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Cutter-workpiece engagement identification in multi-axis milling Aras, Eyyup 2008

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CUTTER-WORKPIECEENGAGEMENTIDENTIFICATIONINMULTI-AXISMILLINGbyEYYUPARASATHESISSUBMITTEDINPARTIALFULFILLEMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYinTHEFACULTYOFGRADUATESTUDIES(MechanicalEngineering)THEUNIVERSITYOFBRITISHCOLUMBIA(Vancouver)July2008?EyyupAras,2008 iiAbstractThisthesispresentscuttersweptvolumegeneration,in-processworkpiecemodelingandCutterWorkpieceEngagement(CWE)algorithmsforfindingtheinstantaneousintersectionsbetweencutterandworkpieceinmilling.Oneofthestepsinsimulatingmachiningoperationsistheaccurateextractionoftheintersectiongeometrybetweencutterandworkpiece.Thisgeometryisakeyinputtoforcecalculationsandfeedrateschedulinginmilling.Giventhatindustrial machined components can have highly complex geometries, extractingintersectionsaccuratelyandefficientlyischallenging.Threemainstepsareneededtoobtainthe intersection geometry between cutter and workpiece. These are the Swept volumegeneration,in-processworkpiecemodelingandCWEextractionrespectively.Inthisthesisananalyticalmethodologyfordeterminingtheshapesofthecuttersweptenvelopesisdeveloped.Inthismethodology,cuttersurfacesperforming5-axistoolmotionsaredecomposedintoasetofcharacteristiccircles.Forobtainingthesecirclesaconceptoftwo-parameter-familyofspheresisintroduced.Consideringrelationshipsamongthecirclesthesweptenvelopesaredefinedanalytically.Theimplementationofmethodologyissimple,especiallywhenthecuttergeometriesarerepresentedbypipesurfaces.Duringthemachiningsimulationtheworkpieceupdateisrequiredtokeeptrackofthematerialremovalprocess.Severalchoicesforworkpieceupdatesexist.Thesearethesolid,facetted and vector model based methodologies. For updating the workpiece surfacesrepresentedbythesolidorfacetedmodelsthirdpartysoftwarecanbeused.Inthisthesismulti-axismillingupdatemethodologiesaredevelopedforworkpiecesdefinedbydiscretevectors with different orientations. For simplifying the intersection calculations betweendiscretevectorsandthetoolenvelopethepropertiesofcanalsurfacesareutilized.A typical NC cutter has different surfaces with varying geometries and during thematerialremovalprocessrestrictedregionsofthesesurfacesareeligibletocontactthein-process workpiece. In this thesis these regions are analyzed with respect to different toolmotions.Laterusingtheresultsfromtheseanalysesthesolid,polyhedralandvectorbasedCWEmethodologiesaredevelopedforarangeofdifferenttypesofcuttersandmulti-axistool motions. The workpiece surfaces cover a wide range of surface geometries includingsculpturedsurfaces. iiiTableofContentsAbstract .............................................................................................................................. iiTableofContents ................................................................................................................ iiiListofTables....................................................................................................................... viiListofFigures.................................................................................................................... viiiListofAlgorithms.............................................................................................................. xivAcknowledgements..............................................................................................................xvChapter1Introduction.......................................................................................................11.1VirtualMachining .......................................................................................................11.2GeometricModeling ...................................................................................................3  1.2.1SweptVolumeGeneration..................................................................................41.2.2In-processWorkpieceModeling........................................................................51.2.3CutterWorkpieceEngagementExtraction.......................................................101.3ProcessModeling......................................................................................................151.4ScopeofthisResearch ..............................................................................................151.5OrganizationofThesis ..............................................................................................16Chapter2LiteratureReview ...........................................................................................182.1SweptVolumeGeneration ........................................................................................182.1.1SweepDifferentialEquationApproach ...........................................................192.1.2JacobianRankDeficiencyApproach ...............................................................202.1.3SweptprofilebasedApproaches......................................................................202.1.4Discussion ........................................................................................................212.2In-processWorkpieceModeling...............................................................................22 iv2.2.1SolidModelerBasedMethodologies ...............................................................222.2.2ApproximateModelerBasedMethodologies ..................................................242.2.3Discussion ........................................................................................................282.3CutterWorkpieceEngagementExtraction................................................................292.3.1SolidmodelBasedMethods.............................................................................302.3.2PolyhedralModelBasedMethods ...................................................................322.3.3DiscreteVectorBasedMethods.......................................................................332.3.4OtherExistingMethods ...................................................................................342.3.5Discussion ........................................................................................................352.4Summary ...................................................................................................................37Chapter3CutterSweptVolumeGeneration .................................................................383.1CanalSurfaces...........................................................................................................383.1.1ExplicitRepresentationofCanalSurfaces.......................................................393.1.2CuttingToolGeometriesasCanalSurfaces ....................................................41 3.2Two-Parameter-FamilyofSpheresinMulti-AxisMilling .......................................423.2.1ApplyingtheMethodologyontheCylinderSurface .......................................463.2.2ApplyingtheMethodologyontheFrustumofaConeSurface .......................483.2.3ApplyingtheMethodologyontheToroidalSurface .......................................503.3ClosedFormSweptProfileEquations ......................................................................523.4Examples...................................................................................................................583.5Discussion .................................................................................................................59Chapter4In-processWorkpieceModeling....................................................................61 4.1ModelingtheIn-processWorkpieceinSolid  andFacettedRepresentation......................................................................................62 4.2ModelingtheIn-processWorkpieceinVectorBasedRepresentation .....................64 4.2.1MillingCutterGeometriesinParametricForm ...............................................64 4.2.1.1CylinderSurface......................................................................................66 4.2.1.2FrustumofaConeSurface......................................................................66 4.2.1.3TorusSurface ..........................................................................................67 v 4.2.2WorkpieceModelandLocalization.................................................................68 4.2.3UpdatingIn-processWorkpieceinMulti-AxisMilling...................................74 4.2.3.1UpdatingwithFlat-Endmill ...................................................................78 4.2.3.2UpdatingwithTapered-Flat-EndMill.....................................................84 4.2.3.3UpdatingwithFillet-Endmill .................................................................87 4.2.4Implementation.................................................................................................944.3Discussion .................................................................................................................97Chapter5FeasibleContactSurfaces ..............................................................................985.1ToolMotionsinMilling............................................................................................995.2MillingCutterGeometries ......................................................................................1025.3CalculatingtheFeasibleContactSurfaces..............................................................1055.3.1EnvelopeBoundaryoftheUpper-Cone.........................................................1085.3.2EnvelopeBoundaryoftheCorner-Torus.......................................................1115.3.3EnvelopeBoundaryoftheLower-Cone.........................................................1135.4AnalyzingtheDistributionofFeasibleContactSurfaces .......................................1145.5Discussion ...............................................................................................................124Chapter6CutterWorkpieceEngagements..................................................................1256.1CutterWorkpieceEngagementsinSolidModels ...................................................1276.1.1EngagementExtractionMethodologyin3-AxisMilling...............................1286.1.2Implementation...............................................................................................1366.1.3EngagementExtractionMethodologyin5AxisMilling...............................1386.2CutterWorkpieceEngagementsinPolyhedralModels ..........................................1436.2.1EngagementExtractionMethodologyin3-AxisMilling...............................1466.2.1.1MappingMforLinearToolpath............................................................1496.2.1.2MappingMforCircularToolpath.........................................................1576.2.1.3MappingMforHelicalToolpath ..........................................................1606.2.1.4Implementation......................................................................................1636.2.2EngagementextractionMethodologyin5-axisMilling ................................1716.3TheCutterWorkpieceEngagementsinVectorBasedModel ................................173 vi  6.3.1IntersectingSegmentAgainstPlane...............................................................173  6.3.2IntersectingSegmentAgainstSphere ............................................................174  6.3.3IntersectingSegmentAgainstCylinder..........................................................175  6.3.4IntersectingSegmentAgainstaCone ............................................................177  6.3.5ObtainingtheCutterWorkpieceEngagements..............................................1796.4Discussion ...............................................................................................................179Chapter7Conclusions....................................................................................................183 7.1ContributionsandLimitations................................................................................1837.2FutureWork ............................................................................................................187Bibliography ......................................................................................................................188AppendixA ........................................................................................................................193AppendixB ........................................................................................................................196 viiListofTablesTable1.1:ComparisonsbetweenCWEextractionmethods ................................................14Table5.1:Theangularrangesofthefeedangle ................................................................116Table5.2:CWEKpointsetsofthegenericcutterunderdifferenttoolmotions .................123Table6.1:Constituentsurfacesofcuttergeometriesaftergeometric decompositionin3-axismilling..............................................................................131Table6.2:Feasibleengagementpointsforcuttersurfaceswithrespectto thetoolmotions.......................................................................................................132 viiiListofFiguresFigure1.1:VirtualMachining ...............................................................................................2Figure1.2:StepsinthegeometricmodelingforextractingCWEs........................................3Figure1.3:Sweptvolumesfrom2?-Axismilling ..............................................................4Figure1.4:Sweepingtheprofilecurvealongtoolpath(3-Axishelicalmilling) .................5Figure1.5:ACSGtree...........................................................................................................6Figure1.6:ACISrepresentationalhierarchy .........................................................................7Figure1.7:Generationofin-processworkpieceandtheremovalvolume ............................8Figure1.8:(a)STLfilestructure,and(b)Atessellatedmechanicalpart..............................9Figure1.9:Updatingtheworkpiecesurfacesrepresentedbyz-vectors.................................9Figure1.10:CutterWorkpieceEngagementparameters .....................................................10Figure1.11:(a)CWEboundariesduringarectangularblock,and (b)Asculpturedsurfacemachining ..........................................................................11Figure1.12:Finalmachinedsurfaceswithcusps ................................................................12Figure1.13:Possiblecutter/facetintersections....................................................................13Figure1.14:Thechordalerrorincutterfacetintersections.................................................13Figure1.15:Z?mapcalculationerrorswhenthegridsizeislarge......................................14Figure1.16:CWEareafortheforceprediction ...................................................................15Figure1.17:CWEareadecompositionsforsculpturedsurfaces..........................................15Figure2.1:Decomposingobjectboundary..........................................................................19Figure2.2:(a)Thepseudo-insertsofthetoroidalEndmill,and(b)thepositionofa pseudo-insertattwotoolpositions............................................................................21Figure2.3:Updatingin-processworkpiece .........................................................................23Figure2.4:3-axiscuttersweptvolumewiththeboundaryfaces.........................................24Figure2.5:SimulationoftheNCmillingbyprojectingpixelsofacomputer graphicsimageontothepartsurface.........................................................................24Figure2.6:Updatingtheworkpiecesurfacesrepresentedbyparallelz-vectors..................26Figure2.7:(a)A3-axisapproximationof5-axissweptvolume,and(b)theextended z-buffermodelofworkpiece.....................................................................................26 ixFigure2.8:DNVandDVVrepresentationoftheworkpiecesurfaces..................................27Figure2.9:Regularmeshdecimationfor3-axisNCmillingsimulation.............................28Figure2.10:ExtractingCWEsbyadvancingsemicircleapproach for2?-axismilling..................................................................................................31Figure2.11:ExtractingCWEsbyadvancingsemicylinderapproach for2?-Axismilling.................................................................................................31Figure2.12:3-axisCWEextractionwithBall-Endmill......................................................32Figure2.13:Procedureforextractingin-cutsegmentsofthecuttingedges........................32Figure2.14:Cutterengagementportion ..............................................................................33Figure2.15:(a)Chordalerrorinthecutterfacetintersection, and(b)theradiusenlargementofthecutter..............................................................33Figure2.16:TheEntry/exitanglecalculationforeachdisc ................................................34Figure2.17:Thepixelbasedengagementextraction...........................................................34Figure2.18:AlineartoolpathandregionexpressedasaBoolean expressionofhalfspaces...........................................................................................35Figure3.1:Geometricdescriptionofacanalsurface ..........................................................40Figure3.2:Sometypicalmillingcuttergeometries.............................................................42Figure3.3:Generatingcuttergeometrieswithamovingsphere .........................................42Figure3.4:(a)ThecharacteristicK,and(b)thegreatScircles...........................................44Figure3.5:Intersectingthecharacteristic(K)andthegreat(S)circles...............................45Figure3.6:IntersectioncasesbetweenKandS ...................................................................45Figure3.7:Themovingsphereofacylindersurfacein5-axismotion ...............................46Figure3.8:Themovingsphereofaconesurfacein5-axismotion.....................................48Figure3.9:Themovingsphereofatoroidalsurfacein5-axismotion................................50Figure3.10:ThecharacteristiccircleKofthetorusintheFrenetFrame ...........................53Figure3.11:ThecharacteristiccircleKinTCS...................................................................54Figure3.12:Limitsofthegrazingpointsinthegeneral......................................................56Figure3.13:ThecharacteristiccircleKbecomesagreatcirclein(a)torus, andin(b)cylindersurfaces ......................................................................................57Figure3.14:Envelopessurfacesof(a)Flat-End,(b)Ball-End,and (c)Fillet-Endcutters..................................................................................................58 xFigure3.15:EnvelopessurfacesofaTaper-Ball-Endmillfromdifferent pointofviewsin5-axismilling.................................................................................59Figure3.16:Differentcuttergeometriesgeneratedbyamovingsphere.............................60Figure4.1:AB-repSolidModelerbasedin-processworkpieceupdate ............................63Figure4.2:APolyhedralModelerbasedin-processworkpieceupdate..............................63Figure4.3:Cuttergeometriesascanalsurfaces:(a)cylinder, (b)frustumofacone,and(c)torus..........................................................................65Figure4.4:Representingtheworkpiecesurfacesby(a)surfacenormalvectors and(b)verticalvectors..............................................................................................69Figure4.5:RepresentingthefeatureshapesinDiscreteNormalvector anddiscreteverticalvectorapproaches.....................................................................70Figure4.6:(a)3-Axismachining,and(b)(3+2)-Axismachining....................................70Figure4.7:Representinginitialworkpiecewithdiscretevectorslocated inXY,XZandYZplanes).......................................................................................71Figure4.8:AABBofatoolmovementinOxycoordinatesystem......................................73Figure4.9:Decomposingthesweptsurfaceintothreeregions).........................................76Figure4.10:EnvelopesurfacesgeneratedbythecylindricalpartofaFlat-Endmill..........79Figure4.11:Possibleintersectionsbetweenavectorandenvelopeplane: (a)nointersection,(b)oneintersectionand(c)vectorliesintheplane ..................80Figure4.12:Parametersetsforupdatingthediscretevector...............................................81Figure4.13:IntersectingtheBottom-Flatsurfacewithadiscretevector............................82Figure4.14:IntersectioncasesbetweentheBottom-Flatsurfaceandadiscretevector......84Figure4.15:Envelopesurfacesgeneratedbythefrustumofaconepart ............................86Figure4.16:Motiontypeswithrespecttothefeedvectorf:(a)descending, (b)horizontaland(c)ascendingmotion ...................................................................87Figure4.17:Theenvelopeparametersofatoroidalsurfaceundertheplungingmotion ....88Figure4.18:Envelope/vectorintersectioncases:twopoints(1), onepoint(2)andnointersection(3)........................................................................91Figure4.19:Therootsofthenonlinearequationf(t)whenthevectorintersects (a)inonepointorin(b)twopoints ..........................................................................92Figure4.20:NCmillingsimulationofaDoormold............................................................95 xiFigure4.21:NCmillingsimulationofanAutohood..........................................................95Figure4.22:NCmillingsimulationresultsforaGearboxcoverwith (a)AFlat-Endmilland(b,c)Ball-Endmills ...........................................................96Figure4.23:NCmillingsimulationfor5-axisimpellermachining.....................................96Figure5.1:Cuttermotionsinmilling:(a)2?-axis,(b)3-axis,(c)5-axis..........................99Figure5.2:Thelocalandreferenceframesofacutter ......................................................100Figure5.3:Geometricdefinitionofthegenericcutter.......................................................103Figure5.4:Boundarypartitionofthegenericcutter..........................................................106Figure5.5:Pointsetsusedindefiningengagements.........................................................107Figure5.6:BoundariesoftheCWEKintheFillet-Endmill...............................................108Figure5.7:ArbitrarypointsIU,ITandILontheupper-cone,corner-torus andlower-conesurfacesrespectively......................................................................109Figure5.8:Instantaneouscuttercontactsurfacesin(a,b)5-axis, andin(c,d)3-axisplungemotions..........................................................................115Figure5.9:Feedangleranges ............................................................................................116Figure5.10:Thecorner-toruswithupperandlowersurfaceboundaries ..........................118Figure5.11:Envelopeboundarysetson(a)thefront,and(b)thebackfaces ofthecorner-torus ..................................................................................................119Figure5.12:Feasiblecontactsurfacesofthetoroidalpartwith respecttothecutterfeedangle................................................................................121Figure6.1:CWEExtractionSteps .....................................................................................125Figure6.2:CWEareafortheforceprediction ...................................................................126Figure6.3:AB-repSolidModelerbasedCWEextraction................................................128Figure6.4:PointsetsusedindefiningEngagements ........................................................129Figure6.5:CWEparametersofanarbitrarypointP..........................................................130Figure6.6:DefiningCWEparametersuandvon(a)torus,(b)sphere, (c)frustumofacone,(d)cylinder,and(e)flatbottom surfacesofcommonmillingcutters ........................................................................130Figure6.7:Geometricdecompositionofthecuttersurfaces .............................................131Figure6.8:DecomposingthepointsetCWEKofthetorusintothreeparts.......................132Figure6.9:Sweptvolumesgeneratedbythekinematicallyfeasible xii engagementpointsets .............................................................................................133Figure6.10:ProcedureforobtainingtheCWEs ................................................................134Figure6.11:CWEsfortheFlatEndmillperformingalinear3-axisdescendingmotion..136Figure6.12:CWEsfortheTaper-Endmillperformingalinear3-axis ascendingmotion.....................................................................................................137Figure6.13:HelicalToolMotionswithaFlat-EndMillandCWEs .................................138Figure6.14:Envelopeboundaryin(a)5-axismilling,and in(b)3-axismillingrespectively ...........................................................................139Figure6.15:Offsettingthefeasiblecontactsurface...........................................................140Figure6.16:CWEstepsin5-axismillingmethodology ....................................................141Figure6.17:5-axisCWEsduringthefirstpassoftheimpellermachining .......................142Figure6.18:5-axisCWEsduringthesecondpassoftheimpellermachining..................142Figure6.19:In-processworkpiecesafterthethirdandfourthpassesrespectively ...........143Figure6.20:Typicalintersectionsbetweenafacetandacuttingtool...............................143Figure6.21:Theedgesoffacetsdeviatefromtherealsurface .........................................144Figure6.22:(a)Constituentsurfacesofmillingcuttersand (b)sometypicalmillingcuttersurfaces .................................................................145Figure6.23:CWEparameters ............................................................................................146Figure6.24:PointsetsCWEK(t),CWE(t)andbCWE(t)usedindefiningengagements....148Figure6.25:CWEcalculationsintheparametricdomainP(phi,d,L)................................148Figure6.26:DescriptionofapointonacuttermovingalongaToolPath........................149Figure6.27:3-axisLinearToolMotionswithaBNEM....................................................150Figure6.28:Engagementregionsofthefront(a)andback(b)contactfaces ...................151Figure6.29:ProcedureforperformingMappingMG:E3arrowrightP(phi,d,L) .............................151Figure6.30:DifferentcutterlocationsforIelementCWEK(t)...................................................154Figure6.31:CylindricalcontactfaceCWEK,C(t)ofBNEM...............................................157Figure6.32:Movingcoordinateframeforcirculartoolpath ............................................158Figure6.33:Parametervaluesofthecuttertooltip...........................................................159Figure6.34:SweepsforHelicalMilling............................................................................160Figure6.35:Removalvolumesof(a)sidefaceand(b)bottomface.................................161Figure6.36:ParametersdescribingahelicaltoolmotionfortheFlat-Endmill................162 xiiiFigure6.37:DifferentcutterlocationsforanengagementpointIelementCWEK(t).................163Figure6.38:ImplementationofCWEextractionmethodology.........................................164Figure6.39:CWEsforBall-Endmillperformingalinear3-axismove............................166Figure6.40:CWEsforFlatEndmillperformingalinear3-axismove.............................166Figure6.41:CWEsforTapered-Flat-Endmillperformingalinear3-axismove..............167Figure6.42:HelicalToolMotionswithaFlat-EndmillandCWEs..................................168Figure6.43:CWEsforFlat-Endmillperformingacircularmove ....................................169Figure6.44:CWEsforBall-Endmillperformingalinear3-axismove............................170Figure6.45:Helicaltoolpathapplicationwithremovalvolumesforeachhalfturn ........170Figure6.46:CWEsoftheHelicalToolMotions................................................................171Figure6.47:CWEextractionstepsfor5-axismilling........................................................172Figure6.48:Intersectingasegmentagainstaplane ..........................................................174Figure6.49:Differentcasesinsegment/sphereintersections: (a)Twointersectionpoints,(b)intersectingtangentially, (c)nointersection,(d)segmentstartsinsidesphere, and(e)segmentstartsoutsidesphere......................................................................175Figure6.50:Asegmentisintersectedagainstthecylindergivenby pointsBandQandtheradiusr...............................................................................176Figure6.51:Aconewithdefiningparameters...................................................................178Figure6.52:Decomposingcuttersurfacesintogrids.........................................................179Figure6.53:RemovalVolumesandCWEsfordifferentresolutions.................................181Figure6.54:Theeffectofthefacetresolutions .................................................................182FigureB.1:(a)CylindricalCWEK,c1(t)and(b)bottomCWEK,c2(t),facesoftheFlatEndMill.......................................................................................196FigureB.2:(a)FrontCWEK,co1(t)and(b)backCWEK,co2(t)conicalfacesofTaperedFlatEndMill................................................................................198FigureB.3:(a)FrontCWEK,t1(t)and(b)backCWEK,t2(t)contactfacesofToroidalEndMill ..............................................................................................200FigureB.4:Cutterinterferenceswithapointinspace.......................................................201 xivListofAlgorithmsAlgorithm4.1:Obtainingtherootsoff(t).....................................................................94Algorithm6.1:ObtainingtheclosedboundariesoftheCWEs....................................135 xvAcknowledgmentsIwouldliketoexpressmysincerethanksandgratitudetomyresearchco-supervisorsDr.DerekYip-HoiandDr.Hsi-Yung(Steve)Fengfortheirinvaluableguidancethroughoutthestudy.AlsoIwouldliketoexpressmydeepestgratitudeandthankstoDr.YusufAltintasforhiscontinuousmoralandfinancialsupportthroughoutthecourseofthisresearch.Iwouldlike to extend my sincere gratitude and appreciation to my colleagues and staff at theMechanicalEngineeringDepartmentfortheirencouragementandhelpatvariousoccasions.My friend (dear brother) Dr. Dimitri Ostafiev deserves special thanks for his varioussupportive roles. I would like to thank my family for their unwavering encouragement. IdedicatethisthesistomysonIbrahimT.Aras. 1Chapter1Introduction Manufacturingisanintegralandindispensablepartoftheeconomy.Asaresultofglobalcompetition,manufacturersarefacingchallengesforbothreducingtheproductioncostsandimproving the product quality at the same time. They are trying to reduce the lead timebeforetheimplementationofanewproductandalsotominimizethecycleoftheproductdevelopment.Manufacturingcontainsdifferentareassuchasforging,casting,machiningetc.Machiningorcuttingofmetalisakeyactivityinmostmanufacturingenvironments.TodaymodernmachinetoolsareCNC(ComputerNumericControl)millingmachinesandlathes.Amicroprocessor in each machine reads the NC-Code program that the user creates andperforms the programmed operations. Traditionally, the NC program is verified andcorrectedbyacostlytimeconsumingprocessofmachiningplasticorwoodenmodels.ForsolvingthisproblemanewapproachVirtualMachining(VM)hasbeenintroduced.1.1VirtualMachiningOneofthetechniquesforadvancingtheproductivityandqualityofmachiningprocessesistodesign,testandproducethepartsinavirtualenvironment.VMisusedforsimulationofthemachiningprocesspriortoactualmachining,therebyavoidingcostlytesttrialsontheshopfloor.Virtualmachiningcanbeconsideredmanufacturinginthecomputer.Figure1.1shows three major components of the VM: Computer Aided Process Planning (CAPP),GeometricmodelingandProcessmodeling.Chapter1.Introduction 2Figure1.1:VirtualMachiningACADmodelistakenasaninputintoCAPP.InCAPP,machiningoperationswhichincludetoolpathsandprocessparameterssuchasspindlespeedandfeedratearegenerated.Thegeneratedtoolpathsandtheprocessparametersareinputtogeometricmodeling. IngeometricmodelingNCtoolpathsareverifiedtoeliminateun-cutmaterialandgougesonthefinal part surfaces, to prevent cutter and workpiece collisions during machining. Cuttingforces are a key input to simulating the vibration of machine tools (chatter) prior toimplementing the real machining process. This simulation can be used to optimizeinstantaneous process parameters to avoid chatter and improve machining quality.Instantaneous cutting forces are determined by the feed rate, spindle speed, and CutterWorkpieceEngagements(CWEs).CWEsareextractedingeometricmodelingforsupportingcuttingforcepredictioninprocessmodeling.Inprocessmodeling,cuttingforces,powerandtorquearepredictedbyutilizingthelawsofthemetalcuttingprocessandCWEsobtainedfrom geometric modeling. Also process parameters can be optimized to obtain bettermachined part quality. The optimized process parameters and the tool paths are sent to aCNC machine for producing the final workpiece. In the following sections geometricmodelingandprocessmodelingareintroduced.Chapter1.Introduction 31.2GeometricModelingIn geometric modeling CWEs are extracted according to the input requirements fromprocess modeling. Figure 1.2 illustrates the steps involved in geometric modeling forextractingCWEs.InputsfromaCAD/CAMsystemincludethegeometricrepresentationoftheinitialworkpiece,thetoolpathandthegeometricdescriptionofthecuttingtool.Figure1.2:StepsinthegeometricmodelingforextractingCWEsInthefirststep,sweptvolumesaregeneratedforthecuttingtoolfollowingatoolpath.Then in the second step these swept volumes are subtracted from the initial workpiecesequentially to obtain the updated workpiece (in-process workpiece). The in-processworkpieceingeometricmodelingisimportantforbothNCtoolpathverificationandCWEextraction.Instep3thein-processworkpiecegeometryisusedtofindtheCWEsforeachtoolpath.TheoutputfromCWEextractionispassedontoprocessmodelingwherecuttingforcesarepredictedandusedtoanalyzeandoptimizetheprocess. ThreemainstepsarethereforeneededtoobtainCWEsingeometricmodelingsquare4 weptVolumeGenerationChapter1.Introduction 4square4 n-processWorkpieceModelingsquare4 WEExtractionInthenextsubsectionseachstepisintroduced.1.2.1SweptVolumeGenerationOneofthesubtasksinidentifyingtheCWEgeometryinvolvesupdatingthein-processworkpiece geometry after each non-self intersecting tool path (or tool path segment). Anaccuratemodelofthein-processworkpieceisthereforeimportanttoensurethatcorrectCWEgeometry is calculated as the simulation progresses and the cutting tool reenters regionspreviouslymilled.Creatingthein-processworkpiecerequiresmodelingandsubtractingtheswept volume generated by each cutter movement along successive tool paths from themodel of the stock, finally yielding the machined surfaces of the final part model.Mathematically,thesweptvolumeisthesetofallpointsinspaceencompassedwithintheobjectenvelopduringitsmotion.Themovingobjectwhichiscalledthegeneratorcanbeacurve,asurfaceorasolidandinthisthesisthegeneratorisarigidmillingcutter.Themotionofthegeneratoriscalledthesweepmotion.Thesimplestsweepmotionsarethetranslationalsweep(Figure1.3(a))androtationalsweepaboutafixedaxis(Figure1.3(b)).Figure1.3:Sweptvolumesfrom2?