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1、time Look-ahead and AdaPtive Parametric Curve InterpolatorJournalofShanghaiJiaotongUniversity(Science),Vo1.E一12,No.3,2007,388393ArticleID:10071172(2007)03038806StudyonReal-timeLook-aheadandAdaPtiveParametricCurveInterpolatorGUOWei(郭伟),DUDaoshan(杜道山),XURongzhen(徐荣珍),LICmngxin(李从心)(NationalDie&.Mo
2、uldCADEng.ResearchCenter,ShanghaiJiaotongUniv.,Shanghai200030,China)Abstract:Theprincipleofrealtimelookaheadwasintroducedandanalysed.AnadaptiveparametriccurveinterpolatorwitharealtimelookaheadfunctionwasdevelopedfornonuniformrationalBspline(NURBS)curvesinterpolation.whichconsideringthemaximumacceler
3、ation/deceleratonofthemachinetoo1.Inordertodealwiththeacceleration/decelerationaroundthefeedratesensitivecorners,thelookaheadfunctionwasdesignedandillustrated.Itcandetectandadjustthefeedrateadaptively.Withthehelpofrealtimelookahead,theaccelerat;on/decelerationcanbelimitedtotherangeofthemachinetoolca
4、pacity.Thus.feedratefluctuationisreduced.ANURBScurveinterpolationexperimentwasprovidedtoverifythefeasibilityandadvantagesoftheproposedinterpolatorwitharealtimelookaheadfunction.Keywords:adaptiveinterpolator;lookahead;nonuniformrationalBsplinecurves;feedratesensitivecorner;chorderrorCLCnumber:TP273Do
5、cumentcode:AInmoderncomputeraideddesign(CAD)systerns,complexshapesurfacesforpartssuchasdies,moulds,aerospaceandcarmodelsareusuallyrepresentedbyparametriccurves.Amongmanyparametriccurverepresentationmethods,nonuniformrationalBspline(NURBS)isoneofthemostpopularindustrystandardtools,whichhasbeenwidelyu
6、sedintheinterpolatordesign.Nowadays,3numberofparametricinterpolatorshavebeenproposedbyinvestigatorsandcommercialcomputernumericalcontrol(CNC)systemmanufacturers.一8】.Bediela1.set(.)asaconstantintheuniforminterpolationalgorithmtoreducespeedfluctuationduringtheinterpolationprocess.Huangelaldevelopedfir
7、storderapproximationinterpolationalgorithmsusingEulerandTaylorsexpansionsrespectively.ThesefirstorderapproximationinterpolationalgorithmsprovideReeeiveddate:20061113Foundationitem:NewHighSpeedCuttingCNCSystemResearch*Email:uniformcurvespeedduringtheinterpolationprocess.Then.thesecondorderapproximati
8、onandspeedcontrolledinterpolationalgorithmswereproposedbyYangelal.yieldingmorepreciseresults.However,thesealgorithmsdonotconsiderchorderrorexplicitlyduringinterpolationprocess.Furthermore,YehandHsudevelopedanadaptiveparametriccurveinterpolationalgorithm引.whicheliminatesthedrawbacksofboththeconstantp
9、arameterintervalandconstantspeedinterpolation.alsoitconsiderschorderror.However.theacceleration/decelerationcapacityofthemachineisnotmentioned.Basedonthepreviousworks.lookaheadfunctionisintroducedandanadaptiveinterpolatorwithrealtimelookaheadfunctionisdevelopedforNURBScurveinthispaper.Thedesignofloo
10、k-aheadmoduleisdiscussedandgivenindetail.Withthehelpofrealtimelookahead.theacceleration/decelerationcanbelimitedtotherangeofthemachinetoolcapacity.Thus.feedratefluctuationisredLaced.LookaheadschemehasbeenusedinmanyStudyonRealtimeLookaheadandAdaptiveParametric?389CNCmachinestominimizethestart/stopmot
11、ionsandtoachieveaconstantfeedrate.Asequenceofinterpolationpointsaretakenintotherealtimelookaheadmoduleforprecalculationbeforetheyarereadytobeexecuted.Thelookaheadwindowmovesforwardwiththeinterpolationprocessuntilthefinalpointismet,asillustratedinFig.1.Theresultingdataarestoredinafirstinfirstout(FIFO
12、)buffercontinuouslyduringtheoperation.LookaheadwindowFig.1IllustrationoflookaheadprincipleOnWithconventionalcontrols.thefeedratechangesexcessively.Withlookaheadscheme,thenextmoveisanticipatedandasmoothfeedrateprofileisgenerated.Figure2showsthecomparisonoffeedrateprofilesoftwomethods.LookaheadConvent
