Feedforward control of sheet bending based on force measurements

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ContentslistsavailableatScienceDirect

Journal

of

Manufacturing

Processes

jo u r n al h om ep age : w w w . e l s e v i e r . c o m / l o c a t e / m a n p r o

Technical

Paper

Feedforward

control

of

sheet

bending

based

on

force

measurements

Jos

Havinga

a,∗

_,

_Ton

_van

_den

_Boogaard

a

_,

_Franz

_Dallinger

b

_,

_Pavel

_Hora

b

a_Faculty_of_Engineering_Technology,_University_of_Twente,_{Drienerlolaan}_5,_Enschede,_The_Netherlands

b_Institute_of_Virtual_{Manufacturing,}_ETH_Zürich,_{Tannenstrasse}_3,_Zürich,_Switzerland

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received2December2016

Receivedinrevisedform4October2017

Accepted6October2017 Keywords: Metalforming Sheetbending Product-to-productvariation Feedforwardcontrol Forcemeasurement Processestimator LASSOregression

a

b

s

t

r

a

c

t

Theaccuracywhichcanbereachedwithfeedbackcontrolinindustrialmetalformingprocessesislimited byproduct-to-productvariationsintheprocess.Thesevariationsmaybecontrolledwithafeedforward controlsystem.Todoso,ameasurementsystemandaprocessestimatorareneededtomeasureand correctforthesevariations.Inthiswork,theuseofforcemeasurementsforfeedforwardcontrolofasteel sheetbendingprocessisinvestigated.Amulti-stagedemonstratorprocessisdevelopedwithcutting,deep drawingandbendingstagesandalargenumberofforcesensors.Severalrelationsbetweenprocessforces andtheproductgeometryarestudiedinanextensiveanalysisofmeasurementdatafromtheproduction process.ItisproposedtobuildamovingwindowprocessestimatorusingtheLASSOregressionmethod. Theparametersoftheregressionmodelareupdatedduringproductionbasedonhistoricaldataofthe productionline.Severalsimulationrunsareperformedtoestimatetheeffectoffeedforwardcontrolon theprocessaccuracy.Thesesimulationrunsarebasedonmeasurementsfromtherealproductionline andminorassumptions.Theproposedapproachleadstoanestimateddecreaseof24%intherootmean squareerroroftheﬁnalanglewithrespecttoasimulationrunwithfeedbackcontrolonly.

1. Introduction:controlofmetalformingprocesses

Metalformingprocessessufferfromvariationsofmaterial prop-ertiesandprocessconditions.Thesedisturbancesaffecttheﬁnal propertiesoftheproduct.For productswithnarrowtolerances, closed-loopcontrolmaybeusedtoensureproductquality.A driv-ingfactorforrecentdevelopmentsincontrolofformingprocesses istheuseofinformationtechnology,enablingforreal-time pro-cessingofincreasingly detailedsensor dataandprocessmodels

[1].

Anextensivereviewoncontrolofmetalformingprocessesis givenintheworkofPolyblanketal.[2].Theyobservedthatcontrol systemsinformingprocessesareusuallydesignedtocontrolthe movementofthetooling,butlacktheabilitytocontroltheﬁnal stateoftheproduct,suchasitsgeometryorstressdistribution. Ontheotherhand,theyforeseeanexpansionintheapplications ofclosed-loopcontrolinmetalforming,basedonthe accessibil-ityofthemaincomponentsneededforprocesscontrol:sensors, actuatorsandprocessmodels.Giventheongoingdevelopmentof computingpower,increasinglydetailedprocessmodelsand

con-∗ Correspondingauthor.

E-mailaddress:joshavinga@gmail.nl(J.Havinga).

trolsystemsmaybedevelopedinpursuitofahigheraccuracyof formingprocesses.

Animportantfactorforthedesignofmanufacturingcontrol sys-temsistherateofchangeoftheprocessdisturbances.Disturbances whichchangeslowlyovertimemaybecontrolledwithfeedback systems,adapting theprocess settings basedonmeasurements fromfinishedproducts.However,product-to-productvariations cannotbeeliminatedwithsuchsystemsandmustbecontrolled withfeedforwardcontrolbasedonmeasurementsfromthe semi-finishedproducts.Inthecurrentwork,theeligibilityoftheuseof forcemeasurementsinafeedforwardestimatorofasheet bend-ingprocessisinvestigated.Thequestioniswhetherthevariations intheprocessarereflectedbytheforcemeasurements.Several researchershaveinvestigatedtheuseofforcemeasurementsin controlofformingprocesses(Section2).Differentmethodshave beendevelopedtoadjustforchangesinproductionconditions(e.g. changesinnominalsheetthicknessorinusedmaterial).However, littleworkhasbeenpublishedoncontrolofproduct-to-product variationsinmetalformingprocesses.Thisisreflectedbythesmall numberofproductsusedfortrainingofcontrolmodelsandfor val-idationofthecontrolmethods:formoststudiesthedatasetsize doesnotexceed100products(Table1).

Tostudyproduct-to-productvariationsinmetalforming,large datasetswithmeasurementsfromeveryproductinthe produc-tionlineare needed.To thebest ofour knowledge,nostudies

https://doi.org/10.1016/j.jmapro.2017.10.011

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[9] data inbetween0.70and0.85mmand

ﬁvelubricationconditions

Helleretal.[10] Airbending Linearmodelwithsemi-analytical

simulationdata

Threematerials,threethicknesses

andfourtargetangles

n/a 72

Longoand

Maccarini[11]

Airbending Linearinterpolationwith

simulationdata

No n/a 5

Grocheetal.[12] Airbending Linearregression Threematerials,threethicknesses

andthreetargetangles

Unclear 30

Fig.1.Demonstratorproduct.

withanextensiveanalysisof a largenumber ofmeasurements havebeenpresentedintheliteratureonmetalformingprocesses. In theMEGaFiTproject[3],a metalforming demonstrator pro-cesshasbeendevelopedtoinvestigatethefeasibilityofcontrolof product-to-productvariationsinmetalforming.The demonstra-torprocessincludesdeepdrawing,forgingandbendingsteps.In thiswork,theuseofforcemeasurementsforcontrolofthe bend-ingstageisinvestigated.Section2isaliteraturestudyontheuse offorcemeasurementsforcontrolofmetalformingprocesses.In Section3thedemonstratorprocessispresented.InSection4 sev-eralcorrelationsbetweenmeasurementdataareshowntodeepen theinsightintothedemonstratorprocess.InSection5aproposal ismade fora model basedonexperimentaldatatobe usedin thecontrolofthebendingprocess.Aseriesofsimulationrunsare performedusingtheproposedmethodincombinationwith mea-surementdataobtainedfromthedemonstratorprocessinorderto assessthepotentialeffectofafeedforwardcontrolsystembasedon forcemeasurements.Finally,conclusionsanddiscussionarefound inSection6.

2. Literature:forcemeasurementsforfeedforwardcontrol

Thefollowingoverviewislimitedtostudieswith experimen-taldataontherelationsbetweenforcemeasurementsandproduct properties,whereaspurelynumericalstudiesareneglected.A sum-maryofthediscussedpapersisgiveninTable1.

Hardtetal.[4]implementedforceandcurvaturemeasurements inarollstraighteningprocess.Theyestimatedthecurvatureafter unloadingbasedona user-deﬁned valueofthe elasticbending stiffnessandforceandcurvaturemeasurements.Inasetofeight

experiments,theyachievedareductionof53–95%forthe maxi-mumlateraldeﬂectionoftheproduct.

