DepositionModelingfor Paint ApplicationonSurfacesEmbeddedin

�����David C. Conner PrasadN. Atkar Alfred A. Rizzi Howie Choset

CMU-RI-TR-02-08

October2002

RoboticsInstituteCarnegie Mellon University

Pittsburgh,Pennsylvania15213

c�

CarnegieMellon University

Abstract

Aspart of an ongoingcollaborativeeffort with theFord Motor Company, our research aimsto developpractical andefficient trajectoryplanningtools for automotivepainting. Not only mustthepaint applicatorpassover all pointson the surface, it mustdo so in a mannerthat ensures the uniformity of the coatingthickness.Thisis non-trivial giventhecomplexity of automotivesurfaces.Thisreportdocumentsour effortsto developanalyticdepositionmodelsfor electrostaticrotating bell (ESRB)atomizers, which haverecentlybecomewidely usedin the automotivepainting industry. Conventionaldepositionmodels,usedin earlierautomatictrajectoryplanningtools,fail to capture thecomplexity of depositionpatternsgeneratedbyESRBatomizers. Themodelspresentedheretakeinto accountboththesurfacecurvatureandthedepositionpatternof ESRBatomizers, enablingplanning tools to optimizeatomizertrajectoriesto meetseveral measuresofquality, such ascoatinguniformity. In addition to thedevelopmentof our models,wepresentexperimentalresultsusedto evaluateour models,andverify theinteractionbetweenthedepositionpattern,trajectory, andsurfacecurvature.

I

Contents

1 Introduction 1

2 Prior Work 12.1 TrajectoryPlanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 DepositionSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Deposition Modeling 53.1 2D DepositionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 PlanarDepositionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 SurfaceProjectionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 1D CollapseModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.1 PlanarThicknessVariationCalculation . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Cylindrical ThicknessVariationCalculation. . . . . . . . . . . . . . . . . . . . . . 17

4 Experimental Validation 244.1 DepositionModelParameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 PlanarDepositionResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 SurfaceDepositionResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 MiscellaneousResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Conclusions 29

6 Acknowledgments 29

References 31

A Revised Asymmetry Term for 2D Deposition Model 33

B Experimental Data 35

III

1 Introduction

Theapplicationof paint in theautomotive industryis a critical stepin theassemblyprocess.Thepaintmustbeappliedin a uniform manner, asexcessive variationin coatingthicknessis visible to thehumaneye andleadsto customerrejection[1]. Thepaintapplicatortrajectoriesmustbeplannedin sucha way asto guar-anteecompletecoverage,while minimizing thicknessvariation.Additionally, thepaintingprocessis subjectto severeeconomicandenvironmentalpenaltiesfor inefficiency. Sincestrict environmentalregulationslimittheamountof exhaustfrom thepaintingprocess,any reductionin thetotal amountof paintsprayeddirectlyimpactsboththeenvironmentandtheeconomics.This limits theamountof over sprayandstart/stops(trig-gering)of thepaintapplicator.

Industrial robotsare widely usedfor automotive paint applicationbecauseof the repeatabilityof thesurfacefinish, along with the removal of humansfrom a hazardousenvironment. The task of applyingautomotive paint hasmoved further beyond humancapabilitieswith the advent of high speedrotatingbellatomizersandelectrostaticcharging,bothusedto increasetransferefficiency. While applyingpaintis purelyrobotic,generatingtrajectoriesfor therobotsis largely a humanendeavor basedon theexperienceof skilledtechnicians.Theplanningtoolswidely available,suchasRobCADTM Paint, arelimited to simplepathson2D silhouettes,whicharethenprojectedbackontothe3D automotivesurface.Althoughtheuseof simulationsoftwarehascut theamountof validationrequiredon vehicles,theprocessstill requiressignificanttrial anderror. Sincethe final pathscannotbe generateduntil the body designis finalized–which is oneof the lastitemsin thedesignprocess–thedevelopmentof goodpaintingplansrepresentsa bottleneckin theconcept-to-customertime-line. Any progressin automatingthis taskultimatelydecreasesthe total time requiredtobringa new conceptto thecustomer.

In additionto timesavings,theautomaticgenerationof trajectoriesallowsfor theevaluationof trajectoriesagainstasetof specifiedcriteria.Trajectoriesthatareplannedfor therobotsmustyield paintdepositionthatisbothcompletein its coverage,andsufficiently uniformsothatthevariationin thicknessis notnoticeableanddoesnot degradethe mechanicalpropertiesof the coating. By planningtrajectoriesthat limit the requiredamountof paint sprayedoff the surface,andthe amountof start/stopcyclesof the paint flow, pathsmoreefficient from thestandpointof totalpaintusagecanbegenerated.

In thisreportwediscussour initial stepstowardsolvingtheproblemof automaticallygeneratingtrajecto-riesfor automotivepaintingon arbitrarysurfaces.Thecomplicateddepositionpatternsgeneratedby rotatingbell atomizershave madepreviouswork in trajectoryplanninginadequate,thereforethis reportoutlinesthedevelopmentof analyticmodelsof depositionpatternsfor this classof paint applicators,anddiscussestheimpactof the structureof thesepatternson pathplanning. Section2 coversrelevantprior work for this re-search,including both depositionmodelingandtrajectoryplanningfor paint application. In Section3, wedevelop analyticmodelsof the depositionpatterngeneratedby high speedrotatingbell atomizers,widelyusedfor automotivepainting.We furtherdevelopananalyticrelationbetweenthestructureof thedepositionpatternandthe variability of the paint thickness.In Section4, we discussexperimentaltestsandmethodsusedto determinevaluesfor theparametersof theanalyticmodels. Furtherresultsof experimentsdesignedto validatetheparameterizedmodelsdevelopedin Section3 arealsopresented.Finally, in Section5 wedrawconclusionsfrom theresultsanddiscussthefuturedirectionof ourwork.

2 Prior Work

Thework thatwepresentin this reportis anoutgrowth of ourprior work in theareaof coverageplanning[2,3]. Our earlier work developedplansfor guaranteeingcompletecoverageof an unknown area,and was

1

later lifted to surfacesembeddedin � �� [4]. While this prior work guaranteedcompletecoverage,it did notnecessarilyyield uniform coverage.Our work now focuseson thetaskof planningtrajectoriesin a way thatguaranteescompletecoverage,while at thesametimeminimizingcoatingthicknessvariation.Theremainderof thissectiondiscussesprior work in two broadareasthatoverlapwith ourcurrentfocus:trajectoryplanninganddepositionmodeling.

2.1 Trajectory Planning

An earlyattemptatautomatedtrajectoryplanningfor paintingrobotswastheAutomaticTrajectoryPlanningSystem(ATPS)[5]. ThesystemtookCAD datain theform of B-splinesor Beziersurfaces,andplannedrobotpathsthat followedin thedirectionof leastcurvature.Thevelocity profile alongthepathwasdeterminedtooptimizecoatinguniformity andtotal paintingtime given the plannedpaths. The paint depositionpatternwasassumedto be circular, with a uniform distribution within the circle. Paint distribution on the surfacewasbasedon theintersectionof asurfacepointwith thespraycone.Thishighly simplifieddepositionmodellimited theutility of themethodin industrialapplication.

A “teachless”spraypaintingsystemwasdevelopedby Asakawa andTakeuchi [6]. A seriesof spraypaintingpointswere input into the CAD data,andthe systemautomaticallygeneratedthe requiredpaths,includingoff thesurface(over spray)points. Thesystemrequireda setof parameterssuchastheellipticaldiametersof the paint patternandthe desireddistancebetweenconsecutive passes,which is known astheindex distance. A depositionmodel was not given; apparentlythe requiredparameterswere determinedexperimentally. No informationwasgivenabouthow theparameterswereestimated,or how to adaptthemto new surfaceshapes.

Researchersat theUniversityof Dortmundproposeda generalizedframework for off-line programmingof robots[7]. In this work they usetheexampleof paintingrobots,andproposea simplebivariateGaussianmodel for the paint deposition. Their work considersthe optimumindex distance,andcalculatesthe dis-tancebasedon theGaussiandepositionpattern.Thework doesnot considersurfaceeffectsor complicateddepositionpatterns.

Shenget al. developedan automatedCAD-guidedplanningsystemfor spraypainting[8]. Their workuseda simplifiedpaintdepositionmodel,without anexplicit dependenceon thesurfacebeingpainted.Thepathplanningalgorithmdependedonauser-definedindex distance,andwasverifiedusingRobCADTM Paintsoftware. In later work, Shenget al. extendedtheir prior work to considerthe effectsof surfacecurvaturein a limited manner[9]. This work formedpatcheswherethesurfacenormalsof triangularelementsusedtoapproximatethegivensurfacewerewithin certainbounds.Theboundwasbasedon a maximumdeviationanglebetweenthesurfaceelementnormalandthepaintgunnormal. Thepatchesandgunorientationwereiteratively solved to give an acceptablepaint depositionpattern. The work doesnot addressstitching theapproximatelyplanarpatchestogether, nordoesit addressgeneratingtrajectoriesoverhighly curvedsections.Thedepositionmodelassumedasimpleparabolicthicknessprofilewith acirculardepositionpattern.

Arikan and Balkan developeda paint depositionsimulationwherethe paint depositionmodel usedabetadistribution, shown in Figure1 [10]. Thepaperconsideredtheeffect of thedistribution patternon theoptimalindex distance,alongwith apreliminaryattemptatconsideringsurfaceeffectsonthedeposition.Thedevelopedspraypatternassumesanaerosolspray, andis not appropriatefor rotatingbell atomizers.

An automatictrajectorygenerationsystemfor unknown partshasbeendevelopedwhich usesscanninglaserrangefindersto detectpartsandtheir salientfeatures[11]. Features—suchasplanes,cylinders,andcavities—weredetectedfrom rangedata,andtrajectorieswerestitchedtogetherbasedonplansfor eachfea-turetype. This work wasonly concernedwith coverage,anddid not addressdepositionmodelsor thicknessvariation.

2

Hertling et al. reporton researchat OdenseUniversityin Denmarkto developanautomated“task curveplanner” for paintingrobotsin a shipyard environment[12]. Part of the motivation is the prohibitive costof robotprogrammingfor small numbersof paintingrunson customparts. Thecited paperfocusedon thedevelopmentof thedepositionmodels,whichfit aseriesof basisfunctionsto theexperimentaldataassumingan elliptical pattern. Hertling et al. report that the observeddepositionpatternswerenot uniform, anddidnot exhibit a parabolicprofile asreportedby other researchers.Although the final work is not publishedat this time, the researchershave demonstratedusingtheir modelsin numericoptimizationto planpathsonflat plates. For moreinformationseehttp://www.mip.sdu.dk/research/Smartpainter/index.html . The workassumesaerosolsprays,with the authorsspecificallyexcludingelectrostaticspraysbecauseof the inherentcomplexity of thedepositionpattern.

