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Analytical estimation of measurement uncertainty in surface plate calibration by the Moody method using differential levels
Precision Engineering 27 (2003) 323–332
Short communication
Analytical estimation of measurement uncertainty in surface plate calibration by the Moody method using differential levels Joseph Drescher∗ UTC, Pratt & Whitney, 400 Main Street M/S 165-09, East Hartford, CT 06108, USA Received 22 March 2002; received in revised form 13 August 2002; accepted 18 December 2002
Abstract Surface plate quality influences the calibration of artifacts, instruments, working gauges and, by extension, the quality of manufactured parts. A statement of uncertainty for the surface plate calibration is needed for conformance testing and for estimating uncertainty in calibrations that rely on the surface plate. An analytical approach to the estimate of uncertainty for a common measurement method is given. The residual height at each position on the surface is expressed in terms of the measurement parameters and the measured angles. Uncertainty of each residual height value is then derived following the common methods for propagation of uncertainty. The maximum uncertainty is at the center of the surface and the calculated bounds on error compare well with values of closure from actual measurements. The uncertainty of height values is proportional to the spacing of the measurement positions and to the squareroot of the number of positions. Using data from many plates of various size and quality, a generalized uncertainty in flatness for the measurement method was estimated. The derivations also allow calculations for specific applications. Finally, a method is proposed for reporting uncertainty in surface slope which is required for estimating uncertainties of items calibrated on a surface plate. © 2003 Elsevier Inc. All rights reserved. Keywords: Measurement uncertainty; Surface plate calibration; Electronic levels; Union Jack Method
1. Introduction Industrial surface plates are calibrated periodically. One of the most common procedures is attributed to K.J. Hume although it is usually called the Moody [1] or Union Jack Method. Flatness is measured using angular deviations along eight lines as shown in Fig. 1. The angles are integrated for height profiles. Slopes and offsets are added to the profiles as required for consistency at the intersections. A reference plane is set to zero at the center of the measured area and oriented such that the values at the ends of each diagonal are equal [2]. Flatness is the range of deviations from the reference plane. Other measurement methods are used which vary in the density of measurement grid and fitting algorithms [3,4]. It has been shown that multiple redundancies together with least-squares plane fitting can result in more accurate flatness measurements [5–7]. The purpose of the work described here was not to develop better measurement methods but to assess the uncertainty in a common method used by industry.
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+1-860-565-2929; fax: +1-860-755-5388. E-mail address: [email protected] (J. Drescher).
0141-6359/03/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0141-6359(03)00032-1
Analytical application of rules for estimating uncertainty [8] in this application is tedious and the solution is not found in the literature. The measurement uncertainty for specific instrumentation has been estimated from results of repeated application of the method to a single surface; the uncertainty derived from variation in flatness results [3]. This estimate overlooks the unknown, systematic sources of uncertainty and it may be difficult to derive an uncertainty statement that applies to a variety of conditions using this method. Monte Carlo simulation has also been performed to estimate measurement uncertainty. A set of measurements is treated as the nominal surface and the variation in results from the simulations becomes the estimate of uncertainty [4,7,9]. The simulation can include the unknown systematic components and effects due to the shape of the surface [9]. This method can produce an estimate of uncertainty which applies well to a specific surface. The analytical approach has advantages. It provides explicit expressions for the sensitivity of results to input variables which may be controlled. It provides a framework with which to analyze the effects of changes to the method or instrumentation. It can lead to a general estimate of uncertainty when required, and the results can be used by a technician using a handheld calculator or tables.
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max(z1 , z2 , z3 ) (z1 + z2 + |z1 − z2 |) /2 + z3 + |(z1 + z2 + |z1 − z2 |) /2 − z3 | = , etc. 2 The max and min functions are, therefore, discontinuous in the Taylor series expansion. While this is no problem for numerical techniques [4], the analytical application of rules for the propagation of uncertainty becomes difficult. To begin the approach, an assumption is made. This is, the variance of either max or min function is equal to the variance of the maximum or minimum values, respectively. That is,
Fig. 1. Lines of measurement for surface plate measurement.
2. Uncertainty analysis The uncertainty for the flatness result begins with the mathematical model: Φ = max(zi,j ) − min(zi,j )
(1)
where Φ is flatness; zi,j is two-dimensional array of residual height data after a plane fit. The definition of flatness is important. It is the perpendicular distance between two parallel planes where: (a) all points of the surface are on or between the two planes and (b) the perpendicular distance between them is a minimum [10,11]. Given a set of height data, the minimum separation condition is not trivial to realize mathematically. A least-squares plane fit is sometimes considered an acceptable approximation with upper and lower planes offset to satisfy condition (a) [12]. The Moody analysis is another approximation of the true definition that is widely accepted for surface plate applications. Both options give estimates of flatness that are higher than true flatness. Uncertainty in the flatness result related to the inexact definition as realized by the Moody analysis is excluded in this analysis. Uncertainty estimates are used to either establish conformance zones within tolerance zones in certification of surface plates [13] or to estimate the uncertainty of measurements performed on the surface plate. Given that the uncertainty related to the realized definition would contribute a bias toward smaller flatness values, it is most appropriate to simply exclude this term. Although it is possible to include biased terms in the uncertainty analysis [8,14], the result would not be useful in this application. Uncertainty related to the discrete sampling of a continuous surface is also excluded. Therefore, a statement is produced here relating a measurement result to the result of a perfect measurement of the same surface following the same measurement and analysis procedure. The function for either maximum or minimum of a series of data is an iterative comparison involving the absolute value. For example, for a string of values, zi max(zi ) = . . . max({max[max(z1 , z2 )], z3 }, z4 ) . . . where, max(z1 , z2 ) =
z1 + z2 + |z1 − z2 | 2
u2 [max(zi,j )] = u2 [(zi,j )max ] u2 [min(zi,j )] = u2 [(zi,j )min ]
(2)
If the expected errors were the same at each measured point, there would be no error in the estimate of uncertainty due to this simplification. However, as will be shown, the uncertainty at a point is a function of its position on the surface. Therefore, there is uncertainty in the evaluation of the max and min functions related to which points are the extremes. The error due to this assumption is proportional to the variation in uncertainty from point to point and proportional to the magnitude of uncertainty relative to the surface deviations. Fortunately, the equations for uncertainty in residual height at a position are not affected by this assumption. After derivation, they are used to check the applicability of Eq. (2). It is also assumed the variance of Φ is the sum of variances of the minimum and maximum values. That is, no correlations exist between errors at the maximum and minimum. u2 (Φ) = u2 [(zi,j )max ] + u2 [(zi,j )min ]
(3)
This is strictly correct if the maximum and minimum height values occur along different lines of measurement (Fig. 1), although error may result from this assumption when the maximum and minimum height values occur along the same line of measurement. This error is highest when the two points are adjacent to one another, decreases with separation between the two points, and goes to zero when either point occurs at the end of a line. A review of test data (32 sets) shows a low probability that the maximum and minimum points will be close to one another on the same line of measurement. Of 32 randomly chosen data sets, only one had both maximum and minimum values along a single line of measurement. In that case also the correlation between max and min was zero because one of the points occurred at the end of the line. Given that maximum and minimum values may generally occur at any location on the surface, the next step in the uncertainty analysis is to establish the uncertainties of residual height values as a function of position. Subscripts d, p, and b are used for the diagonal, perimeter, and bisector height data, respectively. The number of measurement points is designated as n, l, and m for the diagonal, long dimension, and short dimension, respectively. Fig. 1 shows the measurement lines and their relative directions in order from 1 to 8.
