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BayesianBayesi

    Bayesian

1524Yaoeta1./dZhejiangUnivSciA20089(11):1524-1530

    ;JournalofZhejiangUniversitySCIENCEA

    ;ISSN1673-565X(Print);ISSN1862-1775(Online)

    ;?zjueducn~zus;wM,wspringerlinkcom

    ;E?mail:jZUS@zjueducrl

    ;Bayesiannetworksmodelingforthermalerrorof

    ;numericalcontrolmachinetools

    ;XinhuaYAO,Jian-zhongFU,ZichenCHEN

    ;(iCollegeofMechanicalandEnergyEngineering,Zh4iangUniversity,Hangzhou31002ZChina)

    ;(2Zh~iangProvincialKeyLabofAdvancedManufacturingTechnology,Hangzhou31002~China)

    ;E-mail:yaoxinhua@hzcnc.com

    ;ReceivedMay3,2008;revisionacceptedAug.9,2008

    ;Abstract:Theinteractionbetweentheheatsourcelocation.itsintensity,thermalexpansioncoefficient.themachinesystem

    ;configurationandtherunningenvironmentcreatescomplextherma1behaviorofamachinetoo1.andalsomakestherma1error

    ;predictiondifficult.Toaddressthisissue.anove1predictionmethodformachinetoolthermalerrorbasedoffBayesiannetworks

    ;(BNs)waspresented.Themethoddescribedcausalrelationshipsoffactorsinducingthermaldeforrnationbygraphtheoryand

    ;estimatedthethermalerrorbyBayesianstatisticaltechniques.Duetotheeffectivecombinationofdomainknowledgeandsampled

    ;data,theBNmethodcouldadapttothechangeofmrmingstateofmachine,andobtainsatisfactorypredictionaccuracy.Ex

    ;perimentsonspindlethermaldeformationwereconductedtoevaluatethemodelingperformance.Experimentalresultsindicate

    ;thattheBNmethodperforlTISfarbetterthantheleastsquaresfLS1analysisintermsofmodelingestimationaccuracy.

    ;Keywords:Bayesiannetworks(BNs),Thermalerrormodel,Numericalcontrol(N0machinetool

    ;doi:10.163l~zus.A0820337Documentcode:ACLCnumber:TN929.5

    ;rNTR0DUCTION

    ;Thermalerrorisoneofthemostsignificant

    ;factorsaffectingtheprecisionofmachinetools.Itis

    ;believedthat40%-70%oftheerrorinprecisionparts

    ;arisesfromthermalerror(PahkandLee.2002).A

    ;greatdealofworkhasbeencarriedoutoverthelast

;decadeintheestimationandcompensationoftem

    ;peraturedependenterrors(Ramesheta1.,2000).Re

    ;searchershaveemployedvarioustechniquessuchas ;leastsquares(LS)fittingtechniquetomodelthe ;thermalbehavior.

    ;However.somedicultieshaveallalongre

    ;mainedinmodelingfortheinfluenceofpropertiesof ;thermalerror.YangandNi(20051indicatedthatthe ;complexthermalbehaviorsofamachinearecreated ;byinteractionsamongtheheatsourcelocation.heat ;projectsupportedbyNationalNaturalScienceFoundationofChina

    ;(No.50675199)andtheScienceandTechnologyProjectofZhejiang

    ;Province(No2006C11067),China

    ;sourceintensity,thermalexpansioncoefficient.the ;machinesystemconfigurationandtherunning ;environment.Thissituationleadstotwoproblems: ;oneisthatthethermallyinducederrorbecomesa ;time?-varyingnonlinearandnon--stationaryprocess ;(HuangandLiang,2005),andtheotheristhatthe ;variousinteractionsmakeithardtounderstandthe ;relationshipsamongvariousfactors.Theseproblems ;affectthemodelaccuracydirectly.andlimittheuse ;oftraditionalfittingmethodsinmodelingtherrflal ;error.Therefore,newmodelingapproachesare ;neededtoovercomethesedifficulties.

