University of Huddersfield Repository
Abdulshahed, Ali M., Longstaff, Andrew P., Fletcher, Simon and Potdar, Akshay
Thermal error modelling of a gantrytype 5axis machine tool using a Grey Neural Network Model
Original Citation
Abdulshahed, Ali M., Longstaff, Andrew P., Fletcher, Simon and Potdar, Akshay (2016) Thermal
error modelling of a gantrytype 5axis machine tool using a Grey Neural Network Model. Journal
of Manufacturing Systems, 41. pp. 130142. ISSN 02786125
This version is available at http://eprints.hud.ac.uk/29423/
The University Repository is a digital collection of the research output of the
University, available on Open Access. Copyright and Moral Rights for the items
on this site are retained by the individual author and/or other copyright owners.
Users may access full items free of charge; copies of full text items generally
can be reproduced, displayed or performed and given to third parties in any
format or medium for personal research or study, educational or notforprofit
purposes without prior permission or charge, provided:
•
The authors, title and full bibliographic details is credited in any copy;
•
A hyperlink and/or URL is included for the original metadata page; and
•
The content is not changed in any way.
For more information, including our policy and submission procedure, please
contact the Repository Team at: [email protected].
http://eprints.hud.ac.uk/
Contents lists available atScienceDirect
Journal
of
Manufacturing
Systems
j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / j m a n s y s
Thermal
error
modelling
of
a
gantry-type
5-axis
machine
tool
using
a
Grey
Neural
Network
Model
Ali
M.
Abdulshahed
∗,
Andrew
P.
Longstaff,
Simon
Fletcher,
Akshay
Potdar
CentreforPrecisionTechnologies,UniversityofHuddersfield,Queensgate,HD13DH,UKa
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received15February2016
Receivedinrevisedform22August2016 Accepted23August2016
Keywords:
Gantry-type5-axismachinetool Modelling
PSO
Greysystemtheory Artificialneuralnetwork Greyneuralnetwork
a
b
s
t
r
a
c
t
Thispaperpresentsanewmodellingmethodologyforcompensationofthethermalerrorsona gantry-type5-axisCNCmachinetool.Themethodusesa“GreyNeuralNetworkModelwithConvolutionIntegral” (GNNMCI(1,N)),whichmakesfulluseofthesimilaritiesandcomplementaritybetweenGreysystem modelsandartificialneuralnetworks(ANNs)toovercomethedisadvantageofapplyingeithermodelin isolation.AParticleSwarmOptimisation(PSO)algorithmisalsoemployedtooptimisetheproposedGrey neuralnetwork.Thesizeofthedatapairsiscrucialwhenthegenerationofdataisacostlyaffair,since themachinedowntimenecessarytoacquirethedataisoftenconsideredprohibitive.Undersuch circum-stances,optimisationofthenumberofdatapairsusedfortrainingisofprimeconcernforcalibratinga physicalmodelortrainingablack-boxmodel.AGreyAccumulatedGeneratingOperation(AGO),which isabasisoftheGreysystemtheory,isusedtotransformtheoriginaldatatoamonotonicseriesofdata, whichhaslessrandomnessthantheoriginalseriesofdata.Thechoiceofinputstothethermalmodelisa non-trivialdecisionwhichisultimatelyacompromisebetweentheabilitytoobtaindatathatsufficiently correlateswiththethermaldistortionandthecostofimplementationofthenecessaryfeedbacksensors. Inthisstudy,temperaturemeasurementatkeylocationswassupplementedbydirectdistortion mea-surementataccessiblelocations.Thisformofdatafusionsimplifiesthemodellingprocess,enhancesthe accuracyofthesystemandreducestheoverallnumberofinputstothemodel,sinceotherwiseamuch largernumberofthermalsensorswouldberequiredtocovertheentirestructure.TheZ-axisheatingtest, C-axisheatingtest,andthecombined(helical)movementareconsideredinthiswork.Thecompensation values,calculatedbytheGNNMCI(1,N)modelweresenttothecontrollerforliveerrorcompensation. Testresultsshowthata85%reductioninthermalerrorswasachievedaftercompensation.
CrownCopyright©2016PublishedbyElsevierLtdonbehalfofTheSocietyofManufacturing Engineers.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction
Thereisafocusofcurrentresearchonhighproductionrates onsmallmachinetools.However,largemachinetoolsareofgreat importance because of the significant demand for large high-accuracyparts,suchasimpellers,engineblocks,aeroplanesections, aerofoils,etc.Theaccuracyofa gantry-type5-axismachinetool capableofmanufacturinglargepartsisusuallynotashighasthatof small,three-axismachinetoolsbecausethereareagreaternumber oferrorsources,whichareamplifiedbybiggervolumesandlonger axisstrokes.Highaccuracyforsmallermachinesisoftenachievable byimproveddesignor other“erroravoidance” strategies. How-ever,thesamereductionsinerrorarenotalwaystechnicallyor commerciallyviableforlargermachines.
∗ Correspondingauthor.
E-mailaddress:[email protected](A.M.Abdulshahed).
Thermalerrorscanhaveparticularlysignificanteffectsonthe accuracyoflargemachines.Theycomefromthermaldeformations ofthemachineelementscausedbyheatsourcesthatexistwithin thestructure(i.e.ballscrews,bearings,axisdrivemotors,friction onthewaysurfaces,andtheflowsofcoolant)andtheambient temperaturechanges. Thosethermal errors havebeenreported tobeapproximately70%ofthetotalpositioningerroroftheCNC machinetool[1],thisdiffersfrommachine-to-machine.Although thermalerrorsmightbereducedbymakingthemachinefroma materialthathasalowcoefficientofthermalexpansion,anerror compensationsystemisoftenconsideredtobeamoreeconomical methodofdecreasingthermalerrors.Compensationisaprocess wherethethermalerrorpresentataparticulartimeiscorrected byadjustingthepositionofamachine’saxesbyanamountequal totheerroratthatposition.Anextensivestudyhasbeencarried outintheareaofthermalerrorcompensation.Researchershave employedvarioustechniquessuchasafinite-elementmethod[2]
andfinite-differencemethod[3]inmodellingthethermal char-http://dx.doi.org/10.1016/j.jmsy.2016.08.006
0278-6125/CrownCopyright©2016PublishedbyElsevierLtdonbehalfofTheSocietyofManufacturingEngineers.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).
acteristics.However,buildinganumericalmodelcanbea great challengeduetoproblemsofestablishingtheboundaryconditions andaccuratelyobtainingthecharacteristicofheattransfer. There-fore,testingofthemachinetoolisstillrequiredtocalibratethe modelforsuccessfulapplicationofthetechnique.
Incontrast,othertechniquesuseempiricalmodelling,wherethe modelisbasedontheexperimentalmeasurementsofthemachine tool, ratherthan calibrating anexisting model.Different model structureshavebeenusedtopredictthermalerrors inmachine toolssuchasmultipleregressionanalysis[4],typesofartificial neu-ralnetworks[5],fuzzylogic[6],anadaptiveneuro-fuzzyinference system[7,8],Greysystemtheory[9]andacombinationofseveral differentmodellingmethods[10,11].
EarlyworkbyChenetal.[4]usedbothamultipleregression analysis(MRA)modelandanartificialneuralnetwork(ANN)model for thermalerror compensationof a horizontalmachining cen-tre.Tobuildtheirmodels,810datasetswerecollectedfromfive differenttests;eachtest wasrunfor6hforaheatingcycleand thenstoppedfor10hforacoolingdowncycle.Withtheir experi-mentalresults,thethermalerrorwasreducedfrom196to8mm. Wang[10]usedaHierarchy-Genetic-Algorithm(HGA)trained neu-ralnetworkinordertomapthetemperaturechangeagainstthe thermalresponseofthemachinetool.Wang[8]alsoproposeda thermalmodelbyusinganAdaptiveNeuroFuzzyInference Sys-tem(ANFIS)andoptimisedthenumberofsensorsbyGreysystem modelGM(1,m).Ahybridlearningmethod,whichisacombination ofbothsteepestdescentandtheleast-squaresestimatormethods, wasusedinthelearningalgorithms.Experimentalresultsindicated thatthethermalerrorcompensationmodelcouldreducethe ther-malerrortolessthan9munderrealcuttingconditions.Wangin Refs.[10,8]used150minand480minofdataacquisitioninorder tobuildHGAandANFISmodels,respectively.However,both mod-elsrequiretrainingcyclestocalibratethemodelhowtorespond tovariouschangesininputconditions.Eskandarietal.[12] pre-sentedamethodbywhichtocompensateforpositional,geometric, andthermallyinducederrorsofthree-axisCNCmillingmachine usinganofflinetechnique.Thermalerrorsaremodelledbythree empiricalmodels:MRA,ANN,andANFIS.Tobuildtheirmodels,the experimentaldatawerecollectedevery10minwhilethemachine wasrunningfor120min.Theexperimentaldataaredividedinto trainingandcheckingdatasets.Theirvalidatedresultsonafree form,showsignificantaverageimprovementof41%oftheerrors. Abdulshahedet al.[13] proposeda thermal modelby usingan ANFISwithfuzzyc-meansclustering.Differentgroupsofkey tem-peraturepointswereidentifiedfromthermalimagesusinganovel schemabasedonaGM(0,N)modelandFuzzyc-meansclustering. Experimentalresultsindicatedthatthethermalerror compensa-tionmodelcouldreducethethermalerrortolessthan2m.Also, similarworkshavebeencarriedoutbythesameauthorsinRefs.
