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Development of straightness measurement technique using the profile matching method
Int. J. Maeh.ToolsManufagt.Vol.37, No. 2, pp. 135-147,1997 Copyright© 1996Publi.~l by El~vierScienceLtd Printedin GreatBritain.All right~re,fred 0~90-6955/97517.00+ .00
Pergamon
PII: S0890-.6~5(96)00045-4 DEVELOPMENT OF STRAIGHTNESS MEASUREMENT TECHNIQUE USING THE PROFILE MATCHING METHOD H.J. PAHK,t J.S. PARKt and I. YEOt (Received 16 October 1995; in final form 16 April 1996) Abstract--The property of straightness is one of fundamental geometric tolerances to be strictly controlled for guideways of machine tools and measuring machines. Straightness measurement for long guideways is usually difficult to perform, and it requires additional equipment or special treatment with limited applications. In this paper, a new approach is proposed using the profile matching technique for the long guideways, which can be applied to most straightness measurements. An edge of relatively short length is located along a divided section of a long guideway, and the local straightness measurement is performed. The edge is then moved to the next section with several overlap points. After the local straightness profile is measured for every section along the long guideway with overlap, the global straightness profile is constructed using the profile matching technique based on the least-squares method. The proposed technique is numerically tested for two cases of known global straightness profile: arc profile and irregular profile, with and without random error intervention, respectively. The developed technique has been practically applied to a vertical milling machine of the knee type, and demonstrated a good performance. Thus the accuracy and efficiency of the proposed method are demonstrated, and show great potential for a variety of applications for most straightness measurement cases, including straight edges, laser optics and angular measurement equipments. Copyright © 1996 Published by Elsevier Science Ltd
1. INTRODUCTION
The straightness tolerance is one of fundamental geometric tolerances to be strictly controlled for guideways of machine tools and measuring machines. Several methods have been usually used for straightness measurements: laser interferometer with straightness optics, straight edge with displacement sensors such as LVDT (linear variable displacement transducer) and gap sensor, and incremental angular measurement using autocollimator or angular measurement optics, etc. [1, 2]. For the method using the straight edge, linear displacement sensors measure the deviation between the guideway and the edge, and it can give an accurate straightness profile when combined with the reversal technique for removing errors of the reference edge [1, 2]. For long guideways, however, the long straight edge was hardly ever used, because the required long straight edge is usually difficult to manufacture and is prone to deformation due to gravity. Burdekin [ 1] mentioned an overlap technique performing angle measurements, in which a long guideway is divided into several sections, and the local straightness is measured with a short edge along each section, Angle measurement is also performed using a precision level at each edge setup, and then it is combined to give the global straightness profile for the long guideway. The overlap technique with angle measurement, however, has been also limited to the vertical straightness measurement along the horizontal axis due to the limited direction of the level operation. In this paper, a new approach is proposed using the profile matching technique for the long guideways, which is applicable to vertical and horizontal straightness measurements without performing angular measurements. An edge of relatively short length is located along a divided section of a long guideway, and the local straightness measurement is performed. The edge is then moved to the next section with several overlap points. After the local straightness profile is measured for every section along the long guideway, the global straightness profile is constructed using the profile matching technique based on the least-straightness method. The proposed technique is numerically tested for two cases of known global straightness profiles: arc profile and irregular profile, and those tDepartment of Mechanical Design and Production Engineering, Seoul National University, Seoul, Korea, 151-742. 135
136
H.J. Pahk et
al.
profiles with and without random error intervention, respectively. The developed technique has been practically applied to a vertical milling machine of moving table type, and demonstrated a good performance. Thus the accuracy and efficiency of the proposed method are demonstrated, and showed great potential for a variety of applications for most straightness measurements of long guideways, including laser optics and angular measurement equipments.
2. STRAIGHTNESSMEASUREMENTTECHNIQUEUSING STRAIGHTEDGES Straight edges are usually made of steel or granite. The edge sides are used as the reference surface for straightness measurement, where the straightness error can be evaluated by a comparison between a machine axis and the reference surface. The straightness profile of a reference straight edge is also to be measured and can be compensated for precise straightness measurement. The reversal technique is generally used for evaluation of the straightness profile of the reference surface [2].
2.1. Reversal technique Let X be a nominal position along an edge and the straightness profile of a machine axis is to be measured; S(X), Y(X) are the straightness profile along the edge and the machine axis, respectively. When the measurement setup is as in the left-hand side of Fig. 1, the measured profile would be the difference resulting from S(X)-Y(X); when the edge is reversed as in the right-hand side of Fig. 1, the measured profile would be the difference resulting from -S(X)-Y(X). Thus the edge straightness profile S(X) and the straightness profile of machine axis Y(X) can be evaluated. That is, forward measurement profile = S(X)-Y(X), reverse measurement profile = - S ( X ) - Y(X). Thus Y(X) = -(forward measurement profile + reverse measurement profile)/2,
(1)
S(X) = (forward measurement-reverse measurement profile)/2
(2)
Therefore the straightness profile of the edge and the machine axis can be evaluated simultaneously, and the edge profile is stored and can be used for profile compensation for accurate straightness measurement. 2.2. Straightness evaluation methods After the straightness profiles are measured, appropriate straightness evaluation methods are to be applied. There are several ways for evaluation of straightness tolerance, and
Bright edge s(x) Guideway
Y(X)
Measuredprofiles(x)_-~((X)7 -S(X)-Y(X)/ Fig. 1. Reversal technique.