    ;Withthedevelopmentandprogressofuncertain ;theoryandartificialintelligence.somenovelmethods ;forerrorcompensationhavebeenproposed.Fuand ;Chen(20041presentedanartificialneuralnetwclrk ;methodwhichwascombinedwithfuzzylogicto ;predictthermalerror.Anewmethodofprincipal ;componentanalysiswaspresentedtoidentifythe ;structureandparametersofthefuzzymode1.Then, ;theartificialneuralnetworkmethodwasusedto ;

    ;Yaoeta1./dZhejiangUnivSciA20089(11):1524?1530 ;overcometheproblemwhichthefuzzymode1could ;notresolve,becausetheeffectivedatanumberwas ;smallerthantheconsequenceparameternumber.Lin ;Pfa1.(2008)proposedamodelingmethodbasedon ;onlineLSsupportvectormachiner0LS-SVM)to ;implementthermalerrorcompensation.Thedata ;collectedbytemperaturesensorsand1aserposition ;sensorsweretrainedtoconstructthethermalerror

;modelbasedonOLSSVM.Thethermalerrorcould

    ;bepredictedbythismodel,andatthesametimethe ;modelwasmodifiedrecursivelyaccordingtonew ;inputdata.However.acommonshortcomingwith ;thesemodelswasthe”black—box”natureofthem—

    ;selves.Thatistosay.thesemodelsthatpredicted ;thermalerroronlydependedonthetemperaturedata ;collectedonmeasuringpoints,butwithouttakingthe ;relationshipsamongvariousfactorsintoaccount. ;Thus.muchexperientialknowledgecannotbeused ;bythesemodelstoimprovetheircalculationprecision ;andspeed.

    ;ABNisaprobabilisticrepresentationforun

    ;certainrelationshipsandisusefulformodeling ;realworldproblemssuchasdiagnosis,forecasting, ;manufacturingcontrol,etc.,whereinthereexistmul

    ;tiplecauseandeffectdependencyrelations(Hecker. ;manandBreese.1996).Itisadeptinreasoningand ;decisionmakingbywellcombiningexperiential ;knowledgeandsampleddata.Thispaperseeksto ;presentathermalerrormodelfornumericalcontrol ;ONE)machinetoolbasedonBNsandisorganizedas ;follows:inthenextsection.aconceptOfBNsisin

    ;troduced;InSection3,themodelingmethodbasedon ;BNsisproposedandhowtouseittoanalyzethermal ;erroristhenpresented;Totestthevalidityofthe ;method.anexperimentisshowninSection4andthe ;resultisanalyzed;Finally.someconclusionsarepre

    ;sentedinSection5.

    ;BAYESIANNETW0RKS

    ;ABNisagraphicalmodelconsistingofnodes ;whichrepresentcausesandeffectsinrealworld

    ;situationsandasetofedgeseachofwhichconnects ;twonodes.Ifthereexistsacausalrelationshipbe

    ;tweenanytwoofthenodes.theedgewouldbedi

    ;rectional1eadingfromthecausevariabletotheeffect ;variable.Forsuchdirectededges.theedgeissaidto ;befromparenttochild.ABNisalsocalleda’di

    ;1525

    ;rectedacyclicgraph’forthisreasonasalltheedgesof

    ;thegraphpointinaparticulardirectionandthereisno ;waytostartfromonenode,travelalongasetofdi

    ;rectededgesintheproperdirectionandarriveatthe ;samenode.

    ;MathematicalrepresentatiOnsofBayesiannet. ;works

    ;ABNforasetofvariables{….,}con—

    ;sistsofanetworkSthatencodesasetofconditional ;independenceassertionsaboutvariablesinXandaset ;oflocalprobabilitydistributionsassociatedwitheach ;variable(HeckermanandBreese,1996).Together, ;thesecomponentsdefinethejointprobabilitydistri- ;butionforX.