13、ionalmethodof,Fig.2Comparisonoffeedrateprofiles2.1DiagramforInterpretati0nwithaRealtimeLookaheadParametriccurveequationcanbedescribed:c(x,=X(u一(1),一,where,isparametervariable.Thekeyissueofparametercurveinterpolationistocalculate1from.+(J十c+(?+dj?d2ydudu2dud)7I.Jz(+)+l(2)where7istheinterpolationperio
14、dandFisthecuttingtoolmovingspeed.TheEq.(2).isusedinthefollowinginterpolator.AnadaptiveparametriccurveinterpolatorwitharealtimelookaheadfunctionisdesignedandshowninFig.3.Thearchitecturemainlyincludesthreefunctionmodules:theparametercalculationmodule,theadaptivefeedrateadjustmentmoduleandrealtimelooka
15、headmodule.Thefunctionsandtasksofthreemodulesareexplainedasfollows:Fig.3Diagramforinterpretationwitharealtimelookahead(1)Theparametercalculationmoduleisusedtocalculate,+lfromandF.Inputsoftheparametercalculationmodulearecurveproperties(controlpoint,knotvector,weight,commandfeedrateF)andtheparameterva
16、riable,.Outputisthenextstepparametervariableu+1.(2)Theadaptivefeedrateadjustmentmoduleistoadiustfeedratetoreducechorderrorintothesettolerance.Itsinputsincludetheparametervariable.andthemaximalallowablechorderrorwhiletheoutputisadjustedfeedrate.(3)Therealtimelookaheadmoduleistodetectthefeedratesensit
17、ivepositionbycomparingthefeedrateatpoint,andthefeedratepoint,l,andthentodealwiththefeedratetoconfineacceleration/decelerationwithinthecapacityofthemachinetoo1.Inthisproposeddesign.thefeedrateisdeter一葛.I口390GUOWei(郭伟),DUDaoshah(杜道山minedbythreeways.First,thefeedrateisgivenascommandwithchorderrorundert
18、hesettoleranee.Second,thisfeedrateisadjustedadaptivelywhenchorderrorisbiggerthanthesettolerance.Third,thefeedrateafterthestepprocessingaboveiscalculatedbylookaheadfunction.Theinputstotherea1.timeNURBSinterpolatorarethegeomet-ricinformation,suchastheknotvector,weights,controlpointsandcuttingcondmons,
19、forexamplethefeedrate.Theoutputswillbesenttoamotioncontroller.2.2CalculationofChordErrorandAdaptiveFeedrateAsthecuttermovesalongthecurve,twopositionerrorsmayoccurbetweentheinterpolatedcurveandtheidealcurve:radialerrorandchorderror.asshowninFig.4(a).Basically,radialerroriscausedbyroundofferrorincompu
20、tationprocessandcanbeeliminatedwithhighercomputationprecision.Chorderroriscausedbytheinterpolationprocess,andisconsideredasthemainerrorsourceAneararcapproximationisadoptedtocalculatethechorderror,asshowninFig.4(b).C(ui)|口(蜥)Fig.4ChorderrorcalculationFromtheFig.4(a),thechorderrorcanbegivenas:,一P,一/一(
21、厶/2)(3)where,L,一V(,)Tinoneinterpolationinterval;pi一1/K,P,andKaretheradiusofcurvatureandcurvatureatparameter,respectively.Thecurvatureisgivenas:K.=ll竺竺:ldC(u)/dufI3(4)Anadaptivemethodhasbeenproposed,bywhichthefeedrateiskepttobeconstantatmosttime,anditwillbeadaptivelyadjustedintermsofthecurvatureradiu
22、softheinterpolationpointswhenthechorderrorexceedsthetolerance.So,fromEq.(3),assumingtheallowedchorderroris,thefeedratecouldbeadaptivelychangedalongthecurvesubjectingtothefollowinglaw:fF,1屑.therwLjwhereFisthecommandedfeedrate.2.3DesignofReal-timeLook-aheadBecausetheabruptchangeoffeedratewillprobablye
23、xceedtheacceleration/decelerationcapacityofthemachinetool.arealtimelookaheadfunctionisusedtodealwiththeabruptchangeoffeedrate.Thepurposeoflookaheadhastwoaspects:oneistodetectthefeedratesensitivepositionwherethefeedratemaychangeabruptly;theotheristodeterminethestartpointofacceleration/decelerateandfu
24、rthersmooththefeedratew;thinthesensitivezone.Usuallytheacceleration/decelerationadjustmentisnecessarywhenencounteringalittleradiusofcurvaturealongthecurve.Twosituationsmayappear:increasingfeedrateordecreasingfeedrate.AssumingcommandfeedrateisFandthemaximalacceleration/decelerationcapacityisAm.Firstc
25、onsideringthedecelerationsituation,intheworstcasefeedratemaybereducedtozerofromthecurrentfeedrate.Withthemaximaldecelerationcapacity,theminimaldistanceneededtofinishthedecelerationwillbeS一F;/2A.Astructurearrayisdesignedtostoreasequenceoftheformerinterpolationpoints.Theformofthisstructurearrayisshown
