Table Of ContentTheory of Parallel Mechanisms
MECHANISMS AND MACHINE SCIENCE
Volume 6
Series Editor
MARCO CECCARELLI
Forfurther volumes:
http://www.springer.com/series/8779
Zhen Huang (cid:129) Qinchuan Li (cid:129) Huafeng Ding
Theory of Parallel
Mechanisms
ZhenHuang QinchuanLi
RoboticsResearchCenter MechatronicDepartment
YanshanUniversity ZhejiangSci-TechUniversity
Qinhuangdao,Hebei,China Hangzhou,China
HuafengDing
RoboticsResearchCenter
YanshanUniversity
Qinhuangdao,China
ISSN2211-0984 ISSN2211-0992(electronic)
ISBN978-94-007-4200-0 ISBN978-94-007-4201-7(eBook)
DOI10.1007/978-94-007-4201-7
SpringerDordrechtHeidelbergNewYorkLondon
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Preface
In the past decades, parallel mechanisms (PMs) have attracted a lot of attention
fromtheacademicandindustrialcommunities.Comparedwiththemorecommonly
used serial robots, the parallel one has attractive advantages in accuracy, rigidity,
capacity,andload-to-weightratio.ThePMshavebeenandarebeingusedinawide
variety of applications such as motion simulators, parallel manipulators, nano-
manipulators,andmicro-manipulators.Inrecentyears,theresearchandapplication
have evolved from general six-DOF PMs to lower-mobility PMs. The essential
reason is that lower-mobility PMs have similar applications to general six-DOF
PMs,whiletheyaremuchsimplerinstructureandcheaperincost.Theresearchof
lower-mobilityPMshasbecomenewhotpoint.Agreatdealofresearchonlower-
mobility PM has been carried out all over the world, and a large number of new
mechanisms,suchasDelta,Tricept,andmedicalrobots,havebeenbuiltforvarious
applications.
ThisbookintroducesouroriginalresearcheffortsonPMsforthe30years.The
contentsincludemechanismanalysesandsyntheses.
Inmechanismanalysis,theunifiedmobilitymethodologyisfirstsystematically
presented. The search for a general and valid mobility methodology has been
ongoing for about 150 years. Our methodology is proposed based on the screw
theory,whosegeneralityandvalidityhaveonlybeenrecentlyproven.Thisisavery
important progress. The principle of the kinematic influence coefficient and
its new development are described. This principle fits the kinematic analysis of
various parallel manipulators including both 6-DOF and lower-mobility ones.
The singularities are classified from a new point of view, and new progresses in
singularityareintroduced.Theconceptoftheover-determinateinputisresearched,
and in practice, there are many machines that work with over-determinate input,
i.e., their input number is much bigger than their mobility number. To set the
inputstobeaccordanceandoptimumdistributeandtoobtaintheexpectantmotion
acceleration is introduced here. A new method of force analysis of PMs is
presented.Thismethodbased onscrewtheory canremarkablyreduce thenumber
of unknowns and keep the number of simultaneous equilibrium equations not
more than six onevery occasion. Inmechanism synthesis, the synthesisof spatial
v
vi Preface
symmetrical PMs is discussed. The synthesis method of difficult four- and five-
DOF symmetrical mechanisms, which has first been put forward by our group in
2002, is emphatically introduced. The three-order screw system and its space
distribution of kinematic screws for infinite possible motions of lower-mobility
mechanismsarealsoanalyzed.Inthelastchapter,anewtheoryforthetopological
structure analysis and synthesis of kinematic chains is represented. Based on the
array representation of loops in topological graphs of kinematic chains, the basic
loopoperationalgebraandauniquerepresentationareintroduced.Addressingthe
problemofisomorphismidentificationbyfindingauniquerepresentationofgraphs
ispresented.Thisprocessmakesisomorphismidentificationveryeasyandremains
efficientevenwhenthekinematicchainlinksincreaseuptothethirties.Theunique
numerical atlas database is established and developed for use in the numerical
synthesisofmechanisms.