    ;Forajointprobabilitydistribution,fromthe ;chainruleofprobability,wehave

    ;P(,…,)=P(IXl,,…,Xi).(1)-1

    ;ForeveryXitherewillbesomesubset7)

    ;{,x2….,1)suchthatand{X1….,Xi1)\7)

    ;areconditionallyindependent.Thatis,forany, ;(l….,Xi.)=P(l(Xi)),

    ;wherexidenotesthevariableandrepresentsthe ;parentsofnodeinS.CombiningEqs.(1)and(2), ;P(,…,)=nP(XiI()).(3)l

    ;Now,weassumethatthenumberofnodesina ;BNis.theaveragenumberofparentnodesismand ;eachvariableownskvaluesonaverage.Then,the ;computationcomplexityofEq.(3)isnkm,whileitis ;

    ;1toEq.(1).Obviously,thetimecostwilldecline ;byusingEq.f31whenmisfarsmallerthan. ;ParameterlearningforBayesiannetworks ;Theprocesstorefinethelocalprobabilitydis- ;tributionsofaBNusinggivendataiscalled’pa—

    ;rameterlearning’.Givennetstructureandback.

    ;groundknowledgewedenote

    ;=

    ;P(l(Xi),Sh,),(4)

    ;1526Yaoeta1./dZhejiangUnivSciA20089(11):15241530

    ;whereekistheparameterofaBN,representingthe ;probabilityofXiwhosevalueiskandthevalueofits ;parentsisj.Forconvenience,wedefinethevectorof ;parameters

    ;whererandqdenotethenumbersofpossiblevalues ;ofYiandtheirparents,respectively.

    ;Giventhisclassoflocaldistributionfunctions, ;theposteriordistributioncanbecalculatedefficiently ;andinaclosedforillunderthreeassumptions ;(Heckerman,1997):(1)Therearenomissingdatain

    ;SampleD,thatisSampleDiscomplete;(2)The ;parametervectorsaremutuallyindependent;(3) ;Thedistributionof,isaDirichletdistribution. ;Then,theposteriordistributioncanbeobtained ;asfollows:

    ;Nqi

    ;P(OSlD,Sh,)=nP(ID,Sh,)

    ;N,…,N+Nj,

    ;

    ;f(+)]

    ;k=l”.(5)

    ;?兀厂(+)k=1

    ;whereNkisthecoefficientofDirichletdistribution ;indicatingthepriordistributionofvariables.Njkisthe ;numberofcasesinDinwhichXi=xandthevalue ;ofitsparentsisj.

    ;ProbabilisticinferenceforBayesiannetworks ;Ingeneral,thecomputationofaprobabilityofa ;constructedmodelisknownasprobabilisticinference ;Inthissection.probabilisticinferenceinBNsisde

    ;scribedbriefly.

    ;ToaBNwhosestructurehasbeenfixed.ifthere ;areNcasesinsampleddata.theinferencecanbe ;calculatedasfollowsfHeckerman,l997): ;P(ID,S)=JP(IOS,D,S)P(OSID,S)dOs

    ;k+Njk

    ;NI+Njk

    ;Especially,ifthereisonlyonevariabletobe ;calculated,Eq.(6)canbesimplifiedas ;l

    ;=

    ;tq

    ;whereXiisthepredictionvalueof,andshould ;sarisfy

    ;=max{G}x{JD(I(Xi),Sh))

    ;THERMALERR0RMODELINGWITHBAYES.

    ;IANNETW0RKS

    ;Complexbehaviorsofamachinetool(asde. ;scribedinINTRODUCTION,needtobeaddressed ;bythethermalerrormode1inarrivingatanaccurate ;predictionundervaryingoperationconditions.In ;ordertoaccountfortheseeffects.meBN--especially ;dealingwithuncertainrelationships--istherefore ;veryessentia1.

;Modelingprocess

    ;WiththehelpofBayesianstatistics,prior ;knowledgeofadomaincanbecombinedwellwith ;sampleddata.Accordingly,modelingofthermal ;errorsinmachinetoolswillbeimplementedbytwo ;steps:(1)Constructnetworkfrompriorknowledge. ;Hereweshouldidentifythefactorsthatareofcritical ;importanceininducingthermalerrors.analyzethe ;interrelationamongthem.thendescribeallthisin. ;formationbybuildingadirectedacyclicgraph,and ;finallyassesspriorprobabilityofeachvariablein ;graph.(2)Conductprobabilitieslearningandinfer- ;enceinthenetworkbasedonsampleddata.Thepa. ;rametersofnetworkwillberefined,andtheresultof ;modelingcanbecalculatedbyprobabilisticinference. ;Theentiremodelingflowcha~isdescribedas ;Fig.1indetail.