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2.1. Diagonals Each diagonal height vector is adjusted to be zero at its midpoint and to have zero slope between its endpoints. The residual height of the kth point along a diagonal is: k 1 dkr = dk − dn − dn/2 + dn (4) n 2 where for each line of measurement, dk is the height at the kth position derived from angular measurements relative to the arbitrary datum, Superscript, r, denotes the residuals of the slope and offset addition and n, the number of measurement points (spacing increments) along a diagonal. The first-order approximation for the variance is: r 2 r 2 ∂dk ∂dk 2 r 2 u (dk ) = u (dk ) + u2 (dn ) ∂dk ∂dn ∂dkr 2 2 + u (dn/2 ) ∂dn/2 r r ∂dk ∂dk u(dk )u(dn )r(dk , dn ) +2 ∂dk ∂dn r ∂dkr ∂dk u(dk )u(dn/2 )r(dk , dn/2 ) +2 ∂dk ∂dn/2 r ∂dk ∂dkr +2 u(dn )u(dn/2 )r(dn , dn/2 ) (5) ∂dn ∂dn/2 A solution of Eq. (5) is now derived to estimate the standard uncertainty of the residual height values as a function of incremental distance along the line. The sensitivity coefficients, partial derivatives of Eq. (5), may be seen at once. The standard uncertainties, u(x), and the correlation coefficients, r(x, y) require some discussion. 2.1.1. Standard uncertainties dk is the cumulative sum of angles multiplied by foot spacing. dk = Sd1 θ1 + Sd2 θ2 + Sd3 θ3 + · · · + Sdk θk
(6)
The variance is: u2 (dk ) =
k f =1
∂dk ∂Sdf
2 u2 (Sdf ) +
k ∂dk 2 f =1
∂θf
u2 (θf )
∂dk ∂dk +2 u(Sdf )u(θg )r(Sdf , θg ) ∂Sdf ∂θg f =1g=1 k k−1 ∂dk ∂dk u(Sdf )u(Sdg )r(Sdf , Sdg ) +2 ∂Sdf ∂Sdg k k
f =1g=f +1
k k−1 ∂dk ∂dk u(θf )u(θg )r(θf , θg ) +2 ∂θf ∂θg
(7)
f =1g=g+1
where only linear terms of the expansion of Eq. (6) are considered. The following assumptions are made:
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• Error in foot spacing, Sd , along the diagonal is independent of error in measured angles, θ. • Angle measurement errors are uncorrelated with one another. • Variance and covariance terms associated with uncertainty in foot spacing sum to zero. The calculation of heights from angles implies that the carriage feet are knife edges separated by a known distance and that each positioning involves placing the rear knife-edge foot at the exact position of the front knife-edge foot from the previous measurement. With this assumption, errors in foot spacing would accumulate along the line and become part of the height error. However, this error does not accumulate in the actual procedure because positions of the feet at any two adjacent measurement locations may overlap or be separated by a gap. Therefore, this is not a spacing error as defined by its use in Eqs. (6) and (7). Somewhat less qualitatively it can be argued, there is negative correlation between the spacing errors. That is, a short spacing at the kth position is likely followed by a long spacing error at the k + 1 position and it also increases the chances of a long spacing error at the k +2 position, etc. Therefore, the correlation coefficients in the fourth term of Eq. (7) are negative and their sum negates the first term. Errors do occur due to the spacing and positioning of the feet but this is considered to contribute to angular measurement uncertainty. Noting that the nominal foot spacing is constant for a single line, Eq. (7) is simplified to: u2 (dk ) = (Sd )2
k
u2 (θf )
(8)
f =1
2.1.2. Uncertainty of angle measurement The uncertainty of angle measurement at each position consists of the stated accuracy of the instrument calibration, thermal stability, effects of error in placement of the carriage, effects of error in setting of the feet spacing, and effects of the interacting, finite areas of contact. Other effects such as hysteresis, resolution, electrical noise, and vibration are either encorporated in the calibrated accuracy or small relative to these five terms. For the differential electronic levels, none of these terms is proportional to distance along the line of measurement. The standard uncertainty of the electronic level angular measurement is assumed constant. Without subscript, therefore, the variance of the angular measurement uncertainty is: u2 (θ) = u2 (cal) + u2 (thermal) + u2 (placement) + u2 (setting) + u2 (contact area)
(9)
where these five factors are taken as independent. Typical electronic levels are calibrated to a stated accuracy of the greater of ±0.5% of reading or ±0.5 arcsec [15]. For typical measurements of surface plates, ±0.5 arcsec applies. The standard uncertainty of angular measurement due to calibration uncertainty is derived from this value. It
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includes a possible bias error in addition to possible gain error and other uncertainties associated with the electronic level calibration procedure. Because a bias error in the levels does not affect the flatness measurement result the standard uncertainty based on a uniform probability distribution of bounds ±0.5 arcsec is conservative [8]: u2 (cal) =
2 1 12 (1.0 arcsec)
(10)
Error due to temperature is stated as 0.02% of full-scale per [15]. For typical surface plate measurements throughout the year in a typical factory, data predicts air temperature between 21 and 35 ◦ C neglecting extremes [16]. However, because angles are measured relative to an arbitrarily set datum, only the temperature change during a surface plate measurement is considered. A temperature change during the time for a measurement may typically be up to ±2 ◦ C. For many measurements throughout the year and at different times of day, it is estimated that the temperature change during a measurement is equally likely to be any number within this ±2 ◦ C. Therefore, the standard uncertainty of angular measurement due to thermal stability can be calculated from the estimate: ◦C
u2 (thermal) = =
◦ ◦ 2 1 12 (0.0002 per C × 400 arcsec × 4 C) 2 1 12 (0.32 arcsec)
(11)
where ±200 arcsec is a typical scale setting for electronic levels used in this application. The other three terms of the right hand side of Eq. (9) involve the interaction of the measurement device and the actual surface. Errors due to placement and setting error depend on the surface curvature in the vicinity of the contacts. This was approximated using data from 25 surface plates. Twenty-one of the plates were inspection of grade AA and four were of grade A [2]. They ranged in size from 760 to 3050 mm. Differences between adjacent angle measurements were taken and normalized to the particular foot spacing for the 200 measurement lines. A histogram of the results is shown in Fig. 2. For the analysis of placement and setting errors, the feet are considered to be knife edges. The angle measurement
Fig. 3. Angular measurement error due to error in placement of carriage relative to the nominal position.
error due to placement error is shown in Fig. 3. A carriage is positioned by the skill of a technician who aligns a mark on the carriage to similar marks on a scale which is laid parallel to the line of measurement or to marks that have been transferred to the surface from the scale. The accuracy of the alignment is limited by pencil linewidths and parallax error. A type B estimate of standard uncertainty is derived in this case based on a positioning error bounds of ±1 mm. The scale accuracy is negligible. With the ±3 s bound of 0.03 arcsec/mm from the results of Fig. 2 multiplied by the average foot spacing of 140 mm, the bounds on a uniform probability distribution are established. The height error is within h = ±0.03 arcsec/mm × 140 mm × 4.85 × 10−6 rad/arcsec × 1 mm = ±2.0 × 10−5 mm (12) The bounds on angular error due to placement are then α =
h 1 × = ±0.03 arcsec 140 mm 4.85 × 10−6 rad/arcsec (13)
and the standard uncertainty in angular measurement due to placement is: u2 (placement) =
2 1 12 (0.06 arcsec)
(14)
Fig. 4 shows that the analysis of error due to foot space setting is similar to that for error due to placement. This error also depends on the skill of the technician and the ability to align
Fig. 2. Histogram of change in surface slope per unit distance from several lines of measurement on several surface plates of grade AA and grade A. Units are shown in arcsec/mm to normalize for the varied foot spacing in the data from which this was derived.
Fig. 4. Angular measurement error due to error in setting of carriage feet spacing.
J. Drescher / Precision Engineering 27 (2003) 323–332
Fig. 5. Effect of finite contact area on angular measurement for constant curvature.
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between the typical measurement positions. The calculated differences in curvature are shown in the histogram of Fig. 7. The bounds on the change in curvature can be estimated as three times the standard deviation of 9.9 × 10−5 arcsec/mm2 . For the moment consider a general line profile h(x). Then from the histogram of Fig. 7, the approximation can be written: 2 ( h/x2 )i − (2 h/x2 )i−1 3s = max x (1/Ri − 1/Ri−1 ) = max (16) x
marks on a scale with marks on the carriage. However, the risk of parallax is reduced because the setting can take place at a comfortable location and there is no secondary transfer of dimension from the ruler. The error bounds of setting are estimated as ±0.5 mm yielding a h of ±1.0×10−5 mm following Eq. (12) and a of ±0.015 arcsec following Eq. (13) where the error in using the nominal spacing of 140 mm is negligible. The standard uncertainty in angular measurement due to setting is:
where s is the standard deviation of the expected changes in curvature as shown in Fig. 7 and Ri is the radius of curvature at the ith position. Because we are only concerned with relative differences in curvature, the value Ri−1 can be set to infinity in Eq. (16). That is, the value of hfront in Fig. 6 is zero. The value of hrear can then be found by first solving for the radius of curvature, Ri . From Eq. (16): 1 max = [3s(x)] (17) Ri
u2 (setting) =
Solving for the radius of curvature while taking care of units:
2 1 12 (0.03 arcsec)
(15)
The error due to finite contact area is also surface dependent. When the surface is not perfectly flat, the effective contact points (or lines) are unlikely to be at the center of the nominal contact area. However, as shown in Fig. 5 for constant curvature, when the effective contact point is forward of center on the front foot, the effective contact point is behind center on the rear foot and vice versa. No angle measurement error occurs due to contact area for constant curvature. The deviation in height is equal at the front and rear resulting in zero angular deviation compared to the ideal knife edges. In general, the curvature is not equal at the front contact and the rear contact positions, so the deviations in height are also different as pictured schematically in Fig. 6. An angular deviation occurs relative to the ideal, knife-edge situation. The error depends on the difference in curvature. Just as the difference in slope was characterized by relative differences of the measured angles, the same data was “differentiated” again to yield an estimate of the expected change in curvature
Ri =
1 3(0.000099 arcsec/mm2 )(140 mm) arcsec × = 5 × 106 mm 4.85 × 10−6 mm/mm
(18)
This is the radius of curvature corresponding to the largest expected curvature difference between two points a distance of 140 mm apart. The estimate for bounds on h due to the area of contact effect is then calculated as: (w/2)2 hrear = ± (19) Ri where w is the length of the contact. For a typical carriage foot, w is approximately 12 mm. Therefore, hrear = ±
(6 mm)2 = ±7 × 10−6 mm 5 × 106 mm
(20)
Finally, solving for the bounds of uncertainty in angular measurement due to the contact area: (7 × 10−6 mm) × (4.85 × 10−6 mm arcsec/mm) h θ = =± x 140 mm = ±0.01 arcsec
And considering a uniform probability distribution with upper and lower bounds equal to θ, the square of the standard uncertainty in angular measurement due to the contact area is: u2 (contact area) = Fig. 6. Effect of finite contact area on angular measurement for different surface curvature at the front contact and rear contact positions.