Giventhatmanyoftheabovementionedresearcharebasedonthescrewtheory,
thebasicscrewtheoryisfirstintroducedinthebeginningofthisbook.
Using the screw theory to analyze some issues on spatial mechanisms is quite
facile and convenient. This theory is also a good one for various mathematical
instruments. A pair of spatial vectors or dual vectors can be used to construct a
screw. The screw can then be applied to express the following: (1) position and
orientationofaspatialstraightlineingeometry,(2)lineandangularvelocitiesofa
rigidbodyinkinematics,(3)forceandmomentinstatics,(4)constraintforceand
couple, and (5) rotational and translational mobilities in freedom analysis. The
conceptofa screw with six scalarsis then easily used inkinematics and dynamic
analysis. The screw can be facilely transformed into various mathematic forms,
such as for vector, matrix, algebraic, and geometrical analyses. The screw has a
cleargeometricalconcept,anexplicitphysicalmeaning,asimpleexpressingform,
andconvenientalgebraiccalculation.Forthesereasons,thescrewconceptiswidely
appliedinmechanisms,especiallyrecently,toresolvenumerousdifficultforeland
issues. Students, engineers, and practically anyone who has studied linear algebra
caneasilyunderstandthetheory.
The authors gratefully acknowledge the continuous financial support of the
National Natural Science Foundation of China for more than 20 years. This book
can be a textbook for postgraduate students and general scientific technique per-
sonnel. Some more profound chapters can be suitable for doctoral students in the
fieldofmechanicalengineering.
YanshanUniversity ZhenHuang
Qinhuangdaobeachfront
Contents
1 BasicsofScrewTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 EquationofaLine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 MutualMomentofTwoLines. . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 LineVectorsandScrews. . . .. . . . . .. . . . . .. . . . . . .. . . . . .. 6
1.4.1 TheLineVector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.2 TheScrew. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 ScrewAlgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.1 ScrewSum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.2 ProductofaScalarandaScrew. . . . . . . . . . . . . . . . . . . 11
1.5.3 ReciprocalProduct. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 InstantaneousKinematicsofaRigidBody. . . . . . . . . . . . . . . . . 11
1.6.1 InstantaneousRotation. . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6.2 InstantaneousTranslation. . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.3 InstantaneousScrewMotion. . . . . . . . . . . . . . . . . . . .. . 13
1.7 StaticsofaRigidBody. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7.1 AForceActingonaBody. . . . . . . . . . . . . . . . . . . . . . . 14
1.7.2 ACoupleActingonaBody. . . . . . . . . . . . . . . . . . . . . . 15
1.7.3 ATwistActingonaBody. . . . . . . . . . . . . . . . . . . . . . . 15
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 DependencyandReciprocityofScrews. . . . . . . . . . . . . . . . . . . . . . 17
2.1 ConceptofScrewSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Second-OrderScrewSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 LinearCombinationofTwoScrews. . . . . . . . . . . . . . . . 19
2.2.2 SpecialTwo-ScrewSystem. . . . . . . . . . . . . . . . . . . . . . 21
2.3 Third-OrderScrewSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 PrincipalScrews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 SpecialThree-ScrewSystems. . . . . . . . . . . . . . . . . . . . . 26
2.4 GrassmannLineGeometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
viii Contents
2.5 ScrewDependencyinDifferentGeometricalSpaces. . . . . . . . . . 30
2.5.1 BasicConcepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.2 DifferentGeometricalSpaces. . . . . . . . . . . . . . . . . . . . . 31
2.6 ReciprocalScrews. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.1 ConceptofaReciprocalScrew. . . . . . . . . . . . . . . . . . . . 36
2.6.2 DualisminthePhysicalMeaning
