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MoredetailsandexamplesonrobotarmsandkinematicsDenavit-HartenbergNotationINTRODUCTIONForwardKinematics:todeterminewherethe

robot’shandis?(Ifalljointvariablesareknown)InverseKinematics:tocalculatewhateachjointvariableis?(Ifwedesirethatthehandbelocatedataparticularpoint)D

irectKinematicsDirectKinematicswithnomatricesWhereismyhand?DirectKinematics:HERE!DirectKinematics•Positionoftipin(x,y)coo

rdinatesDirectKinematicsAlgorithm1)Drawsketch2)Numberlinks.Base=0,Lastlink=n3)Identifyandnumberrobotjoints4)DrawaxisZiforjointi5)Determinejo

intlengthai-1betweenZi-1andZi6)DrawaxisXi-17)Determinejointtwisti-1measuredaroundXi-18)Determinethejointoffsetdi

9)DeterminejointangleiaroundZi10+11)WritelinktransformationandconcatenateOftensufficientfor2DKinematicProblemsforManipulation•Reliablyposit

ionthetip-gofromonepositiontoanotherposition•Don’thitanything,avoidobstacles•Makesmoothmotions–atreasonablespeedsand–atreasonable

accelerations•Adjusttochangingconditions-–i.e.whensomethingispickeduprespondtothechangeinweightROBOTSASMECHANISMsRobotKinematics:ROBOTSASMECHANISMFig

.2.1Aone-degree-of-freedomclosed-loopfour-barmechanismMultipletyperobothavemultipleDOF.(3Dimensional,openloop,chainmecha

nisms)Fig.2.2(a)Closed-loopversus(b)open-loopmechanismChapter2RobotKinematics:PositionAnalysisFig.2.3Representati

onofapointinspaceApointPinspace:3coordinatesrelativetoareferenceframe^^^kcjbiaPzyxRepresentation

ofaPointinSpaceChapter2RobotKinematics:PositionAnalysisFig.2.4RepresentationofavectorinspaceAVectorPinspace:3coordina

tesofitstailandofitshead^^^__kcjbiaPzyxwzyxP__RepresentationofaVectorinSpaceChapter2Ro

botKinematics:PositionAnalysisFig.2.5RepresentationofaframeattheoriginofthereferenceframeEachUnitVectorismutua

llyperpendicular.:normal,orientation,approachvectorzzzyyyxxxaonaonaonFRepresentationofaFrameattheOriginofaFixed-ReferenceF

rameChapter2RobotKinematics:PositionAnalysisFig.2.6RepresentationofaframeinaframeEachUnitVectorismutuallyperpendicular.:norma

l,orientation,approachvector1000zzzzyyyyxxxxPaonPaonPaonFRepresentationofaFrameinaFixedReferenceFrameThesameasl

astslideChapter2RobotKinematics:PositionAnalysisFig.2.8RepresentationofanobjectinspaceAnobjectcanbe

representedinspacebyattachingaframetoitandrepresentingtheframeinspace.1000zzzzyyyyxxxxobjectPaonPaonPaonF

RepresentationofaRigidBodyChapter2RobotKinematics:PositionAnalysisAtransformationmatricesmustbeinsquareform.•Itismucheasiertocalculatetheinverseofsq

uarematrices.•Tomultiplytwomatrices,theirdimensionsmustmatch.1000zzzzyyyyxxxxPaonPaonPaonFHOM

OGENEOUSTRANSFORMATIONMATRICESRepresentationofTransformationsofrigidobjectsin3DspaceChapter2RobotKinematics:PositionAnalysi

sFig.2.9RepresentationofanpuretranslationinspaceAtransformationisdefinedasmakingamovementinspace.•Apuretranslation.•Apurer

otationaboutanaxis.•Acombinationoftranslationorrotations.1000100010001zyxdddTRepresentationofaPureTranslati

onidentitySamevalueaChapter2RobotKinematics:PositionAnalysisFig.2.10Coordinatesofapointinarotatingframebeforeandafterrotationaroundaxisx.Assumption:

Theframeisattheoriginofthereferenceframeandparalleltoit.Fig.2.11Coordinatesofapointrelativetothereferenceframeandrotatingframeasview

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

reesuccessivetransformationsAnumberofsuccessivetranslationsandrotations….RepresentationofCombinedTransformationsOrder

isimportantx,y,zn,o,anioiaiT1T2T3Fig.2.14ChangingtheorderoftransformationswillchangethefinalresultOrderofTransformationsisimportantx

,y,zn,o,atranslationChapter2RobotKinematics:PositionAnalysisFig.2.15Transformationsrelativetothecurrentframes.Example2.8

TransformationsRelativetotheRotatingFrametranslationrotationMATRICESFORFORWARDANDINVERSEKINEMATICSOFROBOTS•Forposit

ion•FororientationChapter2RobotKinematics:PositionAnalysisFig.2.17Thehandframeoftherobotrelativetotherefe

renceframe.ForwardKinematicsAnalysis:•Calculatingthepositionandorientationofthehandoftherobot.Ifallrobotjointvariablesareknown,onecanca

lculatewheretherobotisatanyinstant..FORWARDANDINVERSEKINEMATICSOFROBOTSChapter2RobotKinematics:PositionAnalysisForwardKinema

ticsandInverseKinematicsequationforpositionanalysis:(a)Cartesian(gantry,rectangular)coordinates.(b)Cylindricalcoordinates.(c)Sph

ericalcoordinates.(d)Articulated(anthropomorphic,orall-revolute)coordinates.ForwardandInverseKinematicsEquationsforPositionChapter2RobotKinematics:P

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

PPPTTForwardandInverseKinematicsEquationsforPosition(a)Cartesian(Gantry,Rectangular)CoordinatesChapter2RobotKinematics:PositionAnalysis

2Lineartranslationsand1rotation•translationofralongthex-axis•rotationofaboutthez-axis•translationoflalongthez-axisF

ig.2.19CylindricalCoordinates.100010000lrSCSrCSCTTcylPR,0,0))Trans(,)Rot(Trans(0,0,),,(rzllrTTcylPRForwardan

dInverseKinematicsEquationsforPosition:CylindricalCoordinatescosinesineChapter2RobotKinematics:Posit

ionAnalysis2Lineartranslationsand1rotation•translationofralongthez-axis•rotationofaboutthey-axis•rotationo

falongthez-axisFig.2.20SphericalCoordinates.10000rCCSSrSSSCSCC

rSCSSCCTTsphPR))Trans()Rot(Rot()(0,0,,,,,yzlrsphPRTTForwardandInverseKinematicsEquationsforPosition(c)Spheri

calCoordinatesChapter2RobotKinematics:PositionAnalysis3rotations->Denavit-HartenbergrepresentationFig.2.21A

rticulatedCoordinates.ForwardandInverseKinematicsEquationsforPosition(d)ArticulatedCoordinatesChapter2RobotKinematics:PositionAnalysisRoll,Pitch,Y

aw(RPY)anglesEuleranglesArticulatedjointsForwardandInverseKinematicsEquationsforOrientationChapter2RobotKinem

atics:PositionAnalysisRoll:Rotationofabout-axis(z-axisofthemovingframe)Pitch:Rotationofabout-axis(y-axis

ofthemovingframe)Yaw:Rotationofabout-axis(x-axisofthemovingframe)aaononFig.2.22RPYrotationsaboutthecurrentaxes.ForwardandIn

verseKinematicsEquationsforOrientation(a)Roll,Pitch,Yaw(RPY)AnglesChapter2RobotKinematics:PositionAnalysisFig.2.24Eulerrotationsaboutthecurrentax