2 1 12 (0.02 arcsec)
(21)
Now substituting into Eq. (9) for the combined standard uncertainty of angular measurement:
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Fig. 7. Histogram of change in curvature per unit distance from several lines of measurement on several surface plates of grade AA and grade A [2].
u2 (θ) =
(1.0 arcsec)2 + (0.32 arcsec)2 + (0.06 arcsec)2 (22) + (0.03 arcsec)2 + (0.02 arcsec)2 1 12
u2 (θ) = 0.092 arcsec2 With the previous assumption that this quantity is constant, Eq. (8) becomes u2 (dk ) = k(Sd )2 u2 (θ)
2.2. Perimeter lines of measurement
(23)
and the remaining variance terms of Eq. (5) can be written for k = n and k = n/2. u2 (dn ) = n(Sd )2 u2 (θ)
(24)
n u (dn/2 ) = (Sd )2 u2 (θ) 2
(25)
2
At k = n/2 the variance is zero. There is no difference between the fitting of perfect data and fitting of data with errors. At either end of the diagonal Eq. (27) becomes: n (28) u2 (dnr ) = u2 (d0r ) = (Sd )2 u2 (θ) 4
Eq. (5) is rewritten here with the substitutions of Eqs. (23)–(25) and collection of terms.
1 k 2 n 2 r 2 2 u (dk ) = (Sd ) u (θ) k + n + − 2 n 2 √ √ 1 k − kn[r(dk , dn )] − 2kn[r(dk , dn/2 )] +2 2 n √ 1 k − n 2[r(dn , dn/2 )] − (26) 2 n 2.1.3. Correlation coefficients Variations in the height values along the diagonal before the endpoint fit represent a one-dimensional random walk with a constant standard deviation, σ, at each position. The variance at the kth position is kσ 2 . And the correlation coefficient between two of these variables, kth and nth for example, can be shown to be (k/n)1/2 , where k < n [17]. The general expression for the variance at position k is, therefore 2 n 2 u2 (θ) n − k (S for k ≤ ) d 4 n 2 u2 (dkr ) = (27) 2 k 3n n (Sd )2 u2 (θ) 2k − − for k ≥ n 4 2
The perimeter profiles are a function of the diagonal ends. The expression for the residual height of the ith point along a perimeter is similar to Eq. (4): i i pri 3 = pi |3 − pl |3 + d0r 1 + dnr 2 − d0r 1 l l
(29)
The additional, numbered subscripts after the vertical lines in this equation refer to the line numbers of the diagram of Fig. 1. The square of the uncertainty for the ith perimeter height value is: u
2
pri 3
=
2 ∂pri 3 ∂pi |3
+
2 ∂pri 3 ∂pl |3
+
u2 pi |3
u
2
pl |3 +
2 ∂pri 3 u2 d0r 1 r ∂d 0 1
2 r ∂pri 3 ∂pi 3 ∂pri 3 r 2 +2 u d n 2 ∂pi |3 ∂pl |3 ∂d r
n 2
× u pi |3 u pl |3 r pi |3 , pl |3
(30)
where all other covariances are zero because there is no correlation between errors along the perimeter with errors along a diagonal. Sensitivity coefficients may be seen at once and the standard uncertainties for the unfit measured height values are derived as before. (31) u2 pi |3 = i(Slp )2 u2 (θ) u2 pl |3 = l(Slp )2 u2 (θ)
(32)
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Also, noting the uncertainty is equal on both ends of both diagonals, Eq. (30) becomes: u
2
pri 3
= i(Slp ) u (θ) +
−i l
For the short bisector (line 7 in Fig. 1), the general expression is:
2
l(Slp ) u (θ) −i n 2 2 + (Sd ) u (θ) + 2 4 l √ √ × i(Slp )u(θ) l(Slp )u(θ)r pi |3 , pl |3 2 2
u
2 2
(33)
Simplification results in the general expression for the square of the standard uncertainty for the residual values at any ith station along either long perimeter after fitting to the diagonal endpoints. i2 n 2 r u (pi ) = i − (34) (Slp )2 u2 (θ) + (Sd )2 u2 (θ) l 4 At i = 0 and l, the first term of Eq. (34) becomes zero and it reduces to the square of the uncertainty at the ends of the diagonal. The expression is maximum at the midpoint, i = l/2. l n u2 (prl/2 ) = (Slp )2 u2 (θ) + (Sd )2 u2 (θ) (35) 4 4 For the short perimeters at any jth position: j2 n 2 r (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) u (pj ) = j − m 4 And at the midpoint of the short perimeter, j = m/2: m n u2 (prm/2 ) = (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) 4 4
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2
(bjr ) =
j2 l j− (Ssp )2 u2 (θ) + (Slp )2 u2 (θ) m 4 n + (Sd )2 u2 (θ) 4
(41)
and at the maximum, j = m/2: r )= u2 (bm/2
m l n (Ssp )2 u2 (θ) + (Slp )2 u2 (θ) + (Sd )2 u2 (θ) 4 4 4 (42)
After the fitting analysis, the standard uncertainty of all the height data points is a maximum at the center as determined by the standard uncertainty of the midpoint of either bisector. This expression is simplified as: r r u2 (bm/2 ) = u2 (bl/2 ) 2 2 2 S S Sd sp lp = u2 (θ) m +l +n 2 2 2
(43) (36)
(37)
2.3. Bisector lines of measurement The bisectors are a function of the perimeter midpoints. The expression for the height of the ith point along the long bisector with slope and offset added is given as Eq. (38). Subscript 4 refers to the long bisector as labeled in Fig. 1: i i r bir 4 = bi |4 − bl |4 + prm/2 + pm/2 − prm/2 6 8 6 l l (38) Following the previous derivations, the general expression for the variance along the long bisector is: i2 2 r u (bi ) = i − (Slp )2 u2 (θ) l m n + (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) (39) 4 4
2.4. Flatness uncertainty The estimated standard uncertainty as a function of position on the surface plate is shown schematically in Fig. 8. According to Eq. (3), the standard uncertainty in flatness is the RSS of the two uncertainty values at the measured high and low spots on the surface plate. The plot of Fig. 8 applies to any surface plate measurement parameters. That is, the uncertainty at the two locations of interest could be calculated using the derivations here and therefore, the task-specific uncertainty can be found analytically. A general statement of uncertainty regarding the flatness measurement with this measurement method and analysis technique may be required. Eq. (43) can be applied using the average values of step size and average numbers of steps along each measurement line. For the data analyzed, the average step size is 140 mm and the average number of steps in a line is 11. The maximum value of
At i = 0 and l, this reduces to the variance at the midpoints of the perimeters and the expression is maximum at the midpoint, i = l/2. r u2 (bl/2 )=
l m n (Slp )2 u2 (θ) + (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) 4 4 4 (40)
Fig. 8. Qualitative plot of uncertainty as a function of location along the various lines of measurement. The maximum uncertainty occurs at the midpoints of the lines which bisect the measured area orthogonally.
