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Force-closure workspace analysis of
cable-driven parallel mechanisms
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Force-closureworkspaceanalysisofcable-driven
parallelmechanisms
CongBangPhama,*,SongHuatYeoa,GuilinYangb,
MustafaShabbirKurbanhusena,I-MingChena
aSchoolofMechanicalandAerospaceEngineering,NanyangTechnologicalUniversity,Singapore637098,Singapore
bMechatronicsGroup,SingaporeInstituteofManufacturingTechnology,Singapore638075,Singapore
Received9November2004;receivedinrevisedform23March2005;accepted27April2005
Availableonline27June2005
Abstract
A cable-driven parallel mechanism (CDPM) possesses a number of promising advantages over the
conventional rigid-link mechanisms, such as simple and light-weight mechanical structure, high-loading
capacity,andlargereachableworkspace.However,theformulationsandresultsobtainedfortherigid-link
mechanisms cannot be directly applied to CDPMs due to the unilateral property of cables. This paperfocusesontheworkspaceanalysisoffullyrestrainedpositioningmechanisms.Becausethecabletensionisthemostessentialissuetoconstrainthemovingplatform,theforce-closureworkspaceismainlystudied.Ageneralapproachisproposedtochecktheforce-closurecondition.Thisconditionisexpressedintermsoftheconvexhullwhichenclosestheorigin.However,suchaconditionisformulatedandexpressedinhighdimensions.Tosimplifytheanalysis,arecursivedimensionreductionalgorithmisproposedtocheckcon-vexhullsinonedimensionspaces.ThisalgorithmisverifiedthroughsimulationresultsofvariousCDPMs.
/C2112005 Elsevier Ltd. All rights reserved.
Keywords:Force-closure;Workspaceanalysis;Cable-drivenmechanism
0094-114X/$-seefrontmatter /C2112005ElsevierLtd.Allrightsreserved.
doi:10.1016/j.mechmachtheory.2005.04.003*Correspondingauthor.
E-mail addresses: [anonimizat] (C.B. Pham), [anonimizat] (S.H. Yeo), glyang@simtech.a-
star.edu.sg(G.Yang),[anonimizat] (M.S.Kurbanhusen), [anonimizat] (I-MingChen).www.elsevier.com/locate/mechmtMechanismandMachineTheory41(2006)53–69
Mechanism
and
Machine Theory

1.Introduction
Currently,mostofmechanismsusedforrobotmanipulatorsareconventionallinkagemecha-
nismsthatconsistofanumberofrigidlinksandjoints.Basedontheirkinematicstructures,these
mechanismsareclassifiedintotwomajortypes:serialandparallelmechanisms [1].Fig.1showsa
cable-drivenparallelmechanism(CDPM)formedbyreplacingallthesupportinglegsofaparallel
mechanismwithcables.ItisseenthattheCDPMisaclosed-loopmechanisminwhichthemoving
platformisconnectedtothebasebyseveralcables.Thebasesuspendingpointscanbemountedat
theextremitiesofthebase.Therefore,thesemanipulatorscanperformmanipulationtasksrequir-ingalargereachableworkspace.Generally,aCDPMhasthefollowingsignificantadvantages:
•Simplelight-weightmechanicalstructure,resultinginlowenergyconsumption.
•Largereachableworkspace,limitedmainlybycablelengthsandcabletensionconstraints.
•Lowmomentinertiaandhighspeedmotion.
•Easyreconfigurabilitybyrelocatingplatformconnectingpointsandbasesuspendingpoints.
TheadvantagesmaketheCDPMapromisingalternativetotherigid-linkmechanismsinmany
industrialapplications,suchasloadliftingandpositioning [2],coordinatemeasurement [3,4],air-
crafttesting[5],hapticdevices [6,7],androbotrehabilitation [8].
Unlikerigidlinks,cablesarecharacterizedbytheunilateralproperty(canpullbutcannotpush
movingplatforms),andthereforetheformulationsandresultsobtainedforthekinematicsanal-
ysis,workspaceanalysis,trajectoryplanning,etc.oftherigid-linkmechanismscannotbedirectly
applied.Hence,onechallengingissueistodeterminetheposeswherebythemovingplatformis
fullyconstrainedbythecables.ForCDPMs,itisknownthatmaintainingpositivecabletension
iscriticalinconstrainingthemovingplatform.Hence,theforce-closureworkspaceofCDPMsisa
setofposeswherebyresultantcabletensionscansustainanarbitraryexternalwrenchactingon
themovingplatform.Oncetheforce-closureworkspaceisobtained,itsarea(inplanarcases)orits
Fig.1. Acable-drivenparallelmechanism.54 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

