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Optical 3-D dynamic measurement system and its application to polymer membrane inflation tests
Optics and Lasers in Engineering 33 (2000) 261}276
Optical 3-D dynamic measurement system and its application to polymer membrane in#ation tests Yong Li , James A. Nemes *, A. Derdouri Mechanical Engineering Department, McGill University, 817 Sherbrooke W., Montreal, QC, Canada H3A 2K6 Industrial Materials Institute, NRC, 75 de Mortagne, Boucherville, QC, Canada J4B 6Y4 Received 12 April 2000; accepted 11 July 2000
Abstract An optical surface measurement system, which is capable of measuring transient surface shape, has been developed by using a high-speed digital camera. The system is based on the grating projection and Fourier transform technique. A calibration procedure is developed to allow the system to generate Cartesian coordinates directly, which are with respect to a "xed coordinate system in 3-D space. The measurement accuracy ($50 lm) is de"ned and veri"ed as the maximum error between measured values and the known values of standard objects both #at and curved. The camera and a grating projector are mounted into a portable sensor head to allow in situ measurements. In addition, external force or pressure signals can be correlated with each measurement through a device called the multi-channel data link. The system is capable of digitizing a 3-D curved surface into an array of points with known xyz coordinates at a sampling rate from 30 to 1000 Hz. As an application, the system is used to measure the transient surface shape during a polymer membrane in#ation test. The measurement results along with the pressure information provide an approach to determine the material parameters used in di!erent material models. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fourier transform; 3-D coordinates measurement; Calibration; Membrane in#ation test
1. Introduction Our physical world, known as three-dimensional (3-D) Euclidean space, is characterized in algebraic terms by establishing a Cartesian coordinate system * three mutually perpendicular numbered lines whose origins coincide at a single point.
* Corresponding author. Fax: 514-398-7365. E-mail address: [email protected] (J.A. Nemes). 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 4 7 - 6
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The position of any point in 3-D space can be represented by an ordered triple of real numbers (x, y, z), known as the Cartesian coordinates of this point. Any 3-D curved surface can be represented by an array of points with known Cartesian coordinates, and then a mathematical model can be formulated to describe the surface shape based on these points. From the Cartesian coordinates, other quantities, such as displacement and curvature, can be calculated. Therefore, the ultimate goal of 3-D measurement techniques, both mechanical and optical, is the determination of the Cartesian coordinates. Unlike the conventional coordinate measurement machine (CMM), optical techniques have many advantages * speed, non-contact and providing whole-"eld information. However, early methods, such as the geometric MoireH techniques [1}3], su!er from some major di$culties. Firstly, the MoireH fringes themselves do not allow a direct determination of whether a contour line showing concentric fringes is a hill or a valley. This shortcoming is solved by the grating projection technique, which is based on phase calculation by the fast Fourier transform [4] or the phase shifting methods [5,6]. Secondly, the measurement results are not Cartesian coordinates with respect to a coordinate system. Fig. 1 shows the direct results from the conventional optical methods. After the light ray passes through a lens assembly B}C, point A on a 3-D curved surface creates a corresponding image point (point D) on the image plane inside a CCD digital camera. Line O O represents the optical axis of the camera, which is perpendicular to the image plane. These optical methods can provide the distance between point A and the reference plane, or the height, with the location of its corresponding point on the image plane * (i, j). The image coordinates (i, j) are with respect to a 2-D image coordinate system, which is dependent on the position
Fig. 1. Direct measurement results from the optical pro"le measurement methods.
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and orientation of the camera. In addition, di!erent points in 3-D space may have the same corresponding point on the image plane. Therefore, to employ a 2-D image coordinate system to describe the 3-D world is an indirect method. As a minimum, it is not straightforward to verify the accuracy. This drawback may not be serious for some applications, however it is for many. If point A is described with respect to a Cartesian coordinate system, which is the intention in this work, its Cartesian coordinates (x, y, z) are needed. Therefore, it is necessary to convert what the optical methods provide (h, i, j) into what is needed (x, y, z). After de"ning the height as the z coordinate, which is the distance between the point measured and the reference plane, the lateral coordinates (x, y) are still lacking. To convert the image coordinates (i, j) into the Cartesian coordinates (x, y) by using a constant scale ratio along with ignoring the lens distortions will create errors for non-#at surfaces since scale ratios are functions of the point's height. In addition, the lens distortion is not negligible when an accurate surface shape is requested. In this paper, a mathematical model is proposed to carry out this conversion while considering lens distortion based on the method used in photogrammetry. In addition, the measurement accuracy is veri"ed in a manner used in industry. All whole-"eld optical measurement techniques, including the MoireH methods and the grating projection methods, are mainly designated for shape measurement of the surfaces in static state. When a surface is moving and changing its form in 3-D space due to a dynamic event, such as impact or vibration, it will be a challenge to measure the surface shape as a function of time. Conventional CMMs and laser scanning systems cannot carry out a whole-"eld measurement in a very short period of time, of the order of milliseconds. So far, there is no standard methodology to measure the transient shape of a structure under dynamic events. In 1982, Chai [7] conducted dynamic measurements on a composite panel subjected to out-of-plane impact by using the shadow MoireH method with a high-speed "lm camera. More recently, Kokidko et al. [8] also employed the shadow MoireH method combined with highspeed photography to measure the deformation of a glass "ber reinforced plastic panel at impact of a projectile. These studies are examples of the extended application of the shadow MoireH method, but all the drawbacks associated with the shadow MoireH method are present. The most detrimental shortcoming that jeopardizes the measurement is the failure to provide 3-D Cartesian coordinates with respect to a "xed coordinate system. In this paper, a prototype system is developed to tackle this challenge by combining the grating projection method with a high-speed digital camera. After calibration, the system is able to provide the Cartesian coordinates at a certain sampling rate. As an example, the system is applied to the study of polymer membrane in#ation.
