The application of ANFIS prediction models for thermal error compensation on CNC machine tools

6.a ANFIS-Grid model output vs the actual thermal drift.. 7.ANFIS-Grid model output vs the actual thermal drift.. b ANFIS-FCM model output vs the actual thermal drift 5 h, Test II.. Abdu

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

Centre for Precision Technologies, University of Huddersfield, HD1 3DH, UK

a r t i c l e i n f o

Article history:

Received 30 October 2012

Received in revised form 14 October 2014

Accepted 13 November 2014

Available online 21 November 2014

Keywords:

CNC machine tool

Thermal error modelling

ANFIS

Grey system theory

a b s t r a c t ThermalerrorscanhavesignificanteffectsonCNCmachinetoolaccuracy.Theerrorscomefromthermal deformationsofthemachineelementscausedbyheatsourceswithinthemachinestructureorfrom ambienttemperaturechange.Theeffectoftemperaturecanbereducedbyerroravoidanceornumerical compensation.Theperformanceofathermalerrorcompensationsystemessentiallydependsuponthe accuracyandrobustnessofthethermalerrormodelanditsinputmeasurements.Thispaperfirstreviews differentmethodsofdesigningthermalerrormodels,beforeconcentratingonemployinganadaptive neurofuzzyinferencesystem(ANFIS)todesigntwothermalpredictionmodels:ANFISbydividingthedata spaceintorectangularsub-spaces(ANFIS-Gridmodel)andANFISbyusingthefuzzyc-meansclustering method(ANFIS-FCMmodel).Greysystemtheoryisusedtoobtaintheinfluencerankingofallpossible temperaturesensorsonthethermalresponseofthemachinestructure.Alltheinfluenceweightingsof thethermalsensorsareclusteredintogroupsusingthefuzzyc-means(FCM)clusteringmethod,the groupsthenbeingfurtherreducedbycorrelationanalysis

AstudyofasmallCNCmillingmachineisusedtoprovidetrainingdatafortheproposedmodelsand thentoprovideindependenttestingdatasets.TheresultsofthestudyshowthattheANFIS-FCMmodel

issuperiorintermsoftheaccuracyofitspredictiveabilitywiththebenefitoffewerrules.Theresidual valueoftheproposedmodelissmallerthan±4␮m.Thiscombinedmethodologycanprovideimproved accuracyandrobustnessofathermalerrorcompensationsystem

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/3.0/)

Thermalerrorsofmachinetools,causedbyinternaland

exter-nalheatsources,areoneofthemainfactorsaffectingCNCmachine

toolaccuracy.Internalheatsourcescompriseallheatsourcesthat

aredirectlycausedbythemachinetoolandcuttingprocess,such

asspindlemotors,frictioninbearings,etc.Externalheatsources

areattributedtotheenvironmentinwhichthemachineislocated,

suchasneighbouringmachines,opening/closingofmachineshop

doors,cyclicvariationoftheenvironmentaltemperatureduring

thedayandnightanddifferingbehaviourbetweenseasons.The

complexthermal behaviourofa machineis createdby

interac-tionbetweenthesedifferentheatsources.Accordingtovarious

publications[1–3],thermalerrorsrepresentupto75%ofthetotal

positioningerroroftheCNCmachinetool.Theresponsetospindle

∗ Corresponding author Tel.: +44 0 1484 472596.

E-mail addresses: Ali.Abdulshahed@hud.ac.uk , aa shahed@yahoo.com

(A.M Abdulshahed), a.p.longstaff@hud.ac.uk (A.P Longstaff), s.fletcher@hud.ac.uk

(S Fletcher).

heatingisconsideredtobethemajorerrorcomponentamongthem [4].Oneofthemethodsemployedtoavoidthisprobleminvolves theuseofthermallystablematerialssuchasfibre-reinforced plas-tics,cementconcrete,etc.intheconstructionofthemachinetool

ortodesignsymmetryandisolateheatsources[4].Althoughthese aregoodpractisestoreducethedeformationoftheCNCmachine toolstructure,theymaketheeliminationoferrorsveryexpensive andcanleadtootherproblems,suchasincreasedvibrationorlower acceleration

Anothertechniqueisreducingthermalerrorsthrough numeri-calcompensation.Compensationisaprocesswherethethermal error present at a particular time and position is corrected

by adjusting the position of a machine’s axes by an amount equal totheerroratthat position.Errorcompensationscanbe more attractive than making physical changes to the machine structure.First,errorcompensationisoftenless expensivethan thedesign effort, manufacturingand runningcosts involved in error avoidance Secondly, error compensation is more adapt-able in that it can accommodate changes in error sources, whichsometimescannotbeaccommodatedbystructuralchange techniques[3]

http://dx.doi.org/10.1016/j.asoc.2014.11.012

1568-4946/© 2014 The Authors Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/3.0/ ).

