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Application of an image processing technique in strain measurement in sheet metal forming
Journal of Materials Processing Technology, 33 (1992) 299-310 Elsevier
299
Application of an image processing technique in strain measurement in sheet metal forming Zheng Tan, Lars Melin and Claes Magnusson Department o[ Materials Science and Production Technology, Lule~ University of Technology, Lule~, Sweden (Received August 13, 1991; accepted March 9, 1992)
Industrial Summary In sheet metal forming, strainsor strainincrements are evaluatedusuallyfrom the shape change of grids marked previously on the surface of the workpiece, the present understanding of formabilityin a forming process being based largelyon the knowledge gained from the strainmeasurement. The theory of square-gridanalysisisa great help in gaining knowledge of sheet metal forming. In the applicationof this theory, nodal points on a sheet metal surface etched with squaregrids have to be measured. Conventional grid measurements are performed manually, which is eithertime-consuming or of low accuracy.It is known that the image processingtechnique isvery powerful in digitizingthe image of an object.A newly developed image processing device which consistsof a video camera, a monitor and a personal computer, has thereforebeen introduced into strain measurement. The computer is equipped with an A D card and with data processing software. With such a device,the measurement of a gridded surface can be performed conveniently, with accuracy and efficiency.The combination of the image processingtechnique and the theory of square-gridstrain analysispermits rapid measurement and strain analysisover the surface of a workpiece marked with a largenumber of nodal points.
I. Introduction
Strain analysis is performed mostly by means of the grid method, in which the configurations of a grid marked previously on the surface of the workpiece are measured before and after deformation. The surface grid marking reduces the strain determination to a simple two-dimensional problem. It is assumed usually that an element of the grid over which the measurement is taken can be regarded as plane and that one of the principal directions is perpendicular to the sheet surface, hence in-plane measurement on the surface coupled with the incompressibility rule will determine the three principal strains. The circle-grid analysis was proposed first by Keeler and Goodwin [ 1,2 ] and Correspondence to: Zheng Tan, Department of Materials Science and Production Technology, Lule~ University of Technology, S-951 87, LuleA, Sweden.
0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
300 has long been applied in sheet metal forming. The usual method is that a pattern of fine circles (usually 5 mm in diameter) is etched on the surface of the sheet metal before pressing. After deformation, these circle-grids will be distorted into ellipses, the major and minor diameters of these distorted circles then being measured to determine the principal surface strains in the workpiece. Circle-grid analysis has been applied primarily to investigate large strains in cases where instability or fracture is concerned. For the problem of 'shape fixability', in which small strains (about 5% engineering strain) are involved and the strain distribution over a large portion of a surface is to be investigated, circle-grid analysis is not effective, since it is difficult to distinguish the major and minor diameters of slightly distorted circle-grids. Sowerby and his co-workers [ 3-5 ] have demonstrated that strain measurement can be simplified substantially if a square or quadrilateral grid is used in the strain analysis. In their theory, it is anticipated that the straining will occur by homogeneous deformation within every element of the grid. Homogeneous deformation is a linear process, and the strains can be determined by measuring the spatial coordinates of the nodal points on the gridded surface. This theory makes computer-aided strain measurement and analysis possible, and it has several advantages, for instance, speed of measurement, flexibility in choosing the size of an element of the grid and the ability to analyze noncoaxial strain path deformation [6 ], if the influence of strain path on the formability is to be studied. Due to its advantages over circle-grid analysis, squaregrid analysis has been applied by several researchers in their work [ 7-9 ]. When the theory of square-grid analysis was presented by Sowerby and his co-workers, it was pointed out that a fully automatic method of measuring the nodal points on the surface of a gridded stamping should be found. Since the strain analysis is performed by evaluating the final configuration of a square or quadrilateral grid with reference to its initial size, to measure the nodal coordinates of such a grid on the deformed metal sheet is therefore the most important task in the strain analysis. It is known that the shape and position of a surface can be described by massive nodal coordinates, if there exist distinguishable nodes on the surface. The image processing technique is very effective in digitizing such a surface: applications of this technique in engineering measurement have been reported [ 10-14 ]. Schedin and Melander [8 ] have utilized an automatic image analyzer in strain measurement by locating the central crosses of circle-grids with a movable cursor on a digitizing table. Sklad et al. [15] have carried out the measurement with a computerized vision device. These devices have obvious shortages in strain measurement, because the image coordinates of the nodal points were used for strain calculation. In this case, image distortion can not be taken into account and, further, the measuring efficiency is poor. A newly developed image processing device is introduced into strain measurement, and consists of a video camera, a monitor and a personal computer equipped with an AD card and data deducing software.
301
With this device, the measurement is done simply by recording two or more images of the surface to be measured from different view positions, reading the image coordinates automatically and calculating the spatial coordinates. 2. Strain analysis
The theory of Sowerby and his co-workers is described briefly for completeness. In their analysis, a 'pure homogeneous deformation' mode is assumed. Strains in a sheet metal forming process can be revealed by the changes in the shape of the grids. Provided that the initial nodal positions of a grid previously marked on the flat blank are known and the corresponding spatial positions of the nodes on the deformed surface are measured, the deformation gradient ratios of the line elements of the grid can then be evaluated easily. To illustrate the theory of finite strain, a triangular grid, shown in Fig. 1, is considered as being deformed in its plane. Actually, there is no need to restrict the grid shape to a triangle or a square. It should be reiterated that the theory of strain analysis describes 'in-plane' deformation under the hypothesis of a pure homogeneous strain mode. In this mode, the new nodal coordinates (x', y' ) of a nodal point within each grid are a linear function of the initial coordinates (x, y). The triangular grid considered consists of the line elements OA and OB: after deformation, they are elongated to OA' and OB', the spatial lengths and positions of which can be described by the nodal coordinates of the triangular pinnacles. If a movable local coordinate system is built up within the element grid, it is possible always to allow the origin to coincide with one of the pinnacles and the z-axis to be normal to the grid surface through transformation of coordinates. Suppose that the initial and final nodal positions are known, and that the two configurations of a grid before and after deformation are superimposed, the in-plane deformation within every triangular grid can be expressed by the following equations: Y A'
B' B 0
X
Fig. 1. A triangular grid element before and after forming, the origin of the axes being superimposed with a node of the grid.
