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MoredetailsandexamplesonrobotarmsandkinematicsDenavit-HartenbergNotationINTRODUCTIONForwardKinemati

cs:todeterminewheretherobot’shandis?(Ifalljointvariablesareknown)InverseKinematics:tocalculatewhateachjointvariableis?(Ifwedesirethatthe

handbelocatedataparticularpoint)DirectKinematicsDirectKinematicswithnomatricesWhereismyhand?DirectKinematics:HERE!DirectKinematics•Posit

ionoftipin(x,y)coordinatesDirectKinematicsAlgorithm1)Drawsketch2)Numberlinks.Base=0,Lastlink=n3)Identi

fyandnumberrobotjoints4)DrawaxisZiforjointi5)Determinejointlengthai-1betweenZi-1andZi6)DrawaxisXi-17)Determinejointtwist

i-1measuredaroundXi-18)Determinethejointoffsetdi9)DeterminejointangleiaroundZi10+11)Writelinktransformatio

nandconcatenateOftensufficientfor2DKinematicProblemsforManipulation•Reliablypositionthetip-gofromonepositiontoanotherposition•Don’thitanything,avoido

bstacles•Makesmoothmotions–atreasonablespeedsand–atreasonableaccelerations•Adjusttochangingconditions-–i.e.whensomethingispickeduprespondtothechangei

nweightROBOTSASMECHANISMsRobotKinematics:ROBOTSASMECHANISMFig.2.1Aone-degree-of-freedomclosed-loopfour-barmechanismMultipletyperobot

havemultipleDOF.(3Dimensional,openloop,chainmechanisms)Fig.2.2(a)Closed-loopversus(b)open-loopmechanismC

hapter2RobotKinematics:PositionAnalysisFig.2.3RepresentationofapointinspaceApointPinspace:3coordinatesrelativetoareferenceframe^^^kcjbiaPzyxRep

resentationofaPointinSpaceChapter2RobotKinematics:PositionAnalysisFig.2.4RepresentationofavectorinspaceAVectorPinspace:3coordi

natesofitstailandofitshead^^^__kcjbiaPzyxwzyxP__RepresentationofaVectorinSpaceChapter2RobotKinemati

cs:PositionAnalysisFig.2.5RepresentationofaframeattheoriginofthereferenceframeEachUnitVectorismutuallyperpendicular.:normal,orientation,appr

oachvectorzzzyyyxxxaonaonaonFRepresentationofaFrameattheOriginofaFixed-ReferenceFrameChapter2Ro

botKinematics:PositionAnalysisFig.2.6RepresentationofaframeinaframeEachUnitVectorismutuallyperpendicular.:normal,orientatio

n,approachvector1000zzzzyyyyxxxxPaonPaonPaonFRepresentationofaFrameinaFixedReferenceFrameThesameaslastslideChapter2

RobotKinematics:PositionAnalysisFig.2.8RepresentationofanobjectinspaceAnobjectcanberepresentedinspacebyattachingaframetoitand

representingtheframeinspace.1000zzzzyyyyxxxxobjectPaonPaonPaonFRepresentationofaRigidBodyChapter2RobotKinematics:PositionAna

lysisAtransformationmatricesmustbeinsquareform.•Itismucheasiertocalculatetheinverseofsquarematrices.•Tomultiplytwomatrices,theirdi

mensionsmustmatch.1000zzzzyyyyxxxxPaonPaonPaonFHOMOGENEOUSTRANSFORMATIONMATRICESRepresentationofTransformationsofrigidobjectsin3

DspaceChapter2RobotKinematics:PositionAnalysisFig.2.9RepresentationofanpuretranslationinspaceAtransformationisdefinedasmakin

gamovementinspace.•Apuretranslation.•Apurerotationaboutanaxis.•Acombinationoftranslationorrotations.

1000100010001zyxdddTRepresentationofaPureTranslationidentitySamevalueaChapter2RobotKinematics:PositionAnalysisFig.2.10Coordinates

ofapointinarotatingframebeforeandafterrotationaroundaxisx.Assumption:Theframeisattheoriginofthereferenceframeandparalleltoit.Fig.2.11Coo

rdinatesofapointrelativetothereferenceframeandrotatingframeasviewedfromthex-axis.RepresentationofaPureRotationab

outanAxisProjectionsasseenfromxaxisx,y,zn,o,aFig.2.13EffectsofthreesuccessivetransformationsAnumberofsuccessivetrans

lationsandrotations….RepresentationofCombinedTransformationsOrderisimportantx,y,zn,o,anioiaiT1T2T3Fig.2.14Changingtheorderoftransformationswillcha

ngethefinalresultOrderofTransformationsisimportantx,y,zn,o,atranslationChapter2RobotKinematics:PositionAnalysisFig.2.15Transformationsrelativeto

thecurrentframes.Example2.8TransformationsRelativetotheRotatingFrametranslationrotationMATRICESFORFORWARDANDINVERSEKINEMATICSOFROB