thermalerrorsinadirectorindirectway.Directcompensationis

simpleyetefficientphilosophy,makinguseofdirectlymeasured

displacementsbetweenatoolandaworkpiece,oftenusing

pro-bing.However,directmeasurementcompensationhasanumber

ofdisadvantages.For instance,itislikelythatsomeofthemost

significantthermalproblemsarecausedbyrapidthermalchanges

Trackingandcorrectingtheserapidmovementswouldrequire

fre-quentmeasurements.Whenatool-mountedprobeisused,each

measurementrequiresabreakinmachining,thereforeintroducing

unacceptabletimedelays.Inaddition,probingmeasurementscan

bepronetoerrorscausedbyswarforcoolantonthesurfaceofthe

workpiece[3].Thiscanbeovercomebyrepeatedmeasurementsor

othermeans,butincursfurthercostintermsofhardwareor

pro-ductiontime.Realistically,direct thermalcompensationismost

applicabletofixedtooling,suchaslathes[2],whereadedicated

sensorcanbeconvenientlylocated

1.1 Thermalmodellingmethods

Therearetwo generalschoolsof thoughtrelated toindirect

thermalerrorcompensation.Thefirstmethodusesprinciple-based

modelssuchasthefiniteelementanalysis(FEA)model[5]andfinite

differenceelementmethod(FDEM)[2].Mianetal.[5]proposed

anovelofflineapproachtomodellingtheenvironmentalthermal

errorofmachinetoolsinordertoreducethedowntimerequired

tocalibratethemodel.BasedonanFEAmodel,themethodwas

foundtoreducethemachinedowntimefromafortnightto12.5h

Theirmodellingapproachwastestedandvalidatedonaproduction

machinetooloveraone-yearperiodandfoundtobeveryrobust

However,buildinganumericalmodelcanbeagreatchallengedue

toproblemsofestablishingtheboundaryconditionsandaccurately

obtainingthecharacteristicsofheattransfer

Thesecondmethodisempiricalmodellingbasedoncorrelation

betweenthemeasuredtemperaturechangesandtheresultant

dis-placementofthefunctionalpointofthemachinetool,whichisthe

changeinrelativelocationbetweenthetoolandworkpiece.Linear

regressionisthesimplestmethodtocorrelatemeasured

tempera-tureswithresultingdisplacement.Aleastsquaresapproachisused

toobtainthecoefficientsthatdeterminetherelationshipbetween

inputsandoutputwithoutusinganyphysicalequation.Although

thismethodcanprovidereasonableresultsforagivenmachinetest

regime,thethermaldisplacementusuallychangeswithvariation

inthemachiningprocessandtheenvironment,whichintroduces

anderrorintothemodel[6].Thelinearregressionmodelisalso

time-consumingandlabourintensivetodesign

Inrecentyears,ithasbeenshownthatthermalerrorscanbe

suc-cessfullypredictedbyartificialintelligencemodellingtechniques

suchasartificialneuralnetworksANNs[7,8],fuzzylogic[9],

adap-tiveneuro-fuzzyinferencesystems[8]andacombinationofseveral

differentmodellingmethods[10]

Theadaptiveneurofuzzyinferencesystem(ANFIS)hasbecome

anattractive,powerful,generalmodellingtechnique,combining

wellestablishedlearninglawsofANNsand thelinguistic

trans-parencyoffuzzylogictheory[11].ByemployingtheANNtechnique

to update theparameters of the Takagi-Sugeno type inference

model,theANFISisgiventheabilitytolearnfromtrainingdata

inthesamewayasanANN.Thesolutionsmappedoutontoafuzzy

inferencesystem(FIS)canthereforebedescribedinlinguisticlabels

(fuzzysets)[12].Thus,thenodesandthehiddenlayersare

deter-minedprecisely bya FISintheANFIS network.Thiseliminates

thewell-knowndifficultyofdeterminingthehiddenlayerofANN

modelsandatthesametimeimprovingitspredictioncapability

ANFISisconsideredbecauseitdoesnotrequirecomplex

mathe-maticalmodel,itisfastandadaptiveandthedevelopedprediction

toolcanbeimplementedquickly,whichisessentialforthermal

errorscompensation.ANFIStechniqueshavealreadybeenapplied

todifferentengineeringareassuchassupporttodecision-making [13,14],modellingtool wearin turningprocess [15], and mod-ellingthermalerrorsinmachinetools[8,16].Abdulshahedetal.[8] comparedtheabilityofANFISandANNstopredictthermalerror compensation inCNC machine tools.The resultsindicated that althoughANNshaveagoodlevelofpredictionaccuracy,theANFIS modelsweresuperiorintermsofforecasting ability.Wang[16] alsoproposedathermalmodelusingANFIS.Experimentalresults indicatedthatthethermalerrorcompensationmodelcouldreduce thethermalerrortolessthan9␮mundercuttingconditions.He usedsixinputswiththreefuzzysetsperinput,producinga com-pleterulesetof729(36)rulesinordertobuildanANFISmodel Clearly,Wang’smodel ispractically limitedtolow dimensional modelling.Itisimportanttonotethataneffectivepartitionofthe inputspacecandecreasethenumberofrulesandthusincreasethe speedinbothlearningandapplicationphases.However,areliable andreproducibleproceduretobeappliedinapracticalmannerin ordinaryworkshopconditionswasnotproposed.Forexample,the numberoffuzzyrulesincreasesexponentiallywhenthenumber

ofvariablesrises.Toovercomethislimitation,fuzzyc-means algo-rithmscouldbeusedtodetermineclusterseffectively,providing betterclusteredinputstopredictionmodel

1.2 Reductionofmodelinputs Intuitively, locatinga largenumberof sensorsonamachine toolstructureshouldenhancetheaccuracyofthethermal error model sinceit increasestheinformation input.However,many researchersaimtoreducethenumberofrequiredtemperature sen-sors.Toolargeanumberofsensorsmightleadtoanincreaseinthe constraintsandcostofthecompensationsystem,aswellaspossibly leadingtopoorrobustnessofthethermalmodelbecauseofincrease

indatanoise.Severalstudieshaveusedstatisticalapproachessuch

asengineeringjudgement,thermalmodeanalysis,stepwise regres-sionandcorrelationcoefficientstoselectthetemperaturesensors forthermalerrorcompensationmodels[17].YanandYang[18] proposedanMRAmodelcombingtwomethods,namelythedirect criterionmethodandindirectgroupingmethod;bothmethodsare basedonsyntheticGreycorrelation.Usingthismethod,the num-beroftemperaturesensorswasreducedfrom16tofourandthe residualrangewasreducedfor69.1%.Hanetal.[19]proposeda correlationcoefficientanalysisand fuzzyc-meansclusteringfor selectingtemperaturesensorsbothintheirrobustregression ther-mal errormodel and ANN model [20]; the number of thermal sensorswasreducedfrom32tofive.However,thesemethods suf-ferfromthefollowingdrawbacks:alargeamountofdataisneeded

inordertoselectpropersensors;andtheavailabledatamust sat-isfyatypicaldistributionsuchasnormal(orGaussian)distribution [21].Therefore,asystematicapproachisstillneededtominimise thenumberoftemperaturesensorsandselecttheirlocationsso thatthedowntimeandresourcescanbereducedwhilerobustness

isincreased

GreysystemtheoryisamethodintroducedbyDenginearly 1980s[22]withtheintentiontostudytheGreysystemsbyusing mathematicalmethodswithpoorinformationandsmalldatasets