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standard uncertainty occurs at the midpoint of the bisectors as: 140 mm 2 2 r 2 u (bl/2 ) = 0.092 arcsec (11)(3) 2 × (5 × 10−6 arcsec−1 )2 = 0.61 m
(44)
To establish worst case bounds, this value can be assigned to both the minimum and maximum surface plate height value. Then, the standard uncertainty in flatness would be calculated as: √ r u(ϕ) = 2u(bl/2 ) = 0.86 m (45) 3. Verification Estimates of flatness measurement uncertainty from participants in an intercomparison of flatness measurements ranged from 0.2 to 0.6 m [18]. This compares well with the results here considering that those estimates were for a specific plate and that most participants used gridlines with multiple redundant measurements at each point. Two additional steps were taken to verify the analysis. The first was to assess the error in the assumption that the uncertainty of the max (or min) function is the same as the uncertainty of the maximum (or minimum) value. As noted in the writing of Eq. (2), error in the flatness uncertainty estimate may occur if an uncertainty zone associated with a measured maximum (or minimum) is encompassed by the uncertainty zones associated with another height value. To check for this condition, two methods of estimating the flatness were compared. Twenty-five sets of data were analyzed according to the Moody method to yield 25 sets of residual height data. The uncertainty at each position was also calculated for each set according to the Eqs. (27), (34), (36), and (39).
In the first estimate, the difference between maximum and minimum height values was added to two times the standard uncertainty associated with the respective maximum and minimum values. These numbers are plotted along the horizontal axis of Fig. 9. The second estimate was obtained by adding two standard uncertainties to each respective height value and separately subtracting two standard uncertainties to each respective height value to create two additional sets of values for each data set. The difference between the overall maximum and minimum from these two data sets was then computed. This is plotted along the vertical axis of Fig. 9. The deviations from the line of unit slope represent the difference between the two methods and the error in the assumption of Eq. (2). Twenty of 25 errors were zero. That is both methods used the same two points and the uncertainty zones at those extremes enveloped all other measured heights together with their respective uncertainty zones. Where this was not the case there was one error at 6.8% difference, one at 1.2% difference, and three at less than 1% difference. The largest error of 6.8% difference occurred on a relatively large plate (1200 mm × 1800 mm) with a nominal flatness of less than 5 m. The second step to verification compared closure errors from actual datasets to the bounds on error as established by Eq. (43) of the analysis. Eq. (43) was used to calculate the uncertainty of the residual height values at the center of 34 surface plates which had been previously measured. These were compared to the two closure errors from each data set. Actual closure values were converted to absolute values for comparison to the estimate and a coverage factor of 3 was used. The results are shown in Fig. 10. The lines on this figure show graphically the distribution of the 68 data points within the 1, 2, and 3 standard uncertainty bands. As the uncertainty estimate represents an envelope for measurement error, Fig. 10 shows agreement between the analysis and experimental
Fig. 9. Results to test the approximation that the variance of max (or min) function is equal to the variance of the maximum (or minimum) values. For 25 sets of surface plate data the maximum difference after expanding the uncertainty at all positions is plotted vs. the difference between maximum and minimum values together with their associated expanded uncertainties.
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Fig. 11. Standard uncertainty of surface slope as a function of point separation distance as defined by the multiple of average foot spacing used in the surface measurement.
of slope can be calculated from the residual (fitted) data. The best numbers to use for the propagation of uncertainty to the calibration of other artifacts depends on the size and form of those artifacts and the design of the instruments used in the calibration. For example, the uncertainty in measuring data. This validates the analysis to some extent although the parallelism of a 500 mm parallel with a height stand depends outliers and broad distribution suggest the model does not partially on the expected slope and range of possible slopes include all the anomolous measurement error sources. between pairs of points separated by 250 mm (assuming the parallel is supported near its Airy points). Meanwhile, the cosine error term of the height stand, may require slope 4. Propagation of uncertainty to other calibrations data from a second set of points separated by 150 mm, for example (if that is the size of its base). Flatness of a surface plate and uncertainty in flatness are It is proposed that standard uncertainty in slope should be of little use except for the qualitative correlations, provided reported as a function of separation distance for the calibraby experience and tradition, between the flatness and the tion of surface plates in addition to the flatness and uncer“goodness” for certain applications. To apply these quantities tainty in flatness. The following formula applies. 2 n 2 (zk − zk−λ )f 1 u(γ)λ = − γ¯ λ 3l + 3m + 2n − 8λ − 1 λSd Fig. 10. Plot of actual closure values from 34 surface plate measurements vs. the uncertainty at the center of the bisector lines as estimated by Eq. (43).
f =1k=λ
+
l 5 (zi − zi−λ )f f =3 i=λ
λSlp
to uncertainty analyses of the instruments and artifacts calibrated on the surface plate requires broad assumptions about the relationship of flatness to surface slope. The preferred outputs of a surface plate calibration for purposes of propagation of uncertainty to the other instruments and artifacts
2 − γ¯ λ
2 1/2 m 8 (zj − zj−λ )f + − γ¯ λ λSsp
where λ is an integer corresponding to the multiple of measurement increments (foot spacing) used for the separation distance in the calculation of the standard uncertainty, f is an index and subscript corresponding to the line of measurement (Fig. 1), and γ¯ λ is the average slope given by:
n m l 2 5 8 (zk − zk−λ )f (zi − zi−λ )f (zj − zj−λ )f 1 γ¯ λ = + + 2n + 3l + 3m − 8λ λSd λSlp λSsp f =1k=λ
is the surface slope and the uncertainty associated with surface slope. A slope can be calculated from any pair of residual height values as the height difference divided by the distance between the two points. With many pairs of height values available, an average surface slope and standard uncertainty
(46)
f =6 j=λ
f =3 i=λ
(47)
f =6 j=λ
This is a type A evaluation of uncertainty which appropriately includes the random portion of the uncertainties of residual height values but excludes the systematic portions. This result is calculated using example data and shown in Fig. 11. More complicated schemes are certainly possible to characterize the surface and may be justified for some
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applications. However, the proposed results can readily be used to assess uncertainty in subsequent calibrations that use the surface together with other mechanical artifacts and equipment. 5. Conclusions For electronic levels, the effects of interaction between carriage feet and the surface to be measured are small relative to the calibrated accuracy and the thermal stability of the instruments. The uncertainty of the height values resulting from the data fit to the reference plane is a function of the position in the plane. Therefore, the uncertainty in flatness depends on the locations of “high” and “low” height values. The maximum uncertainty exists at the midpoints of the bisectors. The uncertainty of height values is proportional to the chosen foot spacing and proportional to the squareroot of the number of steps. A general expression for uncertainty in flatness measurement was derived which bounds a significant collection of data comprised of closure values from actual measurements. A simple method is proposed for reporting of surface slope to facilitate the use of surface plate measurement data in the assessment of uncertainty for calibrations performed on the surface. References [1] Moody JC. How to calibrate surface plates in the plant. The Tool Engineer: official publication of the American Society of Tool Engineers 1955;October:85–91.
[2] Federal Specification GGG-P-463c. Plate, surface, (granite) 1973; September 10. [3] Bruin W, Meijer J. Analysis of flatness measurement and form stability of a granite surface plate. Ann CIRP 1980;29/1:385–90. [4] De Vicente J, Sanchez Perez AM. Uncertainty estimation methods by simulation in flatness measurements. IMEKO Bull 1991;20/1:1056–61. [5] Meijer J. Accuracy of surface plate measurements—general purpose software for flatness measurement. Ann CIRP 1990;39/1:545–8. [6] Haitjema H, Meijer J. Evaluation of surface plate flatness measurements. Eur J Mech Eng 1993;38/4:165–72. [7] Haitjema H. Iterative solution of least-squares problems applied to flatness and grid measurements. In: Ciarline P, Cox MG, Pavese F, Richter D, editors. Advanced tools in metrology, vol. II; 1996. [8] ISO Guide. Guide to the expression of uncertainty in measurement, ISBN 1993;92-67-10188-9. [9] Haitjema H. Uncertainty propagation in surface plate measurements. In: Proceedings of the Fourth International Symposium on Dimensional Metrology in Production and Quality Control. ISM QC, Tampere; 1992. p. 304–20. [10] ANSI Y14.5. Dimensioning and tolerancing; 1994. [11] ISO 8512-2. Surface plates—part 2: granite; 1990. [12] ISO 230-1. Test code for machine tools—part 1: geometric accuracy of machines operating under no-load or finishing conditions; 1996. [13] ISO 14253-1. Geometrical product specifications (GPS)—inspection by measurement of workpieces and measuring equipment—part 1: decision rules for proving conformance or non-conformance with specifications; 1998. [14] Drescher JD. Assessment of machine tool accuracy using ISO guide to the expression of uncertainty in measurement (GUM). In: Proceedings of the ASPE; 1997. [15] Mahr-Federal Calibration specification for electronic levels and electronic amplifier Model EMD-832. [16] Unpublished data. [17] Tartakovsky A. Personal communication; 7/26/2002. [18] Ludicke F. Intercomparison of flatness measurements. EUR 14059 EN; 1992.