RamabhadranandAntonio presenta framework for efficient optimizationof trajectoriesfor paintingap-plications[13]. Theirwork focusesontheorganizationof theoptimizationproblem,assumingageneralformfor thedepositionmodel. In their work, thedepositionmodelis assumedto beeithera bivariateCauchyorGaussiandistribution(Figure1)appliedto aflat panel.Thiswork focusesontheefficiency of theoptimizationtechnique,anddoesnot covernew groundin developingrealisticdepositionmodels.

−40 −20 0 20 400

20

40

60

80

100Parabolic Distribution

position (mm)

Pai

nt T

hick

ness

( µ

m )

−40 −20 0 20 400

20

40

60

80

100

σ =1σ =5σ =10

σ =20

σ =40

Gaussian Distribution

position (mm)

Pai

nt T

hick

ness

( µ

m )

−40 −20 0 20 400

20

40

60

80

100

s=1s=5

s=10

s=20

s=40

Cauchy Distribution

position (mm)

Pai

nt T

hick

ness

( µ

m )

−40 −20 0 20 400

20

40

60

80

100

β=1.5β=2

β=3

β=5

β=10

Beta Distribution

position (mm)

Pai

nt T

hick

ness

( µ

m )

Figure1: Simpledepositionpatternstypically usedin previousresearch.Cauchydistributionsaresimilar toGaussiandistributions,but have thinnerpeaksandfattertails.

3

2.2 Deposition Simulation

Theworkscitedthusfarhaveall usedeitherhighlysimplifieddepositionmodels,or modelsthataredevelopedspecificallyfor aerosolsprayatomizers.Automotive coatingprocessesaremoving increasinglytowardstheuseof electrostaticrotatingbell (ESRB)atomizersin orderto increasetransferefficiencies[14, 15, 16]. In anESRBatomizer, paintfluid is forcedontotheinnersurfaceof ahighspeedrotatingbell, which is maintainedat a high voltageof 50-90 kV relative to the groundedsurfacebeingpainted. Most modernsystemsusenegative polarity at the bell [1]. Figure2 shows a schematicof the atomizerconfiguration.The paint flowbreaksupat theedgeof thebell, formingacloudof droplets,asit is expelledradiallydueto centrifugalforceimpartedto thepaintby therotatingbell. Eachpaintdropletis chargeddueto thechargeon thebell. If theparticlecharge is above the Rayleighlimit, the dropletwill breakapart,further atomizingthe paint spray.High velocity shapingair, andoften a chargedpatterncontrol ring, is usedto force the chargedparticlestowardsthesurface.Electrostaticforcesandaerodynamiceffectsinfluencethetrajectoriesof eachparticle.

-- --

-

Charged

Paint�

Particles

Shaping�

Air

Flow

Charged

Rotating Bell

Charged Pattern

Control Ring

Paint

�Flow

- --

Figure2: Electrostaticrotatingbell atomizerwith paintparticletrajectoryandshapingair flow linesshown.

Earlywork in modelingtheelectricaleffectsof theseESRBsystemswasperformedbyElmoursi[15]. Thesimulationassumeda uniform dropletsizeanddid not considertheevaporationof solventduringtransport.Furthermore,neitheraerodynamiceffects nor the interdependencebetweenthe droplet trajectoryand theelectricfield were taken into consideration.The modeldid considerthe effect of spacecharge dueto thedistributedchargeddropletson the electricfield. Sincethe spatialcharge distribution is dependenton theelectricfield, whichis in turndependentonthespatialchargedistribution,aniterativesolutiontechniquewasusedto arriveatamutuallyconsistentsolution.It wasfoundthatincreasingthechargedensityof theparticlesincreasedtheelectricfield, therebyincreasingthedeposition.However, therewasatradeoff in thatincreasedchargedensitiesincreasedtheexpansionof thespraycloud.For chargedensitiesthatweretoo low, theclouddid not expand,leaving a noticeabledoughnutshape.High chargedensitiescausedtheouterportionsof thefan to be exposedto weaker fields, therebylimiting the deposition. In additionto the voltage,the chargedensityis affectedby thebell speed,paintdelivery rate,andpaintresistivity.

Ellwood andBraslaw developeda finite elementmodel of the depositioncharacteristicsof the ESRBatomizers,whichextendedthework of Elmoursiby includingmomentumeffects[14]. Theimpactof variousparticlesizeswasincludedby modelinga largenumberof trial trajectoriesfor severalparticlesizeclasses.Becausethe particlecharging time scalesareon the orderof the time scaleof the atomizationprocess,thepaintparticlesareoftenincompletelycharged.Thesimulationusedaconstantcharge-to-massratio,basedonprior work in the literature.For eachtrial, themomentumbalanceon eachparticlewascalculatedbasedontheaerodynamicdragforcesandelectrostaticforcesdueto thechargedparticlemoving throughtheelectric

4

field. Themodelaccountedfor theexchangeof momentumbetweenthepaintdropletsandthegasphasedueto solventevaporation.Thespacechargedensityis calculatedasanensembleaverageof all of thetrials overall sizeclasses.A streamline,upwind,Petrov-Galerkinfinite-element(SUPG)formulationis usedto solvethe coupleddynamicandelectricalequationsof the continuousgasphaseandelectricalfield basedon thediscreteparticletrajectories.Thesimulationtheniteratively solvesfor theparticletrajectoriesassuminganelectricfield, andthe electricfield assumingparticletrajectories,until the solutionsconverge. Simulationswith andwithout the patterncontrol ring were conducted.The patterncontrol ring exerts force onto theparticlesthroughanelectricfield sufficient to directthesprayplumetowardsthesurface,therebyincreasingthetransferefficiency of theprocess.Oneeffectpredictedby thesimulation,andconfirmedby experimentalobservation, is a doublering phenomenonin the depositionpattern,wherethe paint is depositedin twoconcentricrings.

HuangandLai alsoconductedstudiesof thespraytransportfrom theESRBatomizerusingfinite elementtechniques[16]. Fewer detailsof their derivationaregiven in their preliminarypaper, but accordingto theauthorsthesimulationsshow “consistenttrendswith experimentalobservation.”

3 Deposition Modeling

The depositionmodeldevelopedin this reporthastwo primary purposes:i) to capturethe structureof thedepositionpatternfor useby planningtools,andii ) to supportsimulationsthataccuratelypredicttheresultsof specificatomizertrajectories.Thesetwo purposesleadto contradictorycriteria for evaluatingthemodel.First, themodelmustbeaccurateenoughto capturethestructureof thedepositionandaccuratelypredictthedepositionon a varietyof surfaceshapes.However, themodelmustbetractablefrom theperspective of thesimulationandplanningtools,sincethemodelwill beusedby theplanner.

Furthermore,it is desiredthat themodelbeof ananalyticform thatadmitsa closedform calculationofpartialderivatives.This will enableanalyticlocal optimizationwith respectto quality measures.Thedesirefor simple,analyticmodelshasled us to rejectexplicit finite-elementcomputationof fluid dynamicsandelectro-staticeffectsof thetypepresentedin [14, 15]. Althoughthesetechniquesmaygeneratemoreaccuratesimulationsof paintdeposition,andcanthereforebe justifiedduringa final pathvalidationandrefinement,the computationalexpenseis not justified during preliminary developmentof the path planningtools. Adiscreterepresentation,which couldmodelarbitrarydistributions,wasalsoconsideredbeforebeingrejecteddueto thedesirefor analyticrepresentations.

Thepatternof paintdeposition,or film build, generatedby ESRBatomizersis a functionof thespecificatomizer, processparameters,shapeof the surface,andrelative orientationof the atomizerto the surface.For the ABB Micro-Micro Bell Atomizerstudiedin this report,the overall shapeof the depositionpatternis roughlycircularwhenthebell is orientednormalto a flat panelandtheatomizeris stationary. Thepaintdepositionpatterngivesthemeasuredpaint thicknessover two dimensions;we refer to this asthe2D depo-sition pattern. As theatomizerpassesover thesurface,themajority of thepaintemittedby theatomizerisdepositedon thesurface,althoughsomepaint is entrainedin theshapingair andlost. We refer to thepaintthicknessprofile orthogonalto thedirectionof travel, which is equivalentto thatobtainedby integratingthe2D depositionmodelalongthe directionof travel, asthe 1D collapse. Figure3 shows the relationshipbe-tweenthe2D depositionpatternandtheresulting1D thicknessprofile. Our approachhasbeento modelthe2D depositionpatternandthe1D collapseseparatelybecausetheintegralof the2D depositionpatternis nottractable.

5

Figure3: Paintingflat panelshowing therelationshipbetweenthe2D Depositionpatternandthe integratedthicknessprofile (1D collapse).

3.1 2D Deposition Model

We werelooking for a 2D depositionmodelthatassignsthe rateof paint depositionor depositionflux at agivenpoint on anarbitrarysurface,givena specificpathlocationandorientationof theatomizer. Themodelwe developed,denoted����������� , is of the form ������� ��� ��!#"%$&�('*)��+%� ,�� � , where �.-/� ���0�1!2" is apoint andunit surfacenormalon thesurfacebeingpaintedand �3-1'4)��+�� is a pathlocationandorientationof thebell atomizer. We refer to �������5�6� asthe2D depositionmodel, or simply thedepositionmodel. Thedepositionmodel,which is dependenton thepaintflow rateandotherprocessparameters,is scaledto givethedepositionflux. The total thicknessat a givenpoint on thesurfaceis dependenton this depositionflux,thepathfollowedby theatomizerover thesurface,andthespeedat which thepathis traversed[10, 12, 13].

Sinceparameterizingthedepositionmodelfor arbitrarysurfacesis difficult atbest,andsinceexperimentaldatafor planarsurfacesis readilyavailable,we developedananalyticmodelfor depositionflux on a planarsurface.Throughrecourseto differentialgeometry, theplanardepositionis mappedontoanarbitrarysurfacein a way that preserves the total paint volume. We refer to the analytic model for the planarsurfaceastheplanar depositionmodel. Theplanarsurfaceis referredto asthedepositionmodelplane, andis shownin Figure4. The depositionmodelplaneis orientednormal to the atomizera fixed distance798 from theatomizerpathlocation � alongtheatomizernormal :; . We assumethat thepathlocation � is thetool centerpointframe(TCPF)specifiedby theplanner, andusedby therobotcontrolprogram.TheTCPFspecifiesboththelocationof thetool centerpoint,andtheorientationof thetool in space.Often,theTCPFis specifiedontheautomotivesurfaceandnot on thepaintatomizerat theendof therobot. We assumethepaint is emittedfrom atheoreticalemissionpoint < locatedalongtheatomizernormalat thedistance79= from thetool centerpoint. Thedistancefrom theemissionpoint to thedepositionmodelplaneis givenby 7?>@7 8�A 79= . GiventheTCPF � , 7 8 , and 79= thepaintemissionpoint anddepositionmodelplaneareuniquelyspecified.

