Error analysis and compensation for the

IntJAdvManufTechnol(2004)23:495–500DOI10.1007/s00170-003-1662-6

ORIGINALARTICLE

S.-H.YangÆK.-H.KimÆY.K.ParkÆS.-G.Lee

Erroranalysisandcompensationforthevolumetricerrorsofaverticalmachiningcentreusingahemisphericalhelixballbartest

Received:24July2002/Accepted:29January2003/Publishedonline:14February2004ÓSpringer-VerlagLondonLimited2004

AbstractMachiningaccuracyisdirectlyin uencedbythequasi-staticerrorsofamachinetool.Sincemachineer-rorshaveadirecte ectuponboththesurface nishandgeometricshapeofthe nishedworkpiece,itisimpera-tivetomeasurethemachineerrorsandtocompensateforthem.Alasermeasurementsystemtoidentifygeometricerrorsofamachinetoolhasdisadvantages,suchasahighcost,alongcalibrationtimeandtheusageofavolumetricerrorsynthesismodel.Inthisstudy,wepro-posedanovelanalysisofthegeometricerrorsofamachinetoolusingaballbartestwithoutusingacom-plicatederrorsynthesismodel.Also,astatisticalanalysismethodwasemployedtoderivegeometricerrorsusingahemisphericalhelixballbartest.Accordingtotheexperimentalresult,weobservedthatgeometricerrorsoftheverticalmachiningcentrewerecompensatedby88%.KeywordsMachinetoolerrorsÆVolumetricerrorsynthesismodelÆErrorcompensationÆHemisphericalhelixballbartestÆRegressionanalysis

1Introduction

Asthegeometryandfunctionsofaproductbecomecomplicatedanddiverse,theaccuracyofamachinetoolandthee ciencyofamachiningprocesshavebecomecrucialfactors.Accuracyisthemostsigni cantelementinevaluatingthecapabilityofamachinetool.FortheimprovedcapabilityofaCNCmachinetool,therefore,

S.-H.Yang(&)ÆK.-H.KimSchoolofMechanicalEng.,

KyungpookNationalUniversityE-mail:syang@knu.ac.kr

Y.K.Park

SchoolofMechanicalandAutomotiveEng.,CatholicUniversityofDaegu

S.-G.Lee

CollegeofMechanicalandIndustrialSystemEngineering,KyungHeeUniversity

anerrorcompensationtechniqueisaverye ectivemethod.Thus,thedevelopmentofanaccurateandeconomicerrormeasurementsystemisessential.

Quasi-staticerrors,a ectingthedimensionalaccu-racyorthetoleranceofamachinedproduct,occupy40–70%ofmachinetoolerrors.[1]Quasi-staticerrorsarede nedastherelativepositionerrorsbetweentheworkpieceandthetool.Geometricerrorsandthermalerrorsofamachinetoolbelongtoquasi-staticerrorswhichareassociatedwiththestructureofthemachinetoolitself.Thegeometricerrorsofamachinetoolareduetotheshapeerrorofmechanicalelementswhicharekinematicallylinked,andthemisalignmentofmechan-icalelements.Intheend,theerrorisexpressedasavolumetricerror;thediscrepancyofthetoolandtheworkpiece,atthetipofatool.

Geometricerrorsofamachinetoolcanbemeasuredbyalasersystem.Itsprimaryforteistheaccuracyoferrormeasurement.However,thelasersystemrequiresalargecapitalinvestmentduetoitshighcost,andtheinstallationiscomplicated.Also,di cultiesinmea-surementandthusaconsiderableamountoftimeforeachmeasurementareunavoidable.Finally,anexperi-encedandskilfuloperatorisabsolutelyrequired.In1982,toovercometheseproblemsassociatedwiththelasersystem,Bryan[2,3,4]proposedthetelescopingballbarsystemfortheaccuracyevaluationofamachinetool.Since1982,theballbarsystemhasbeenfrequentlyadoptedinmanyresearches[5,6,7,8,9,10,11,12,13].Theballbarsystemhasalsobeenwidelyappliedinindustrywheretheaccuracyofamachinetoolshouldbecheckedregularlyandfrequently.

Amathematicalmethodexpressingthevolumetricerrorofa3-axismachinetoolinavector,aderivationofamodelusing21geometricerrorcomponentsandamethodofmeasuringthe21errorcomponentswithalaserinterferometersystemanderrorcompensationhavebeenproposedbymanyresearchers.Also,thee ective-nessofthemethodshasbeenproved[9,10,15,16,17].However,theaforementionedmethodsevaluatetheaccuracyofamachinetoolestimateandcompensatefor

496

thevolumetricerrorsofamachinetoolusinganerrorsynthesismodelaftermeasuringtheerrorcomponentsofeachaxis.Therefore,insteadofdirectlyusingmeasureddataascompensatingvalues,theycouldcompensatethevolumetricerrorsonlyafterconstructingacomplexerrorsynthesismodel.Inaddition,alloftheerrorestimationmethodsusingaballbarhavefocusedoneacherrorcomponentusingmeasureddata[11,12,13,14].