[11,14,15].
Wangetal.[9]proposedasystematicmethodologyforthe ther-malerrorcompensationofamachinetool.Thethermalresponse was modelled using a Grey model based onGrey system the-orytopredictthethermal errorswithonly30minofmeasured data.Unfortunately,theirmodellackstheabilityofself-learning, self-adaption,self-organisation,andconsiderationoffeedback cor-rection. Therefore, their model obtained under one particular operatingconditionisnotrobustunderotheroperationconditions. Gomez-Acedoetal.[16]proposedaparametricstatespacemodel forthecompensationofthermaldistortionsinlargemachinetools. Onlytwo-temperature sensorsand spindle speedwere usedas modelinputs.Asmallnumberofthermalsensors,however,might leadtopoorpredictionaccuracy.
Whilst empirical models can be good at predicting thermal errors,theyrequirealargeamountofdatawithdifferentworking conditionstodeterminethegoverninglawsofthesystem.
How-ever,arealisticgoverninglawmaynotexistevenwhen alarge amountofdatahasbeenmeasured.Furthermore,theprocessof obtainingsuchdatacantakeseveralhoursforinternalheatingtests andseveraldaysormorefortheenvironmentaltest.
The growing complexity of manufacturing systems drives researchtodeveloptechniquestoimitatetheunderlying function-alityofthesystem.Inthepast,themodelhadtobekeptassimple aspossible.Forinstance,althoughtheANNmodelsaremore accu-ratethantheregressionmodels,thecalibrationoftheregression modelcoefficientsissimpler(leastsquaresapproach). Neverthe-less,thereisstillastrongargumentforsimplicity,wherepossible, toavoidover-constrainingthesystemandintroducinginstability. Extensiveresearchhasalsoexploredanumberofmetamodels,e.g. polynomialmodels,radialbasisfunction(RBF),andANNmodels. Metamodelinginvolves(i)choosinganexperimentaldesign, (ii) choosingamodel,andthen(iii)training/calibratingthemodelto theexperimentaldata[17].Thereareseveraloptionsforeachof thesestepsasillustratedinRef.[17].Hussainetal.[18]haveused ametamodelingtechniquebasedonradialbasisfunctions,which exploredusingfactorialandLatinhypercubedesigns.Theresulting metamodelwastestedonsevendifferentdatasets,obtainedfrom knowninput-outputrelationships.Simulationresultsindicatethat thefactorialdesignsgenerallyprovidedbetterfitcomparedwith LatinhypercubedesignsformetamodelsusingRBF,exceptinsome instancesnearthecentreofdesignspace.
Properlydesigned experimentsshould beused toobtainan accuratemodel.Thenumberofsamplescanvarygreatlydepending onthecomplexityofthesystemunderconsideration[19]. How-ever,manyotherstatisticalmodelshavebeentrainedsuccessfully withsmallamountsoftraining data[20,21,19].Buragohainand Mahanta[19]haveproposedanANFISbasedmodellingmethod where the number of data samplesemployed for training was minimisedbyapplicationofanengineeringstatisticaltechnique calledfullfactorialdesign.FurthermoreinRefs.[20,21]theyhave appliedanothermethodcalledV-Foldtechnique.Although,their techniqueswereabletoconstructamodelwithasmallnumberof trainingsamples(asfewas7),theystillusedalltheexperimental samplesinordertoselecttheoptimalones.Datatransformation canalsochangethesmoothnessand comparabilityof thedata. Forinstance,HuangandChu[22]haveproposedadata transfor-mationtechniquetosimplifythefuzzymodellingprocedures.The transformationmethodallowsthewholerawdatatobemapped toanotherdomainsuchthatthereisnoneedtoadjustthe mem-bershipfunctions,andthefuzzificationprocessis simplytaking placeonthefixedones. ShmiloviciandAguilar-Martin[23]have alsoutilisedBox-Coxtransformtoimprovethequalityofthefuzzy model,beforeparameteroptimisationoccurs.Therefore, optimisa-tioninthenumberoftrainingpatternsanddatadomainusedfor trainingareofprimeconcerninthefieldofmodelling.
Tosupplementtheproposedmodel,weusetheAGOtoincrease thelinearcharacteristicsandreducetherandomnessfromthe mea-suringsamples.Thissimplebuteffectivetechniqueallowsusto buildthethermalmodelundertheconditionofsmalltrainingdata. Inshort,theproposedmodelincorporatestheAGOmethodintothe modellingprocesstoimproveitspredictionaccuracyand robust-nesswithminimalefforts.
Thehysteresiseffectisdefinedasasystemthathasmemory, wheretheeffects ofthecurrentinputtothesystemare experi-encedwithacertaindelayintime[24].Duetovaryingthermaltime constant,thermaleffectsonCNCmachinetoolshavethe charac-teristicofmemorisingthepreviousthermalstatus.Therefore,the errorsin amachinetoolarenot onlydependentonthecurrent thermalstatusmeasuredatthesurface,butalsoinfluencedbythe previousconditionsofthemachine.Thehysteresisbehaviourwill introduceerrorineachcycle,whichinaworstcasescenariocanbe seeninlargemachinetoolswithbiggervolumes,longerstrokesand
heaviercuttingloads[25].Thishysteresisphenomenonmakesthe static/instantaneousmodellingapproachlessrobust.The charac-terisationofstructuralmaterialwhichexhibitsthermalhysteresis needsaspecialconsideration.Thisismoreevidentwhentherateof temperaturechangeislowascomparedwiththespeedofresponse ofthermaldisplacementandalsowheresurface-mountedsensors donotreflecttheslower-changinginternaltemperature.Therefore, mostoftheabove-mentionedmethodsrequirealargeamountof measureddataduringheating/coolingcycles.Methodsthatrequire acalibratedmodeltopredictthermalerrorsareexpectedtobe con-foundedbytheverylargevarietyofworkingconditionsthatexist inamachine tool.Furthermore,attentionis oftendrawn tothe prohibitivedowntimerequiredtoconducttheexperimentsinan ordinarymachineshop[26].
Accurate and reliable measurements of key variables of the machinetoolareveryimportant.Theinformationfromthese vari-ableswillbeusedformodeltraining/calibration;therefore,they shouldcontainthemostrelevantfeedbackinformation.Inmostof thethermalerrormodelsofmachinetools,temperaturesensorsare usedasinputstoestimatethermaldeformation[9,10,12].However, spindlespeed,axisfeedrate,machiningtimeandotherparameters ofthemachinecanalsobetakenintoconsiderationbecausethey areresponsibleformajorheatsources[27].Insomecases[28,29]
nodirecttemperaturemeasurementistakenandonlythe spin-dlespeedandfeedrateareusedasinputs.However,thisstrategyis limited,becausethemodelobtainedunderoneparticularoperation conditionisnotrobustunderotheroperationconditions.Therefore, errorreductionneedsgreaterunderstandingofthemachinetool propertiesanderrorsources.Thisresultsintheneedforamachine toolstructuralmonitoringsystem.
Fibre Bragg Grating (FBG) sensors are used for strain mea-surementpurposes[30].Theyhaveseveraladvantagesoverother sensorsintermsofsensitivityandquality[30]andcouldbe embed-dedinafuture,commercialisedsystem.Inliterature,thecommon applicationsofFBGaredamagedetection,structurehealth moni-toringandstrainmeasurementinharshenvironments[31,32].FBG canbeemployedtoobservethechangeinthestrainofthestructure withrespecttovariationintemperaturetoprovideanewresponse ofthesystem.Byusingthesesensors,themodellingprocesscan becomesimpler,morerobustandmoreefficientsincethe num-berofthermalsensorscanbereducedandtheeffectsofthermal hysteresisminimised.
Huangetal.[33]usedFBGtoinvestigatetheeffectof temper-aturevariationsofaheavy-dutymachinetoolontheshopfloor. ThevariationsofambienttemperatureweremeasuredbytheFBG sensorsand thespindlethermalshifterrorsweremonitoredby laserdisplacementsensorssimultaneously.Experimentalresults indicate that thespindle thermal errors have a similar change trendfollowingtheambienttemperature.Basedonacquireddata byFBGsensorsandthermal error,theauthorssuggestedthat a thermalerrorcompensationmodelcouldbebuiltbyusingseveral modellingtechniquessuchasmultiplelinearregressions,neural network,andothersystemidentificationmethods;however,no implementationhasbeendoneinthisregard.
Thissectionhashighlightedthatmanythermal errormodels ofmachinetoolsusedtemperaturesensorsasinputstoestimate thermaldeformations.Thedevelopmentofacompensationsystem usingotherparametersofthemachineisdiscussedand investi-gatedinthiswork.
This paper develops an error compensation system for the gantrytype5-axismachinetool.Anovelpredictionmodel“Grey NeuralNetworkModelwithConvolutionIntegral(GNNMCI(1,N))” isproposed, which makesfull useof thesimilaritiesand com-plementaritybetweenGreysystemmodelsand artificialneural networkstoovercomethedisadvantageofapplyingeitheraGrey modeloranartificialneuralnetworkindividually.Itsmost
signifi-Fig.1. TheconceptofGreysystem.
cantadvantageisthatitneedsasmallamountofexperimentaldata foraccurateprediction,andtherequirementforthedata distribu-tionisalsolow.AParticleSwarm Optimisation(PSO)algorithm isalsoemployedtooptimisetheGreyneuralnetwork.Different physicalinputswillbeappliedtotheproposedmodel,whichare capableofsimplifyingthesystempredictionmodel.Thisisbecause differentphysicalinputs(temperatureandstrain)havedifferent correlationefficiencyandtheireffectiveandcooperativefusionis expectedtoproduceabetterpredictionresults.Theexperimental resultsshowthattheproposedmodelhasanexcellentperformance intermsoftheaccuracyofitspredictiveabilityandreductionof machinedowntimewhencomparedagainsttraditionalandother self-learningtechniques.