2. Fundamental concepts A Ronchi grating projected onto a surface is modulated as the surface is deformed. As shown in Fig. 2, the deformed grating image will be equispaced straight lines if the surface is #at. In the case of a cylindrical surface, it will give an image of straight lines
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Fig. 2. Grating images on #at and cylindrical surfaces: (a) setup for a #at surface; (b) setup for a cylindrical surface; (c) deformed grating image on the #at surface; (d) deformed grating image on the cylindrical surface.
with varying spacing or an image of equispaced curved lines, depending on the projection angle. It should, therefore, be possible to calculate the surface shape directly from the grating pattern formed on the surface if the projection and viewing geometry are "xed and known. In dealing with the deformed grating image, the geometric MoireH methods use a second grating to &"lter' the deformed grating image to generate the MoireH fringes. However, to calculate the phase directly from a deformed grating image, a more e$cient way, can be used to obtain surface shape information. Among several phase-computing methods, the phase shifting method can generate the phase information, but there is a di$culty for dynamic measurements since at least three frames of deformed grating images are needed and image acquisition takes time. In our system, the Fourier transform method is used, which needs only one frame of deformed grating image to obtain the phase information. Unlike the geometric MoireH methods, no MoireH fringes are generated by the interference between a reference grating and a deformed grating image. Therefore, most of the di$culties associated with the geometric MoireH methods are avoided. Details of the Fourier transform method are given by Takeda et al. [4]. In this paper, we will focus on how to obtain the Cartesian coordinates, how to verify the accuracy, and how to integrate the system with a high-speed CCD digital camera.
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3. System con5guration As shown in Fig. 3, the measurement system consists of a high-speed CCD digital camera and its control unit (Motion Corder SR-1000 by Eastman Kodak Company, Maximum 1000 pictures per second), a light source, a grating projector, a multichannel data link (MCDL) also by Eastman Kodak Company, and a personal computer. Actually, the high-speed camera and the grating projector are mounted inside an aluminum box, called the sensor head. The light source illuminates a Ronchi grating inside the projector, and the grating is projected onto the surface to be measured by a 16 mm lens. The deformed grating images on the specimen surface are digitized and saved into the control unit of the high-speed camera system at a certain frame rate (30}1000 fps). In the mean time, the MCDL acquires external analog and digital signals simultaneously with each image frame. The image sequence stored is replayed to select interesting frames to represent a dynamic event. The selected image frames are then transferred into a computer and saved as standard image "les via an SCSI-II connection. After each deformed grating image being processed by the computer, an array of points (as many as 200,000) with known xyz coordinates is generated, which represents the surface shape inside the "eld of view (FOV) of the camera. Since a high-speed camera is used, usually a sequence of deformed grating images during a dynamic event is saved successively at a certain frame rate. By processing these deformed grating images, the surface shape correlated with external signals (pressure or force) is known as a function of time since the time interval between image frames is known. In conclusion, the system has the ability to digitize the whole surface inside the "eld of view into an array of points with known xyz coordinates, at a certain sampling rate (30}1000 Hz).
Fig. 3. Con"guration of the high-speed 3-D coordinate measurement system.
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For the current system con"guration, the "eld of view is approximately 100;100 mm, and the depth of view (outside this range the grating image will be blurred) is around 50 mm. The distance from the sensor head to the surface to be measured, called the working distance, is approximately 200}250 mm. The dimensions of the sensor head are approximately 335(L);200(W);80(H) mm. The image acquisition time is dependent on the electronic shutter speed (as low as 50 ls) and the framing rate of the camera (30}1000 fps). The time needed to process one image frame saved on the hard drive is approximately 15 s for the current computer (Pentium 200 MHz). The accuracy for xyz coordinates calculation is $50 lm, and this is the maximum error between the measured value and the actual dimension. More details of the accuracy will be described later. By changing the lens speci"cations and the geometry of the sensor head, the "eld of view and the working distance can be modi"ed to "t a particular application. 4. System calibration As described previously, the grating projection method has been limited in allowing the evaluation of the height information with respect to a reference plane after transferring the phase into the height. Moreover, the location of a point in 3-D space is expressed with respect to the column and row indices (i, j) of the corresponding image point on the image plane. Based on the similar methods used in photogrammetry and machine vision, a mathematical model is proposed to convert the information directly from the grating projection method (h, i, j) into the Cartesian coordinates (x, y, z). First, a camera model is introduced. 4.1. Camera model A camera model is used to set up the relationship between the 2-D image coordinate system and the 3-D global Cartesian coordinate system. In this case, a 2-D image point (i, j) has to be back projected into a 3-D ray * a ray in 3-D space that the corresponding 3-D object point must lie on. With the help of the height information, the location of the 3-D object point on this ray can be determined. As shown in Fig. 4,
Fig. 4. Camera model with lens distortion.
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a pinhole camera model with lens distortion is considered. Let P be an object point in 3-D space, and (x , y , z ) be its coordinates with respect to a "xed global Cartesian coordinate system (GCS). Let another Cartesian coordinate system, called the camera coordinate system (CCS), have its XY plane parallel to the image plane, such that the X-axis is parallel with the horizontal direction of the image, and the Y-axis is parallel with the vertical one. The origin of the CCS is located at the lens center and the Z-axis is aligned with the optical axis of the lens, which is perpendicular to the image plane. Let (x , y , z ) be the coordinates of the 3-D point P with respect to the CCS. If there is no lens distortion, the corresponding image point of P on the image plane would be Q. However, due to the e!ect of lens distortion, the actual image point is Q . Let (i, j) denote the 2-D image coordinates (in pixels) of the actual image point Q with respect to the computer image coordinate system (ICS). The origin of the ICS is the intersection point of the image plane and the optical axis. The distance between the image plane and the lens center is denoted as f. The overall transformation from (x , y , z ) to (i, j) can be divided into the following four steps. (1) Translation and rotation from the GCS to the CCS: The transformation from the global coordinate system to the camera coordinate system can be expressed as
x r y r " z r 1 0
r r r 0
r r
t
t
0
1
r
t
x y . z 1
(1)
Here (t , t , t ) is a translation vector, and r }r is a 3;3 rotation matrix. (2) Perspective projection from a 3-D object point in the CCS to a 2-D image point on the image plane: Let (i , j ) be the 2-D image coordinates of the undistorted image S S point Q lying on the image plane. Then, we have x i "f S z
and
y j "f . S z
(2)
(3) Radial lens distortion: After considering the radial lens distortion, the 2-D image coordinates (i, j) of the distorted image point Q can have i "i(1#kr) and j "j(1#kr). S S
(3)
Here r"i;i#j;j and k is the lens distortion coe$cient. Based on the experience of some of the researchers [9], Eq. (3) is adequate enough to model lens distortion for industrial vision applications. More elaborate modeling not only would not help but would also cause numerical instability. (4) Scaling of 2D image coordinates: The horizontal and vertical pixel spacings, d and d (millimeter/pixel), are used to scale from pixels to millimeters. S T
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Fig. 5. Schematic of the experimental setup of camera calibration.