Barakatcompensatedfortheerrorsofa3-dimen-sionalcoordinatemeasuringmachine(CMM)usingasimpli ingastandardstatisticaltechnique,Yangetal.[17]showedthatonlylineardis-placementerrorsandsquarenesserrorsamong21errorcomponentshavemajore ectsonthevolumetricerrors.Also,theyproposedagenuine3-dimensionalballbarmeasurementinplaceofconventionalplanemeasurements,andthustheyreducedmeasurementtimeandsetuperrors.Inthisresearchwork,aballbartestwithahelicalpathonahemisphereisemployedtoeasilyestimatethevolumetricerrors.Thus,thegeo-metricerrorsofaverticalmachiningcentrecanbeobtainedwitheaseinindustry.Multipleregressionanalysiswasemployedandtheerrorateachpointwascompensatedaccordingtopositioncompensation.Asaresult,about88%compensationwasachieved.

2Thevolumetricerrorsynthesismodelofaverticalmachiningcentre

2.1Thederivationofavolumetricerrorsynthesismodel

Thechieferrorsamongtheerrorsofamachinetoolarequasi-staticerrors(geometricandthermalerrors).Thegeometricerrorsofamachinetoolarefroma awofthemachinetoolitselfinmanufacturing,amisalignmentofmechanicalelements,staticdeformationand/orwear.Thermalerrorsarefromthermaldistortionsofmachinecomponentsassociatedwithheatsources,suchasmo-tors,hydraulicsystems,bearingsandambienttempera-ture.Theseerrors,eitheraloneortogether,a ectthe nalmachinetoolerrors.

Theoretically,a3-axismachinetoolcontains21geometricerrorcomponents:3translationerrorsand3rotationerrorsof3axes,and3squarenesserrorsintotalbetweenthreesetsofpairaxes.Thevolumetricerrorsynthesismodelcompoundingallerrorcomponentsofaverticalmachinetoolisderivedasthefollowingformula,andwasillustratedindetailbyYangetal.[15].Dx¼DxtþDxg¼DSxÀDyxxþDyzx

ÀdxxþezxÀÀdxyþdxzÀeysLþezzOyzyÀOyxyþTyÀyÁTyþeyzðTzÀLÞ

þezyÀOyzyþTyÀyÁþeyxðOyxzÀOyzzÀTzþLÀzÞþeyyðLÀOyzzÀTzÀzÞÀSzxz

Dy¼DytþDyg¼DSyÀDyxyþDyzyÀdyxÀdyy

þdyzþexsLþezyðÀOyzxÀTxÞþezzTxþexzðLÀTzÞþezxðOyxxÀOyzxÀTxþxÞþexyðOyzzþTzÀLþzÞþexxðOyzzÀOyxzþTzÀLþzÞþSxyxÀSyzzDz¼DztþDzg¼DSzÀDyxzþDyzzÀdzxÀdzyþdzz

ÀeyzTxþeyyðOyzxþTxÞþexzTy

þeyxðOyzxÀOyxxþTxÀxÞþexyÀÀOyzyÀTyþy

Á

þexxÀOyxyÀOyzyÀTyþy

Áð1Þ

where:Dx,Dy,Dz:

thecoordinatesofworkpieceintheX-axisslidesystemC

Dxg,Dyg,Dzg:thegeometricerrorsofama-chinetoolineachdirectiondxx,dyy,dzz:thelineardisplacementerrors

alongtheX-,Y-,andZ-axes

dyx,dzx,dxy,dzy,dxy,dyz:thestraightnesserrorsOyxx,Oyxy,Oyxz:thedistancesofX-axisoriginOx

relativetoY-axisoriginOyinthreedirections

Oyzx,Oyzy,Oyzz:thedistancesoftheZ-axisorigin

OzrelativetoY-axisoriginOyinthreedirections

exx,eyy,ezz:therollerrorsalongX-,Y-,and

Z-axes

eyx,ezx,exy,ezy,exz,eyz,:theangularerrorsSxy,Syz,Sxz:thesquarenesserrorsbetween

thepairaxes(oraxispairs)