InGreysystemtheory,GM(h,N)denotesaGreymodel,whereh

istheorderofdifferenceequationandNisthenumberof vari-ables.TheGM(h,N)modelcanbeusedtodescribetherelationship betweentheinfluencingsequencefactorsandthemajorsequence factorofasystem.Furthermore,weightsofeachfactorrepresent theirimportancetothemajorsequencefactorofthesystem.Its mostsignificantadvantageisthatitneedsonlyasmallamountof experimentaldataforaccurateprediction,andtherequirementfor thedatadistributionisalsolow[21]

Fig 1.Basic structure of ANFIS.

Inthispaper,theGM(1,N)modelandfuzzyc-means

cluster-ingareusedtodeterminethemajorsensorsinfluencingthermal

errorsofasmallverticalmillingmachine(VMC),whichis

capa-bleofsimplifyingthesystempredictionmodel.Thenweusedthe

ANFIStobuildtwothermalpredictionmodelsbasedonselected

sensors:ANFISbydividingthedataspaceintorectangular

sub-spaces(ANFIS-Grid)andANFISbyusingfuzzyc-meansclustering

methodwithANFIS(ANFIS-FCM).Thiscombinedmethodologycan

helptoimproverobustnessoftheproposedmodel,andreducethe

effectofsensoruncertainty

introducedbyJang[11].AccordingtoJang,theANFISisaneural

networkthatisfunctionallythesameasaTakagi-Sugenotype

infer-encemodel.ANFIShasbecomeanattractive,powerfulmodelling

technique,combiningwellestablishedlearninglawsofANNsand

thelinguistictransparencyoffuzzylogictheorywithinthe

frame-workofadaptivenetworks.Fuzzyinferencesystems(FIS)areone

ofthemostwell-knownapplicationsoffuzzylogictheory.Inthe

fuzzyinferencesystems,themembershipfunctionstypicallyhave

tobemanuallyadjustedbytrialanderror.TheFISmodelperforms

likeawhitebox,meaningthatthemodeldesignerscandiscover

howthemodelachieveditsgoal.Ontheotherhand,artificialneural

networks(ANNs)canlearn,butperformlikeablackboxregarding

howthegoalisachieved.ApplyingtheANNtechniquetodevelop

theparametersofafuzzymodelallowsustolearnfromagiven

setoftrainingdata,justlikeanANN.Atthesametime,the

solu-tionmappedoutintothefuzzymodelcanbeexplainedinlinguistic

termsasacollectionof“IF–THEN”rules

2.1 ANFISarchitecture

ThearchitectureofANFISisshowninFig.1.Fivelayersareused

toconstructthismodel.Eachlayercontainsseveralnodesdescribed

bythenodefunction.Adaptivenodes,denotedbysquares,

rep-resent the parameter sets that are adjustable in these nodes

Conversely,fixednodes,denotedbycircles,representthe

param-etersetsthatarefixedinthemodel.Simple ANFISarchitecture,

whichusestwovariables(T1andT2)asinputsandoneoutput(F:

thermaldrift),willbedescribedinthissectioninordertoexplain

theconceptoftheANFISstructure

intoafuzzysetbymeansofmembershipfunctions(MFs).It

con-tainsadaptivenodeswithnodefunctionsdescribedas:

whereT1 and T2 are theinputnodei, AandB arethe

linguis-tic labels associated with this node, (T ) and (T ) are the

membershipfunctions(MFs),TherearemanytypesofMFsthat canbeused.However,aGaussianshapedfunctionwithmaximum andminimumequalto1and0isusuallyadapted.Parametersin thislayeraredefinedaspremiseparameters

circleandlabelledby

,withthenodefunctiontobemultiplied

byinputsignalstoserveasoutputsignal

O2,i=wi=Ai(T1)·Bi−2(T2), fori=1,2 (3) wheretheO2,iistheoutputofLayer2.Theoutputsignalwi repre-sentsthefiringstrengthoftherule

markedbyacircleandlabelledbyN,withnodefunctionto nor-malisethefiringstrengthbycomputingtheratiooftheithnode firingstrengthtosumofallrules’firingstrength

O3,i= ¯w= wi

wheretheO3,iistheoutputofLayer3.Thequantity ¯w isknownas thenormalisedfiringstrength

byasquare,withnodefunctionasfollowing:

wheref1andf2arethefuzzyif–thenrulesasfollows:

• Rule1.IFT1isA1andT2isB1,THENf1=p1T1+q1T2+r1

• Rule2.IFT1isA2andT2isB2,THENf2=p2T1+q2T2+r2 wherepi,qiand riaretheparametersset,referredtoasthe consequentparameters

byacircleandlabelledby

,withnodefunctiontocalculatethe overalloutputby:

O5,i=

iw¯i·fi=



iwifi

ThesimplestlearningruleofANFISis“back-propagation”which computeserrorsignalsrecursivelyfromtheoutputlayer(Layer5) backwardtotheinputnodes(Layer1).Thislearningruleisexactly thesameastheback-propagationlearningruleusedinthecommon feed-forwardneuralnetworks[8,23].Althoughthismethodcanbe appliedtoidentifytheparametersinanANFISnetwork,themethod

isgenerallyslowandlikelytobecometrappedinlocalminima[11] Differentlearningtechniques,suchasahybrid-learningalgorithm [14]orgeneticalgorithm(GA)[24],canbeadoptedtosolvethis trainingproblem.BetterperformanceofANFISmodelshasbeen shownbyadoptingarapidhybridlearningmethod,which inte-gratesthegradientdescentmethodandtheleast-squaresmethod

tooptimiseparameters[23,25,26].Thusinthispaper,thehybrid learningmethodisusedforconstructingtheproposedmodels 2.2 Extractionoftheinitialfuzzymodel

Inordertostartthemodellingprocess,aninitialfuzzymodel hastobederived.Thismodelisrequiredtoselecttheinput vari-ables,inputspacepartitioningorclustering,choosingthenumber andtypeofmembershipfunctionsforinputs,creatingfuzzyrules, andtheirpremiseandconclusionparts.Foragivendataset, differ-entANFISmodelscanbeconstructedusingdifferentidentification methodssuchasgridpartitioning,andfuzzyc-meansclustering (FCM)[23]

AThe ANFIS-Grid partition method is the combination of grid partitionandANFIS.Thedataspacedividesintorectangular sub-spaces usingaxis-paralleledpartitionsbasedonapre-defined

lim-itationofthismethodisthatthenumberofrulesrisesrapidly

asthenumberofinputs(sensors)increases.Forexample,ifthe

numberofinputsensorsisnandthepartitionedfuzzysubsetfor

eachinputsensorism,thenthenumberofpossiblefuzzyrules

ismn.Whilethenumberofvariablesraises,thenumberoffuzzy

rulesincreasesexponentially,whichrequiresalargecomputer

memory.AccordingtoJang[11],gridpartitionisonlysuitable

forproblemswithasmallnumberofinputvariables(e.g.fewer

than6).Inthispaper,theproposedthermalerrormodelhasfive

inputs.ItisreasonabletoapplytheANFIS-Gridpartitionmethod

BTheANFIS-fuzzyc-meansclusteringisthemostcommonmethod

offuzzyclustering[25].Essentially,itworkswiththeprincipleof

minimisinganobjectivefunctionthatdefinesthedistancefrom

anygivendatapointtoaclustercentre.Thisdistanceisweighted

bythevalueofMFsofthedatapoint[25].IntheFCMmethod,

whichisproposedtoimproveANFISperformance,thedataare

classifiedintopertinentgroupsbasedontheirdegreesofMFs.In

thisclusteringmethod,itisassumedthatthenumberofclusters,

nc,isknownoratleastfixed.ItdividesagivendatasetX={x1, ,

xn}intocclusters.Moredetailcanbefoundinthenextsection

Inordertoobtainasmallnumberoffuzzyrules,afuzzyrule

generationtechniquethatintegratesANFISwithFCMclustering

canbeused,wheretheFCMisusedtosystematicallyidentifythe

fuzzyMFsandfuzzyrulebaseforANFISmodel.Inthispaper,to

identifypremisemembershipfunctions,thetwoaforementioned

methodswereusedandcompared

2.3 Fuzzyc-meansclustering

Fuzzyc-means(FCM)isasoftclusteringmethodinwhicheach

datapointbelongstoacluster,withadegreespecifiedbya

mem-bershipgrade.Dunnintroducedthisalgorithmin1973[28]andit

wasimprovedbyBezdek[29].FCMalgorithmisthefuzzymode

ofK-meansalgorithmanditdoesnotconsidersharpboundaries

betweentheclusters[30,31].Thus, thesignificantadvantageof

FCMistheallowanceofpartialbelongingsofanyobjecttodifferent

groupsoftheuniversalsetinsteadofbelongingtoasinglegroup

totally

FCMpartitionsacollectionofnvectorsxi,i=1,2, ,ninto

fuzzygroups,anddeterminesaclustercentreforeachgroupsuch

thattheobjectivefunctionofdissimilaritymeasureisreduced

i=1,2, ,carearbitrarilyselectedfromthenpoints.Thesteps

oftheFCMmethodarenowbrieflyexplained:firstly,thecentresof

eachclusterci,i=1,2, ,carerandomlyselectedfromthendata

patterns{x1,x2,x3, ,xn}.Secondly,themembershipmatrix()

iscomputedwiththefollowingequation:

where,

ij:thedegreeofmembershipofobjectjinclusteri;

M:thefuzzinessindexvaryingintherange[1,∞];and

dij=||ci−xj||:theEuclideandistancebetweenciandxj

Thirdly,theobjectivefunctioniscalculatedwiththefollowing

equation.Theprocessisstoppedifitfallsbelowacertainthreshold:

J(U,c1,c2, ,cc)=

c



Ji= c



c



Finally,thenewcfuzzyclustercentresci,i=1,2, ,care cal-culatedusingthefollowingequation:

ci=

n

j=1mijxj

n

j=1m ij

(9)

Inthispaper,theFCMalgorithmwillbeusedtoseparatewhole training datapairs intoseveralsubsets (membershipfunctions) withdifferentcentres.EachsubsetwillbetrainedbytheANFIS,as proposedbyParketal.[32].Furthermore,theFCMalgorithmwill

beusedtofindtheoptimaltemperaturedataclustersforthermal errorcompensationmodels[33]

influenceonpredicationaccuracyandrobustnessofathermal pre-dictionmodel.Oneofthedifficultissuesinthermalerrormodelling

istheselectionofappropriatelocationsforthetemperature sen-sors,which isa key factorintheaccuracy ofthethermal error model.ThisstudyadoptsGreysystemtheorytoidentifytheproper sensorpositionsforthermalerrormodelling