Combining these four steps together, we have x r #y r #z r #t , (1!kr)id "f S x r #y r #z r #t x r #y r #z r #t . (1!kr)jd "f T x r #y r #z r #t
(4)
Thus the relationship between the 2-D image coordinates (i, j) and the global 3-D Cartesian coordinates (x , y , z ) is established. 4.2. Camera calibration The purpose of the camera calibration is to determine the constants involved in Eq. (4) and the constants used to convert phase information into the height information, by giving a set of 3-D calibration points and their corresponding 2-D image coordinates. In this work, a set of monoview non-coplanar points is used. As shown in Fig. 5, the camera is mounted on a translation stage, and a metal block, called the calibration target, is placed in front of the camera. Several images are taken at di!erent locations of the translation stage to generate a set of points with known 3-D coordinates (x, y, z) and their corresponding 2-D image coordinates (i, j). Using N pairs of 2-D/3-D coordinates in Eq. (4), the coe$cients can be determined by using the pseudo-inverse method. After the calibration, the whole system is portable and can be taken to any site to perform in situ measurements. As long as the camera and the geometry between the camera itself and the projector remain unchanged, there is no need to re-calibrate. To have the 3-D Cartesian coordinates as the ultimate results is very useful since it provides a practical tool for verifying the measurement accuracy.
5. Accuracy tests Unlike the geometric MoireH methods and the phase-measuring techniques, the way to de"ne the measurement accuracy in this system is more similar to what is used in
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Fig. 6. Surface #atness test.
Fig. 7. Measured point cloud of a #at surface.
industry. Standard objects with known geometry are measured with the optical system. Then the accuracy of the optical system is de"ned by comparing the measured values with the standard values. Objects with #at surface and curved surface are used in the accuracy tests. 5.1. Flat surface test An accurately machined #at surface is placed in front of the sensor head at di!erent z values (!22}22 mm) in the depth of view, as shown in Fig. 6. For each position, the surface is measured by the optical system to generate a point cloud with known xyz coordinates. Fig. 7 shows a point cloud with a reduced number of points. A leastsquare plane "tting is carried out to evaluate the #atness of the data, and the results are given in Table 1. Compared with the original surface #atness ((20 lm), the results compare very well. It is worth knowing that approximately 68% of the points are within one standard deviation from the "tted plane, and about 97% points are within two standard deviations. Points having larger error, which are due to isolated
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Table 1 Flat surface test results Surface Number location z (mm) of points !22.00 !18.00 !14.00 !10.00 !6.00 !2.00
154,775 154,784 154,806 154,809 154,812 154,812
Std. deviation (lm) 11.02 11.06 10.59 10.88 10.54 10.40
Surface location Number z (mm) of points 2.00 6.00 10.00 14.00 18.00 22.00
154,811 154,811 154,812 154,841 154,812 154,810
Std. deviation (lm) 10.82 11.05 11.28 11.95 12.99 14.02
Fig. 8. Cylindrical surface test.
noise, are relatively small in numbers. Their e!ects can be limited by a "ltering process. 5.2. Cylindrical surface test The accuracy test done on a cylindrical surface is slightly di!erent from that on a #at surface since for the curved surfaces the lateral coordinates, x and y, will play a role. Errors with x and y coordinates will a!ect the data deviation. In the test, a cylindrical surface with a diameter of 75.810$0.004 mm is placed approximately in the center of the depth of view, as shown in Fig. 8. The measurement result, which is shown in Fig. 9, is evaluated for the deviation at di!erent y locations (!25}25 mm). The measured points at y"0.0 mm and the standard cylindrical surface are shown in Fig. 10. Table 2 summarizes the deviation results for all y values. Again, the accuracy of the optical system is veri"ed.
6. Membrane in6ation test In recent decades, thermoforming has been used extensively to fabricate plastic products for a wide range of industries. Concurrently, tremendous advances in nonlinear "nite element analysis and computing power have made it possible to simulate the thermoforming process, resulting in savings in product design. However,
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Fig. 9. Measured point cloud on a cylindrical surface.
Fig. 10. Measured points at y"0.0 and the standard cylindrical surface. Table 2 Cylindrical surface test results y (mm) !25.0 !20.0 !15.0 !10.0 !5.0 0.0 Std. deviation (lm) 30.5 25.7 31.5 36.8 32.3 28.7
5.0 33.0
10.0 34.5
15.0 33.9
20.0 35.8
25.0 30.4
in order for the simulation to be reliable, accurate constitutive descriptions of the polymer are needed. Several hyperelastic constitutive theories are commonly employed, including the Mooney-Rivlin and Ogden models. Regardless of the model used, parameters in the model must be experimentally determined.