6

Ω=Ωd-Ωe

e

Deposition Model PlaneB

z

Ωd

ΩepC

Figure 4: The atomizerpath location uniquely determinesboth the emissionpoint and depositionmodelplane.As drawn thevalueof 79= is negativesincetheemissionpoint is behindtheatomizerpathlocation � .3.1.1 Planar Deposition Model

We denotethe planardepositionmodelas DE��F#�G>HDE��I2��J*� , where D(�K� � " ,L� � , and F3>M�NIO��J*� is a pointon thedepositionmodelplane. Thepoint F is itself a functionof thesurfacepoint � andtheatomizerpathlocation � , asshown in Figure5. Becausetheemissionpoint < andthedepositionmodelplanearedirectlyrelatedto the pathlocation � , the point F is a functionof both � and � , suchthat DK��F#�P>QDK��F��R���5�6�5� . Theorientationof the x-y depositionmodel planeaboutthe z-axis of the bell atomizeris determinedby theorientationof theatomizerassembly, andis independentof thedirectionof atomizertravel.

Ω

eS

sTq

Deposition Model PlaneU

SurfaceV

eW sXz

eW sXnV

W

Figure5: Projectionof depositionmodelonto arbitrarysurface. Although the vectorsare in reality threedimensional,this simplefigureconveysthebasicresults.(Note:Thepathlocation � is not shown.)

Theplanardepositionmodelusestwo Gaussians—oneoffset1D Gaussianrevolvedaroundtheorigin and

7

one2D centeredGaussian—andascalingfunctionthatgeneratesanasymmetryin themodel.Theasymmetryis requiredbecausesignificantasymmetrywasfoundin preliminaryexperiments.Althoughtherotatingbellis axially symmetric,theshapingair nozzlesandpatterncontrol ring arenot necessarilysymmetric,whichgive rise to theasymmetriesfound in thepatterns[1]. Theresultingplanardepositionmodel,similar to theasymmetricvolcanoshown in Figure3, is givenbyDE��I2�5J4�Y>[Z]\]���5^ A Z " � _6�NIO��J*�a`%\b��I2�5J4�#cdZ " ` " �NI2�5J*�#�]� (1)where Z]\e-.� �f scalesthedistribution to give thepaintdepositionflux in unitsof thicknesspersecondandZ " -hg i*�j^lk weightstherevolvedGaussianagainstthecenteredGaussian.To accountfor asymmetryin thedepositionpattern,therevolvedoffsetGaussian,`%\P�m� � " ,n� � , is scaledby thefunction _(�m� � " ,o� � . Wedefine _ to be _6�NIO��J*�Y>p�q^�cdZ �0r�sut ��vxwyv tEz �NJ{�5IK� A}| �5�{�where Z � -hg i*�j^lk weightstheasymmetryscalingfunctionfor therevolvedGaussian1. Thephaseangle, | ,allows theasymmetryto belocalizedrelative to theatomizerreferenceframe.

Lookingattheindividualcomponentsof (1), thenotionof revolvinganoffsetGaussianfor ` \ is somewhatambiguous.Figure6 shows two possiblechoices.In thefirst thevaluesfor Id-~g i*�}k aredeterminedandthenrotatedabouttheverticalaxisshown in the left figure. This yieldstheresultshown by the lower curvein thefigureon theright. Theresultingsweptvolumeis notdifferentiableat theorigin. This is dueto thelost“tail” to the left of theaxisof rotation. Thesecondchoiceis to accountfor the tail beingsweptalongwiththecurveandaddingto theresult.This resultsin adifferentiablefunctionasshown by theuppercurvein theright handsideof Figure6. Usingthis technique,therevolvedoffsetGaussian,`%\ , is definedto be

` \ �NI2�5J*�> ^ #m A% I " c}J " A(b "z� "\ c l4 A% I " cJ " c "zx "\

�where is the offset radius, \ is the standarddeviation of the Gaussian,and normalizesthe depositionsuchthatintegralof ` \ over I and J equalsone.Thescalingfactor is givenby > zxz� "\ l4 A "zx "\E c \ z 5 z \ 1} Thesecondexponentialin `%\ accountsfor the“tail” of theGaussianthatcrossestheaxisof revolution.

ThecenteredGaussian,` " ��� � " ,¡� � , alsonormalized,is givenby` " ��I2��J*�> ^z# "" m A IK"�c}J4"zx "" �where " is thestandarddeviationof thecenteredGaussian.3.1.2 Surface Projection Model

The planarmodel definedabove appliesto depositionon flat panels,with the atomizerorientednormalto the surfaceand locateda fixed offset distancefrom the depositionmodel plane. The next stepin the

1Thescalingfunctionhasbeenmodifiedbasedon theexperimentalresultsgivenin Section4. Thereaderis referredto AppendixAfor thedefinitionof thelatestmodel.

8

0 Distance

Thi

ckne

ss P

rofil

e

0 Distance

Thi

ckne

ss P

rofil

e

Figure6: (l) OffsetGaussianbumpshown with ^j¢�i%£ rotation. (r) Curvesshowing resultof consideringthetail of theGaussianin therevolvedsurface(upper)andignoringthetail portion(lower).

modeldevelopmentis to extendtheplanarmodelto arbitrarysurfaces,offsetdistances,andorientations.Wesimplify the modelusinga simplegeometricprojection,which ignoresthe electrostaticandfluid dynamiceffectsof thepaintspray, just asin theprior work cited in Section2. However, our derivationpreservesthetotalpaintvolume[9, 10, 12]. We chosethisprojectionmodelbecausetypical carpaintingapplicationskeepthebell atomizerat a roughlyconstantoffsetdistance,androughlynormalorientationrelative to thesurfacebeingpainted.

The projectionmodel,shown in Figure5, is developedby assumingthat all of the paint emits from apoint source,calledthe emissionpoint < , which is constrainedto lie alongthe bell-to-surfacevector :; asdescribedabove.Note,thisemissionpoint is a theoreticalemissionpoint,notnecessarilycoincidentwith thebell atomizercenterpoint.

Thedepositionmodelplaneis embeddedin � � � orthogonalto the :; vectoradistance7 from theemissionpoint. Furthermore,weassumethatthex-y frameof theplanardepositionmodelis alignedwith thex-y planeof theatomizerreferenceframe.A vectorfrom theemissionpoint to a point � on thesurfacepassesthroughthe depositionmodelplaneat point F¤>¦¥4��������� . We abusenotationanduse F to refer to either the 2Dposition �NIO��J*� on thedepositionmodelplaneor the3D position ��I2��J{�79� relative to theemissionpoint. Thepoint F is a functionof boththepathlocationandthesurfacepoint. It is assumedthat theplanardepositionDK��F#�P>QDK��IO��J*� , asdefinedin (1), is known for a givenpoint F§>M�NIO��J{�y79� on the depositionmodelplanedefinedin Figure5.

A differentialelementonthedepositionmodelplanegivesapaintsolidsvolumeof ¨©>~DK��F#�DRIDRJ . In thegeneralcase,asthisdifferentialelementis projectedontothesurfaceaboutpoint � , theareaof theprojectionis differentfrom thatof the differentialelement.In orderfor the total volume,andthereforethe total paintmass(assumingconstantsolidsdensity),to remainunchanged,the paint thicknessmustchange.We willderivetherequiredrelationshipbetweenthethicknesson theplanarmodelandtheprojectedthicknesson thesurfaceusingtheconceptof areamagnificationasdefinedin differentialgeometry[17].

We simplify the derivation by usingtwo steps:first we mapthe depositionfrom the depositionmodelplaneto theemissionpoint in away thatpreservesvolume,thenmapfrom theemissionpoint to thesurface2.Let ªK«}>¬���*� | �-]k A O®xz � O®xz gm�¯k A O®xz � O®xz g�°&±²!2" – i.e., the openlower hemispherecenteredat the

2Themapis actuallyto a spherewith infinitesimalradiuscenteredat theemissionpoint.

9

emissionpoint asdeterminedby � , thepathlocation3. If we define³>?vxwyv tKz �NI2�79� and | >[vxwyv tEz �NJ{�y79� ,the point in the ª « parameterspacecorrespondsto the unit vector :[7 . Themappinģd�YªE«¯,¹µ« is givenas¸]��*� | �>p��7¯wv t *�7ºwv t | �y79�   (2)For the map ¸»�*ªK«Q, µ« definedabove, the coordinatevector fields are definedas )¼?>L½{¸ ® ½K@>¾ 7 r ¿ "l�5i*�yi�À , )�ÁP>~½K¸ ® ½ | > ¾ i*�7 r ¿ " | �yi�À , with normalorientationgivenby Â[>p��i*�yi*�j^� .

In orderto conservepaintvolume,thethicknessmustdecreaseastheareaincreases,andviceversa.Thisallowsthecalculationof thedepositionin the ª « parameterspace,DÄÃÅK��*� | ���ª « ,Æ� � , basedontheplanardepositionmodelfor thedepositionmodelplaneandtheareamagnificationfactorof themap ¸ at point F .Thedepositionin the ªE« parameterspaceis givenbyDOÃÅK��*� | �>~ÇÈÊÉx��¸��4DK��F#�Ë� (3)wheretheareamagnificationÇaÈ É ��¸�� is givenbyÇaÈ É ��¸��>~Ì w ) ¼) Á >[7 " r j¿ " r ¿ " |   (4)

Now considertheprojectionfrom theemissionpoint to a point � on thesurfacebeingpainted.Let Íδbe the tangentplaneattachedto thesurfaceat point � . This tangentplaneis definedby thepoint � andthesurfacenormalvector :Ï .