The Grey systems theory is a methodology that focuses on studyingtheGreysystemsbyusingmathematicalmethodswith

aonlyfewdatasetsandpoorinformation.Thetechniqueworkson uncertainsystemsthathavepartialknownandpartialunknown information.Itsmostsignificantadvantageisthatitneedsasmall amount of experimental data for accurate prediction, and the requirementforthedatadistributionisalsolow[21].Thereare manytypesofGreymodels;theGreyGM(1,N)modelwillbeused

inthiswork

3.1 TheGM(1,N)model Thefirst-orderGreymodel,GM(1,N),isamultivariableGrey modelformulti-factorforecasting.GM(1,N)meansaGreymodel thathasNvariablesincludingonedependentvariableandN−1 independentvariables.AssumethatthereareNvariables,xi(i=1,

2, ,N),andeachvariablehasninitialsequencesas:

x(0)i ={x(0)

i (1),x(0)i (2), ,x(0)i (n)} (i=1,2, ,N)

the smoothness of the sequence, the accumulative generation operation (AGO) is applied to convert the sequences to be strictlymonotonicincreasingsequences.Forsimplification,letus definethefirst-orderaccumulativegenerationoperation(1-AGO) sequenceforx(0)i as:

x(1)i ={x(1)

i (1),x(1)i (2), ,x(1)i (n)}, where,

x(1)i (k)=

k



j=1

x(0)i (j) (k=1,2, ,n)

Then,theGM(1,N)modelcanbeexpressedbythefollowing Greydifferentialequation[21]:

x(0)1 (k)+az(1)1 (k)=

N



j=2

bjXj(1)(k)

=b2x(1)2 (k)+b3x(1)3 (k)+···+bNx(1)N (k), (10)

Fig 2.Block diagram of the proposed system.

Inwhich,z(1)1 (K)isdefinedas:

z1(1)(k)=0.5x(1)1 (k−1)+0.5x(1)1 (k) k=2,3,4, ,n

wherethecoefficientsaandbjarecalledthesystemdevelopment

parameterandthedrivingparameters,respectively

FromEq.(10),wecanwrite:

x(0)1 (2)+az1(1)(2)=b2x(1)2 (2)+···+bNx(1)N (2),

x(0)1 (3)+az1(1)(3)=b2x(1)2 (3)+···+bNx(1)N (3),

x1(0)(n)+az(1)1 (n)=b2x(1)2 (n)+···+bNx(1)N (n)

(11)

Eq.(5)canbewritteninthematrixformas:

⎡

⎢

⎢

⎢

⎣

x(0)1 (2)

x(0)1 (3)

x(0)1 (n)

⎤

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎣

z1(1)(2) x2(1)(2) ··· x(1)N (2)

z1(1)(3) x2(1)(3) ··· x(1)N (3)

z1(1)(n) x2(1)(n) ··· x(1)N (n)

⎤

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎣

a

b2

bN

⎤

⎥

⎥

Thecoefficientsofthemodelcanthenbeobtainedusingthe

least-squareestimatemethodas:

where,ˆ =

⎡

⎢

⎣

a

b2

bN

⎤

⎥

⎦, Y=

⎡

⎢

⎢

x(0)1 (2)

x(0)1 (3)

x(0)1 (n)

⎤

⎥

⎥,

B=

⎡

⎢

⎢

z(1)1 (2) x1(1)(2) ··· x(1)N (2)

z(1)1 (3) x1(1)(3) ··· x(1)N (3)

z(1)1 (n) x(1)2 (n) ··· x(1)N (n)

⎤

⎥

⎥

Therefore,theinfluencerankingfromtheindependentvariables

tothedependentvariablecanbeknownbycomparingthemodel

valuesofb2∼bN

Toobtainrobustmodels,alltheinfluenceweightingofthermal

sensorsisclusteredintogroupsusingFCM.Then,onesensorfrom

eachclusterisselectedtorepresentthetemperaturesensorsof

thesamecategoryaccordingtoitsinfluencecoefficientwiththe

Fig 3.Location of thermal sensors on the machine.

thermaldrift.Therefore,byselectingfivesensors,theANFISmodels canbebuilteasilytopredictthethermaldrift

Thewholeblockdiagramoftheproposedsystemisshownin Fig.2,wherevariablesT1toTNrepresentthetemperaturedata cap-turedfromthetemperaturesensors,andthethermaldriftobtained fromnon-contactdisplacementtransducers(NCDTs)

4.1 Setupofmeasurementsystem Fig.3showstheblockdiagramofathree-axisverticalmilling machine(VMC).Themotorsfortheaxesaredirectlycoupledtoa ballscrewthatissupportedbybearingsateachend.Thespindleis rotatedbyaDCmotormountedonthetopofthespindlecarrier Thespindlespeedcanbecontrolledfrom60rpmto8000rpm.In ordertoobtainthetemperaturedataofthismachinetool,atotal

of76thermalsensorsareplacedonthemachine.Thesensorscan

beclassifiedintodifferentcategoriesaccordingtotheirpositions

asillustratedinTable1 Themachinetoolissubjectedtocontinuouslychanging oper-ationconditions Itis rarelymaintainedatsteady stateandthe heatgeneratedinternallywillvarysignificantlyasthespindle rota-tionspeedischanged.Whenthisiscombinedwiththeeffectof ambientchanges,theresultisthecomplexthermalbehaviourof themachine.Fivenon-contactdisplacementtransducers(NCDTs) are used to measure the displacement of a precision test bar,

0 10 20 30 40 50 60 0

5 10 15

Time (Minutes)

Temperature sensor T11 (Test II, 4000 rpm) Temperature sensor T11 (Test III, 4000 rpm) Temperature sensor T11 (Test V, 8000 rpm) Temperature sensor T11 (Test VI, 8000 rpm)

0 10 20 30 40 50 60

20

22

24

26

28

30

32

34

36

38

40

Time (Minutes)

Temperature sensor T11 (Test II, 4000 rpm) Temperature sensor T11 (Test III, 4000 rpm) Temperature sensor T11 (Test V, 8000 rpm) Temperature sensor T11 (Test VI, 8000 rpm)

Fig 4.(a) Absolute temperature of the selected sensor in different tests (b) Magnitude of temperature changes in different tests.

