Improved Machine Tool Linear Axis Calibration Through Continuous Motion Data Capture

University of Huddersfield Repository

Miller, Jonathan, Longstaff, Andrew P., Parkinson, Simon and Fletcher, Simon

Improved Machine Tool Linear Axis Calibration Through Continuous Motion Data Capture

Original Citation

Miller, Jonathan, Longstaff, Andrew P., Parkinson, Simon and Fletcher, Simon (2017) Improved 

Machine Tool Linear Axis Calibration Through Continuous Motion Data Capture. Precision 

Engineering, 47. pp. 249­260. ISSN 0141­6359 

This version is available at http://eprints.hud.ac.uk/29386/

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ContentslistsavailableatScienceDirect

Precision

Engineering

j ou rn a l h om epa g e :w w w . e l s e v i e r . c o m / l o c a t e / p r e c i s i o n

Improved

machine

tool

linear

axis

calibration

through

continuous

motion

data

capture

J.E.

Miller

,

A.P.

Longstaff,

S.

Parkinson,

S.

Fletcher

CentreforPrecisionTechnologies,SchoolofComputingandEngineering,UniversityofHuddersfield,Queensgate,HuddersfieldHD13DH,UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received15February2016

Receivedinrevisedform25August2016 Accepted31August2016

Availableonline2September2016 Keywords:

Machinetoolcalibration Continuousmotionmeasurement Quasi-staticmeasurement Geometricerror Manufacturing

a

b

s

t

r

a

c

t

Machinetoolcalibrationisbecomingrecognisedasanimportantpartofthemanufacturingprocess.The currentinternationalstandardsformachinetoollinearaxescalibrationsupporttheuseofquasi-static calibrationtechniques.Thesetechniquescanbetimeconsumingbutmoreimportantlyacompromisein qualityduetothepracticalrestrictiononthespatialresolutionoftargetpositionsontheaxisundertest. Continuousmotioncalibrationtechniqueshavethepotentialtodramaticallyincreasecalibration qual-ity.Throughtakingseveralmeasurementvaluespersecondwhiletheaxisundertestisinmotion,itis possibletomeasureinfargreaterdetail.Furthermore,sincemachinetoolsnormallyoperateindynamic mode,thecalibrationdatacanbemorerepresentativeifitiscapturedwhilethemachineisinmotion.The drawbacktomeasuringtheaxiswhileinmotionisthepotentialincreaseinmeasurementuncertainty. Inthefollowingpaper,differentmethodsofcontinuousmotioncalibrationarediscussed.Atime-based continuousmotionsolutionisproposedaswellasanoveloptimisationandcorrelationalgorithmto accu-ratelyfusethedatatakenfromquasi-staticandcontinuousmotionmeasurements.Themeasurement methodallowsforminimalquasi-staticmeasurementstobetakenwhileusingacontinuousmotion mea-surementtoenhancethecalibrationprocesswithvirtuallynoadditionaltimeconstraints.Theproposed methoddoesnotrequireanyadditionalmachineinterfacing,makingitamorereadilyaccessiblesolution forwidespreadmachinetoolusethanothertechniqueswhichrequirehardwarelinkstotheCNC.The resultofwhichmeansashortercalibrationroutineandenhancedresults.Thequasi-staticand continu-ousmotionmeasurementsshowedcorrelationtowithin1␮matthequasi-staticmeasurementtargets. Anerrorof13␮mwasdetailedonthecontinuousmotion,butwasmissedusingthestandardtest.On alarger,lessaccuratemachine,thequasi-staticandcontinuousmotionmeasurementswereonaverage within3␮mofeachotherhowever,showedastandarddeviationof4␮mwhichislessthan1%ofthe overallerror.Finally,ahighfrequencycyclicerrorwasdetectedinthecontinuousmotionmeasurement butwasmissedinthequasi-staticmeasurement.

©2016TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Measurementofthegeometricerrorsofmachinetoolsis becom-ing a moreimportant partof maintaining thepartaccuracy of manufacturedcomponents.Themeasurement datacanbeused toevaluatecapabilitywhenbuyingamachineandtomonitorits performanceduringitsoperationallifetime.Furthermore,by mea-suring theerrors it is possibletoperform diagnostictasks and remedialcautionintheformofmechanicaladjustmentor numer-icalcompensation.Inorder formachinetool owners tobeable tomanufacturewithconfidence,theymusthaveassurancethat theirmachinesareworkingwithinarequiredtolerance.Forhigh

∗Correspondingauthor.

E-mailaddress:[email protected](J.E.Miller).

quantity,lowvalueproduction,thetolerancesarelikelytobelarger thanthatoflowquantity,highvalueproductionsuchasaerospace wheretolerancesareatamicrometrelevel.

Inordertoensurethatpartsmanufacturedarewithintolerance (rightfirsttimeproduction)ahighqualitycalibrationprocessmust beinplacetoensurethattheproductionmachineiscapable.The calibrationprocesswillencompassarangeofdifferenttechniques. Forexample,alaserinterferometermeasurementsystemcanbe usedtomeasuremanyofthegeometricerrorsshowninFig.1[1], whilelineardisplacementsensorsandartefactsareusedfor oth-ers.Thetelescopicballbarsystem[2]providesapopularwayof assessingmachinetoolperformanceduringcontouring.

Thereareatotalof21geometricerrors(Fig.1)forathreeaxis machinetoolsuchasthec-framemachinewhicharefound exten-sivelyinindustry(Fig.2).Machinetoolownerswillsystematically calibratetheirmachinesbased onmanufacturingrequirements. http://dx.doi.org/10.1016/j.precisioneng.2016.08.010

Fig.1. Six-degrees-of-freedomandsquarenesserror.

Thecurrentinternationalstandardsformachinetoolcalibrationof linearpositioningdeviationsdefinedbyISO[3]supportstheuseof quasi-staticcalibration.Quasi-staticmeasurement(QS-M)involves commandingthemachinetomovefrompointtopointalongthe lengthoftheaxisundertest.Themachineiscommandedtomove toatargetposition,whereitwillremainstationaryforanominal timeperiodwhileameasurementistaken.