volumes(inspatialcases)canbeusedasaprimarycriteriatodesignacable-drivenparallelmech-
anismsuchthattheforce-closureworkspacematchestherequiredworkspace.
Becauseoftheadvantagesanduniquefeaturesofcables,CDPMshavereceivedagreatatten-
tion in robotics literature. The first general classification was given by Ming and Higuchi [9].
Basedonthenumberofcables( m)andthenumberofdegreesoffreedom( n),theCDPMswere
classified into three categories, i.e. the incompletely restrained positioning mechanisms
(m<n+1),the completelyrestrainedpositioningmechanisms( m=n+1)andthe redundantly
restrained positioning mechanisms ( m>n+1). With such classification, a number of different
workspaceshavebeenstudied.OneoftheearlyworksisthatfortheNISTROBOCRANE [2],
which is similar to a Gough–Stewart platform parallel manipulator, but replacing the parallel
linkswithanumberofcables.Tadokoroalsoanalyzedthereachableworkspaceforacable-dri-
ven,six-DOFmotionbaseforvirtualsensationofacceleration [10],andexploredthereachable
workspace of a redundant eight-wire mechanism to derive optimal wire configuration [11].
Dynamic workspace analysis has been performed by Gosselin and Barrette [12]in such a way
that the motion of a moving platform is incorporated into a set of wrenches called a pseudo-
pyramid. The statically reachable workspace is defined by Agrawal and coworkers [13]as the
set all end-effector poses that can be reached statically. Additionally, Verhoeven et al. has
exploredthe controllableworkspaceand otherworkspacepropertiesfor tendonbased Stewart
platformsin[14–16].Recently,Ebert–Uphoffhasreviewedsomebasicworkspaceterminologies
for cable-driven robots [17]and alsogeometrically explored the wrench-feasible workspace for
cable-driven robots by visualizing a net wrench set as a hyper-parallelogram [18]. A similar
researchissuetermedthecable-forceregionisalsoexploredbyOsumietal. [19].However,incon-
straining positive cable tensions, these analyses are almost based on the null space approach
through pseudo-inverse matrices or graphical approaches that are only convenient in some
specificcases.
Theobjectivesofthispaperaretodevelopageneralalgorithmtoexaminetheforce-closure
condition which is sufficient to fully restrain the moving platform, and to generate the force-
closure workspace for CDPMs. Here, the research is focused on fully restrained CDPMs
(mPn+1).Ageneralrecursiveapproachisproposedtochecktheforce-closurecondition.This
conditionisexpressedintermsoftheconvexhullwhichenclosestheorigin.Recursivityimplies
thatcheckingtheconvexhullisrealizedin( n/C01)-dimensionalspacesinsteadofinan n-dimen-
sionalspace.Thisisdonebyreducingonerowandonecolumnoftheoriginalsystematatimevia
Gaussianeliminations.Thisalgorithmisverifiedthroughapplyingittoanalyzetheworkspaceof
both the completely restrained 4-3 cable-driven planar parallel mechanism (4-3-CDPPM) as
shown inFig. 2(a) and the redundantly restrained 8-6 cable-driven spatial parallel mechanism
(8-6-CDSPM)asshownin Fig.2(b).
Inouranalysis,thefollowingassumptionsaremade:
•Each motor controls the length of exactly one cable; therefore, there are mmotors and m
cables.
•Duetothefactthatcablescanonlypullbutcannotpush,allcablesmustremaininpositive
tensionatalltimes.
•Allcablesareassumedtobeinelasticandextendinstraightlinepathsfromthebasesuspending
pointBitotheplatformconnectingpoint Pi.C.B.Phametal./MechanismandMachineTheory41(2006)53–69 55

