Stiffness Analysis of Cable-Driven Parallel Robots

发布时间：2024-11-08

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StiffnessAnalysisofCable-DrivenParallelRobots

AmirMoradi

Athesissubmittedtothe

DepartmentofMechanicalandMaterialsEngineering

inconformitywiththerequirementsfor

thedegreeofDoctorofPhilosophy

Queen’sUniversity

Kingston,Ontario,Canada

April,2013

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Abstract

Theaimofthisthesisisthesti nessanalysisofcable-drivenparallelrobots.Cable-drivenparallelrobotshavedrawnconsiderableattentionbecauseoftheiruniqueabilitiesandadvantagessuchasthelargeworkspace,lightweightofcableactuators,easydisassemblyandtransportationoftherobot.Themobileplatformofacable-drivenparallelrobotisattachedtothebasewithmultiplecables.

Oneoftheparametersthatshouldbestudiedtomakesurearobotisabletoexecuteataskaccuratelyissti nessoftherobot.Inordertoinvestigatethesti nessbehaviourofarobot,thesti nessmatrixcanbecalculatedasthe rststep.Becausecablesactintension,keepingthepositivetensionincablesbecomesachallenge.Inordertohaveafullycontrollablerobot,anactuationredundancyisneeded.Thesecomplexitiesareaddressedinthethesisandsimulations.

Inthisthesis,thecompleteformofthesti nessmatrixisconsideredwithoutneglectinganytermsincalculationofthesti ness.Somesti nessindicessuchassingle-dimensionalsti nessbasedonsti nessellipse,directionalsti nessandconditionnumberofthesti nessmatrixareintroducedandcalculatedandsti nessmapsoftherobotaredeveloped.Inaddition,theissueofunitinconsistencyincalculatingthesti nessindexisaddressed.

Oneoftheareaswhichisalsoaddressedinthisthesisisfailureanalysisbasedonthesti nessofrobot.Thee ectofthefailureinoneormorecablesormotorsismodelledandsti nessmapsaredevelopedforthefailuresituation.Itisshownthatbychangingthe

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anchorpositionandmobileplatformorientation,theloststi nessafterfailureofacableormotorcanberetrievedpartially.Optimumanchorpositionandmobileplatformorientationareidenti edtomaximizetheareaofthesti nessmap.

Conditionnumberofthesti nessmatrixwhilerobotisfollowingatrajectoryisopti-mized.Inaddition,whenonecablefailsduringthepathplanning,therecoveryoftherobotisstudied.Finally,theseanalysesonsti nessandfailureprovidethedesignerwiththenecessaryandvaluableinformationabouttheanchorpositionsandactuatortoques.

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Acknowledgements

Iwouldliketoexpressmysinceregratitudeandappreciationtowardmysupervisor,Pro-fessorRonAndersonforhiskindandnever-failingsupport.Icouldnothaveimaginedcompletingmythesiswithouthiscontinuedguidance,endlessencouragementanduniqueteachingmethods.Ithasbeenaprivilegetoworkwithhimwhichcannotbedescribedinwords.

Iwouldliketothankallgeneralsta whohelpedmeduringmystudyatQueen’sUniversity,speciallysta membersatDepartmentofMechanicalandMaterialsEngineering.

IwouldalsoliketothankmydearfriendDr.MaryamAgahiwithwhomIhadthechanceofworkinginthesamelaboratoryandhavinghelpfuldiscussionsthroughoutmyresearch.Iappreciateherpurefriendshipthroughherkindheart.ManythankstomydearfriendsspeciallymyfriendsinKingstonwhosekindsupporthasbeenmysourceofenergyandstressrelief.

Lastbutnotleast,Iwishtothankmylovelyparentsandsisterswhohavealwaysbeenatmysideevenphysicallyfaraway.Theirunconditionalloveandsupporthaveenabledmetopursuemygoalsandalsokeepcon dentandstrongduringhardships.

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TableofContentsAbstractAcknowledgementsTableofContents

ListofTables

ListofFigures

Notation

Chapter1:Introduction..............................