Müller-Duysing[5]developedacontrolsystemforairbending basedonmeasurementoftheintermediateangleandtheforce dur-ingbending.Theintermediateangleatthreedifferentpositions ofthepunchandthebendingforceattwodifferentpositionsof thepunchwereusedasinputofalinearregressionmodel.After optimizing theregression coefﬁcients based onmeasured data, theRoot-Mean-SquareError(RMSE)ofthepredictionerrorofthe modelwas0.17◦overasetof44productswiththreedifferent mate-rials.TheRMSEofthepredictionerrorofthebestmodelwithout forcemeasurementwas0.24◦.

Yang et al. [6] created an experimental database of an air bendingprocess withforce andangle measurementsand other information suchas sheet thickness, ambienttemperature and materialname.Forthecontrolofanewworkpiece,datafromthe databasewasselectedbasedonafuzzylogicsystem.Usingthis selection,theycomparedtheexperimentalforcecurveswiththe measuredforcecurveanddeterminedthemaximumforceneeded toachievethedesiredangle.Anaccuracyof0.1◦wasreachedwhen thedatabase wasﬁlled withinformation similar tothat of the processedworkpiece.Anaccuracyof0.25◦ wasreachedwhena databasewithalargevarietyofexperimentaldatawasusedin combination withthemodiﬁed fuzzymodels.The uncontrolled accuracyisnotmentionedinthepaper.

Astudyonairbendingwiththreedifferentsheetthicknesses wasperformedbyForcelleseetal.[7].Theyfittedtheforcecurveto aregressionmodelwithfiveparametersandrelatedtheparameters tothefinalangleofthestripusingneuralnetworkmodels.They studiedtheeffectof thetraining setsizeof theneuralnetwork modelsonthebendingaccuracyandfoundstandarddeviationsin therangeof0.11–0.23◦fortheanglewiththedifferenttrainingset sizesandthicknesses.Theuncontrolledaccuracyisnotreported.

Nastran and Kuzman [8] examined a wire bending process which is precededby a straightening process. Theyobserved a strongcorrelationbetweentheforceontherollersduring straight-eningandtheﬁnalgeometryafterbending.Theydevelopedamodel torelatethevariationoftheforcetovariationoftheyieldstressand proposedtocontrolthepositionoftherollerstoaffecttheamount ofplasticworkduringstraighteningandkeeptheyieldstressofthe incomingwireforthebendingprocessataconstantlevel.

Controlof a steelchannel forming process by adjusting the binderforcetrajectorywasinvestigatedbyViswanathanetal.[9]. Thecoefﬁcientsofathirdorderpolynomialﬁtoftheforcecurve

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Fig.2.Technicaldrawingofdemonstratorproduct,includingnumberingofthe

ﬂaps.Theforgedproﬁleinthecenterofthecupisnotdrawn.

Fig.3.Tooling.

wereused asinput ofa two-layer neuralnetwork. After train-ingthenetworkwithfivedifferentsteelsheetsandfourdifferent lubricationconditions,itwasusedtocontrolthespringbackofthe product.Newtestswereperformedwiththesamesteelgrades asusedfortrainingtheneuralnetwork.Springbackvalueswithin specifications(10◦–12◦)werefound.However,twooutofthefour experimentswithothersteelgradesresultedinproductsoutof specification.

Helleretal.[10]usedforceandanglemeasurementsinanair bendingprocesswithpartialunloading.Usingsimulationdataand alinearrelationbetweenthemeasuredforceatthefullyloadedand thepartiallyunloadedstate,theydeterminedthemaximumforce neededtoachievethedesiredangle.Induplicateexperimentswith threedifferentmaterials,threedifferentsheetthicknessesandfour differenttargetangles,theyfoundanaverageerrorof0.4◦.

LongoandMaccarini[11]usedanﬁniteelementmodelto esti-matematerialpropertiesbasedontheforcecurveinairbending. Theestimatewasusedtopredicttheangleafterspringback.They

foundgoodagreementintheaveragepredictedspringbackangle overasetofﬁveexperiments.

Grocheetal.[12]usedthemaximumblankingforcefroma pre-viousprocessstageandthesheetthicknesstopredicttheangleafter springbackinairbending,basedonalinearregressionmodelbuilt withhistoricaldatafromtheprocess.Thefeedforwardcontrolloop wascombinedwithafeedbackcontrollerandtheapproachshowed improvedperformancewithrespecttotheuseoffeedbackcontrol only.

Thesigniﬁcance of force measurementsfor control ofmetal forming processes is evidently shown throughout the above-mentioned studies. However, force measurements have been mainlyusedtoadaptthemanufacturingprocesstochangesin pro-cessconditions.Inthiswork,theuseofforcemeasurementsfor predictingproduct-to-productvariationsisstudied.

3. Demonstratorprocess

Ademonstratorprocesshasbeendesigned toinvestigatethe feasibility of product-to-product control in metal forming. The demonstratorproductisshowninFig.1.Threetypesofforming processeshavebeenusedintheprocess:deepdrawing,forgingand bending.Theproductisproducedinanindustrialenvironmentata productionrateofoneproductpersecond.Itisformedfroma280 or300␮mthickand38mmwidestainlesssteelstripanditremains attachedtothestripduringallprocesssteps.Themaindimensions oftheproductareshowninFig.2.

Thetooling consistsof fourmodules:cutting,deepdrawing, forgingandbending(Fig.3),mountedonanindustrialstamping press.Everymodulehasseveralpositionsinwhich theforming stepsareperformed.Theorderandnumberingoftheprocesssteps isshowninFig.4.Duringeachstrokeofthepressallproductsmove oneposition.Therefore,thetotalproductiontimeofoneproductis aroundtwentyseconds.Thestripremainsalignedinthetoolingby theuseofpilotholesandpins.

Thefirstmodulehasthreecuttingstages.Thesecondmodule hasasingledeepdrawingstage,withfourblankholdersections whicharecontrolledindependentlywithahydraulicsystem.This enablescontroloftheroundnessoftheflange,whichisinfluenced bytheanisotropyofthematerial.Thehydraulicsystemis auto-maticallyre-pressurizedtothemaximumpressurepmaxwhenits

pressuredropsbelowtheminimumpressurepmin.Thethird

mod-ulehastwocuttingstagesandoneforgingstagetoformsix200␮m wideparallelribsinthebottomofthecup.Theheightoftheribs dependsontheappliedforgingforce[13].

Inthelastmodule,threeflapsarecutoutinthebottomofthe cup.Thereafter,theflapsarebentto50◦ inacloseddie(Fig.5a), andfinallytheyarebentbacktoadesiredangleinanopenbending stage(Fig.5b).Duringthefirstbendingstage,thebendingforces ofallflapsaremeasured.Theforce–displacementcurvefromthis bendingstageisstrongly nonlineardue tothegeometryofthe tooling.Thebendingforcemeasurementsandtheinterpretations thereofarediscussedinSection4.3.Inthesecondbendingstage,the amountofbackbendingcanbeadaptedfromproducttoproduct. AsshowninFig.5b,ahorizontaltranslationwhichisimposedbya servomotorconvertsintoverticalmovementofthebackbending punchthroughaslantedgroovewitharatioof1to10.Theactuator doesnotmoveduringcontactbetweenthepunchandtheproduct. Inthelaststageofthebendingmodule,apictureofthesideofflap 2istakentodeterminetheangle(Fig.6).Thestandarddeviationof theanglemeasurementerrorisdeterminedtobelessthan0.01◦.

