The Grand 4R Four-Bar Based Inherently Balanced Linkage Architecture for synthesis of shaking force balanced and gravity force balanced mechanisms

van der Wijk, Volkert

DOI

10.1016/j.mechmachtheory.2020.103815

Publication date

2020

Document Version

Final published version

Published in

Mechanism and Machine Theory

Citation (APA)

van der Wijk, V. (2020). The Grand 4R Four-Bar Based Inherently Balanced Linkage Architecture for

synthesis of shaking force balanced and gravity force balanced mechanisms. Mechanism and Machine

Theory, 150, [103815]. https://doi.org/10.1016/j.mechmachtheory.2020.103815

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Research

paper

The

Grand

4R

Four-Bar

Based

Inherently

Balanced

Linkage

Architecture

for

synthesis

of

shaking

force

balanced

and

gravity

force

balanced

mechanisms

Volkert

van

der

Wijk

Mechatronic System Design, Department of Precision and Microsystems Engineering, Faculty of Mechanical, Maritime, and Materials

Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 6 October 2019 Revised 19 January 2020 Accepted 27 January 2020 Keywords:

Shaking force balance Gravity balance Inherent balance Principal vectors Four-bar deployable linkage Center of mass

a

b

s

t

r

a

c

t

Consideringbalancingasstartingpointinthedesignofmechanismsandmanipulatorsis knownasinherentbalancing.Inherentlybalancedlinkagearchitecturesthenformthe ba-sisfromwhichbalancedmechanismsolutionsaresynthesized,needingnocountermasses contrarytobalancingofgivenmechanisms.Inthispaperanewandadvancedinherently balanced linkagearchitecture with the 4Rfour-bar linkage as abasis is presented,the Grand4R Four-BarBased Inherently BalancedLinkage Architecture.With24 links itis 26 timesoverconstrainedyetmovablewithstationarycenterofmass.Itisshownthatall the-oriesfortracingthecenterofmassofafour-barlinkagearefoundinside.Itisshownalso howfromthisarchitectureavarietyofnewnormallyconstrained2-DoFbalancedlinkages arederivedbyremovingaselectionoflinks.Thisisdoneforthesituationsthatalllinks aremasssymmetric,forwhich32solutionsarepresented,andthatalllinkshavea gen-eralmassdistribution.Asanexample,thebalanceconditionsfortheTWIN-4B,asolution oftwosimilar4Rfour-barlinkages,oneinsidetheother,arederivedanditisshownhow thissolutioncanbetransformedintoaspatialinherentlybalanceddoubleBennettlinkage. © 2020TheAuthor.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Balanceddevices are usedin a variety of applications.Shaking force balance together withshaking moment balance, whenmechanismsmovewithoutresultantreactionforcesandreactionmomentsatthebaseeliminatingbasevibrations,is foundinhighspeedrobotmanipulators[1–4],amongothersforlowsettling time[5],dynamicdecoupling,andimproved stability[6].Itisalsoknownforreducing noiseinmachinery[7]andforincreasing ergonomiccomfortinhandtools[8]. Gravityforcebalance,whengravitydoesnotinfluencethemotionofthemechanism,isappliedinroboticmanipulatorsfor exampleto lower actuatortorques [9,10], toincrease payload capacity[11], andtoimprove thecalibration accuracy[12]. Forlargedeployablestructuressuchasinkineticarchitecturegravityforcebalanceisimportantforstability,safety,andlow actuationpower[13].

Although shaking force balance is aboutdynamic (inertial) forces and gravity force balance is about forces within a staticforce fieldforwhichthey aretwocompletelydifferentphenomena, theysharethesamedesignsolutions.Namely, a

E-mail address: v.vanderwijk@tudelft.nl https://doi.org/10.1016/j.mechmachtheory.2020.103815

0094-114X/© 2020 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Fig. 1. (a) Principal vectors a 1 , a 21 , a 23 , and a 3 describe the motion of the link CoMs with respect to the common CoM in S through principal points P i ; (b) Closed principal vector linkage where the principal vectors have been transformed into rigid links with revolute pairs and mass (both drawn to scale). shakingforcebalancedmechanismisgravityforcebalancedaswell.Thereversehoweverisnottrue,gravityforcebalanced mechanismsarenotshakingforcebalancedinthedirectionperpendiculartothegravityforceandtheyarealsonotshaking force balanced ifelementsother than (counter-)masses are usedsuch assprings [14].Since thispaperconsiders balance solutionswithsolelymasses,forcebalanceisusedtorepresentbothshakingforcebalanceandgravityforcebalance.

Themotivationofthisworkoriginatedfromresearchontheshakingforcebalancingofmulti-degree-of-freedom (multi-DoF) mechanisms andmanipulators. It wasobserved that common balance methods andapproaches for 1-DoF linkages

[15,16] lead, becauseof the required counter-masses andcounterrotations, to a significant addition of mass, inertia, and complexitywhenappliedtomulti-DoFlinkages[17–22]andalsotosignificantelastodynamicproblems[23].Thebetter bal-ancesolutionsshowedsomeadvantageouskinematicorgeometricproperties.Supportedbytheideathatthekinematicsof multi-DoFmanipulatorsisveryflexible-toperformaspecifictasknumerouskinematicsolutionsofamulti-DoF(parallel) manipulatorcanbefound-thisledtotheinherentdynamicbalanceapproachwherethepropertiesofbalanceformthe ba-sisandarethestartingpointinthedesignofbalancedmechanisms,whichiscontrarytothecommonapproachofaimingat findingabalancesolutionforagivenmechanismdesign[13].Thebalancepropertiesarecomposedin,so-called,inherently balanced linkagearchitectures:movablecompositionsoflinkswhichsolelyincludethekinematicandgeometric relations forbalance.Fromtheselinkagearchitecturesthen,asanextstep,designsolutionsforanintendedtaskcanbederivedina varietyofwaysbypossiblyalteringanything-forexampleby eliminating,relocating,andresizinglinks,introducinggears orsliders-andaslongastherelationsforbalancearemaintainedallresultsremaininherentlybalanced.

The methodofprincipalvectors[24] was foundfundamentalinexpressingthepropertiesforbalance andforcreating the inherentlybalanced linkages. First open-chainprincipal vector linkages were developed,showinga strong relationto pantographlinkagesandtheirpropertiesofsimilarity[25].Promisingresultswereobtainedbysynthesisofa2-DoF dynam-icallybalancedgrasperandtheDUAL-Vhighspeed3-DoFdynamicallybalancedmanipulator[3]andalsobynewdesignsof largebalancedstructuressuchasamovablebridgeandadeployableroofstructure[13].Subsequently,variousclosed-chain principalvector linkages[26] andspecific multi-looplinkagearchitecturesbasedonold graphicalmethodsfortracingthe common centerof mass(CoM) of linkages were developed [13,27].Investigating these latterlinkage architecturesled to theawarenessthatall ofthemshouldbepartofasingleextensivelinkagearchitecture,aGrandInherentlyBalanced Link-ageArchitecture. The preliminaryresultsforthe Grand4R Four-Bar Based InherentlyBalancedLinkage Architecture were presentedin[28].

ThegoalofthispaperistopresentthecompleteGrandInherentlyBalancedLinkageArchitecturebasedonthe4R four-bar linkageand toshow in detail howit leadsto a variety ofinherently balanced mechanismsolutions. First the Grand Architectureisshownandexplainedformasssymmetriclinks,i.e.linkswiththeirCoMonthelinethroughthejoints. Sub-sequentlyitisshownhowavarietyofinherentlybalancedmechanismsolutionscanbederivedfromtheGrandArchitecture andhowthe knowntheoriesfortracing thecommonCoM ofafour-bar linkagearerelated. Then theGrandArchitecture isgeneralizedformassasymmetriclinksanditisshownhowtheinherentlybalancedmechanismsolutionsfromthe gen-eralizedarchitecture differfromthesolutions fromtheGrandArchitecture withmasssymmetriclinks. Foroneinherently balanced mechanismsolution,named theTWIN-4Blinkage,itisexplainedhowthe balanceconditionscanbe derived.At the end an example ofa spatial inherently balanced linkage solutionis given by transforming the TWIN-4B into a spa-tialinherentlybalanceddoubleBennettlinkage.TheAppendixpresents322-DoFinherentlybalancedmechanismsolutions obtainedfromtheGrandArchitectureformasssymmetriclinks,togetherwiththeirbalanceconditions.

