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Force Response and Stability of Deployable Mechanisms for Small Satellites

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Title:
Force Response and Stability of Deployable Mechanisms for Small Satellites
Creator:
Hansoge, Amrith Nagesh
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (125 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
WIENS,GLORIA JEAN
Committee Co-Chair:
CRANE,CARL D,III
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Artificial satellites ( jstor )
Jacobians ( jstor )
Kinematics ( jstor )
Matrices ( jstor )
Mechanical forces ( jstor )
Mechanical springs ( jstor )
Simulations ( jstor )
Software ( jstor )
Spring constant ( jstor )
Stiffness ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
deployablemechanisms -- equilibrium -- equilibriumstability -- force -- forceresponse -- incremental -- response -- rotational -- rotationalsprings -- satellites -- small -- smallsatellites -- stability
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, M.S.

Notes

Abstract:
In the past, deployable mechanisms for large satellites were developed and over time, highly efficient and multi-featured deployable structures were witnessed. However, with concerns over cost and effort, the focus has significantly shifted to small satellites with much simpler platforms. Currently, the small satellite community is still risk-adverse in that mechanisms used to deploy sensors and antennae in space have been largely restricted to simple one arm pin jointed members or telescopic mechanisms. However, with the advancements in sensor types and capabilities, and reduction in their size and power needs, interest is growing in using more hierarchical deployment schemes for sensor platforms that efficiently fit within small satellites. Furthermore, tape-spring boom technology is currently being downsized to dimensions associated with small satellites which offer a potential dual-means of adding stiffness and a passive means of actuation. The objective of this thesis is to demonstrate the possible applications of tape-spring boom technology to deployable structures for small satellites, mainly rigid-link, pin-jointed and spring-loaded mechanisms. The goal is to enhance the positioning integrity of the deployed structure serving as a sensor platform and to retain a level of simplicity of the deployment mechanism. In this thesis, simple deployable structures like the 6-bar pin jointed mechanisms are presented, which offer multiple platform capabilities within one deployment, hence, leading us back towards the multi-functionality aspect, one of the crucial features that the full sized satellites possessed. Furthermore, these 6-bar mechanisms are designed to actuate by means of a single tape-spring boom. A theoretical, quasi-static approach for determining the displacement response of a deployable structure is implemented for demonstrating the deployment pattern. The stowed structure has pre-loaded rotational springs at the joints and is simultaneously actuated by a boom. Since the pre-loads change continuously as the system deploys, nonlinear iterative matrix methods are used to solve this problem. The stability of these mechanisms is studied at their equilibrium points. Moreover, a tape-spring boom, which is bi-stable in nature, offers further stiffness to the structure in its deployed state. Integrating these bi-stable booms within a deployable mechanism and by looking at the characteristics of the Hessian of the potential energy function, it is also shown how this sufficiently rigid boom affects the stability of the deployed structure. Herein, the force method of matrix analysis for deployable structures is used for analyses. To further validate and confirm the theoretical quasi-static approach predicts the deployment patterns, MSC/Adams dynamic simulations were conducted. At the end, the possibilities of the system failing due to insufficient actuation force by the boom, the condition where the boom does not reach its second stable position, is also briefly discussed. In summary, this thesis demonstrates boom integration which offers enhanced stiffness behavior of deployed mechanisms, simultaneously providing compact stowed configurations and a means of actuation. ( en )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2014.
Local:
Adviser: WIENS,GLORIA JEAN.
Local:
Co-adviser: CRANE,CARL D,III.
Statement of Responsibility:
by Amrith Nagesh Hansoge.

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UFRGP
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Copyright Hansoge, Amrith Nagesh. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
969976945 ( OCLC )
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LD1780 2014 ( lcc )

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SangameshR.Deepake-mail:sangu@mecheng.iisc.ernet.inG.K.Ananthasureshe-mail:suresh@mecheng.iisc.ernet.in DepartmentofMechanicalEngineering, IndianInstituteofScience, Bangalore560012,IndiaPerfectStaticBalanceofLinkages byAdditionofSpringsButNot AuxiliaryBodiesAlinkageofrigidbodiesundergravityloadscanbestaticallycounter-balancedbyaddingcompensatinggravityloads.Similarly,gravityloadsorspringloadscanbecounterbalancedbyaddingsprings.Inthecurrentliterature,amongthetechniquesthatadd springs,someachieveperfectstaticbalancewhileothersachieveonlyapproximatebalance.Further,allofthemaddauxiliarybodiestothelinkageinadditiontosprings.We presentaperfectstaticbalancingtechniquethataddsonlyspringsbutnotauxiliary bodies,incontrasttotheexistingtechniques.Thistechniquecancounter-balanceboth gravityloadsandspringloads.Thetechniquerequiresthateveryjointthatconnectstwo bodiesinthelinkagebeeitherarevolutejointorasphericaljoint.Apartfromthis,the linkagecanhaveanynumberofbodiesconnectedinanymanner.Inordertoachieve perfectbalance,thistechniquerequiresthatallthespringloadshavethefeatureofzerofree-length,asisthecasewiththeexistingtechniques.Thisrequirementisneither impracticalnorrestrictivesincethefeaturecanbepracticallyincorporatedintoany normalspringeitherbymodifyingthespringorbyaddinganotherspringinparallel. [DOI:10.1115/1.4006521]1IntroductionAlinkageissaidtobestaticallybalancedifitisinstaticequilibriumin allits conÞgurations.Inthispaper,alltheloadsona linkageareassumedtobeconservative.Hence,staticbalanceis equivalenttoinvarianceofthenetpotentialenergyofalltheloads withrespecttoallconÞgurationsofthelinkage.Thispapergives anewtechniquetostaticallybalancearevolute-jointedlinkage loadedbyconstantforces(e.g.,gravity)and/orzero-free-length springs.Althoughthetechniqueisdetailedinthispaperforplanar linkages,itextendsto spatial sphericaland/orrevolute-jointed linkages. Theneedforstaticbalanceofgravityloadsinstructuresand machinesiswellknown.Hence,anumberoftechniquesare developedforstaticbalanceofgravityloads[ 1 Ð 4 ].Theneedfor staticbalanceofinherentspringloadsisnotascommon.This needisparticularlyfeltincompliantmechanismswhereanelasticallydeformablestructureisusedbutitsstiffnessisnotalways desired.Thisworkismotivatedbysuchpracticalapplications. TechniquesintheliteratureforstaticbalancingaloadedlinkagemaybeclassiÞedintoapproximatebalancingtechniquesand perfectbalancingtechniques.Theperfectbalancingtechniques canbefurthersubdividedinto Category1:Bothoriginalloadsandbalancingloadsareconstantweights. Category2:Originalloadsareconstantweightsandbalancing loadsare zero-free-length springloads. Category3:Bothoriginalandbalancingloadsare zero-freelength springloads. ThesecategoriesareillustratedinFig. 1 .ThetoprowofFig. 1 showsthesecategoriesforalever.Whilethestaticbalanceofa leverincategory1isknownforalongtime,thebalancingofa leverundercategories2and3wasdiscoveredrelativelyrecently, aswouldbeevidentfromtheliteraturesurveygivenlater.The bottomrowofFig. 1 showsthesecategoriesforamultibodylinkage.Formultibodyrevolute-jointedlinkages,staticbalancing techniquesareknownonlyforcategories1and2.Furthermore, formultibodylinkagesundercategory2,beyond3Rseriallinkage,allthemethodsreportedsofarintheliteratureuseauxiliary bodies.ThebottomrowofFig. 1 undercategory2illustratesone suchreportedmethod[ 2 ]whereauxiliarybodiesarehighlighted ingraycolor. Thispaperdealswithperfectstaticbalancingofmultibody revolute-jointedlinkages.Itshowsthatjustasbalancingunder category1canbedonewithoutauxiliarybodies,balancingunder categories2and3canalsobedonewithoutusingauxiliary bodies.Thebackgroundforthisworkispresentednext. 1.1Zero-Free-LengthSpringsandPerfectStatic Balancing. Zero-free-lengthsprings,incontrasttonormal springs,havezero-lengthbetweenitsendpointswhenthespring forceiszero.Whenaspringisanchoredtotwobodieshavingrelativemotion,thespringforceasafunctionofitstwoanchor pointsisofinterest.AsillustratedinFig. 2 ,thisfunctionhappens tobelinearinazero-free-lengthspringbutnonlinearinapositive-free-lengthspringinspiteofbothspringshavingalinear force-deßectionrelationship.Appendix A showsthatthenonlinearityassociatedwithnonzero-free-lengthspringsprevents Fig.1Threecategoriesofperfectstaticbalancingtechniques shownonaleverandamultibodylinkage ContributedbytheMechanismsandRoboticsCommitteeofASMEforpublicationintheJOURNALOFMECHANISMSANDROBOTICS.ManuscriptreceivedFebruary26, 2011;ÞnalmanuscriptreceivedFebruary22,2012;publishedonlineApril25,2012. Assoc.Editor:FrankC.Park.JournalofMechanismsandRobotics MAY2012,Vol.4 /021014-1 CopyrightVC2012byASME

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perfectstatic-balancewhenonlynormallyavailablepositive-freelengthspringsarepresent.Herder[ 5 ]hasdocumentedafewpracticalarrangementstodecreasethefree-lengthofanormalspring allthewaytozeroandeventoanegativevalue. Ifanormalpositive-free-lengthspringcannotbeincorporated intoanyofthearrangementsofHerder[ 5 ],asisthecasewith loadingsprings,thenbyaddinganappropriatenegative-freelengthspringinparalleltoit,azero-free-lengthspringcanberealizedoutofbothofthem.Inthisway,evennormalspringloads canbebroughtundertheambitoftechniquesundercategory3. Thus,thetechniquepresentedinthispaperispracticalandgeneral enoughtohandlenormalspringloadsinadditiontoweightsand zero-free-lengthspringloads. 1.2LiteratureSurvey 1.2.1PerfectStaticBalancingofaLever. ThethreecategoriesofperfectstaticbalancingofalevercanbeseeninFig. 1 . Thetechniqueundercategory1isknownfromhistorictimesand ispopularlyknownastheleverprincipleofArchimedes.The discoveryofzero-free-lengthspringsandthetechniqueundercategory2arecreditedtoLucienLaCoste(seeRef.[ 1 ]).Weconsider thisdiscoverytobeground-breakingsinceitshowedfortheÞrst timethataweightcanbestaticallybalancedbyaspring. Recently,usingadifferentialbevelgear,thistechniqueisadapted forstaticbalancingabodyhavingthespatialrollandpitchmotion aboutapoint[ 6 ].Thetechniquesundercategory3foraleverare discussedindetailbyHerder[ 5 ]. 1.2.2PerfectStaticBalancingofaMultibodyLinkage. Most ofthestaticbalancingtechniquesformultibodylinkagesunder category1havebeenknownforalongtime.Thetechniqueshown inFig. 1 undercategory1isonesuch.Anotherexampleofthe techniquesunderthiscategoryisinRef.[ 7 ]. Undercategory2,StreitandShin[ 2 ]showedthatinprinciple anyplanarlinkageloadedbygravityloadscanbebalancedusing zero-free-lengthsprings.Theyalsoprovidedadifferenttechnique tostaticallybalanceserialrevolute-jointedplanarlinkages.Itis thistechniquethatisillustratedusingthree-revolute-jointedlinkageinFig. 1 undercategory2.Thesameworkalsoprovides anothertechniqueforalinkagewithrevolute-sliderjointpairs. AlthoughStreitandShin[ 2 ]providetechniquesforplanarlinkages,thetechniquescanbeextendedtospatiallinkages.Rahman etal.[ 8 ]provideonesuchextensiontoanthropomorphicrobots. Recentworkonstaticbalancingofspatiallinkagesincludesthat ofLinetal.[ 9 ].References[ 3 ]and[ 4 ]provideadifferentclassof techniquesundercategory2forrevolute-jointedlinkages.All thesetechniquesuseauxiliarybodiesinadditiontoextrazerofree-lengthsprings. Undercategory3,foramultibodylinkage,thereisatechnique whichisapplicableonlyforafour-barlinkageandatwo-revolute-jointedlinkage[ 10 ].Later,werecognizedtwomoremethods forthesamelinkagesinRef.[ 11 ]. Undercategories2and/or3,Refs.[ 12 ]and[ 13 ]aswellasa methodinRef.[ 11 ]donotuseauxiliarylinks.Thecurrentpaper hasevolvedoutofRef.[ 12 ]andallowsamoregeneralclassof solutionsincomparisontoRef.[ 12 ].Reference[ 13 ]derivesequationsgoverningstaticbalanceofgravityloaded2Rand3Rlinkagesandprovidessolutionstotheequationswithoutusing auxiliarylinks. 1.2.3OtherStaticBalancingTechniques. Amongthebalanc-ingtechniquesoutsidetheambitoftheaforementionedthreecategories,mostareapproximatebalancingtechniquesandafew, althoughperfectbalancingtechniquesusingordinarysprings,use camsandpulleystomodulatethebehaviorofsprings.Further,all thosetechniquesbalanceagainstgravityloads.Agrawaland Agrawal[ 14 ]presentedanapproximatestaticbalancingmethod usingnonzero-free-lengthsprings.Gopalswamyetal.[ 15 ]gave anapproximatestaticbalancingtechniquewheretorsionalsprings wereusedasbalancingelements.Thereisalotofliteratureon staticbalancingofparallelmanipulatorsandonesuchworkis Ref.[ 16 ].Thebalancingtechniquesthatmodulatethebehaviorof springsincludethetechniquesinRefs.[ 17 ]and[ 18 ],whereapulleyofvaryingradiuswasused,andthetechniqueinRef.[ 19 ], whereacamwasused. 1.3PracticalRelevance. Theutilityoftechniquesundercategory2iswellrecognized.Thesetechniquesareappliedinstatic balancingofrobots,anglepoiselamps,andßightsimulators.Ifa robotoraßightsimulatorisstaticallybalanced,thentheactuators donothavetoworkagainstthegravityloadsactingonthelinks oftherobotorthecockpitofßightsimulator.Thisgreatlyreduces theforce/torquerequirementoftheactuatorsandalsosupposedly makestheactuatorcontroleasy.Further,anadvantageofbalancingtechniquesincategory2overcategory1isthattheinertia addedtothelinkageisminimal.Thenewtechniquethatthispaper presentsunderthiscategorywillprovideonemoreoptiontoadesignerseekingtostaticallybalancelinkagesundergravityloads usingsprings. Theutilityofanytechniqueundercategory3isnotdirect. Therearehardlyanypracticalproblemswherealinkageunder zero-free-lengthspringsisrequiredtobestaticallybalanced againstit.However,therearesituationswherealinkageunder elasticloadsotherthanthezero-free-lengthspringloadsis requiredtobebalanced.Forexample,thebalancingoftheelastic forcesofacosmeticcoveringinahandprosthesis(seeRefs.[ 10 ] and[ 20 ])andtheinherentelasticforcesinacompliantmechanism (seeRef.[ 21 ])isdesired.Unlikeinahandprosthesis,thereisno inherentlinkageinacompliantmechanism,whichisamonolithic elasticpiecethattransmitsforceormotionbyvirtueofelasticdeformation.However,itisestablishedthatcompliantmechanisms withßexuraljointsandcertainkindsofslendersegmentscanbe modeledasrigid-linkageswithtorsionalspringsandtension springs(seeRefs.[ 22 ]and[ 23 ]).Sincetheperfectstaticbalance ofthesetypesofelasticforcesonlinkagesisnotdemonstrated, onewouldlookforagoodapproximatestaticbalance.Ifsuch elasticforcesareapproximatelymodeledaszero-free-length springs,thenthetechniquesundercategory3wouldofferinsights andalsoastartingpointforoptimizationtechniquestobalance suchelasticloads. 1.4OrganizationofthePaper. Inordertoshowthefeatures ofzero-free-lengthspringsandconstantloadsthatmakeperfect staticbalancepossiblewiththem,perfectstaticbalanceofoneof thesimplestlinkages:arigidbodyonafulcrum,i.e.alever,isdiscussedinSec. 2 .Section 3 showsthateventhoughtheprinciples ofstaticbalanceofalevercanbeextendedtoarigidbodyfreely movinginaplane,staticbalancingthetranslationcomponentof therigidbodyisnotpossibleinmostpracticalconditions.Based onaresultinSec. 3 ,itisshowninSec. 4 thatanassemblageof Fig.2Differencebetweenzero-free-lengthspringandnormal spring021014-2/ Vol.4,MAY2012 TransactionsoftheASME

