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MoredetailsandexamplesonrobotarmsandkinematicsDenavit-HartenbergNotationINTRODUCTIONForwardKinematics:todetermin

ewheretherobot’shandis?(Ifalljointvariablesareknown)InverseKinematics:tocalculatewhateachjointvariableis?(Ifwedesirethatthehan

dbelocatedataparticularpoint)DirectKinematicsDirectKinematicswithnomatricesWhereismyhand?DirectKinematics:HERE!DirectKinematics

•Positionoftipin(x,y)coordinatesDirectKinematicsAlgorithm1)Drawsketch2)Numberlinks.Base=0,Lastlink=n3)Identifyandnumberrobotjoints4)Drawax

isZiforjointi5)Determinejointlengthai-1betweenZi-1andZi6)DrawaxisXi-17)Determinejointtwisti-1measuredaroun

dXi-18)Determinethejointoffsetdi9)DeterminejointangleiaroundZi10+11)WritelinktransformationandconcatenateOftens

ufficientfor2DKinematicProblemsforManipulation•Reliablypositionthetip-gofromonepositiontoanotherposition•D

on’thitanything,avoidobstacles•Makesmoothmotions–atreasonablespeedsand–atreasonableaccelerations•Adjusttochangingconditions-–i.

e.whensomethingispickeduprespondtothechangeinweightROBOTSASMECHANISMsRobotKinematics:ROBOTSASMECHANISMFig.2.1Aone-

degree-of-freedomclosed-loopfour-barmechanismMultipletyperobothavemultipleDOF.(3Dimensional,openloop,chainmechanisms)Fig.2.2

(a)Closed-loopversus(b)open-loopmechanismChapter2RobotKinematics:PositionAnalysisFig.2.3Representationofapointinspa

ceApointPinspace:3coordinatesrelativetoareferenceframe^^^kcjbiaPzyxRepresentationofaPointinSpaceC

hapter2RobotKinematics:PositionAnalysisFig.2.4RepresentationofavectorinspaceAVectorPinspace:3coordinatesofitstailandofitshe

ad^^^__kcjbiaPzyxwzyxP__RepresentationofaVectorinSpaceChapter2RobotKinematics:PositionAnalysisFig.2.5Representationofaframeattheori

ginofthereferenceframeEachUnitVectorismutuallyperpendicular.:normal,orientation,approachvectorzzzyyy

xxxaonaonaonFRepresentationofaFrameattheOriginofaFixed-ReferenceFrameChapter2RobotKinematics:PositionAnalysisFig.2.6

RepresentationofaframeinaframeEachUnitVectorismutuallyperpendicular.:normal,orientation,approachvector

1000zzzzyyyyxxxxPaonPaonPaonFRepresentationofaFrameinaFixedReferenceFrameThesameaslastslideChapter2RobotKinematics:PositionAnalysisFig.2.8Rep

resentationofanobjectinspaceAnobjectcanberepresentedinspacebyattachingaframetoitandrepresentingtheframeinspace.100

0zzzzyyyyxxxxobjectPaonPaonPaonFRepresentationofaRigidBodyChapter2RobotKinematics:PositionAnalysisAtransformationmatricesmust

beinsquareform.•Itismucheasiertocalculatetheinverseofsquarematrices.•Tomultiplytwomatrices,theirdimensionsmustmatch.1000zzzzy

yyyxxxxPaonPaonPaonFHOMOGENEOUSTRANSFORMATIONMATRICESRepresentationofTransformationsofrigidobjectsin3D

spaceChapter2RobotKinematics:PositionAnalysisFig.2.9RepresentationofanpuretranslationinspaceAtransformationisdefinedasmakingamovementinspac

e.•Apuretranslation.•Apurerotationaboutanaxis.•Acombinationoftranslationorrotations.1000100010001zyxdddTRepresentationofaPureTransl

ationidentitySamevalueaChapter2RobotKinematics:PositionAnalysisFig.2.10Coordinatesofapointinarotatingframebeforeandafterrotationaroundaxisx.Ass

umption:Theframeisattheoriginofthereferenceframeandparalleltoit.Fig.2.11Coordinatesofapointrelativetothereferenceframeandrotatingframeasview

edfromthex-axis.RepresentationofaPureRotationaboutanAxisProjectionsasseenfromxaxisx,y,zn,o,aFig.2.13

