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DESCRIPTIONANDSMOOTHINGOFNCMOTIONPATHBASEDONTHECUBICTRIGONOMETRICINTERPOLATIONSPLINEJIANMINGTAO1,A,AIPINGSONG1,BANDDANPINGYI1,C1COLLEGEOFMECHANICALENGINEERING,YANGZHOUUNIVERSITY,YANGZHOU225127,CHINAATAOJIANMING6126COM,BAPSONGSINACOM,CYIDANPING6163COMKEYWORDSSPLINECURVEINTERPOLATIONADJUSTABLESHAPETRAJECTORYDESCRIPTIONSMOOTHPATHABSTRACTINORDERTOBETTERDESCRIBETHECOMPLEXMOTIONPATHOFNCMACHININGANDREALIZETHESMOOTHTRANSITIONBETWEENPATHSEGMENTS,AKINDOFCUBICTRIGONOMETRICINTERPOLATIONSPLINECURVEWASPUTFORWARDBASEDONASETOFSPECIALBASISFUNCTIONTHESPLINECURVEWHICHWITHADJUSTABLESHAPESATISFIESTHE1CCONTINUITYANDITCANACCURATELYDESCRIBESOMECOMMONENGINEERINGCURVESSUCHASSTRAIGHTLINE,CIRCULARARCANDFREECURVEACCORDINGTOTHEGIVENINFORMATIONOFCONTROLPOINTS,DIFFERENTSHAPESOFINTERPOLATIONSPLINECURVECANBEGOTTENBYCHANGINGTHEADJUSTMENTCOEFFICIENTSTHROUGHSELECTINGPROPERCONTROLPOINTSANDSHAPEADJUSTMENTCOEFFICIENTSNEARTHECORNER,INSERTTHESPLINECURVECANREALIZETHESMOOTHTRANSITIONATTHECORNEROFADJACENTNCMOTIONPATHSEGMENTS,WHICHCANENSURETHESTABILITYOFMOTIONPATHANDTHECONTINUOUSOFFEEDSPEEDMEANWHILE,ITALSOCANREDUCETHEIMPACTTONCMACHINEINTRODUCTIONHIGHSPEEDNCMACHININGISTHEIMPORTANTWAYTOIMPROVEEFFICIENCYANDQUALITYOFPARTSMACHININGDURINGTHEHIGHSPEEDOPERATIONPROCESSOFMACHINETOOL,ITNEEDSTOENSURETHESTABILITYOFMACHINETOOLMOVEMENTSTOAVOIDGENERATINGLARGERIMPACTSWHICHWILLAFFECTTHEQUALITYOFPARTSMACHINING,MEANWHILETOPROTECTTHEMACHINETOOLFEEDINGSYSTEM1INACTUALMACHININGPROCESS,NCMOTIONPATHSOFTENCONSISTOFANUMBEROFSTRAIGHTLINESANDCIRCULARARCSATPRESENT,TODEALWITHTHESPEEDATTHECORNEROFADJACENTNCMACHININGPATHSEGMENTS,THEMAINMETHODISTOSLOWDOWNTHESPEEDTOZEROATTHEENDOFTHECURRENTPROCESSINGSEGMENT,ANDTHENACCELERATETOCOMMANDSPEEDINTHESTARTINGPOSITIONOFTHENEXTPROCESSINGSEGMENTUSINGTHISWAYTOGETTHROUGHEACHCORNERWITHZEROSPEEDCANAVOIDLARGERIMPACTSTONCMACHINEHOWEVER,THISWAYWILLCAUSEFREQUENTSTARTANDSTOPOFSPEEDDURINGTHEMACHININGPROCESS,ANDITWILLSERIOUSLYAFFECTTHEIMPROVEMENTOFPARTSMACHININGEFFICIENCY2THEREFORE,ITNEEDSTOSTUDYNEWMOTIONCONTROLMETHODATTHECORNER,WHICHCANMAKETHETRANSFERSPEEDDOESNOTDROPTOZERO,ANDREACHTHEPURPOSEOFREALIZINGTHEHIGHSPEEDSMOOTHTRANSITIONBETWEENTWOADJACENTPATHSEGMENTSTHUSTOIMPROVETHEMACHININGEFFICIENCYANDLIMITTHEIMPACTLOADSINDEALINGWITHTHESPEEDATTHECORNEROFADJACENTMACHININGPATHSEGMENTS,LITERATURES3,4PROPOSEDTOADDCIRCULARARCORQUADRATICCURVEFORCORNERTRANSITION,INORDERTOMAKETHESPEEDDOESNOTDROPTOZEROTOACERTAINEXTENT,THISMETHODIMPROVEDTHEMACHININGEFFICIENCYANDREDUCEDTHEIMPACTSTOMACHINETOOLSBUTITLACKSTHECONTROLOFACCELERATIONANDTHEERRORCONTROLISNOTSTRICTWHENTHECORNERISLARGER,THEIMPROVEMENTOFTRANSFERSPEEDWILLBELIMITED,SOTHISMETHODDOESNOTWELLMEETTHENEEDSOFHIGHSPEEDMACHININGLITERATURE5PUTFORWARDASPEEDCONTROLMETHODWITHLOOKAHEADTHISMETHODLIMITEDTHEIMPACTSFORMEDBYTHECHANGEOFVELOCITYVECTORATTHECORNERTHROUGHTHELIMITEDSPEEDDROPPINGATTHECORNERTOPURSUETHEMAXIMUMMACHININGEFFICIENCYBUTITNEEDSALARGEAMOUNTOFPRECOMPUTATIONANDITREQUIRESHIGHERNUMERICALCONTROLSYSTEMLITERATURE6PROPOSEDAVECTORMETHODTOREALIZETHESMOOTHTRANSITIONBETWEENTWOPATHSEGMENTSTHROUGHTHEANTICIPATORYANALYSISOFTHEMOTIONPARAMETERS,THISMETHODCANIMPROVETHETRANSFERSPEEDWHENTHECORNERISLARGERBUTWHENTHECORNERISSMALLER,THETRANSFERSPEEDISSTILLNOTSMOOTHENOUGHAPPLIEDMECHANICSANDMATERIALSVOLS3653662013PP515521ONLINEAVAILABLESINCE2013/AUG/16ATWWWSCIENTIFICNET2013TRANSTECHPUBLICATIONS,SWITZERLANDDOI104028/WWWSCIENTIFICNET/AMM365366515ALLRIGHTSRESERVEDNOPARTOFCONTENTSOFTHISPAPERMAYBEREPRODUCEDORTRANSMITTEDINANYFORMORBYANYMEANSWITHOUTTHEWRITTENPERMISSIONOFTTP,WWWTTPNETID58241131203,YANGZHOUUNIVERSITY,YANGZHOU,CHINA27/02/14,062335ONTHEBASISOFCUBICHERMITEINTERPOLATIONFUNCTION7,8,THISPAPERPUTSFORWARDAKINDOFADJUSTABLESHAPECUBICTRIGONOMETRICPOLYNOMIALINTERPOLATIONSPLINECURVE,WHICHCANBEUSEDTODESCRIBETHENCMACHININGPATHS,SUCHASSTRAIGHTLINE,CIRCULARARCANDFREECURVESEGMENTS,ANDITHASPROPERTIESOFSIMPLECALCULATION,FLEXIBLESTRUCTURE,ETCTHESPLINECANALSOBEUSEDTODESCRIBETHESMOOTHTRANSITIONCURVESATTHECORNERANDREALIZETHEHIGHSPEEDSMOOTHTRANSITIONBETWEENNCMACHININGPATHSEGMENTS,WHICHMEETSTHENEEDSOFMODERNNUMERICALCONTROLSYSTEMFORHIGHSPEED,STABILITYANDFLEXIBILITYCUBICTRIGONOMETRICINTERPOLATIONSPLINECURVEBASISFUNCTIONOFSPLINEFORANYGIVENVALUEOFVARIABLESANDK,PARAMETERUSATISFIES01UTHEFOLLOWINGFORMULAISCALLEDASETOFCUBICTRIGONOMETRICSPLINEBASISFUNCTION3320,33321,33322,33,32222211222222221111111122222222111BUCSSSBUKCKSKSCKSKBUKKCKKSSKCKSKB3322222UKCKSKSKC1INFORMULASISSTANDFORSIN2U,CISSTANDFORCOS2UINTERPOLATIONSPLINECURVESUPPOSETHATTHEGROUPOFGIVENCONTROLPOINTSAREIQ0,1,NI，,THENTHESPLINECURVESEGMENTSCANBEDEFINEDAS3,30IIJJJPUQBU,01U,0,1,3IN，2THECURVEPUWHICHISCOMPOSEDOFALLSMALLCURVESEGMENTSIPUISCALLEDTHECUBICTRIGONOMETRICINTERPOLATIONSPLINECURVEFROMEXPRESSION2,ITISEASYTOKNOWTHATTHEWHOLECURVEPUFORMEDBY2NSEGMENTSOFSMALLCURVESTONUMBERISEGMENTCURVEIPUEXISTS121121320,01,1IIIIIIIIIIIIIIPQPQQKQQPQPQQKQQ3ANDTONUMBER1ISEGMENTCURVE1IPUEXISTS111213212132430,01,1IIIIIIIIIIIIIIPQPQQKQQPQPQQKQQ4COMPARINGFORMULA3WITH4,ITCANBEOBVIOUSLYFOUNDTHATTHEADJACENTCURVESIPUAND1IPUHAVETHEFOLLOWINGCONNECTIONS11121321010IIIIIIIIIPPQPPQQKQQ5ACCORDINGTOFORMULAS3,4AND5,ATHEOREMCANBEGOTTEN,THATISTHESPLINECURVEPUINTERPOLATESTHEGROUPOFGIVENCONTROLPOINTSFROM0QTO2NQANDITALSOSATISFIES1CCONTINUITYSPREADOUTTHESPLINECURVEEXPRESSION2INTOAPOLYNOMIALFORM,THENCANGET516MACHINEDESIGNANDMANUFACTURINGENGINEERINGII3323321322222211222222221111111122221IIIUCSSSKCKSKSCKSKKKCKKPQQQ3233232222112222ISSKCKSKKCKSKSKCQ633233213222222112222222211111111222211IIIXXCSSSXKCKSKSCKSKXKKCKK3233233323321222212222222221122222211111IIISSKCKSKXKCKSKSKCYYCSSSYKCKSKSC3322332322111222222221112222IIKSKYKKCKKSSKCKSKYKCKSKSKC7INFORMULAS7AND8SISSTANDFORSIN2U,CISSTANDFORCOS2U,01UABFIG1ADJUSTABLESHAPEINTERPOLATIONSPLINECURVEFROMABOVECANKNOW,ANYFOURGIVENCONTROLPOINTSCOMPLETELYDETERMINEASEGMENTOFADJUSTABLESHAPECUBICTRIGONOMETRICINTERPOLATIONSPLINECURVETHECURVEINTERPOLATESTHEFIRSTTWOCONTROLPOINTS,ANDTHELATTERTWOCONTROLPOINTSAREUSEDTOCALCULATETHEENDPOINTSLOPESOFTHECURVESEGMENTSETTINGIQ,1IQ,2IQ,3IQRESPECTIVELYASA,B,C,DFOURASSUREDPOINTS,ADOPTINGAUTOLISPPROGRAMMINGINTHEAUTOCADCANDRAWASEGMENTOFADJUSTABLESHAPECUBICTRIGONOMETRICINTERPOLATIONSPLINECURVE,ASSHOWNINFIG1CHANGINGTHEVALUEOFANDKCANREALIZETHESHAPEADJUSTMENTOFTHECURVESEGMENTTHEFOURCURVESINFIG1ARESPECTIVELYCORRESPONDINGTOTHESPLINECURVESEGMENTSWHENKTAKESACERTAINVALUEOF05,0,05,1AND15WHILETHEFOURCURVESINFIG1BRESPECTIVELYCORRESPONDINGTOTHESPLINECURVESEGMENTSWHENTAKESACERTAINVALUEOF1,03K,05,06AND07FIG2RINGINTERPOLATIONSPLINEFIG3STRAIGHTLINEINTERPOLATIONSPLINEFURTHERMORE,RINGFORMINTERPOLATIONSPLINECURVECANBEGENERATEDWHENTHECONTROLPOINTSAREREUSEDASSHOWNINFIG2,WHENTHECONTROLPOINTSSEQUENCEISB,B,C,D,IEPOINTBISREUSED,THENTHROUGHPROGRAMMINGCANGENERATETHERINGSPLINECURVESEGMENTSTHETHREERINGCURVESINFIGAPPLIEDMECHANICSANDMATERIALSVOLS3653665172RESPECTIVELYCORRESPONDINGTOTHESPLINECURVESEGMENTSWHENKTAKESACERTAINVALUEOF1,1,0,AND1KEEPINGTHETHREEPOINTSA,B,CLOCATEINONESTRAIGHTLINE,THENTHESTARTINGPOINTVECTOROFTHERINGSPLINEISINTHESAMEDIRECTIONWITHAB,ANDTHEENDPOINTVECTORISINTHESAMEDIRECTIONWITHBDTHISFEATURECANBEWELLUSEDTODEALWITHTHECORNERSMOOTHTRANSITIONBETWEENADJACENTPATHSEGMENTSINNCMACHININGENGINEERINGAPPLICATIONOFTHECUBICTRIGONOMETRICINTERPOLATIONSPLINECURVECOMPAREDWITHTHETRADITIONALFERGUSONCURVE,BEZIERCURVEANDBSPLINECURVE,THEADJUSTABLESHAPECUBICTRIGONOMETRICINTERPOLATIONSPLINECURVEHASBETTERPROPERTIESITNOTONLYHASTHEPROPERTIESOFINTERPOLATIONANDSHAPEADJUSTABLE,BUTALSOBECAUSETHEEXPRESSIONOFTHESPLINECURVECONTAINSTRIGONOMETRICPOLYNOMIAL,BYCHOOSINGPROPERCONTROLPOINTSANDSHAPEADJUSTMENTCOEFFICIENTS,THESPLINECURVECANACCURATELYDESCRIBETHECOMMONENGINEERINGCURVESSUCHASSTRAIGHTLINE,CIRCULARARCANDFREECURVE,ETCTHEREFORE,THESPLINECURVECANBEUSEDTODESCRIBETHECOMPLEXNCMOTIONPATHSMEANWHILE,THROUGHSELECTINGPROPERCONTROLPOINTSNEARTHECORNEROFADJACENTMACHININGPATHSEGMENTSANDSHAPEADJUSTMENTCOEFFICIENTS,ITCANGENERATETHERINGFORMORCIRCULARARCSPLINECURVEWHICHCANREALIZETHEHIGHSPEEDSMOOTHTRANSITIONAROUNDTHECORNERSTRAIGHTLINEDESCRIPTIONOFNCMOTIONPATHSUPPOSETHETWOENDPOINTSOFALINESEGMENTAREBANDC,TAKINGTHEFOURCONTROLPOINTSASSEQUENCEASB,C,B,C,THENASTRAIGHTLINEFROMBTOCCANBECONSTRUCTEDASSHOWNINFIG3ANALYZEFORMULA7,WHEN22IIIIXXYY,3131IIIIXXYY,0AND1K,FORMULA7TURNSINTOTHELINEARPARAMETEREQUATION311311IIIIIIXXXCXYYYCY8INFORMULACISSTANDFORCOS2U,01UCIRCULARARCDESCRIPTIONOFNCMOTIONPATHUSINGTHESPLINECURVEBYTAKINGFOURSPECIALCONTROLPOINTSANDSHAPEADJUSTMENTCOEFFICIENTSCANACCURATELYDESCRIBETHECIRCULARARCSUPPOSETHETWOENDPOINTSOFACIRCULARARCSEGMENTAREAANDB,TAKINGFOURCONTINUOUSCONTROLPOINTSAS,0AA,0,BA,2CAAAND2,DAA,SHAPEADJUSTMENTCOEFFICIENTSAS1,1K,THENACIRCULARARCFROMATOBCANBECONSTRUCTEDPUTTHEABOVEFOURCONTROLPOINTSCOORDINATESANDSHAPEADJUSTMENTCOEFFICIENTSINTOFORMULA7,THEPARAMETRICEQUATIONOFCIRCULARARCCANBEGOTTENXACYASII9INFORMULASISSTANDFORSIN2U,CISSTANDFORCOS2U,01UFIG4SHOWSTHECIRCULARARCSEGMENTWITHCENTERANGLEOF90,WHICHCORRESPONDINGTOTHECHANGEOFPARAMETERUFROM0TO1WHENUSINGTHESPLINECURVETODESCRIBECIRCULARARCWITHCENTERANGLELESSTHAN90,ITCANBEDONETHROUGHTHECONTROLOFTHEVALUEOFPARAMETERUANDWHENTODESCRIBECIRCULARARCWITHCENTERANGLEMORETHAN90,USINGAFEWCIRCULARARCSEGMENTSJOININGTOGETHERTOREALIZEFREECURVEDESCRIPTIONOFNCMOTIONPATHUSINGTHISCUBICTRIGONOMETRICINTERPOLATIONSPLINECANEASILYDESCRIBETHEFREECURVEFROMABOVEWEKNOWTHATNGIVENCONTROLPOINTSCANCONSTRUCT3NSEGMENTSOFSPLINECURVESASSHOWNINFIG5,THROUGHA,B,C,D,E,F,G,H,I,JTENCONTROLPOINTSCANGENERATEASEGMENTOFFREECURVEWHICHFORMEDBYSEVENSEGMENTSOFSMALLSPLINECURVESANDTHEWHOLEFREECURVESATISFIES1CCONTINUITY,WHICHENSURESTHESMOOTHNESSOFTHESPLINECURVE518MACHINEDESIGNANDMANUFACTURINGENGINEERINGIIFIG4CIRCULARARCINTERPOLATIONSPLINEFIG5FREECURVEINTERPOLATIONSPLINEPLANNINGOFNCMOTIONPATHFIG6SHOWSAPARTOFTHECURRENTCOMMONLYUSEDMOTIONPATHINNCMACHININGTHEMOVINGPATHOFTOOLCENTERISABCDDURINGTHEMOTIONPATH,POINTSBANDCARESHARPTURNINGPOINTSWHENTHECORNERANGLEFORMEDBYADJACENTMACHININGPATHSEGMENTSISLARGER,ITWILLCAUSEOVERSIZEDCHANGESOFMOVEMENTVELOCITYVECTOR,WHICHWILLBEEASIERTOCAUSELARGERIMPACTSTONCMACHINETOOLSTHISPHENOMENONISESPECIALLYMOREOBVIOUSUNDERTHECIRCUMSTANCEOFHIGHSPEEDMACHININGACCORDINGTOTHEPROPERTIESOFTHESPLINECURVE,INSERTINGTHESPLINEATTHECORNEROFADJACENTMACHININGPATHSEGMENTS,CANGENERATETHETRAJECTORIESOFCORNERTRANSITIONANDREALIZETHESMOOTHTRANSFERBETWEENMACHININGPATHSEGMENTS,SOASTOACHIEVETHEPURPOSEOFSMOOTHINGCORNERTRANSITIONSPEEDFIG6NCMOTIONPATHBEFORESMOOTHINGFIG7NCMOTIONPATHAFTERSMOOTHINGFIG7SHOWSTHEDESCRIBEDNCMOTIONPATHBASEDONTHEADJUSTABLESHARPCUBICTRIGONOMETRICINTERPOLATIONSPLINECURVEAMONGTHEM,POINTS1CAND2CARETHEINTERSECTIONSOFTOOLOUTERCONTOURLINEWITHSEGMENTSBCANDCDWHENTHETOOLCENTERISATPOINTCSEGMENTSAB,1BC,AND2CDARESTRAIGHTLINES,INSERTARINGFORMSPLINETRANSITIONCURVEOUTSIDETHECORNERPOINTBANDCIRCULARARCSPLINETRANSITIONCURVEINSIDETHECORNERPOINTCSUCHPLANNEDNCMACHININGMOTIONPATHISSMOOTH,ANDTHEREISNOSPEEDDROPANDRISEDURINGTHECORNERTRANSITION,WHICHCANGUARANTEETHESMOOTHTRANSFERAROUNDTHECORNER,ANDEASYTOREALIZETHEHIGHSPEEDSMOOTHMACHININGBETWEENNCMOTIONPATHSEGMENTSSMOOTHINGOFEXTERNALCORNERASSHOWNINFIG8,INORDERTOSMOOTHTRANSFERTHEMACHININGSEGMENTSBETWEENMLANDNL,ARINGFORMSPLINECURVEISINSERTEDATTHEEXTERNALCORNERPOINTB,WHICHISGENERATEDBYTHEFOURCONTROLPOINTSWITHSEQUENCEASB,B,1B,2BANDSHAPEADJUSTMENTCOEFFICIENTAS1KAMONGTHEM,POINT1BISONTHEEXTENSIONCORDOFSEGMENTMLANDPOINT2BISONTHESEGMENTNLSELECTINGTHECONTROLPOINTSASTHISWAYCANENSURETHATTHESTARTINGPOINTVECTOROFTHESPLINETRANSFERCURVEISINTHESAMEDIRECTIONWITHSEGMENTML,ANDTHEENDPOINTVECTORISINTHESAMEDIRECTIONWITHSEGMENTNL,WHICHCANREALIZETHEEXTERNALCORNERSMOOTHTRANSITIONBETWEENNCMOTIONPATHSEGMENTSCHANGETHEVALUEOFSHAPEADJUSTMENTCOEFFICIENTCANGENERATEDIFFERENTSIZESOFSPLINECURVES,SOTHROUGHCHOOSINGPROPERVALUEOFCANCONTROLTHECHANGEOFACCELERATIONDURINGTHECORNERMOVEMENTPROCESSAPPLIEDMECHANICSANDMATERIALSVOLS365366519FIG8EXTERNALCORNERSMOOTHTRANSITIONFIG9INTERNALCORNERSMOOTHTRANSITIONSMOOTHINGOFINTERNALCORNERASSHOWNINFIG9,ACIRCULARARCSPLINECURVEISINSERTEDATTHEINTERNALCORNERPOINTCTOSMOOTHTRANSFERTHESEGMENTSBETWEENNLANDOLTHEFOURCONTINUOUSCONTROLPOINTSARE1C,2C,3C,4CANDSHAPEADJUSTMENTCOEFFICIENT1KAMONGTHEM,POINTS1CAND2CARETHEINTERSECTIONSOFTOOLOUTERCONTOURLINEWITHSEGMENTSNLANDOLWHENTHETOOLCENTERISATPOINTCPOINT3CISONTHEEXTENSIONCORDOFSEGMENTNL,POINT4CISONTHESEGMENTOL,ANDTHEIRCOORDINATESAREDETERMINEDBYPOINTS1CAND2CSIMILARLY,THESTARTINGPOINTVECTOROFTHISCURVEISINTHESAMEDIRECTIONWITHSEGMENTNLANDTHEENDPOINTVECTORISINTHESAMEDIRECTIONWITHSEGMENTOL,WHICHCANREALIZETHEINTERNALCORNERSMOOTHTRANSITIONOFNCMOTIONPATHSEGMENTSCHANGETHEVALUEOFSHAPEADJUSTMENTCOEFFICIENTCANCHANGETHESIZEOFTHETRANSFERCURVES,SOTHROUGHCHOOSINGPROPERVALUEOFCANCONTROLTHECHANGEOFACCELERATIONDURINGTHECORNERMOVEMENTPROCESS,MEANWHILE,ITALSOCANCONTROLTHEMACHININGERRORCONCLUSIONTHISPAPERGIVESOUTASETOFSPECIALBASISFUNCTION,ANDTHECONSTRUCTEDCURVEBASEDONITISCALLEDCUBICTRIGONOMETRICINTERPOLATIONSPLINECURVEWHICHSATISFIES1CCONTINUITYTHECURVEHASTHEFOLLOWINGADVANTAGES1CHANGINGONECONTROLPOINTONLYAFFECTSFOURSEGMENTCURVESWHICHARERELATEDTOIT,ANDHASNOEFFECTSTOOTHERPARTSOFTHEWHOLECURVE,SOTHECURVEHASGOODPROPERTYOFLOCALITY2EACHSEGMENTOFSPLINEHASTWOSHAPEADJUSTMENTCOEFFICIENTSANDK,ANDTHESHAPEOFTHECURVECANBEEASILYCONTROLLEDBYCHANGINGTHEVALUESOFANDK,SOTHECURVEHASTHEPROPERTYOFSHAPEADJUSTABLE3SELECTINGPROPERCONTROLPOINTSANDSHAPEADJUSTMENTCOEFFICIENTS,THESPLINECANPRECISELYDESCRIBESOMECOMMONENGINEERINGCURVESSUCHASSTRAIGHTLINE,CIRCULARARCANDFREECURVE,SOTHESPLINEHASTHEFLEXIBILITYOFCURVESTRUCTURING4THESPLINEUSINGTHEFORMOFPOLYNOMIALEXPRESSIONTOAVOIDTHERATIONALFORM,SOITNEEDSFEWERAMOUNTSOFCALCULATIONANDSTORAGESPACETHEREFORE,USINGTHISSPLINETODESCRIBETHECOMPLEXNCMOTIONPATHSHASOBVIOUSADVANTAGES,ANDITALSOCANBEUSEDTODEALWITHTHECORNERSMOOTHTRANSITIONBETWEENADJACENTNCMOTIONPATHSEGMENTSBYINSERTINGTHEADJUSTABLESHARPSPLINEATTHECORNERCANEASILYREALIZETHEHIGHSPEEDSMOOTHTRANSITIONBETWEENNCPATHSEGMENTS,WHICHCANMEETTHENEEDSOFMODERNNUMERICALCONTROLSYSTEMFORHIGHSPEED,STABILITYANDFLEXIBILITYREFERENCES1ERKORKMAZK,ALTINATASYHIGHSPEEDCN