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rigidbodiesinaplanewithzero-free-lengthspringandconstant loadinteractionsbetweenthebodiescanalwaysbestatically balancediftheassemblageformsarevolute-jointedlinkage. Section 5 arguesthatthetechniqueforplanarrevolute-jointed linkagesextendsforspatialsphericaland/orrevolute-jointedlinkages.ConcludingremarksareinSec. 6 .2BalancingaLeverConsideraleverpivotedtotheground,asshowninFig. 3 .The conÞgurationoftheleverwithrespecttotheglobalframeofreference( X Y )canbedescribedby h ,whichistheanglefromthe globalframetothelocalframeofreferenceofthelever. Figure 3 alsoshowstwokindsofload:(1)aspringattached betweenapointoftheleverandapointoftheglobalframeand (2)aconstantforceactingatapointonthelever.Here,aconstant forcemeansthattheforcehasaconstantdirection withrespectto theglobalframe andaconstantmagnitude.AcompletespeciÞcationofthespringloadwouldinvolve(1)thespringconstant, denotedby k ,(2)thelocalcoordinateoftheanchorpointonthe lever,denotedby a ¼½ axayT,and(3)theglobalcoordinateof theanchorpointontheglobalreferenceframe,denotedby b ¼½ bxbyT.AcompletespeciÞcationoftheconstantforcewould involve(1)theforcecomponentswithrespecttotheglobalframe, denotedby f ¼ fxfyT,and(2)thelocalcoordinateofthepoint ofactionoftheforceonthelever,denotedby p ¼ pxpy½T. 2.1PotentialEnergyasaFunctionoftheConfiguration Variable. ByreferringtoFig. 3 ,thepotentialenergyoftheconstantloadis PEc¼ fTr þ R h ðÞ p ðÞ (1) where r ¼ rxryTisthecoordinateoftheoriginofthelocal frameontheleverwithrespecttotheglobalframeand R isthe rotationmatrixfunctiongivenby R w ðÞ¼ cos w sin w sin w cos w foranyangle w (2) Thepotentialenergyofthespringis PEs¼ k 2 l l0ðÞ2¼ k 2 l2 kl0l þ k 2 l2 0(3) where l0isthefreelengthofthespringand l isthemagnitudeof d ,thedisplacementofone-endpointofthespringwithrespectto theother.This d ,referringtoFig. 3 ,is d ¼ r þ R h ðÞ a ðÞ b (4) Since l2¼ dTd ,thepotentialenergyinexpression(3)maybe rewrittenas PEs¼ k 2 dTd kl0 dTd p þ k 2 l2 0(5) Ifthefree-lengthofthespringiszero,thenonlytheÞrsttermin Eq. (5) remainsandhence,wecallitasPEs ,zero,i.e. PEs ; zero¼ k 2 dTd ¼ k 2 r þ R h ðÞ a ðÞ b ðÞTr þ R h ðÞ a ðÞ b ðÞ ¼ k 2 ð rTr þ aTRTð h Þ R ð h Þ a þ bTb 2 rTb þ 2 rTR ð h Þ a 2 bTR ð h Þ a Þ ¼ k 2 rTr þ aTa þ bTb 2 rTb þ 2 rTR ð h Þ a 2 bTR ð h Þ a * RTð h Þ R ð h Þ¼ I (6) Sincetheremaininglasttwotermsofthepotentialenergyin Eq. (5) arenonzeroonlyiffree-length l0isnonzero,wename thesetermsasPEs ,nonzero,i.e. PEs ; nonzero¼ kl0 dTd p þ k 2 l2 0(7) FromEq. (6) ,itfollowsthat dTd ¼ 2 kPEs ; zero.Substitutingthisin Eq. (7) leadstothefollowingexpressionforPEs ,nonzero. PEs ; nonzero¼ l0 2 kPEs ; zerop þ k 2 l2 0(8) IntheexpressionsofthepotentialenergyinEqs. (1) , (6) ,and (8) , astheconÞgurationofthelevervaries, f , p , a , b , k , l0remainconstantsand r ismadeaconstantbychoosingtheoriginofthelocal frameonthelevertocoincidewiththepivotpoint.ThedependencyoftheexpressionsontheconÞgurationisduethematrix R ( h ) which,byexaminingthedeÞnitionof R inEq. (2) ,canbesplitas R ð h Þ¼ 10 01 cos h þ 0 1 10 sin h ¼ I cos h þ R p 2 sin h (9) Thisformof R ( h )indicatesthatPEcinEq. (1) andPEs ,zeroin Eq. (6) canbewrittenasalinearcombinationofsin h ,cos h ,and 1(forconstants).ThecoefÞcientsofsin h ,cos h and1arepresented,forclarity,inatabularforminTable 1 .Thus,wenow havepotentialenergyofconstantandspringloadsexpressedas functionsofconÞgurationvariable h . 2.2InvarianceofPotentialEnergyWithRespecttothe ConfigurationVariable 2.2.1TrivialConditions. Thepotentialenergyofthespring onthelevercanhaveconstantpotentialenergyonlyundertrivial conditions:(1)thespringstiffnessiszero( k ¼ 0),(2)theanchor pointontheleverisatthehingepoint( a ¼ 0),and(3)theanchor pointontheglobalframeisatthehingepoint( b ¼ r ).Similartrivialconditionsfortheconstantloadsare(1)theloadiszero( f ¼ 0 ) and(2)theloadactsatthepivotpoint( p ¼ 0 ).Itisonlyunder thesetrivialconditionsthatthecoefÞcientsofcos h andsin h becomezeroinTable 1 . 2.2.2TheDiscoveryofLucienLaCoste. Eventhoughanontrivialspringandanontrivialconstantloadcannotbeindividually instaticbalance,theytogethercanbe,asdemonstratedinFig. 4 . ThiswasÞrstrecognizedbyLucienLaCoste(seeRef.[ 1 ])inthe contextofhavingapendulumofinÞniteperiod.Figure 4 showsa leverundertheactionofaweight W thatisbalancedbyazeroFig.3AleverunderaconstantloadandaspringloadJournalofMechanismsandRobotics MAY2012,Vol.4 /021014-3

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free-lengthspringofspringconstant k anchoredabovethepivot oftheleverataheightof h .AsshownintheÞgure,underthecondition W ¼ kh ,thepotentialenergyisinvariantwithrespectto conÞgurationvariable h . 2.2.3SeveralZero-Free-LengthSpringsandConstant Loads. Thebalancingcondition W ¼ kh oftheexampleinFig. 4 willnowbegeneralizedtoaleverunderseveralconstantloads andzero-free-lengthspringloads.Sinceseveralloadsarenow beingconsidered,letbothconstantloadsandzero-free-length springloadsbeorderedtoallowindexing.Thenotation ai, bi, ki, hasthesamemeaningas a , b ,and k inFig. 3 otherthanthatitcorrespondsto i thspring. fiand pialsohavesimilarmeaning.Further,letthenumberofconstantloadsbe ncandthenumberof zero-free-lengthspringloadsbe ns. Sincethepotentialenergyofeachoftheconstantloadsandthe zero-free-lengthspringloadsarealinearcombinationofcos h ,sin h and1,theirnetpotentialenergyisalsoalinearcombinationof cos h ,sin h and1.Further,sincecos h andsin h and1arelinearly independentfunctionsof h ,theirlinearcombinationisaconstant ifandonlyifthecoefÞcientsofnonconstantfunctions,i.e.,cos h andsin h arezero.Writing,withthehelpofTable 1 ,thecoefÞcientsofcos h andsin h ofthenetpotentialenergyofalltheloads andequatingthemtozeroleadtothefollowingequations: Xnci ¼ 1fy ; ipy ; iþ fx ; ipx ; i þ Xnsi ¼ 1kiay ; iryþ ax ; irx ay ; iby ; i ax ; ibx ; i ¼ 0 (10) Xnci ¼ 1fx ; ipy ; i fy ; ipx ; i þ Xnsi ¼ 1kið ax ; iry ay ; irx ax ; iby ; iþ ay ; ibx ; iÞ¼ 0 (11) whicharetheconditionsforconstantpotentialenergy(orstatic balance)ofseveralconstantandzero-free-lengthspringloadsona lever.Theseconditionsareapplicabletoallthethreecategoriesof Fig. 1 .Further,bychoosingappropriateloadparameters,itispossibletosatisfytheconditions inpractice ,aswasthecaseinthe exampleofFig. 4 . 2.2.4NormalPositive-Free-LengthSprings. Asfarasnormallyavailablepositive-free-lengthspringsareconcerned,the squarerootterminEq. (8) posesasevererestrictiononstatic balancing,asexplainedindetailinAppendix A .Hence,forthe remainderofthispaper,allthespringloadsareofzero-free-length withtheunderstandingthatapositive-free-lengthspringcanbe broughtintotheambitofzero-free-lengthbycombiningitwithan appropriatenegative-free-lengthspring. Ournextaimistoderiveasetofconditionsforthestaticbalanceofarevolute-jointedmultibodylinkageloadedbyconstant loadsandzero-free-lengthspringloads.Beforethat,itisusefulto considerthestaticbalanceofasinglerigidbodymovingfreelyin aplane.3BalancingofaRigidBodyinaPlaneConsidertherigidbodyshowninFig. 3 .Anappropriatesetof conÞgurationvariablesforthebodyis f r , h g .Itmaybenotedthat r inFig. 5 ,incontrasttoFig. 3 ,isanindependent variable becausethebodyisfreetomoveintheplane. Theloadsonthebodyareasetofzero-free-lengthspringloads andconstantloads,andbothsetsofloadsareexertedbytheglobal frameofreferenceasshowninFig. 5 .Thenotations nc, ns, ai, bi, ki, fi,and pihavethesamemeaningasinSec. 2 .Thepotential energyoftheloadsisalsothesameasinSec. 2 exceptthat rxand ryarenowindependentvariables.InTable 1 ofSec. 2 ,whenlinearlyindependentfunctionsof f r , h g arepulledoutasbasisfunctions,Table 2 isobtained.AsisevidentfromTable 2 ,the potentialenergyoftheloadsisnowalinearcombinationofthe followingbasisfunctions:cos h ,sin h , rxcos h , rycos h , rxsin h , rysin h , r2 x, r2 y, rx, ry,and1. Table1Potentialenergyoftheweightandthezero-free-lengthcomponentofthespringactingontheleverisalinearcombination ofcos h ,sin h ,and1. Coefficients BasisWeightZero-free-lengthcomponentofspringload cos h fTp ¼ ( fypyþ fxpx) k ( r b )Ta ¼ k ( ayryþ axrx ayby axbx) sin h fTR p 2 p ¼ð fxpy fypxÞ k r b ðÞTR p 2 a ¼ k ð axry ayrx axbyþ aybxÞ 1 fTr ¼ fyry fxrxþ k 2rTr þ aTa þ bTb 2 rTb ¼þ k 2ð r2 y 2 byryþ r2 x 2 bxrxþ b2 yþ b2 xþ a2 yþ a2 xÞ Fig.4Staticbalancingofaweightbyaspring Fig.5Abodythatisfreetomoveinaplane021014-4/ Vol.4,MAY2012 TransactionsoftheASME

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Thenetpotentialenergyof ncconstantloadsand nsspringloads isalsoalinearcombinationsofthesamebasisfunctions.Furthermore,cos h ,sin h , rxcos h , rycos h , rxsin h , rysin h , r2 x, r2 y, rx, ry, and1arelinearlyindependentfunctionsof f r , h g .Hence,froma reasoningsimilartotheoneinSec. 2 ,forthenetpotentialenergy tobeindependentoftheconÞgurationvariables,thecoefÞcients ofallthebasisfunctionsotherthan1havetobezero.However,it isnotpracticaltomakethecoefÞcientsofallthesefunctionsas zerosbecauseofthefollowingreasons:€Thereareonlygravityloads:Gravityisthemostimportant practicallyseeninstanceofaconstantload.Whenalltheconstantloadsaregravityloads, fi¼ migi,where miisthemass and g istheaccelerationduetogravity.Further,thecoefÞcientof rxand rybecome gxPngi ¼ 1mi and gyPngi ¼ 1mi . Since mi> 0, 8 i , Pngi ¼ 1mi> 0.Also,sincetheacceleration duetogravityisnonzero,both gxand gycannotbezero. Hence,thecoefÞcientsofboth rxand rycannotbezero.€Therearezero-free-lengthspringloads,possiblywithgravity loads:Inthiscase,thecoefÞcientsofboth r2 xand r2 yare Pnsi ¼ 1ki.Sincethespringconstantsofallthespringsconsideredherearepositive,( ki> 0, 8 i ), Pnsi ¼ 1kicannotbezero. Hence,thecoefÞcientsof r2 xand r2 ycannotbezero. However,asshowninAppendix B.1 ,thereisnosuchpractical difÞcultyinmakingthecoefÞcientsofall h -dependentfunctions, i.e.,cos h ,sin h , rxcos h , rycos h , rxsin h ,and rysin h aszero. Setting h -dependenttermstozeroamountstothefollowingsetof independentconstraints: Xnci ¼ 1ð fy ; ipy ; iþ fx ; ipx ; iÞ Xnsi ¼ 1kið ay ; iby ; iþ ax ; ibx ; iÞ ¼ 0 (12) þ Xnci ¼ 1ð fx ; ipy ; i fy ; ipx ; iÞ Xnsi ¼ 1kið ax ; iby ; i ay ; ibx ; iÞ ¼ 0 (13) Xnsi ¼ 1kiaiðÞ¼ 0 (14) ItisshowninAppendix B.1 thatiftheseconstraintsarenotsatisÞedbytheloads,thenbyaddingnotmorethantwozero-freelengthsprings,theseconstraintscanbesatisÞed.Anumerical exampletodemonstratethesameisgiveninFig. 6 . Inspiteofbeingabletomakethepotentialenergyoftheloads onthelinkindependentof h ,thedependencyon r stillremains. InSec. 4 ,weshowthatifthebodyisjoinedtoanappropriate linkage,thenbyaddingextraloadstootherpartsofthelinkage, the r -dependenttermsofthepotentialenergycanbebalanceout. BeforeweproceedtoSec. 4 ,itmaybenotedthatthepotential energyofaconstantloadorazero-free-lengthspringloadfalls underthefollowinggeneralform: U ¼ rTu þ j rTr þ rTR h ðÞ v þ wTR h ðÞ q þ c (15) where h and r aretheconÞgurationvariablesoftherigidbodyon whichtheloadacts.Inthecaseofconstantloads,bycomparing Eq. (1) withEq. (15) ,wehave u ¼ f ; j ¼ 0 ; v ¼ 0 ; w ¼ f ; q ¼ p ; and c ¼ 0 (16) andinthecaseofzero-free-lengthspringloads,bycomparing Eqs. (6) and (15) ,wehave Table2Potentialenergyofweightandspringactingonalinkmovinginaplane. Coefficientsofthebasis BasisWeightSpringloadGeneralizedpotential(seeEq. (15) cos h ( fypyþ fxpx) k ( aybyþ axbx)( qywyþ qxwx) sin h þ ( fxpy fypx) k ( axby aybx)( qxwy qywx) rxcos h 0 kaxvxrycos h 0 kayvyrxsin h 0 kay vyrysin h 0 kaxvxr2 x0 k 2j r2 y0 k 2j rx fx kbxuxry fy kbyuy10 þ k 2ð a2 xþ a2 yþ b2 xþ b2 yÞ c Fig.6Arigidbodymovingfreelyinaplaneunderaconstantloadismadetohave h -independentpotentialenergybyadditionoftwozero-free-lengthspringsJournalofMechanismsandRobotics MAY2012,Vol.4 /021014-5

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u ¼ k b ; j ¼ k 2 ; v ¼ k a ; w ¼ b ; q ¼ k a ; and c ¼ k 2 bTb þ aTa (17) Laterinthepaper,weencounterpotentialenergyfunctionsthat areoftheformgiveninEq. (15) ,buttheycannotbeattributedto zero-free-lengthspringloadsorconstantloads actingonthebody . Hence,thereisaneedtogeneralizeconstraints (10) Ð (14) tothe formgiveninEq. (15) .Suchageneralizationispossiblebecause ascanbeseeninthelastcolumnofTable 2 ,thepotentialgivenin Eq. (15) isalinearcombinationofthebasisfunctionsgivenin Table 2 justasinthecaseofconstantandzero-free-lengthspring loads.Thefollowingpropositionstatesthegeneralization. Proposition3.1.Ifthereare n functionsoftheform Ui¼ rTuiþ jirTr þ rTR h ðÞ viþ wT iR h ðÞ qiþ ci; i ¼ 1 n (18) with r and h asthevariables,then Pn i ¼ 1Uiisindependentof h if andonlyifthefollowingconstraintsaresatisÞed: Xn i ¼ 1qy ; iwy ; iþ qx ; iwx ; i ¼ 0 (19) Xn i ¼ 1qx ; iwy ; i qy ; iwx ; i ¼ 0 (20) Xn i ¼ 1vi¼ 0 (21) WhentheseconstraintsaresatisÞed, Pn i ¼ 1Uidependsonlyon r in thefollowingform: Xn i ¼ 1Ui¼ Xn i ¼ 1rTuiþ jirTr þ ci ¼ rT Xn i ¼ 1uiþ rTr Xn i ¼ 1jiþ Xn i ¼ 1ci(22) Furthermore,if r happenstobeaconstant(asinalever)withonly h beingthevariable,then Pn i ¼ 1Uiisindependentof h (andhence aconstant)ifandonlyifthefollowingconstraintsaresatisÞed: Xn i ¼ 1vy ; iryþ vx ; irxþ qy ; iwy ; iþ qx ; iwx ; i ¼ 0 (23) Xn i ¼ 1vx ; iry vy ; irxþ qx ; iwy ; i qy ; iwx ; i ¼ 0 (24) Proof .Theproofisalongthesamelinesasthederivationof Eqs. (10) Ð (14) . Itmaybenotedthatinspiteofconsideringageneralformof potentialinEq. (18) ,theinabilitytomakethenetpotentialenergy independentof r remainsbecauseofthefollowingreason.Inall thecasesthatweconsidernext, ji 0and ji> 0foratleastone valueof i .Hence,the r -dependentterm, Pn i ¼ 1jirTr ,cannotbe zerointheexpressionfor Pn i ¼ 1Ui.4NewStaticBalancingTechniquesfor Revolute-JointedLinkagesIfthereisasinglerigidbodywithloadsexertedbyareference frame,thenthenetpotentialenergyoftheloadsdependsonthe conÞgurationofthebodywithrespecttothereferenceframe.If thereareseveralsuchbodies,thenthenetpotentialenergyofall theloads onallthebodies dependsontheconÞgurationof allthe bodies .ThisdependencyontheconÞgurationof allthebodies can bereducedtothatofa singlebody providedthebodiesareconnectedbyrevolutejoints(tobeginwith,say,inaserialoratreestructuredmanner)andtheloadsarezero-free-lengthspringloads andconstantloads.Ifthissinglebodyisthereferenceframeitself, thenthenetpotentialenergyisaconstant(implyingstaticbalance)sincetheconÞgurationofthereferenceframewithrespect toitselfisalwaysÞxed.Thisresultfollowsasaconsequenceof thepropositionthatispresentednext. 4.1ThePotentialEnergyofLoadsonaBodyTransformed asaFunctionofAnotherBody. Wearenowconsideringseveral rigidbodies,eachofthemwithitsown r , h , nc, ns, ai, bi, ki, pi,etc. Todistinguishthesequantitiesbelongingtodifferentrigidbodies, wenumbertherigidbodiesandputthenumberasasuperscriptto thesesymbols.Hence r , h , nc, ns, ai, bi, ki, pi,and fiofbody j are nowrepresentedas rj, hj, nj c, nj s, aj i, bj i, kj i, pj i,etc. Proposition4.1.ThenetsumofasetoffunctionsoftheconÞgurationvariablesofabody l intheformgiveninEq. (15) ,i.e. Ul i¼ rlTul iþ jl irlTrlþ rlTR hl vl iþ wlTiR hl ql iþ cl i; i ¼ 1 nl(25) canbeexpressedasafunctionofthesameformbutofbody j ,i.e. Xnli ¼ 1Ul i¼ Uj i¼ rjTuj iþ jj irjTrjþ rjTR hj vj iþ wjTiR hj ql iþ cj i(26) providedthefollowingconditionsaresatisÞed: Condition1:ThereisapointthatisrigidlyÞxedtobothbody l andbody j .Suchapointiscalledasacommon pointofbodies l and j . Condition2:Theoriginofthelocalcoordinateframeofbody l isatthecommonpoint. Condition3:Thesumofthesetoffunctionsofbody l isdependentonlyon r intheformgiveninEq. (22) . Proof .Letthelocalcoordinateofthecommonpointrequired bycondition1inbody l be sl jandinbody j be sj l.Thecommonalityofthepointcanbewrittenasfollows: rlþ R hl sl j¼ rjþ R hj sj l(27) Condition2impliesthat sl j¼ 0 .Substituting sl j¼ 0 intoEq. (27) leadsto rl¼ rjþ R hj sj l(28) Condition3impliesthatthesumofthesetoffunctionsofbody l canbewrittenas Xnli ¼ 1Ul i¼ rlTXnli ¼ 1ul iþ rlTrl Xnli ¼ 1jl i(29) TheconstanttermisomittedinEq. (29) sinceitisinconsequential forthediscussion. Substitutionof rlfromEq. (28) intoEq. (29) andsimpliÞcation usingthefactthat RT( h ) R ( h )isidentityleadtothefollowing expressionfor Pnli ¼ 1Ul i: 021014-6/ Vol.4,MAY2012 TransactionsoftheASME