302
XA' =FllXA A-F12YA
YA' =F21XA 4-F22YA
XB' =Fllxn +F12yB
YB' ----F21XB4"F22YB
(1)
where Fii are the components of possible deformation gradient or elongation ratios of the line elements 0A' and OB' with projection onto the initial lines OA and OB. Equations (1) can be written in the form of a matrix also, as YA' YS'
=
F21 F22
YA YB
In the following deduction, tensor notation is introduced, which makes the deduction succinct. Equation (2) can be rewritten in tensor form
x=F.X
(3)
Transforming (3) yields F:x.X
-1
(4)
where X - 1 is the inverse matrix of X and IP is the deformation gradient tensor with the components Fii (i= 2, j = 2), which can be considered as strains and evaluated from the initial and final nodal coordinates. For finite strains, the deformation gradient tensor is generally unsymmetric, i.e., F12-~F21. In this case, a symmetric deformation gradient tensor S for describing pure homogeneous deformation is introduced, which produces the same shape change as does F [4,16], so that F can be written as
F=R.S
(5)
where R is a rigid body rotation tensor which relates the respective deformation gradient tensors in homogeneous deformation and pure homogeneous deformation. Multiplying F with its transposed tensor/P t yields
Ft.F=S 2
(6)
Since R is orthogonal and S is symmetric, it must be true that R t . R = I and S t = S in deducing (6) from (5), where I is a unit tensor. It is known that establishing the principal axes and the principal elongation ratios in the homogeneous deformation is ambiguous: in order to eliminate the ambiguity, a symmetric second-order tensor known as the Green deformation tensor is then used. The eigenvalues of this tensor are identical to the square of the principal elongation ratios arising in the pure homogeneous mode. The product of the deformation gradient tensor IP and its transposed tensor F t is defined as the Green deformation tensor C C:.~'v t* F :
~2
(7)
303 Upon expanding (7), the components of C are ell =F121 +F~I
C12-~-C21--~F11F12+F21F22
(8)
C22 =F~2 +F~2 The components of tensor C are invariants to rigid body rotations. The use of (5) and (7) yields the same principal elongation ratios for both the actual and the pure homogeneous deformation modes mathematically. The eigenvalues of C are derived as the principal elongation ratios squared, which are given by
22
)~11,~22 -
C11 "~C22- ~ ( C l l - C22)2 2
~
+C~2
(9)
where ~ 11and)122 are principal elongation ratios of the final length to the initial length of a pair of line elements within a grid in the principal directions. It follows that the principal logarithmic surface strains are el ,e2 = In (~ 12,~22)
(10)
The third principal strain e3, which is normal to the sheet surface, is obtained by assuming constancy of volume, el + e2+ ~3-- 0. The orientation of the first principal axis with respect to the x axis is determined from tan 28=
2 C12 Cll - C22
(11)
3. Measuring nodal points o n a gridded surface
The theory of square-grid strain analysis makes itpossible to investigate the principal strains in a workpiece by measuring, after deformation, the nodal coordinates of the square-grid pattern etched previously on the surface. Measuring massive nodal points on a surface with high accuracy and efficiency is not an easy task. The existing measuring methods and tools are not so satisfactory for square-grid strain analysis. Automatic methods to measure or digitize a gridded surface are therefore expected. A powerful image processing technique which is called close-range photogrammetry has been introduced into strain measurement by the authors [17 ]. By means of this technique, a large number of points on a surface can be measured or digitized simultaneously, with very limited h u m a n interaction. A newly developed image processing device is suggested to be used in strain measurement which, as is shown in Fig. 2, consists of a video camera, a monitor and a personal computer. The measuring procedure is:taking and storing the
304
Fig. 2. The newly developed image processing device.
.••0 View2 v2
~~
fl, f2: focal point
~
~
y
View,
x Fig. 3. Stereo images used for triangulating the spatial coordinates of nodes on a surface.
