Lead screw type comparators are calibrated soas to reduce errors to one micron on eachaxis for ballistic camera plate measurements.

RALPH A. GUGEL

RCA Service Co.-kfissile Test ProjectPatrick A ir Force Base, Florida

Comparator Calibration

(A bstraci on page 857)

(1.1)

INTRODUCTION

THIS ~APER extracts major items of infor-matIOn from RCA Data Reduction TR­

63-1, Photogrammetric Data Reduction Anal­ysis-Calibration of Comparators for Bal­listic Camera Data, by Ralph Gugel, Decem­ber, 1963. The paper is a sequel to the pre­vious. paper published by Rosenfield (2), inthat It presents the statistical analysis neces­sary to validate the results obtained fromRosenfield's mathematical model.

The au thor wishes to express his appreci­ation to Dr. L. Lasman and Mr. George H.Rosenfield for their assistance in preparingthis report, and to Mr. James Duncan whoprogrammed the calibrations for an auto­matic computer reduction. Also, he wishes toacknowledge the efforts of the film readerswho contributed to the calibration of thecomparators described in the appendices.

NONPERPENDICULARITY OF AXES

. Rosenfield (2) describes non perpendicular­Ity as "the correction angle e through whichthe secondary guide way must be rotated tobe perpendicular to the principle guide way."The development presented below is based onthe method of inversion so that a calibratedstandard is not necessary. This method makesuse of the principle that, in the absence ofnonperpendicularity error, the X-coordinatesrelative to a fixed coordinate system on theplate will be equal in magnitude but oppositein sign when the plate is inverted about theY-axis of the comparator.

Two points whose X-coordinates are widelyseparated are selected as reference points toestablish the grid coordinate system. Thesepoin ts are used to align the plate with theX-(principle) axis of the comparator. It issuggested that a minimum of five verticallines of 11 poin ts each equally spaced over the

majority of the comparator measuring formatshould be selected to read as calibrationpoints to give suf-ficient redundancy andstrength to the solution. A minimum of twoindependent sets of observations are made oneach grid point, including the reference points,in order to determine the setting error. Theplate is then inverted about the secondary(Y) axis, realigned and the readings repeated.

Each set of readings is corrected for tem­perature fluctuations and the comparatorerrors previously calibrated. The averagevalue of the observations and the setting errorare computed independently for the directand inverted positions. The average values ofthe observations in each position are thentranslated to the grid origin by the equations

x.' = 2x - x,. - Xl, 2

y.' _ 2y - Yr - y,,- 2

RALPH A. GUGEL

853

854 PHOTOGRAMMETRIC E GINEERING

in which

X," = x' cos{3 - Y' sin {3

V," = - x' sin /3 + y' cos{3. (1.2)

FIG.!. Relationship between comparator co­ordinates (X, V), unrotated grid coordinates(X', V'), and rotated grid coordinates (X", V').

The coordinates for each point are then ro­tated to a rectangular system paral1el to thecomparator X-axis by the equations (Figure1)

(1.8)

(1.6)

(1.5)0% = an + alY.

v" = X" + Y" sin.Vy = y" cos.

n

'LXi17ii_I

a, = sine. = - ---

The value for cos e may then be computedby

when the angle e is very small.If the method of calibrating a comparator

using a grid as given by Hal1ert (5), is reducedto only the portion concerning perpendicular­ity, it can be seen that it is equivalent to themethod described above. I t has been foundfrom experience that either of these two meth­ods provides better results than the methodpreviously used at AMR. The previousmethod is described by Rosenfield (2) and wasderived from a method developed by lug (3).This method, as employed at AMR, had twobasic weaknesses. Firstly the computation ofthe angle of non perpendicularity is not basedupon a least squares adjustment of the obser­vations on al1 the data points, but instead isbased upon an averaging technique using anumber of individually computed angles ofnonperpendicularity. Secondly, the errorpropagation is weak since the reading vari­ance is automatical1y increased by the doubledifferencing method of the analysis. Thisearlier method has therefore been replaced by

cos. = [1 - sin' .J'/'. (1. 7)

With the values for sine e and cos e computed,the coordinates of the calibration points cor­rected for nonperpendicularity error may becomputed by

The constan t term represen ts the averageerror in the uncorrected data. The linearcoefficien t represen ts the sine of the correctionangle e through which the secondary guideway must be rotated to be perpendicular tothe principle guide way. Actually the linearcoefficient represents the tangent of the cor­rection angle e, but for the smal1 angles asso­ciated with nonperpendicularity of axes itmay be assumed that tan e equals sine e equalse in radians.