-AxismillingIn 2 ? Axis and 3 Axis milling, the swept volumes of the cutters can be generated bysweeping the profile curve along the tool path. Figure 1.4 illustrates the swept volumegenerated by a helical tool motion. If a cutter takes a complex tool motion such astranslationalplusrotational(nonfixedrotationalaxis),thecorrespondingsweepoperationisChapter1.Introduction 5calledageneralsweep.Thiskindoftoolmotionsappearsin5-Axismilling.Thereisagreatchallengeinsweptvolumegenerationfor5-Axistoolmotionswherecutterswithdifferentsurface geometries move along 3- dimensional spatial curves with changing tool axisorientations.Althoughimportantresearchhasbeendoneonsweptvolumes,today?sCADsystemsdonotincludesweptvolumegenerationasacomponent.Figure1.4:Sweepingtheprofilecurvealongtoolpath(3-Axishelicalmilling)For swept volume generation some methods have been developed. Unfortunately in thesemethods swept volume computations have been done with complex differential equationsthat require numerical solutions. These limit their practicality and therefore, few methodshavebeenproposedtodetermineefficientlythesweptprofileinNCmachining.Butthesemethodologiesareeithercuttergeometryspecificortheyprovideapproximatesolutions.Theefficiencyofthesemethodologiesstillneedstobeimproved.1.2.2In-processWorkpieceModelingForNCverificationandCWEextractionanaccuratein-processworkpiecerepresentationisneeded.Themodelingofthein-processworkpieceandthecalculationofCWEgeometryinvolves trade-offs between computational efficiency and the accuracy of the result.DeterminingthecorrectcombinationofthesetwofactorsisanopenquestionthatinvolvesdevelopingandunderstandingoftherequirementsformillingprocessmodelingforwhichtheCWEgeometryisaninput.Severalchoicesformodelingthein-processworkpieceexist.ThetwomostcommonaremathematicallyaccuratesolidmodelingthatareusedinCADsystemsChapter1.Introduction 6and approximate modeling such as those used in computer graphics: facetted and z-Mapmodels.Solidmodelingoffersthebestchoiceforhighlyaccuratemodelingofthegeometricconditions.Howeverchallengesexisttomakingthisapproachbothefficientandgenerallyrobusttohandledegenerategeometricconditionsthatcanoccurwhenlargenumbersoftoolpaths need to be simulated. Further, the presence of accurate geometric and topological(connectivity between geometry) information can potentially be exploited to develop?intelligent? approaches for CWE geometry extraction. These would search for patterns(features)intheremovalvolumeswheretheengagementgeometryisconstantorchangingina predictable way. This is not possible when using approximate models where accurategeometryandrelationshipsarenotmaintainedinthedatastructure.Theuseofsolidmodelstherefore has unstudied potential. The approaches to In-process Workpiece Modeling arediscussedinthefollowingsubsections.square4 SolidModeling Solidmodelingtechniquealsocalledvolumetricmodelingisusedinmanyapplicationssuchasgeometricdesign,NCcodegenerationandvisualization.Theuseofsolidmodelsformanufacturing is becoming more widespread with developing computer technology. Themost popular solid modeling representation schemes are the constructive solid geometry(CSG)andboundaryrepresentation(B-rep)schemes. IntheCSGapproachacomplexsurfaceiscreatedfromsimplersolidscalledprimitivesbyusingBooleanoperators:Intersection(intersection),Union(union)andDifference(SF100000).Theprimitivesareparameterizedsolidsthatareeitherregulargeometricshapessuchasspheres,conesandcubes or complex application specific features such as drill/counter-bored holes. ThehierarchicalrepresentationoffeaturesandtheBooleanoperationsiscapturedinabinarytreecalled the CSG tree where leaves represent primitives and nodes represent Booleanoperations(Figure1.5). Figure1.5hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe hierarchical representation of features and the Boolean operations in a binary tree fordescribingtheCSGtree[35].Chapter1.Introduction 7 In solid modeling, boundary representation (B-rep) is a methodology for representingshapes using their limits. If it is compared with the CSG representation which uses onlyBooleanoperationsandprimitiveobjects,theB-rephasamuchrichersetofoperationsandbecauseofthisforCADsystemsitismoreappropriate.B-rep models contain two parts:geometry and topology. The geometry information in the B-rep model is composed ofcurve/surfaceequationsandpointcoordinates.ThetopologyinformationistheconnectivitybetweengeometricentitiesFACE,EDGEandVERTEX.square4 TheshapeofaFACEisdefinedbyasurfacewhichhasaboundaryrepresentedbyconnectedEDGEs.square4 TheshapeofanEDGEisdefinedbyacurvewhichhasaboundaryrepresentedbytwoVERTEXs.square4 ApointrepresentsthelocationoftheVERTEX.OtherelementsintheB-reprepresentationaretheSHELL(asetofconnectedFACEs),theLUMP(collectionofSHELLs)andtheLOOP(acircuitorlistofEDGEsboundingaFACE).Thereareseveralwaysofviewingthisdatastructure.ForexampleitcanbeconsideredasatreeorahierarchywithBODYasaroot.ABODYcanhaveLUMPswhicharecomprisedofSHELLsformedfromagroupofFACES. Inthisthesisbothforupdatingthein-processworkpieceandforextractingtheCWEs,underthesolidmodelersection,aB-repmethodologyisused.Forthisreasonacommercialgeometric modeler ACIS [4] is utilized. ACIS is an object oriented geometric modelingtoolkitdesignedforuseasageometricengine.Figure1.6showsthedatastructureusedinACIS. Figure1.6hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheACISrepresentationalhierarchy[4].In the B-rep based approach, the in-process workpiece can be generated by subtracting asweptvolumeofthecutterfromtheworkpiece.Figure(1.7)illustratesthegenerationofthein-processworkpieceintheithtoolmotion.iGS Representsthesweptvolumeintheithtoolmotionand 1-i isthein-processworkpiecebeforetheithtoolmotion.SubtractingiGS fromChapter1.Introduction 81W i.e. )*( 1 iWSW i =- updates the in-process workpiece for the next tool motion andalsointersectingiGS with 1-i generatestheremovalvolume i .Figure1.7:Generationofin-processworkpieceandtheremovalvolumesquare4 FacettedModels Anotheralternativeformodelingthein-processgeometrythatisstartingtoreceivemoreattentionarePolyhedralModels.Thesemayofferagoodcompromisebetweenmanageablecomputational speed, robustness and accuracy. These models have become pervasive insupporting engineering applications. They are found in all CAD applications as facettedmodelsforvisualizationandareusedextensivelyinsimulation,CAEandrapidprototyping.Inthismodelingapproachworkpiecesurfacesarerepresentedby afinitesetofpolygonalplanescalledfacets.Themostcommonlyusedshapesarethetrianglesandbecauseofthisthetermfacetisusuallyunderstoodtomeantriangularfacet.Convertingthemathematicallyprecisemodelstothetriangulatedmodeliscalledtessellation.Fortessellationtheoriginalsurfacesofthemodelaresampledforsetsofpointsandthenthesepointsareconnectedforconstructingtriangles.TheSTL(StereolithographyTessellationLanguage)formatforrapidprototyping is the most well-known file format for the triangulated models.  In the STLmodel each facet is described by three vertices and a normal direction of the triangle asshowninFigure1.8.ThenormalvectorisdirectedoutwardfromthesurfaceandtheverticesChapter1.Introduction 9areorderedwithrespecttotherighthandrule.Forthenon-planarsurfacesusingagreaternumberoffacetsgivesabetterapproximationofthetessellationtotheoriginalsurface. facetnormal ZYX n  outerloop  vertex 111 ZYX v     vertex 222 ZYX v   vertex 333 ZYX v  endloop endfacet     (a)  (b)Figure1.8:(a)STLfilestructure,and(b)atessellatedmechanicalpart.square4 Z-MapModels In these modeling techniques the workpiece geometry is broken into a set of evenlydistributeddiscretevectorswhicharecalledz-directionvectorsorZDVs(Figure1.9).Thelengthoftheeachvectorrepresentsthedepthoftheworkpiece.ThespacingoftheZDVsisadjustedaccordingtoadesiredlevelofaccuracy.Thez-Mapapproachhasthreesub-tasks:Discretizationofworkpiece,Localizationandintersection.Inthediscretization,thedesignsurfaceistransformedintoasufficientlydensedistributionofsurfacerays.Thelocalizationprocessfindsthepossiblesubsetofraysforeachtoolmotion.Finallyintheintersectiontaskthecutvaluesbetweenz-directionvectorsandthetoolsweptenvelopearefoundforupdatingthe workpiece surfaces. In this methodology the toolpath envelope is modeled as a set ofgeometricprimitivessuchascylinderandplane.Thusthecomputationalcostforcalculatingthe intersections between the cutter and the workpiece is reduced by doing simpleline/primitiveintersectionformulti-Axismachining. Figure1.9hasbeenremovedduetocopyrightrestrictions.Theinformationremovedistheupdatingtheworkpiecesurfacesrepresentedbyz-vectors[7].Chapter1.Introduction 101.2.3CutterWorkpieceEngagementExtraction Oneofthestepsinsimulatingmachiningoperations(VirtualMachining)istheaccurateextractionoftheintersectiongeometrybetweenthecuttingtoolandtheworkpieceduringmachining. Given that industrial machined components can have a highly complexworkpiece and cutting tool geometries, extracting Cutter/Workpiece Engagement (CWE)geometry accurately and efficiently is challenging. CWEgeometry is a key input to forcecalculations and feed rate scheduling in milling operations. This geometry defines theinstantaneousintersectionboundarybetweenthecuttingtoolandthein-processworkpieceateachlocationalongatoolpath.FromtheCWE,thecutterfluteentry/exitangles( Entry andExitphi1 ) and depth of cut d are found (Figure 1.10) and are in turn used to calculate theinstantaneouscuttingforcesintheradial,tangentialandfeeddirections.TheprimarytaskinfindingtheCWEgeometryisfindingtheboundaryoftheengagementregion.Thismayalsobemultipleregions.Therepresentationofthisboundarywilldependonthemathematicalrepresentationoftheworkpiece?ssurfaces.Figure1.10:CutterWorkpieceEngagementparameters.InCWE extractionthetaskisrelatively simple whencylindricalendmillsandsimpleworkpieces are used (Figure 1.11(a)). However in practice this is often not the case. Theforce prediction and feedrate scheduling are largely affected by accuracy of the predictedarea. The most complicated CWE calculations occur during the machining of sculpturedChapter1.Introduction 11surfaces(Figure1.11(b)).ForcesinsculpturedsurfacemachiningaredifficulttodeterminebecauseofthecontinuouslychangingCWEgeometrythatcanoccurateachfeedstep.Thismakes feedrate scheduling for the complex part quite difficult. For this reason whenmachiningcomplexpartsthefeedrateistypicallysettoaconstantvalueoveranumberofcuttertoolpathsbasedontheworstcaseengagementgeometrythatthecutterencounters.Furthermoreifthecuttingtoolgeometryisalsocomplexandthemachiningprocessinvolvesrotationalaxesi.e.4or5axismachiningthenfindingtheengagementgeometrybecomesanevenmorechallengingtask.(a)(b)Figure1.11:(a)CWEboundariesduringarectangularblock,and(b)asculpturedsurfacemachiningThe approaches for finding the CWE geometry can be classified into two majorcategoriesbasedonthemathematicalrepresentationoftheworkpiecegeometry.TheseareSolidmodelingapproaches,andDiscretemodelingapproaches.TheseapproachestoCWEextractionarediscussedinthefollowingsubsections.square4 SolidModelingApproachtoCWEExtraction Forsolidmodelsthesetaketheformofsimpledualformalgebraic/parametricsurfaces(planes, spheres, cylinders, cones, tori) or pure parametric surfaces (Bezier, B-spline,NURBS).Whentheworkpieceandcuttingtoolarerepresentedbyparametricsurfacesthenthe solid modeling approach applies numerical parametric surface intersection algorithmswhicharebasedonsubdivision,andcurvetracing(marching)methods[10,11,59,65].Ontheotherhandiftheworkpieceandcuttingtoolarerepresentedbythenaturalquadricsurfacesthen analytical methods that are either geometric or algebraic may be appliedChapter1.Introduction 12[33,48,55,56,63]. While solid modelers have been recognized as one approach to findingCWE geometry, for limited applications, computationalcomplexity and robustness remainissuesthatneedtobeaddressediftheapproachistobeviablefromapracticalperspective.Otherlimitationscomeforthesizeofthedatastructurethatisnecessaryinparticularforcapturing relationships between topology. These relationships are preserved when using asolidmodelerforsmallsurfaceartifactssuchascuspsthataregeneratedduringmachining(Figure1.12).Thisresultsinadatastructurethatislargeandthatgrowsasthesimulationprogressesparticularlywhenballandbullnosecuttersarebeingused.Figure1.12:Finalmachinedsurfaceswithcuspssquare4 DiscreteModelingApproachestoCWEExtractionDiscrete modeling approaches have been used in verifying the correctness of NC toolpaths. Some of these have been extended to extracting CWEs in support of physicalsimulationoftheprocessthatstartswiththecalculationofthecuttingforces.Anumberofapproachesinthisareacanbeclassifiedintotwogroups:CWEextractioninpolyhedralandinz-Mapandvectorbasedmodels.CWEextractioninpolyhedralmodelsPolyhedralmodelsusefacets(inthisthesistriangularfacets)andtheyaresupportedbymanyCAD softwares. Since each facet in the model is planar with linear boundaries, theintersection algorithms that need to be applied are simpler than those used in the solidmodelingapproach.Acuttingtoolintersectswithafacetinadifferentways(Figure1.13).ForobtainingtheCWEarea,facetsoftheremovalvolumewhichcontainlinearboundariesareintersectedwiththesurfaceofthecutterandthentheintersectionpointsareconnectedtoChapter1.Introduction 13eachother.TheuseoftheremovalvolumeasthebasisforCWEextractiongreatlyreducesthesizeofthedatastructurethatmustbemanipulated.      Figure1.13:Possiblecutter/facetintersectionsThefacetingalgorithmthatgeneratesthismodelapproximatessurfacestoaspecifiedchordalerror.Ascanbeseenfromthe2Dview(Figure1.14)thisresultsinfacetsthatlieoutsidethetoolenvelopata given locationeventhoughthecuttingtoolisincontactwiththeactualremovalvolumesurface.ThisfacetshouldbeconsideredinfindingtheCWEboundarybutwouldbedifficulttodetectsinceitdoesnotintersectwiththetoolgeometry.Theseerrorscanbereducedbyincreasingboththeresolutionofthepolyhedralmeshandtheradiusofthecutter.Figure1.14:ThechordalerrorincutterfacetintersectionsCWEextractioninZ-MapandvectorbasedmodelsThe z-Map and vector based models represent the workpiece using directional vectorsemanatingfromagridonaworkpiecesurface.TheseareupdatedasthecuttingtoolsweepsChapter1.Introduction 14throughdifferentregionsofthegridtocapturethenewheightsoftheworkpiecei.e.thein-process workpiece boundary. The engagements are determined by finding intersectionsbetweenthecuttingtoolgeometryandvectorsalongthenormalsatdiscretepointsonthesurface. These intersections are calculated by a vector/surface intersection instead ofsurface/surface intersection in solid modeling. Although this approach results in a shortercomputation time than a solid modeler based approach for example, the accuracy of thisapproach greatly depends on the resolution of workpiece. In geometric simulation, thedominantapproachesarethevectorbasedsolutions.Thoughmathematicallymoretractablethan the solid modeler approach, as shown in Figure 1.15 these techniques suffer frominaccuraciesduetotherasterizationeffectcommontomanydiscretizedproblems[91].CuttingtoolareaVectorintersectwithcutterActualintersectionFigure1.15:Z?mapcalculationerrorswhenthegridsizeislargeThoughaccuracyisimprovedbyincreasingtheresolutionoftheunderlyinggrid,thiscomeswiththeexpenseoflargermemoryandcomputationalrequirements. There is always a tradeoff between computational efficiency and accuracy in theseapproaches.Table1.1showsthecomparisonofthesolidmodelerandthediscretemodelerapproachesfortheCWEextractions.Table1.1:ComparisonsbetweenCWEextractionmethodsChapter1.Introduction 151.3ProcessModelingPhysicalsimulationneedsCWEsasinputstopredictcuttingforces.Inprocessmodeling,the cutting forces are predicted by utilizing CWEs obtained from geometric modeling. InordertoproperlydefinetheinputformatfromCWEforforceprediction,aforcemodelmustbeidentified.Numerousmodelshavebeenproposedintheliteraturee.g.Refs.[5,18,30,70].InthisthesiswecalculatetheCWEsforsupportingtheforcepredictionmodeldescribedin[5]. This model finds the Cartesian force components by analytically integrating thedifferentialcuttingforcesalongthein-cutportionofeachcutterflute.InthismodeltheCWEarea with a fixed axialdepth of cut is defined bymapping the engagement region on thecuttersurfaceontothephi -zplanewhichrepresentstheengagementangle(phi )versusthedepth(z)ofcutrespectively(Figure1.16). Figure1.16hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheCWEareafortheforceprediction[5].AsshowninFigure1.16,theinputsfromCWEstothisforcemodelaretheentry )st andthe exit )ex angles of the cutter with respect to the feed vector and the axial location)maxmin d ofthecutterengagementarea.ThisforcemodelislimitedtoaCWEareawithaboundarydefinedbyfourconnectedlines.Forgeneralengagementconditions,theCWEareacan be discretized into smaller engagement zones (Figure 1.17) and the overall result isobtainedbyintegratingtheforcesobtainedfromthesezones. Figure1.17hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheCWEareadecompositionsforsculpturedsurfaces[89].1.4ScopeofThisResearchFromtheaboveitcanbeseenthattosupportthedevelopmentofVirtualMachininganumberofchallengesexistinperformingCutterWorkpieceEngagement(CWE)calculationsandmakingthesemoreefficientandrobust.Theseare:Sweptvolumegeneration,In-processChapter1.Introduction 16work piece modeling and CWE extractions. Considering these challenges together, theobjectivesofthisthesisare:square4 Todevelopacomputationallyefficientandgenericsweptvolumemethodologyformulti-axismillingoperations.Atoolpathmaycontainhundredsorthousandsoftoolmotionswhichmake the computational cost for characterizing the geometry of each tool swept volumeprohibitivelyexpensive.Thislimitationmotivatesresearchinthisthesisintomethodologiesthat provide computationally simple analytical solutions to the swept volume generationproblem.square4 To develop an efficient and robust in-process workpiece update methodology. Duringmachiningsimulationforeachtoolmovementthemodificationoftheworkpiecegeometryisrequiredtokeeptrackofthematerialremovalprocess.BecauseNCverificationandCutterWorkpieceEngagement(CWE)extractiondirectlydependonmaterialremovalanaccuratein-processworkpieceupdateisneeded.square4 To develop a methodology for identifying regions of the cutter surfaces that have thepotentialtoengagewiththeworkpieceduringmachining.AtypicalNCcutterhasdifferentsurfaceswithvaryinggeometriesandduringthematerialremovalprocessrestrictedregionsof these cutter surfaces are eligible to contact the in-process workpiece. Identifying theseregionsiscriticaltosimplifyingtheCWEextractioncalculationsforawiderangeofcuttersperformingmulti-axesmachining.square4 To develop solid (B-rep), polyhedral and vector based multi-axes CWE extractionmethodologiestosupportthecalculationofcuttingforcesinmilling.Thesemethodologiesshouldbedevelopedusingarangeofdifferenttypesofcuttersandtoolpathsdefinedby3,5-axestoolmotions.Theworkpiecesurfacesshouldcoverawiderangeofsurfacegeometriessuchasthesculpturedsurfaces.1.5OrganizationofThesis Henceforththethesisisorganizedasfollows:AreviewofrelatedliteratureispresentedinChapter2,followedbyanewgenericsweptvolumemethodologyformulti-axesmillinginChapter 3. In Chapter 4 efficient in-process update methodologies are presented. FeasibleengagementregionsofthecuttersurfacesduringthemachiningareanalyzedinChapter5,Chapter1.Introduction 17followed by the CWE extraction methodologies for solid, polyhedral and vector basedrepresentations in Chapter 6. The conclusions and possible future research directions arediscussedinChapter7.AppendicesclarifyingsomeofthecomputationaldetailsareprovidedfollowingtheBibliography. 18Chapter2LiteratureReview AsisshowninChapter1,VirtualMachininghastwomainparts:thegeometricmodelingand the process modeling. The process modeling requires Cutter Workpiece Engagement(CWE)calculationsfromthegeometricmodelingtopredictthecuttingforcesinmilling.Thisbecomesachallengingtaskwhenthegeometryofthecutterandtheworkpiecearecomplexin multi-axis machining. An extensive amount of research has focused on geometric andphysical simulations of the machining process. The important contributions from theseresearchworksareintegratedinthevirtualmachiningenvironment.Inthischaptersomeoftheimportantresearchcontributionsinthisfieldwillbereviewed.Specificallyresearchintoswept volume generation, in-process workpiece modeling and CWE extractionmethodologiesisreviewed.2.1SweptVolumeGeneration Mathematically,thesweptvolumeisthesetofallpointsinspaceencompassedwithintheobjectenvelopduringitsmotion.Asweptsurfaceistheboundaryofthesweptvolume.Theswept surfaces and volumes are frequently used in graphical modeling, computer aideddesign,NCmachiningverificationandrobotanalysisetc. As mentioned in chapter (1),sweptvolumegenerationisconsideredoneoftheimportantstepsinvirtualmachining,sinceremovalvolumegenerationandthein-processworkpieceupdaterequiresweptvolumes. Themathematicalformulationofthesweptvolumeproblemhasbeeninvestigatedusingjacobian rank deficiency method [1,2], sweep differential equation (SDE) [15], envelopetheory[53,61,82],implicitmodeling[75]andMinkowskisums[25].AbdelMaleketal.[3]presentedacomprehensivesurveyandreviewonthemethodologiesforthesweptvolumegeneration.Althoughinthepastdecadestheproblemofthesweptvolumegenerationhasbeenstudiedwidely,theproblemisstillnotconsideredtobesufficientlywellsolved.ThebasicideainNCverificationandsimulationistoremovethecuttersweptvolumefromtheworkpiecestockandthustoobtainthefinalmachinedsurfaces.ForNCmachiningChapter2.LiteratureReview 19somesweptvolumegenerationmethodshavebeendeveloped.Wangetal.[83]computedtheboundary points for the swept volume of APT-type cutter using the sweep envelopedifferentialequationsmethod(SEDE)[15].Abdel-Maleketal.[1]andBlackmoreetal.[15]solved systems of the implicit equations numerically for obtaining the swept envelopes.Wang et al. [82] derived the tangency condition in which the velocity of a point on theenvelopesurfacemustbetangenttotheenvelopesurface.Unfortunately in these methods swept volume computations have been done withcomplex differential equations that require numerical solutions and these limit theirpracticality.Therefore,fewmethodshavebeenproposedtodeterminethesweptprofileinNC machining efficiently. In the following sub-sections some of the dominant techniqueswillbediscussed.2.1.1SweepDifferentialEquationApproach TheSweepDifferentialEquation(SDE)methodanditsvariants[13,14,15,16,83]weredevelopedforrepresentingandanalyzingsweptvolumes.Thekeyelementofthisapproachisthesweepdifferentialequation(SDE).Inthismethodtheboundaryofthesweptvolumeofanobjectcanberepresentedtobethesubsetoftheunionofa)thegrazingpointsontheboundaryoftheobjectduringtheentiresweepatwhichthevectorfieldoftheSDEneitherpointsintooroutoftheobjectinteriorb)theingresspointsatthebeginningofthesweepandc) the egress points on the object boundary (Figure 2.1). The SDE method has beenimplementedfor3DsweptvolumerepresentationsgeneratedbyaFlat-EndandaBall-Endmill. But the computational cost increased in 3D problems seriously and this affected thespeedoftheimplementation.Inordertoovercomethiscomputationaldifficulty,Blackmoreetal.[15]developedanextensionoftheSDEmethodthattheycalledthesweepenvelopedifferentialequation(SEDE)approach.TheSEDEalgorithmisusedtocalculatethesweptvolumegeneratedbyageneral7-parameterAPTtoolin5-axisNCmillingprocess. Figure2.1hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthedecomposingobjectboundary[15].Chapter2.LiteratureReview 202.1.2JacobianRankDeficiencyApproach TheJacobianRankDeficiency(JRD)methodwaspresentedbyAbdelMaleketal.[1].Thisapproachisbasedonthesingularitytheoryindifferentialgeometry.AJacobianrankdeficiencyconditionisimplementedinordertodetermineallentitiesthatappearinternalandexternaltothesweptvolume.Aperturbationmethodisthenintroducedinordertoselectthoseentitiesthatareboundarytothesweptvolume.Theyshowedthattheimplicitsurfaceisdefinedwhenallthe3x3sub-Jacobiansaresimultaneouslyzero.Withthisapproachtheexactboundary envelope of a swept volume in a closed form can be generated.  Although thepresentedformulationisvalidforanynumberofparametersinanentity,itbecomesmoredifficulttoimplementbecausethenon-linearequationsresultingfromthedeterminantsofthesub-jacobiansalsoincreasesandsystembecomesmoredifficulttosolve.2.1.3SweptProfileBasedApproachesChungetal.[21]developedamethodologyforrepresentingthecuttersweptsurfaceofageneralized cutter in a single valued form. They obtained analytical expressions of thegeneratingcurvefordifferentcuttergeometries.Howeverthismethodologyislimitedto3-axismillingwithlineartoolmotions.Sheltamietal.[67]proposedamethodthatisbasedonidentifyinggeneratingcurvesalongthetoolpathandconnectingthemintoasolidmodelofthesweptvolume.Ithasbeenshownthatateachinstanceintimethereexistsacurveonthecuttersurfacethatdescribesthecontributionofthetoolpositiontothefinalbottomsweptsurface.Thiscurverepresentstheimprintofthetoolonthemachinedsurface.Howeverthistechnique has not yet been extended to include turns or twist of the general 5-axis toolmotions.Alsoitisassumedthatthiscurvecouldbeapproximatedbyacircle.Howeverthisassumptionloosesitsvalidityin5-axistoolmotions.Rothetal.[75]presentedageometricmethodforgeneratingsweptvolumeofatoroidalendmillperforming5-axistoolmotions.Thistechnique,calledimprintorcrossproductmethod,usesvectoralgebraforobtainingthepointsonthesweptenvelopeandthuseliminatestheuseofthecomplicatedSEDEequations.Themethodisbasedondiscretizingthetoolintopseudo-inserts(Figure2.2)andidentifyingimprintpointsusingamodifiedprincipleofsilhouettes.Forobtainingtheimprintcurvetheimprint points of each pseudo-insert are connected by a piecewise linear curve. Later thecollectionofimprintcurvesisjoinedtoapproximatethesweptsurface.Chapter2.LiteratureReview 21 Figure2.2hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe(a)Thepseudo-insertsofthetoroidalEndmill,and(b)thepositionofapseudo-insertattwotoolpositions[75]. Pottman and Peternell [61] described an explicit geometric method for computing thecharacteristicpointsofamovingsurfaceofrevolution.Inthismethodtheplane,normaltothe velocity vector is intersected with a circle chosen from the surface of revolution. Thevalidintersectionsgeneratethecharacteristicpointsontheenvelopesurface.LaterMannetal.[52]extendedtheimprintmethodtosimulatethemillingoperationswithdifferentcuttergeometries. They described that the imprint method can be used with cylinder and toruscuttergeometriesforobtainingthesweptprofiles.Chiouetal.[20]developedasweptprofilemethodologyforageneralizedcutterin5-axisNCmachiningbyanalyzingthemachinetoolmotionsbasedonthemachineconfigurations.Inthisworkthesweptenvelopeofacutterisconstructedbyintegratingtheintermediatesweptprofiles.LaterDuetal.[86]introducedanewB-RepbasedapproachwhichispartlyderivedfromWang?smethod[82].InthisworkinstantaneousprofilesoftheFillet-Endcutterarecalculatedbyintroducingthebasisofthemoving frame. This approach approximates the swept profile of a Fillet-End mill in theparametricformbyinterpolatingasetofpointsonthecuttersurface.2.1.4DiscussionFromtheabovediscussionitcanbeseenthatsweepdifferentialequationandjacobianrankdeficiencyapproachesgeneratethemostpreciseboundariesofasweptvolumeintheparametricorimplicitform.Butbecausetheycontainnumericalcalculationstheirapplicationto NC machining is limited. On the other hand, the nature of the swept profile basedapproachespromisesanapproximationtothesweptvolume.Thoughaccuracyisimprovedby decreasing the distance between consecutive tool locations and by using more grazingpointsforthesweptprofile,thiscomesattheexpenseoflongersimulationruns.Atoolpathmaycontainhundredsorthousandsoftoolmotionswhichmakethecomputationalcostforcharacterizing the geometry of each tool swept volume prohibitively expensive. ThislimitationmotivatesresearchinthisthesisintomethodologiesthatprovidecomputationallyChapter2.LiteratureReview 22simpleanalyticalsolutionstothesweptvolumegeneration.Alsointheliteraturethesweptprofilebasedapproacheshaveconcentratedmainlyonthesimplermillingcuttergeometries.A comprehensive analytical solution in 5-axis milling has not yet been described for thegeneralsurfaceofrevolutionwhichcoversthebroadestrangeofcuttergeometries.Inthisthesisananalyticalmethodologyfordeterminingtheshapeofthesweptenvelopesgenerated by a general surface of revolution is developed. In this methodology, cuttersurfacesperforming5-axistoolmotionsaredecomposedintoasetofcharacteristiccircles.Forobtainingthesecirclesanewconcepttwo-parameter-familyofspheresisintroduced.Inthisconceptthecenterofamovingsphereisafunctionoftwoparametersrepresentingthecuttersurfaceandthetoolmotion.Foragiven5-axistoolmotionamemberfromthisfamilyof spheres generates two circles: a characteristic and a great circle. Considering therelationshipbetweenthesecirclesananalyticalformulawhichdescribesthesweptenvelopeis developed. The implementation of the methodology is simple, especially with cuttergeometriesrepresentedbypipesurfacessuchasthetorusandcircularcylinderwithfewercalculationsbeingused.2.2In-processWorkpieceModelingForNCsimulationandCWEextractionanaccuratein-processworkpiecerepresentationis needed. NC simulation usually implies cutting simulation and verification. The cuttingsimulationisforvisualizationofthecuttingprocessandtheverificationisforcomparisonofthe machined surface with the design surface. The modeling of the in-process workpiecegeometry involves trade-offs between computational efficiency and the accuracy of theresult.Severalchoicesformodelingthein-processworkpieceexist.Thetwomostcommonare mathematically accurate Solid modeler based methodologies that are used in CADsystems and approximate modeler based methodologies such as those used in computergraphics.Inthefollowingsub-sectionstheseapproacheswillbediscussed.2.2.1SolidModelerBasedMethodologies Solidmodelingtheorywasdevelopedinthelate1960sandearly1970s.CurrentlythemostpopularschemesusedinsolidmodelersaretheBoundaryrepresentation(B-rep)andConstructiveSolidGeometry(CSG).IntheB-repmethodologyanobjectisrepresentedbyChapter2.LiteratureReview 23bothitsboundariesdefinedbyFaces,Edges,Verticesandtheconnectivityinformation.Inthe CSG representation the Boolean operations and the simple primitives are saved in abinarytreedatastructure.Solidmodelingmethodologiesofferthepossibilityofdoingbothsimulation and verification. For simulation the swept volumes of the tool movements aresubtracted from the in-process workpiece model. Verification is performed by Booleandifferencesbetweentheprocessedworkpieceandthedesignpart.Researchershaveinthepastinvestigatedthepotentialofsolidmodelersforsupportingthemachiningprocess[68-71,73,76,80].ThecomputationalcomplexityduetotheBooleanoperationswasidentifiedasoneofthedifficultiesinapplyingthismethodology.Developmentsincomputerprocessorspeeds and new computational technologies such as parallel computing now make this amoreviableprospect[72]. Voelcker and Hunt [80] did an exploratory study of the feasibility of using CSGmodelingsystemforNCsimulation.IthasbeenreportedthatthecostofsimulationintheCSG approach is proportional to the fourth power of the number of tool movements. AtypicalNCprogramcancontainmorethan10000toolmovements.SpenceandAltintas[70]developeda2?-axisinstantaneouscutter/workpieceimmersionsimulationmethodbasedonaCSGmodel.ThisworkwaslatertransferredtotheB-repmodeler[9,64].LaterSpenceetal. [71] presented integrated solid modeler based solutions for machining. In this work,geometric andmillingprocesssimulationcombinedwithonlinemonitoring andcontrolisillustratedfor2?-axispocketmillingapplication(Figure2.3).ForthispurposetheACISsolidmodelerkernelisutilized. Figure2.3hasbeenremovedduetocopyrightrestrictions.Theinformationremovedistheupdatingin-processworkpiece[71]. Elbestawi et al. [37] developed an improved process simulation system for Ball-Endmillingofsculpturedsurfaces.Theworkpiece,cutterandCWEgeometriesaremodeledusinga geometric simulation system which uses the commercial solid modeler (ACIS) as ageometricengine.LaterImanietal.[38]presentedamethodforgeometricallysimulatinga3-axisBall-Endmillingoperation.Theydevelopedanadvancedsweeping/skinningtechniqueforgeneratingapreciseB-repmodeloftheBall-Endmillsweptvolume(Figure2.4).InthisChapter2.LiteratureReview 24worksemifinishingoperationsaresimulatedbyperformingconsecutiveBooleanoperationsbetween the in-process workpiece and the cutter swept volumes. The critical geometricinformationsuchasthe instantaneouschip geometry andthescallopheightsareextractedfromthein-processworkpiece. Figure2.4hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe3-axiscuttersweptvolumewiththeboundaryfaces[38].2.2.2ApproximateModelerBasedMethodologies Forincreasingefficiencyandrobustnessintheprocessoftheworkpieceupdateanumberofapproximateapproacheshavebeendeveloped.Theapproachescanbeclassifiedintothreemajor categories: Image space (or view based) methods, Object based methods andpolyhedralmodelbasedmethod.square4 Imagespace(orviewbased)methods Chappel [19] developed a method called the ?point vector technique?. The designsurface is approximated by a set of points and the vectors originating from these pointsextend until they reach the boundary of the original stock. For simulating the machining,eachvectorontheworkpiecesurfaceisintersectedwiththecuttersweptvolumesandthevector length is shortened if it intersects the swept volume. Oliver and Goodman [58,74]presentedanapproachsimilartoChappel?s.Inthisapproachtheworkpieceisdiscretizedintoarrays containing the surface coordinates and the corresponding normal vectors. Thediscretizationoftheworkpiecesurfacesaredoneaccordingtothecomputergraphicsimageoftheworkpiecesurfaces.Eachpixelonthescreenisprojectedontoworkpiecesurfaceandthesepixelpointsonthesurfacebecometheapproximationoftheworkpiece(Figure2.5).Inthis work the vector - cutter swept volume intersections are performed with a similarapproachtoChappel?s. Figure2.5hasbeenremovedduetocopyrightrestrictions.TheinformationremovedisthesimulationoftheNCmillingbyprojectingpixelsofacomputergraphicsimageontothepartsurface[58].Chapter2.LiteratureReview 25Later Wang [81,82] developed an image space approach. For each pixel on the screen avectorisgeneratedandusingascan-linealgorithmtheintersectionsbetweenvectorsandthetoolpath envelope are calculated. The z-buffer of the workpiece is modified by applyingBooleandifferenceoperationswiththez-bufferofthetoolsweptvolumes.AlsoVanHook[78]developedanapproachcalledthe?extendedz-buffer?.Thepixelimagesofthecutterarepre-computed and then the cutter is subtracted from the workpiece as it moves along atoolpath.Thismethodisappliedto3-axismillinginwhichtheorientationofthecutterisfixed.LaterAtherton[6]extendedVanHook?smethodtohandle5-axismilling. The main limitations with the image space methodologies are: Because they are viewdependant,errorsnotvisibleinthechosenviewingdirectionareundetected.Fordetectingtheerrorsinanotherviewofthepart,theentiresimulationmustbestartedagain.Alsosmallmachiningerrorsi.e.lessthan0.1mmareunlikelytobedetectedbyavisualinspectionofthecomputergraphicsimage.Despitetheirlimitationstheviewbasedapproachesarethebestapproximation methodologies for determining how much material is being removed for agiventoolmovement.Eachpixelcanrepresentavolumeofmaterialand,materialremovalcanbeapproximatedwithaccuracydependentonthesizeoftheobjectandthenumberofthepixels[41].square4 Objectbasedmethods The Object based in-process workpiece update methodologies are developed in [23,24,32,39,40].Inthesemethodologiesasetofpointsarechosenontheworkpiecesurfacesand using these points actual surfaces are approximated. These methodologies use similarcalculation techniques from Chappel?s [19] and Oliver?s [58] approaches. Using the toolmovements the vectors on the surface of the workpiece are updated. Some of theadvantageous of these approaches are [41]: a) Material removal simulation is efficientlyaccomplished by intersecting the surface normals with the tool swept volumes. b) Themachining verification is easily done by the updated vector length. Also using the colorcontourmapoftheworkpiecesurface,outoftoleranceareascanbeseen.c)Theusercanspecifytheaccuracyofthesimulation.Chapter2.LiteratureReview 26 Jerardetal.