images of the gridded surface to be measured; pre-processing the images; reading the nodal coordinates of the points on the images automatically; calculating the 3D coordinates from the recorded 2D image coordinates by photogrammetric theory; and calculating the strains by square-grid analysis theory. To obtain the 3D spatial coordinates, triangulating a single common point from two separate images must be carried out, that is, a pair (or more) of images of a gridded surface are taken successively from two different camera positions, the images being stored in the computer for further data processing. Figure 3 illustrates schematically the perspective projection of spatial nodal points on the images through the video camera. After the pre-processing of the stored pictures, with the lines thinned, the nodal points are digitized to read their image coordinates, so that the corresponding 3D spatial coordinates can be calculated from the stereo image pair. The theory concerned is the projective theory of photogrammetry, and the linear transformation from the 2D image coordinates to the 3D object coordi-
305 nates is described in a homogeneous coordinate system. Mathematically, the translation, rotation, scaling and perspective projection can be expressed in a single transformation matrix [18-20]
[wuI IA11A12A13A141 wv w
=
A21 A31
A22 A32
A23
A33
A24 A34
"
X
Y Z
(12)
1
where w may be termed a scale factor; u and v are the 2D image coordinates of a nodal point; x, y and z are the corresponding 3D coordinates in the objective space; and Aij are the coefficients of the perspective projection transformation matrix defining the rigid rotation, scaling and perspective projection. Expanding (12) and substituting the expression for w into the first two equations yields
(A1, -A31u)x+ (A12 -A32u)y+ (A13-A33u)z= (A34u-A14)
(13)
(A21 - A31v )x + (A22-A32v)y+ (A23 -A33 v )z= (A34v - A24 ) In solving eqns. (13), there are two possible approaches: (1) the image coordinates u, v and the transformation matrix are known, whilst the spatial coordinates are unknown; or (2) the spatial and image coordinates are known, whilst components Aij of the transformation matrix are unknown. For case (1), the spatial coordinates x, y and z are the unknowns to be determined from known image coordinates u, v and the transformation matrix, so that there are three unknowns in the two equations: if two images for the nodal points have been obtained from different view positions and the 2D image coordinates measured, then four equations (two per image) are generated, so that any point may have its 3D spatial coordinates calculated. For case (2), there are 12 unknowns Aij (i=3, j = 4 ) in the two equations, twelve equations with twelve unknowns being needed. To obtain the twelve components of the transformation matrix the image system has to be calibrated. Several methods can be applied in the calibration: a fairly simple interactive calibration process can be designed by choosing six non-coplanar points, called 'control points', which must be covered on the two images. These points can be measured directly from the workpiece by a conventional method in order to obtain their locations in the objective space: by so doing, an objective coordinate system is defined. When the images have been digitized and all the image coordinates obtained, both the 3D spatial coordinates and the corresponding 2D image coordinates of these control points are then imput in the computer to calculate A~j. Since the projection is considered in a homogeneous coordinate system, one of the unknowns may be set arbitrarily, it being common practice to set A34 = 1. Case (2) provides the basis for the calibration of the system [19,21].
306 Provided that calibration has been done and the image coordinates measured, with Alij as the coefficients of the transformation matrix for image 1 and A2/i as those for image 2, if ul and Vl are the image coordinates of a nodal point on image 1 and u2 and v2 are those of the same point on image 2, the spatial coordinates (x, y, z) of the point can be calculated from the following equations
(A111-A131ul)xq- (Al12
-AI~2ul)y+
(Al13
--A133ul)z=A134ul -Al14
(14)
(A121 -A131 v 1)x+ (A122-A132v 1)Y+ (A123-A133 v 1)z--~A134vi -A124 and (A211 - A 2 3 1 u 2 ) x + (A212 - A 2 a 2 u 2 ) y + (A213 - A 2 3 3 u 2 ) z = A 2 3 4 u 2 - A 2 1 4
(15)
(A221 - A231 v 2 ) x + (A222 - A 232 v 2 )Y + (A223 - A23~ v 2 ) z = A234 v 2 - A224
Equations (14) and (15) can be written also in matrix form, as (16)
PX=D
where the matrices are
Am-A131ul AII2-AI32ul AII3-AI33ul p=
A121-A131vl
A122-A132Vl
A123-Ala3v,
(17)
A2,I-A23,u2 A2,2-A232u2 A213-A2~3u2 A221-A231v2 A222-A232v2 A223-A233v2 X=
D~---
(xyz) t
A134Ul--All4 A134Vl-A124 A234u2--A2~4
(18)
A 234v e - A ~e4
Equations (16) are over-determined, with four equations for three unknowns. To solve the over-determined system, both sides of (16) can be multiplied with pt, which is a transposed matrix of P . In this way, a least-squares solution is utilized PtPX=ptD
(19)
Finally, the spatial coordinates (x, y, z) can be obtained from X-- (ptp)-1pt D
(20)
4. S t r a i n calculation
A workpiece etched originally with a I0 × 10 m m square grid, as shown on the monitor screen in Fig. 2, has been measured to show the features of the
307
image processing theory and of the image system. The nodal points are first measured with an optical measuring microscope on the surface, the results of which will be used for comparison. Six non-coplanar points among the measured nodal points are chosen as control points for the calibration of the image system. The acquisition of image coordinates of the nodal points is carried out with the image processing device, shown in Fig. 2, the device being programmed so that the nodal points can be digitizedand numbered automatically. It takes less than half an hour to complete the image taking, the pre-processing, and the digitizing.A pre-processed image taken from the monitor screen is shown in Fig. 4. After the digitizationof the images, the image coordinates as well as the spatial coordinates of the control points are input into the personal computer to calculate the spatial coordinates of all the points. The measured and calculated three-dimensional coordinates are listed in Table 1, which is composed of 10 columns. Column one, tabled as PN, denotes the point number, whilst columns 2, 3, 4 and columns 5, 6, 7 contain the threedimensional coordinates measured directlyon the surface and those calculated from the image coordinates, respectively. The last three columns give the deviations. Finally, the root-mean-square errors are given. After the three-dimensional coordinates of nodes have been calculated,which provides a description of the deformed surface, square-grid strain analysis theory is applied to analyze the strain values. Since in-plane strain measurement is assumed in the theory, additional coordinate transformation from the global to the local coordinates for each grid must be carried out. As it is known that the symmetric Green deformation tensor provides the same principal strain values with respect to any local coordinates, setting-up of the local axes can just be in consideration of convenience. After the calculation and transfor-
Fig. 4. A pre-processed image of the specimen with thinned lines and numbered nodes.