Commandingthemachine tomovefrom point topoint and dwellinguntilthemachinehassettledistimeconsuming,a prob-lemthatisexacerbatedifafinespatialtargetintervalisselected, anexampleofwhichisprovidedinSection3.Manufacturerswant theirmachinestobeinoperation,minimisingmachinedown-time forthemeasurement process.Inorder toachievethis, acoarse targetinterval might beselected.However,this couldresult in someerrorsoftheaxisremainingundetectedduetotheeffects ofsignalaliasing;thetrueerrorformwillnotbesufficiently rep-resentedifthereisinsufficientspatialsamplerate.Furthermore, whenamachinetoolisinoperation,itislikelythattheaxiswill

Fig.2.Three-axismachinetool.

not bestationary. Dueto this, quasi-staticcalibrationcouldbe seenasaninadequatemethodformeasuringtheperformanceof machinetoolsduringtheirintendedoperation;itisareasonable compromisetoindicatemachinetoolcapabilitybutwillmisssome potentiallyvitalinformation.

Precisionmachiningcanbeimprovedbyapplyingnumerical compensation,forwhichtworequirementsmustbemet:precise errorpredictionandaccuratecompensation(correctionoferrorin themachine’scontroller)[4].AsQS-Mtechniquesdonotmeasure themachinetoolinitsusualmodeofoperationitisreasonableto arguethatthesetechniquesdonotentirelymeetthefirst require-ment,makingthesecondnotpossible.

Bytakingseveralreadingspersecondonatimebasiswhilethe axisisinmotion,continuousmotioncalibrationhasthepotential tosignificantlyreducetheoverallcalibrationtimewhileincreasing calibrationqualitybyremovingtheeffectsofsignalaliasing.

Duringquasi-staticcalibration,theCNCpartprogramprovides thenominalpositionsforthemeasurement.Theerroristhen sim-plycalculatedasthedifferencebetweenthenominalsfromthe partprogramandtheactualsmeasuredbythelaser.The techni-calchallengeforcontinuousmotionmeasurementsisintheability toconvertthemeasurement,whichistypicallyintheformofa displacementmeasurementinthetimedomain(linearpositioning measurement),toageometricerrorinthepositiondomain,since thereisnoexplicitnominalpositionwhenusingthismethod.

Itistheaimofthispapertodemonstratethecorrelationbetween thequasi-staticandcontinuousmotionerrorofmachinetoolsand howacontinuousmotionmeasurementcanbeusedtoenhancethe calibrationprocesswhilereducingtherequirednumberoftargets foraquasi-staticmeasurement.Differenttechniquesforcalculating thegeometricerrorfromalinearpositioningmeasurementhave beendevelopedandwillbediscussedinthefollowingsection.

2. Background

CNCmachinetoolscontainarange ofdifferenterrors.These errors can be defined as: kinematic errors;thermo-mechanical errors;errorsinducedbyloadsonthemachine;andcontinuous motionforces.Allofthesecontributetotheoverallgeometric per-formanceofthemachine[5,6].

Kinematicerrorsareduetothemachine’simperfectgeometry suchasaxismisalignmentanderrorsinthemachine’smeasuring system[5].Barakat[7]describeskinematicerrorsinrelationtoa coordinatemeasuringmachine(CMM)astheerrorappearingin theabilityoftheCMMtoreachtheexactspecifiedpositionbythe controller.Duetothesimilaritiesinthekinematicchainand homo-geneouscoordinatesystembetweenmachinetoolsandcoordinate measuringmachinesthesamedefinitioncanbeappliedtomachine tools.

Thermo−mechanicalerrors arecaused byinternal and exter-nalheatsourcesresultinginthermo-elasticdeformationsofthe machinetoolcausinggeometricalinaccuracies[8].Relative move-mentbetweenthevariouselementsofthemachinecausesheatto begeneratedatthecontactzonesanditisthisheatthatleadsto thedeformationofthemachineelements[9].Thermo-mechanical errorsaresaidtobethecauseof70%ofmachinetoolgeometric inaccuracy[10]

Loadsonthemachinearecausedbyinternalandexternalforces. Forexample,thelocationandweightoftheworkpiececouldaffect themachine’sangularprofileandsoimpairtheoverallaccuracy [5,11].

The effectof the three error sources mentioned exist when themachineisstationary.Dynamicerrorsaretheadditionalerrors occurringwhenthemachineismovingataprogrammedfeedrate. Thedynamicerrorsresultfromvaryingalterationsofthemachine

componentsunderdynamicforces[12]aswellaselectronicand control-relatederrors.

Asindustrystrivesforhigheraccuraciesandfasterproduction cycles,theimportanceofanefficientcalibrationprocessbecomes evenmorecritical[13,14].Machinetoolownerswillsystematically calibratetheirmachinesbasedontheirindividualrequirements andthecalibrationprocessissupportedbyarangeofliterature [15,3,16–18].

In order to reduce calibration times,new methods of mea-surement have been introduced. A displacement measurement approachwasintroducedtocombatthelengthoftimeitwouldtake toidentifythe21possiblegeometricerrorsofa3-axismachine.The methodinvolvestakingpositionmeasurementsalong15linesof themachineworkingzone,allowingtheindividualaxiserrorstobe inferred[19,20].Thismethodisfasterthantraditionaltechniques butisaquasi-staticmeasurementmethod.

A novel method of artefact probing was also introduced in order torapidlyverify theaccuracy of machine tools[21].The methodworksbyfixinga referenceartefactsomewherewithin themachine’sworkingvolume.Aftera fullcalibrationhasbeen performed,abaselineroutineisperformedwheretheartefactis probedandsetsmachinevariablesforeachpoint.Afterthis, when-evera rapid verification isneeded, themachine canperform a verification routingwhere the artefactis probedagain andthe deviationfromthebaselineresultisreported.Thismethodis effec-tiveinmeasuringasmallareaofthemachine’sworkingvolume. However,artefactmethodsdonotmeasuretheentiremachine,a problemparticularlyforlargemachines,orgivespatiallydetailed characterisationofalltheindividualerrors.Furthermore,artefact measurementatdifferentspeedstoevaluatedynamiceffectsisnot atrivialextrapolation.

The“Chase-the-ball”technique[22]deliversaquickcalibration forfiveaxismachinetools.Themeasurementsaretakenwhilethe machineisstoppedtoavoidbacklashinfluencescausedbya rever-salerrorwhenthemachinechangesdirectionofmotion.Adynamic R-testwasalsointroducedforthecalibrationofrotaryaxes[23]. Thesemethodsallusemultipleaxismovementsasawaytoreduce calibrationtime.