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rjTXnli ¼ 1ul i ! z}|{uj iþ rjTrjXnli ¼ 1jl i ! z}|{jj iþ rjTR hj 2 sj lXnli ¼ 1jl i ! z}|{vj iþ Xnli ¼ 1ul i !Tz}|{wj T iR hj sj l z}|{qj i¼ Xnli ¼ 1Ul i¼ Uj i(30) Again,theconstanttermisomittedinEq. (30) .Itmaybereadily recognizedthatthesumofthesetoffunctionsonbody l , Pnli ¼ 1Ul i, asseeninEq. (30) isindeedoftheformgiveninEq. (26) . 4.2Proposition4.1astheRecursiveRelationofan IterativeStaticBalancingAlgorithm. Wenowshowthat Proposition4.1canbetreatedasarecursiverelationthatcanbe incorporatedintoaniterativeproceduretoachievestaticbalance ofalinkage.Forthepurposeofthissection,werestrictthelinkage onwhichtheiterativeprocedurecanbeappliedtohavethefollowingfeatures: (1)Thelinkageshouldbetree-structured(i.e.,noclosedloops). Thisfeatureisnecessarysincearecursiverelationrequires atree-structuretopropagate. (2)Allthejointsofthetree-structureshouldberevolutejoints. Thisfeatureisnecessarytosatisfycondition1ofProposition4.1. (3)Wewantalltheloadstohavepotentialenergyfunctionsof theformgiveninEq. (25) ofProposition4.1.Whilewe knowthatzero-free-lengthspringsandconstantloadsdo havethisform(seeEqs. (16) and (17) ),thefactthatthereare severalbodiesinvolvedrequiresattention.TheconÞgurationvariables( rl, hl)ofdifferentbodies(i.e.,ofdifferent l ) shouldbewithrespecttoacommonglobalframeofreference.Henceconstantloadsonallthebodiesshouldbeconstantwithrespecttoacommonglobalreferenceframeand anyzero-free-lengthspringshouldhaveitsoneanchorpoint onthesamecommonglobalframewhiletheotheranchor pointcanbeonanyofthebodiesconstitutingthelinkage. (4)Thecommonreferenceframeshouldbeoneofthebodies ofthelinkage,i.e.,itshouldjointobody/bodiesofthelinkagebyrevolutejoint/joints. 4.2.1TheIterativeStaticBalancingAlgorithm. Wenow presenttheiterativealgorithmandprovethatitleadstostatic balance. PreparatorySteps (1)Assignthereferencebodyastherootnodeofthetreestructure(bodiesarerepresentedasnodesandjointsaslines joiningthenodes).Withthisassignment,foreverylink/ bodyotherthantheroot,thereisaparentbody.Further,everylinkotherthanaterminallinkhasoneormorechildren. (2)Choosealocalframeofreferenceoneverylinktocoincide withthecenterofrevolutejointbetweenthelinkandits parent.Foreverylink k , rkand hkdecidetheconÞguration ofitslocalframewithrespecttheframeoftheroot. (3)Givethistree-structurewiththegivenconstantandzerofree-lengthspringloads(togetherreferredtoasoriginal loads)asaninputtothefollowingiterativeprocedure. IterativeProcedure. Entrycondition:Ifthetree-structurecontainsonlytherootnode,thenexitfromtheiterativeprocedure. Otherwise,proceedtostep1. Step1:Anyterminalnode l ,hasassociatedwithitthefollowing threekindsofpotentialenergyfunctions:(1)duetooriginalloads onbody l ,(2)duetoassociationthathappenedinstep3ofpreviousiterations,and(3)additionalloadsonbody l .Letthenumber ofsuchfunctionsberepresentedby nl o, nl c, nl a,respectively.The Þrsttwokindsoffunctionsareknownfromthegivenproblemand previousiterations,respectively,andthetaskinthisstepistoÞnd theadditionalloadssothat Case(a):Equations (19) Ð (21) aresatisÞedif l isnotachild (i.e.,notÞrstgenerationdescendant)oftheroot. Case(b):Equations (23) Ð (24) aresatisÞedif l isachildofthe root.Notethatinthiscase rlisaconstantbecauseofthewaylocal frameischoseninthepreparatorysteps. Thistaskmakessenseonlyiftheallkindsofpotentialenergy functionsfallundertheformofEq. (15) with( r , h )being( rl, hl). TheÞrstandthethirdkindofpotentialenergyfunctionsdofall undertheformbecauseofthekindofloadswearerestrictingto (seeEqs. (16) and (17) ).Thesecondkindofpotentialenergy functionsconformtotheformbecauseofstep2whichisadirect consequenceofProposition4.1. ThecriticalroleofProposition 4.1inenablingthisiterativeproceduremaybenoted. Further, Appendix B.2 assertsthatthetaskofthisstepisalwaysfeasible. Itmaybenotedthatthereareseveralsetofadditionalloadsthat satisfytheseequations.Thisnonuniquenesscallsfordiscretionof thedesignerinchoosingasuitablesetofadditionalloads. Step2:Incase(a),express Pnl oþ nl cþ nl ai ¼ 1Ul iintheformgivenin Eq. (15) where r and h aretheconÞgurationvariablesofthe parent ofnode l .Thisispossiblesincecondition1(becauseofrevolutejoint),condition2(becauseofpreparatorysteps)and condition3(becauseofstep1)ofProposition4.1aresatisÞed.In case(b),recognizethat Pnl oþ nl cþ nl ai ¼ 1Ul iisaconstantasperProposition3.1. Step3:Associate Pnl oþ nl cþ nl ai ¼ 1Ul iwiththeparentlinkof l andfor energyconservation,disassociate Ul i, i ¼ 1 nl oþ nl cþ nl afrom node l .Becauseofthisassociation, np ð l Þ c( p ð l Þ denotesparentnode of l ,and nk cdenotesthenumberofpotentialenergyfunctionsassociatedwithnode ksofar atstep3)getsincrementedbyoneand theassociatedfunctioncanbewrittenas Up ð l Þ nc¼ Xnl oþ nl cþ nl ai ¼ 1Ul i(31) Step4:Withallpotentialenergyfunctions"robed"fromnode l to itsparent,deletethis terminal node l . Iterator:Oncesteps1Ð4arecompleted,anewtrimmedtreestructureresultswheretheparentsofthenodesdeletedinstep4 hasadditionalpotentialenergyfunctionsassociatedwiththem. Followthisiterativeprocedureagainwiththistrimmedtreestructureastheinput. Witheveryiteration,thetree-structureshrinksanditeventually getsreducedtothesinglerootnode.Anyofthe n0 cpotential energyfunctions(ofthesecondkind)associatedwiththisreduced rootisfromoneofthechildrenoftheroot.Asperstep1,this associationisthroughcase(b).Anyfunctionassociatedthrough case(b)isaconstantasrecognizedinstep2.Thus,thesumof these n0 cpotentialenergyfunctionsisalsoconstant.Further,the sumofthese n0 cpotentialenergyfunctionsisactuallythesum potentialenergyoforiginalloadsandadditionalloadsonallthe descendantsoftheroot.ThiscanbeveriÞedbyrecursivesubstitutioninEq. (31) asexempliÞedinEq. (32) .Therefore,theoriginal loadsareinstaticbalancewiththeadditionalloads. Illustrationofthealgorithmona4Rlinkageunderconstant loads:Figure 7 showsa4Rlinkagewherefourrevolutejointsconnectthegroundandfourotherbodiesserially.Thegroundexerts constantgravitationalforceoneachofthefourbodies.Hencewe takethegroundastherootandnumberthebodiesaccordinglyas showninFig. 7( b ) .Thelocalframeofreferencearelocatedon eachofthebodiesasperthepreparatorystep2.Theconstant loadsonbodies1Ð4arerepresentedas C1 1, C2 1, C3 1,and C4 1,respectively.Theirdetails(pointofaction p andforcevector f )arepresentedinitemnumber1,4,7,and10ofthetableinFig. 7 . Nowwegivethetree-structuretotheiterativeprocedure.The terminalnodeofthetreeis4.Thereisapotentialenergyfunction associatedwiththenodedueto C4 1whichisrepresentedas JournalofMechanismsandRobotics MAY2012,Vol.4 /021014-7

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U4 1C4 1 .Thereisnopotentialenergyfunctionofthesecondkind ð n4 c¼ 0 Þ .Twozero-free-lengthsprings Z4 1and Z4 2areadded sothatthefunctions U4 1C4 1 , U4 2Z4 1 ,and U4 3C4 2 satisfy Eqs. (19) Ð (21) aspercase(a)ofstep1.Allthedetailsofthe springs,constantloadsaswellastheirpotentialenergyinthe standardform(seeEq. (15) )arepresentedinthetableofFig. 7 . Now,asperstep2,thesumof U4 1C4 1 , U4 2Z4 1 ,and U4 3Z4 2 is transformedasa U3 2ð r3; h3Þ inaccordancewithEq. (30) ofProposition(4.1).Thisisfollowedbymakingofanewtree-structureby deletingnode4andassociating U3 2withnode3ofthenewtreestructure.ThiscompletestheÞrstiteration.Theseconditeration actsonthenewtree-structure.ThetableinFig. 7 andFig. 7( a ) giveallthedetailsofalltheiterations.Attheendoffouriterationsweareleftwithasinglerootnodehavingconstantfunction U0 1associatedwithit.Byfollowingthedashedarrowedlineofthe Fig. 7( a ) inthereverseorder,itmaybeveriÞedthat U0 1¼ U1 1C1 1 þ U2 1C2 1 þ U3 1C3 1 þ U4 1C4 1 þ U4 2Z4 1 þ U4 3Z4 2 þ U3 3Z3 1 þ U2 3Z2 1 (32) Hence C4 1, C3 1, C2 1,and C1 1areinstaticbalancewith Z4 1, Z4 2, Z3 1, and Z2 1. Toverifythestaticbalance,thislinkagealongwiththeloads wasmodeledinADAMS.Withzerodamping,apulseofenergy wasinitiallyintroducedtothesystem.Whenthedynamicsimulationofthesystemwascarriedoutitwasnoticedthatthenetkineticenergywasconstantovertime.Thisimpliesthattherewas nopotentialgradientalongthepaththatthelinkagetookinthe dynamicsimulation. InFig. 7( c ) ,joint1andbody1areeliminatedtomodifythis4R exampleintoa3Rexample.Joint2nowjoinsbody2withthe groundat r2¼ 0 5 32 .Restofthebodiesandtheirnumberingis unchanged.TheÞrsttwoiterationsforthisexampleareidentical tothe4Rexample.Thethirditerationisthelastsincenode2now isachildoftheroot.Asrequiredatthisiteration,itmaybeveriÞedthatEqs. (23) and (24) aresatisÞedwith r2¼ 0 5 32 .Onecan haveasimilarmodiÞcationof4Rexampleintoa2Rexampleas showninFig. 7( d ) . Fig.7Detailsofstaticallybalancedgravityloaded4RlinkageanditsmodiÞcationinto3R and2Rlinkage.021014-8/ Vol.4,MAY2012 TransactionsoftheASME

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Illustrationofthealgorithmona2Rlinkageunderazero-freelengthspringload:JustasFig. 7 haseverydetailofthe4Rexample,Fig. 8 haseverydetailofthisexample.Theexplanationis alsoalongthesamelinesofthepreviousexample.Inthisexample,thegivenoriginalloadis Z2 1andthebalancingloadsare Z2 2and Z2 3. Tobuttressthefactthatzero-free-lengthspringsarepractical,a prototypeofthisexamplewasmade,asshowninFig. 8( b ) .To Fig.8Detailsofstaticbalanceofa2Rlinkageunderspringload Fig.9Detailsofstaticbalanceofa4Rtree-structurelinkageunderaconstantloadanda springloadJournalofMechanismsandRobotics MAY2012,Vol.4 /021014-9

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realizezero-free-lengthsprings,pulley-stringarrangementwas used,thedetailsofwhichcanbefoundinHerder[ 5 ]. Illustrationofthealgorithmona4Rtree-structurelinkage underbothconstantloadandzero-free-lengthspringload:While theprevioustwoexampleshadserialarchitecture,thisexample hasbranchesemanatingformthesamenode,asshownin Fig. 9( a ) .Theoriginalloadsactingonitare C3 1and Z4 1.Insteadof takingoriginalloadstobeexclusivelyconstantloadsorexclusivelyzero-free-lengthspringloads,herewehavetakenacombinationofbothtypesofloads.Theseoriginalloadsarebalancedby addingsprings Z3 1, Z3 2, Z4 2, Z2 1,and Z1 1atvariousiterationsinthe iterativealgorithm.Apictorialdepictionoftheiterationsonthese linkagesisgiveninFig. 9( b ) .Alltheremainingdetailsaregiven inthetableofthesameÞgure. Toverifythestaticbalance, h sofbodies1Ð4arevariedinthe followingform: h1¼ p 4þ p sin2 p t ðÞ , h2¼ p 12þ p sin2 p t ðÞ , h3¼ p 1 : 7þ p sin2 p t ðÞ , h4¼ p 1 : 3þ p sin2 p t ðÞ .Thepotentialenergyvariation oforiginalloads C3 1and Z4 1aswellasthebalancingloads,i.e., Z3 1, Z3 2, Z4 2, Z2 1,and Z1 1,areplottedinFig. 10 .Thesumofall thesevariationsisalsoplottedandithasturnedouttobeaconstant.ThisveriÞesthestaticbalance. 4.3StaticBalancingofAnyRevolute-JointedLinkages WithAnyKindofZero-Free-LengthSpringandConstant LoadInteractionWithintheLinkage. Inthestaticbalancing methodforlinkagesprovidedinSec. 4.2 ,otherthanthefactthat thelinkagetobebalancedhastoberevolute-jointedandthatload interactionsareofzero-free-lengthspringorconstantloads,there weretwomorerestrictionsasfollows: (1)Itshouldbepossibletoconsiderthattheloadsonallthe bodiesareexertedbyacommonreferencebody(orframe) ofthelinkage. (2)Thelinkageshouldhaveatree-structure(i.e.,without closedloops). WhentheÞrstrestrictionisviolatedasinFig. 11( a ) ,itisalways possibletobreaktheloadinteractionsintoasuperpositionofseveralloadsetswitheachsetcomplyingtotheÞrstrestriction.For example,theloadinteractioninFig. 11( a ) isbrokenintotwoload setsinFigs. 11( b ) and 11( c ) .Thereferencebodyineachofthese setsisindicatedbyanasterisksymbol(*)initsrespectiveÞgure. Furthermore,inaloadset,ifthereareclosedloops,thenthe closedloopscanbebrokenbyrelaxingcertainjointconstraints. Figures 11( c ) and 11( d ) illustratesbreakingofclosedloops respectivelyinFigs. 11( b ) and 11( c ) .Withclosedloopsbroken, eachoftheloadsetscomplywiththetworestrictionsandtheycan bestaticallybalancedbyaddingbalancingloadsasperSec. 4.2 . Onceeachoftheloadsetsisbalanced,thejointconstraintsthat wererelaxedforbreakingclosedloopscanbereimposedwithout disturbingthestaticbalance.Inotherwords,whenthepotential energythatisafunctionoftheconÞgurationspaceisaconstant,it remainsastheconstantevenwhentheconÞgurationspaceisrestricted(duetore-attachmentofthebrokenjoints). Oncetheconstraintsarereimposed,thelinkagesinalltheload setsarethesameastheoriginallinkageandtheloadsonallthe setscanbesuperposed.Sinceeachloadsetisinstaticbalance,the superpositionisalsoinstaticbalance.Inotherwords,thesumof severalconstantpotentialenergyfunctionsduetoseveralload setsisalsoaconstant.Thissuperpositioncontainsalltheoriginal loadsonthegivenlinkage.Theremnantloadsinthissuperpositionaretheadditionalloadsthatbalancetheoriginalloads.Inthis way,additionalloadsthatstaticallybalanceanyrevolute-jointed linkagewithzero-free-lengthspringandconstantloadinteractions betweenthebodiesofthelinkagecanalwaysbefound.5StaticBalanceofSpatialLinkagesHaving Zero-Free-LengthSpringandConstantLoad InteractionsWithintheLinkageThestaticbalancingtechniqueofSec. 4 wasbasedonProposition4.1,whichwaspresentedfortwoplanarbodies,withtherevolutejointsonlyservingtosatisfythecondition1ofthe proposition.Wecanhaveanalogousstaticbalancingtechniquefor sphericalandrevolute-jointed spatial linkageswithzero-freelengthspringandconstantloadinteractionsbetweenthebodiesof thelinkageprovidedthat (1)thepotentialenergyforzero-free-lengthspringandconstantloadshasthesameformasgiveninEq. (25) , (2)Proposition4.1istrueevenifthetwobodies( l and j )are freetomoveinspace, (3)sphericalandrevolutejointsensurecondition1ofProposition4.1,and (4)analogoustoconstraints (19) Ð (21) whichenablesatisfying condition3oftheProposition4.1,thereareconstraint equationsforspatialcasethatabodycansatisfyinpractice, possiblybyadditionofextrazero-free-lengthspringloads. TheÞrstoneistruesinceinthederivationofthepotential energyofconstantloadsinEq. (1) andzero-free-lengthspring loadsinEq. (6) wouldrequirenomodiÞcationevenif r weretobe consideredasaspatialglobalcoordinate(3 1matrix), a , b , p weretobeconsideredasspatiallocalcoordinates,and R ( h )were tobeconsideredasthespatialrotationmatrixofthebodieswith h possiblyrepresentingEulerangles.That RTR istheidentity Fig.10PotentialEnergyvariationofspringloads,constant loads,andtheirsum Fig.11Breakingaproblemasasuperpositionofseveralproblemwitheachproblembeingstaticbalanceofrevolute-jointed tree-structuredlinkagewithloadsexertedbytherootbody021014-10/ Vol.4,MAY2012 TransactionsoftheASME