    ;Networkdesign

    ;Thermalerrorsinmachinetoolsarecausedby ;severalfactors,suchasmachiningtime,coolingliq- ;uidandworkingenvironment.ABNinvolvingmore ;factorswil1havehigherprecisionbutincreased ;complexity(Loeta1.,1999).Basedonanoverall ;cOnsideratiOnofvariousfactors.anetworkwith16 ;?

    ;}

    ;,-l

    ;U

    ;=

    ;-}

    ;,_l

    ;U

    ;=

    ;.

    ;}

    ;,-l

    ;U?

    ;=

    ;

    ;,

    ;?

    ;d兀川

    ;==

    ;

;

    ;=

    ;==

    ;==

    1530 ;Yaoetal/dZhejiangUnivSciA20089(11):1524

    ;…………___________I_______-_-…………?_-_____________

    ;Fig.1ModelingflowchartofBNmethod

    comprising13therma1variables.rotate ;factors---

    ;speedofspindle,feedrateandcoolingliquid_has

    ;beenconstructedinexperiments.However.toexpress ;clearlyandcompactly,anetworkwith4nodesistaken ;asanexampleherein.Threetemperaturevariables ;whichconsistofambienttemperatureTo,frontbearing ;temperatureT1andelectricmachinetemperatureT2, ;togetherwiththermaldeformationinaxialdirection ;D0andinradialdirectionD,fc}rlT1anetwork.Here

    ;variablesTo,T1andT2representthealterativevalues ;oftemperature.Accordingtocausalrelationships ;amongvariables,thenetworkcanbeconstructedas ;Fig.2.Obviously,anyvariableinthisnetworkand ;{….,1}\)areconditionallyindependent.

    ;Fig.2Networkforthermalerrorsprediction ;Datapreprocess

    ;ThevariablesofBNsshouldbediscrete,soitis ;1527

    ;essentialtodiscretizevariablesoftemperatureand ;thermaldeformation.Supposethattherangeofa ;variableisE=[1owi,f]’thenitshouldbedivided

    ;intoafinitenumberofnonoverlappingintervalswith

    ;equalintervalwidthw,thatis

    ;and

    ;=

    ;{[Go,G1]u[C/1,Ci2]k-)…u[G,1,C},

    ;?,=:G1,1,2….,k,

    ;wherelowo<1<G2<…<G.k-1<C=upi.Afterthis ;transition.itcanbeseenthatthevariableownsk ;discretevalues.

    ;Theissueabouthowtodeterminethevalueofw ;willbediscussedhere.Inprinciple.theintervalwidth ;shouldsatisfytworequirements:f1)Precisionre

    ;quirement.AsBNsestimatethevalueofavariable ;accordingtotheprobabilityofeachinterva1.interval ;sizelimitstheprecisionofmodeling.Furthermore.the ;middlevalueoftheintervalwhichhasmaximal

    ;probabilitywillbeusedinthispaperastheresultof ;theprediction.intervalwidththereforeshouldhave ;thesamemagnitudeasprecisionrequirement.(2) ;Performancerequirement.AsmentionedinSection2, ;thecomputationcomplexityofBNis.Whenthe ;structureofanetworkisfixed.thenumberofnodes ;andtheaveragenumberofparentnodesmarealso ;fixed.Thusthecomputationcomplexitywillincrease ;swiftlywhenthenumberofintervalskgrows.Obvi

    ;ously.settinglargesizewil1bebeneficialtoreducethe ;numberofintervalsandreducethecomputation ;complexity.

    ;Tosumup.thesizeofintervalsisrelativeto ;modelingprecisionandcomputationcomplexity,and ;itsvalueshoulddependoncertainapplications.To ;someapplicationswithhighprecisionrequirements,

    ;usinghierarchicalmeanstotruncatetherangeof ;variablesgraduallywillbeanidea1method. ;MODELVALIDAT10N

    ;AseriesofexperimentsonanNCmachining

    ;centerhavebeenconductedtoverifythemethod ;basedonBNs.Withsmarttemperaturesensorsand ;laserpositionsensors(KEYENCELK150H),we

    ;havecollectedthetemperatureondifferentpositions ;ofthemachinetoolandthermaldeformationof ;spindle,respectively(Fig.3).