Continuous motion calibration techniques for individual machinetoolaxiscalibrationhavealsobeenexplored. Postleth-waite[24]developedatime-basedapproach.Thisapproachdoes notrequireknowledgeofthemachine’snominalpositionsinceit infersitfromthechangeinfeedrate.Thesystemworksbyfitting anendpointstraightlinethroughthemeasurementdatawhenthe machinewasestimatedtobeatanominallyconstantvelocity.The processhingesonthefactthatwhenamachineisprogrammedto moveataconstantfeedrate,anyerrorwillbemanifestedinsmall fluctuationsinaxisvelocity.Thismethoddoesnotprovidea fit-tingtechnique andthetechniqueproposedwillpotentiallylose thegradientcomponentoftheaxiserror.

ApositionbasedmeasurementsystemwasdevelopedbyCastro [25].Themethodofcalibrationusesthemachine’sencodersignal pulsesasthetriggerreference,thusenablingtotakemeasurements “onthefly”.Thesystemworksbyselectingtargetsatfine inter-vals.Providedthattheintervalisanexactmultipleoftheencoder, eachtimethemachinegetstotheintervalpositiontheencoderwill sendapulsetotheattachedcomputerinformingthemeasurement systemtotakeareading.

Thisworkhaslimitationswhichaffectitsuse.Forexample,in ordertoidentifycyclic errors,targetsmust bechosenat afine enoughincrement.However,ifthetargetintervalistoosmallthe systemwillgointoerror.Itisworthnotingthatinthetestscarried out,atargetintervalof1mmwassufficienttopickupcyclicerrors. Ontheflycalibrationhasalsobeenachievedthroughtheuseof atrackinginterferometer.Schwenke[26]usedtheanalogue sig-nalsfromthemeasurement system(encoder/scales), wherethe

analoguedatafromallfiveaxesaredigitizedbycountercardsand processedinarealtimeoperatingsystem.

Thistechniquehastheabilitytospeedupthecalibrationprocess aswellassignificantlyincreasingthemeasurementpointdensity. Thetechnicalchallengewiththismethodofmeasurementisthe synchronous acquisitionof themachine’saxis positionand the measurement signal[26].Furthermore,suchtechniquesrely on specialisthardwareandsoftwareaswellasinvasivemodification ofthemachine’sfeedbackloop,thisis notalwayspracticaland requiresapotentiallyhightime-investmenttoinitiate.

The developmentof anauto-alignment laser interferometer, inwhichmultiplemachineaxescouldbemovedsimultaneously, allowsforfastercalibrationtimes[27].However,suchsystemsthat relyonmeasurementsandfeedbackfrommultipledeviceshavean increaseduncertaintyontheoverallmeasurementduetotiming errorbetweenthedifferentmeasurementdevicesortheindividual inaccuracyofsystemcomponents[28–30].

Whencalibratingamachinetool,itisalsoimportantto con-sidertherelationshipbetweenthedifferenterrorsonthemachine. Thekinematicerrorsofamachinetoolinfluencetheuncertaintyof measurementoftheothererrors.Forexample,straightness devi-ationsofalinearaxiscancompromisetheresultofasquareness measurement.Furthermore,whenmeasuringlinearpositioningof theX-axis,theresultswillbedependentonthatofthechosen posi-tionoftheotherstationaryaxes[31]duetotheinherentangular errormotionsoftheslidesandtheAbbéoffsetsinvolved[15].

Asconfirmedbytheontheflycalibrationtechnique[26], mea-suringcontinuouslywillprovidesignificantlymoremeasurement pointsandproduceadetailedmeasuredsignal.Thishasthe poten-tialtoallowforafargreateranalysisoftherelationshipsbetween thedifferentgeometricerrors.Additionally,theidentificationof anyerrorcausedbyelectricalcompensationwithinthemachine’s controllerwillbeclearlyvisible,aswilltheeffectsofthe compen-sationontheaxis.

Thefocusofthispaperistoidentifyanaccurateandreliable fit-tingtechniquetocontinuousdata.Itiswellknownthatcontinuous dataacquisitionprovidesafarhigherconcentrationof measure-mentpoints.However,thetechnicalchallengeisensuringthatthe measurementdataacquiredcanbereliablyconvertedintoerrorin thepositiondomainwithoutsecondarysignals.

3. Quasi-staticcalibration

Thetotalnumberoftargetpositionsselectedforaquasi-static measurementhasadirectinfluenceoverthequalityofthe mea-surement signal. ISO 230:2:2014 [3] requires a minimum of 5 targetspermeter for anaxisupto2min length.Dueto com-mercialpressure,machinetoolownerswanttomaximizemachine toolavailability. Itisthereforepossiblethatmanymachinetool ownerswillcalibratetheirmachinesaimingtomeettheminimum permittedstandard.

Fig.3showstheresultsofsuchameasurementperformedon athreeaxismachinetoolmovingfrom0mmto450mm.Whena coarsetargetresolutionischosen(90mmstepsize)the measure-mentsignalshowsbidirectionalpositioningerrorof8␮m.

Increasingthetotalnumberoftargetsto18andreducingthe spatialresolutionfrom90mmto25mmimprovesthe measure-mentsignaland showsbidirectionalpositioningerrorof16␮m. Notonlydoestherangeinerrorincrease,butthemeasurement signalstartstoshowacyclicerrorthatcouldnotbeenseeninthe previouslowresolutionmeasurementduetosignalaliasing.

Selectingafinespatialresolution(5mmstepsize)increasesthe numberoftargetpositionsdramatically(200targetspermeter). Selecting such a fine spatial resolution gives a highly detailed measurement signalthatshowsanothersmall increasein error

Fig.3.Quasi-staticmeasurementresolutioncomparison.

range,howevershowsa detailedpictureofthemachine’scyclic behaviour.

Amachinetoolcalibrationengineercould,forthesakeof expe-diency,takeonlythetestat90mmstepsize.Fromthis,theymay concludethattheaxisundertestisperformingwellanddecidenot tocompensate.Whenmeasuringata25mmstepsizewhatlooks tobeaballscrewpitchorcyclicerrorisbecomingevident,machine toolownersmaychoosetotryandcompensatethiserrororeven replacetheballscrew.Itisnotuntilameasurementata5mmstep sizeisselectedthatwecanclearlyseethattheerrorisunlikelytobe mechanical,butisduetonumericalcompensationinthemachine controller.Becausemechanicalerrorsareusuallygradual,sudden jumpsinaxispositionasseeninFig.3indicatesaneffectfroman electricalcontrolloop.Inthiscaseitwasfoundtobetheresults ofpoorlyappliednumericalcompensation.Suchbehaviourofthe axiscouldnotreadilybeidentifiedusingatraditionalquasi-static measurement,reinforcingthevalueoftheproposedtechniquesfor diagnosticpurposes.

