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NBS Technical Note 1246 I ^ i^ ^

A New Statistical Model for the

Calibration of Force Sensors

Charles P. Reeve

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

June 1988

.'•^^

U.S. Department of Commerce

C. Waiiam Verity, Secretary

National Bureau of Standards

Ernest Ambler, Director
\

National Bureau of Standards U.S. Government Printing Office For sale by the Superintendent

Technical Note 1246 Washington: 1988 of Documents,

Natl. Bur. Stand. (U.S.), U.S. Government Printing Office,

Tech. Note 1246 Washington, DC 20402

46 pages (June 1988)

CODEN: NBTNAE

1
Contents

1 Introduction 1

2 Preliminaries 2

2.1 Previous Methods 2

2.2 Loading Sequences 3

2.3 Sources of Error 4

2.4 Preparation for Measurement 6

3 Features of This Method 6

4 Statistical Analysis 8

4.1 Statistical Model . 8

4.2 Analysis of Difference Data 11

4.3 Analysis of Mean Data 13

4.4 Summary of Error Contributions 15

4.5 Confidence and Prediction Intervals 16

4.6 Inverse Prediction 18

5 Example 21

6 Simulation Results 24

7 Optimality Considerations 25

8 Conclusion 27

References 28

A Definitions of the F and t Distributions 30

B Output of FCAL88 Computer Analysis of Example Data 32

111
List of Figures

1 Mean data with fitted straight line 5

2 Residuals from linear fit to mean data 5

3 Residuals from quadratic fit to mean data 5

4 Residuals from cubic fit to mean data 5

5 Difference data with fitted polynomial—run 1 7

6 Difference data with fitted polynomial—run 2 7

7 Difference data with fitted polynomial—run 3 7

8 Combined difference data with fitted polynomials 7

9 Single use inverse prediction on same machine 19

10 Multiple use inverse prediction on same machine 19

11 Multiple use inverse prediction on different machine 19

12 Three types of 95% confidence intervals (and original data) 19

13 Probability density function of F(5,18) distribution 31

14 Cumulative distribution function of F(5,18) distribution 31

15 Probability density function of t(10) distribution 31

16 Cumulative distribution function of t(10) distribution 31

List of Tables

1 Estimators of known sources of error 15

2 Comparison of two fits to the example data 22

3 Partial listing of 95% confidence intervals for the example data 23

4 Original and transformed load values for the example data 24

5 Computed 'best' polynomial degrees for 10,000 simulated data sets 25

6 Comparison of unsealed variances of the fitted coefficients 6k for several load

grids with n = 10 26

IV
A New Statistical Model for the Calibration of

Force Sensors

Charles P. Reeve

Abstract

The National Bureau of Standards has been calibrating force sensors for many

years. The objective in these calibrations is to determine the functional relationship

between the applied load and the sensor response. In a typical calibration several runs

are made in which identical sequences of known loads are applied to the sensor. The

sensor is rotated in the loading machine between runs. The previous method of analysis

incorporated a quadratic polynomial model which was fit to the pooled data. The new

method presented here fits separate polynomials to the mean data and between-run

differences. The 'best' degrees of these polynomials are automatically determined by

algorithms which incorporate statistical tests. As a result, error contributions from

several sources are quantized. Methods for computing confidence intervals for the

'true' sensor response and for a new observation are given, and methods of inverse

prediction (measurement of an unknown load) based on these intervals are illustrated.

Key words: calibration curve; confidence interval; force sensor; inverse prediction; least

squares polynomial; load cell; proving ring; statistical model

1 Introduction

The National Bureau of Standards (NBS) has been calibrating force measuring devices for

many years. The current status of these calibrations has been described by Mitchell [13]. In

that paper the generic term 'force sensor' is used to include '... instruments and systems that

are variously referred to by names such as load cell, proving ring, force gage, force link, force

transducer, load ring, ring dynamometer, compression dynamometer, tension dynamometer,

and crane scale.' Hereafter the generic term will be used except in reference to a specific

type of sensor. These sensors are normally used to measure forces in the 10 to 1,000,000

pound-force (Ibf) range. Their response curves typically look linear to the eye as shown in

figure 1. The objective in calibrating a force sensor is to determine the functional relationship

between the applied load and the sensor response. This is accomplished in the laboratory

by applying a series of known loads to the sensor and observing its response on a readout
instrument. The calibration is usually complicated by the interaction of the sensor with the

loading machine as discussed by Mitchell and Pontius [15]. Also, the sensor may creep or drift

under sustained load (see Mitchell and Baker [14]) and it may be sensitive to atmospheric

conditions such as temperature and pressure. A general discussion of the above problems

and others is given by Pontius and Mitchell [19].

A series of interlaboratory comparisons of several force sensors over a seven year period

has been reported by Peterson, Jenkins, and Mitchell [18] (following a preliminary report

by Peterson and Bloss [17]). In general the force sensors exhibited reasonable short term

stability, but there was some evidence of long term drift which indicates the need for their

periodic recalibration.

The purpose of this paper is to present a new statistical model for the force sensor

calibration process which takes into account errors from several known sources. The model,

based on variable-degree polynomials, is designed to extract as much information as possible

from data taken in the traditional manner. By quantifying the various kinds of errors, the

underlying processes may be better understood as more calibration results are analyzed by

the new model.

In developing the new model many sets of force sensor data were analyzed. As more data

are collected and analyzed the model may need to be modified. For this reason the methods

in this paper should not be construed as the ultimate answer to the force sensor calibration

problem.

The mathematical and statistical aspects of the model are described in considerable

detail in section 4. A nontechnical outhne of the new model is given in section 3. Some

prehminary discussions of other aspects of force sensor cahbration are condensed from the

previously referenced papers and given in section 2. The reader would do well to be familiar

with those references, especially [13], beforehand.

An example with real data is given in section 5, and the results of a simulation study

based on this example are reported in section 6. A question of optimality in the choice of

applied loads is considered in section 7. The FORTRAN computer program FCAL88 has been written to perform the analyses

described in this paper. The program is self contained and has been successfully run on both

mainframe and personal computers. A copy of the program can be made available to the

user upon request.

2 Preliminaries

2.1 Previous Methods

In 1946 Wilson, Tate, and Borkowski [25] published a paper on the calibration of proving

rings in which 'calibration factors', the ratio of the load to the corresponding deflection, were computed. Corresponding to these factors were 'specification limits' which served as uncertainty bounds. The factors were not quite constant with load, thus indicating a slight
nonlinearity in the proving ring response. Errors from various sources and loading sequences were also discussed.

In 1964 Hockersmith and Ku [11] developed a statistical model for proving rings in which

the response (the pooled runs) was taken to be a quadratic function of the applied load.

A method for computing a confidence band for the 'true' underlying response function was

given. This model proved satisfactory and was incorporated into the calibration process for

both proving rings and load cells.

2.2 Loading Sequences

Many force sensors can be calibrated in both tension and compression modes. The response

is expected to be somewhat different in each mode. Due to hysteresis effects the response

may also depend on whether the loads are applied in ascending or descending order. The

device may thus have several distinct cahbration curves. The ASTM Standard [1], currently most frequently used at NBS, requires at least thirty

measurements over the working range of the device during calibration. At NBS the calibration often consists of two runs at fifteen loads or three runs at ten loads. In the latter case the

applied loads are typically at 10%, 20%, 30%, ..., 90%, and 100% of capacity. Between runs

the sensor is rotated in the loading machine. This rotation allows errors from misalignment

to be sampled.