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currently three methods are implemented in this system depending on the reference straight line: end-points straightness, least-squares straightness and minimum-zone straightness. Since the straightness profile is evaluated as the deviation of a given profile from a reference straight line, the measured profile is adjusted to give the straightness profile. Let Yj be a global profile, and 8Yj be a straightness profile at Xj position; the straightness profile 8Yj can be evaluated as 6Yj = YFAXj,
(3)
and straightness tolerance,
ST, is
ST = max (~Yj)-min (SYj),
(4)
where A is the best-fit slope of the reference straight line for the measured profile. There are currently three methods for the evaluation of the best-fit slope, end-points fit, leastsquares fit and minimum-zone fit. 2.2.1. End-pointsfit. This method has been used traditionally on the shop floor, and the constant A of Equation (3) is evaluated from a reference straight line passing the two end-points of the measured profile. That is,
A = (YM- Y,)I(XM-XI),
(5)
where (X~, Y0, (XM, YM) are the end-point data of the measured profile, respectively. The end-points fit can give the slope easily, but it has the disadvantage that the measured data on the middle part are not used for the evaluation of the slope A. 2.2.2. Least-squaresfit. As a way for overcoming the disadvantage of the end-points fit, the slope A can be evaluated using the points on the middle part as well as the end points. The slope A of Equation (3) can be evaluated using the least-squares technique, which is
A= ( N ~XjYj- ~Xj~Yj) N ~ ~ - ( ~X/) j=l
j=l
j=l
""
j=l
'j=l
"
2),
(6)
"
where N is the total number of data points on the measured profile. 2.2.3. Minimum-zonefit. International standards (ISO) [4] and many national standards define the straightness tolerance based on the minimum-zone tolerance, in which the straightness tolerance is evaluated as the minimum distance between the two parallel straight lines enclosing the measured datum. The slope A can be determined so that the straightness tolerance [Equation (4)] is minimum, and there are several methods for minimum-zone fit. As a practical method, Burdekin and Pahk [3] proposed the enclose tilt technique, where the slope can be evaluated using Equation (7) if three enclosing points, P1, P2 and P3 are found with alternate signs.
A = (Yea- Ym)l(Xea-XeO•
(7)
Therefore the straightness profile and the straightness tolerance can be evaluated from Equations (3) and (4). 3. PROFILEMATCHINGTECHNIQUEFOR STRAIGHTNESSPROFILE
3.1. Concept of profile matching The concept of profile matching for straightness measurement can be explained as follows. Consider a long guideway of arc profile for the span length L, as shown in Fig. 2,
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H.J. Pahk et al. L
[--
Global straightness D A
B
C ~ D
A
Local straightness with overlap
B Reconstructed global straightness
Fig. 2. Concept of profile matching for straightness measurement.
where the global straightness profile is to be measured. In order to measure the global straightness profile, the guideway can be divided into three local sections: OAB, ABC and BCD, whose length is l, allowing overlap. Assume AB is the overlap region between the OAB and ABC local sections, and BC is the overlap region between the ABC and BCD local sections. When the straightness profile is measured along the OAB section with a straight edge of length l, the measured local straightness profile is like the OAB profile in the second part of Fig. 2. Similarly, when the straightness profile is measured for the ABC local section, the measured local straightness profile is like the ABC profile in the second part of Fig. 2. The overlap region, AB, is the physically identical region in the guideway, and thus the AB region must show the identical profile in the OAB and ABC profiles. Therefore the profile of ABC local section can be transformed such that the AB region of OAB section and the AB region of ABC section can be matched. A best profile matching method can be designed such that the AB region of the OAB section and the AB region of the ABC section can be matched. Thus the global profile OABC can be constructed as in the third part of Fig. 2. Similarly, the local profile BCD can be transformed such that the overlap BC region can be matched with that of the ABC profile, and thus the global profile OABCD can be constructed. In this paper, the profile matching technique method is proposed using the least-squares method, and a detailed description is given as follows. 3.2. Least-squares method In Fig. 3(a), consider a straight edge located at Oi position (i=1, 2, 3...) along the guideway, and define a local coordinate system, O~iY~, where O~Xi indicates a direction parallel to the edge surface, and II,-indicates a local straightness profile. When the edge is moved to the next position O~+1,the local profile is similarly measured as Y;+l, and the overlap region can be defined as shown in Fig. 3(b). The profile matching can be performed using the data on the overlap region. Let Yi. ,, Yi+,. k (k=l,2...N) be the kth point data on the overlap region of Yi, Y~+~profiles, respectively. As the straightness profiles are physically identical for the overlap region, the difference in the profiles on the overlap region is due to the differences in the setup of the edge at the Oi, O~+l position, respectively. Thus Yi.k-Yi+ I., = AiXk + Bi for k = 1,2...N,
(8)
where Ai, Bi are coefficients for defining the slope and offset of the edge at the i+1 locations, and Xk is the distance from the beginning point to the kth point on the overlap region. The A~, Bi coefficients can be evaluated based on the least-squares technique as follows.