ofReciprocalScrews. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 ReciprocalScrewSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 ReciprocalScrewandConstrainedMotion. . . . . . . . . . . . . . . . . 41
2.8.1 ThreeSkewLinesinSpace. . . . . . . . . . . . . . . . . . . . . . 42
2.8.2 ThreeLinesParalleltoaPlaneWithout
aCommonNormal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8.3 ThreeNon-concurrentCoplanarLines. . . . . . . . . . . . . . 44
2.8.4 ThreeCoplanarandConcurrentLineVectors. . . . . . . . . 44
2.8.5 ThreeLineVectorsConcurrentinSpace. . . . . . . . . . . . . 44
2.8.6 ThreeLineVectorsParallelinSpace. . . . . . . . . . . . . . . 45
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 MobilityAnalysisPart-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 TheConceptandDefinitionofMobility. . . . . . . . . . . . . . . . . . . 47
3.2 MobilityOpenIssue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Gr€ubler-KutzbachCriterion. . . . . . . . . . . . . . . . . . . . . . 49
3.2.2 MobilityOpenIssue. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 MobilityPrincipleBasedonReciprocalScrew. . . . . . . . . . . . . . 53
3.3.1 MechanismCanBeExpressedasaScrewSystem. . . . . . 53
3.3.2 DevelopmentofOurUnifiedMobilityPrinciple. . . . . . . 54
3.3.3 TheModifiedG-KFormulas. . . . . . . . . . . . . . . . . . . . . 55
3.4 ConstraintAnalysisBasedonReciprocalScrew. . . . . . . . . . . . . 57
3.4.1 TheCommonConstraint. . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.2 ParallelConstraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.3 Over-Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.4 TheGeneralizedKinematicPair. . . . . . . . . . . . . . . . . . . 59
3.5 MobilityPropertyAnalyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 TranslationandRotation. . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.2 RotationalAxis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.3 InstantaneousMobilityandFull-CycleMobility. . . . . . . 63
3.5.4 Full-FieldMobility. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5.5 ParasiticMotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.6 Self-motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 MobilityAnalysisPart-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 MobilityAnalysisofSimpleMechanisms. . . . . . . . . . . . . . . . . 71
4.1.1 OpenChainLinkage. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.2 RobervalMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Contents ix
4.1.3 RUPURMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.4 Herve´ Six-BarMechanism. . . . . . . . . . . . . . . . . . . . . . . 77
4.1.5 Spatial4PMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.6 DelassusH-H-H-HMechanism. . . . . . . . . . . . . . . . . . . 79
4.1.7 Herve´’sCCCMechanism. . . . . . . . . . . . . . . . . . . . . . . 80
4.2 MobilityAnalysisofClassicalMechanisms. . . . . . . . . . . . . . . . 81
4.2.1 BennettMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.2 Five-BarGoldbergLinkage. . . . . . . . . . . . . . . . . . . . . . 84
4.2.3 Six-BarGoldbergLinkage. . . . . . . . . . . . . . . . . . . . . . . 86
4.2.4 MyardLinkagewithSymmetricalPlane. . . . . . . . . . . . . 87
4.2.5 BricardwithSymmetricalPlane. . . . . . . . . . . . . . . . . . . 88
4.2.6 AltmannAbb.34Mechanism. . . . . . . . . . . . . . . . . . . . . 91
4.2.7 AltmannSix-BarLinkage. . . . . . . . . . . . . . . . . . . . . . . 94
4.2.8 WaldronSix-BarLinkage. . . . . . . . . . . . . . . . . . . . . . . 95
4.3 MobilityAnalysisofModernParallelMechanisms. . . . . . . . . . . 97
4.3.1 4-DOF4-URUMechanism. . . . . . . . . . . . . . . . . . . . . . 97
4.3.2 3-CRRMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.3 ZlatanovandGosselin’sMechanism. . . . . . . . . . . . . . . . 100
4.3.4 Carricato’sMechanism. . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.5 DeltaMechanism.. . . . . .. . . . .. . . . . .. . . . .. . . . .. . 103
4.3.6 H4Manipulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.7 Yang’sMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4 MobilityAnalysisofHobermanSwitch-PitchBall. . . . . . . . . . . 114
