280 × 960 with an element size of 3.75 μm2 . The environmental temperature was 20 ◦ C ± 0.2 ◦ C. The camera was used without a lens to avoid image distortion. Steel balls of 15 mm and 25 mm were used and the plano-convex lenses used were of 25 mm and 50 mm focal length. The steel balls were moved in 25 μm horizontal steps along the y-axis (as seen in Figure 7). Figures 8(a) and 8(b) are the offset variance charts of the interference fringe centers in the horizontal and vertical directions observed from the camera. Table I shows the pixel-displacement conversion constant k1 after calibration. Figure 8(c) shows the residual error after k1 was calibrated.
FIG. 8. Experimental results of steel ball offset 100 μm in the negative y-direction. Changes of the interference fringe center: (a) in the horizontal direction at the camera; (b) in the vertical direction at the camera; and (c) residual error after calibration.
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095115-6
H.-W. Lee and C. H. Liu
Rev. Sci. Instrum. 85, 095115 (2014)
FIG. 9. Illustration of the proposed method applies to assembling of a laser tracer. The module is fixed with the bearing on the right hand side and the laser beam passes through the center of bearing. Using the ball center alignment module, the bearing center can be adjusted to approach the ball center.
V. CONCLUSIONS
A steel ball center determination method based on the Michelson interferometer was proposed in this study. The experiment suggests that a camera without a lens would magnify the ball offset more than 140%. The actual magnification varies with the diameter of the ball, the focal length of the lens, and the distance between the camera and plano-convex lens. Changing the distance between the camera and the plano-convex lens changes the value of the pixeldisplacement conversion constant. The larger the radius of steel ball the lower the measurement sensitivity. Ideally, the pixel-displacement conversion constant can be doubled by using a 2X telecentric lens with the camera and if lens distortion is ignored. According to this idea, we designed a ball center alignment device for assembly for laser tracer, as seen in Figure 9. In the future, we suggest that this device can be applied on the CMM stylus in a step ball gauge, for ball/hole plate measurement and also on roundness measuring machines after measurement precision and accuracy has been improved. ACKNOWLEDGMENTS
The work was supported by Ministry of Science and Technology, Taiwan (MOST 102-2218-E-005-012).
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