1、CoordinateTransformationand/ResolvedMotionRateControlArobotismadeupoflinksconnectedbyjoints/Inkinematicanalysis/itisconsideredtobeachainoflinksandjoints/Atoneendofthechainisthesupportingbase；attheotherendistheendeffectororhand/Controllingarobotrequiresthattheendeffectororhandbemovedtoaspecificpointi
2、nspacetocarryoutatask/Inperformingthetask/therobotsendeffectormustmovethroughaparticulardesignatedpath/Thissectiondiscussesasimplifiedmathematicalmethodthatisusefulindescribingtherelationshipoftheendeffectortotherobotbasecoordinates/Boththepositioninspacendtheorientationordirectionoftherobotsendeffe
3、ctormustbedescribedandcontrolled/Determiningthepositionandorientationofanyjointinarobotrelativetoitsbasecoordinatesetrequiresthetransformationofcoordinatesthroughallotherjointsbetweenthebasereferenceandthejointwhosecoordinatesarebeingdetermined/Iftherobothassixjoints/ordegreesoffreedom/itisnecessary
4、tosetupsixcoordinatetransfers/oneforeachjoint/Eachtransformationrelatesthecoordinatesofonejointtothecoordinatesofthepreviousjointinthechainoflinksandjoints/1/Joint-to-WorldTransformationsWorldcoordinatesaredefinedasthebasereferencecoordinatesoftherobot/Thesecoordinatesaretakenthroughthebasejointofth
5、erobotorataknowndistancefromit/Basecoordinates/byconvention/aredefinedasx0/y0/andz0inthe0coordinateframe/Jointcoordinatesaredefinedasthesetofcoordinatescenteredonaparticularjint/Inaslidingorprismaticjoint/onecoordinateofthecoordinatesetisalongthedirectionofmotion/Inarotaryorrevolutejoint/onecoordina
6、teisparalleltotheaxisofthejoint/Figure6/23showstherelationshipbetweenthesuccessivejointsinarobot/JointI-1couldbetakenasthebasereferencejointbecauseitcouldbeanyjoint/Thenextjointtowardtheendeffector(alongthechainoflinks)hasthenumber1assigned；eachsuccessivejointhasanassignednumberidentificationthatiso
7、negreaterthanthe/precedingjoint/Linksbetweenjointsareassignedinasimilarway/TheseassignmentsareshowninFigure6/23/Inthisparticularcase/aCartesian(x/y/z)systemisused/IfwechoosetosetIequalto1/thefirstlinkislink1/andthefirstjointhasthecoordinatesofx0/y0/z0andcanbeconsideredfixedinthereferencebase/2/Coord
8、inateFrameParametersEachcoordinatesetisalsocalledaframe/Everyframeisdeterminedbyfourparametersthatdescribehowitrelatestoapreviousframe/ThismathematicalapproachwasfirstdevelopedbyDenavitandHartenbergin1955/Itprovidesthenecessaryparametersinhomogeneousmatricestoperformthetransformationsbetweencoordina
9、tesystemsinaremarkablysimpleway(DenavitandHartenberg10/asdescribedinSection6/6/2/Therearetwodistancesandtwoanglesineachsetoffourparameters/Theseparametersarecalledthetasub-i(Os)ssub-i(s1)/sub-i(a1)/andalphasub-i(a1)/asshowninFigure6/23/Framesareorientedanddeterminedbytworules/1/Thez1axisliesalongthe
10、axisofmotionofthe/thjoint/2/TheX/axisisnormal(perpendicular)tothe/axisandispointingawayfromit/Inacartesiancoordinatesystem/thexaxisisalwaysnormal(perpendicular)tothezaxis/Thus/rule2abovemeansthatthex1axisisnormaltoboththez/axisandtheZaxisandgoesfromthezi_1axistotheZ1axis/Traceoutthisrelationshipinfi