EffectsofthreesuccessivetransformationsAnumberofsuccessivetranslationsandrotations….RepresentationofCombin

edTransformationsOrderisimportantx,y,zn,o,anioiaiT1T2T3Fig.2.14Changingtheorderoftransformationswillchangethefinal

resultOrderofTransformationsisimportantx,y,zn,o,atranslationChapter2RobotKinematics:PositionAnalysisFig.

2.15Transformationsrelativetothecurrentframes.Example2.8TransformationsRelativetotheRotatingFrametranslationrotationMATRICESFORFORWARD

ANDINVERSEKINEMATICSOFROBOTS•Forposition•FororientationChapter2RobotKinematics:PositionAnalysisFig.2.17Theh

andframeoftherobotrelativetothereferenceframe.ForwardKinematicsAnalysis:•Calculatingthepositionandorientationoft

hehandoftherobot.Ifallrobotjointvariablesareknown,onecancalculatewheretherobotisatanyinstant..FORWAR

DANDINVERSEKINEMATICSOFROBOTSChapter2RobotKinematics:PositionAnalysisForwardKinematicsandInverseKinematicsequationforpositionanalysis:(a)Cartes

ian(gantry,rectangular)coordinates.(b)Cylindricalcoordinates.(c)Sphericalcoordinates.(d)Articulated(anthropomor

phic,orall-revolute)coordinates.ForwardandInverseKinematicsEquationsforPositionChapter2RobotKinematics:PositionAnalys

isIBM7565robot•Allactuatorislinear.•AgantryrobotisaCartesianrobot.Fig.2.18CartesianCoordinates.1000100010001

zyxcartPRPPPTTForwardandInverseKinematicsEquationsforPosition(a)Cartesian(Gantry,Rectangular)CoordinatesChapt

er2RobotKinematics:PositionAnalysis2Lineartranslationsand1rotation•translationofralongthex-axis•rotationofaboutthez-axis•translationoflalongthe

z-axisFig.2.19CylindricalCoordinates.100010000lrSCSrCSCTTcylPR,0,0))Trans(,)Rot(Trans(0,0,),,(rzllrTTcylPRForwa

rdandInverseKinematicsEquationsforPosition:CylindricalCoordinatescosinesineChapter2RobotKinematics:Po

sitionAnalysis2Lineartranslationsand1rotation•translationofralongthez-axis•rotationofaboutthey-axis•rotationofalongthez-axisFig.2.20SphericalC

oordinates.10000rCCSSrSSSCSCCrSCSSCCTTsphPR))Trans()Rot(Rot()(0,0,,,,,yzlrsphPRTTForwa

rdandInverseKinematicsEquationsforPosition(c)SphericalCoordinatesChapter2RobotKinematics:PositionAnalysis3rotations->D

enavit-HartenbergrepresentationFig.2.21ArticulatedCoordinates.ForwardandInverseKinematicsEquationsforPosition(d)ArticulatedCoordinatesCh

apter2RobotKinematics:PositionAnalysisRoll,Pitch,Yaw(RPY)anglesEuleranglesArticulatedjointsForwardandInverseKinematicsEquationsforOrientati

onChapter2RobotKinematics:PositionAnalysisRoll:Rotationofabout-axis(z-axisofthemovingframe)Pitch:Rotationofab

out-axis(y-axisofthemovingframe)Yaw:Rotationofabout-axis(x-axisofthemovingframe)aaononFig.2.22RPYro

tationsaboutthecurrentaxes.ForwardandInverseKinematicsEquationsforOrientation(a)Roll,Pitch,Yaw(RPY)Angle

sChapter2RobotKinematics:PositionAnalysisFig.2.24Eulerrotationsaboutthecurrentaxes.Rotationofabout-axis(z-ax

isofthemovingframe)followedbyRotationofabout-axis(y-axisofthemovingframe)followedbyRotationofabout-axis(z-axisofthemovingframe).aoaForwardandI

nverseKinematicsEquationsforOrientation(b)EulerAnglesChapter2RobotKinematics:PositionAnalysis)()(,,,,noazyxcartHRRPYPPPTT)()(,,