The coefficien ts of the error model may becomputed using a least squares adjustment 011

al1 the poin ts read by

tx.\_1

Go = --­n

(1.4)- Yli" + Y2/'Yi =-----

2

y" y'

y \

~:"I\\ Plx,yl

t-- -~ y"

\ r---\ ,~y'

i\ -1.

~ ;:;'"'X..'""_x

x'~

(,,~,:0 tj

!\\

! \\\\\

x0 ,.I"<1poIO.. ld. Wo W

where Xi represents the error due to non per­pendicularity of the measuring axes of thecomparator at a distance 17i from the center ofthe plate. (See Figure 2.)

The errors may then be fitted to an errormodel of the form

. yr - YlSill {3 = -;c----=---=-----,---

[(Xr - Xl)' + (Yr - Yl)'J1/'

cos/3 = [1 - sin'{3J 1/' (1.3)

It is obvious that, in the absence of non­perpendicularity error, the rectangular coor­dinates X", Y" for the image i wil1 be un­altered by the inversion of the plate about theY-axis except for the sign of the X"-coordi­nate. The average of the X"-coordinates of apoint for the direct and inverted positionsshould then be zero for each poin t. An y devia­tion from zero is due to non perpendicularityof the measuring axes of the comparator. Ifthe comparator coordinates in the directposition are denoted by Xli", Yli" and thecoordinates in the inverted position are de­noted by X 2i", Y,." then

X "+X"X,=_h_'_~

2

COMPARATOR CALIBRATION 855

the method described above to calibrate thecomparators at AMR.

METHODS OF ANALYZING THE RESULTS

The quali ty of the data used to determinethe coefficien ts of the error models is deter­mined by the precision of the readings and thequality of the standards used. The quality ofcalibrations cannot be expected to be anybetter than the quality of the data fitted tothe error models. Therefore, a satisfactorycalibration has been achieved when there is nosignificant difference between the variance ofthe data fitted to the error model and thevariance of the residuals remaining after theadjustment. An F-test can be used to deter­mine whether a satisfactory calibration hasbeen achieved.

Some of the error models used for the cali­brations are capable of producing zero resid­uals. If the adjustment is allowed to proceedbeyond the poin t where the two variances areapproximately equal, then the error modelcould be correcting for errors that are due onlyto reading error. This could actually introduceerrors in the corrected coordinates. AI thoughthe magnitude of these errors would not ex­ceed the standard error of the data fitted tothe error model, it definitely is not desirableto introduce them. Consequently, it is advis­able to check the variance ratio prior to theadjustment, and during each step in the ad­justment, so that this undesirable overadjust­ment does not occur.

The F-test to determine the proper stop­ping poin t is constructed as shown below. Thefollowing notation is introduced:

Sl = the variance of the residuals.Sr2 = the variance of the data fitted to the

error model.

If (S.2/Sr 5) > F(a,iJ)(0.95) , one continues withadjustment; otherwise he stops the adjust­ment.

F(a,iJ)(0.95) is a tabular value from thestatistical F distribution and is a function ofthe degrees of freedom associated with thevariance of the residuals (a), the variance ofthe data fi tted to the error model (m, and theprobability level (0.95). The degrees of free­dom associated with these variances are afunction of the error model, the number ofpoints involved, the number of sets of read­ings, and the operations performed on thereadings to provide the data fitted to the errormodels. The following sections provide thedetails on computing the necessary values forthe F-test for each type of calibration.

----l..-----It-----~-x"

FIG. 2. Determination of Nonperpendicularity.

PERIODIC LEADSCREW ERROR

The data to be fttted to (-he error model arethe differences between the average measuredcoordinates of points in the initial positionand their respective average measured coordi­nates in the reset position. If there were noerror in the measurements or the leadscrew,these differences would be a constant equal tothe distance the scale had been moved for thereadings in the reset position. In the calibra­tion procedure it is first necessary to deter­mine whether the variance of these differencesfrom a constan t reflect anything other thanreading error. The computations for thisanalysis are as follows.

The variances of the readings in each posi­tion are gi ven by

L: L: (d2ik)2j k

(TX22 = ----- . (2.1)n(N 2 - 1)

where the subscript 1 denotes the initial posi­tion, the subscript 2 denotes the reset posi­tion, d is the deviation of each reading from itsrespective average coordinate,

i=I,2,' " n; it is the number of pointsread,

j = 1,2,' " Nt; N l is the number of sets ofreadings on each point in the initialposition, and

k = 1,2, ... , N 2 ; N 2 is the number of sets ofreadings on each point in the resetposition.