2. Materialsandmethods
2.1. ModellingthethermalerrorusingaGreyneuralnetwork TheGreysystemtheory,establishedbyDenginRef.[34],isa methodologythatfocusesonsolvingproblemsinvolving incom-pleteinformationorsmallsamples.Thetechniquecanbeapplied touncertainsystemswithpartiallyknowninformation by gen-erating,mining,andextractingusefulinformationfromavailable datasothatsystembehavioursandtheirhiddenlawsofevolution canbeaccuratelydescribed.ItusesaBlack–Grey–Whitecolourto describecomplexsystems[35],theconceptsofaGreysystemcan beillustratedasinFig.1.Agreynumberisakindoffigurethatwe onlyknowtherangeofvalues,anddonotknowanexactvalue. Thisnumbercanbeanintervalorageneralnumbersetto repre-sentthedegreeofuncertaintyofinformation.GM(1,N)isthemost widelyusedimplementationinliterature[36],whichcanestablish afirst-orderdifferentialequationfeaturedbycomprehensiveand dynamicanalysisoftherelationshipbetweensystemparameters. TheAccumulatedGeneratingOperation(AGO)isthemost impor-tantcharacteristicoftheGreysystemtheory,anditsbenefitisto increasethelinearcharactersandreducetherandomnessofthe samples.BasedontheexistingGM(1,N)model,Tien[36]proposed aGMC(1,N)model,whichisanimprovedGreypredictionmodel. ThemodellingvaluesbyGM(1,N)arecorrectedbyincludinga con-volutionintegral.Traditionally,thesemodelshavebeencalibrated bytheleastsquaremethod.However,duetothenonlinearityofthe problem,theleastsquaresolutionmaynotmeettheexpectation.
Comparedwithotherempiricalmodels,artificialneural net-workshaveastrongcapacityforprocessinginformation,parallel processing,andself-learning.However,theyhavesome disadvan-tagessuchas:theneedforalargenumberoflearningsamples; thelongtrainingcomputationtime;andthe“blackbox”resultsare non-interpretable,meaningthatanon-physicallyrealisticsolution canbereachedbutnotidentified.Inaddition,theworking condi-tionsofmachinetoolsareingeneralcomplexandsusceptibleto unexpectedperturbationoninputsignals.Therefore,ANNmodels inisolationhavesignificantdrawbacksasamodellingapproachfor thermalerrorcompensation[37].
Becausethewayofpresentinginformationforneuralnetwork andGreymodelshassomecommonalityinformat,thetwo
meth-odscanbefused.Twolevelscanbeadded;aninitialGreylevel willprocesstheinputinformationandawhiteninglevelafterto processtheoutputinformationtoobtaingoodresults[38]. There-fore,theGrey meaningiscontainedintheneuralnetwork.The advantagesofbothcanbeusedtobuildahigh-performance neu-ralnetworkmodelwithaminimumamountoftrainingdata.The maindifferencebetweenGreyneuralnetworkmodellingand con-ventionalneuralnetworkmodellingisthatthehiddenlayersand theirnodesaredeterminedpreciselybytheGreysystemtheory intheGreyneuralnetworkmethodwhiletheconventional neu-ralnetworkmethodologydefinesthemthroughtedioustrialand errorwork.Although,radialbasisfunction(RBF)methodhasbeen widelyusedintime-seriespredictionwithfewtrainingdataset,it isstilldifficulttoselectanappropriatenetworkstructure. There-fore,theproposedmethoddrivesfromGreysystemtheorywhich isarelativelysimilartofuzzymathematicaltools.
Currently,themostneuralnetworksbasedonstandard back propagation(BP)learningalgorithm.AsstandardBPalgorithmuses gradientdecentmethoditiseasytofallintolocalminimaandhas poorgeneralisationperformance. ThePSO algorithm was intro-ducedbyEberhartandKennedyinRef.[39]asanalternativeto otherevolutionarytechniques.ThePSOalgorithmisinspiredbythe behavioursofthenaturalswarms,suchastheformationofflocks ofbirdsandschooloffish.TheadvantagesofPSOalgorithmisthat itdoesnotrequiretheobjectivefunctiontobedifferentiableas ingradientdecentmethod,whichmakesfewassumptionsabout theproblemtobesolved[40].Furthermore,ithassimplestructure anditsoptimisationmethodillustratesaclearphysicalmeaning. PSOconsistsofapopulationformedbyindividualscalledparticles, whereeachonerepresentsapossiblesolutionoftheproblem.Each particletriestosearchthebestpositionwithtimeinD-dimensional space(solutionspace).Duringswimorflight,eachparticleadjusts its”flying”or“swimming”inlightofitsownexperienceandits companions’experience,includingthecurrentposition,velocity andthebestpreviouspositionexperiencedbyitselfandits com-panions.Therefore,insteadofusingthestandardalgorithms,anew method,PSOalgorithmisemployedtooptimisetheGreyneural networkparametersinthisstudy.
2.2. GNNMCI(1,N)architecture
The fusion model of Grey system and neural network is employedinthemodellingofthethermalerrorofmachinetools. Themodelcanrevealthelong-termtrendofdataand,by driv-ingthemodelbytheAGO,ratherthanrawdata,canminimizethe effectofsomeoftherandomoccurrences.Therefore,thefirststep forbuildingGNNMCI(1,N)istocarryout1-AGO(first-order Accu-mulatedGeneratingOperation)tothedata,soastoincreasethe linearcharacteristicsandreducetherandomnessfromthe measur-ingsamples(seeAppendixA).Tounderstandthispropertyinmore detail,Fig.3showsoriginal(temperaturechanges)andconverted seriesofdata.ThePSOalgorithm,withcapabilitytooptimise com-plexnumericalfunctions[39],isadoptedtotraintheGNNMCI(1, N)model.Finally,anIAGO(inverseAccumulatedGenerating Oper-ation)isperformedtopredictthethermalerrorandgeneratethe finalcompensationvalues.Themodelfullytakestheadvantagesof neuralnetworksandGreymodels,andovercomesthe disadvan-tagesofthem,achievingthegoalofeffective,efficientandaccurate modelling.Themodellingdetailisdescribedasfollows:
TheGreypredictionmodelwithconvolutionintegralGMC(1,N)
[36]is: dX1(1)
dt +b1X (1)
1 =b2X2(1)+b3X3(1)+...+bNXN(1)+u, (2.1)
Fig.2.ThemappingstructureofGNNMCI(1,6).
whereX1(1)isthe1-AGOdataofthepredictedseriesandXi(1),i= 2,3,...,n are thecorresponding 1-AGO data of the associated series,b1isthedevelopmentcoefficient,bi(i=2,3,...,N) the driv-ingcoefficient,anduistheGreycontrolparameter.Therefore,time responsesequencescanbeobtained.
ˆx(1)1 (k+1)=x1(1)(1) e−b1k+u (t−1)× k
=1 e−b1 k−+1 2 .1 2[f () +f (−1)], (2.2)where u (t−1) is the unit step function; and f ()=
Nj=2bjXj(1)()+uk=1,2,...,n.
Tocalculatethecoefficientsbiandu,theneuralnetworkmethod canbeusedtomapEq.(2.2)toaneuralnetwork.Then,the neu-ralnetworkmodelistraineduntiltheperformanceissatisfactory. Finally,theoptimalcorrespondingweightsareusedastheGrey neuralnetworkweightstopredictthethermalerror,similar pro-cedurecanbeseeninRef.[38].
WecanprocessEq.(2.2)further.Let; G=u (t−1)× k
=1 e−b1 k−+1 2 .1 2[f ()+f (−1)] . (2.3)WecanrewriteEq.(2.2)as: ˆx(1)1 (k+1)=
x(1)1 (1)
e−b1k+G. (2.4)
ThenEq.(2.4)canbeconvertedintoEq.(2.5)asfollows: ˆx(1)1 (k+1)=
x(1)1 (1) e−b1 k 1+e−b1k+G 1 1+e−b1k
1+e−b1k, ˆx(1) 1 (k+1)=
x(1) 1 (1)
1− 1 1+e−b1k +G 1 1+e−b1k
1+e−b1k, =
x1(1)(1)−x(1)1 (1) 1 1+e−b1k+G 1 1+e−b1k
1+e−b1k. (2.5)
MapEq.(2.5)intoaneuralnetwork,andthemappingstructure isshowninFig.2.
Wherekistheserialnumberofinputparameters;
In this study, x(1)1 (k+1) is chosen as a dependentvariable (networkoutput) and x(1)2 (k+1),x(1)3 (k+1) ,...,x(1)N−1(k+1),as independentvariables,(Nisthenumberofnetworkinputs);
w11,w21,w22,...,w2n;w31,w32...w3n aretheweights ofthe network;
LayerA,layerB,layerC,andlayerDarethefourlayersofthe network,respectively.