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matrix,whichwasusedinthederivation,istrueinthespatialcase also. ThesecondoneistruesinceProposition4.1reliesonEq. (27) , that RTR istheidentitymatrix,andmatrix-algebraicmanipulations.Allofthemarevalidinthespatialcasealso. Thethirdoneistruesinceasphericaljointensuresacommon pointbetweenthetwobodiesitjoinsandarevolutejointensures acommonlinebetweenthetwobodiesitjoins. Inthefourthone,theconstraints (19) Ð (21) wereobtainedby expressingthepotentialenergyasalinearcombinationofasetof basisfunctionsinvolving r and h andsettingthecoefÞcientof h dependentbasisfunctionstozero.Inasimilarvein,inthespatial casealso,thepotentialenergywouldbealinearcombinationofa setofbasisfunctionsinvolving r androtationdeÞning,say,Euler angles.BysettingthecoefÞcientsofEulerangle-dependentterms tozero,theanalogousspatialconstraintscanbeobtained.The claimthatwearenotsubstantiatinginthispaper,forthesakeof brevity,isthattheseanalogousconstraintequationscanalsobe satisÞedinpractice,ifnecessarywiththeadditionofextrazerofree-lengthsprings. Thus,thestaticbalancingtechniquepresentedinthepaperfor revolute-jointedplanarlinkagesextendstosphericalandrevolutejointedspatiallinkages.6ConclusionWepresentedatechniquetostaticallybalanceanyplanar revolute-jointedlinkagehavingzero-free-lengthspringandconstantloadinteractionsbetweenthebodiesofthelinkage.The techniqueinvolvesonlyadditionofzero-free-lengthspringsbut notanyextralink,unlikespring-aidedperfectstaticbalancing techniquescurrentlyintheliterature.Thetechniqueextendsto spatialsphericalandrevolute-jointedlinkagesaswell.ThetechniquereliesonarecursiverelationtoiterativelyremovethedependenceofthepotentialenergyontheconÞgurationvariablesof thebodiesofthelinkage.Recognizingtherecursiverelationalong withtheminimalconditionsthatenableitconstitutesthemain contributionofthispaper.AppendixA:PerfectStaticBalanceand Positive-Free-LengthSpringsThisisanappendixtoSec. 2 .Here,thedifÞcultyinachieving perfectstaticbalanceofaleverbyusingnormallyavailablepositive-free-lengthspringsisdiscussed. The dTd terminthepotentialenergyexpressionofaspring giveninEq. (5) wasseentobealinearcombinationofsin h ,cos h and1whenexpandedasinEq. (6) .Hence,bywriting dTd as a sin h þ b cos h þ c ,thepotentialenergyexpressionofthespring becomes: k 2 a sin h þ b cos h þ c ðÞ kl0 a sin h þ b cos h þ c ðÞ p þ k 2 l2 0 (33) TheÞrstterminEq. (33) isthezero-free-lengthpartandthesecondtermisthefree-lengthpart.Ifthefree-lengthispositive,i.e., l0> 0,thenthefree-lengthpartisnegative.Thefree-lengthpartof thespringisnonconstant(i.e., k = 0andnotboth a and b iszero) exceptfortrivialsituationswherespringconstantiszeroorthe springisattachedtothepivotofthelever.Whenthereareseveral butÞnitepositive-free-lengthandnontrivialsprings,thenetcontributionofthefree-lengthpartisnegative,anditisalsonot knownhavethepossibilityofbeingaconstant,unlikethezerofree-lengthpart.Furthermore,thefree-lengthpartisalsonot knowntobeinthefunctionspacespannedbysin h andcos h . Hence,thepossibilityoffree-lengthpartcancelling(moduloa constant)withzero-free-lengthpartisalsoruledout.Thus,with severalpositive-free-lengthsprings,thereisnowaythenetpotentialenergycouldbecomeaconstant.AppendixB:ConstraintsCanbeSatisfied,IfNotasIt Is,byAdditionofExtraZero-Free-LengthSpring LoadsAppendixB.1:SatisfyingConstraints (12) – (14) . Thisappendixdemonstrateshowbyaddingextrazero-free-lengthspring loadsconstraints (12) Ð (14) canbesatisÞed.Todifferentiate betweenoriginalloadsandbalancingloads,let ns,oand ns,brespectivelyrepresentthenumberoforiginalandbalancingzerofree-lengthspringloadswith ns,oþ ns,b¼ ns.Also,letthespring loadsbeindexedsuchthattheÞrst ns,oloadsareoriginalloads withtheremainingbeingbalancingloads.Similarmeaning appliesfor nc,oand nc,b. Case1:Originalloadsviolatetheconstraint(14),andbalancing loadsareonlyzero-free-lengthsprings.Letustrytosatisfyallthe constraintsbyaddingasinglezero-free-lengthspring.Asperthe notation,thisspringgetstheindex i ¼ ns,oþ 1.Theconstraint(14) canbewrittenasfollows: kns ; oþ 1ans ; oþ 1¼ Xns ; oi ¼ 1kiai(34) wheretheknownquantitiesrelatedtooriginalloadsareonthe righthandside.Equation (34) givestheuniquesolutionof kiaifor i ¼ ns,oþ 1totheconstraint (14) .Furthermore,theconstraints (12) and (13) canberewrittenas kiax ; ikiay ; i kiay ; ikiax ; i bx ; iby ; i i ¼ ns ; oþ 1¼ Xnci ¼ 1ð fyipy ; iþ fx ; ipx ; iÞ Xns ; oi ¼ 1kið ay ; iby ; iþ ax ; ibx ; iÞ þ Xnci ¼ 1ð fx ; ipy ; i fy ; ipx ; iÞ Xns ; oi ¼ 1kið ax ; iby ; i ay ; ibx ; iÞ 2 6 6 6 6 4 3 7 7 7 7 5 (35) The2 2matrixonthelefthandsideoftheequationsisknown since kiaifor i ¼ ns,oþ 1isalreadysolvedinEq. (34) .Furthermore,thematrixisnonsingularsincetherighthandsideof Eq. (34) thatisthesameas kns,o þ 1ans,o þ 1is nonzeroasperthe descriptionthiscase .Wetake bx ; iby ; i½Tfor i ¼ ns,oþ 1asthe inverseofthe2 2matrixtimestherighthandsideofthe Eq. (35) sothattheconstraints (12) and (13) canalsobesatisÞed. Thus,theoretically,withasingleadditionalzero-free-length spring,allthreeconstraints (12) Ð (14) canbesatisÞed. Case2:Originalloadssatisfy (14) ,butviolateatleastoneofthe constraints (12) and (13) .Balancingloadsareonlyzero-freelengthsprings. Ifweproceedalongthesamelinesasthepreviouscase,thenin Eq. (35) ,the2 2matrixonthelefthandsidebecomessingular zero-matrixwhereastherighthandsideisnonzerobythe descriptionofthecase.Thus,inthiscase,withasinglebalancing zero-free-lengthspring,itisnotpossibletosatisfytherelatedconstraint.However,itmaybeveriÞedthatbyaddingtwobalancing springs,alltheconstraintscanbesatisÞed. Thecases1and2coverallpossibletypesofconstraintviolation.HenceweassertthatiftheconstraintsarenotsatisÞedasit is,thenbyaddingaminimumofonezero-free-lengthspringin case (1) (thecomponentrelatedtooriginalloadsinconstraint (14) isnonzero)andtwozero-free-lengthspringincase (2) (thecomponentrelatedtooriginalloadsinconstraint (14) iszero),theconstraintscanbesatisÞed. JournalofMechanismsandRobotics MAY2012,Vol.4 /021014-11

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AppendixB.2:SatisfyingConstraints (19) – (21) InAppendix B.1,iftheconstraints (12) Ð (14) arerespectivelysubstitutedby constraints (19) Ð (21) ,thereisgoingtobenochangeexceptfor therighthandsideofEq. (34) whichtakestheform Pnoi ¼ 1vi, andtherighthandsideofEq. (35) whichtakestheform ½ Pn i ¼ 1qy ; iwy ; iþ qx ; iwx ; i Pn i ¼ 1qx ; iwy ; i qy ; iwx ; i T.Thereis nochangeinthelefthandsidesinceherealso,theadditionalloads arezero-free-lengthsprings.Hence,analogoustotheconclusion ofAppendixB.1,weconcludethatiftheconstraints (19) Ð (21) are notsatisÞedasitis,thenbyadditionofaminimumofonezerofree-lengthspringincasetheconstraint (21) isoriginallyviolated andaminimumoftwozero-free-lengthspringsincasetheconstraint (21) isnotoriginallyviolated,thethreeconstraintscanbe satisÞed.References[1]LucienLaCoste, www.http://en.wikipedia.org/wiki/Lucien_LaCoste [2]Streit,D.A.,andShin,E.,1993,"EquilibratorsforPlanarLinkages," Trans. ASMEJ.Mech.Des. , 115 (3),pp.604Ð611. [3]Fattah,A.,andAgrawal,S.K.,2006,"Gravity-BalancingofClassesof IndustrialRobots," Proceedingsofthe2006IEEEInternationalConferenceon RoboticsandAutomation ,May15Ð19,Orlando,Florida,pp.2872Ð2877. [4]Agrawal,S.K.,andFattah,A.,2004,"Gravity-BalancingofSpatialRobotic Manipulators," Mech.Mach.Theory , 39 (12),pp.1331Ð1344. [5]Herder,J.L.,2001,"Energy-FreeSystems:Theory,ConceptionandDesignof StaticallyBalancedSpringMechanisms,"Ph.D.thesis,DelftUniversityof Technology,Delft,Netherlands. [6]Cho,C.,Lee,W.,andKang,S.,2010,"StaticBalancingofaManipulatorWith HemisphericalWorkSpace," 2010IEEE/ASMEInternationalConferenceon AdvancedIntelligentMechatronics(AIM) ,pp.1269Ð1274. [7]Zhang,D.,Gao,F.,Hu,X.,andGao,Z.,2011,"StaticBalancingandDynamic ModelingofaThree-Degree-of-FreedomParallelKinematicManipulator," 2011IEEEInternationalConferenceonRoboticsandAutomation(ICRA) ,pp. 3211Ð3217. [8]Rahman,T.,Ramanathan,R.,Seliktar,R.,andHarwin,W.,1995,"ASimple TechniquetoPassivelyGravity-BalanceArticulatedMechanisms," ASMEJ. Mech.Des. , 117 (4),pp.655Ð658. [9]Lin,P.-Y.,Shieh,W.-B.,andChen,D.-Z.,2010,"DesignofaGravityBalancedGeneralSpatialSerial-TypeManipulator,"ASMEJ.Mech.Rob., 2 , 031003. [10]Herder,J.L.,1998,"DesignofSpringForceCompensationSystems," Mech. Mach.Theory , 33 (1-2),pp.151Ð161. [11]Deepak,S.R.,andAnanthasuresh,G.K.,2012,"StaticBalancingofaFour-Bar LinkageandItsCognates," Mech.Mach.Theory , 48 (0),pp.62Ð80. [12]Deepak,S.R.,andAnanthasuresh,G.K.,2009,"StaticBalancingofSpringLoadedPlanarRevolute-JointLinkagesWithoutAuxiliaryLinks," 14th NationalConferenceonMachinesandMechanisms ,Dec.17Ð18,NITDurgapur,India. [13]Lin,P.-Y.,Shieh,W.-B.,andChen,D.-Z.,2010,"AStiffnessMatrixApproach fortheDesignofStaticallyBalancedPlanarArticulatedManipulators," Mech. Mach.Theory , 45 ,pp.1877Ð1891. [14]Agrawal,A.,andAgrawal,S.,2005,"DesignofGravityBalancingLegOrthosisUsingNon-ZeroFreeLengthSprings," Mech.Mach.Theory , 40 ,pp. 693Ð709. [15]Gopalswamy,A.,Gupta,P.,andVidyasagar,M.,1992,"ANewParallelogram LinkageConÞgurationforGravityCompensationUsingTorsionalSprings," Proceedingsofthe1992IEEEInternationalConferenceonRoboticsand Automation . [16]Gosselin,C.M.,andWang,J.,2000,"StaticBalancingofSpatialSix-Degreeof-FreedomParallelMechanismsWithRevoluteActuators," J.Rob.Syst. , 17 (3),pp.159Ð170. [17]Ulrich,N.,andKumar,V.,1991,"PassiveMechanicalGravityCompensation forRobotManipulators," Proceedingsofthe1991IEEEInternationalConferenceonRoboticsandAutomation ,Vol.2,pp.1536Ð1541. [18]Endo,G.,Yamada,H.,Yajima,A.,Ogata,M.,andHirose,S.,2010,"APassive WeightCompensationMechanismWithaNon-CircularPulleyandaSpring," 2010IEEEInternationalConferenceonRoboticsandAutomation(ICRA) ,pp. 3843Ð3848. [19]Koser,K.,2009,"ACamMechanismforGravity-Balancing," Mech.Res. Commun. , 36 (4),pp.523Ð530. [20]deVisser,H.,andHerder,J.L.,2000,"Force-DirectedDesignofaVoluntary ClosingHandProsthesis,"J.Rehabil.Res.Dev.,37 ,pp.261Ð271. [21]Herder,J.L.,andvandenBerg,F.P.A.,2000,"StaticallyBalancedCompliant Mechanisms(SBCMS),anExampleandProspects," ProceedingsASMEDETC 26thBiennialMechanismsandRoboticsConference ,Paperno.DETC2000/ MECH-14144. [22]Howell,L.L.,andMidha,A.,1995,"ParametricDeßectionApproximationsfor End-Loaded,Large-DeßectionBeamsinCompliantMechanisms," ASMEJ. Mech.Des. , 117 ,pp.156Ð165. [23]Howell,L.L.,2001, CompliantMechanisms ,Wiley,NewYork.021014-12/ Vol.4,MAY2012 TransactionsoftheASME



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FORCE RESPONSE AND STABILITY OF DEPLOYABLE MECHANISMS FOR SMALL SATELLITES By AMRITH N. HANSOGE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2014

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© 2014 Amrith N. Hansoge

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To my p arents, Nandini and Nagesh R. Hansoge, and my b rother, Lalith fo r everything they have given me

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4 ACKNOWLEDGMENTS The motivating idea for this work is attributed to the work done on tape spring booms by Dr. Tom Murphey and his team at AFRL, Albuquerque, New Mexico. However, credits for subsequent brainstorming to test the practical implementation of this technology, solely goes to my adviser, Dr. Gloria J. Wiens. I feel truly blessed to have been given this opportunity by her to work on this nascent and exciting technology. I am really grateful to her for believing in my abilities, sticking by me even while she was away on federal government duties and inspiring and ebbing me on to go that one step further. I would also like to thank my committee member, gritties of my research, when sought. Secondly, no amount of gratitude can repay the unconditional support, trust and love that my parents and family have given me over the years and I owe a big thank you to each one of them. I would also like to thank my lab mates Paul, Karl, JoAnna, Leopoldo and Alonso for all the technical discussions and fun times we have had both in the lab and off it. In hindsight, it indebted to my friends here in Gainesville and back in India for their enormous support and gracious love during some very testing times in these first two years of my living in a new country. Lastly, I would like to thank God Almighty for giving me the power and the will to drive on and achieve my dreams in all honesty and good faith.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LIST OF FIGURES ................................ ................................ ................................ ......................... 8 LIST OF OBJECTS ................................ ................................ ................................ ....................... 12 LIST OF ABBREVIATIONS ................................ ................................ ................................ ........ 13 NOMENC LATURE ................................ ................................ ................................ ...................... 14 ABSTRACT ................................ ................................ ................................ ................................ ... 16 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 18 Literature Review ................................ ................................ ................................ ................... 18 Mechanism: Kinematic Configuration ................................ ................................ ................... 31 2 STRUCTURAL COMPUTATIONS OF MECHANISMS ................................ .................... 35 Equilibrium, Compatibility and Flexibility Matrices ................................ ............................. 35 Part 1 of Computation: Boom as a Force ................................ ................................ ................ 40 Rotational Spring Moments Forces Equivalence ................................ ......................... 41 Effect of Rotational Springs ................................ ................................ ............................ 44 Force Response Predictor Technique ................................ ................................ .............. 45 Part 2 of Computation: Boom as Member ................................ ................................ .............. 45 Kinematic Bifurcations ................................ ................................ ................................ ........... 47 Overall Stiffness Matrix ................................ ................................ ................................ ......... 49 3 EQUILIBRIUM STABILITY OF MECHANISMS ................................ .............................. 50 Potential Energy Function ................................ ................................ ................................ ...... 50 Eq uilibrium Conditions Jacobian and Hessian Matrices ................................ ..................... 52 Stability of multiple degrees of freedom mechanisms ................................ ........................... 53 4 CASE STUDIES AND VARIATIONS OF THE SIX BAR MECHANISMS ...................... 54 Equilibrium Matrices ................................ ................................ ................................ .............. 54 Configuration 1: ................................ ................................ ................................ ............... 54 Configuration 2: ................................ ................................ ................................ ............... 55 Configuration 3: ................................ ................................ ................................ ............... 55