The location of the temperature sensors.

8–32 Strip 1 Sensors (placed on the carrier)

33–61 Strip 2 Sensors (placed on the carrier)

representingthetool,intheX,YandZaxes.Theconfigurationis

showninFig.3

Inthiswork,avarietyofheatingandcoolingtestsarecarried

outindifferentambientconditionsanddifferentspindlespeedsof

theVMC(seeTable2).Briefappraisalofthemethodologyshows

thevariationconsideredinthisstudy.ComparingTestIandTestVI

showsthatahigherspindlerotationspeedcausesalargerthermal

errorforthesametimeduration.WhereascomparingTestIIwith

TestIIIandTestVwithTestVI,itcanbeseenthatthesamespindle

rotationspeed,andthesametimeduration,gaverisetodifferent

thermalerror.Thiswasduetochangeoftheambientconditionsand

hysteresiseffect.Moredetailofthesedifferencescanbeobserved

byexaminingaselectedtemperaturesensoronthespindlecarrier

(T11);Fig.4(a)showsdifferentinitialconditionsofthemachineand

Fig.4(b)showsthedifferentmagnitudeoftemperaturechangesin

differenttests.Anexampleofheatingandcoolingtestisillustrated

asfollows:theverticalmillingmachinewasexaminedbyrunning

atitshighestspindlespeedof8000rpmfor1htoexcitethelargest

thermalbehaviour.Thetemperaturesensorsattheselectedpoints

-20 0 20 40 60 80

Time (Minutes)

X axis

Y axis

Z axis

Fig 5. Thermal drift of the spindle (spindle speed 8000 rpm).

onthemachinetoolandthethermaldisplacementofthespindleare measuredsimultaneously;thethermaldisplacementofthevertical millingmachineisshowninFig.5.Themaximumdisplacementof theX-axisis3␮m,theY-axisis79␮mandtheZ-axisis22␮m.The X-axisthermaldisplacementismuchsmallerthanthatoftheY-axis andtheZ-axisduetothemechanicalsymmetryofthemachineand thereforeisnotinvestigatefurtherinthispaper;onlytheY-axisand Z-axiserrorsareconsidered

4.2 Influenceweightingofsensorsatvariouscriticalpoints Theselectionoftemperaturevariablesisakeyfactortothe accu-racyofthethermalerrormodel,whichwillbeadverselyaffected

The various heating and cooling tests.

Y-direction (␮m)

Test name

Table 3

The clustering result.

GROUP 5 T1–T3, T6, T7, T64–T67, T70, T71, T73–T75

ifthereisinsufficientcoverageofthetemperaturedistribution.At

thesametime,thecalibration/trainingtimeandtherelativecost

ofthesystemwillincreaseifthenumberofinputvariablesislarge

Therefore,thelocationofsuitabletemperaturesensorsshouldbe

determinedbeforethemodellingprocess

Byapplying theGrey modelGM (1,N) ontheexperimental

datafromone of abovementioned tests(Test VI),theinfluence

coefficientscanbeobtainedasfollows:

SupposethatT1∼T76representsthemajorvariables(inputs)

x(0)2 ∼x(0)

n and the measurement of the NCDT sensors in the

Y-direction is the target variable (output) x(0)1 The influence

coefficientscanbeobtainedbyEq.(13),as b2 ∼ b76 .Thegreater

theinfluenceweight,thegreatertheimpactonthethermalerror,

and themore likely it is that the temperature variablecan be

regardedasapossiblemodellingvariable

Next,theinfluenceweightingsareclusteredtofiveclustersby

usingfuzzyc-meansclusteringanalysis(seeTable3).Afterward,

onesensorfromeachclusterisselectedaccordingtoitsinfluence

weightwiththethermaldisplacementtorepresentthe

tempera-turesensorsofthesamecategory.InthiscasetheyareT18,T55,T63,

T68andT71.Thesetemperaturesensorsarelocatedonthespindle

carrier(Strip1andStrip2),spindleboss,ambientnearthecolumn,

andambientnearthebase,respectively

Forthepurposeofcomparison,anothertestwascarriedouton

thewell-knownk-meansclustering.Thesoftclusteringapproach

producesmorereasonableresultsthanthehardclustering

How-ever,FCMrequiresmoreiterationsthank-means,becauseofthe

fuzzycalculations

4.3 ANFISmodelsdesign

Oneofthemainconcernswithdesigningathermalerror

com-pensationmodelusingANFIS,oranyotherself-learningalgorithm,

iswhetherthetrainingdatathatwasmeasuredatoneparticular

operatingconditionoftheCNCmachinetoolwouldbesufficient

totrainthemodelfullyforotheroperationalconditions.Inother

words,isthemeasureddatasufficientforthemodeltobeapplicable

foralloperatingconditions?