Where, the corresponding neural network weights can be assignedasfollows:
0 20 40 60 80 100 120 140 160 180 -1 0 1 2 3x 10 4 Val u es of new s e ri es Time (h) 0 20 40 60 80 100 120 140 160 18-20 0 2 4 6 T e m p era ture c han ge, ºC Values of new series Temperature change
Fig.3.TheoriginaldatavsAGOconverteddata.
Let us assume that d1=f (b2) ,d2=f (b3) ,...,dN−2= f (bN−2) ,dN−1=f (u).
w11=b1,w21=−x(1)1 (1) ,w22=d1,w23=d2,...,w2N−1=dN−2, w2N=dN−1,w31=w32=w33=...=w3N=1+e−b1k
Thebiasvalueofx(1)1 (k+1) is: =
−x(1)
1 (k+1)
(1+e−b1k) (2.6)
ThetransferfunctionofLayerBis asigmoidfunctionf (x)= 1
1+e−x,thetransferfunctionsofotherlayer’sneuronareadoptedas alinearfunctionf (x)=x.
2.2.1. GNNMCI(1,N)learningalgorithm
ThelearningalgorithmofGNNMCI(1,N)canbesummarisedas follows:
Step1:Foreachinputseries,(k,Xj(1)(k)),(k=2,3,...,N)theoutput ofeachlayeriscalculated.
LayerA:a=w11k; LayerB:b=f(w11k)=1 1 +e−w11k; LayerC: c1=bw21,c2=x2(k) bw22,c3=x3(k) bw23,...cn−1= xn(k) bw2n−1,cn=bw2n; and LayerD:d=w31c1+w32c2+...w3n−1cn−1+w3ncn−.
Step2:Inordertoavoidtheentrapmentinalocalminimum,a PSOalgorithmisadoptedtotraintheGNNMCI(1,N)model.Here, aparticlereferstoaweightinthemodelthatchangesitsposition fromonemovetoanotherbasedonvelocityupdates.Theflowchart forPSOimplementationisgiveninFig.4,andthemathematical descriptionofPSOalgorithmisasfollows;supposethatthesearch spaceisD-dimensional,thenthecurrentpositionandvelocityof theithparticlecanberepresentedbyWi= [wi1,wi2,...,wiD]Tand Vi= [
v
i1,v
i2,...,v
iD]Trespectively,wherei=1,2,...,MandMis thenumberofparticlesintheswarm.Particleicanrememberthebestpositionsofar,whichisknown as the local best position Pbesti= [pbest1,pbest2,...,pbestiD]T. It can also obtain the best position that the whole swarm establish, known as the global best position Gbesti= [gbest1,gbest2,...,gbestiD]T. The first position and velocity of particleiare randomly initialisedby the uniformly distributed
Fig.4. FlowchartforPSOimplementation.
variables. Afterwards, Particle i adjusts its velocity of iteration k+1accordingtothelocalandglobalbestpositions,aswellasthe velocityandpositionofiterationk,asfollows:
Vi(k+1)=ωVi(k)+C1∗R∗ (Pbesti(k)−Wi(k))
+C2∗R∗ (Gbesti(k)−Wi(k)) (2.7) whereωistheinertiafactorwhichisusedtomanipulatetheimpact ofthepreviousvelocitiesonthecurrentvelocity.C1 and C2 are
FBG sensor on front
FBGs mounted on the cross-beam
FBGs sensors on rear
Fig.5. Ageneraloverviewofthe5-axismillingmachineandlocationoftheFibreBraggGratings(FBGs).
theself-confidencefactorandtheswarm-confidencefactor, respec-tively.Risauniformlydistributedrandomrealnumbersthatcan takeanyvaluebetween0and1.
Withtheupdatedvelocity,thepositionofparticleiinthe iter-ationk+1canbeobtainedasfollows:
Wi(k+1)=Wi(k)+Vi(k+1) (2.8) Thefitnessofparticleismeasuredusingafitnessfunctionthat quantifiesthedistancebetweentheparticleanditsoptimal solu-tionasfollows: f (Wi)= N
k=1ˆx(0)(k)−x(0)(k)2 (2.9) wherefisthefitnessvalue, ˆx(0)(k) isthetargetoutput;and,x(0)(k) is thepredictedoutputbasedonconnectionweight(particle) updat-ing.
Step3:updatethevelocityandpositionofeachparticlebased onEqs.(2.7)and(2.8).
Adjustingtheconnectionweightsbetweenlayers: –AdjustingtheconnectionweightsfromLAtoLC. –Adjustthebiasvalue.
=
−x(1)
1 (k+1)
(1+e−w11k)
Step4:Ifthevalueoftheerrormeetstherequirementofthe model,orapre-determinednumberofepochsarepassed,thenthe networktrainingwillendifnot,thenreturntoStep3.
Step5:ExporttheoptimalsolutionWi.
3. Experimentalsetupandapproach
Inthisstudy,themachineunderinvestigationisa5-axisgantry millingmachineasshowninFig.5.Themachineis constructed ofthreelinearaxesX,Y,Z,andtworotaryaxesBandC.The tool-carryingspindleismountedontheBaxisandforthisconfiguration, allaxesmovethetool.ThemaximumspeedsalongtheXaxis,Yaxis, andZaxisofthemachiningcentreare75m/min,75m/min,and 70m/min,andthetravelsare2.5m,1.2m,and0.7m,respectively. Thespindlehasamaximumrotationalspeedof3200revolutions perminute.Thismachinehaslinearscalefeedbackforthethree axesanddirectlymountedrotaryencoderfortheBandCaxes.
Thefirststepinmodellingthethermalerrorsofthismachine wastoperformaninitialassessmenttoidentifymachinestructural elementsandheatsourcesthatcontributemostsignificantlytothe machineerrors.Athermalimagingcamerawasusedtorecord tem-peraturedistributionsacrossthemachinestructureduring“dry” operations,i.e.withoutcoolantpresent.Thetwomaincontributors tothermal error weredue toC-axis rotationand Z-axis move-mentoftheram.Thesetwoerrorsarethereforeanalysedinthis paper.MATLABprocessingroutineshavebeendevisedinRef.[41]
togenerate“virtual”temperaturesensorsfromthethermographic images,whichwereusedtoidentifytheoptimalpositiontoinstall surface-mounttemperaturesensorsonthesurfaceofthe struc-ture(seeFig.6).Fromrelated workonthisaspect[13]and the initialtests,atotaloftwelvetemperaturesensorswereplacedon themachine.Sixsensorswerelocatedonornearthemajorheat sources:onemeasuredthesurfacetemperatureoftheramnear theC-axismotor(T1);one(T2)measuredthesurfacetemperature ofthelowerbearingoftheballscrew;twomonitoredthe gradi-entfromtheendoftheram(T3,andT4);andtwomeasuredthe
Thermal imaging camera
A general overview of the experimental setup Location of temperature/strain sensors
FBG sensors on rear
Fig.6. Strategyoftemperaturesensorplacementonramstructure.
surfacetemperatureoftheZ-axismotor(T5,andT6).Anothersix temperaturesensorswereplacedaroundthemachinetopickup theambienttemperaturechanges.Fourlaserdisplacement sen-sorswereusedtomeasurethedisplacementofatestbar(attached tothespindle)causedbythethermaldistortionofthemachine: twomeasureddisplacementofthetestbarintheY-axis and Z-axisdirections(thisstudydidnotconsidertheX-axisdirectiondue tosymmetry ofthemachine);two measuredanytilt.Ageneral overviewoftheexperimentalsetupisshowninFig.7.
Toimprovetheaccuracyofthepredictedmodel,andtoavoid theneedfora largenumberof temperaturesensors,additional feedbackinformation issuppliedbyFibreBraggGratings (FBGs) asshowninFigs.5 and6.Thiscandetectthechangeinlength by measuring the detectable strain. However, the FBG sensor itself is also affected by temperature by a factor that equates to8.64m/m/◦C. One methodof compensating temperature is touseanunconstrainedgratingtomeasuretemperature. Never-theless,this wasunviable forthis application becauseit would requireadditionalgratingstobemounted,incurringadditionalcost andrequiringadditionalmountingspace,whichwasnotreadily available.Instead,thelow-costtemperaturesensorsusedforthe temperature-basedmodelwereusedtocorrectforchangeinthe gratingtemperature.ThreeFBGsensorswereplacedontheram structureinorder tomeasure thedistortion ofeach sideofthe structure.AnotherfourFBGsensorswereplacedonthecross-beam
Fig.7. Proceduresformeasuringthethermalerrors.
structuretomonitorthethermalresponsewithchangeinthe ambi-enttemperature.Theseon-linemeasureswillbeusedasinputto theproposedmodelinordertopredictthegrowthoftheramalong theZ-axisdirection.
3.1. Hysteresiseffect
Fig.8showstestresultsfromacycleoftwohoursheating-up andanothertwohoursforcoolingdowntestdetailwillbegivenin Section4.Resultsshowthatthetemperatureofthemachinetool (T2Ramrear)changedwithacertaindelayrelativetovariationin themachinedisplacementandFBGsensors(FBG-1,andFBG-2).