2. CompositionoftheGrand4RFour-barBasedInherentlyBalancedLinkageArchitecture

The motions ofthe CoMs of moving mechanismelementswith respectto their common CoMcan be described with vectorsofconstant magnitudeknown astheprincipalvectors [13,24].Forinstance, themotionofthe linkmassesof four elementsinaclosedchaincanbedescribedwithrespecttotheircommonCoMwiththefourprincipalvectorsa1,a21,a23,

anda3 [13,26]. Thisis illustratedin Fig.1a wherefourlinks withlengthsli forma planar linkagewithrevolute pairs in jointsA0,A1,A2,andA3.Eachlinkherehasamassmi andismasssymmetricwithrespecttothelinethroughits joints, whichmeansthatthelinkCoMSiislocatedalongthelinethroughthejoints.Thefourprincipalvectorsdescribeaspecial invariant linkpoint Piin threeofthe fourlinks, namedthe principalpoints, withwhichthey formthree parallelograms.

Fig. 2. Linkage architecture based on the combination of the four different principal vector sets. 8 links come together with revolute pairs in S and they can also be merged into the four single links B 1 SB 6 , B 2 SB 5 , B 3 SB 8 , and B 4 SB 7 .

TheprincipalpointscanbedescribedwithrespecttothecommonCoMofallfourlinks, whichisinS,withtheprincipal vectorsas:

P1= −a 21eiθ2+ a3eiθ3 P2 =a1eiθ1+a3eiθ3

P3 = a1eiθ1+ a23eiθ2 (1)

withthe principal dimensionsa1=



a1



, a21=



a21



, a23=



a23



, and a3=



a3



which are the distances betweenthe

pointsP1andA1,P2 andA1,P2andA2,andP3andA2,respectively,andareconstantsforanymotionofthemechanism.The

distancesbetweenthepointsP1andS1,P2andS2,andP3andS3arenamedp1,p2,andp3,respectively.Theseequationsare

linearcombinationsofthetime dependentunit vectorofeach rotationalDoF

θ

i.Theyholdforboththesituationthatlink 4iseliminatedandtheresultingopenchainhasthreeDoFsaswellasforthedepictedclosedchainwithtwoDoFswhere

θ1

,

θ2

,and

θ

3havedependency.

Because oftheir constant lengths, the principal vectorscan be transformed into realrigid linksas shown inFig. 1b. Theseprincipalvectorlinkshavelengthsa1,a21,a23,anda3andhaverevolutepairsinP1,P2,P3,B1,B2,andS.Shereisthe

commonCoMofall10linksandisaninvariantlinkpointinbothlinksB1SandB2S.Thecompletelinkageisforcebalanced

aboutSandwhenSiskeptasafixedpointinthebasethenaforcebalancedlinkagewithtwoDoFsisobtained.

ThelinkageinFig.1bisreferredtoasaclosed-chainprincipalvectorlinkageofwhichallthelinksaremasssymmetric. Themassesoftheprincipalvectorlinksm11,m12,m13,m31,m32,andm33,havetheirCoMalongthelinethroughtheirjoints

atadistancep11,p12,p13,p31,p32,andp33,respectively,fromthejointsasindicatedbythearrows.Thebalanceconditions

forwhichthe commonCoM isinjointSforall motionscanbederived fromtheconditionsofthegeneralversionwhere thelinkshaveagenerallylocatedlinkCoM[13,26]as:

m1p1 =

(

m2+ m3+ m4+ m31+ m32

)

a1− m 4 e4 l4 l1+ m12p12+ m13p13 m2p2 =



m1+m4 e4 l4



a21 −



m3+m4



1 −e4 l4



a23 +m11p11− m31p31 m3p3 =

(

m1 +m2+m4 +m11+m12

)

a3 − m4



1 −e4 l4



l3 +m32p32+m33p33 (2)

wherea21 anda23 are relatedbyl2 asa21+a23=l2. IntheAppendix thelinkageofFig. 1bisshownaslinkagesolution

1 withan alternative notation of these balance conditions. To give a realistic image, Fig. 1 wasdrawn to scale for m1,

m2,m3,andm4 locatedhalfwaytheir linkswithvalues proportional totheir link lengths(ml₁1 = m₂ l₂ = m₃ l₃ = m₄ l₄) withthe other massesequalto zero.Ifthemassof theother linksisincluded thenS shiftsa littlecloserto link2 forwhichthe parallelogramsbecomeslightlysmaller.

InFig.1bonlyoneofthefourpossiblewaystoconstructprincipalvectorlinksinaclosedchainoffourlinksisshown, whichcanbeseenasthattheparallelogramsarecentered aboutlink2.Itisalsopossibletoconstructtheparallelograms aboutlink1,link3,orlink4ofwhichtheresultslooksimilar. Whenallfourofthemarecombinedintothesamelinkage, thentheresultinFig. 2isobtainedwhereeightlinksconnect withrevolutepairs inS.Thislinkagearchitecture isbased onthe four differentprincipal vector sets andhas three principalpoints ineach of thefour-bar links. For clarityof the illustrationthemassesoftheprincipalvectorlinkswerenotdrawn.

WiththeeightlinksconnectinginSthelinkagearchitectureis6timesoverconstrainedyetmovable.Sinceeachopposite pairoflinksmovesasifitisasinglerigidlink,theeightlinkscanbetransformedintothefoursinglelinksB1SB6,B2SB5,

B3SB8,andB4SB7withSastheircommonjoint.Inthatcasethelinkagearchitectureis10timesoverconstrainedyetmovable.

Inthe architecture ofFig.2 linksP31B2 and P43B7 can be extended toa commonjoint C1.AlsolinksP11B1 andP41B4

canbe extended toa commonjointC2,linksP22B3 andP12B6 canbe extended toa commonjointin C3 andlinksP33B5

andP23B8 can beextended toa commonjointinC4.InFig. 3theseextensionsareincluded withrevolutepairs injoints

Ci.RevolutepairsinjointsD1,D2,D3 andD4 arealsoincluded,whichispossiblesincethelinksintersectatinvariantlink

Fig. 3. The Grand 4R Four-Bar Based Inherently Balanced Linkage Architecture including the extended links with joints C i and the joints D i . The common CoM of all 24 links in S for all motions.

Fig. 4. (a) Inherently balanced linkage with S as a coupler point of a similar four-bar linkage, connected through double contour linkage P 43 C 1 P 31 ; (b)

Inherently balanced linkage with three pantographs, shown with the two possible pantographs in the center which form a Burmester’s focal mechanism.

TheresultingarchitectureinFig.3willbereferred toastheGrand4RFour-barBasedInherentlyBalancedLinkage Archi-tecture.Itshowsthecompletecompositionofthekinematicconditionsforinherentbalancewitha4R four-barlinkageas abasis.WiththefourlinkspassingthroughSthearchitectureis26timesoverconstrainedyetmovable.AlsohereSisthe commonCoMofall24linksforanymotion.Inthenext sectionitisshownthatbecauseofthehighnumberof overcon-straints awide variety ofnormallyconstrainedinherently balancedmechanismsolutions can bederived fromthisGrand Architecture.

3. Synthesisofnormallyconstrainedinherentlybalancedlinkagesolutions

TheGrandArchitectureinFig.3canbeusedasthestartingpointinthedesignofabalancedmechanismormanipulator foranintendedtask.EverymechanismthatisderivedfromtheGrandArchitecturewithoutviolatingthekinematicgeometry has inherent balance properties.This means that then with a specific combinationof mass parameter values,calculated fromthebalanceconditions,theresultingmechanismisbalancedwithrespecttojointS,thebasepivot,withouttheneed ofadditionalmassesorelements.Withthisapproachbalancingisconsideredbeforethe(final)designofthemechanismof whichthe topologythen dependsprimarily onadvantageous balancecapability insteadofother reasonsforthechoiceof theinitialmechanismtopology.