Development of straightness measurementtechnique
139
(a) Yi
A Yi+l
I o+/x Oi
~-~ Xi+l
(b)
YI
Yi
01
0 i
Y ÷1
Oi+ I
Yi + I,k
_
Xl Xi X i + l
Overlap -1 (c) Yi,k
Yi + l,k Ai Xk + Bi
Fig. 3. (a) Global and local coordinate system; (b) measured profile in the local coordinate system;(c) profile matching on the overlap region. The sum of squares, Ei, between the Y; and Y;+~ profiles in the overlap region, is N
(9)
Ei = ~ (Yi,~-Yi + l.k-AiXk-Bi) 2. k=l
Applying the variational principle, and assigning ~Eil~A i -- O, igEJ3Bg = O, N
N
(10)
A, ~ X~k + Bi ~ X~ = ZX~(Yi,k- Yg+ l.k), k=l
k=l
N
(11)
Ai E Xk + BiN = ~(Yi.k- Yi + i.k). k=l
Therefore the coefficients Ai and Bi can be calculated from Equations (10) and (11), that is,
Ai =
Xk(Yz,k--Yi+ i,k)-I
B,=
Xk k=l
(Yi.t-Yi+ J.k I N k=l
;
k=i
2 , k=l
(12)
;
( ~dX~t~(Y,.k-Y,+,.D-~'~Xt(Y,.k-Y,+I.D N N ~iXkt/ lN ~ X~k- t Nk~lXkl.21 k=l "k=l
k=l
=
=1
=
(13)
Thus the coefficients A; and Bi can be evaluated, and the relationship between the profiles Y; and Yi+~ is constructed. From Equation (8) the converted profile (conversion from Y;+l to Y;), CYi+l, is
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(14)
CYi+ j = Yi+ l + A i X + Bi,
where X is the distance from Oi÷.. As the Yi.~ profile is converted to the Yi profile in the form of CYi÷. the augmented Yi profile, AYi, can be constructed as the sum of existing profile Y~and the converted profile CY~+I. Thus
(15)
AYi = Yi + C Y I . l for i = 1,2,3...
Equations (14) and (15) provide a useful relationship for the profile matching. In this way, every local profile can be converted to the previous local profile, and thus transformed down to the first local profile, i.e. a global profile. After all local profiles are transformed to the first local profile in the global coordinate system, appropriate straightness evaluation methods can be applied for the straightness profile evaluation. The developed profile matching technique has been implemented around a microcomputer environment, and the flowchart of the computer program for the developed technique is shown in Fig. 4. 4. STRAIGHTNESS M E A S U R E M E N T SIMULATION
In order to test the proposed straightness measurement technique using the profile matching, two known straightness profiles are used: arc profile and irregular profile. The
Input the local profile data. Yj for j = 1,2,3..
Assign i = total number of local profile - 1
Retrieve Yi,k,Yi + l,k on the overlap region for k = 1,2,3 ......
Caculate the slope A i and offset B i for the overlap region
I ?ransform the Yi + I profile data into the profile Yi profile data
No
Fig. 4. Flowchart for computer program of profile matching.
Development of straightness measurement technique
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known global profiles are divided into several local sections, and the local profiles are simulated from the global profiles. For the local profile data, the proposed profile matching technique is used, and the global profiles are reconstructed from the local profiles. The reconstructed profiles are then compared with the original global profiles. Random scattering is added to the local profiles--this is to simulate random errors in the practical measurement procedures. The effect of the size of the overlap region is also considered in terms of the number of data points on the overlap region, and the deviation between the original and the reconstructed profile is observed as the number of overlap points changes. 4.1. Arc profile of straightness A concave straightness profile of 10 p~m magnitude over 1200 mm length is considered for practical application, and the profile is divided into several local profiles. The local profiles are then combined to give the global straightness profile, using the proposed profile matching technique. In order to simulate the practical measurement cases, random errors of 0.1 /~m maximum magnitude (which is typical for conventional mechanical measurement) are added to the local profiles. The local profiles with random error disturbance are then combined to give the global profile using the profile matching technique, and the effect of size of the overlap region is also investigated. When no random errors are involved, the reconstructed profiles are found as identical to the original profiles. Figure 5(a) shows the local arc profiles, where 0.1 /~m random errors are intervened, and four points are considered for the overlap region. Figure 5(b) shows the reconstructed and original profiles, demonstrating very good similarity between the two profiles in the case of random error intervention. In order to investigate the effect of overlap size, 30 independent measurement sets are simulated for the local arc profiles with varying random error distribution, and they are combined to give corresponding global profiles, allowing various number of data points on the overlap region. The maximum deviation between the original and reconstructed profiles is recorded for the number of overlap points. The mean deviation between the profiles is also recorded for the number of overlap points. Figure 5(c) shows the maximum and mean deviation when the number of overlap points is between two and six. The (a)
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maximum deviation approaches 0.2 /xm, and the mean deviation approaches about 0.06 /xm when the number of overlap points is four or more. 4.2. Irregular profile of straightness An irregular profile of 10 /xm straightness tolerance over 1200 mm length is also considered for simulation. The divided local profiles are simulated from the global profile, and the identical reconstructed profile is obtained using the proposed profile matching technique. When no random errors are involved, the reconstructed profiles are found identical to the original profiles. In order to simulate the practical measurement cases, random errors of 0.1 /xm maximum magnitude (which is typical for the conventional straightness
Development of straightness measurementtechnique
143
measurement) are added to the local profiles. The local profiles with random error disturbance are then combined to give the global profile using the profile matching technique, and the effect of the size of the overlap region is also investigated. Figure 6(a) shows the local irregular profiles, and Fig. 6(b) shows the reconstructed and original profiles, demonstrating negligible deviation between the profiles for the case of four points overlapping.