InfactFig.3isfromapopulartypeofmachinetoolwhichhas averycommon(industrystandard)controllerwhichissoldinthe manythousands.Fromextensivetestingonthemachine,theshape oftheplotisnota“malfunction” assuch, butratheraproblem withthewaythenumericalcompensationisbeingappliedonthis seriesofmachines;thereisaresolutionlimitationwithinthe con-trollercoupledwiththeresponsivenessofthecontrolloop,that causesquantisationofthecompensatorymoves.Thisphenomenon illustratesafairlycommon,real-worldexampleofcompensation causingstepswhich arenotnormallypickedupbyquasi-static measurement. Thenew ISO/TR 16907:2015[32] recognises the importanceofthe“leastincrementstep”andwhilethepresented figuremaybeanextremeexample,itisbynomeansunusual.The issuehighlightedbythisexampleisthatmachinetooluserscanbe presentedwithQS-Mgraphsshowingexcellentperformance.Itis notuntilahightargetresolutionisselectedthatthetrue perfor-manceoftheaxisisuncovered.

Fig.3highlightsthelimitationsofchoosingacoarsetarget reso-lution.Choosinga5mmstepsizeprovidesahighlydetailedsignal, howevertookover1htocomplete.Torepeatsuchameasurement forallcomponentsofeachaxisofthemachinewouldnotbefeasible tomanymachinetoolownersduetotimeconstraints.

Thelengthoftimetocompleteaquasi-staticmeasurementcan beestimatedusingthefollowingformula

t=

wherelisthetotallengthofthemeasurement,fristheprogrammed feedrate,nisthetotalnumberoftargetsanddwisthedwelltime. Thisfunctiongivesanapproximatetimedurationfora measure-mentasitdoesnottakeintoaccountaccelerationordeceleration foreachtarget.

However,basedontheaboveformula(Eq.(1))selectinga5mm stepsize ata programmedfeedrate of500mm/min foran axis 500mminlengthwithadwelltimeof4sgivesatotaltimeofover 7minperdirection,completingfivebi-directionalrunstakesthe totaltime to76mincompared to23minand 13minfor25mm and90mmrespectively.Repeatingthis testforthetwoangular measurements(pitchandyaw)meansthatforasingleaxisa con-servativeestimateforthetotaltesttimeisnearly4h.

Itisunlikelythatsuchafinespatialresolutionwillbechosenfor aquasi-staticmeasurement.Iftheresultsofameasurementshow anunusualpattern,itispossiblethatcalibrationengineerswilldo anothermeasurementatafinerresolutioninordertodiagnosea cycliceffect.However,testingtheentiremachineatsuchahigh resolutionwillnotbefeasibleformanymachinetoolownersdue tothelengthoftimethatthemachinewouldbeoutofoperation. Furthermore,ambientandinternallygeneratedthermaleffectswill havesignificantlymoreinfluenceonmeasurementoversuchalong cycle.

4. Continuousmotioncalibration

Inthispaper,continuousmotionmeasurementsreferto mea-suringthemachinetoolaxisdisplacementcontinuouslyovertime. Themachine’snominalpositioncanbedefinedasthedesired posi-tionoftheaxiswithzeroerror.

Although highly effective, position-based measurement sys-tems that rely onspecialist hardware and software as wellas modificationofthemachine’sfeedbackloopareundesirableasa generalapproachduetothecostofimplementation,implications onwarrantiesiffeeddrivesystemsareinterruptedandtheneed foramachinespecificsolution.Thefocusofthisinvestigationwas thereforedevelopingatime-basedapproachthatcanbeused uni-versallyacrossallmachineswithnomodificationstothemachine feedbackloop.

Whenmeasuringcontinuouslyonlydisplacementismeasured. Themeasuredsignalwillcontaingeometricerror,howeverthis errorisnotvisiblefromadisplacementagainsttimegraph(Fig.4)

sinceitisdominatedbythechangeinnominalposition.Theerror iscalculatedbyfirstlyremovingperiodsofaccelerationand decel-erationfromthedisplacementsignal.Thisensuresthatthefitted lineonlyappliestothepartofthesignalwheretheaxisundertest wastravellingatanominallyconstantvelocity.Withacceleration ratesonmodernCNC’s,thelossofdataisnormallylessthan5mm ateachendoftravel.Althoughalimitation,itcanbearguedthat asitisunlikelyforpartstobemachinedinthisregionofthe work-ingvolumethereforeremovingthisperiodofthesignalwillhave minimaleffectonthecalibrationquality.Theoverallprocessisas follows;

1.Identifythesectionofdatacapturedwheretheaxisundertest isnominallymovingwithconstantvelocity.

2.Calculatetheconstantvelocitybylinearleastsquares(the gra-dientofthemeasurementwithrespecttotime).

3.Removethevelocitytoconverttothepositiondomain(removes progressiveerror).

4.Reconstructtheprogressive(scale)errorlostduringtheleast squarefittingusingacoarsequasi-staticmeasurement. 5.Applyasmall(sub-micron)shifttothecontinuousmotiondata

inordertodatumthedatawiththequasi-staticmeasurement.

4.1. Velocityalgorithm

Thealgorithmdevelopedinordertoidentifytheperiodofthe signalwheretheaxisundertestwastravellingatanominally con-stantvelocitypresentedanontrivialtask.Theprocesshingeson thefactthatthemachinetoolistraversingthelengthoftheaxisat anominallyconstantvelocity.Thefirststepofthealgorithmwas todifferentiatethedisplacementdatatoproduceavelocitygraph inthetimedomain.

v

ıt (2)

Themeanvelocityaswellasthedirectioniscalculated.Iftheaxis undertestistravellingatapositivevelocitythefirstseriesof sam-pleswithvalueslessthanthemeanofthewholesignalareremoved. Likewise,thelastseriesofsampleswhosevalueislessthanthe meanareremoved.Iftheaxisundertestismovingwithnegative velocitythenthesameprocessisappliedforsamplesgreaterthan themean.