Themappingfrom ª « to thetangentplane ͳ´ , denotedÐa�*ª « ,¹Í³´ , is givenbyÐÊ��*� | �>~Ñ��Òwv t *��wyv t | �j^�]� (5)where Ñ is givenby ÑÓ> ÔÕ :×

Combiningwith (3), we have DÄÜÝb���b�Y> ÇaÈ É �R¸��ÇaÈ Ù ��л� DK�RF#�   (7)Givenapathframe � , asurfacepoint � , andthevectors:; and :~D Ü Ý����Y> 7" Õ :

æ ��Iä%��Jxäj��> ÷�øùyú0û ï é êìëí ï é êôë ¾ ��^ A Z " � _6��Iä%��Jxä�cJ*��`%\x�NIEä��5Jxä�c}J*�ceZ " ` " �NIä%��J�ä�c}J*� À D�J   (10)Unfortunately, the complexity of the analyticmodelrendersthe calculationof an analyticintegral for (10)intractable.Instead,we directly definea 1D collapsemodelusingthreeseparateGaussians.In this model,chosenbecauseit fit theexperimentaldatawell, two Gaussiansareoffsetfrom thecenterlineto allow asym-metriesin the depositionpatternto be modeled,while the third Gaussianis centered. The complete1Dcollapsemodelis givenby ü �NI{�ý> \�ÿþ "�� \� ø � \ l4 A ��� í�� ø �" � ø c\� � " l4 A ��� f � �" �

c\�� � � 4 A ��" � �   (11)Thethreeexponentialtermsthatconstitutethis modelareeachnormalized,and ó representstheoffsetsand ó thestandarddeviations.Thegains � ó areusedto specifythepaintdepositionthicknessfor eachGaussian,andaredifferent from the constantsdefinedfor the 2D depositionmodel. Figure8 shows the componentGaussiansandthe compositefilm build for a particularsetof parametervaluesfor (11). Note this modelassumesa particulardirectionof travel, andwill most likely have differentparametervaluesfor differentdirectionsof travel. Experimentaldeterminationof theparametervaluesis discussedin Section4.

−500 0 5000

5

10

15

20

25

30

35

40

Position (cm)

Film

Bui

ld (

mic

rons

)

CompositeGaussians

Figure8: Asymmetric1D collapsemodelwith componentGaussiansshown.

3.2.1 Planar Thickness Variation Calculation

In order to control the amountof variation in the coatingthickness,the trajectoryplannermustknow therelationshipbetweenthedepositionpatternof theatomizerandthedepositionon thesurfacebeingpainted.For paintingspecialists,the knowledgeis intuitive basedon yearsof experience.To automatethe processof generatingthesetrajectories,we needa computableunderstandingof therelationshipbetweendepositionpatternsandthicknessvariation.

12

Typically, the depositionpatternis narrow comparedto the width of the surfacebeingpainted,andre-quiresmultiple passesto completelycover thesurfaceasshown in Figure9. Thedistancebetweenconsecu-tive passesis known asthe index distance; theprocessof changingpaintinglanesis referredto asindexing.A naturalquestionwhenplanningpathsfor thepaintingrobot is, ‘what is theappropriateindex distancetocontrolvariation?’Initially restrictingourselvesto aflat panel,wewill answerthisquestionby lookingat thepaint depositionthicknessprofile perpendicularto the directionof travel usingthe 1D collapsemodel. Weassumethat therobot is moving in a straightline, andthat thepathis sufficiently long sothateffectsduetochangingdirectionsarenegligible4.

}}}Pass #1�

Pass #2

Pass #3�

Index Distance

Figure9: Paintingaflat panelwith 3 passes.

To developanunderstandingof how thedepositionpatternandindex distanceinteractto determinethick-nessvariation,we will assumeaninfinite planepaintedby aninfinite numberof passeswith theatomizerata consistentorientationrelative to the planeandmoving at a constantspeed.For a given location I in theinteriorof theplane,alonga line perpendicularto thedirectionof travel, thetotal thicknessis givenbyæ ��I2���I{��> ��ó�� í �

ü �NI³c����I{�{� (12)where�dI is theindex distance,and ü ��� � is the1D collapsemodelfor thegivenspeedandorientationrelativeto thedirectionof travel. In this case,the i è�� passis assumedto bealonga centerlineof theplane,with thevariable I beingmeasuredrelative to this centerline,perpendicularto thedirectionof travel. Looking at thethicknessmeasurementsaswevary I , themeasurementpatternrepeatsitself with aperiodequalto theindexdistance.

Thethicknessvariation,overoneindex distance,is givenby " > ^�dI ç�� �í � � ¾ æ ��I2���dIK� A æ À " D�I2� (13)4Thequestion,‘What is sufficiently long?’, is answeredrelative to thediameterof thedepositionpattern.

13

whereæ

is theaveragethicknessover theinterval, which is definedasæ > ^�dI ç �!�í �!� æ �NI{�9D�I   (14)The limits of integrationwerechosendueto the periodicity of the variationcalculation,but areotherwisearbitrary. For this reason,boththelimits of integrationandthe ^ ® �dI termareconsideredto beconstants.

Thenormalizedvariationoveroneindex, with respectto theaveragethicknessæ

, is givenby "" > ^�dI ç æ ��IO���dIK�æ �#�I{� A ^ " D�Iâ>$�I û æ �NIO���dIK� " D�I¾ û æ �NIO���dIK�9D�I À " A ^   (15)Note,thelimits of integrationhavebeendroppedfor compactness,but arethesameas(14). Theintegralmaybeexpanded,usingequations(12)and(14), to give "" >%�dI & �ó�� í � & �' � í � û

ü ��I³c��(�dIK� ü ��I³c*) �dIK�D�I& �ó�� í � & �' � í � ûü �NI³c��(�dIK�D�I û ü ��IÎc+),�I{�9D�I A ^   (16)

Theintegraltermsin thedenominatoraretractable,andleadto termsinvolving theerrorfunction, j� ��� � . Theresultbeinggivenby û ü ��I³c+),�dIK�D�I > � " \ ) 5 � � í�� ø f '�- �þ " � ø �yc " ) 5 � � f � f '(- �þ " � �yc � ) 5 � � f '�- �þ " � � �   (17)The integral term in the numeratorresultsin a morecomplex solution,but an analyticsolutiondoesexist.The equationsare sufficiently complex so that a numericalintegration is warrantedin the calculationofthe numeratorterm in the summation.Numericalintegrationwasusedto generatethe plots in this andthefollowing sub-sections.

With regard to the infinite summations,the infinite extentsof the Gaussiandistributions usedin thedepositionmodelsarean idealization. The actualdepositionpatternsareclearly finite, so the summationsmaybetruncatedbasedonthenumberof passesthatinteractwith agivenpointonthesurface.For simulationpurposes,wechoseto truncatethesummationsbasedona5- calculation.Giventhe1D collapsemodel,wecalculatetheinteractionwidth as . >0/0v � \Yc21 \b� " c21 " ��1 � �Ë�choosingthe interactionwidth basedon both theoffset radiusandthestandarddeviations. Thesummationlimits, bothpositiveandnegative,arethencalculatedas3 > ¿ s�4ìsut65 .�dI   (18)Usingequations(16), (17), and(18),alongwith basicnumericintegration,thevariationasa functionof theindex distancefor a givensetof 1D collapseparametervaluescanbecalculated.

A typicaldeviation,definedasthesquarerootof variation,versusindex distancecurveis shown in Figure10. As expectedlarge indicesyield high variation. If the model was only a single centerGaussian,the

14

variationversusindex distancecurvewould beanisotone(monotonicallyincreasing)function. However, incaseswheretherearesignificantoffsetGaussianterms,thereis a local minimum in variationversusindexdistancecurve. The existenceof this “sweetspot” may allow the useof larger index distancesto generatepaintcoveragewith acceptablevariation,while reducingtotal cycle time. However thevariationtendsto besensitive to changesin index distanceat this spot,so for tight tolerances,keepingindex distancessmallerthanthelocal minimummaybeadvisable.Knowledgeof how thethicknessvariationchangeswith changesin index distanceis helpful to thetrajectoryplanner, whetherin determininganabsoluteindex distanceor inevaluatingthesensitivity of agivenindex distance.

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

Index Distance (mm)

Nor

mal

ized

Dev

iatio

n (

σ N )

Figure10: Normalizeddeviationvs. index distancefor a typical 1D collapsemodelon a flat surface.

Theability to evaluatethethicknessvariationasa functionof modelparametervalues,aswell asatom-izer orientationandsurfaceeffects,allows us to analyzethe effectsof changeson any of theseoperationalvariables.For example,considerthe effect of changingthe offsetdistance,which impactsthe 1D collapseparametervalues.Figure11showsthevariationsurfacegeneratedby varyingboththeindex distanceandtheoffset distance.The closedform solutionsgiven above allow us to analyzethe effect on coatingthicknessfrom varyingany of theparametervalues.

With theanalyticrepresentationfor variationgivenin (16), we areableto evaluatethestructuraleffectsof agivenmodelonvariationaswevarytheindex distances.By evaluatingthepartialderivativewith respectto index distance,we canemploy root finding techniquesto find local minimumof thevariation. We beginby takingthepartialderivativeof (15) to give7 � 87 - � > z �I û:9 ���=�?A@ �CB �D��E= �!� 8 �¾ û 9 ���

1

1.5

2

0

200

400

600

8000

0.1

0.2

0.3

0.4

0.5

Bell Offset(Zb/Ω)

Index Distance (mm)

Nor

mal

ized

Dev

iatio

n (

σ N )

Figure11: Normalizeddeviation vs. index distanceandoffsetdistancefor a typical depositionmodelon aflat surface. I!J is theactualoffsetdistancefrom thesurfaceto emissionpoint,while 7 is thenominaloffsetdistanceto thedepositionmodelplane.

partialdifferentiationoperators,we obtain7 � 87 - � > z �dI & ê &LK û NM ê � - �� =�O K @ �!��E= �!� 8 �& ê &LK û M ê � - � 8 � û M K � - � 8 �A z �dI & ê &PK�&$Q û � M ê � - � M K � - ��� 8 � û =�O Q @ �D��E= �!� 8 �& ê &0K(&%Q û M ê � - � 8 � û M K � - � 8 � û M Q � - �� 8 �   (20)Thesummationsarebetweennegativeandpositive

3, asdefinedin (18). Lookingat eachintegral term,first

notethat ½ ü ' �#�I{�½R�dI >2) ½ü ' �G�dIK�½EI �

which impliesthat ç ½ü ' �G�dIK�½S�I D�I]>2)

ü ' �T�I{�  Theterm ç ü ó �T�I{� ½

ü ' �G�dIK�½S�I D�Idoesnot have an analyticsolution(at leastnonesolvableby MathematicaTM), but caneasilybe solved bynumericintegration.Theterm ç ü óE�G�dIK� ü ' �T�dIK�D�Ihasananalyticsolution,but asmentionedbefore,thesolutionis socomplicatedthatnumericintegrationisalsoadvised. Becauseof the linearity, the summationtermscanbe moved inside the integral to improvethe efficiency of the overall numericsolution. Denotingnumericintegrationby “Int( )”, andmaking the

16

substitutionsgivenabove,wecanrewrite (20)as7 � 87 - � > z �dIVUXW�Y & ê & K M ê � - � =�O K @ �!��E= �!� & ê & K û M ê � - � 8 � û M K � - � 8 �A z �dI UXW�Y ¾ & ê & K M ê � - � M K � - � À & Q[Z M Q � - ��& ê & K & Q û M ê � - � 8 � û M K � - � 8 � û M K � - � 8 � � (21)wheretheintegral termsin thedenominatorsareevaluatedaccordingto (17).