Firstit isshownhownormallyconstrained2-DoFlinkagesolutions canbederived fromtheGrandArchitecturewhich relatetoknowntheoryonthemotionofthecommonCoM,followedbyexamplesofother2-DoFlinkagesolutions.When fromtheGrandArchitectureinFig.3allinternallinksareremovedbutlinksP31C1,C1P43,B7B4,andB4P41,thenthenormally

constrainedlinkageinFig.4aisobtainedwhereS,aninvariantpointinlinkB7B4,isthecommonCoMofall8links.Inthis

solutionafour-barP43B7B4P41canbeobservedthatissimilartoandsmallerthanthefourbarA0A1A2A3,sharinglinkA3A0.

ThisisrelatedtothetheoryofKreutzinger[29]whofoundthat thecommonCoMofthethreemovinglinksofafour-bar linkage tracesa couplercurve of asimilar four-bar linkagewhich is alsosimilar to a couplercurve ofthe linkageitself. WhereKreutzingerusedthissimilarfourbarlinkageasagraphicaltoolforanalysis,herethesimilarfourbarisareallinkage ofwhichthemassesareincludedaswell.

TheextensionoflinkP43B7toC1andlinkC1P31togetherconstrainthefour-barlinkageP43B7B4P41tomovesynchronously

withfour-barlinkageA0A1A2A3.Infact,linksP43C1andC1P31formtogetherwithlinksA0A3andA2A3anothersimilar

four-barlinkage.ThisisrelatedtothetheoryofShchepetil’nikov[30]whonameditadoublecontourtransformationwherelinks P43C1andC1P31arethedoublecontouroflinksA0A1andA1A2andsubsequentlylinksP41B4andB4B7arethedoublecontour

Fig. 5. (a) Inherently balanced linkage with a single pantograph to two opposite four-bar links; (b) Inherently balanced linkage with design variation of the pantograph in Fig. 5 a.

Fig. 6. (a) Design variation of the inherently balanced linkage in Fig. 1 b; (b) Inherently balanced linkage solution with a single pantograph to two connect- ing four-bar links.

andlesscumbersomeascomparedtothemethodofprincipalvectors,whichwasalsoinventedforgraphicalanalysisofthe motionofthecommonCoMofasystemofrigidbodies[31].

WithintheGrandArchitecture inFig. 3numeroussimilar four-bar linkagescan be found. Ofthesolution inFig.4a 8 versionscanbefoundwhichareessentiallyequal,twowithrespecttoeachlinkofthefour-barA0A1A2A3,aleftandaright

one.ForinstancetheversionwithlinksP11B1,B1B6,P12C3,andC3P22isleftorientedandwithrespecttofour-barlinkA0A1.

WhenwithintheGrandArchitecturealllinksareremovedbutlinksP11D1,P23D1,D1C4,C4D3,B8S,B5S,P33D3,andP41D3,

thenthesolutioninFig.4bisobtained.Thissolutioncanberegardedasacombinationofthreebalancedpantographswhere pantographA0A1A2P11D1P23tracesthecommonCoMofitsfourlinksinD1.SimilarlythepantographA2A3A0P33D3P41traces

thecommonCoMofitsfourlinksinD3.ThethirdpantographD1C4D3B5SB8connectstothemasscentersofthetwoother

pantographsandtracesthecommonCoMofallthreepantographsinS.Thissolutionisrelatedtothegraphicalmethodof FishertodeterminethecommonCoMofallbodysegmentsofahumaninmotion[24].

Itisnotedthat inFig.4blinksP23D1andD1C4 aretwoseparate linksandalsoP33D3andD3C4 aretwoseparate links.

Sincethesizeofapantographdoesnotaffectitspropertiesofsimilarityorscaling,thelinklengthsofthecentralpantograph canbe chosenfreely suchthat linkD1C4 isno longerinlinewithlink P23D1 andlink D3C4 isnolonger inlinewithlink

P33D3.Thisholdsalsoforthetwootherpantographsthatthereforeneednottobeembeddedwithinthelinksofthefour-bar

linkage.

Forthesolution inFig. 4bthere exist4 versionswithin the GrandArchitecture.Forthe central pantograph alsolinks D1C2,C2D3,B1S,andSB4 could bechosenforwhichalongthediagonalA1A3two pantographversionsarefound.Similarly

two versions of three pantographs can be found along the diagonal A0A2. It is interesting to note that the two central

pantographstogetherformawellknownBurmester’sfocallinkagewithSasthefocalpoint[32,33].Thismeansthatwithin theGrandArchitecturetwoBurmester’sfocallinkagesarefound.

Byremovingall internallinksfromtheGrandArchitecturebutthelinksP21B1,B1B6,B6P42,P42B5,B5B2,andB2P21 the

inherently balanced solution in Fig. 5a is obtained. These links form a single pantograph withthe common CoM of all 10linksin S,thejoint betweenlinksB1B6 andB5B2.This solutionisrelated tothe theory ofDobrovol’skii [34] wherea

singlepantograph connected tothe two opposite four-bar linksis used asa graphicaltool foranalysisof the motionof thecommonCoMofthe four-barlinkage.From Fig.5aitis foundthat theconnectionpointsofthepantograph withthe four-barlinkageareprincipalpoints,aninterestingnewfinding.

Dobrovol’skiipresentedthepantograph intheformshowninFig. 5b,whichis adesignvariationofthepantograph in

Fig.5awhereE1 isajointbetweentheextendedlinksP21B1 andP42B5andjointsB2 andB6areremoved.Asexplainedfor

thepantographsinFig.4b,alsothispantographmayhaveanyshapeforthesamepropertyofsimilarityorscaling,keeping theratioP21S:P42Sconstant.

Fig.6a shows avariation of the principal vector linkagein Fig. 1b,also an inherently balanced linkage solution.This solutionisrelatedtothetheoryofArtobolevskii[35]inwhichtheprincipalvectorsareusedasagraphicaltoolinaslightly differentwaywiththeactiveuseofonlyoneprincipalpoint,P11.

Fig. 7. Inherently balanced linkage (a) with centered similar four-bar connected by a parallelogram; (b) with similar four-bar and a connecting link to a side.

Fig. 8. Inherently balanced linkage (a) with centered similar four-bars; (b) without any similar four-bar or parallelogram.

Herewith the inherentlybalanced linkage solutions derived from the GrandArchitecture that are relatedto a known theoryhavebeenfound.Howevertherestillarenumerousotherinherentlybalancedlinkagesolutionsthatcanbederived. Forinstanceby removingallinternallinksfromtheGrandArchitecturebutlinksP11C2,C2P41,B1S,andSB4 thesolutionin

Fig.6b isobtained. Thissolutionshowsthat thecommonCoM ofall8 linksisinSforall motions,whichis ajointofa singlepantograph connectedinprincipalpointsP11 andP41 withthefour-barlinkage.Thismeans thatin additiontothe

solutioninFig.5b,asinglepantographcanalsobeconnectedtotwoconnectingfour-barlinksforinherentbalance. InFig.7aaninherentlybalancedsolutionisshownthatisobtainedwhenfromtheGrandArchitectureallinternallinks areremovedbutlinksP12B6,B6S,SB7,andB7P43.Theresultisasmallsimilarfour-barlinkagethatisconnectedtotheouter

four-bar linkagebymeans ofa parallelogram. Fig.7bshowstheinherently balanced solutionobtainedby removing from theGrandArchitectureallinternallinksbutlinksP11B1,B1B6,B6P12,andB6P42.Theresultisasimilarfour-barlinkageasin

Fig.4a,howeverhereitisconstrainedwiththeouterfour-barthroughlinkB6P42,formingaparallelogram.

Fig.8ashowstheresultofaninherentlybalancedlinkageobtainedbysolelykeepinglinksP23C4,C4P33,D1C2,C2D3,B1S,

andSB8 inside theGrandArchitecture.Thisisadifferentcombinationofsimilarfour-barlinkageswhichismorecentered

as compared to the solution in Fig. 4a. The result in Fig. 8b is obtained by removing all internal links from the Grand ArchitecturebutlinksP21B1,B1B6,B6P42,P13B7,B7B4,andB4P32.ThecommonCoMofall10linksisinS,whichisthejoint

between linksB1B6 andB7B4. Withrespect to the results so far, this inherently balanced linkage has noparallelograms

neithersimilarfour-barlinkages.

Intotal32differentinherentlybalancedlinkagesolutionshavebeenderivedfromtheGrandArchitectureinFig.3,which arepresentedintheAppendixtogetherwiththeirbalanceconditions.