(a)
LOACL PROFILE(IRREGULAR) 30
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PROFILE
1000
100
....
10
=.
m-~Wl
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"
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v
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Continued.
In order to investigate the effect of overlap size, 30 independent measurement sets are simulated for the local irregular profiles with varying random error distribution, and they are combined to give corresponding global profiles, allowing various number of data points on the overlap region. The maximum deviation between the original and reconstructed profiles is recorded for the number of overlap points. The mean deviation between the profiles is also recorded for the number of overlap points. Figure 6(c) shows the maximum and mean deviation between the original and reconstructed profiles when the number of overlap points is between two and six. The maximum deviation approaches about 0.2 /xm, and the mean deviation approaches to 0.06 /xm as the number of overlap points is three or more. 5. PRACTICALAPPLICATIONTO A CNC MILLINGMACHINEOF MOVINGTABLETYPE The developed method has been applied to a practical CNC milling machine of moving table type, whose working dimension was 600x450x300 mm, and the horizontal straightness error is to be evaluated. A LVDT (linear variable displacement transducer) sensor was mounted at the spindle and the straight edge was aligned on the table along the horizontal direction. In order to obtain the global straightness profile and the edge profile, the machine was measured back and forth for the 450 mm travel along the horizontal direction, while the LVDT was measuring the deviation between the machine profile and the edge profile. Figure 7(a) shows the measured data from the LVDT sensor for the forward and backward directions, where those blank square dots, small and large filled square dots indicate the repetitive measurement results. Applying the reversal technique, the edge profile and the machine profile have been obtained, giving convex profile of magnitude 7.5 /~m for the straight edge and a concave profile of magnitude 26.4 /xm for the machine profile as shown in Fig. 7(b) and (c). The calibrated straightness data of the edge profile was stored in the computer memory and the local profile measurement performed; the length of local profile was 120 mm (12 steps in 10 mm unit steps). The local profile was calculated as the measured profile minus the edge profile. Figure 8(a) shows the local profile, and the global profile was generated from the local profiles using the proposed profile matching technique. Figure 8(b) shows the reconstructed global straight-
145
Development of straightness measurement technique (a)
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_t
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-15 -20
10 9 8 7 5 4 3 2
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Fig. 7.
Continued.
Local P r o f i l e 2.5
2
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1.5
0.5
-0.5
[
"-
""
-1 Distance Fig. 8. (a) Local straightness profile; (b) reconstructed and original profiles; (c) maximum and mean deviation versus number of overlap points.
ness profile and the original global profile when the number of overlap points is seven, where the reconstructed profile shows 26.7 /~m straightness error (concave), demonstrating closeness between the reconstructed profile and the original profile. Figure 8(c) shows the maximum and mean deviations between the two profiles when the number of overlap points increase. The deviation data were remarkably reduced when the number of overlap points is five and more. Therefore the proposed technique has been verified through the experimental work. 6. CONCLUSIONS
1. An efficient method for straightness measurement is proposed, using the profile matching technique based on the least-squares method for the overlap regions of local straightness profiles. The proposed method is found to be useful for the global straightness evaluation when local straightness profiles are measured, and the accuracy and efficiency are demonstrated from a numerical simulation for the straightness measurement data. 2. From the numerical simulation of random error scattering, a general trend is found in that the reconstructed profile is much closer to the original profile when the number of data points in the overlap regions increases.
Development of straightness measurement technique (b)
147
7 Point Overlap
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Fig. 8. Continued. 3. The proposed technique can be applied for most straightness measurement situations using the straight edge, laser interferometer, angle measurement equipments, etc. Therefore the global straightness profile can be obtained for the long object by the proposed technique, using local straightness profiles measured by equipment of relatively short working ranges. 4. The developed method has been practically applied to a CNC milling machine of moving table type, and showed good agreement between the original profile and the reconstructed profile. Thus the efficiency of the proposed technique has been demonstrated. Acknowledgement--The authors are grateful to the NSDM-ERC of Pusan National University for the partial
funding to the research. REFERENCES [1] M. Burdekin, Instrumentation and techniques for the evaluation of machine tool errors, machine tool accuracy. Technology of Machine Tools, Vol. 5, p. 9.7.1. University of California (1980). [2] R. Hoeken, Machine tool metrology,ASPE Tutorial, ASPE Annual Meeting, Florida (1992). [3] M. Burdekin and H. Pahk, The application of a micro computer to the on-line calibration of the flatness of engineering surfaces, Proc. lnstn Mech. Engrs B 203, 127 (1989). [4] Applied metrology--limits, fits and surface properties,/SO Standards Handbook, P.46, International Organization for Standardization, Switzerland (1988).