    ;1528Yaoeta1./dZhejiangUnivSciA20089(11):15241530

    ;Fig.3Thermalerrormeasurement

    ;Modelingtest

    ;Themachinerunscontinuouslyfor2.5hinan ;experimentandthedataiScollectedonceevery ;minute.ThisexperimentiSrepeatedseveraltimes ;undersimilarconditionsandthirtysetsofdataare ;collected.Inordertoexplainhowtoprocessthedata, ;thethermalerroratthemomentt=-20miniStakenas ;anexample.Table1citesaportionofdata. ;Table1Portionofexperimentaldatawhent=-20min ;Somecharacteristicvaluesofstatisticsareshown ;inTable2.Basedonit.togetherwiththeprecision ;requirements.theintervalsofeachvariablearesetas ;Table3.Thenthenumberofsampleddatainevery ;intervalcanbeobtained.andtheprobabilitydistribu

    ;tionforvariablesD0andD1canbecalculatedac. ;cordingtoEq.(7).Theparameterinthisformula

    ;indicatestheprioridistributionwhichshouldbeof- ;feredbyexpertsingenera1.HereNkissetas,.The ;resultofthecalculationcanbeseeninTable4,where ;Pk=(1,2….,6),j…etheprobabilityofeach

    ;intervalofthethermaldeformation.Toaxialthermal ;deformationDo,fortheinterval[11.0-12.0)hasthe ;maximalprobability,themiddlevalueofit(i.e.,l1.5 ;um)iSthepredictivevalueofthermaldeformation. ;Similarly,itcanbeestimatedthatthevalueofD1is ;6.4pm.

    ;ExperimentalresuItanalysis

    ;TofurtherconfirmthevalidityoftheBNmethod, ;themodelingperformanceofitiScomparedwiththe ;LSmethodintermsofmodelingaccuracy.The ;modelingresultsforaxialthermaldeformationfrom ;twomethodsandtheactua1valueofmachineare ;showninFig.4.andtheresultsforradialthermal ;deformationareshowninFig.5.Furthermore.the ;Table2Valuesofstatisticalcharacteristic ;(.C)

    ;71l(.c)

    ;(.C)

    ;Do(m)

    ;4

    ;4

    ;6

    ;D1(m)6

    ;0-0.OO1)/[o.0010.002)/

    ;f0.0020.0031

    ;1.350-1.400)/1.400-1.550)/

    ;1.550-1.700)/1.700-1.850)

    ;1.400-1.700)/[1.7002.ooo)/

    ;2.ooo2.300)/[2.3002.600)

    ;f8.09.0)/[9.010.o)/

    ;10.011.0)/11.0-12.0)/

    ;12.0-13.0)/13.0l4.01

    ;[5.4-5.8)/[5.8-6.2)/[6.2-6.6)/

    ;[6.6-7.0)/[7.0-7.4)/[7.4-7.8)

    ;Table4Predictiveresultofthermaldeformation ;ya0eta1./090100l10l2O130140150

    ;Time(min)

    ;Fig.5Modelingvalidationresultsinradialdirection.(a)Thermaldeformation;(b)Residual

    error

    ;1529

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    ;加巧0

    ;一骞II.暑盘一时IIIII(__IIT11).10一砖拿一?

    ;1530Yaootal/JZhejiangUnivSciA20089(11):15241530

    ;modelingresultsarecomparedbasedonthemean ;absolutepercentageerror(MAPE)asdefinedbelow: ;MAPE=l00

    ;_()/(}

    ;,

    ;=

    ;150,(8)

    ;wheredlisthemodelingvalue,whiledtheactual ;value.

    ;WithregardtoMAPEshowninTable5,theBN ;modelperformsfarbetterthantheLSmethodas ;expected,improvingtheaccuracybyapproximately ;80%.

    ;Table5ComparisonofMAPEbetweentheBNandthe ;LSmethods

    ;MethodMAPE(%

    ;AxialdirectionRadialdirection

    ;C0NCLUS10N

    ;Inthispaper.anovelmethodbasedontheBN ;mode1isdevelopedtoesti?

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