Theremainingsectionsofthispaperareorganizedasfollows:inSection2,kinematicmodelling
andstaticequilibriumofthemovingplatformareformulated.Force-closureconditionandrecur-
siveconvexhullcheckingalgorithmfollowedbyforce-closureworkspacegenerationarepresented
inSection3.Section4illustratessomeresultsofforce-closureworkspaceobtainedbytherecur-
sivealgorithm.ThepaperissummarizedinSection5.
2.Kinematic modellingandstatic equilibrium
AschematicdiagramofafullyrestrainedCDPMisshownin Fig.3,inwhichthemovingplat-
formisconnectedtothebasethroughdrivingcables, li¼BiPi/C131/C131!ði¼1;2;…;mȚ.Fig. 2. Two examples of symmetric cable-driven parallel mechanisms: (a) 4-3-CDPPM: completely restrained
positioningmechanism,(b)8-6-CDSPM:redundantlyrestrainedpositioningmechanism.
BiPi li
{B}{P}
P
O
Fig.3. KinematicdiagramofaCDPM.56 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

Letframe{B}bethebaseframeandframe{ P}bethemovingplatformframe(attachedatthe
centerofmass Pofthemovingplatform).Theframe{ P}isknownwithrespecttotheframe{ B}
bythekinematictransformationmatrix TB,Pasfollows:
TB;P¼Rp
01/C20/C21
ð1Ț
wherep={xyz}Tisthepositionvectorofpoint Pwithrespecttothebaseframe{ B}and
R¼ca/C1cbð/C0sa/C1ccțca/C1sb/C1scȚð sa/C1scțca/C1sb/C1ccȚ
sa/C1cbðca/C1ccțsa/C1sb/C1scȚð /C0 ca/C1scțsa/C1sb/C1ccȚ
/C0sb cb/C1sc cb/C1cc2
643
75
representstheorientationofframe{ P}withrespecttoframe{ B},inwhich a,bandcarethe
Z/C0Y/C0XEulerangles[20].
Inordertosustainanyexternalwrench( fp,mp)appliedonthemovingplatform,allcablesmust
beabletocreatetensionforcestoachieveequilibriumofthemovingplatform.Referringto Fig.4,
theequilibriumconditionsforforceandtorqueatthemovingplatformareasfollows:
Xm
i¼1tițfp¼0 ð2Ț
Xm
i¼1ri/C2tițmp¼0 ð3Ț
whereti¼tiui¼/C0tili
lirepresentsthetensionforcethatactsintheoppositedirectionof li.Substi-
tutingtiintoEqs.(2)and(3),thefollowingequationisobtained:
A/C1T¼BwithTP0 ð4Ț
Piui
titm
Pm
riP
mpfp
Fig.4. Freebodydiagramofthemovingplatform.C.B.Phametal./MechanismandMachineTheory41(2006)53–69 57