1.1

1.2

1.3BackgroundandMotivation...........................ProposedWork..................................ContributionsandThesisOutline........................iiiiivviiiixxiv1268

15Chapter2:LiteratureReview...........................2.1TerminologiesandTheoriesinRobotics....................2.1.12.1.22.1.32.1.42.1.5PositionAnalysisofRobots.......................JacobianAnalysisofRobots.......................Pseudo-inverseofJacobianMatrix...................StaticsandSti nessAnalysisofRobots................DynamicsofRobots...........................

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2.2LiteratureReview................................

Chapter3:Sti nessFormulation.........................

3.1Modelling.....................................

3.1.1Kinematics................................

3.1.1.1JacobianMatrixof3DOFPlanarRobot...........

3.1.1.2JacobianMatrixof2DOFTranslationalRobot.......

3.1.2ForceAnalysis..............................

3.1.3Sti nessMatrixFormulation......................

3.1.3.1Sti nessMatrixof3DOFPlanarRobot...........

3.1.3.2Sti nessMatrixof2DOFTranslationalRobot.......

3.2Sti nessIndices..................................

3.2.1SingleDimensionalSti nessBasedonSti nessEllipse........

3.2.1.1UnitInconsistentSti nessMatrices.............

3.2.2DirectionalSti ness...........................

3.2.3ConditionNumberofSti nessMatrix.................

3.3Conclusion....................................

Chapter4:Sti nessandOtherQualityMaps.................

4.1Sti nessMaps...................................

4.1.1Sti nessMapsof3DOFPlanarRobots................

4.1.1.1TwoCableAttachmentsonMobilePlatform........

4.1.1.2SymmetricalFourCableAttachmentsonMobilePlatform

4.1.1.3CrossedFourCableCon guration..............

4.1.2Sti nessMapsof2DOFTranslationalRobots.............

4.2OptimumLayoutoftheRobottoMaximizetheAreaofSti nessMaps...

4.3PotentialEnergyMaps..............................

4.4De ectionMaps..................................

1625262626282930313334353638404042434445474950525556

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4.5

4.6ConditionNumberMaps.............................Conclusion....................................5859

100Chapter5:FailureAnalysis............................5.1FailureFormulation5.1.15.1.25.2...............................FailureofaCable............................FailureofaMotor............................E ectofFailureontheSti nessMaps.....................5.2.15.2.2Sti nessMapsof3DOFPlanarRobotafterFailure..........Sti nessMapsof2DOFTranslationalRobotafterFailure......5.35.4RetrievingLostSti nessofRobotafterFailure................OptimumLayoutsofRobotsafterFailure...................5.4.15.4.25.4.3OptimizingAnchorPosition.......................OptimizingMobilePlatformOrientation................OptimizingAnchorPositionandMobilePlatformOrientation....5.5Conclusions....................................Chapter6:TrajectoryPlanning.........................6.16.2Optimumcon guration.............................OptimumTrajectoryPlanning.........................6.2.16.2.26.36.4StraightLinePath............................CircularPath...............................E ectofRedundancyonOptimumTrajectoryPlanning...........FailureRecoveryinTrajectoryPlanning....................6.4.16.4.2StraightLinePath............................CircularPath...............................6.5Conclusions....................................

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Chapter7:Conclusions...............................101

7.1

7.2Summary.....................................FutureWork...................................101103Bibliography......................................106

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ListofTables4.1GAfunctionparameters.............................

6.1GAfunctionparameters.............................

5481

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ListofFigures

1.1AKUKAPickandPlaceserialrobotinafactory[48]............

1.2Aparallelrobotformedbysixlinearactuators(hexapod)controlsthemove-mentfortheAirbusA3200FullFlightSimulator(FFS)atBalticAviationAcademy[9]....................................

1.3TerracottaAncientGreekdolls,exhibitedintheNationalArchaeologicalMu-seuminAthens,room56,5th/4thcenturyBC[73]..............

1.4NISTRoboCranestripingpainto aU.S.AirForceC-130[3]........

1.5SkycamHDatanESPNonABC-broadcastUniversityofCalifornia,Berkeleyfootballgame[74].................................

3.1Coordinatesandvariablesforthe3DOFplanarcable-drivenparallelrobot.

3.2Coordinatesandvariablesforthe2DOFtranslationalcable-drivenparallelrobot........................................