It is necessary to make measurements at zero load in order to define the basehne for the

readout of the sensor. Since conditions under zero load can be different than under nonzero

loads, those readings have to be treated differently. The manner in which they are treated

depends somewhat on the characteristics of the loading machine. Some deadweight machines

require weights to be added sequentially while others require a return to zero between loads.

In general a run will consist of a series of readings at zero and nonzero loads. From these

readings a set of deflections can be computed for each of the nonzero loads adjusted for

instrument drift to a zero baseline. The manner of adjustment depends on the theorized

drift behavior of the instrument between zero loads. If the instrument is assumed not to

drift then an initial zero load reading is subtracted from the following nonzero load readings.

If a linear drift is assumed then nonzero load readings are adjusted proportionally according

to 'before' and 'after' zero load readings. It will be assumed that all loads are equally spaced

in time so that drift corrections can be easily computed. It is imperative that the same

loading sequence be used for each run. In [19] a quadratic curve is said to be fit to the y—yo data. This means that readings y were

taken at various loads and the reading of the instrument under zero load yo was subtracted

from it. In cases where a particular yo value or a combination of yo values is subtracted from

more than one y value the resulting y — yo values are correlated. Previous fitting procedures

have ignored this correlation by assuming that all the y — yo were independent. An attempt

to incorporate these correlations into the new model was made. However, the computational

complexities introduced were felt to outweigh the aesthetic value in 'correctly' modeling the
observations. As a result the y — yo values will continue to be assumed independent in the

new method of analysis.

2.3 Sources of Error

The method of analysis in [1,11,19], discussed earher, is to pool all the runs and fit a quadratic curve to the combined data. Upon examining hundreds of these fits to load cell data the

two most often seen phenomena are a serpentine (S'-shaped) curve in the residuals, shown in

figure 3, and a fanning out of the residuals from each run as shown to some extent in figure

8. The serpentine curve is thought to be the result of model error, that is, the quadratic curve insufficiently modeling the actual underlying sensor characteristic. At this point there

is no solid theoretical basis for stating what the response should be.

In this study many mathematical models were tried in hopes of finding a general curve which would adequately fit the vast majority of calibration data. These models include

quadratic and higher polynomials, half power polynomials, negative power polynomials, and

exponential, sinusoidal, logarithmic, and Bessel functions. In specific cases each of these

has provided a good fit, but none has been found satisfactory in general. The most success was had with quadratic, cubic, and quartic polynomials. The cubic term was found to be

the most effective way of removing the serpentine curve in many cases. The decision was

thus made to base the new method on seeking higher order polynomial terms which will

significantly decrease the residual error. The fanning out of residuals is due to between-run error, referred to in [15] as consisting

of 'zero shift' and 'load proportional' components. This error is believed to be due to changes

in mechanical alignment (or misalignment) when the sensor is rotated, with the magnitude

of the error being dependent on the sensor/ machine combination. Ideally, the net applied

load is parallel and symmetric with respect to the axis of the sensor, and the sensor response

function is constant over time. In reality, however, small components of non-axisymmetric

forces exist whenever a load is applied. The sensor response depends on the magnitude of

those forces, the sensitivity of the sensor to those forces, and the orientation of the sensor in

the loading machine. Between-run errors appear to take the form of separate polynomials of

the same degree but possibly different coefficients.

A study by Mitchell, Seifarth, and Reeve [16] showed that eccentric (off-center) loading

results in sinusoidal differences in response as a function of angle of orientation of the load

eccentricity. By making runs with the sensor equiangularly spaced around the circle, the

sinusoidal components of error tend to cancel out when the runs are averaged.

Other sources of error are generally of smaller magnitude and easier to model. The ran- dom fluctuations in the output under repeated loads of the same magnitude are another

kind of error. These may be due to many separate physical processes which are ordinarily

not measured during a calibration, for example, fluctuations in atmospheric conditions, vi- brations, and oscillations of the dead weights. The effect of all these short term variations

combine to give what looks like random 'noise' and can be called within-run error.
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When the instrument readout is too coarse a roundoff error results. This type of error can easily mask the effects of other errors. Finally, errors in the accepted dead weight values whether due to calibration error or air

buoyancy effects are a source of systematic error. A term to account for this may be added

at the end of the calibration. The value for this error at NBS is estimated not to exceed

0.002%. This term is not added in the ASTM method of analysis.

2.4 Preparation for Measurement

The force sensor is mounted in the loading machine in either the tension or compression mode.

Several 'warm-up' or 'pre-loading' runs are then made in order to stabilize the response of

the device and reduce hysteresis effects.

3 Features of This Method

The purpose of this section is to describe in nontechnical terms how the new method of

analysis works. The reader who is interested in the details of the statistical model and

analysis will find those in section 4. The five main features of the method are:

1. A polynomial response model is fit to the mean data (the average of all the runs).

Subject to certain limitations the degree of this polynomial is determined from the

data. The upper limit on the degree of the polynomial is set by the user. Figures 2,

3, and 4 show residuals from linear, quadratic, and cubic fits respectively to the mean

data from the example in appendix B (note the different vertical scales). The cubic fit was determined to be 'best'. In this case the upper limit on the degree was five.

2. A polynomial response model is fit to each set of difference data (the difference between

the individual runs and the mean data). Subject to certain limitations the degree

of these polynomials is automatically determined from the data. The degree of each

polynomial is constrained to be the same, and as before the upper limit on that degree

is set by the user. For the example the difference data is shown in figures 5, 6, and

7 with a straight line fit to each run. The combined data are shown in figure 8. The

linear fit was determined to be 'best'. The upper limit on the degree was three.

3. A rough error budget is computed for each of the five sources of error discussed in

section 2.3. This summary may prove to be useful in correlating the error structure of

the data with the underlying physical processes. The error budget for the example is shown in table 2 in section 5 in the k*—3 column.

4. Three types of confidence intervals are computed and their proper usage is discussed.

These intervals provide the user with a reahstic estimate of the uncertainty associated with the calibrated force sensor response function, and they allow the reahstic

uncertainty of further use, such as inverse prediction, to be computed.

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by the cahbrated force sensor. The uses of these methods are graphically illustrated in

figures 9, 10, and 11.

4 Statistical Analysis

4.1 Statistical Model

The sensor response to axial forces may be considered fixed throughout the calibration, but

the response to non-axial forces must be considered random because it changes unpredictably

whenever the sensor is re-oriented in the loading machine. The fixed response to axial loads

Li is modeled by the A;-degree polynomial

vk{Ld = J:m (1)

where k is usually two or three. During the j^^ orientation of the sensor [j^^ run), the random

response to non-axial forces is modeled by the A-degree polynomial

h qhj{Li) = Yl UtjL\

t=o where h is usually zero or one and j indexes the runs. Though assumed to be drawn randomly

from a population, the {wtj|i = 0,1,..., /i} are treated as fixed effects for the duration of

the j^^ run. In some cases the random effects may prove to be not significant, in which case h is taken to be —1 and the {ufj} do not exist.