856 PHOTOGRAMMETRIC ENGINEERING

(2.3)

(2.2)

where 8S, is the standard error of the scalecalibration.

The constant error in the measured co­ordinates is computed by

(2.6)

(2.5)Ux2 = ----'---

n(N - 1)

where d is the deviation of each measuredcoordinate from the average measured co­ordinate for'each poin t,

i = 1 2, . .. n; n is the number of points read and

= 1 2, . .. N; N is the number of sets of readings.

The variance of the data fitted to the errormodel is then

t (l5 - D i )2i-I

5,u2 =-----n - 1

where D is the difference between the averagecoordinate in the reset and initial position foreach of the n points and D is the averagedifference between points.

If

(5,0'/5T') > F{ (n - 1), [n(N! + N. - 2) Jl (0.95),

UXJ 2 UX22

5 T2 = --+--.

N, N2

To determine whether any error other thanreading error is reflected in the uncorrecteddifferences compute

The variance of the differences in the aver­age coordinates is

then the data reflect period leadscrew errorover and above the reading error and the datashould be fi tted to the error model. If theratio of the two variances is not greater thanthe F value then no significant periodic erroris presen tin the leadscrew and the adj ust­ment is not necessary.

If the data are fitted to the sine wave errormodel the variance of the residuals becomes

(2.7)

where D is the deviation of the average mea­sured coordinate from the given calibratedcoordinate for each scale graduation.

The variance of the data about the con­stant error is then

t Vx i 2

SJlI2=~,n-3

(2.4)(2.8)

(2.9)

where Vxi represents the residual for eachpoint resulting from the least squares adjust­ment and the F-test becomes

(5.12/5;) > F{<n - 3) [n(N! + N 2 - 2) Jl (0.95).

The residuals should be tested after eachiteration during the least squares adjustment.

SCALE ERROR AND SECULAR LEADSCREW

ERROR

The data to be fitted to the error model arethe deviations of the average measuredcoordinates of the scale graduations from thegi ven calibra ted coordinates of the scale. Ifthere were no errors in the measuremen ts, theleadscrew, or the given coordinates of thescale graduations, these deviations would allbe zero. In the calibration procedure it is firstnecessary to determine whether the varianceof these deviations reflects anything otherthan reading error and error in the givencoordinates of the scale graduations. Thecomputations necessary for this analysis areas follows.

The variance of the readings is computedby

where V" is the difference between the aver­age measured coordinate for each point andthe constan terror A o.

If

(SU0 2/ST2) > F{ (n - 1), [n(N - 1) Jl (0.95)

then the residuals reflect scale error and/orsecular leadscrew error and the data should befi tted to a polynomial. If the ratio of the twovariances is not greater than the tabular F­value then no significant error other thanreading error and error in the calibrated scaleis present in the data and the adj ustmen t isnot necessary.

If the data are fitted to the polynomial thevariance of the residuals becomes

n

LV:ri2

SV12=~n-p

where p is the number of coefficients com­puted for the polynomial and V" is theresidual for each point resulting from theleast squares adj ustmen t.

The F-test for the polynomial fit becomes

(S.N5;) > F{ (n - p), [n(N - 1)]J (0.95).

L

COMPARATOR CALIBRATION 857

The degree of the polynomial to which thedata are fitted should be increased by steps ofone and the residuals tested as above aftereach step until the F-test indicates no sig­nificant errors remain or until a second degreepolynomial has been fitted. Experience hasshown that the higher order terms of the poly­nomial do not significantly reduce the errors.If significant errors still remain after the datahave been fitted to a second degree poly­nomial, the data should then be subjected to aharmonic analysis to correct the remainingerrors.

When the data are subjected to the harmonic

If there was no error in the readings, thestandard, or the comparator ways, the directand inverted readings could both be expectedto produce means of zero. Since these errorsdo exist, the direct and inverted readings willproduce different means and it is necessary tocompu te the reading variance independen tlyfor the direct and inverted readings. Thevariance of the deviations fi tted to the errormodel is then a function of the variances ofthe two subgroups.