where
A¼u1
r1/C2u1u2
r2/C2u2/C1/C1/C1um
rm/C2um/C20/C21
2Rn/C2m:structurematrix
T¼t1t2/C1/C1/C1tm fgT2Rm:cabletension
B¼/C0fpmp/C8/C9T2Rn:externalwrench
Thedirectionalunitvector ui(i=1,2, …,m)inthestructurematrixdependsonthepostureof
themovingplatformandisobtainedbythevectorloop-closureequationas:
ui¼OB/C131!
i/C0OP/C131!/C0PPi/C131!
kOB/C131!
i/C0OP/C131!/C0PPi/C131!k¼bi/C0p/C0Rrp
i
kbi/C0p/C0Rrp
ikð5Ț
wherethesuperscript pisusedtodenotethatthequantityiswritteninframe{ P}.Hence,the
structurematrix Aisexpressedwithrespecttothepostureofthemovingplatform.
3.Force-closureworkspace
Forparallelmechanismswithrigidlinks,itsworkspaceisthespacewheretheinverse/forward
kinematicsolutionsexist.HoweverforaCDPM,itsworkspaceisthespacewheresetsofpositive
cabletensionsexistbecausetheyareneededtoconstrainthemovingplatformallthetimeregard-
lessofanyexternalwrench.Inotherwords,ifthereisatleastonesetofpositivecabletensionsata
specificposeformingaforceclosure,thenthisposebelongstotheforce-closureworkspace.Gen-
erally,force-closureworkspaceisasetofposesthattheforce-closureconditionissatisfied.
3.1.Force-closurecondition
ACDPMissaidtohaveaforce-closureinaparticularposeifandonlyifanyarbitraryexternal
wrench applied at the moving platform can be sustained through appropriate tension forces
(t1,t2,…,tm)inthecables.Withtheassumptionthattheactuatortorquesareofunlimitedmag-
nitude,fromEq. (4),theconditionofbeingfullyrestrainedismathematicallydescribedas
8Fp2Rn:9t1;t2;…;tm2½0;1Ț :Xm
i¼1tisi¼/C0Fp ð6Ț
wheresi¼ui
ri/C2ui/C26/C27
:theithcolumnvectorofthestructurematrix A.
Eq.(6)implies that the set of vectors tisimust positively span Rn. Subsequently, this set is
positivelydependent(bychoosing Fp=0)suchthat:
rankðAȚ¼nand ð7Ț
Xm
i¼1tisi¼0 ð8Ț
Fromtheconvextheory [21],thefollowingtheoremshowsthatthesetofallconvexcombina-
tionsinthelefthandsideofEq. (8)isactuallyaconvexhull.58 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

Theorem 1. Let C ={x1,x2,…,xm}be a finite collection of points in Rn. Then the convex hull of the
xi(i = 1,2, …,m), is the set of all their convex combinations, i.e.
H¼coðCȚ¼ xjx¼Xm
i¼1kixi;Xm
i¼1ki¼1;06ki2Rforall i()
Eq.(8)isequivalentlyrepresentedasfollows:
Xm
i¼1tisi¼Xm
i¼1ti
Pm
i¼1tisi¼Xm
i¼1kisi¼0 with ki¼ti
Pm
i¼1tið9Ț
FromthenotationofTheorem1,Eq. (9)formsaconvexhullbasedonthestructurematrix A.
Furthermore,theoriginbelongstothatconvexhullbecausetheoriginistheconvexcombination
ofsi.Hence,thefollowingpropositionshowsanutilizationofconvexhullinsatisfyingtheforce-
closurecondition.
Proposition 1. A moving platform with n degrees of freedom is fully restrained if and only if the
convex hull formed by column vectors of the structure matrix A,co{s1,s2,…,sm}, contains a
neighborhood of the origin.
Proof. Necessity :Becausethe platformisfullyrestrained,asmallexternalwrenchO( e)canbe
expressedbyEq. (6)suchthat:
Xm
i¼1tisi¼OðeȚð 10Ț
DividingEq. (10)byPm
i¼1tigives:
Xm
i¼1ti
Pm
i¼1tisi¼OðeȚ
Pm
i¼1ti’OðeȚð 11Ț
It is obvious that the neighborhood of the origin belongs to the convex hull (according to
Theorem1).
Sufficiency:AstheneighborhoodoftheoriginO( e)isenclosedbytheconvexhull,itcanbe
expressed:
Xm
i¼1tisi¼OðeȚwithXm
i¼1ti¼1and 8ti>0 ð12Ț
Eq.(12)impliesthattherealwaysexistsasetofpositivetensionsbeingabletosustainasmall
externalwrenchinanydirection.Therefore,themovingplatformisfullyrestrained. h
3.2.Recursivealgorithmcheckingforce-closurecondition
As discussed above, the force-closure condition is satisfied if the n-dimensional convex hull,
formedbythestructurematrix,enclosestheorigin.However,ifthenumberoftask-spacedimen-
sionsnisgreaterthanthree,itisimpossibletorepresenttheconvexhullgeometrically.Therefore,C.B.Phametal./MechanismandMachineTheory41(2006)53–69 59

ageneralalgorithmisproposedtocheckwhetherthemovingplatform,ataparticularpose,isfully
restrained.Inotherwords,itiswhethertheconvexhullenclosestheoriginofthetask-spaceframe.
To solve the above problem ingeneral, the resultedidea is that if the size of the task-space
dimensioncanbereducedintoone( n=1),i.e.theconvexhullbecomesaclosedlinesegment,
itiseasiertochecktheoriginenclosure,i.e.whetherthislinesegmentpassestheoriginasillus-
tratedinFig.5.Therefore,thedimensionreductionofastructurematrix A(nrows,mcolumns)
isdescribedintwostepsasfollows:
•Step1:Foreachcolumnvectorinaspaceof Rn,thereisauniquehyperplanepassingthrough
theoriginandbeingorthogonaltothisvector.Thishyperplaneisasubspaceof Rn/C01.Bypro-
jectingtheothercolumnvectorsontothishyperplane,projectedvectorsareexpressedinthis
newsubspaceof Rn/C01. Inaddition,thevectorwhichisperpendicular tothishyperplanecan
beignoredduetoitszeroprojectedvalue.Therefore,thereareonly m/C01columnvectorsin
thesubspaceof Rn/C01.Totally,fromtheoriginalspacein Rn,thisstepresultsin msubspaces
inRn/C01formedcorrespondingto mcolumnvectors.
•Step2:Foreachsubspaceformedabove,thereareonly m/C01columnvectors.Therefore,the
convexhullwhichisformedbythese m/C01vectorsisalsorequiredtoenclosetheoriginofthe
subspaceinordertohaveaforce-closure.Generally, mnewconvexhullsarecheckedforsat-
isfactionofforce-closure.Ifallofthemenclosestheoriginoftherespectivesubspaces,theori-
ginalsystemformsaforce-closure.Otherwise,ifanyofthemdoesnothavetheforce-closure,thismeansthatacertainexternalwrenchcannotberesistedinthissubspace.Consequently,the
originalproblemdoesnotsatisfytheforce-closurecondition.
Theseproceduresareillustratedin Fig.6.Itcanbeseenfrom Fig.6(a)thatifthethreevectors
(OS/C131!
i)areprojectedontothelinewhichisperpendiculartoanyvector OS/C131!
i,therealwaysexistboth
positivecomponentsandnegativecomponents.Thismeansthatanyexternalforcecanberesisted
bysomeofthesecabletensions.Ontheotherhand,asshownin Fig.6(b),ifthethreevectorsare
projectedontothelinewhichisperpendicularto OS/C131!
1orOS/C131!
3,allcomponentshavethesamesign.
Thismeansthatallcabletensionscannotresisttheexternalforceinthesamedirectiononthepro-
jectedline.Forbothcases,theoriginalconvexhullsareinthespaceof R2.Whenprojectedonto
theline,theoriginalproblemisequivalenttothreenewconvexhullswhichareexpressedasline
segmentsinthesubspaceof R1.Asaresult,checkingforceclosureinthespaceof R1issimpleasit
onlyinvolvescheckingthesignofcomponents.
Asmentionedinpreviousdiscussion,thefirststepistoreducethespacedimensionandthesec-
ondstepistochecktheconvexhullsformedinthesubspaces.Ifthedimensionofthesubspacesis
O+ f
fp)(
1−f)(
2−f)(
j−f
)(
1+f)(
2+f)(
i+f
Fig.5. Co( fð/C0Ț
1;fðțȚ
1;…;fð/C0Ț
j;fðțȚ
i)isaclosedlinesegmentpassingtheorigin.60 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