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6 Case Studies ................................ ................................ ................................ ............................ 55 Case Study 1a: Variable Spring Stiffness Constants ................................ ....................... 56 Case Study 1b: Variable Spring Preloads ................................ ................................ ........ 57 Stability Plots ................................ ................................ ................................ ................... 58 Two more var iations of the Six bar Mechanism ................................ ............................. 69 Case Study 2: Deployment Patterns under constant parameters ................................ ..... 74 5 SIMULATION EXAMPLES IN MSC ADAMS ................................ ................................ ... 82 Comparison of assumptions between models ................................ ................................ ......... 82 Configuration 1 ................................ ................................ ................................ ................ 83 Configuration 2 ................................ ................................ ................................ ................ 86 Configuration 3 ................................ ................................ ................................ ................ 88 Configuration 4 ................................ ................................ ................................ ................ 90 Configuration 5 ................................ ................................ ................................ ................ 92 Configuratio n 6 ................................ ................................ ................................ ................ 94 6 DISCUSSIONS AND CONCLUSION ................................ ................................ ................ 101 Discussions ................................ ................................ ................................ ........................... 101 Future Work ................................ ................................ ................................ .......................... 102 APPENDIX OVERAL L FULL SIZED EQUILIBRIUM MATRIX ................................ ............................... 104 TRANFORMATION (ROTATION JACOBIAN) MATRIX ................................ ..................... 106 SAMPLE CODE IN MATLAB FOR CONFIGURATION 4 ................................ ..................... 107 Main File ................................ ................................ ................................ ............................... 107 Part 1 Computation Function File ................................ ................................ ......................... 110 Part 2 Computation Function File ................................ ................................ ......................... 115 Function to calculate the angles and link lengths after computations ................................ .. 119 Jacobian and Hessian Matrices Function File ................................ ................................ ...... 120 LIST OF REFERENCES ................................ ................................ ................................ ............. 122 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 125

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7 LIST OF TABLES Table page 4 1 Constant Parameters of the Structure ................................ ................................ ................. 56 4 2 Parameter Values for Variable Spring Stiffness Sweep Algorithm (superscripts represent global joint numbers. Refer Figure 1 16) ................................ ........................... 56 4 3 Parameter Values for Variable Spring Preload Sweep Algorithm (superscripts represent global joint numbers. Refer Figure 1 16) ................................ ........................... 57 4 4 Parameter Values for Symmetric Loading Deployment Pattern (superscripts represent global joint numbers. Refer Figure 1 1 6) ................................ ........................... 74 4 5 Total Load Values for Configuration 1 ................................ ................................ .............. 75 4 6 Total Load Values for Configuration 2 ................................ ................................ .............. 76 4 7 Total Load Values for Configuration 3 ................................ ................................ .............. 77 4 8 Total Load Values for Configuration 4 ................................ ................................ .............. 78 4 9 Total Load Values for Configuration 5 ................................ ................................ .............. 79 4 10 Total Load Values for Configuration 6 ................................ ................................ .............. 80 5 1 Parameter Values for Software Simulation and Theoretical Calculations of the Deployment Patterns All Configurations ................................ ................................ ........ 83 5 2 Parameter Values for Software Simulation of an unstable set Configuration 4, with fixed length of boom for Part 2 of simulation ................................ ................................ .... 98 5 3 Parameter Values for Software Simulation of a stable set Configuration 4, with fixed length of boom for Part 2 of simulation ................................ ................................ .... 99

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8 LIST OF FIGURES Figure page 1 1 Double Ri ng Antenna ................................ ................................ ................................ ........ 20 1 2 Space Applications of Deployable Structures ................................ ................................ .... 20 1 3 Hoberman 2D mechanism deployment pattern A to C ................................ ...................... 21 1 4 Retractable Roof ................................ ................................ ................................ ................ 21 1 5 Two Irathane and AI alloy cylinders ................................ ................................ ................. 22 1 6 Solid Surface Deployment Antenna ................................ ................................ ................... 22 1 7 Solar Sail for small satellites ................................ ................................ .............................. 24 1 8 Carbon Mandrel Deployable Reflector for small satellites ................................ ................ 25 1 9 SAR Structure for small satellites ................................ ................................ ...................... 25 1 10 STEM ................................ ................................ ................................ ................................ . 26 1 11 MARSIS Boom ................................ ................................ ................................ .................. 26 1 12 CFRP Boom ................................ ................................ ................................ ....................... 27 1 13 Elastic Hinge Boom ................................ ................................ ................................ ........... 28 1 14 TRAC Boom ................................ ................................ ................................ ...................... 28 1 15 SIMPLE Bo om ................................ ................................ ................................ ................... 29 1 16 Typical arrangement of a six bar mechanism with the integrated boom ........................... 32 1 17 Variations in boom arrangement within a six bar mechanism ................................ .......... 33 2 1 Single Beam Memb er ................................ ................................ ................................ ........ 37 2 2 Force Moment Equivalent System ................................ ................................ .................... 42 4 1 Starting position for Configurations 1 6 ................................ ................................ ............ 58 4 2 Double Sarrus Mechanism (six bar in 2D) on a 2U frame Singular Configurations/Bifurcation Points ................................ ................................ ..................... 58 4 3 Stab ility Plot Varying Spring Constants, Constant Initial Preloads Configuration 1 ................................ ................................ ................................ ................................ .......... 59

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9 4 4 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 1 ................................ ................................ ................................ .................. 59 4 5 Stability Plot Constant Spring Constants, Varying Initial Preloads Configurat ion 1 ................................ ................................ ................................ ................................ .......... 60 4 6 2D Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 1 ................................ ................................ ................................ .................. 60 4 7 Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 2 ................................ ................................ ................................ ................................ .......... 6 1 4 8 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 2 ................................ ................................ ................................ .................. 61 4 9 Stability Cube ................................ ................................ ................................ ..................... 63 4 10 Stabi lity Plot Constant Spring Constants, Varying Initial Preloads Configuration 2 ................................ ................................ ................................ ................................ .......... 64 4 11 2D Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 2 ................................ ................................ ................................ .................. 64 4 12 Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 3 ................................ ................................ ................................ ................................ .......... 65 4 13 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configurati on 3 ................................ ................................ ................................ .................. 65 4 14 Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 3 ................................ ................................ ................................ ................................ .......... 66 4 15 2D Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 3 ................................ ................................ ................................ .................. 66 4 16 Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 4 ................................ ................................ ................................ ................................ .......... 67 4 17 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 4 ................................ ................................ ................................ .................. 67 4 18 Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 4 ................................ ................................ ................................ ................................ .......... 68 4 19 2D Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 4 ................................ ................................ ................................ .................. 68 4 20 Two more variations ................................ ................................ ................................ .......... 69

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10 4 21 Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 5 ................................ ................................ ................................ ................................ .......... 70 4 22 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 5 ................................ ................................ ................................ .................. 70 4 23 Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 5 ................................ ................................ ................................ ................................ .......... 71 4 24 2D Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 5 ................................ ................................ ................................ .................. 71 4 25 Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 6 ................................ ................................ ................................ ................................ .......... 72 4 26 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 6 ................................ ................................ ................................ .................. 72 4 27 Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 6 ................................ ................................ ................................ ................................ .......... 73 4 28 2D Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 6 ................................ ................................ ................................ .................. 73 4 29 Displacement Response and Stable final Equilibrium Configuration 1 ......................... 75 4 30 Displacement Response and Unstable final Equilibrium Configuration 2 ..................... 76 4 31 Displacement Response and Unstable final Equilibrium Configuration 3 ..................... 77 4 32 Displacement Response and Stable final Equilibrium Configuration 4 ......................... 78 4 33 Displacement Response and Stable final Equilibrium Configuration 5 ......................... 79 4 34 Displacement Response and Unstable final Equilibrium Configuration 6 ..................... 80 5 1 Representation of the boom in an ADAMS model ................................ ............................ 82 5 2 Stabilizing Boom Length Configuration 1 ................................ ................................ ...... 84 5 3 Stabilizing Joint Angles Configuration 1 ................................ ................................ ........ 84 5 4 Joint Angles from quasi static analysis MATLAB Configuration 1 ........................... 85 5 5 Deployment Pattern in MATLAB Configuration 1 ................................ ........................ 85 5 6 Stabilizing Boom Length Configuration 2 ................................ ................................ ...... 86 5 7 Stabilizing Joint Angles Configuration 2 ................................ ................................ ........ 86

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11 5 8 Joint Angles from quasi static analysis MATLAB Configuration 2 ........................... 87 5 9 De ployment Pattern in MATLAB Configuration 2 ................................ ........................ 87 5 10 Stabilizing Boom Length Configuration 3 ................................ ................................ ...... 88 5 11 Stabilizing Joint Angles Configuration 3 ................................ ................................ ........ 88 5 12 Joint Angles from quasi static analysis MATLAB Configuration 3 ........................... 89 5 13 Deployment Pattern in MATLAB Configuration 3 ................................ ........................ 89 5 14 Stabilizing Boom Length Configuration 4 ................................ ................................ ...... 90 5 15 Stabilizing Joint Angles Configuration 4 ................................ ................................ ........ 90 5 16 Joint Angles from quasi static analysis MATLAB Configuration 4 ........................... 91 5 17 Deployment Pattern in MATLAB Configuration 4 ................................ ........................ 91 5 18 Stabilizing Boom Length Configuration 5 ................................ ................................ ...... 92 5 19 Stabilizing Joint Angles Configuration 5 ................................ ................................ ........ 92 5 20 Joint Angles from quasi static analysis MATLAB Configuration 5 ........................... 93 5 21 Deployment Pattern in MATLAB Configuration 5 ................................ ........................ 93 5 22 Stabilizing Boom Lengths Configuration 6 ................................ ................................ .... 94 5 23 Stabilizing Joint Angles Configuration 6 ................................ ................................ ........ 94 5 24 Joint Angles from quasi static analysis MATLAB Configuration 6 ........................... 95 5 25 Deployment Pattern in MATLAB Configuration 6 ................................ ........................ 95 5 26 Stabilizing Joint Angles (before and after perturbation) ................................ .................... 97 5 27 Stabilizing Boom Length (before and after perturbation) gives leverage to the system ... 97 5 28 Configuration 4 Unstable remains unstable with fixed length boom model .................. 99 5 29 Configuration 4 Stable remains stable with fixed length boom model ......................... 100

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12 LIST OF OBJECTS Object page 5 1 ADAMS Simulation Reaching Equilibrium Configuration 1 ................................ ...... 84 5 2 ADAMS Simulation Reaching Equilibrium Configuration 2 ................................ ...... 86 5 3 ADAMS Simulation Reaching Equilibrium Configuration 3 ................................ ...... 88 5 4 ADAMS Simulation Reaching Equilibrium Configuration 4 ................................ ...... 90 5 5 ADAMS Simulation Reaching Equilibrium Configuration 5 ................................ ...... 92 5 6 ADAMS Simulation Reaching Equilibrium Configuration 6 ................................ ...... 94 5 7 MSC/ADAMS Simulation with perturbation using BISTOP function .............................. 97 5 8 MSC/ADAMS simulation of an unstable set of Configuration 4 with fixed boom length and pe rturbation ................................ ................................ ................................ ...... 99 5 9 MSC/ADAMS simulation of a stable set of Configuration 4 with fixed boom length and perturbation ................................ ................................ ................................ ............... 100 5 10 MSC/ADAMS Static Analysis of the stable set Configuration 4 ................................ .. 100

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13 LIST OF ABBREVIATIONS BMAB JCM Breadth Multi Angulated Beam Joint Connectivity Matrix SIMPLE Self contained Linear Meter class Deployable Boom SLE STEM TRAC Scissor like Element Storable Tubular Extendible Member Triangular Retractable and Collapsible

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14 NOMENCLATURE A Equilibrium Matrix B Compatibility (or Kinematic) Matrix C b j m h r Flexibility Matrix Number of Bars Number of Joints Number of degrees of freedom (or degree of kinematic indeterminacy) Number of states of self stress (or degree of static indeterminacy) Number of Rows of the Equilibrium Matrix Rank of Equilibrium Matrix Number of rotational springs Nodal displacement vector, where joints are taken in order and the x component precedes the y component E P t q Bar elongation vector Nodal Force Vector Internal Axial Forces Vector Spring Moment Vector Spring Rotation Vector Length of link b K Diagonal Spring Stiffness Matrix M A matrix whose columns contain a set of joint displacement components that constitute independent inextensional mechanisms R A matrix whose columns contain a set of spring rotation angles for the corresponding mechanisms in M Q Generalized coordinates of the pin joints Lagrange Multiplier (typically, the axial forces in the bars)

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15 c J H Kinematic constraint equations Number of Kinematic constraint equations Jacobian Matrix between the Cartesian coordinates x and y and the Polar coordinates Jacobian Matrix of the Potential Energy Function with respect to each Cartesian variable Hessian Matrix of the Potential Energy Function Force vector equivalent to the preloads in the springs Force applied by the boom Force applied at ea ch step Fraction of total load P applied at each step Generalized displacement vector Position Vector of joints in a configuration space Angle that each link makes with the horizontal (x) axis in the global coordinate system Rotation Matrix (3x3)

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16 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science FORCE RESPONSE AND STABILITY OF DEPLOYABLE MECHANISMS FOR SMALL SAT E LLITES By Amrith N. Hansoge August 2014 Chair: Gloria J. Wiens Major: Mechanical Engineering In the past, deployable mechanisms for large satellites were developed and over time, highly efficient and multi featured deployable structures were witnessed. However, with concerns over cost and effort, the focus has significantly shifted to small satel lites with much simpler platforms. mechanisms used to deploy sensors and antennae in space have been largely restricted to simple one arm pin jointed members or telescopic mechanisms . However, with the advancements in sensor types and capabilities, and reduction in their size and power needs, interest is growing in using more hierarchical deployment schemes for sensor platforms that efficiently fit within small satellites. Furthermo re, tape spring boom technology is currently being downsized to dimensions associated with small satellites which offer a potential dual means of adding stiffness and a passive means of actuation . The objective of this thesis is to demonstrate the possible applications of tape spring boom technology to deployable structures for small satellites, mainly rigid link, pin jointed and spring loaded mechanisms. The goal is to enhance the positioning integrity of the deploye d structure serving as a sensor platform an d to retain a level of simplicity of the deployment mechanism.

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17 In this thesis, simple deployable structures like the 6 bar pin jointed mechanisms are presented, which offer multiple platform capabilitie s within one deployment , h ence , lead ing us back towards the multi functionality aspect one of the crucial features that the full sized satellites possessed. Furthermore, these 6 bar mechanisms are designed to actuate by means of a single tape spring boom. A theoretical , quasi static approach for determining the displacement response of a deployable structure is implemented for demonstrating the deployment pattern . The stowed structure has pre loaded rotational springs at the joints and is simultaneously actuated by a boom. Since the pre loads change continuously as the system deploys, nonlinear iterative matrix methods are used to solve this problem. The s tability of these mechanisms is studied at their equilibrium points. Moreover, a tape spring boom which is bi stable in nature offers f urther stiffness to the structure in its deployed state. Integrating these bi stable booms within a deployable mechanism and by looking at the characteristics of the Hessian of the potential energy function, it is also shown how this sufficiently rigid boo m affects the stability of the deployed structure. Herein, the force method of matrix analysis for deployable structures is used for analyses. To further validate and confirm the theoretical quasi static app roach predict s the deployment patterns , MSC/Adams dynamic simulations were conducted . At the end, the possibilities of the system failing due to insufficient actuation force by the boom the condition where the boom does not reach its second stable position is also briefly discussed. In summary, this thesis demonstrates boom integration which offers enhanced stiffness behavior of deployed mechanisms, simultaneously providing compact stowed configurations and a means of actuation.

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18 CHAPTER 1 INTRODUCTION Literature Review W ith the advancements in sensor types and capabilities, and reduction in their size and power needs, interest is growing in using hierarchical deployment schemes for sensor platforms that efficiently fit within small satellit es. Furthermore, to kee p with the concepts of small satellites, the meter long boom technology is currently being downsized to dimensions associated with small satellites. Tape spring booms depending on their configurations can possess significant axial and flexural stiffness i n their deployed configuration and therefore the boom can add to the stiffness characteristics of the deployed system by behaving like a rigid member. T herefore, t o enhance the positioning integrity of such a deployed mechanism (aka a deployed structure) an d retain a level of simplicity, the integration of tape spring booms within deployable rigid link mechanisms for small satellite applications is explored in this thesis. The integration of these booms offers the potential of enhanced stiffness behavior of the deployed mechanism while simultaneously providing compact stowed configurations and a passive means of actuation. Th e objective of this thesis is to demonstrate the possible applications of tape spring boom technology to deployable structures for sm all satellites, mainly rigid link, pin jointed and spring loaded mechanisms. This work aims to demonstrate the quasi static deployment pattern of a typical six bar mechanism and its variations, viz. the presence of triple revolute joints, ternary links, mu lti angulated links and fixed joints. Secondly, this work also seeks to study the equilibrium stability characteristics of such mechanisms. Furthermore, simulations of identical systems in a dynamics solver, MSC/ ADAMS is used to validate and confirm the th eoretical quasi static approach presented in this work to predict the deployment patterns.