The following notation will be used in the statistical analysis which incorporates these

polynomial models. Let

n = the number of distinct nonzero loads applied per run,

r = the number of runs (sequences of n loads),

Li = the i*'' nonzero load in the loading sequence

{i = l,2,...,n),

yij = the observed (net) value of the instrument output

when load Li is applied during run j [j = 1, 2, . . . , r),

^t — the coefficient of L] in the fixed response

(^ = 0,1,...,/:),

Utj — the coefficient of Lj in the random response

(^ = 0,l,...,A;^- = l,2,...,r),

k = the degree of the fixed response polynomial,
h = the degree of the random response polynomials,

k-min = the lower bound on k (usually 1),

hmin — the lower bound on h (usually —1),

kmax — the upper bound on fc, hmax — the upper bound on /i, and

tij = the random 'noise' in the instrument output

when the i^^ load is apphed during the j^^ run.

The statistical model then takes the form

Vij = Pk{Li) + qhj{Li) + €ij = f3o + i3iLi + ...-{- pkL\ + uo, + wi,L, + . . . + UhjL^i + e.-j (2)

where the polynomial degrees k and h [k > h) are to be determined from the data by

statistical tests. The {e,j} are assumed to be independent error values from a distribution

with mean zero and variance a^. (Recall from section 2.2 that the y,j are assumed to be

independent even though correlations may have been introduced through corrections for

instrument drift). The vectors of random coefficients (uqj Uij . . . u^j)^ {j = 1, 2, . . . , r) are assumed to be independent with expectation (0 ... 0)^ and variance S^, an {h-\-l) x (A + 1)

symmetric positive definite matrix. (The 't' superscript indicates matrix transposition.)

The 'best' degree polynomial fit to data may be determined either by starting with a low

degree polynomial and adding higher order terms which are significant, or by starting with

a high degree polynomial and deleting higher order terms which are not significant. After

due consideration the method of 'deleting terms', as recommended by Hager and Antle [9], was determined to be the most appropriate for computing the 'best' values of both k and h

(also see Hoel [12]). This requires the user to establish the upper bounds kmax and hmax (on

k and h respectively) for the initial fits. A convenient way to analyze model (2) is to work with the independent sets of observations {yi} and {zij} derived from the {yij} by

r

Vi = ^Vijlr (i = 1,2, ...,n) and

Zi] = Vi] - Vi (i = 1,2, ...,r).

These sets can be identified as the mean data and the difference data respectively.

In matrix notation the m,odel for the mean data is y=:X;9+XiU + 6 (3)

where

y = {h y2 Vn)^,
p =

u

(^0 Pi .

{uo Ui . - Y with ut = ^utj/r,

= (ei €2 e„)^ with tt = ^ctj/r,

and

X = (Xi X2) -

/ 1

1 L2

h+l

V 1 Ln

Li I Li

Lfi I L^+'

Li I L^^

L', LU

In statistical terminology (3) is a 'mixed' model because of the presence of fixed (^) and

random (u, e) effects. It follows that E(y) = X;3 and Var(y) = (XiE^Xf+crel)/^ = ^ where

I is the n X n identity matrix and V is an n x n covariance matrix which is positive definite

when al > [E(-) and Var(-) denote the statistical expectation and variance]. Giesbrecht and

Burns [8] show that, when estimates of S^ and a^ are available, the generalized least squares

estimate of the fixed effects is ^ = (X^V-iX)-iX^V-iy with Y^r{^) = ±^ = (X^V-^X)"!

where V = (XiSyXf + alT)/r. In the 'large sample' case the estimate /3 is known to have a

smaller variance than the ordinary least squares estimate. (Methods for testing hypotheses,

computing confidence intervals, and approximating degrees of freedom are also given in [8]. For more discussion of the mixed model see Cunningham and Henderson [5], Henderson [10],

and Searle [21;22, ch. 9-10].)

In section 4.2 it will be shown that the variance estimate S^^ with r— 1 degrees of freedom

is obtainable from the difference data. Current practice at NBS is to make three runs, thus

Su is based on only two degrees of freedom. Although the number of runs could be increased

some, it is not feasible to make r — 1 'large'. Since bad estimates of /3 could result from using

Su based on very few degrees of freedom, the above generalized least squares approach will

be set aside in favor of ordinary (unweighted) least squares.

For the purpose of computation the {ufj} are treated as fixed effects, thus model (3) for

the mean data can be re-expressed as

y = X« + 6 (4)

where

e = (^0 + Uo f3i + ui = {9q 9i . .. 6k)

Ph + Uh /3h+i (3h+2 • •• /^kY

(5)

and € is as before. In this formulation the mean effect w^, treated as a fixed effect, gets

'absorbed' into the effect /?<(< = 1,2, ..., /i) thus only the net effect 6t can be estimated. It

10
follows that

E(y) = X^, and

Var(y) = lal/r.

This model will be analyzed in section 4.3.

The model for the difference data is

where

Zi =

Z2

V z, /

/ ^1. \

/ Xi

Xi

V Xi J

d2

Vd, J

+

62

, d, =

V Znj J

( Uoj — Uo \

Uij — Ui

\ Uhj - Uh I

1 ^j —

e2j — £2

(6)

(7)

and the zero matrices (0) are n x [h -{- 1). Treating the differences dj as fixed effects, it follows that

E(zj) = Xidj and

Var(z,) - Ial{r-l)/r (j = 1,2, . .

. , r). (8)

This model is analyzed in the next section.

4.2 Analysis of Difference Data

Applying the usual method of unweighted least squares (see Cameron [2, pp. 24-26]) to (8),

for a given h > 0, yields the least squares estimates

-Tv \-ivT d,-(Xi^Xi)-^X,^z, (i = l,2,...,r). (9)

Recall from section 4.1 that the between-run effects modeled by {utj} may not always be

significant. When that is the case there are no {dj} to estimate, since model (7) is degenerate,

and h is taken to be —1.

The total residual sum of squares, incorporating the r systems in (7) and (8), is

CO (h\ - ! ^U(^i - Xid,)'"(Z; - X,d,) (h > 0)

so that

E[SSM] = {n-h-l){r-l)a

(10)

(11)

11
Note that the factor r — 1 appears in (11) instead of r since the means {e,} were subtracted

from the error values {cjj} in (7).

The following algorithm is used to determine from the diiference data the 'best' degree,

h*, of the polynomials qhj{Li) modehng the {utj} effects. Statistical hypothesis testing at a level of significance Qh is used to determine whether, for a given h^ the terms {uhjL'l\j =

1,2, ... ,r} are needed. The /i^ax-degree model is fit first, then insignificant higher degree

terms are dropped r at a time. The constraint hmin ^ h* < hmax is applied where hmin = — 1- Algorithm H.

1. oGZ fl ^ ri-max2. \i h = hmin then step 5. Else,

3. Set Ve = {n — h — l)(r — 1) and compute the statistic ^ ^

[SSe(fe-l)-SSe(/^)]/(r-l)

SSe{h)/l>e

which has the F(r — 1, i/g) distribution under the hypothesis Uhi — Uh2 = . . . — Uhr = 0. Use (9) and (10) as appropriate.