Thisisarecursiveprocess,afterthefirstiterationandthefirst setsofsamplesareremoved.Anewmeanvalueiscalculatedas wellasthestandarddeviation.Duetothenoiseinthesignal,the machineis presumedtobeat aconstant velocity whenallthe samplesarewithintwostandarddeviations ofthemean.Ifthis conditionisnotmet,theprocessisrepeated,continually remov-ingpartsfromthebeginningandendofthesignaluntilthetest conditionismet.Once thetestconditionis met,datacaptured forafurther1msisremovedfromthestartandendofthe sig-naltoensurethatanyspikescausedbychangesinaccelerationare removed.

Theprocesshingesonthefactthatthemachineistraversingthe lengthoftheaxisatanominallyconstantvelocity.Furthermore,in somecasesthetestconditioncouldbechangedtomakethe algo-rithmmorerobust.Forhighprecisionmachines,thetestcondition couldhaveatightertolerancethantwostandarddeviationofthe mean.

4.2. Continuousmotionerror

Oncethepartofthesignalwheretheaxiswastravellingata nominallyconstantvelocityhasbeenidentified,thegradientofthe

bestfitlinecanthenbecalculatedusingalinearleastsquaresfit showninthefollowingformula;

m=

whereXandYisthemeanofthetimedataanddisplacementvector respectively.Thisfunctionevaluatestothemeanvelocity.

Thevariable mis thegradientand mean velocitycalculated usingxrepresentingthetimedomaindataandyrepresentingthe measurementdata.

Oncethecoefficientshavebeenfoundtheyareevaluatedanda bestfitvector(y)thatcanbeusedastheaxisnominalpositionfor calculatingcontinuousmotionerror(Eq.(4)).

d= d− y (4)

whereyisthebestfitvector,disthecontinuousmotion measure-mentdataanddisthecalculatedcontinuousmotionerror.

4.3. Scaleerror

Althougheffective,usingleastsquarefittingtogeneratea refer-encevectorwillremoveanyprogressive(scale)errorfromwithin thecontinuousmotionmeasurementasshowninSection5.

Inordertoreapplythescaleerroraquasi-staticmeasurementis required.Fromthecontinuousmotiondata,itispossibletoextract twopossibletargetpositionsatthestartandendofthesignalas thecommandedstartandstoppositionsareknown.

Althoughitmayseemintuitivetousethesetwopointsto reap-plythescaleerror,andinitiallydoesworkforthesimplecase,itis notarobustmethodbecauseeffectivelythereconstructionisbased upononlytwomeasurementpoints.Thisintroducesaweakness, whichwouldthenbeamplifiedbychoosingtwopointswhichmay notcorrelatetothetime-baseddataset.Inmanycasesmachine toolaxeshaverapidlychangingerrorprofilesattheextremesof travel.Theeffectofthisphenomenoncanbedemonstratedwith simulateddata.

Fig.5shows asimulatederrorsignalwhich containsa large endeffect.Asperiodsofaccelerationanddecelerationarebeing removed,thecontinuousmeasurementdata willnot detectthe endeffectthatisbeingdetectedinthequasi-staticmeasurement. Forsimplicity,accelerationtoconstantvelocitytookonesecond, removingtheendeffectcompletelyfromthecontinuousmotion

Fig.6. Simulatedsignal.

signal.Theactualaxiserrorduringtheconstantvelocityperiodis showninFig.6.

ItisclearthatapplyingthesimulatedendpointfromFig.5would beincorrect.Astheendpointfitrepresentstheamountoferrorover thefullaxislengthratherthantheerrorovertheconstantvelocity period.Performingtheleastsquarefitontheconstantvelocitypath andapplyingthegradientfromtheendpointfitwillgivetheresults showninFig.7.

Itispossibletoreconstructthescaleerrorusingjusttwo quasi-staticmeasurementpointsprovidingthatthemeasurementpoints aretakenwithintheconstantvelocitypath.Thiswillgiveamore accuratefit totheend pointmethodhowever,it willintroduce uncertaintyduetopotentialaliasing.Theresultofthismethodof fittingisshowninFig.8.

Inordertoreapplythescaleerrortomoreaccuratelyreconstruct thesignalanoptimisationalgorithmhasbeendeveloped.The algo-rithmsearchesfortheminimumrootmeansquareerrorbetween thequasi-staticandcontinuousmotionmeasurementsignalusing thesimplexsearchmethod[33].

Fig.7. Reconstructionwithincorrectgradient.

Fig.8.Reconstructionthroughendpointfit.

Thegoalofthisprocessistoconvergeonthebestcoefficients fortheslope(mx+c)thatminimisestherootmeansquareerror between the quasi-static and dynamic error values. Once the coefficientshavebeenfoundandthegradientcalculated,itcanbe easilyapplied.

d= d+sgradient (5)

Reconstruction of the signal to reintroduce the scale error requiresQS-Mtoperformtheaboveoperation.Iftheaxisunder testisbeingmeasuredaccordingtostandards,agood-qualityfit canbeachievedusingthedatabeingcapturedasdemonstratedin Fig.9.

Using just two quasi-static measurement points hasgreater uncertaintythan usingmultiplepointsfor correlation.Whereas when usingthe optimizationtechnique proposedin thepaper, uncertaintyisminimisedasmoredataisbeingusedtoreconstruct thesignal.

Comparingthereconstructedcontinuousmotionerrorwiththe actualerrorshowedthattheendpointmethodshowninFig.7

hasarootmeansquareerrorof3.03␮m.UsingtwoQS-Mpoints fromwithintheconstantvelocitypathshowninFig.8hadaroot meansquareerrorof0.48␮m.Whereastheoptimizationtechnique showninFig.9therootmeansquareerrorwas0.14␮m.

4.4. Axislocation

Thefinalsteptoconverttothepositiondomainistofindthe datumpointwhichallowsforthereconstructionofthenominal discrete“command”values.

Firstly,theactualpositionoftheaxisatthefirst(non-zero)target position(quasi-staticmeasurement)iscalculatedsimplyby

1=1+ı1 (6)

where1isthetargetpositionatindex1andı1iscorresponding errorvalueforthetarget.Notethatthisistrueforalltargetpositions whereınandnwouldbefortargetpositionandmeasuredvalue

atthenthtargetnumber.