308 TABLE 1 Comparison of the measured a n d calculated nodal coordinates ( m m ) PN
x
y
z
x'
y'
z'
1 2 5 6 7 8 13 14 17 18 19 20 21 23 24 26 34 37 44
17.463 17.633 18.603 18.798 19.280 27.885 26.797 27.218 55.364 56.079 55.829 54.877 65.632 66.930 66.322 63.332 87.254 83.558 90.056
156.253 165.984 196.008 206.130 216.290 216.190 165.153 155.577 183.308 193.875 204.382 214.426 214.565 193.433 182.825 162.121 204.924 160.631 152.089
1.700 3.948 6.150 5.979 4.705 10.530 9.023 5.274 37.071 37.592 35.979 31.775 32.465 37.335 37.577 29.768 31.899 25.146 13.186
17.465 17.437 18.878 19.137 19.279 28.615 26.957 27.490 55.351 56.165 55.862 54.939 65.851 66.948 66.266 63,153 87.248 83.397 90.056
156.252 165.822 195.876 206.194 216.290 216.233 164.987 155.329 183.328 194.015 204.480 214.640 214.636 193.406 182.961 161.599 204.934 160.432 152.088
1.690 3.778 6.687 6.363 4.710 10.634 8.770 5.377 37.236 38.318 36.515 31.889 32.269 37.128 37.448 29.698 31.974 25.525 13.182
R M S error
x-x'
y-y'
z-z'
-0.002 0.196 -0.275 -0.339 0.001 -0.730 -0.160 -0.272 0.013 -0.086 -0.033 -0.062 -0.219 -0.018 0.056 0.179 0.006 0.161 0.000
0,001 0,162 0.186 -0.064 0,000 -0,043 0,166 0,248 -0,020 -0,140 -0,098 -0,214 -0,071 0.027 -0.136 0.522 -0,010 0.199 0.001
0.010 0.170 0.537 -0,384 -0.005 -0.104 0.253 -0.103 -0.165 -0.726 -0.536 -0.114 0.196 0.227 0.129 0,070 -0.075 -0.379 0,004
0.227
0.173
0.296
TABLE 2 Strainvaluesat severalpointsobtainedby square-gridanalysistheory PN
~
f~
f-~m
16 17 18 19 22 23 24 25 26
0.1746 0.1398 0.1436 0.1886 0.1641 0.1663 0.1414 0.1475 0.1978
0.1664 0.1202 0.1438 0.1925 0.1639 0.1423 0.1434 0.1444 0.2313
0.0082 0.0196 -0.0002 -0.0039 0.0002 0.0240 -0.0020 -0.0031 0.0335
R M S error
0.0156
mation of coordinates, square-grid analysis theory is applied to obtain the strains. Some results of the strain analysis are given in Table 2, where ~ is the equivalent strain at each node calculated from the precisely measured threedimensional coordinates, and em is the mean value of the equivalent strains
309
calculated from image coordinates obtained in two measurements. It can be seen that the results calculated from the nodal coordinates measured with the conventional device and those measured by means of the image processing device are quite close to each other, but that the measuring efficiency of the latter is much higher. The equivalent strain is an important parameter, the values of which are slightly different in the pure homogeneous deformation and in the real one [5]. In Table 2, only part of the results are listed. 5. Conclusions
It has been shown, in some detail, how the proposed image processing technique can be applied to strain measurement in sheet metal forming. Compared with other, conventional, measuring methods, this technique is powerful in digitizing a gridded surface. The results in the experimental evaluation indicate that the combination of strain analysis theory with the computerized vision system provides an effective means of strain measurement. The effort of measuring does not grow with the number of points, and the accuracy can be improved by increasing the number of images taken from extra camera positions. Once a gridded surface has been digitized and the strains at nodal points have been calculated, the values of el and e2 may be plotted in graphical forms, for instance, a bivariable strain distribution with a forming limit diagram superimposed can provide an intuitional graphic tool for investigating the safety of a forming operation. A very important factor affecting the accuracy is the choice of control points and the parallactic angle. It is suggested to maintain a parallactic angle about 60 to 110 degrees and also that there should be a certain amount of redundancy between the image pair, in order to obtain results of sufficient accuracy and reliability. Acknowledgements The authors would like to thank the Swedish Board of Technical Development (STU) for the financial support of this work. We are also grateful to Wei-Xing Wang and Stefan Dahlhielm, IPACS Company, for their valuable help in the image technique.
References 1 2 3 4
S.P. Keeler, SAE Congress, Detroit, MI, 1968, Paper 680092, pp. 1-9. G.M. Goodwin, SAE Congress, Detroit, MI, 1968, Paper 680093, pp. 380-387. R. Sowerby, E. Chu and J.L. Duncan, J. Strain Anal., 17 (1982) 95-101. R. SowerbyandP.C. Chakravarti, J. Strain Anal., 18 (2) (1983) 119-123.
