机器人学导论(英) 第三讲

Review•Point•Vector•Matrix•Calculus•Linearalgebra•Dotproduct•Crossproduct•Position/Orientation•Rotationmatrix•DirectioncosineReview•Coordinatesystem•Frame•Mapping•Translation•Rotation•Homogeneoustransorm•Crossproduct•Operator•Translationaloperator•RotationaloperatorReview•Transformationoperator•Compoundtransformation•InvertingatransformChapter3ManipulatorKinematicsCh3.1Introduction•Whatiskinematics?Kinematicsisthesciencesofmotionthattreatsthesubjectwithoutregardtotheforcesthatcauseit.•Example:Withtheknowledgeofmanipulator’slinklengthandjointangles,howtocomputethepositionandorientationofthemanipulatorsInkinematics,weconsiderabouttheposition,thevelocity,theacceleration,andthehigherderivativesoftheposition.Thestudyofkinematicsreferstoallthegeometricalandtime-basedpropertiesofthemotion.•ForwardkinematicsGiventhejointvariablesoftherobot,determinethepositionandorientationoftheend-effector.[]12...TnθθθΘ=),,,,,(αβγzyxY=Ch3.1IntroductionCh3.2LinkDescription•KinematicChainArobotcanbetreatedasasetofrightbodies(rigidlinks)connectedtogetheratvariousjoints.Ch3.2LinkDescription•Lowerpair:describetheconnectionbetweenapairofbodieswhenrelativemotionischaracterizedbytwosurfaceslidingoveronanother.•Jointtype(lowerpair):1.Revolute2.Prismatic3.Cylindrical4.Planar5.Screw6.Spherical•Weconsiderthejointwith1degree-of-freedom(DOF).AjointwithmDOFcanbemodeledasmjointsof1DOFconnectedbym-1linksofzerolength.Ch3.2LinkDescription•Theactionofjointcanbedescribedbyasinglerealnumber:theangleofrotationinthecaseofarevolutejoint,orthedisplacementinthecaseofaprismaticjoint.Ch3.2LinkDescription•Numberofthelinksstartsfromtheimmobilebase(link0),thefirstmovingbodyislink1andsoon,outtothefreedendofthearm,whichislinkn.…link1link2link0link6Ch3.2LinkDescription•Inkinematics,alinkisconsideredasarigidbodythatdefinetherelationshipbetweentwoneighboringjointaxes.1−iaaxisi-1axisilinki-11−iα1−iα:twistangle•Twoparameters:linklengthandlinktwistangleareusedtodefinetherelativelocationofthetwoaxes.Ch3.2LinkDescription•Linklength:thedistancebetweentwoaxesismeasuredalongalinethatismutuallyperpendiculartobothaxes.Thislineisuniqueexceptinthecasethattwoaxesareparallel.axisi-1axisilinki-11−ia1−ia:linklengthCh3.2LinkDescription•Imaginethataplane,whosenormalisthemutuallyperpendicularlinejustconstructed,wecanprojecttheaxisi-1andaxisiontothatplaneandmeasuretheanglefromaxisi-1toaxisibytheright-handrule.axisi-1axisilinkn-11−iα:twistangle1−iaCh3.3Link-ConnectionDescription•Intermediatelinksinthechain:Neighboringlinkshasacommonjointaxisbetweenthem.Thedistancealongthiscommonaxisfromonelinktoanotherisdefinedas“linkoffset”.Itwillbevariableforprismaticjoint.axisi-1axisilinki-11−iαid1−iaiθiaid:linkoffset•Theanglethatrotatesaboutthiscommonaxisbetweenonelinkanditsneighboringlinkisdefinedas“jointangle”.Itwillbevariableforrevolutejoint.Ch3.3Link-ConnectionDescriptionaxisi-1axisilinki-11−iαid1−iaiθiaiθ:jointangleCh3.3Link-ConnectionDescription•Linklengthandlinktwistangledependonjointaxesiandi+1.•Forfirstlinkinthechain,wesetand.00=a00=α•Forlastlinkinthechain,wesetand.0=na0=nα•Ifjoint1isrevolute,issettobe0,zeropositionofwillbechosenarbitrarily.1d1θ•Ifjoint1isprismatic,issettobe0;zeropositionofwillbechosenarbitrarily.1θ1dCh3.3Link-ConnectionDescription•Linkparameters¾Forrevolutejoint,thejointangleiscalled“jointvariable”,andtheotherthreequantitiesarefixedlinkparameters.¾Forprismaticjoint,thelinkoffsetiscalled“jointvariable”,andtheotherthreelinkquantitiesarefixedlinkparameters.θd•Anyrobotcanbedescribedkinematicallybygivingthevaluesoffourquantitiesforeachlink,i.e.,Denavit-Hartenbergnotation[]iiiidaθαvariableconstant•Aframeisattachedtoeachlinktodescribethelocationofalinkrelativetoitsneighbors.•Thelinkframearenamedbynumberaccordingtothelinktowhichtheyareattached.Step1:frame{i}link{i}link1link2link0link6{0}{1}{2}{6}Ch3.4AffixingFramestoLinksCh3.4AffixingFramestoLinks•Step2:intermediatelinkinthechain:iZˆjointaxisiframe{i}axisi-1axisilinki-11−iαid1−iaiθia1ˆ−iZ1ˆ−iX1ˆ−iYiZˆiXˆiYˆlinkiCh3.4AffixingFramestoLinks•Step3:directionof-along,fromjointitojointi+1iXˆiaaxisi-1axisilinki-11−iαid1−iaiθia1ˆ−iZ1ˆ−iX1ˆ−iYiZˆiXˆiYˆlinki•Step4:directionofcanbedeterminedbyright-handruleˆiYCh3.4AffixingFramestoLinks•Howaboutifintersect?isnormaltotheplaneofand.iZˆ1ˆ+iZiXˆiZˆ1ˆ+iZ•FirstlinkinthechainsSelectionofframe{0}isarbitrarily,willbeselectedalong.0ˆZ1ˆZ•LastlinkinthechainsForrevolutejointischosenalongwhen,theoriginofframe{n}ischosentoset.nXˆ0=nd1ˆ−nX0=nθForprismaticjoint,thedirectionofischosetomakeandtheoriginof{n}ischosenattheintersectionofandjointaxisnwhen.0=nθnXˆ1ˆ−nX0=ndCh3.4AffixingFramestoLinksaxisi-1axisilinki-11−iαid1−iaiθia1ˆ−iZ1ˆ−iX1ˆ−iYiZˆiXˆiYˆlinki•Summaryoflinkparameters:thedistancefromtomeasuredalong:theanglefromtomeasuredabout:thedistancefromtomeasuredalong:theanglefromtomeasuredaboutiXˆiaiαiZˆ1ˆ+iZiZˆ1ˆ+iZiXˆidiθ1ˆ−iXiXˆiZˆ1ˆ−iXiXˆiZˆCh3.4AffixingFramestoLinks•Summaryoflink-frameattachmentprocedure:Step1:identifythejoinaxis(revolutejoint/prismaticjoint)Step2:identifythecommonperpendicularbetweentwoneigh