4. If Gf(r_i,i/^)(iVi) < 1 — a/i set h = h—1 then step 2 (the left hand side of the inequaUty

is the cumulative distribution function of the F distribution with parameters r — 1 and

i^e as defined in appendix A). Else,

5. Set h* = h,

u, = {n-h*-l){r-l) (12)

<7,2 = SSe(/.*)/z/e, (13)

and, if h* > 0,

±u = td,dj/{r-l). (14)

i=i

6. Done.

When h* > the estimated mean squared error for between-run effects is computed from

(10) and (11) to be

.2 SSe(-l)-SSe(/^*)

The statistic F^, — ^l/crl then has the F[{r— l){h* + 1), Ve] distribution under the hypothesis

{utj = 0\t = 0,1,..., h*]j = 1,2, ...,r}. It may seem strange to calculate this statistic

when the fact that h > already implies that the between-run effects are significant as determined by the statistic Fh in step 3 of algorithm H. The usefulness of Fb is that it tests the significance of the r{h* -f 1) nonzero {wfj} effects as a whole whereas F^ tests only r of

the {utj} at a time. The level of significance of Fj, may then be displayed as the significance

of the between-run effects.

12
4.3 Analysis of Mean Data

Applying the method of unweighted least squares to (4) and (6), for a given k > 0, yields

9 = (X^X)-iX^y (16)

where

Var(^) = S^ = (X^X)-V,Vr. (17)

After fitting the effects the remaining components of error in (4) are the mean noise effects

€ and possible 'model error' due to the exclusion of significant higher order terms. In the

manner of Draper and Smith [4, pp. 26-30] the model error /z, at load Li is taken to be

fii = Ti - E[pk{Li)] (i = 1, 2, . .

. , n)

where Ti is the 'true' device response to the axial load Li and pk{Li) is as in (1). If the model

is correct then ^, = (i = 1,2, . . . ,n). The individual model error values {/ij} cannot be

estimated (since the true values {r,} are unknown), but their variance cr^ = ^7=i A'?/'^ ^^^ be estimated as will be shown later (/i = Z^iLi f^i/''^ is assumed to be zero).

After obtaining least squares estimates for by (16), under the /^-degree model, the

residual sum of squares in (4) is

SSe^(A;) = (y-X0)^(y-X^) (18)

so that

The following algorithm is used to determine from the mean data the 'best' degree, k*, of

the polynomial pk{Li) modeling the 'true' response curve. Statistical hypothesis testing at

levels of significance am and ak is used to determine whether, for a given k, the term OkL'- is needed. The A;^a2.-degree model is fit first, then insignificant higher degree terms are dropped

one by one. The term OkL'- is dropped if either 6k or the model error under the {k — 1)-

degree model is determined to be not significant. The constraint max{/i*, kmin} ^ k* < kmax

is applied where kmin = 1- It is fully expected that for some force sensors the model error cannot be eliminated.

Algorithm K.

i. oet K '= rZrnax'

2. \ik < max{/i*, kmin} then step 7. Else,

3. Set Vem = n — k — 1 and compute the statistic

Frr. =

^l

which has the F{vemi ^e) distribution under the hypothesis cr^ = 0. Use (16) and (18).

13
4. If GF(^^^^t,^){Fm) < 1 - a^ or k = k^ax set k = k - I then step 2. Else,

5. Set fc = A; + 1, Vgrn = n — k — \^ and compute the statistic

which has the F(l, Vem) distribution under the hypothesis 6^ = 0.

6. If GF(i^t,^^)[Fk) < 1 — Qjt set k = k — 2 then step 2. Else,

7. Set A;* = k,

^em — n — k* — I, and (19)

aL = rSSem{k*)/uem- (20)

8. Done.

Having determined h* and k*, the 'design' matrix X is fixed and is computed by

(16). The estimated variance of 9 is obtained by substituting the proper estimate of a^ into

(17). When a^^ is significantly larger than cr^, indicated by a large value of Fm in step 3

of algorithm /i, then a^^ should be used so that the eifects of significant model error are included in the estimated variance of 9 , thus

±^ = {X'Xr'crlJr. (21)

(Note that model errors are actually biases, and the quantity al^ contains the mean squared

model error as a reasonable estimate of bias effects.)

When (7gj^ is not significantly larger than a^ the two values may be pooled to give

where

Vep = fe -\- t^em = r{n - h* - 1) + h* - fc*, (22)

thus

S^ = {X-^Xy^a^/r. (23)

Recalling from (5) that

r A + wt {o<t<h*)

' \ (3t {h* <t< k*) (24)

and that E(w<) = 0, an estimate of /9 is h ^9 (25)

with

Vrr(^) = S^ = S, + i(^" °) (26)

where S^ is computed from either (21) or (23) as appropriate, S^ is as in (14), and the zero matrices (0) are of the proper size to make the rightmost matrix in (26) (A;* + 1) x {k* + 1)-

14
4.4 Summary of Error Contributions

To recapitulate from section 2.3, the five types of error considered are

1

roundoff,

2. within-run,

3. model,

4. between-run, and

5. systematic.

A review of past calibrations of force sensors at NBS has shown that any of these sources of

error can be dominant. It would seem to be useful then, for a given calibration, to estimate

the relative contribution of error from each of the sources. One measure of error from a

given source is a 'cr-like' value to represent the 'average' contribution of that source to a

single observation. Estimates of such values can be computed from the data for the first four sources. For the fifth source an absolute limit {E) is used for combined effects of all 'systematic' errors as described in section 2.3. It is not a 'cr-like' value.

In table 1 the sources of error are listed along with their 'cr-hke' estimators (5). The

5-values are for rough comparison of sources of error only. They are not intended to be

'corrected' values of standard errors previously computed. (In the past some methods,

which are valid under certain conditions, have been given for 'correcting' error variances

for roundoff error. See Eisenhart [6].) Hopefully the 5-values will aid the experimenter

in understanding the characteristics of the force sensor-machine combination and perhaps

indicate areas where improvements could be made.

Table 1: Estimators of known sources of error Error type Error 'estimator' Reference eqs.

roundoff Sr = ^^i:u^l (27)

within-run 5e = ^max{0,a2-52} (13,27,28)

model S^ = -ymaxjO, 5-2^ - a^} (13,20,28,29)

between-run S, = yj{h* -f l)al/n (15)

systematic E = combined limits

At this point the method of determining roundoff error must be addressed. Often data

from a digital readout is inherently rounded. In recording the data the user may round

even further. Traditionally (see Snedecor and Cochran [24, p. 81]) the rounding error, /), is treated as a uniformly distributed random variable in the interval [—A/2, A/2] where A is the smallest possible difference between two unequal numbers. It follows that E(p) = and

15
Var(p) = A^/12. For the sake of generality suppose that the rounding interval of the readout

device is A, at load L,- and that pij is the rounding error corresponding to the observation

yij. Then the 'mean squared' rounding error is

In models (4) and (7) the {tij} values were said to represent the 'noise' in the measurement

system. In fact, each e,j also includes the roundoff error pij.

The quantities in (13) and (20) can be split into components

al

^ 5,2 + 5,2 (28)

^L - 52 + 52 + 5^ (29)

to yield estimates of 5^ and 5, free of roundoff error. The variance for between-run effects,

(jj, in (15) reflects the 'average' variance per nonzero Utj. Multiplying this by [h* + 1)/?^

yields an 'average' variance per observation, S^.

4.5 Confidence and Prediction Intervals

The final step in the calibration process is to compute uncertainty bounds for the quantities

of interest. In the present case those bounds are

1. a confidence interval for the 'true' sensor response at a given load,

2. a confidence band for the 'true' sensor response curve over its entire range, and

3. a prediction interval for a new, independent observation with the sensor randomly

oriented in the same machine.