Pergamon
PII: S0890-.6~5(96)00045-4 DEVELOPMENT OF STRAIGHTNESS MEASUREMENT TECHNIQUE USING THE PROFILE MATCHING METHOD H.J. PAHK,t J.S. PARKt and I. YEOt (Received 16 October 1995; in final form 16 April 1996) Abstract--The property of straightness is one of fundamental geometric tolerances to be strictly controlled for guideways of machine tools and measuring machines. Straightness measurement for long guideways is usually difficult to perform, and it requires additional equipment or special treatment with limited applications. In this paper, a new approach is proposed using the profile matching technique for the long guideways, which can be applied to most straightness measurements. An edge of relatively short length is located along a divided section of a long guideway, and the local straightness measurement is performed. The edge is then moved to the next section with several overlap points. After the local straightness profile is measured for every section along the long guideway with overlap, the global straightness profile is constructed using the profile matching technique based on the least-squares method. The proposed technique is numerically tested for two cases of known global straightness profile: arc profile and irregular profile, with and without random error intervention, respectively. The developed technique has been practically applied to a vertical milling machine of the knee type, and demonstrated a good performance. Thus the accuracy and efficiency of the proposed method are demonstrated, and show great potential for a variety of applications for most straightness measurement cases, including straight edges, laser optics and angular measurement equipments. Copyright © 1996 Published by Elsevier Science Ltd
1. INTRODUCTION
The straightness tolerance is one of fundamental geometric tolerances to be strictly controlled for guideways of machine tools and measuring machines. Several methods have been usually used for straightness measurements: laser interferometer with straightness optics, straight edge with displacement sensors such as LVDT (linear variable displacement transducer) and gap sensor, and incremental angular measurement using autocollimator or angular measurement optics, etc. [1, 2]. For the method using the straight edge, linear displacement sensors measure the deviation between the guideway and the edge, and it can give an accurate straightness profile when combined with the reversal technique for removing errors of the reference edge [1, 2]. For long guideways, however, the long straight edge was hardly ever used, because the required long straight edge is usually difficult to manufacture and is prone to deformation due to gravity. Burdekin [ 1] mentioned an overlap technique performing angle measurements, in which a long guideway is divided into several sections, and the local straightness is measured with a short edge along each section, Angle measurement is also performed using a precision level at each edge setup, and then it is combined to give the global straightness profile for the long guideway. The overlap technique with angle measurement, however, has been also limited to the vertical straightness measurement along the horizontal axis due to the limited direction of the level operation. In this paper, a new approach is proposed using the profile matching technique for the long guideways, which is applicable to vertical and horizontal straightness measurements without performing angular measurements. An edge of relatively short length is located along a divided section of a long guideway, and the local straightness measurement is performed. The edge is then moved to the next section with several overlap points. After the local straightness profile is measured for every section along the long guideway, the global straightness profile is constructed using the profile matching technique based on the least-straightness method. The proposed technique is numerically tested for two cases of known global straightness profiles: arc profile and irregular profile, and those tDepartment of Mechanical Design and Production Engineering, Seoul National University, Seoul, Korea, 151-742. 135
136
H.J. Pahk et
al.
profiles with and without random error intervention, respectively. The developed technique has been practically applied to a vertical milling machine of moving table type, and demonstrated a good performance. Thus the accuracy and efficiency of the proposed method are demonstrated, and showed great potential for a variety of applications for most straightness measurements of long guideways, including laser optics and angular measurement equipments.
2. STRAIGHTNESSMEASUREMENTTECHNIQUEUSING STRAIGHTEDGES Straight edges are usually made of steel or granite. The edge sides are used as the reference surface for straightness measurement, where the straightness error can be evaluated by a comparison between a machine axis and the reference surface. The straightness profile of a reference straight edge is also to be measured and can be compensated for precise straightness measurement. The reversal technique is generally used for evaluation of the straightness profile of the reference surface [2].
2.1. Reversal technique Let X be a nominal position along an edge and the straightness profile of a machine axis is to be measured; S(X), Y(X) are the straightness profile along the edge and the machine axis, respectively. When the measurement setup is as in the left-hand side of Fig. 1, the measured profile would be the difference resulting from S(X)-Y(X); when the edge is reversed as in the right-hand side of Fig. 1, the measured profile would be the difference resulting from -S(X)-Y(X). Thus the edge straightness profile S(X) and the straightness profile of machine axis Y(X) can be evaluated. That is, forward measurement profile = S(X)-Y(X), reverse measurement profile = - S ( X ) - Y(X). Thus Y(X) = -(forward measurement profile + reverse measurement profile)/2,
(1)
S(X) = (forward measurement-reverse measurement profile)/2
(2)
Therefore the straightness profile of the edge and the machine axis can be evaluated simultaneously, and the edge profile is stored and can be used for profile compensation for accurate straightness measurement. 2.2. Straightness evaluation methods After the straightness profiles are measured, appropriate straightness evaluation methods are to be applied. There are several ways for evaluation of straightness tolerance, and
Bright edge s(x) Guideway
Y(X)
Measuredprofiles(x)_-~((X)7 -S(X)-Y(X)/ Fig. 1. Reversal technique.