3.3Comparingthedistributionofthesingledimensionalsti nessindexanddi-rectionalsti nessforanexamplesti nessmatrix................

4.1Cross-sectionof7×7wirerope[80]......................

4.2Example3DOFplanarfour-cable-drivenparallelrobots............

4.3Sti nessmapsoftheshowninFigure4.2(a)robotwithoutgravity......

4.4Sti nessmapsoftherobotshowninFigure4.2(a)withgravity........

4.5Sti nessmapsalongXdirectionoftherobotshowninFigure4.2(a)withoutgravity.......................................

334552628394344454646

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4.6

4.7Sti nessmapsoftherobotshowninFigure4.2(b)withgravity........Sti nessmapsaboutZdirectionoftherobotshowninFigure4.2(b)without

gravityat( =5 ).................................4748

4.8Sti nessmapsaboutZdirectionoftherobotshowninFigure4.2(b)without

gravityat( =0 ).................................48

514.9Sti nessmapsoftherobotshowninFigure4.2(c)withgravity........4.102DOFtranslationalthree-cable-drivenparallelrobot..............4.11Sti nessmapsoftherobotshowninFigure4.10withoutgravity.......

4.12Sti nessmapsoftherobotwithoutgravityfortherequiredminimumsti ness.51

4.13Sti nessmapsoftherobotshowninFigure4.10withgravity.........

4.14Sti nessmapsoftherobotwithgravityfortherequiredminimumsti ness.

4.15Sti nessmapsoftherobotwiththeoptimumlayout..............

4.16Sti nessmapsoftherobotfortherequiredminimumsti nesswiththeopti-

mumlayout.....................................

4.17ElasticpotentialenergymapsoftherobotinFigure4.2(a)withgravityat

( =0 ).......................................

4.18De ectionmapsoftherobotwheretheunitexternalforceinXdirectionis

applied.......................................

4.19De ectionmapsoftherobotwheretheunitexternalforceinYdirectionis

applied.......................................

4.20De ectionmapsoftherobotwheretheunitexternalmomentaboutZdirec-

tionisapplied...................................

4.21ConditionnumbermapsoftherobotshowninFigure4.10..........

4.22ConditionnumbermapsoftherobotshowninFigure4.2(a).........

5.1

5.2Example3DOFplanarfour-cable-drivenparallelrobots............Sti nessmapsoftherobotshowninFigure5.1(a)withgravityat( =5 ).

575859636357575555525354

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5.3Sti nessmapsoftherobotshowninFigure5.1(a)afterfailureofcable1..5.4Sti nessmapsoftherobotshowninFigure5.1(a)afterfailureofmotor1..

5.5Sti nessmapsoftherobotshowninFigure5.1(b)withgravityat( =0 ).

5.6Sti nessmapsoftherobotshowninFigure5.1(b)afterfailureofcable1..

5.7Sti nessmapsoftherobotshowninFigure5.1(c)afterfailureofcable1..

5.8Sti nessmapsoftherobotshowninFigure4.10afterfailureofcable1...

5.9Regeneratedsti nessmapsofthecasestudyinFigure5.3afterapplyingan

externalmomentof2.5Nm............................

5.10Regeneratedsti nessmapsofthecasestudyinFigure5.6whenorientation

ofthemobileplatformis =5 .........................

5.11Regeneratedsti nessmapsofthecasestudyinFigure5.7afterrelocating

theanchorpositionofcable4...........................

5.12Variationofthesti nessmapareawiththepositionofanchor4.......

5.13Sti nessmapsoftherobotafterfailureofcable1withtheoptimumanchor

positionforcable4................................

5.14Sti nessmapsoftherobotafterfailureofmotor1withtheoptimumanchor

positionforcable4................................

5.15Sti nessmapsoftherobotshowninFigure4.10afterfailureofacablewith

theoptimumlayout................................

5.16Sti nessmapsaboutZdirectionoftherobotshowninFigure5.1(a)after

failureofonecablewithgravityat( =5 )...................

5.17Variationofsti nessmapareawithchangesinorientationofmobileplatform

afterfailureofacable...............................

5.18Sti nessmapsoftherobotaboutZdirectionafterfailurewithoptimum

orientationofmobileplatform..........................