Ideally,anANFISmodelistrainedbyatrainingsetthatincludes

manytrainingpairscollectedfromalllikelyconditions.However,

therecostofmachinedowntimetocapturethetrainingdataisa

significantconcern,becausetheimpactonproductivitycanhavea

highpenalty.Forthisreason,reducingthenumberoftrainingpairs

requiredisveryattractive

TestIVwasconsideredtovalidatethemethodofreducingthe

numberoftrainingcycles.Measurementsofthermalerrorand

cor-respondingtemperatureswererecordedwhilethemachinewas

runthrougharangeofdutycycleasfollows:Itwasallowedtorun

atspindlespeed4000rpmfor120min,andthenpausedfor60min

beforerunningforanother120min;andthenstoppedfor180min

Hence,thedataobtainedfromthistestisdividedintothreeparts

whichweretraining,checking,andtestingdataset.Thechecking

datasetwasusedforover-fittingmodelvalidation,whilethe

test-ingdatasetwasusedtoverifytheaccuracyandtheeffectivenessof

thetrainedmodel

FivetemperaturesensorsfromSection4.2wereusedasinput

variablestothemodelsandthethermaldisplacementinthe

Y-directionwaschosenasatargetvariable.TheGaussianfunctions

Performance of ANFIS-FCM models with various numbers of n c Models Number of

clusters (n c )

Convergence epochs

RMSE of testing dataset

Table 5

Linguistic rules.

Linguistic rules

1 If (T18 is T18cluster1) and (T55 is T55cluster1) and (T63 is T63cluster1) and (T68 is T68cluster1) and (T71 is T71cluster1) then (out1 is out1cluster1)

2 If (T18 is T18cluster2) and (T55 is T55cluster2) and (T63 is T63cluster2) and (T68 is T68cluster2) and (T71 is T71cluster2) then (out1 is out1cluster2)

3 If (T18 is T18cluster3) and (T55 is T55cluster3) and (T63 is T63cluster3) and (T68 is T68cluster3) and (T71 is T71cluster3) then (out1 is out1cluster3)

areusedtodescribethemembershipdegreeoftheseinputs,dueto theiradvantagesofbeingsmoothandnon-zeroateachpoint[8] AftersettingtheinitialparametervaluesintheANFISmodels,the inputmembershipfunctionswereadjustedusingahybridlearning scheme

Extensivesimulationswereconductedtodeterminethe opti-mumstructure oftheFISmodelsthrough variousexperiments TheoptimalnumberofMFswasdeterminedbyassigning differ-entnumbersofMFsfortheANFIS-Gridmodel,anddifferentvalues

tothenumberofclusters(nc)fortheANFIS-FCMmodel, respec-tively.ToofewMFswillnotallowanANFISmodeltobemapped well.However,toomanyMFswillincreasethedifficultyof train-ingandwillleadtoover-fittingormemorisingundesirableinputs suchasnoise.Thepredictionerrorsweremeasuredseparatelyfor eachmodelusingtherootmeansquareerror(RMSE)indexwith thetestingdataset.Anexampleofselectingtheoptimumstructure fortheANFIS-FCMmodelispresentedasfollows:

Inthismodellingmethod,theoptimumsizeoftheFISmodelwas determined,andtheresultsareshowninTable4.Differentnumbers

ofepochswereselectedforeachmodelbecausethetrainingprocess onlyneedstobecarriedoutuntiltheerrorsconverge.Ascanbe seeninTable4,itcannotsimplybestatedthatbetterresultswill

beobtainedwithmoreclusters.ItwasfoundthattheFISmodel withthree(nc=3)clustersexhibitedthelowestRMSEvalue(1.7)for thetestingdataset.Consequently,thisFISmodelwiththreerules wasconsideredtobetheoptimal.Thecorrespondingrulesofthe optimummodelareprovidedinTable5

Similarly,theoptimumFISmodel forANFIS-Gridmodel was determinedbyarbitrarilyvaryingthenumberofMFsfrom2to4 TheFISmodelwiththreeMFsperinput(243rules)wasfoundto

betheoptimum

In this section,theaim is tousethestructure of theANFIS modelsdescribedintheprevioussectiontoderiveathermalerror compensationsystem Withthepurposeof evaluating the pre-dictionperformanceofthemodelsgeneratedusingdatasetTest

IV,theremainingdatasetsTestI,TestII,TestIII,TestV,andTest

VI were used to run the models The experimentaltests were carriedoutthroughoutdifferenttime durations,different ambi-enttemperaturesand differentspindlerotation speedsinorder

tovalidatetherobustnessofthemodellingmethod.The perfor-manceofthemodelsusedinthisstudywascomputedusingthree performance criteria,including root-mean-square error(RMSE), correlationcoefficient(R2)andalsotheresidualvalue

0 20 40 60 80 100 120 -5

0 5 10 15 20 25 30 35 40

Time (Minutes)

ANFIS-FCM Thermal Error Residual value

0 20 40 60 80 100 120

-5

0

5

10

15

20

25

30

35

40

Time (Minutes)

ANFIS-GRID Thermal Error Residual value

Fig 6.(a) ANFIS-Grid model output vs the actual thermal drift (b) ANFIS-FCM model output vs the actual thermal drift (2 h, Test I).

-5 0 5 10 15 20 25 30 35 40

Time (Minutes)

ANFIS-FCM Thermal Error Residual value

-5

0

5

10

15

20

25

30

35

40

Time (Minutes)

ANFIS-GRID Thermal Error Residual value

Fig 7.ANFIS-Grid model output vs the actual thermal drift (b) ANFIS-FCM model output vs the actual thermal drift (5 h, Test II).