310 E. Chu, R. Sowerby, R. Soldaat and J.L. Duncan, 13th IDDRG Cong. Proc., Melbourne, Australia, 1984, pp. 9-21. 6 T.C. Hsu, J. Strain Anal., 1 (3) (1966) 216-222. 7 W.R. Thorpe and T.J. Nihill, J. Mech. Work. Technol., 9 (1984) 5-20. 8 E. Schedin and A. Melander, J. Appl. Metalwork., 4 (2) (1986) 143-156. 9 V.K. Jain, L.E. Matson and H.L. Gegel, J. Mater. Shap. Technol., 5 (1988) 243. 10 K.W. Wong, Cong. Proc. on Close-range Photogrammetric Systems, Champaign, IL, USA, 1975, pp. 598-611. 11 M.A.R. Cooper, Photogramm. Rec., 9 (53) (1979) 601-619. 12 T. Bednarski and A. Majde, Soc. Francoise Photogramm. Bull., 42 (1971) 55-62. 13 J.L. Posdamer and M.D. Altschuler, Comput. Graphics Image Process., 18 (1982) 1-17. 14 C.S. Fraser, Photogramm. Eng. Remote Sensing, 52 (10) (1986) 1627-1635. 15 M.P. Sklad, C.B. Harris and B.A. Yungblud, 16th IDDRG Cong. Proc., Boliinge, Sweden, 1990, 63-7O. 16 C.Y. ChoiandT.C. Hsu, J. Strain Anal., 6 (1) (1971) 62-69. 17 Zheng Tan, Licentiate Thesis, Lule~ University of Technology, Sweden, 1990: 14L. 18 E.H. Thomapson, Photogramm. Rec., 7 (37) (1971) 39-45. 19 F.I. Parke, Comput. Graphics, 1 (1975) 5-7. 20 W.M. Newman, Principles of Interactive Computer Graphics, 2nd ed., McGraw-Hill, New York, 1979. 21 J.N. Hatzopaulos, Photogramm. Eng. Remote Sensing, 51 (10) (1985) 1583-1588. 5
299
Application of an image processing technique in strain measurement in sheet metal forming Zheng Tan, Lars Melin and Claes Magnusson Department o[ Materials Science and Production Technology, Lule~ University of Technology, Lule~, Sweden (Received August 13, 1991; accepted March 9, 1992)
Industrial Summary In sheet metal forming, strainsor strainincrements are evaluatedusuallyfrom the shape change of grids marked previously on the surface of the workpiece, the present understanding of formabilityin a forming process being based largelyon the knowledge gained from the strainmeasurement. The theory of square-gridanalysisisa great help in gaining knowledge of sheet metal forming. In the applicationof this theory, nodal points on a sheet metal surface etched with squaregrids have to be measured. Conventional grid measurements are performed manually, which is eithertime-consuming or of low accuracy.It is known that the image processingtechnique isvery powerful in digitizingthe image of an object.A newly developed image processing device which consistsof a video camera, a monitor and a personal computer, has thereforebeen introduced into strain measurement. The computer is equipped with an A D card and with data processing software. With such a device,the measurement of a gridded surface can be performed conveniently, with accuracy and efficiency.The combination of the image processingtechnique and the theory of square-gridstrain analysispermits rapid measurement and strain analysisover the surface of a workpiece marked with a largenumber of nodal points.
I. Introduction
Strain analysis is performed mostly by means of the grid method, in which the configurations of a grid marked previously on the surface of the workpiece are measured before and after deformation. The surface grid marking reduces the strain determination to a simple two-dimensional problem. It is assumed usually that an element of the grid over which the measurement is taken can be regarded as plane and that one of the principal directions is perpendicular to the sheet surface, hence in-plane measurement on the surface coupled with the incompressibility rule will determine the three principal strains. The circle-grid analysis was proposed first by Keeler and Goodwin [ 1,2 ] and Correspondence to: Zheng Tan, Department of Materials Science and Production Technology, Lule~ University of Technology, S-951 87, LuleA, Sweden.
0924-0136/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
300 has long been applied in sheet metal forming. The usual method is that a pattern of fine circles (usually 5 mm in diameter) is etched on the surface of the sheet metal before pressing. After deformation, these circle-grids will be distorted into ellipses, the major and minor diameters of these distorted circles then being measured to determine the principal surface strains in the workpiece. Circle-grid analysis has been applied primarily to investigate large strains in cases where instability or fracture is concerned. For the problem of 'shape fixability', in which small strains (about 5% engineering strain) are involved and the strain distribution over a large portion of a surface is to be investigated, circle-grid analysis is not effective, since it is difficult to distinguish the major and minor diameters of slightly distorted circle-grids. Sowerby and his co-workers [ 3-5 ] have demonstrated that strain measurement can be simplified substantially if a square or quadrilateral grid is used in the strain analysis. In their theory, it is anticipated that the straining will occur by homogeneous deformation within every element of the grid. Homogeneous deformation is a linear process, and the strains can be determined by measuring the spatial coordinates of the nodal points on the gridded surface. This theory makes computer-aided strain measurement and analysis possible, and it has several advantages, for instance, speed of measurement, flexibility in choosing the size of an element of the grid and the ability to analyze noncoaxial strain path deformation [6 ], if the influence of strain path on the formability is to be studied. Due to its advantages over circle-grid analysis, squaregrid analysis has been applied by several researchers in their work [ 7-9 ]. When the theory of square-grid analysis was presented by Sowerby and his co-workers, it was pointed out that a fully automatic method of measuring the nodal points on the surface of a gridded stamping should be found. Since the strain analysis is performed by evaluating the final configuration of a square or quadrilateral grid with reference to its initial size, to measure the nodal coordinates of such a grid on the deformed metal sheet is therefore the most important task in the strain analysis. It is known that the shape and position of a surface can be described by massive nodal coordinates, if there exist distinguishable nodes on the surface. The image processing technique is very effective in digitizing such a surface: applications of this technique in engineering measurement have been reported [ 10-14 ]. Schedin and Melander [8 ] have utilized an automatic image analyzer in strain measurement by locating the central crosses of circle-grids with a movable cursor on a digitizing table. Sklad et al. [15] have carried out the measurement with a computerized vision device. These devices have obvious shortages in strain measurement, because the image coordinates of the nodal points were used for strain calculation. In this case, image distortion can not be taken into account and, further, the measuring efficiency is poor. A newly developed image processing device is introduced into strain measurement, and consists of a video camera, a monitor and a personal computer equipped with an AD card and data deducing software.