morethanone,thesesubspacesarefurtherreducedintoothersubspacesbyrepeatingstep1and
soon.Asshownin Fig.7,oneoriginalsystemin Rnisdecomposedto msubsystemsin Rn/C01.Then
each subsystem in Rn/C01is decomposed to ( m/C01) subsystems in Rn/C02. Therefore, using more
cables considerately increases the number of subsystems in R1. Typically, for a 4-3-CDPPM,
thenumberofsubsystemsin R1is12whereasthatfora8-6-CDSPMis6720.
Mathematically,itisnotedthatprojectingcolumnvectorsontoahyperplanewhichisperpendic-
ulartothecolumnvector jisaGaussianeliminationforthecorrespondingcolumnvector.Asare-
sult,aconvexhullcheckingprocedureisproposedthatisbasedonarecursivealgorithm.This
algorithmisnamed‘‘FCC[ A(n,m)]’’(Force-ClosureCheck).Generally,bygivingtheoriginalstruc-
turematrix A(nrows,mcolumns)tothealgorithm,thisalgorithmwillreducethespacecontinu-
ouslyuntilallsubspacesareone-dimensional.Theflowchartofthisalgorithmisdescribedin Fig.8.
AsshowninFig.8,therecursivealgorithmFCCconsistsoftwoprocedures:
•One-dimensioncase: Thisisthesimplestcaseinwhich
(a) Thereisatleastonerowconsistingofthesamesigncomponents.Thisimpliesthatthe
convexhulldoesnotenclosetheorigin.Thealgorithmreturnsavalue /C2120/C213,or
Fig.7. Diagramofdecomposingspacedimension.Fig.6. Twosamplesofconvexhull.(a) O2co(S1,S2,S3).(b)O62co(S1,S2,S3).C.B.Phametal./MechanismandMachineTheory41(2006)53–69 61

(b) Thenumberofrowsisone.Ifthesignsamongcomponentsarechanged,thelinesegment
enclosestheorigin.Thealgorithmreturnsavalue /C2121/C213.Otherwise,itreturnsavalue /C2120/C213,i.e.
thesignsareunchanged.
•Higherdimensioncase: Thisisthecasesuchthatthenumberofrowsisgreaterthanoneandthe
signsarechangedineachrow.ReducingspacedimensionisdonebyGaussianeliminationuntil
one-dimensional systems. If there is one subsystem which does not satisfy the force-closure
condition,thealgorithmreturnsavalue /C2120/C213.
ForconvenientprogramminginMatlab,theGaussianeliminationisalwaysdoneforthefirst
column.Therefore,acolumnindex jisusedtoincreasethecolumnorder,thenthiscolumnis
swapped with the first column. To demonstrate this algorithm, detailed checking steps of two
examplesshownin Fig.6arepresentedinAppendix A.FCC[A(n,m)]
Column order, j = 1FCC = 0
FCC = 1If sign is changed
in each row?
Gaussian elimination
of the first columnj = j + 1, then
swap two columns 1 and j
FCC = 0 FCC = 1FCC[A(n–1,m–1)] = 1? If the last column
is done ( j = m)?If the number of
rows is one?YesNo
Yes
No
NoYes
YesNo
If
Fig.8. Recursivealgorithm‘‘force-closurecheck’’—FCC.62 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