5.19Variationofthesti nessmapareawithorientationofmobileplatformand

positionofanchor4afterfailureofcable1...................

6464656566676868697171727374757677

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5.20Sti nessmapsoftherobotafterfailureofcable1withtheoptimummobile

platformorientationandanchorpositionforcable4..............

6.16.278Coordinateandvariablesofcableiof2DOFplanarcable-drivenparallelrobot.80Optimumcon gurationswherecable1to4arerespectivelyshownbylines

(,,,).Mobileplatformandanchorpositionsare

.....................81respectivelyshownbymarkers( , ).

6.3Followingastraightlinewhilekeepingtheconditionnumbercloseto1.Cable

1to3arerespectivelyshownbylines(,,).Mobileplatform

83andanchorpositionsarerespectivelyshownbymarkers( , )........

6.4Conditionnumberofsti nessmatrixwhilefollowingthestraightlineshown

inFig.6.3.

6.5

6.6....................................8484Areaofsti nessellipsewhilefollowingthestraightlineshowninFig.6.3..AnchorpositionofcableswhilefollowingthestraightlineshowninFig.6.3.

Anchorpositionofcable1to3arerespectivelyshownbylines(,,

)........................................

6.7Followingacircularpath(counterclockwise)whilekeepingthecondition

numbercloseto1.Cable1to3arerespectivelyshownbylines(

,,85).Mobileplatformandanchorpositionsarerespectivelyshown

86bymarkers( , ).................................

6.8Conditionnumberofsti nessmatrixwhilefollowingthecircularpathshown

inFig.6.7.

6.9....................................8788Areaofsti nessellipsewhilefollowingthecircularpathshowninFig.6.7..

6.10AnchorpositionofcableswhilefollowingthecircularpathshowninFig.6.7.

Anchorpositionofcable1to3arerespectivelyshownbylines(,,

88)........................................

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6.11Conditionnumberofsti nessmatrixwhilefollowingthecircularpathby4

cable-drivenrobot.................................

6.12Areaofsti nessellipsewhilefollowingthecircularpathby4cable-driven

robot........................................

6.13Anchorpositionofcableswhilefollowingthecircularpathby4cable-driven

robot.Anchorpositionofcable1to4arerespectivelyshownbylines(,

,,)................................

6.14Con gurationofcableswhenfailureoccursattf=5s.Cable1to4are

respectivelyshownbylines(,,,)............

6.15Conditionnumberwhilefollowingthestraightlinepathandfailureoccurs

atinstanttf=5s.................................

6.16Areaofsti nessellipsefollowingthestraightlinepathandfailureoccursat

instanttf=5s...................................

6.17Anchorpositionofcablesfollowingthestraightlinepathandfailureoccurs

atinstanttf=5s.Anchorpositionofcable1to4arerespectivelyshown

bylines(,,,).......................

6.18Con gurationofcableswhenfailureoccursattf=10s.Cable1to4are

respectivelyshownbylines(,,,)............

6.19Conditionnumberwhilefollowingthecircularpathandfailureoccursat

instanttf=10s..................................

6.20Areaofsti nessellipsefollowingthecircularpathandfailureoccursatin-

stanttf=10s...................................

6.21Anchorpositionofcablesfollowingthecircularepathandfailureoccursat

instanttf=10s.Anchorpositionofcable1to4arerespectivelyshownby

lines(,,,).........................

8990909192939596979899

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Notation

ai=[aiTx,aiy]

Ac=πD2/4

ei=[cos( +θi),sin( +θi)]T

F=[Fx,Fy]T

fcThepositionvectorofanchorAiThecross-sectionalareaofthecableAnchorpointofcableiThelengthofthelinesegmentiAttachmentpointonthemobileplatformCompliancematrixThenominal(outer)diameterofthecablewhichcanbemeasuredbycaliperTheunitvectorinthedirectionfrompointPtotheattach-mentpointonthemobileplatformBiTheequivalentYoung’smodulusofelasticityofthecable,whichcanbeidenti edbytensiontestMagnitudeoftheexternalforceappliedonthemobileplat-formVectorofforceCartesianforcesappliedonthemobileplatform,whichisalsocalledthewrenchactingonthemobileplatformCentrifugalforceappliedtothemobileplatform

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