es.Rotationofabout-axis(z-axisofthemovingframe)followedbyRotationofabout-axis(y-axisofthemovingframe)followedbyRotationofabout-axis(z-axisofthe

movingframe).aoaForwardandInverseKinematicsEquationsforOrientation(b)EulerAnglesChapter2RobotKinematics:Po

sitionAnalysis)()(,,,,noazyxcartHRRPYPPPTT)()(,,,,EulerTTrsphHRAssumption:RobotismadeofaCartesianandanRPYsetofjoints.Assumption:Robo

tismadeofaSphericalCoordinateandanEulerangle.AnotherCombinationcanbepossible……Denavit-HartenbergReprese

ntationForwardandInverseKinematicsEquationsforOrientationRoll,Pitch,Yaw(RPY)AnglesForwardandInverseTransformationsforrobotarmsF

ig.2.16TheUniverse,robot,hand,part,andendeffecterframes.StepsofcalculationofanInversematrix:1.Calculatethedet

erminantofthematrix.2.Transposethematrix.3.Replaceeachelementofthetransposedmatrixbyitsownminor(adjointmatrix).4.D

ividetheconvertedmatrixbythedeterminant.INVERSEOFTRANSFORMATIONMATRICESIdentityTransformations1.WeoftenneedtocalculateINVERSEMATRICES2.

Itisgoodtoreducethenumberofsuchoperations3.WeneedtodothesecalculationsfastHowtofindanInverseMatrixBofmatrixA?InverseHomogeneousTransformation

HomogeneousCoordinates•Homogeneouscoordinates:embed3Dvectorsinto4Dbyaddinga“1”•Moregenerally,thetransformationmatrixThastheform:Fact

orScalingTrans.Perspect.VectorTrans.MatrixRot.Ta11a12a13b1a21a22a23b2a31a32a33b3c1c2c3sfItispresentedinmoredetailont

heWWW!ForvarioustypesofrobotswehavedifferentmaneuveringspacesForvarioustypesofrobotswecalculatedifferentforwardandinve

rsetransformationsForvarioustypesofrobotswesolvedifferentforwardandinversekinematicproblemsForwardandInverseKinematics:SingleLinkExampleForw

ardandInverseKinematics:SingleLinkExampleeasyDenavit–HartenbergideaDenavit-HartenbergRepresentation:Fig.2.25AD-Hrepresentationofageneral-purp

osejoint-linkcombination@Simplewayofmodelingrobotlinksandjointsforanyrobotconfiguration,regardlessofitsseq

uenceorcomplexity.@Transformationsinanycoordinatesispossible.@Anypossiblecombinationsofjointsandlinksa

ndall-revolutearticulatedrobotscanberepresented.DENAVIT-HARTENBERGREPRESENTATIONOFFORWARDKINEMATICEQUATIONSOFROBOTChapter2RobotKinematic

s:PositionAnalysis⊙:Arotationanglebetweentwolinks,aboutthez-axis(revolute).⊙d:Thedistance(offset)onthez-axis,betweenlinks(prismatic).⊙a:Thelengthofe

achcommonnormal(Jointoffset).⊙:The“twist”anglebetweentwosuccessivez-axes(Jointtwist)(revolute)Onlyanddarejointvar

iables.DENAVIT-HARTENBERGREPRESENTATIONSymbolTerminologies:Linksarein3D,anyshapeassociatedwithZialwaysOnlyrotationOnlytranslationOnlyoffsetO

nlyoffsetOnlyrotationAxisalignmentDENAVIT-HARTENBERGREPRESENTATIONforeachlink4linkparametersChapter2RobotKinematics:PositionAnalysis⊙:Arotationangl

ebetweentwolinks,aboutthez-axis(revolute).⊙d:Thedistance(offset)onthez-axis,betweenlinks(prismatic).⊙a:Thelengthofeachcommonnormal(Jointoffset

