FRONTISPIECE. Full front view of the Multilaterative Comparator.

DUANE C. BROW 1*

D. Brown Associates, Inc.Melbourne, Fla. 32901

Computational Tradeoffs in theDesign of a ComparatorA simple and economical one-micron plate comparator results

from the exploitation of the computer as a primary principle

of the systems design.

(Abstract on next page)

INTRODUCTION

T HE SELF-CALIBRATING MultilaterativeComparator began about four years ago

as a thought experiment intended to prove apoi n t. Previous to this, in our work in thedata reduction of missile and satellite track­ing systems, we had developed a reductioncalled EM BET (Error Model Best Esti mate ofTrajectory). EMBET is designed to reconcile

* Presented at the Semi-Annual Convention ofthe American Society of Photogrammetry, St.Louis, Mo., October 1967 under the title "Compu­tational Tradeoffs in the Design of a One MicronPlate Comparator." (The Appendix of the originalpaper is not reproduced here. Anyone who is in­terested in the Appendix should consult the author.-Editor)

the conRicting trajectories produced bydifferent tracking systems or combinationsof tracking systems. Such conRicts are theresult of unknown systematic errors in thevarious observational channels. Previous re­ductions had ignored the existence of un­known systematic errors and had deter­mined the trajectory by means of indepen­dent point-by-point least-squares adjust­ments of the tracking observations. Thus, ifthe coordinates X, V, Z of n trajectorypoints were to be determined from m ~3

observational channels per point, such re­ductions would entail simply the formationand sol u tion of n independen t sets of normalequations of order 3X3. Accordingly, thesolution for any given point would be un-

lR5

186 PHOTOGRAl\IMETRIC ENGINEERING

affected by, and likewise would have no ef­fect on, the solution for any other point.

The DIBET reduction, on the other hand,recognizes that systematic errors do existin the observations and that, though un­kno\\"n, they can (for the most part) bemathematically modelled. I t then attemptsto reCO\'er in a single, massi\'e least-squaresadjustment the coordinates of all trajectorypoints and the unkno\\'n parameters of theerror models of the various observationalchannels. Inasmuch as each trajectory pointintroduces three unknowns, the normal equa­tions arising from the simultaneous recoveryof n trajectory points and p error coeffi­cients are of order (p+Jn) X(p+Jn). Henceif several hundred trajectory points are to bedetermined, the normal equations can be­come very large indeed. However, the normalequations have a highly patterned structurethat is exploited in the E\IBET reduction to

many tracking systems were unduly compli­cated and that many functions performed byhardware could be performed equally well,if not better, by software soundly based onprinciples of self-calibration. In due course,we began preachi ng the gospel of self-cali­bration through designed redundancy-thatsimpler, more effecti\'e and less expensivesystems could be developed if the power ofself-calibration by means of software wereduly exploi ted as a pri mary pri nci pie of sys­tems design. To our way of thinking, ad­vanced concepts of data reduction shoulddirect develop men t rather than emerge asaf tel' thoughts.

I t was for the purpose of providing a con­crete demonstration of these principles, then,that we originally undertook the conceptualdevelopment of the Mul'Lilaterative Com­parator. As conceived, it was to be a radicaldeparture from conventional comparators.

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ABSTRACT: Pr·inciples of self-calibration through designed rednndancy havebeen successfully applied to the design of a unique, large format (245 X 245mm.), lightweight (10 kg.), self-wlibrating and self-checking comparator ofone-micron accuracy. The comparator achie-ues exrreme instrumental simplicityand enhanced accuracy through judicious tradeofJs of mechanical and computa­tional factors, the general availability of modern digital computers being adecisive element of the design.

develop a special algori thm that renders theirsolution practical no matter how manypoints are to be carried. By virtue of thisalgorithm, the computational effort for theformation and solution of the normal equa­tions increases only linearly with the numberof points being carried. Moreover, the largestmatrix operation to be perfonlled is the in­version of a matrix of order equal to thenumber of error parameters bei ng recovered(i.e., a pXp matrix).

Inasmuch as the error coefficients are re­covered without recoul'se to external stan­dards of calibration, the EMBET reduction canbe said to consi tu te a process of self-calibra­tion. The practical effectiveness of E\1BET

depends mainly on three factors: (a) theadequacy of the error models; (b) the degreeuf obseryational redundancy; (c) the strengthof the geometry. \"here these can be ade­quately controlled, self-calibration can suc­cessfully suppress the ultimate effect of sys­tematic errors to insignificance.

