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1、Chapter11TrajectoryPlanning:Continuous-PathOperations11.1IntroductionAsafollow-uptoChapter6,wherewestudiedtrajectoryplanningforpick-and-placeoperations(PPO),westudyinthischaptercontinuous-pathoperations.InPPO,thepose,twist,andtwist-rateoftheEEarespecifiedonly
2、atthetwoendsofthetrajectory,thepurposeoftrajectoryplanningthenbeingtoblendthetwoendposeswithasmoothmotion.Whenthisblendingisdoneinthejoint-variablespace,theproblemisstraightforward,asdemonstratedinChap•ter6.ThereareinstancesinwhichtheblendingmustbemadeinCarte
3、sianspace,inwhichadvancednotionsofinterpolationinwhatisknownastheim•agespaceofspatialdisplacements,asintroducedbyRavaniandRoth(1984),areneeded.Theimagespaceofspatialdisplacementsisaprojectivespacewiththreedualdimensions,whichmeansthatapointofthisspaceisspeci•
4、fiedbyfourcoordinates—similartothehomogeneouscoordinatesintroducedinSection2.5—oftheformXi4-e^i,fori=1,2,3,4,whereeisthedualunity,whichhasthepropertythate^=0.Theforegoingcoordinatesarethusdualnumbers,theirpurposebeingtorepresentbothrotationandtranslationinone
5、singlequantity.InfollowingRavaniandRoth'swork,GeandKang(1995)pro•posedaninterpolationschemethatproducescurvesintheimagespacewithsecond-ordergeometriccontinuity,whicharereferredtoasG^curves.Theseinterpolationtechniquesliebeyondthescopeofthebookandwillbeleftasi
6、de.Theinterestedreaderwillfindacomprehensiveandup-to-datereviewofthesetechniquesin(SrinivasanandGe,1997).43011.TrajectoryPlajining:Continuous-PathOperationsThepurposeofthischapteristodevelopmotioninterpolationtechniquesinCartesianspacethatproducesmoothmotions
7、inbothCartesianandjointspaces.MotioninterpolationinjointspacewasdiscussedinChapter6,thepresentchapterbeingdevotedtomotioninterpolationinCartesianspace.Tothisend,weresorttobasicnotionsofdifferentialgeometry.11.2CurveGeometryContinuous-pathroboticsapplicationsa
8、ppearinoperationssuchasarc-welding,flame-cutting,deburring,androuting.Intheseoperations,atoolisrigidlyattachedtotheend-effectorofaroboticmanipulator,thetoolbeingmeanttotraceacontinuousand