4. Generalizationformassasymmetriclinks

When the linkCoM is not located alongthe line throughthe joints ofthe linkbut ata certain offsetfrom thisline, then the link is mass asymmetric. This is the generalcase while mass symmetric links can be seen as a specific case, howeversimplerandmorecommoninrealapplications.Fig.9presentstheGrand4RFour-BarInherentlyBalancedLinkage Architecture whereall linksare mass asymmetric andSis the commonCoM forall motions.As compared to the mass-symmetriccaseinFig.3,herethe principalpointsdonot layalong thelinethroughthe jointsandvariousinternal links havebecome triangularelements,such aslinkP12B6C3 whereB6 isnot onthelinethrough P12 andC3.SegmentP12B6 is

paralleltolineA0P42andsegmentB6C3 isparalleltolinkB4P32 andtolineP41A3.AnotherdifferenceisthatjointsDidonot existforthegeneralcase. Thishasasa directconsequencethat thegeneralarchitectureisonly 18timesoverconstrained andthat thenumberof normallyconstrainedinherentlybalanced linkagesolutions that canbe derived fromthe general architectureissignificantlylower.

TheGeneralGrandArchitecturewasdrawntoscalewiththeparametervaluesinTable1asexplainedinFig.10which iswithout includingthemass oftheinternal links. Thecomposition ofthegeneralarchitecture is analogousto themass symmetricarchitecturebycombiningfourgeneralclosed-chainprincipalvectorlinkageswhichwereobtainedandprecisely

Fig. 9. Grand 4R Four-bar Based Inherently Balanced Architecture for general mass distributions of the links.

Table 1

Parameter values of the General Grand Architecture drawn in Fig. 9 and Fig. 10 .

(cm) (kg) (cm) (cm)

l1 = 70 m1 = 8 . 0 e1 = 0 . 55 · l 1 f1 = 0 . 15 · l 1

l2 = 65 m2 = 6 . 7 e2 = 0 . 43 · l 2 f2 = 0 . 10 · l 2

l3 = 55 m3 = 6 . 5 e3 = 0 . 52 · l 3 f3 = 0 . 23 · l 3

l4 = 100 m4 = 5 . 5 e4 = 0 . 48 · l 4 f4 = 0 . 07 · l 4

Fig. 10. Illustration of the link CoM parameters for the architecture in Fig. 9 (drawn to scale).

Fig. 11. Inherently balanced linkage solutions with mass asymmetric links (a) generalized from Fig. 4 a; (b) generalized from Fig. 6 b.

calculatedin [13,26]. Alsoherethereare eight linkscoming together injointS, whichalso can bemerged into thefour ternaryelementsB1SB6,B2SB5,B3SB8,andB4SB7withSascommonjoint.

Theinherently balanced linkagesolutions thatcan be derived fromtheGeneral GrandArchitectureare comparableof topologywiththesolutionsobtainedfromthemasssymmetriccase.ForinstanceFig.11ashowsthesolutionofFig.4afor linkswithagenerallylocatedCoM.AllthreelinksP43B7C1,P31C1,andB7SB4 haveanangleinsteadofbeingstraightlinksas

inFig.4a.

InFig. 11b thesolution forgeneralmassdistributions of thelinkage inFig.6b isshown. Alsohere linksP23B8C4 and

P33B5C4 have an angle at B8 and B5, respectively. Fig. 12a shows an inherently balanced solution comparableto Fig. 8b

Fig. 12. Inherently balanced linkage solutions with mass asymmetric links (a) generalized from Fig. 8 b; (b) closely related to Fig. 7 a with two additional links.

Fig. 13. (a) TWIN-4B inherently balanced linkage with link parameters; (b) 3-DoF open loop model with mass parameters and equivalent masses substi- tuting links 2 and 6.

isshownthatisclosesttothesolutioninFig.7a.SinceforgeneralmassdistributionsjointsDidonotexist,twoadditional linksP13B8 andP21B1arerequiredtoconstrainthemotionnormally.Inthissolutionalllinksarestraight.

5. Derivingthebalanceconditions:exampleoftheTWIN-4B

Afterthechoiceofaninherentlybalancedlinkagesolutionasstartingpointinthedesign,thenextstepistoobtainthe relations amongthe massparameter values forwhichthe linkageis balanced. Theserelations are knownasthe balance conditions.In thissection it isshown, as an example,how thebalance conditionsare derived forthe linkagein Fig.7a, whichwillbereferredtoasTWIN-4Blinkage.

InFig.13theTWIN-4Blinkageisshownwithitsdesignparameters.Thebalanceconditionsinfactconsistoftwogroups ofconditions.Thefirstgroupofbalanceconditionsarethekinematicbalanceconditionsandfollowfromthepropertiesof similarityasdeterminedbytheGrandArchitectureandcanbewrittenfromFig.13aas:

a1 l1 = a2 l2 = a3 l3 = a4 l4 (3) for similarity of the two four-bars and

|

A0P12

|

=

|

P43D4

|

=a5 and

|

P12D4

|

=

|

A0P43

|

=a6 defining the parallelogram

A0P12D4P43.Sincethesepropertiesofsimilarityarecompletelydefinedbythedesignofthelinks,theyturnouttobepurely

geometric conditions.Thesecond group ofbalanceconditionsare themassbalance conditionswhichdeterminethe rela-tionsamongthelinkmassvaluesandtheircenteredlocationineachlink.Theseconditionscanbederivedwithamethod wherethe linkageobtainsthree relativedegreesoffreedom andwritingthe linearmomentumequationsofeach relative DoFindividually[13,26].Thistakesadvantageofthecharacteristicthatthelinearmomentumofaforcebalancedmechanism isconstant(zero)foranyofitsmotions.

The first step is to model the linkage as an open loop linkage withthree degreesof freedom. This can be done in multipleways suchasthe oneillustrated inFig. 13bwherelink 2(A1A2) anditsparallel link6 (B7S) areeliminated and

substituted withequivalentmasses. The mass m2 of link 2is modeled withequivalentmassesma2 andmb2 located inA1

andA2,respectively,wheremassequivalenceisdeterminedbyma2+mb2=m2 andma2e2=mb2

(

l2− e2

)

.Themassm6oflink

6 is modeled similarly with equivalent massesma

6 and mb6 located in S and B7,respectively, where mass equivalence is

determined byma

6+mb6=m6 andma6p6=mb6

(

a2− p6

)

.Anotheroptionwhich issimilar istosubstitute link3(A2A3) and

itsparallellink7(SB6).Thesearethetwosimpleoptions,sincesubstitutingeitherlink1(A0A1)orlink4(A3A0)withtheir

parallel linkwoulddisconnecttheinner linksfromtheouter links. Inthatcasethegeometryofthegrandarchitectureis neededasgraphicaltoolforanalyzingtheindividualmotionsofeachrelativeDoF,whichismorecomplicated.Asshownin

Fig.13b,parallelogramsnotrelatedtothesubstitutedlinksdonotneedtobeopened.

The second stepthen istowrite thelinear momentumequationsforthe individual motionofeach ofthethree DoFs andequalthemtothelinearmomentumofthetotalmassmoving inS,thedesiredcommonCoM.Theindividual motion meansrotationofoneofthethreeprincipalelements(herelinks1,3,and4)aboutanypointwithitsparallellinksrotating

Fig. 14. (a) Motion of the first relative DoF; (b) Motion of the second relative DoF; (c) Motion of the third relative DoF. With the linear momentum equations of these motions the mass balance conditions are obtained readily.

synchronouslywhileallotherlinkssolelytranslateorareimmobile[13,26].InFig.14aonepossibilityofthemotionofthe firstDoFisillustratedwherelink1rotatesaboutA0,link5rotatesaboutP43,links3and4are immobileandlinks7and

8havesolelytranslationalmotion.ThelinearmomentumL1 ofthismotioncanbewrittenwithrespecttoreferenceframe

x1y1,whichisalignedwithA0A1 asillustrated,andmustbeequaltothelinearmomentumofthetotalmassmovinginS: L1 ˙

θ

1 =



_m 1

(

a5+ p1

)

+ ma2l1 + m5

(

a5 + p5

)

+ mb 6

(

a5 +a1

)

+

(

ma6+m7 +m8

)

a5 0



=



mtota5 0



(4)

withthetotal mass mtot=m1+m2+m3+m4+m5+m6+m7+m8 andthe equivalentmassesma₂=m2

(

1− e2/l2

)

,ma₆=

m6

(

1− p6/a2

)

,andmb6=m6p6/a2.