The bounds in 2 and 3 may further be used for inverse prediction as discussed in section

4.6. Seber [23, ch. 5] serves as a reference for the computations which follow. In each of the

above cases formulas are given for two-sided 100(1 — S)% bounds where 5 is a probability in

the exclusive range (0,1).

Note that one item missing from the above list is the vector of fixed effects y3. In most

regression applications a confidence interval for (3 would be of interest, but here it is not.

If such an interval were of interest it could be formed using the quantities defined in (25)

and (26), but the question of the appropriate number of degrees of freedom to assign to the

variance in (26) would be difficult. A frighteningly complicated approximation is given in

[8]. Incorporating (24) the 'true' sensor response to the axial load L can be expressed

16
Taking E{ut) = the predicted sensor response is

k*

t=o = x'^e (30)

where A = (Ai As)^ = (1 L . . . L'^* | 1^'+' . . . L^'f, and

Var[pfc.(L)] = A^i^A +Aii„Ai/r - t/i + f/2. (31)

A confidence interval for the 'true' response at load L, pk*{L), is then

Pk*iL) ± ti_5/2{ui)\/y8,T[pk>iL)]

where Vi is the number of degrees of freedom associated with the estimated variance and

ti_s/2{^i) is a quantile of Student's t distribution with Ui degrees of freedom (see appendix

A). Since the estimated variance in (31) is a linear combination of two variances, an 'effective'

number of degrees of freedom is computed by the Welch-Satterthwaite formula (see Gaylor

and Hopper [7]) to be _

(U^ + U.r

"'-mive + UiKr-D ^'''

where

as in (19), if ^^ was computed by (21)

as in (22), '\i H q was computed by (23)

Note that \ih* = —1 then 17 ^ is degenerate, thus U2 = and ui = uq. This type of confidence

interval is appropriate, for example, if the sensor response at nominal maximum load is to

be used as a check standard. This type of interval is illustrated by the dotted lines in figure

12 for the example in appendix B.

A confidence band for the entire 'true ' response curve is taken to be the set (over L) of

all individual confidence intervals for Pk*{L) of the form

Pk*{L)±K^ (33)

where

Ki = yj{k* + l)F,_s{k* + l,i/i)Var[p,.(i:)] (34)

and Fi_s{k* -{- !, I'l) IS a. quantile of the F distribution with k* -\-l and Vi degrees of freedom

as described in appendix A. This type of band is illustrated by the dashed line in figure 12.

A third quantity of interest is a confidence interval for a new observation y* obtained

when the sensor is re-oriented in the same measuring machine and a load L* applied. The

new observation can be written

y*=Pk.{L*) + xfu + e*

17
where A* = (A^ A^) = [l L* . . . L*^' \ L*('^*+i) . . . L*'=*)^, u is a random {h* + l)-vector

from a distribution with mean zero and variance E „, and e* is an independent deviate from

a distribution with mean zero and variance a^. It follows that

E(?/*) = pAL*) and (35)

Var(y*) = Varb,.(L*)] + A*^17,At+a,2 = xfEuK + (^l- (36)

Replacing the expectation and variance of y* by their estimates and incorporating (30) and

(31) gives

y* = pk*{L*) = A*^^ and

Var(y*) = Var[p,*(L*)] + A^i,At + a^' = A*^f^A* + (l + i)Aff .a; + <7,2 (37) = V1 + V2 + V3.

As before the 'effective' number of degrees of freedom for the estimated variance in (37) is

(V. + V.

+

v^r . , "' Vfl„e + V}l{r-\) + V}lu,

^^'•'

where vq is as in (32) and i/g is as in (12). As before, if h* — —1 then l^u is degenerate and

(38) is computed with V2 = 0. A confidence interval for a new observation y* as defined

above is then

Pk>{r)±K2 (39)

where

K2 = h-si2{v;)\[^Mf)- (40)

This type of interval is illustrated by the soHd line in figure 12.

4.6 Inverse Prediction

The calibrated force sensor is ordinarily used to measure an unknown load L* from a single

new observation y* on some machine, not necessarily the one on which it was cahbrated.

The desired result is an estimate of the magnitude of the unknown load and a confidence

interval for it. This application, known as inverse prediction, consists of

• computing the calibration curve and the appropriate confidence band(s) for the sensor,

• obtaining one or more new observations at unknown load(s), and

• estimating each unknown load and its confidence interval.

18
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19
Three cases of inverse prediction with a calibrated force sensor will be considered here:

• a single use case on the same machine,

• a multiple use case on the same machine, and

• a multiple use case on a different machine.

The term multiple use simply means that the same calibration curve will be used for inverse

prediction more than once. This situation is beheved most likely to occur in practice, thus

the multiple use methods (described later in this section) are recommended.

The single use case will be given only for use on the same machine. A confidence interval

for a new observation y* can be obtained for any L by (39). So, given y*, a 100(1 — S)%

confidence interval for the unknown load L* is the set of L's for which the confidence interval

calculated by (39) encloses y*, i.e., for which

Pk'iL)-K2<y*<Pk'{L)-\-K2

where K2 is a^ in (40). This set will be connected when the upper and lower bounds of the

confidence band are monotonic. The estimate of L* is L* which satistifes

Pk*{L*) = y*.

If k* > 1 then L* may have to be computed iteratively. Figure 9 illustrates the graphical

procedure for determining L* and the confidence interval {L*_,L'^) for L*. Again, this type

of confidence interval for L* is valid only when a single inverse prediction is made after a given calibration.

In the m,ultiple use case [same machine) a well known but complicated method of inverse

prediction is that of SchefFe [20]. In a forthcoming paper Carroll, Sacks, and Spiegelman [3],

hereafter designated CSS, present a modification of the SchefFe method which is far simpler

to implement and generally leads to shorter intervals. The first part of the CSS method is to compute the 100(1 — 6)% confidence band for the entire response curve as in (33) and

(34). This band is illustrated in figure 11. The second part is to compute a 100(1 — a)%

confidence interval for the 'true' response pk*{L*) based on the new observation y*. The

estimated variance of y* is computed from (36) to be

Var(y*) = xfs^Xl-\-al = W^-^W2 (41)

which is 'effectively' based on

_

{w, + w^r ,,,.

20
degrees of freedom. If 17 „ is degenerate then Wi = and u^ = u^. Notice that the variance of

Pk*{L*) does not enter into this part of the computation. An appropriate confidence interval

for pk*{L*), incorporating (35) and (41), is

y" ± Ks (43)

where

Ks = ii-a/2(^3*)\/Var(r) (44)

Note that the variance term Wi in (41) is a function of L* (through A*) which is unknown.

In this case L* may be approximated by Pk}{y*) in the vector A^.

A confidence interval for the unknown load L*, using the CSS procedure, is then the set of all X's for which

Pk*{L) - lU - Ks <y*< PAL) + K, + K3 (45)

where Ki and K3 are as in (34) and (44) respectively. This procedure assures that, for a given cahbration process, the probability is 1 — ^ that the expected proportion of calculated

confidence intervals for L* that actually cover L* will be at least 1 — a in the long run. The

probabihty 1 — ^ refers to the probability that the calibration of the sensor is 'good', that

is, the confidence band covers the 'true' underlying response curve. Figure 10 illustrates the

graphical procedure for determining L* and the confidence interval (Z,1,L^) for L* in the

case under consideration.