3.2.2 Cylindrical Thickness Variation Calculation

The developmentof the above variation calculationswas presentedfor planarsurfaces. To calculatethevariationon the surfaceof a cylinder, onemight imaginetaking the planar1D collapse,andprojectingitaroundthecylinder. This doesnot work dueto thedistortionof thedepositionpatternasit is projectedontothecurvedsurfaceof thecylinder, aswell aspaintlostdueto overspray. Ourapproachwill beto first calculatethe2D depositiononthecylinderusingthe2D depositionmodeldevelopedin Section3.1,andthencalculatethe 1D collapseof the depositionon the cylinder. We cannumericallyintegrate(8) to find numericvaluesfor the paint thicknessprofile on the cylinder orthogonalto the directionof travel. Therearetwo primarydirectionsto travel: eitheraroundthe cylinder andindex alongthecylinder, or alongthe cylinder indexingaroundthecylinderasshown in Figure12

Figure 12: (l)Painting along the cylinder axis, and indexing aroundthe cylinder. (r) Painting aroundthecylinder, andindexing alongthecylinderaxis.

Considerthe arrangementshown in Figure 13, wherewe are painting the outsideof a cylinder. Thecylinder is centeredalongthey-axis,with IE"�cd¶�"e>0\G"M , where \ M is theradiusof thecylinder. We assumethatthepaintatomizeris orientednormalto thesurface,locatedalongthey-axisatanangle from thex-axisin thexz-plane.Specifyingtheatomizerpathlocationas ��\]J ¿ ö r *��JAJ�^\]J r5sut �� , wheretheradiusof thepathlocationis definedto be \ J >$\ M c&798 , theorientationof theatomizerframeis givenas � A ¿ ö r *�yi*� A r5sìt �� .We refer to a point � on the surfaceasbeing locatedat �G\ M ¿ ö r[_ �5J Ù �^\ M r�sut`_ � , with an inward pointingnormalof � A ¿ ö r[_ �yi*� A r5suta_ � .

Whenpaintinga cylinder, the radiusof the cylinder hasa direct impacton the depositionof paint onthe surface. This is dueto both the curvatureeffectsandthe paint lost dueto over spray. Smallercylinderradii resultin morepaintloss,asdepictedin Figure14. Note,thatourgeometricprojectionmodelprecludes

17

xbyczd

zd pexb pe yc pe

θ Ψ

xfzgzg ph

θi

Ψ

RcjRbkΩl dm

xf phΩl

enRen

Figure13: Painting a cylindrical surface. The atomizerlocationis specifiedby the modelparameters,theangle , andthedistancealongthecylinderaxis J J . A point on thecylindersurfaceis specifiedby theangle_ andthedistancealongthecylinderaxis J Ù .paintingmorethana 180degreearcof thecylinder. Theactuallimitation canbecalculatedgiventheradiusof thecylinder \ M andthelocationof theemissionpoint, \ = >$\ J A 7 = , where \ J is theradiusof thepathlocationand 7 = is thedistancefrom thepathlocationto theemissionpointalongtheatomizernormal.At thelimit, theanglebetweenthevectorfrom theemissionpoint to a givensurfacepoint, andthesurfacenormalvectoris 90 degrees,asdepictedin Figure14. Thelimiting angleis givenby_ ñôóuõó è > ¿ ö r í \ \ M\ =  

Whentraveling aroundthe cylinder, we will assumethat we arepaintinga half cylinder, and that theindexing occurson thebacksideof thecylinder. In this way paintdoesnot fall on thehalf cylinder we arepainting(dueto thegeometricprojection),andindexing effectsarenegligible. The1D collapseis calculatedby integratingaroundthe cylinder, from A _ ñôóuõ»ó è to c _ ñôóuõó è , creatinga profile alongtheextrusionaxisofthecylinder. Figure15 shows theresulting1D collapseprofilesfor a seriesof cylinder radii. Giventhe1Dcollapseon thecylinder, the variationcalculationsareidenticalto theplanarcasewhenindexing alongthecylinder axisof extrusion.Figure16 shows theresultingthicknessdeviation surfaceasbothcylinder radiusandindex distancearevaried.

Whentraveling alongthe cylinder axis, the 1D collapseis calculatedby integratingalongthe cylinderaxis, resultingin a profile on a circular slice of the cylinder. We assumethat indexing aroundthe cylinderoccursat a distanceaway from wherewe measurethethicknessprofile, so that the indexing operationdoesnot impactthethicknessprofile. Figure18showstheresulting1D collapseprofilesfor variouscylinderradii.

The1D collapsethicknessvaluesarefoundby numericallyintegratingthedepositionasprojectedontocylindersof variousradii. Unfortunately, for smallcylinder radii the1D collapsemodeldevelopedin Section3.2doesnotprovideagoodmodelto thenumericallyobtaineddata.In otherwords,thebestparameterfit for

18

overospraypzone

ψq limitrzs

x

Figure14: Lostpaintdueto overspray, assuminggeometricprojection.

Rt

=100 mm

Rt

=3162.3 mm

Rt

=1000 mm

Rt

=316.2 mm

R=1x106 mm

R=10000 mm

R=31.6 mmR=10 mm

Figure 15: 1D collapsethicknessprofiles, obtainedby integrating the depositionmodel after projec-tion onto the cylinder, where the atomizer is painting around the cylinder and indexing along theaxis of extrusion. The graphic also shows the results for a seriesof cylinders with radii of \ M >^��^jiFu��^��^jiAv�^i�i�i�i*�y+*^�w�i*�j^ji�i�i4�5+^xw*�j^i�i*�5+*^   w�5v t Ìâ^iy/z/ . The thicknessprofile correspondingto the largestradiuscloselymatchesthat of the flat panelprofile. The thicknessdecreasesasthe radii de-creasebecauseof theincreasingamountof paintlost to overspraydueto thegeometricprojectiondevelopedin Section3.1.2.Theprofiledoesmaintainthebasicshapeevenasthethicknessdecreases.

19

0

500

1000

0

200

400

600

8000

0.05

0.1

0.15

0.2

0.25

Rc (mm) Index

Distance (mm)

Nor

mal

ized

Cyl

inde

r D

evia

tion

(σ)

Figure16: Normalizedthicknessdeviation asa function of both cylinder radiusand index distancewhenpaintingaroundthe cylinder andindexing alongthe cylinder. As it moves,the atomizeris alwaysorientednormalto thecylindersurface,at a constantoffset.

0

500

1000

0

200

400

600

8000

20

40

60

80

100

Rc (mm) Index

Distance (mm)

Ave

rage

Thi

ckne

ss (

mic

rons

)

Figure17: Averagethicknessvaluesfor cylinderpaintingcorrespondingto Figure16.

(11) doesnot provide a goodmatchto thedata,asshown in Figure19. Onecoulddefinea new 1D collapsefor this specialcaseor solve the variationcalculationsnumericallyusingthe 2D model. Sincethe current1D collapsefits well until the cylinder radii is muchsmallerthanthedepositionpatternwidth, we will notseekto definea new modelat this time. Solving thevariationcalculationsusingnumericintegrationof the2D depositionmodelis not useddueto thecomputationalinefficienciesinvolved.For thesereasons,we willkeepthe current1D collapse,andfocuson cylinderswith larger radii. It is expectedthat for the surfacesof interest,namelyautomotive surfaces,small radii of curvaturewill occurascharacterlineson larger lesscurvedsurfaces,which will dominatetheplanning.Oncethe1D collapseis found,thevariationcalculationsareidenticalwith the index distanceassumedto be arc lengthon the cylinder surface. The correspondingdeviationsurfaceis shown in Figure20.

Sofarwehave focusedon paintingtheexternal(convex) surfaceof cylinders;we now switchto paintingtheinside(concave)surfaceof a cylinder asshown in Figure21. We will restrictthediscussionto cylinders

20

Rt

=100 mm

R=1000 mm

Rt

=316.2 mm

R=1x106 mm

Rt

=10 mm

Figure 18: 1D collapsethicknessprofiles when painting along the cylinder axis of extrusion and in-dexing aroundthe cylinder. The graphic also shows the results for a seriesof cylinders with radii of\ M >¡^� ^jiFux�^ji�i�i�5+*^w*�j^ji�i�5v t Ìâ^jiz/y/ . The maximumthicknessdoesnot changesignificantly, buttheprofile getsthinnerasthecylinder radiusdecreases.

−100 −50 0 50 1000

5

10

15

20

25

30

Arc Length R ⋅ ψ (mm)

Inte

grat

ed D

epos

ition

(m

icro

ns) Integrated

1D model

Figure19: Attemptedfit of 1D collapsemodelto the numericallyintegratedcollapsewhenpaintingalongthecylinder axisof extrusionfor \ M >p^ji�i mm. For thedepositionparametervaluesusedin thesetests,thecurrent1D collapsemodelprovidesagoodfit if thecylinder radiusis 500mm or greater.

21

400

600

800

1000

0

200

400

600

8000

0.2

0.4

0.6

0.8

Rc (mm) Index

Distance (mm)

Nor

mal

ized

Cyl

inde

r D

evia

tion

(σ)

Figure20: Normalizedthicknessdeviation asa function of both cylinder radiusand index distancewhenpaintingalongthecylinder axisof extrusionandindexing aroundthecylinder. As it moves,theatomizerisalwaysorientednormalto thecylindersurface,at aconstantoffset.

whoseradii exceedtheoffsetdistanceof theatomizeremissionpointfrom thesurface,otherwisetheemissionpoint is beyondthesurfacefocal point andindexing is ill defined.Note thatall of thepaint sprayedon theinsidewill fall on the surface,unlike the convex surfacewherepaint is lost dueto over spray. The actualcalculationsof depositionandvariationare identical to convex painting,only the definition of the surfacenormalschange.

deposition model plane{z

x

Figure21: Painting the inside(concave) surfaceof a cylinder. We assumethat the radiusof thecylinder islargerthantheoffsetof theatomizeremissionpoint from thesurface.

When painting aroundthe inside of the cylinder, the depositionprofile getstaller and sharperas theradiusdecreases,asshown in Figure22. Whenpaintingalongtheinsideof thecylinder, smallerradii tendtomagnifythepeaksdueto theoffsetGaussianterms,asshown in Figure23. The1D collapsemodelgenerallyprovidesa goodfit to the integratedcollapsefor cylinderswith radii larger thanthesurfaceoffsetdistance,whetherpaintingalongor aroundthecylinder.