In the calibration procedure it is first of allnecessary to determine whether the varianceof the data to be fitted to the error model

ABSTRACT: The procedures jor calibrating a prect~wn leadscrew type com­parator are described and the results oj the calibration of the ballistic plate com­parators in use at the A tlantic ~Missile Range are given. The errors to be cali­brated are: (1) periodic leadscrew error, (2) scale error and secular leadscrewerror, (3) weave and curvature oj the ways and (4) nonperpendicularity of axes.The standard error oj the plate coordinates obtained jrom N[ann comparators422C66 and 422D49 and Wild/Stereo Comparator STK-l were reduced from2.2, 4.9 and 5.2 microns to 1.2, 1.2 and 1.9 microns respectively as a result oj thecalibrations. Each calibration is described in detail giving the reading pro­cedures, calibrated standards necessary, error models fit and methods oj analyz­ing the results.

(2.10)

analysis it is necessary to start the adj ustmen tusing the first two harmonics because only thecosine coefficient of the last harmonic is used.

For the harmonic analysis the variance ofthe residuals becomes

n-!

L Vx i 2

51''2.2 = i"""l

n - p - 2q + 1

where q is the number of harmonics used incorrecting the data.

The F-test becomes

(5,,2'15;) > F[(n - P - 2q + 1), n(N - 1)](0.95).

Again, the harmonic analysis should beapplied by adding one harmonic at a step andthe residuals tested after each step to deter­mine when to stop the adj ustment.

WEAVE AND CURVATUlm OF THE WAYS

The da ta to be fi tted to the error model arethe deviations of the average measured co­ordinates from a straight line. Since thecalibration does not use a calibrated standardthe proced ures used employ direct and in­verted measurements to remove errors in thestandard. The data fitted to the error modelare the result of averaging the readings in thedirect and inverted positions.

reflects anything other than reading error.The compu tations necessary for this analysisare as follows,

The variances of the readings in each posi­tion are computed by

L L (d'i})2

UX12 =n(N, - 1)

L L (d2i/.)2

UX22 =k (2.11)

n(N2 - 1)

where the subscript 1 denotes the directposi tion, the su bscript 2 denotes the invertedposition, d is the deviation of each readingfrom the average coordinate for that point,

i= 1,2, ... ,n; n is the number of pointsread,

j = 1,2, .. ',N,; N l is the number of sets ofreadings on each point in the directposi tion

k = 1,2, "', N 2; N 2 is the number of sets ofreadings on each point in the invertedposition.

The variance of the data to be fitted to theerror model is then

858 PHOTOGRAMMETRIC ENGINEERING

where X is the average measured coordinatefor each point.

The variance of the da ta abou tAo is then

use a calibrated standard, the procedures usedemploy direct and inverted measurements toremove errors in the standard. The datafitted to the error model are the result ofaveraging the readings in the direct andinverted positions.

Since the individual means of the direct andinverted readings for a point are opposite insign, it is necessary to compute the readingvariance independen tly for the direct andinverted readings. The variance of the devia­tions fi tted to the error model is then a func­tion of the variance of the two su bgrou ps.

In the calibration procedure it is first of allnecessary to determine whether the varianceof the data to be fitted to the error modelreflects anything other than reading error.The compu tations necessary for this analysisare as follows.

The variances of the readings in each posi­tion are computed by

(2.14)

(2.13)

(2.12)

tXiAo=~

n

i_ISvc 2 = ----

n - 1

To determine if the variance of the datareflects anything other than reading errorcompute the ratio of the two variances.

If

(S,02/S,') > 1'1 (11- 1), [1I(N1+ N 2 - 2)]j (0.95)

1 (UX,2 UX22)S; =- --+-- .4 N 1 N 2

Compute the constant error in the data tobe fi tted to the error model by

(2.15)

(2.16)

then the residuals reflect error due to weaveand curvature of the ways and the datashould be subjected to the harmonic analysis.If the ratio of the two variances is not greaterthan the tabular F-value then no significanterror other than reading error is present in thedata and no adjustment is necessary.

When the data are subjected to the har­monic analysis the variance of the residualsbecomes

tV.l:i2

S.,2 = _;_-1 _

n - 2q + 1

where V" represents the residual for eachpoint resulting from the adjustment and q isthe number of harmonics used in correctingthe data.

The F-test for the harmonic analysis be­comes

(S,.I'/s;»FI (1I-2q+l), [n(N,+N2-2)]j (0.95).