In the multiple use case {different machine from the one on which the sensor was calibrated) the previous analysis may be applied using the variance estimates S u and a"^ appropriate to that machine. If these quantities are available then they, along with their degrees

of freedom which correspond to r — 1 and i/g, may be substituted into (41) and (42) to give

Ks as in (44). The prediction of L* and its confidence band then follow as before. In the

likely event that both of these quantities are not available, an approximate 100(1 — a)%

confidence interval for pk*{L*) of the form

y* ± K,

corresponding to (43) may be used where K^ is based on statistical evidence or 'engineering

judgement'. This interval is used in conjunction with (33) to provide an estimate L* and a

confidence interval {L*_^L*^) for L* as illustrated in figure 11. The probability statement for

the coverage is the same as in the previous case (fig. 10).

In CSS two modifications are discussed which will shorten the inverse prediction intervals

slightly, but for the sake of simphcity they will not be used here.

5 Example

In the process of developing the model described in this paper, many sets of force calibration

data were analyzed. The complete analysis of one of these data sets will now be illustrated.

The example chosen is typical in that

21
• three runs were made,

• the between-run error appears to be well modeled by linear pieces, and

the residuals from a quadratic fit appear to have a serpentine form.

The example was analyzed by the new method with hmax = 3, kmax — ^i och = 0.01,

am = 0.10, and a^ = 0.025. At each stage of the fitting procedure to determine h* the

probabiHty of overfitting was ah = 0.01, and the corresponding probability of overfitting

when determining k* was am<yk — 0.0025. The analysis was performed by the FORTRAN

program FCAL88 which incorporates the new method described in this paper. The output

from the program is given in appendix B. The net observations {yij} appear on the first page

of that output. The degrees of the 'best fitting' polynomials were determined to he h* = 1 and k* = 3. Figures 1 through 12 are based on this analysis of the example. In figures 9

through 11 the confidence intervals have been magnified by a factor of 2000 vertically for

the purpose of illustration.

The same observations were re-run with h^ax = k^ax = 2 in order to limit the fits to

nothing higher than a quadratic. In that case h* = 1 and k* — 2 were computed (the

computer output is not shown). Some results from the two analyses are compared in table

2 below.

Table 2: Comparison of two fits to the example data

Quantity k*=2 k*=3 Reference

h* 1 1 (Alg. H)

<3-e 0.120- 10-4 0.120- 10-4 Eq. (13)

<^em 0.754-10-4 0.204-10-4 Eq. (20)

00 1.03347 1.03347 Eq. (16)

0, 0.96743 0.96767 Eq. (16)

02 0.32926 - 10-3 0.32575 - 10-3 Eq. (16)

03 -0.33087 - 10-3 Eq. (16)

Sr 0.2887-10-^ 0.2887-10-^ (Table 1)

Se 0.1167-10-4 0.1167-10-4 (Table 1)

Sm 0.1301 - 10-3 0.3329-10-4 (Table 1)

Sk 0.2263 • 10-4 0.2263 - 10-4 (Table 1)

E 0.2013- 10-4 0.2013- 10-4 (Table 1)

Note that FCAL88 produces a two-page table with four kinds of 95% confidence/prediction

intervals. This allows inverse prediction to be done by interpolation on the appropriate

columns. For example, suppose an unknown load L* was applied (on the same machine)

and that an independent (net) observation y* = 1.47901 was obtained. The estimate of L*

linearly interpolated from the Predicted response and Load columns is L* = 221,757.2

22
(see table 3 which reproduces part of the two-page table from appendix B). Using columns

Table 3: Partial listing of 95% confidence interval? for the example data

Load

Predicted

response

Single use Multiple use

[3] [4

(lower) (upper)

[7] [8] (lower) (upper)

219,000 1.46061 1.46051 1.46071 1.46044 1.46078

222,000 1.48063 1.48053 1.48074 1.48046 1.48081

[3] and [4] the single use 95% confidence interval for L* is (221,740.9 , 221,772.2), and using

columns [7] and [8] the corresponding multiple use interval is (221,730.4 , 221,782.7). If uncertainty due to systematic errors is to be accounted for, each of these confidence intervals

would be increased by (0.002%)(221, 757.2) « 4.4 in each direction. (For this example the

load unit is pounds-force.)

For numerical stability FCAL88 linearly transforms the load values Li to the interval

[—1,1] before any computations begin (a good practice in polynomial regression). This

transformation is invisible to the user except for the computed values of S^ ('ESTIMATED

COVARIANCE MATRIX FOR BETWEEN-RUN EFFECTS', p. B-3), ('ESTIMATES OF POLYNOMIAL

RESPONSE CURVE COEFFICIENTS FROM MEAN DATA', p. B-7), and S^- ('ESTIMATED COVARIANCE MATRIX FOR POLYNOMIAL COEFFICIENTS', p. B-7). These three quantities are highly

dependent on the extreme values of the load grid,

Lmin = mmjLi, L2, . .

. , Ln} and Lmax = max{Li, L2, . .. , Ln] ,

which transform to —1 and 1 respectively. Consequently, values of Sl^, ^, and S^ obtained

from different calibrations are comparable only if Lmin and Lmax are the same in each case. The linear transformation to [—1, 1] is

2(L — Lmtn) -, X = 1 , T — T

and the inverse transformation is

J- V"^ I ^ )\^max ^min) |^ j-

(46)

(47)

The original and transformed load values for the example data are shown in table 4. The

virtue in transforming is seen when examining, for example, S^ as defined by (21) or (23).

This quantity contains the factor

U = (X^X)-^ (48)

which is called the unsealed variance of 0. Note that U depends only on the pre-selected

load grid, by (3), and not on the observed data, therefore it is an appropriate means for

23
Table 4: Original and transformed load values for the example data

Original Transformed

i L^ (Ibf) Xi

1 10,000 -1.000000

2 30,000 -0.862069

3 60,000 -0.655172

4 90,000 -0.448276

5 120,000 -0.241379

6 150,000 -0.034483

7 180,000 +0.172414

8 210,000 +0.379310

9 240,000 +0.586207

10 270,000 +0.793103

11 300,000 +1.000000

comparison. Using the Li values to form X, equation (48) yields

Ui

0.116-10+°' -0.295-10-°*

-0.295 - 10-°* 0.104 • 10-°^ 0.194-10-°^ -0.772-10-'*

-0.368 - 10-'^ 0.156 • 10-1^

0.194- 10-°^ -0.368-10-'^

-0.772-10-'* 0.156 10-'^ 0.610-10-'^ -0.128-10-24

-0.128-10-2* 0.277-10-3° (49)

Similarly, using the corresponding Xi values to form X,

U.

0.214-10+°° -0.149 -10-°2 -0.149 -10-°2 0.157-10+°' -0.296 - 10+°° 0.160 10-°2 -0.164 10-°2 -0.186-10+°'

-0.296- 10-°°

0.160- 10-°2 0.712-10+°°

0.272- 10-°'

-0.164 -10-°2 -0.186- 10+°'

0.272 - 10-°' 0.257-10+°' (50)

Note that the lower right entry in the right hand side of (49) approaches the underflow limit on some computers, a potentially unpleasant situation. A much better situation occurs when

computing with the transformed loads, as illustrated in (50). Transitions to and from the

original loads are easily made with (46) and (47).