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19 A deployable structure is a structure that can be transformed, with the addition of an ener gy input, from a stowed, compact configuration to a n expanded form . The ea rliest known deployable structure to humans is the Umbrella. Over the years, more so in the last century, numerous types of d eployable structures have been designed and developed by researchers and manufacturers worldwide. Despite the large variety, they c an be categorized broadly into two groups as: Strut Structures : scissor hinged, tensile, pneumatic and sliding mechanisms, etc. , Surface (membrane) Structures : folde d, inflatable, telescopic, etc. Deployable structures have been extensively documented in Merchan [1]. Some of the structures which have found importance are the double ring antenna (Figure 1 1) (Xu [2]), the Hoberman mechanisms (Figure 1 3) (Kassabian et al [3]), solar arrays, solar sails, astro mesh antenna (Figure 1 2) (Mikulas et al [4]), r etractable roof structures (Figure 1 4 Escrig [5]), foldable cylinders and solid surface deployable antenna (Figures 1 5 and 1 6) (Guest [6]). Important foundational work on deployable Tensegrity structures was done by Knight [7] and Tibert [8]. Murphey [9 ] has subtly elucidated the need for space deployable structures, the current technologies in use, and the challenges that lie ahead in this field. In this thesis, the main focus is on the pin jointed bar structures belonging to the first category listed a bove. A d eployable structure for space applications is really important because there always exist space constraints in launch rockets and thus cannot be transported to space in full size. The d eployability of a structure definitely results in an extra cos t due to more sophisticated, expensive, movable connections, locking mechan isms, and deployment mechanisms but it is generally balanced by the it s greater potential for adaptability, mobility, and labor saving construction.

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20 A B Figure 1 1. Double Ring Antenna. A) Expanded and B) Compressed . Source : Xu, Y., Guan, F., ring 554. A B C D Figure 1 2. Space Ap plications of Deployable Structures. A) Solar Arrays, B) Solar Sail, B) Astro Mesh Antenna and C) TRW PAMS. Source : Mikulas, M. M., Murphey, T., D and 2 D /ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL, USA, pp. AIAA 2008 2243.

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21 A C Figure 1 3. Hoberman 2D mechan ism deployment pattern A to C. Source : Kassabian, P. E., Bldgs, 134, pp. 45 56. A B Figure 1 4. Retractable Roof A to B. Source s : Kassabian, P. E., Pellegrino, S., 1999, 56. Escrig 1(2), pp. 79 91.

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22 A B Figure 1 5. Two Irathane and AI alloy C y linders. A) vertical and B) h orizontal . Source : Guest, ssertation, University of Cambridge, England. Figure 1 6. Solid Surface Deployment Antenna . Source : England.

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23 Pellegrino [10], quotes, Often th e study of the deployment process is the most challenging phase in the development of new deployable structures, and poses severe constraints on what can or cannot be realized. Therefore, it is impossible, to approach the field of deployable structures wit h a single, general concept or theory. Instead, the study of successful structures, and of the concepts behind them, with great care, it is possible to evolve new concepts that expand the boundaries of the feasible regions in this design space. Since the turn of the century, we have arrived at a situation where the core concepts and theories to handle complex macro sized deployable structures are well established now and the focus is now, like the fields of electronics and computer science, slowly shifting to cost effective, compact, small and micro level space systems. In the past, deployable mechanisms for large satellites were developed and over time, highly efficient and m ulti featured deployable structures were witnessed. However, with concerns over cost and effort, the focus has significantly shifted to small satellites and has led to an acceleration of work to efficiently package the deployable mechanisms within these sm all satellites. Notable work has been made by Knight [7], Pellegrino et al [11] (Figure 1 9) , Barrett [12] (Figure 1 8) and Murphey [9] on deployable solar sails, reflectors and downsizing the large scale mechanisms to fit into small satellites. However, d ue to the simultaneous focus given to reduce the actuation efforts as well, there has been rapid development in composite boom technology too . Significant success in boom technology has already been realized by STEM (Storable Tubular Extendible Member) (Fi gure 1 7) by Ri mrott [13], by Pellegrino [14] , S olar Sail by Fernandez et al [15] (Figure 1 7) , CTM (Collapsible Tube Mast) by SENER, DLR (Figure 1 11) , TRAC (Triangular Retractable and Collapsible) by Banik [16] (Figure 1 14), Roybal et al [17] and Thomas [18] Elastic Hinge boom by Beavers [19] (Figure 1 13) and CFRP boom by Sickinger et al [20] (Figure 1 12) . The Air Force Research Laboratory ( AFRL ) at Albuquerque, N ew M exico has developed one such actuation boom recently called SIMPLE,

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24 which is a bi stab le tape spring made of the composite material NiTi (Figure 1 15 ) (Jeon and Murphey [21]) . But then again, the reduction in satellite sizes has resulted in mono function missions, which, in turn has led to diminishing the multi function capabilities of the same. A B Figure 1 7. Solar Sail for small satellites . A) Top View and B) Side View. Source : Fernandez, J. M., Lappas, V. J., Daton Concept using bi Vol. 69(1 2), pp. 78 85.

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25 Figure 1 8. Carbon Mandrel Deployable Reflector for small satellites . Source : Barrett, R., 2007, Satellites, Logan, UT, USA, pp. 109. A B Figure 1 9. SAR Structure for small satellites . Source : Pellegrino, S., Kukathasan, S., Tibert, G., CUED/D STRUCT/TR190, Defence Evaluation Research Agency and the University of Cambridge , England.

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26 Figure 1 10. STEM . A) Single STEM, B) and C) Bi STEM . Source : Mallikarachchi, C., 2011, Walled Composite Deployable Booms with Tape Dissertation, University of Cambridge, England. Figure 1 11. MARSIS Boom . A) Hi ng e and B) Stowed Configuration. Source : Mallikarachchi, Walled Composite Deployable Booms with Tape Ph.D. Dissertation, University of Cambridge, England.

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27 A B C D E Figure 1 12. CFRP Boom. A D) De ployment and E) Close up view. Source : Mallikarachchi, C., Walled Composite Deployable Booms with Tape Dissertation, University of Cambridge, England.

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28 Figure 1 13. Elastic Hinge Boom . Source : Beavers, F. L., Munshi , N. A., Lake, M. S., Maji, A., and Materials Conference, Denver, Colorado, USA, pp. 1 5. Figu re 1 14. TRAC Boom . Source : Force Institute of Technology, OH, USA.

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29 C Figure 1 15. SIMPLE Boom. A) Hub, B) Fully Deployed and C ) Stowed . Source : Parera, P. M., Licentiate Thesis in Engineering Mechanics, Technical Reports from Royal Institute of Technology, Department of Mechanics, KTH University, Stockholm, Sweden. In this thesis , simple deployable structures like the 6 bar pin jointed mechanisms are presented, as shown in Fig ures 1 1 6 and 1 17 which offer multiple platforms . These, therefore, lead us back towards the multi functionality aspect one of the c rucial features that the full sized satellites possessed. To keep up with the concepts of small satellites, the current meter long booms are also appropriately downsized theoretically into dimensions associated with small satellites. Furthermore, these 6 b ar mechanisms are designed to actuate by means of a single boom.

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30 In this work , a theoretical outline to compute, quasi statically, the force displacement response of a deployable structure is presented. The stowed structure has pre loaded rotational spring s at the joints and is simultaneously actuated by a boom. The pre loaded rotational springs have two functions: one, to aid the deployment of the system, and two, to provide joint stiffness at deployed equilibrium positions. Since the pre loads change con tinuously as the sys tem deploys, nonlinear incremental matrix methods are used to solve this problem. O ne assumption made in this work is that the boom possesses significant axial and flexural stiffness in its deployed configuration and therefore the boom also adds to the stiffness characteristics of the system by behaving like a rigid member. This is because the tape spring strands within a boom can always be chosen to have certain dimensions and can always be arranged in certain ways that increases its ax ial and flexural stiffness using the concepts of mechanics of materials. The number of tape spring strands can also be increased to obtain a more rigid boom . In fact, research on improving the cross sectional inertia is very much an active field and a few notable examples are STEM (Storable Tubular E xtendible Member) by Rimrott [13] and fur ther developed by Pellegrino [14] by implementing nested booms, Solar Sail by Fernandez et al [15] , CTM (Collapsible Tube Mast) by SENER, DLR, TRAC (Triangular Retractabl e and Collapsible) by Banik [16], Murphey et al [17] and Thomas [18] demonstrates the adoption of TRAC geometry to tape springs. Therefore, the structural computation procedure is divided into two parts: 1. Until the boom reaches its deployed stable configura tion the boom is modeled only as a force acting on its connecting points with the mechanism. 2. The mechanism stops expanding as the stable length of the deploying boom is reached, at which time boom length is locked. Hence, there on the system t ries to reach the equilibrium configuration based on the residual forces due to the springs.

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31 Since incremental techniques are being employed, this analysis can also be used to check if the applied force is sufficient for the necessary deployment or not. Su ch a situation arises, however, only when: The springs oppose deployment, i.e., when they cross their neutral positions, or when they pass through bifurcation points leading to change in displacement mode direction, which in turn may lead to an increasing load set. The load of equipment and sensors placed on the platforms and/or atmospheric loads are greater than the vectorial sum of boom force and spring preloads. This work aims to demonstrate the quasi static deployment pattern of a typical six bar mechan ism and its variations, viz. the presence of triple revolute joints, ternary links, multi angulated links and fixed joints. Secondly, this work also seeks to study the equilibrium stability characteristics of such mechanisms. Furthermore, simulations of id entical systems in a dynamics solver, MSC/ADAMS is used to validate and confirm the theoretical quasi static approach presented in this work to pr edict the deployment patterns. Mechanism: Kinematic Configuration A typical arrangement of a tape spring boo m within a pin jointed rigid body mechanism is illustrated in Figure 1 16 . To accommodate the stowage of such a mechanism in a 3U with pin joints located 0.1m apart . Link s III and IV are each 0.05m long and all other rigid links are 0.3m long. Links III and IV form a ternary link, in which joint 4 (located between links III and IV) is gid link between joints 3 and 5; Joint 4 serves as on e of the two anchor points for the tape spring boom. It may be noted that the link lengths can further be generalized and still be accommodated in a 3U small satellite frame, but for simplicity, certain symmetric features have been used.

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32 Figure 1 1 6. Typical arrangement of a six bar mechanism with the integrated boom . In Figure 1 17 , the tape spring boom is connected within the six bar mechanism in four different ways. The joints, bars and springs are numbered counter clockwise. The two end joints of the boom do not have any rotational springs connected to them. Also, the mid joints of the ternary links in configurations 2 and 4 do not have any rotational springs. It should be noted here that the procedure explained in this work is explained using sample scaled down dimensions of the structure and the boom. It is assumed throughout th e procedure that the boom is sufficiently rigid upon reaching its deployed bi stable configuration . In the end, it will be shown through graphical representations how the orientation and the stability of the structures at equilibrium change with changes in joint stiffness and pre loads as well as with the configuration selected for the integration of the boom with the deployment mechanism . 1 v i x Tape spring boom, link VII y i i i i ii i v v 2 3 4 5 6 I II V VI III IV

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33 A B C D Figure 1 17. Variations in boom arrangement within a six bar mechanism . A) Configuration 1, B) Configuration 2 , C) Configuration 3 and D) Configuration 4. As can be seen from Figure 1 1 6 , all links and joints are numbered in ascending order in the counter clockwise direction with the boom being numbered last in all cases. The boom properties are used only in Part 2 of the computation. The link angles are measured in the counter clockwise direction from the horizontal. The rotation within each joint and seen by the spring is equal to the rotation of the higher numbered bar minus the rotation of the lower numbered ba r at that joint. All joint variables are denoted by subscripts and all link variables are hangeably in this thesis . Parameters in bold notations are vectors. The material presented in this work is divided in the following way. In Chapter 2, a theoretical outline on the techniques to build the necessary structural matrices and the implementation of the Predictor Technique to generate the force displacement response of the

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34 system is pres ented. Further, the possibilities of kinematic bifurcations and a simple way to build the stiffness matrix of the same structures are briefly discussed. Chapter 3 consists of techniques to develop the potential energy function for a given structural system and using it to determine the stability characteristics of the system at equilibrium. In Chapter 4, the techniques presented in the previous chapters are demonstrated through a few examples. Chapter 5 contains software simulations (MSC/ADAMS) of a few str uctures based on the assumptions made in this work. These results are compared and contrasted with the theoretical ones from Chapter 4; validating and confirm ing of the theoretical quasi static approach to predict the deployment patterns . Finally, in Chap ter 6, other important results and aspects are discussed and directions for future work are also presented. Appendices A and B provide details of two key matrices used in the analysis. Appendix C contains the annotated source code used for generating the results presented in Chapter 4.

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35 CHAPTER 2 STRUCTURAL COMPUTATIONS OF MECHANISMS Equilibrium, Compatibility and Flexibility Matrices As described at the end of the previous chapter, here, due to the non linear nature of systems with springs, an incremental technique is adopted to compute the positions and orientations of the rigid bars and the rotational displacements within the springs at the joints. The computation is divided into two parts: Part 1 involves boom deployment and the boom is assum ed only as a source of external applied force until it reaches its stable state. It is assumed in all of our examples that the boom is able to deploy fully to its stable state. The uncertain case of partial deployment is only discussed in brief in Chapter 6. In Part 2, the boom is considered as a stable member and hence is modeled as a bar member. The computation is allowed to run until the system reaches a steady state, deployed configuration. In simulation, the steady state is considered reached when the incremental displacements become less than or equal to1e 6 m. Furthermore, at steady state and equilibrium, the work done by residual forces, if any, should be equal to zero or within the tolerance limits. For the modeling the deployment, Pellegrino and Ca lladine [24], Kwan and Pellegrino [25] suggested a method based on the force method to build the Equilibrium, the Compatibility and the Flexibility matrices for structures . The force method was established long ago and has been explained in great detail by McGuire and Gallagher [26] and Kanchi [27]. For the force method each link may be first modeled as a beam as shown in Figure 2 1. However, after assembling the overall equilibrium matrix for a system, as will be showed later in the chapter, the beam mode l can be reduced to a bar model which will not have bending stresses and rotational degrees of freedom. Using this model, the equilibrium, compatibility and flexibility equations for a rigid body pin jointed structure can be written as,

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36 ( 2 1) ( 2 2 ) And (2 3 ) where, matrices A, B and C are the equilibrium, compatibility and flexibility matrices respectively, and t is the vector of internal stresses, P is the vector of external loads, d is the vector of displacements at the joints, e is the vector of strains and is the vector of initial strains. is assumed to be zero in all forthcoming analyses. For systems in equilibrium, generally have (2 4 ) For ease of understanding, let us consider Configuration 4 in Figure 1 14 as the reference mechanism f or all computations from here on. The Equilibrium, Compatibility and Flexibility equations for a single beam member in local coordinate system (refer Figure 2 1) is given by, ( 2 5 ) (2 6 )

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37 (2 7 ) Figure 2 1. Single Beam Member . A) Free body Diagram of a beam member, B) Internal Forces and C) Displacement Coordinates of a beam member. Source : Kwan, A. S. K., elements for Deployable 254. As opposed to the conventional way of assuming an axial force, a bending moment and a shear force as the three independent stresses, here, the axial force and two bending moments are chosen as the three independent stresses. Proceeding further to obtain the overall structural

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38 matrix, Eq. 2 5 is pre multiplied by the below rotation matrix ( ) and Eq. 2 6 is post multiplied by the same rotation matrix on both sides of the equations to convert them into the global coordinate system. (2 8 ) (2 9 ) These transformed member matrices may then be assembled into, for example, the overall equilibrium matrix ( A ). To aid this process, especially when the complexity of connections in a deployable structure is high, a suitable Joint Connectivity Matrix (JCM) may be used to simplify tedium and computer computations of the same. The JCM can be built in the following way: The columns represent members of the structure and the rows represent revolute based on the numbering of end nodes of members in the local coordinate sy stems and the revolute joint that each of these end nodes correspond to. For example, local coordinate node 1 of Link 1 is numbered as Joint 1 (this is because the other half of the binary revolute joint is the node on the ground) in the global coordinate system. Node 2 in the local coordinate forms one part of the binary revolute joint (Joint 2) in the global coordinate system while the other part is formed by node 1 of Link 2. All unary revolute joints are entered in separate rows. Howeve r, one may observ e in row 6 of Eq. 2 10 , there are two entries in the same unary joint. This is because the ternary link (Links 3 and 4) is modeled as two separate members with a common mid node. Therefore, this is a representation of the fixed joint between the two member s of the ternary link. All other entries are marked as zeros. The JCM assists in splitting up the transformed member

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39 matrices and placing them suitably in the overall equilibrium matrix corresponding to the appropriate end node numbers. ( 2 10 ) For Configuration 4, the overall full sized equilibrium equation is given in Appendix A. One of the advantages of using the force method to analyze structures is that without any estimate the number of degrees of freedom (or degree of kinematic indeterminacy) and states of self stress (or degree of static indeterminacy) (Pellegrino and Calladine [24]). Therefore, the degrees of freedom m and the states of self stress h of a structural system can be calculated as, ( 2 11 ) (2 12 ) Where, r is the rank of the overall equilibrium matrix and and are the number of rows and columns of the overall equilibrium matrix. The number of degrees of freedom thus obtained is later ascertained and used as a validation instrument when independent m odes matrices M (in terms of displacements) and R (in terms of rotations of joints) are computed. Caution needs to be administered while computing the equilibrium matrix involving mechanisms that have ternary links which is very well explained by Kwan and Pellegrino [25] and Lu, et al [28] in their work.

