Table Of ContentScrewTheoryandItsApplicationtoSpatialRobotManipulators
Discoverafreshtakeonclassicalscrewtheoryandunderstandthegeometryembed-
ded within robots and mechanisms with this essential text. The book begins with a
geometricalstudyofpoints,lines,andplanesandslowlyprogressesthereadertowards
amasteryofscrewtheorywithsomecutting-edgeresults,allwhileusingonlybasic
linearalgebraandordinaryvectors.Itfeaturesadiscussionofthegeometryofparallel
andserialrobotmanipulators,inadditiontothereciprocityofscrewsandasingularity
study.Familiarizingthereaderwithscrewgeometryinordertostudythestaticsand
kinematics of robots and mechanisms, this is a perfect resource for engineers and
graduatestudents.
Carl D. Crane III is a professor in the department of mechanical and aerospace
engineeringattheUniversityofFlorida.HeisafellowoftheASME.
MichaelGriffisisaseniorlecturerattheUniversityofFlorida.
JosephDuffy was a great kinematician who passionately promoted screw theory. He
wasthedirectoroftheCenterforIntelligentMachinesandRoboticsattheUniversity
of Florida and a graduate research professor, who taught screw theory. He received
countlessawards,includingtheASMEMachineDesignAwardin2000.
Published online by Cambridge University Press
Screw Theory and Its Application
to Spatial Robot Manipulators
CARL D. CRANE III
UniversityofFlorida
MICHAEL GRIFFIS
UniversityofFlorida
JOSEPH DUFFY
UniversityofFlorida
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Informationonthistitle:www.cambridge.org/9780521630894
DOI:10.1017/9781139019217
©CarlD.Crane,MichaelGriffis,andJosephDuffy2022
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Firstpublished2022
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ToAnne,Sherry,andKaren.
Thewomenwholovedus,andwhoenduredus…
whileweplayedwithrobots.
Joseph,Carl,andMichael
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Contents
Preface pagexi
1 GeometryofPoints,Lines,andPlanes 1
1.1 Introduction 1
1.2 ThePositionVectorofaPoint 1
1.3 TheEquationofaPlane 3
1.3.1 SampleProblem 6
1.3.2 SampleProblem 7
1.4 ProjectionofaPointontoaPlane 8
1.5 TheEquationofaLine 9
1.5.1 SampleProblem 14
1.6 TwoPlanesDetermineaLine 16
1.7 ThePencilofPlanesthroughaLine 20
1.7.1 ThePlaneDefinedbyaLineandaPoint 20
1.7.2 ThePlaneThatContainsaLineandIsParalleltoa
SecondLine 21
1.7.3 ThePlaneDefinedbyaPairofParallelLines 22
1.8 ALineandaPlaneDetermineaPoint 23
1.9 DeterminationofthePointonaLineThatIsClosesttoaGivenPoint 24
1.10 TheMutualMomentofTwoLines 25
1.10.1 NumericalExample 28
1.11 DeterminationoftheUniquePerpendicularLinetoTwoGivenLines 28
1.12 APairofIntersectingLines 31
1.12.1 Approach1 32
1.12.2 Approach2 33
1.13 Summary 33
1.14 Problems 33
2 CoordinateTransformationsandManipulatorKinematics 36
2.1 Introduction 36
2.2 RelativePoseofTwoCoordinateSystems 36
2.3 TransformationsofPoints 38
2.4 InverseofaTransform 40
2.5 StandardTransformations 41
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viii Contents
2.6 GeneralTransformations 43
2.6.1 DeterminationofEquivalentRotationMatrix 43
2.6.2 DeterminationofAxisandAngleofRotation 46
2.7 TransformationofDirectionVectors 47
2.8 TransformationofLines 48
2.9 TransformationsofPlanes 50
2.10 SpatialLinksandJoints 51
2.10.1 SpatialLink 52
2.10.2 RevoluteJoint(R) 52
2.10.3 PrismaticJoint(P) 53
2.10.4 CylindricalJoint(C) 53
2.10.5 ScrewJoint(H) 54
2.10.6 HigherOrderJoints 55
2.11 LabelingofaKinematicChain 57
2.11.1 Step1:LabeltheJointAxisVectors 57
2.11.2 Step2:LabeltheLinkVectors 58
2.11.3 Step3:LabeltheJointAnglesandTwistAngles 59