6 Simulation Results

A simulation study was performed using the estimated quantities computed for the example

(by the new method) as a basis. Specifically, 10,000 sets of {yij} values were generated using

(2) where (/?o $i $2 $3) were substituted for (/?o /3i ^2 f^s), the {cij} were generated from

a normal distribution with mean zero and variance a^, and the {(woj ^ij)} were generated

from a multivariate normal distribution with mean zero and variance I! y,- No allowance was made for model error. Each set of simulated observations was run under the FCAL88

program. The number of occurrences of each computed (/i*,fc*) pair are given in table 5

24
Table 5: Computed 'best' polynomial degrees for 10,000 simulated data sets

h*

k*

1 2 3 4 5

-1 1683 2

1118 1

1 6949 23 17

2 104 1 3 99 3

above. The levels of significance used were ah = 0.01, a^ = 0.10, and ak = 0.025 as before.

Note that the entries in italics represent cases of overfitting which are controlled by the three

levels of significance. For example, the number of entries in the /i* = 3 row is 102 which

approximates the expected number 10,0000/1 = 100, and the number of entries in the /:* = 5

column is 20 which approximates the expected number 10,000a^Q'fc = 25. Similar numbers

of entries would be expected for the cases h* = 2 and k* = i respectively.

Underfitting is little cause for concern in this simulation. The fact that no cases oi k* = I or 2 were observed indicates that the coefficient ^^3 is highly significant in the example. On

the other hand, the coefficients of the linear terms in the between-run eifects are much less significant since they were determined to be not significant about 28% of the time.

The greatest value in this simulation exercise may be to forewarn the user that repeated

calibrations of the same force sensor may result in several different (A*, k*) pairs, even though

the underlying 'true' sensor response curve does not change.

7 Optimality Considerations

In a typical NBS calibration of a force sensor the applied loads are uniformly spaced between 10% and 100% of sensor capacity (see section 2.2). The question naturally arises as to whether a different grid of appHed loads, with the same number of points, would produce

'better' results in some sense. Seber [23, pp. 231-234] discusses several types of optimahty

criteria for polynomial regression of a fixed degree. Of those criteria the one of most importance in the present application would seem to be the minimization of the variance of

9k^ the fitted coefficient of the highest power of the load. In algorithm K, 6k is tested for

significance for one or more values of k. For a given load grid and polynomial degree k the 'design' matrix X is computed as in

(3). This matrix is then used in computing the vector of fitted coefficients as in (16) and

its variance as in (21) or (23). As discussed in section 5, the factor (X-^X)"^ is the unsealed

variance of 0, thus the unsealed variance of 0^ is the {k -\- 1 ^ k -\- I) entry of (X-^X)~^ (lower

right corner). This quantity is computed for polynomial fits of degree 2, 3, 4, and 5 for

each of seven different load grids as shown in table 6. (In this paper it is assumed that no

25
Table 6: Comparison of unsealed variances of the fitted coefficients 6^ for several load grids

with n = 10

Unsealed variance of 6^

Load grid (% of sensor capacity) k = 2 k = 3 k = A k^b

1 10.0 10.0 10.0 55.0 55.0

55.0 55.0 100.0 100.0 100.0 0.417 _ „ _

2 10.0 10.0 32.5 32.5 32.5

77.5 77.5 77.5 100.0 100.0 0.741 1.63 _

3 10.0 10.0 23.2 23.2 55.0

55.0 86.8 86.8 100.0 100.0 0.714 3.00 7.00

^ 10.0 18.6 18.6 41.1 41.1

68.9 68.9 91.4 91.4 100.0 0.800 3.20 12.79 25.6

5 10.0 12.7 20.5 32.5 47.2

62.8 77.5 89.5 97.3 100.0 0.740 3.01 12.19 49.3

^ 10.0 15.0 20.0 30.0 45.0

65.0 80.0 90.0 95.0 100.0 0.874 3.11 11.63 40.4

^ 10.0 20.0 30.0 40.0 50.0

60.0 70.0 80.0 90.0 100.0 0.777 2.69 10.21 43.7

^ Optimal load grid in 10%-100% range for 2""^ degree fit

2 Optimal load grid in 10%-100% range for 3'"'^ degree fit 3 Optimal load grid in 10%-100% range for 4*'' degree fit

^ Optimal load grid in 10%-100% range for 5*'' degree fit

^ Optimal load grid in 10%-100% range for 9^'' degree fit ^ Load grid ^ rounded to nearest 5%

^ Uniform load grid

polynomial higher than a

5*'' degree will be fit.) In order for the comparisons to be fair, each

grid contains the same number of points. The first five of these grids are optimal for the

indicated polynomial degrees. They are the so-called Chebyshev points,

Xi — — cosf — ) [i 0,1,..., A;),

linearly transformed from the interval [—1,1] to the interval [10%, 100%] times sensor ca- pacity. Note that in the case n> k -\-\ optimality is achieved by replicating measurements

at certain loads. In all cases the unsealed variance of d]^ was computed from the equivalent

loads in [—1, 1]. The first four grids provide the optimal values shown in boldface. The first three are of no

use since there are not enough distinct loads to fit all polynomials of interest. Examination

of the last four shows that none is consistently better than the others over the degrees

26
considered. By a slim margin the uniform grid appears to be best. Based on this brief

analysis there seems to be no evidence that any other load grid should be preferred over the

uniform.

8 Conclusion

Force sensors are not characterized by a single number but by a response curve over a wide range of loads. The precise form of this curve depends on the details of design and

construction of the particular sensor. After trying various mathematical models for the

response curve, it was concluded that variable degree polynomials generally provided the

best fits. A new statistical model for caHbrating force sensors, based on automatic procedures for

determining the degrees of polynomials, has been presented. It attempts to model several

known sources of error and quantize their relative contributions. Methods for computing

several types of confidence and prediction intervals have been given. The three levels of

significance used in determining the polynomial degrees may be adjusted to suit the users needs as may be the levels for the confidence/prediction intervals. Usage of the calibration curves for inverse prediction has also been discussed and illustrated. A brief optimality study

has shown that the traditional uniform spacing of applied loads cannot be improved on when

allowing polynomial fits of S*'* degree or less. This new method of analysis is not hkely to solve all the problems of force sensor calibration. The author believes, however, that this method will prove to be a useful tool in

developing a better understanding and a better characterization of their behavior.

Acknowledgement. The author wishes to thank two colleagues for their significant contributions to this paper. Stefan D. Leigh of the Statistical Engineering Division fit various

functions to many sets of force calibration data, and Richard A. Mitchell of the Automated

Production Technology Division provided valuable guidance in acquainting me with the force

calibration process.

27
References

[1] American Society for Testing and Materials, 'Standard Practice of Calibration of ForceMeasuring Instruments for Verifying the Load Indication of Testing Machines', E74-83,

Annual Book of ASTM Standards, 1984.

[2] Cameron, J.M., 'The Use of the Method of Least Squares in Calibration', National

Bureau of Standards, NBSIR 74-587, September 1974.

[3] Carroll, R.J., Sacks, J., and Spiegelman, C.H., 'A Quick and Easy Multiple Calibration

Curve Procedure', to appear in Technometrics, May 1988.

[4] Draper, N.R. and Smith, H., Applied Regression Analysis, John Wiley & Sons, Inc.,

1966.