Furthermore,Fig.9showshysteresisplotofdifferentsensors,it canbeclearlyseenthattheFBGsensorslocatedonthemachineram exhibitlowerhysteresis.Forexample,(FBG-1,andFBG-2)sensors respondinanalmostlinearfashion,whetherthemachineisbeing heatedorcooled.Itcanalsobeobservedthatthetemperatureat thepointofmeasurement (T2Ramrear)possessslightlyhigher hysteresisbehaviourrelativetoothersensors;thereisalatencyof approximately10min.ByusingFBGsensors,theeffectofthermal hysteresiscouldbeminimised.Therefore,theapplicationofFBG sensorscouldallowforamoreaccuratepredictionofthermalerror. 3.2. Errorcompensationmodel
Themodeldesignersoftenwanttoknowwhichheatsources haveadominanteffectandwhichexertlessinfluenceonthermal responseofthemachinetool.Poorlocationandasmallnumberof thermalsensorswillleadtopoorpredictionaccuracy.However, a large number of thermal sensors mayhave a negative influ-enceonamodel’srobustnessbecauseeachthermalsensormay bringnoisetothemodelaswellasbringingusefulinformation. Furthermore,issuesrelatingtosensorreliabilityarecommercially sensitive;thefewersensorsinstalledthefewerpotentialfailures. Theoptimalsensorlocationswereselectedbasedonourworkin Refs.[15,13].TheMatlabsoftwarehasbeenusedsuccessfullyin numerousotherapplications[42–46].Thus,thethermal compensa-tionmodelisdesignedandsimulatedintheMATLABenvironment. Theintegratedmodelwasdesignedasfollows:
Step1:A1-AGO(first-orderAccumulatedGenerating Opera-tion)isappliedtotherawdatatoincreasethelinearcharacteristics andreducetherandomnessfromthemeasuringsamples.
Step2:TheGNNMCI(1,N)modelistrainedwithaPSOalgorithm asdiscussedinSection2.2.1.
0 50 100 150 200 250 -2 0 2 4 6 8 Tempe ra ture c h an ge , C Time (min) 0 50 100 150 200 2500 20 40 60 80 100 Th erm a l di sp la c e ment ( µ m)
Thermal error (Y direction) FBG-1 (rear) FBG-2 (rear) T2 Ram bottom rear
Heati
ng up
Coolin
g down
Hysteresis effect
Fig.8.Datafusion(temperatureandstrain).
-1 0 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 70 80 90
Temperature change ºC - Strain
Th e rm a l d is p lac em en t ( m) FBG-1 (rear, normalised) FBG-2 (rear, normalised)
T2 Ram bottom rear
Fig.9. Hysteresisplotfromdifferentsensors.
Step3:AnIAGO(InverseAccumulatedGeneratingOperation) isperformedtocalculatethethermalerrorandgeneratethefinal compensationvalue.
To demonstratethe modelling of thermal error using GNN-MCI(1,N) model, five independence variables (temperature and strain)were selectedbased ontheirinfluence coefficientvalue usingtheGreymodel[13].ThreeFBGsensors(FBG-1,FBG-2,and FBG-3) were placed on the ram structure in order to measure thedistortionofeach sideofthestructure.AnotherFBG-4 sen-sorwasplacedonthecross-beamstructuretomonitorthethermal responsewithchangeintheambienttemperature.Additionally, thereisanothertemperaturesensorwasplacedontheram struc-ture(T2Ramrear).Theseon-linemeasureswillbeusedasinputto theproposedmodelinordertopredictthegrowthoftheramalong theZ-axisdirection.
In this paper, two compensation procedures were used to predictthethermalerrors.Thefirstmethodwastoobtainthe GNN-MCI(1,N)modelatthefirststageofthetestregime,andthentouse thismodeltopredictthemachinemovementduringtheremainder
Table1
Thetrainingdatafromfirst5readings.
Numberof sample FBG-1 FBG-2 FBG-3 FBG-4 Temp Displacement 0 0 0 0 0 0 0 1 −0.0072 0.2091 0.0573 0.086 0.0620 −0.2282 2 −0.0217 0.3173 −0.1719 −0.344 0.1870 −0.5766 3 0.2031 0.2019 0.0143 0.0502 0.3120 −0.8127 4 0.1378 0.4903 −0.2220 0.0215 0.5000 −1.1718
ofthesametestorforotherregimes.Theotherwastoobtainthe modelparametersduringashorttest,andthenpredictthethermal displacementforallothertests.Theadvantageofusingashorttest tocalibratethemodelisthatitreducesnon-productivedowntime ofthemachine.Thepotentialdisadvantageisthelackaccuracyof themodelduetolowtrainingexperience.
InordertooptimisetheGNNMCI(1,N)parameters(weights), theexperimentaldatasetsweredividedintotrainingset(and after-warddirectvalidation),validationset(crossvalidation),andtesting set.Anexampleoftrainingdatasetfromashorttestoffive sam-plesisillustratedinTable1;fourFBGsensorsandonetemperature sensorareusedasinputs,andY-axisdisplacementasoutput.
InthePSOalgorithm,thenumberoftheparticlesissettobe90 whilsttheself-confidencefactorandtheswarm-confidencefactor areC1=1.5andC2=2,respectively.Theinertiaweightwastaken asadecreasinglinearfunctioniniterationindexkfrom0.9to0.4, whichwerethesameasthosesuggestedbyotherpapers[47,48]
andthesevaluesdidnotdependontheproblems.After100 train-ingepochs,thetotalerrorwasatacceptablelevel.TheGreyneural networkweightsobtainedusingPSOalgorithmare:
Weights w11 w21 w22 w23 w24 w25 w26 w27
Values 0.1244 0.001 0.8787 1.5390 0.8830 0.1567 0.7005 1.7403 Weights w31 w31 w32 w33 w34 w35 w36 w37
Values 1.5855 1.5855 1.5855 1.5855 1.5855 1.5855 1.5855 1.5855 Trainingandvalidationerrorsdiminishthroughtheinitialphase oftrainingstage.Thefirsttestwastocheckwhetherthemodel isabletoreproducethetrainingdatasetthathasbeenusedfor trainingstage(directvalidation).Subsequently,crossvalidationhas beenappliedtocheckthemodelvalidity.Whenthevalidationerror becomesminimum,themostappropriatemodelisachieved.The predictionresultofthenextsixvaluesofthermalerrorsderived
(a)
(b)
0 10 20 30 40 50 60 70 -5 0 5 10 15 20 25 30 35Time (Minutes)
Te m pe ra tu re -S tra in (C º-m ic ron ) FBG-1 FBG-2 FBG-3 FBG-8 Temp 0 10 20 30 40 50 60 70 -50 -40 -30 -20 -10 0 10
Time (Minutes)
T he rm a l d isp la ce m e n t (m ic ro n ) GNNMCI(1,N) Output Thermal Error Residual value
Fig.10.(a)Temperatureandstrainasmodelinputs.(b)GNNMCI(1,6)modeloutputvstheactualthermalresponse.
Table2
ThemodellingvaluesofthermalerrorsbasedonGNNMCI(1,6)model.
Numberofsample Modelinputs(AGO) GNNMCI(1,6)model(m) Thermalerror(m) Residualvalue(m)
FBG-1 FBG-2 FBG-3 FBG-4 Temp(◦C) 5 0.4642 0.618 0.1432 0.1361 0.5620 −0.9017 −1.4144 0.5127 6 0.5512 0.8581 −0.0430 −0.0573 0.6870 −1.2606 −1.7823 0.5217 7 0.6165 0.8220 −0.2578 −0.1433 0.8120 −1.5017 −2.0300 0.5283 8 0.5802 1.0528 −0.1934 −0.3081 0.8750 −1.8693 −2.4049 0.5356 9 0.9429 1.4494 −0.0573 −0.1720 1.0000 −2.5568 −2.6566 0.0998 10 0.9211 1.1321 −0.0358 −0.2938 1.1250 −2.7040 −3.0366 0.3326
bytheseweightsbasedonthisGNNMCI(1,6)modelarelistedin
Table2.
ThefinalGNNMCI(1,6)modelbeingtrainedandvalidatedin thisworkhasbeentestedbynewunseendataset.Theindependent variablesareshowninFig.10(a).Simulationresultsshowthatthe thermalerrorintheZdirectioncanbesignificantlyreducedtoless than±5musingtestingdataset(seeFig.10(b)).Furthermore,this resultshowsthatthePSOalgorithmcanactasanalternative train-ingalgorithmforGreyneuralnetworkthatcanbeusedforthermal errorcompensation.
Themodelling approach mentioned in this section is a pre-liminaryworkwithascopetobeextended inthenextsections byconsideringavarietyofmodellingmethodssuchasamodular approach.
4. Resultsanddiscussion
Several experiments were conducted on the 5-axis milling machine.Theprimarymotivationoftheseexperimentswasto com-pensatethedeformationtakingplaceintheramofthemachinein theZ-axisdirectionasaresultofheatinducedbyrotationofthe C-axisandbymotionoftheZ-axis.TheZ-axisheatingtest,C-axis heatingtest,andthecombined(helical)movementareconsidered inthispaper.Detailedproceduresandresultsareasfollows: 4.1. Case1:Z-axisheatingtest
Inthistest,theramreciprocatesataspeedof70m/min10times beforedwellingfor10s(toallowstablemeasurement)toexcite thethermalbehaviourintheram.Thiscycleisrepeatedforthetwo hours“heating”cycle.Theaxesremainstationaryforasubsequent twohourscoolingcycle.Thetemperaturevariationismeasuredby thetemperaturesensorsandthechangeinthestrainoftheram
andcrossbeamaremeasuredwithFBGsensors.Thedataisgiven inFig.11(a).Theheatsourcesontheramstructurearefrictionin thetwosupportbearingsoftheZ-axisballscrew,frictioninthe ball-nutandthepowerlossoftheZ-axismotor.Additionally,thereisan effectfromchangeinambienttemperatureonthewholestructure ofthemachine.Laserpositionsensorswereusedtomeasurethe growthoftheramalongtheZ-axisdirection.Itcanbeseenthatthe riseintemperaturemeasuredbytheselectedsensorscorrelatesto anerrorintheZ-axisofmorethan100m.