Short communication
Analytical estimation of measurement uncertainty in surface plate calibration by the Moody method using differential levels Joseph Drescher∗ UTC, Pratt & Whitney, 400 Main Street M/S 165-09, East Hartford, CT 06108, USA Received 22 March 2002; received in revised form 13 August 2002; accepted 18 December 2002
Abstract Surface plate quality influences the calibration of artifacts, instruments, working gauges and, by extension, the quality of manufactured parts. A statement of uncertainty for the surface plate calibration is needed for conformance testing and for estimating uncertainty in calibrations that rely on the surface plate. An analytical approach to the estimate of uncertainty for a common measurement method is given. The residual height at each position on the surface is expressed in terms of the measurement parameters and the measured angles. Uncertainty of each residual height value is then derived following the common methods for propagation of uncertainty. The maximum uncertainty is at the center of the surface and the calculated bounds on error compare well with values of closure from actual measurements. The uncertainty of height values is proportional to the spacing of the measurement positions and to the squareroot of the number of positions. Using data from many plates of various size and quality, a generalized uncertainty in flatness for the measurement method was estimated. The derivations also allow calculations for specific applications. Finally, a method is proposed for reporting uncertainty in surface slope which is required for estimating uncertainties of items calibrated on a surface plate. © 2003 Elsevier Inc. All rights reserved. Keywords: Measurement uncertainty; Surface plate calibration; Electronic levels; Union Jack Method
1. Introduction Industrial surface plates are calibrated periodically. One of the most common procedures is attributed to K.J. Hume although it is usually called the Moody [1] or Union Jack Method. Flatness is measured using angular deviations along eight lines as shown in Fig. 1. The angles are integrated for height profiles. Slopes and offsets are added to the profiles as required for consistency at the intersections. A reference plane is set to zero at the center of the measured area and oriented such that the values at the ends of each diagonal are equal [2]. Flatness is the range of deviations from the reference plane. Other measurement methods are used which vary in the density of measurement grid and fitting algorithms [3,4]. It has been shown that multiple redundancies together with least-squares plane fitting can result in more accurate flatness measurements [5–7]. The purpose of the work described here was not to develop better measurement methods but to assess the uncertainty in a common method used by industry.
∗ Tel.:
+1-860-565-2929; fax: +1-860-755-5388. E-mail address: [email protected] (J. Drescher).
0141-6359/03/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0141-6359(03)00032-1
Analytical application of rules for estimating uncertainty [8] in this application is tedious and the solution is not found in the literature. The measurement uncertainty for specific instrumentation has been estimated from results of repeated application of the method to a single surface; the uncertainty derived from variation in flatness results [3]. This estimate overlooks the unknown, systematic sources of uncertainty and it may be difficult to derive an uncertainty statement that applies to a variety of conditions using this method. Monte Carlo simulation has also been performed to estimate measurement uncertainty. A set of measurements is treated as the nominal surface and the variation in results from the simulations becomes the estimate of uncertainty [4,7,9]. The simulation can include the unknown systematic components and effects due to the shape of the surface [9]. This method can produce an estimate of uncertainty which applies well to a specific surface. The analytical approach has advantages. It provides explicit expressions for the sensitivity of results to input variables which may be controlled. It provides a framework with which to analyze the effects of changes to the method or instrumentation. It can lead to a general estimate of uncertainty when required, and the results can be used by a technician using a handheld calculator or tables.
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max(z1 , z2 , z3 ) (z1 + z2 + |z1 − z2 |) /2 + z3 + |(z1 + z2 + |z1 − z2 |) /2 − z3 | = , etc. 2 The max and min functions are, therefore, discontinuous in the Taylor series expansion. While this is no problem for numerical techniques [4], the analytical application of rules for the propagation of uncertainty becomes difficult. To begin the approach, an assumption is made. This is, the variance of either max or min function is equal to the variance of the maximum or minimum values, respectively. That is,
Fig. 1. Lines of measurement for surface plate measurement.
2. Uncertainty analysis The uncertainty for the flatness result begins with the mathematical model: Φ = max(zi,j ) − min(zi,j )
(1)
where Φ is flatness; zi,j is two-dimensional array of residual height data after a plane fit. The definition of flatness is important. It is the perpendicular distance between two parallel planes where: (a) all points of the surface are on or between the two planes and (b) the perpendicular distance between them is a minimum [10,11]. Given a set of height data, the minimum separation condition is not trivial to realize mathematically. A least-squares plane fit is sometimes considered an acceptable approximation with upper and lower planes offset to satisfy condition (a) [12]. The Moody analysis is another approximation of the true definition that is widely accepted for surface plate applications. Both options give estimates of flatness that are higher than true flatness. Uncertainty in the flatness result related to the inexact definition as realized by the Moody analysis is excluded in this analysis. Uncertainty estimates are used to either establish conformance zones within tolerance zones in certification of surface plates [13] or to estimate the uncertainty of measurements performed on the surface plate. Given that the uncertainty related to the realized definition would contribute a bias toward smaller flatness values, it is most appropriate to simply exclude this term. Although it is possible to include biased terms in the uncertainty analysis [8,14], the result would not be useful in this application. Uncertainty related to the discrete sampling of a continuous surface is also excluded. Therefore, a statement is produced here relating a measurement result to the result of a perfect measurement of the same surface following the same measurement and analysis procedure. The function for either maximum or minimum of a series of data is an iterative comparison involving the absolute value. For example, for a string of values, zi max(zi ) = . . . max({max[max(z1 , z2 )], z3 }, z4 ) . . . where, max(z1 , z2 ) =
z1 + z2 + |z1 − z2 | 2
u2 [max(zi,j )] = u2 [(zi,j )max ] u2 [min(zi,j )] = u2 [(zi,j )min ]
(2)
If the expected errors were the same at each measured point, there would be no error in the estimate of uncertainty due to this simplification. However, as will be shown, the uncertainty at a point is a function of its position on the surface. Therefore, there is uncertainty in the evaluation of the max and min functions related to which points are the extremes. The error due to this assumption is proportional to the variation in uncertainty from point to point and proportional to the magnitude of uncertainty relative to the surface deviations. Fortunately, the equations for uncertainty in residual height at a position are not affected by this assumption. After derivation, they are used to check the applicability of Eq. (2). It is also assumed the variance of Φ is the sum of variances of the minimum and maximum values. That is, no correlations exist between errors at the maximum and minimum. u2 (Φ) = u2 [(zi,j )max ] + u2 [(zi,j )min ]
(3)
This is strictly correct if the maximum and minimum height values occur along different lines of measurement (Fig. 1), although error may result from this assumption when the maximum and minimum height values occur along the same line of measurement. This error is highest when the two points are adjacent to one another, decreases with separation between the two points, and goes to zero when either point occurs at the end of a line. A review of test data (32 sets) shows a low probability that the maximum and minimum points will be close to one another on the same line of measurement. Of 32 randomly chosen data sets, only one had both maximum and minimum values along a single line of measurement. In that case also the correlation between max and min was zero because one of the points occurred at the end of the line. Given that maximum and minimum values may generally occur at any location on the surface, the next step in the uncertainty analysis is to establish the uncertainties of residual height values as a function of position. Subscripts d, p, and b are used for the diagonal, perimeter, and bisector height data, respectively. The number of measurement points is designated as n, l, and m for the diagonal, long dimension, and short dimension, respectively. Fig. 1 shows the measurement lines and their relative directions in order from 1 to 8.