UX,2 = ----'---n(NI - 1)

L L (d2ik)2; k

UX 22 = -----n(N2 - 1)

where the subscript 1 denotes the direct posi­tion, the subscript 2 denotes the invertedposition, d is the deviation of each readingfrom the average coordinate for that point,

i=1,2, "', n; n is the number of pointsread,

j = 1,2, .. ',N,; N 1 is the number of sets ofreadings on each point in the directposition,

k = 1,2, "',N 2 ; N 2 is the number of sets ofreadings on each point in the invertedposition.

The variance of the data due to reading isthen

The total variance in the data to be fi ttedto the error model is then

where 5p2

, 5.2 and 5w2 are the standard errors

of the periodic error, secular error and weaveof the ways calibrations respectively.

The error in the comparator coordinatesdue to non perpendicularity of the measuringaxes is the average of the direct and invertedX-coordinates for each point. These errorsare computed by

If this F-test shows significant errors overand above the reading error then the datashould be subject to the harmonic analysis.

Again, the analysis should be started byevaluating the first two harmonics in the firststep and adding one harmonic at a time untilthe F-test indicates a satisfactory calibrationhas been achieved.

NONPERPENDlCULARLTY OF AXES

The data to be fitted to the error model arethe deviations of the average measured X -co­ordinates from a straight line perpendicularto the X-axis. Since the calibration does not

. 1 (UX,2 UX22)S,"2 = - --+-- .

4 Nt N 2

SrI2=Sr02+Sp2+S.2+SW 2

(2.17)

(2.18

COMPARATOR CALIBRATION 859

(5v02/5rI2) > F[n, n(N, + N2 - 2»)(0.95)

(SvN5rI 2) > F[(n - 2), n(N, + N 2 - 2»)(0.95).

(2.22)

S MMARY

The calibration proced ures and the statis­tical analysis presented here wil1 producecalibration coefficien ts for correcting readingsmade on any leadscrew type comparator. Al­though some better methods may exist, thesemethods wil1 produce calibrations whosequality is limited only by the mechanicallimits of the comparator, the reading precis­ion, and the standard error of the calibratedscale, with a minimum of expense required forcalibrated standards. The number of sets ofreadings to be made depends on the settingprecision that can be achieved and the de­sired quality of the calibration.

The methods described have been used atthe Atlantic Missile Range to reduce com­parator errors to one micron in each axis ofour ballistic plate comparators.

REFERENCES

1. Bennet, J. M., "Method for Determining Com­parator Screw Errors with Precision," Journalof the Optical Society of America, Volume 51, No.10, October 1961, pp. 1133-1138.

2. Rosenfield, G. H., "Calibration of a PrecisionCoordinate Comparator," PHOTOGRAMMETRICENGINEERING, Volume 29, No. I, January 1963,pp.161-174.

3. Zug, R. 5., "High Altitude Range Bombing bythe Aberdeen Bombing Mission, Using BallisticCameras," Research Services Division,ReportNo.1, Ordnance Research and DevelopmentCenter, Aberdeen, Md., 10 December 1945, pp.112-117.

4. Collen, F. c., Appendix A, "Calibration of aPrecision Coordinate Comparator," PHOTO­GRAMMETlUC ENGINEERING, Volume 29, No. I,January 1963, pp. 161-174.

5. Hallert, Bertil, P., "Determination of the Geo­metrical Quality of Comparators for Image Co­ordinate Measurement," GIMRADA ResearchNote No.3, 1 August 1962. Geodesy, Intelli­gence, and Mapping, Research and Develop­ment Agency, U. S. Army, Corps of Engineers,Fort Belvoir, Virginia.

(2.19)

(2.20)

(2.21)

2

tXi2

i=lSl'02= __-.

n

XIiI+X2i"Xi =

where X/' is the average coordinate in thedirect position and X 2" is the average co­ordinate in the inverted position for eachpoint.

The variance of the uncorrected errors isthen

where Vx represents the residual for eachpoint resulting from the adjustment.

The F-test for the error model fit becomes

To determine if the variance of the uncor­rected data reflects anything other than read­ing error compute the ratio of the two vari­ances.

If

If this F-test shows significant errors overand above the reading error then the calibra­tion is unsatisfactory and must be repeated.

The standard error of the correction anglef is then

then the residuals reflect error due to non per­pendicularityof the axes and the data shouldbe fi tted to the error model. If the ra tio of thetwo variances is not greater than the tabularF-value then no significant error other thanreading error is present in the data and noadj ustment is necessary.

When the data are subjected to the rotationto correct for nonperpendiculari ty error thevariance of the residuals becomes