ThesimulationresultshowsthattheGNNMCI(1,6)modelcan predicttheerroraccuratelyandalsocantrackthesuddenchanges ofthermalerrorprecisely(themaximumresidualisapproximately 16m,a85%improvementseeFig.11(b)),evenwithsuchashort trainingperiod.Indeed,thegreatestlossinmodelaccuracyoccurs overonehourafterthe“heating”cycle.Themajorityofthis ther-malerrorderivesfromareactiontoambientchanges,forwhich themodelhasnotbeentrained.Thiseffectmaynotbesignificant inpracticesinceitcouldbearguedthatthemachinewillnotbe pro-ducingpartsiftheaxesarenotbeingused.Nevertheless,thisissue willbeaddressedunderfurtherworkforthosesituationswhere themachiningregimeexcitesdifferentpartsofthestructureduring variousoperations.
4.2. Case2:C-axisheatingtest
In this test,the C-axisrotates at 2500rpm tentimes before dwellingfor10s(formeasurement)toexcitethethermalbehaviour inthemachineram.Thiscycleisrepeatedforthetwohours “heat-ing”cycle.Theaxesremainstationaryforasubsequenttwohours coolingcycle.DatacollectedfromtemperaturesensorsandFBGs sensorsareshowninFig.12(a).Theheatsourcesinthistestare thefrictionintheC-axisbearingsandlossfromthemotorlocated insidetheramstructure(nearthelocationoftemperaturesensor
(a)
(b)
0 50 100 150 200 250 -120 -100 -80 -60 -40 -20 0 20 Time (Minutes) T he rm a l d isp la ce m en t ( m ic ro n ) GNNMCI(1,N) Output Thermal Error Residual value 0 50 100 150 200 250 300 -4 -2 0 2 4 6 8 10 Time (Minutes) Te m pe ra tu re -S tr a in FBG-1 (rear) FBG-2 (rear) FBG-3 (front) FBG-4 (fcross beam) T3 Ballscrew bearingFig.11.(a)Temperatureandstrainasmodelinputs.(b)GNNMCI(1,6)modeloutputvstheactualthermalresponse.
(a)
(b)
0 50 100 150 200 250 300 350 -2 0 2 4 6 8 10 12Time (Minutes)
Te m p e rat u re-S tr a in FBG-1 (rear) FBG-2 (rear) FBG-3 (front) FBG-4 (cross beam)
T2 Ram bottom rear
0 50 100 150 200 250 300 350 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10
Time (Minutes)
T he rm a l di s pla ce me n t GNNMCI(1,N) Output Thermal Error Residual value
Fig.12.(a)Temperatureandstrainasmodelinputs.(b)GNNMCI(1,6)modeloutputvstheactualthermalresponse. T1).Therefore,T1isthehighesttemperature(risingby7◦C).The
maximumvalueofT2islowerthan(4◦C),andthevalueofT3and T4arethelowest(1◦C)becausetheyarerelativelyfurtherfromthe heatsource.TheZ-axisthermalerrorwasgreaterthan80m.
AswiththeZ-axisheatingtest,themodelweightswereobtained atthefirststageofthetestregime.Simulationresultsshowthatthe GNNMCI(1,6)modelcanprovideagoodpredictionresult.Fig.12
(b),presentsthecomparisonbetweenthermaldisplacementsfrom theactualmeasureddataandtheoutputofthemodel.Itcanbe seenthatthepredictionabilityofthemodelisexcellent,andthat themodelshowsareductionfrom80mto±8m.
4.3. Case3:combinedaxis(helical)test
Inthistest,theC-axisrotateswhiletheZ-axisisalsooscillated simultaneously(helicaltest).Thepurposewastovalidatethe com-pensationmodelforthethermalerrorthatwastrainedfromthe previoustwocases(Case1,Case2).Thiswastodemonstratethat thethermalmodelcouldbebuiltupinamodularformandsois extensibletotheremainderofthestructure.
Thefourhoursvalidationtestwasagainequallydividedinto twostagesofheatingandcoolingcycles.Fig.13(a)describesthe temperature/strainchangeduringthetestregime,whichinduces thermalexpansionintheZ-axisdirectionofapproximately95m. Themodelweightswereobtainedfromthepreviousindependent
C-axisandZ-axistests.Fig.13(b)showsareductioninerrorfrom 95mto±9m,withthelossinperformanceagainbeing preva-lentquitesometimeaftertheheatingpartofthecycle.Thisstudy validatesthemodularapproach,whichmeansthatthecombining trainingdatacanbesuperimposedoneachotherinonemodel. 4.4. Comparisonwithothermodels
InordertoassesstheabilityoftheGNNMCI(1,N)modelrelative tothatofaneuralnetworkmodel,andaconventionalGMC(1,N) model,twomodelswereconstructedusingthesameinput vari-ablestotheGNNMCI(1,N)modelwithfiveinputs.AFeed-forward MultilayerPerceptron (MLP) hasbeenwidely usedANNmodel forthermalerrorcompensation[5,49],sothiswasselectedasthe benchmark.Inthismodel,70%ofthedatasetwasassignedasthe trainingset,whiletheremaining30%wasusedfortestingthe per-formanceofthemodelprediction.Usually,ANNmodelhavethree layers:Input,hiddenandoutputlayer.AnANNmodelwiththree layerswasusedinthisstudy:theinputlayerhasfiveinput vari-ablesandtheoutputlayerhasoneneuron(thethermalresponse intheZ-axisdirection).Although,anANNmodelisabletolearn fromrelationshipsbetweeninputsandoutput,theoptimal num-ber ofneurons in hiddenlayer hastobe found.Therefore, the selectionofthisnumberisatrialanderrorprocessthatmaybe changedduringtheoptimisationprocess.Thenumberofneuronsin
(a)
(b)
0 50 100 150 200 250 300 -100 -80 -60 -40 -20 0 20 Time (Minutes) T he rm al di s p la c e m e n t GNNMCI(1,N) OutputThermal Error
Residual value 0 50 100 150 200 250 300 -2 0 2 4 6 8 10 12 Time (Minutes) T e m p e rat ur e-S tr a in FBG-1 (rear) FBG-2 (rear) FBG-3 (front) FBG-4 (cross beam)
T2 Ram bottom rear
Fig.13.(a)Temperatureandstrainasmodelinputs.(b)GNNMCI(1,6)modeloutputvstheactualthermalresponse.
Fig.14.(a)ANNmodeloutputvstheactualthermalresponse.(b)GMC(1,6)modeloutputvstheactualthermalresponse.
hiddenlayerwasvariedfrom1to15withdifferenttransfer func-tionsnamelylogarithmicsigmoidandtangentsigmoid.Westarted withonehiddenneurons,andthentheANNistrainedandtested. Thenumberofhiddenneuronsisthenincreased,andthetraining processisrepeatedwhiletheoverallresultsofthetrainingand test-ingareimproved.Thus,afteraseriesofsimulationstofindthebest architecture,anANNmodelwith10neuronsinthehiddenlayer andlogarithmicsigmoidwasconstructedtopredictthethermal responseintheZ-axisdirection.
Generally,ANNsaretrainedbyadjustingtheweightstoreach fromaparticularinputtoaspecifictargetusingasuitablelearning algorithm until theANN outputmatches the target. The train-ingprocess stops whenthe errorfallsbelow a pre-determined valueorthemaximumnumberofepochsreaches.Among super-vised gradient-based training method, Levenberg–Marquart is commonlyusedbecauseofitsintegrationofadvantagesofGauss methodand steepestgradient descentalgorithm.ThebestANN architectureisillustratedinTable3below.
AnotherGreymodel wasdeveloped byusingthe traditional LeastSquares(LS)methodinordertoevaluatethemodel param-eters.TheunknownvariablesoftheEq.(2.2)weredeterminedby thetraditionalleastsquaresmethod.Asimilarmodelhasbeenused earlierbyWangetal.[9]forthermalerrorcompensationonCNC machinetool.
Table3
Feed-forwardmultilayerperceptronarchitecture.
Typesofnetworks Threelayerfeed-forwardback-propagation Hiddenlayerneurons 10neurons
inputlayer 5neurons
Outputlayer 1neuron
Transferfunction logarithmicsigmoid Trainingalgorithm LevenbergMarquart
Performancefunction MSE
Maximumnumberepochs 5000
Performancegoal 0.0001
Learningrate 0.1
Momentumconstant 0.90
The three comparison models were further verified by the unseencombinedaxistest(Section4.3),notusedduringthe train-ing,validationandtestingstages.PredictiveresultsusingtheANN and Grey model areshown in Fig. 14 (a)and (b), respectively, andcanbecomparedtoFig.13(b),whichistheresultfromthe methodproposedinthispaper.Theperformanceofeachofthe three thermal predictionmodels ispresented and compared in
Table4,wherethethreemodelsarevalidatedbythesametesting dataset.Accordingtothepredictiveresultsandevaluationcriteria valuesinTable4,itisveryclearthattheGNNMCI(1,N)modelhas asmallerRootMeanSquareError(RMSE),residualvalue(±9m),
Table4
Performancecalculationoftheusedmodels.