While the precedingresultson cylindrical surfacesarenot directly applicableto automotive surfaces,

22

R=254 mm

R=3162.3 mmR=1000 mmR=316.2 mm

Rt

=1|

x106 mm

Rt

=10000 mm

Figure 22: Integrated 1D thicknessprofiles for painting around the inside of a cylinder, and index-ing along the cylinder. The graphic shows the results for a seriesof cylinders with radii of \ M >^Û� ^iFux�j^²� ^jiAv�^i�i�i�i*�y+*^w z �j^ji�i�i*�y+4^�w   z �5v t Ì z 1 ^P�1^jiAux�y+*^�w z �j^ji�i�i*�y+*^�w   z �5v t Ì z 1

they do provide insight into the relationshipbetweenthe depositionpatternandthe surfacecurvature. Oursubsequentwork will focuson moregeneral“automotive-like” surfaces.In Section4, we focuson resultsofexperimentsdesignedto evaluateourmodelsandthetheoreticalresultspresentedin thissection.

4 Experimental Validation

The 2D depositionand1D collapsemodelswerevalidatedby conductinga seriesof testsat ABB ProcessAutomationin Auburn Hills, Michiganon September26-27,2001.TheexperimentsusedanABB S3robotwith anABB 50 mm Micro-Micro Bell atomizerattachedto applya solventbasedautomotivepaintto phos-phatecoatedtestpanels. The operatingconditionsof the applicationprocesswere80-90kV electrostaticvoltage,150cc/minpaint flow, 250 l/min shapingair flow, anda bell speedof +�iYi�i�i RPM. The total filmthicknessof the oven curedtestpanelswasmeasuredwith an Elcometer355 coatingthicknessmeasuringdevice. Five measurementsweretakenfor eachdatapoint, with thelow andhigh discardedandtheaverageof the remainingthreerecorded.The averagephosphatethicknesswasthensubtractedfrom the total filmthicknessto give thepaintthickness.

4.1 Deposition Model Parameterization

In orderto parameterizethe 1D collapseand2D depositionmodels,experimentaldatawasgatheredfromflat panelspaintedby threepassesasshown in Figure5. Threedifferentindicesweretested:525mm, 577mm, and625mm. We choseto parameterizeour modelsusinga 577mm index distancefor this threepasstestbecausethe observedvariationwassufficient to discernthe structuraleffectsof the offset radii neededfor the modelparameterization.Figure24 shows thestepsthatwe followed. Using the577mm index testdata,weusedastandardnumericoptimizationroutinein MATLABTM to determinethebestparametervalues( � \ � � " � � � � b\ � " � \ � " � � ) for the1D collapsemodeldefinedin (11). Themaximumof thetwo offsetradiiandthemaximumstandarddeviationfrom the1D collapsemodelwerethenusedto initialize theoptimizationof the2D depositionmodelparametervalues.

Given an initial parameterizationof the 2D depositionmodel,we calculatedthe 1D collapsethicknessvaluesfor the parameterized2D modelusingnumericintegration. The integrated1D collapsevalueswerethencomparedto the experimentaldata. Numericoptimizationwasusedto find the 2D depositionmodelparametervalues( Z]\�5Z " �5Z � � | � � \x� " ) thatminimizedthesumsquarederrorbetweentheexperimentaldataandthenumericallyintegrated1D collapse.Theparameterizedmodels,both2D and1D collapse,wereshown previously in Figure3. Figure25showstheresultingprofile (1D collapse)obtainedfrom asimulationusing the 2D depositionmodelagainstthe datato which it wasfit. The simulationresultswereobtainedthroughnumericevaluationof our depositionmodels,andmatchtheexperimentaldatawell.

4.2 Planar Deposition Results

Usingthe2D depositionmodelparameterizedby the577mmindex threepasstest,thedepositionsgeneratedby 525 and 625 mm index testswere simulated. The resultsare shown in Figure 26. The model givesa good predictionof both averagefilm build and the structureof the variation for theseflat panel tests.Most importantly, themodelcapturedboththeasymmetriesandthestructuralvariationdependenceon indexdistance.

24

Fit 1D collapse to 577mm index 3-pass

e xperimental dataF

it 2

D deposition to5

77 mm index 3-passe xperimental dataS

imulate 577 mmindex using 2

D model

a nd compareresulting profile to

3

-pass experimentald

ata

S

imulate 525 mmindex using 2

D model

a nd compareresulting profile to

5

25 mm 3-passe xperimental data

S

imulate 625 mmindex using 2

D model

a nd compareresulting profile to

6

25 mm 3-passe xperimental data

Figure24: Stepsin fitting andverifying modelperformance.

−1000 −500 0 500 10000

5

10

15

20

25

30

35

40

Distance (mm)

Film

Bui

ld (

mic

rons

)

simulationdata

Figure25: Flatpaneltestresultsusinga577mmindex distance(avg. error= 6� microns,standarddevi-ation= F 6 microns).Thedepositionsimulationusestriangulatedsurfaceelementsto modelthethicknessdepositionata givenpointon thesurface.

4.3 Surface Deposition Results

Giventherelativelygoodresultsof theflat paneltests,theprojectionof theplanardepositionmodelontoarbi-trary surfaceswastested.A representativeautomotivesurfacewasobtainedby usinga truck door. Figure27shows a CAD modelof the truck door used,with an examplepathshown. The door hasa line of convex

25

−1000 −500 0 500 10000

5

10

15

20

25

30

35

40

Distance (mm)

Film

Bui

ld (

mic

rons

)

simulationdata

−1000 −500 0 500 10000

5

10

15

20

25

30

35

40

Distance (mm)

Film

Bui

ld (

mic

rons

)

simulationdata

Figure26: Flat paneltestresults:(l) 525mm index test,and(r) 625mm index test.Both (l) and(r) usedthemodelparameterizedby thedatafrom the577mm index test.Both simulationscapturethevariationduetothestructureof thedepositionpattern.(averageerror: l = 6 F andr = :A microns,standarddeviations:l = 6 [ andr = A A microns).curvaturenearthe middle,with a pronouncedconcave curvatureon the bottomthird of the door. A seriesof testswereconductedusingboth horizontalandvertical passesover the door. For the horizontalpasses,film build measurementsweretakenin four verticalcolumnsof dataspreadacrossthedoor, numberedtop tobottom. For theverticalpasses,themeasurementsweretakenfrom six rows spreadvertically over thedoorspaningleft to right acrossthe door. For the first horizontaltest,resultsfor a typical columnareshown inFigure27. Thesimulateddepositionfor eachpassindividually is alsoshown.

Nearthetop of thedoor, in therelatively flat portion,thesimulationgivessomewhatreasonableresults.However, thesimulationhasdifficulty predictingpaintthicknessin thehighly curvedsectionnearthebottomof thedoor. Clearlythepassalongthelowerportionof thedoordepositsmorepaintthanthesimulationpre-dicts. It is theorized,asshown in Figure28,thatwhenthesurfacecurvesawayfrom theatomizer, electrostaticeffectsdominateinvalidatingthegeometricprojectionmodeldescribedin Section3.

Similar testswereconductedfor verticalpaintingmotions,with comparableresults.For datacollectedon theupperrelatively flat portionof thedoor, thesimulationresultswerereasonable.However, for resultson thelowerportionof thedoor, thesimulationagainpredictedtoo little paintdeposition.

4.4 Miscellaneous Results

It wasalsodesiredto verify thatourdepositionmodelsscalewith applicatorspeed.To thisend,two additionaltestswereconducted.Thesetestsusedasinglepass,with therobotpaintinghorizontally, at tip speedsof 100mm/secand250mm/sec.It wasintendedto comparetheseresultsto the50 mm/sec3-passresults.The250mm/sectestresultedin significantspatteringat thenominalpaintflow ratesbeingused.Sincethesensorwasnotdesignedto measurediscretedropsof paint,theresultsweredeemedinadmissible.Theresultfor the100mm/sectestis shown in Figure30. As shown, thesimulationpredictedmuchgreaterpaintdepositionthanactuallymeasured.It hasbeentheorizedthat the transferefficienciesincreasewhenpaintingwet surfaces,

26

0 200 400 6000

10

20

30

40

50

Vertical Position from Top (mm)

Film

Bui

ld (

mic

rons

)

total pass #1pass #2pass #3data

Figure27: (l) Doorwith horizontalpaintpathshown. Therobotpaintsleft to right startingat theleft of pass#1, thentravelsright to left alongpass#2, finishingby going left to right alongpass#3. (r) Simulationof ahorizontalpaintingmotionover thedoor, with depositionby individualpassesshown.

Deposition Plane

Surface

Figure28: Theorizedwarpingof paintparticletrajectoriesdueto electrostaticeffects.

becausesomepaint initially bouncesoff of thedry surfaces.Slower tip speedsallow morewet paintfilm tobuild up, therebyincreasingtheaveragetransferefficiency [1, 18].

Also noticethat thedatasetin Figure30 exhibits anasymmetry, while thesimulationdoesnot. Duringthis test,theorientationof theatomizerwasconsistentwith theprevious5773-passtest,while thedirectionof travel was orthogonalto the 577 mm index test usedin the model parameterization.The depositionmodeldevelopedin Section3 hasa hemisphericasymmetry. This testimpliestheneedfor a morelocalizedasymmetryterm in the 2D model,asis definedin AppendixA. Sincethe 1D collapsemodelis dependent

27

−600 −400 −200 0 200 400 6000

5

10

15

20

Horizontal Position (mm)

Film

Bui

ld (

mic

rons

)

simulationdata

−600 −400 −200 0 200 400 6000

5

10

15

20

Horizontal Position (mm)

Film

Bui

ld (

mic

rons

)

simulationdata

Figure29: Verticalpaintingmotionon door: (a)upperflat portion(b) lowercurvedportion

−1000 −500 0 500 10000

5

10

15

20

Distance (mm)

Film

Bui

ld (

mic

rons

)

simulationdata

Figure30: SingleHorizontalpasswith V=100mm/sec.Simulationpredictshigherpaintdepositionthandatashows for this higherspeed,likely dueto changesin wet/drytransferefficiency.

on the directionof travel anyway, no changeto the 1D collapsemodelis needed.With the additionof thismorelocalizedasymmetrycomponent,themodelerrorwould needto becalculatedagainstbothhorizontalandverticalmotionsduringtheparameteroptimization.

28

5 Conclusions

Theresultsof ourexperimentalstudyallow usto concludethatourmodelscapturetherelevantstructureof theplanardepositionpattern,andthedependenceof thethicknessvariationon thatstructure.It is alsoapparentthattheinteractionof thepaintdropletsemittedfrom theatomizerandthesurfacecurvaturehasasignificantimpacton theactualdepositionpatternon curvedsurfaces.Thesepreliminaryconclusionsalsoindicatetheneedfor additionaltestsregardingthe dependenceof the depositionpatternandtransferefficiency on thespeedof theatomizerasit movesrelative to thesurface.