[40]describeda3-axismachiningmethodologyforsimulatingthegeometricaspects of numerically controlled machining. The methodology is based on a discreteapproximationoftheobjectintoasetofsurfacepoints.Vectorspassedthroughthesepointsare intersected with the tool movements (Figure 2.6). For increasing the efficiency thelocalizationmethodologydescribedin[41]isused. Figure2.6hasbeenremovedduetocopyrightrestrictions.Theinformationremovedistheupdatingtheworkpiecesurfacesrepresentedbyparallelz-vectors[40]. FussellandJerardintheirrecentresearch[32]usedanextendedz-Buffermodelandadiscreteaxialslicemodelforrepresentingtheworkpieceandthecuttingtool(Ball-Endmill)respectively. Inthisworka3-axisapproximationofthe5-axistoolmovementisusedtosimplifythecalculationswhilemaintainingadesiredlevelofaccuracy(Figure2.7-a).Sincethetoolsweptvolumeismodeledasasetofgeometricprimitivessuchasnaturalquadricsurfaces and planes, the intersection calculations are simplified to a set of line/surfaceintersections.Itisassumedthatallprimitivesarelinearandthecurvaturesinthetoolpathandthetoolorientationsareneglected.Forthisreason5-axistoolpathsaresubdividedintomany3-axistoolpaths. Although approximating the 5-axis tool moves with 3-axis linear toolmovesisaprimarydrawbackinthiswork,thecomparisonsbyQuinn[62]totheexact5-axistoolmoveswiththe3-axisapproximationmethodshowthatthe3-axismethodisfasterandrobust.Anotherdrawbackinthisworkisthatalldiscretevectorsoftheworkpiecemodellieinonedirectionregardlessofsurfacenormaldirections,wherethedirectionsarealongtheverticalz-axisofaCartesiancoordinatesystem(Figure2.7-b). Figure2.7hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe(a)A3-axisapproximationof5-axissweptvolume,and(b)theextendedz-buffermodelofworkpiece[32]. Baeketal.presentedaz-Mapupdatemethodforlinearly[7]andcircularly[8]movingAPT-type tools in 3-axis milling. Machining process is simulated through numericallycalculatingtheintersectionpointsbetweenthez-Mapvectorsandthetoolsweptenvelope.Chapter2.LiteratureReview 27Theintersectionpointsareexpressedasthesolutionofasystemofnonlinearequations.Thissystemofequationsistransformedintoasingle-variablefunctionwhosezeroisasolutionofthesystem.Alsointhisworktheyshowhowtocalculatethecandidateintervalinwhichtheuniquezeroexists.Inthismethodologyboththetoolrotationaxisandthez-Mapvectorsarerestrictedtoliealongtheverticalz-axisofaCartesiancoordinatesystem. Park et. al. [43] developed a hybrid cutting simulation methodology via the DiscreteNormalVector(DNV)andtheDiscreteVerticalVector(DVV)models.IntheDNVapproachaworkpiceconsistsofdiscretevectorswhosedirectionvectorsaresurfacenormalvectors,wherealldirectionsarenotnecessarilyidentical.Alsospacingmayvarydependinguponthesurfacelocalproperties.Ontheotherhand,intheDVVapproachalldiscretevectorsoftheworkpiecemodellieinonlyonedirectionregardlessofsurfacenormaldirections,wherethedirectionsarealongtheverticalz-axisofaCartesiancoordinatesystem.TheDNVapproachgenerates better surface quality especially when the workpiece model has vertical walls,sharpedges,andoverhang(Figure2.8).ButontheotherhandintheDNVrepresentationthelocalization of the cutter swept envelope is difficult and this decreases the computationalefficiency.Themainideainthisworkistotakefulladvantageofeachtype?sstrengthsintermsofspeed.Butontheotherhand,inthishybridmethodologythecuttersweptvolumesaregeneratedbyasolidmodeler.Atypicaltoolpathmaycontainthousandsoftoolmotions.In this case using the solid modeler for the cutter swept volume generation increases thecomputationalcostduringthesimulation. Figure2.8hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheDNVandDVVrepresentationoftheworkpiecesurfaces[43].square4 Polyhedralmodelbasedmethod Another alternative for updating the in-process workpiece in NC machining is apolyhedral model based method. In the polyhedral model the workpiece surfaces arerepresentedbyafinitesetofpolygonalplanescalledfacets.Themostcommonlyusedshapesaretrianglesandconvertingthemathematicallyprecisemodelstothetriangulatedmodelsiscalledtessellation.InthisapproachinordertoachieverealtimeNCsimulationthenumberof polygons has to be reduced. But this results in poor image quality. Ong et. al. [49]Chapter2.LiteratureReview 28developedanapproachtoreducethenumberofpolygonswithoutmuchlossinimagequality(Figure 2.9). They proposed an adaptive regular mesh decimation algorithm which uses aquadtree to represent the milling surface. This algorithm can automatically adjust thepolygondensitytoapproximatethemillingsurfaceduringthesimulationinrealtime. Figure2.9hasbeenremovedduetocopyrightrestrictions.Theinformationremovedistheregularmeshdecimationfor3-axisNCmillingsimulation[49].2.2.3Discussion During the machining simulation for each tool movement the modification of theworkpiecegeometryisrequiredtokeeptrackofthematerialremovalprocess.BecausetheNCverificationandCutterWorkpieceEngagement(CWE)extractiondirectlydependentonthematerialremovalanaccuratein-processworkpiecemodelingisneeded. Fromtheliteraturereviewitcanbeseenthatseveralchoicesformodelingthein-processworkpieceexist.ThetwomostcommonaremathematicallyaccuratesolidmodelingthatareusedinCADsystemsandapproximatemodelingsuchasthoseusedincomputergraphics:facettedanddiscretevectormodels.Eachmethodologyhasitsownstrengthsandweaknessesintermsofcomputationalcomplexity,representationaccuracyandtherobustness.Itcanbeseenfromtheliteraturethatforupdatingtheworkpiecesurfacesrepresentedbythesolidorfaceted models third party softwares can be used. For example the ACIS solid modelingkernelisoneofthesesoftwares. But on the other hand for the workpiece geometries represented by discrete vectorsaccurate and computationally efficient in-process workpiece update methodologies areneeded.Intheliteraturethediscretevectorshaveorientationsalongthez-axisofthestandardbasesofR3.Butwhenthein-processworkpiecehassomefeatureslikeverticalwallsandsharpedges,representingtheworkpiecewithonedirectionalvectorsgenerateslessaccurateresultsinvisualizationofthefinalproductandCWEextractions.Alsoinrecentworks[7,8]the tool axis has a fixed orientation along the z-axis of the standard bases of R3 and theworkpieceupdatesareperformednumericallyforthecommonmillingcutters.Stillneededare in-process workpiece update methodologies in which the workpiece geometries arerepresentedbydiscretevectorshavingdifferentorientationsandthetoolrotationaxeshaveChapter2.LiteratureReview 29arbitraryinclinationsinspace.Alsointhesemethodologiesforthegivencuttergeometrythecalculations,ifitispossible,mustbeanalyticforcomputationalefficiency. Inthisthesisthediscretevectorswiththeirorientationsinthedirectionsofx,y,z-axesofthestandardbasesofR3areused.Thereforeinthisrepresentationmorevectorsindifferentdirections are used and thus the quality in the visualization of the final product has beenincreased.Atypicalmillingtoolpathcontainsthousandsoftoolmovementsandduringthemachining simulation for calculating the intersections only a small percentage of all thediscretevectorsisneeded.Forthispurposeforlocalizingthetoolenvelopeduringsimulationthe Axis Aligned Bounding Box (AABB) is used. For simplifying the intersectioncalculationsthepropertiesofcanalsurfacesareutilized.Inthedevelopedmethodologiesthetoolmotionsin(3+2)-axismillinginwhichthecuttercanhaveanarbitraryfixedorientationin space are considered. The 5-axis tool motions can be approximated by (3+2)-axis toolmotionsandanexampleisgivenforillustratingthissituation.Thecalculationsforthemajorcuttergeometriessuchasthesphere,cylinder,frustumofaconeandflat-bottomsurfacesaremadeanalytically.Becauseofthecomplexityofthetorusshapethecalculationsaremadebyusing the numerical root finding methods. For this purpose a root finding analysis isdevelopedforguaranteeingtheroot(s)inthegiveninterval.2.3CutterWorkpieceEngagementExtraction ThegoalinCWEextractionistoobtainengagementconditionsbetweenmillingcutterandthein-processworkpieceforsupportingprocessmodeling.FromtheCWEsinstantaneousin-cutsegmentsorengagedportionsofthecuttingedgesareobtained.Forextractingthein-cutsegmentsCWEboundariesareintersectedwiththecutteredges eitherin2Dorin3DEuclidian space. In both approaches edge/edge intersection is performed. In the 2D-spaceapproach,firstthecuttercontactfaceisintersectedwiththein-processworkpieceandCWEboundaryisobtainedin3DEuclidianspace.ThenthisCWEboundaryandthecutteredgesare mapped into two dimensional space. Finally for obtaining the in-cut segments, eachcutting edgeisintersectedwiththeCWEboundaryinthisspace[70,89]. Inthe3D-spaceapproach intersections between CWE boundary and cutter edges are performed in 3DEuclidianspace[26,37].Inthisthesisforextractingin-cutsegmentsthe2D-spaceapproachisutilized.Chapter2.LiteratureReview 30 Intheliterature,basedonthemathematicalrepresentationoftheworkpiecegeometry,theCWE extraction methodologies can be classified into three major categories: Solid modelbased,polyhedralmodelbasedanddiscretevectorbasedmethods.AlsosomeotherexistingCWEextractionmethodsarediscussedinsection(2.3.4).2.3.1Solidmodelbasedmethods Some research on Solid model based CWE extraction methodologies for supportingprocess modeling has been reported [26,,37,47,70,89]. In these CWE extractionmethodologies the initial workpiece geometry is represented by using the B-rep model.Geometric and topological algorithms of this model are utilized for both updating the in-processworkpieceandextractingtheCWEgeometry.Forupdatingthein-processworkpiecegeometryfirstthesweptvolumeofthecutterisgeneratedforagiventoolpathsegmentandthenthisvolumeissubtractedfromthecurrentworkpiecestateusingregularizedBooleansubtraction. Cutter workpiece engagement calculations are performed using this newworkpiece state until the start of the following tool path segment. If the toolpath has selfintersections then for obtaining the correct CWEs it may be necessary to decompose thegiven toolpath into non-intersecting smaller segments. From the literature the CWEextractions using a solid modeler can be classified into two groups:  2 ?-axis and 3-axismethods.square4 2?-Axismethods ForcalculatingCWEsin2?-axismillingwithaFlat-Endmill,aB-repbasedapproachhas been presented by Spence et al. [70]. Figure 2.10(a) illustrates that a Flat-End millengages with the in-process workpiece Wi at a constant depth of cut d along the entiretoolpath.ItisshowninFigure2.10(b)thatengagementsaredeterminedbyperformingtheintersectionsbetweenanadvancingsemicircleCimovinginthetangentialdirectionofthetoolpath and the boundaries of Wi on an engagement plane PLZ-Top. The engagementgeometryiscalculatedasentry( st )andexit( ex )anglesforeachstepoftheFlat-Endmill(Figure2.10(c)).ThesteptakenbetweeneachCWEcalculationisthefeedperrevolutionofthe cutter. Although this methodology is an important step in applying solid modelingtechnologytotheCWEextractionproblem,itonlygeneratestheconstantdepthofcut.Chapter2.LiteratureReview 31 Figure2.10hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheextractingCWEsbyadvancingsemicircleapproachfor2?-axismilling[70]. LaterYip-Hoietal.[89]presentedanotherB-repbasedapproachforextractingCWEsin2 ? - axis milling. He extended Spence?s methodology by performing the regularizedBooleanintersectionoperationbetweenasemicylindricalcuttersurfaceandthein-processworkpiece(Figure2.11).Inthisworkforimprintingtheengagementregionboundariesonthe cutter surface, the solid modeler?s (ACIS) surface/surface intersection algorithms areused.Thentheengagementboundariesaremappedfrom3DEuclidianspaceinto2Dspacedefinedbytheengagementangleandthedepthofcutrespectively.Latertheseengagementregionsaredecomposedintosmallerrectangularregionsforfulfillingtherequirementoftheforcemodeldescribedin[5].Theadvantageofthismethodologyisthatitcanbeappliedtoawiderrangeofworkpiecegeometriesi.e.theinitialworkpiecegeometryisnotrectangularprismatic.Ontheotherhanditisstilla2?-axismethodologywithaFlat-Endmillandtheengagementboundariesarelimitedtostraightlinesandcirculararcs. Figure2.11hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheextractingCWEsbyadvancingsemicylinderapproachfor2?-Axismilling[89].square4 3-Axismethods Elbestawi et al. [37] developed a Ball-End milling process simulation system. In thisworktheBallEndmillperforms2-axisoranascending(withdifferentup-hillangles)3-axismotion.Thegeometriesoftheworkpiece,thecutterandCWEsaremodeledusing(ACIS)asa geometric engine. The engaged portion of the cutting edge (the instantaneous in-cutsegment)iscomputedintwosteps:Inthefirststeptheboundaryofthecontactfacebetweenthe spherical portion of the Ball-End mill and the in-process workpiece is obtained. ThisboundaryconsistsofonecircularedgeandtwoB-splineedges(Figure2.12).Inthesecondstep the intersection points between interpolated cutting edge and boundary curves of thecontactfacearefound.Thisisperformedbyusingedge/edgeintersectionswithaprescribedtolerance.Finallythecuttingedgeisdecomposedintoin-cutandout-cutsegments.Chapter2.LiteratureReview 32 Figure2.12hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe3-axisCWEextractionwithBall-Endmill[37]. Also El-Mounayri et al. [26] presented a geometric approach for 3-axis machiningprocesssimulationusingaBall-Endmill.Inthisworksolidmodelsareusedforrepresentingthe in-process workpiece and the removal volume. The cutting edges are represented byBeziercurvesin3DEuclidianspace.Forobtainingthein-cutsegmentsofthecuttingedges,the Bezier curves are intersected with the removal volume (Figure 2.13). Later thesesegmentsareusedforevaluatingtheinstantaneouscuttingforces. Figure2.13hasbeenremovedduetocopyrightrestrictions.Theinformationremovedistheprocedureforextractingin-cutsegmentsofthecuttingedges[26].2.3.2PolyhedralModelBasedMethods Althoughtheyareapproximate,polyhedralmodelsprovidetheadvantageofsimplifyingthe workpiece surface geometry to planes which consist of linear boundaries. Thus theintersection calculations reduce to line / surface intersections. These can be performedanalyticallyforthegeometryfoundoncuttingtools.Someresearch[88,90]onpolyhedralbasedCWEextractionmethodologiesforsupportingtheprocessmodelinghasbeenreported. Yao[88]presentedgeometricalgorithmsforestimatingcutterengagementvaluesfor3-axis Ball-End milling processes of tessellated free-form surfaces. In this work 3D surfacegeometry is represented by an STL (?Stereo lithography?) model in which surfaces aretessellatedorbrokendownintoaseriesofsmalltriangles.ItisassumedthatthetoolpathisalinearsegmentandonlythefrontieroftheBall-Endmillhasengagementwiththeworkpiece(Figure 2.14). The engaged portion of the cutter is estimated in three steps: First findingtrianglesthatintersectthecutter,thenintersectingthosetriangleswiththecutterforobtainingtheintersectioncurvesegmentsandfinallyusingthosecurvesegmentstoformtheclosedengagementregion.Chapter2.LiteratureReview 33 Figure2.14hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthecutterengagementportion[88]. Later Yip-Hoi et al. [90] presented a polyhedral model based CWE calculationmethodology in 2 ?-axis and 3-axis millings. For reducing the amount of data andcalculations this methodology works on removal volumes instead of the in-processworkpiece. To further reduce the number of intersections an R*-tree based localizationtechniqueisappliedforlocalizingthefacetsthathavepotentialintersectionswiththecuttingtool. The methodology first calculates the intersection points between facet edges and thecutter,andthenusingamarkcirclemethoditcreatesintersectionsegmentsbetweentheseintersection points. Finally an undirected graph is used for connecting those intersectionsegmentstoformtheboundaryoftheCWEarea.Asexplainedinsection(1.2.3)thereisacordalerrorinthepolyhedralmodelrepresentations.Inthiscasethecutterdoesnotintersectthe facets on the side-wall surfaces of the removal volume. In this work for solving thisproblemthecutterradiusisenlargedslightly(Figure2.15).Thereforetheentryanglesareslightlylargerwhiletheexitanglesarelessthanwhattheyshouldbe. Figure2.15hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe(a)Chordalerrorinthecutterfacetintersection,and(b)theradiusenlargementofthecutter[90].2.3.3DiscreteVectorBasedMethods SomeresearchondiscretevectorbasedCWEextractionforsupportingprocessmodelinghasbeenreported[18,31,32,42,91].Intheseapproachestheworkpieceisbrokenintoasetofevenly distributed parallel lines which have directions along the vertical z axis of theCartesiancoordinatesystem.Multiplezvaluesarestoredinonevector,allowingextensionsto multi-axis milling with multiple tool passes. The spacing between vectors can bedeterminedbasedonthedesiredaccuracy,localsurfacecurvatureoftheworkpieceandthetoolsize.CWEsareextractedaccordingtorequirementsofthemechanisticforcemodel.Themechanisticforcemodeldividesthecuttingtoolintoasetofaxialdiscelements.Chapter2.LiteratureReview 34 Therearetwoprimaryconcernsforeachdiscelement.Firstconcerniswhetherornotthediscelementisincontactwiththein-processworkpieceandthesecondconcernforadiscengagedwiththein-processworkpieceistodefinethelimitsoftheengagementintermsofentry/exitangles.Theentryandexitanglesarecalculatedbysolvingforthenormaldirectioncomponentofeachintersection,andstoringthemaximumandminimumnormalpositionsfoundduringagiventoolmove(Figure2.16).  Figure2.16hasbeenremovedduetocopyrightrestrictions.TheinformationremovedistheEntry/exitanglecalculationforeachdisc[31].2.3.4OtherExistingMethods SofaritisshownthattherearemainlythreemajorapproachesforextractingCWEs.Alsothere are some other existing methodologies which are not classified under majorapproaches. For example Stori et al. [84] presented a metric based approach for toolpathoptimization in high speed machining. For obtaining cutter engagements a pixel basedsimulationprocedureisused(Figure2.17).InthisworkaFlat-Endmillfollowsa2Dlineartoolpath.Boththelocalin-processgeometryandthecutterisdiscritized.Anyun-machinedareaisrepresentedbya0,andthemachinedareaisrepresentedbya1.Theinstantaneousengagementsareestimatedbycheckingthestatusofthein-processworkpiecegeometryforthepixelsthatoverlapthecircumferenceofthecuttingtoolinthegivencutterlocationpoint. Figure2.17hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthepixelbasedengagementextraction[84]. AlsoGuptaetal.[34]developedananalyticalmethodforcomputingcutterengagementfunctions in 2.5D machining. In this work the analytical approaches are described fordetermining the cutter engagements utilizing the engagement functions for individual halfspacesthatcomprisetheworkpiecegeometry(Figure2.18).Theedgesoftheworkpiecearerepresentedashalfspaces.Theformulaeareobtainedfordeterminingthecutterengagementfunctionsinthecases:circularcutandlinearhalfspace,andcircularcutandcircularhalfspace.Chapter2.LiteratureReview 35 Figure2.18hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe?AlineartoolpathandregionexpressedasaBooleanexpressionofhalfspaces?[34].2.3.5Discussion The purpose of Cutter Workpiece Engagement (CWE) extraction is to identifyengagementconditionsbetweenthecutterandthein-processworkpieceduringthemillingoperation for predicting the cutting forces. From the above discussion it can be seen thatResearch work on CWE extraction can be classified into three main approaches: Solidmodelerbased,polyhedralmodelerbasedandvectorbasedmethodologies.Thereisalwaysatradeoffbetween computationalefficiency andaccuracyinthese approaches. For examplethepolyhedralandvectorbasedapproachesgenerateapproximatedCWEresultsandbecauseofthistheyrequireashortercomputationaltimethandoesthesolidmodelerbasedapproach.But on the other hand in the solid modeler based approach the most accurate CWEs areobtainedwiththecostofhighercomputationaltime.ThesethreemainapproachestoCWEextractionarediscussedinthefollowingparts.square4 In the literature solid modeling approaches are concentrated on simple workpiecesurfaces such as cylinders and planes. But the most complicated CWE calculations occurduring the machining of sculptured surfaces. Because of continuously changing surfacegeometry, modeling CWEs in sculptured surface machining is not so easy. In the givenapproaches the cutters perform either 2 ? -axis machining or 3-axis machining withascendingmotioninwhichonlythefrontpartofthecutterremovesmaterial.Butsometimeswithrespecttothemotiontypethebacksideorthebottom-flatpartofacuttermayhavecontactwiththein-processworkpiece.ForexampleinaplungingmotionofaFlat-Endmillthebottomflatsurfaceremovesmaterial.ThereforeCWEextractionwithdifferentsurfacesof a cutter must be addressed. Also there is a needed for a 5-axis CWE extractionmethodologyinwhichcutterhaschangingorientations.square4 Fromtheliterature,itcanbeseenthatforthepolyhedralmodelbasedCWEextractionssofar little work has been conducted. Also there is a robustness issue in CWE extractionsbecauseofthechordalerror.Chapter2.LiteratureReview 36square4 In the literature, the proposed discrete vector based methodologies represent the in-processworkpiecebyusingdiscretevectorswhichlieinonlyonedirectionregardlessofthesurface normal direction. But when the orientation of the cutter axis is parallel to thesevectors,forsomecuttergeometries,extractingCWEsbecomesdifficulti.e.3-axismachiningwithaFlat-Endmill. From the above discussion it can be seen that comprehensive methods for extractingCWEsusingsolid,polyhedralanddiscretevectorapproachesareyettobefullydeveloped.Inthis thesis solid, polyhedral and vector based CWE methodologies are developed using arange of different types of cutters and tool paths defined by 3, 5-axes tool motions. Theworkpiecesurfacescoverawiderangeofsurfacegeometriessuchasthesculpturedsurfaces.In3-axissolidandpolyhedralmodelapproachestoreducethesizeofthedatastructurethatneedstobemanipulatedtheremovalvolumeisusedinsteadofthein-processworkpiece.For5-axismachiningitistheworkpieceitself. Inthe3-axissolidmodelermethodologythecuttersurfacesaredecomposedintosub-constituent surfaces with respect to the feed vector direction. Then these surfaces areintersectedwiththeirremovalvolumesforobtainingtheboundarycurvesoftheclosedCWEarea. Decomposing the cutter surfaces allows CWEs for different parts of a given cuttergeometry,i.e.bottomflatorbacksideofacuttertobemoreeasilyextracted.Duringthematerialremovalprocessonlycertainpartsofthecuttersurfacesareeligibletocontactthein-processworkpiece.Inthisthesisforrepresentingtheseregionsofthecutterboundaryaterminologyfeasiblecontactsurfaces(FCS)isintroduced.Thewordfeasibleisusedbecausealthoughthesesurfacesareeligibletocontactthein-processworkpiece,theymayormaynotremove material depending on the cutter position relative to the workpiece. Because in asolid model the envelope boundary for 5-axis tool motions are approximated by splinecurves,applyingthesamemethodologydescribedforthe3-axismillinggeneratesnonrobustresults.Becauseofthisinthe5-axismethodologyanoffsetbodyobtainedfromthefeasiblecontactsurfaceisintersectedwiththein-processworkpieceatagivencutterlocationpoint.Thenface/faceintersectionsareperformedforobtainingtheboundariesoftheCWEs. For addressing the chordal error problem in polyhedral models a 3-axis  mappingtechnique is developed that transforms a polyhedral model of the removal volume fromChapter2.LiteratureReview 37Euclideanspacetoaparametricspacedefinedbylocationalongthetoolpath,engagementangle and the depth-of-cut. As a result, intersection operations are reduced to first orderplane-plane intersections. This approach reduces the complexity of the cutter/workpieceintersectionsandalsoeliminatesrobustnessproblemsfoundinstandardpolyhedralmodeling.Because in 5-axis tool motions the direction of the tool axis vector continuously changesapplyingthemappingdescribedforthe3-axismillingincreasesthedistortionsofthefacets.Therefore an approach similar to that of the 5-axis solid modeler is developed. In thisapproachtheonlydifferenceistheapplicationoftheface-faceintersections. Inthediscretevectorapproachthecuttersurfacesareslicedalongthetoolrotationaxisandthenthesurfaceofeachsliceisdecomposedintobuckets.Thesizesofthebucketsareadjusted with respect to the resolution of discrete vectors. Later during the simulationdiscretevectorsareintersectedwiththecuttersurfacesforrepresentingthematerialremovalandthebucketswhichcontaintheintersectionpointsareplotted2.4Summary In this chapter, a review of the literature in swept volume generation, in-processworkpiece modeling and CWE extraction are presented. It has been shown that because atoolpathcontainsthousandsoftoolmotionsthesweptvolumegenerationbecomescostlyintermsoflargememoryandcomputationalrequirements.ForthisreasonforNCmachiningthesweptvolumeconstructionmustbeeasyandthegeometricinformationfordefiningthevolume must require less memory. The accuracy of the in-process workpiece modelingeffectsthequalityoftheCWEs.Especiallyintheobjectbasedin-processworkpieceupdatemethods using discrete vectors lying only in one direction generates problems in bothworkpiecesurfacequalityandCWErepresentations.FortheCWEextractionsthereisaneedto develop 3 to 5-axis machining approaches for different types of cutter and workpiecesurfacesinsolid,polyhedralandvectorbasedmodels. 38Chapter3CutterSweptVolumeGeneration Thischapterpresentsananalyticalmethodologytocalculatecutterprofilesformodelingthesweptvolumesofgeneralsurfacesofrevolutionwithrespectto5-axistoolmotions.Inthismethodologythegeometryofacutterisconsideredasacanalsurface.Brieflyacanalsurfaceisanenvelopeofamovingspherewithvaryingradius.Inthismethodology,cuttersurfacesperforming5-axistoolmotionsaredecomposedintoasetofcharacteristiccircles.Forobtainingthesecirclesanewconcepttwo-parameter-familyofspheresisintroduced.Inthisconceptthecenterofamovingsphereisafunctionoftwoparametersrepresentingthecuttersurfaceandthetoolmotion.Ontheotherhandtheradiusofthemovingsphereisonlyafunctionofoneparameter.Foragiven5-axistoolmotionamemberfromthisfamilyofspheresgeneratestwocircles:acharacteristicandagreatcircle.Consideringtherelationshipbetweenthesecirclesananalyticalformulawhichdescribesthesweptenvelopeisdeveloped.Also applying this methodology, the implicit envelope equations of the common cuttergeometriesperforming5-Axistoolmotionsarederived. Theparametricrepresentationsofcanalsurfacesinone-parameterfamilyofspheresarepresentedinsection(3.1).Theconceptofthetwo-parameter-familyofspheresisintroducedin section (3.2), followed by the closed form swept profile equations in section (3.3).Examples from the implementation of the methodology are shown in section (3.4), andfinallythechapterendswiththediscussioninsection(3.5).3.1CanalSurfacesAcanalsurfaceafii10038istheenvelopeofamovingspherewithvaryingradius,definedbyatrajectoryofitscenter(spinecurve)m(t)andaradiusfunctionr(t),wheretelementR.Whentheradiusfunctionr(t)hasaconstantvalue,thecanalsurfaceisalsocalledasapipesurface.Canal surfaces have wide applications in different areas such as shape reconstruction,construction of blending surfaces, transition surfaces between pipes and robotic pathplanning [28,66,83]. Some examples of the canal surfaces are pipe surfaces, tori, naturalChapter3.CutterSweptVolumeGeneration 39quadricsanddupincyclides[50,60].Forsimulationandcomputergraphics,amodelwhichcontainscanalsurfaceshassomeadvantageoussuchasthemodelconstructioniseasy,thegeometricinformationfordefiningthemodelrequireslessspaceandefficiencyishighforrenderingthemodel[45,57,87].Definingequationsofacanalsurfaceareexpressedas[60]   022 =            (3.1)   0minuteminuteminute t         (3.2)where?(t)representsone-parameterfamilyofspheresandP=(x,y,z)isapointonthespheresurface (see Figure 3.1). A canal surface can contain a one parameter set of so calledcharacteristic circles K(t). The characteristic circles are obtained by the followingintersection   )minute               (3.3)where the plane ?' is perpendicular to the vector ofmminute . The reality of a canal surface ingeneral depends on )minute and the length ofmminute . If the envelope is real then the followingconditionholds   022 greaterequalminuteminute t              (3.4)3.1.1ExplicitRepresentationofCanalSurfacesFig. 3.1 illustrates a canal surface with definingparameters. In this figure for a givenparametervaluet,amovingspheretouchesthecanalsurfaceafii10038atacharacteristiccircleK(t).AcanalsurfacecanbeparameterizedbyusingK(t).ForparameterizationthecenterC(t)andtheradiusR(t)ofthecharacteristiccircleareused.Thecentercanbeexpressedas   )( )()(cos)()()( ttttrtt mmmC minuteminute+= phi1          (3.5)Chapter3.CutterSweptVolumeGeneration 40where ) isanangle(Fig.3.1)betweentwovectors )minute and ),anditsatisfiesthefollowingdotproductequation   )minuteminute       (3.6)Figure3.1:GeometricdescriptionofacanalsurfaceUsingEqs.(3.1),(3.2)and(3.6)thefollowingtrigonometricequationisderived   )()()()( )())(()(cos ttrttP ttPt mmm m?m minuteminute-=minute- minute-=phi1         (3.7)Bysubstitutingtheequivalentof ) intoEq.(3.5)thecenterofthecharacteristiccircleisexpandedasfollows   2)( )()()()()( tttrtrtt mmmC minuteminuteminute-=           (3.8)AlsotheradiusR(t)ofthecharacteristiccircleisobtainedbyChapter3.CutterSweptVolumeGeneration 41   )())()(sin)()( 22tttrttrtRmmminuteminute=        (3.9)where the equivalent of ) is derived from Eq.(3.7). A canal surface afii10038 can beconsideredasasetofcharacteristiccircles,andthusbyusingEqs.(3.8)and(3.9)afii10038canbeexpressedas   ))       (3.10)wherethetaisanangleofthecharacteristiccircleanditchangesintherangesof0to2pi.AlsoinEq.(3.10),M(t)andB(t)aretheunitnormalandbinormalvectorsoftheFrenettrihedronandtheyareontheplane?'(t)whichcontainsthecharacteristiccircle.Forthecanalsurfacethefrenettrihedronisformedwiththefollowingcomponents   )( )(ttmmminuteminute , )()( )()( tt tt mm mm minuteminute?minute minuteminute?minute , )    (3.11)whereT(t)istheunittangentvectorinthedirectionofmminute .ItisassumedinEq.(3.11)thatthespinecurvemisbiregular,inotherwordsalongmthefollowingconditionisfulfilled   0              (3.12)3.1.2CuttingToolGeometriesasCanalSurfaces AccordingtotheAutomaticallyProgrammedTool(APT)definition[46]millingcutterscanbedefined.Thecuttergeometriesarerepresentedbynaturalquadricsandtorus.Naturalquadricsconsistofthesphere,circularcylindersandthecone.Togetherwiththeplane(adegeneratequadric)andtorustheseconstitutethesurfacegeometriesfoundonthemajorityofcuttersusedinmilling.Forexampleaballnoseendmillisdefinedbytwonaturalquadricsurfaces?sphericalandcylindrical.OtherexamplesareshowninFig3.2.Chapter3.CutterSweptVolumeGeneration 42Figure3.2:SometypicalmillingcuttergeometriesAlsothecuttersshown inFig.3.2 canbeconsidered ascanalsurfaces.Forexample,ifaspheremovesalongastraightlineandtheradiusofthesphereincreaseslinearly,a canalsurfacesocalledafrustumofaconeisgenerated(Fig.3.3(a)).Ontheotherhandifaspherewithaconstantradiusfollowsaspinecurvetrajectory,itenvelopesaspecialcanalsurfacecalled a pipe surface. Two common pipe surfaces cylinder and torus are built when thespheremovesalonglinearpathandacircularpathrespectively(Fig.3.3(b)-(c))Figure3.3:Generatingcuttergeometrieswithamovingsphere3.2Two-Parameter-FamilyofSpheresinMulti-AxisMilling Itisshowninsection(3.1)thatamovingsphere?(t)followsaspinecurvetrajectoryanditenvelopesacanalsurfaceafii10038,wheretisaspinecurveparameter.ButwhenacutterChapter3.CutterSweptVolumeGeneration 43performsamillingsimulation,amovingsphereofthecuttergeometryfollowstwodifferenttrajectories: the first one is the spine curve and the second one is the toolpath trajectoryrepresented by the parameter uelementR. In this case the center of the moving sphere can berepresentedbytwoparameters.Thetwo-parameter-familyofspherescanbedefinedby  ?(t,u):?(P-m(t,u))2?r(t)2=0          (3.13)where ?(t,u) represents two-parameter family of spheres and P is a point on the spheresurface.NotethatinEq.(3.13),althoughthecenterofthesphere m is afunctionoftwoparameterstandu,theradiusrisonlyafunctionoftheparametert.Thisisbecausethecutterradiusonlychangesalongthespinecurveforagivencutterlocation.Let?tand?ubethe partial derivatives of ?(t,u) with respect to t and u. These partial derivatives can beexpressedas  ?t(t,u):?(P-m(t,u))?mt+rrt=0         (3.14)  ?u(t,u):?(P-m(t,u))?mu=0           (3.15)where mt and mu are the partial derivatives of m(t,u) with respect to correspondingparameters.Notethatwhenthecuttergeometryispipesurfacertbecomeszero.Eqs.(3.14)and (3.15) represent two planes with the surface normals mt and mu respectively. In thissection it will be shown that under a tool motion the two-parameter-family of spheresgenerates so called grazing points on the cutter envelope surface. Grazing points are thediscretepointsthatthesurfaceofthecutterwillleavebehindasitmovesinthespace.Thefollowingremarksandthepropertymotivatefindinggrazingpoints.Remark3.1:AcharacteristiccircleKembeddedin?(t,u)isasolutionofthesystemofEqs.(3.13)and(3.14)   )t             Chapter3.CutterSweptVolumeGeneration 44This is an intersection of a sphere ? with a plane ?t which has a normal vector in thedirectionofmt.Remark3.2:Bydefinitionagreatcircleisasectionofaspherethatcontainsadiameterofthesphere[44].AgreatcircleSisasolutionofthesystemofEqs.(3.13)and(3.15)   )u Alsothisisanintersectionofasphere?withaplane?uwhichhasanormalvectorinthedirectionofmu.Itcanbeconcludedfromremarks(3.1)and(3.2)thatthecharacteristicandthegreatcircleslieontheplanes?tand?urespectively.ThesecirclesareillustratedinFig.3.4foragivenparametersettandu.   Figure3.4:(a)ThecharacteristicK,and(b)thegreatScirclesProperty3.1:(i)Intersectingcharacteristicandgreatcirclesgeneratespointsonthecutterenvelopesurface(seeFig.3.5)andalsothesepointsarethesolutionofthesystemofEqs.(3.13-3.15)(ii)AlternativelyifapointP=(x,y,z)isoneoftheseintersectionpointsthenitsatisfiesthefollowingsystemofenvelopeequations[82] 0              (3.16)   0),,,( =partialdiffpartialdiff u uzyxf       (3.17)Chapter3.CutterSweptVolumeGeneration 45where ) implicitly represents the family of cutter surfaces with respect to thetoolpathparameteru.Figure3.5:Intersectingthecharacteristic(K)andthegreat(S)circlesAccordingtoproperty(3.1)ifapointPliesonthecutterenvelopesurfacethenPsatisfiesbothcircleequationsi.e.PelementKintersectionS.TherelativepositionsoftwocirclesKandSmaybeclassifiedintothreecases:(i)theyintersectattwodifferentpoints,(ii)theyintersectatonepointtangentiallyand(iii)theydon?tintersect(Fig.3.6).Figure3.6:IntersectioncasesbetweenKandS.Thecommoncuttergeometriesaredefinedbycylinder,frustumofaconeandtorussurfaces.Inthefollowingsubsectionsusingthosegeometrieswhichperform5-axismotionthevalidityofproperty(3.1)willbeanalyzed.Alsoforthesegeometriestheimplicitenvelopeequationswillbederived.Laterinsection(3.3)themethodologywillbegeneralizedforasurfaceofrevolution.5-axistoolmotionsarerepresentedbythetrajectoryofthetoolcenterpointF(u),andtheinstantaneousorientationofthetoolaxisA(u)whichisalwayscoincidentwiththeChapter3.CutterSweptVolumeGeneration 46toolrotationaxis.UsingF(u)andA(u),aToolCoordinateSystem(TCS)onthetoolcanbedefinedbyasetofmutuallyorthogonalunitvectorsn,d,andeas A ,if 0partialdiffpartialdiff uA then |/| / uupartialdiffpartialdiff partialdiffpartialdiff AA and n  or                      (3.18) if 0partialdiffpartialdiff uA and 0partialdiffpartialdiffuF then|uupartialdiffpartialdiff?partialdiffpartialdiff?= FAFAe and n AccordingtoEq.(3.18)thepositionandtheorientationofthecuttermustbepiecewisedifferentiable.Andalsoplungingmotionalongthecutterrotationalaxisisnotallowed.3.2.1ApplyingtheMethodologyontheCylinderSurface Fig.3.7illustratesacylindersurfacewithamemberoftwo-parameter-familyofsphereswhichisalsocalledmovingsphere.InthisfigureFandFurepresentthetoolcenterpointwhichisdefinedatthetoolbottomsurfaceandthevelocityvectoratthispointrespectively.Figure3.7:Themovingsphereofacylindersurfacein5-axismotionThecenterofthemovingspherecanbeexpressedbyChapter3.CutterSweptVolumeGeneration 47  m(t,u)=F(u)+tn(u)             (3.19)wherenisafunctionoftheparameteru.Aftertakingpartialderivativesofmwithrespecttoparameterstandu,thefollowingequationsareobtained.  mt(t,u)=n(u)               (3.20)  mu(t,u)=Fu(u)+tnu(u)  (3.21)wheresubscriptsrepresentthecorrespondingderivatives.Becausethecylindersurfaceisapipesurface,theradiusofthemovingsphereisconstanti.e.r(t)=randthusrt=0.Whenm,mt, mu, r and rt are substituted into Eqs. (3.13-3.15), they yields the following expandedversionsrespectively. (P?F(u))?