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Although a number of di!erent types of experiments can be employed for parameter determination, the membrane in#ation test, used initially by Treloar [10] o!ers a number of advantages, prominent among which is that a biaxial stress "eld is produced, which is similar to that encountered in many thermoforming operations. In the membrane in#ation test, a sheet of the thermoplastic material being studied is clamped between two plates, both of which have circular concentric holes. The exposed polymer sheet is heated on both sides by IR heating elements until the surface temperature uniformly reaches the softening point, which is normally close to the heat de#ection temperature (HDT). An in#ation medium, which is generally air or nitrogen gas, is introduced under pressure into a chamber mounted on the bottom side of the in#ation "xture, causing the membrane to deform into a bubble. In addition, by controlling the #ow rate of the pressurizing medium, the rate of deformation of the polymer can be varied, which is usually done, so that the resulting strain rates approximate those in the forming operation under consideration. As a result, #ow rates in the membrane test are often chosen such that the in#ation test time is on the order of 1 s, which is a typical time for many thermoforming operations. The in#ation test is usually recorded with a video camera, which is synchronized with a data acquisition system monitoring the chamber pressure. A marking grid made of concentric circles and several diameters is sometimes drawn on the membrane sheet to monitor the deformation rates in the azimutal and longitudinal directions. Side-mounted cameras are used to record bubble height and a camera may be placed above the membrane to record the deformation of the grid in the pole area. By assuming the deformed bubble shape to be spherical and under a state of biaxial stress, analytical expressions relating forces in the longitudinal and azimutal directions to the bubble height and applied pressure can be obtained as shown by Yang and Feng [11]. Then, for a given choice of constitutive model, material parameters can be determined by solving the equations numerically as demonstrated by Derdouri et al. [12]. The optical measurement system, described in Sections 3}5, is used here to measure the deformed surface of three di!erent polymers at di!erent #ow rates and temperatures during the membrane in#ation test. A schematic of the experimental con"guration is shown in Fig. 11. The optical sensor head is mounted above the membrane looking downward to measure the surface deformation. The 110 mm square sheet of polymeric material is clamped between two plates that leave a 63.5 mm diameter opening at its center exposed to infrared heaters. The MCDL is used to synchronize the deformed grating image with the instantaneous pressure. The optical measuring system is operated at 250 frames per second with a shutter speed of 1/250 s. Maximum measuring time is 2.2 s. Due to the high accuracy, high speed, and full-"eld capability of the measurement system, a number of interesting features from the tests are revealed, which had previously not been observed. Fig. 12 shows a typical measurement result in this case for a 1.5 mm thick acrylonitrile butadiene styrene (ABS) sheet at 1503C, with a #ow rate of 3.0 l/s at 160 ms after the start of in#ation. Each white dot represents a measured point, which is a subset of the &200,000 measured pixels. The coordinate system, which is associated with the sensor head, is also shown in the "gure. By viewing a sequence of images, the in#ation
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Fig. 11. Setup of the membrane in#ation test: (1) thermoplastic membrane; (2) pressure chamber.
Fig. 12. Measurement result of an ABS membrane during the in#ation test.
history of the membrane can be visualized. This is shown more clearly by looking at a cross section through the pole at selected instances in time as shown in Fig. 13. Careful examination of the membrane surface shows that the material near the supporting ring exhibits very little deformation, resulting in a non-spherical shape. This can be seen more clearly in Fig. 14, which shows a cross section through the pole, plotted along with an idealized sphere of the same height. Deviation from the idealized shape can be attributed to temperature gradients within the membrane. Despite
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Fig. 13. Membrane in#ation at di!erent times: from bottom up at time 20, 40, 80, 120, and 160 ms.
Fig. 14. Bubble pro"le at 160 ms after in#ation.
precautions to obtain a uniform temperature distribution, some gradients are inevitable, and may become signi"cant. Temperature conduction into the constraining plates results in a portion of the material near the rim being at a temperature signi"cantly below the value at the center of the membrane, resulting in a material region which is much sti!er. The presence of thermal gradients was con"rmed through use of an infrared camera. Also of interest, is the shape of the surface after heating, but prior to in#ation. The temperature increase results in a signi"cant decrease in material sti!ness, which may cause the thin membrane to sag under its own weight. In addition, the membrane is also constrained radially, preventing thermal expansion, and possibly resulting in membrane buckling, which also would be downward due to gravity. Either type of thermal deformation was previously undetectable since the constraining plates obscure the view of side-mounted cameras, and it is usually too small to be detected by vertically mounted cameras. Using the system described here, however, the thermal deformation is clearly detectable. For the ABS material, the deformed shape is similar to a simple "rst buckling mode, with the maximum out of plane displacement at the center dependent on the test temperature. The measurements indicate a maximum displacement of 0.75 mm at 1303C, 1.67 mm at 1503C and 5.41 mm at 1803C. The thermally deformed shape of high-density polyethylene (HDPE) is considerably more
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Fig. 15. Thermal warpage of an HDPE sheet at 1403C.
Fig. 16. Bubble pro"les of the HDPE sheet at 0 and 220 ms.
complex than ABS and is similar to a higher buckling mode, as shown in Fig. 15. If the temperature distribution is considered to be axisymmetric, the resulting thermal deformation should be similar to that seen for ABS. Therefore, the observed thermally deformed shape of HDPE must be due to either a non-uniformity in the membrane, such as its thickness, or due to anisotropy, which could arise from sheet processing. Whatever the cause, it has a pronounced e!ect on the bubble shape throughout the in#ation, as seen in Fig. 16, which shows a section of the HDPE material at 220 ms after the start of in#ation. Clearly, measurement of the initial thermal deformation, which is possible using the optical measurement system described here, provides better insight into the subsequent in#ation response of the polymer.
7. Conclusions A high-speed optical surface measurement system, which is capable of measuring transient surface shape, has been developed by using a high-speed digital camera. The system generates Cartesian coordinates directly, which are with respect to a "xed
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coordinate system. The measurement accuracy has been veri"ed with #at and curved standard objects. As a test application, the system is used to measure the transient surface during the membrane in#ation experiment. With its high accuracy and full-"eld capabilities a number of features of the test are revealed. The measurement system provides the entire surface deformation history rather than only the history of the pole height as is commonly done. These results raise the possibility of using the additional measurement data in conjunction with "nite element simulations to more accurately determine material parameters in constitutive models or to serve as a basis for comparison of the models themselves.
Acknowledgements This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) through an equipment grant. The authors are very grateful to M. Plourde of IMI/NRC for her invaluable assistance in the in#ation tests.