26、aspointi3,F,distancestep,distanceentire.Where,isvariableparameterofthei-thinterpolationpointinthestructurearray;Fisitsfeedrate;distancestepisthedistancebetweenthispointandthenextinterpolationpoint,alsodistancestepFT.Tistheinterpolationsampletime;distanceentireisthedistancebetweenthecurrentpointandth
27、efirstinterpolationpointinthestructurearraypointO,F,distancestep,0.Intermsofabovedefinition,thefollowingequationsaresatisfied:point门.distanceentirepointi-1.disStudyonReal?timeLook?aheadandAdaptiveParametric?-39ltanceentire+point一1.distancesteppointO.distancestep+point1.distancestep+point1.distancest
28、epIhestructurearraydoesnotneedtostorealloftheinterpolationpointsaccordingtotwolimitations.OneisthatthedistancefromthefirstpointtothelastpointisnotsmallerthanS,i.e.pointr.distanceentireS;theotheristhatthedistancefromthesecondpointtothelastpointissmallerthanS,i.e.(point.distanceentirepointEo.distances
29、tep)<S.Whenafeedratesensitivepointislocated,thedistanceneededtofinishthedecelerationcanbecalculatedaccordingtothemaximaldecelerationcapacityandthemaximalallowablefeedratefromtheEq.(5).Inthestructurearray,thepointfromwherethedecelerationstartscanbefoundbyusingabisectionmethod.Thesearchingcondition
30、isthatthedistancebetweenthispointandthefeedratesensitivepointisnotsmallerthanthedistancecalculatedabove.Then,reinterpolationcanbedonebytracingbacktothispointfromthecurrentinterpolationpoint.i.e.thefeedratesensitivepoint.Duringreinterpolating,theactualdecelerationiscalculatedbasedonthefollowinginform
31、ation:thefeedrateatthispoint,thefeedrateatthefeedratesensitivepointandthedistancebetweentwopoints.Thusthefeedrateisadiustedintermsoftheactualdecelerationbeforearrivingthefeedratesensitivepoint.Inthedecelerationzone.afterlookaheadfunction,feedrateissmallerthantheresultofadaptiveadustment,thus,thechor
32、derrorcanbeassured.Butinthecaseofacceleration,thefeedrateispossiblylargerthantheresultofadaptiveadjustmentwiththesamemethodasthedecelerationtreatment.Whenencounteringafeedratesensitivepointintheaccelerationzone,inordertokeepthechorderrorwithinthetolerancerange,thefeedratecalculationissubjecttoF(,)一F
33、(一1)+ATInsummaryofaboveanalysis,thecompleteacceleration/decelerationadjustmentprocessesoflookaheadfunctionatthefeedratesensitivepositionareasfollows(showninFig.5):墨qFig.5Diagramoflookaheadaccelerationanddeceleration(1)DetectF(,)一F(一1)一thefeedratesensitivepoint.If>A,thepointoncurveisafeedratesensi
34、tivepointandlookaheadfunctionisthereforenecessary.MarkthispointasA.Else,jumptoStep(10)andexitlookaheadfunction.(2)DeterminedecelerationoraccelerationadjustmentforpointA.IfF(,)一F(ul1)>0,itisanaccelerationsensitivepoint.ThefeedrateiscalculatedbyF(u)一F(u一1)+A7.Then,jumptoStep(8).IfF(,)一F(一1)<0,it
35、isadecelerationsensitivepoint,gotoStep(3).(3)CalculatethedistanceneededtodecreasefeedrateFtoF(,)withthedecelerationA,markthisdistanceasSp.Obviously,Sp5.(4)FindapointBbyusingbisectionmethodinthestructurearraytoassurethedistanceSbbetweenpointAandpointBisnotsmallerthanSp,i.e.SbS.(5)GobacktopointBandrei
36、nterpolate.Theaetualdeceleration.口,iscalculatedbasedontheparameters:thefeedrateatthispointstoredinthestructurearray,thefeedrateatthefeedratesensitivepoint,F(,),andthedistancebetweentwopoints,Sb.(6)ReinterpolatefrompointB.Thefeedrateisadjusteddependingontheactualdeceleration口,F(,)一F(,一1)一日?T.(7)Reint
37、erpolateuntilachievingpointA.(8)Aftereachinterpolation,updatethestructurearray.Anewcomponent,pointi3,F,distancestep,distanceentire,isaddedintothestructurearray.392GUOWei(郭伟),DUDaoshah(杜道山),etal(9)Checkthestructurearray,andassure:thedistancefromthefirstpointtothe1astpointiSnotsmallerthanS,i.e.point门.