Models Performanceindices
R RMSE E Residual
GNNMCI(1,N)model 0.98 4.60 0.94 ±9m
ANNmodel 0.95 6.82 0.92 ±15m
GMC(1,N)model 0.83 15.26 0.60 ±26m
higherefficiencycoefficient(E),andhighercorrelationcoefficient (R)comparedtotheANNandGreymodel.TheANNmodel per-formedbetterthantheGreymodelforpredictingthermalerrorin Z-direction.ItcanbealsoobservedfromTable4thatthemodels developedusingtheartificialintelligencetechniquesoutperformed thestatisticalmodel(Greymodelwith(LS)).However,althoughthe ANNmodeldoesreducetheresidualvaluetolessthan±15m,it requiresalargeamountofhighqualitydatasettotrainthemodel. Furthermore,itisworthnotingthattheANNmodelneedsaproper optimisationtopredicteffectively.Forinstance,theANNmodel needs10neuronsinthehiddenlayer,whichwasdifficultto opti-mise.Therefore,theresultsobtainedfromtheproposedGNNMCI(1, N)modelexhibit betterperformancethanconventionalmodels, withfarfewertrainingsamples.Consequently,thispaperdevelops asimple,lesscomputationallyintensiveandlower-costapproach withahighadaptationrate.
This work develops an error compensation model for the gantry type 5-axis machine tool. The machine operates in a non-temperaturecontrolledenvironment.Changesintemperature causethemachinetochangeshapeandresultinalossofaccuracy. Intheinitialworkonthismachine,onlytemperaturesensorswere usedasinputstothemodel.Themodelestablishedbyonly tem-peraturesensorsonthismachinehashighresidualvaluedueto complexityofthethermalbehaviour,asaresultofbiggervolumes, andlongerstrokes.Themodelwasimprovedbyfusionofboth tem-peraturesensorsanddirectstrainmeasurementfromFBGsensors. Additionally,anothermodelwasbuiltupoftwocomponent mod-ules.Thevalidationofcombinedthermalinputswasshowntobe aseffectiveaswhentheindividualelementswerevalidated.
Unliketheexistingdeterministicmodels,theproposedmethod iseasily extensibletootherphysicalvariables. Thismeansthat alternativeoradditionalsensorscanbedeployedwithminimal retraining required. Furthermore, other machine or machining parameterscanbeacquireddirectlyfromthecontroller to pro-videsomefeedforwardinformationandtominimisetheeffectsof thermalhysteresis.Exampleisthespindlespeedoraxisfeedrate, althoughothersignificantfactorscanalsobeconsidered.Itisworth notingthatchangestomotorbehaviouroveritslifetimewillaffect thethermaloutputatagivenspeed.Forthisreason,theinclusionof theprimaryparametersisnon-trivialwhenlookingforlong-term accuracyfromthemodelanditcanbemorerobustonlytoinclude thederivedvaluesthatdirectlyaffectaccuracy.
Oneofthemajorproblemsforthermalerrormodellingisthe complexwayinwhichthemachinetooldistortsduetothe environ-mentalchangecombinedwithdutycycleeffects.Itwillneverreach atruethermalequilibriumcondition.Futurestudieswillalso con-centrateontheinvestigationbylookingatapplyingthismodelling techniquetoamachinetoolunderdifferentconditions;different EnvironmentalTemperatureVariationerror(ETVE)tests(summer andwinter),andmorecomplexdutycycles.
5. Conclusions
Thisresearchworkproposesathermalerrormodellingmethod basedontheGreysystemtheoryandthelearningabilityofthe artificialneuralnetworkinasinglesystem.Thenumberofsensors usedinthismodelwasminimisedbyfusionofbothtemperature
sensorsanddirectstrainmeasurementfromFibreBraggGratings (FBG)sensors.Wehaveshownthatamodelconsistingofa com-binationofadirectstrainmeasurementandtemperaturesensors canminimisethehysteresiseffectwithmuchmoresensitivity.The modelwasbuiltupoftwocomponentmodulesandsoisshownto beextensibletotheremainderofthestructurebyaddingfurther models.Thisisimportantwherechangestothestructureare pos-sible,sinceitmeansthatonlythatpartofthemodelneedstobe retrained.Italsomeansthatforgreaterprecision,otherstructural elementscanbeconveniently includedin themodel depending upontheamountofprecisionrequired.Thecompensationsystem usingGNNMCI(1,N)modelhasbeenfoundtobeflexible,quick andefficienttoimplement,andhasbeenusedtoreducethermal errorsfromheatingoftheCandZaxesofagantrymachinebyover 85%usingaquickheatingtestforcalibratingthemodel.This dra-maticallyreducestheamountofexperimentaldata,andsoreduces thedowntimeneededforimplementingthecompensationmodel. Theproposedmodelwascomparedtotwootherarchitecturesand demonstratedbetterperformancethanANNmodelandGreymodel withfarfewertrainingsamples.
Itcanthereforebeconcludedthatthethermalerror compensa-tionmodelusingGNNMCI(1,N)introducedinthisstudycanbe appliedin modularformtoany CNCmachine tool becausethe modeldoesnotrelyonaparametricmodelofthethermalerror behaviour.Inaddition,thismodelisopentoextensiontoother physicalinputs,meaningthatalternativesensorscanbedeployed with minimal retraining required. There is still largeroom for enhancementoftheproposedmodelbyincludingmore machin-ingparametersfromthecontrollertoprovidesomefeedforward information,andtrydifferenthybridAItoolstooptimisethemodel parametersforthermalerrormodelling.Futurestudieswillalso concentrateonvalidatingtheproposedmodelwithdifferentCNC machinetoolconfigurationsundersophisticatedoperation condi-tions.
Acknowledgements
Theworkcarriedoutin thispaperispartiallyfundedbythe EU Project (NMP2-SL-2010-260051) “HARCO” (Hierarchical and Adaptive smaRt COmponents for precision production systems application).Theauthorsdowishtothankallthepartnersofthe consortium.
The authors alsogratefully acknowledge theUK’s Engineer-ingandPhysicalSciencesResearchCouncil(EPSRC)fundingofthe EPSRCCentreforInnovativeManufacturinginAdvancedMetrology (GrantRef:EP/I033424/1).
AppendixA. Accumulationgenerationoperation(AGO)
Accumulationgenerationisatechniqueusedtouncovera devel-opmenttendency existing in theprocess of accumulatingGrey quantitiessothatthefeaturesandlawsofintegrationhiddenin theraw data can bediscovered [50]. Thedynamic characteris-ticofproposedmodelresultsfromtheaccumulationgeneration operation.Thetechniquetransformstheoriginaldatatofirstorder 1-AGOdata,whichreducestherandomnessofthesamples,so mak-ingiteasiertodesigntheGreyneuralmodel.Theoutputvalueof themodelcanbeassociatedwithInverseAccumulatedGenerating Operation,abbreviatedasIAGO,theprocedureofAGOandIAGOis summarisedasfollows:
Step1:considertheoriginalseriesas:
X(0)=x(0)(1) ,x(0)(2) ,...x(0)(k−1) ,x(0)(k) . (A.1)
Step2:fromtheoriginalseries,selectingthefirstvalueasthe firstvalueofthenewseries,selectingthefirstvalueplusthesecond
oneoftheoriginalseriesasthesecondvalueofthenewseries, selectingthesumofthefirstthreevaluesoftheoriginalseriesas thethirdvalueofthenewseries,andsoon,asfollows:
X(1)=x(1)(1) ,x(1)(2) ,...x(1)(n−1) ,x(1)(n) . (A.2) Bysodoing,weobtainthenew1-AGOseriesX(1)ofthe orig-inaldataX(0),whichhavemoreregularseriesforthebenefitof modellinginsteadofmodellingwithoriginaldata.
Step3:1-IAGO can beappliedto obtaintheoriginal series, selectingthefirstvalueasthefirstvalueofthenewseries, select-ingthesecondvalueminusthefirstoneoftheoriginalseriesasthe secondentryofthenewseries,selectingthethirdvalueminusthe secondoneoftheoriginalseriesasthethirdvalueofthenewseries, andsoon.Themathematicalexpressionsareasthefollowing: X(0)=x(1)(k)−x(1)(k−1) , (A.3) wherek=2,3,...,n.x(0)(1)=x(1)(1).
Therefore, by applying AGO transformation, the following important advantages can be obtained: (i) removing extreme fluctuationandnoisesothatthenewseriesismorestablefor mod-elling,(ii)thenewserieshasalinearcharacteristicwhichmakes iteasiertomodelinsteadofmodellingwiththeoriginaldata,(iii) andithasthecharacteristicofdeterminingrealisticgoverninglaws fromtheavailabledata[50,51].Theemphasisistodiscoverthetrue propertiesofthesystemundertheconditionofsmalltrainingdata.
References
[1]RameshR,MannanM,PooA.Errorcompensationinmachinetools—areview: PartII:thermalerrors.IntJMachToolsManuf2000;40:1257–84.
[2]MianNS,FletcherS,LongstaffAP,MyersA.EfficientestimationbyFEAof machinetooldistortionduetoenvironmentaltemperatureperturbations. Pre-cisEng2013;37:372–9.
[3]BossmannsB,TuJF.Athermalmodelforhighspeedmotorizedspindles.IntJ MachToolsManuf1999;39:1345–66.