Dxt,Dyt,Dzt:thethermalerrorsofamachine

toolineachdirection

DSx,DSy,DSz:thespindledriftsatstandard-lengthtooltipalongX-,Y-,and

Z-axes

exs,eys:thespindletiltsalongXandY

axes

Dyxx,Dyxy,Dyxz:thethermaldriftsoforiginOx

relativetoOyinX,YandZdirections

Dyzx,Dyzy,Dyzz:thethermaldriftsoforiginOz

relativetoOyinX,YandZdirections

Tx,Ty,Tz:thecoordinatesofstandard-lengthtooltipinspindlecarrier

coordinatesystemD

L:thetooloffsetdistancewhena

toollengthischanged

x,y,z:eachtraveldistancesofslidein

X-,Y-andZ-axescoordinatesystem

497

2.2Thederivationofa3-dimensionalballbarequation

A3-dimensionalballbarmeasurementofahemispher-icalhelixincludingallthemotionsofx,y,z-axes(di erentfromconventionalballbartestsmeasuringonly2-dimensionalplanes)wereperformed.Consideringthe3-dimensionalpositionofaballbar,akinematicequationincludinggeometricerrorofeachpointwasderived(Figs.1,2).Although(x,y,z)isthepositionoftheballbarsystemwith(x0,y0,z0)astheoriginandra-diusR,therealÀpositionÁoftheballbarsystembecomes000

(x¢,y¢,z¢)withx0;y0;z0astheoriginbecausethegeo-metricerrorisinvolved.

Theradiuserror,DRmeasuredbythelinearvariabledi erentialtransformer(LVDT)becomes

qÀÁ2ÀÁ2ÀÁ2 0000000DR¼RÀR¼xÀx0þyÀy0þzÀz0ÀR

ð2Þ

and

SquaringbothsidesofEq.3andneglectingthesec-ondandhigher-ordertermsproduces:RDR¼ðxÀx0ÞðDxÀDx0ÞþðyÀy0ÞðDyÀDy0Þ

þðzÀz0ÞðDzÀDz0Þ2.3Statisticalanalysis

Thevolumetricerrorcanbeexpressedasafunctionofthesliderpositiononlyineachaxis.Therefore,multipleregressionanalysisisusedtoexpressEq.1asaregres-sionmodel.

Dx¼a1Xþa2Zþa3

Dy¼b1Xþb2Yþb3Zþb4Dz¼c1Zþc2

SubstitutingEq.5intoEq.4andexpressingitasamatrix,wecanobtainalinearequationinEq.6:b¼Ax

ð6Þð5Þð4Þ

qRþDR¼ðxÀx0þDxÀDx0Þ2þðyÀy0þDyÀDy0Þ2þðzÀz0þDzÀDz0Þ2

where:bT¼½RDRi

ð3

Þ

Fig.1Amachinecoordinate

system

3

ðxiÀx0Þ2

6ðxiÀx0ÞðziÀz0Þ7

76

76yðÀyÞðxÀxÞi0i07AT¼676ðyiÀy0Þ2

76

4ðyÀyÞðzÀzÞ5

i0i0

2

ðziÀz0Þ23a16a27676b17

7x¼66b27674b35a1

2b:A:x:i:

Ann-dimensionalradiuserrorvectoroftheballbarmeasurement

Ann·6matrixdenotingthepositionofmeasure-ment

coef cientvectorofamodel

1,2,....,npositionsofmeasurementsupton

Usingtheleastsquaremethod,themodelcoe cientvectorxcanbeestimatedinEq.7:

ÀÁÀ1^x¼ATAATbð7Þ

Fig.2Aschematicdiagramfortheballbartest

InEq.7,constants(a3,b4,c2)cannotbeobtainedbe-causetheballbariseliminated,beingexpressedasa

498

combinationofrelativepositionerrors.However,sincethereisnoerrorattheoriginofamachinetool,itdoesnotloseanygenerality.

3Theexperimentsandcompensation3.1Theexperimentalmethods

Experimentsusingtheballbarsystemproceededasfollows.

1.SetpointÀ,X=300mmandY=300mm,onthebedastheoriginforballbarmeasurement(Fig.3).2.Withtheradiusof250mm,proceedwithahemi-sphericalhelixballbarmeasurement(Fig.4).

3.Measureerrorvaluesat32pointsalongthehelicaltoolpathonthehemisphere.

4.Repeatsteps1.through3.aswechangetheoriginfromÀto`and´.(Fig.3)PointsÀand`areforerrorestimation,andpoint´istoverifyexperi-mentalresultsandtocompensatetheerrorsbasedontheestimatederrorsbyÀand`.

5.CalculatexinEq.7usingDRofthemeasuredpoints.Averticalmachiningcentre(KIAKV60)wasused.Also,theexperimentswereperformedunderthecoolconditiontoguaranteethatonlygeometricerrorsshouldbemeasuredwithoutanythermalerrors.