Alargeamountofdataisstoredduringmanufacturing.Process forces,blankholderpressure,ﬂapangleandincomingsheet thick-nessaremeasuredandstoredduringproduction.Anoverviewof thesensorsisgiveninTable2.Relevantinformationisobtained

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Fig.4.Productionstepsofdemonstratorproduct.

Table2

Overviewofsensorsinthedemonstratorprocess.

Processstage(nr) Description Sensor Range Samplingrate Amount

Thickness Vollmer 0–1mm 1

Cutting(1) Pilotholecuttingforce KistlerSlimLine9132B29 0–7kN 5kHz 1

Deepdrawing(7) Punchforce KistlerSlimLine9133B29 0–14kN 5kHz 1

Deepdrawing(7) Blankholderpressure 5kHz 4

Forging(12) Punchforce KistlerSlimline9134B29 0–26kN 5kHz 1

Bending(16) Punchforce KistlerSlimLine9130B39 0–3kN 5kHz 3

Bending(19) Flapangle AVTStingrayCCD 20–80◦ ₁

Fig.5. Thetwoformingstepsfromthebendingmodule.(a)Overbendingstage (stage16).(b)Backbendingstage(stage18).

fromthesensorsduringthefractionofthestrokewherethereis contactbetweenthetoolingandtheproduct,whereasnosignal ismeasuredduringtherestofthestroke.Thenumberofrelevant samplingpointsforeachprocessstagedependsonthesensor sam-plingrate,theexactgeometryofthetooling,thestampingrate(1 stokepersecond)andthetotalstrokelengthofthepress(38mm). Thenumberofsamplingpointsduringcontactbetweentoolingand productisroughly100forthecuttingstage,700forthe deepdraw-ingstage,350fortheforgingstageand450forthebendingstage. Ineachtest,allrawdataarestoredandpost-processedwithofﬂine proceduressuchasﬁlteringandaligningofforcemeasurements. Moredetailsonthepost-processingcanbefoundinHavinga[14]. Anoverviewofpost-processedsensorsignalscanbefoundinFig.7. Fourtestswithdifferentsettingswereperformed.Anoverview ofthetestsisgiveninTable3.Atleast3000productshavebeen producedineverytest.Twodifferentnominalsheetthicknesses havebeenused.Thevaluespminandpmaxdenotethelowerand

upperboundforthepressureofthehydraulicsystemforthedeep drawingblankholders.Foreverysheetthickness,onetestwithand onetestwithoutbackbendingwasperformed.Somechangeshave beenmadetothetoolingfortestnumber4.Thepurposeofthese changeswastominimizethedisturbancestotheprocesscausedby thetooling.Thestabilityofthehydraulicsystemofthedeep draw-ingstageispoor,aswillbediscussedinSection4.4.Thestability wasimprovedbyreducingthebandwidthofthepressureofthe hydraulicsystem.Furthermore,theforgingpunchwasremovedto decreasetheloadsonthetooling.

Fig.6. Pictureofﬂapfrominlinecamera(step19).

4. Testresults

Severallongtestrunshavebeenperformedwiththe demon-stratorprocess.Allforcemeasurementsduringdeformationofthe productarestoredforallproducts.Hence,thousandsofdatapoints arestoredperproduct.

Theobjectiveof thisworkistodecreaseproduct-to-product variationinthedemonstratorprocessusingfeedforwardcontrol basedonforcemeasurements.Todoso,therelationsbetweenforce measurementsandthefinalangleoftheflaphavetobeidentified.In Section5,thisisdonewithregressionanalysis.However,itishard togaininsightaboutthemeasurementsbasedontheseregression modelsonly.Themeasurementsandtheirrelationswiththefinal angleoftheflaparethekeyfactorforfeedforwardcontrolofthe demonstratorprocess.Therefore, severalobservationsaboutthe measurementsarediscussedinthissection.

Thissectionstartswiththedeﬁnitionofa scaledcorrelation measureinSection4.1.Thismeasureisusedforanalysisofthe data.Thereafter,thefollowingaspectsofthedataarediscussed:the amountofvariationoftheangle(Section4.2),thebendingforces (Section4.3), variation inthedeep drawingstage (Section4.4), theeffectofthicknessvariation(Section4.5)andthecorrelation betweentheforcesandtheangle(Section4.6).

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Table3

Overviewofdatasets.

Datasetnumber Nominalthickness Backbending Blankholderpmin/pmax Datasetsize

1 280␮m Yes 100/130bar 3507products

2 280␮m No 100/130bar 3268products

3 300␮m Yes 100/130bar 3142products

4 300␮m No 100/105bar 3530products

Fig.7.Typicalsensorsignalsforallprocessstagesforasingleproduct.Thedeepestpointofthepressat500msisindicatedwithadashedline.Theplottedtimerangeofeach curveindicatestherangeofdatawhichhasbeenusedintheprocessestimatorfromSection5.(a)Cutting.(b)Deepdrawing.(c)Blankholderpressure(fourblankholders).

(d)Forging.(e)Bending(threepunches).

4.1. Scaledcorrelation

Oneofthetoolsforidentifyingcorrelationsbetweentwosignals isthecross-correlationplot.Inthecaseofcontrolofmetalforming processes,thelong-termcorrelationisofnointerest,because long-termvariationscanbecontrolledwithfeedbackcontrol.Therefore, ascaledcorrelationmeasureisusedtodeterminetheshort-term correlation[15].Thescaledcorrelationcoefﬁcientisbasedonthe Pearsoncorrelationcoefﬁcient.ThePearsoncorrelationbetween twosignalsXandYis:

(X,Y )=cov(X,Y ) XY

(1) ThecovariancebetweenXandYiscov(X,Y),andthestandard deviationsofXandYareXandYrespectively.Forthe

demonstra-torprocess,XandYmaybemeasurementssuchasthethicknessor themaximumbendingforceforeachproduct.

Thescaledcorrelationfactoris determinedasfollows: abin widthnb ischosen, and thecomplete datasetisdivided intoK

non-overlappingbinswitha widthof nb products.ThePearson

correlationcoefficientisdeterminedineverybin,andthescaled correlationcoefficientsisdefinedastheaverageofthesevalues:

s= 1 K K

i=1 i (2)

whereKistheintegerpartofthenumberofproductsinthedataset dividedbythebin widthnb.Inthis work,a binwidthnbof 10

productsisused.

Thescaledcorrelationfactorsareusedincross-correlationplots. Cross-correlationisdeﬁnedas:

s()=s(X(t),Y (t+)) (3)

Thefactorisashiftinthenumberingoftheproducts.If–as anexample–thedatasetsizeis1003products,Xisthethickness andYisthemaximumbendingforce,s(3)isthescaledcorrelation

betweenthethicknessofproducts1to1000andthemaximum bendingforceofproducts4to1003.Inthecaseofauto-correlation, XandYareequal.