301
With this device, the measurement is done simply by recording two or more images of the surface to be measured from different view positions, reading the image coordinates automatically and calculating the spatial coordinates. 2. Strain analysis
The theory of Sowerby and his co-workers is described briefly for completeness. In their analysis, a 'pure homogeneous deformation' mode is assumed. Strains in a sheet metal forming process can be revealed by the changes in the shape of the grids. Provided that the initial nodal positions of a grid previously marked on the flat blank are known and the corresponding spatial positions of the nodes on the deformed surface are measured, the deformation gradient ratios of the line elements of the grid can then be evaluated easily. To illustrate the theory of finite strain, a triangular grid, shown in Fig. 1, is considered as being deformed in its plane. Actually, there is no need to restrict the grid shape to a triangle or a square. It should be reiterated that the theory of strain analysis describes 'in-plane' deformation under the hypothesis of a pure homogeneous strain mode. In this mode, the new nodal coordinates (x', y' ) of a nodal point within each grid are a linear function of the initial coordinates (x, y). The triangular grid considered consists of the line elements OA and OB: after deformation, they are elongated to OA' and OB', the spatial lengths and positions of which can be described by the nodal coordinates of the triangular pinnacles. If a movable local coordinate system is built up within the element grid, it is possible always to allow the origin to coincide with one of the pinnacles and the z-axis to be normal to the grid surface through transformation of coordinates. Suppose that the initial and final nodal positions are known, and that the two configurations of a grid before and after deformation are superimposed, the in-plane deformation within every triangular grid can be expressed by the following equations: Y A'
B' B 0
X
Fig. 1. A triangular grid element before and after forming, the origin of the axes being superimposed with a node of the grid.
302
XA' =FllXA A-F12YA
YA' =F21XA 4-F22YA
XB' =Fllxn +F12yB
YB' ----F21XB4"F22YB
(1)
where Fii are the components of possible deformation gradient or elongation ratios of the line elements 0A' and OB' with projection onto the initial lines OA and OB. Equations (1) can be written in the form of a matrix also, as YA' YS'
=
F21 F22
YA YB
In the following deduction, tensor notation is introduced, which makes the deduction succinct. Equation (2) can be rewritten in tensor form
x=F.X
(3)
Transforming (3) yields F:x.X
-1
(4)
where X - 1 is the inverse matrix of X and IP is the deformation gradient tensor with the components Fii (i= 2, j = 2), which can be considered as strains and evaluated from the initial and final nodal coordinates. For finite strains, the deformation gradient tensor is generally unsymmetric, i.e., F12-~F21. In this case, a symmetric deformation gradient tensor S for describing pure homogeneous deformation is introduced, which produces the same shape change as does F [4,16], so that F can be written as
F=R.S
(5)
where R is a rigid body rotation tensor which relates the respective deformation gradient tensors in homogeneous deformation and pure homogeneous deformation. Multiplying F with its transposed tensor/P t yields
Ft.F=S 2
(6)
Since R is orthogonal and S is symmetric, it must be true that R t . R = I and S t = S in deducing (6) from (5), where I is a unit tensor. It is known that establishing the principal axes and the principal elongation ratios in the homogeneous deformation is ambiguous: in order to eliminate the ambiguity, a symmetric second-order tensor known as the Green deformation tensor is then used. The eigenvalues of this tensor are identical to the square of the principal elongation ratios arising in the pure homogeneous mode. The product of the deformation gradient tensor IP and its transposed tensor F t is defined as the Green deformation tensor C C:.~'v t* F :
~2
(7)
303 Upon expanding (7), the components of C are ell =F121 +F~I
C12-~-C21--~F11F12+F21F22
(8)
C22 =F~2 +F~2 The components of tensor C are invariants to rigid body rotations. The use of (5) and (7) yields the same principal elongation ratios for both the actual and the pure homogeneous deformation modes mathematically. The eigenvalues of C are derived as the principal elongation ratios squared, which are given by
22
)~11,~22 -
C11 "~C22- ~ ( C l l - C22)2 2
~
+C~2
(9)
where ~ 11and)122 are principal elongation ratios of the final length to the initial length of a pair of line elements within a grid in the principal directions. It follows that the principal logarithmic surface strains are el ,e2 = In (~ 12,~22)
(10)
The third principal strain e3, which is normal to the sheet surface, is obtained by assuming constancy of volume, el + e2+ ~3-- 0. The orientation of the first principal axis with respect to the x axis is determined from tan 28=
2 C12 Cll - C22
(11)
3. Measuring nodal points o n a gridded surface
The theory of square-grid strain analysis makes itpossible to investigate the principal strains in a workpiece by measuring, after deformation, the nodal coordinates of the square-grid pattern etched previously on the surface. Measuring massive nodal points on a surface with high accuracy and efficiency is not an easy task. The existing measuring methods and tools are not so satisfactory for square-grid strain analysis. Automatic methods to measure or digitize a gridded surface are therefore expected. A powerful image processing technique which is called close-range photogrammetry has been introduced into strain measurement by the authors [17 ]. By means of this technique, a large number of points on a surface can be measured or digitized simultaneously, with very limited h u m a n interaction. A newly developed image processing device is suggested to be used in strain measurement which, as is shown in Fig. 2, consists of a video camera, a monitor and a personal computer. The measuring procedure is:taking and storing the
304
Fig. 2. The newly developed image processing device.