As practical experience with EMBET wasgained, it became increasingly apparent that

It was to be capable of determining coor­dinates within a large format (245 X245 mm.)to accuracies on the order of one micron. Yetinstead of being massive (200 to 300 kg.,typical of conventional comparators), it wasto be lightweight (about 10 kg.). Instead ofbeing bulky, yet delicate, it was to be com­pact, rugged and readily transportable in ahand-held carrying case. Insteetd of requiringprecisely linear and precisely orthogonal waysit was to do away with these troublesomeelements altogether. Instead of requiring acontrolled environment, it was to be rela­tively insensitive to environment and hencewell-suited to field measurements of ballisticcamera plates. In addition, it was to be fullyself-checking and self-calibrating, simple inconstruction, simple to operate, capable ofbeing digitized, and immune to effects ofconstan t personal biases or even to system­atically changing personal biases. More­over, it was to provide sound estimates of theaccuracies of final plate coordinates as well asother measures suitable as indices for qualitycontrol. Finally, it was to be comparable in

C01llPlJTATIONAL TRADEOFFS IN THE DESIGN OF A CO.\IP,\RATOR lS7

range) rij from four fixed external points (i= 1,2,3,4) whose coordinates X{, Y{ we assu metemporarily to be known. This gi\'es rise tothe following set of four observational equa­tions:

Because the external points, or tracking stu,tions as we shall call them, are considered tobe of known location, and the ranges ri, an'considered to be kn')\I'n by \·irtue of measure,ment, the a\)o\'e constitutes a S\'stem of fourequations in but [\\'0 unkno\\'n~, the desiredcoordinates Xi, Yj. \Ve may thus perform asimple least-squares adjustment to minimizethe effects of random measuring errors. A by­p,'oduct of such an adj u tment is the errorpropagation associated with the adjustedcoordinates. [f the plate format were squareand if the tracking stations were located ashort distance from the midsides of the for­mat, \I'e would find from numerical simula­tion that, for a specified standard deviation(T of ranging, the standard deviations of XI'

Yj obtained from a least-squares adjustmentwould range in magnitude from 0.71 (T forcentrally located points to 0.85 (T for pointsnear the edges of the format. This means that,throughou t the format, basic measuring ac­curacies are geometrically enhanced, ratherthan diluted, and that geometrical variationof accuracies of plate coordinates is accept­ably small.

The foregoing is, of course, an abstractionand it does represent a severe limiting case int hat the coordinates of all tracking stationsare assu med to be perfectly known. Yet hadwe been unable to obtain satisfactory resultsfrom this primitive limiting case, there wouldhave been no point to proceeding furtheralong the above lines.

The next logical matter to be consideredis the effect of abandoning the assumptionthat the coordinates of the tracking stationsare perfectly known. "ow if these coordinateswere considered to be known, the coordinatesystem is thereby implicitly defined. Inabandoning this knowledge, we gain theprerogative of defining the coordinate systemto be employed. Several logical choices pre­sent themselves. \~Tecould,forexample,choosethe origin to coincide with some particularimage on the plate and define the po itivey-axis to pass through some other image. Thesystem then becomes uniquely defined once

over-all measuring speed wi th conven tionallarge-format comparators and was to costsu bstan tially less.

About two years ago, it became apparentthat an actual need and market existed forsuch a de\·ice. Accordingly, we undertook anin-house project to turn the concept (shortlyto be described) into reality. Before de\'elop­ing an engineering test model, we performeda series of computer simulations to ascertaintheoretical feasibility of the conceptual ap­proach. The si mu Ja tions demonstrated tha tif the approach could be successfully mech­anized, the recovery of plate coordinates andother para meters could indeed be performedto the required accuracies. Guided by theresults of the computer simulations, we thendesigned an engi neeri ng test model. The engi­neering test model was completed a year ago.Insamuch as its performance matched ourfondest expectations, we are now manufactur­ing the comparator in quantity.

In this paper we shall trace the develop­ment of the comparator, placing particularemphasis on the dominant role played bydata reduction.

THE CONCEPT

Our point of departure was to consider theproblem of measuring the precise coordinatesof a set of points on a plate as being equiva­lent to a two-dimensional tracking problem.At its most primitive and most abstract level,the problem was formulated in terms of a fOlll'station ranging system as in Figure 1. Herewe imagine that the coordinates Xj, Yj of anarbitrary point on a plate are to be deter­mined from measurements of distance (or

'..