InFig.14bthemotionofthesecond DoFisillustrated wherelink4rotatesaboutP43,link 8rotatesaboutD4,links1,

3,and7have translationalmotionandlink 5isimmobile. Thelinear momentumL2 ofthismotioncan be writtenwith

respecttoreferenceframex2y2,whichis alignedwithA0A3 asillustrated, andmustbeequalto thelinearmomentumof

thetotalmassmovinginS:

L2 ˙

θ

2 =



₋

₍

_m 1 + ma2

)

a6 + m4p4 + m8

(

a4 − p 8

)

+

(

mb 2+m3

)(

l4− a6

)

+

(

ma6+ m7

)

a4 0



=



mtota4 0



(5) withmb 2=m2e2/l2.

InFig.14cthemotionofthe thirdDoFis illustratedwherelink3rotatesaboutA3,link7rotatessynchronouslyabout

B6,andallotherlinksareimmobile.ThelinearmomentumL3ofthismotioncanbewrittenwithrespecttoreferenceframe

x3y3,whichisalignedwithA3A2 asillustrated,andmustbeequaltothelinearmomentumofthetotalmassmovinginS: L3 ˙

θ

3 =



mb 2l3 + m3e3+ m7

(

a3− p 7

)

+ ma6a3 0



=



mtota3 0



(6) Withthelinearmomentumequations(Eqs.4–6)thesecondgroupofbalanceconditionsisreadilyknown.Aftersubstituting mtot,theycanberewrittenas:

(

m2+ m3 + m4

)

a5 − m 1p1 − m a2l1 − m 5p5 − m b6a1 = 0 (7)

(

m1+ m2 + m3 + m4+ m5 + mb6

)

a4 +

(

m1+ m2 + m3

)

a6

−

(

mb

2+ m3

)

l4− m 4p4+ m8p8= 0 (8)

m7 = 0 . 89 p7 = 4 . 46

m8 = 4 . 25 p8 = 21 . 23

By combining the kinematic balance conditions with thesemass balance conditionsthe force balance conditions of the TWIN-4BinFig.13arefoundas:

(

m2 + m3 + m4

)

a5− m 1p1−



m2



1 −e2 l2



+ m6 p6 l2



l1 − m 5p5 = 0 (10)

(

m1 +m2 +m3+m4 +m5

)

a4 +

(

m1+m2 +m3

)

a6 −



m2 e2 l2 + m3− m 6 p6 l2



l4 − m 4p4 + m8p8 = 0 (11)

(

m1 + m2 + m3+ m4 + m5+ m8

)

l3 l4 a4−



m2 e2 l2 − m 6 p6 l2



l3 − m 3e3+ m7p7 = 0 (12)

wherea2=l2a1/l1,a2=l2a4/l4,a2=l2a3/l3anda3=l3a4/l4 weresubstitutedtogetherwiththeexpressionsforthe

equiv-alentmassesma

2,mb2,ma6,andmb6.

Whendesigningthelinkage,itispossibletochoose,forinstance,therelativesizeofthesimilarfour-barswithascaling factor

λ

.Thena1=

λ

l1,a2=

λ

l2,a3=

λ

l3,a4=

λ

l4andfromtheforcebalanceconditions(10–12)parametersa5,a6,andp7

canbecalculated,respectively,as:

a5 = m1p1+ m2

(

1 −el22

)

l1+ m5p5+ m6 p6 l2l1 m2 + m3+ m4 (13) a6 = −

(

m1 + m2+ m3 + m4+ m5

)

λ

l4+

(

m2e_l2 2 + m3− m 6 p6 l2

)

l4 + m4p4 − m 8p8 m1+ m2 + m3 (14) p7 = −

(

m1+ m2 + m3 + m4+ m5 + m8

)

λ

l3 +

(

m2e_l2 2 − m 6 p6 l2

)

l3 + m3e3 m7 (15) Ontheotherhanditisalsopossibletheleavetherelativesizeofthesimilarfour-barsdependentbycalculatinga4 from

(12)as: a4 = l4 l3



₍

_m 2el22− m 6 p6 l2

)

l3 + m3e3− m 7p7 m1+ m2 + m3+ m4 + m5 + m8

(16) Thescalingfactorofthesimilarfour-barsthenisdeterminedby

λ

=a4/l4withwhicha1,a2,a3,a5,anda6 canbeobtained

asinthefirstcase.

Asa numericalexample,Table2showsasetofparametervaluesforwhichtheTWIN-4Bisinherentlybalanced andS isthecommonCoMforanymotionofthelinkage.Theresultsrepresenttherealisticsituationwhenthemassofeachlink isdirectlyrelatedtothelengthofeachlink(mass perlengthisequalforalllinks)andwhenthelinkCoMofeach linkis halfwaythelink.Thescaling factoroftheinner four-barwithrespect totheouterfour-barhereis

λ

₌0_.16.Itis possible toscalethelinkagegeometrywithoutaffectingthebalance,i.e.alllengthsinTable2could bemultipliedby,forinstance, sevenortenandthedesignwouldremainbalancedwiththesamemassvalues.SimilarlyallmassvaluesinTable2could bemultipliedbyanynumberwithoutaffectingthebalanceforthesamegeometry,linklengthsandlinkCoMpositions.

6. SynthesisexampleofaspatialinherentlybalanceddoubleBennettlinkage

Theprincipalvectorsarenotlimitedtoplanarmotions:they describethemotionofthelinkCoMs withrespecttothe commonCoMforanyspatialmotion.Forinstance,thefour-barlinkageinFig.1acouldhavesphericaljointsinA0,A1,A2,

Fig. 15. The Inherently Balanced Double Bennett Linkage with its common CoM in fixed joint S (b) can be obtained from the planar TWIN-4B linkage by having equal opposite link lengths as in (a) and by twisting the revolute pairs out of plane according to the Bennett conditions.

Table 3

Numerical example of parameter values for inherent balance of the Double Bennett Linkage in Fig. 15 .

(cm) (kg) (cm) (cm) l1 = 70 m1 = 7 . 00 p1 = 17 . 31 a1 = 13 . 75 l2 = 90 m2 = 9 . 00 e2 = 45 . 00 a2 = 17 . 67 l3 = l 1 m3 = m 1 e3 = 35 . 00 a3 = a 1 l4 = l 2 m4 = m 2 p4 = 22 . 43 a4 = a 2 m5 = 3 . 14 p5 = −1 . 97 a5 = 17 . 69 m6 = 1 . 77 p6 = 8 . 84 a6 = 22 . 57 m7 = 1 . 37 p7 = 6 . 87 m8 = 4 . 02 p8 = 20 . 12

motiontheprincipalvectorsdefineplanes,theprincipal vectorplanes,withdifferentorientations[36].Infact,the planar situationisjustaspecificcaseofthegeneralspatialsituationwhenalltheprincipalvectorsarewithinthesameplaneand thedifferentprincipalvectorplanesobtainthesameorientationandunitetoasingleplane.ThismeansthatalsotheGrand Architecturein Fig.3can be transformedinto a spatiallymoving inherently balanced linkagearchitecture. The challenge, however,istofindtherightcombinationofjointstoensure thecorrectspatialmotions,notgainingovermobilityandon theother hand not lockingthe linkageimmobile. It isbeyondthe scope ofthis articleto studythisin detail,butas an examplewithsolelyrevolutepairs itisillustrated howtheTWIN-4B linkagecanbe transformedinto aspatialinherently balanceddoubleBennettlinkage.