,,EulerTTrsphHRAssumption:RobotismadeofaCartesianandanRPYsetofjoints.Assumption:RobotismadeofaSphericalCoordinateandan

Eulerangle.AnotherCombinationcanbepossible……Denavit-HartenbergRepresentationForwardandInverseKinematicsEquationsforOrienta

tionRoll,Pitch,Yaw(RPY)AnglesForwardandInverseTransformationsforrobotarmsFig.2.16TheUniverse,robot,hand,part,ande

ndeffecterframes.StepsofcalculationofanInversematrix:1.Calculatethedeterminantofthematrix.2.Transposethematrix.3.Replaceeachelementofthet

ransposedmatrixbyitsownminor(adjointmatrix).4.Dividetheconvertedmatrixbythedeterminant.INVERSEOFTRANSFORMATIONMATRIC

ESIdentityTransformations1.WeoftenneedtocalculateINVERSEMATRICES2.Itisgoodtoreducethenumberofsuchoperations3.Weneedtodothesecalculations

fastHowtofindanInverseMatrixBofmatrixA?InverseHomogeneousTransformationHomogeneousCoordinates•Homogeneouscoordinates:embed3

Dvectorsinto4Dbyaddinga“1”•Moregenerally,thetransformationmatrixThastheform:FactorScalingTrans.Perspect.VectorTrans.MatrixRot.Ta11a12a13b1a2

1a22a23b2a31a32a33b3c1c2c3sfItispresentedinmoredetailontheWWW!ForvarioustypesofrobotswehavedifferentmaneuveringspacesForvari

oustypesofrobotswecalculatedifferentforwardandinversetransformationsForvarioustypesofrobotswesolvedifferentforwardandinversekinematicpr

oblemsForwardandInverseKinematics:SingleLinkExampleForwardandInverseKinematics:SingleLinkExampleeasyDenavit–Hart

enbergideaDenavit-HartenbergRepresentation:Fig.2.25AD-Hrepresentationofageneral-purposejoint-linkcombination@Simp

lewayofmodelingrobotlinksandjointsforanyrobotconfiguration,regardlessofitssequenceorcomplexity.@Transform

ationsinanycoordinatesispossible.@Anypossiblecombinationsofjointsandlinksandall-revolutearticulatedrobots

canberepresented.DENAVIT-HARTENBERGREPRESENTATIONOFFORWARDKINEMATICEQUATIONSOFROBOTChapter2RobotKinematics:PositionAnalysis⊙:Arotationanglebetw

eentwolinks,aboutthez-axis(revolute).⊙d:Thedistance(offset)onthez-axis,betweenlinks(prismatic).⊙a:Thelengtho

feachcommonnormal(Jointoffset).⊙:The“twist”anglebetweentwosuccessivez-axes(Jointtwist)(revolute)Onlyanddarejointvariables.DENAVIT-HARTENBERG

REPRESENTATIONSymbolTerminologies:Linksarein3D,anyshapeassociatedwithZialwaysOnlyrotationOnlytranslationOn

lyoffsetOnlyoffsetOnlyrotationAxisalignmentDENAVIT-HARTENBERGREPRESENTATIONforeachlink4linkparametersChapter2RobotKinematics:PositionAnalysis⊙

:Arotationanglebetweentwolinks,aboutthez-axis(revolute).⊙d:Thedistance(offset)onthez-axis,betweenlinks

(prismatic).⊙a:Thelengthofeachcommonnormal(Jointoffset).⊙:The“twist”anglebetweentwosuccessivez-axes(