J. Drescher / Precision Engineering 27 (2003) 323–332
2.1. Diagonals Each diagonal height vector is adjusted to be zero at its midpoint and to have zero slope between its endpoints. The residual height of the kth point along a diagonal is: k 1 dkr = dk − dn − dn/2 + dn (4) n 2 where for each line of measurement, dk is the height at the kth position derived from angular measurements relative to the arbitrary datum, Superscript, r, denotes the residuals of the slope and offset addition and n, the number of measurement points (spacing increments) along a diagonal. The first-order approximation for the variance is: r 2 r 2 ∂dk ∂dk 2 r 2 u (dk ) = u (dk ) + u2 (dn ) ∂dk ∂dn ∂dkr 2 2 + u (dn/2 ) ∂dn/2 r r ∂dk ∂dk u(dk )u(dn )r(dk , dn ) +2 ∂dk ∂dn r ∂dkr ∂dk u(dk )u(dn/2 )r(dk , dn/2 ) +2 ∂dk ∂dn/2 r ∂dk ∂dkr +2 u(dn )u(dn/2 )r(dn , dn/2 ) (5) ∂dn ∂dn/2 A solution of Eq. (5) is now derived to estimate the standard uncertainty of the residual height values as a function of incremental distance along the line. The sensitivity coefficients, partial derivatives of Eq. (5), may be seen at once. The standard uncertainties, u(x), and the correlation coefficients, r(x, y) require some discussion. 2.1.1. Standard uncertainties dk is the cumulative sum of angles multiplied by foot spacing. dk = Sd1 θ1 + Sd2 θ2 + Sd3 θ3 + · · · + Sdk θk
(6)
The variance is: u2 (dk ) =
k f =1
∂dk ∂Sdf
2 u2 (Sdf ) +
k ∂dk 2 f =1
∂θf
u2 (θf )
∂dk ∂dk +2 u(Sdf )u(θg )r(Sdf , θg ) ∂Sdf ∂θg f =1g=1 k k−1 ∂dk ∂dk u(Sdf )u(Sdg )r(Sdf , Sdg ) +2 ∂Sdf ∂Sdg k k
f =1g=f +1
k k−1 ∂dk ∂dk u(θf )u(θg )r(θf , θg ) +2 ∂θf ∂θg
(7)
f =1g=g+1
where only linear terms of the expansion of Eq. (6) are considered. The following assumptions are made:
325
• Error in foot spacing, Sd , along the diagonal is independent of error in measured angles, θ. • Angle measurement errors are uncorrelated with one another. • Variance and covariance terms associated with uncertainty in foot spacing sum to zero. The calculation of heights from angles implies that the carriage feet are knife edges separated by a known distance and that each positioning involves placing the rear knife-edge foot at the exact position of the front knife-edge foot from the previous measurement. With this assumption, errors in foot spacing would accumulate along the line and become part of the height error. However, this error does not accumulate in the actual procedure because positions of the feet at any two adjacent measurement locations may overlap or be separated by a gap. Therefore, this is not a spacing error as defined by its use in Eqs. (6) and (7). Somewhat less qualitatively it can be argued, there is negative correlation between the spacing errors. That is, a short spacing at the kth position is likely followed by a long spacing error at the k + 1 position and it also increases the chances of a long spacing error at the k +2 position, etc. Therefore, the correlation coefficients in the fourth term of Eq. (7) are negative and their sum negates the first term. Errors do occur due to the spacing and positioning of the feet but this is considered to contribute to angular measurement uncertainty. Noting that the nominal foot spacing is constant for a single line, Eq. (7) is simplified to: u2 (dk ) = (Sd )2
k
u2 (θf )
(8)
f =1
2.1.2. Uncertainty of angle measurement The uncertainty of angle measurement at each position consists of the stated accuracy of the instrument calibration, thermal stability, effects of error in placement of the carriage, effects of error in setting of the feet spacing, and effects of the interacting, finite areas of contact. Other effects such as hysteresis, resolution, electrical noise, and vibration are either encorporated in the calibrated accuracy or small relative to these five terms. For the differential electronic levels, none of these terms is proportional to distance along the line of measurement. The standard uncertainty of the electronic level angular measurement is assumed constant. Without subscript, therefore, the variance of the angular measurement uncertainty is: u2 (θ) = u2 (cal) + u2 (thermal) + u2 (placement) + u2 (setting) + u2 (contact area)
(9)
where these five factors are taken as independent. Typical electronic levels are calibrated to a stated accuracy of the greater of ±0.5% of reading or ±0.5 arcsec [15]. For typical measurements of surface plates, ±0.5 arcsec applies. The standard uncertainty of angular measurement due to calibration uncertainty is derived from this value. It
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includes a possible bias error in addition to possible gain error and other uncertainties associated with the electronic level calibration procedure. Because a bias error in the levels does not affect the flatness measurement result the standard uncertainty based on a uniform probability distribution of bounds ±0.5 arcsec is conservative [8]: u2 (cal) =
2 1 12 (1.0 arcsec)
(10)
Error due to temperature is stated as 0.02% of full-scale per [15]. For typical surface plate measurements throughout the year in a typical factory, data predicts air temperature between 21 and 35 ◦ C neglecting extremes [16]. However, because angles are measured relative to an arbitrarily set datum, only the temperature change during a surface plate measurement is considered. A temperature change during the time for a measurement may typically be up to ±2 ◦ C. For many measurements throughout the year and at different times of day, it is estimated that the temperature change during a measurement is equally likely to be any number within this ±2 ◦ C. Therefore, the standard uncertainty of angular measurement due to thermal stability can be calculated from the estimate: ◦C
u2 (thermal) = =
◦ ◦ 2 1 12 (0.0002 per C × 400 arcsec × 4 C) 2 1 12 (0.32 arcsec)
(11)
where ±200 arcsec is a typical scale setting for electronic levels used in this application. The other three terms of the right hand side of Eq. (9) involve the interaction of the measurement device and the actual surface. Errors due to placement and setting error depend on the surface curvature in the vicinity of the contacts. This was approximated using data from 25 surface plates. Twenty-one of the plates were inspection of grade AA and four were of grade A [2]. They ranged in size from 760 to 3050 mm. Differences between adjacent angle measurements were taken and normalized to the particular foot spacing for the 200 measurement lines. A histogram of the results is shown in Fig. 2. For the analysis of placement and setting errors, the feet are considered to be knife edges. The angle measurement
Fig. 3. Angular measurement error due to error in placement of carriage relative to the nominal position.
error due to placement error is shown in Fig. 3. A carriage is positioned by the skill of a technician who aligns a mark on the carriage to similar marks on a scale which is laid parallel to the line of measurement or to marks that have been transferred to the surface from the scale. The accuracy of the alignment is limited by pencil linewidths and parallax error. A type B estimate of standard uncertainty is derived in this case based on a positioning error bounds of ±1 mm. The scale accuracy is negligible. With the ±3 s bound of 0.03 arcsec/mm from the results of Fig. 2 multiplied by the average foot spacing of 140 mm, the bounds on a uniform probability distribution are established. The height error is within h = ±0.03 arcsec/mm × 140 mm × 4.85 × 10−6 rad/arcsec × 1 mm = ±2.0 × 10−5 mm (12) The bounds on angular error due to placement are then α =
h 1 × = ±0.03 arcsec 140 mm 4.85 × 10−6 rad/arcsec (13)
and the standard uncertainty in angular measurement due to placement is: u2 (placement) =
2 1 12 (0.06 arcsec)
(14)
Fig. 4 shows that the analysis of error due to foot space setting is similar to that for error due to placement. This error also depends on the skill of the technician and the ability to align
Fig. 2. Histogram of change in surface slope per unit distance from several lines of measurement on several surface plates of grade AA and grade A. Units are shown in arcsec/mm to normalize for the varied foot spacing in the data from which this was derived.
Fig. 4. Angular measurement error due to error in setting of carriage feet spacing.
J. Drescher / Precision Engineering 27 (2003) 323–332
Fig. 5. Effect of finite contact area on angular measurement for constant curvature.
327
between the typical measurement positions. The calculated differences in curvature are shown in the histogram of Fig. 7. The bounds on the change in curvature can be estimated as three times the standard deviation of 9.9 × 10−5 arcsec/mm2 . For the moment consider a general line profile h(x). Then from the histogram of Fig. 7, the approximation can be written: 2 ( h/x2 )i − (2 h/x2 )i−1 3s = max x (1/Ri − 1/Ri−1 ) = max (16) x
marks on a scale with marks on the carriage. However, the risk of parallax is reduced because the setting can take place at a comfortable location and there is no secondary transfer of dimension from the ruler. The error bounds of setting are estimated as ±0.5 mm yielding a h of ±1.0×10−5 mm following Eq. (12) and a of ±0.015 arcsec following Eq. (13) where the error in using the nominal spacing of 140 mm is negligible. The standard uncertainty in angular measurement due to setting is:
where s is the standard deviation of the expected changes in curvature as shown in Fig. 7 and Ri is the radius of curvature at the ith position. Because we are only concerned with relative differences in curvature, the value Ri−1 can be set to infinity in Eq. (16). That is, the value of hfront in Fig. 6 is zero. The value of hrear can then be found by first solving for the radius of curvature, Ri . From Eq. (16): 1 max = [3s(x)] (17) Ri
u2 (setting) =
Solving for the radius of curvature while taking care of units:
2 1 12 (0.03 arcsec)
(15)
The error due to finite contact area is also surface dependent. When the surface is not perfectly flat, the effective contact points (or lines) are unlikely to be at the center of the nominal contact area. However, as shown in Fig. 5 for constant curvature, when the effective contact point is forward of center on the front foot, the effective contact point is behind center on the rear foot and vice versa. No angle measurement error occurs due to contact area for constant curvature. The deviation in height is equal at the front and rear resulting in zero angular deviation compared to the ideal knife edges. In general, the curvature is not equal at the front contact and the rear contact positions, so the deviations in height are also different as pictured schematically in Fig. 6. An angular deviation occurs relative to the ideal, knife-edge situation. The error depends on the difference in curvature. Just as the difference in slope was characterized by relative differences of the measured angles, the same data was “differentiated” again to yield an estimate of the expected change in curvature
Ri =
1 3(0.000099 arcsec/mm2 )(140 mm) arcsec × = 5 × 106 mm 4.85 × 10−6 mm/mm
(18)
This is the radius of curvature corresponding to the largest expected curvature difference between two points a distance of 140 mm apart. The estimate for bounds on h due to the area of contact effect is then calculated as: (w/2)2 hrear = ± (19) Ri where w is the length of the contact. For a typical carriage foot, w is approximately 12 mm. Therefore, hrear = ±
(6 mm)2 = ±7 × 10−6 mm 5 × 106 mm
(20)
Finally, solving for the bounds of uncertainty in angular measurement due to the contact area: (7 × 10−6 mm) × (4.85 × 10−6 mm arcsec/mm) h θ = =± x 140 mm = ±0.01 arcsec
And considering a uniform probability distribution with upper and lower bounds equal to θ, the square of the standard uncertainty in angular measurement due to the contact area is: u2 (contact area) = Fig. 6. Effect of finite contact area on angular measurement for different surface curvature at the front contact and rear contact positions.