4.2. Anglevariation

ThemeasuredanglesfromthedifferenttestsareshowninFig.8. Ineachplot,thescalingofthehorizontalaxisinbetweenproducts number1000and1050ischanged,togiveanimpressionofthe amountofproduct-to-productvariation.Product-to-product vari-ationrepresentsasigniﬁcantpartofthetotalvariationinalltests. Thetestswithbackbending(Fig.8aandc)showmorevariation thanthetestswithoutbackbending(Fig.8bandd).Suddenjumps intheangleareobservedinthetestswithbackbending.However, thisdoesnotimplythatthejumpsarecausedbydisturbancesfrom thebackbendingstage.InSection4.6itwillbeshownthatsomeof thesejumpscanbepredictedwiththeforcesfromtheoverbending stage.

4.3. Bendingforce

InFig.9,thebendingforcesofdataset3areshownforallthree flaps.Thedifferencesbetweentheaverageforcecurvesperflapare relatedtotheexacttoolinggeometryandalignmentateach ben-der.Theshapeofthebendingforcecurvehasbeenstudiedwitha finiteelementmodel.Duringthefirstpartofthestroke,free bend-ingoccurs.Ataround485ms,theslopeoftheforcecurveincreases becausethecontactareabetweenthepunchandtheflapstarts moving.Asecondincreaseoftheslopeisobservedafter490ms, whenthetipoftheflaptouchesthedie.Theforcereachesits

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max-Fig.8.Anglesofﬂap2.(a–c)PreviouslypublishedinVanDenBoogaardetal.[16].(a)Constantbackbending.280␮msheetthickness.Standarddeviation0.326◦_._(b)_No_back

bending.280␮msheetthickness.Standarddeviation0.176◦.(c)Constantbackbending.300␮msheetthickness.Standarddeviation0.238◦.(d)Nobackbending.300␮m

sheetthickness.Standarddeviation0.135◦.

Fig.9.Bendingforceofsubsetfromdataset3(300␮mwithbackbending),for

bendersone( ),two( )andthree( ).

imumvaluejustbeforethedeepestpointofthepressduetostrain ratesensitivity of thematerial. Theexact length of thebender stronglyinﬂuencesthemaximumbendingforce.Theincreaseof theforceofbender3around450msiscausedbycontactbetween thebenderandthedieduetomisalignmentofthetooling.Fig.9

alsogivesanimpressionoftheamountofvariationofthe bend-ingforcesduringthetest.InSection5.5itwillbeinvestigatedhow thesevariationsrelatetothevariationoftheﬁnalangleoftheﬂap. 4.4. Deepdrawingvariation

Thehydraulicsystemfortheblankholdersofthedeep draw-ingstagehasastrongeffectonthestabilityofthewholeprocess. Thestabilityofthehydraulicsystemispoorduetoseveralissues, suchas thepositioningofthevalvesand sensors,thelengthof thetubesandthelimitedsizeoftheoilreservoir.Therefore,the completesystemisre-pressurizedeveryfewseconds,causing a periodicvariationintheblankholderpressure.Thesystemis re-pressurizedtopmaxwhenthepressuredropsbelowpmin.Mosttest

runswereperformedwithapressurerangeof102–130bar,causing thesystemtore-pressurizeaftereveryeightproducts.Thepressure

Fig.10.Pressurefromblankholder2for100subsequentproductswithdifferent

settingsforthehydraulicsystem.(a)Dataset2,pmin/pmax=102/130bar.(b)Dataset

4,pmin/pmax=100/105bar.

Fig.11.Autocorrelationofmaximumpressureofblankholder2( ),maximum

forgingforce( )andmaximumforceofbender2( )fordataset2(280␮m

withoutbackbending).

ofoneblankholderofdataset2isshowninFig.10a.Less varia-tionwasobservedintestnumber4duetothenarrowerrangeof 100–105bar,whichcausedthesystemtore-pressurizesafterevery twoproducts(Fig.10b).

InFig.11,theautocorrelationoftheblankholderpressureof dataset2isshown.Theperiodof8productsforpressurizingthe systemcanbeclearlyobserved.Inthesameﬁgure,itcanbeseen thatotherprocessstepsarealsoaffectedbythevariationofthe

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Fig.12.Scaledcross-correlationbetweenthicknessandforcesforallfourdatasets.

(a)Cross-correlationbetweenthicknessandmaximumcuttingforce.(b)

Cross-correlationbetweenthicknessandmaximumforceofbender2.

hydraulic system:the autocorrelation of the maximum forging forceandofthemaximumbendingforcehavethesame periodic-ityof8products.Onecausefortheinteractionbetweentheprocess stepsisthecomplianceofthestampingpress.Variationoftheforce atonepositioninthepressaffectsthedeformationofthetooling andconsequentlyaffectstheotherstagesofproduction.Thiscan beveriﬁedwiththecross-correlationplotsbetweenthemaximum deepdrawing,forgingandbendingforces.

InFig.11,theautocorrelationoftheblankholderpressureof dataset2isshown.Theperiodofeightproductsforpressurizing thesystemcanbeclearlyobserved.Inthesameﬁgure,itcanbe seenthatotherprocessstepsarealsoaffectedbythevariationof thehydraulicsystem:theautocorrelationofthemaximumforging forceandofthemaximumbendingforcehavethesameperiodicity ofeightproducts.Acausefortheinteractionbetweentheprocess stepsisthecomplianceofthestampingpress.Variationoftheforce atonepositioninthepressaffectsthedeformationofthetooling andconsequentlyaffectstheotherstagesofproduction.Thiscan beveriﬁedwiththecross-correlationplotsbetweenthemaximum deepdrawing,forgingandbendingforces[17].

4.5. Thicknessvariation

In the demonstratorprocess, the thickness of the incoming sheetismeasuredfor everyproduct.Hence,theeffect ofsheet thicknessontheprocessforcescanbeinvestigated.First,the cross-correlationbetweenthethicknessandthemaximumcuttingforce isshowninFig.12aforalldatasets.Asexpected,thecorrelation betweenthicknessandmaximumcuttingforceispositive:athicker sheetrequiresahighercuttingforce.Thecross-correlationbetween bendingforceandthicknessgivesasimilarresult(Fig.12b).Hence, partofthevariationoftheprocessforcescanbeattributedto vari-ationofthethicknessofthesheet.However,thecorrelationvalue oflessthan0.4showsthattheeffectofthethicknessonthe cut-tingandbendingislimitedwithrespecttothetotalvariationinthe process.

4.6. Correlationbetweenforcesandangle

Theobjectiveofthisworkistoinvestigatetheuseofforce mea-surementsforcontrolofthebendingstage.Todetecttherelations betweensmallvariationsintheprocessforces andtheangleof theflap,thecorrelationsshouldbeidentifiedusinglargedatasets. Twoexamplesofweakbutclearcorrelationsbetweenforcesand theanglecanbefoundinFig.13.Thesecorrelationshaveavalue of−0.32and0.18.Thesizeofthedatasetsenablesidentification ofthesecorrelationswithstatisticalsignificance:thep-valuesare 2_·10−85and6_·10−27respectively.