TheBennettlinkageisaspatialfour-barlinkagewithsolelyrevolutepairsasjoints[37].Itispossibletochangethetwo similarfour-barsoftheTWIN-4BlinkageinFigs.7aand13aintotwosimilarBennettlinkagesbyincludingthesameBennett conditionsforeachofthem.Then thefirststepisto includetheBennettconditionthatopposite linkshaveequallength, whichisillustratedinFig.15awherel1=l3,l2=l4,a1=a3,anda2=a4.Thisresultsinalinkagewiththreeparallelograms,

onewithsidesl1andl2,asimilaronewithsidesa1anda2,andathirdonewithsidesa5anda6.Thenextstepistoinclude

thetwistanglesoftheaxes ofrotationofthejoints,whichareagainequalforeachpairofoppositelinksandarerelated with sin

α

l1 = sin

β

l2 (17) Thisisillustratedfor

α

=40◦ and

β

=55.73◦inFig.15bwhichshowstheresultinglinkagewiththetwosimilarBennett linkages.TheparallelogramA0P12D4P43 remainsplanarandconnectsthetwosimilarBennettssuchthattheymovesimilarly

atall times.Byincluding theBennett conditions,the force balance conditions(10–12) do not changeanddetermine the commonCoM ofthisInherently BalancedDouble BennettLinkagetobe in(fixed)jointSforall motions.Table3showsa numericalexampleofaset ofparametervaluesforinherentbalance.Sincethevaluesdonotdepend onthetwistangles, theyholdforanytwistofthefour-bars.SimilarlyasfortheTWIN-4BvaluesinTable2,alsoherethevaluesrepresentthe realisticsituationshowninFig.15bwherethemassofeachlinkisdirectlyrelatedtothelengthofeachlinkandwherethe linkCoMofeachlinkishalfwaythelink.HerethescalingfactoroftheinnerBennettwithrespecttotheouterBennettis

λ

=0.20.

7. Discussion

TheGrandArchitectureinFig.3wasformedfromfourclosed-chainprincipalvectorlinkageswithsomelinksextended topivots Ci asillustrated inFig. 2.Other link extensionswithextra jointsare also possible,forinstance linksP21B1 and

such asgears andsliders.Alsonewlinksmay beadded that movesimilarly to other linksinthe GrandArchitecture.As long asthe kinematicpropertiesoftheGrandArchitectureare maintained,anychangeispossible withoutdisruptingthe inherentbalance.Thisnextstepinthesynthesisisstillatopicunderinvestigation.

A disadvantage ofthe presented inherently balanced linkage solutions could be their limited range ofmotion dueto singularitiesoftheparallelograms.Exchanginglinkswithothermechanismelementsmayberequiredtoenlargetherange ofmotion.Alsothe choice oftheclosed-chain four-barlinkageasbasis ofthe grandarchitecture limitedthe resultsthat couldbefoundto2-DoFsolutions.Tofindothersolutions,forinstancewith3-DoFor4-DoF,otherlinkagesneedtobeused asbasisofagrandarchitecturewithwhichtheapproachinthispapercanbefollowed.

Thispaperdoesnot considertheshakingmomentbalance,howeverinherentlyshakingmoment-balancedmechanisms canbeobtainedfrominherentlyforce-balancedlinkagesolutionsbyreducing theDoFsoftheforce-balancedlinkage solu-tion.Forinstancefroma 3-DoFforce-balancedlinkagesolutiona2-DoF force-andmoment-balancedlinkagesolutioncan bederived byconstrainingtherelativemotionsaccordingtotheinertiaparameters.Thiscanbeaccomplishedforinstance byintroducingsliderelements[38].

Almostallinherentlybalancedlinkagesolutionsinthispaperwereobtainedbysolelyremovinglinksfromtheinsideof theGrandArchitecture. Thisresultsinlinkages thatare force balancedwithout theneedof countermasses,whichisrare for commonbalance methods since forthem the useof countermasses isthe main issue.The link CoM of all linkscan be atornearby their naturallocations,which isperfectlyaccording tothe philosophyofinherent balance.Whenlinks of thecircumscribedfour-barareeliminatedinsteadofinnerlinks,thenlinkagesolutionsareobtainedwithlinksthat extend freelyasshownforlinkagesolution10intheappendix.Theseextendedlinkscouldbe regardedaskindofcountermasses, howevertheycanalsobeappliedasusefullinksinanapplication.Thiscategoryofinherentlybalancedlinkagesolutionsis leftforfuturework.

Alltheinherently balancedlinkagesolutions consistofatleast8links.Foraparallel mechanismorrobotmanipulator designthesolutionswheretwolinksare connectedinthecommonCoM Sisusefulsincethen,todrivethe2-DoFsofthe linkage,the two actuators can be mountedat thebase. Forother applications such asdeployable structures itmight be morepracticaltohaveasinglelinkinS.

Theforcebalanceconditionsshowedtoconsistoftwoparts:(1)kinematicbalanceconditionsgivenbytheGrand Archi-tectureand(2)massbalanceconditionsdeterminingthemassparametervaluesofthelinks.Thisisdifferentfromcommon balancemethodswhichonlyconsiderthesecondpart.Itisbecauseofthekinematicbalanceconditionsthatadvantageous balancesolutionsareobtainedwiththepresentedinherentbalanceapproach.

ItwasshownthatthesolutionsthatcanbederivedfromtheGrandArchitecturewithmassasymmetriclinksinFig.9is limitedascomparedtothesolutionsthatcanbederivedfromtheGrandArchitecturewithmasssymmetriclinksinFig.3. However,nothingrestrictsthedesignertochangethelinksofamasssymmetricinherentlybalanced linkagesolutioninto mass asymmetriclinks, that’s very well possible.Alsothen the massbalance conditionscan be obtainedfromthelinear momentumequationswhichformassasymmetriclinksismoreextensiveasexplainedin[26].

From the resultof the inherently balanced double Bennett linkage in Fig. 15b, together withthe Grand Architecture inFig.3,itcanbederived thatthecommonCoMofanormalBennettlinkagetracesacouplercurveofasimilarBennett linkagewhichisalsosimilartoacouplercurveoftheBennettlinkageitself.ThisisanextensionofthetheoryofKreutzinger explainedinSection3whichshowsthesamepropertyforplanarfour-barlinkages[29].Itwillbeinterestingtoinvestigate thispropertyalsoforotherspatialinherentlybalancedlinkagesolutions.

8. Conclusions

Inthis papertheGrand4R Four-Bar Based InherentlyBalancedLinkage Architecture waspresentedandit wasshown howfromthishighlyoverconstrainedmechanismavariety of32normallyconstrained2-DoFinherently balancedlinkage solutionscouldbederivedbyremovingspecificcombinationsofredundantlinks.Thesesolutionsallhavespecifickinematic propertiesofsimilarity withadvantageousbalance capabilitysuch asnoneed ofcountermasses.Forone linkagesolution namedtheTWIN-4B itwasshownhowthe balanceconditionsregardingthemassparameterscan bederived withlinear momentumequationsandthebalanceconditionsforall32solutionswerepresented.Thedifferencebetweensolutionswith masssymmetriclinksandmassasymmetriclinkswasexplained.FromtheGrandArchitecturewithmassasymmetriclinks fewersolutionscanbederivedascomparedtotheGrandArchitecturewithmasssymmetriclinks,while,interestingly,allthe masssymmetriclinksolutionscanalsohavemassasymmetriclinks.Itwasshownhowaplanarinherentlybalancedlinkage solutioncouldbetransformedintoaspatialinherentlybalancedlinkagesolutionbyexampleoftheTWIN-4Bwhichresulted, afterincludingtheBennettconditions,intoaninherentlybalanceddoubleBennettlinkage.Alltheseresultstogetherpresent

Acknowledgment

ThispublicationwasfinanciallysupportedbytheNetherlandsOrganisationforScientificResearch(NWO,NWO15146).

Appendix:Overviewof2-DoFinherentlybalancedlinkageswiththeirconditionsforshakingforcebalanceand gravityforcebalance

This appendix presents 32 2-DoF inherently balanced linkage solutions withmass symmetric linksderived from the GrandArchitecture in Fig.3 together with their balance conditions.The massbalance conditions are comparableto and derivedasEqs.(7)–(9)inSection 5,whicharewithoutthekinematicbalanceconditions.Thekinematicbalanceconditions whichdeterminethesimilar motionsofparallel linksandthe similarityoffour-barsare givenforalllinkagesolutions in

Table4.Assuch,thedesignercandirectlyuseeachofthese32linkagesasastartingpointintheirdesignofaforcebalanced device.