2 1 12 (0.02 arcsec)
(21)
Now substituting into Eq. (9) for the combined standard uncertainty of angular measurement:
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J. Drescher / Precision Engineering 27 (2003) 323–332
Fig. 7. Histogram of change in curvature per unit distance from several lines of measurement on several surface plates of grade AA and grade A [2].
u2 (θ) =
(1.0 arcsec)2 + (0.32 arcsec)2 + (0.06 arcsec)2 (22) + (0.03 arcsec)2 + (0.02 arcsec)2 1 12
u2 (θ) = 0.092 arcsec2 With the previous assumption that this quantity is constant, Eq. (8) becomes u2 (dk ) = k(Sd )2 u2 (θ)
2.2. Perimeter lines of measurement
(23)
and the remaining variance terms of Eq. (5) can be written for k = n and k = n/2. u2 (dn ) = n(Sd )2 u2 (θ)
(24)
n u (dn/2 ) = (Sd )2 u2 (θ) 2
(25)
2
At k = n/2 the variance is zero. There is no difference between the fitting of perfect data and fitting of data with errors. At either end of the diagonal Eq. (27) becomes: n (28) u2 (dnr ) = u2 (d0r ) = (Sd )2 u2 (θ) 4
Eq. (5) is rewritten here with the substitutions of Eqs. (23)–(25) and collection of terms.
1 k 2 n 2 r 2 2 u (dk ) = (Sd ) u (θ) k + n + − 2 n 2 √ √ 1 k − kn[r(dk , dn )] − 2kn[r(dk , dn/2 )] +2 2 n √ 1 k − n 2[r(dn , dn/2 )] − (26) 2 n 2.1.3. Correlation coefficients Variations in the height values along the diagonal before the endpoint fit represent a one-dimensional random walk with a constant standard deviation, σ, at each position. The variance at the kth position is kσ 2 . And the correlation coefficient between two of these variables, kth and nth for example, can be shown to be (k/n)1/2 , where k < n [17]. The general expression for the variance at position k is, therefore 2 n 2 u2 (θ) n − k (S for k ≤ ) d 4 n 2 u2 (dkr ) = (27) 2 k 3n n (Sd )2 u2 (θ) 2k − − for k ≥ n 4 2
The perimeter profiles are a function of the diagonal ends. The expression for the residual height of the ith point along a perimeter is similar to Eq. (4): i i pri 3 = pi |3 − pl |3 + d0r 1 + dnr 2 − d0r 1 l l
(29)
The additional, numbered subscripts after the vertical lines in this equation refer to the line numbers of the diagram of Fig. 1. The square of the uncertainty for the ith perimeter height value is: u
2
pri 3
=
2 ∂pri 3 ∂pi |3
+
2 ∂pri 3 ∂pl |3
+
u2 pi |3
u
2
pl |3 +
2 ∂pri 3 u2 d0r 1 r ∂d 0 1
2 r ∂pri 3 ∂pi 3 ∂pri 3 r 2 +2 u d n 2 ∂pi |3 ∂pl |3 ∂d r
n 2
× u pi |3 u pl |3 r pi |3 , pl |3
(30)
where all other covariances are zero because there is no correlation between errors along the perimeter with errors along a diagonal. Sensitivity coefficients may be seen at once and the standard uncertainties for the unfit measured height values are derived as before. (31) u2 pi |3 = i(Slp )2 u2 (θ) u2 pl |3 = l(Slp )2 u2 (θ)
(32)
J. Drescher / Precision Engineering 27 (2003) 323–332
Also, noting the uncertainty is equal on both ends of both diagonals, Eq. (30) becomes: u
2
pri 3
= i(Slp ) u (θ) +
−i l
For the short bisector (line 7 in Fig. 1), the general expression is:
2
l(Slp ) u (θ) −i n 2 2 + (Sd ) u (θ) + 2 4 l √ √ × i(Slp )u(θ) l(Slp )u(θ)r pi |3 , pl |3 2 2
u
2 2
(33)
Simplification results in the general expression for the square of the standard uncertainty for the residual values at any ith station along either long perimeter after fitting to the diagonal endpoints. i2 n 2 r u (pi ) = i − (34) (Slp )2 u2 (θ) + (Sd )2 u2 (θ) l 4 At i = 0 and l, the first term of Eq. (34) becomes zero and it reduces to the square of the uncertainty at the ends of the diagonal. The expression is maximum at the midpoint, i = l/2. l n u2 (prl/2 ) = (Slp )2 u2 (θ) + (Sd )2 u2 (θ) (35) 4 4 For the short perimeters at any jth position: j2 n 2 r (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) u (pj ) = j − m 4 And at the midpoint of the short perimeter, j = m/2: m n u2 (prm/2 ) = (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) 4 4
329
2
(bjr ) =
j2 l j− (Ssp )2 u2 (θ) + (Slp )2 u2 (θ) m 4 n + (Sd )2 u2 (θ) 4
(41)
and at the maximum, j = m/2: r )= u2 (bm/2
m l n (Ssp )2 u2 (θ) + (Slp )2 u2 (θ) + (Sd )2 u2 (θ) 4 4 4 (42)
After the fitting analysis, the standard uncertainty of all the height data points is a maximum at the center as determined by the standard uncertainty of the midpoint of either bisector. This expression is simplified as: r r u2 (bm/2 ) = u2 (bl/2 ) 2 2 2 S S Sd sp lp = u2 (θ) m +l +n 2 2 2
(43) (36)
(37)
2.3. Bisector lines of measurement The bisectors are a function of the perimeter midpoints. The expression for the height of the ith point along the long bisector with slope and offset added is given as Eq. (38). Subscript 4 refers to the long bisector as labeled in Fig. 1: i i r bir 4 = bi |4 − bl |4 + prm/2 + pm/2 − prm/2 6 8 6 l l (38) Following the previous derivations, the general expression for the variance along the long bisector is: i2 2 r u (bi ) = i − (Slp )2 u2 (θ) l m n + (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) (39) 4 4
2.4. Flatness uncertainty The estimated standard uncertainty as a function of position on the surface plate is shown schematically in Fig. 8. According to Eq. (3), the standard uncertainty in flatness is the RSS of the two uncertainty values at the measured high and low spots on the surface plate. The plot of Fig. 8 applies to any surface plate measurement parameters. That is, the uncertainty at the two locations of interest could be calculated using the derivations here and therefore, the task-specific uncertainty can be found analytically. A general statement of uncertainty regarding the flatness measurement with this measurement method and analysis technique may be required. Eq. (43) can be applied using the average values of step size and average numbers of steps along each measurement line. For the data analyzed, the average step size is 140 mm and the average number of steps in a line is 11. The maximum value of
At i = 0 and l, this reduces to the variance at the midpoints of the perimeters and the expression is maximum at the midpoint, i = l/2. r u2 (bl/2 )=
l m n (Slp )2 u2 (θ) + (Ssp )2 u2 (θ) + (Sd )2 u2 (θ) 4 4 4 (40)
Fig. 8. Qualitative plot of uncertainty as a function of location along the various lines of measurement. The maximum uncertainty occurs at the midpoints of the lines which bisect the measured area orthogonally.