5. Processestimator

Severalrelationsinthemeasurementdatahavebeendiscussed inSection4.Althoughthecorrelationsbetweenmeasurementdata

Fig.13.Angleversusforcedatafordataset4.(a)Angleversusdeepdrawingforce

at445ms,withcorrelationvalue−0.32.(b)Angleversusbender2forceat505ms

withcorrelationvalue0.18.

andthefinalangleareweak,theycanbeusedinafeedforward esti-mator.Inthissection,theeffectoffeedforwardcontrolwithforce measurementsisestimatedbasedonprocesscontrolsimulations whichhavebeenperformedusingexperimentaldataobtainedfrom thedemonstratorprocess.Agenerallinearizedmodelofametal formingprocessisgiveninSection5.1.Followingthismodel,the equationsforfittingaregressionmodelbasedonhistoricaldataare giveninSection5.2,andtheregressionmethodusedforfittingis discussedinSection5.3.Theassumptionsandprocedurefor sim-ulatingprocesscontrolbasedonexperimentallyobtainedprocess dataaregiveninSection5.4andtheresultsarepresentedinSection

5.5.

5.1. Processcontrol

Asystemdiagramforcontrolofmetalformingprocessesisgiven inFig.14.Theprocessmayberegardedasadiscretesystem,where everyproductisonesample.Theprocessisdividedintotwoparts, whereP(1)_is_the_part_of_the_process_which_provides_measurements

forfeedforwardcontrolandP(2)_is_the_part_of_the_process_which_is

affectedbythecontrolsystem.Bothpartsconsistofoneormore processstages.Thepropertiesoftheproductaftertheﬁrstpart oftheprocessaredenotedbyy1 ∈Ry1.Thecontrolparameters u∈Ru_act_on_the_second_part_of_the_process._The_ﬁnal_properties

oftheproductaredenotedwithy2 ∈Ry2andtheerrorinproduct

propertieswithe ∈Ry2_._The_ﬁrst_part_of_the_process_delivers_a_set ofmeasurementsm ∈Rm_to_be_used_in_a_feedforward_loop._The

disturbancesd1 ∈Rd1andd2 ∈Rd2actonbothpartsoftheprocess

P(1)_and_P(2)_{respectively.}

The sourcesof variation canbe seen asdisturbances tothe system.Some important sources of variation in metal forming arevariation ofmaterialproperties, sheetthickness,lubrication propertiesandtoolwear.Thesesourcesarerecognizedbymany researchersintheﬁeldofrobustoptimizationandcontrolof form-ingprocesses.Anoverviewofdifferentstudiesonthesesources ofvariation is given intheworkof Hazra etal. [18].However, therearemanyothersourcesofvariationwhichreceiveless atten-tionin studiesontheaccuracyofmetalforming processes.Col

[19]gives anoverviewoffactorsaffectingformingprocessesin anindustrialenvironmentbasedonhisprofessionalexperience. Henamesalargenumberofinﬂuencingfactorssuchasvariation ofblankholderforces,guidanceoftherams,theeffectof tempera-tureonlubrication,localizationoflubricant,toolingpositioningand pressstiffness.Inmanycases,itisunfeasibletoperformthe exten-sivemeasurementsneededtoquantifythesesourcesofvariation andtheireffectontheformingprocess.

Variationsintheformingprocessevolve overtime,although differentsourcesofvariation mayhavedifferentdynamics:tool wearincreasesslowlyduringlifetimeofthetooling,whereasthe temperatureofthetoolingincreasesrapidlyduringrunning-in.For controlofformingprocesses,itisessentialtohavedirectorindirect measurementsofthedisturbancestothesystem.Inotherwords, importantsourcesofvariationshouldbeobservable.

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Fig.14.Systemdiagramforcontrolofametalformingprocess.Thecomponentsofthefeedforwardsystemareshowningray.The ˆhat representsestimatedvalues.

Duetoelasticspringback,thefinalstateoftheproductisonly reachedafterreleaseoftheproduct.Consequently,thefinalstate canonlybedirectlymeasuredwhentheproductisfinishedandno controlactionscanbeappliedanymore.Inthecontroldiagramthis isindicatedwiththedelayblockz−N,whereNisthemeasurement delayinnumberofproducts.Themeasurementdelayisatleastone product,butitmaybeseveralproducts,dependingonthespeedand positionofthemeasurementsystem.

Whencompensatingforlong-termvariationsinamass produc-tionprocess,it maybesufficienttomeasurethefinalstateofa productandcompensatetheerrorfortheupcomingproductswith afeedbackloop.Thefeedbackloopisshowninblackinthecontrol diagram.However,product-to-productvariationscannotbe elim-inatedinsuchanapproach.Othermeasurementsshouldbeused toestimatetheeffectofproduct-to-productvariationsonthefinal stateoftheproduct.Thesemaybeeitherdirectmeasurementson theintermediatestateoftheproduct,orindirectmeasurements whichcarryinformationabouttheprocessandtheproduct,such asforcemeasurements.Thesemeasurementscanbeusedina feed-forwardlooptocontroltheprocess.

Todeterminetherelationbetweenthemeasurementsmand theﬁnalpropertiesoftheproducty2,themathematicalequations

oftheprocessaredeterminedandlinearized.Firstly,theprocess isdescribed withtheunknownfunctionsf,g,and h,which are functionsofthedisturbancesd1andd2andthecontrolparameters u:

m=f(d1) (4)

y1=g(d1) (5)

y2=h(y1,d2,u) (6)

Theseequationsarelinearizedaroundtheirnominalvaluesd1, d2anduusingTaylorexpansion:

m_≈f(d₁)₊(d1−d1)·

∇

f(d1) (7) y₁≈g(d₁)+(d1−d1)·

∇

g(d1) (8) y₂≈ h(g(d 1),d2,u) +((d1−d1)·

∇

g(d 1))·

∇

y1h(g(d 1)) +(d2−d2)·

∇

d2h(d 2) +(u−u₎_·

_∇

uh(u) (9)

Thederivativesintheequationsabovearerenamedto:

P(1)m =

∇

f(d1) (10) P(1)y1 =

∇

g(d 1) (11) P(2)_y₁ =

∇

y1h(g(d 1)) (12) P(2)_d 2 =

∇

d2h(d 2) (13) P(2)u =

∇

uh(u) (14)

Thechangesofthemeasurementsmandtheﬁnalproduct prop-ertiesy2withrespecttotheirnominalvaluescannowbewritten

as:

m=P(1)_md1 (15)

y₂=P(2)_y₁P(1)_y₁d1+P(2)_d₂d2+P(2)u u (16)

Thetargetpropertiesy2areinﬂuencedbythreecomponents:

twoarerelatedtothedisturbancesd1andd2,andoneisrelated

tothecontrolparametersu.ThecomponentP(2)_d

2d2 cannotbe estimatedinthefeedforwardloopbecausethereareno measure-mentsavailablefromP(2)_._Hence,_the_goal_of_the_feedforward_loop_is

toestimatethecomponentP(2)y1P

(1)

y1d1fromEq.(16)basedonthe measurementsm.UsingEq.(15)andassumingthatP(1)m

−1 exists,it follows: P(2)y1P (1) y1d1=P (2) y1P (1) y1P (1) m −1 m (17)

Hence,therelationbetweenthevariationinthemeasurements mandthevariationoftheﬁnalpropertiesoftheproducty2

duetothedisturbanced1isgivenbythematrixP(2)y1P

(1) y1P (1) m −1 .This matrixhastobeestimatedtobeabletousefeedforwardcontrol basedonthemeasurementsm.

Different approaches may be used to determinethe matrix

P(2)y1P (1) y1P (1) m −1

.Thismaybedoneeitherofﬂine(e.g.usinga numer-ical or analytical model) or online based on historical data of theprocess.Thelatteroptionisindicatedinthecontroldiagram (Fig.14)by thedashedlines,indicating which data isrequired toestimatetherelationbetweenthemeasurementsandtheﬁnal propertiesoftheproduct.