Inherently balanced linkage Mass balance conditions

1 (m2 + m 3 + m 4 + m 8 + m 9) a 1 − m1 p 1 − m 4el44 l 1 + m 6 p 6 + m 7 p 7 = 0 (m1 + m 3 + m 4) a 2 − m 2 p 2 + m 5 p 5 − (m3 + m 4(1 −el44)) l 2 − m 8 p 8 = 0 (m1 + m 2 + m 4 + m 5 + m 6) a 3 − m 3 p 3 − m4(1 −el44) l 3 + m 9 p 9 + m 10 p 10 = 0 2 (m1 + m 2 + m 3 + m 4 + m 9) a 1 − m1 e 1 − m 2(1 −el22) l 1 − m 5 p 5 − m6 a 3 − m 8 p 8 + m 10 p 10 = 0 (m1 + m 2 + m 3) a 5 + m 4 p 4 − (m1 + m 2(1 −el22)) l 4 = 0 (m1 + m 2 + m 3 + m 4 + m 10) a 2 − m3 e 3 − m 2el22 l 3 − m 5 a 4 − m6 p 6 − m 7 p 7 + m 9 p 9 = 0 3 (m1 + m 2 + m 3 + m 4 + m 6) a 1 − m1 e 1 − m 2(1 −el22) l 1 − m 5 p 5 + m7 p 7 = 0 (m1 + m 2 + m 3) a 4 + m 4 p 4 − (m1 + m 2(1 −el22)) l 4 = 0 (m1 + m 2 + m 3 + m 4) a 3 − m 3 e 3 − m2el22 l 3 − m 5 a 2 + m 6 p 6 − m 8 p 8 = 0

5 (m1 + m 3 + m 4 + m 11(1 −pa118)) a 1 − m2 p 2 − m 3(1 −el33) l 2 + m 6 p 6 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m11(1 −pa118)) a 3 − m 1 p 1 + m 5 p 5 + m7 p 7 + (m2 + m 3 + m 4 + m11(1 −pa118)) a 2 −(m3 e3 l3 + m 4 + m11(1 −pa118)) l 1 − m 9 p 9 − (m10 + m 11pa118 + m 12) a 4 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m7 + m 11(1 −pa118)) a 5 − m 4 e 4 − m3el33 l 4 − m 11(1 − p11 a8)(l4 − a 7) − m10 p 10 − m 11pa118 a 6 − m12(a6 + p 12) + m 8 p 8 = 0 6 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m10(1 −pa106)) a 1 − m 1 e 1 + m 7 p 7 − (m4el44 + m 10(1 − p10 a6)) l 1 − m 9 p 9 + m10(1 −pa106) a 5 = 0 (m1 + m 3 + m 4 + m 10(1 −pa106)) a 4 − (m1 + m 4el44 + m 10(1 − p10 a6)) l 2 + m2 p 2 + m 6 p 6 − m 8 p 8 − (m9 + m 10pa106) a 2 = 0 (m1 + m 2 + m 4 + m 10(1 −pa106)) a 3 − m3 p 3 − m 4(1 −el44) l 3 + m 5 p 5 = 0 7 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m7 + m 8 + m 10pa106) a 1 − m 2 e 2 − m3(1 −el33) l 2 − m 5 p 5 − (m6 + m 10pa106) a 2 + m 9 p 9 = 0 (m2 + m 3 + m 4 + m 5 + m 6 + m10pa106) a 4 −(m2 + m 3(1 − e3 l3)+ m5 + m 6 + m 10pa106) l 1 − m 1 p 1 + (m5 + m 10pa106) a 3 + m 6 p 6 + m 8 p 8 = 0 (m1 + m 2 + m 3 + m 5 + m 6 + m10pa106) a 5 − m 4 p 4 − m3el33 l 4 + m 7 p 7 = 0 8 (m2 + m 3 + m 4) a 1 − m 1 p 1 − m 4el44 l 1 − (m5 + m 7 + m 10pa106) a 3 + m7 p 7 + m 8 p 8 = 0 m2 p 2 + (m3 + m 4(1 −el44)) l 2 − (m1 + m 3 + m 4) a 2 −(m1 + m 3 + m4 + m 5 + m 7 + m 10pa106) a 4 − m5 p 5 − m 6 p 6 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m7 + m 8 + m 10pa106) a 5 − m 3 e 3 − m4(1 −el44) l 3 + m 9 p 9 = 0

a − m p = 0 10 (m2 + m 3(1 −ea36) + m 7(1 − p7 a6)) a 1 − m1 p 1 − m 3(1 −ae36) l 1 − m 3 e3 a6 a 5 + m 4 p 4 − m7(1 −pa67)(l1 − a 5) − m 8 p 8 = 0 (m1 + m 3(1 −ea36) + m 7(1 − p7 a6)) a 2 − m2 p 2 + m 5 p 5 = 0 (m1 + m 2 + m 3(1 −ea36) + m 4 + m5 + m 7(1 −ap67)) a 3 + m 6 p 6 − (m3ea36 + m 7 p7 a6 + m 8) a 4 = 0 11 (m2 + m 3 + m 4) a 1 − m 5pa55 a 6 − m1 p 1 − m 2(1 −le22) l 1 + m 8 p 8 = 0 (m1 + m 2 + m 3 + m 5pa55) a 2 − m 4 p 4 − (m2el22 + m 3) l 4 + m 7 p 7 = 0 (m1 + m 2 + m 3 + m 4 + m 5ap55 + m7 + m 8) a 3 − m 2el22 l 3 − m 3 e 3 − m5(1 −pa55) a 4 − m 6 p 6 = 0 12 (m1 + m 2 + m 3 + m 4 + m 8) a 2 − m1 e 1 − m 4el44 l 1 − m 5 p 5 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 8) a 3 − (m1 + m 2 + m 3 + m 4) a 5 + (m1 + m 2 + m 4el44) l 2 − m 2 e 2 − m6 p 6 − m 7 a 4 − m 8 p 8 = 0 (m1 + m 2 + m 4) a 1 − m 3 p 3 − m4(1 −el44) l 3 − m 7 p 7 = 0 13 (m2 + m 3 + m 4) a 1 − m 1 p 1 − m4el44 l 1 + m 9 p 9 − m 5 p 5 = 0 m2 p 2 + m 6 p 6 − m 8 p 8 − m 5 a 4 − (m1 + m 3 + m 4) a 2 + m 7 a 5 + (m3 + m 4(1 −el44)) l 2 = 0 (m1 + m 2 + m 3 + m 4 + m 8 + m 9) a 3 − m3 e 3 − m 4(1 −el44) l 3 − m7 p 7 + m 10 p 10 = 0

(m1 + m 3 + m 4 + m 9 + m 10pa106) a 3 − m1el11 l 2 − m 2 p 2 + m 6 p 6 − m8 p 8 − m 10(1 −pa106) a 4 = 0 15 (m1 + m 2 + m 3 + m 4 + m 5 + m 6pa66 + m7 + m 8 + m 9) a 1 − m 1 e 1 − m2(1 −el22) l 1 − m 5 p 5 + m 10 p 10 = 0 (m1 + m 2 + m 3 + m 5) a 4 −(m6pa66 + m7) a 2 + (m1 + m 2 + m 3) a 5 − (m2el22 + m 3) l 4 − m 4 p 4 − m 8 p 8 = 0 (m1 + m 2 + m 4 + m 5 + m 6pa66 + m7) a 3 − m 2el22 l 3 − m 3 p 3 − m6ap66 a 7 − m 7 p 7 + m 9 p 9 = 0 16 (m1 + m 2 + m 3 + m 4) a 1 − m1 e 1 − m 4el44 l 1 − m 5 p 5 − m8 p 8 + m 10 p 10 = 0 (m1 + m 3 + m 4) a 2 + m 2 p 2 − (m1 + m 4el44) l 2 − m 5 a 5 − m6 p 6 + m 7 a 4 = 0 (m1 + m 2 + m 3 + m 4 + m 10) a 3 − m3 e 3 − m 4(1 −el44) l 3 − m7 p 7 − m 8 a 6 + m 9 p 9 = 0 17 (m1 + m 2 + m 3 + m 4 + m 5 + m7(1 −pa76)) a 1 − m 3(1 − e3 l3) l 2 − m2 e 2 + m 6 p 6 − m 7pa67 a 4 + m8 p 8 − m 10 p 10 = 0 (m2 + m 3 + m 4 + m 7(1 −pa76)) a 2 − (m3el33 + m 4 + m 7(1 − p7 a6)) l 1 + m1 p 1 − m 5 p 5 = 0 (m1 + m 2 + m 3 + m 7(1 −pa76)+ m8) a 3 −(m3el33 + m 7(1 − p7 a6)) l 4 − m4 p 4 + (m7(1 −ap76) − m 10) a 5 − m9 p 9 = 0 18 (m1 + m 2 + m 3 + m 4 + m 5 + m7ap76 + m 10 p10 a7) a 1 − m 2 e 2 − (m3(1 −el33) + m 10 p10 a7) l 2 − (m6 + m 7(1 −pa76) − m 10 p10 a7) a 4 + m6 p 6 = 0 (m1 + m 2 + m 3 + m 4 + m 7pa76 + m10pa107) a 2 − m 1 e 1 − (m2 + m 3(1 −el33) + m 10 p10 a7) l 1 + m5 p 5 − m 8 p 8 = 0 (m1 + m 2 + m 3 + m 4 + m 7pa76 + m8 + m 10pa107) a 3 − m 4 e 4 − (m3el33 + m 7 p7 a6) l 4 + m 9 p 9 + (m7pa67 − m 10(1 − p10 a7)) a 5 = 0