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currently three methods are implemented in this system depending on the reference straight line: end-points straightness, least-squares straightness and minimum-zone straightness. Since the straightness profile is evaluated as the deviation of a given profile from a reference straight line, the measured profile is adjusted to give the straightness profile. Let Yj be a global profile, and 8Yj be a straightness profile at Xj position; the straightness profile 8Yj can be evaluated as 6Yj = YFAXj,
(3)
and straightness tolerance,
ST, is
ST = max (~Yj)-min (SYj),
(4)
where A is the best-fit slope of the reference straight line for the measured profile. There are currently three methods for the evaluation of the best-fit slope, end-points fit, leastsquares fit and minimum-zone fit. 2.2.1. End-pointsfit. This method has been used traditionally on the shop floor, and the constant A of Equation (3) is evaluated from a reference straight line passing the two end-points of the measured profile. That is,
A = (YM- Y,)I(XM-XI),
(5)
where (X~, Y0, (XM, YM) are the end-point data of the measured profile, respectively. The end-points fit can give the slope easily, but it has the disadvantage that the measured data on the middle part are not used for the evaluation of the slope A. 2.2.2. Least-squaresfit. As a way for overcoming the disadvantage of the end-points fit, the slope A can be evaluated using the points on the middle part as well as the end points. The slope A of Equation (3) can be evaluated using the least-squares technique, which is
A= ( N ~XjYj- ~Xj~Yj) N ~ ~ - ( ~X/) j=l
j=l
j=l
""
j=l
'j=l
"
2),
(6)
"
where N is the total number of data points on the measured profile. 2.2.3. Minimum-zonefit. International standards (ISO) [4] and many national standards define the straightness tolerance based on the minimum-zone tolerance, in which the straightness tolerance is evaluated as the minimum distance between the two parallel straight lines enclosing the measured datum. The slope A can be determined so that the straightness tolerance [Equation (4)] is minimum, and there are several methods for minimum-zone fit. As a practical method, Burdekin and Pahk [3] proposed the enclose tilt technique, where the slope can be evaluated using Equation (7) if three enclosing points, P1, P2 and P3 are found with alternate signs.
A = (Yea- Ym)l(Xea-XeO•
(7)
Therefore the straightness profile and the straightness tolerance can be evaluated from Equations (3) and (4). 3. PROFILEMATCHINGTECHNIQUEFOR STRAIGHTNESSPROFILE
3.1. Concept of profile matching The concept of profile matching for straightness measurement can be explained as follows. Consider a long guideway of arc profile for the span length L, as shown in Fig. 2,
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[--
Global straightness D A
B
C ~ D
A
Local straightness with overlap
B Reconstructed global straightness
Fig. 2. Concept of profile matching for straightness measurement.
where the global straightness profile is to be measured. In order to measure the global straightness profile, the guideway can be divided into three local sections: OAB, ABC and BCD, whose length is l, allowing overlap. Assume AB is the overlap region between the OAB and ABC local sections, and BC is the overlap region between the ABC and BCD local sections. When the straightness profile is measured along the OAB section with a straight edge of length l, the measured local straightness profile is like the OAB profile in the second part of Fig. 2. Similarly, when the straightness profile is measured for the ABC local section, the measured local straightness profile is like the ABC profile in the second part of Fig. 2. The overlap region, AB, is the physically identical region in the guideway, and thus the AB region must show the identical profile in the OAB and ABC profiles. Therefore the profile of ABC local section can be transformed such that the AB region of OAB section and the AB region of ABC section can be matched. A best profile matching method can be designed such that the AB region of the OAB section and the AB region of the ABC section can be matched. Thus the global profile OABC can be constructed as in the third part of Fig. 2. Similarly, the local profile BCD can be transformed such that the overlap BC region can be matched with that of the ABC profile, and thus the global profile OABCD can be constructed. In this paper, the profile matching technique method is proposed using the least-squares method, and a detailed description is given as follows. 3.2. Least-squares method In Fig. 3(a), consider a straight edge located at Oi position (i=1, 2, 3...) along the guideway, and define a local coordinate system, O~iY~, where O~Xi indicates a direction parallel to the edge surface, and II,-indicates a local straightness profile. When the edge is moved to the next position O~+1,the local profile is similarly measured as Y;+l, and the overlap region can be defined as shown in Fig. 3(b). The profile matching can be performed using the data on the overlap region. Let Yi. ,, Yi+,. k (k=l,2...N) be the kth point data on the overlap region of Yi, Y~+~profiles, respectively. As the straightness profiles are physically identical for the overlap region, the difference in the profiles on the overlap region is due to the differences in the setup of the edge at the Oi, O~+l position, respectively. Thus Yi.k-Yi+ I., = AiXk + Bi for k = 1,2...N,
(8)
where Ai, Bi are coefficients for defining the slope and offset of the edge at the i+1 locations, and Xk is the distance from the beginning point to the kth point on the overlap region. The A~, Bi coefficients can be evaluated based on the least-squares technique as follows.