5.2. Modelﬁtting

In this work, it is proposed to usehistorical data from the manufacturingprocesstodeterminetherelationsbetween pro-cessmeasurementsandﬁnalpropertiesoftheproducts.Atevery momentduringproduction,itisexpectedthatthemostrecent pro-ducedproductshavethehighestresemblancewiththecurrently producedproducts.Hence,itispreferabletousethemostrecent datafortheprocessestimator.Todoso,amovingwindowprocess estimatorisproposed,whichiscontinuouslyupdatedbasedonthe npmostrecentlyproducedproducts.Twodifferentcasesare

con-sidered:thecasethatthematrixP(2)u fortheeffectofthecontrol

parametersontheﬁnalproductpropertiesisknown,andthecase thatitisunknown.Inthelattercase,estimationofP(2)u isincluded

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intheprocedure.Fortheformercase,startwithcombiningEqs. (16)and(15)to: y2−P(2)u u=P(2)y1P (1) y1P (1) m −1 m+P(2)_d 2d2 (18)

Inthisequation,y2 aretheﬁnalpropertiesoftheproduct,u

thecontrol parameters, m the forcemeasurements and d2 the

unobservabledisturbances.Themeanvaluesinthedatasetwith npproductsaresubtractedfromthedatatodeterminethe

varia-tions.Theaboveequationholdsforasingleproduct.Expandingthe equationtoallnpproductsgives:

Y2−P(2)u U=P(2)y1P (1) y1P (1) m −1 M+P(2)_d 2D2 (19)

ThedatasetofhistoricaldatacontainsY2,UandM.Itis

assumedthatP(2)_d

2D2actsasGaussianwhitenoiseonthesystem. Thesystemofequationscanberewrittentoaregressionproblem intheformZ=ˇ1M+

1,with:

Z=Y2−P(2)u U (20) ˇ1=P(2)y1P (1) y1P (1) m −1 (21)

1=P(2)_d₂D2 (22)

Regressionmethodscanbeusedtodeterminethecoefﬁcient matrixˇ1,establishingtheprocessestimator.Thereafter,the

pro-cessestimatorcanbeusedinthecontrolsystemtointerpretnew measurementsm.Notethatitisnotneededtoknowthesizeofthe disturbancevectord1toestablishtherelationbetweenZandM.

Inthecasethatthetransfermatrixfromcontrolparameterto targetpropertiesP(2)u isunknown,Eq.(19)iswrittenas:

Y2=

P(2)y1P (1) y1P (1) m −1 P(2)u

_M U

+P(2)_d 2D2 (23)

ThisequationcanberewrittentotheformY2=ˇ2X+

2and ˇ2canbedetermined,with:

ˇ₂=

P(2)y1P (1) y1P (1) m −1 P(2)u

(24) X=

M U

(25)

2=P(2)_d₂D2 (26)

Theseequationsarebasedonthefollowingassumptions. Vio-lationsoftheseassumptionsmayaffectthequalityoftheprocess estimator.

1.Thesystemislinearintherangeofvariationofd1andu.

2.Theinverse mapping P(1)m −1

exists. Different disturbances d1

resultindifferentmeasurementsm.

3.TheknowntransfermatrixP(2)_u valueiscorrect.

4.Thereisenoughdataavailableforestimationofthecoefﬁcient matrixˇ1orˇ2.Therequiredamountofdataalsodependson

themeasurementnoiseandthemagnitudeoftheunobservable disturbancesd2withrespecttotheobservabledisturbancesd1.

5.Disturbances d1 and d2 are uncorrelated. Violation of this

assumption implies that the variations of d2 can be partly

observedinthemeasurementsm,leadingtoanimprovement ofthefeedforwardcontrolsystem.

6.Disturbanced2actsasGaussianwhitenoiseonthesystem.This

maybeincorrectbecausethevariationofdisturbanced2may

becorrelatedintimeordeterminedbyadifferentprobability distribution.

Theseassumptionsdonotrestricttheﬁttingprocedureofthe linearregressionmodelwiththedatasetofY2,UandM.Itis

pro-posedtousetheLASSOregressionmethodforﬁttingthelinear model.TheLASSOmethodisdiscussedinthefollowingsection. 5.3. LASSOregression

The derivation in this section follows the case that P(2)_u is known(Eqs.(19)–(22)),whereasthecasethatP(2)_u isunknown(Eqs.

(23)–(26))canbederivedequivalently.Inthedatasetwhichisused toﬁtthemodel,npisthenumberofproducts,nmthenumberof

measurementsperproduct,ny2 thenumberofoutputproperties perproductandnuthenumberofcontrolparameters.Theny2rows oftheoutputmatrixZfromEq.(20)canbeﬁttedseparately.Hence, thesystemtobesolvedhastheformz=ˇ1M,wherezhassize

(1×np),ˇ1hassize(1×nm)andMhassize(nm×np).

EveryrowoftheinputmatrixMstandsforoneofthe mea-surementsminusthemeanvalueofthatmeasurementoverthe ﬁttingdataset.Forthedemonstratorprocess,theseare measure-mentsfromoneforcesensoratdifferentsamplingpointsofthe forcecurves.Thisisdifferentthanwhathasbeendoneinthestudies discussedinSection1.Theusualapproachistoextractoneorfew characteristicsfromtheforcecurves.However,weproposetouse thedatafromthefullforcecurvetomaximizetheinformation gath-eredduringproduction.MatrixMisexpectedtobemulticollinear becausedifferentmeasurementsareexpectedtocorrelate.

Whenthenumberofmeasurementsperproductismuchsmaller thanthenumberofproductsin thedataset(nmnp), the

coef-ﬁcientsˇ1 maybesolvedwiththeOrdinaryLeastSquares(OLS)

method.However,whenusingdatafromforcemeasurementsina processestimator,nmmayeasilyexceednp.Inthatcase,nounique

solutioncanbefoundwithOLS.Hence,otherregression meth-odsshouldbeusedtofindasolutionwithoutoverfittingthedata. Severalmethodshavebeendevelopedforfittingdatawithmore independentvariablesthandatapoints,suchasstepwise regres-sionmethods,whereasubsetofthedataisselectedtofitthemodel. Othermethods,suchasridgeregression,LASSOregression[20]and leastangleregression[21],restrictthemagnitudeofthecoefficient matrixˇ1topreventoverfitting.

Inthiswork,theLASSOregressionmethodisusedforfitting theprocessestimator.Thefirststepofthefittingprocedureis nor-malizationofthedata.Theoutputvectorzandeverycolumnof theinputmatrixMisnormalizedinsuchawaythatitsmean becomeszeroanditsstandarddeviationbecomesone.Thereafter, thecoefficientvectorˇ1isfoundbasedonthefollowing

minimiza-tion: min ˇ1∈Rm

⎛

⎝

1 2np np

i=1

zi−ˇ1mi

2 + nm

j=1

ˇ1j

⎞

⎠

(27)

Theparameterrestrictsthemagnitudeofthecoefﬁcientsˇ1.

Ifiszero,Eq.(27)reducestoOLS.Eq.(27)isaconvexproblemand canbesolvedwithstandardoptimizationtechniques.Anefﬁcient solverfortheLASSOmethodhasbeendevelopedbyEfronetal.