(P?F(u))?2t{(P?F(u))?n(u)}+t2?r2=0(3.22) (P?F(t)?tn(u))?n(u)=0       (3.23) (P?F(u)?tn(u))?(Fu(u)+tnu(u))=0          (3.24) Accordingtoproperty(3.1)ifapointPsatisfiesthesystemofEqs.(3.22?3.24)thenitliesattheintersectionofthecharacteristicandthegreatcircles.Thecharacteristiccircles(K)arethesolutionofsystemofEqs.(3.22)and(3.23),andthegreatcircles(S)arethesolutionofsystemofEqs.(3.22)and(3.24).ForsolvingthissystemofequationstheparametertfromEq.(3.23)isextractedas t=(P?F(u))?n(u)                (3.25) andthenitispluggedintoEq.(3.22).Thisyieldsthefollowingimplicitrepresentationofthecylindersurfacein3DEuclidianspace (P?F(u))?(P?F(u))?{(P?F(u))?n(u)}2?r2=0       (3.26)Chapter3.CutterSweptVolumeGeneration 48wherePisapositionofapoint(x,y,z)onthecylindersurface.Theaboveequationcanalsobedenotedbyf(x,y,z,u)=0.ByperformingdotproductoperationsEq.(3.24)isexpandedas (P?F(u))?Fu(u)?t{Fu(u)?n(u)+(P?F(u))?nu(u)}=0      (3.27)Noticethatinthisexpansionn(u)?nu(u)=0.SubstitutingEq.(3.25)intoEq.(3.27)yields (P?F(u))?Fu(u)?(P?F(u))?n(u){Fu(u)?n(u)+(P?F(u))?nu(u)}=0 (3.28)Eq. (3.28) is the partial derivative of the Eq. (3.26) with respect to u and also it can bedenotedby 0 .Thustheproperty(3.1)holdsforthecylindricalsurface.3.2.2ApplyingtheMethodologyontheFrustumofaConeSurface The5-axismotionofaconicalsurfacewithitsmovingsphereisillustratedinFig.3.8.InthisfigureV(u)andalpharepresenttipoftheconeandtheconehalfanglerespectively.AlsothetoolcenterpointF(u)isthecenterofthelowerbase.Figure3.8:Themovingsphereofaconesurfacein5-axismotionChapter3.CutterSweptVolumeGeneration 49Thetipoftheconecanbeexpressedby(seeFig.3.8) )tan ualpharb n                (3.29)whererbistheradiusoflowerbase.Alsofromthisfigure,thecenterofthemovingspherecanbeobtainedby m(t,u)=V(u)+tn(u)               (3.30)Thepartialderivativesofmwithrespecttotanduarederivedas mt(t,u)=n(u)                 (3.31) mu(t,u)=Vu(u)+tnu(u)              (3.32)Forthefrustumofaconesurfacetheradiusofthemovingspherelinearlyincreases.Thisradiusanditsderivativecanbeexpressedbyr(t)=tsinalphaandrt(t)=sinalpharespectively.Bysubstitutionsofm,mt,mu,randrt,thesystemofEqs.(3.13-3.15)isexpandedrespectivelyas (P?V(u))?(P?V(u))?2t{(P?V(u))?n(u)}+t2cos2alpha=0      (3.33) (P?V(u)?tn(u))?n(u)+tsin2alpha=0          (3.34) (P?V(u))?Vu(u)?t{Vu(u)?n(u)?(P?V(u))?nu(u)}=0 (3.35)ForsolvingthesystemofEqs.(3.33-3.35),theparametertfromEq.(3.34)isextractedas alpha uuP 2cos )())(( n?V-                 (3.36)SubstitutingEq.(3.36)intoEq.(3.33)yieldstheimplicitrepresentationf(x,y,z,u)=0ofthefrustumofaconesurfacein3DEuclidianspace (P?V(u))?(P?V(u))?(1+tan2alpha){(P?V(u))?n(u)}2=0      (3.37)Chapter3.CutterSweptVolumeGeneration 50Also plugging Eq. (3.36) into Eq. (3.35) yields the following partial derivative0 oftheEq.(3.37)withrespecttou (P?V(u))?Vu(u)+(1+tan2alpha){(P?V(u))?n(u)]{Vu(u)?n(u)?         (P?V(u))?nu(u)}=0       (3.38)Thustheproperty(3.1)holdsforthefrustumofaconesurfacealso.3.2.3ApplyingtheMethodologyontheToroidalSurface It is shown in previous two sections that for the cylinder and the frustum of a conesurfaces the spine curve or trajectory of the moving sphere is linear. But in case of thetoroidal surface, the moving sphere follows a circular trajectory. Figure 3.9 illustrates atoroidalsurfacewiththemovingsphere.Figure3.9:Themovingsphereofatoroidalsurfacein5-axismotionThecenterofthemovingspherecanbeexpressedas m(t,u)=F(u)+Rcostd(u)+Rsinte(u)          (3.39)wheretwoorthogonalunitvectorsdandedefinetheplaneofthecirculartrajectoryin3DEuclidianspace,alsoRandFarethemajorradiusandthecenterofthetorus.TakingthepartialderivativesofmwithrespecttotanduyieldsChapter3.CutterSweptVolumeGeneration 51 mt(t,u)=?Rsintd(u)+Rcoste(u)           (3.40) mu(t,u)=Fu(u)+Rcostdu(u)+Rsinteu(u) (3.41)Becausethetorusisapipesurfacetheradiusofthemovingsphereisconstanti.e.r(t)=randthusrt=0.Whenm,mt,mu,randrtarepluggedintoEqs.(3.13-3.15),thefollowingexpandedversionsareobtainedrespectively. (P?F(u))?(P?F(u))?2Rcost(P?F(u))?d  ?2Rsint(P?F(u))?e+R2?r2=0(3.42) (P?F(u)?Rcostd?Rsinte)?(?Rsintd+Rcoste)=0(3.43) (P?F(u)?Rcostd?Rsinte)?(Fu(u)+Rcostdu+Rsinteu)=0(3.44)SolvingEq.(3.43)fortheparametertyieldsthefollowingtwoexpressions2}))({(}))(({))cos?eduut--=    (3.45)2}))({(}))(({))sin?eeuut--=           (3.46) PluggingcostandsintintoEq.(3.42)yieldsthefollowingimplicitrepresentationf(x,y,z,u)=0ofthetoroidalsurfacein3DEuclidianspace.22}))?eur-=   (3.47)Theaboveequationcanbearrangedbysquaringthebothsidesandthisyields0}22=-?eur (3.48)Chapter3.CutterSweptVolumeGeneration 52ForexamplewhenP=(x,y,z),F=(0,0,0),d=[100]Tande=[010]TarepluggedintotheEq.(3.48),thewellknownformulaofatorusisobtainedintheoriginoftheEuclidianspaceasfollows (x2+y2+z2+R2?r2)2?4R2(x2+y2)=0  Also plugging cost and sint into Eq. (3.44) yields the following partial derivative0 oftheEq.(3.47)withrespecttou. ( ) 022 =partialdiffpartialdiff ?euu     (3.49)Thustheproperty(3.1)holdsforthetoroidalsurfacealso.3.3ClosedFormSweptProfileEquations In section (3.1) a canal surface has been described explicitly by K(t,theta), where theparametertrepresentsthespinecurveorthetrajectoryofthemovingspherecenter.Laterinsection(3.2),amethodologybasedonthetwo-parameter-familyofspheresconcepthasbeenintroduced. For describing the motion of the moving sphere, a new parameter u whichrepresents a toolpath has been introduced. In this section by utilizing the two-parameterfamilies of spheres, the closed form swept profile equations for the general surface ofrevolution will be derived. In general the radius R(t,u) and the center C(t,u) of thecharacteristiccirclearedefinedbythefollowingequations. u),())(),( 22tttrutRttmm -=           (3.50) 2),( ),()()(),(),( ut uttrtrututttt mmmC -= (3.51)wherethecenterofthemovingspherem(t,u)definedbytwoparameters.Notethattheseequations are very similar to those of canal surfaces defined in section (3.1.1), where theChapter3.CutterSweptVolumeGeneration 53centerofthemovingsphereisthefunctionoft.AcuttersurfaceconsideredasasetofthecharacteristiccirclescanbedefinedbyusingEqs.(3.50)and(3.51)as ))21 u    (3.52)where [ ].InEq.(3.52),twoorthogonalunitvectorsw1(t,u)andw2(t,u)definetheplaneofthecharacteristiccircleK.Ingeneralthecuttergeometrieshaveeithercircularorlinearspinecurves.Forexample,thetoroidalsurfacehasacircularspinecurve(Figure3.10).Inthiscasew1(t,u)andw2(t,u)representrespectivelythenormalM(t,u)andbi-normalB(t,u)unitvectorsoftheFrenetframe,i.e.theframeformedbythefollowingthreevectorstutuupartialdiffpartialdiffpartialdiffpartialdiff= ),()) mmT ,tutututuu2222)))partialdiffpartialdiff?partialdiffpartialdiffpartialdiffpartialdiff?partialdiffpartialdiff= mmmM , ) (3.53)whereT(t,u)isthetangentvectoralongthespinecurvem(t,u).Figure3.10:ThecharacteristiccircleKofthetorusintheFrenetFrameOntheotherhand,cylindricalandconicalsurfaceshavelinearspinecurves(Figure3.11).Inthis case the Frenet Frame components are not applicable for them because the curvaturevanishesandthefollowingconditiondoesnotholdChapter3.CutterSweptVolumeGeneration 54 0),(),( 22notequalpartialdiffpartialdiffpartialdiffpartialdiff t utt ut mm                (3.54)Forthissituationw1(t,u)andw2(t,u)inEq.(3.52)representdandecomponentsoftheToolCoordinateSystem(TCS)respectively.Figure3.11:ThecharacteristiccircleKinTCSThe points on the cutter surface that lie on the moving sphere are embedded in thecharacteristiccircleK(t,u,theta).Thusthenormalvectorofthecuttersurfacecanbeexpressedas(seeFigures3.10and3.11) )  (3.55)Accordingtoproperty(3.1)ifapointisontheenvelopesurfacethenitisembeddedinthecirclesKandS.Thusthefollowingequationholds 0u     (3.56)wheremuisthevelocityvectorofthemovingspherecenter.PluggingEq.(3.55)intoEq.(3.56)yields ( ) 0u            (3.57)Chapter3.CutterSweptVolumeGeneration 55AfterreplacingKwithitsequivalentfromEq.(3.52)andtakingthedotproduct,Eq.(3.57)takesthefollowingtrigonometricform 0012 =     (3.58)where ( ) ( ) ( ))u ,12 m  ( ) ( ) ( ))u ,21 m  ( ) ( )( ) ( ) ( ) )utAutt ,,, 20mm= Equation(3.58)canbewritteninthefollowingform ( ) 002221 =              (3.59)where ( )1212 , A- .SolvingEq.(3.59)forthetayieldsbraceexbraceexbraceexbraceleftbtbraceexbraceexbraceexbraceleftmidbracelefttp-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+=--betabeta222102222101sinsinAAAA    Under the condition of 202221 A , the inverse function )1 x- is in the range of]. In this case, there exist real solutions for 2,1theta and the characteristic circle Kcontains one or two grazing points. Plugging 2,1 into Eq. (3.52) yields the closed formequationofthecutterenvelopesurface ))22,112,1 u    (3.60)Chapter3.CutterSweptVolumeGeneration 56Notethat,undertheconditionof 202221 A ,theEq.(3.58)doesnothavearealsolution.Whenthisisthecase,certaincrosssectionsofthecutterdonotcontainanygrazingpoint(Figure3.12).Figure3.12:LimitsofthegrazingpointsinthegeneralsurfaceofrevolutionAsmentionedbeforeinsection(3.1.2)cylinderandtorussurfacesarecalledpipesurfaces.Bydefinitionapipesurfaceisanenvelopeofthefamilyofsphereswithaconstantradius.Inthissurfacetypethecentersofthecharacteristiccircleandthemovingsphereequaltoeachother.Eq.(3.14)whichrepresentsaplanewithanormaldirectionvectormtcontainsanrtterm. For a pipe surface because the radius of the moving sphere r is constant, this termbecomes equal to zero, i.e.  rt = 0. In this case the characteristic circle obtained byintersectingEq.(3.13)andEq.(3.14)becomesagreatcircleofthemovingsphere,lyinginthenormalplaneofthespinecurve.Alsothisgreatcircleisperpendiculartothevectormt.Thusamovingsphereofapipesurfaceunderacuttermotioncontainstwogreatcircles:Thefirstonewhichisperpendiculartovectormtrepresentsthecuttersurfacegeometryandthesecondonewhichisperpendiculartovectormurepresentsthetoolmotion.Bothgreatcirclessharethesamecentermalongaspinecurve(Figure3.13)Chapter3.CutterSweptVolumeGeneration 57       (a)(b)Figure3.13:ThecharacteristiccircleKbecomesagreatcirclein(a)torus,andin(b)cylindersurfacesIfacuttergeometryisapipesurface,thereisanalternativesimplersolutionforobtaininggrazing points on the cutter envelope surface. The following property motivates thisalternativesolutionforpipesurfaces.Property3.2:Asurfacenormalvectorofamovingsphere,generatedbythevectorproductofmtandmu,passesthroughagrazingpointontheenvelopesurface.Accordingtoproperty(3.2),thenormalvectorofamovingsphereNwhichpassesthroughagrazingpointP1andthecenterofthemovingspherem,isperpendiculartobothmtandmu(Figure3.13(a-b)).ThusNcanbeexpressedbythefollowingequation. ( ) ( )ut ,     (3.61)Usingequation(3.61),pointsP1andP2ontheenvelopesurfacecanbecalculatedas ( ) |||| NN,m1 rutP += , ( ) |||| NN,m2 rutP -=       (3.62)Chapter3.CutterSweptVolumeGeneration 58Note that the angle between two grazing points P1 and P2 is 1800 for cutter geometriesrepresentedbypipesurfaces.Adetailedproofoftheproperty(3.2)ispresentedinAppendixA.1.3.4Examples ThepresentedmethodologyhasbeenimplementedusingtheACISsolidmodelingkernel.Figure3.14showstheenvelopesurfacesofFlat-End,Ball-EndandFillet-Endcuttersin5-axismilling.Foreachillustrationthreecuttergeometriesareshownforthecutterlocations:atthebeginning, atthemiddleandattheendofthetoolpathrespectively.Alsoforthesecutterlocationstheircorrespondingsweptprofilescurvescoloredbyredareshown.Inthesefigurescutterenvelopesurfacesaregeneratedbythetwo-parameter-familyofspheres.FortheFlat-EndandtheBall-Endmillsamemberofthisfamilyisshownbyaredsphere.OntheotherhandfortheFillet-Endmilltwospheresareshown,greensphereisforthetoruspartandtheredsphereisforthecylinderpartrespectively.Fortheseexamplesasingletoolpathsegmentisused.InanotherexampleaTaper-Ball-Endmillisperforming5-axistoolmotions(Figure3.15).Thecutterenvelopesurfacesareshownfromdifferentpointofviews.Inthiscasethetoolpathcontainsanumberofsegments.     (a)(b)(c)Figure3.14:Envelopessurfacesof(a)Flat-End,(b)Ball-End,and(c)Fillet-Endcutters.Chapter3.CutterSweptVolumeGeneration 59Figure3.15:EnvelopesurfacesofaTaper-Ball-Endmillfromdifferentpointsofviewin5-axismilling3.5Discussion A typical NC toolpath may contain thousands of tool motions which make thecomputationalcostforcharacterizingthegeometryofeachtoolsweptvolumeprohibitivelyexpensive. In this chapter an analytical approach for determining the shape of the sweptenvelopes generated by a general surface of revolution has been presented. The cuttergeometrieshavebeenmodeledascanalsurfacesgeneratedbytheone-parameterfamilyofspheres.Formodelingthe5-axistoolmotionsanewparameterhasbeenintroducedforthecenterofthemovingsphere.Fordescribingthecutterenvelopesurfacesthetwo-parameter-familyofsphereshasbeenintroduced.Analyticallyithasbeenprovenforageneralsurfaceofrevolutionthatanypointontheenvelopesurfacealsobelongstoamemberfromthetwo-parameter-family of spheres.  In this methodology describing the radius function and thespinecurveofthemovingspheredifferentcuttersurfacescanbeobtained.Fromthissensethemethodologyisindependentofanyparticularcuttergeometry.Theimplementationofthemethodologyissimple,especiallyifthecuttergeometriesarepipesurfacessuchastorusandcircularcylinderfewercalculationsareused.AlthoughtheexamplesfromtheapplicationofChapter3.CutterSweptVolumeGeneration 60themethodology areshownforthe commonmilling geometriesdescribedby7-parameterAPTlikecutters[46],thedevelopedmethodologycanbeappliedtorarecuttergeometriesalso (Figure 3.16). Later in the next chapter it will be shown that using the methodologydevelopedinthischapterbringscomputationalefficiencytotheupdateprocessofin-processworkpiecesrepresentedbydiscretevectors.Thesphericalsurfacehasnotbeenconsideredindevelopments because simply the swept profile of a sphere is the great circle which ispassing through the center of the sphere and having a normal along the direction of themotion. Self intersections in this research have not been considered. Modifications to themethodologywillneedtobedevelopedforthisspecialcase. Figure3.16:Differentcuttergeometriesgeneratedbyamovingsphere 61Chapter4In-processWorkpieceModeling In this chapter the methodologies for modeling the in-process workpiece in millingoperations are presented. During the machining simulation for each tool movement themodificationoftheworkpiece geometryisrequiredtokeeptrackofthematerialremovalprocess.BecausetheNCverificationandCutterWorkpieceEngagement(CWE)extractionaredirectlydependentonthematerialremovalanaccuratein-processworkpiecemodelingisneeded. Several choices for modeling the in-process work piece are exist. The two mostcommon are mathematically accurate solid modeling that is used in CAD systems andapproximatemodelingsuchasthoseusedincomputergraphics:facettedanddiscretevectormodels.Intheworkpieceupdateprocessfirstthesweptvolumeofagiventoolmovementinthecutterlocationdatafileismodeledandthenthisvolumeissubtractedfromthein-processworkpiece.Forupdatingtheworkpiecesurfacesrepresentedbythesolidorfacetedmodelsthirdpartysoftwarescanbeused.ForexampleinthesolidmodelerbasedapproachthecuttersweptvolumescanbegeneratedbyusingtheACISsolidmodelingkernelandthenusingtheBooleansubtractionoperationdefinedinthiskernelthecuttersweptvolumesaresubtractedfrom the in-process workpiece. These update methodologies are briefly introduced in thischapter. On the other hand for the workpiece geometries represented by discrete vectorsefficient in-process workpiece update methodologies for different cutter geometriesperforming multi-axis machining are needed. In the literature [7,8,18,24,32,41,91] thediscretevectorshaveorientationsalongthez-axisofthestandardbasesofR3.Butwhenthein-process workpiece has features like vertical walls and sharp edges, representing theworkpiecewithonedirectionalvectorsgenerateslessaccurateresultsinvisualizationofthefinal product [43] and CWE extractions. For increasing the accuracy more vectors withdifferentorientationsareneeded.Butinthiscaselocalizingthecutterenvelopeduringthesimulation becomes difficult and therefore the computational time increases. Therefore anefficientlocalizationmethodologyisrequired.Alsoinrecentworks[7,8]thetoolaxishasafixedorientationalongthez-axisofthestandardbasesofR3andtheworkpieceupdatesareperformednumericallyforthemillingcuttergeometries.ThereforeinthischapterthevectorChapter4.In-processWorkpieceModeling 62based update methodologies are developed for the vectors having different orientations.These methodologies allow multi-axis milling in which the tool may have an arbitraryorientationinspace.Alsoforthecuttergeometriesdefinedbyacircularcylinder,frustumofacone,sphereandplanetheupdatingcalculationsareperformedanalytically.Forthetorusgeometrynumericalapproacheswithguaranteedrootfindingresultsareutilized. The common milling cutter geometries are presented in section (4.2.1). In theserepresentationscuttersurfacesaredefinedparametricallyattheoriginofthestandardbasesofR3.Inthissectionforderivationsthepropertiesofcanalsurfacesgiveninchapter(3)areutilized.Thevectorbasedworkpiecemodelandthelocalizationofthecutterenvelopearepresentedinsection(4.2.2).Inthefirstpartofthissectionaworkpiecemodelingstrategyisintroducedandinthesecondpartalocalizationmethodologyforthecutterenvelopesurfacesispresented.Theworkpieceupdatemethodologiesinmulti-axismillingfordifferentcuttergeometriesarepresentedinsection(4.2.3).Intheintroductionpartofthissectionthegeneralformulasfordefiningthecutterenvelopesarederivedandtheninthefollowingsubsectionsthese formulas are utilized in the intersection calculations of different cutter geometries.Examplesfromtheimplementationofthemethodologiesareshowninsection(4.2.4),andfinallythechapterendswiththediscussioninsection(4.3).4.1ModelingtheIn-processWorkpieceinSolidandFacettedrepresentation Asolidmodelerbasedin-processworkpieceupdateisillustratedinFigure(4.1).InputsfromaCAD/CAMsystemincludethegeometricrepresentationoftheinitialworkpiece,thetoolpathandthegeometricdescriptionofthecuttingtool.Inthefirststep,thesweptvolumesofthecuttingtoolaregeneratedforagiventoolpath.Theninthesecondstepthesesweptvolumes are subtracted from the initial workpiece sequentially for obtaining the updatedworkpiece (in-process workpiece). In this research for modeling the in-process workpieceandthesweptvolumetheB-repbasedgeometricmodelerACISkernelandC++isutilized.Chapter4.In-processWorkpieceModeling 63Figure4.1:AB-repSolidModelerbasedin-processworkpieceupdate.Inthisresearchforupdatingthein-processworkpiecerepresentedbytessellatedmodelsaprototype system is assembled using existing commercial software applications and C++implementations.Thissystemisshownin(Figure4.2).ThecuttersweptvolumeisgeneratedbytheACISsolidmodelerandthenthissweptvolumeisexportedasaSTLformatusingACIS functions. A Boolean intersection between this swept volume and the current in-process workpiece is performed using the polyhedral modeling Boolean operatorsimplementedinacommercialapplication,MagicsX[51].ThisprototypesystemcreatesSTLmodelofthein-processworkpiece.Figure4.2:APolyhedralModelerbasedin-processworkpieceupdate.Chapter4.In-processWorkpieceModeling 644.2ModelingtheIn-processWorkpieceinVectorBasedRepresentationInthissectionthemulti-axismillingupdatemethodologiesaredevelopedforworkpiecesdefined by discrete vectors with different orientations. For simplifying the intersectioncalculationsbetweendiscretevectorsandthetoolenvelopethepropertiesofcanalsurfacesareutilized.4.2.1MillingCutterGeometriesinParametricForm Themethodologiesforupdatingthein-processworkpiecerepresentedbydiscretevectorsaredevelopedforcutterswithnaturalquadricsandtoroidalsurfaces.Naturalquadricsconsistofthesphere,circularcylindersandthecone.Togetherwiththeplane(adegeneratequadric)and torus these constitute the surface geometries found on the majority of cutters used inmilling.Inthischapterthemajorcuttergeometries(circularcylinder,frustumofaconeandtorus) are defined parametrically by using the properties of the canal surfaces defined inchapter(3).Theparametricrepresentationofacanalsurfacegeneratedbyaone-parameterfamiliesofspheresisgivenbythecharacteristiccircleas ))21 t   (4.1) where thetaelement[0,2pi]. C(t) and R(t) represent the center and radius of the characteristic circlerespectivelyandtheyareexpressedasfollows 2)))()()()(tttrtrttmmmCminuteminuteminute-=  and          (4.2) )( )()(22tttrtRmmminuteminute= (4.3)wherefortelementR,m(t)andr(t)arethecenterofthemovingsphereandtheradiusfunctionrespectively. The center m(t) is located on the spine curve (the trajectory of the movingsphere).AlsoinEq.(4.1)twoorthogonalunitvectorsw1(t)andw2(t)definetheplaneofthecharacteristic circle K(t, theta). Figure (4.3) illustrates the major cutter geometries as canalChapter4.In-processWorkpieceModeling 65surfaceswithrespecttothestandardbasisofR3denotedby(i,j, k) unitvectorsandtheoriginO.Inthisfigurethecuttergeometrieshaveeithercircularorlinearspinecurves.Iftheyhavelinearspinecurves(SeeFigure4.3,a-b)w1(t)andw2(t)representthexandyaxisofthelocalcoordinatesystemrespectively.Ontheotherhandiftheyhaveacircularspinecurve (see Figure 4.3 -c) w1(t) and w2(t) represent respectively the normal M(t) and bi-normalB(t)unitvectorsoftheFrenetframe,i.e.theframeformedbythefollowingthreevectors )( )(ttmmminuteminute , )()( )()()( tt ttt mm mmM minuteminute?minute minuteminute?minute= , )    (4.4)whereT(t)istheunittangentvectorinthedirectionof ) .ItisassumedinEq.(4.4)thatthe spine curve represented by m(t) is biregular, in other words along m(t) the followingconditionisfulfilled 0               (4.5)   (a)(b)(c)Figure4.3:Cuttergeometriesascanalsurfaces:(a)cylinder,(b)frustumofaconeand(c)torus).InthefollowingsubsectionstheparametricrepresentationsofthesurfacegeometriesshowninFigure(4.3)arefoundbyusingtheEqs.(4.1-4.5).Laterinsection(4.2.3)theywillbeusedforupdatingthein-processworkpiece.Chapter4.In-processWorkpieceModeling 664.2.1.1CylinderSurface The cylinder surface can be parameterized by using the characteristic circle given inEq.(4.1).TwoofthevariablesinthisequationC(t)andR(t)dependonthecenterandtheradius function of a moving sphere. From Figure (4.3 -a), the center of a one-parameterfamilyofspherescanbewrittenasbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=tt 00) k                  (4.6)wheretheparameterttakesvaluesbetweenzeroandtheheightofthecylinder,i.e.telement[0,h].The partial derivative of the Eq. (4.6) with respect to t equals to the unit vector k andthus 1.Asexplainedinchapter(3),thecylindersurfaceisalsoapipesurfacewithaconstantradiusr.Thereforethederivativeoftheradiusfunctionisequaltozeroi.e. 0 .ThusthecenterandtheradiusofthecharacteristiccirclecanberepresentedbyC(t)=tkandR(t)=rrespectively.PluggingthecenterandtheradiusintoEq.(4.1) yieldsthecylindersurfacegeometryinthefollowingparametricform ( )bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=trrrcylinder thetathetatheta sincossin j         (4.7)4.2.1.2FrustumofaConeSurface AsillustratedinFigure(4.3-b),theoriginofthestandardbasisOislocatedabovethecutterbottomcenterwithdistanceequaltoRbtanalpha,whereRbandalpharepresenttheradiusofthe cutter bottom and the cone half-angle respectively. The center of a moving sphere isobtainedby k ,wherethelimitsoftfromFigure(4.3-b)aredefinedby bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttpelementalpha2,0ht                  (4.8)Chapter4.In-processWorkpieceModeling 67where h represents the height.  For the frustum of a conesurface the radius of a movingspherer(t)isgivenby alphacosbR                 (4.9)Whenm(t),r(t)andtheirderivativeswithrespecttotarepluggedintoEqs.(4.2)and(4.3),thecenterandtheradiusofthecharacteristiccircleisobtainedasfollows C(t)=(tcos2alpha?Rbtanalpha)k              (4.10-a) R(t)=tsinalphacosalpha+Rb (4.10-b)             PluggingEqs.(4.10-a,b)intoEq.(4.1)yieldsthefrustumofaconesurfacegeometryinthefollowingparametricform( ) ( ) ( )( )( )( )bracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp-++=+alphathetathetathetatansincossin22bbbbRRRR j (4.11)4.2.1.3TorusSurface InFigure(4.3-c)atorussurfaceisshownwiththeoriginOinthecenterofthetorus.Foratorusthespinecurveisacircleandthereforethecenterofamovingsphereisobtainedbybracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=0sincossin ttt rhorhorho   (4.12)where rho isthemajorradiusofthetorusandtelement[0,2pi].ThepartialderivativeofEq.(4.12)withrespecttotisobtainedasfollowsChapter4.In-processWorkpieceModeling 68               (4.13)Asapipesurfacetorushasaconstantradiusrandtherefore 0 .When ) , ) ,r(t)and )arepluggedintoEqs.(4.2)and(4.3),thecenterandtheradiusofthecharacteristiccircleareobtainedasfollows                (4.14-a) r                    (4.14-b)AlsofordescribingthecharacteristiccircleofthetorusthecomponentsoftheFrenetframeareneeded.ForthispurposethepartialderivativeofEq.(4.13)withrespecttotisobtainedasfollows                (4.15)Plugging ) , ) and )intoEq.(4.4)yields T(t)=?sinti+costj,M(t)=k,B(t)=costi+sintj      (4.16)FinallypluggingEqs.(4.14-a,b)and(4.16)intoEq.(4.1)yieldstorussurfacegeometryinthefollowingparametricform( ))( )( )bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp++=+thetathetathetathetacossincossinrttttorus j  (4.17)4.2.2WorkpieceModelandLocalization Therearetwomainapproachesforrepresentingthein-processworkpieceusingdiscretevectors [43]: Discrete Normal Vector and Discrete Vertical Vector approaches. In theChapter4.In-processWorkpieceModeling 69DiscreteNormalVectorapproach(Figure4.4-a)aworkpiceismadeupofdiscretevectorswhose directions are the surface normals of the workpiece. In this representation alldirectionsarenotnecessarilyidentical.Alsospacingmayvarydependinguponthesurfacelocalproperties.Ontheotherhand,intheDiscreteVerticalVectorapproach(Figure4.4-b)alldiscretevectorsoftheworkpiecemodellieinonlyonedirectionregardlessofsurfacenormaldirections,wherethedirectionsarealongtheverticalz-axisofthestandardbasisofR3.(a)(b)Figure4.4:Representingtheworkpiecesurfacesby(a)surfacenormalvectorsand(b)verticalvectors.The workpiece model can have vertical walls and sharp edges (Figure 4.5). The DiscreteNormal Vectorapproachcanrepresentthosesurfacefeatureswell with respecttoa giventolerance.Butbecauseinthisapproachthedirectionsofdiscretevectorsarenotidentical,localizingthecutterenvelopesurfaceduringthemachiningsimulationbecomesdifficultandthereforethecomputationaltimeincreases.Duringthemachiningsimulationthemosttimeconsuming process is the localization of the cutter envelope [43]. Although the DiscreteVerticalVectorapproachgiveslessaccurateresultsintherepresentationoftheverticalwallsandthesharpedges,itiscomputationallyfasterandthelocalizationofthecutterenvelopeiseasy.Chapter4.In-processWorkpieceModeling 70Figure4.5:RepresentingthefeatureshapesinDiscreteNormalvectoranddiscreteverticalvectorapproaches. For some workpiece geometries 3-axis machining is not suitable for updating thesurfaces.Forexample,ifthepartisdeep(Figure4.6-a)a3-axistoolpathisnotsufficientenoughforfinishing.Norcantheybeusedformillinghardmaterialwithlongcutterswithoutgenerating a bad surface finish and long machining times. For solving this problem thespindleofthecutteristiltedandashortercuttingtoolisused(Figure4.6-b).Thistypeofthemachining is called as (3+2)-axis milling in which a cutter can have an arbitrary fixedorientationinspace.Inthisthesisthediscretevectorupdatemethodologiesaredevelopedfor(3+2)-axis milling. Also this methodology can be adapted to 5-Axis milling with someapproximations. Figure4.6hasbeenremovedduetocopyrightrestrictions.Theinformationremovedisthe(a)3-Axismachining,and(b)(3+2)-Axismachining[36].Inthisresearchtheworkpieceismodeledbydiscretevectorshavingorientationsinthedirectionsofx,y,z-axesofthestandardbasisofR3.ThereforethisrepresentationcanbeseenasanenhancedversionoftheDiscreteVerticalVectorapproach.Inthisrepresentationmorevectorsindifferentdirectionsareusedandthus,especiallywhentheworkpiecehasfeatureslikeverticalwallsandsharpedges,thequalityinthevisualizationofthefinalproductandCWE area is increased. Also the localization advantage of the Discrete Vertical Vectorapproach is preserved. In machining simulations the initial workpiece geometry is mostlyrepresented by a rectangular stock. The modeling this stock with discrete vectors is easybecause for a given direction all discrete vectors have the same height and the sameChapter4.In-processWorkpieceModeling 71orientation.Butontheotherhandiftheinitialworkpiecesurfaceshavearbitraryshapesthensomestepsmustbetakenforrepresentingtheinitialworkpieceusingdiscretevectors.Inthisresearchtheworkpieceisinitiallyrepresentedbyatessellated(triangularfacets)model.Fortransforming this tessellated model into a discrete vector model first the Axis AlignedBoundingBox(AABB)oftheinitialworkpiecemodelisobtained(Figure4.7-a).Forthispurposethemaximumandminimumx,y,zcoordinatesofthefacetsareused.TheAABBisarectangularsix-sidedbox(in3D,foursidedin2D)categorizedbyhavingitsfacesorientedinsuchawaythatitsfacenormalsareatalltimesparallelwiththeaxesofR3.ThenfromthefacesrepresentedbyOxy,OxzandOyzcoordinatesystems(seefigure4.7-b)raysareshotthroughthetessellatedmodelinthedirectionsofthez,yandx-axesofR3respectively.Forfindingtheportionsoftherayswhichrepresenttheworkpiece,triangle/rayintersectionsareperformed.Ifarayismadeupofseveralsublinesthenusingthelinkedlistdatastructuretheselinesareconnectedtoeachother.Alsolaterduringthemachiningsimulationifalineispartitionedintosmallerlinesagainthelinkedlistdatastructureisusedforconnectingthem.Notethatforlocalizationpurposes(itwillbeexplainedlater)thediscretevectorsarelocatedinthebuckets.     (a)(b)Figure4.7:RepresentinginitialworkpiecewithdiscretevectorslocatedinXY,XZandYZplanes Forupdatingthein-processworkpiecethediscretevectorsareintersectedwiththetoolenvelope. It would be computationally expensive to calculate the intersections of all thediscretevectorsforeachtoolmovement.Itisthereforedesirabletolocalizethecalculationsbyeliminatingfromconsiderationthevectorswhichhavenopossibilityofintersectingthetool envelope for the given toolpath segment. In this research for localizing the cutterenvelopeduringthesimulationtheboundingboxconceptisutilized.ThereareseveraltypesChapter4.In-processWorkpieceModeling 72of bounding boxes in 2D space such as a circle, Axis Aligned Bounding Box (AABB),OrientedBoundingBox(OBB),convexhulletc.Thereisalwaysatradeoffbetweenabetterbound and a faster test in these approaches. For this research the AABB approach in 2Dspace is utilized. The best feature of the AABB is its fast overlap check, which simplyinvolves direct comparison of individual coordinate values. In this work for a given toolmovementthreeAABBswhicharetheprojectionsofthetoolenvelopeontoOxy,OxzandOyzcoordinatesystemsaregenerated.InthefollowingsectionthestepsforobtainingtheAABBofthetoolenvelopeinOxycoordinatesystemaregiven.TheAABBsinOxzandOyzcoordinatesystems can beobtainedfollowingthesamesteps. ThelocalizationofthetoolenvelopeinOxycoordinatesystemhasfoursteps.ForthesestepspleaserefertoFigure(4.8).InthisfigureaTaper-Ball-EndmillismovingbetweenthecutterlocationsPSandPE.Step1:Findingtheboundingcylinderin3D:InthisstepatthecutterlocationpointPStheboundingcylinderofthecutterhavinganorientationinthedirectionofthetoolaxisisfound.Thedimensionsoftheboundingcylinderaredefinedbytheheightandthelargestradiusofthecutter.ForexamplefortheTaper-Ball-Endmillshownin(Figure4.8)thetopcircle(ctop)hasthelargestradiusr.Thereforeusingthesameradiusrthebottomcircle(cbottom)oftheboundingcylindercanbeobtained.Thecenterofthecbottomislocatedatthetooltippointofthecutter.Thereisasymmetryplanewhichcontainsthetoolrotationaxisandpassesthroughthecentersofthectopandcbottom.Thenormalvectorofthisplane(Nplane)canbeobtainedby ]plane ? .IntersectingthisplanewithctopandcbottomoftheboundingcylindergeneratesfourpointsI1,I2,I3,andI4respectivelyin3Dspace.Chapter4.In-processWorkpieceModeling 73Figure4.8:AABBofatoolmovementinOxycoordinatesystem.Step 2: Finding the bounding box of the cutter in 2D: Assuming I1 and I4 are theoutermostpoints,twocirclesctopandcbottomandthosefourintersectionpointsareprojectedontoOxycoordinatesystem.ThisprojectiongeneratestheboundingboxofthecutterwhichhasthecornerpointsP1,P2,P3,andP4.Thesecornerpointscanbecalculatedbybracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp+++0//221 yynnr bracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp++-0//221 yynnr                      (4.18)bracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp+++0//224 yynnr bracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp++-0//224 yynnr wherenxandnyarethecomponentsofthetoolrotationaxisvectorn.Chapter4.In-processWorkpieceModeling 74Step3:FindingtheAABBofthetoolmovement:Inthisstagefirstusingthesamesteps(1and2)theboundingboxofthecutterforthelocationofPEisfound.Nowthere aretwoboundingboxeswitheightcornerpointsinOxycoordinatesystem.Thenusingtheminimumandmaximumxandycoordinatesofthoseeightcornerpoints(P1toP8)theAABBofthetool movement is obtained. Note that because the tool performs a motion with a constantorientation, the projections of the intermediate tool motions between PS and PE are alsocontainedinthesameAABBofthetoolmovement.Step4:Findingthesetofbuckets:InthissteptheoverlapcheckisperformedbetweenthebucketsandtheAABBofthetoolmovement.ThebucketsinOxycoordinatesystemcanbethoughtasAABBs.ForfindingtheoverlappingbucketstheAABB?AABBintersectionisperformed.Thealgorithmforthistestcanbefoundin[27].Ifanoverlapisfoundthenthediscretevectorsintheoverlappingbucketareusedfortheintersectiontestforupdatingthein-processworkpiece.Thusonlyasmallpercentageofallthediscretevectorsareexaminedfor each tool movement. The intersections will be shown in the next section for differentcuttergeometries.4.2.3UpdatingIn-processWorkpieceinMulti-AxisMilling Inthissectionthein-processworkpiecesurfacesareupdatedbyusingdifferentmillingcutters.Althoughinsection(4.2.2)thediscretevectorshaveorientationsalongthex,yandz-axesofthestandardbasisofR3,inthissectiontheupdateformulaswillbederivedforthegeneral case in which the discrete vectors have arbitrary orientations. Also for thecalculations(3+2)-axismillinginwhichacuttercanhaveanarbitraryfixedorientationinspaceisconsidered.Inmulti-axismillingtoolmotionsare representedbythetrajectoryofthetoolcontrolpointF(u)andtheinstantaneousorientationofthetoolaxisA(u)whichisalwayscoincidentwiththetoolrotationaxis,whereuelement[0,1].Insection(4.2.1)themajorcuttergeometriesareparametricallydefinedwithrespecttothestandardbasisofR3whichhasanoriginatO.Inthissectionthecontrolpointsofthesegeometriesaredefinedwithrespecttothelocationwhich corresponds to O. For example the location of the control point for the cylindricalsurfaceisatthebottomradialcenter,forthefrustumofaconesurfaceitisabovethebottomradialcenterwithdistanceequalstoRbtanalphaandforthetoroidalsurfaceitisatthecenterofChapter4.In-processWorkpieceModeling 75thetorus.Acomplextoolpathcanbeconsideredasthecombinationoflinesegments,whereeachsegmentrepresentsthelineardistancebetweentwoconsecutivecutterlocationpoints.Foralineartoolpath(linesegment)thecontrolpointcanbeexpressedwithrespecttothetoolpathstartPS(xS,yS,zS)andendPE(xE,yE,zE)coordinatesasfollows F(u)=PS+(PE-PS)u            (4.19)wherethedifferenceof(PE-PS)iscalledastheinstantaneousfeeddirectiondenotedbyf.Note that for the (3+2)-axis machining the instantaneous tool orientation A(u) is fixedbetweenthetoollocationsPSandPE.Twosetsofcoordinatesystemsareusedinthispaper.A Work Piece Coordinate System (WCS) which is coincident with the standard basis ofR3denotedbyunitvectors[ijk],andalocalToolCoordinateSystem(TCS)denotedbyunitvectors[den].UsingFandA,onecandefineTCSonthetooldefinedbyasetofmutuallyorthogonalunitvectorsn,d,ande.TheTCSisdefinedas |zyx n (4.20) if 0partialdiffpartialdiffuF then|]uueeezpartialdiffpartialdiff?partialdiffpartialdiff?