2.11.4 Step4:LabeltheOffsetandLinkLengths 59
2.11.5 Step5:CompilationofMechanismParameters 62
2.12 StandardLinkCoordinateSystems 63
2.13 Summary 64
2.14 Problems 65
3 StaticsofaRigidBody 66
3.1 Introduction 66
3.2 TheCoordinatesofaForce 66
3.3 TheCoordinatesofaCouple 68
3.4 TranslationofaForce:EquivalentForce/CoupleCombination 69
3.5 ADynameandaWrench 71
3.5.1 SampleProblem 74
3.6 TransformationofScrewCoordinates 75
3.6.1 SampleProblem 76
3.7 ForwardandReverseStaticAnalysisofIn-ParallelPlatformDevices 80
3.8 ForwardandReverseStaticAnalysisofaSerialManipulator 82
3.8.1 SampleProblem 86
3.8.2 SampleProblem 90
3.9 TheResultantofaPairofWrenchesActingUponaRigidBody 90
3.10 TheCylindroid 94
3.11 CircularRepresentationoftheCylindroid 99
3.12 MotorProduct 102
3.13 Problems 105
4 VelocityAnalysis 108
4.1 Introduction 108
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Contents ix
4.2 TimeDerivativeofaVector 108
4.2.1 Case1:PointsAandB EmbeddedinBody0 109
4.2.2 Case2:PointAEmbeddedinBody0andPointB
EmbeddedinBody1 109
4.2.3 Case3:PointsAandB EmbeddedinBody1 110
4.2.4 Case4:PointAEmbeddedinBody1andPointB
EmbeddedinBody2 112
4.2.5 Case5:DerivativesofaVectorwithRespecttoTwo
CoordinateSystems 114
4.3 DefinitionofVelocityState 116
4.4 ScrewInterpretationofVelocityState 119
4.4.1 ExampleProblem1 121
4.4.2 ExampleProblem2 123
4.4.3 ExampleProblem3 124
4.4.4 ExampleProblem4 124
4.5 SerialChainofThreeRigidBodies 125
4.6 SerialChainofMultipleRigidBodies 128
4.7 ForwardandReverseVelocityAnalysesforSerialManipulators 129
4.7.1 ExampleProblem 131
4.7.2 ExampleProblem 136
4.8 Problems 138
5 ReciprocalScrews 142
5.1 Introduction 142
5.2 DefinitionofReciprocalScrews 142
5.3 ReciprocalProduct 144
5.3.1 AxesoftheTwoScrewsIntersect 144
5.3.2 AxesArePerpendicular 145
5.3.3 AxesAreParallel 145
5.3.4 BodyConstrainedinTranslation 145
5.4 ReciprocalScrewSystems 147
5.4.1 ASingleScrew:OneDegreeofFreedom 147
5.4.2 TwoScrews:TwoDegreesofFreedom 148
5.4.3 ThreeScrews:ThreeDegreesofFreedom 150
5.4.4 FourScrews:FourDegreesofFreedom 153
5.4.5 FiveScrews:FiveDegreesofFreedom 154
5.4.6 SixScrews:SixDegreesofFreedom 154
5.4.7 ExampleProblem1 155
5.4.8 ExampleProblem2 156
5.4.9 ExampleProblem3 157
5.4.10 ExampleProblem4 157
5.4.11 ExampleProblem5 158
5.5 VelocityAnalysisofSerialManipulatorsUsingReciprocalScrews 159
5.5.1 ExampleProblem 161
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x Contents
5.6 ForwardandReverseVelocityAnalysisforParallelMechanisms 163
5.6.1 ReverseVelocityAnalysis 163
5.6.2 ForwardVelocityAnalysis 164
5.6.3 ExampleProblem 165
5.7 Problems 166
6 SingularityAnalysisofSerialChains 168
6.1 Introduction 168
6.2 SingularConfigurations 168
6.3 SymbolicExpansionofScrewCoordinates 169
6.4 SelectionofCoordinateSystem 171
6.5 ExampleProblems 172
6.5.1 CincinnatiMilacronT3-776Manipulator 172
6.5.2 G.E.P60Manipulator 175
6.6 Problems 180
7 AccelerationAnalysisofSeriallyConnectedRigidBodies 181
7.1 Introduction 181
7.2 GeneralCaseofTwoBodies 181
7.3 GeneralCaseofThreeBodies 183
7.4 GeneralCaseofFourBodies 186
7.5 GeneralCaseofnBodies 188
7.6 TwoBodiesConnectedbyaScrewJoint 189
7.7 TwoBodiesConnectedbyaRevoluteJoint 191
7.8 TwoBodiesConnectedbyaPrismaticJoint 192
7.9 ForwardAccelerationAnalysisforSerialManipulators 193
7.9.1 SampleProblem 194
7.10 ReverseAccelerationAnalysisforSerialManipulators 196
7.10.1 SampleProblem 198
7.11 Problems 199
A DerivationofCylindroidEquations 202
Bibliography 207
Index 208
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