FIG. 1. Geometry of idealized, four station,two-dimensional trilateration.

1",/ = (Xj - x1')2 + (Yj - y{)2

1"2/ = (Xj - x{) '}, + (Yi - Y2c)2

1';1) '}, = (.rj - :\"/) 2 + (Vj _ Y:{) '},

'4i2 = (.rj - X.{)2 + (Vj _ y.{)2.

(1 )

li)i) 1'1iOTOGRAMMETRIC ENGINEEI{ING

we decide whether it is to be right or lefthanded. Alternatively, we could choose theorigin to be at, say, Station 1 thereby render­ing its coordinates Xl', yl' equal to 0, 0. Inaddition, we could define the system to beright handed with the positive y-axis passingthrough Station 3 thereby rendering its x­coordinate X3C equal to zero. This, too, woulduniquely define the coordinate system. Instill another choice, the system may be de­fined as right handed with the positive y­axis passing through points 1 and 3, therebyrendering X{=xo<=O, and with the originchosen so that the y-coordinates of Stations 2and 4 are eq ual in magni tude bu t of opposi tesign (thus y{= -y/). This has the advantageover the previous choice of placing the originnear the center of the plate and is the partiCLt­lar choice we decided to adopt.

Returning to the set of four Equations 1arising from the measurements of the j-thpoint, we now regard not only Xi, Yi as un­known but also the following five coordinatesof the tracking stations: yl', X{, y{, yo<, x,c (x,c=X3c=0 and y/= -y{ by definition of thecoordinate system). Accordingly, we nowhave a system of four equations in seven un­knowns, and a unique solution does not exist.On the other hand, if we direct our attentionto the entire system of equations generatedby the measurements of n distinct points, wesee that it involves 4n equations in a total5+2n unknowns (the five station coordinatesand the Xi, Yi for each of the n points). Henceif n ~3 there will be more equations than un­knowns, and a solution can be attempted.The system of normal eq uations generatedby the simultaneous adjustment of the ob­servations from all n points is of order 2n+5and hence increases with increasing n.

However, the normal equations are ofprecisely the same form as those arising in anEMBET reduction; in fact, they may be viewedas a special, four-station, two-dimensionalcase of a general EMBEI' reduction for an m­station ranging system. In our problem, theerror parameters to be recovered consist ofthe five unknown coordinates of the trackingstations. Accordingly, by the application ofthe EM BET algorithm, the general system ofnormal eq uations can efficien tly be collapsedto a 5X5 system involving only the stationcoordinates as unknowns. After the stationcoordinates have been determined, the coor­dinates of the poin ts Xi, Yi can be establishedthrough the solution of n independent 2X2systems of red uced normal eq ua tions. \lifeshall not go into details of the solution at thispoint. The important matter is that the for-

mation and solution of the normal equationspresents no practical difficulties, no matterhow many points are to be carried in the re­d uction. Also, the rigorous error propagationemerges from the reduction.

I t remains to be established whether or notthe recovery of tracking station coordinatesleads to significant dilution of accuracies inthe recovery of image coordinates. By numer­ical simulation, we find that when about25 well-distributed points are carried in thesolution, the expected accuracies of the Xi,Yi are degraded only by from 5 to 15 percentover what they would have had all coordinateof the tracking stations been perfectly known.Moreover, the more points carried in the re­duction, the less the dilution of accuraciesentailed by the need for recovering stationcoordinates.

The above result was of signular impor­tance to the development of the comparator,for it mean t that, by virtue of a data red uc­tion tradeoff, the design did not need to beconcerned wi th problems relating to preciselocation of points that would correspond totracking stations.

The fact that the coordinates of trackingstations need not be known also has a vitalconsequence relating to the practical mech­anization of the measuring process. As acrude first approxi mation, we could envisionthe process as one in which a transparentscale is pivoted about its zero point so thatit can sweep over the entire plate. With theaid of a device to interpolate between divi­sions of the scale, one could then determinethe distances from the tracking station (i.e.,the pivot) to all points of interest on theplate. After all points had thus been mea­sured from one station, the scale could be re­moved to a second station (pivot), where­upon the process could be repeated. In thismanner the required measurements could beobtained from all four stations.

