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IntJAdvManufTechnol200016635–6422000SpringerVerlagLondonLimitedThePreProcessingofDataPointsforCurveFittinginReverseEngineeringMingChihHuangandChingChihTaiDepartmentofMechanicalEngineering,TatungUniversity,Taipei,TaiwanReverseengineeringhasbecomeanimportanttoolforCADmodelconstructionfromthedatapoints,measuredbyacoordinatemeasuringmachineCMM,ofanexistingpart.AmajorprobleminreverseengineeringisthatthemeasuredpointshavinganirregularformatandunequaldistributionaredifficulttofitintoaBsplinecurveorsurface.Thepaperpresentsamethodforpreprocessingdatapointsforcurvefittinginreverseengineering.TheproposedmethodhasbeendevelopedtoprocessthemeasureddatapointsbeforefittingintoaBsplineform.TheformatofthenewdatapointsregeneratedbytheproposedmethodissuitablefortherequirementsforfittingintoasmoothBsplinecurvewithagoodshape.Theentireprocedureofthismethodinvolvesfiltering,curvatureanalysis,segmentation,regressing,andregeneratingsteps.Themethodisimplementedandusedforapracticalapplicationinreverseengineering.TheresultofthereconstructionprovestheviabilityoftheproposedmethodforintegrationwithcurrentcommercialCADsystems.KeywordsCurvefittingPreprocessingofdatapointsReverseengineering1.IntroductionWiththeprogressinthedevelopmentofcomputerhardwareandsoftwaretechnology,theconceptofcomputeraidedtechnologyforproductdevelopmenthasbecomemorewidelyacceptedbyindustry.ThegapbetweendesignandmanufacturingisnowbeinggraduallynarrowedthroughthedevelopmentofnewCADtechnology.Inanormalautomatedmanufacturingenvironment,theoperationsequenceusuallystartsfromproductdesignviageometricmodelscreatedinCADsystems,andendswiththegenerationofmachininginstructionsrequiredtoconvertrawmaterialintoafinishedproduct,basedonthegeometricmodel.TorealisetheadvantagesofmoderncomCorrespondenceandoffprintrequeststoMingChihHuang,DepartmentofMechanicalEngineering,TatungUniversity,40ChungshanNRoad,3rdSection,Taipei104,Taiwan.EmailmindyKmgher.ttit.edu.twputeraidedtechnologyintheproductdevelopmentandmanufacturingprocess,ageometricmodeloftheparttobecreatedintheCADsystemisrequired.However,therearesomesituationsinproductdevelopmentinwhichaphysicalmodelorsampleisproducedbeforecreatingtheCADmodel1.Whereaclaymodel,forexample,indesigningautomobilebodypanels,ismadebythedesignerorartistbasedonconceptualsketchesofwhatthepanelshouldlooklike.2.Whereasampleexistswithouttheoriginaldrawingordocumentationdefinition.3.WheretheCADmodelrepresentingtheparthastoberevisedowingtodesignchangeduringmanufacturing.Inallofthesesituations,thephysicalmodelorsamplemustbereverseengineeredtocreateorrefinetheCADmodel.Incontrasttothisconventionalmanufacturingsequence,reverseengineeringtypicallystartswithmeasuringanexistingphysicalobjectsothataCADmodelcanbededucedinordertoexploittheadvantagesofCADtechnologies.Theprocessofreverseengineeringcanusuallybesubdividedintothreestages,i.e.datacapture,datasegmentationandCADmodellingand/orupdating1,2.Aphysicalmockuporprototypeisfirstmeasuredbyacoordinatemeasuringmachineoralaserscannertoacquirethegeometricinformationintheformof3Dpoints.Themeasuredresultsarethensegmentedintotopologicalregionsforfurtherprocessing.Eachregionrepresentsasinglegeometricfeaturethatcanberepresentedmathematicallybyasimplesurfaceinthecaseofmodelreconstruction.CADmodellingreconstructsthesurfaceofaregionandcombinesthesesurfacesintoacompletemodelrepresentingthemeasuredpartorprototype3.Inpracticalmeasuringcases,however,therearemanysituationswherethegeometricinformationofaphysicalprototypeorsamplecannotbemeasuredcompletelyandaccuratelytoreconstructagoodCADmodel.Somedatapointsofthemeasuredsurfacemaybeirregular,havemeasurementerrors,orcannotbeacquired.AsshowninFig.1,themainsurfaceofmeasuredobjectmayhavefeaturessuchasholes,islands,orroughnesscausedbymanufacturinginaccuracy,consequentlytheCMMprobecannotcapturethecompletesetofdatapointstoreconstructtheentiresurface.636M.C.HuangandC.C.TaiFig.1.Thegeneralproblemsinapracticalmeasuringcase.Measurementofanexistingobjectsurfaceinreverseengineeringcanbeachievedbyusingeithercontactprobingornoncontactsensingprobingtechniques.Whatevertechniqueisapplied,therearemanypracticalproblemswithacquiringdatapoints,forexamples,noise,andincompletedata.Withoutextensiveprocessingtoadjustthedatapoints,theseproblemswillcausetheCADmodeltobereconstructedwithanundesiredshape.InordertorebuildtheCADmodelcorrectlyandsatisfactorily,thispaperpresentsausefulandeffectivemethodtopreprocessthedatapointsforcurvefitting.Usingtheproposedmethod,thedatapointsareregeneratedinaspecifiedformat,whichissuitableforfittingintoacurverepresentedinBsplineformwithouttheproblemspreviouslymentioned.Afterfittingallofthecurves,thesurfacemodelcanbecompletedbyconnectingthecurves.2.TheTheoryofBsplineMostofthesurfacebasedCADsystemsexpressshapesrequiredformodellingbyparametricequations,suchasinBe´zierorBsplineforms.ThemostusedistheBsplineform.BsplinesarethestandardforrepresentingfreeformcurvesandsurfacesincurrentcommercialCADsystems.BsplinecurvesandBe´ziercurveshavemanyadvantagesincommon4.Controlpointsinfluencethecurveshapeinapredictable,naturalway,makingthemgoodcandidatesforuseinaninteractiveenvironment.Bothtypesofcurvearevariationdiminishing,axisindependent,andmultivalued,andbothexhibittheconvexhullproperty.However,itisthelocalcontrolofcurveshapewhichispossiblewithBsplinesthatgivesthetechniqueanadvantageovertheBe´ziertechnique,asdoestheabilitytoaddcontrolpointswithoutincreasingthedegreeofthecurve.Consideringtherealworldapplicationsrequirement,theBsplinetechniqueisusedtorepresentcurvesandsurfacesinthisresearch.ABsplinecurveisasetofbasisfunctionswhichcombinestheeffectsofn1controlpoints.AparametricBsplinecurveisgivenbypuOni0piNi,ku0u11Picontrolpointsn1numberofcontrolpointsNi,kutheBsplinebasisfunctionsuparameterForBsplinecurves,thedegreeofthesepolynomialsiscontrolledbyaparameterkandisusuallyindependentofthenumberofcontrolpoints,andtheBsplinebasisfunctionsaredefinedbythefollowingexpressionNi,1uH1ifuiuui10otherwise2andNi,kuuuiuikuiNi,k1uuik1uuik1ui1Ni1,k1u3Wherekcontrolsthedegreek1oftheresultingpolynomialsinuandthusalsocontrolsthecontinuityofthecurve.ABsplinesurfaceisdefinedinasimilarwaytoatensorproductinaBsplinecurve.ItisalsopossibletodefineaBsplinesurfacehavingdifferentdegreesintheuandvdirectionsSu,vOni0Omj0pijNi,puNi,qv0u143.CurveFittingGivenasetofdatapointsmeasuredfromexistingobject,curvefittingisrequiredtopassthroughthedatapoints.Theleastsquaresfittingtechniqueisthemostusedalgorithmwhichaimsatapproximating,basedonaniterativemethod,asetofdatapointstoformaBspline5–7.GivenasetofdatapointsQk,k0,1,2,...,n,thatlieonanunknowncurvePforcertainparametervaluesuk,k0,1,2,...,nitisnecessarytodetermineanexactinterpolationorbestfittingcurve,P.Tosolvethisproblem,theparametervaluesukforeachofthedatapointsmustbeassumed.Theknotvectorandthedegreeofthecurvearealsodetermined.Thedegreeinpracticalapplicationsisgenerally3order4.TheparametervaluescanbedeterminedbythechordlengthmethodQkPukOni0piNi,pukk0,1,...,n5u00,uiOij1uQjQj1u.Onj1uQjQj1u.6Giventheparametervalues,aknotvectorthatreflectsthedistributionoftheseparametershasthefollowingformU{0,0,...,0,V1,V2,...,Vn,1,1,...,1}p1p1Vj1pOjp1ijuij1,2,...,np7PreProcessingofDataPointsforCurveFitting637Fig.2.Curvefittingwithunequaldistributionofdatapoints.Itcanbeprovedthatthecoefficientmatrixistotallypositiveandbandedwithabandwidthoflessthanp,therefore,thelinearsystemcanbesolvedsafelybyGaussianeliminationwithoutpivoting.Ni,pukui,k0,...,nEquation5canbewritteninamatrixformQNP8whereQisanm11matrix,Nisanm1n1matrix,andPisann11matrix.Sincem.n,thisequationisoverdetermined.ThesolutionisPNTN1NTQ94.TheRequirementforFittingaSetofDataintoaBSplineCurveInordertoproduceaBsplinecurvewithagoodshape,somecharacteristicsarerequiredtofitthedatapointsetintoacurvepresentedinBsplineform.First,thedatapointsmustbeinawellorderedsequence.WhenapplyingtheprogramtofitasetofdatapointsintoaBsplinecurve,thedatapointsmustbereadonebyoneinaspecifiedorder.Ifthedatapointsarenotinorder,thiswillcauseanundesiredtwistoranoutofcontrolshapeoftheBsplinecurve.Secondly,anevendispersionofthedatapointsisbetterforcurvefitting.Inthemeasuringprocedure,somefactors,suchasthevibrationofthemachine,thenoiseinthesystem,andtheroughnessofthesurfaceofthemeasuredobjectwillinfluencetheresultofthemeasurement.Allofthesephenomenawillcauselocalshakesinthecurvewhichpassesthroughtheproblempoints.Therefore,asmoothgradationofthelocationofthedatapointsisnecessaryforgeneratingahighqualityBsplinecurve.HavingthedatapointsequallydistributedisimportantforimprovingtheresultofparameterisationforfittingaBsplinecurve.AsthemathematicalpresentationshowsinEq.9,thecontrolpointsmatrixPisdeterminedbythebasisfunctionsNanddatapointsQ,wherethebasisfunctionsNaredeterminedbytheparametersuiwhicharecorrespondtothedistributionofthedatapoints.Ifthedatapointsaredistributedunequally,thecontrolpointswillalsobedistributedunequallyandwillcausealackofsmoothnessofthefittingcurve.Asmentionedabove,inpracticalmeasuringcases,themainsurfaceFig.3.Curvefittingwithequaldistributionofdatapoints.Fig.4.Theprocedureofdatapointspreprocessing.ofaphysicalsampleoftenhassomefeaturessuchasholes,islands,andradiusfillets,whichpreventtheCMMprobefromcapturingdatapointswithequaldistribution.Ifacurveisrebuiltbyfittingdatapointswithanunequaldistribution,asshowninFig.2,thegeneratedcurvemaydifferfromtherealshapeofthemeasuredobject.Figure3illustratesthatasmootherandmoreaccuratereconstructionmaybeobtainedbyfittinganequallyspacedsetofdatapoints.5.ThePreProcessingofDataPointsToachievetherequirementsforfittingasetofdatapointsintoaBsplinecurveasmentionedabove,itisveryimportantandnecessarythatthedatapointsmustbepreprocessedbeforecurvefitting.Inthefollowingdescription,ausefulandeffectivemethodforpreprocessingthedatapointsforcurvefittingispresented.Theconceptofthismethodistoregressasetofmeasuringdatapointsintoanonparametricequationinimplicitorexplicitform,andthisequationmustalsosatisfythecontinuityofthecurvature.Foraplanecurve,theexplicitnonparametricequationtakesthegeneralformyfx.Figure4638M.C.HuangandC.C.TaiFig.5.Curvatureiscalculatedbythreediscretepointsonacircle.illustratesanoverviewoftheproceduretopreprocessthedatapointsforreverseengineering.Datapointfilteringisthefirststepindisplacingtheunwantedpointsandthenoisypoints.TheoriginaldatapointsmeasuredfromaphysicalprototypeoranexistingsamplebyaCMMareindiscreteformat.Whenthemeasuredpointsareplottedinadiagram,thenoisypointswhichobviouslydeviatefromtheoriginalcurvecanbeselectedandremovedbyavisualsearchbythedesignerforextensiveprocessing.Inaddition,thedistinctdiscontinuouspointswhichapparentlyrelatetoasharpchangeinshapemayalsobeseparatedeasilyforfurtherprocessing.ManyapproacheshavebeendevelopedforgeneratingaCADmodelfrommeasuredpointsinreverseengineering.Acomplexmodelisusuallyconstructedbysubdividingthecompletemodelintoindividualsimplesurfaces8,9.EachoftheindividualsurfacesdefinesasinglefeatureinaCADsystemandacompleteCADmodelisobtainedbyfurthertrimming,blendingandfilleting,orusingothersurfaceprocessingtools.Whenthedesignerisgivenasetofunorganiseddatapointsmeasuredfromanexistingobject,datapointsegmentationisrequiredtoreconstructacompletemodelbydefiningindividualsimplesurfaces.Therefore,curvatureanalysisforthedatapointsisusedforsubdividingthedatapointsintoindividualgroups.InordertoextracttheprofilecurvesforCADmodelreconstruction,inthisstep,datapointsaredividedintodifferentgroupsdependingupontheresultofcurvaturecalculationandanalysisofthedatapoints.Foreach2Dcurve,yfx,thecurvatureisdefinedaskd2ydx2F1SdydxD2G3/2f†1f¢23/210Ifthedataisexpressedindiscreteform,foranythreeconsecutivepointsinthesameplaneX1,Y1X2,Y2X3,Y3,thethreepointsformacircleandthecentreX0,Y0canbecalculatedasseeFig.5X0abcdY0efgdwhereaX1X2X2X1Y3Y2bX2X3X3X2Y2Y1cY1Y3Y2Y1Y3Y2d2X2X1Y3Y2X3X2Y2Y1Fig.6.Thefilletofthemodel.Fig.7.Thecurvaturechangeofthefillet.eY1Y2Y2Y1X3X2fY2Y3Y3Y2X2X1gX1X3X2X1X3X2And,thecurvaturekofX2,Y2canbedefinedask1r1X0X22Y0Y2211Figure6illustratesanexampleinwhichthecurvaturesofaplanecurveconsistingofadatapointsetarecalculatedusingthepreviousmethod.Thecurvatureofthecurvedeterminedbythedatapointsetchangesfrom0to0.0333,asshowninFig.7.Thisindicatesthatthereisafilletfeaturewitharadius30inthedatapointsset.Thus,thesepointscanbeisolatedfromtheoriginaldatapoints,andformasinglefeature.Bycurvatureanalysis,thetotalarrayofdatapointsisdividedintoseveralgroups.Eachofthesegroupsisasegmentedformoftheoriginaldatapointswhichisdevoidofanysharpchangeinshape.Aftersegmentation,individualgroupsofdatapointsareseparatelyregressedintoexplicitnonparametricequations,andthenthedatapointscanberegeneratedfromtheregressionequationinawellorderedsequence,withappropriatespacingandanequaldistributionsothatbetterfittingcanbeachieved.TheformatofthenewdatapointsetisvalidforfittingintoasinglesimpleBsplinecurvewithoutinnerconstraints,whichcanbeappliedforfurthereditingandmodifying,suchastrimmingandextending.Bycombiningindividualcurvestoconstructallofthesurfaces,designersmayeffortlesslyachieveacompleteCADmodelconformingtothedesignintent.Additionally,someregressionerrorsareintroducedbytheregressionoperationbetweenthemeasuredpointsandtheregressionequation.Inthefollowingexample,theorderoftheregressionequationisdiscussed,becauseitbearsacloserelationshiptotheregressionerrors.Givenasetofexistingdatapoints,thesetisregressedusingadifferentorderoftheregressionorder2,3,4,5.Figure8illustratestherelationshipbetweentheorderoftheregressionequationsandtheregressederrorscalculatedbytherootmeansquarer.m.s.method.This
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