38、distanceentiren;thedistancefromthesecondpointtothelastpointissmallerthanS,i.e.(pointF-i-I.distanceentirepointEo.distancestep)<5.(10)Exit1ookaheadfunction3.1MouldExampleTheproposedinterpolationalgorithmistestedinacastingmouldmachiningexperiment.OneofthecuttingpathsofcastingmouldshowninFig.6isuseda
39、sanexample.O?04o.020O.25,O.5,O.5,O.75,1,1,1.ThesamplingtimeininterpolationiS:T一2ms.ThefeedratecommandiS:F一0.2m/s.Themaxima1allowablechorderroriS:E一0.5t*m.Themaxima1allowableacce1eration/dece1erationis:Am一0.5m/s.3.2InterpolationResultsAnalysisFigures79aretheinterpolationresultsupontheaboveNURBScurve,
40、includingfeedrateprofile,acce1eration/dece1erationprofileandthechorderrorprofile.Thereare902interpolationstepsandtheconsumedtimeiS1.804seconds.Figure7showsthatthefeedratemaintainsnearthepointofthecommandedfeedrate(0.2m/s)atthemosttime.ComparethefeedrateprofileinFig.7andthechorderrorprofileinFig.9,it
41、canbeseenthatthereiSaveryclosecorrelationbetweenthechorderrorandfeedrate.Whenthe,g一,勺Fig.7Feedrateprofileoffinishmachining00.51.01.52.0,/SFig.8Acceleration/decelerationprofileoffihishmachining00.51.01.52.O,/SFig.9Chorderrorprofileoffinishmachiningchorderrorreachesthemaximumpermissiblevalue,thefeedra
42、teiSadaptedandsloweddownSOthatthechorderrorneverexceedsthemaximumva1ue.Figure8showstheacce1eration/dece1erationprofile.Aftertheacce1eration/dece1erationadjustmentinlookaheadfunction,thechangeoffeedrateiSmoreflexibleandtheacce1eration/dece1erationprofileisimprovedandconfinedwithinthedesiredrange.Inth
43、ispaper,theprincipleofrealtimelookaheadisintroduced.Anadaptiveparametriccurveinterpolatorwitharealtimelookaheadfunctionisdeveloped.Inordertodealwiththeacceleration/642O24oooIs.g一0甍l.u65432OOOOOOO0唇T1/j0己pJ0UStudyonRealtimeLookaheadandAdaptiveParametric?393decelerationaroundthefeedratesensitivecorner
44、s.thelookaheadfunctionisdesignedandillustratedindetails.Itcandetectandadiustthefeedrateadaptively.ANURBScurveinterpolationexperimentisprovidedtoverifythefeasibilityandadvantagesoftheproposedinterpolatorwitharealtimelookaheadfunction.Througharealtimelookaheadfunction,theacceleration/decelerationcapac
45、ityofmachinetoolismet.Aquitegoodperformanceisachievedbyusingtheproposedinterpolator.Inaddition,adesiredchorderrorisobtainedwhilefeedrateprofileissignificantlysmoothedtoo.12HwangJ.InterferencefreetoolpathgenerationintheNCmachiningofparametriccompoundsurfacesJ.ComputerAidedDesign.1992.24(12):667677.Ko
46、renY.LoCC,ShpitalniM.CNCinterpolators:algorithmandanalysisJ.ASMEManufacturing(iencesandEngineering,1993.64:8392.3456789BediS.AiiI,QuanN.AdvancedinterpoiationtechniquesforCNCmachinesEJ.ASME.IournalofEngineeringforIndustry.1993,115:329336.HuangJT,YangDCH.Ageneralizedinterpolatorforcommandgenerationofparametri