Shifting of the scale from pivot to pivot isobviously a cumbersome procedure and isone that would be awkward to mechanize.Fortunately, because the coordinates of thepivot points need not be known, a geometri­cally equivalent process becomes feasible. J tconsists of pivoting the scale about a singlepoint and rotating the plate by nominally90° between one set of measurements andanother. This generates a total set of mea­surements exactly equivalent to the set thatwould have been produced from a fixed plateand four separate pivots as discussed above.To see this more clearly, we may imagine afixed coordinate system X, y attached to the

COMl'UTATIONAL TJ<ADEOFFS IN THE DESIGN OF A COMl'AI{ATOI{ 1~9

.~~ (O,r;)

FIG.2. IIluslrating geometrical equivalent ofMultilaterative Comparator.

plate. If the plate is in Position 1, the pivotwill occupy a position having coordinates x{,y{, from which the ranges to the points ofinterest would be measured. Now we imaginethe plate rotated and possibly translated toa new position. The coordinates Xj, Yj of thepoints on the plate will not be changed by thisprocess, bu t the pivot will occupy a new posi­tion X2e, Y2e , from which ranges would again bemeasured. However, if the plate had not beenmoved to its new position, precisely the sameset of measurements could have been ob­tained by measuring from a new pivot at x/,Y2e. Accordingly, the two processes are equiv­alent.

By fixing the pivot and rotating the platebetween sets of measurements, we are muchcloser to a process that is feasible to mecha­nize. However, as we ultimately aim at mi­cron accuracies, the physical difficulties ofpivoti ng precisely abou t the zero mark of thescale, or about any precisely known point forthat matter, are formidable indeed. \Ve mayattempt to get around such difficulties byassuming that the location of the pivot rela­tive to the zero mark is completely unknownand is to be recovered in the reduction. Geo­metrically, then, our system becomes theequivalent of that pictured in Figure 2. Herewe have let a and fl denote the radial andtangential components of the offset of thepivot point relative to the zero mark of thescale. \\'e ha\'e also recognized that sincethere is, in reality, only one pivot point andone measuring arm, the same a and fl applyto all four measuring stations. The measuredrange rij consists now of the distance from the

zero mark of the scale to the point Xj, yj, andthe distance from the tracking station to Xj,

Yj is given by the hypotenuse of a right tri­angle having sides of length Y;j+a and fl,respectively. Considering this and consideringour choice of coordinate system, we now ob­tain the following set of observational equa­tions from the measurements of thej-th point

(r,) + "')' + {3' = x,' + (Yj - )',')'

(r'j + "')' + {3' = (.1'j - ~',e)' + (Yj - )'{)'(2)

(raj + "')' + (3' = x,' + (Yj - y,')'

(r4j + "')' + (32 = (x) - X4')' + ()'j + )'",)'.Except for the additon of the two new param­eters a and fl, this system is of the same formas the one just considered in which coordi­nates of the tracking stations were unknown.The measurements from n points again gen­emte 4n equations, but now the number ofunknowns is increased from 2n+5 to 2n+7.The solution has the same properties as beforeexcept that the reduced normal equations arenow of order 7X7 instead of 5X5. From thenumerical simulation of a 2S-point pattern wefind that the addition of the offset parametersa and fl introduce essentially no further dilu­tion of the accuracies of the recovered coor­dinates Xj, Yj.

In view of the foregoing, we were able toconclude that the design of the comparatorneed not be concerned with the establish­ment of the precise offset of the pivot fromthe zero mark of the scale, Here too, a majorsimplification was achieved by means of acomputational tradeoff. An incidental, butimportant, benefit gained by the introduc­tion of a as an unknown is that it servesdouble duty by also compensating auto­matically for any unknown, constant personalbias. Thus if one were to contaminate a givenset of measured ranges by the application of acommon additive constant, one would obtainprecisely the same coordinates Xj, Yj fromindependent reductions of both the originaland the contaminated sets of measurements;the values obtained for a, however, woulddiffer by an amount equal to the additiveconstant.

Enough background has been presented atthis point to convey a general understandingof the principles underlying the developmentof the 1ultilaterative Comparator. Thoseinterested in finer details of the mathematicaldevelopment are referred to an appendixwhich can be obtained on request from theauthor. \Ve shall now direct our attention toconsideration of the comparator as it evolvedfrom the above considerations, But before sodoing, we would emphasize that the software

190 PHOTOGRAMMETRIC ENGINEERING

FIG. 3. Multilaterative Comparator in normal operating position.

for the comparator had been developed,tested, exercised, refined and retested, wellbefore the go-ahead was given to the designof the hardware.

IMPLEMENTATION OF THE CONCEPT

The comparator is pictured alone in theFrontispiece and in its operational position inFigure 3. The measuring arm A pivots abouta point B created by a single captive, top­loaded, self-seating ball bearing having adiameter of 0.75 inches. The micrometer endC of the measuring arm is supported by apair of rollers that run on the guide rail D.The measuring arm can be locked at any de­sired point on the guiderail by releasing

either of the dual, spring loaded brake lev­ers E.

A closeup of the measuring arm with itsside panels removed is shown in Figure 4.The Bausch and Lomb zoom macroscope(10 X ->30 X) F is mounted on a carriage Gwhich rolls on sleeve bearings along a pair ofstainless steel rods and can be Jocked at anydesired point by releasing the spring loadedbrake lever H.

The scale I is 6 mm. thick, 29 mm. wideand 292 mm. long. The bottom side is gradu­ated at one-millimeter intervals over a lengthof 261 mm. Every even division is numbered.The ends of the scale are attached to sup­ports which ride on sleeve bearings on a pair

H, F,

FIG. 4. MeaslIring arm with side panels removed.

COMPUTATIONAL TRADEOFFS IN THE DESIGN OF A COMPARATOR 191

of stainless steel rods. The lower support ofthe scale is free and the upper su pport isattached to the nut of the micrometer screwwhich is 2 centimeters in length. By turningthe micrometer dru m J, one can translate thescale radially from the pivot by a preciselyknown amount. Automatic stops in themicrometer head limit this translation to atmost one millimeter. The amount of trans­lation can be read by a vernier to the nearesthalf micron.

The plate to be measured is mounted in theplate holder J( (Fron tispiece), the four cornersof which rest freely on four pads machined tobe in the same plane as the upper guiderail.Different plate holders are made to accom­modate plates having the following nominaldimensions: 9.5 X9.5 inches, 8 X 10 inches,290 X 215 m m. Each plate holder is pre-ad­justed to accommodate a standard platethickness of 0.240 inches. This leaves a gap of75 to 100 microns between the upper surfaceof the plate and the bottom surface of thescale. Such a gap is sufficiently small to per­mi t both the plate and the grad uations of thescale to be sharply in focus at the upper mag­nification of 30 X. Standard plates thinnerthan 0.240 inches can be accommodated bymeans of adj ustable footscrews fixed to thecorners of the plateholder. On each edge ofthe plateholder a pair of V-blocks, which havebeen mated to a pair of positioning pins perm­an en tly affixed to the mainframe, serve to fixthe plateholder in each of the four standardmeasuring positions.

Illumination of the plate is provided by abuilt-in light table containing three 8-wattfluorescent lamps. A pair of folding legs builtinto the back of the mainframe support thecomparator in its normal operating positionat an angle of 45° from the horizontal.

Because the stainless steel rods of themeasuring arm were selected to have verynearly the same coefficien t of expansion asglass, the comparator is relatively insensitiveto variations in temperature. The glass em­ployed for the scale is of the same type as isused for Kodak plates.

OPERATION OF THE COMPARATOR

By rotating the measuring arm and trans­lating the microscope, one can quickly bringany desired point on the plate into the fieldof the microscope. To meaSlJl"e a point, onefirst brings it within a circular reticle havinga diameter of 400 microns at plate scale.There is no need for precise cen tering 01 theimage within the reticle, for a lateral offset ofas much as 100 microns will cause an error inradial distance of only 0.1 micron in the worst

case. The amount of parallactic error causedby the gap bet""een the scale and the platedepends on the effecti,"e focal length of theobjective; for the B&L Macroscope, a 100­micron radial offset will cause an error of 0.3micron in radial distance.

The image as seen within the reticle willlie between a pair of millimeter graduationsof the scale. The whole millimeter reading istaken directly from the lower graduation. Toobtain the fractional reading, one rotates themicrometer dru m to translate the lowergradualionuntil it precisely bisects the image.The fractional reading is then made from themicrometer drum to the nearest half micron.The reading is recorded opposite a pointnumber and the process is repeated for allthe points to be measured. When all of themeasurements have been performed for agiven position of the plate, the measuringarm is moved out of the way to a nearlyhorizontal position and the plate holder isrotated 90° and replaced on the comparator.The points are again measured and theirreadings are recorded opposite the point num­ber. The operation is completed when theplate has thus been measured in all four posi­tions.