Themodelswedevelopedaccuratelypredictdepositiononplanarsurfaces,wheretheatomizeris orientednormalto thesurface.Additionally, ouranalytic1D collapsemodeleffectivelypredictsthedependenceof thethicknessvariationon theindex distancebetweenpasses.Althoughtheexperimentalresultsfrom depositionon the curvedsurfaceof thedoor point to shortcomingswith the simplegeometricprojectiondevelopedinSection3, the experimentsdo confirm the interactionof the surfacecurvaturewith the planardepositionpattern.

Despitetheshortcomingsof our2D depositionmodel,themodelsareusefulfor ourresearch.By usingananalyticmodel,weareableto developourunderstandingof theinteractionbetweenthesurface,thedepositionpattern,andtheatomizerpath.Thisenablesourexplorationof pathplanningtechniquesthatinfluenceoverallqualitymeasuressuchasthicknessvariation,cycle time,andefficiency. Sincethemainfocusof our researchis on path planning,we will continueto usetheseanalytic modelsduring the developmentphaseof ourplanningtools. Sinceour planningtools rely only on the structureof the depositionon the surface,andnot on theunderlyingmodel,theneedfor moreexpensive modelsor experimentaldatais delayeduntil theimplementationstage.

During our next roundof experiments,we will validatethemodificationsto theasymmetrytermsin ourplanardepositionmodel. This will requirethreepasstestsin both the horizontalandvertical directions,with theresultingmodelfit from thecombinationof bothdatasets.Experimentswill beconductedto morerigorouslyevaluatethedependenceof thedepositionrateon thespeedof theatomizer. For this we will needto differentiatebetweeneffectsdueto speedalone,andeffectsdueto a prior build up of wet paint on thesurface.We will alsobegin thepreliminaryvalidationof someof our pathplanningtechniqueswith respectto openingsin thesurfaceandsurfacecurvature.

6 Acknowledgments

Thiswork wassupportedby theNationalScienceFoundationthroughgrantIIS-9987972andtheFordMotorCompany.

The authorsgratefully acknowledgethe assistanceof the Ford Motor Company andABB ProcessAu-tomationfor theirassistancein conductingtheseexperiments.Wewould liketo specificallyacknowledgeDr.JakeBraslaw, our Fordcollaborator, who hasbeenextremelyhelpful throughoutthiseffort.

29

30

References

[1] JacobBraslaw. personalcommunication,2001.

[2] ErcanU. Acar, Howie Choset,Alfred A. Rizzi, PrasadN. Atkar, andDouglasHull. ExactCellularDecompositionsin Termsof Critical Pointsof MorseFunctionsfor Sensor-basedCoverageTasks.TheInternationalJournalof RoboticsResearch, acceptedfor publication2001.

[3] ErcanU. Acar andHowie Choset. Sensor-basedCoverageof Unknown Environments: IncrementalConstructionof ExactCellular Decompositionsin Termsof Critical Pointsof MorseFunctions. TheInternationalJournalof RoboticsResearch, acceptedfor publication2001.

[4] PrasadN. Atkar, Howie Choset,Alfred A. Rizzi, andErcanU. Acar. ExactCellularDecompositionofClosedOrientableSurfacesEmbeddedin . In IEEEInt’l. Conf. onRoboticsandAutomation, Seoul,Korea,May 2001.

[5] Suk-HwanSuh,In-KeeWoo, andSung-KeeNoh. Developmentof An AutomatedTrajectoryPlanningSystem(ATPS)for SprayPainting Robots. In IEEE Int’l. Conf. on Roboticsand Automation, Sacre-mento,California,USA, April 1991.

[6] NaokiAsakawaandYoshimiTakeuchi.TeachlessSpray-Paintingof SculpturedSurfaceby anIndustrialRobot.In IEEEInt’l. Conf. onRoboticsandAutomation, Albuquerque,New Mexico,USA,April 1997.

[7] EckhardFreund,Dirk Rokossa,andJürgenRoßmann.Process-OrientedApproachto anEfficient Off-line Programmingof IndustrialRobots.In IECON’98: Proceedingsof the24thAnnualConferenceoftheIEEE IndustrialElectronicsSociety, volume1, 1998.

[8] WeihuaSheng,Ning Xi, Mumin Song,Yifan Chen,andPerryMacNeille. AutomatedCAD-GuidedRobotPathPlanningfor SprayPaintingof CompoundSurfaces.In IEEE/RSJInt’l. Conf. on IntelligentRobotsandSystems, 2000.

[9] HepingChen,WeihuaSheng,Ning Xi, Mumin Song,andYifan Chen. AutomatedRobotTrajectoryPlanningfor SprayPaintingof Free-FormSurfacesin AutomotiveManufacturing.In IEEE Int’l. Conf.on RoboticsandAutomation, Washington,D.C.,USA, May 2002.

[10] M. A. SahirandTunaBalkan. ProcessModeling, Simulation,andPaint ThicknessMeasurementforRoboticSprayPainting. Journalof RoboticSystems, Vol. 17(9),2000.

[11] AndreasPichler, MarkusVincze,HenrikAnderson,OleMadsen,andKurt Häusler. A Methodfor Auto-maticSprayPaintingof Unknown Parts.In IEEEInt’l. Conf. onRoboticsandAutomation, Washington,D.C.,USA, May 2002.

[12] PeterHertling,LarsHøg,RuneLarsen,JohnW. Perram,andHenrikGordonPetersen.TaskCurvePlan-ning for PaintingRobots— Part I: ProcessModelingandCalibration. IEEE Transactionson RoboticsandAutomation, Vol. 12(2),April 1996.

[13] RamanujamRamabhadranand JohnK. Antonio. Fast Solution Techniquesfor a Classof OptimalTrajectoryPlanningProblemswith Applicationsto AutomatedSprayCoating. IEEE TransactionsonRoboticsandAutomation, Vol. 13(4),August1997.

31

[14] Kevin R. J.EllwoodandJ.Braslaw. A Finite-ElementModel for anElectrostaticBell Sprayer.Journalof Electrostatics, Vol. 45(1),1998.

[15] AlaaA. Elmoursi.ElectricalCharacterizationof Bell-TypeElectrostaticPaintingSystems.IEEETrans-actionson IndustryApplications, Vol. 28(5),October1992.

[16] HuaHuangandMing-ChaiLai. Simulationof SprayTransportfrom RotaryCupAtomizerusingKIVA-3V. In ICLASS2000, Pasadena,California,USA, July2000.

[17] J.A.Thorpe.ElementaryTopicsin DifferentialGeometry. Springer-Verlag.,New York, NY, 1979.

[18] Jeff Petty. personalcommunication,2001.

32

A Revised Asymmetry Term for 2D Deposition Model

Theplanardepositionmodeldevelopedin Section3.1.1failedto capturethebidirectionalasymmetryexhib-ited in our experiments.Therevision to this modelpresentedhereis designedto allow themodelto capturethemorelocalizedasymmetryevidentfrom thebidirectionaltests.

For convenience,thebasicform of theplanardeposition,first givenin (1) is repeatedhere:[�^� !¡$¢¤£¤¥ a ¢§¦x :¨©�ª¥6 «£NG^� ¬®¢§¦,«F¦A�^6   Theoriginalasymmetryfunction,givenin Section3.1.1,wasdefinedto be¨©�^6 D¡¯ ¬�¢ y°^±�² G³¾�¿ #À>Á ° �¹ à  2   ¦

34

B Experimental Data

35

36

AB

B F

lexi

ble

Aut

omat

ion

Pro

cess

Val

idat

ion

Lab

Tes

ting

Pro

gram

For

d / C

arne

gie

Mel

lon

Tes

ts

Pai

nt-M

ater

ial

Per

cent

-Sol

ids-

Vol

.E

stim

ated

Dat

e9/

27/2

001

Spe

cific

-Wei

ght-

(lbs/

gal)

Bel

l92

3P

erce

nt-S

olid

s-W

eigh

t

Vis

cosi

ty-(

For

d-#4

-Cup

)17

.89-

sec.

@80

0 F

Tes

t-P

anel

-#S

etpt

..-F

luid

-F

low

-cc/

min

Sha

ping

-Air-

Nl/m

inB

ell-S

peed

-K

RP

MB

ell-C

up-S

ize

Tar

get-

Dis

tanc

eH

.V.-

Not

es

9-26

-1�&

�215

025

030

5010

90

9-26

-3�&

�415

025

030

5010

90

9-26

-5�&

�615

025

030

5010

90

9-26

-7�&

�815

025

030

5010

90

Con

veyo

r�S

peed

�(F

PM

)S

tatio

nary

Gun

�Trig

ger�

On�

Leng

th�(

Inch

es)

Man

ual�T

rig.

Boo

th�A

ir�F

low

�(F

PM

)

Boo

th�T

empe

ratu

re�(

F)

Boo

th�H

umid

ity�(

%R

H)

Tor

eado

r�R

ed�(

New

�Bat

ch�-

�Jun

e�'0

1)

�Usi

ng�2

N14

08�5

0mm

�Bel

l�Cup

�and

�"A

nnul

ar�S

lit"�

S.�A

.�Rin

g�w

/200

mm

�rep

ulsi

on�r

ing.

��App

licat

or�in

dex�

577m

m,�t

ip�s

peed

�50

mm

/sec

,�3�v

ertic

al�s

trok

es.

�Usi

ng�2

N14

08�5

0mm

�Bel

l�Cup

�and

�"A

nnul

ar�S

lit"�

S.�A

.�Rin

g�w

/200

mm

�rep

ulsi

on�r

ing.

��App

licat

or�in

dex�

625m

m,�t

ip�s

peed

�50

mm

/sec

,�3�v

ertic

al�s

trok

es.

�Usi

ng�2

N14

08�5

0mm

�Bel

l�Cup

�and

�"A

nnul

ar�S

lit"�

S.�A

.�Rin

g�w

/200

mm

�rep

ulsi

on�r

ing.

��App

licat

or�in

dex�

577m

m,�t

ip�s

peed

�25

0mm

/sec

,�3�v

ertic

al�s

trok

es.

�Usi

ng�2

N14

08�5

0mm

�Bel

l�Cup

�and

�"A

nnul

ar�S

lit"�

S.�A

.�Rin

g�w

/200

mm

�rep

ulsi

on�r

ing.

��App

licat

or�in

dex�

525m

m,�t

ip�s

peed

�50

mm

/sec

,�3�v

ertic

al�s

trok

es.

For

d-C

MU

9-26

-01

Mat

rix8/

8/20

02

For

d�/�C

arne

gie�

Mel

lon�

Tes

tsO

vera

ll�P

anel

Cen

ter�

60"

Cen

ter�

48"

Cen

ter�

36"

For

dF

ord

For

dF

ord

Dat

e�9/

27/2

001

Film

�Avg

.S

td.�D

ev.

Eva

l.F

ilm�A

vg.

Std

.�Dev

.E

val.

Film

�Avg

.S

td.�D

ev.

Eva

l.F

ilm�A

vg.