[21].Afterﬁttingtheregressionmodel,itcanbeusedtopredict thevariationintheoutputpropertiesy2ofanewproductgiven

itsmeasurementm.Forthedemonstratorprocess,theforce mea-surementisusedtopredictthevariationintheﬁnalangleofthe ﬂap.

5.4. Simulationofcontrolwithforcemeasurements

Inordertoestimatetheeffectivenessofacontrolsystemfor thebendingstageofthedemonstratorprocess,itisessentialto useexperimentaldatawhichisobtainedfromthedemonstrator

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Fig.15.Simulationﬂowchart.

processitself.Therefore,asimulationisperformedwhichusesall relevantexperimentaldatafromatest(i.e.variationsinprocess forces as wellas anglemeasurements)to estimatehow differ-entcontrollerswould haveperformediftheywould havebeen usedforthatspeciﬁctest.Theonlyassumptionswhichhavebeen madeinthesimulationsarerelatedtotheactuationsystem.Itis assumedthattheactuationsystemisperfectlylinearandperfectly known.Obviously,theexactperformanceoftherealactuation sys-tem(adjustmentofthebackbendingpunchpositioninstage18,see

Fig.5b)shouldbeinvestigatedinordertodeﬁnetheexactcontroller performance.Anoverviewofthesimulationprocedureisgivenin

Fig.15.ThesimulationshavebeenperformedusingMATLAB. Itmustbenotedthatthetraditionalapproachforvalidation, wherethedataissplitinaﬁttingandavalidationfraction, can-notbeappliedwiththeproposedapproach.Themovingwindow processestimatorimpliesthattherelationbetweenmeasurements andﬁnalpropertiesoftheproductisestimatedbasedonthedata fromthenpmostrecentlyproducedproducts.Hence,thechange

inﬁnalproductpropertiesofasingleproductispredictedwitha modelwhichneverincludesdatafromthatproductitself. There-fore,theessentialconditionthatdatashouldnotbeusedforﬁtting andpredictionsimultaneouslyisnotviolatedwithinthese simula-tionruns.

Theﬁrststepsofthesimulationarerelatedtopost-processingof themeasureddatafromthedemonstratorprocess.Datafromthe testrunswithbackbendingareusedforthesimulation(datasets 1and3fromTable3).Thesensordataisalignedandﬁltered[14]

andthereafterresampled.Eachsensorsignalisresampledwithin

angleinthedataset.

ThedisturbancesfromEq. (28)aretransferredtosimulation runsofthecontrolsystem.Furthermore,itis assumedthat the effectofthecontrolparameterontheangleP(2)u isconstant(i.e.

independentofu,d1andd2)andperfectlyknown.Another

assump-tionisthatthecontrolsystemiffastenoughtoadjustthecontrol parameterutoanypositionwithinonestrokeofthepress.These aretheonlyassumptionsneededtodeterminetheeffectofa con-trolsystemontheaccuracyoftheprocess.TheexactvalueofP(2)u

isnotrelevantforthesimulations,becausetheeffectofPu(2)can

becompensatedwithscalingofthecontrollerparameters.Inother words,simulationswithdifferentvaluesforPu(2)cangiveexactly

equalresultsifthecontrollerparametersarescaledaccordingly. Therefore,avalueof1istakenforP_u(2).Theprocessequationsfor thesimulationrunscanbefoundbyrewritingEq.(9)usingEq.(28)

to: ˆ ˛i=˛

u_,_d 1,d2

+P(2) u ui+P(2)y1P (1) y1d1 +P (2) d2d2= ¯˛+P(2) u ui+ (˛i−¯˛) (29) where ˛ˆi is the predicted angle withcontrol, and ˛i is the

measuredanglefromtherealtest.Thecontrolparameteruiis

determinedwiththecontrolsystemandhasafeedbackcomponent uFB

i aswellasafeedforwardcomponentuFFi .Forthefeedback

loop,aPIcontrollerisusedwiththefollowingformulation: uFB i =Kpei−N+Ki i−N

j=1 ej (30)

ThemeasurementdelayNisthreeproducts.Thegainfactorsfor theproportionalandintegralactionareKpandKirespectively.The

errorofthei-thproductisdenotedbyei,with¯˛astargetangle.

Forthefeedforwardloop,theprocessestimatorisﬁtbasedon thedatasetofmostrecentnpproductsfollowingtheapproach

dis-cussedinSection5.2.Theforcemeasurementsaregatheredinthe measurementmatrixM.Forproducti,themeasurementvectormi

containsitsforcemeasurementsfromallprocessstages.The pro-cessestimatoriscontinuouslyupdatedaftereveryﬁveproducts andthefeedforwardcomponentuFF_i isdeterminedforeach prod-uctusingtheLASSOregressionmodelandtheforcemeasurements

mifromthatspeciﬁcproduct.Ifnpischosentoolow,thedataset

sizeisnotsufﬁcienttodetectthecorrelationsbetweenthe mea-surementsandtheﬁnalangle.Ontheotherhand,onlydatawhich isrelevantforthecurrentstateoftheprocessshouldbeincluded inthemodel.Someofthedisturbancesind1andd2maychange

slowlyovertime.Hence,itisexpectedthatrecentdatahasmore resemblancewiththecurrentstateoftheprocess.Therefore,itis expectedthatthequalityofthemodelwilldecreaseifnpischosen

toolarge.

Toperformasimulationrun,fourparametershavetobechosen: thedatasetsizenp,theLASSOparameter(Eq.(27))andthegain

factorsKpandKi.Differentdatasetsizesnphavebeenused,

vary-ingfrom100to1000.Theparameterhasbeenvariedinbetween 0.005and0.5.Foreverycombinationofnpand,thegainfactors

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Fig.16. RMSEoftheangleinthesimulationrunsforalldatasets,withoutcontrol,

withfeedback(FB)andwithfeedbackandfeedforward(FB+FF).

Kp andKihavebeenoptimizedbasedontheRMSEoftheresult.

Obviously,thisisanunfeasibleapproachforimplementationina productionline,becausetheRMSEcanonlybedetermined after-wards.Therefore,furtherstudyisneededtodevelopaprocedure forselectionofthegainfactorsandthemodelparametersnpand

.Itisexpectedthattheseparameterscanbeselectedandupdated onlinewithalearningalgorithm.

5.5. Results

Theprocedurefromtheprevioussectionisappliedtothetwo datasetswithconstantbackbending.TheRMSEfortherun with-outcontrol,thesimulationrunwithfeedbackcontrolandthebest simulationrunwithfeedforwardcontrolbasedonforce measure-mentsisgiveninTable4andvisualizedinFig.16.Theﬁrst1002 productsarenotusedintheRMSEcalculationbecausethe feedfor-wardloopisactivatedatproduct1003inthecasewiththelargest datasetsizenp.Fortherunswithcontrol,thegainfactorsofthePI

controllerhavebeenoptimized.AsigniﬁcantreductionoftheRMSE isachievedwithfeedbackcontrol:34%and26%fordatasets1and 3respectively.Thatisexpected,becausethelong-termvariationin thesedatasetsisstrong,ascanbeseeninFig.8aandc.

