CoordinateTransformationand.ResolvedMotionRateControlArobotismadeupoflinksconnectedbyjoints.Inkinematicanalysis,itisconsideredtobeachainoflinksandjoints.Atoneendofthechainisthesupportingbase；attheotherendistheendeffectororhand.Controllingarobotrequiresthattheendeffectororhandbemovedtoaspecificpointinspacetocarryoutatask.Inperformingthetask,therobotsendeffectormustmovethroughaparticulardesignatedpath.Thissectiondiscussesasimplifiedmathematicalmethodthatisusefulindescribingtherelationshipoftheendeffectortotherobotbasecoordinates.Boththepositioninspacendtheorientationordirectionoftherobotsendeffectormustbedescribedandcontrolled.Determiningthepositionandorientationofanyjointinarobotrelativetoitsbasecoordinatesetrequiresthetransformationofcoordinatesthroughallotherjointsbetweenthebasereferenceandthejointwhosecoordinatesarebeingdetermined.Iftherobothassixjoints,ordegreesoffreedom,itisnecessarytosetupsixcoordinatetransfers,oneforeachjoint.Eachtransformationrelatesthecoordinatesofonejointtothecoordinatesofthepreviousjointinthechainoflinksandjoints.1.Joint-to-WorldTransformationsWorldcoordinatesaredefinedasthebasereferencecoordinatesoftherobot.Thesecoordinatesaretakenthroughthebasejointoftherobotorataknowndistancefromit.Basecoordinates,byconvention,aredefinedasx0,y0,andz0inthe0coordinateframe.Jointcoordinatesaredefinedasthesetofcoordinatescenteredonaparticularjint.Inaslidingorprismaticjoint,onecoordinateofthecoordinatesetisalongthedirectionofmotion.Inarotaryorrevolutejoint,onecoordinateisparalleltotheaxisofthejoint.Figure6.23showstherelationshipbetweenthesuccessivejointsinarobot.JointI-1couldbetakenasthebasereferencejointbecauseitcouldbeanyjoint.Thenextjointtowardtheendeffector(alongthechainoflinks)hasthenumber1assigned；eachsuccessivejointhasanassignednumberidentificationthatisonegreaterthanthe.precedingjoint.Linksbetweenjointsareassignedinasimilarway.TheseassignmentsareshowninFigure6.23.Inthisparticularcase,aCartesian(x,y,z)systemisused.IfwechoosetosetIequalto1,thefirstlinkislink1,andthefirstjointhasthecoordinatesofx0,y0,z0andcanbeconsideredfixedinthereferencebase.2.CoordinateFrameParametersEachcoordinatesetisalsocalledaframe.Everyframeisdeterminedbyfourparametersthatdescribehowitrelatestoapreviousframe.ThismathematicalapproachwasfirstdevelopedbyDenavitandHartenbergin1955.Itprovidesthenecessaryparametersinhomogeneousmatricestoperformthetransformationsbetweencoordinatesystemsinaremarkablysimpleway(DenavitandHartenberg10,asdescribedinSection6.6.2.Therearetwodistancesandtwoanglesineachsetoffourparameters.Theseparametersarecalledthetasub-i(Os)ssub-i(s1),sub-i(a1),andalphasub-i(a1),asshowninFigure6.23.Framesareorientedanddeterminedbytworules:1.Thez1axisliesalongtheaxisofmotionofthe/thjoint.2.TheX,axisisnormal(perpendicular)tothe,axisandispointingawayfromit.Inacartesiancoordinatesystem,thexaxisisalwaysnormal(perpendicular)tothezaxis.Thus,rule2abovemeansthatthex1axisisnormaltoboththez,axisandtheZaxisandgoesfromthezi_1axistotheZ1axis.Traceoutthisrelationshipinfigure6.23tomakesurethatthispointisclear.Notethatthecoordinatesystemisattachedtothecorrespondinglinkeventhoughitszaxisdirectionisdeterminedbythedirectionofthejoint.Theithframemoveswiththeithlink；thenthframe(assignedtothehandorendeffector)moveswiththehand.Therearen+1framesonannjointarm.Thefourparametersforeachithjointaredefinedasfollows(checkeachoftheseinFigure6.23):1.Theparameter0,istheanglefromtheaxistothex1axismeasuredaroundthe,axisgoinginthepositivedirectionspecifiedbytheright-handrule.Theright-handrulespecifiesthatmovingfromthexaxistotheyaxisaroundzintheclockwisedirection(thedirectionofadvanceofaright-handedscrew)willbepositive.Movingfromytozclockwisearoundxisalsopositive,asisthemovementfromztoxclockwisearoundy.Goinginthecounterclockwisedirectiongivesanegativevaluefortheangle.2.TheparameterS,isthedistancefromtheoriginofthe(I-1)thframetotheintersectionoftheaxiswiththex1axis.Thisdistanceismeasuredalongthez,_1axis.3.Theparametera1istheshortestdistancefromtheZ,axistothez,axis.NotethatthisdistanceismeasuredalongalinewhichmakesanangleOwiththeXaxis.Thetwoslashes(1/)throughthesetwolinesinfigure6.23indicatethattheyareparallel.Ifthecenterofthecoordinatesforthez.axisliesonthez1_axis,whichisoftentrue,thevalueofa1iszero.IntheStanfordarm,forexample,illustratedinFigure6.24allthea1valuesarezero.4.Theparameteraistheoffsetanglefromthez,.axistotheziaxisaboutthex,axis,againusingtheright-handrule.Threebackslashes()oneachlineareusedtoindicatethatthereferencelineforcrsparalleltothezaxis.Inthesimplerobotconfigurationsthatwewillconsider,thedistancesbetweenaxesmaybecomezero.Inmostcasesalso,theangleswillbeeitherzeroormultiplesof90degrees.Anypointintheithcoordinateframe(x,y,z1)canbetransformedtothe(i-1)thsystembyusingthehomogeneousmatrixdefinedbytheparameterspreviouslydeterminedinthematrixequationwhereA(,_1)iSthe4X4homogeneousmatrixderivedfromthevaluesofthefourjointparameters.Multiplyingthecoordinatevectorofthe(I-1)thframebytheA(l_1)matrixgeneratesthecoordinatevectoroftheithframe.ThevalueofthegeneralizedAmatrixisgivenbyPaul321asTransformationofhandcoordinatestoworldcoordinatescanbedonebysuccessivelymultiplyingtogethertheindividualhomogeneousmatrices,theAmatrices,representingthetransformationsbetweencoordinateframe.TheseAmatricesareobtainedbysubstitutingintheprecedingmatrixthevaluesfoundforthefourparametersinTable6.5.TheTmatrixisthetotaltransformationmatrixandisdefinedbywherethesubscri