= FAFAe and nzyx d   AccordingtoabovedefinitionsitisnotallowedforthecuttertoplungealongitsrotationalaxisA(u).AlsoitisassumedthatthedaxisoftheTCSisalignedwiththeinstantaneousfeeddirection.Inthisworkthediscretevectorswhichrepresenttheworkpiecearedefinedby ( ) vaba -                (4.21)wherevelement[0,1]andalso ( )aaaa z and ( )bbbb z arethestartandendpointsofavectorrespectively.Ifthecuttersurfacesparameterizedinsection(4.2.1)movewithrespecttoTCSapointonthecuttersurfacecanbedescribedinWCSasfollowsChapter4.In-processWorkpieceModeling 76 )MM n         (4.22)where the subscript M represents the cutter surface type i.e. cylinder, frustum and torus.When a cutter moves along a toolpath three types of surfaces are generated: The ingresssurface )M theta- atu=0,theintermediateportion(alsocalledastheenvelopesurface))0 uM theta at0<u<1andtheegresssurface )M theta+ atu=1(seeFigure4.9).Updatingthein-processworkpieceatthestartandendofthetoolpathsegmentiseasy,forthispurposethediscretevectorsareintersectedwiththetoolgeometryatPSandPErespectively.Inthefollowingsubsectionstheformulaswillbedevelopedforupdatingthein-processworkpieceintheintermediateportion )0 <M theta .Figure4.9:DecomposingthesweptsurfaceintothreeregionsCutter surface geometries given in section (4.2.1) are canal surfaces and therefore theirsurfacenormalNcanberepresentedintheWCSasfollows )M m        (4.23)wherem(t,u)isthecenterofamovingspherewhichisdefinedintheWCSasfollowsChapter4.In-processWorkpieceModeling 77 )     (4.24)Apointontheenvelopesurfaceisembeddedin )M theta andalsothesurfacenormal(N)passingthroughitisperpendiculartothevelocityvectoratthecenterofthemovingsphere.In(3+2)-axismillingwithlineartoolpaththevelocityvectorisequaltothefeeddirectionvector(f).Thereforeforanarbitrarypointontheenvelopesurfacethefollowingdotproductequationholds 00 =M theta               (4.25)PluggingEq.(4.23)intotheaboveequationyields ( ) 0M theta        (4.26)Theaboveequationissolvedforthetawithrespecttothegivenvaluesofparametersuandt.There are three possible solutions for theta: two distinct real solutions theta1,2 ,  two equal realsolutions 21 theta ,andnorealsolution.PluggingthevaluesofthetaintoEq.(4.22)yieldsthelocationsofthepointsontheenvelopesurface )2,12,12,1 thetaMM n    (4.27)where P1,2 represents two points which correspond to theta1,2 respectively. As explained inchapter(3),thecircularcylinderandtorusarepipesurfaces.Bydefinitionapipesurfaceisdescribedasanenvelopeofthefamilyofthesphereswithaconstantradius.Thespinecurve(alsocalledastrajectory)ofamovingsphereisalineforthecylinderandacircleforthetorus.Ithasbeenproveninchapter(3)thatforthecircularcylinderandtorussurfacesthereisanalternativesolutioninfindingthelocationofanarbitrarypointontheenvelopesurface.InthisalternativesolutionthenormalvectorpassingthroughapointontheenvelopesurfacecanbeobtainedbyChapter4.In-processWorkpieceModeling 78 t utu ut partialdiffpartialdiffpartialdiffpartialdiff ),(),( mm              (4.28)Thenusingtheaboveequationthecoordinatesofanarbitrarypointontheenvelopesurfacewithrespecttothegivenvaluesofuandtarefoundby ||),(|| ),(2,1 ut utNN               (4.29)whereristheradiusofthemovingsphere.InEq.(4.29)thesigns(m )indicatesthattherearetwooppositepointsonthecutterenvelopesurfacewithanangulardifferenceequalsto1800.Theformulasdevelopedinthissectionwillbeusedinthefollowingsubsectionsfordifferentcuttergeometries.4.2.3.1UpdatingwithFlat-Endmill AFlat-Endmillismadeupofacylindricalsurfaceatthesideandaflatsurfaceatthebottom.Dependingonthefeedvectordirectionthecylindricalandflatsurfacesengagewiththe workpiece within [0,pi] and [0,2pi] angular ranges respectively. In the following twosubsectionstheanalyticalformulaswillbederivedforthesesurfaces.Cylindricalsurface AsexplainedbeforethecircularcylinderisapipesurfaceandalocationofapointontheenvelopesurfacecanbefoundwitheitherEq.(4.27)orEq.(4.29).BecauseofitssimplicitytheEq.(4.29)willbeusedinderivations.ThecenterofthemovingsphereforthecylindricalsurfacecanbeobtainedbypluggingEqs.(4.6)and(4.19)intoEq.(4.24) ( )bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp+tuSES 00]            (4.30)Chapter4.In-processWorkpieceModeling 79ForfindingthenormalvectordefinedinEq.(4.28)thepartialderivativesofEq.(4.30)withrespecttouandtaretakenasfollows ( )fm =partialdiffpartialdiff SE Pu ut ),( ,  nm =partialdiffpartialdiff t ut ),(           (4.31)PluggingEqs.(4.30)and(4.31)intoEq.(4.29)yieldsthecoordinatesofpointsonthecylinderenvelopesurfaceasfollows nS +2,1 m             (4.32)Whenthevaluesoftelement[0,h]anduelement[0,1]aresubstitutedintoEq.(4.32)twoplanarenvelopesurfacesdenotedby 0cylinder aregenerated(seeFigure4.10).Thesesurfacesspannedbytwolinearlyindependentvectorsfandnhavetheiroriginsatthelocations eS + and eS - .The feed (f) and tool axis (n) vectors are linearly independent because according to TCSdefinitiongiveninEq.(4.20)thecutterdoesnothaveaplungingmotionalongitsrotationalaxis.Thereforethedirectionsofthesetwovectorsarenotopposite.Figure4.10:EnvelopesurfacesgeneratedbythecylindricalpartofaFlat-EndmillChapter4.In-processWorkpieceModeling 80Apointatwhichavectorintersectstheenvelopes( 0cylinder )isdescribedbysettingthevectorlineI(v)giveninEq.(4.21)equalstoP1,2giveninEq.(4.32).Thusthisyieldsalinearsystemofthreeequationsinthreevariablest,uandvasfollows nSaba +     (4.33)TheEq.(4.33)canbewritteninamatrixformsas ABtvu1=bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp                 (4.34)where1bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp---=zyxeeeAmmm,3bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp---=bbbzyxB There are three possible intersection cases between an envelope plane and a vector: nointersection,vectorintersectsinonepointandvectorliesintheplane.AllthesecasesareillustratedinFigure(4.11-a,b,c).     (a)(b)(c)Figure4.11:Possibleintersectionsbetweenavectorandenvelopeplane:(a)nointersection,(b)oneintersection,and(c)vectorliesintheplane.If the vector is parallel to the envelope plane or if it lies in the envelope plane then thecolumns of the matrix B will be linearly dependant. In this case because the matrix BChapter4.In-processWorkpieceModeling 81becomes singular there will be no real solution for Eq. (4.34). If the vector intersects theenvelopeplanes(seeFigure4.12),therealparametervaluesu1,2,v1,2andt1,2areobtainedfromEq.(4.34).Iftheseparametervaluessatisfytherangesdefinedbytelement[0,h],uelement[0,1]andvelement[0,1]thenthevectorisupdated.Figure4.12:ParametersetsforupdatingthediscretevectorBottom-Flatsurface Figure(4.13)illustratestheBottom-FlatsurfaceofaFlat-Endmill.Thissurface(alsoitcanbethoughtasadisc)isenclosedbyacircledefinedinWCS.Becauseadiscisnotacanalsurface, in this section a different approach is used for the derivations. The parametricrepresentationforacircleofradiusRcenteredatF(u)isgivenby ( ) e            (4.35)    wheretelement[0,2pi],d=[dxdydz]ande=[exeyez]aretwoorthogonalunitvectorsoftheTCSwhichdefinetheplaneofthecircle.Chapter4.In-processWorkpieceModeling 82Figure4.13:IntersectingtheBottom-Flatsurfacewithadiscretevector.ApointatwhichavectorintersectstheBottom-FlatsurfaceisdescribedbysettingthelineI(v)fromEq.(4.21)equalstoP(u,t)giveninEq.(4.35).Thusthisyieldsanonlinearsystemofthreeequationsinthreevariablesu,tandvasfollows ( ) xxSESaba e   (4.36-a) ( ) yySESaba e (4.36-b) ( ) zzSESaba e (4.36-c)Forthegeneralsolutionofthissystem,firsttheequivalentofthevariablevfromEq.(4.36-b)issubstitutedintoEqs.(4.36-a)and(4.36-c)respectively.Thustwononlinearequationswithvariablesuandtareobtained.Thentheequivalentofufromoneofthesetwoequationsispluggedintoanotherforeliminatingu.Thereforethefinalnonlinearequationonlydependsonthevariabletanditcanbesolvedanalytically.Butasexplainedinsection(4.2.2),becauseofthelocalizationapproachadoptedtheworkpieceisrepresentedbydiscretevectorshavingorientationsinthedirectionsofx,y,z?axesofthestandardbasisofR3.Inthissectiontheformulasarederivedforthediscretevectorshavingorientationsinthedirectionofthez-axisofR3.Forthexandydirectionssimilarstepscanbeapplied.Havingorientationsalongthez-axis of R3 eliminates v from Eqs. (4.36-a) and (4.36-b). Further for eliminating u theChapter4.In-processWorkpieceModeling 83equivalentofufromEq.(4.36-a)ispluggedintoEq.(4.36-b)andthisyieldsthefollowingequationwithvariablet 0              (4.37)wheretheconstantcoefficientsaregivenby ( )ySExSE d  ( )ySExSE e  )SaSEaSSE x Eq.(4.37)canbewritteninthefollowingform ( ) 022 =               (4.38)where ( )12- .SolvingEq.(4.38)fortyieldsbraceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+=--betabeta212211sinsinBCtBCt    (4.39)If in Eq. (4.39) 222 C then the inverse function (.)1- is in the range of].Inthiscase,thereexistrealsolutionst1,2.Forupdatingthediscretevectorthevalues of u and v must be checked also. Therefore t1,2 are plugged into Eq. (4.36-a) forobtaininguvalues.LaterthevaluesoftanduarepluggedintoEq.(4.36-c)andthisyieldsthevaluesofv.Iftheparametervaluesareintherangesdefinedbyuelement[0,1]andvelement[0,1],thenthevectorisupdated.AlsonotethatwhentheBottom-Flatsurfacemovesalongalineartoolpath,anellipticalcylinderisgenerated(plungingorhorizontalmotionsareexcluded).InFigure (4.14) three possible intersection cases between this cylinder and a vector areillustratedforagiventoolpathdefinedbetweenPSandPE.Chapter4.In-processWorkpieceModeling 84Figure4.14:IntersectioncasesbetweentheBottom-Flatsurfaceandadiscretevector4.2.3.2UpdatingwithTapered-Flat-EndMill ATapered-Flat-Endmillismadeupofafrustumofaconesurfaceatthesideandaflatsurface at the bottom. With respect to the feed vector both surfaces engage with theworkpiecewithin[0,2pi]angularranges.Inthissectiontheanalyticalformulasforupdatingthein-processworkpiecearederivedforthefrustumofaconesurface.Theformulasforthebottom flat part can be used from section (4.2.3.1). When Eq. (4.11) is plugged into Eq.(4.22)thefrustumofaconesurfaceinWCSisobtainedasfollows( )( )( )bracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp-+++alphathetathetathetatansincos]2bbbfrustumRRRu n      (4.40)For finding the center of a moving sphere in WCS, k from section (4.2.1.2) ispluggedintoEq.(4.24)andthisyields ( )bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp+tu 00]             (4.41)Thelocationsofarbitrarypointsontheenvelopesurfacesgeneratedbythefrustumofaconeare obtained by Eq. (4.27). For this purpose it is needed to calculate theta1,2 for the givenparametervaluesofuandt.Eqs(4.40)and(4.41)arepluggedintoEq.(4.26)andthisyieldsChapter4.In-processWorkpieceModeling 85 (tsinalphacosalpha+Rb)costheta(d?f)-(tsin2alpha+Rbtanalpha)(n?f)=0 (4.42)Noticethate?f=0.WhentheEq.(4.42)issolvedforthetathefollowingequationisobtained parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp??= -ffalphatheta tancos 12 m (4.43)PluggingEqs.(4.43),(4.11)and(4.19)intoEq.(4.27)yieldsthefollowingenvelopesurfaceequationforthefrustumofacone Gfrustum(t,theta1,2,u)=PS+(PE-PS)u+(tsinalphacosalpha+Rb)costheta1,2d     +(tsinalphacosalpha+Rb)sintheta1,2e+(tcos2alpha?Rbtanalpha)n(4.44)Theaboveequationcanberearrangedasfollows Gfrustum(t,theta1,2,u)=O1,2+uf+kS1,2   (4.45)where O1,2=PS+Rbcostheta1,2d+Rbsintheta1,2e?Rbtanalphan S1,2=sinalphacostheta1,2d+sinalphasintheta1,2e+cosalphan k=tcosalphaWhenthevaluesof ]2 alpha anduelement[0,1]aresubstitutedintoEq.(4.45)twoplanarenvelope surfaces are generated (see Figure 4.15). These envelope surfaces spanned bylinearlyindependentvectorsfandS1,2haveoriginsatO1,2.Chapter4.In-processWorkpieceModeling 86Figure4.15:Envelopesurfacesgeneratedbythefrustumofaconepart.BysettingavectorlineI(v)fromEq.(4.21)equaltoGfrustum(t,theta1,2,u),alinearsystemofthreeequationsinthreevariablesareobtainedasfollows 2,12,1 Saba +            (4.46)TheEq.(4.46)canbewritteninamatrixformsas ABtvu1=bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp                 (4.47)where 332,1132,1 )] ?? - baa I         (4.48)IfthecolumnsofthematrixBarelinearlyindependentthenEq.(4.47)generatestherealparameter values u1,2 , v1,2   and t1,2. If these parameter values satisfy the ranges definedby ]2 alpha ,uelement[0,1]andvelement[0,1]thenthevectorisupdated.Chapter4.In-processWorkpieceModeling 874.2.3.3UpdatingwithFillet-Endmill AFillet-Endmillismadeupofatoroidalsurfaceatthelowerside,aflatsurfaceatthebottomandacylindricalsurfaceattheupperside.Dependingonthefeedvectordirectioneachoneofthesesurfacesupdatesthein-processworkpiece.ForexampleinFigure(4.16)three possible motion types are illustrated with respect to the feed vector f: descending(plunging), horizontal and ascending motion respectively. In these figures redlines on thecylindricalsurfacesandredcurvesonthetoroidalsurfacesrepresentthegrazingpointsatwhich the discrete vectors may intersect the envelope surface 0torus . In this section forcalculationsthedescendingmotionshowninFigure(4.16-a)willbeusedandsolutionsoftheothermotiontypescanbeobtainedbyusingthesamesteps.InthissectiontheworkpieceupdateformulaswillbederivedforthetoroidalsurfaceoftheFillet-Endmill.Theformulasforthecylindricalandbottomflatpartscanbeusedfromsection(4.2.3.1).(a)(b)(c)Figure4.16:Motiontypeswithrespecttothefeedvectorf:(a)descending,(b)horizontal,and(c)ascendingmotion.ForatoroidalsurfacethecenterofamovingsphereinWCSisobtainedbypluggingEqs.(4.12)and(4.19)intoEq.(4.24)andthisyields( )bracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp+0sincos] ttuSES rhorhon          (4.49)ThepartialderivativesofEq.(4.49)withrespecttouandtareobtainedasfollows ( )fm =partialdiffpartialdiff SE Pu ut ),( , em tt ut cos),( rhopartialdiffpartialdiff      (4.50)Chapter4.In-processWorkpieceModeling 88As explained before, the toroidal surface is a pipe surface with a constant radius and acircular spine curve. An alternative solution has been given in Eq. (4.29) for finding thelocation of an arbitrary point on the envelope surface generated by torus. In this sectionbecauseofitssimplicity,incalculationsthisalternativesolutionwillbeused.Thesigns( m )in Eq. (4.29) represent two opposite grazing points P1 and P2 with an angular differenceequalsto1800(seeFigure4.17).Figure4.17:TheenvelopeparametersofatoroidalsurfaceundertheplungingmotionBecauseforthedescendingmotionthegrazingpointP2alwayslieswithinthebodyoftheFillet-Endmill,forupdatingthein-processworkpieceP1whichcorrespondstothe(+)signinEq. (4.29) will be used. Thus an arbitrary point on the envelope surface of a torus withrespecttothedescendingmotioncanberepresentedby))||)),(1 ututuuurutPttmmNNm??=+=         (4.51)where mu(t,u) and mt(t,u) are the partial derivatives of m(t,u) with respect to u and t.Pluggingm(t,u),mu(t,u)andmt(t,u)fromEqs.(4.49)and(4.50)intoEq.(4.51)yields ( ) ( ) ( )( ) ( )efdf efdf ?+?- ?+?- tt ttSES cossin cossin1 rhorho rhorho (4.52)Chapter4.In-processWorkpieceModeling 89For simplifying the calculations the cross products in Eq. (4.52) will be denoted bydzyx k and ezyx q . A point at which a vector intersects thecutterenvelopeisdescribedbysettingthevectorlineI(v)describedinEq.(4.21)equalstoP1giveninEq.(4.51).Thusthisyieldsanonlinearsystemofthreeequationsinthreevariablest,uandvasfollows( ) qk  tt qtkt xxxxSESaba cossin cossin +- +- (4.53-a)( ) qk  tt qtkt yyyySESaba cossin cossin +- +- (4.53-b)( ) qk  tt qtkt zzzzSESaba cossin cossin +- +- (4.53-c)Forsolvingtheabovesystemofequations,firsttheequivalentofthevariablev fromEq.(4.53-b)ispluggedintoEqs.(4.53-a)and(4.53-c)respectively.Thuswiththisprocesstwononlinearequationswithvariablesuandtareobtained.Latertheequivalentofufromoneofthese equations is plugged into another for eliminating u. Therefore the final nonlinearequationdependsonthevariabletonly.Butinthisresearch,asexplainedinsection(4.2.2),for representing the in-process workpiece the directions of discrete vector lines are takenalongthex,yandzaxesofthestandardbasisofR3.Thereforeinthefollowingpartsthenonlinear system given by Eqs. (4.53-a,b,c) will be solved for discrete vectors havingorientationsalongthez-axisofR3andthesimilarstepscanbeusedforthediscretevectorspointinginthedirectionofxandyaxesofR3.  Whenthediscretevectorsareorientedinthez-axisdirectionofR3,thevariablevfromEqs.(4.53-a)and(4.53-b)areeliminatedandthusthesetwoequationsareonlydependentonuandt.FurtherforeliminatingutheequivalentofufromEq.(4.53-a)ispluggedintoEq.(4.53-b)andthisyieldsthefollowingnonlinearfunctionwithvariablet ( ) ( ) ( ) 0            (4.54)whereChapter4.In-processWorkpieceModeling 90 qk tt xxqyyqrdxxdyytA SEySExySExSE cossin )()()()()( +- ---+---= rhorho  qk tt yykxxkrexxeyytB SExSEyySExSE cossin )()()()()( +- ---+---= rhorho  )aSSESaSE x Thefollowingpropertymotivatesfindingthesolutionofthenonlinearfunctionf(t)Property 4.1: For a given linear toolpath the swept envelope of the torus ( 0torus ) has aconvexshapeandadiscretevectorintersectsthisenvelopeatmosttwopoints.Accordingtoproperty(4.1)therearethreepossibleintersectioncasesbetweenanenvelopesurfacegeneratedbyatorusandadiscretevector:vectorintersectsintwopoints,vectorintersectsinonepointandnointersection.AllthreecasesareillustratedinFigure(4.18).Inthisfiguretheenvelopesurface 0torus hastwoparts,thelowerenvelope( 0 , Lowertorus ,coloredby green) and the upper envelope ( 0 , Uppertorus , colored by gray) surfaces, i.e.0,0,0UpperGGG union= . As explained before the cutter is performing a descendingmotion.Thereforeforupdatingthein-processworkpiecethelowerenvelopesurfaceistakenintoaccount.Theupperenvelopesurfaceisnotconsideredinthecalculationsbecauseduringthe tool motion it is enclosed within the cylindrical part of the Fillet-End mill. Thus theintersection points shown in Figure (4.18) are located on the lower envelope surface( 0 , Lowertorus ).Chapter4.In-processWorkpieceModeling 91Figure4.18:Envelope/vectorintersectioncases:twopoints(1),onepoint(2)andnointersection(3).The real solutions of Eq. (4.54) correspond to the intersection points described by theproperty(4.1).Forsolvingthenonlinearfunctionf(t)giveninEq.(4.54)aparameterintervalfortisrequired.AsillustratedinFigure(4.19-a,b),forobtaining 0 , Lowertorus theparameterttakesvaluesintherangedefinedby ].Inthesefigurestheenvelopeboundarymade up of grazing points (colored by red) intersects with discrete vectors in one or twopoints.ItcanbeseenfromFigure(4.19-a)thatifthereisoneintersectionpoint(I1)whichcorresponds to t1 the function values at pi /2 and  3 pi /2 have opposite signs i.e.0 .Ontheotherhandiftherearetwointersectionpoints(I1andI2)whichcorrespond to t1 and t2 then the function values atpi /2 and 3pi /2 have the same sign i.e.0 .Chapter4.In-processWorkpieceModeling 92      (a)          (b)Figure4.19:Therootsofthenonlinearequationf(t)whenthevectorintersectsin(a)onepointorin(b)twopoints.ThusthefirststepforsolvingtheEq.(4.54)istocheckthesignsoffunctionf(t)atpi /2and3 pi /2. If the signs are opposite then there is only one root in the interval defined by].Thereforeoneofthenumericalrootfindingmethods,i.e.Bisection,canbeusedtocalculatethesingleroott1.Butontheotherhandifthesignsoffunctionf(t)atpi /2and3pi /2arethesamethenthenumericalmethodscannotbeapplieddirectly.Forthiscasethefollowingtheoremmotivatesfindingtworootst1andt2.Theorem 4.1: (Rolle's Theorem)  Let the function f be continuous on [a, b], anddifferentiableon(a,b),andsupposethatf(a)=f(b).Thenthereissomecwitha<c<bsuchthatf'(c)=0.Thefunctionf(t)giveninEq.(4.54)hascoefficientsA(t)andB(t)whicharealsoafunctionof t . These coefficients have a denominator given by q . Expanding thisdenominatoryieldsChapter4.In-processWorkpieceModeling 93 ( ) ( )22222222 cos zyxzyx q     (4.55)Notethatfortheaboveequationthedotproductbetweenkandqequalstozeroi.e. k?q=(f?d)?(f?e)=(f?f)(d?e)-(f?e)(d?f)=0ItcanbeseenfromEq.(4.55)thatthedenominatorcannotbezero.Thereforethefunctionf(t)giveninEq.(4.54)iscontinuous.Also because A(t), cost, B(t), sint and C are differentiable at ) , the nonlinearfunction f(t) given in Eq.(4.54) is also differentiable in the open interval givenby ) . Furthermore, if the function f(t) has two roots t1 and t2 in the interval] then 021 = . Therefore the Rolle?s Theorem holds for the Eq.(4.54)andthusthereisavaluecbetweentworootst1andt2forwhichf'(c)=0.Ifthecvalueisfound,theintervalgivenby ]ispartitionedintotwosubintervalsdefinedby] and ] .Now 0 and 0 andthusbyapplyingoneofthenumericalrootfindingmethods,i.e.Bisection,intothesetwosubintervalstherootst1andt2canbefound.Inthisresearchforfindingcvalue,aMatlab[54]functionfminbndisused.fminbndfindstheminimumofafunctionofonevariablewithinafixedinterval.Thealgorithmusedinfminbndisbasedongoldensectionsearchandparabolicinterpolation.See[29] and [17] for details about the algorithm. Note that when the signs of f(t) atpi /2 and3pi /2 are the same i.e. 0 , then each one of ) and ) has aminussignorviceversa.Ifeachonehasaminussignthenforfindingthelocalminimumvalueatcusingfminbndthenonlinearfunctionf(t)ismultipliedbyminus.Thisprocessonlyreflectsthefunctionf(t)withrespecttot-axisin(t,f(t))graphanditdoesnotchangethelocationsoftheroots.Thewholeprocessforfindingtherootsofthenonlinearequationf(t)isgiveninAlgorithm(4.1).Chapter4.In-processWorkpieceModeling 94Algorithm4.1:Obtainingtherootsoff(t).If the function f(t) has root(s) then for updating the discrete vector the values of theparametersuandvmustbechecked.Theroot(s)obtainedfromAlgorithm(4.1)arepluggedintoEq.(4.53-a)forfindingthevalue(s)ofthetoolpathparameteru.Thentheroot(s)anduvalue(s)arepluggedintoEq.(4.53-c)forobtainingthevectorparameterv.Iftheparametervaluesareintherangesofuelement[0,1]andvelement[0,1],thenthevectorisupdated.Otherwiseitcanbeconcludedthatthereisnointersectionbetweenthecutterenvelopeandthediscretevector.4.2.4Implementation The presented methodologies have been implemented by using C++ software. Figure(4.20)and(4.21)showthesimulationoftheDoormoldandAutohoodrespectively.TheinitialworkpiecesaregivenasrectangularstocksandforthemachiningaBall-Endmillwith10 mm radius in Figure (4.20) and 5 mm radius in Figure (4.21) is used. In the bothmachiningsimulationsonelayeroftoolpahts(coloredbyred)areused.Thetoolpathfilesforthese two examples are taken from VERICUT [79]. For visualizing the final workpiecesdiscretevectorsaretessellatedbyusingMatlab.Chapter4.In-processWorkpieceModeling 95Figure4.20:NCmillingsimulationofaDoormold.Figure4.21:NCmillingsimulationofanAutohood.In the next example the machining simulation of the gearbox cover is illustrated (Figure4.22).FordesigningthegearboxcoverandthengeneratingthetoolpathsUGSNX3[77]isused.Forthismachiningsimulationtherearethreelayersoftoolpaths.InthefirstlayeraFlat-End mill with 5mm radius updates the in-process workpiece (Figure 4.22 -a). In thesecond and third layers the Ball-End mills with radiuses 5mm and 2.5 mm respectivelyperform the simulation (Figure 4.22 -b, c). For visualizing the in-process workpiece aftereachprocessedlayerMatlabisusedfortessellation.Thegreenlinesinthesefiguresrepresentthetoolpaths.Chapter4.In-processWorkpieceModeling 96(a)(b)(c)Figure4.22:NCmillingsimulationresultsforaGearboxcoverwith(a)AFlat-Endmilland(b,c)Ball-Endmills.Inthelastexamplethedevelopedmethodologyisappliedtothe5-axisimpellermachining(seeFigure(4.23)).TheTaper-Ball-Endmillisused.InFigure(4.23-a)theinitialworkpiecewithtoolpathsandinFigure(4.23-b)thefinishedcavityofoneblade areshown. Inthisexample5-axistoolmovesareapproximatedby(3+2)-axistoolmoves.Thisisachievedbysubdividingeach5-axistoolpathintoasmany(3+2)-axistoolpathsasnecessarytomaintainthe desired level of accuracy. For minimizing the error introduced by (3+2)-axisapproximation the average of the tool orientations at the beginning and at the end of atoolpathsegmentisusedforthefixedtoolorientationonthissegment.(a)(b)Figure4.23:NCmillingsimulationfor5-axisimpellermachining.Chapter4.In-processWorkpieceModeling 974.3DiscussionIn this chapter discrete vector based in-process workpiece update methodologies havebeenpresented.Thediscretevectorswiththeirorientationsinthedirectionsofx,y,z-axesofR3havebeenused.Thereforeinthisrepresentationmorevectorsindifferentdirectionshavebeenusedandthus,especiallywhentheworkpiecehasfeatureslikeverticalwallsandsharpedges, the quality in the visualization of the final product has been increased. Also thelocalization advantage of the Discrete Vertical Vector approach has been preserved. Atypical milling tool path contains thousands of tool movements and during the machiningsimulation for calculating the intersections only the small percentage of all the discretevectorsisneeded.ForthispurposeforlocalizingthetoolenvelopeduringthesimulationtheAxisAlignedBoundingBox(AABB)hasbeenused.ThebestfeatureoftheAABBisitsfastoverlapcheck,whichsimplyinvolvesdirectcomparisonofindividualcoordinatevalues.Asexplainedinthischapterforsomeworkpiecegeometries3-axismachiningisnotsuitableforupdatingtheworkpiece surfaces. Becauseofthisinthedevelopedmethodologiesthetoolmotionsin(3+2)-axismillinginwhichthecuttercanhaveanarbitraryfixedorientationinspace have been considered. The exact 5-axis milling motions are not preferable in theworkpieceupdatesimulationsbecause allthecalculationsrequireusing thenonlinearrootfinding algorithms and therefore the computational time becomes high. Under thisconsideration 5-axis tool motions can be approximated by (3+2)-axis tool motions. Anexample has been given in Figure (4.23) for illustrating this situation. In this chapter forsimplifyingtheintersectioncalculationsthepropertiesofcanalsurfacesdefinedinsection(3)areutilized.Thecutterenvelopeandvectorintersectioncalculationsforthecylinder,frustumofaconeandFlat-Bottomsurfaceshavebeenmadeanalytically.Becauseofthecomplexityofthetorusshapethesecalculationshavebeen madebyusingthenumerical rootfindingmethods.Forthispurposearootfindinganalysishasbeendevelopedforguaranteeingtheroot(s)inthegiveninterval.Theintersectionformulasforthespheresurfacehavenotbeendeveloped because when a sphere moves along a linear toolpath it envelopes a cylinderwhichhasanorientationinthedirectionofthefeedvector.Thereforetheupdateformulasforthesphericalpartofacutterarereducedtocylinder-vectorintersection. 98Chapter5FeasibleContactSurfacesIn this chapter the feasible contact surfaces of NC cutters under varying tool motions arepresented.AtypicalNCcutterhasdifferentsurfaceswithvaryinggeometries.ForexampleaFlat-End mill has a cylindrical part at the side and a flat part at the bottom. During thematerialremovalprocessonlycertainpartsofthecuttersurfacesareeligibletocontactthein-process workpiece. In this chapter for representing these so-called certain parts aterminologyfeasiblecontactsurfaces(FCS)isintroduced.Thewordfeasibleisusedbecausealthoughthesesurfacesareeligibletocontactthein-processworkpiece,theymayormaynotremovematerialdependingonthecutterpositionrelativetotheworkpiece.WhentheFCScontactthein-processworkpiecetheCutterWorkpieceEngagements(CWEs)aregenerated.SinceCWEsaresubsetsofFCS,formulatingtheFCShelpsustobetterunderstandtheCWEgenerationprocess.TheboundariesoftheFCSaredefinedbythecuttergeometryandtheenvelopeboundaryset.Theenvelopeboundarysetcontainspointsthatthecuttersurfacewillleave behind as it moves along a toolpath. Throughout this chapter it is assumed that thecutterfollowsarigidmotion.Simplyarigidmotionisatransformationovertimethatdoesnotchangetheshapeofthecutter,onlyitslocationandorientation.Thereforeforthegiventoolmotionsthecuttergeometrystaysfixedbutontheotherhandtheenvelopeboundarysetmaychange. Two factors effect the construction process of the FCS: the cutter geometry and themotiontype.ForexampleforaFlat-Endcutter,in2?-axismillingonlythecylindricalparthasaFCS,butontheotherhandin3-axisplungemillingboththecylindricalandtheflatparts have a FCS. The general tool motions in milling are presented in section (5.1) andfollowed by the milling cutter geometries in section (5.2). For obtaining the FCS theenvelopeboundarysetsofagenericcutterperforming5-axistoolmotionsareformulatedinsection(5.3).Forthispurposeatangencyfunctiondefinedbythesurfacenormalandthetoolvelocityisutilized.Laterinsection(5.4)thedistributionofthefeasiblecontactsurfaceswithrespecttocuttervelocityisanalyzed.Thenfinallythechapterendswiththediscussioninsection(5.5).Chapter5.FeasibleContactSurfaces 995.1ToolMotionsinMilling Asmentionedintheintroduction,toolmotionsareoneoftheinputsintheconstructionoffeasiblecontactsurfaces(FCS).Lateritwillbeshowninsection(5.4)withexamplesthatthetypeofthetoolmotionisakeycomponentinthedecisionof?whetherornotcuttersurfacescontaintheFCS?.Inthefirstpartofthissectionthemainmillingmotionswiththeirtypicalcharacteristics are introduced and then in the second part the formulae which define thelocationandthevelocityofanarbitrarypointonthecuttersurfacearederived. Therearemainlythreetypesofcuttermotionsinmilling:2?-axis,3-axisand5-axis.In2 ? - axis motion which is the simplest, a cutter simultaneously translates along twoCartesianaxeswithafixedtoolaxisvector(Figure5.1-a).Ontheotherhanda3-axistoolmotionwhichhasafixedtoolaxisvectorallowsthecutterthreedegreesoffreedom.ThesethreedegreesoffreedomcorrespondtosimultaneoustranslationsalongthreeCartesianaxes(Figure5.1-b).Oneofthesedegreesallowsthecuttertoslideupanddown.In5-axistoolmotion,acutterhasfiveaxesofmovement,threeofwhicharetranslationalandtheothertworotational(Figure5.1-c). (a)(b)(c)Figure5.1:Cuttermotionsinmilling:(a)2?-axis,(b)3-axis,(c)5-axis. In this work two kinds of coordinate systems are used: The local - Tool CoordinateSystem (TCS) and the reference - Machine Coordinate System (MCS). The three unitCartesiancoordinatesoftheTCSaredenotedbyxL,yL,andzL.TheTCSispositionedatthetooltipFwithzLalongtherotationaxisvectorn(Figure5.2).OntheotherhandtheMCSisrepresentedbythebasisvectorsi,j,andkalongxM-yM-zMrespectively.ThepositionandtheorientationoftheTCSarespecifiedwithrespecttotheMCS.LetthebasisoftheTCSbedefinedas nL ,if 0then ||L nn&& and LLL z        Chapter5.FeasibleContactSurfaces 100   if 0,then              (5.1) nL , ||FFL VnVy??= if 0F notequal? Vn and Lzyx ?= where F isthevelocityintheoriginofTCS.AccordingtoEq.(5.1)thepositionandtheorientationofacutter(G)mustbepiecewisedifferentiableandalsoplungingmotionalongthecutterrotationaxisisnotallowed.Figure5.2:ThelocalandreferenceframesofacutterAsmentionedintheintroduction,throughoutthischapteritisassumedthatthecutterfollowsarigidmotioninE3.Thusatoolmotioncanbedescribedanalyticallyby r=d+Rp                  (5.2)where p and r are the position vectors of an arbitrary point on G in the local and thereferencecoordinatesystemrespectively.Risarotationmatrixanddisatranslationvector.Therotationmatrixisanorthogonalmatrixforarigidbodytransformation.IngeneraldandRarefunctionsoftimet.ThereforeEq.(5.2)determinesthelocationofapointinMCSatagiventime.ThevelocityofapointonG canbeobtainedbydifferentiatingEq.(5.2)withrespecttot p&& + (5.3)Chapter5.FeasibleContactSurfaces 101PluggingtheequivalentofpfromEq.(5.2)intoEq.(5.3)yields ( )- -1&&   (5.4)where the multiplication of two matrices ( 1-& ) is called as the angular velocity matrixdenotedbyOmega.Omegaisaskewsymmetricmatrix.Thiscanbeprovenasfollows:SinceRisanorthogonalmatrix,RRT=IandR-1=RT.BydifferentiatingRRT=I,thefollowingequationisobtained. 0TT =&&                  (5.5)ThesecondterminEq.(5.5)canbewrittenas ( ) ( )T1TT1T -- = R&&& .Andalsothefirstterminthisequationisequalto 1-& .ThusEq.(5.5)takesthefollowingform ( )T11 -- - R&&                  (5.6)Theaboveequationprovesthat Omegaisaskewsymmetricmatrix.ThegeneralformofOmegaisgivenbybracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttpOmegaOmegaOmega=000XXY              (5.7)where X , Y and Z are three components of the angular velocity in MCS. The skewsymmetric matrix Omega can also be represented as a vector. For example by using thecomponentsof Omegatheangularvelocityvectoromegacanberepresentedby kZYX Omega                (5.8)Chapter5.FeasibleContactSurfaces 102ItcanbeeasilyverifiedthatthemultiplicationofOmegawithanarbitraryvector,letsaym,inMCSisequaltothecrossproductbetweenomegaandthisvector,i.e. m                 (5.9)ThereforeEq.(5.4)whichdefinesthevelocityofanarbitrarypointonGcanberewrittenas ( ) p&&              (5.10)Laterinsection(5.3)theinstantaneousvelocityofthetoolaxiswillberequired.Itisderivedfromthefollowingequation k                    (5.11)DifferentiatingEq.(5.11)yields n-1&&               (5.12)FromEq.(5.9),theaboveequationtakesthefollowingform n                   (5.13)Thustheinstantaneousvelocityofthetoolaxiscanbecalculatedintermsofangularvelocityandtherotationaxisvector.5.2MillingCutterGeometries Theconstructionoffeasiblecontactsurfacesrequiresthemodelingofthetoolgeometry.Inthissectionforobtainingavarietyofcuttershapesaparametricmodelofagenericcutterisused(Figure5.3).Thegeometryofthegenericcuttercanbedescribedbythefollowingparameters[46]Chapter5.FeasibleContactSurfaces 103R:Majorradius,rc:Minorradiusofthecornertorus.hc:Distancebetweencuttertippointandthecenterofthecornerradius.alpha: Anglefromtoolrotationaxistothecutterbottom,(0<alpha<=<pi/2).beta: Anglebetweenupperconesideandtoolrotationaxis,(-pi/2<beta<pi/2).r: Cutterradius,h:Theheightofthetool.Byappropriatelychoosingtheseparametersavarietyofcuttershapescanbeobtained.Figure5.3:GeometricdefinitionofthegenericcutterThegenericcuttercanbedecomposedintoUpper-Cone(GU),Corner-Torus(GT)andLower-Cone (GL) parametric surfaces and the boundaries between these surfaces are C1 -continuous.TherepresentationsofthesesurfaceswithrespecttoTCSaregivenby ( )( )( )parenrightexparenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftexparenleftbtparenlefttp+++=uuuuccctbetathetathetathetasinsincos,        (5.14-a) ( )( )( )parenrightexparenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftexparenleftbtparenlefttp-++=phithetathetaphicossincos,cccTrrrt       (5.14-b) ( )parenrightexparenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftexparenleftbtparenlefttp=llllccct thetathetatheta sincos,   (5.14-c)   Chapter5.FeasibleContactSurfaces 104wherethetaelement[0,2pi],cuelement[0,hu],clelement[0,hl],phi element[(pi/2-alpha),(pi/2-beta)]andalsotherearefollowingrelationshipsbetweenthegeometricparameters. betac , ( ) alphacl r  alphaclc r , betaccu r AlsoEqs.(5.14,a-c)canbewrittenintheMCSas: ( )( ) ( )LLsin)yzthetabetauuMcc++         (5.15-a) ( )( ) ( )LLsin)yzthetaphiccMrr++      (5.15-b) ( )LLsincosyxthetathetallMcc++  (5.15-c)   wheretelementRisatoolpathparameterandMindicatesthatrepresentationsareintheMCS.ThenormalofacuttersurfacecanbedescribedintheTCSas qGsGqGsG partialdiffpartialdiffpartialdiffpartialdiffpartialdiffpartialdiffpartialdiffpartialdiff /       (5.16)wheresandqarethegeometricparameters.AlsothenormalcanbewrittenintheMCSforeachpartofthegenericcutteras ( ) LLLU cos yM    (5.17-a) ( ) LLLT sin yM  (5.17-b) ( ) LLLL cos yM (5.17-c)whereQisanarbitrarypointonthecuttersurface.Chapter5.FeasibleContactSurfaces 1055.3CalculatingtheFeasibleContactSurfaces TwoinputsfordescribingtheFeasibleContactSurfaces(FCS),thetoolmotionsandthemilling tool geometries have been presented in the previous two sections. The formulaedeveloped in these sections will be used for describing the FCS. A typical milling cuttercontainsvaryinggeometriesandduringthemachiningeachoneofthesegeometriesbehavesdifferentlyandinturnthiseffectstheconstructionoftheFCS.Thiscanbeexplainedwithasimpleexample.ForexampleaFlat-Endmillhasacylindricalpartatthesideandaflatpartat the bottom. In a 2 ?