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(a) Yi
A Yi+l
I o+/x Oi
~-~ Xi+l
(b)
YI
Yi
01
0 i
Y ÷1
Oi+ I
Yi + I,k
_
Xl Xi X i + l
Overlap -1 (c) Yi,k
Yi + l,k Ai Xk + Bi
Fig. 3. (a) Global and local coordinate system; (b) measured profile in the local coordinate system;(c) profile matching on the overlap region. The sum of squares, Ei, between the Y; and Y;+~ profiles in the overlap region, is N
(9)
Ei = ~ (Yi,~-Yi + l.k-AiXk-Bi) 2. k=l
Applying the variational principle, and assigning ~Eil~A i -- O, igEJ3Bg = O, N
N
(10)
A, ~ X~k + Bi ~ X~ = ZX~(Yi,k- Yg+ l.k), k=l
k=l
N
(11)
Ai E Xk + BiN = ~(Yi.k- Yi + i.k). k=l
Therefore the coefficients Ai and Bi can be calculated from Equations (10) and (11), that is,
Ai =
Xk(Yz,k--Yi+ i,k)-I
B,=
Xk k=l
(Yi.t-Yi+ J.k I N k=l
;
k=i
2 , k=l
(12)
;
( ~dX~t~(Y,.k-Y,+,.D-~'~Xt(Y,.k-Y,+I.D N N ~iXkt/ lN ~ X~k- t Nk~lXkl.21 k=l "k=l
k=l
=
=1
=
(13)
Thus the coefficients A; and Bi can be evaluated, and the relationship between the profiles Y; and Yi+~ is constructed. From Equation (8) the converted profile (conversion from Y;+l to Y;), CYi+l, is
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(14)
CYi+ j = Yi+ l + A i X + Bi,
where X is the distance from Oi÷.. As the Yi.~ profile is converted to the Yi profile in the form of CYi÷. the augmented Yi profile, AYi, can be constructed as the sum of existing profile Y~and the converted profile CY~+I. Thus
(15)
AYi = Yi + C Y I . l for i = 1,2,3...
Equations (14) and (15) provide a useful relationship for the profile matching. In this way, every local profile can be converted to the previous local profile, and thus transformed down to the first local profile, i.e. a global profile. After all local profiles are transformed to the first local profile in the global coordinate system, appropriate straightness evaluation methods can be applied for the straightness profile evaluation. The developed profile matching technique has been implemented around a microcomputer environment, and the flowchart of the computer program for the developed technique is shown in Fig. 4. 4. STRAIGHTNESS M E A S U R E M E N T SIMULATION
In order to test the proposed straightness measurement technique using the profile matching, two known straightness profiles are used: arc profile and irregular profile. The
Input the local profile data. Yj for j = 1,2,3..
Assign i = total number of local profile - 1
Retrieve Yi,k,Yi + l,k on the overlap region for k = 1,2,3 ......
Caculate the slope A i and offset B i for the overlap region
I ?ransform the Yi + I profile data into the profile Yi profile data
No
Fig. 4. Flowchart for computer program of profile matching.
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known global profiles are divided into several local sections, and the local profiles are simulated from the global profiles. For the local profile data, the proposed profile matching technique is used, and the global profiles are reconstructed from the local profiles. The reconstructed profiles are then compared with the original global profiles. Random scattering is added to the local profiles--this is to simulate random errors in the practical measurement procedures. The effect of the size of the overlap region is also considered in terms of the number of data points on the overlap region, and the deviation between the original and the reconstructed profile is observed as the number of overlap points changes. 4.1. Arc profile of straightness A concave straightness profile of 10 p~m magnitude over 1200 mm length is considered for practical application, and the profile is divided into several local profiles. The local profiles are then combined to give the global straightness profile, using the proposed profile matching technique. In order to simulate the practical measurement cases, random errors of 0.1 /~m maximum magnitude (which is typical for conventional mechanical measurement) are added to the local profiles. The local profiles with random error disturbance are then combined to give the global profile using the profile matching technique, and the effect of size of the overlap region is also investigated. When no random errors are involved, the reconstructed profiles are found as identical to the original profiles. Figure 5(a) shows the local arc profiles, where 0.1 /~m random errors are intervened, and four points are considered for the overlap region. Figure 5(b) shows the reconstructed and original profiles, demonstrating very good similarity between the two profiles in the case of random error intervention. In order to investigate the effect of overlap size, 30 independent measurement sets are simulated for the local arc profiles with varying random error distribution, and they are combined to give corresponding global profiles, allowing various number of data points on the overlap region. The maximum deviation between the original and reconstructed profiles is recorded for the number of overlap points. The mean deviation between the profiles is also recorded for the number of overlap points. Figure 5(c) shows the maximum and mean deviation when the number of overlap points is between two and six. The (a)
LOCAL PROFILE(ARC) 15
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ARC PROFILE 15 10
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maximum deviation approaches 0.2 /xm, and the mean deviation approaches about 0.06 /xm when the number of overlap points is four or more. 4.2. Irregular profile of straightness An irregular profile of 10 /xm straightness tolerance over 1200 mm length is also considered for simulation. The divided local profiles are simulated from the global profile, and the identical reconstructed profile is obtained using the proposed profile matching technique. When no random errors are involved, the reconstructed profiles are found identical to the original profiles. In order to simulate the practical measurement cases, random errors of 0.1 /xm maximum magnitude (which is typical for the conventional straightness
Development of straightness measurementtechnique
143
measurement) are added to the local profiles. The local profiles with random error disturbance are then combined to give the global profile using the profile matching technique, and the effect of the size of the overlap region is also investigated. Figure 6(a) shows the local irregular profiles, and Fig. 6(b) shows the reconstructed and original profiles, demonstrating negligible deviation between the profiles for the case of four points overlapping.