Despi te the fact that the plate is measuredin four positions, the total measuring time isabout the same as that required when doublesettings are performed on a conven tional two­screw comparator. This is attributable mainlyto two factors: (a) each setting involves onlya single radial bisection, rather than bisectionin two directions; (b) slewing is very rapid,for the microscope can be moved into mea­suring position from any point on the plateto any other point in a matter of about fiveseconds. An experienced operator can per­form the complete measurement of a platecontaining 30 images within one hour; ifdouble sl:'ttings are required (i.e., a total of 8measurements per point), the time is in­creased to about 90 minutes.

CALI BRATTON

A systems analysis of the comparator showsthat aside from stability of the plate duringmeasuring, only three elemen ts of the systemare critical to the attainment of the desiredmeasuring accuracy of one micron. They arethe scale, the screw and the pivot. All otherelemen ts req uire tolerances from one to twoorders of magnitude less demanding. Forexample, for one micron accuracy the linear­ity of the translation of the microscope needbe good only to abou t ±300 microns (actu­ally, it is 10 times better than this). Likewise,the axis of the microscope need remai n

192 PHOTOGRAMylETRIC ENGINEERING

Scale to be Calibrated,

Master Scale

FIG. 5. Alignment device used in calibration of scales against master scale.

parallel to itself only to within ±0.6° (again,the actual tolerance is more than 10 timesbetter than this). Similarly, because the scaleis translated by at most one millimeter, thedirection of the translation must be parallelto the axis of the scale only to within ± go, atolerance that is easily bettered by a factorof 100.

The scale is indeed a critical item becauseeach measurement will inherit the error of thegraduation to which it is referred. This meansthat, in the absence of other errors, one­micron accuracies demand a scale that iseither accurate to one micron or else is cali­brated to an accuracy of one micron. Thescales employed in the production models ofthe comparator are manufactured to thefollowing specifications: cu mulative non­linearity of spacing is not to exceed one mi­cron over the total length of the scale; errorin spacing of successive divisions is not toexceed 0.5 micron. Above and beyond this,we com pare each scale wi th one of the scalesselected to serve as a master scale. In makingthis comparison, the scale to be tested isplaced in face-to-face contact with the mas­ter, a spacing of nominally 10 microns (or onelinewidth) being set between correspondingdivisions.

To facilitate the alignment of the twoscales, the ends of the master are cemen tedinto position in the device pictured in Figure5. The scale to be tested is then aligned withthe aid of positioning screws, whereupon it isclamped vertically at both ends to preventfurther move men t relative to the master.The center-to-center spacing between corre­sponding divisions of the two scales is easilymeasured to an accuracy of ± 0.2 microns bymeans of a vratson Image Shearing Microm­eter" employed at a magnification of 200 X.Jf the spacings were perfectly constant for all

divisions, the scale to be tested would per­fectly match the master. Therefore, the vari­ation of the spacing about the mean is themeasure of the discrepancy between the twoscales. In this manner all scales are referredto the master scale with an accuracy of ± 0.2microns. The master scale itself has been cali­brated in terferometrically to a certified ac­curacy of ± 0.5 micron by the ationalBureau of Standards. By virtue of the com­parative process just described, all scales in­herit the absolute accuracies of the masterwith virtually no dilution, for rms errors of0.5 and 0.2 micron combine vectorially tor(0.5)2+(0.2)2]1=0.54 micron. The computerprogram is designed to apply automaticallythe calibrated corrections for a given scale.

The second critical element of the com­parator, the micrometer screw, is calibratedto an accuracy of ±0.3 micron over its mea­suring range of one millimeter. This is ac­complished with the aid of a microscope scalecalibrated at 50-micron intervals over alength of one milli meter to an accuracy of±0.2 micron. Inasmuch as the screw is cali­brated when mounted in the comparator, thecalibration also accounts for any periodicerrors resul ting from eccentrici ty of the mea­suring drum or from progressive errors in thegraduation of the drum (the former mayamoun t to as much as ± 0.5 micron; thelatter, by virtue of a dividing accuracy of± 15 seconds of arc, should not exceed ±0.12micron). The calibrated corrections for thescrew are au tomatically applied by the com­puter program.