OTS•Forposition•FororientationChapter2RobotKinematics:PositionAnalysisFig.2.17Thehandframeoftherobotrelativetothereferencef

rame.ForwardKinematicsAnalysis:•Calculatingthepositionandorientationofthehandoftherobot.Ifallrobotjointvariabl

esareknown,onecancalculatewheretherobotisatanyinstant..FORWARDANDINVERSEKINEMATICSOFROBOTSChapter2RobotKinematics:PositionAnalysi

sForwardKinematicsandInverseKinematicsequationforpositionanalysis:(a)Cartesian(gantry,rectangular)coordin

ates.(b)Cylindricalcoordinates.(c)Sphericalcoordinates.(d)Articulated(anthropomorphic,orall-revolute)coordinates.Forwar

dandInverseKinematicsEquationsforPositionChapter2RobotKinematics:PositionAnalysisIBM7565robot•Allactuatorislinear.•AgantryrobotisaCart

esianrobot.Fig.2.18CartesianCoordinates.1000100010001zyxcartPRPPPTTForwardandInverseKinematicsEquationsforPosi

tion(a)Cartesian(Gantry,Rectangular)CoordinatesChapter2RobotKinematics:PositionAnalysis2Lineartranslationsand1ro

tation•translationofralongthex-axis•rotationofaboutthez-axis•translationoflalongthez-axisFig.2.19CylindricalCo

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

dInverseKinematicsEquationsforPosition:CylindricalCoordinatescosinesineChapter2RobotKinematics:PositionAnalysis2Lineartranslationsand1rota

tion•translationofralongthez-axis•rotationofaboutthey-axis•rotationofalongthez-axisFig.2.20SphericalCoordinates.

10000rCCSSrSSSCSCCrSCSSCCTTsphPR))Trans()Rot(Rot()(0,0,,,,,yzlrsphPRTTForwardandInverseKinematicsEquationsf

orPosition(c)SphericalCoordinatesChapter2RobotKinematics:PositionAnalysis3rotations->Denavit-Hartenbergrepresent

ationFig.2.21ArticulatedCoordinates.ForwardandInverseKinematicsEquationsforPosition(d)ArticulatedCoordinatesChapter2RobotKinematics:Pos

itionAnalysisRoll,Pitch,Yaw(RPY)anglesEuleranglesArticulatedjointsForwardandInverseKinematicsEquationsforOrientati

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

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

ardandInverseKinematicsEquationsforOrientation(a)Roll,Pitch,Yaw(RPY)AnglesChapter2RobotKinematics:Po

sitionAnalysisFig.2.24Eulerrotationsaboutthecurrentaxes.Rotationofabout-axis(z-axisofthemovingframe)followedbyRotationofabout-axis(y-axis

ofthemovingframe)followedbyRotationofabout-axis(z-axisofthemovingframe).aoaForwardandInverseKinematicsEquations

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

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

e.AnotherCombinationcanbepossible……Denavit-HartenbergRepresentationForwardandInverseKinematicsEquationsforOrientationRoll,Pitch,

Yaw(RPY)AnglesForwardandInverseTransformationsforrobotarmsFig.2.16TheUniverse,robot,hand,part,andendeffecterframes.Stepsof

calculationofanInversematrix:1.Calculatethedeterminantofthematrix.2.Transposethematrix.3.Replaceeachelementofthetransposedmatrixbyitsown

minor(adjointmatrix).4.Dividetheconvertedmatrixbythedeterminant.INVERSEOFTRANSFORMATIONMATRICESIdentityTransformations1.WeoftenneedtocalculateINVERS

EMATRICES2.Itisgoodtoreducethenumberofsuchoperations3.WeneedtodothesecalculationsfastHowtofindanInverseMatrixBofm

atrixA?InverseHomogeneousTransformationHomogeneousCoordinates•Homogeneouscoordinates:embed3Dvectorsinto4Dbyaddinga“1”•Moregenerally,thetransformation

matrixThastheform:FactorScalingTrans.Perspect.VectorTrans.MatrixRot.Ta11a12a13b1a21a22a23b2a31a32a33b3c1c2c3sfItispresentedinmo

redetailontheWWW!ForvarioustypesofrobotswehavedifferentmaneuveringspacesForvarioustypesofrobotswecalculatedifferentforwardandinvers

etransformationsForvarioustypesofrobotswesolvedifferentforwardandinversekinematicproblemsForwardandInverseKinematic

s:SingleLinkExampleForwardandInverseKinematics:SingleLinkExampleeasyDenavit–HartenbergideaDenavit-HartenbergRep

resentation:Fig.2.25AD-Hrepresentationofageneral-purposejoint-linkcombination@Simplewayofmodelingrobotlinksandjointsfora

nyrobotconfiguration,regardlessofitssequenceorcomplexity.@Transformationsinanycoordinatesispossible.@An

ypossiblecombinationsofjointsandlinksandall-revolutearticulatedrobotscanberepresented.DENAVIT-HARTENBERGREPR