Fig.3Thesetupoftheoriginpointintheballbartest

Fig.4Ameasurementpointalongthehemispherical

helix

Table1Resultsofthegeometricerroranalysis

Thevalueofthemodelparameter

aa1Àb2À1.3844EÀ05b1À3.98543EÀ041.81592E2.17282EÀÀ0405b22.73096EÀ04c31

À7.60534EÀ

05

3.2Theexperimentalresults

Equation7wassolvedusingDRobtainedineachballbartest,andparametersoftheregressionmodela1,a2,b1,b2,b3andc1were nallyobtained.Table1showstheaveragevaluesfromthetwoexperimentswiththeoriginÀand`,respectively.

ToevaluatethevalidityofthevaluesinTable1,wecanmeasurethegeometricerrorsusingalasermeasurementsystemandcomparethemwiththevaluesinTable1.AnalternativeisthatafterwepredictarbitraryerrorsinanimaginaryballbartestusingthevaluesinTable1andtheregressionmodel,wecancomparetheerrorswithrealerrorsbytheactualballbarmeasurement.Asanexample,thevolumetricerrorsofaballbartestwithanoriginof(850,300)werepredictedusingTable1andtheregressionmodel.Thesepredictederrorswerecomparedwiththemeasuredvaluesfromarealballbartest,andtheresultisshowninFig.5andTable2.Theaveragediscrepancyislessthan10lmasinTable2,anditshowsthatthevaluesfromthesimpli edmodelarevalid.3.3Compensationandanalysis

ThevolumetricerrorwasobtainedusingtheparametersinTable1andEq.5.Basedonthis,Dx0,Dy0,Dz0at

the

Fig.5MeasuredvaluesfromarealballbartestTable2Asummaryofthepredictionresults

Absoluteresidualoftwovalues

Maximumerror(mm)0.0187Meanerror(mm)

0.0085

Table3Neworiginpointsaftercompensation

OriginpointbeforeNeworiginpointcompensation

aftercompensationXposition(mm)850.000849.983Yposition(mm)300.000300.039Zposition(mm)

12.748

12.749

Fig.6Compensationvaluesofeach

axis

Fig.7Theresultusingthepositioncompensationmethod

originandDx,DyDzathemisphericalhelixballbarmeasurementscanbeestimated.Also,bysubtractingtheseerrorsfromtheorigin(850,300),wecan ndaneworigin,i.e.,thepositioncompensatedorigin(Table3).AlsoasinFig.6,newmeasurementpointswereob-tainedbycalculatingtheX,YandZ-axiscompensatedvaluesateachmeasurementpoint.Aftercompensatinginthisway,weperformedtheballbarmeasurementforthenewpoint(´).Figure7showstheresultofthepo-sitioncompensationmethod.

Theballbarmeasurementerrorsaftercompensationaregreatlyreducedcomparedwiththeerrorsbeforecompensation.Themaximumerrorof78.5°lmandtheaverageerrorof32.2°lmwerereducedtothemaximum

499

Table4Asummaryofthecompensationresults

Without

With

compensationcompensationMaximumerror(mm)—0.07850.0084(89.3%)absolutevaluesMeanerror(mm)—0.0322

0.0039

(87.9%)

absolutevalues

errorof8.4°lmandtheaverageerrorof4°lmwithcompensation.Inconclusion,weachieved88%oftheerrorreduction(Table4).

Tosummarisetheresults,wepredictedandmeasuredthegeometricerrorsofamachinetoolusingtheballbartest.Toovercometheshortcomingsofaconventionalballbartestfora2-dimensionalplanemeasurement,weusedtheballbartestofahemisphericalhelixandregressionanalysisfortheevaluationofgeometricer-rors.Asaresult,wecouldsuccessfullycompensatethemachinetoolerrors.

4Conclusions

Theobjectiveofthisresearchworkwastheerroranal-ysisofaverticalmachiningcentreusingthehemi-sphericalhelixballbartestandaregressionmodel.Basedontheresultsofexperiments,wesummarisetheconclusionsasfollows.

1.Avolumetricerrorsynthesismodelofa3-axisverti-calmachinetoolwasderivedusingahomogeneoustransformationmatrixandasimpli edregressionmodelwasproposed.

2.Insteadofahigh-pricedlasermeasurementsystem,asimplealgorithmtoestimatethegeometricerrorsofamachinetoolwasproposedusingtheconvenient3-dimensionalballbarmeasurementsystem.

3.Aregressionmodelcouldreplaceacompletevolu-metricerrorsynthesismodelforthegeometricerrorcompensationofamachinetool.Applyingtheregressionmodeltothehemisphericalhelixballbarmeasurement,wecouldalsodecreasethevolumetricerrorsofaverticalmachiningcentreby88%.

4.Withoutrelyingonacomplicatedvolumetricerrorsynthesismodel,wecandirectlyusetheballbarmeasurementvaluesascompensatedvalueswithasimpleregressionequation.

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