Thenextstageinvolvesiteratingthroughthecontinuousmotion displacementdataandlocatingtheindexofthesampleclosestto

1,wewillcalltheindexi.Thequasi-statictargetposition1has tobeinanareaofthecontinuousmotionmeasurementsignalwhen theaxisundertestwastravellingatanominallyconstantvelocity, asanyperiodofaccelerationhasbeenremoved.Ififallswithin

aperiodofaccelerationforthecontinuousmotionmeasurement, thenextstaticmeasurementisselectedandtheindexcalculated.

Itispresumedthatbecausetheaxisundertestwascommanded tomovetoposition1butwasactuallyatposition1(quasi-static measurement),thatthesamplethatcorrespondsto1inthe con-tinuousmotionsignal(i)correspondstothenominalposition1.

Thedeviationfromtheactualpositionandthenominaltarget posi-tionisthen calculatedforthecontinuousmotionmeasurement signal.

l=i−i (7)

Anewshiftedcontinuousmotiontargetvectorcanbe gen-eratedbyapplyingthesmallshiftltothecontinuousmotion

displacementdata.

=−l (8)

Thisfullydefinesthenominalcommandvalueintheposition domainforeachmeasurementpoint.

5. Validation

Inordertovalidatethetechniquedescribedabove,continuous motionmeasurementswerecompared toquasi-static measure-mentswithasmallstepsize.However,themethodisalsoequally effectivewhencomparingagainstquasi-staticmeasurementwith amuchlargerstepsize,sothatitcanbeusedonmachineswhere afinespatialresolutionisnotnecessary(i.e.theerroroftheaxisis notrapidlychangingwithrespecttoposition.)

Thealgorithmandmathematicalprocessingwasdevelopedin MATLABforpracticality.However,thesolutioncaneasilybe devel-opedforanyprogramminglanguage.

In order to validate the technique presented in Section 4, themethod wasvalidated againsta range ofquasi-static mea-surements.Thequasi-staticmeasurementscomplywiththeISO standardfortargetspermeter,howevernorandomtargetswere selected.Arandomelementissometimesusedtohelpindicateany cyclicerror.ThefirsttestwasperformedonathreeaxisC-body machinetool.Theaxisundertestwascommandedtomovefrom 0mmto450mmat500mm/min.Thecontinuousmotiondatawas sampledatafrequencyof100Hz.

Periodsofaccelerationanddecelerationwereremovedfromthe signalleavingonlythepartofthesignalwheretheaxisundertest

Fig.10.Continuousmotionerror.

wastravellingatanominallyconstantvelocity.Theleastsquares linewascalculatedand thecontinuousmotionerrorcalculated throughEq.(4)(Fig.10).

Theaxiswasthenmeasuredusingquasi-statictechniques.Three differentspatialresolutionswereselected(asshowninFig.3).

Comparingthehighresolutionquasi-staticmeasurement(5mm stepsize)showedcorrelationtowithin2_␮m(Fig.11).Howeverthis deviationisduetothegradientbeingremovedfromthecontinuous motionerrorsignal.Thequasi-staticmeasurementcontains5 bi-directionalrunsasrecommendedbyISO[3].This,aswellasthe smallstepsizeiscausingthermaldriftofapproximately4␮m.

By applying the gradient (mx+c) calculated through the optimisationalgorithmtothecontinuousmotionerrorshows sub-micrometercorrelationbetweenthecontinuousmotionerrorand thequasi-staticmeasurement(Fig.12).

Highresolutionquasi-staticmeasurementsaretimeconsuming. Thequasi-staticmeasurement,showninFig.12tookapproximately 1htocomplete. Due tothelengthof time it tooktocomplete themeasurement,theeffectsofthermaldriftareclearly distort-ingthemeasurement.Thisisanotherlimitationtothequasi-static measurementtechnique.Thealgorithmdevelopedgivestheuser

Fig. 11.Comparison between continuous motion and quasi-static (gradient removed).

Fig.12.Comparisonbetweencontinuousmotionandquasi-staticmeasurement error.

theoptiontofitthecontinuousmotionmeasurementtakinginto account thermo-mechanicaleffects by averaging theoptimised coefficientsforeachquasi-staticbi-directionalrun.Inthiscase,just thefirstbi-directionalrunwaschosentoeliminatethermaldrift (Fig.12).

Additionally,theusercanselectjustthefirstbi-directionalrun toobservetheeffectofthermo-mechanicalerrorsasdemonstrated inFig.13.

Sofar,fortherequirement ofdemonstrating thecorrelation betweenthequasi-staticandcontinuousmotionerrorsofmachine toolsthequasi-staticmeasurementshavebeenofa highspatial resolution.Itisunlikelythatmachinetoolownerswillusesucha highspatialresolutionduetothelengthoftimethemachinetool willbeoutofproduction.

5.1. Enhancingthecalibrationprocess

Continuousmotionmeasurementsare not yet supportedby theinternationalstandard for machinetool calibration[3].The proposedmethodcanworkasavaluable enhancementtool for

Fig.13.Comparisonbetweencontinuousmotionandquasi-staticmeasurementon thefirstbi-directionalrun.

Fig.14.Comparisonbetweencontinuousmotionandlowresolutionquasi-static measurement.

machinetoolcalibrationengineers.Performingaquasi-static mea-surementthatmeetstheminimumISOstandardprovidesenough informationtofitandfuseacontinuousmotionmeasurementas demonstratedinFig.14.

Additionally,itisworthrememberingthatacontinuousmotion measurementforamediumsizedmachinetoolwilltakelessthat a minute to complete. Therefore performing a calibration that exceeds the ISO standard (10 points per meter) can be easily enhancedusing theproposedmethodasshown inFig.15 with virtuallynotimeconstraintstothecalibrationprocess.

Themorequasi-staticmeasurementsaretakenthemore accu-ratethecontinuousmotionmeasurementwillfit.However,asseen bycomparingFigs.14and15,thecontinuousmotionmeasurement fittedusingthelowresolutionquasi-staticmeasurementis accu-ratetowithinapproximately2␮mofthefittedcontinuousmotion measurementforahigherresolutionmeasurement.

Comparingthecontinuousmotionmeasurement signalsthat have been fitted against quasi-static measurements of differ-enttargetresolutionsalsoshowstherobustnessofthemethod. Thecontinuousmotionsignalfitted againstthehighresolution

Fig.15.Comparisonbetweencontinuousmotionandmediumresolution quasi-staticmeasurement.

Table1 RMSEvalues.