-axis milling only the cylindrical part can contact in-processworkpieceandalsoonlythefrontofthecylindercanremovematerial.Fromthisexampleitcanbeseenthatbottomflatpartandthebackofthecylindercannotremovematerial.Inthisexample the front part of the cylinder represents the feasible contact surface. The morecomplicatedsituationoccurswhenacutterperformsamulti-axismachining.InthisworkforcalculatingFCSofthegenericcutterperforming5-axistoolmotionsatangencyfunctionisutilized. The tangency function f is defined with respect to surface normal ( )M N andinstantaneousvelocity ( )M V as[82] ( ) ( ) ( )MM V  (5.18)wherepandtrepresentthecuttergeometryandthetoolpathparameterrespectively.Atanyinstant the surface boundary of a cutter can be partitioned into three sub boundaries withrespecttotangencyfunction:forwardboundary(egresspoints),envelopeboundary(grazingpoints) and backward boundary (ingress points). Figure (5.4) illustrates these threeboundaries.Thuscuttersurfacesattimetareenclosedwithinthesethreeboundaries ( ) ( ) ( ) ( )MMMM ,0 +- union       (5.19)where( )M ,+ istheforwardboundarywith ( ) 0 ( )M ,0 istheenvelopeboundarywith ( ) 0,and( )M ,- isthebackwardboundarywith ( ) 0Chapter5.FeasibleContactSurfaces 106Figure5.4:Boundarypartitionofthegenericcutter.The cutting tool can contact the in-process workpiece through the forward and theenvelope boundaries and therefore the combination of these two boundaries defines thefeasible contact surfaces of a cutter.The word feasible is used because these surfaces areeligibletocontactthein-processworkpiece.Althoughtheyareeligibletocontactthematerialremoval depends on the cutter position relative to the workpiece. The Cutter WorkpieceEngagements(CWEs)aretheimprintsofFCSonthein-processworkpieceandforagiveninstantboththein-processworkpieceandthecuttersurfacemustshareapointfromFCS.ToclearlyseetherelationshipbetweenFCSandCWEsthefollowingterminologyisintroduced.GeneralTerminologyCWEK(p,t):Thesetofallpointsonacuttersurfaceatlocationtwhere ( ) 0.Thesearealsocalledaskinematicallyfeasibleengagementpoints.CWE(p,t):Thesetofallpointsonacuttersurfaceatlocationtthatareengagedwiththe workpiece.Accordingtheaboveterminologythefeasiblecontactsurfacescanberepresentedbytheset CWEK where ( ) ( ) ( )MMK ,0+ . Also the relationships among these setscanbewrittenas( GK CWE ),where G denotesthesetofallpointsonacuttersurface.Chapter5.FeasibleContactSurfaces 107 A typical NC cutter has different surfaces with varying geometries and during themachiningtheengagementcharacteristicsofeachsurfacebecomedifferent.Inthisworkforclearly seeing this, the FCS are distributed among actively cutting surfaces. Under thisconsiderationthesetCWEKcanbedividedintoconstituentsubsets.ForexampleinFigure(5.5)aFlat-Endmillismovingalongahelicaltoolpathforenlargingahole.Becauseoftheplungingeffectboththeside(cylindrical)andthebottom(flat)faceshavecontactswiththein-processworkpiece.ForthiscasethesetCWEKcanbewrittenas BottomKSideKK CWE ,, union            (5.20)wherethesetsCWEK,SideandCWEK,BottomrepresenttheFCSgeneratedbythecylindricalandtheflatpart respectively.Alsointhisfigurethecutterworkpiece engagementsasthesubsetsofFCSareshown.Figure5.5:PointsetsusedindefiningengagementsItwillbeshowninthenextchapterthatforobtainingthecutterworkpieceengagementsthegeometriclimitsofthesetCWEKarerequired.ThesegeometriclimitsaredeterminedbytheChapter5.FeasibleContactSurfaces 108cuttersurfaceboundariesandtheenvelopeboundary.Figure(5.6)illustratesthesegeometriclimitsforthetoroidalpartofaFillet-Endmill.Figure5.6:BoundariesoftheCWEKintheFillet-EndmillBecause the cutter performs a rigid motion defined in section (5.1) the cutter surfaceboundariesstayfixed.Theseboundariescanbecalculatedbyusingtheformulaedefinedinsection(5.2).Butontheotherhand,thecontentoftheenvelopeboundarychanges.ThereforeformodelingtheFCSthecalculationoftheenvelopeboundaryistheprimaryconcerninthiswork.Inthefollowingsubsectionstheenvelopeboundaryofthegenericcutterperforming5-axistoolmotionswillbecalculated.Asnotedbeforeinsection(5.2),thedefinedgeometryofa generic cutter can be decomposed into three sub surfaces: upper-cone, corner-torus andlower-cone. Therefore the full envelope boundary of the cutter can be generated bycombining the partial envelope boundaries which are described on the individual cuttersurfaces.5.3.1EnvelopeBoundaryoftheUpper-Cone In Figure (5.7) a general end mill performing 5-axis machining is shown with threearbitrarypointsondifferentsurfaces.Fromthisfigure,thevelocityofanarbitrarypointIUontheupper-conesurfacecanbeexpressedinMCSby ( )arrowright? UUM IFU              (5.21)wherethevectorarrowrightUIF isdefinedinTCS.WhentheEq.(5.14-a)ispluggedintotheaboveequation,thevelocitybecomesasChapter5.FeasibleContactSurfaces 109( ) ( ) ( ) ( ) ( )( ) ( )LLsincosyx??thetathetauuMcc (5.22)Figure5.7:ArbitrarypointsIU,ITandILontheupper-cone,corner-torusandlower-conesurfacesrespectively.Forobtainingtheenvelopeboundaryoftheupper-cone,Eqs.(5.22)and(5.17-a)arepluggedintothefollowingequation ( ) ( ) ( ) 0UU =UMUMuU I         (5.23)ThenbyperformingvectoroperationsEq.(5.23)takesthefollowingform( ) ( ) ( ) ( )( ) ( )( ) ] ( )]sinFLL=++VyybetabetabetauuUcc   (5.24)Notethatfortheaboveexpansionthescalartripleproductisevaluatedas ) andalsoEq.(5.13)isutilizedforthefollowingoperationChapter5.FeasibleContactSurfaces 110 ( ) 0|||| =?n nomegannL &&& Eq.(5.24)canbewritteninaclosedform ( ) 0UUUuU C           (5.25)where ( ) ( ) ( ) ( ) ( )LLFL sin yuccuU c  ( )FL VU  ( )FL VU In ( x ) type equations, the combination of cosine and sine functions can bewrittenasasinglesinefunctionas ( )             (5.26)where the amplitude 22 b , the phase ( ) and 0 . The derivation ofEq.(5.26)canbeshownbyutilizingthecomplexdomainoperationsas( ) ( ) ]parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp++=parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp-+=bracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftbtbracketlefttp+bracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftbtbracketlefttp++-+parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp--parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttpparenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp ---baxbaabxbaeeeieeeeabxiabxiixbiixbiixixarctan2121212sincos2arctan2222arctan2222 (5.27)Chapter5.FeasibleContactSurfaces 111Notethatinthelastlineoftheaboveequationcosinerepresentationistransformedintosinerepresentationundertheconsiderationofarctanfunctionasbraceexbraceexbraceleftbtbraceexbraceexbraceleftmidbracelefttp>parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp-<parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp--=01210121)yyyyypipiThustheEq.(5.25)canbewritteninthefollowingform ( ) 022 =UUUU C              (5.28)where ( )UUU B .SolvingEq.(5.28)forthetayields( )braceexbraceexbraceexbraceleftbtbraceexbraceexbraceexbraceleftmidbracelefttp-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+=--UUUUUUuBCBCtkappakappatheta2121sinsin,            (5.29)When Eq. (5.29) is plugged into Eq. (5.15-a), for a given cutter location point theenvelopeboundaryoftheupper-conesurfaceisobtained.5.3.2EnvelopeBoundaryoftheCorner-Torus ThevelocityofarbitrarypointITonthetoroidalsurface(Figure5.7)canbeexpressedinMCSby ( )arrowright? TTM IFT               (5.30)PluggingEq.(5.14-b)intoEq.(5.30)yields ( ) ( ) ( ) ( ) ( )( ) ( )LLsincosyx??thetathetaccMrr (5.31)Chapter5.FeasibleContactSurfaces 112For obtaining the envelope boundary of the corner-torus, Eqs. (5.31) and (5.17-b) arepluggedintothefollowingequation ( ) ( ) ( ) 0TT =TMTMT I         (5.32)andconsideringthevectoroperationstheaboveequationyields ( ) 0TTTT C           (5.33)where ( ) ( ) ( )LLFL cos ycT h  ( )FL VT  ( )FL VT Eq.(5.33)canbewritteninthefollowingform ( ) 022 =TTTT C              (5.34)where ( )TTT B .SolvingEq.(5.34)forthetayields ( )braceexbraceexbraceexbraceleftbtbraceexbraceexbraceexbraceleftmidbracelefttp-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+=--TTTTTTBCBCtkappakappaphi2121sinsin,            (5.35)When Eq. (5.35) is plugged into Eq. (5.15-b), for a given cutter location point theenvelopeboundaryofthecorner-torussurfaceisobtained.Chapter5.FeasibleContactSurfaces 1135.3.3EnvelopeBoundaryoftheLower-Cone ThevelocityofapointILonthelower-conesurface(Figure5.7)isgivenby ( )arrowright? LLM FIFL               (5.36)TheEq.(5.36)isexpandedbysubstitutionofEq.(5.14-c)as ( ) ( ) ( )( )LLsincosyx??thetathetallMcc      (5.37)Eqs.(5.37)and(5.17-c)arepluggedintothefollowingequation ( ) ( ) ( ) 0LL =LMLMlL I        (5.38)andafterthevectoroperationstheenvelopeboundaryofthelower-coneisobtainedintheclosedform ( ) 0LLLlL C          (5.39)where ( ) ( ) ( )LFLL cos yllL c  ( )FL VL  ( )FL VL TheEq.(5.39)istransformedintothefollowingform ( ) 022 =LLLL C      (5.40)where ( )LLL B ,anditissolvedforthetaasChapter5.FeasibleContactSurfaces 114 ( )braceexbraceexbraceexbraceleftbtbraceexbraceexbraceexbraceleftmidbracelefttp-parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+--parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+=--LLTLLLlBCBCtkappakappatheta2121sinsin,            (5.41)When Eq. (5.41) is plugged into Eq. (5.15-c), for a given cutter location point theenvelopeboundaryofthelower-conesurfaceisobtained.5.4AnalyzingtheDistributionofFeasibleContactSurfaces In the previous section the generic cutter surfaces have been partitioned into threeboundary point sets by utilizing the tangency function. Then for each surface patch theenvelopeboundarywhichdefinesthelimitsofthefeasiblecontactsurfaces(FCS)hasbeenformulatedbycalculatingtheparametertheta.Foragivencuttergeometry,anychangeinthetool tip velocity VF and the angular velocityomegawill effect the range of theta and in turn thelocationofthepointsetCWEKwhichrepresentthefeasiblecontactsurfaces.ThereforethecuttermotiontypehasdirecteffectintheconstructionprocessofFCS. In 5-axis machining the cutter rotation axis can be tilted in any direction. In generalduringmachiningthesidefaceofthecutterhasinstantcontactwiththein-processworkpieceand the engagement angle covers the angular range of [00, 1800]. For example in Figure5.8(a-b)Taper-EndandBall-Endmillsareperforming5-axismotionsandonlythesidefacesofthesecutterscontainthepointsetCWEK.Butontheotherhandin3-axismillingthetoolrotationaxisisfixedandwithrespecttothetooltipvelocity,morecuttersurfacesmaybeinvolvedinmachining.Thisisillustratedfor3-axisplungemotioninFigures(5.8-c)foraFlat-Endmilland((5.8-d)foraBall-Endmillrespectively.Chapter5.FeasibleContactSurfaces 115 (a)(b)(c)(d)Figure5.8:Instantaneouscuttercontactsurfacesin(a,b)5-axis,andin(c,d)3-axisplungemotionsBecause3-axismotioncancovermorecontactsurfaces,inthissectionthedistributionoftheFCSonthecutterwillbeanalyzedwithrespecttothismotiontype.FordefiningthelimitsoftheFCStwoboundariesareused:thesurfaceboundariesandtheenvelopeboundary.Thesurfaceboundariesarefixedforagiventoolmotionandtheycanbecalculatedfromsection(5.2).Forcalculatingtheenvelopeboundary,theformulaedevelopedinsection(5.3)willbeutilizedbyconsideringthattheangularvelocityiszero.AlsothetooltipvelocityVFwillbedenotedbyfwhichisshortforthefeed.InFigure(5.9-a)thegenericcutterprofileandthefeedvectorfareshown.InitiallythefeedvectoriscoincidentwiththezLaxisoftheTCS.From this figure the feed anglepsi which is measured from the zL axis in the clock wisedirectioncanbeobtainedas parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp= -LL?coszzpsi                 (5.42) Any change in the direction of feed vector will effect the distribution of the FCS on thecutter.Forclearlyseeingthis,thegenericcutterispartitionedintotwosections(Figure5.9-b):Frontface(F)andBackface(B).AnunboundedplanewhichpassesthroughyLandzLaxesofTCSseparatesthesetwofaces.Chapter5.FeasibleContactSurfaces 116     (a)    (b)Figure5.9:FeedanglerangesFigure(5.9-a)illustratesthatthefeedanglepsi takesdifferentvaluesinfivedifferentregions(shaded area). The angles which separate these regions are defined by the geometricparametersalphaandbetaofthegenericcutterfromsection(5.2).AlsothesecondandthefourthcolumnsofTable(5.1)showthefeedanglerangescorrespondingtothesefiveregions.Inthistablethefirstandthethirdcolumnsshowalphanumericsymbolsi.e.F1,B2etc.fordifferentrangesofthefeedangle.LettersFandBdenotethefrontandthebackfacesofthecutter(Figure5.9-b)respectively.Table5.1:TheangularrangesofthefeedangleAccordingtothefeedanglerangesgiveninTable(5.1)thecuttercanhavethreetypesofmotion:Ascendingtype(F1toF4),Horizontaltype(F5)andDescendingtype(B1toB4).WithrespecttothesetoolmotionsthefollowingpropositionscanbegivenChapter5.FeasibleContactSurfaces 117Proposition5.1:Whenthecutterhasascendingandhorizontaltypemotionswhichcovertherange of [ ]0 , the point set CWEK is located on the front face (F) of the cutter.ThereforeonlythefrontfacecancontainthefeasiblecontactsurfacesduringthemachiningProposition 5.2: When the cutter has descending motion which covers the rangeof ( ]00 180 ,thepointsetCWEKislocatedonboththefrontandthebackfacesofthecutter.Thereforebothfacescancontainthefeasiblecontactsurfacesduringthemachining.Note that feed angles ( beta ) and ( beta0 ) are excluded from these propositions.becauseintheformeronethetopcircleoftheupper-conetouchestheworkpieceandinthelatter one all cutter surfaces totally plunge into the workpiece. These propositions will beprovenforthecorner-torussurfaceofthegenericcutterbyconsideringthefeedangleintherangesofF1,F3,B2andB4fromTable(5.1).TheresultsfortheothersurfacepatchescanbefoundinTable(5.2).Theresultsofthesepropositionswillbeusedinthenextchapterduringthecutterworkpieceengagementidentifications.Proof:Ithasbeenshowninsection(5.2)thattwoofthegeometricparametersfordescribingthe corner-torus are theta and phi . Figures (5.10-a, b) illustrates thatphi is measured in CCWdirectionfromthetoolrotationaxisandthetaismeasuredinCCWdirectionfromthexLaxisofTCS.Itcanbeseenfromthesefiguresthatthephi angleofanarbitrarypointPmustliewithintherangeof ]00 beta i.e. )00 betaT            (5.43)When the lower ( )0L and the upper ( )0U limits ofphi from Eq. (5.43) arepluggedintoEq.(5.35),thefollowingexpressionsareobtained: ( )( ) parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp= -LLxz alphatheta tancos 12 m  (5.44-a)Chapter5.FeasibleContactSurfaces 118 ( )( ) parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp= -LLxz betatheta tancos 12 m   (5.44-b)wheresuperscriptsLandUrepresentthelowerandtheupperlimitrespectively.NotethattheangularvelocityinEq.(5.35)becomeszeroin3-axismachining.(a)(b)Figure5.10:Thecorner-toruswithupperandlowersurfaceboundaries.square4  Cutterhasafeedangle( beta ) InthiscasetheanglebetweenthefeedvectorfandzLaxisequalstobetaandthecutterhasanascendingmotion.ForthisfeedanglethedotproductsinEqs.(5.44,a-b)areevaluatedasfollows betaLL z  )0LL beta PluggingtheaboveequationsbackintoEqs.(5.44-a,b)yields parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp= -betaalphathetatantancos 1L2 m              (5.45-a) 0U2,1 0             (5.45-b)Chapter5.FeasibleContactSurfaces 119Note that when the lower and the upper limits of phi are equal to each other i.e.)00 beta ,thecorner-torusdisappearsfromthegenericcutter.Forthecorner-torusgeometry to be present, the inequality ) must hold and thus under thisconsideration ) . In calculations these inequalities will be used frequently. TheratioinsidetheparenthesisofEq.(5.45-a)isgreaterthanoneandthustheinversecosinefunctiondoesnothavearealsolution.InEq.(5.45-b)thetahasrepeatedrootsofzeros. Fromtheaboveresultsitcanbeconcludedthatwhenthegenericcuttermoveswiththefeed angle equal to beta, the envelope boundary of the corner-torus contains only one pointwhichcorrespondstoangles ( )0U and 0U2,1 0 .InFigure(5.11-a),F1representsthispointwhichislocatedonthefrontfaceofthecorner-torus.Thusthisprovestheproposition(5.1)forthismotiontype.     (a)(b)Figure5.11:Envelopeboundarysetsonthe(a)front,andthe(b)backfacesofthecorner-torus.square4  Cutterhasafeedangle( alpha ) Inthiscasethecutterisperforminganascendingmotionwithafeedangleequalstoalpha.AfterevaluatingEqs.(5.44-a,b)withalpha,thefollowingequationsareobtained 0L2,1 0                 (5.46-a) parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp= -alphabetathetatantancos 1U2 m                (5.46-b)Chapter5.FeasibleContactSurfaces 120TheratioinsidetheparenthesisofEq.(5.46-b)ispositiveandlessthanone.Thereforeforagiven parameter set ( )0U and U2,1 the upper surface boundary of the corner-toruscontainstwopointswhicharemembersoftheenvelopeboundary.Thesearetwosymmetricpointswithrespecttothefeedvectorandtheyarelocatedonthefrontfaceofthecorner-torus.TherepeatedrootsinEq.(5.46-a)andthelowerlimitofphi , ( )0L generateapointonthelowersurfaceboundaryofthecorner-torus.Thissinglepointwhichisalsoamemberoftheenvelopeboundaryislocatedonthefrontface.Fromtheseresultsitcanbeseenthattheupperandthelowersurfaceboundariesofthecorner?toruscontainenvelopeboundarypointsonthefrontface.Thereforeitcanbeconcludedthatchangingphi between( )0L and ( )0U generatesnewenvelopeboundarypointsonthefrontface.F3inFigure(5.11-a)showsanexampleoftheenvelopeboundaryforthismotion. ThusforthismotiontypetheenvelopeboundaryonthefrontfaceandtheportionoftheuppersurfaceboundaryonthefrontfacedrawthelimitsofthepointsetCWEK.Thusthisprovestheproposition(5.1)forthismotiontype.F3inFigure(5.12)illustratesthepointsetCWEKwithshadedarea.Chapter5.FeasibleContactSurfaces 121Figure5.12:Feasiblecontactsurfacesofthetoroidalpartwithrespecttothecutterfeedangle.square4  Cutterhasafeedangle( alpha0 ) In this and the following case the cutter has a plunging motion with respect to feedvector. When the feed angle alpha0 is plugged into Eqs. (5.44-a, b), the followingresultsareobtained 0L2,1 180               (5.47-a) parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp-=-alphabetathetatantancos 12 mU              (5.47-b)InEq.(5.47-a),thelowerlimitofthetahastworealsolutions 0 .Thesetwosolutionsandthelowerlimitofphi , ( )0L generateonlyonepointonthelowersurfaceboundaryoftheChapter5.FeasibleContactSurfaces 122corner-torus.Thispointwhichalsobelongstotheenvelopeboundaryislocatedonthebackface.TheratioinsidetheparenthesisofEq.(5.47-b)isgreaterthan-1andthereforetherearetwodistinctrealsolutionsfortheupperlimitoftheta.Thesesolutionsandtheupperlimitofphi ,( )0U producetwodistinctpointsontheuppersurfaceboundaryofthecorner-torus.These points which are members of the envelope boundary are located on the back face.Thereforebychangingphi betweenitslimitsotherenvelopeboundarypointsaregeneratedonthebackface.InFigure(5.11-b),B2representstheenvelopeboundaryforthismotiontype.ThelimitsofthepointsetCWEKaredefinedbytheenvelopeboundaryonthebackfaceandtheportionoftheuppersurfaceboundaryonthefrontface.Thusthisprovestheproposition(5.2).B2inFigure(5.12)illustratesthepointsetCWEKwithshadedarea.square4  Cutterhasafeedangle( beta0 )WiththisfeedangleEqs.(5.44,a-b)takethefollowingform parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp-=-betaalphathetatantancos 1L2 m               (5.48-a) 0U2,1 180            (5.48-b)BecausetheratioinEq.(5.48-a)islessthan-1,thereisnorealsolutionforthelowerlimitoftheta.TheupperlimithastwosolutionsinEq.(5.48-b).Thesesolutionsandtheupperlimitofphi ,( )0U produceonlyonepointontheuppersurfaceboundaryofthecorner-torus.B4inFigure(5.11-b)showsthispointonthebackface. For this motion it can be said thateven there is only one envelope boundary point, the whole surface of the corner-toruscontainspointsfromthesetCWEK.B4inFigure(5.12)illustratesthepointsetCWEKwithshadedarea.ForclearlyseeingthissituationthetangencyfunctioninEq.(5.18)isutilizedforthecornertorus.Underthe3-axismachiningthetangencyfunctionofthecorner-torustakesthefollowingform ( ) ( ) ( )( )LLsinsinyxphiphi++          (5.49)Chapter5.FeasibleContactSurfaces 123WhentheEq.(5.49)isevaluatedforthefeedangle beta0 ,ityields ( ) ( )         (5.50)where ]andthus theta element[-1,1].NotethatbecausethefeedvectorfandyLaxisofTCSareorthogonal,theirdotproductisequaltozero.AccordingtoEq.(5.43)itcanbesaidthat  beta and phi . When these inequalities are evaluated in Eq. (5.50) thetangency function becomes ( ) 0 , where ]00 beta . Therefore the wholesurface of the corner-torus contains the point set CWEK. Thus this proves the proposition(5.2).B4inFigure(5.12)illustratesthepointsetCWEKwithshadedarea. Alsotheupperandthelowerconesurfacepatchesofthegenericcuttermayhavefeasiblecontactsurfaces(FCS).ThedistributionofFCSonthesesurfacescanbeanalyzedusingthesame steps taken for the corner-torus. Table (5.2) shows the distribution of FCS on eachsurfacepatchofthegenericcutterwithrespecttothemotiontypesdefinedinTable(5.1).CWEK,U,CWEK,TandCWEK,Lrepresentthefeasibleengagementpointsetsoftheupper-cone,thecorner-torusandthelower-conesurfacerespectively.Ifforagivenfeedangle,thesurfacepatchdoesnotcontainanypointfromtheFCS,thesymbol{}isusedforrepresentingthissituation.Forexample,inTable(5.2)forthefeedangleinF2onlytheupper-coneandthecorner-torussurfaceshaveFCSandthusduringthematerialremovalonlythesesurfacescancontactthein-processworkpiece.Table5.2:CWEKpointsetsofthegenericcutterunderdifferenttoolmotions.Chapter5.FeasibleContactSurfaces 1245.5Discussion In this chapter the engagement behaviors of NC cutter surfaces under varying toolmotionshavebeenpresentedbyintroducingthefeasiblecontactsurfaces(FCS)terminology.ThedescriptionoftheFCShasbeenmadebyutilizingthekinematicallyfeasibleengagementpointscontainedinthesetCWEK.InatypicalNCcuttertheconstituentsurfacegeometriesshowdifferencesandbecauseofthisduringthemachiningtheengagementcharacteristicsforthesesurfacesbecomedifferent.UnderthisconsiderationthesetCWEKhasbeenpartitionedamong the constituent surfaces of a cutter. As mentioned in this chapter, the CutterWorkpieceEngagements(CWEs)arethesubsetsoftheFCSandforcalculatingCWEstheboundariesoftheFCSmustbeidentified.Inthisworktheseboundarieswhicharelimitsofthe set CWEK have been defined by the cutter surface boundaries and the envelopeboundaries.Undertherigidtoolmotionsthecuttersurfaceboundariesstayfixed.Butontheotherhandtheenvelopeboundariesmay change.Formodelingtheenvelopeboundariesatangencyfunctiondefinedbythesurfacenormalandthetoolvelocityhasbeenutilized.LaterbychangingthetoolvelocitydirectionthedistributionsoftheFCSonthecutterhavebeenanalyzed. Also it has been shown with examples that only the certain parts of the cuttersurfaceshavecontactswiththein-processworkpiece.TheresultsfromthischapterwillbeusedintheCWEsgeneration. 125Chapter6CutterWorkpieceEngagementsInthischapterthemethodologiesforobtainingCutterWorkpieceEngagements(CWEs)inmillingarepresented.Cuttingforcesareakeyinputtosimulatingthevibrationofmachinetools(chatter)priortoimplementingtherealmachiningprocess.Theseforcesaredeterminedbythefeedrate,spindlespeed,andCWEs(capturesthedepthofcut).OfthesefindingtheCWE is most challenging due to the complex geometry of the in-process workpiece andvarying tool motions. The CWE geometry defines the instantaneous intersection boundarybetween the cutting tool and the in-process workpiece at each location along a tool path.Figure (6.1) summarizes the steps involved in CWE extraction. Inputs from CAD/CAMinclude the tool paths in the form of a CL Data (cutter location data) file, geometricdescription of the cutting tool and a geometric representation (B-rep, polyhedral, vectorbasedmodel)oftheinitialworkpiece.Thekeystepswhicharethesweptvolumegenerationandthein-processworkpieceupdatehavebeenpresentedinthepreviouschapters.CAD/CAMModelofSweptVolumeModelofIn-processWorkpieceCWEExtractorCWEExtractorEndofToolPath?EndofToolPath?NextToolPath?NextToolPath?ModelofCuttingToolDoneDoneDone13noyesyesnoIn-ProcessGeometryGeneratorIn-ProcessGeometryGeneratorSweptVolumeGeneratorSweptVolumeGenerator21 CLDataInitialWorkpieceCutter/Workpiece EngagementsProcessModeling&OptimizationCuttingForcePredictionStabilityAnalysisSpeedandFeedOptimizationStaticDeflectionAnalysisStaticDeflectionAnalysisCWEExtractionModelofRemovalVolumeFigure6.1:CWEExtractionStepsChapter6.CutterWorkpieceEngagements 126In this chapter the CWEs are calculated for supporting the force prediction modeldescribedin[5].ThismodelfindstheCartesianforcecomponentsbyanalyticallyintegratingthedifferentialcuttingforcesalongthein-cutportionofeachcutterflute.InthismodelCWEarea with a fixed axialdepth of cut is defined bymapping the engagement region on thecuttersurfaceontoX-Yplanewhichrepresentstheengagementangleversusthedepthofcutrespectively(Figure6.2). Figure6.2hasbeenremovedduetocopyrightrestrictions.TheinformationremovedisCWEareafortheforceprediction,[5]. In this chapter the methodologies for finding the CWEs are developed based on themathematical representation of the workpiece geometry. The workpiece geometries aredefined by a solid, a polyhedral and a vector based modeler respectively. Althoughpolyhedralandvectorbasedapproachesrequireashortercomputationaltimethandoesthesolid modeler based approach, the accuracy of these approaches depends greatly on theresolutionoftheworkpiece.Thereisalwaysatradeoffbetweencomputationalefficiencyandaccuracyintheseapproaches.Forexamplethesolidmodelerhasanadvantageinaccuracybecauseitprovidesanaccurategeometricrepresentationfortheworkpiece.ThereforeinthesolidmodelerbasedapproachthemostaccuraterepresentationsoftheCWEsareobtained.Butontheotherhandthismodelerusesnumericaltechniquesthatarelimitedprimarilybyefficiencyandrobustness. TheCutterWorkpieceEngagementsinSolidModelsarepresentedinsection(6.1).Thissectioncontainstwomethodologies:EngagementExtractionMethodologyin3-AxisMilling(section 6.1.1) and Engagement Extraction Methodology in 5-Axis Milling (section 6.1.3)respectively. In the 3-axis milling methodology the cutter surfaces are decomposed withrespecttothefeeddirectionasexplainedinchapter(5),andthenthesedecomposedsurfacesareintersectedwiththeirremovalvolumesforobtainingtheclosedCWEarea.Inthe5-axismethodologythesimilarapproachisutilizedwithonedifference.Inthiscasethein-processworkpiece is used instead of cutter removal volumes. The developed solid modeler basedmethodologiesareimplementedusingACISsolidmodelingkernel.Thebothmethodologiesaresupportedbyexamples.Chapter6.CutterWorkpieceEngagements 127 TheCutterWorkpieceEngagementsinPolyhedralModelsarepresentedinsection(6.2).Also this section contains two methodologies. In the 3-axis CWE methodology (section6.2.1)thechordalerrorproblemdescribedinchapter(1)isaddressed.ForfindingtheCWEsapolyhedralrepresentationoftheremovalvolumeismappedfromEuclideanspaceintoaparametric space. Thus CWE calculations are reduced to line-plane intersections. In thismethodologytheformulasaredevelopedforthelinear,circularandhelicaltoolpaths.Forthe5-axismillingadirecttriangle/surfaceintersectionapproach(section6.2.2)isdeveloped.Inthis approach for generating CWE area, without doing mapping the cutter surface isintersectedwiththetriangularfacetsobtainedfromthein-processworkpiece. The CutterWorkpiece Engagements in Vector Based Models are presented in section (6.3). In thismethodology the workpiece geometry is broken into a set of evenly distributed discretevectors and also the cutter is discretized into slices perpendicular to the tool axis. ForgeneratingCWEstheintersectionsareperformedbetweendiscretevectorsandcutterslices.Finallythechapterendswiththediscussioninsection(6.4). 6.1CutterWorkpieceEngagementsinSolidModelsThis section presents Solid modeling methodologies for finding Cutter WorkpieceEngagements(CWEs)generatedduring3and5-axismachiningoffree?formsurfacesusingarangeofdifferenttypesofcuttingtoolsandtoolpaths.Figure(6.3)summarizesthestepsinvolved in CWE extraction using B-rep based solid modeler. Inputs from CAD/CAMinclude the tool paths in the form of a CL Data (cutter location data) file, geometricdescriptionofthecuttingtoolandageometricrepresentation(B-rep)oftheinitialworkpiece.Key steps include swept and removal volume generation for each tool path. Althoughcomputationalcomplexityandrobustness,forlimitedapplications,remainissuesthatneedtobe addressed, solid modelers have been recognized as one approach to finding CWEgeometry. Solid modelers have an advantage in accuracy because they generate the exactmathematical representation for the intersections. The methodologies in this section havebeenimplementedusingacommercialgeometricmodeler(ACIS)whichisselectedtobethekernel around which the geometric simulator is built. In these approaches, in-processworkpiece updating and cutter/workpiece engagement extraction are performed usinggeometricandtopologicalgorithmswithinthesolidmodelerkernel.InthesesituationstheChapter6.CutterWorkpieceEngagements 128solidmodelingapproachappliesnumericalsurfaceintersectionalgorithmswhicharebasedonsubdivision,andcurvetracing(marching)methods.Figure6.3:AB-repSolidModelerbasedCWEextraction6.1.1EngagementExtractionMethodologyin3-AxisMilling This section presents a B-rep Solid modeler based methodology for finding CWEsgeneratedduring3-axismachining.ForthispurposecuttersurfacesaredecomposedwithrespecttothetoolfeeddirectionandthentheyintersectedwiththeirremovalvolumesforobtainingtheboundarycurvesoftheclosedCWEareain3DEuclidianspace.Latertheseboundary curves are mapped from Euclidean space to a parametric space defined by theengagementangleandthedepth-of-cutforagiventoolgeometry. As explained in chapter (5), a typical NC cutter has different surfaces with varyinggeometriesandduringthemachiningtheengagementcharacteristicsofeachsurfacebecomedifferent. Under this consideration in this section the kinematically feasible engagementpoints CWEK are divided into constituent sub sets. The primary task in finding the CWEgeometryisfindingtheboundaryoftheengagementregion.TheboundarysetofCWEisChapter6.CutterWorkpieceEngagements 129representedbybCWE.Thisdefinesthegeometryrequiredforinputtoprocessmodeling(i.e.forceprediction).Thus )K           (6.1)Thesethreesetsareillustratedin(Figure6.4)fortheFlat-Endmillperformingahelicaltoolmotion. In this figure the sets bCWE, CWE and CWEK are shown separately for the side(cylindrical)faceandthebottom(flat)faceoftheFlat-Endmill.Figure6.4:PointsetsusedindefiningEngagements Fortheforcemodeldescribedin[5]in-cutsegmentsofthecuttingedgesareneeded.Inthissectionforobtainingin-cutsegmentsbothbCWEandthecuttingedgesaremappedinto2Dspacedefinedbytheengagementangle(u)andthedepthofcut(v).Thenthecurve/curveintersectionisperformedbetweenbCWEandthecuttingedges.Thecommonmillingcuttershave different surface geometries i.e. a Ball-End mill is defined by two natural quadricsurfaces ? spherical and cylindrical. As a result of this the parameterization of theengagementareadiffersforeachcuttersurface.IntheFlat-Endmill,forexample,thedepthofcutvforanarbitrarypointPelementCWEisthedistancebetweenthelocationofPandthecuttertooltipalongthetoolrotationaxis,andtheengagementangleuismeasuredfromtheyLaxisofTCS(Figure6.5).Chapter6.CutterWorkpieceEngagements 130Figure6.5:CWEparametersofanarbitrarypointP. Theengagementparametersfordifferentcutter surfacesareillustratedinFigure(6.6).BecausetheFlat-End,theTaper-EndandtheFillet-Endcuttershaveaflatsurfaceattheirbottom,alsointhisfiguretheengagementparametersofthecutterbottomsurfaceareshown.Asexplainedinchapter(5), whenthefeedangleisintherangeof(B1toB4)thecutterbottomsurfacecanhavecontactwiththeworkpieceduringthemachining.   (a)(b)(c)(d)(e)Figure6.6:DefiningCWEparametersuandvon(a)torus,(b)sphere,(c)frustumofacone,(d)cylinder,and(e)flatbottomsurfacesofcommonmillingcutters.Ithasbeenshowninchapter(5)forthe3-axismachiningthataccordingtothefeedanglerangesthecuttercanhavethreetypesofmotion:Ascendingtype(F1toF4),Horizontaltype(F5) and Descending type (B1 to B4). These motion types effect the construction of thekinematicallyfeasibleengagementpointsetCWEKandconsequentlythischangestheangularrangesoftheengagementparameteruasfollowssquare4 Ifthefeedangleisintherangeof(F1toF5)onlythefrontfaceofthecuttercanhavecontact with the workpiece and the engagement angle u covers ] range i.e.pi .Chapter6.CutterWorkpieceEngagements 131square4 Ifthefeedangleisintherangeof(B1toB4),boththefrontandthebackfacesofthecuttercanhavecontactwiththeworkpiece.Thistimetherangeoftheengagementangleubecomes: pi Thetwocasesaboveshowthatbasedonthekinematicsofthe3-axismachiningthecuttersurfaces differently contribute to the CWE extraction. For this reason cutter surfaces arebrokendownintotheirconstituentsubsurfaces.Ithasbeenshowninchapter(5)that,thefrontandthebacksurfacesareseparatedbyanunboundedplanewhichpassesthroughyLandzLaxesofTCS.AlsoifthecutterhasaflatbottomsurfacethenthissurfaceliesonanotherunboundedplanewhichpassesthroughxLandyLaxesofTCS.ThisdecompositionisshownfortheTaper-EndandFlat-EndmillsinFigure(6.7),whereGBack,GBottomandGFrontrepresenttheconstituentsubsurfacesofthecuttergeometryG.Table(6.1)showsthedecompositionofthesurfacegeometriesforthecommonmillingtools.Figure6.7:GeometricdecompositionofthecuttersurfacesTable6.1:Constituentsurfacesofcuttergeometriesaftergeometricdecompositionin3-axismilling.Each constituent surface of a cutter geometry given in the Table (6.1) generates itskinematically feasible engagement point set with respect to the cutter feed vector. ForexamplewhenaFillet-Endmillhasadescendingmotiondescribedby(B1toB4)allthreeChapter6.CutterWorkpieceEngagements 132constituent surfaces of the toroidal part: GBack ,  GBottom and GFront generate the setsCWEK,Back, CWEK,Bottom and   CWEK,Front respectively (see Figure 6.8). Thus the followingequationcanbewrittenforatoroidalpartperformingthismotion FrontKBottomKBackKK CWE ,*,*, union        (6.2)where * andlater * representregularizedBooleanunionandintersectionsetoperationsrespectively.Figure6.8:DecomposingthepointsetCWEKofthetorusintothreeparts.Notethatdependingonthemotiontypesomeofthesesubsetscanbeempty.Forexampleintheascendingmotion(F1toF5)onlythefrontpartofthecutterengageswiththeworkpieceandthereforeonlythesetCWEK,Frontisconsideredasafullset.Table(6.2)showsthesethreesets for different cutter geometries with respect to the feed angle ranges. The symbol ?representsanemptysetforagivenfeedanglerange.Table6.2:FeasibleengagementpointsforcuttersurfaceswithrespecttothetoolmotionsChapter6.CutterWorkpieceEngagements 133Thefollowingproperties(6.1)and(6.2)motivatefindingtheCutterWorkpieceEngagementsin3-axismilling.Property6.1:GivenatoolpathTifortheithtoolmotionandthecuttersurfacegeometry:a) Sweeping the point sets CWEK,Back, CWEK,Bottom and CWEK,Front   along Ti generates thesweptvolumesofthecorrespondingcuttersurfacesi.e.SVi(Back),SVi(Bottom)andSVi(Front)respectively.ForagiventoolmotionandthecuttersurfacegeometrytheconstituentpartsofthesetCWEK arechosenfromtheTable(6.2)forsweeping.ForexampleFigure(6.9) illustrates this for the Fillet-End mill which follows a linear toolpath in descendingmotion(B1toB4).(a)(b)(c)Figure6.9:Sweptvolumesgeneratedbythekinematicallyfeasibleengagementpointsets.b)Intersectingthesweptvolumesfromtheproperty1(a)withthein-processworkpieceWi-1generatestheremovalvolumesasfollows )*1 Backiii arrowright-            )*1 Bottomiii arrowright-          (6.3) )*1 Frontiii arrowright- whereRVi(.)representsthecorrespondingremovalvolumeofeachsweptvolume.Notethatfor the ith tool motion the total removed material from the in-process workpiece can bedenotedasfollows )** Frontiiii union (6.4)Chapter6.CutterWorkpieceEngagements 134Property 6.2:  The intersection of constituent surfaces GBack, GBottom and GFront with theircorrespondingremovalvolumes (.)i generatesthecutterworkpieceengagementboundarybCWEiforthegivencutterlocationasfollows)})})}***FrontBottomBackiiiintersectionunionunion(6.5)The3-axisCWEextractionmethodologyissummarizedinFigure6.10.TheinputsareithtoolpathsegmentTi,in-processworkpieceWi-1andtheconstituentsurfacesofthecuttergeometryG.Figure6.10:ProcedureforobtainingtheCWEsThe reported method has been implemented using a commercial geometric modeler(ACIS)whichisselectedtobethekernelaroundwhichthegeometricsimulatorisbuilt.Inthedescribedmethodologyfor generatingthebCWEs,theface?faceintersectionsbetweencutter surfaces and the removal volume surfaces are performed. For this purpose a givenremoval volume is decomposed into its faces by using ACIS function api_get_faces. Thefaces obtained from this function are intersected with the cutter surface which is alsoChapter6.CutterWorkpieceEngagements 135represented as a FACE Entity. These intersections are performed by using the functionapi_fafa_int. For obtaining the cutter surface i.e. GFront, first the circular wires which areperpendiculartothetoolaxisaregeneratedalongthetoolaxisandthentheyareskinnedbyusing the function api_skin_wires.  It has been shown in chapter (1) that ACISrepresentationalhierarchycontainsBODY,LUMPandSHELLetc.ThereforeabodycanberepresentedbyLUMPs,aLUMPcanberepresentedbySHELLsandsoon.Thefunctionapi_fafa_intgeneratesintersectioncurvesasaBODYEntity.Forobtainingthepropertiesofthesecurvesi.ethestartandendcoordinates,theseBODYsaredecomposedintoLUMPs,SHELLs,WIREsandEDGEsrespectively.TheAlgorithm(6.1)forthisprocessisshownbelowintheC++format.TheinputsareRVi(Front),GFrontandacutterlocationpointPCL.ItgeneratestheclosedboundarysetoftheCWEareaforthegivencutterlocationpoint.Input:RVi(front),GFront,PCL(xCL,yCL,zCL)Output:bCWEiatPCLENTITY_LISTface_list;//ContainerofremovalvolumefacesFACE*ff=api_get_faces(RVi(front),face_list);//pointsthefirstfaceoftheface_listFACE*cutter_face=GFront;BODY*int_curve=NULL;//amemberfrombCWELUMP*rem_vol_lump=NULL;SHELL*rem_vol_shell=NULL;WIRE*rem_vol_wire=NULL;EDGE*rem_vol_edge=NULL;while((ff=(FACE*)face_list.next())!