4.4.1 StructureAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.4.2 Three-LinkChain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.3 Eight-LinkLoop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.4 DoubleLoop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.5 Three-LoopChain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.6 TheWholeMechanism. . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 Six-HoleCubiformMechanism. . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5.1 Double-HoleLinkage. . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.5.2 Four-HoleLinkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5.3 Five-HoleLinkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5.4 TheWholeSix-HoleMechanism. . . . . . . . . . . . . . . . . . 132
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5 KinematicInfluenceCoefficientandKinematicsAnalysis. . . . . . . . 135
5.1 ConceptofKIC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2 KICandKinematicAnalysisofSerialChains. . . . . . . . . . . . . . 138
5.2.1 PositionAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2.2 First-OrderKIC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.3 Second-OrderKIC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3 KinematicAnalysisofParallelMechanism. . . . . . . . . . . . . . . . 144
5.3.1 First-OrderKICandMechanismVelocityAnalysis. . . . . 146
5.3.2 Second-OrderKICandMechanismAccelerations. . . . . . 150
x Contents
5.4 VirtualMechanismPrincipleofLower-Mobility
ParallelMechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.4.1 VirtualMechanismPrinciple. . . . . . . . . . . . . . . . . . . . . 155
5.4.2 KinematicAnalysisBasedonVirtual
MechanismPrinciple. . . . . . . . . . . . . . . . . . . . . . . . . . . 157
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6 Full-ScaleFeasibleInstantaneousScrewMotion. . . . . .. . . . . . . .. 163
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 DeterminationofPrincipalScrews.. . . . .. . . . .. . . . . .. . . . .. 165
6.2.1 TheRepresentationofPitchandAxes. . . . .. . . . . . . . .. 165
6.2.2 PrincipalScrewsofaThird-OrderScrewSystem. . . . . . 167
6.3 Full-ScaleFeasibleInstantaneousScrewsofthe3-RPS
Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3.1 VirtualMechanismandJacobianMatrix. . . . . . . . . . . . . 171
6.3.2 UpperPlatformIsParalleltotheBase. . . . . . . . . . . . . . 173
6.3.3 TheUpperPlatformRotatesbyanAngleaAbout
Linea a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
2 3
6.3.4 GeneralConfigurationofthe3-RPSMechanism. . . . . . . 177
6.4 Full-ScaleFeasibleInstantaneousScrew
ofa3-UPUMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.1 MobilityAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4.2 First-OrderInfluenceMatricesandKinematic
Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.4.3 InitialConfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.4.4 TheSecondConfiguration. . . . . . . . . . . . . . . . . . . . . . . 186
6.5 Full-ScaleFeasibleInstantaneousScrewofa3-RPS
PyramidMechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6.5.1 First-OrderInfluenceCoefficientMatrix . . . . . . . . . . . . 189
6.5.2 PrincipalScrewsandFull-ScaleFeasibleMotions. . . . . . 191
6.6 A3-DOFRotationalParallelManipulatorWithout
IntersectingAxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.6.1 AnOpenProblemofthePMswithIntersectingAxes. . . 201
6.6.2 A3-DRevoluteMechanismWithoutIntersectingAxes. . 203
6.6.3 TheOrientationWorkspace. . . . . . . . . . . . . . . . . . . . . . 207
6.6.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.6.5 DiscussionsAbouttheDifferencesBetween
theSPMsandthe3-RPSCubicPM. . . . . . . . . . . . . . . . 214
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7 SpecialConfigurationofMechanisms. . . . . . . . . . . . . . . . . . . . . . . 217
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.2 ClassificationoftheSpecialConfiguration. . . . . . . . . . . . . . . . . 219
7.2.1 SingularKinematicsClassification. . . . . . . . . . . . . . . . . 220
7.2.2 ClassificationoftheSingularityviaaLinearComplex. . . 223