References [1] Takasaki H. MoireH topography. Appl Opt 1970;9(6):1467}72. [2] Meadows DM, Johnson WO, Allen TB. Generation of surface contours by MoireH patterns. Appl Opt 1970;9(4):942}7. [3] Pirodda L. Shadow and projection MoireH techniques for absolute or relative mapping of surface shapes. Opt Engng 1982;21(4):640}9. [4] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringe-pattern analysis for computerbased topography and interferometry. J Opt Soc Am 1982;72(1):156}60. [5] Reid GT, Rixon RC, Messer HI. Absolute and comparative measurements of three-dimensional shape by phase measuring MoireH topography. Opt Lasers Technol 1984;16:315}9. [6] Srinivasan V, Liu HC, Halioua M. Automated phase-measuring pro"lometry of 3-D di!use objects. Appl Opt 1985;24:185}8. [7] Chai H. The growth of impact damage in compressive loaded laminates. PhD thesis, California Institute of Technology, 1982. [8] Kokidko D, Gee L, Chou SC, Chiang FP. Method for measuring transient out-of-plane deformation during impact. Int J Impact Engng 1997;19(2):127}33. [9] Tsai RY. A versatile camera calibration technique for high-accuracy 3D machine vision metrology using o!-the-shelf TV cameras and lenses. IEEE J Robotics Automat 1987;RA-3(4):323}44. [10] Treloar LRG. Stress}strain data for vulcanised rubber under various types of deformation. Trans Faraday Soc 1944;40:59}70. [11] Yang WH, Feng WW. On axisymmetrical deformation of nonlinear membranes. ASME J Appl Mech 1970;37:1002}11. [12] Derdouri A, Connolly R, Khayat R, Verron E, Peseux B. Material constants identi"cation for thermoforming simulation. Proceedings of ANTEC'98, 1998. p. 672}675.
Optical 3-D dynamic measurement system and its application to polymer membrane in#ation tests Yong Li , James A. Nemes *, A. Derdouri Mechanical Engineering Department, McGill University, 817 Sherbrooke W., Montreal, QC, Canada H3A 2K6 Industrial Materials Institute, NRC, 75 de Mortagne, Boucherville, QC, Canada J4B 6Y4 Received 12 April 2000; accepted 11 July 2000
Abstract An optical surface measurement system, which is capable of measuring transient surface shape, has been developed by using a high-speed digital camera. The system is based on the grating projection and Fourier transform technique. A calibration procedure is developed to allow the system to generate Cartesian coordinates directly, which are with respect to a "xed coordinate system in 3-D space. The measurement accuracy ($50 lm) is de"ned and veri"ed as the maximum error between measured values and the known values of standard objects both #at and curved. The camera and a grating projector are mounted into a portable sensor head to allow in situ measurements. In addition, external force or pressure signals can be correlated with each measurement through a device called the multi-channel data link. The system is capable of digitizing a 3-D curved surface into an array of points with known xyz coordinates at a sampling rate from 30 to 1000 Hz. As an application, the system is used to measure the transient surface shape during a polymer membrane in#ation test. The measurement results along with the pressure information provide an approach to determine the material parameters used in di!erent material models. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fourier transform; 3-D coordinates measurement; Calibration; Membrane in#ation test
1. Introduction Our physical world, known as three-dimensional (3-D) Euclidean space, is characterized in algebraic terms by establishing a Cartesian coordinate system * three mutually perpendicular numbered lines whose origins coincide at a single point.
* Corresponding author. Fax: 514-398-7365. E-mail address: [email protected] (J.A. Nemes). 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 4 7 - 6
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The position of any point in 3-D space can be represented by an ordered triple of real numbers (x, y, z), known as the Cartesian coordinates of this point. Any 3-D curved surface can be represented by an array of points with known Cartesian coordinates, and then a mathematical model can be formulated to describe the surface shape based on these points. From the Cartesian coordinates, other quantities, such as displacement and curvature, can be calculated. Therefore, the ultimate goal of 3-D measurement techniques, both mechanical and optical, is the determination of the Cartesian coordinates. Unlike the conventional coordinate measurement machine (CMM), optical techniques have many advantages * speed, non-contact and providing whole-"eld information. However, early methods, such as the geometric MoireH techniques [1}3], su!er from some major di$culties. Firstly, the MoireH fringes themselves do not allow a direct determination of whether a contour line showing concentric fringes is a hill or a valley. This shortcoming is solved by the grating projection technique, which is based on phase calculation by the fast Fourier transform [4] or the phase shifting methods [5,6]. Secondly, the measurement results are not Cartesian coordinates with respect to a coordinate system. Fig. 1 shows the direct results from the conventional optical methods. After the light ray passes through a lens assembly B}C, point A on a 3-D curved surface creates a corresponding image point (point D) on the image plane inside a CCD digital camera. Line O O represents the optical axis of the camera, which is perpendicular to the image plane. These optical methods can provide the distance between point A and the reference plane, or the height, with the location of its corresponding point on the image plane * (i, j). The image coordinates (i, j) are with respect to a 2-D image coordinate system, which is dependent on the position
Fig. 1. Direct measurement results from the optical pro"le measurement methods.
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and orientation of the camera. In addition, di!erent points in 3-D space may have the same corresponding point on the image plane. Therefore, to employ a 2-D image coordinate system to describe the 3-D world is an indirect method. As a minimum, it is not straightforward to verify the accuracy. This drawback may not be serious for some applications, however it is for many. If point A is described with respect to a Cartesian coordinate system, which is the intention in this work, its Cartesian coordinates (x, y, z) are needed. Therefore, it is necessary to convert what the optical methods provide (h, i, j) into what is needed (x, y, z). After de"ning the height as the z coordinate, which is the distance between the point measured and the reference plane, the lateral coordinates (x, y) are still lacking. To convert the image coordinates (i, j) into the Cartesian coordinates (x, y) by using a constant scale ratio along with ignoring the lens distortions will create errors for non-#at surfaces since scale ratios are functions of the point's height. In addition, the lens distortion is not negligible when an accurate surface shape is requested. In this paper, a mathematical model is proposed to carry out this conversion while considering lens distortion based on the method used in photogrammetry. In addition, the measurement accuracy is veri"ed in a manner used in industry. All whole-"eld optical measurement techniques, including the MoireH methods and the grating projection methods, are mainly designated for shape measurement of the surfaces in static state. When a surface is moving and changing its form in 3-D space due to a dynamic event, such as impact or vibration, it will be a challenge to measure the surface shape as a function of time. Conventional CMMs and laser scanning systems cannot carry out a whole-"eld measurement in a very short period of time, of the order of milliseconds. So far, there is no standard methodology to measure the transient shape of a structure under dynamic events. In 1982, Chai [7] conducted dynamic measurements on a composite panel subjected to out-of-plane impact by using the shadow MoireH method with a high-speed "lm camera. More recently, Kokidko et al. [8] also employed the shadow MoireH method combined with highspeed photography to measure the deformation of a glass "ber reinforced plastic panel at impact of a projectile. These studies are examples of the extended application of the shadow MoireH method, but all the drawbacks associated with the shadow MoireH method are present. The most detrimental shortcoming that jeopardizes the measurement is the failure to provide 3-D Cartesian coordinates with respect to a "xed coordinate system. In this paper, a prototype system is developed to tackle this challenge by combining the grating projection method with a high-speed digital camera. After calibration, the system is able to provide the Cartesian coordinates at a certain sampling rate. As an example, the system is applied to the study of polymer membrane in#ation.