[5] Cunningham, E.P. and Henderson, C.R., 'An Iterative Procedure for Estimating Fixed

Effects and Variance Components in Mixed Model Situations', Biometrics 24(1), March

1968, pp. 13-25.

[6] Eisenhart, C, 'Effects of Rounding or Grouping Data', Ch. 4, pp. 186-223 in Eisenhart, C, Hastay, M.W., and Wallis, W.A., Selected Techniques for Statistical Analysis,

McGraw-Hill Book Company, 1947.

[7] Gaylor, D.W. and Hopper, F.N., 'Estimating the Degrees of Freedom for Linear Combinations of Mean Squares by Satterthwaite's Formula', Technometrics ll(4)i November

1969, pp. 691-706.

[8] Geisbrecht, E.G. and Burns, J.C., 'Two-Stage Analysis on a Mixed Model: LargeSample Asymptotic Theory and Small-Sample Simulation Results', Biometrics 41(2),

June 1985, pp. 477-486.

[9] Hager, H. and Antle, C, 'The Choice of Degree of a Polynomial Model', Journal of the

Royal Statistical Society B 30(3), 1968, pp. 469-471.

[10] Henderson, Charles R., 'Analysis of Covariance in the Mixed Model: Higher-Level,

Nonhomogeneous, and Random Regressions', Biometrics 38(3), September 1982, pp.

623-640.

[11] Hockersmith, Thomas E. and Ku, Harry H., 'Uncertainties Associated with Proving

Ring Calibration', Precision Measurement and Calibration - Mechanics, National Bureau of Standards, SP 300, Vol. 8, pp. 239-246 (also. Instrument Society of America,

Preprint No. 12.3-2-64, 19th Annual ISA Conference and Exhibit, October 1964).

28
[12] Hoel, Paul G., 'On Testing for the Degree of a Polynomial', Technometrics 10(4).,

November 1968, pp. 757-767.

[13] Mitchell, Richard A., 'Force Calibration at the National Bureau of Standards', National

Bureau of Standards, Technical Note 1227, August 1986.

[14] Mitchell, R.A. and Baker, S.M., 'Characterizing the Creep Response of Load Cells',

VDI Berichte, Nr. 312, 1978, pp. 43-48.

[15] Mitchell, Richard A. and Pontius, Paul E., 'Force Sensor-Machine Interaction', Proceedings of the 27"^ International Instrumentation Symposium, Instrument Society of

America, 1981, pp. 225-232.

[16] Mitchell, R.A., Seifarth, R.L., and Reeve, C.P., 'Eccentric Loading Sensitivity of Force

Sensors', Proceedings of the \l^^ Conference of IMEKO TC-3 on Measurement of Force

and Mass, Amsterdam, The Netherlands, May 1986.

[17] Peterson, R.W. and Bloss, R.L., 'Interlaboratory Comparison of Force Cahbration Using ASTM Method E74-74, Phase P, National Bureau of Standards, NBSIR 76-1145,

August 1976.

[18] Peterson, Robert W., Jenkins, Linwood, and Mitchell, Richard A., 'Interlaboratory

Comparison of Force Calibrations Using ASTM Method E74-74', National Bureau of

Standards, Technical Note 1211, April 1985.

[19] Pontius, P.E. and Mitchell, R.A., 'Inherent Problems in Force Measurement', Experimental Mechanics 22(3), March 1982, pp. 81-88.

[20] SchefFe, H., 'A Statistical Theory of Calibration', Annals of Statistics 1(1), January

1973, pp. 1-37.

[21] Searle, S.R., 'Topics in Variance Component Estimation', Biometrics 27(1), March 1971,

pp. 1-76.

[22] Searle, S.R., Linear Models, John Wiley & Sons, 1971.

[23] Seber, G.A.F., Linear Regression Analysis, John Wiley & Sons, 1977.

[24] Snedecor, George W. and Cochran, William G., Statistical Methods, Sixth Edition, The

Iowa State University Press, 1976.

[25] Wilson, Bruce L., Tate, Douglas R., and Borkowski, George, 'Proving Rings for Calibrating Testing Machines', National Bureau of Standards, Circular C454, August 1946.

29
A Definitions of the F and t distributions

The probability density function (p.d.f.) of the F{m,n) distribution may be expressed as — m—

2

2 gF{m,n){^)

r(T)r(f)(i + ^)

2

where x > 0, m > 0, n > 0, and T{a) = J^ x°^~^e~^dx. This function is plotted in figure 13

for the case m = 5 and n = 18.

The cumulative distnbution function (c.d.f.) of the F{m,n) distribution may be ex- pressed as GF(m,n){^) - / gF(m,n){u)du

Jo where a: > 0. This function is plotted in figure 14 for m and n as above. Note that

G'F(m,n)(0) = and GF(m,n){oo) = 1- The p-quantile of the F{m,n) distribution, denoted by Fp{m,n), is the x value which

satisfies GF(^rn,n){x) = p.

' When an F test is performed either the c.d.f or a quantile must be computed. Since

the c.d.f. is easier to compute on a machine, that quantity has been incorporated into the

F tests of algorithms H (section 4.2) and K (section 4.3).

In similar fashion the p.d.f of the t{n) distribution is

2

where —oo < a: < oo and n > 0. This curve is plotted in figure 15 for the case n=10. The

c.d.f is

Gt{n){x) = / gt{n){u)du

•^ — CO where —oo < x < oo. Note that Gt(n)i — oo) — 0, Gt{n){0) = 0.5, and G((„)(oo) = 1. This

function is plotted in figure 16 for n as above. The p-quantile, denoted by tp{n), is the x

value which satisfies Gt(„)(ar) = p. In this paper the t distribution is used only in computing confidence intervals (sections

4.5 and 4.6), thus only quantiles are required.

30
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41

*-U.S. Government Printing Office : 1988 - 201 -597/82S78
NBS-n4A iREv. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS/TN-1246

2. Performing Organ. Report No. 3. Publication Date

June 1988

4. TITLE AND SUBTITLE

A New Statistical Model for the Calibration of Force Sensors

5. AUTHOR(S)

Charles P. Reeve

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Same as #6

10. SUPPLEMENTARY NOTES

[

1 Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less (actual summary of most significant information. If document includes a si gnificant

bi bliography or literature survey, mention it here) The National Bureau of Standards has been calibrating force sensors for many years.

The objective in these calibrations is to determine the functional relationship

between the applied load and sensor response. In a typical calibration several

runs are made in which identical sequences of known loads are applied to the sensor.

The sensor is rotated in the loading machine between runs. The previous method of

analysis incorporated a quadratic polynomial model which was fit to the pooled data.

The new method presented here fits separate polynomials to the mean data and

between-run differences. The "best" degrees of these polynomials are automatically

determined by algorithms which incorporate statistical tests. As a result, error contributions from several sources are quantized. Methods for computing confidence

intervals for the "true" sensor response and for a new observation are given, and

methods of inverse prediction (measurement of an unknown load) based on these

intervals are illustrated.

12. KEY WORDS (Six to twelve entries; o/phobet/col order; capitalize only proper names; and separate key words by semicolons)

calibration curve; confidence interval; force sensor; inverse prediction; least

squares polynomial; load cell; proving ring; statistical model

13. AVAILABILITY

[^Xj Unl imi ted

I I For Official Distribution. Do Not Release to NTIS

[X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

46

15. Price

USCOMM-DC 6043-P80


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