(m1 + m 2 + m 3 + m 4 + m 8 + m10(1 −pa108)) a 3 − m 3 e 3 − m4(1 −el44) l 3 − m 7 p 7 + m 9 p 9 − m10pa108 a 4 = 0 20 (m1 + m 2 + m 3 + m 4 + m 5 + m 6pa67 + m7) a 1 − m 1 e 1 − m 2(1 −el22) l 1 − m5 a 4 + m 8 p 8 − m 10 p 10 = 0 (m1 + m 2 + m 3 + m 5 + m 6ap67 + m 7) a 2 − (m2el22 + m 3 + m 6 p6 a7 + m 7) l 4 − m4 p 4 − m 5 p 5 + m 6ap76 a 5 + m 7 p 7 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6pa67 + m7 + m 8) a 3 − m 2el22 l 3 − m 3 e 3 − (m6ap67 + m 7) a 6 − m 9 p 9 − m 10 a 8 = 0

21 Same conditions as for 22

22 (m1 + m 2 + m 3 + m 4 + m 6ap69) a 1 − m1 e 1 − m 4(1 −el44) l 1 − m 5 p 5 − m6(1 −pa96)(a4 + a 5) − m 10 p10 a10 a 4 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m10ap1010) a 2 + (m1 + m 3 + m 4 + m6pa69) a 6 − m 2 p 2 −(m3 + m 4 e4 l4 + m6pa69) l 2 + m 7 p 7 − m 8 a 7 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m7 + m 8 + m 10pa1010) a 3 − m3 e 3 −(m4el44 + m 6 p6 a9) l 3 + m6pa69 a 8 − m 8 p 8 + m 9 p 9 = 0

23 Same conditions as for 22

(m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m7 + m 8 + m 10pa105) a 3 − m 3 e 3 − m4el44 l 3 − m 8 p 8 + m 9 p 9 = 0 25 (m2 + m 3 + m 4) a 5 − m 1 p 1 − m2(1 −el22) l 1 − m 5 p 5 − m 6 p6 a2 a 1 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m6ap62) a 4 + (m1 + m 2 + m 3) a 6 − (m2el22 + m 3) l 4 − m 4 p 4 + m 8 p 8 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6pa26 + m8) a 3 − m 2el22 l 3 − m 3 e 3 + m 7 p 7 = 0 26 (m2 + m 3 + m 4 + m 5 + m 6) a 1 − m 1 p 1 − m2(1 −el22) l 1 − m 5 p 5 + m 9 p 9 = 0 (m1 + m 2 + m 3 + m 4) a 2 −(m2el22 + m 3) l 4 − m4 e 4 + m 5 a 5 − m 6 p 6 − m 7 a 4 + m 8 p 8 = 0 (m1 + m 2 + m 4 + m 7 + m 8) a 3 − m 2el22 l 3 − m3 p 3 − m 7 p 7 + m 10 p 10 = 0 27 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m10(1 −pa106)) a 1 − m 4 e 4 − m3el33 l 4 + m 7 p 7 + m 8 p 8 − m10(1 −pa106) a 4 = 0 (m2 + m 3 + m 4 + m 10(1 −pa106)) a 2 − (m8 + m 10pa106) a 5 + m 1 p 1 + m 5 p 5 − (m3el33 + m 4 + m 10(1 − p10 a6)) l 1 − m9 p 9 = 0 (m1 + m 3 + m 4 + m 10(1 −pa106)) a 3 − m2 p 2 − m 3(1 −el33) l 2 + m 6 p 6 = 0 28 (m1 + m 2 + m 3 + m 5ap58 + m 9 + m 10) a 1 − m1(1 −el11) l 4 − m 4 p 4 − m 5 p5 a8 a 5 + m8 p 8 − m 10 p 10 = 0 (m1 + m 2 + m 3 + m 4 + m 5pa58 + m 7 + m 8 + m9 + m 10) a 4 + (m1 + m 2 + m 4 + m 5pa58 + m7 + m 9 + m 10) a 2 −(m1el11 + m 2 + m5ap58 + m 9 + m 10) l 3 − m 3 p 3 + (m5pa85 + m 10) a 6 − m 7 p 7 + m 9 p 9 = 0 (m1 + m 3 + m 4 + m 7 + m 8) a 3 − m 1el11 l 2 − m2 p 2 − m 5(1 −ap58) a 7 − m 6 p 6 = 0

m1l1 l 2 − m 2 p 2 + m 6 p 6 = 0 30 (m2 + m 3 + m 4 + m 9pa79) a 1 − m1 p 1 − m 4el44 l 1 − m 5 p 5 − m8 p 8 − m 9(1 −pa97) a 4 = 0 (m1 + m 3 + m 4 + m 9pa79) a 2 + (m1 + m2 + m 3 + m 4 + m 5 + m 9) a 5 − (m3 + m 4(1 −el44) + m 9 p9 a7) l 2 − m2 p 2 + m 6 p 6 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m9) a 3 − m 3 e 3 − m 4(1 −el44) l 3 + m7 p 7 − m 9pa97 a 6 = 0 31 (m2 + m 3 + m 4) a 1 − m 1 p 1 − m4(1 −el44) l 1 + m 6 p 6 + m 8 p 8 = 0 (m1 + m 2 + m 3 + m 4 + m 6 + m 7 + m8) a 2 −(m1 + m 4(1 −el44)) l 2 − m2 e 2 + m 5 p 5 + m 10 p 10 = 0 (m1 + m 2 + m 3 + m 4 + m 5 + m 6 + m8) a 3 − m 3 e 3 −(m4el44 + m 8) l 3 − m7 p 7 + m 9 p 9 = 0 32 (m1 + m 2 + m 3 + m 8 + m 9 + m12pa127) a 1 − m 3 e3 l3 l 4 − m4 p 4 + m 5 p 5 + m 7 p 7 = 0 (m2 + m 3 + m 4 + m 12pa127) a 2 + (m11 + m 12(1 −pa127)) a 5 − (m3el33 + m 4) l 1 + m 1 p 1 − m 5 a 4 − m6 p 6 + m 9 p 9 + m 10 p 10 = 0 (m1 + m 3 + m 4 + m 6 + m 7) a 3 − m2 p 2 − m 3(1 −le33) l 2 + m 8 p 8 + m11 p 11 − m 12pa127 a 6 = 0 Table 4

Kinematic balance conditions of the 32 inherently balanced linkage solutions.

Linkage Kinematic balance conditions

1 none 2 a1+a3 l1 = a5 l4 = d1 l2 , a2+a4 l3 = l4−a5 l4 3 a2+a3 l3 = l4−a4 l4 , a4 l4 = a1+d1 l1 = d2 l2 4 l1−a1 l1 = a2+a4 l2 = a3+d1 l3 = l4−d2 l4 5 a1 l2 = a8 l3 , a3 l1 = a5 l4 , a2 l1 = a7 l4 , a4 l1 = a6 l4 = l1−a2−a3 l1 − a8 l3 6 a1+a5 l1 = a2+a4 l2 = l3−a3 l3 = l4−a6 l4 7 a2−a1 l2 = l1−a4−a3 l1 = a5 l4 = a6 l3 8 a3 l1 = a4 l2 = a5 l3 = a6 l4 9 a2 l4 = a3+a4 l3 = l2−d2 l2 , l4−a2 l4 = a1+d1 l1