The primary requirement of the thirdcritical element of the comparator, the pivot,is that it provide a stationary point of rota­tion. This point is the center of a 0.75-inchball bearing having a sphericity of ± 0.2.1micron. The center of the ball is located in the

CO~IPUTATIOXAL TRADEOFFS IN THE DESIGN OF A COMPARATOR 193

plane of the scale thereby making the mea­surements insensitive to any up-and-dowllmotion of the far end of the scale. Wobble ofthe bearing is restricted by means of a spring­loaded thrust bearing that provides a con­trolled loading against the thrust bracket.The thrust is directed precisely along theaxis of rotation. With the comparator in itsnormal operating position and the thrust onthe pivot set at 3 kg, a radial wobble amoun t­ing to as much as 2 microns can be detectedover the sweep of the measuring arm; with athrust of 6 kg, there is no discernible radialwobble (the lateral component of wobble hasno effect on the measurements). To provide acomfortable margin of safety, the thrust onprod uction models is set at 10 kg.

RESULTS

Various production have models of thecomparator undergone extensive testing overthe past year. I n com pari sons em ployi ngplates with well defined images measuredboth on a conventional, two-screw compara­tor and on the Multilaterative Comparator,an rms agreement of final coordinates rangingbetween ± 2 and ± 3 microns is usually ob­tained after allowance has been made for atranslation and rotation of the one coordinatesystem to conform to the other. If one were toassu me that the discrepancies between theresults are equally attributable to both com­parators, one would conclude that the rmsaccuracy of the coordinates produced by theMultilaterative Comparator generally rangebetween ± 1.4 and ±2.1 microns, a rangecompatible with typical setting accuracies.Thus external evidence indicates that theMultilaterative Comparator is capable ofproducing accuracies at least comparablewith those produced by conventional, one­micron comparators.

A comparison has also been made againsta standard consisting of a 23X23 cm. Zeissgrid on which a representative sample of 33grid intersections had been calibrated by themanufacturer to a stated accuracy of 0.8micron (rms). The discrepancy (after allow­ance for translation and rotation) betweencomparator coordinates and grid coordinatesof the 33 calibrated points amounted to 1.50microns (rms). Of this, about 1.3 microns(i.e., [(1.5)2- (0.8)2]t can be attributed to thecomparator.

In addition to external evidence concerningaccuracies, there is internal evidence arisingas a by-product of the reduction of each plate.We refer here to the ranging residuals result­ing from the adjustment. By virtue of the

redundancy of the measuring process, eachpoint generates four observational residuals.These may be regarded as closures of quadri­lateration. As such, they should be consistentwith the errors to be expected from combinederrors of setting and errors of the compara­tor. Both the random and the systematicerrors of the comparator will be reflected inthe residuals. This means that if uncompen­sated systematic errors are sizeable in com­parison wi th random errors, they will domi­nate the determination of the residuals andwill reveal their presence in a plot of residualvectors. Thus the residuals from the adjust­ment provide a truly meaningful indication oftotal measuring accuracy, and the rms errorof the residuals, represen ting as it does an rmserror of closure, provides a particularly suita­ble criterion for quality control.

The rms errors of closure on productionmodels have been found to range typicallyfrom 1.5 to 2.0 microns for double settings onpoints marked by a Wild PUG III. The set ofresiduals obtained from measurements of a5 X5 array of PUC points evenly spaced at45 mm. intervals is listed in Table 1. Inas­much as double settings we,-e made on eachpoint, the rms error of setting was also deter­mined. The rms error of an individual settingwas found to be 1.7 microns which meant thatthe rms error of the mean of each pair ofsettings amounted to 1.2 microns. Inasmuchas the rms error of closure turned ou t to be1.6 microns (Table 1), this suggests that therms contribution of the comparator itself ison the order [(1.6)2- (1. 2) 2] h,,; 1.1 microns.

By-pmducts of the reduction are estimatesof the standard deviations of the plate coor­dinates. In the above example, 80 percent ofthe standard deviations in x and y ranged be­tween 1.1 and 1.3 microns; 2 points had stan­dard deviations as large as 1.5 micmns. Thesefigures consider the total error propagation;that is, the combined effect of the propagationof random measuring errors and the errors re­maining in the recovered values of the param­eters of the com para tor. I n general, thestandard deviations of the adj us ted x, y­coordinates will be somewhat smaller thanthe rms error of closure.