=NULL){//iteratesuntilallfacesofthe//removalvolumeisprocessesapi_fafa_int(cutter_face,ff,int_curve);rem_vol_lump=int_curvearrowrightlump();while(rem_vol_lump!=NULL){rem_vol_shell=rem_vol_lumparrowrightshell();while(rem_vol_shell!=NULL){rem_vol_wire=rem_vol_shellarrowrightwire();while(rem_vol_wire!=NULL){rem_vol_edge=rem_vol_wirearrowrightcoedge()arrowrightedge();//processtheedgehereforobtainingbCWEsrem_vol_wire=rem_vol_wirearrowrightnext();}rem_vol_shell=rem_vol_shellarrowrightnext();}rem_vol_lump=rem_vol_lumparrowrightnext();}ff=ffarrowrightnext();}Algorithm6.1:ObtainingtheclosedboundariesoftheCWEsChapter6.CutterWorkpieceEngagements 1366.1.2Implementation Figures(6.11)and(6.12)showFlat-EndandTaper-Endmillsremovingmaterialalongalinearrampingtoolpathi.e.thetoolmovesinallthreeaxessimultaneously.Theremovalvolumesassociatedwiththedifferentcuttersurfacesareseparated.InthecaseoftheFlat-End mill this gives thematerial removed by the cylindrical at the side and the flat at thebottomsurfaces.PlotsofCWEsatdifferentCutterLocations(CLs)areillustrated.Theseareobtained by intersecting the constituent surfaces of the cutter with their correspondingremovalvolumes.TwoformatsareusedforplottingtheCWEs.ThefirstisanXYplotofdepth-of-cutv(asmeasuredfromthetooltippoint)versusengagementangleu.Thesecondplotshowstheengagementareaofthecutterbottomsurfaceinapolarformat:cutterbottomradius r versus engagement angle u. In Figure (6.11) the Flat-End mill is performing adescending motion in which the feed angle is in the range of (B1 to B4). Thus both thecylindricalandthebottomsurfaceremovematerial.ForthismotiontypeoftheFlatEndmillonly the CWEK,Front  and the CWEK,Bottom point sets are active. The XY plots show theengagementbetweenzeroandpiforthefrontface,andpolarplotsshowtheengagementbetweenzeroand2piforthebottomflatface.Figure6.11:CWEsfortheFlatEndmillperformingalinear3-axisdescendingmotion.InFigure(6.12)becausetheTaper-Endmillisrampingupinwhichthefeedangleisintherangeof(F1toF5),onlythepointsetCWEK,Frontisactiveduringthematerialremoval.ThusChapter6.CutterWorkpieceEngagements 137for this motion type, only the front face of the cutter contributes to the CWE extractionprocedurewiththeengagementangleuchangesbetweenzeroandpiradian.Figure6.12:CWEsfortheTaper-Endmillperformingalinear3-axisascendingmotion.Inthelastexample(Figure6.13)aFlat-Endmillisfollowingahelicaltoolpath(plungingcutting)forenlargingahole.InthisexampleboththesideandthebottomfacesoftheFlat-Endmillareremovingmaterial.Thecutterperformsthreehalfturnseachcorrespondingtoa01800 - rangei.e.thirdturnhasthestartingangle 0360 andendingangle 0540 .InthismotionoftheFlat-EndmillthepointsetsCWEK,FrontandCWEK,Bottomareactive.ThetotalremovedmaterialRV(Front)andRV(Bottom)areshownforeachhalfturnwithdifferentcolors.Forthe cutter location CL1 both types of plots are shown because the bottom of the cutter isremovingmaterialatthislocationalso.ForCL2onlytheside(cylindrical)faceofthecutterhasengagementwiththeworkpiece.Chapter6.CutterWorkpieceEngagements 138Figure6.13:HelicalToolMotionswithaFlat-EndMillandCWEs6.1.3EngagementExtractionMethodologyin5-AxisMilling ThissectionpresentsaB-repSolidmodelerbasedmethodologyforcalculatingCWEsin5-axis milling operation. Many of the steps defined in the previous section for the 3-axisCWEmethodologyareapplicableforthe5-axisCWEmethodologyinthissection.Thereisonlyonemaindifferencebetweenthesetwomethodologies.In5-axisCWEmethodologyforobtaining bCWEs the feasible contact surface of a cutter will be intersected with the in-process workpiece instead of the removal volume which is used in the 3-axis CWEmethodology. At any given instance of the 5-axis tool motion the bottom center and topcenter of the rigid cutter may move in directions that do not lie in the same plane. Forexample in Figure (6.14-a), the top velocity vector VTop and the bottom velocity vectorVBottom point to the different directions. An arbitrary velocity V on the tool axis can becalculatedbylinearlyinterpolatingVTopandVBottomasfollows ( ) BottomTop 1 V               (6.6)whereuelement[0,1].Ontheotherhandin3-axismillingthetopandbottomcentersofthecuttermoveinthesamedirection(Figure6.14-b). As explained in chapter (5), a cutter contacts in-process workpiece through the set ofCWEK.Mostofthepointsinthissetlietowardsthefrontofthecutterandaremachinedawayasthetoolleavesitscurrentposition.OnlythosepointsforwhichthemotiondirectionChapter6.CutterWorkpieceEngagements 139isperpendiculartothecuttersurfacenormalareleftbehindonthemachinedsurfaceasacurve.Asexplainedinchapter(5),thesepointsdefinetheenvelopeboundaryofthecutterwhichdescribesthegeometriclimitsofthesetCWEK.(a)(b)Figure6.14:(a)Envelopeboundaryin5-axismilling,andin(b)3-axismillingrespectively. Asexplainedinchapter(5),fordefiningtheboundariesofthefeasiblecontactsurfaces(FCS)envelopeboundarysetisused.AsillustratedinFigure(6.14-b)fortheFlat-Endmillperforming3-axismachining,becausethetopandthebottomvelocityvectorsofthecutterpointtothesamedirectiontheenvelopeboundaryhasalinearcharacteristic.Butontheotherhandinthe5-axistoolmotionsthevelocityvectoronthecutteraxiscontinuouslychangesandasaresulttheenvelopeboundaryloosesitslinearity(seeFigure6.14-a).Inthiscasetheenvelopeboundarycurvesareapproximatedbysplines.ForexampleinthecaseofaFlat-End mill, these boundaries are represented by the helix like curves [12]. Because of theapproximationintheenvelopeboundarycurvestheCWEmethodologydescribedforthe3-axis milling does not properly work for the 5-axis milling. The intersections obtainednumerically by the B-rep solid modeler become non robust. For solving this problem amethodologybasedontheintersectionsbetweentheFCS andthein-processworkpieceispresentedinthissection.ThismethodologyisexplainedforanimpellermachiningusingaTaper-Ball-Endmillanditcanalsobeappliedtoothermillingcuttersperforming5-axistoolmoves. InFigure(6.15-a)aTaper-Ball-EndmillisshownforagivenCutterLocation(CL)point.At this CL point the feasible contact surface defined by the envelope and cutter surfaceboundaries is constructed (Figure 6.15-b). In the next stage (Figure 6.15-c) we offset thissurfaceinthe(+/-)directionsofthevelocityvectorVwithaninfinitesimalamount( 2).ThisprocessmakesthissurfaceavolumewhichiscalledasBODY.TominimizetheerrorChapter6.CutterWorkpieceEngagements 140introducedbythisoffsettingthelocationofthevectorVistakeninthemiddleofthetoolaxis.ThiscausesageometricerroratbothVTopandVBottominequalmagnitudeandzeroerrorhalfwaybetweenthesetwopoints.(a)(b)(c)Figure6.15:Offsettingthefeasiblecontactsurface AsillustratedinFigure(6.16)therearethreemainstepsforgeneratingtheCWEs.InthefirststeptheBODYobtainedbyoffsettingthefeasiblecontactsurfaceisintersectedwiththein-processworkpieceatagivencutterlocationpoint.Thisintersectiongeneratesaremovalvolume.Inthesecondsteptheremovalvolumeisdecomposedintoitsconstituentfacesandbetweenthesefacesandthefeasiblecontactsurfaceface/faceintersectionsareperformed.Eachoneoftheseintersectionsgeneratesacurve.ThefullboundaryoftheCWEarea,bCWEisthecombinationofeachindividualcurveobtainedfromface/faceintersections.Finallyinthelaststep,bCWErepresentedin3DEuclidianspaceismappedinto2Dspacedefinedbytheimmersion(engagement)angleandthedepthofcut.Notethatthese3stepsareperformedforeachcutterlocationpointonatoolpathsegment.Chapter6.CutterWorkpieceEngagements 141Figure6.16:CWEstepsin5-axismillingmethodology Theimplementationofthe5-axisCWEmethodologyisillustratedinFigures(6.17)to(6.19) respectively. For this implementation the solid modeler kernel (ACIS) and C++ isused.TheACISfunctionapi_get_facesdecomposesthefacesofthein-processworkpiece.Then each one of these faces is intersected with the feasible contact surface utilizing thefunctionapi_fafa_int.ForobtainingtheclosedboundariesoftheCWEsthesameproceduredescribed in Algorithm (6.1) is followed. In Figures (6.17) and (6.18) for the given threecutter location points XY plots of depth-of-cut versus immersion angle are plotted. Theseexamplesareillustratedforthefirstandsecondpassesofthecutter.Alsointhesefigurestheremovalvolumesforeachpassesareshown.Fromthesetwofiguresitcanbeseenthattheamount of the removed material in the first pass is more than that of the second pass. InFigure (6.19) the in-process workpiece is shown after the third and fourth passesrespectively.Chapter6.CutterWorkpieceEngagements 142Figure6.17:5-axisCWEsduringthefirstpassoftheimpellermachiningFigure6.18:5-axisCWEsduringthesecondpassoftheimpellermachiningChapter6.CutterWorkpieceEngagements 143Figure6.19:In-processworkpiecesafterthethirdandfourthpassesrespectively.6.2CutterWorkpieceEngagementsinPolyhedralModels Polyhedral models provide the advantage of simplifying the workpiece surfacegeometry to planes which consist of linear boundaries. Thus the intersection calculationsreducetoline/surfaceintersections.Thesecanbeperformedanalyticallyforthegeometryfoundoncuttingtools.ForobtainingCWEarea,facetswhichcontainlinearboundariesareintersectedwiththesurfaceofthecutterandthentheintersectionpointsareconnectedtoeachother.Ifthecuttersurfacehasthesecondorderequation,e.g.cylinder,coneorspherenaturalquadricsurfaces,eachline?surfaceintersectiongivestworoots.Ifthecuttersurfacehasthe fourthorder equatione.g.torus, eachlinesurfaceintersection givesfourroots. Inmost cases only one of these roots are needed for obtaining the CWEs and the rest isredundant.CWEextractionalgorithmsmustberobustenoughtohandlethecompletesetofintersectioncasesbetweenthecuttingtoolandatriangularfacet(seeFigure6.20).Figure6.20:TypicalintersectionsbetweenafacetandacuttingtoolAmajorconsiderationinCWEextractionistheformofthein-process geometry.Beyondtherebeingdifferencesbasedonthetypeofmodelbeingusedthereisalsoachoiceofusingthe in-process workpiece or the removal volume. Either can be generated by applying4Intersectionpoints1TangentpointCuttingtool2Intersectionpoints1tangentand2intersectionpointsChapter6.CutterWorkpieceEngagements 144Booleanoperationsbetweenthesweptvolumeforatoolpathandtheinitialworkpieceorin-processworkpiecegeneratedbytheprevioustoolpath(subtractionorintersection).Theuseoftheremovalvolumeinsteadofthein-processworkpiecehasanumberofadvantages:square4 The size of the geometry model that must be manipulated by the CWE extractionalgorithmsissignificantlysmallerfortheremovalvolume.square4 TheuseoftheremovalvolumemodelbettersupportsparallelcomputationstrategiesforCWEextraction.TheCWEsforeachremovalvolumecanbeextractedindependently.Anagentbasedmethodologythatdoesjustthisisdescribedin[85].For these reasons the removal volume is used in the research described in this chapter tocapturethein-processgeometry. Sinceapolyhedralmodelisanapproximaterepresentationoftheexactanalyticalsurfacefromwhichitwasgenerateditwouldseemthatsimilaraccuracyissuestothosefoundinthediscretemodelingapproachesexist.Oneexampleisillustratedin(Figure6.21).Shownisthepolyhedralmodelofaremovalvolumemachinedfromtheworkpiecebyacylindricalendmill.Thefacetingalgorithmthatgeneratesthismodelapproximatessurfacestoaspecifiedchordalerror.Ascanbeseenfromthe2Dviewthisresultsinfacetsthatlieoutsidethetoolenvelopatagivenlocationeventhoughthecuttingtoolisincontactwiththeactualremovalvolumesurface.ThisfacetshouldbeconsideredinfindingtheCWEboundarybutwouldbedifficulttodetectsinceitdoesnotintersectwiththetoolgeometry.Figure6.21:TheedgesoffacetsdeviatefromtherealsurfaceInthissectionCWEsarecalculatedforthecommonmillingcuttersperforming3-axisand5-axistoolmovements.IncalculationsthecuttingtoolgeometriesarerepresentedimplicitlyChapter6.CutterWorkpieceEngagements 145by natural quadrics and toroidal surfaces. Natural quadrics (Figure 6.22-a) consist of thesphere,circularcylinderandthecone.Togetherwiththeplane(adegeneratequadric)andtorustheseconstitutethesurfacegeometriesfoundonthemajorityofcuttersusedinmilling.For example a ball nose end mill (BNEM) is defined by two natural quadric surfaces ?sphericalandcylindrical.OtherexamplesareshowninFigure(6.22-b).(a)            (b)Figure6.22:(a)Constituentsurfacesofmillingcuttersand(b)sometypicalmillingcuttersurfaces.IndevelopingtheCWEmethodologiesforthepolyhedralmodelstheimplicitequationsofthecuttergeometrieswillbeused.Theseare Sphere: ( ) ( ) 02 =        (6.7-a) Cylinder: ( ) ( ) ( ) ] 022 =  (6.7-b) Cone: ( ) ] ( ) ( ) 022 = (6.7-c) Torus: [ ] 0222222 =C (6.7-d) Plane: 0       (6.7-e)Chapter6.CutterWorkpieceEngagements 146where P represents the position of an arbitrary point on the cutter?s surface and thecoordinatesofCare(xC,yC,zC).6.2.1EngagementExtractionMethodologyin3-AxisMilling ThissectionpresentsamethodologyforcalculatingCWEsfrompolyhedralmodelsthat addresses the chordal error problem described in section (6.2) and which reduces theproblem to line-plane intersections for the common milling cutter geometries and movetypes. In this methodology for finding the CWEs in 3-Axis milling, a polyhedralrepresentation of the removal volume is mapped from Euclidean space into a parametricspace. The nature of the swept geometry and the goal of engagement extraction pointstowardsapreferredparameterization.AsshowninFigure(6.23),theengagement(immersion)angle(phi),depthofcut(d)forpointsonthe cuttersurfaceandthetooltipdistance(L(t))makeupthisparameterization,P(phi,d,L(t))where(0<=<t<=<1).Figure6.23:CWEparametersThis mapping as will be seen has the effect of reducing the cutting tool geometry to anunbounded plane. Thus the boundary of the CWE is found by performing first orderintersectionsbetweentheplanarrepresentationofthecuttingtoolandtheplanarfacetsinthepolyhedralrepresentationoftheremovalvolume.Chapter6.CutterWorkpieceEngagements 147GeneralTerminologyTi:ToolpathfortheithtoolmotionRVi:RemovalvolumegeneratedforithtoolmotioninE3.M:AtransformationthatmapsRVifromE3toP(phi,d,L)i.e.M:E3arrowrightP(phi,d,L)PiRV :ThemapofRViinP(phi,d,L).N:Surfacenormalofapointoncutter.CWEK(t):Thesetofallpointsonthesurfaceofacutteratlocationtalongatoolpathwheref?Ngreaterequal0.Thesearekinematicallyfeasibleengagementpointsforagiveninstantaneousfeedvectorf.CWE(t):Thesetofallpointsonthesurfaceofacutteratlocationtalongatoolpaththat areengagedwiththeworkpiece.bCWE(t):PointsontheboundaryofCWE(t).InfindingengagementgeometrythefollowingpropertyoftherepresentationsforremovalvolumesintheparametricspaceP(phi,d,L)motivatefindingthemappingM:E3arrowrightP(phi,d,L).Property6.3:Given Pi arepresentationoftheremovalvolumeRVifortoolpathTiinP(phi,d,L),itsintersectionwithanunboundedplaneQgeneratesaclosedsetofpointsCWE(t)thatconstitutesallpointsinengagementwiththeworkpieceatlocationtalongthetoolpath. OfinterestistheboundarysetofCWE(t),bCWE(t).Thisdefinesthegeometryrequiredforinputtoprocessmodeling(i.e.forceprediction).Thus bCWE(t)propersubsetCWE(t)propersubsetCWEK(t)             (6.8)ThesethreesetsareillustratedinFigure(6.24).Chapter6.CutterWorkpieceEngagements 148Figure6.24:PointsetsCWEK(t),CWE(t)andbCWE(t)usedindefiningengagements ThepropertygivenabovepointtoanovelapproachforfindingtheengagementgeometryassumingthatthemappingMcanbeconstructed:Given Pi generatedbyapplyingMtoRVi,CWE(t)andbCWE(t)canbefoundbyintersecting Pi withanunboundedplaneforeachcutterlocationdefinedbyt(Figure6.25).TheuseofanunboundedplaneinfindingengagementseliminatestheproblemhighlightedinFigure(6.21)wherethechordalerrorinthepolyhedralrepresentationofaremovalvolumeintroducesuncertaintyintheintersectioncalculation. Further the reduction of the cutter surface geometry to a first order formsimplifiesintersectioncalculationsparticularlywhentheremovalvolumeispolyhedral.Figure6.25:CWEcalculationsintheparametricdomainP(phi,d,L).Chapter6.CutterWorkpieceEngagements 1496.2.1.1MappingMforLinearToolpath InthissectionderivationsforthelineartoolpathwillbeshownfortheBallNoseEndMill (BNEM) and formulas for the other cutter types can be found in Appendix A-2. ABNEMismadeupofahemi-sphericalandacylindricalsurface.Basedonthekinematicsof3-axis machining subsets of these surfaces denoted CWEK,S(t) and CWEK,C(t) respectivelywillcontributetothesetCWEK(t)asdefinedinchapter(5)i.e. CWEK(t)=CWEK,S(t)unionCWEK,C(t)Transformation formulas will be derived for the mapping of the hemi-spherical surface,Msphere.Themethodologyandmanyofformulaedevelopedapplyequallytothemappingforthecylindricalsurfaceofthecutter,Mcyl.Thiswillbefurtherexplainedlaterinthissection.DerivationofMspherefortheBNEM Inadditiontothegeometricdefinitionofthesurfacethelocationofapointonthecuttergeometryasitmovesalongatoolpathisalsorequired.Todothisthefollowingtermsareintroduced(seeFigure6.26).Figure6.26:DescriptionofapointonacuttermovingalongaToolPathWCS: WorkpieceCoordinateSystem(i,j,k).Thisiswherethe geometryoftheremovalvolumeandtoolpathsaredefined.TCS: ToolCoordinateSystem(u,v,w)positionedatthetooltipwithwalongthecutteraxis.Itisassumedthatthexaxisofthelocaltoolcoordinatesystemisalignedwiththefeedvectordirection.Chapter6.CutterWorkpieceEngagements 150T(t): Thepositionofthetooltippoint(xT,yT,zT,t)attalongtoolpathTiintheWCS.I: Thepositionofapoint(xI,yI,zI)onthecuttersurfacethatbelongstothesetCWEK(t)attalongtoolpathTiintheWCS.F(t): Thepositionofareferencepoint(xR,yR,zR,t)onthecuttingtoolaxisattalong  toolpathTi.Notethatinthissectionthecomponentsofthe(TCS)aredefinedbyu,vandwwhicharedifferentthanthosedefinedinchapter(5).Thereasonforthisistomaintaintheclarityinthevariablesymbolizationi.e.touselesssubscripts.Thepossible3-axislineartoolmotionsareshown in Figure (6.27): horizontal, ascending and descending (from left to right)respectively.Figure6.27:3-axisLinearToolMotionswithaBNEMFormotionsAandBtheengagementangleofapointat ), tSK mustliewithin[0,pi]i.e. pi, tSK .AtCboththefrontandthebacksidesofthehemi-sphericalsurfaceofthecutterhaveengagementsandthetotalengagementareacoversthefull[0,2pi]range i.e. pi, <=<SK .The mapping will be developed for the most generalcaseCwherethehemi-sphericalsurfaceengagestheworkpieceintworegions?thefrontcontactfaceCWEK,s1(t)andthebackcontactfaceCWEK,s2(t)where 2,1,, ) sKsKSK CWE Figure(6.28)showstheengagementregionsandparameters(phi,d,L)forapointatIonthehemi-sphericalsurface.Chapter6.CutterWorkpieceEngagements 151 (a)(b)Figure6.28:Engagementregionsofthe(a)frontand(b)backcontactfacesThemappingmethodologyissummarizedinFigure(6.29)andconsistsof4steps.Theinputsare a point on the removal volume I(xI,yI,zI)elementE3, the implicit representation of the cuttersurfacegeometryGandtheparametricformofthetoolpathTi.Figure6.29:ProcedureforperformingMappingMG:E3arrowrightP(phi,d,L).Step1:InthissteptheparametervaluetalongTiisfound.Foralineartoolpath,thetooltipcoordinateswithrespecttotoolpathstartVS(xS,yS,zS)andendVE(xE,yE,zE)pointsaregivenby,Chapter6.CutterWorkpieceEngagements 152 ( ) tSES V , 1         (6.9)ForaBNEMthereferencepointatFischosentobethecenterofthesphere.Itslocationcanbeexpressedusingthecuttingtooltipcoordinatesas, ( ) n            (6.10)where r and n are the radius of the hemi-sphere and unit normal vector of the tool axisrespectively. When the hemi-spherical surface moves along a tool path, a family GS(t) ofsurfacesisgenerated.AnexpressionforthisfamilyofsurfacesisobtainedbysubstitutinginthecoordinatesofthereferencepointatFfromEq.(6.10)intotheimplicitformofaspheregivenbyEq.(6.7-a). ( ) ( ) 02 =S F     (6.11)WhenthepointatPbelongstothesetCWEK,S(t),Eq.(6.11)canberewrittenas, ( ) ( ) 02, =SK F            (6.12)GivenapointatIthatisknowntobeanengagementpoint,Eq.(6.12)canbeexpandedtotakeonthefollowingquadraticform, A2t2+A1t+A0=0               (6.13)wherethecoefficientsA2,A1andA0aregivenby,  22 SE V  ( ) ( )]SES V1              (6.14) 220 rS Chapter6.CutterWorkpieceEngagements 153SolvingfortinEq.(6.13)givesthepositionofthecuttertooltipwhenthepointatIisanengagementpointbetweenthecutterandtheworkpiece.InfindingthecorrecttoolpositionforanintersectionpointIthefollowingpropertyisused.Property 6.4: Given a linear tool path Ti and the cutter surface geometry G, for 3-axismachining,a) ThemaximumnumberoftoolpositionswherethecuttertouchesapointinspaceisequaltothedegreeofthecuttersurfacegeometryG,i.e.sphereisadegreeoftwosurface.b) Ifapointbelongstotheremovalvolumethenthereisatleastonecutterlocationwherethecuttersurfacetouchesthispoint.c) IfthepointisontheboundaryofthesweptvolumeSViofthecutterthereisasingletoolpositionwherethecuttersurfacetouchesthispoint. ThesolutionforEq.(6.13)givestworealroots(Property6.4-a,b)t1andt2wheret1<=<t2.TheserootsrepresenttwopossiblecutterlocationswheretheintersectionpointIliesonthesphericalsurfaceasshowninFigure(6.30-a).Todifferentiatebetweenthetwochoicesandmakethecorrectselectionfort,thesignofthedotproductsf?N1andf?N2betweenthefeedvectorfandthesurfacenormalsN1andN2forthetwopositionsofthecutterisused.AnegativevalueindicatesthatIisintheshadowofthecutterandsocannotbeamemberofCWEK(t).Forexample,asshowninthefigureatlocationt2thedotproductisnegativewhileatt1,itispositive.ItiseasytoseethatforlineartoolmotionsthesmallerrootofEq.(6.13)shouldalwaysbeused. ForthespecialcasewhereIliesontheboundaryofthesweptvolumeofthecuttersuchthatthescalarproductf?N1,N2=0,thecuttingtoolhasonlyonecutterlocation(Property6.4-c)asshowninFigure(6.30-b).ThesolutiontoEq.(6.13)yieldsarepeatedrootandalsoIsatisfiesthefollowingsystemofequationsasshownat[82] 0III =S  , 0),,,( III =partialdiffpartialdiff t tzyxGS          (6.15)Chapter6.CutterWorkpieceEngagements 154         (a) (b)Figure6.30:DifferentcutterlocationsforIelementCWEK(t).Step 2: Given the value of the tool path parameter t = tI when the cutter surface passesthroughI,atransformationiscreatedtomaptheglobalcoordinatesystem(WCS)toalocaltoolcoordinatesystem(TCS)atthislocationonthetoolpath.WiththewaxisofcutterbeingoneoftheaxesoftheTCSthevaxis(seeFigure6.28)isobtainedfromthecrossproductofwandtheinstantaneousfeeddirectionvectorf(t=tI).Thethirdaxisuisperpendiculartothefirsttwo. |                (6.16) | where kSESESE z           (6.17)UsingEqs.(6.16)thetransformationofapointintheTCStooneintheWCSisgivenby, [ ] Itt =                   (6.18)where [ ]IITTwvuJtwvutttzzzt ===bracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttpequivalencebracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp=10001000) (6.19)Chapter6.CutterWorkpieceEngagements 155andI'arethecoordinatesofIintheTCS.Step3:Tofindphi1(t)andd(t)thecoordinatesofanengagementpointintheTCSisrequired.ThisisobtainedfromEq.(6.18)as, [ ] 1 I-=tt                   (6.20)Giventhat[ ] Itt= isorthogonal,itsinverseisdefinedby, [ ]II10001twwwttzyx=-=bracketrightexbracketrightexbracketrightexbracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftexbracketleftexbracketleftexbracketleftbtbracketlefttp?=TJ  (6.21)Expandingequation(6.20)using(6.21)givesthecoordinatesofI'( III , zminute ).Theseare, )II t   )II t  (6.22) )II t  Step 4: Given the coordinates of I' on the hemi-spherical surface of the BNEM, its toolengagementanglephi1(t)isobtainedusingsphericalcoordinates.TheEngagementangleforthefrontcontactfaceCWEK,s1(t)isgivenby parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttpminuteminute== -2I2III cos)( yxyttphi1      (6.23-a)andforthebackcontactfaceCWEK,s2(t)isgivenbyChapter6.CutterWorkpieceEngagements 156 parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttpminuteminute-== -2I2III cos2)( yxytt piphi1     (6.23-b)ThedepthoftheengagementpointforthesphericalpartistheanglefromT(t)toI'suchthat: parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp minute-== -rzttd I1I 1cos)(      (6.24)andfinallytheparameterL(t)ofIisobtainedusing, SE VI                (6.25)DerivationofMcylfortheBNEMThemappingforthecylindricalsurfaceoftheBallendcutterisobtainedfollowingthesamestepsjustdevelopedforthehemisphericalcuttersurface.Itcanbeseenthatexceptforthefirst,allstepsandequationstoaccomplishthemappingremainthesame.Forthefirststep,thegeometryofthecylinderchangesEqs.(6.11)and(6.12)to,[ ] 022 =C n    (6.26)and, [ ] 022, =CK n  (6.27)respectively(seeFigure6.31).Whererandnaretheradiusofthecylindricalsurfaceandtheunitnormalvectorofthetoolrotationaxisrespectively.AswithEq.(6.13),Eq.(6.27)resultsin a quadratic equation in t when I and F are substituted. The solution to this and theremaining steps in the mapping procedure lead to the representation of a point I on thecylindricalsurfaceoftheBNEMin(phi1,d,L)coordinates.DetailedtransformationformulasforthecylindricalendmillcanbefoundintheAppendixA.2.Chapter6.CutterWorkpieceEngagements 157 Figure6.31:CylindricalcontactfaceCWEK,C(t)ofBNEM6.2.1.2MappingMforCircularToolpath In this section we highlight the mapping procedure for a circular tool path. The stepssummarized in (Figure 6.29) can be followed with some important differences. One inparticularisthatthefeeddirectionchangesasafunctionofthetoolpathparametricvariablet.ThisimpactsthetransformationfromtheWCStotheTCS.TheparametricrepresentationforthetooltippointforacirculartoolpathofradiusRcenteredat )CCC z isgivenby ( ) e   (6.28)where [ ] (see Figure 6.32). d(xd, yd, zd) and e(xe, ye, ze)  are two orthogonal unitvectorsdefiningtheplaneofthecirculartoolpath.Thus,thisrepresentationforTisgeneraleventhoughcircularinterpolationonmostmillingmachinesisrestrictedtotheXY,XZandYZplanes.Chapter6.CutterWorkpieceEngagements 158Figure6.32:MovingcoordinateframeforcirculartoolpathAswithlineartoolpathsthelocaltoolcoordinatesystemTCSdefinedwithu,vandwhasitsoriginatthetooltippoint.ThereferencepointFisexpressedintermsofthecuttingtooltipcoordinatesusingEq.(6.10).Followingthesamestepsoutlinedinsection(6.2.1.1)Eq.(6.28)issubstitutedintoEq.(6.10)thenintotheimplicitequationofthehemisphericalsurfaceofthecutter(6.7-a)withthecoordinatesofapointIbelongingtotheCWEK(t)obtainedfromtheremovalvolume.Theresultingequationcanbeexpressedintermsoftheparametertinthefollowingform 0012 =   (6.29)whereA2,A1andA0areconstantsforthecurrentintersectionpointIgivenbythefollowingexpressions ( ) ( ) ( )]ZCIYCIXCI n ,   (6.30) ( )2 , ( )1 , 2220 r   Eq.(6.29)intcanbewritteninthefollowingform 002221 =             (6.31)Chapter6.CutterWorkpieceEngagements 159where ( )1212 , A- .Thisleadstothefollowinggeneralequationfort ( ) pin + ,  parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+-= -22210sinAAalpha       (6.32)wherenisinteger.Definen1tobethesmallestnsuchthatt>t0.Thenn1canbecalculatedas parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttpbracketrightexbracketrightexbracketrighttpbracketleftexbracketleftexbracketlefttp -bracketrightexbracketrightexbracketrighttpbracketleftexbracketleftexbracketlefttp +=2122,2min01 pialphapialpha tn   (6.33)Aftern1isfounditcanbeincrementedbyunitstepstocalculatethenexttooltippointTforthe given I. There can be up to two cutter location points in the interval for a givenengagement point I (Figure 6.33). As explained in section (6.2.1.1) the minimum root isselected.IfIisontheenvelopesurfaceofthecutter(Property6.4-c),Eq.(6.29)givestworepeatedroots.HavingsolvedfortheparametertvaluewhenIisanengagementpoint,itissubstituted into equation (6.28) to find the tool tip coordinates. The direction of the feedvectoratthislocationisgivenbykjT))eezydtd++  (6.34)Figure6.33:ParametervaluesofthecuttertooltipChapter6.CutterWorkpieceEngagements 160Givenfandt,Eqs.(6.23-a,b)and(6.24)areusedtofind ( ) parametersforIandL(t)ofIisobtainedusing tI =      (6.35)themappingforthecylindricalportionofthecutterfollowsthesamestepsusingtheimplicitsurfaceforacylinder.6.2.1.3MappingMforHelicalToolpath Aspecialcaseof3-axismachiningishelicalmillingshowninFigure(6.34).Inthisoperationacuttingtoolisfeedalongitstoolaxis(Z-axis)asacircleisinterpolatedbytheothertwo(XandYaxes).Thecutterfollowsahelicaltrajectory.Thisoperationisusefulfor(1)contourmillingofcylindricalprotrusionsorforenlargingofpre-machinedorpre-formedholes,(2)forholemachiningintosolidstock. Figure6.34:SweepsforHelicalMillingInthissectionderivationsofthemappingMforthehelicaltoolpathwillbeshownfortheFlat-End mill. The methodology and many of formulae developed apply equally to themapping for the other types of the cutter geometries. A Flat end mill is made up of acylindricalsurfaceforitssideandaflatsurfaceforitsbottom.Basedonthekinematicsof3-axismachiningthesesurfacesdenotedCWEK,c1(t)andCWEK,c2(t)respectivelywillcontributetothesetCWEK(t)asdefinedbeforei.e. CWEK(t)=CWEK,c1(t) unionCWEK,c2(t)Chapter6.CutterWorkpieceEngagements 161As explained in section (5), during machining each cutter surface contributes to CWEsdifferently.Becauseofthistheremovalvolumesgeneratedbythesideandflatsurfacesofthe Flat-End mill are separated in this research. The Side face (Figure 6.35-a) has anengagementintherangeofzerotopi,andthebottomface(Figure6.35-b)intherangeofzeroto2pi.           (a)           (b)Figure6.35:Removalvolumesof(a)sidefaceand(b)bottomface.In this section the mapping methodology to be described is applied to the removalvolumegeneratedbythecuttersidefaceonly.ThisisbecausedirectlyintersectingaplanewhichisperpendiculartothetoolrotationaxiswiththeremovalvolumeofthebottomfacegivestheCWEsofthebottomface.ForthecylindricalsurfacetoolreferencepointFequalstothetooltippointT(seeFigure6.36) andtheparametricrepresentationforthetooltippointforahelicaltoolpathofradiusRcenteredatC(xc,yc,zc)is  ( )parenrightexparenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftexparenleftbtparenlefttp+++=ttttCCCT )sin(),           (6.36)wheretelement[0,2pi]and2picisaconstantgivingtheverticalseparationsofthehelix?sloopswherec<0.Chapter6.CutterWorkpieceEngagements 162Figure6.36:ParametersdescribingahelicaltoolmotionfortheFlat-EndmillFollowing the same steps outlined in section (6.2.1.1), Eq.(6.36) is substituted into theimplicit equation of the cylindrical surface of the cutter (6.7-b) with the coordinates of apointIbelongingtotheCWEK(t)obtainedfromtheremovalvolume.Theresultingequationcanbeexpressedintermsoftheparametertinthefollowingform 0012 =             (6.37)where ( ) RCI -2  ( ) RCI -1       ( ) ( ) 22220 rCICI - Eq.(6.37)intcanbesolvedbyapplyingtheEqs.(6.32)and(6.33).Therecanbeuptotwocutter location points in the interval for a given engagement point I (Figure 6.37). AsexplainedinProperty(6.4)theminimumrootisselected.Chapter6.CutterWorkpieceEngagements 163Figure6.37:DifferentcutterlocationsforanengagementpointIelementCWEK(t).AswithlineartoolpathsthelocaltoolcoordinatesystemTCSdefinedwithu,vandwhasitsoriginatthetooltippoint.ThedirectionofthefeedvectorfortheparametertvaluefromEq.(6.36)isgivenby kjiTf ctRtRdtdtt I ++-=== )(cos)(sin)(  (6.38)Usingt,theparameterL(t)ofIisobtainedby 22 cII +        (6.39)Givenfandt,theEq.(6.23-a)isusedtofindphi1 parameterforIandfinally,thedepthoftheengagementpointasdefinedbythedistancefromT(t)toI'measuredalongthetoolaxisvectorissimplythez-coordinateofI'. )III t    (6.40)6.2.1.4Implementation Inthisresearchthemappingsdevelopedintheprevioussectionsareapplieddirectlytoapolyhedralrepresentationoftheremovalvolume.ThespecificrepresentationutilizedistheSTL(?StereoLithography?)formatthoughotherrepresentations(VRML,.jt,.hsfetc.)canaseasilybeadopted.Asexplainedinchapter(1),intheSTLrepresentationthegeometryofsurfacesisrepresentedbysmalltrianglescalledfacets.ThesefacetsaredescribedbythreeChapter6.CutterWorkpieceEngagements 164vertices and the normal direction of the triangle. To test the methodology for CWEextraction, a prototype system has been assembled using existing commercial softwareapplications and C++ implementations of the mapping procedure described above. Thissystemisshownin(Figure6.38).SimulationoftheCLDataisperformedusingVERICUTacommercial NC verification application. This application uses a voxel-based model tocapturechangestothein-processworkpiece.STLrepresentationsofin-processworkpiecestatescanbegeneratedfromthevoxelrepresentationpriortoanytoolpathmotionintheCLData file. Not currently available through the programmable APIs provided forcustomization is a function for outputting STL representations of removal volumes.However,thisisprobablyaneasyextensiontoimplement.Thus,toobtainremovalvolumesforagiventoolmotion,(ACIS)isutilizedtomodelaB-repsolidoftheassociatedsweptvolume.Figure6.38:ImplementationofCWEextractionmethodologyThesesweptvolumesareexportedtoSTLformat.ABooleanintersectionbetweenthissweptvolume and the current in-process workpiece output from Vericut is performed using thepolyhedral modeling Boolean operators implemented in a third commercial application,MagicsX[51].ThoughcurrentlyamanualprocessthisprototypesystemcreatesSTLmodelsChapter6.CutterWorkpieceEngagements 165ofremovalvolumesgeneratedduringthemachiningofcomplexcomponents.ItutilizesthesameCAD/CAMdatageneratedformachiningthepart. In this section examples of the application of the mapping M to removal volumesgeneratedbydifferenttypesofcuttingtoolsandtoolmotionsarepresented.Thefirstsetofexamples is designed to demonstrate the generality of the approach with respect to toolgeometryandlinear,circularandhelicaltoolmotions.ThesecondsetcomesdirectlyfromapplyingtheprototypesystemdescribedintheprevioussectiontoCAD/CAMdatacreatedinmachiningagearboxcover.Thisdemonstratesthepracticalityofthemethodology.ExampleSet1 Inthisexamplesetdifferentcuttingtoolsandlinear,circularandhelicaltoolmotionsarepresentedtodemonstratethegeneralityoftheapproach.Figures(6.39)to(6.41)showBall,FlatandTapered-FlatEndmillsremovingmaterialalongalinearrampingtoolpathi.e.thetool moves in all three axes simultaneously. In each case the original and transformedremoval volumes are given. The removal volumes associated with the different cuttersurfacesareseparated.Inthecaseoftheflatendmillthisgivesthematerialremovedbythecylindrical (Side Face) and flat (Bottom Face) cutter surfaces. Plots of CWEs at differentCutterLocations(CL)alongthetoolpathareillustrated.Theseareobtainedbyintersectingthetransformedremovalvolumewithaplanerepresentingthecuttingtool.TwoformatsforplottingtheCWEsareused.ThefirstisanXYplotofdepth-of-cut(asmeasuredfromthetooltippoint)versusengagementangle.Thesecondplotshowstheengagementareaofthecutterbottomsurface(Figure6.40)inapolarformat. InthelinearrampingexampleoftheBall-Endmilltheplotsshowthatengagementofthecutteroccursoverthefullrangei.e.[0,2pi].Betweenzeroandpitheengagementisduetothefrontcontactface(theCWEinthiszoneisdividedtoillustratethis)andbetweenpiand2pi due to the back contact face. For the Flat-End mill XY plots show the engagementbetweenzeroandpiforthesidefaceandpolarplotsshowtheengagementbetweenzeroand2piforthebottomface.BecausetheTapered-Flat-Endmillisrampinguponlyitsfrontfacehasengagementwiththeworkpieceanditchangesbetweenzeroandpi.Chapter6.CutterWorkpieceEngagements 166Figure6.39:CWEsforBall-Endmillperformingalinear3-axismoveFigure6.40:CWEsforFlatEndmillperformingalinear3-axismoveChapter6.CutterWorkpieceEngagements 167Figure6.41:CWEsforTapered-Flat-Endmillperformingalinear3-axismoveThenextexampleshowsaFlat-Endmillremovingmaterialalongahelicaltoolpath(Figures6.42 ? a,b,c).  In each case the original and transformed removal volumes are shown. InFigure (6.42-d) the transformed removal volumes for the side face are shown and for thegivenCLs(cutterlocationpoints)CWEsareobtainedbyintersectingaplanerepresentingthecuttingtoolwiththetransformedremovalvolume.TwoformatsforplottingtheCWEsareused.  