Std

.�Dev

.E

val.

Row

�#1

1.07

280.

3138

29.2

5%1.

1613

0.20

6717

.80%

1.23

900.

1176

9.49

%1.

2839

0.09

337.

27%

Tes

t�Num

ber�

#�9-

26-1

�&�2

Row

�#2

1.05

370.

3056

29.0

1%1.

1383

0.20

7318

.21%

1.22

050.

1111

9.10

%1.

2631

0.08

746.

92%

Row

�#3

1.07

600.

3134

29.1

3%1.

1652

0.20

3417

.45%

1.24

530.

1012

8.13

%1.

2883

0.07

135.

53%

cc/m

in.�

150

Sha

ping

,�(N

l/min

)25

0R

PM

(x�1

000)

30B

ell�C

up,�m

m50

Tar

get�D

ist.,

�in.

10H

.V.,�

KV

90

Row

�#1

23

Pnl

�Pos

.F

ilm�T

hkF

ilm�T

hkF

ilm�T

hk0 1 2

0.33

0.32

30.

380.

420.

374

0.43

0.46

0.43

50.

510.

500.

506

0.58

0.58

0.58

70.

690.

650.

658

0.79

0.76

0.72

90.

880.

870.

8510

0.98

0.91

0.98

111.

040.

951.

0312

1.11

1.02

1.08

131.

101.

101.

0914

1.09

1.07

1.09

151.

051.

051.

0616

1.01

1.02

1.03

171.

030.

991.

0718

1.06

1.07

1.11

191.

171.

141.

1920

1.28

1.13

1.27

211.

291.

131.

3222

1.31

1.22

1.37

231.

291.

321.

3424

1.27

1.27

1.32

251.

281.

231.

3226

1.30

1.28

1.33

271.

301.

331.

3328

1.30

1.32

1.33

291.

261.

311.

2930

1.23

1.28

1.25

311.

251.

261.

2532

1.27

1.26

1.26

331.

231.

261.

2034

1.20

1.15

1.14

351.

161.

051.

1836

1.16

1.12

1.18

371.

161.

121.

1838

1.12

1.12

1.23

391.

191.

201.

2940

1.27

1.26

1.36

411.

311.

331.

3442

1.35

1.33

1.33

431.

341.

331.

3244

1.34

1.30

1.30

451.

311.

271.

2946

1.28

1.27

1.28

471.

341.

281.

3048

1.39

1.32

1.32

491.

391.

361.

3450

1.39

1.36

1.36

511.

421.

371.

3552

1.44

1.35

1.35

531.

391.

331.

3354

1.34

1.30

1.31

551.

281.

271.

2456

1.23

1.21

1.17

571.

161.

141.

1758

1.09

1.12

1.16

591.

101.

101.

1660

1.10

1.14

1.16

611.

111.

181.

1362

1.11

1.10

1.09

631.

041.

021.

0364

0.96

0.91

0.97

650.

820.

800.

8566

0.69

0.68

0.74

670.

590.

570.

6168

0.49

0.49

0.48

690.

440.

400.

4470

0.39

0.38

0.40

710.

36

�Usi

ng�2

N14

08�5

0mm

�Bel

l�Cup

�and

�"A

nnul

ar�S

lit"�

S.�A

.�Rin

g�w

/200

mm

�re

puls

ion�

ring.

��App

licat

or�in

dex�

525m

m,�t

ip�s

peed

�50m

m/s

ec,�3

�ve

rtic

al�s

trok

es.

02

46

810

1214

1618

2022

2426

2830

3234

3638

4042

4446

4850

5254

5658

6062

6466

6870

72

1

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

Film

Thk

.

Pan

el P

ositi

on

Row

#

525m

m In

dex

50m

m/s

ec. t

ip s

peed

1.50

-1.6

01.

40-1

.50

1.30

-1.4

01.

20-1

.30

1.10

-1.2

01.

00-1

.10

0.90

-1.0

00.

80-0

.90

0.70

-0.8

00.

60-0

.70

0.50

-0.6

00.

40-0

.50

0.30

-0.4

00.

20-0

.30

0.10

-0.2

00.

00-0

.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

01

23

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

2627

2829

3031

3233

3435

3637

3839

4041

4243

4445

4647

4849

5051

5253

5455

5657

5859

6061

6263

6465

6667

6869

7071

72

Pan

el P

ositi

on

F i l m B u i l d

1 2 3

For

d-C

MU

9-26

-01

Pan

el�#

926-

1&2

8/8/

2002

For

d�/�C

arne

gie�

Mel

lon�

Tes

tsO

vera

ll�P

anel

Cen

ter�

60"

Cen

ter�

48"

Cen

ter�

36"

For

dF

ord

For

dF

ord

Dat

e�9/

27/2

001

Film

�Avg

.S

td.�D

ev.

Eva

l.F

ilm�A

vg.

Std

.�Dev

.E

val.

Film

�Avg

.S

td.�D

ev.

Eva

l.F

ilm�A

vg.

Std

.�Dev

.E

val.

Row

�#1

1.07

730.

1909

17.7

2%1.

1376

0.09

268.

14%

1.15

520.

0781

6.76

%1.

1590

0.08

187.

06%

Tes

t�Num

ber�

#�9-

26-3

�&�4

Row

�#2

1.06

560.

1878

17.6

2%1.

1243

0.08

417.

48%

1.14

090.

0651

5.70

%1.

1514

0.06

986.

06%

Row

�#3

1.06

090.

1880

17.7

2%1.

1224

0.07

636.

80%

1.13

900.

0577

5.07

%1.

1409

0.06

245.

47%

cc/m

in.�

150

Sha

ping

,�(N

l/min

)25

0R

PM

(x�1

000)

30B

ell�C

up,�m

m50

Tar

get�D

ist.,

�in.

10H

.V.,�

KV

90

Row

�#1

23

Pnl

�Pos

.F

ilm�T

hkF

ilm�T

hkF

ilm�T

hk0 1 2

0.57

0.53

30.

620.

620.

604

0.68

0.73

0.67

50.

800.

830.

806

0.92

0.91

0.92

71.

010.

981.

008

1.11

1.06

1.08

91.

171.

141.

1410

1.24

1.17

1.20

111.

211.

201.

1712

1.18

1.16

1.14

131.

161.

131.

1114

1.14

1.10

1.08

151.

181.

061.

1116

1.21

1.11

1.15

171.

241.

161.

1518

1.26

1.18

1.15

191.

231.

191.

1620

1.19

1.16

1.17

211.

141.

131.

1222

1.09

1.12

1.06

231.

081.

101.

0824

1.06

1.09

1.11

251.

061.

091.

1026

1.07

1.10

1.10

271.

131.

121.

1328

1.19

1.16

1.16

291.

191.

211.

1830

1.19

1.22

1.19

311.

241.

241.

2432

1.29

1.22

1.30

331.

241.

201.

2434

1.19

1.14

1.19

351.

121.

081.

1336

1.12

1.12

1.13

371.

121.

121.

1338

1.06

1.12

1.07

391.

121.

151.

1240

1.18

1.22

1.17

411.

231.

281.

1842

1.28

1.24

1.19

431.

221.

191.

1744

1.16

1.14

1.15

451.

081.

081.

1046

1.01

1.08

1.04

471.

041.

081.

0548

1.08

1.05

1.05

491.

121.

031.

1050

1.17

1.10

1.15

511.

191.

161.

1452

1.21

1.15

1.13

531.

201.

151.

1454

1.20

1.20

1.15

551.

191.

261.

1856

1.18

1.19

1.20

571.

161.

131.

1758

1.14

1.11

1.14

591.

071.

091.

1260

1.01

1.09

1.09

611.

051.

091.

1162

1.09

1.11

1.13

631.

101.

121.

1064

1.10

1.08

1.06

651.

021.

051.

0066

0.94

0.95

0.95

670.

830.

860.

8268

0.73

0.74

0.70

690.

630.

620.

6070

0.53

0.53

0.49

710.

43

�Usi

ng�2

N14

08�5

0mm

�Bel

l�Cup

�and

�"A

nnul

ar�S

lit"�

S.�A

.�Rin

g�w

/200

mm

�re

puls

ion�

ring.

��App

licat

or�in

dex�

525m

m,�t

ip�s

peed

�50m

m/s

ec,�3

�ve

rtic

al�s

trok

es.

02

46

810

1214

1618

2022

2426

2830

3234

3638

4042

4446

4850

5254

5658

6062

6466

6870

72

1

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

Film

Thk

.

Pan

el P

ositi

on

Row

#

577m

m In

dex

50m

m/s

ec. t

ip s

peed

1.50

-1.6

01.

40-1

.50

1.30

-1.4

01.

20-1

.30

1.10

-1.2

01.

00-1

.10

0.90

-1.0

00.

80-0

.90

0.70

-0.8

00.

60-0

.70

0.50

-0.6

00.

40-0

.50

0.30

-0.4

00.

20-0

.30

0.10

-0.2

00.

00-0

.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

01

23

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

2627

2829

3031

3233

3435

3637

3839

4041

4243

4445

4647

4849

5051

5253

5455

5657

5859

6061

6263

6465

6667

6869

7071

72

Pan

el P

ositi

on

F i l m B u i l d

1 2 3

For

d-C

MU

9-26

-01

Pan

el�#

926-

3&4

8/8/

2002

For

d�/�C

arne

gie�

Mel

lon�

Tes

tsO

vera

ll�P

anel

Cen

ter�

60"

Cen

ter�

48"

Cen

ter�

36"

For

dF

ord

For

dF

ord

Dat

e�9/

27/2

001

Film

�Avg

.S

td.�D

ev.

Eva

l.F

ilm�A

vg.

Std

.�Dev

.E

val.

Film

�Avg

.S

td.�D

ev.

Eva

l.F

ilm�A

vg.

Std

.�Dev

.E

val.

Row

�#1

0.99

820.

1303

13.0

5%1.

0337

0.08

288.

01%

1.03

080.

0913

8.85

%1.

0178

0.09

909.

73%

Tes

t�Num

ber�

#�9-

26-5

�&�6

Row

�#2

1.00

180.

1306

13.0

3%1.

0311

0.09

519.

22%

1.02

760.

1014

9.87

%1.

0047

0.10

049.

99%

Row

�#3

0.99

870.

1261

12.6

3%1.

0248

0.10

019.

77%

1.01

270.

1014

10.0

1%0.

9819

0.09

729.

90%

cc/m

in.�

150

Sha

ping

,�(N

l/min

)25

0R

PM

(x�1

000)

30B

ell�C

up,�m

m50

Tar

get�D

ist.,

�in.

10H

.V.,�

KV

90

Row

�#1

23

Pnl

�Pos

.F

ilm�T

hkF

ilm�T

hkF

ilm�T

hk0 1 2