ESENTATIONOFFORWARDKINEMATICEQUATIONSOFROBOTChapter2RobotKinematics:PositionAnalysis⊙:Arotationanglebetweentwolinks,aboutthez-axis(revolute).⊙d:Th

edistance(offset)onthez-axis,betweenlinks(prismatic).⊙a:Thelengthofeachcommonnormal(Jointoffset).⊙:The“twist”an

glebetweentwosuccessivez-axes(Jointtwist)(revolute)Onlyanddarejointvariables.DENAVIT-HARTENBERGREPRESENTATIONSymbolTermin

ologies:Linksarein3D,anyshapeassociatedwithZialwaysOnlyrotationOnlytranslationOnlyoffsetOnlyoffsetOnlyrotationAx

isalignmentDENAVIT-HARTENBERGREPRESENTATIONforeachlink4linkparametersChapter2RobotKinematics:PositionAnalysis⊙:A

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

(prismatic).⊙a:Thelengthofeachcommonnormal(Jointoffset).⊙:The“twist”anglebetweentwosuccessivez-axes(Jointtwist)(revolute)Onlya

nddarejointvariables.DENAVIT-HARTENBERGREPRESENTATIONSymbolTerminologies:ExamplewiththreeRevoluteJointsi(i-1)a(i-1)dii0000010a0012-90a1d22Z

0X0Y0Z1X2Y1X1Y2d2a0a1Denavit-HartenbergLinkParameterTableTheDHParameterTableApplyfirstApplylastDenavit-HartenbergRepresent

ationofJoint-Link-JointTransformationNotationforDenavit-HartenbergRepresentationofJoint-Link-JointTransformationAlphaapp

liedfirstFourTransformationsfromoneJointtotheNext•Orderofmultiplicationofmatricesisinverseoforderofapplyingthem•HereweshoworderofmatricesJoint

-Link-JointDenavit-HartenbergRepresentationofJoint-Link-JointTransformation•AlphaisappliedfirstHowtocreat

easinglematrixAnEXAMPLE:Denavit-HartenbergRepresentationofJoint-Link-JointTransformationforType1LinkFinalmatrixfromprevioussl

idesubstitutesubstituteNumericorsymbolicmatricesTheDenavit-HartenbergMatrixforanotherlinktype1000cosαcosαsinαcos

θsinαsinθsinαsinαcosαcosθcosαsinθ0sinθcosθi1)(i1)(i1)(ii1)(iii1)(i1)(i1)(ii1)(ii1)(iiidda•SimilaritytoHomegeneous:Ju

stliketheHomogeneousMatrix,theDenavit-HartenbergMatrixisatransformationmatrixfromonecoordinateframetothenext.•Usinga

seriesofD-HMatrixmultiplicationsandtheD-HParametertable,thefinalresultisatransformationmatrixfromsomeframetoyourinit

ialframe.Z(i-1)X(i-1)Y(i-1)(i-1)a(i-1)ZiYiXiaidiiPutthetransformationhereforeverylink1.InDENAVIT-HARTENBERGREPRESENTATIONwemustbeabletofindparameter

sforeachlink2.SowemustknowlinktypesLinksbetweenrevolutejointsln=0Type3LinkJointn+1Jointndn=0Linknxn-1xnln=0

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

ndInverseTransformationsonMatricesStartpoint:•Assignjointnumberntothefirstshownjoint.•Assignalocalreferenceframefor

eachandeveryjointbeforeorafterthesejoints.•Y-axisisnotusedinD-Hrepresentation.DENAVIT-HARTENBERGREPRESENTATIO

NPROCEDURES.1٭Alljointsarerepresentedbyaz-axis.•(right-handruleforrotationaljoint,linearmovementforprismaticjoint)2.Thecommonnormal

isonelinemutuallyperpendiculartoanytwoskewlines.3.Parallelz-axesjointsmakeainfinitenumberofcommonnorm

al.4.Intersectingz-axesoftwosuccessivejointsmakenocommonnormalbetweenthem(Lengthis0.).DENAVIT-HARTENBERGREPRESENTATIONProceduresfora

ssigningalocalreferenceframetoeachjoint:Chapter2RobotKinematics:PositionAnalysis⊙:Arotationaboutthez-axis.⊙d:Thedistanceonthez-axis.⊙a:Thelengthof

eachcommonnormal(Jointoffset).⊙:Theanglebetweentwosuccessivez-axes(Jointtwist)Onlyanddarejointvariables.DENAVIT-HARTE

NBERGREPRESENTATIONSymbolTerminologiesReminder:Chapter2RobotKinematics:PositionAnalysis(I)Rotateaboutthezn-axisanableof

n+1.(Coplanar)(II)Translatealongzn-axisadistanceofdn+1tomakexnandxn+1colinear.(III)Translatealongthexn-axisadistanceofan+1to

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

necessarymotionstotransformfromonereferenceframetothenext.