[4]ChenJ,YuanJ,NiJ.Thermalerrormodellingforreal-timeerrorcompensation. IntJAdvManufTechnol1996;12:266–75.
[5]ChenJ,ChiouG.Quicktestingandmodelingofthermally-inducederrorsofCNC machinetools.IntJMachToolsManuf1995;35:1063–74.
[6]LeeJ-H,LeeJ-H,YangS-H.Developmentofthermalerrormodelwithminimum numberofvariablesusingfuzzylogicstrategy.KSMEIntJ2001;15:1482–9.
[7]AbdulshahedA,LongstaffAP,FletcherS,MyersA.ComparativestudyofANN andANFISpredictionmodelsforthermalerrorcompensationonCNCmachine tools.In:LasermetrologyandmachineperformanceX.2013.p.79–88.
[8]WangKC.Thermalerrormodelingofamachiningcenterusinggreysystem theoryandadaptivenetwork-basedfuzzyinferencesystem.In:Cybernetics andintelligentsystems.2006.p.1–6.
[9]WangY,ZhangG,MoonKS,SutherlandJW.Compensationforthethermalerror ofamulti-axismachiningcenter.JMaterProcessTechnol1998;75:45–53.
[10]WangKC.Thermalerrormodelingofamachiningcenterusinggreysystem the-oryandHGA-trainedneuralnetwork.In:Cyberneticsandintelligentsystems. 2006.p.1–7.
[11]AbdulshahedA,LongstaffAP,FletcherS,MyersA.ApplicationofGNNMCI(1,N) toenvironmentalthermalerrormodellingofCNCmachinetools.In:The3rd internationalconferenceonadvancedmanufacturingengineeringand tech-nologies.2013.p.253–62.
[12]EskandariS,ArezooB,AbdullahA.Positional,geometrical,andthermalerrors compensationbytoolpathmodificationusingthreemethodsofregression, neuralnetworks,andfuzzylogic.IntJAdvManufTechnol2013;65:1635–49.
[13]AbdulshahedAM,LongstaffAP,FletcherS,MyersA.Thermalerrormodellingof machinetoolsbasedonANFISwithfuzzyc-meansclusteringusingathermal imagingcamera.ApplMathModel2015;39:1837–52.
[14]AbdulshahedA,LongstaffAP,FletcherS.Aparticleswarmoptimisation-based GreypredictionmodelforthermalerrorcompensationonCNCmachinetools. In:LasermetrologyandmachineperformanceXI,LAMDAMAP2015.2015.p. 369–78.
[15]AbdulshahedAM,LongstaffAP,FletcherS.TheapplicationofANFIS predic-tionmodelsforthermalerrorcompensationonCNCmachinetools.ApplSoft Comput2015;27:158–68.
[16]Gomez-AcedoE,OlarraA,OriveJ,LopezdelaCalleLN.Methodologyforthe designofathermaldistortioncompensationforlargemachinetoolsbased instate-spacerepresentation withKalmanfilter.IntJ MachToolsManuf 2013;75:100–8.
[17]Simpson TW, Poplinski J, Koch PN, AllenJK. Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 2001;17:129–50.
[18]HussainMF,BartonRR,JoshiSB.Metamodeling:radialbasisfunctions,versus polynomials.EurJOperRes2002;138:142–54.
[19]BuragohainM,MahantaC.AnovelapproachforANFISmodellingbasedonfull factorialdesign.ApplSoftComput2008;8:609–25.
[20]BuragohainM,MahantaC.ANFISmodelingofnonlinearsystembasedonVfold technique.IEEEinternationalconferenceonindustrialtechnology,2006(ICIT 2006)2006:2178–83.
[21]BuragohainM,MahantaC.ANFISmodellingofnonlinearsystembasedon sub-tractiveclusteringandV-foldtechnique.2006annualIEEEIndiaconference 2006:1–6.
[22]HuangY-P,ChuH-C.Simplifyingfuzzymodelingbybothgrayrelational anal-ysisanddatatransformationmethods.FuzzySetsSyst1999;104:183–97.
[23]ShmiloviciA,Aguilar-MartinJ.Improvingfuzzysystemsidentificationwith datatransformations.IntJApproxReason1999;22:93–107.
[24]HassaniV,TjahjowidodoT,DoTN.Asurveyonhysteresismodeling, identifica-tionandcontrol.MechSystSignalProcess2014;49:209–33.
[25]TanB,MaoX,LiuH,LiB,HeS,PengF,etal.Athermalerrormodelforlarge machinetoolsthatconsidersenvironmentalthermalhysteresiseffects.IntJ MachToolsManuf2014;82–83:11–20.
[26]LongstaffAP,FletcherS,FordDG.Practicalexperienceofthermaltestingwith referencetoISO230Part3.In:LasermetrologyandmachineperformanceVI. 2003.p.473–83.
[27]LiX,DuR.Analysisandcompensationofworkpieceerrorsinturning.IntJProd Res2002;40:1647–67.
[28]Brecher C, Hirsch P, Weck M. Compensation of thermo-elastic machine tooldeformationbasedoncontrolinternaldata.CIRPAnn:ManufTechnol 2004;53:299–304.
[29]GebhardtM,MayrJ,FurrerN,WidmerT,WeikertS,KnappW.Highprecision grey-boxmodelforcompensationofthermalerrorsonfive-axismachines.CIRP Ann:ManufTechnol2014;63:509–12.
[30]BosettiP,BruschiS.EnhancingpositioningaccuracyofCNCmachinetools bymeansofdirectmeasurementofdeformation.IntJAdvManufTechnol 2012;58:651–62.
[31]GrattanKTV,SunT.Fiberopticsensortechnology:anoverview.SensActuators A:Phys2000;82:40–61.
[32]KreuzerM.StrainmeasurementwithfiberBragggratingsensors.Darmstadt, Germany:HBM;2006.
[33]HuangJ,ZhouZ,LiuM,ZhangE,ChenM,PhamDT,etal.Real-timemeasurement oftemperaturefieldinheavy-dutymachinetoolsusingfiberBragggrating sensorsandanalysisofthermalshifterrors.Mechatronics2015;31:16–21.
[34]DengJ-L.Controlproblemsofgreysystems.SystControlLett1982;1:288–94.
[35]LiuS,ForrestJ,YingjieY.Abriefintroductiontogreysystemstheory.In: ProceedingofIEEEinternationalconferenceongreysystemsandintelligent services2011.2011.p.1–9.
[36]TienT-L.AresearchonthegreypredictionmodelGM(1,n).ApplMathComput 2012;218:4903–16.
[37]WhiteA,PostlethwaiteS,FordDG.Ageneralpurposethermalerror com-pensationsystemforCNCmachinetools.In:Lasermetrologyandmachine performanceV.2001.p.3–13.
[38]YuanJ,ZhongL,LiX,LiJ.Modelingofgreyneuralnetworkanditsapplications. In:KangL,etal.,editors.AdvancesinComputationandIntelligence,vol.5370. Berlin,Heidelberg:Springer;2008.p.297–305.
[39]EberhartRC,KennedyJ.Anewoptimizerusingparticleswarmtheory. Pro-ceedingsofthesixthinternationalsymposiumonmicromachineandhuman science1995:39–43.
[40]Donoso-BravoA,MailierJ,MartinC,RodríguezJ,Aceves-LaraCA,WouwerAV. Modelselection,identificationandvalidationinanaerobicdigestion:areview. WaterRes2011;45:5347–64.
[41]FletcherS,LongstaffAP,MyersA.Measurementmethodsforefficientthermal assessmentanderrorcompensation.In:Proceedingsofthetopicalmeeting: thermaleffectsinprecisionengineering.2007.
[42]Valipour M.Optimization ofneural networksforprecipitationanalysisin a humid region to detectdrought and wet year alarms. Meteorol Appl 2016;23:91–100.
[43]ValipourM,SefidkouhiMAG,EslamianS.Surfaceirrigationsimulationmodels: areview.IntJHydrolSciTechnol2015;5:51–70.
[44]MahdizadehKhasraghiM,GholamiSefidkouhiMA,ValipourM.Simulationof open-andclosed-endborderirrigationsystemsusingSIRMOD.ArchAgronSoil Sci2015;61:929–41.
[45]ValipourM,BanihabibME,BehbahaniSMR.ComparisonoftheARMA,ARIMA, andtheautoregressiveartificialneuralnetworkmodelsinforecastingthe monthlyinflowofDezdamreservoir.JHydrol2013;476:433–41.
[46]ValipourM.Sprinkleandtrickleirrigationsystemdesignusingtaperedpipes forpressurelossadjusting.JAgricSci2012;4:125.
[47]NakaS,GenjiT,YuraT,FukuyamaY.Ahybridparticleswarmoptimizationfor distributionstateestimation.IEEETransPowerSyst2003;18:60–8.
[48]ShiY,EberhartRC.Parameterselectioninparticleswarmoptimization. Evolu-tionaryprogrammingVII1998:591–600.
[49]LiJ,ZhangW,YangG,TuS,ChenX.Thermal-errormodelingforcomplex physi-calsystems:the-state-of-artsreview.IntJAdvManufTechnol2009;42:168–79.
[50]LiuS,LinY,ForrestJYL.Greysystems:theoryandapplications,vol.68.Berlin, Heidelberg:Springer-Verlag;2010.
[51]AzzehM,NeaguD,CowlingPI.Fuzzygreyrelationalanalysisforsoftwareeffort estimation.EmpirSoftwEng2010;15:60–90.