Theeffectoffeedforwardcontrolbasedonforcemeasurements isverydifferentforthetwodatasets.Fordataset1withasheet thicknessof280␮m,thereductionofRMSEwithrespecttotherun withfeedbackcontrolisonly2%.Theimprovementfortherunwith asheetthicknessof300␮m(dataset3)is24%.Thatindicatesthat theforcemeasurementscarryinformationwhichcanbeusedto controltheprocess.However,insufficientcorrelationbetweenthe forcemeasurementsandtheflapanglecouldbefoundfordataset1 (280␮m)tobeabletoreachasignificantimprovementwith feed-forwardcontrol.Apossiblecauseisthatthetoolingisdesignedfor asheetthicknessof300␮m,andusinganominalsheetthicknessof 280␮mleadstoadditionalinstabilityoftheprocessandaweaker correlationbetweenprocessforcesandfinalgeometry.

Toillustratetheeffectoftheprocessestimatorontheprocess accuracy,a zoomoftheresultsofdataset3isshowninFig.17. Aroundproduct2810,aseveredropintheangleisobservedin thedatasetwithoutcontrol.Thefeedbackcontrollerreactstothe disturbancewithadelay.Aroundproduct2840,theanglereturns totheoriginalvalue,causinganovershootintheresponseofthe feedbackcontroller.Incontrast,thefeedforwardcontrollerisable tointerpret thechanges intheprocessforces andpredictspart ofthevariationsin theﬁnal angle.Whenthedropintheangle occursaroundproduct2810,theimprovementduetothe feed-forwardestimatorisweak.However,duetothemovingwindow forﬁttingtheprocessestimator,thisdataisaddedtothemodel, increasingthepredictiveabilityofthemodel.Whentheangleof theuncontrolledrunreturnstotheoriginalvaluearoundproduct 2840,thefeedforwardestimatorisabletopredictthisbasedon theforcemeasurements,preventingtheovershootwhichoccursif onlyfeedbackcontrolisused.Severalproductslater,around prod-uct2885,theanglesuddenlydropsfortwosubsequentproducts.

Fig.17. Zoomofresultsofdataset3(300␮m).Theblackdashedlineindicatesthe

targetangle.Theprocessestimatorisupdatedaftereveryproductindicatedwitha

blackmarker.

Fig.18.RMSEofsimulationrunswithfeedforwardandoptimizedPIcontrollerasa

functionofmodelparameters.

Thefeedforwardestimatorpredictsthedropintheanglebasedon theforcemeasurementandcompensatesforit.

InFig.18,theRMSEofthesimulationrunswithdifferentmodel parametersnpandandoptimizedPIcontrollerareshown.Only

thepartoftheparameterspacewhichgivesanimprovementwith respecttotherunwithonlyfeedbackcontrolisshown.Thebest parameter setis marked witha white diamond. For dataset 1 (Fig. 18a), onlya small regionofthe parameterspacegives an improvementwithrespecttofeedbackcontrol.Asmall improve-ment withrespectto only feedbackcontrol is achieved witha processestimatorwhichisﬁttedwithalargedataset(np=1000)

andwithastrongconstraintontheregressionparameters(=0.3). Incontrasttotheresultsfromdataset1,itcanbeseenthata largepartoftheparameterspaceresultsin animprovementin accuracyfordataset3(Fig.18b).Hence,thesensitivityofthe feed-forwardsystemtothemodelparametersnpandisnotstrong.That

isimportant,becausetheseparametershavetobechosenbefore runningthecontrolsystemonarealproductionprocess.Thebest resultsarefoundwithnp=600and=0.01.Itisnoticeablethatthe

bestresultsarenotfoundwiththemaximumvalueforthedataset size(np=1000).Itisexpectedthatrecentdatahasmore

resem-blancewiththecurrentstateoftheprocess,leadingtoadecrease inaccuracyoftheprocessestimatoriftoooldproductsareincluded inthemodelﬁtting.However,theaccuracydecreasewith increas-ingdatasetsizeissmallandnoclearconclusionscanbedrawn basedonthecurrentresults.

Thescraprateresultsfordataset3areshowninFig.19.The per-centageofproductsthatdonotmeetthespecificationsisshown asafunctionoftheanglespecification.Atightanglespecification resultsinhighprocessscraprate.AsobservedinFigs.8cand17, themaximumerrorsmaybewellover2◦ forshortperiods dur-ingproduction.However,themostsignificantimprovementsofthe controlsystemsarereachedforanglespecificationsunder0.5◦.The improvementduetofeedbackcontrolwithrespecttonocontrolis mainlyvisibleinthedecreaseofthescraprateatlargevaluesof theanglespecification.Ontheotherhand,thescraprateforthe

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Fig.19.Scraprateplotfordataset3(300␮m).

feedforwardcontrolrunissignificantlylowerthanthescraprate forthefeedbackcontrolrunforallvaluesoftheanglespecification. Therelevanceoftheuseofafeedforwardcontrolsystemdepends ontheactual process specification.For example,if theprocess specificationis0.3◦,thescraprateswithoutcontrol,withfeedback controlandwithcombinedfeedforwardandfeedbackcontrolare estimatedtobe9.7%,8.4%and3.8%respectively.Inthatcase,the useoffeedforwardcontrolreducesthescrapratewithmorethan halfwithrespecttotheuseoffeedbackcontrolonly.

6. Conclusion

In this work,largeexperimentaldatasets froman industrial demonstratorprocesshavebeenusedtoinvestigatethefeasibility ofcontrolofasheetbendingprocessbasedonforcemeasurements. Thedemonstratorprocesshascutting,deepdrawing,forgingand bendingstages.InSection4,itwasshownthatweakcorrelations canbefoundbetweentheforcemeasurementsandthefinalangle ofoneoftheflapsofthedemonstratorproduct.Basedonthese observations,aprocedureisproposedforimplementing feedfor-wardcontrolwithanestimatorwhichusesforcemeasurementsto predicttheeffectoftheprocessdisturbancesonthefinalangleof theflap.TheLASSOregressionmethodisusedtofitprocessmodels basedonexperimentaldatawithmoremodelparametersthan dat-apoints.Amovingwindowisusedforfittingtheregressionmodel basedonthesetofnpmostrecentlyproducedproducts.The

pro-posedcontrollerhasafeedbackloopwithPIcontrol.Theeffectof theproposedapproachisestimatedbasedonsimulationruns.The angleandforcemeasurementsoftheexperimentaldatasetsare usedasinputforthesimulationruns.Basedonfewassumptions, thisdataisusedtomimicaproductionprocesswithacontrol sys-tem.Forthedatasetwithanominalsheetthicknessof300␮m,a reductionoftheRMSEof24%isestimatedduetothefeedforward loop.Hence,itisshownthatforcemeasurementscarryvaluable informationwhichcanbeusedtoincreasetheaccuracyofmetal formingprocessesandcontroltheeffectofshorttermvariationsin theprocess.

Theuseofforcemeasurementsincontrolisoneofthepaths towardsmoreaccuratemetalformingprocesses.Feedforward

con-tageoftheinformationwhichishiddenintheforcemeasurements. Weforeseethatfurtherstepsmaybetakenthroughthe develop-mentofself-learningcontrolsystemsformetalforming,leadingto robustcontrolwhiletakingmaximumadvantageofthedatawhich becomesavailableduringproduction.

Acknowledgements

Theworkleadingtotheseresultshasreceivedfundingfrom theEuropeanCommunity’sSeventhFrameworkProgrammeunder grantagreementnoFP7-285030.WethanktheMEGaFiTteamfor allthecontributionswhichhaveledtotheseresults.

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