3.3.Generatingforce-closureworkspace
Informulatingtheforce-closureworkspace,asetoffullyrestrainedposturesneedstobeob-
tained.Positionalandrotationalrangesarediscretizedintosegments.Ateachoftheircombina-
tions,acorrespondingstructurematrix Aisdetermined.Byapplyingtheforce-closurealgorithm,
returningvalue /C2121/C213meansthisposturebelongstotheforce-closureworkspace,andreturningvalue
/C2120/C213impliesthisposturedoesnotbelongtotheforce-closureworkspace.Althoughthisworkspace
generationtakestimetofinish,thisapproachisverysimpleandeffectivetoobtaintheworkspace
areaorvolumewhennecessary.Iftheresolutionislargeenough,itsareaoritsvolumeisclosedtoanalyticalresultswhicharedifficulttocalculateforgeneralcable-drivenparallelmechanisms.
Inthispaper,twotypicalmechanismsareusedtoverifytherecursiveforce-closurealgorithm.
Firstly,thealgorithmisappliedtothecompletelyrestrainedcable-drivenplanarparallelmecha-
nism(3degreesoffreedom,4cables).Forthismechanism,theorientationoftheplatformisfixed,
asetofpositions( x,y)ofthecenterofmassofthemovingplatform Pdefinestheforce-closure
workspace.Secondly,thealgorithmisappliedtotheredundantlyrestrainedcable-drivenspatial
parallelmechanism(6degreesoffreedom,8cables).Similarly,thethreeorientationsarekeptcon-
stant,andtheforce-closureworkspaceisdefinedbythesetofpositions( x,y,z)ofthecenterof
massofthemovingplatform P.
4.Simulationandresults
In this section, the recursive force-closure algorithm is applied to generate the force-closure
workspace for the two typical mechanisms shown in Fig. 2. Simulated results shows the effect
oftherecursivealgorithm.
Inthefollowingillustrations,themechanismshavethesymmetricdesign.Fortheplanarmech-
anismshownin Fig.9(a),thebaseisasquareof1 ·1(m),andthemovingplatformisarectangle
thathasthedimensionof0.3 ·0.2(m).Forthespatialmechanismshownin Fig.9(b),thebaseisa
unitcubeof1 ·1·1(m),andthemovingplatformisaparallelepipedthathasthedimensionof
0.3·0.2·0.1(m).Thebaseframeislocatedatthepoint B
1.Thecoordinatesofverticesarede-
scribedinFig.9.Thelocalframeislocatedatthecenter Poftherectangle(intheplanarcase)and
Fig.9. Geometricalparameters.(a)Asymmetric4-3-CDPPMand(b)Asymmetric8-6-CDSPM.C.B.Phametal./MechanismandMachineTheory41(2006)53–69 63

Fig.10. Two-dimensionalforce-closureworkspacesofthe4-3-CDPPMs.(a) /=0/C176,(b)/=+3 /C176and(c) /=/C03/C176.64 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

Fig.11. Three-dimensionalforce-closureworkspacesofthe8-6-CDSPM.(a) a=0/C176,b=0/C176,c=0/C176,(b)a=+3 /C176,b=0/C176,
c=0/C176and(c) a=/C03/C176,b=0/C176,c=0/C176.C.B.Phametal./MechanismandMachineTheory41(2006)53–69 65