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standard uncertainty occurs at the midpoint of the bisectors as: 140 mm 2 2 r 2 u (bl/2 ) = 0.092 arcsec (11)(3) 2 × (5 × 10−6 arcsec−1 )2 = 0.61 m
(44)
To establish worst case bounds, this value can be assigned to both the minimum and maximum surface plate height value. Then, the standard uncertainty in flatness would be calculated as: √ r u(ϕ) = 2u(bl/2 ) = 0.86 m (45) 3. Verification Estimates of flatness measurement uncertainty from participants in an intercomparison of flatness measurements ranged from 0.2 to 0.6 m [18]. This compares well with the results here considering that those estimates were for a specific plate and that most participants used gridlines with multiple redundant measurements at each point. Two additional steps were taken to verify the analysis. The first was to assess the error in the assumption that the uncertainty of the max (or min) function is the same as the uncertainty of the maximum (or minimum) value. As noted in the writing of Eq. (2), error in the flatness uncertainty estimate may occur if an uncertainty zone associated with a measured maximum (or minimum) is encompassed by the uncertainty zones associated with another height value. To check for this condition, two methods of estimating the flatness were compared. Twenty-five sets of data were analyzed according to the Moody method to yield 25 sets of residual height data. The uncertainty at each position was also calculated for each set according to the Eqs. (27), (34), (36), and (39).
In the first estimate, the difference between maximum and minimum height values was added to two times the standard uncertainty associated with the respective maximum and minimum values. These numbers are plotted along the horizontal axis of Fig. 9. The second estimate was obtained by adding two standard uncertainties to each respective height value and separately subtracting two standard uncertainties to each respective height value to create two additional sets of values for each data set. The difference between the overall maximum and minimum from these two data sets was then computed. This is plotted along the vertical axis of Fig. 9. The deviations from the line of unit slope represent the difference between the two methods and the error in the assumption of Eq. (2). Twenty of 25 errors were zero. That is both methods used the same two points and the uncertainty zones at those extremes enveloped all other measured heights together with their respective uncertainty zones. Where this was not the case there was one error at 6.8% difference, one at 1.2% difference, and three at less than 1% difference. The largest error of 6.8% difference occurred on a relatively large plate (1200 mm × 1800 mm) with a nominal flatness of less than 5 m. The second step to verification compared closure errors from actual datasets to the bounds on error as established by Eq. (43) of the analysis. Eq. (43) was used to calculate the uncertainty of the residual height values at the center of 34 surface plates which had been previously measured. These were compared to the two closure errors from each data set. Actual closure values were converted to absolute values for comparison to the estimate and a coverage factor of 3 was used. The results are shown in Fig. 10. The lines on this figure show graphically the distribution of the 68 data points within the 1, 2, and 3 standard uncertainty bands. As the uncertainty estimate represents an envelope for measurement error, Fig. 10 shows agreement between the analysis and experimental
Fig. 9. Results to test the approximation that the variance of max (or min) function is equal to the variance of the maximum (or minimum) values. For 25 sets of surface plate data the maximum difference after expanding the uncertainty at all positions is plotted vs. the difference between maximum and minimum values together with their associated expanded uncertainties.
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Fig. 11. Standard uncertainty of surface slope as a function of point separation distance as defined by the multiple of average foot spacing used in the surface measurement.
of slope can be calculated from the residual (fitted) data. The best numbers to use for the propagation of uncertainty to the calibration of other artifacts depends on the size and form of those artifacts and the design of the instruments used in the calibration. For example, the uncertainty in measuring data. This validates the analysis to some extent although the parallelism of a 500 mm parallel with a height stand depends outliers and broad distribution suggest the model does not partially on the expected slope and range of possible slopes include all the anomolous measurement error sources. between pairs of points separated by 250 mm (assuming the parallel is supported near its Airy points). Meanwhile, the cosine error term of the height stand, may require slope 4. Propagation of uncertainty to other calibrations data from a second set of points separated by 150 mm, for example (if that is the size of its base). Flatness of a surface plate and uncertainty in flatness are It is proposed that standard uncertainty in slope should be of little use except for the qualitative correlations, provided reported as a function of separation distance for the calibraby experience and tradition, between the flatness and the tion of surface plates in addition to the flatness and uncer“goodness” for certain applications. To apply these quantities tainty in flatness. The following formula applies. 2 n 2 (zk − zk−λ )f 1 u(γ)λ = − γ¯ λ 3l + 3m + 2n − 8λ − 1 λSd Fig. 10. Plot of actual closure values from 34 surface plate measurements vs. the uncertainty at the center of the bisector lines as estimated by Eq. (43).
f =1k=λ
+
l 5 (zi − zi−λ )f f =3 i=λ
λSlp
to uncertainty analyses of the instruments and artifacts calibrated on the surface plate requires broad assumptions about the relationship of flatness to surface slope. The preferred outputs of a surface plate calibration for purposes of propagation of uncertainty to the other instruments and artifacts
2 − γ¯ λ
2 1/2 m 8 (zj − zj−λ )f + − γ¯ λ λSsp
where λ is an integer corresponding to the multiple of measurement increments (foot spacing) used for the separation distance in the calculation of the standard uncertainty, f is an index and subscript corresponding to the line of measurement (Fig. 1), and γ¯ λ is the average slope given by:
n m l 2 5 8 (zk − zk−λ )f (zi − zi−λ )f (zj − zj−λ )f 1 γ¯ λ = + + 2n + 3l + 3m − 8λ λSd λSlp λSsp f =1k=λ
is the surface slope and the uncertainty associated with surface slope. A slope can be calculated from any pair of residual height values as the height difference divided by the distance between the two points. With many pairs of height values available, an average surface slope and standard uncertainty
(46)
f =6 j=λ
f =3 i=λ
(47)
f =6 j=λ
This is a type A evaluation of uncertainty which appropriately includes the random portion of the uncertainties of residual height values but excludes the systematic portions. This result is calculated using example data and shown in Fig. 11. More complicated schemes are certainly possible to characterize the surface and may be justified for some
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applications. However, the proposed results can readily be used to assess uncertainty in subsequent calibrations that use the surface together with other mechanical artifacts and equipment. 5. Conclusions For electronic levels, the effects of interaction between carriage feet and the surface to be measured are small relative to the calibrated accuracy and the thermal stability of the instruments. The uncertainty of the height values resulting from the data fit to the reference plane is a function of the position in the plane. Therefore, the uncertainty in flatness depends on the locations of “high” and “low” height values. The maximum uncertainty exists at the midpoints of the bisectors. The uncertainty of height values is proportional to the chosen foot spacing and proportional to the squareroot of the number of steps. A general expression for uncertainty in flatness measurement was derived which bounds a significant collection of data comprised of closure values from actual measurements. A simple method is proposed for reporting of surface slope to facilitate the use of surface plate measurement data in the assessment of uncertainty for calibrations performed on the surface. References [1] Moody JC. How to calibrate surface plates in the plant. The Tool Engineer: official publication of the American Society of Tool Engineers 1955;October:85–91.
[2] Federal Specification GGG-P-463c. Plate, surface, (granite) 1973; September 10. [3] Bruin W, Meijer J. Analysis of flatness measurement and form stability of a granite surface plate. Ann CIRP 1980;29/1:385–90. [4] De Vicente J, Sanchez Perez AM. Uncertainty estimation methods by simulation in flatness measurements. IMEKO Bull 1991;20/1:1056–61. [5] Meijer J. Accuracy of surface plate measurements—general purpose software for flatness measurement. Ann CIRP 1990;39/1:545–8. [6] Haitjema H, Meijer J. Evaluation of surface plate flatness measurements. Eur J Mech Eng 1993;38/4:165–72. [7] Haitjema H. Iterative solution of least-squares problems applied to flatness and grid measurements. In: Ciarline P, Cox MG, Pavese F, Richter D, editors. Advanced tools in metrology, vol. II; 1996. [8] ISO Guide. Guide to the expression of uncertainty in measurement, ISBN 1993;92-67-10188-9. [9] Haitjema H. Uncertainty propagation in surface plate measurements. In: Proceedings of the Fourth International Symposium on Dimensional Metrology in Production and Quality Control. ISM QC, Tampere; 1992. p. 304–20. [10] ANSI Y14.5. Dimensioning and tolerancing; 1994. [11] ISO 8512-2. Surface plates—part 2: granite; 1990. [12] ISO 230-1. Test code for machine tools—part 1: geometric accuracy of machines operating under no-load or finishing conditions; 1996. [13] ISO 14253-1. Geometrical product specifications (GPS)—inspection by measurement of workpieces and measuring equipment—part 1: decision rules for proving conformance or non-conformance with specifications; 1998. [14] Drescher JD. Assessment of machine tool accuracy using ISO guide to the expression of uncertainty in measurement (GUM). In: Proceedings of the ASPE; 1997. [15] Mahr-Federal Calibration specification for electronic levels and electronic amplifier Model EMD-832. [16] Unpublished data. [17] Tartakovsky A. Personal communication; 7/26/2002. [18] Ludicke F. Intercomparison of flatness measurements. EUR 14059 EN; 1992.