.••0 View2 v2
~~
fl, f2: focal point
~
~
y
View,
x Fig. 3. Stereo images used for triangulating the spatial coordinates of nodes on a surface.
images of the gridded surface to be measured; pre-processing the images; reading the nodal coordinates of the points on the images automatically; calculating the 3D coordinates from the recorded 2D image coordinates by photogrammetric theory; and calculating the strains by square-grid analysis theory. To obtain the 3D spatial coordinates, triangulating a single common point from two separate images must be carried out, that is, a pair (or more) of images of a gridded surface are taken successively from two different camera positions, the images being stored in the computer for further data processing. Figure 3 illustrates schematically the perspective projection of spatial nodal points on the images through the video camera. After the pre-processing of the stored pictures, with the lines thinned, the nodal points are digitized to read their image coordinates, so that the corresponding 3D spatial coordinates can be calculated from the stereo image pair. The theory concerned is the projective theory of photogrammetry, and the linear transformation from the 2D image coordinates to the 3D object coordi-
305 nates is described in a homogeneous coordinate system. Mathematically, the translation, rotation, scaling and perspective projection can be expressed in a single transformation matrix [18-20]
[wuI IA11A12A13A141 wv w
=
A21 A31
A22 A32
A23
A33
A24 A34
"
X
Y Z
(12)
1
where w may be termed a scale factor; u and v are the 2D image coordinates of a nodal point; x, y and z are the corresponding 3D coordinates in the objective space; and Aij are the coefficients of the perspective projection transformation matrix defining the rigid rotation, scaling and perspective projection. Expanding (12) and substituting the expression for w into the first two equations yields
(A1, -A31u)x+ (A12 -A32u)y+ (A13-A33u)z= (A34u-A14)
(13)
(A21 - A31v )x + (A22-A32v)y+ (A23 -A33 v )z= (A34v - A24 ) In solving eqns. (13), there are two possible approaches: (1) the image coordinates u, v and the transformation matrix are known, whilst the spatial coordinates are unknown; or (2) the spatial and image coordinates are known, whilst components Aij of the transformation matrix are unknown. For case (1), the spatial coordinates x, y and z are the unknowns to be determined from known image coordinates u, v and the transformation matrix, so that there are three unknowns in the two equations: if two images for the nodal points have been obtained from different view positions and the 2D image coordinates measured, then four equations (two per image) are generated, so that any point may have its 3D spatial coordinates calculated. For case (2), there are 12 unknowns Aij (i=3, j = 4 ) in the two equations, twelve equations with twelve unknowns being needed. To obtain the twelve components of the transformation matrix the image system has to be calibrated. Several methods can be applied in the calibration: a fairly simple interactive calibration process can be designed by choosing six non-coplanar points, called 'control points', which must be covered on the two images. These points can be measured directly from the workpiece by a conventional method in order to obtain their locations in the objective space: by so doing, an objective coordinate system is defined. When the images have been digitized and all the image coordinates obtained, both the 3D spatial coordinates and the corresponding 2D image coordinates of these control points are then imput in the computer to calculate A~j. Since the projection is considered in a homogeneous coordinate system, one of the unknowns may be set arbitrarily, it being common practice to set A34 = 1. Case (2) provides the basis for the calibration of the system [19,21].
306 Provided that calibration has been done and the image coordinates measured, with Alij as the coefficients of the transformation matrix for image 1 and A2/i as those for image 2, if ul and Vl are the image coordinates of a nodal point on image 1 and u2 and v2 are those of the same point on image 2, the spatial coordinates (x, y, z) of the point can be calculated from the following equations
(A111-A131ul)xq- (Al12
-AI~2ul)y+
(Al13
--A133ul)z=A134ul -Al14
(14)
(A121 -A131 v 1)x+ (A122-A132v 1)Y+ (A123-A133 v 1)z--~A134vi -A124 and (A211 - A 2 3 1 u 2 ) x + (A212 - A 2 a 2 u 2 ) y + (A213 - A 2 3 3 u 2 ) z = A 2 3 4 u 2 - A 2 1 4
(15)
(A221 - A231 v 2 ) x + (A222 - A 232 v 2 )Y + (A223 - A23~ v 2 ) z = A234 v 2 - A224
Equations (14) and (15) can be written also in matrix form, as (16)
PX=D
where the matrices are
Am-A131ul AII2-AI32ul AII3-AI33ul p=
A121-A131vl
A122-A132Vl
A123-Ala3v,
(17)
A2,I-A23,u2 A2,2-A232u2 A213-A2~3u2 A221-A231v2 A222-A232v2 A223-A233v2 X=
D~---
(xyz) t
A134Ul--All4 A134Vl-A124 A234u2--A2~4
(18)
A 234v e - A ~e4
Equations (16) are over-determined, with four equations for three unknowns. To solve the over-determined system, both sides of (16) can be multiplied with pt, which is a transposed matrix of P . In this way, a least-squares solution is utilized PtPX=ptD
(19)
Finally, the spatial coordinates (x, y, z) can be obtained from X-- (ptp)-1pt D
(20)
4. S t r a i n calculation
A workpiece etched originally with a I0 × 10 m m square grid, as shown on the monitor screen in Fig. 2, has been measured to show the features of the
307
image processing theory and of the image system. The nodal points are first measured with an optical measuring microscope on the surface, the results of which will be used for comparison. Six non-coplanar points among the measured nodal points are chosen as control points for the calibration of the image system. The acquisition of image coordinates of the nodal points is carried out with the image processing device, shown in Fig. 2, the device being programmed so that the nodal points can be digitizedand numbered automatically. It takes less than half an hour to complete the image taking, the pre-processing, and the digitizing.A pre-processed image taken from the monitor screen is shown in Fig. 4. After the digitizationof the images, the image coordinates as well as the spatial coordinates of the control points are input into the personal computer to calculate the spatial coordinates of all the points. The measured and calculated three-dimensional coordinates are listed in Table 1, which is composed of 10 columns. Column one, tabled as PN, denotes the point number, whilst columns 2, 3, 4 and columns 5, 6, 7 contain the threedimensional coordinates measured directlyon the surface and those calculated from the image coordinates, respectively. The last three columns give the deviations. Finally, the root-mean-square errors are given. After the three-dimensional coordinates of nodes have been calculated,which provides a description of the deformed surface, square-grid strain analysis theory is applied to analyze the strain values. Since in-plane strain measurement is assumed in the theory, additional coordinate transformation from the global to the local coordinates for each grid must be carried out. As it is known that the symmetric Green deformation tensor provides the same principal strain values with respect to any local coordinates, setting-up of the local axes can just be in consideration of convenience. After the calculation and transfor-
Fig. 4. A pre-processed image of the specimen with thinned lines and numbered nodes.