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40 It may be noted here that the only internal force which is a bending moment is present at the mid node of the ternary link. Another important deduction from these equations, especially the compatibility equation is the dependence of rotational degrees of freedom on the rigid body displacements. As mentioned in Chapter 1, all members are assumed as inextensional rigid bars and hence the right hand side of the compatibility equation is always equal to zero. This means, (2 13 ) (2 14 ) (2 15 ) The importance of this result becomes clear later in the chapter when the inextensional modes of the mechanism in terms of joint displacements are expressed in terms of joint rotations and subsequently again when pre load moments are expressed as forces. Part 1 of Computation: Boom as a Force Here, the displacement response of the system is computed until the tape spring boom reache s its deployed stable configuration. Since the boom is unstable during its deployment (in transition between its two stable modes), the boom does not add to the stiffness characteristics of acting in its pre determined direction of deployment, at the second anchor point of the boom on the structure. The overall assembled equilibrium matrix may be condensed by techniques given by [29] where rows and columns are eliminated in equal numbers base d on the importance of certain variables in a particular computation. The final form of the equilibrium matrix at each incremental step of Configuration 4 for Part 1 of the computation is given by,

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41 (2 16 ) where, (2 17 ) The internal axial forces vector is ( 2 18 ) Where, (2 19 ) And, make with the horizontal axis of the global coordinate system and are the link lengths. Rotational Spring Moments Forces Equivalence The external force vector mentioned above is calculated as follows. Since, all members in the system are assumed to be rigid bars connected by pin joints, external moments cannot be applied to the system and there is no provision for th em in the equations. Therefore, by using the Principle of Virtual Work, the force vector that is equivalent to the preloads in the rotational springs is first calculated as,

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42 Figure 2 2. Force Moment Equivalent System . (2 20 ) where, is the infinitesimal rotation vector of the springs, is the infinitesimal displacement vector of the nodes (joints), is the moment vector due to the preloads in the rotational springs and is the force vector that balances the moments due to preloads (See Figure 2 2) to keep the system at equilibrium. It may be recalled from Eq. 2 1 3 to 2 15 how rotations are dependent on Cartesian displacements. Therefore, using the relation (2 21 ) in Eq. 2 20 yields (2 22 ) Since the displac ement vector is non zero, Eq. 2 22 is satisfied if (2 23 )

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43 Therefore, the equivalent system of forces can be calculated. However, it must be noted here that the equivalent system of forces that generates the same effect as the preloads is the negative of thus calculated. The Jacobian Matrix can be calculated using the following general express ion for the rotation of a bar, (2 24 ) The rotation of a joint (or the spring) is equal to the rotation of the higher numbered bar minus the rotation of the lower numbered bar at the joint under consideration. The general matrix expression in terms of variables for the Transformation Jacobian is given in Appendix B. Now, let the force applied by the boom be denoted by , then the total force applied on the system would be (See Figure 2 2), (2 25 ) Before proceeding any further, it is important to point out that the since this is a nonlinear constrained system, the forces should be applied in small increments to maintain the kinematic constraints of the system, i.e., only a frac tion of total force P computed at each step should be applied to the next updated configuration. One assumption that is made here is that the system moves slowly and hence can be treated as a quasi fraction of the total force P , we have (2 26 ) and Trial Method. However, in future work, an adaptive technique may be easily developed that can suitably modify the value of time with a view of red ucing the computation time. Another reason why the adaptive technique is important is when the applied force by the boom and/or the initial preloads are

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44 t he final displacement values. This means there will exist residual loads at a particular finite time of computation. However, in the current work, necessary precautions have been taken to ensure that the boom forces and the preloads are relatively high to start off and that displacement convergence issues do not arise. Effect of Rotational Springs The independent inextensional displacement modes of a mechanism can be obtained from the Singular Value Decomposition (SVD) of the equilibrium matrix A . ( 2 27 ) The last m columns of matrix U represent the independent inextensional displacement modes of the mechanism in Cartesian coordinates and which, as a result, constitute the columns of matrix M as well. (2 28 ) Matrix R is the inextensional modes matrix corresponding to M but in terms of rotations. (2 29 ) The most general kinematically admissible state of small displacement of the assembly is found by taking an arbitrary linear combination of the inextensio nal displacement modes of the mechanism. Therefore, by introducing a vector of independent Lagrangian variables , one may relate the displacements and rotation of joints to independent modes matrices M and R as, (2 30 ) And (2 31 )

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45 Proceeding further, and using the Principle of Virtual Work again as described in Pellegrino and Calladine [30], for every incremental step one can write (2 32 ) (2 33 ) Where (2 34 ) Combining Eq. 2 30 , 2 31, 2 32, 2 33 , and 2 34 , yields (2 35 ) One can now solve for and subsequently solve for and to obtain the incremental displacements and rotations of the joints. Force Response Predictor Technique Moving f orward, the equilibrium matrix is now updated and prepared for the next computation step. The above procedure is repeated until the distance between the two anchor points of the boom reach the pre determined deployed length of the boom. The updated configu ration is written as, (2 36 ) Part 2 of C omputation: Boom as Member Once the boom reaches its deployed length, the force applied by the boom ceases and the overall structure stops deploying further as well. The lengths of all members are fixed from this moment on. However, based on the initial preloads, the system might still experience moments due to the preloads at that configuration. Hence, the system will further try to move in certain directions, sway sideward until i t reaches an equilibrium state where either the preloads reduce to zero or where they balance vectorially and cancel out the effects of each other on the system. The

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46 system also has an updated JCM, which now, includes the boom as an additional member and h ence an additional column within it. The JCM for Configuration 4 is given as, ( 2 37 ) (2 38 ) (2 39 ) Where, (2 40 ) One may observe from rows 6 7 that node 2 of the boom now forms a binary revolute joint with the mid node of the ternary link. Another change to Part 2 as compared to Part 1 is the

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47 addition of boom parameters to the Equilibrium, Compatibil ity and Flexibility matrices. Thus, in general, the equilibrium matrix increases in size by one column. For Configuration 4, the final form of equilibrium matrix is expressed as given in Eq. 2 3 8 . ( make with the horizontal axis of the global coordinate system and are the link lengths. From here on, the computation procedure is the same as given in Part 1. Again, since the rive the system faster to an equilibrium state. Also, due to possible round off errors and residuals, the computation may be stopped after the desired lower incremental limit for displacements is reached, which is 1e 6 m in this case. The system can be ass umed to be in equilibrium when the norm of the displacement vector is less than a micro meter. Another way to test the equilibrium condition of a system is to check whether the work done by the residual forces, if any is zero or not. It is explained furthe r in Chapter 3. Kinematic Bifurcations Due to the vast nature of the theory on kinematic bifurcations alone, this thesis does not cover kinematic bifurcations in its body of work. However, for the sake of completion, a well established technique to tackle bifurcation points numerically during deployment is briefly discussed here. One of the definitions of a kinematic bifurcation point is that it is a point in the configuration space of a mechanism where the number of independent inextensional displacement m odes change suddenly, and in the case of multi DOF mechanisms, it is also a point where the mechanism might enter and exit the bifurcation in a different displacement mode. In the second part of Kumar and Pellegrino [31], the concept of second order compat ibility equations is used to find the actual number of kinematic paths that represent

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48 potential finite motions out of the infinite number of first order inextensional mechanisms that are possible. To obtain the second order compatibility equations, one has to first subtract the compatibility equation for a bar under zero strain conditions from the compatibility equation of the same bar under strain. The equation can then be represented as, (2 41 ) Where according to nota tions in [31], is the i th row of compatibility matrix C, is a matrix with 2 d and m columns, obtained from by selecting the rows that correspond to bar i , is a set of m arbitrary coefficients and W is a coefficient matrix to the displacement variables. Also, it is known that for a set of bar extensions e to be compatible, the following condition has to be satisfied. (2 42 ) Where , , is the third matrix obtained from the SVD of the equilibrium matrix and represents the last h columns containing the sets of kinematically incompatible extensions (also known as states of self stress). Therefore, the second order compatibility equations can be finally written as, , (2 43 ) Where (2 44 ) The kinematic paths out of a bifurcation point can be obtained by finding the intersections of the corresponding h quadric surfaces Eq. 2 43 . The solutions for these equations g any well known computer solvers of the

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49 modern day like MATLAB or MAPLE. Also, the incremental fraction can be assumed to play that the value of is adap tive in nature. Additionally, care must be taken to ensure the sign of taking the dot product of the position vectors of the mechanism in the previous and curren t configurations respectively. (2 45 ) Overall Stiffness Matrix In Chapter 5, although stiffness of structures is plotted graphically, the theory presented here to obtain the overall stiffness matrix from the equilibrium, compatibility and flexibility matrices is another cog in theory towards achieving completion of the same. Since the structures that are tackled in this thesis also contain joint stiffness in the form of rotational springs, one has to ensure that the contributions of th ese springs are also included in the overall stiffness matrix. For the pin jointed bar structure alone, the stiffness matrix in terms of the Cartesian displacements of the joints can be written as, (2 46 ) Where, [C] is the assembled fl exibility matr ix of the system based on Eq. 2 7 . Now the stiffness matrix joints alone in terms of Cartesian displacements of the joints can be written as, (2 47 ) Where, [K] is the diagonal matrix containing the stiffness coefficient s of the springs. Thus, the overall stiffness matrix for the system can now be written as, (2 48 )

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50 CHAPTER 3 EQUILIBRIUM STABILITY OF MECHANISMS Potential Energy Function The simplest way to test if a mechanism has reached equilibrium is to check for the component of the external loads vector P in the subspace of loads which cannot be carried. It should be equal to zero. (3 1 ) In energy terms, the work done by fo rces, which may be residual at this point, should be equal to zero. It may be noted here that any structural system can have multiple equilibrium configurations, however, that aspect has not been dealt with in the current body of work as the main aim of th is thesis is to outline a basic framework towards solving such systems. Once the system reaches e quilibrium and satisfies Eq. 3 1 , one can proceed further to discuss the instantaneous characteristics of the system at equilibrium, viz., Stable, U nstable and Neutral. Pellegrino (Tarna i) [10] also quoted by Lengyel [32] gave an expression for the potential energy function of a finite mechanism based on the Hellinger Reissner principle and Lagrange multipliers. The potential energy function is written as, ( 3 2 ) Where, is the applied load atic constraint equations. The k inematic constraint equations for Configuration 4 can be written as, ( 3 3 )

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51 ( 3 4 ) (3 5 ) (3 6 ) (3 7 ) (3 8 ) (3 9 ) (3 10 ) The last constraint equation is based o n the collinearity of nodes of a ternary link (Nagaraj et al [33] ). The third expression in the last constraint equation is zero for the case of Configuration 4. However, in case of angulated members and fixed joints, as shown in a later chapter through ex amples, the last expression has a non zero value. The Lagrange Multipliers are typically the internal forces of the system and can be calculated from the following equation (Pe llegrino [34]) only when Eq. 3 1 is satisfied. (3 11 ) Where, and are the i th columns of matrices U and V obtained from the SVD of the equilibrium matrix respectively, P is the external force vector at equilibrium, is the diagonal element of matrix S and is the vector of h free parameters associated with the self stress states of the system. In the current work, this is zero for all the co nfigurations mentioned in Figure 1 17 .

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52 Equilibrium Conditions Jacobian and Hessian Matrices The first variation of t he potential energy function when equated to zero gives the conditions for equilibrium. Lu, et al. [35] further suggested a method to determine the stability of a mechanism under a given set of constant external loads based on this potential energy functio n. The positive and negative definiteness of the second variation of the function (or the Hessian) give the stability and instability respectively at equilibrium. ( 3 12 ) and, (3 13 ) where, (3 14 ) (3 15 ) The J and H are the Jacobian and Hessian Matrices respectively. Since, this thesis deals with bars, the elastic deformations are very close to zero and so are the incremental inte rnal forces. Hence, from [35] , for stability of the mechanism, (3 1 6) Likewise, for unstable and neutral equilibriums respectively, (3 17 ) And (3 18 )

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53 Stability of multiple degrees of freedom mechanisms In the case of multiple DOF systems, the above expressions reporting stability are not scalars but m by m matrices. Therefore, these matrices have to be again tested for positive and negative definiteness by pre and post multip lying Eq. 3 16 to 3 18 by a column vector of length m consisting of positive numbers to determine its stability characteristics.

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54 CHAPTER 4 CASE STUDIES AND VARIATIONS OF THE SIX BAR MECHANISMS As mentioned in the Introduction Chapter , a tape s pring boom can be made to possess sufficient bending stiffness when the geometry and the arrangement of tape springs within the boom are appropriately chosen. Therefore, the moduli of elasticity and bending are assumed to be high enough to avoid buckling a nd other failure modes and are suitably adopted from Parera [23]. In this thesis , the main focus is on demonstrating the above dis cussed theory on simple rigid link and pin jointed systems and hence in this chapter, two case studies have been presented to show the implications of the boom and rotational springs on the mobility and stability of the system. But before that, the final forms of the equilibrium matrices for the other three configurations (1, 2 and 3) (See Figure 1 17) are presented next. Equilibrium Matrices The Equilibrium Matrices for configurations 1, 2 and 3 at the end of Part 2 of computations are calculated in the sa me way as given in Pellegrino and Calladine [24] and are as follows: Configuration 1: ( 4 1 )

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55 Configuration 2: (4 2 ) Co nfiguration 3: (4 3 ) Since this work is still mainly theoretical and under development, the following section presents a few examples in which the parameters of the boom, bars and springs are appropriately assumed so as to bring the theoretical simulation as cl ose to practical applications as possible. Case Studies Three out of six spring constants will be varied in Case Study 1 a and likewise in Case Study 1b , initial preloads of three springs will be varied and scatter plots between the three

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56 variables will be discussed. The specific values assumed considering the 3U small satellite frame in the computations are as follows (See Fig ure 1 1 6 ): Table 4 1 . Constant Parameters of the Structure Bar and Boom Dimensions and Material Properties Value Bar Len gths for I, II, V, VI 0.3 m Bar Lengths of III and IV 0.05 m Stable deployed length of the tape spring boom 0.3 m Cross sectional area of bars 1e 4 Cross sectional area of boom 10e 4 70 GPa 63 GPa Flexural Modulus of boom 56 GPa Area Moment of Inertia of bars 1.4583e 05 Area Moment of Inertia of boom 1.1222e 05 Case Study 1 a: Variable Spring Stiffness Constants Table 4 2 . Parameter Values for Variable Spring Stiffness Sweep Algorithm (superscripts represent global joint numbers. Refer Fig ure 1 1 6 ) Preloads, Stiffness Coefficients and Force Parameters Value varied between 40N/rad and 60N/rad varied between 40N/rad and 60N/rad

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57 Table 4 2. Continued Preloads, Stiffness Coefficients and Force Parameters Value varied between 40N/rad and 60N/rad Force applied by the boom on the system 100 N 0.005 (for Part 1) and 0.5 (for Part 2) Boom Length for Configurations 1 5 0.3 m Lengths of Boom 1 and Boom 2 for Configuration 6 0.3 m and 0.28 m Case Study 1 b: Variable Spring Preloads Table 4 3 . Parameter Values for Variable Spring Preload Sweep Algorithm (superscripts represent global joint numbers. Refer Figure 1 16) Preloads, Stiffness Coefficients and Force Parameters Value varied between 20 and 40 degrees varied between 11 0 and 13 0 degrees varied between 20 and 40 degrees Force applied by the boom on the system 100 N 0.005 (for Part 1) and 0.5 (for Part 2) Boom Length for Configurations 1 5 0.3 m Lengths of Boom 1 and Boom 2 for Configuration 6 0.3 m and 0.28 m

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58 Stability Plots Fig ure 4 1 shows the starting configuration for analyses. Although the mechanism starts from a fully st owed position as shown in Figure 4 2 in an actual model, the partially deployed starting point in the simulation is chosen to avoid dealing with: Kinematic bifurcations and, ill conditioning of matrices due to very small and very large angles made by the links with the horizontal at configu rations very close to the initially stowed one and at the bifurcat ion points (Figure 4 2) . This research work only focuses on outlining the procedure to tackle systems without any singular configurations. However, one may refer to the theory on kinematic bifurcations for deployable structu res by Kumar and Pell egrino [ 31 ] and suitably adopt the technique for force displacement response problems containing bifurcation issues . Figure 4 1 . Starting position for Configurations 1 6 . A B Figure 4 2 . Double S arrus Mechanism (six bar in 2D) on a 2U frame Singular Configurations/Bifurcation Points A) Fully Stowed and B) Partially Deployed , May 04, 2014. Courtesy of Nikhil Londhe .

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59 Figure 4 3 . Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 1 . (Units: X, Y, Z = N/rad) A B C D E Figure 4 4. 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 1 A) =60N/rad, B) =55N/rad, C) =50N/rad, D) =45N/rad, and E) =40N/rad . (Units: X, Y = N/rad)

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60 Figure 4 5 . Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 1 . (Units: X, Y, Z = degrees) A B C D E Figure 4 6 . 2D Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 1 A) =4 0 deg , B) =35 deg , C) =3 0 deg , D) = 25 deg , and E) =2 0 deg . (Units: X, Y = degrees)

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6 1 Figure 4 7 . Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 2 . (Units: X, Y, Z = N/rad) A B C D E Figure 4 8 . 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 2 A) =60N/rad, B) =55N/rad, C) =50N/rad, D) =45N/rad, and E) =40N/rad . (Units: X, Y = N/rad)

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62 Since the above demonstrated 6 bar mechanism has 2 DOFs, it is difficult to predict intuitively the stability characteristics. Therefore, graphical methods, albeit only up to three dimensions, are a very convenient way to study the effects of springs at the joints. Figures 4 3 to 4 8 , 4 10 to 4 1 9 and 4 21 to 4 28 are scatter plots between: Case Study 1 a varying spring constants and, Case Study 1b varying initial spring preloads according to the data mentioned in the previous paragraph. All equilibrium configurations that end up being stable are marked with blue are unstable are marked with a red cross sweeps through the given bounds of values, it calculates for each set of values the sign defi niteness of the stability matrix. It may be reminded here that a few variables have been assumed constant througho ut all computations due to the three dimensional limit of graphical representations . Configurations 1 and 4 largely achieve stable equilibrium configurations when the spring constants are varied, while Configurations 2 and 3 hardly do. Due to the symmetric nature of Configuration 4 about the boom, it is typically stable when both the spring constants and the initial preloads are symmetric about the boom. Configurations 1, 2 and 4 show more or less similar behavior when the initial spring preloads are varied between the given bounds, while Configuration 3 is largely unstable at equilibriums again. Another interesting use of these plots is towards the practical implementation of such a model. Since, actual components generally possess some amount of dimensioning or measuring errors, the systems might not demonstrate the equilibrium stability behavior at the exact same points shown in the plots. How ever, through suitable interpolation techniques and by looking at the system behavior at neighboring points, one may be able to predict the equilibrium stability characteristics of practical models with good accuracy .