ofthecube(inthespatialcase).Therefore,asetofreachablepositionsofthiscenter Patacertain
orientationofthemovingplatformconstitutestheforce-closureworkspace.
Asshownin Fig.10,theforce-closureworkspaceisgeneratedatvariousorientationsofthe
movingplatform.The2-dimensionalworkspacesobtainedat /=0/C176,+3/C176and/C03/C176areillustrated
inFigs.10(a)–(c),respectively.
Inordertoverifythevalidityoftherecursivealgorithmforredundantlyrestrainedpositioning
mechanisms, the algorithm is also applied to a 8-6-CDSPM. Subsequently, the force-closure
workspacesareillustratedin Fig.11.Bygiving c=0/C176(theorientationabout x)and b=0/C176(the
orientation about y), the three-dimensional workspaces obtained at a=0/C176,+ 3/C176and/C03/C176(the
orientationabout z)areillustratedin Figs.11(a)–(c),respectively.
AsshowninFigs.10and11 ,itisrealizedthattheforce-closureworkspaceisalwayssmaller
thanandinsidetheconvexhullformedbythebase.Thisisusefultosettheinitialpossibledis-
placementrangesfordiscretizing.Moreover,with c=b=0/C176,the8-6-CDSPMlookedfromthe
topisactuallyequivalenttothe4-3-CDPPM.Therefore,thecrosssectionsofthethree-dimen-
sionalworkspacesin Fig.11areexactlythesameasthetwo-dimensionalworkspacesin Fig.10
duetotheirsimilardimensions.
Fromthepreviousillustrations,itcanbeseenthattherecursivealgorithmcanbegenerallyap-
plied to completely restrained positioning mechanisms and redundantly restrained positioning
mechanisms.These results canbe verifiedthrough the null spaceapproach—thehomogeneous
solutionofunderdeterminedlinearsystems [9].
5.Conclusion
Thegeneralalgorithmtogeneratetheforce-closureworkspaceofcable-drivenparallelmecha-
nismsisthemajorsubjectofthispaper.Asetofdiscretepostureswhichsatisfytheforce-closure
conditionconstitutestheforce-closureworkspace.Therecursivealgorithmhasbeenappliedsuc-
cessfullytochecktheforce-closureandthengeneratetheworkspaceofa4-3-CDPPM,whichisa
completely restrained positioning mechanism, and a 8-6-CDSPM, which is a redundantly re-strained positioning mechanism. Due to the generic formulation approach, the proposed algo-
rithm can be used for both completely restrained positioning mechanisms ( m=n+1) and
redundantlyrestrainedpositioningmechanisms( m>n+1).Asthecomputationalresolutionis
increased,theresultedworkspacesaremoreprecise.
AsdiscussedinSection3.2theconvexhullcanalsobevisualizedifthetask-spacedimensionisless
thanorequaltothree.However,thisgeometricalvisualizationbecomesineffectiveasthenumberof
cablesincreasesbecauseoftheconvexhullformulation.Ontheotherhand,theproposedalgorithm
canstillbeappliedforincompletelyrestrainedpositioningmechanisms( m<n+1)bycombining
externalforcesandcabletensionsinstantaneously.Inbrief,thisapproachcanbegenerallyused
tocreatetheforce-closureworkspaceforvariouscable-drivenparallelmechanisms.
Acknowledgements
The authors would like to thank the School of Mechanical and Aerospace Engineering at
NanyangTechnologicalUniversityforsponsoringthisresearchprogram.66 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

AppendixA.Force-closurechecking stepsof examplesin Fig.6
Thesequentialprocedurestochecktheconvexhull(orforce-closure)forsamplecasesin Fig.6
areshowninFigs.A.1andA.2 .InFig.A.1,bothpositiveandnegativecomponentsexistineach
Fig.A.1. (a)Theoriginalstructurematrix.(b)Step1:Thesignsarechangedinthesecondrow.(c)Step2:Thesignsare
changedinthesecondrow.(d)Step3:Thesignsarechangedinthesecondrow.Computationalstepsforthecasein
Fig.6(a).C.B.Phametal./MechanismandMachineTheory41(2006)53–69 67

sub-matrix (the second row), whereas there exists at least one sub-matrix which composes all
negativeorallpositivecomponentsasseenin Fig.A.2.Fig.A.2. (a)Theoriginalstructurematrix.(b)Step1:Thesignsarethesameinthesecondrow.(c)Step2:Thesignsare
changedinthesecondrow.(d)Step3:Thesignsarethesameinthesecondrow.ComputationalstepsforthecaseinFig.6(b).68 C.B.Phametal./MechanismandMachineTheory41(2006)53–69

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