The first is an XY plot of the depth of cut (DOC) and immersion angle which isgeneratedbythesidefaceoftheflatendmill.Theengagementofthissurfaceofthecutteroccursoverthe[ ]rangeascanbeseen.Thesecondisapolarplotofthecutterradiusversustheimmersionanglewhichisgeneratedbythebottomsurfaceoftheflatendmill.Theengagementforthebottomfaceofthecutteroccursoverthe[ ]range.ForCL1andCL2bothtypesofplotsareshownbecausethebottomofthecutterisremovingmaterialattheselocations.Fortheothercutterlocationsonlythesidefaceoftheendmillisengagingtheworkpiece.Chapter6.CutterWorkpieceEngagements 168Figure6.42:HelicalToolMotionswithaFlat-EndmillandCWEsThefinalexample(Figure6.43)demonstratesthetransformationasappliedtoacircular2?Dtool path. These results together show that the mapping methodology reduces variousChapter6.CutterWorkpieceEngagements 169combinations of cutter and tool path geometry to a generic form to which a singleintersectionoperatorcanbeapplied.Figure6.43:CWEsforFlat-EndmillperformingacircularmoveExampleSet2 TheexamplesinthissectionarecreatedusingthestepsoutlinedinFigure(6.38).TheFigure(6.44)showsmodelofagearboxcovertobemachinedfromrectangularstock.TheCADmodelandtoolpathsformachiningwerebothcreatedusingUnigraphicsNX3.Inthisfigure, it is shown in-process workpiece and the removal volume for the next tool pathgeneratedbyVericut,ACISandMagics.TheremovalvolumegeneratedbytheballnoseendmillclearlyshowsthecomplicatedremovalvolumeshapesandresultingCWEsthatcanbegeneratedwhenmachiningcomplexparts.Chapter6.CutterWorkpieceEngagements 170Figure6.44:CWEsforBall-Endmillperformingalinear3-axismoveFigure(6.45)showsahelicalmillingoperationforenlargingahole.InthisexampleboththesideandthebottomfacesoftheFlat-Endmillareremovingmaterial.Inthisfiguresidefaceremovalvolumesareshown.Thecutterperformsfourhalfturnseachcorrespondingtoa00?1800rangei.e.thirdturnhasthestartingangle3600andendingangle5400alongthetoolpath.Figure6.45:Helicaltoolpathapplicationwithremovalvolumesforeachhalfturn.ItcanbeseenfromFigure(6.46)thatstartingfromthefirstturnthematerialremovalrateisconstantlyincreasing.Thenthisratebecomesconstantforthethirdandthefourthturns.Chapter6.CutterWorkpieceEngagements 171Figure6.46:CWEsoftheHelicalToolMotions6.2.2EngagementExtractionMethodologyin5-AxisMilling In3-axismillingcuttertranslatesalongatoolpathwithafixedtoolaxisvector.Forthe3-axisCWEmethodologydescribedintheprevioussectionthemappingisperformedwithrespect to the Tool Coordinate System (TCS) in which the z ?axis was fixed. In thismethodologyitisassumedthatthesizeofthefacetsoftheremovalvolumemustbesmallenough.Thusduringthemappingoftheremovalvolumethetriangledeformationstayssmallenough.Laterthiswillbediscussedinthediscussionsection. On the other hand in 5-axistoolmotionsthedirectionofthetoolaxisvectorcontinuouslychanges.Whenthemappingdescribedintheprevioussectionisappliedtotheremovalvolumeobtainedfromthe5-axistoolmotions,thedeformationonthetrianglesbecomesbig.Oneofthesolutionsforreducingthe deformation is to use much smaller triangular facets but this brings a heavycomputational load to CWE extractions. Therefore, in this research, the mappingmethodologyisnotappliedforobtainingCWEsinthe5-axismilling.Wedevelopedanewmethodologyforthe5-axisCWEextractions.ThismethodologyisexplainedinFigure(6.47)fortheimpellermachiningbyusingaTaper-Flat-Endmillandithas3mainsteps:Step1: Inthisstagetherearetwocomponents:thein-processworkpieceandtheBODY.Bothofthemarerepresentedbytriangularfacetshavingverticesandnormalvectors.ForagivenCutterLocation(CL)pointtheBODYisgeneratedasasolidusingthemethodologydescribedinsection(6.1.3)andthenitisexportedtoSTLformatforobtainingthetessellatedrepresentation. A Boolean intersection between this tessellated BODY and the current in-Chapter6.CutterWorkpieceEngagements 172process workpiece is performed using the polyhedral modeling Boolean operator. ThisintersectionprocesscreatesanSTLmodeloftheremovalvolumeforagivenCLpoint.Step2:Inthisstagetherearetwocomponentsalso:TheremovalvolumeintheSTLformatandthefeasiblecontactsurfaceinthesolidformat.Theanalyticalsurface/lineintersectionsareperformedbetweenthefeasiblecontactsurfaceandthetrianglesoftheremovalvolume.ForexampleinFigure(6.47)-steps2,trianglesoftheremovalvolumeareintersectedwiththeconeandspherepartsofthefeasiblecontactsurface.Step3:InthislaststeptheconnectionoftheintersectionpointsobtainedinthepreviousstepgeneratestheboundaryoftheCWE,bCWE.Thenthisboundarydescribedin3DEuclidianspaceismappedinto2Dspacedefinedbytheengagementangleandthedepthofcut.Figure6.47:CWEextractionstepsfor5-axismilling.Although this 5-axis CWE methodology in polyhedral models is explained by using theTaper-Flat-Endmill,themethodologyappliesequallyfortheothercuttersurfacesalso.IftheChapter6.CutterWorkpieceEngagements 173cuttersurfacehasthesecondorder equation,e.g.cylinder,coneorspherenaturalquadricsurfaces,eachline?surfaceintersectiongivestworoots.Ifthecuttersurfacehasthefourthorderequatione.g.torus,eachlinesurfaceintersectiongivesfourroots.InmostcasesonlyoneoftheserootsareneededforobtainingthebCWEsandtherestisredundant.DuringtheimplementationtheACISgeometrickernelwithC++isusedforcreatingandtessellatingtheBODYandtheMagicsisusedforthepolyhedralBooleanintersectionsandupdatingthein-process workpiece. These three steps are performed for each tool path segment. Afterprocessing a toolpath segment for CWE extractions, the material is removed from the in-processworkpieceinthevicinityofthetoolpathsegment.6.3TheCutterWorkpieceEngagementsinVectorBasedModelIn this section for the CWE calculations the cutting tool geometries are representedimplicitlybynaturalquadricsandtheplane(adegeneratequadric).Naturalquadricsconsistofthesphere,circularcylinderandthecone.6.3.1IntersectingSegmentAgainstPlane GivenapointPwithnormalnonaplanepi ,thefollowingequationcanbewrittenforallpointsXonthisplane ( ) 0      (6.41)Intheaboveequationthevector(X-P)isperpendiculartothevectorn.WhenthedotproductinEq.(6.41)isdistributedacrossthesubtractiontheimplicitrepresentationoftheplanetakestheformof d where P .Letasegmentisgivenby ( ) vaba - for1.ForfindingtheintersectionpointI(seeFigure6.48)betweenthesegmentandtheplanetheequivalentofI(v)issubstitutedforXintheplaneequationandthenthisequationissolvedforv ( ) ) daba =                (6.42)             Chapter6.CutterWorkpieceEngagements 174Isolating v in the above equation yields ( ) ( ))aba I . The expression for vcannowbeinsertedtheparametricequationofthesegmentI(v)forfindingtheintersectionpointsI ( ) ( ))]( )ababaa I           (6.43)Figure6.48:Intersectingasegmentagainstaplane6.3.2IntersectingSegmentAgainstSphere Let a segment is given by ( ) vaba - for 1. Let the sphere surfacedefinedby( ) ( ) 2 ,whereCisthecenterofthesphere,andrisradius.TofindthevvalueatwhichthesegmentintersectthespheresurfaceI(v)issubstitutedforX,giving ( ) ) ( ) ) 2abaaba =        (6.44)Let Ca - and ( ) abab I then the Eq.(6.44) takes the form of( ) ( ) 2 .Expandingthedotproductyieldsthefollowingquadraticequationinv ( ) ( ) 022 k r              (6.45)ThesolutionsofthisquadraticequationisgivenbyChapter6.CutterWorkpieceEngagements 175 ( ) ( ) ( )22,1 2           (6.46)Solution of the above equation gives three outcomes with respect to thediscriminant ( ) ( ) ]2 2 .If 0 therearetworealrootsforwhichsegmentintersectsthespheretwice(Figure6.49-a),If 0 therearetwoequalrealrootsforwhichthesegmenthitsthespheretangentially(Figure6.49-b),andif 0 therearenorealrootswhichcorrespondstosegmentmissingthespherecompletely(Figure6.49-c).Althoughforthecasegivenby 0 therearetworealroots,oneofthemcanbefalseintersection(Figure6.49-d,e).Inthiscasethesegmentcanstartinsideoroutsidesphere.OneofthevvaluesfromEq.(6.45),itsvalueisintherangeof ],givestheintersectionpoint.(a)(b)(c)(d)(e)Figure6.49:Differentcasesinsegment/sphereintersections:(a)Twointersectionpoints,(b)intersectingtangentially,(c)nointersection,(d)segmentstartsinsidesphere,and(e)segmentstartsoutsidesphere.6.3.3IntersectingSegmentAgainstCylinder AcylinderwithanarbitraryorientationcanbedescribedbyanaxiswhichpassesthroughpointsBandQ,andbyaradiusr(seeFigure6.50).LetXdenotesapointonthecylindersurface.Theprojectionofthevector(X-B)ontothecylinderaxisdefinedthevector(Q-B)yields ( ) n               (6.47)where the unit vector ( ) B . Applying Pythagorean Theorem to triangledefinedbythepointsB,XandPyieldsChapter6.CutterWorkpieceEngagements 176 ( ) ( ) 222 r          (6.48)   PluggingEq.(6.47)intoEq.(6.48)yieldsthefollowingimplicitrepresentationforacylindersurface ( ) ( ) ( ) } 022 =           (6.49)Figure6.50:AsegmentisintersectedagainstthecylindergivenbypointsBandQandtheradiusr.The intersection of a segment ( ) vaba - with the cylinder can be found bysubstitutingI(v)forXintoEq.(6.49)andsolvingforv.( ) ) ( ) ) ( ) ) } 022 =abaabaaba n (6.50)Setting Ba - and ab I intheaboveequationtheaboveequationcanbewritteninthefollowingform A2v2+2A1v+A0=0              (6.51)where ( )22 n  ( ) ( )1 Chapter6.CutterWorkpieceEngagements 177 ( ) 220 r SolvingEq.(X)forvgives20212 AAv -= m           (6.52)The sign of ( )0221 A determines the number of real roots in the above equation.If thesign is positive, there are two real roots, which correspond to the line intersecting thecylinderintwopoints.Ifthesignisnegative,therearenorealroots,whichcorrespondstothelinenotintersectingthecylinder.When( )0221 A equalstozerotherearetwoequalrealrootssignifyingthatthelinetangentiallytouchingthecylinder.6.3.4IntersectingSegmentAgainstaCone Thecone(seeFigure6.51)isdefinedbyavertexV,unitaxisdirectionvectornandhalf-anglealpha ,where ( ).LetPdefinethepointsontheconesurface.Thehalf-anglealpha isbetweenP-Vandn,thereforethefollowingexpressioncanbewritten alphaparenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttpVV?n                (6.53)Theaboveequationissquaredforeliminatingthesquarerootcalculationsandthisyieldsthefollowingquadraticequation ( )) ( )22 cos V              (6.54)Chapter6.CutterWorkpieceEngagements 178Figure6.51:AconewithdefiningparametersTheEq.(6.54)representsadoubleconewhichhastheoriginalconeanditsreflectionintheopposite direction of n. For the intersection calculations the original cone is needed. Foreliminating the reflected cone the constraint ( ) 0 is taken into account also. Thequadratic cone equation given in Eq.(6.54) can be written in a quadratic form as( ) ( ) 0T M , where I2T . Therefore the original cone which hastheunitaxisvectorncanbedefinedby ( ) ( ) 0T M ,and ( ) 0          (6.55)Let the line segment is defined by ( ) vaba - , where velementR. For obtaining theintersectionpointsbetweenthelinesegmentandthecone )issubstitutedintoEq.(6.55)andthisyieldsthefollowingquadraticequationinonevariable A2v2+2A1v+A0=0              (6.56)where ( ) ( )TT abab I2  ( ) ( )T Mab -1  ( ) ( )T M0 Chapter6.CutterWorkpieceEngagements 179Asexplainedinsection(6.3.3)thesignof ( )0221 A determinesthenumberofrealrootsintheaboveequation.6.3.5ObtainingtheCutterWorkpieceEngagements InthediscretevectorapproachcalculatingtheCWEsarestraightforward.Byintersectingthelinesegmentswiththecuttergeometriestheintersectionpointsareobtained.Formappingtheintersectionpointsinto2Ddomainrepresentedbytheengagementanglevs.depthofcut,cuttersurfacesaredecomposedintothegrids(Figure6.52).Duringsimulationifanyoneofthesegridscontainsanintersectionpoint,thenitismappedinto2Dspace.Theresolutionofthe grids on the cutter surfaces is adjusted with respect to the resolution of the discretevectorsoftheworkpiece.Figure6.52:Decomposingcuttersurfacesintogrids.6.4Discussion The methodologies presented in this chapter target the important problem of findingCutter Workpiece Engagements (CWEs) in the milling operations. The CWEs have beencalculated for supporting the force prediction model which requires the CWE area in theformat described by the engagement angle versus the depth of cut respectively. Thedevelopedmethodologiescanbeclassifiedintothreecategoriesbasedonthemathematicalrepresentationoftheworkpiecegeometry:Solidmodelerbased,polyhedralmodelerbasedand vector based methodologies. There is always a tradeoff between computationalefficiencyandaccuracyintheseapproaches.Forexamplethepolyhedralandvectorbasedapproaches generate approximatedCWE results andbecauseofthistheyrequire ashorterChapter6.CutterWorkpieceEngagements 180computationaltimethandoesthesolidmodelerbasedapproach.ButontheotherhandinthesolidmodelerbasedapproachthemostaccurateCWEsareobtained. Insection(6.1)theB-repsolidmodelerbasedCWEmethodologiesforthe3and5-axismillinghavebeendevelopedusingarangeofdifferenttypesofcuttersandtoolpaths.Inthe3-axismethodologythedecomposedcuttersurfaceshavebeenintersectedwiththeirremovalvolumes for obtaining the boundary curves of the closed CWE area. For this purpose aremoval volume has been decomposed into its constituent faces and then the face/faceintersectionhavebeenperformedbetweenthefeasiblecontactsurfaceandtheseconstituentfaces at a given cutter location point. Examples have been presented to show that theapproachgeneratesproperengagements.AlsoitcanbeseenfromtheexamplesthatinthismethodologythecomputationalloadintermsoftherequiredBooleanoperationsishigherforthedescendingmotionsdescribedinchapter(5).ForexampleintheascendingmotionofaFlat-Endmillonlythecylindricalpartremovesthematerialfromin-processworkpieceandbecauseofthisoneBooleanintersectionisneededforobtainingtherequiredremovalvolumeinthismethodology.ButontheotherhandthedescendingmotionofthesamecutterrequirestwoBooleanintersectionsoneforthecylindricalpartandanotherforthebottomflatpart.Forthe5-axismillingwehadtodevelopanothermethodology.Asexplainedinsection(6.1.3),becausetheenvelopeboundary for5-axistoolmotionsareapproximatedbysplinecurvesapplyingthesamemethodologydescribedforthe3-axismillinggeneratesnonrobustresults.Becauseofthisin5-axismethodologyaBODYobtainedfromthefeasiblecontactsurfacehasbeenintersectedwiththein-processworkpieceatagivencutterlocationpoint.Thentheface/faceintersectionshavebeenperformedforobtainingtheboundariesoftheCWEs.Inthismethodologyforminimizingtheerrorintroducedbyoffsettingthefeasiblecontactsurfaceinfinitesimally,theoffsetdirectionvectorhasbeenchosenatthemiddleofthetoolrotationaxis. Insection(6.2)thepolyhedralmodelerbasedCWEmethodologiesforthe3and5-axismilling have been developed. For the 3-axis milling a mapping methodology has beendevelopedthattransformsapolyhedralmodeloftheremovalvolumefromEuclideanspacetoaparametricspacewhichisdefinedbythecutterengagementangle,depthofcutandcutter location. To reduce the size of the data structure that needs to be manipulated theremovalvolumehasbeenusedinsteadofthein-processworkpiece.ThisapproachalsohasChapter6.CutterWorkpieceEngagements 181thepotentialofbeingimplementedusingaparallelprocessingstrategy[85].Thismappingmethodology brings some important advantages over other approaches [88,90]. First, thecomplexity in the CWE calculations is reduced to first order analytical plane-planeintersections. When compared to other polyhedral modeling approaches it has greaterrobustnessbecauseitaddressesthechordalerrorproblemfoundinintersectingpolyhedralmodels.InthismethodologyithasbeenassumedthatthesizeofthefacetsintheremovalvolumeissufficientlysmallsoasnottointroducesignificanterrorsintheCWEboundaries.Tostudythis,acomparisonhasbeenmadebetweenengagementsobtainedfromintersectingtheremovalvolumeusingaB-repsolidmodeler(themostaccurateapproach)andthosefromthe3-axispolyhedralapproachdescribedinthischapterusingdifferentfacetingresolutions.ExamplesoftheoriginalandtransformedremovalvolumesareshowninFigure6.53(a)and(b)respectively.CWEsareobtainedforcutterlocationsCL1toCL29.ForCL5theCWEsfromthe solid modeler (Figure 6.53(c)) and polyhedral modeler at facet resolutions of 2 mm(Figure6.53(d)),and6mm(Figure6.53(e))areshown.TocomparetheeffectofthesizeofthefacetstheCWEareaisdecomposedusingaQuadTree[22]spatialdatastructure(Figure6.53(f)).Figure6.53:RemovalVolumesandCWEsfordifferentresolutionsChapter6.CutterWorkpieceEngagements 182The area of the CWE is obtained by accumulating the square areas that lie within theQuadTreerepresentationoftheCWEboundary.ForeachCLpointtheareasobtainedfromthedifferentresolutionsarecomparedwiththatoftheB-repsolidmodeler(Figure6.54(a)).Thegraphshowsa4%errorat6mmandlessthan1%at2mm.Figure6.54(b)showsthefacetsizevs.intersectiontimei.e.thetimeforobtainingtheCWEareaforagivencutterlocationpoint.Whiletheabsolutevaluewouldvarydependingontheimplementation,thetrendshouldremainthesame.Thisshowsthatthereisasmallincreaseintheintersectiontimeastheresolutionisdecreasedfrom6mmto2mmafterwhichitincreasessignificantly.Boththeerrorandintersectiontimeresultspointto2mmbeingapracticallimitforthefacetsizeinthisexample.Wepointoutthatthislimitwillvarydependingonthecuttersize.Anexpressionneedstobedevelopedtocalculatetheresolutionthatconsidersthisparameter.(a) (b)Figure6.54:TheeffectofthefacetresolutionsBecause in 5-axis tool motions the direction of the tool axis vector continuously changesapplyingthemappingdescribedforthe3-axismillinghasincreasedthedistortionsofthefacets.Thereforeanapproachsimilartothatofthe5-axissolidmodelerhasbeendeveloped.Inthisapproachtheonlydifferencewastheapplicationoftheface-faceintersections.Theseintersections have been performed between triangular facets of a BODY and a solidrepresentationofthefeasiblecontactsurface. Andfinallyinsection(6.3)avectorbasedCWEmethodologyhasbeendeveloped.Inthismethodologythe cutter hasbeendiscretizedintoslicesperpendiculartothetoolaxis.Forobtaining the CWEs the intersections have been performed between discrete vectors andcutterslices.Theintersectioncalculationsarestraightforward. 183Chapter7Conclusions7.1ContributionsandLimitations In this thesis new methodologies have been proposed to facilitate Cutter WorkpieceEngagement (CWE) extractions in milling process modeling. This includes an analyticalmethodologyfordeterminingtheshapesofthecuttersweptenvelopesinmulti-axesmilling,methodologiesforupdatingthein-processworkpiecesurfaces,analysisofthefeasiblecuttingfaces and finally algorithms for extracting CWEs. More specifically the contributions aresummarizedasfollows:square4 Ananalyticalapproachfordeterminingtheenvelopeofasweptvolumegeneratedbyageneralsurfaceofrevolutionperformingmulti-axesmachininghasbeendeveloped. In this approach the cutter geometries are represented using canal surfaces and fordescribing the cutter envelope surfaces the two-parameter-family of spheres has beenintroduced. Analytically it has been proven that for cutter surfaces performing 5-axis toolmotionsanypointontheenvelopesurfaceisalsoamemberofthepointsetgeneratedfromthetwo-parameter-familyofspheresformulation.Laterthemethodologyisgeneralizedforcutters with general surfaces of revolution which performs 5-axis tool motions. In thismethodology, by describing the radius function and the trajectory of the moving sphere,differentcuttersurfacescanbeobtained.Inthissensethemethodologyisindependentofanyparticular cutter geometry. The implementation of the methodology is simple. Especiallywhenthecuttergeometriesarepipesurfaces,fewercalculationsareneededfordescribingthecutterenvelopesurfaces.Althoughexamplesfromtheapplicationofthismethodologyhavebeenshownforcommonmillingcuttergeometriesdescribedbythe7-parameterAPTmodel,this methodology can be also applied to rare cutter geometries. In some cases of 5-Axismillingthecuttersweptvolumemaybeself-intersecting,whichrequiresspecialprocessingtohandle the topological and geometric problems due to the complex tool motions. Selfintersections in this research have not been considered. Modifications to the methodologywillneedtobedevelopedforthisspecialcase.Chapter7.Conclusions 184square4 A discrete vector model based in-process workpiece update methodology has beendeveloped. During machining simulation, for each tool movement modification of the in-processworkpiecegeometryisrequiredtokeeptrackofthematerialremovalprocess.Inthisthesisin-process workpiece modeling (or updating) methodologies have been developed using adiscrete vector representation. These vectors having orientations in the directions of thex,y,z-axesofR3areintersectedwithtoolenvelopes.Withthisrepresentationmorevectorsindifferentdirectionsareusedwhencomparedtootherdiscretevectorapproaches.Thereforeespeciallywhentheworkpiecehasfeatureslikeverticalwallsandsharpedges,thequalityinthevisualizationofthefinalproducthasbeenincreased.AlsothelocalizationadvantageoftheDiscreteVerticalVectorapproachhasbeenpreserved.Forsimplifyingtheintersectioncalculations the properties of the canal surfaces have been utilized. For cutter geometriesdefinedbyacircularcylinder,frustumofacone,sphereandplanethevectorintersectioncalculationsforupdatinghavebeenmadeanalytically.Becauseofthecomplexityofthetorusgeometry the calculations in this case have been made by using a numerical root findingmethod. For this purpose a root finding analysis has been developed for guaranteeing theroot(s) in the given interval. A typical milling tool path contains thousands of toolmovements and during the machining simulation for calculating the intersections only asmall percentage of all the discrete vectors is needed. For this purpose a localizationmethodology,basedontheAxisAlignedBoundingBox(AABB)ofeachtoolmovement,hasbeen developed. The best feature of the AABB is its fast overlap check, which simplyinvolvesdirectcomparisonofindividualcoordinatevalues.Asexplainedinchapter(4),forsomeworkpiecegeometries3-axismachiningisnotsuitableforupdatingthesurfaces.Alsoexact5-axismillingtoolmotionsarenotpreferableinworkpieceupdatesimulationsbecausethe calculations require using the nonlinear root finding algorithms and therefore thecomputationaltimebecomeshigh.Thereforeinthedevelopedmethodologiestoolmotionsusing(3+2)-axismillingareconsideredinstead. Using(3+2)-axistoolmotionsinwhichacutter can have an arbitrary fixed orientation in space, 5-axis tool motions can beapproximated.Anexamplehasbeengivenforillustratingthissituation.TheDiscreteNormalVector (DNV) approach has not been considered in this research. The DNV approach canrepresenttheworkpiecesurfacefeatureswellwithrespecttoagiventolerance.ButbecauseChapter7.Conclusions 185in this approach the directions of discrete vectors are not identical, localizing the cutterenvelopesurfaceduringmachiningsimulationbecomesdifficult.Iftheworkpiecesurfacesare represented by DNVs then an efficient localization methodology will need to bedeveloped.square4 TheengagementbehaviorsofNCcuttersurfacesundervaryingtoolmotionshavebeenanalyzed. A typical NC cutter has different surfaces with varying geometries and during thematerialremovalprocessrestrictedregionsofthesesurfacesareeligibletocontactthein-process workpiece with respect to the tool motions. In this thesis for representing theseregions the terminology feasible contact surfaces (FCS) has been introduced. The wordfeasiblehasbeenusedbecausealthoughthesesurfacesareeligibletocontactthein-processworkpiece,theymayormaynotremovematerialdependingonthecutterpositionrelativetothe workpiece. When the FCS contact the in-process workpiece the Cutter WorkpieceEngagements(CWEs)aregenerated.SinceCWEsaresubsetsoftheFCS,formalizingtheFCS helps us to better understand the CWE generation process. The FCS have beenformulizedbyusingthecuttersurfaceandthecutterenvelopeboundaries.Thecuttersurfaceboundaries are fixed, but on the other hand the cutter envelope boundaries may changedepending on the tool motion. For modeling the cutter envelope boundaries a tangencyfunctiondefinedbyusingthesurfacenormalandthetoolvelocityhasbeenutilized.Laterbychanging the tool velocity direction the distributions of the FCS on the cutter have beenanalyzed. The results from these analyses are later used in the development of the CWEextractionmethodologies.square4 Methodologies for obtaining Cutter Workpiece Engagements (CWEs) in milling havebeendeveloped.  AmajorstepinsimulatingmachiningoperationsistheaccurateextractionoftheCWEgeometriesatchangingtoollocations.Thesegeometriesdefinetheinstantaneousintersectionboundariesbetweenthecuttingtoolandthein-processworkpieceateachlocationalongatoolpath.ThemethodologiespresentedinthisthesistargettheimportantproblemoffindingCWEs in milling operations. TheCWEs are calculated for supporting the force predictionChapter7.Conclusions 186modelrequirestheCWEareaintheformatdescribedbytheengagementangleversusthedepthofcut.Inthesemethodologiesawiderangeofcuttergeometries,toolpathsincluding5-axis tool motions and workpiece surfaces have been used. The workpiece surfaces coverdifferentsurface geometriesincludingsculpturedsurfaces.ThedevelopedCWEextractionmethodologies can be classified into three categories based on the mathematicalrepresentationoftheworkpiecegeometry:Solidmodelerbased,polyhedralmodelerbasedand vector based methodologies. There is always a tradeoff between computationalefficiencyandaccuracyintheseapproaches. In3-axissolidandpolyhedralmodelbasedapproachesdevelopedinthisthesistoreducethesizeofthedatastructurethatneedstobemanipulatedtheremovalvolumehasbeenusedinstead of the in-process workpiece.  In the 3-axis solid modeler methodology the cuttersurfaces have been decomposed into different regions with respect to the feed vectordirection.ThenthesesurfaceregionshavebeenintersectedwiththeirremovalvolumesforobtainingtheboundarycurvesoftheclosedCWEarea.Decomposingthecuttersurfacesinthis way allows CWEs to be obtained for different parts of a given cutter geometry, e.g.bottomflatorbacksideofacutter.Usingasolidmodelerbasedrepresentationtheenvelopeboundariesgeneratedby5-axistoolmotionsareapproximatedbysplinecurve.Applyingthesolid model based methodology described for 3-axis tool motions to the removal volumeobtainedfrom5-axistoolmotionscangenerateun-expectedresultsbecauseofthenon-robustsurface/surface intersections. Therefore in this thesis for 5-axis tool motions in-processworkpiecehasbeenusedinsteadoftheremovalvolume.Inthisapproachthefeasiblecontactsurface generated at a given cutter location point has been offsetted linearly with aninfinitesimal amount. As a result of this linear offsetting a surface volume has beengenerated. Then this volume has been intersected with the in-process workpiece. Laterface/faceintersectionshavebeenperformedforobtainingtheboundariesoftheCWEs.Inthis5-axissolidmodelbasedmethodologybylinearlyoffsettingthefeasiblecontactsurfaces,theCWEextractionshavebecamemorerobust. For addressing the chordal error problem in polyhedral models a 3-axis  mappingtechnique has been developed that transforms a polyhedral model of the removal volumefrom Euclidean space to a parametric space defined by location along the tool path,engagementangleanddepth-of-cut.Asaresult,intersectionoperationshavebeenreducedtoChapter7.Conclusions 187first order plane-plane intersections. This approach reduces the complexity of thecutter/workpiece intersections and also eliminates robustness problems found in standardpolyhedral modeling. Because in 5-axis tool motions the direction of the tool axis vectorcontinuously changes applying the mapping described for the 3-axis milling increases thedistortions of the facets. Therefore a CWE extraction approach similar to the 5-axis solidmodelerbasedapproachhasbeendeveloped.InthepolyhedralmodelbasedCWEextractions2mmwasthepracticallimitforthefacetsize.Stillneededisanexpressionforfindingtheoptimalfacetsize.7.2FutureWork Thefutureresearchworkbasedonthisthesisissummarizedinthefollowing:square4 InthisthesisforthepolyhedralmodelbasedCWEextractionmethodologiesaprototypesystem has been assembled using existing commercial software applications and C++implementations.ForfutureworkastandalonepolyhedralbasedCWEextractionapproachthatreplacesthevariouscommercialcomponentswillbedeveloped.square4 The CWE areas must be decomposed for integrating with the force prediction model.Moreefficientdecompositiontechniques(e.g.Quadtrees)willbeinvestigatedasawayforrepresenting the engagement geometry. 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Asmentionedinchapter(3),thecylinderandtorussurfacesarecalledaspipesurfaces.Bydefinitionapipesurfaceisanenvelopeofthefamilyofsphereswithaconstantradius.Inthissurfacetypethecentersofthecharacteristiccircleandthemovingsphereequaltoeachother. In this section because of its complexity the property (3.2) is proven for the torussurfaceandforthecylindricalsurfacesimilarstepscanbeused.WhenthecomponentsoftheFrenetframegiveninEq.(3.53)arepluggedintoEq.(3.52),thetorussurfaceisobtainedasasetofthecharacteristiccirclesinthefollowingform ))    (A.1)where [ ] . The radius of the moving sphere for the pipe surfaces is constant.Thereforethepartialderivativeoftheradiuswithrespecttothetoolpathparametertequalstozero,i.e. ( ) 0t .ThusunderthisconditiontheradiusandthecenterofthecharacteristiccirclefromEqs.(3.50)and(3.51)becomeas r , )              (A.2)Pluggingtheequationsgivenin(A.2)into(A.1)yields ))       (A.3)For obtaining a point (so called grazing point) on the cutter envelope surface Eq.(A.3) ispluggedintoEq.(3.55) andthentheresultantequationrepresentingthesurface normalispluggedintoEq.(3.56).Theseyields ( ) ( ) 0uu theta             (A.4)AppendixA.Obtainingthegrazingpointsforcuttergeometriesrepresentedbypipesurfaces. 194ThetrigonometrictermsfromEq.(A.4)areextractedasfollows( ) ( )22cos BBuu+= , ( ) ( )2sinBMuu+=          or             (A.5) ( ) ( )22cos BBuu+= ,  ( ) ( )2sinBMuu+=    Plugging theta and theta fromEq.(A.5)intoEq.(A.3)yields ( ) ( )( ) ( )parenrightexparenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp+=2)BBmuurututK mtheta       (A.6)ForsimplifyingtheEq.(A.6)thefollowingcrossproductrepresentationisused c        (A.7)UndertherulegiveninEq.(A.7)thenominatorandinsidetheparenthesisofEq.(A.6)canbewrittenasfollows ( ) ( ) )uuu           (A.8)Also from Eq.(A.7) the length of the nominator inside the parenthesis can be written asfollows ( ) ( ) ( ) ( )22 Buuuuu + (A.9)It can be seen that Eq.(A.9) equals to the denominator given  inside the parenthesis ofEq.(A.6).PluggingequivalentsofnominatoranddenominatorgiveninsidetheparenthesisofEq.(A.6).fromEqs.(A.8)and(A.9)yieldsAppendixA.Obtainingthegrazingpointsforcuttergeometriesrepresentedbypipesurfaces. 195 parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp??=||)),(),,(BBmuurututK mtheta           (A.10)Intheaboveequationthecrossproductofthenormalandbi-normalunitvectorsareequaltothetangentvectorofthespinecurvei.e. B .ThusPluggingtheequivalentofofTfromEq.(3.53)intoEq.(A.10)yieldstwograzingpointsP1,2 parenrightexparenrightbtparenrighttpparenleftexparenleftexparenleftbtparenlefttp??==||),(),,( 2,1 tutrutPutKmmm mtheta          (A.11)Thustheproperty(3.2)isprovenforthetoruswhichisalsoapipesurface. 196AppendixBMappingparametersformillingcuttergeometriesB.1DerivationofMcyl A Flat end mill is made up of a cylindrical surface CWEK,c1(t) at the side and a flatsurface CWEK,c2(t) at the bottom (Figure B.1-a,b). Based on the kinematics of the 3-axismachiningthesesurfaceswillcontributetothesetCWEK(t)asdefinedinsection(6.2.1)i.e.2)( cCWECWEtCWE union= .Forthecylindersurfacetheengagementangleofapointat)1, tcK must lie within [0,pi] i.e. ( pi1, tcK ) and for the flatsurfaceitmustliewithin[0,2pi]i.e.( pi2, <=<cK ). (a)(b)FigureB.1:(a)CylindricalCWEK,c1(t),and(b)bottomCWEK,c2(t),facesoftheFlatEndMillB.1.1ObtainingCWEParametersfortheCylindricalSurface ForthecylindricalsurfacetoolreferencepointFequalstothetooltippointT. ( ) ( ) tSES V             (B.1)Forthefirststep,thegeometryofthecylinderchangesEq.(6.11)and(6.12)to,{ } 022 =C n         (B.2)AppendixB.Mappingparametersformillingcuttergeometries. 197and, { } 022, =CK n          (B.3)respectively.AswithEq.(6.13),Eq.(B.3)resultsinaquadraticequationintwhenIandFaresubstituted.Inthiscasethecoefficients 12 , A and 0 aregivenby,( ) ( ) ( )]SISISI z ( )]222 SESE V   ( ) ( )] ( ) ( )SESE V1          (B.4)( ) 2220 r   TworealrootsoftheEq.(B.3)representtwocutterlocationsaccordingtoProperty(6.4)theminimumofthemistakentobethecorrecttoolpositionasexplainedinsection(6.2.1.1).UsingEq.(6.23-a)and(6.25)theparametersphi1 andLareobtained.Finally,thedepthoftheengagementpointasdefinedbythedistancefromT(t)toI'measuredalongthetoolaxisvectorissimplythez-coordinateofI'.)I T       (B.5)B.1.2ObtainingCWEParametersfortheFlatSurface CWEparametersfortheFlatsurfaceareobtainedwithoutdoingmapping.Anunboundedplanewhichisperpendiculartothetoolrotationaxisisintersectedwiththeoriginalremovalvolumeofthebottomsurfaceforagivencutterlocationpoint.Engagementanglephi1 isfoundby Eqs.(6.23-a) and (6.23-b). The depth of the cut for the bottom surface is the distancebetweentheintersectionpointIandthecenterofthebottomsurfacesuchthat2Iyxd minute+minute=                  (B.6)AppendixB.Mappingparametersformillingcuttergeometries. 198B.2DerivationofMconeConical part of a Tapered Flat End Mill is made up of a front CWEK,co1(t) and backCWEK,co2(t) conical contact faces at the side and a flat surface CWEK,co3(t) at the bottomrespectively(seeFigureB.2).Basedonthekinematicsofthe3-axismachiningthesesurfaceswillcontributetothesetCWEK,Co(t)asdefinedinsection(6.2.1)i.e.)3,2,1,, tcoKcoKcoKCoK union (B.7)ForthegiventoolmotiontypeCinFigure(6.27),thecutterhasengagementswithallitssurfaces and the total engagement area covers the full [0,2pi] range i.e.pi, <=<CoK . (a)(b)FigureB.2:(a)FrontCWEK,co1(t),and(b)backCWEK,co2(t)conicalfacesofTaperedFlatEndMillCWEparametersfortheflatsurfaceareusedfromB.1.2.FortheTaperedFlatEndMilltoolreferencepointFequalstothetooltippointT,Eq.(B.1). Forthefirststep,thegeometryoftheconechangesEqs.(6.11)and(6.12)to,( ){ } 022 =Co        (B.8)andAppendixB.Mappingparametersformillingcuttergeometries. 199 ( ){ } 022, =CoK  (B.9)wherealpha ,nandVaretheconehalfangle,unitnormalvectorofthetoolrotationaxisandconevertexcoordinates(seeFigure6.22-a)respectively.TherelationshipbetweenVandFisgivenbynalphartan      (B.10)whereristheradiusoftheconebottomsurface.AswithEq.(6.13),Eq.(B.9) resultsinaquadraticequationintwhenIandFaresubstituted.Inthiscasethecoefficients 12 , A and0A aregivenby,bracketrightexbracketrightbtbracketrighttpbracketleftexbracketleftbtbracketlefttp parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp +-parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp +-parenrightexparenrightbtparenrighttpparenleftexparenleftbtparenlefttp +-=Znrzznyynxxalphaa( ) ( )] 2222 tan SESE V (B.11)( ) ( ) ( )( ) ( )]SESE V21 tan ( ) ( )2220 tan a       TworealrootsoftheEq.(B.9)representtwocutterlocationsaccordingtoproperty(6.4)andthe minimum of them is taken to be the correct tool position as explained in the section(6.2.1.1).UsingEqs.(6.23-a,b),(6.25)and(B.5)theparameters(phi1 ,d,L)areobtained.  B.3DerivationofMtorus AToroidalportionoftheendmillismadeupoffrontCWEK,t1(t)andbackCWEK,t2(t)contactfacesatthesideandaflatsurfaceCWEK,t3(t)atthebottomrespectively(seeFigure(B.3)).Basedonthekinematicsofthe3-axismachiningthesesurfaceswillcontributetothesetCWEK,T(t)asdefinedinsection(6.2.1)i.e.AppendixB.Mappingparametersformillingcuttergeometries. 200)3,2,1,, ttKtKtKTK union .(B.12) (a)(b)FigureB.3:(a)FrontCWEK,t1(t),and(b)backCWEK,t2(t)contactfacesofToroidalEndMillCWEparametersfortheflatsurfaceareusedfromB.1.2.ThereferencepointatFischosentobethecenterofthetorus.Itslocationcanbeexpressedbythecuttingtooltipcoordinatesas,( ) n       (B.13)whererandnaretheradiusoftube(Figure(B.3-a))andunitnormalvectorofthetoolaxisrespectively.Forthefirststep,thegeometryofthetoruschangesEqs.(6.11)and(6.12)to,( ) ( ) } ( ) } 0222222 =CT F  (B.14)and( ) ( ) } ( ) } 0222222, =CTK F  (B.15)respectively.Eq.(B.15)resultsinaquartic4thorderpolynomialequationintwhenIandFaresubstituted.001223344 =   (B.16)AppendixB.Mappingparametersformillingcuttergeometries. 201Inthiscasethecoefficients 01234 , A aregivenby,( )SE V , ( )S 44 a=A , ( )baa ?-=23 4A ( ) ( ) ( ) 22222222 4 SE z (B.17)( ) ( ) ( )SESI z22221 b ( ) [ ]22222220 ) rSI - The Eq.(B.16) gives four roots and according to the property (6.4) toroidal surface hasmaximum four locations for I. For example, two CWE points 1 and 2I are shown inFigure(B.4). Along the tool path, toroidal surfacetouches the point 1 at four interferencelocations i.e. 4321 T,T,T,T thus for  1 the Eq.(B.16) gives four distinct real roots such that4tttt <<< andasexplainedinsection(6.2.1.1)minimumofthemistakentobethetoollocationwhencuttertouchesthispoint.FigureB.4:CutterinterferenceswithapointinspaceToroidalsurfacetouches 2 attheinterferencelocation 5 andbecause 2 isontheenvelopesurfaceofthecutter,Eq.(B.16)givesrepeatedrealrootsforthispoint.UsingEqs.(6.23-a,b)and(6.25),(phi1 ,L)parametersareobtained.ThedepthoftheengagementpointisobtainedbyAppendixB.Mappingparametersformillingcuttergeometries. 202parenrightexparenrightbtparenrighttpparenleftbtparenlefttp minute- rzI1   (B.18)

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