(a)
LOACL PROFILE(IRREGULAR) 30
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H.J. Pahk et
144
(c)
al.
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PROFILE
1000
100
....
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=.
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Fig. 6.
Continued.
In order to investigate the effect of overlap size, 30 independent measurement sets are simulated for the local irregular profiles with varying random error distribution, and they are combined to give corresponding global profiles, allowing various number of data points on the overlap region. The maximum deviation between the original and reconstructed profiles is recorded for the number of overlap points. The mean deviation between the profiles is also recorded for the number of overlap points. Figure 6(c) shows the maximum and mean deviation between the original and reconstructed profiles when the number of overlap points is between two and six. The maximum deviation approaches about 0.2 /xm, and the mean deviation approaches to 0.06 /xm as the number of overlap points is three or more. 5. PRACTICALAPPLICATIONTO A CNC MILLINGMACHINEOF MOVINGTABLETYPE The developed method has been applied to a practical CNC milling machine of moving table type, whose working dimension was 600x450x300 mm, and the horizontal straightness error is to be evaluated. A LVDT (linear variable displacement transducer) sensor was mounted at the spindle and the straight edge was aligned on the table along the horizontal direction. In order to obtain the global straightness profile and the edge profile, the machine was measured back and forth for the 450 mm travel along the horizontal direction, while the LVDT was measuring the deviation between the machine profile and the edge profile. Figure 7(a) shows the measured data from the LVDT sensor for the forward and backward directions, where those blank square dots, small and large filled square dots indicate the repetitive measurement results. Applying the reversal technique, the edge profile and the machine profile have been obtained, giving convex profile of magnitude 7.5 /~m for the straight edge and a concave profile of magnitude 26.4 /xm for the machine profile as shown in Fig. 7(b) and (c). The calibrated straightness data of the edge profile was stored in the computer memory and the local profile measurement performed; the length of local profile was 120 mm (12 steps in 10 mm unit steps). The local profile was calculated as the measured profile minus the edge profile. Figure 8(a) shows the local profile, and the global profile was generated from the local profiles using the proposed profile matching technique. Figure 8(b) shows the reconstructed global straight-
145
Development of straightness measurement technique (a)
Forward-m~sured
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-25
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(b)
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Prof i I e
"%~.
_t
i : ,S ................ ~
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-15 -20
10 9 8 7 5 4 3 2
Di s t a n c e Fig. 7. (a) Straightness measured data; (b) edge profile; (c) machine profile.
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Fig. 7.
Continued.
Local P r o f i l e 2.5
2
\
1.5
0.5
-0.5
[
"-
""
-1 Distance Fig. 8. (a) Local straightness profile; (b) reconstructed and original profiles; (c) maximum and mean deviation versus number of overlap points.
ness profile and the original global profile when the number of overlap points is seven, where the reconstructed profile shows 26.7 /~m straightness error (concave), demonstrating closeness between the reconstructed profile and the original profile. Figure 8(c) shows the maximum and mean deviations between the two profiles when the number of overlap points increase. The deviation data were remarkably reduced when the number of overlap points is five and more. Therefore the proposed technique has been verified through the experimental work. 6. CONCLUSIONS
1. An efficient method for straightness measurement is proposed, using the profile matching technique based on the least-squares method for the overlap regions of local straightness profiles. The proposed method is found to be useful for the global straightness evaluation when local straightness profiles are measured, and the accuracy and efficiency are demonstrated from a numerical simulation for the straightness measurement data. 2. From the numerical simulation of random error scattering, a general trend is found in that the reconstructed profile is much closer to the original profile when the number of data points in the overlap regions increases.
Development of straightness measurement technique (b)
147
7 Point Overlap
i i ,,r
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Fig. 8. Continued. 3. The proposed technique can be applied for most straightness measurement situations using the straight edge, laser interferometer, angle measurement equipments, etc. Therefore the global straightness profile can be obtained for the long object by the proposed technique, using local straightness profiles measured by equipment of relatively short working ranges. 4. The developed method has been practically applied to a CNC milling machine of moving table type, and showed good agreement between the original profile and the reconstructed profile. Thus the efficiency of the proposed technique has been demonstrated. Acknowledgement--The authors are grateful to the NSDM-ERC of Pusan National University for the partial
funding to the research. REFERENCES [1] M. Burdekin, Instrumentation and techniques for the evaluation of machine tool errors, machine tool accuracy. Technology of Machine Tools, Vol. 5, p. 9.7.1. University of California (1980). [2] R. Hoeken, Machine tool metrology,ASPE Tutorial, ASPE Annual Meeting, Florida (1992). [3] M. Burdekin and H. Pahk, The application of a micro computer to the on-line calibration of the flatness of engineering surfaces, Proc. lnstn Mech. Engrs B 203, 127 (1989). [4] Applied metrology--limits, fits and surface properties,/SO Standards Handbook, P.46, International Organization for Standardization, Switzerland (1988).