2. Fundamental concepts A Ronchi grating projected onto a surface is modulated as the surface is deformed. As shown in Fig. 2, the deformed grating image will be equispaced straight lines if the surface is #at. In the case of a cylindrical surface, it will give an image of straight lines
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Fig. 2. Grating images on #at and cylindrical surfaces: (a) setup for a #at surface; (b) setup for a cylindrical surface; (c) deformed grating image on the #at surface; (d) deformed grating image on the cylindrical surface.
with varying spacing or an image of equispaced curved lines, depending on the projection angle. It should, therefore, be possible to calculate the surface shape directly from the grating pattern formed on the surface if the projection and viewing geometry are "xed and known. In dealing with the deformed grating image, the geometric MoireH methods use a second grating to &"lter' the deformed grating image to generate the MoireH fringes. However, to calculate the phase directly from a deformed grating image, a more e$cient way, can be used to obtain surface shape information. Among several phase-computing methods, the phase shifting method can generate the phase information, but there is a di$culty for dynamic measurements since at least three frames of deformed grating images are needed and image acquisition takes time. In our system, the Fourier transform method is used, which needs only one frame of deformed grating image to obtain the phase information. Unlike the geometric MoireH methods, no MoireH fringes are generated by the interference between a reference grating and a deformed grating image. Therefore, most of the di$culties associated with the geometric MoireH methods are avoided. Details of the Fourier transform method are given by Takeda et al. [4]. In this paper, we will focus on how to obtain the Cartesian coordinates, how to verify the accuracy, and how to integrate the system with a high-speed CCD digital camera.
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3. System con5guration As shown in Fig. 3, the measurement system consists of a high-speed CCD digital camera and its control unit (Motion Corder SR-1000 by Eastman Kodak Company, Maximum 1000 pictures per second), a light source, a grating projector, a multichannel data link (MCDL) also by Eastman Kodak Company, and a personal computer. Actually, the high-speed camera and the grating projector are mounted inside an aluminum box, called the sensor head. The light source illuminates a Ronchi grating inside the projector, and the grating is projected onto the surface to be measured by a 16 mm lens. The deformed grating images on the specimen surface are digitized and saved into the control unit of the high-speed camera system at a certain frame rate (30}1000 fps). In the mean time, the MCDL acquires external analog and digital signals simultaneously with each image frame. The image sequence stored is replayed to select interesting frames to represent a dynamic event. The selected image frames are then transferred into a computer and saved as standard image "les via an SCSI-II connection. After each deformed grating image being processed by the computer, an array of points (as many as 200,000) with known xyz coordinates is generated, which represents the surface shape inside the "eld of view (FOV) of the camera. Since a high-speed camera is used, usually a sequence of deformed grating images during a dynamic event is saved successively at a certain frame rate. By processing these deformed grating images, the surface shape correlated with external signals (pressure or force) is known as a function of time since the time interval between image frames is known. In conclusion, the system has the ability to digitize the whole surface inside the "eld of view into an array of points with known xyz coordinates, at a certain sampling rate (30}1000 Hz).
Fig. 3. Con"guration of the high-speed 3-D coordinate measurement system.
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For the current system con"guration, the "eld of view is approximately 100;100 mm, and the depth of view (outside this range the grating image will be blurred) is around 50 mm. The distance from the sensor head to the surface to be measured, called the working distance, is approximately 200}250 mm. The dimensions of the sensor head are approximately 335(L);200(W);80(H) mm. The image acquisition time is dependent on the electronic shutter speed (as low as 50 ls) and the framing rate of the camera (30}1000 fps). The time needed to process one image frame saved on the hard drive is approximately 15 s for the current computer (Pentium 200 MHz). The accuracy for xyz coordinates calculation is $50 lm, and this is the maximum error between the measured value and the actual dimension. More details of the accuracy will be described later. By changing the lens speci"cations and the geometry of the sensor head, the "eld of view and the working distance can be modi"ed to "t a particular application. 4. System calibration As described previously, the grating projection method has been limited in allowing the evaluation of the height information with respect to a reference plane after transferring the phase into the height. Moreover, the location of a point in 3-D space is expressed with respect to the column and row indices (i, j) of the corresponding image point on the image plane. Based on the similar methods used in photogrammetry and machine vision, a mathematical model is proposed to convert the information directly from the grating projection method (h, i, j) into the Cartesian coordinates (x, y, z). First, a camera model is introduced. 4.1. Camera model A camera model is used to set up the relationship between the 2-D image coordinate system and the 3-D global Cartesian coordinate system. In this case, a 2-D image point (i, j) has to be back projected into a 3-D ray * a ray in 3-D space that the corresponding 3-D object point must lie on. With the help of the height information, the location of the 3-D object point on this ray can be determined. As shown in Fig. 4,
Fig. 4. Camera model with lens distortion.