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

ENTATIONSymbolTerminologies:ExamplewiththreeRevoluteJointsi(i-1)a(i-1)dii0000010a0012-90a1d22Z0X0Y0Z1X2Y1X1Y2d2a0a1Denavit-Ha

rtenbergLinkParameterTableTheDHParameterTableApplyfirstApplylastDenavit-HartenbergRepresentationofJoint-Link-JointTransfor

mationNotationforDenavit-HartenbergRepresentationofJoint-Link-JointTransformationAlphaappliedfirstFourTransformationsfromoneJointtotheNe

xt•Orderofmultiplicationofmatricesisinverseoforderofapplyingthem•HereweshoworderofmatricesJoint-Link-JointDenavit-HartenbergRe

presentationofJoint-Link-JointTransformation•AlphaisappliedfirstHowtocreateasinglematrixAnEXAMPLE:De

navit-HartenbergRepresentationofJoint-Link-JointTransformationforType1LinkFinalmatrixfrompreviousslidesubstitutesubstituteN

umericorsymbolicmatricesTheDenavit-HartenbergMatrixforanotherlinktype1000cosαcosαs

inαcosθsinαsinθsinαsinαcosαcosθcosαsinθ0sinθcosθi1)(i1)(i1)(ii1)(iii1)(i1)(i1)(ii1)(ii1)(iiidda•SimilaritytoHomegeneous:JustliketheHomogeneous

Matrix,theDenavit-HartenbergMatrixisatransformationmatrixfromonecoordinateframetothenext.•UsingaseriesofD-HMatrixmultiplicationsan

dtheD-HParametertable,thefinalresultisatransformationmatrixfromsomeframetoyourinitialframe.Z(i-1)X(i-1)Y(i-1)(i-1)a(i-1)ZiYiXiaidiiPutthetra

nsformationhereforeverylink1.InDENAVIT-HARTENBERGREPRESENTATIONwemustbeabletofindparametersforeachlink2.SowemustknowlinktypesL

inksbetweenrevolutejointsln=0Type3LinkJointn+1Jointndn=0Linknxn-1xnln=0dn=0Type4LinkOriginscoinciden-1Jointn+1JointnPartofdn-1Linknxn-1yn-1xnnLi

nksbetweenprismaticjointsForwardandInverseTransformationsonMatricesStartpoint:•Assignjointnumberntothefirstshownjoint.•Assignalocalref

erenceframeforeachandeveryjointbeforeorafterthesejoints.•Y-axisisnotusedinD-Hrepresentation.DENAVIT-HARTENBERGREPR

ESENTATIONPROCEDURES.1٭Alljointsarerepresentedbyaz-axis.•(right-handruleforrotationaljoint,linearmovementforpr

ismaticjoint)2.Thecommonnormalisonelinemutuallyperpendiculartoanytwoskewlines.3.Parallelz-axesjointsmakeain

finitenumberofcommonnormal.4.Intersectingz-axesoftwosuccessivejointsmakenocommonnormalbetweenthem(Lengthis0.).DENAVIT-HARTENBERGREPRESENTATIONProc

eduresforassigningalocalreferenceframetoeachjoint:Chapter2RobotKinematics:PositionAnalysis⊙:Arotation

aboutthez-axis.⊙d:Thedistanceonthez-axis.⊙a:Thelengthofeachcommonnormal(Jointoffset).⊙:Theanglebetweentwosuccessivez-axes(Joint

twist)Onlyanddarejointvariables.DENAVIT-HARTENBERGREPRESENTATIONSymbolTerminologiesReminder:Chapter2RobotKinematics:PositionAnalysis(I)Rotateab

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

ongthexn-axisadistanceofan+1tobringtheoriginsofxn+1together.(IV)Rotatezn-axisaboutxn+1axisanangleof

n+1toalignzn-axiswithzn+1-axis.DENAVIT-HARTENBERGREPRESENTATIONThenecessarymotionstotransformfromonereferenceframetothene

xt.