THE COMPUTER PROGRAM

If developed 15 years ago, the Multilatera­tive Comparator would have been little morethan an interesting academic curiosity, for noone could have tolerated its computationalrequirements. Accordingly, the digital com­puter, so commonplace today, is to be re­garded as an integral part of the comparator

194 PHOTOGRAMMETRIC ENGINEERIKG

TABLE 1. RA!\GING RESIDUALS FROM ADJUSTMENT

OF ARRAY OF 25 POINTS MEAS RED ON

!VI ULTILATERATIVE COMPARATOR

Ranging Residuals

Point Position Position Position Position1 2 3 4

1 1.1 It 1. Sit -0.7It o.11t2 0.3 0.9 -0.6 0.73 -0.3 -0.4 -0.2 -0.24 0.5 0.0 0.5 -0.55 -0.1 0.6 0.5 0.3

6 -0.1 0.6 0.5 0.37 1.7 -0.6 1.3 -1.48 -0.2 -0.1 -0.1 -0.99 -2.0 -2.4 -0.4 -1.3

10 -1.8 1.4 -3.8 3.0

11 0.8 -0.9 1.8 -0.912 1.7 0.8 1.2 0.813 0.2 0.1 0.2 1.014 0.3 0.4 0.5 0.415 0.1 0.5 0.6 -0.5

16 -.1 .0 0.4 -1.5 -0.817 -0.8 1.7 0.2 .1.6.18 0.9 -0.9 1.0 -0.319 -1.8 0.2 -.1.3 -0.920 1.0 -2.1 2.5 -1.1

21 -0.9 1.0 -0.8 -0.122 0.3 0.0 0.0 0.323 0.2 -0.5 -0.3 -0.324 0.3 -0.6 -0.4 -0.825 -0.2 0.2 -0.3 -0.1

Degrees of Freedom=2(25)-7=43Grand RMS=I.6microns

system. For this reason, a fully documentedFortran program is provided wi th the com­parator. The program is designed so that aminimum of modification is required to adaptit to almost any computer. One version of theprogram is designed specifically to run on aminimal computer configuration having cardinput and an available core memory equiva­len t to abou t 8,000 24-bi t words. Anotherversion is designed for medium- to large-scalecomputers. Both versions feature automaticediting and rigorous error propagation.Typical running time on an IBM 7094 forthe reduction of a plate containing 50 mea­sured images is well under 1 minute.

FURTHER DEVELOPMENTS

Optional accessories that have been de­veloped for the comparator include: (a) auto-

matic digital readout on punched cards to aleast cou n t of 0.5 micron; (b) a u ni versa I stagedesigned to accommodate both nonstandardplates and cut film; (c) a dual focus micro­scope designed to focus simultaneously on thescale and on the specimen mounted on theuniversal stage. Productivi ty experiencedwith the digitized version of the comparatorhas been found to be about double that ex­perienced wi th the standard model.

v\·e have under development a compactsemi·automatic setting device that retrofitsto the dual focus microscope. This device isexpected to provide an rms repeatability ofsetting to better than 0.5 micron on points ofmoderately high contrast.

CONCLUSIONS

The successful development of the Multi.laterative Comparator provides a strikingdemonstration of the power and effectivenessof self-calibration through designed re­dundancy. It clearly shows that through suit­able internal contradiction, one can achievecali bra tion. I t also gran ts a proper role to anew, key factor of systems design-the nowcommonplace availability of digital com­puters. By virtue of our success with theMultilaterative Comparator, we now haveunder development a number of other devicesthat are specifically designed to exploit prin­ciples of self-calibration to achieve improvedperformance with drastically simplified hard­ware.

REFERENCES

Brown, D. C. "A Treatment of Analytical Photo­grammetry with Emphasis on Ballistic CameraApplications," AFMTC TR 57-22, August 1957,Air Force Missile Test Center, Patrick Air ForceBase, Florida.

Brown, D. c., "A Solution to the General Problemof Multiple Station Analytical Stereotriangula­tion," A FMTC TR 58-8, February 1958, AirForce Missile Test Center, Patrick Air ForceBase, Florida.

Brown, D. C. "An Advanced Reduction and Cali­bration for Photogrammetric Cameras," A FCRLTR 64-40, January 1964, Air Force CambridgeResearch Laboratories, Bedford, Massa.chusetts.

Brown, D. c., "Advanced Techniques for the Re­duction of Geodetic SECOR Observations,"Final Report, July 1966, U. S. Army ContractNo. DA-44-009-AMC-937 (X), GIMRADA,Fort Belvoir, Virginia.

Brown, D. c.; Davis, R. G.; Johnson, F. c.; "ThePractical and Rigorous Adjustment of LargePhotogrammetric Nets," RADC TDR-64-092 ,October 1964, Rome Air Development Center,Rome, 1 ew York.