11、gure6/23tomakesurethatthispointisclear/Notethatthecoordinatesystemisattachedtothecorrespondinglinkeventhoughitszaxisdirectionisdeterminedbythedirectionofthejoint/Theithframemoveswiththeithlink；thenthframe(assignedtothehandorendeffector)moveswiththehand/Therearen+1framesonannjointarm/Thefourparameter
12、sforeachithjointaredefinedasfollows(checkeachoftheseinFigure6/23)/1/Theparameter0/istheanglefromtheaxistothex1axismeasuredaroundthe/axisgoinginthepositivedirectionspecifiedbytheright-handrule/Theright-handrulespecifiesthatmovingfromthexaxistotheyaxisaroundzintheclockwisedirection(thedirectionofadvan
13、ceofaright-handedscrew)willbepositive/Movingfromytozclockwisearoundxisalsopositive/asisthemovementfromztoxclockwisearoundy/Goinginthecounterclockwisedirectiongivesanegativevaluefortheangle/2/TheparameterS/isthedistancefromtheoriginofthe(I-1)thframetotheintersectionoftheaxiswiththex1axis/Thisdistance
14、ismeasuredalongthez/_1axis/3/Theparametera1istheshortestdistancefromtheZ/axistothez/axis/NotethatthisdistanceismeasuredalongalinewhichmakesanangleOwiththeXaxis/Thetwoslashes(1/)throughthesetwolinesinfigure6/23indicatethattheyareparallel/Ifthecenterofthecoordinatesforthez/axisliesonthez1_axis/whichis
15、oftentrue/thevalueofa1iszero/IntheStanfordarm/forexample/illustratedinFigure6/24allthea1valuesarezero/4/Theparameteraistheoffsetanglefromthez//axistotheziaxisaboutthex/axis/againusingtheright-handrule/Threebackslashes()oneachlineareusedtoindicatethatthereferencelineforcrsparalleltothezaxis/Inthesimp
16、lerobotconfigurationsthatwewillconsider/thedistancesbetweenaxesmaybecomezero/Inmostcasesalso/theangleswillbeeitherzeroormultiplesof90degrees/Anypointintheithcoordinateframe(x/y/z1)canbetransformedtothe(i-1)thsystembyusingthehomogeneousmatrixdefinedbytheparameterspreviouslydeterminedinthematrixequati
17、onwhereA(/_1)iSthe4X4homogeneousmatrixderivedfromthevaluesofthefourjointparameters/Multiplyingthecoordinatevectorofthe(I-1)thframebytheA(l_1)matrixgeneratesthecoordinatevectoroftheithframe/ThevalueofthegeneralizedAmatrixisgivenbyPaul321asTransformationofhandcoordinatestoworldcoordinatescanbedonebysu
18、ccessivelymultiplyingtogethertheindividualhomogeneousmatrices/theAmatrices/representingthetransformationsbetweencoordinateframe/TheseAmatricesareobtainedbysubstitutingintheprecedingmatrixthevaluesfoundforthefourparametersinTable6/5/TheTmatrixisthetotaltransformationmatrixandisdefinedbywherethesubscr
19、iptsfortheAmatricesindicatetheinitialandfinalcoordinatesystems/respectively/UsingtheTmatrix/wecanconvertthehandcoordinatevectortotheworldcoordinatevector/ThismatrixwasfirstappliedtotheStanfordArm/designedbyV/Sheinmanin1969/TheStanfordArmisshowninFigure6/24/Nodesintheschematicdiagramrepresentthesixjo
20、intsofthearm/Fiveofthejointsarerotary(hingejoints)withanglesofrotationO020405/and06Theremainingjoint/atposition3/hasaprismatic(sliding)jointwithalength/s3/thatisvariable/Notethatthreeoftherotaryjointscoincide/sothatS4andS5arezero/Thesethreejointsactasaball-and-socketjointtosupportthehand/Therearefour
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