25mmfit 90mmfit

5mmBenchmark 0.6␮m 2.5␮m

Fig.16.Comparisonbetweenfittedcontinuousmotionsignals.

quasi-staticmeasurement is thebench mark for this test as it demonstratesthebestfit.

Bycalculatingtherootmeansquareerror(RMSE)ofthehigh resolutionfitcontinuousmotionerrorwiththelowresolutionfit continuousmotionerrorshows theimpactofchoosinga coarse targetresolutiononfittingthecontinuousmotionsignal.

ThesignalsusedtocalculatethevaluesinTable1areshown belowinFig.16.

5.2. Furthervalidation

Thecontinuousmotionmeasurementtechniqueproposedwas alsovalidatedonahighspeed gantrymachine. Thelongeraxis (2500mm) and the reduction of machine rigidity provided an increased challenge for continuous motion measurement tech-niques.Thetestconditionswereasfollows.

Feedrate 5000mm/min

Startposition 0mm

Endposition 2450mm

Samplerate 1kHz

Overrun 1mm

Themachinetraversedthelengthoftheaxisbeforecomingtoa halt,thereversalerrorwasremovedbyapplyingtheoverrunbefore areverseruncouldbeperformed.

Twoquasi-staticmeasurementswerealsoperformedfor corre-lation.Thetestconditionswereasfollows.

Feedrate 5000mm/min

Startposition 0mm

Endposition 2450mm

Stepsize 25mm&175mm

Numoftargets 99&15

Numofruns 1

A continuous motion linear positioning measurement was taken.Thecontinuousmotionerrorwascalculatedusingtheleast squarefittingtechnique, thegradient wasfittedusingthe opti-misationalgorithmandshiftedtocorrelatewiththequasi-static measurement.

Fig.17.Comparisonbetweendynamicandmediumresolutionquasi-static.

Once more, a lower resolution quasi-static measurement (175mm step size) was selected to validate the measurement methodagainstamorerealisticquasi-staticmeasurement.Fig.17 showshowtheproposedmethodcanbeusedtoenhancethe cal-ibration process, removingthe effects of aliasingwithminimal additionaltimeconstraints

Oncethegradienthasbeenre-applied(Fig.17),thecontinuous motionerrorandquasi-staticerrorwereonaveragewithin3␮m ofeachother.Ahighresolutionquasi-staticmeasurementwasalso takenandusedforthecontinuousmotionfit(Fig.18).

ThedifferencebetweenthetwosignalsshowninFig.18(high resolution quasi-static and continuous motion error) shows a standarddeviationof4␮m.ThisisshowninFig.19.Thelarge dif-ferenceatthestartofthesignaliscausedbyacyclicerrorinthe continuousmotionmeasurementerrorwhichsoondisappears.Itis likelythatthisisrelatedtopost-accelerationoftheopticandmay betrimmedfromthesignal.

The difference between the two continuous motion signals showninFig.20was1␮m(RMSE).Consideringtheoverallerror

Fig.19.Differencebetweencontinuousmotionandquasi-staticerrorvalues.

Fig.20.Fittedcontinuousmotionsignals.

oftheaxisisinexcessof700␮m,thisresultshowsthe capabili-tiesofthetechniqueproposedinthispaperanditseffectiveness inenhancingthecurrentcalibrationprocesssupportedby interna-tionalstandards.

6. Reproducibility

Tovalidatethemethodfurther,asetofrepeatabilitytestswere performed.Thetestswereperformedonagantrymachine,thetest parameterswereasfollows.

Feedrate 5000mm/min

Startposition 0mm

Endposition 2450mm

Samplerate 1kHz

Overrun 1mm

Atotalof20testswereperformed,10testswereperformedwith theaxisundertesttravellinginapositivedirection,10testswere performedwiththeaxistravellinginanegativedirection.Thetests wereperformedusingtheRenishawXL-80LaserInterferometer, thedatawaspost-processedusingthemethoddescribedinthis paper.

Fig.21. Repeatabilitytest(forwardmotion).

Fig.22. Repeatabilitytest(forwardmotion).

Zoomingintothefigureovera30mmperiodanddatumingthe errorvaluetozero(Fig.21)showstheresultsinbetterdetail. Zoom-inginovera5mm(Fig.22)periodshowsrepeatabilitytobeinthe regionof6␮m.Consideringthelengthoftheaxis(over2000mm) aswellasthelengthoftimetakentoperform10measurements ineachdirection(approximately20min)increasestheprobability ofthermalexpansion.Theresultsaresatisfactoryinvalidatingthe measurementtechnique.

Thequasi-staticmeasurementtakenonthesameaxisshowed unidirectionalrepeatabilityofthemachine’saxisinthepositive directiontobe11␮m.Thereproducibilityofthecontinuousmotion errormeasurementsignaliswellwithintherepeatabilityofthe axis.

7. Uncertaintyanalysis

Thegreatestuncertaintyinthemajorityoflasermeasurement systemsarisesfromdeviationsinenvironmentalconditionssuch asairtemperatureandhumidity.Themeasurementdeviceusedin thisinvestigationissensitivetoenvironmentalchange.A0.26◦C increaseinairtemperatureresultsin0.25␮muncertaintyovera distanceof1meter.Theenvironmentaldeviationsformtheideal

20◦Ccanbecompensatedhoweverthisrequiresadditional sen-sorswhichhaveuncertaintyassociatedwiththemalso.Thelaser deviceandenvironmentalcompensationsystemusedinthis anal-ysishasanexpandeduncertaintyof0.49␮mover1meterwitha coveragefactorofk=2[34].Additionally,ISO230:2(AppendixA) [3]providesformulasforcalculatingtheuncertaintyof measure-mentwhenusingalaserinterferometerandexamplesfordifferent environments.

The uncertainty with regard to feedrate has been assessed experimentally. A typicalthree axis machine (Fig.2) wasused for this analysis. A quasi-static measurement was performed, followed by a range of continuous motions tests taken at dif-ferent feedrates. The first continuous test was performed at 500mm/min andtheresultscanbeseenin Fig.23.The results showsub-micrometre correlation betweenthe quasi-staticand continuous motionerror.However, thecontinuous motion sig-nal detected a high frequency cyclic error which would have remainedundetectedusingatraditionalquasi-staticmeasurement technique.

Inordertoassesshowincreasingthevelocityeffectedthefitting processtherootmeansquareerrorbetweenthefitted continu-ousmotiondataandquasi-staticmeasurementwascalculated.The resultsofwhichcanbeseenTable2.