308 TABLE 1 Comparison of the measured a n d calculated nodal coordinates ( m m ) PN
x
y
z
x'
y'
z'
1 2 5 6 7 8 13 14 17 18 19 20 21 23 24 26 34 37 44
17.463 17.633 18.603 18.798 19.280 27.885 26.797 27.218 55.364 56.079 55.829 54.877 65.632 66.930 66.322 63.332 87.254 83.558 90.056
156.253 165.984 196.008 206.130 216.290 216.190 165.153 155.577 183.308 193.875 204.382 214.426 214.565 193.433 182.825 162.121 204.924 160.631 152.089
1.700 3.948 6.150 5.979 4.705 10.530 9.023 5.274 37.071 37.592 35.979 31.775 32.465 37.335 37.577 29.768 31.899 25.146 13.186
17.465 17.437 18.878 19.137 19.279 28.615 26.957 27.490 55.351 56.165 55.862 54.939 65.851 66.948 66.266 63,153 87.248 83.397 90.056
156.252 165.822 195.876 206.194 216.290 216.233 164.987 155.329 183.328 194.015 204.480 214.640 214.636 193.406 182.961 161.599 204.934 160.432 152.088
1.690 3.778 6.687 6.363 4.710 10.634 8.770 5.377 37.236 38.318 36.515 31.889 32.269 37.128 37.448 29.698 31.974 25.525 13.182
R M S error
x-x'
y-y'
z-z'
-0.002 0.196 -0.275 -0.339 0.001 -0.730 -0.160 -0.272 0.013 -0.086 -0.033 -0.062 -0.219 -0.018 0.056 0.179 0.006 0.161 0.000
0,001 0,162 0.186 -0.064 0,000 -0,043 0,166 0,248 -0,020 -0,140 -0,098 -0,214 -0,071 0.027 -0.136 0.522 -0,010 0.199 0.001
0.010 0.170 0.537 -0,384 -0.005 -0.104 0.253 -0.103 -0.165 -0.726 -0.536 -0.114 0.196 0.227 0.129 0,070 -0.075 -0.379 0,004
0.227
0.173
0.296
TABLE 2 Strainvaluesat severalpointsobtainedby square-gridanalysistheory PN
~
f~
f-~m
16 17 18 19 22 23 24 25 26
0.1746 0.1398 0.1436 0.1886 0.1641 0.1663 0.1414 0.1475 0.1978
0.1664 0.1202 0.1438 0.1925 0.1639 0.1423 0.1434 0.1444 0.2313
0.0082 0.0196 -0.0002 -0.0039 0.0002 0.0240 -0.0020 -0.0031 0.0335
R M S error
0.0156
mation of coordinates, square-grid analysis theory is applied to obtain the strains. Some results of the strain analysis are given in Table 2, where ~ is the equivalent strain at each node calculated from the precisely measured threedimensional coordinates, and em is the mean value of the equivalent strains
309
calculated from image coordinates obtained in two measurements. It can be seen that the results calculated from the nodal coordinates measured with the conventional device and those measured by means of the image processing device are quite close to each other, but that the measuring efficiency of the latter is much higher. The equivalent strain is an important parameter, the values of which are slightly different in the pure homogeneous deformation and in the real one [5]. In Table 2, only part of the results are listed. 5. Conclusions
It has been shown, in some detail, how the proposed image processing technique can be applied to strain measurement in sheet metal forming. Compared with other, conventional, measuring methods, this technique is powerful in digitizing a gridded surface. The results in the experimental evaluation indicate that the combination of strain analysis theory with the computerized vision system provides an effective means of strain measurement. The effort of measuring does not grow with the number of points, and the accuracy can be improved by increasing the number of images taken from extra camera positions. Once a gridded surface has been digitized and the strains at nodal points have been calculated, the values of el and e2 may be plotted in graphical forms, for instance, a bivariable strain distribution with a forming limit diagram superimposed can provide an intuitional graphic tool for investigating the safety of a forming operation. A very important factor affecting the accuracy is the choice of control points and the parallactic angle. It is suggested to maintain a parallactic angle about 60 to 110 degrees and also that there should be a certain amount of redundancy between the image pair, in order to obtain results of sufficient accuracy and reliability. Acknowledgements The authors would like to thank the Swedish Board of Technical Development (STU) for the financial support of this work. We are also grateful to Wei-Xing Wang and Stefan Dahlhielm, IPACS Company, for their valuable help in the image technique.
References 1 2 3 4
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