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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
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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
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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
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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
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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.
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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
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position (mm)
Pai
nt T
hick
ness
( µ
m )
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σ =1σ =5σ =10
σ =20
σ =40
Gaussian Distribution
position (mm)
Pai
nt T
hick
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( µ
m )
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s=1s=5
s=10
s=20
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position (mm)
Pai
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( µ
m )
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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.
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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
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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.
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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.
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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
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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.
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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 � 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 � ��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
-
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32
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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 à2
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511.
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3552
1.44
1.35
1.35
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331.
3354
1.34
1.30
1.31
551.
281.
271.
2456
1.23
1.21
1.17
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141.
1758
1.09
1.12
1.16
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1660
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4446
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6062
6466
6870
72
1
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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
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525m
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1.00
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1.30
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1.50
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01
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
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3435
3637
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4243
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4647
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5859
6061
6263
6465
6667
6869
7071
72
Pan
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271.
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1.18
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46
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6062
6466
6870
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1
0.00
0.10
0.20
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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
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1.30
1.40
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01
23
45
67
89
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1213
1415
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2021
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4243
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4647
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5051
5253
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5859
6061
6263
6465
6667
6869
7071
72
Pan
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