The continuous motion error was reconstructed using both theproposedoptimisationtechniqueandbyusingjusttwoQS-M pointsintheconstantvelocityregion.Thereconstructed continu-oussignalswerecomparedagainstthefirstbi-directionalrunofa QS-M.Wheretheoptimisationroutinewasused,theRMSEatthe QS-Mpointswassub-micrometre,whereaswhenjusttwoQS-M pointswereusedtheRMSEwasonaverage1.5␮m.

Fig.23.Quasi-staticandcontinuousmotionmeasurement.

Table2 RMSEvalues.

Feedratemm/min RMSE (optimised) RMSE(Two QS-Mpoints) 500mm/min 0.5␮m 1.5␮m 2000mm/min 0.5␮m 1.5␮m 4000mm/min 0.6␮m 1.4␮m 6000mm/min 0.6␮m 1.4␮m 8000mm/min 0.7␮m 1.5␮m 10,000mm/min 0.9␮m 1.5␮m 8. Conclusion

Measurement of machine tool axis geometric errors for assessing performance isan importantpart ofprecision manu-facture.Themeasurementdatacanbeusedtoevaluatecapability whenbuyingamachineandtomonitoritsperformanceduringits operationallifetime.Furthermore,bymeasuringtheerrorsitis pos-sibletoperformdiagnostictasksandremedialactionintheform ofmechanicaladjustmentornumericalcompensation.

The accepted standard quasi-static measurement (QS-M) method for positioning accuracy directly measures the error betweenanominal(programmed)andtheactual(measured) posi-tionbymovingtodiscretetargets,waitingfortheaxistosettleand thenmeasuringbeforemovingtothenexttarget.Suchanapproach providesagoodindicationoftheaxisperformance,butdoesnot representamachinetoolinitsnormalmodeofoperationwhere theaxesareofteninmotion.QS-Mtechniquesalsoprovide calibra-tionengineerswiththechallengeofselectingtheoptimalnumber oftargetpositions;thenumberoftargetpositionsisacompromise betweenthecalibrationqualityandthemanufacturingdowntime required for thecalibrationprocess. Asmanufacturesstrive for increasedmachine availability,aneffective, rapidmeasurement processbecomesmoreimportant.

Thispaperintroducesa measurementtechnique toestablish axis accuracy at a high spatial resolution without incurring a time penalty.Thepositioniscapturedwhiletheaxisismoving at anominally constantvelocity. Thismethoddoesnot require additionalhardwareorinterfacingwiththecontrollerandrelies onlyonuniversallyavailablemotioncommands,butrequiresdata processingtoconvertfromthetime-basedmeasurementintoan errormeasurementinthepositiondomain.Theinitialconversion byleast-squaresfittinghastheunwantedeffectofremovingany progressiveerroroftheaxis(scaleelongation)fromthe measure-ment.A QS-Misrequiredtoreconstructtheoverall profile,but acoarsestep-sizeusingasfewastheISOminimum5targetsis showntobesufficient,thereforehaslittleimpactupontheoverall measurementtime.

Measuringcontinuouslyprovidescalibrationengineerswithfar greaterknowledgeofthemachine’sperformance.Anysmallcyclic errors or discontinuitiesfrommechanical damage or electronic controlparameterscanbedetectedbythehighspatialsampling; theQS-Mmethodisrelativelyveryunder-sampledsowouldmiss theeffectsduetosignalaliasing.Thebettersamplingresolution allows anengineer tointerpret whetheran axisisprogressive, cyclic,likelytobemechanicalorelectrical,etc.Thisallowsa diag-nosticfunctionwhichishighlybeneficialtomachinetoolusers. Anycorrectionstothemachine’sgeometrybyrepaircanbemade withfarmoreinformeddecisionsthanwouldbepossiblewitha quasi-staticmeasurement.Thedataproducedusingthismethod canalsobeconvertedintoaxiscompensationtablesinthesame wayasfromatraditionalQS-Mmeasurement.Thebenefitofthe newmethodbeingthat,wherethenumericalcontrollerhasthe capacity,amuchfinercompensationcanbeachieved,reducingthe residualerror.

Threecase studiesusingseparatemachineswithlarge varia-tionsinperformance andaxislengthareprovidedinthepaper. Inthefirstcasestudy,thetestsperformedshowedcorrelationto withinonemicro-metrebetweenthereconstructedmeasurement fromtheproposedtechniquewithatypicalQS-Mstep-sizeanda benchmarkmeasurementoftheerrorusingaveryfinestep-size forvalidation. Thefittingtechniquewasstillaccuratetowithin 2.5␮m(RMSE)whenselectingtheminimumtargetpositionsas recommended by international standards. On the second, less accuratemachine,whenacoarse175mmstepsizewasselected the fitting technique was accurateto within 1␮m of the high resolution fit. Finally, a third case study is introduced to

evaluatetheuncertaintyofthemethod.Thecontinuousmotion sig-nalandquasi-staticmeasurementwereaccuratetowithin0.5␮m at500mm/minand0.9␮mat10000mm/min.Inadditiontothis,a highfrequencycyclicerrorwasdetectedinthecontinuousmotion signalwhichwouldhaveotherwisebeenmissed.Thisgood cor-relationshowsthattheproposedtechniqueisaneffectivewayof measuringthegeometricperformanceofamachinetooltoenhance measurementstakenaccordingtothecurrentsupportedstandard Theproposedtechniqueislimitedtolinearmeasurementofthe machineonly.Itcanthereforebeusedtoevaluateorcompensatefor measurementparalleltoanindividualaxis,ortwoormorelinear axesthatareprogrammedwithlinearinterpolation.Thisapproach iscommensuratewiththetestsspecifiedintheprevailing stan-dards,suchasISO230-2:2014[3].Thistechniqueisnotreadily applicabletoothertestswhereacurvedtrajectoryisdesiredso shouldbesupplementedwithothermethods, suchasthe dou-bleballbar,ifevaluationofcircularityisrequired(ISO230:4:2005 [16]).Themeasurementmethodwillleadtofurtherresearchinto speed-relatederrorsofmachinetoolaxes.

Acknowledgements

TheauthorsgratefullyacknowledgetheUK’sEngineeringand PhysicalSciencesResearchCouncil(EPSRC)fundingoftheEPSRC CentreforInnovativeManufacturinginAdvancedMetrology(Grant Ref:EP/I033424/1)andIndustrialPartnersRenishawPLC.

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