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a pinhole camera model with lens distortion is considered. Let P be an object point in 3-D space, and (x , y , z ) be its coordinates with respect to a "xed global Cartesian coordinate system (GCS). Let another Cartesian coordinate system, called the camera coordinate system (CCS), have its XY plane parallel to the image plane, such that the X-axis is parallel with the horizontal direction of the image, and the Y-axis is parallel with the vertical one. The origin of the CCS is located at the lens center and the Z-axis is aligned with the optical axis of the lens, which is perpendicular to the image plane. Let (x , y , z ) be the coordinates of the 3-D point P with respect to the CCS. If there is no lens distortion, the corresponding image point of P on the image plane would be Q. However, due to the e!ect of lens distortion, the actual image point is Q . Let (i, j) denote the 2-D image coordinates (in pixels) of the actual image point Q with respect to the computer image coordinate system (ICS). The origin of the ICS is the intersection point of the image plane and the optical axis. The distance between the image plane and the lens center is denoted as f. The overall transformation from (x , y , z ) to (i, j) can be divided into the following four steps. (1) Translation and rotation from the GCS to the CCS: The transformation from the global coordinate system to the camera coordinate system can be expressed as
x r y r " z r 1 0
r r r 0
r r
t
t
0
1
r
t
x y . z 1
(1)
Here (t , t , t ) is a translation vector, and r }r is a 3;3 rotation matrix. (2) Perspective projection from a 3-D object point in the CCS to a 2-D image point on the image plane: Let (i , j ) be the 2-D image coordinates of the undistorted image S S point Q lying on the image plane. Then, we have x i "f S z
and
y j "f . S z
(2)
(3) Radial lens distortion: After considering the radial lens distortion, the 2-D image coordinates (i, j) of the distorted image point Q can have i "i(1#kr) and j "j(1#kr). S S
(3)
Here r"i;i#j;j and k is the lens distortion coe$cient. Based on the experience of some of the researchers [9], Eq. (3) is adequate enough to model lens distortion for industrial vision applications. More elaborate modeling not only would not help but would also cause numerical instability. (4) Scaling of 2D image coordinates: The horizontal and vertical pixel spacings, d and d (millimeter/pixel), are used to scale from pixels to millimeters. S T
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Fig. 5. Schematic of the experimental setup of camera calibration.
Combining these four steps together, we have x r #y r #z r #t , (1!kr)id "f S x r #y r #z r #t x r #y r #z r #t . (1!kr)jd "f T x r #y r #z r #t
(4)
Thus the relationship between the 2-D image coordinates (i, j) and the global 3-D Cartesian coordinates (x , y , z ) is established. 4.2. Camera calibration The purpose of the camera calibration is to determine the constants involved in Eq. (4) and the constants used to convert phase information into the height information, by giving a set of 3-D calibration points and their corresponding 2-D image coordinates. In this work, a set of monoview non-coplanar points is used. As shown in Fig. 5, the camera is mounted on a translation stage, and a metal block, called the calibration target, is placed in front of the camera. Several images are taken at di!erent locations of the translation stage to generate a set of points with known 3-D coordinates (x, y, z) and their corresponding 2-D image coordinates (i, j). Using N pairs of 2-D/3-D coordinates in Eq. (4), the coe$cients can be determined by using the pseudo-inverse method. After the calibration, the whole system is portable and can be taken to any site to perform in situ measurements. As long as the camera and the geometry between the camera itself and the projector remain unchanged, there is no need to re-calibrate. To have the 3-D Cartesian coordinates as the ultimate results is very useful since it provides a practical tool for verifying the measurement accuracy.
5. Accuracy tests Unlike the geometric MoireH methods and the phase-measuring techniques, the way to de"ne the measurement accuracy in this system is more similar to what is used in
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Fig. 6. Surface #atness test.
Fig. 7. Measured point cloud of a #at surface.
industry. Standard objects with known geometry are measured with the optical system. Then the accuracy of the optical system is de"ned by comparing the measured values with the standard values. Objects with #at surface and curved surface are used in the accuracy tests. 5.1. Flat surface test An accurately machined #at surface is placed in front of the sensor head at di!erent z values (!22}22 mm) in the depth of view, as shown in Fig. 6. For each position, the surface is measured by the optical system to generate a point cloud with known xyz coordinates. Fig. 7 shows a point cloud with a reduced number of points. A leastsquare plane "tting is carried out to evaluate the #atness of the data, and the results are given in Table 1. Compared with the original surface #atness ((20 lm), the results compare very well. It is worth knowing that approximately 68% of the points are within one standard deviation from the "tted plane, and about 97% points are within two standard deviations. Points having larger error, which are due to isolated
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Table 1 Flat surface test results Surface Number location z (mm) of points !22.00 !18.00 !14.00 !10.00 !6.00 !2.00
154,775 154,784 154,806 154,809 154,812 154,812
Std. deviation (lm) 11.02 11.06 10.59 10.88 10.54 10.40
Surface location Number z (mm) of points 2.00 6.00 10.00 14.00 18.00 22.00
154,811 154,811 154,812 154,841 154,812 154,810
Std. deviation (lm) 10.82 11.05 11.28 11.95 12.99 14.02
Fig. 8. Cylindrical surface test.
noise, are relatively small in numbers. Their e!ects can be limited by a "ltering process. 5.2. Cylindrical surface test The accuracy test done on a cylindrical surface is slightly di!erent from that on a #at surface since for the curved surfaces the lateral coordinates, x and y, will play a role. Errors with x and y coordinates will a!ect the data deviation. In the test, a cylindrical surface with a diameter of 75.810$0.004 mm is placed approximately in the center of the depth of view, as shown in Fig. 8. The measurement result, which is shown in Fig. 9, is evaluated for the deviation at di!erent y locations (!25}25 mm). The measured points at y"0.0 mm and the standard cylindrical surface are shown in Fig. 10. Table 2 summarizes the deviation results for all y values. Again, the accuracy of the optical system is veri"ed.
6. Membrane in6ation test In recent decades, thermoforming has been used extensively to fabricate plastic products for a wide range of industries. Concurrently, tremendous advances in nonlinear "nite element analysis and computing power have made it possible to simulate the thermoforming process, resulting in savings in product design. However,
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Fig. 9. Measured point cloud on a cylindrical surface.
Fig. 10. Measured points at y"0.0 and the standard cylindrical surface. Table 2 Cylindrical surface test results y (mm) !25.0 !20.0 !15.0 !10.0 !5.0 0.0 Std. deviation (lm) 30.5 25.7 31.5 36.8 32.3 28.7
5.0 33.0
10.0 34.5
15.0 33.9
20.0 35.8
25.0 30.4
in order for the simulation to be reliable, accurate constitutive descriptions of the polymer are needed. Several hyperelastic constitutive theories are commonly employed, including the Mooney-Rivlin and Ogden models. Regardless of the model used, parameters in the model must be experimentally determined.