Jointtwist)(revolute)Onlyanddarejointvariables.DENAVIT-HARTENBERGREPRESENTATIONSymbolTerminologies:Examplewit

hthreeRevoluteJointsi(i-1)a(i-1)dii0000010a0012-90a1d22Z0X0Y0Z1X2Y1X1Y2d2a0a1Denavit-HartenbergLinkParameterTableTheDH

ParameterTableApplyfirstApplylastDenavit-HartenbergRepresentationofJoint-Link-JointTransformationNotationf

orDenavit-HartenbergRepresentationofJoint-Link-JointTransformationAlphaappliedfirstFourTransformationsfromoneJointtotheNext•Orderofmultipli

cationofmatricesisinverseoforderofapplyingthem•HereweshoworderofmatricesJoint-Link-JointDenavit-HartenbergRepresentation

ofJoint-Link-JointTransformation•AlphaisappliedfirstHowtocreateasinglematrixAnEXAMPLE:Denavit-HartenbergRepresentationofJoint-Li

nk-JointTransformationforType1LinkFinalmatrixfrompreviousslidesubstitutesubstituteNumericorsymbolicmatricesTheDenavit-HartenbergMatrixforanoth

erlinktype1000cosαcosαsinαcosθsinαsinθsinαsinαcosαcosθcosαsinθ0sinθcosθi1)(i1)(i1)(ii1)(iii1)(i1)(i1)(ii1)(ii1)(iii

dda•SimilaritytoHomegeneous:JustliketheHomogeneousMatrix,theDenavit-HartenbergMatrixisatransformationmatrixfromonecoordinateframetot

henext.•UsingaseriesofD-HMatrixmultiplicationsandtheD-HParametertable,thefinalresultisatransformationmatrixfromso

meframetoyourinitialframe.Z(i-1)X(i-1)Y(i-1)(i-1)a(i-1)ZiYiXiaidiiPutthetransformationhereforeverylink1.InDENAVIT-HARTENBERGREPRESENTATIONwemustbeab

letofindparametersforeachlink2.SowemustknowlinktypesLinksbetweenrevolutejointsln=0Type3LinkJointn+1Jointndn=

0Linknxn-1xnln=0dn=0Type4LinkOriginscoinciden-1Jointn+1JointnPartofdn-1Linknxn-1yn-1xnnLinksbetweenprismaticjoin

tsForwardandInverseTransformationsonMatricesStartpoint:•Assignjointnumberntothefirstshownjoint.•Assignaloca

lreferenceframeforeachandeveryjointbeforeorafterthesejoints.•Y-axisisnotusedinD-Hrepresentation.DENAVIT-HARTENBERGREPRESENTATIONPROCED

URES.1٭Alljointsarerepresentedbyaz-axis.•(right-handruleforrotationaljoint,linearmovementforprismaticjoint)2.Thecommon

normalisonelinemutuallyperpendiculartoanytwoskewlines.3.Parallelz-axesjointsmakeainfinitenumberofcommonnormal.4.Intersecting

z-axesoftwosuccessivejointsmakenocommonnormalbetweenthem(Lengthis0.).DENAVIT-HARTENBERGREPRESENTATIONProcedu

resforassigningalocalreferenceframetoeachjoint:Chapter2RobotKinematics:PositionAnalysis⊙:Arotationaboutthez-axis.⊙d:Thedistan

ceonthez-axis.⊙a:Thelengthofeachcommonnormal(Jointoffset).⊙:Theanglebetweentwosuccessivez-axes(Jointtwist)Onlyanddarejointva

riables.DENAVIT-HARTENBERGREPRESENTATIONSymbolTerminologiesReminder:Chapter2RobotKinematics:PositionAnalysis(I)Rotateab

outthezn-axisanableofn+1.(Coplanar)(II)Translatealongzn-axisadistanceofdn+1tomakexnandxn+1colinear.(III)Translatealongthexn-axisadistanceofan+1tobri

ngtheoriginsofxn+1together.(IV)Rotatezn-axisaboutxn+1axisanangleofn+1toalignzn-axiswithzn+1-axis.DENAVIT-HARTENBERGRE

PRESENTATIONThenecessarymotionstotransformfromonereferenceframetothenext.