5.1 Samespindlespeedunderdifferentoperationconditions

ThepredictionmodelsestablishedusingthedatasetfromTest

IVareusedtoforecastthethermalerrorofTestI,TestII,andTest

III,respectively.Inallexperiments,themachinewasexaminedby

runningthespindleataspeedof4000rpm,butthedurationand

ambienttemperatureisdifferentbetweeneachtestanddifferent fromthetrainingdata,asillustratedinTable2.Thisis representa-tiveofamachinethatmanufacturessimilarparts,butinvarying factoryconditions.Thetemperaturesensorsattheselectedpoints

onthemachinetoolandthethermaldisplacementofthetestbar aremeasuredsimultaneously

0 50 100 150 200 250 300 -5

0 5 10 15 20 25 30 35 40

Time (Minutes)

ANFIS-FCM Thermal Error Residual value

0 50 100 150 200 250 300

-5

0

5

10

15

20

25

30

35

40

Time (Minutes)

ANFIS-GRID Thermal Error Residual value

0 20 40 60 80 100 120 0

10 20 30 40 50 60 70

Time (Minutes)

ANFIS-FCM Thermal Error Residual value

0 20 40 60 80 100 120

0

10

20

30

40

50

60

70

Time (Minutes)

ANFIS-GRID Thermal Error Residual value

Fig 9. ANFIS-Grid model output vs the actual thermal drift (b) ANFIS-FCM model output vs the actual thermal drift (5 h, Test V).

0 20 40 60 80 100 120 140

-10

0

10

20

30

40

50

60

70

80

Time (Minutes)

ANFIS-GRID Thermal Error Residua l value

0 20 40 60 80 100 120 140 -10

0 10 20 30 40 50 60 70 80

Time (Minutes)

ANFIS-FCM Ther mal Err or Residua l value

Fig 10.ANFIS-Grid model output vs the actual thermal drift (b) ANFIS-FCM model output vs the actual thermal drift (5 h, Test VI).

PredictiveresultsforthethreetestsusingANFIS-Gridmodeland

ANFIS-FCMmodelareshowninFigs.6–8.Resultsshowthatthese

twomodelsarecompetitive.Theperformanceofeachofthetwo

thermalpredictionmodelsispresentedinTable6.Theybothcan

predictthenewobservationsandreducetheresidualvaluetoless than±5␮mforeachtest.ItisclearthattheANFIS-FCMmodelhas

asmallerRMSE,residualvalueandhighercorrelationcoefficient thantheANFIS-Gridmodel

Performance calculation of the used models.

Table 7

Performance calculation of the used models.

5.2 Differentspindlespeedunderdifferentoperationconditions

ThepredictionmodelsestablishedusingthedatasetfromTest

IV were furthertested to represent a machine that has

differ-entmanufacturingparameters,alsoinvaryingfactoryconditions

Themachine wasrun at itshighest spindle speed of8000rpm

for one hourto excite more thermal response than duringthe

trainingdata,andthenpausedforanotherhourforcooling(see

TestVandTestVI).PredictiveresultsusingtheANFIS-Gridmodel

and ANFIS-FCMmodelare shownin Figs 9and 10 The

evalu-ation criteriavalues areprovided inTable7.Theresidualerror

obtainedusingtheANFIS-FCMmodelwasagainbetterthanthe

ANFIS-Gridmodel.Inaddition,theANFIS-FCMmodelhasalower

RMSEandslightlyhighercorrelationcoefficientthanthe

ANFIS-Gridmodel.ThisindicatesthattheANFIS-FCMmodelisa good

modellingchoiceforpredictingthethermalerrorofthemachine

tools

Thispaperproposesathermalerrormodellingmethodbased

ontheadaptive neuro fuzzyinferencesystem(ANFIS) in order

toestablishtherelationshipbetweenthethermalerrorsandthe

temperaturechanges.Theproposedmethodologyhastheability

toprovidea simple,transparentandrobustthermal error

com-pensationsystem.Ithastheadvantagesoffuzzylogictheoryand

the learning ability of theartificial neural network in a single

system.Theoptimallocationsforthetemperaturesensorswere

determinedthroughtheGreymodelandfuzzyc-means

cluster-ing.Afterclusteringintogroups,one sensorfromeachgroupis

selectedaccordingtoitsinfluencecoefficientvaluewiththe

ther-mal drift By this method, the number of temperature sensors

wasreducedfrom76possiblelocationstofive,whichsignificantly

minimisedthecomputationaltime,costandeffectofsensor

uncer-tainty

TwotypesofANFISmodelhavebeendiscussedinthispaper:

usinggrid-partitioningandusingfuzzyc-meansclustering.Both

modelswereconstructedandtestedonaCNCmillingmachine.The

resultsfromthetwosetsofvalidationtestsshowthatboth

ANFIS-basedmodels,derivedfromasingleheating-and-coolingcycle,can

improvetheaccuracyofthemachinetoolbyover80%for

vary-ingambientconditions,heatingdurationsandspindlespeeds.The

ANFIC-FCMproducedbetterresults,achievingupto94%

improve-ment in error with a maximum residual error of ±4␮m This

compares favourably with other compensation methods based

uponparametricorself-learningtechniques,suchassimilartests

bytheauthorsusingartificialneuralnetworks[8],asdiscussedin

Section1

Inadditiontothebetterabsoluteaccuracy,theANFIS-FCMhas

beenshowntohavetheadvantageofrequiringfewerrules,inthis

case requiringonlythree rules asopposed tothe243foundto

beoptimalfortheANFIS-Gridmodel.Thisisasignificantbenefit,

since the latter methodis significantly more laborious to

con-struct

Therefore,it canbeconcluded thattheANFIS-FCMmodel is

avalidandpromisingalternativeforpredictingthermalerrorof

machinetoolswithoutincreasingcomputationoverheads

Acknowledgements

TheauthorsgratefullyacknowledgetheUK’sEngineeringand

PhysicalSciencesResearchCouncil(EPSRC)fundingoftheEPSRC

CentreforInnovativeManufacturinginAdvancedMetrology(Grant

Ref:EP/I033424/1)

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