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Although a number of di!erent types of experiments can be employed for parameter determination, the membrane in#ation test, used initially by Treloar [10] o!ers a number of advantages, prominent among which is that a biaxial stress "eld is produced, which is similar to that encountered in many thermoforming operations. In the membrane in#ation test, a sheet of the thermoplastic material being studied is clamped between two plates, both of which have circular concentric holes. The exposed polymer sheet is heated on both sides by IR heating elements until the surface temperature uniformly reaches the softening point, which is normally close to the heat de#ection temperature (HDT). An in#ation medium, which is generally air or nitrogen gas, is introduced under pressure into a chamber mounted on the bottom side of the in#ation "xture, causing the membrane to deform into a bubble. In addition, by controlling the #ow rate of the pressurizing medium, the rate of deformation of the polymer can be varied, which is usually done, so that the resulting strain rates approximate those in the forming operation under consideration. As a result, #ow rates in the membrane test are often chosen such that the in#ation test time is on the order of 1 s, which is a typical time for many thermoforming operations. The in#ation test is usually recorded with a video camera, which is synchronized with a data acquisition system monitoring the chamber pressure. A marking grid made of concentric circles and several diameters is sometimes drawn on the membrane sheet to monitor the deformation rates in the azimutal and longitudinal directions. Side-mounted cameras are used to record bubble height and a camera may be placed above the membrane to record the deformation of the grid in the pole area. By assuming the deformed bubble shape to be spherical and under a state of biaxial stress, analytical expressions relating forces in the longitudinal and azimutal directions to the bubble height and applied pressure can be obtained as shown by Yang and Feng [11]. Then, for a given choice of constitutive model, material parameters can be determined by solving the equations numerically as demonstrated by Derdouri et al. [12]. The optical measurement system, described in Sections 3}5, is used here to measure the deformed surface of three di!erent polymers at di!erent #ow rates and temperatures during the membrane in#ation test. A schematic of the experimental con"guration is shown in Fig. 11. The optical sensor head is mounted above the membrane looking downward to measure the surface deformation. The 110 mm square sheet of polymeric material is clamped between two plates that leave a 63.5 mm diameter opening at its center exposed to infrared heaters. The MCDL is used to synchronize the deformed grating image with the instantaneous pressure. The optical measuring system is operated at 250 frames per second with a shutter speed of 1/250 s. Maximum measuring time is 2.2 s. Due to the high accuracy, high speed, and full-"eld capability of the measurement system, a number of interesting features from the tests are revealed, which had previously not been observed. Fig. 12 shows a typical measurement result in this case for a 1.5 mm thick acrylonitrile butadiene styrene (ABS) sheet at 1503C, with a #ow rate of 3.0 l/s at 160 ms after the start of in#ation. Each white dot represents a measured point, which is a subset of the &200,000 measured pixels. The coordinate system, which is associated with the sensor head, is also shown in the "gure. By viewing a sequence of images, the in#ation
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Fig. 11. Setup of the membrane in#ation test: (1) thermoplastic membrane; (2) pressure chamber.
Fig. 12. Measurement result of an ABS membrane during the in#ation test.
history of the membrane can be visualized. This is shown more clearly by looking at a cross section through the pole at selected instances in time as shown in Fig. 13. Careful examination of the membrane surface shows that the material near the supporting ring exhibits very little deformation, resulting in a non-spherical shape. This can be seen more clearly in Fig. 14, which shows a cross section through the pole, plotted along with an idealized sphere of the same height. Deviation from the idealized shape can be attributed to temperature gradients within the membrane. Despite
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Fig. 13. Membrane in#ation at di!erent times: from bottom up at time 20, 40, 80, 120, and 160 ms.
Fig. 14. Bubble pro"le at 160 ms after in#ation.
precautions to obtain a uniform temperature distribution, some gradients are inevitable, and may become signi"cant. Temperature conduction into the constraining plates results in a portion of the material near the rim being at a temperature signi"cantly below the value at the center of the membrane, resulting in a material region which is much sti!er. The presence of thermal gradients was con"rmed through use of an infrared camera. Also of interest, is the shape of the surface after heating, but prior to in#ation. The temperature increase results in a signi"cant decrease in material sti!ness, which may cause the thin membrane to sag under its own weight. In addition, the membrane is also constrained radially, preventing thermal expansion, and possibly resulting in membrane buckling, which also would be downward due to gravity. Either type of thermal deformation was previously undetectable since the constraining plates obscure the view of side-mounted cameras, and it is usually too small to be detected by vertically mounted cameras. Using the system described here, however, the thermal deformation is clearly detectable. For the ABS material, the deformed shape is similar to a simple "rst buckling mode, with the maximum out of plane displacement at the center dependent on the test temperature. The measurements indicate a maximum displacement of 0.75 mm at 1303C, 1.67 mm at 1503C and 5.41 mm at 1803C. The thermally deformed shape of high-density polyethylene (HDPE) is considerably more
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Fig. 15. Thermal warpage of an HDPE sheet at 1403C.
Fig. 16. Bubble pro"les of the HDPE sheet at 0 and 220 ms.
complex than ABS and is similar to a higher buckling mode, as shown in Fig. 15. If the temperature distribution is considered to be axisymmetric, the resulting thermal deformation should be similar to that seen for ABS. Therefore, the observed thermally deformed shape of HDPE must be due to either a non-uniformity in the membrane, such as its thickness, or due to anisotropy, which could arise from sheet processing. Whatever the cause, it has a pronounced e!ect on the bubble shape throughout the in#ation, as seen in Fig. 16, which shows a section of the HDPE material at 220 ms after the start of in#ation. Clearly, measurement of the initial thermal deformation, which is possible using the optical measurement system described here, provides better insight into the subsequent in#ation response of the polymer.
7. Conclusions A high-speed optical surface measurement system, which is capable of measuring transient surface shape, has been developed by using a high-speed digital camera. The system generates Cartesian coordinates directly, which are with respect to a "xed
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coordinate system. The measurement accuracy has been veri"ed with #at and curved standard objects. As a test application, the system is used to measure the transient surface during the membrane in#ation experiment. With its high accuracy and full-"eld capabilities a number of features of the test are revealed. The measurement system provides the entire surface deformation history rather than only the history of the pole height as is commonly done. These results raise the possibility of using the additional measurement data in conjunction with "nite element simulations to more accurately determine material parameters in constitutive models or to serve as a basis for comparison of the models themselves.
Acknowledgements This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) through an equipment grant. The authors are very grateful to M. Plourde of IMI/NRC for her invaluable assistance in the in#ation tests.
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