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63 Consider for example, the set of value lie in the cubic space between 8 other points in one of the stability plots. The stability of the point in consideration maybe ascertained by taking the planar stability contours along all planes containi ng the six faces of the cube. The planar contour plot looks like as given in Figure 4 9B for the plane =40N/rad of the stability graph of Configuration 4 varying spring constants and constant pr eloads. Now consider the planes that coincide with the c ube ( See Figure 4 9A ) . If all planes that coincide with the cube show a completely stable contour, then any point within the cube, depending on the fineness of the sweep, will represent a stable configuration set of parameters. While it is still an ongoing effort to categorically state about stability behavior within the cube, one may reduce the size of the cubes, i.e., increase the data points in the sweep algorithm to obtain a more accurate and compact cube that represents stability. Therefore, the stabil ity cube is still a function of the coarseness/fineness of the sweep. On the other hand, even if one of the sides shows unstable contour, then the stability characteristics of the desired set of parameters depends on how far the point lies from the plane o f instability. Figure 4 9 . Stability Cube. A) A plane of the sweep plot that coincides with one of the planes of a cube , and B) Contour along the plane =40 N/rad . (0 to 1 Unstable, 0 to +1 Stable)

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64 Figure 4 10 . Stability Plot Constant Spring Constants, Varying Initial Preloads Configuration 2 . (Units: X, Y, Z = degrees) A B C D E Figure 4 11 . 2D Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 2 A) =40 deg, B) =35 deg, C) =30 deg, D) =25 deg, and E) =20 de g . (Units: X, Y = degrees)

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65 Figure 4 12 . Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 3 . (Units: X, Y, Z = N/rad) A B C D E Figure 4 13 . 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 3 A) =60N/rad, B) =55N/rad, C) =50N/rad, D) =45N/rad, and E) =40N/ra d . (Units: X, Y = N/rad)

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66 Figure 4 1 4 . Stability Plot Constant Spring Constants, Varying Init ial Preloads Configuration 3 . (Units: X, Y, Z = degrees) A B C D E Figure 4 15 . 2D Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 3 A) =40 deg, B) =35 deg, C) =30 deg, D) =25 deg, and E) =20 de g . (Units: X, Y = degrees)

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67 Figure 4 1 6 . Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 4 . (Units: X, Y, Z = N/rad) A B C D E Figure 4 17 . 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 4 A) =60N/rad, B) =55N/rad, C) =50N/rad, D) =45N/rad, and E) =40N/ra d . (Units: X, Y = N/rad)

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68 Figure 4 1 8 . Stability Plot Constant Spring Constants, Varying Init ial Preloads Configuration 4 . (Units: X, Y, Z = degrees) A B C D E Figure 4 19 . 2D Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 4 A) =40 deg, B) =35 deg, C) =30 deg, D) =25 deg, and E) =20 de g . (Units: X, Y = degrees)

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69 The 2D plots that follow the 3D plots for each case study and configuration are the planar plots along the five discrete values of in the 3D plots. Similar analyses can be carried out for the spring preloads. Two mo re variations of the Six bar Mechanism Given below are two more variations in the above mentioned six bar mechanisms. This rigor is demonstrated to give a more complete representation of the capability of the theory developed above for planar mechanisms. Configuration 5 consists of a fixed joint of the boom with Bar III and the other joint remains a revolute joint placed at any point across the base. This arrangement reduces the degrees of freedom of the system with the boom to just one. It virtually creat es a ternary link of a triangular shape. The fixed joint of the boom acts as the center of BMAB (breadth multi angulated beam [2 8 ] ). Configuration 6 consists of two booms deploying simultaneously and have slightly different stable lengths, just to make sur e the system is kinematically inextensional. This configuration too reduces the degrees of freedom of the system to one. In other words, this arrangement too behaves as a triangular ternary link. Following is the case study analyses as described above for these two configurations. Configuration 5 shows a high degree of stability in both case studies whereas Configuration 6 largely demonstrates unstable behavior. It is a very interesting observation because both possess a single degree of freedom and Config uration 5 has one member less in it. A B Figure 4 20 . Two more variations. A) Configuration 6: 2 booms and B) Configuration 5 Fixed joint at end of the boom .

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70 Figure 4 21 . Stability Plot Varying Spring Constants, Constant In itial Preloads Configur ation 5 . (Units: X, Y, Z = N/rad) A B C D E Figure 4 2 2 . 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 5 A) =60N/rad, B) =55N/rad, C) =50N/rad, D) =45N/rad, and E) =40N/ra d . (Units: X, Y = N/rad)

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71 Figure 4 23 . Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 5 . (Units: X, Y, Z = degrees) A B C D E Figure 4 24 . 2D Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 5 A) =40 deg, B) =35 deg, C) =30 deg, D) =25 deg, and E) =20 de g . (Units: X, Y = degrees)

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72 Figure 4 2 5 . Stability Plot Varying Spring Constants, Constant In itial Preloads Configuration 6 . (Units: X, Y, Z = N/ rad) A B C D E Figure 4 2 6 . 2D Stability Plot Varying Spring Constants, Constant Initial Preloads Configuration 6 A) =60N/rad, B) =55N/rad, C) =50N/rad, D) =45N/rad, and E) =40N/ra d . (Units: X, Y = N/rad)

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73 Figure 4 2 7 . Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 6 . (Units: X, Y, Z = degrees) A B C D E Figure 4 2 8 . 2D Stability Plot Constant Spring Constants, Varying In itial Preloads Configuration 6 A) =40 deg, B) =35 deg, C) =30 deg, D) =25 deg, and E) =20 de g . (Units: X, Y = degrees)

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74 These stability plots have shown, albeit the small range of values, how the stability characteristics of a particular system vary with changes in the spring stiffnes s coefficients and/or spring preloads. Case Study 2 : Deployment Patterns under constant parameters Case Study 2 presents the force displacement response of a system for a certain pre determined set of values of all variables. The plots include both phases of deployment the boom length increases up to its stable mode (0.3 m) and then the system tries to align and adjust itself based on the residual preloads in the system. Figures 4 2 9 to 4 34 demonstrate the displacement response sweeping from the initial position to the final position for all configurations. Table 4 4. Parameter Values for Symmetric Loading Deployment Pattern (superscripts represent global joint numbers. Refer Figure 1 16) Preloads, Stiffness Coefficients and Force Parameters Value Force applied by the boom on the system 100 N 0.005 (for Part 1) and 0.5 (for Part 2) Boom Length for Configurations 1 5 0.3 m Lengths of Boom 1 and Boom 2 for Configuration 6 0.3 m and 0.28 m

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75 The total sum of forces for both parts 1 and 2 for each configuration is calculated and tabled. Also, b ased on t he theory described in this thesis , for the given set of parameters in Table 4 4, configurations 1, 4 and 5 have stable equi libriums and configurations 2, 3 and 6 have unstable equilibriums. Figure 4 2 9 . Displacement Response and Stable final Equilibrium Configuration 1 . Table 4 5 . Total Load Values for Configuration 1 Load Parameter at Movable Joints Total Value for Part 1 (N) Total Value for Part 2 (N) F2x 39.5047484565855 91.6269938674998 F2y 5.08015273137026 6.24532679739183 F3x 68.7285634766640 45.6629809110884 F3y 148.249870613046 6.61170907384658 F4x 14.3854058731495 63.2224465481451 F4y 162.700825991927 201.073928625588 F5x 35.9948427405222 118.960462086233 F5y 4.83809643362620 6.38908957175098

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76 Figure 4 30 . Displacement Response and Unstable final Equilibrium Configuration 2 . Table 4 6 . Total Load Values for Configuration 2 Load Parameter at Movable Joints Total Value for Part 1 (N) Total Value for Part 2 (N) F2x 55.1232129971104 36.8828912504320 F2y 6.14328548753461 0.0908938566827326 F3x 30.4902245474598 18.6086504495068 F3y 290.198569636050 108.613069249872 F4x 37.0234531587119 0.119123678720540 F4y 257.199079500282 151.257413496695 F5x 22.5467564780751 18.4077371332232 F5y 297.043452168493 108.217476571705 F6x 50.9208015400668 37.0142577388073 F6y 5.94748107850004 0.0580978159116978

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77 Figure 4 3 1 . Displacement Response and Unstable final Equilibrium Configuration 3 . Table 4 7 . Total Load Values for Configuration 3 Load Parameter at Movable Joints Total Value for Part 1 (N) Total Value for Part 2 (N) F2x 52.8098795621861 151.533798720193 F2y 8.22934720097372 40.8855439922862 F3x 92.2429457882067 84.7295855592401 F3y 180.204300371292 295.837948489157 F4x 26.3934813821005 66.8047054736634 F4y 124.505333726613 11.9785281778302 F5x 52.8098952438046 151.534887873357 F5y 8.22934814247169 40.8855116012209

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78 Figure 4 3 2 . Displacement Response and Stable final Equilibrium Configuration 4 . Table 4 8 . Total Load Values for Configuration 4 Load Parameter at Movable Joints Total Value for Part 1 (N) Total Value for Part 2 (N) F2x 52.5678194014016 0.0370972703949745 F2y 1.41191426605847e 12 2.42861286636753e 17 F3x 26.2839097007006 0.0185486351974872 F3y 288.611759048534 0.106284246221071 F4x 5.33256339251098e 15 0 F4y 235.439776978666 0.148341576118900 F5x 26.2839097006999 0.0185486351974863 F5y 288.611759048534 0.106284246221071 F6x 52.5678194014014 0.0370972703949715 F6y 9.89890738822830e 13 5.17988429926675e 15

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79 Figure 4 3 3 . Displacement Response and S table fin al Equilibrium Configuration 5 . Table 4 9 . Total Load Values for Configuration 5 Load Parameter at Movable Joints Total Value for Part 1 (N) Total Value for Part 2 (N) F2x 31.7443907732231 84.5803707365725 F2y 3.03726282127015 43.5569669164494 F3x 10.4717648486235 11.1575085975829 F3y 264.612531594037 221.849076950998 F4x 77.7613909380982 168.886785487179 F4y 158.393619609289 462.376356130035 F5x 20.2470814186384 192.047709179042 F5y 267.747780377088 468.435693064661 F6x 23.4604011024709 215.876270356398 F6y 7.40915502385306 110.981580007092

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80 Figure 4 3 4 . Displacement Response and Uns table fin al Equilibrium Configuration 6 . Table 4 10 . Total Load Values for Configuration 6 Load Parameter at Movable Joints Total Value for Part 1 (N) Total Value for Part 2 (N) F2x 39.5047484565855 91.6269938674998 F2y 5.08015273137026 6.24532679739183 F3x 68.7285634766640 45.6629809110884 F3y 148.249870613046 6.61170907384658 F4x 14.3854058731495 63.2224465481451 F4y 162.700825991927 201.073928625588 F5x 35.9948427405222 118.960462086233 F5y 4.83809643362620 6.38908957175098 The plots in Case Study 2 displayed the deployment pattern for the given set of parameters (See Table 4 4). This helps in predicting motion of such systems before building the actual models. Also Tables 4 5 to 4 10 consist of the total values of forces at each node (joint),

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81 after adding the incremental forces at each step of the computation. This gives an idea of the amount of force re quired to displace the system to the desired configuration . This total force includes the forces applied by the boom as well as the torsion springs in the system. Therefore, if the total initial force applied by the boom and those b y the springs is less than the values given in Tables 4 5 to 4 10 for the given parameters in Table 4 4 , then one may come to the conclusion that th e applied force will not take the system s to the desired configuration. The set of values of parameters given in Table 4 4 is just an example set and can be given any values based on the type of system one wants to explore.

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82 CHAPTER 5 SIMULATION EXAMPLES I N MSC ADAMS Comparison of assumptions between models In this Chapter, the six bar configurations mentioned in this thesis are simulated in a commercial software, MSC ADAMS and the deployment patterns are compared with those obtained from the theoretical a nalyses in MATLAB as a part of the validation process. Apart from the conventional techniques of modeling rigid link mechanisms on MSC ADAMS, the following two features were incorporated to the models developed for this work: As mentioned in Chapter 1, du ring the first part of deployment, the boom is modeled as a force and hence, the ADAMS model is constructed as shown in Figure 5 1. At the two anchor points of the boom, two very small lengths of rigid links are connected via revolute joints (or fixed join t in the case of Configuration 5) to the base and joint 3 respectively. Further, these two small links are constrained to move along a translation joint with each other. Note that ALL anchor points shown in the figures that follow are the tiny dot like ele ments, one at the base and the other at either the joints or the mid node between Link III and IV. Figure 5 1 . Representation of the boom in an ADAMS model . For the second part of deployment, after the distance between the two anchor points reaches the theoretical stable length of deployment, further motion in that direction is restricted by the use of bi directional collision dampers (BISTOP function) in the model. This is a simpler and still a valid way to model the boom over modeling the non isotropic behavior of the composite material that it is made of. It is again mentioned here that the boom is assumed to have a sufficiently high flexural rigidity by way of careful arrangement of tape spring strands within the boom and that it would not come across buckling situations in any of the analyses presented in this work. All simulations were carried out dynamically but the joints were carefully damped to help achieve steady states quickly while at the same time care was taken to make sure it was not high e nough to affect the very motion of the system along the independent modes of the system. Therefore, this kind of damping meant that the system moved in a quasi static manner, the very approach used in the theoretical calculations described in this thesis.

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83 In the following pages, graphs for each configuration are listed in the following order. Graphs displaying the stabilization of boom lengths and the joint angles with time in MSC/ADAMS Motion objects captured from the software Deployment patterns from MATLAB for the given set of values of parameters Angles plotted from the quasi static analysis in MATLAB Table 5 1 . Parameter Values for Software Simulation and Theoretical Calculations of the Deployment Patterns All Configurations Preloads, Stiffness Coefficients and Force Parameters Value Force applied by the boom on the system 100 N 70 GPa Boom Length for Configurations 1 5 0.3 m Lengths of Boom 1 and Boom 2 for Configuration 6 0.3 m and 0.28 m Note that since the angles plotted in MSC/ADAMS are relative rotations, the angles graphs from MSC/ADAMS and MATLAB, the following expression may be used to confirm the closeness of results from the theoretical quasi static analysis and software simulation. Joint Relative Rotation Angle in MSC/ADAMS I nitial Joint Angle as given in Figure 1 20 = Joint Angle from the theoretical quasi static analysis in MATLAB (5 1) Table 5 1 gives the set of parameters assumed for this set of analyses.

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84 Configuration 1 Figure 5 2. Stabilizing Boom Length Configuration 1 . Figure 5 3. Stabilizing Joint Angles Configuration 1 . Object 5 1. ADAMS Simulation Reaching Equilibrium Configuration 1 (.avi file 702 KB) Figur es 5 2 and 5 3 demonstrate the stabilizing characteristics of the boom length and the joint angles of the dynamic simulation in MSC/ADAMS. Also, using Eq. 5 1 , it can be seen that the equilibrium angles predicted by the quasi static theoretical approach (F igure 5 4) and the

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85 software simulation (Figure 5 3) are within 2 3 degrees of each other. Figure 5 5 and Object 5 1 provide a confirmation of the same through visuals of the deployment pattern in MATLAB and MSC/ADAMS respectively. The time axis in Figure 5 4 is only for representation purposes as we know this analysis is quasi static. Figure 5 4 . Joint Angles from quasi static analysis MATLAB Configuration 1 . Figure 5 5. Deployment Pattern in MATLAB Configuration 1 .

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86 Configuration 2 Figure 5 6. Stabilizing Boom Length Configuration 2 . Figure 5 7. Stabilizing Joint Angles Configuration 2 . Object 5 2 . ADAMS Simulation Reaching Equilibrium Configuration 2 (.avi file 2 MB) Figures 5 6 and 5 7 demonstrate the stabilizing characteristics of the boom length and the joint angles of the dynamic simulation in MSC/ADAMS. Also, using Eq. 5 1 , it can be seen that the equilibrium angles predicted by the quasi static theoretical approach (Figure 5 8) and the

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87 software simulation (Figure 5 7) are within 2 3 degrees of each other. Figure 5 9 and Object 5 2 provide a confirmation of the same through visuals of the deployment pattern in MATLAB and MSC/ADAMS respectively. The time axis in Figure 5 8 is only for representation purposes as we know this analysis is quasi static. Figure 5 8. Joint Angles from quasi static analysis MATLAB Configurati on 2 . Figure 5 9. Deployment Pattern in MATLAB Configuration 2 .

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88 Configuration 3 Figure 5 10. Stabilizing Boom Length Configuration 3 . Figure 5 11. Stabilizing Joint Angles Configuration 3 . Object 5 3 . ADAMS Simulation Reaching Equilibrium Configuration 3 (.avi file 1 MB) Figures 5 10 and 5 11 demonstrate the stabilizing characteristics of the boom length and the joint angles of the dynamic simulation in MSC/ADAMS. Also, using Eq. 5 1 , it can be seen that the equilibrium angles predicted by the quasi static theoretical approach (Figure 5 12) and the software simulation (Figure 5 11) are within 2 3 degrees of ea ch other. Figure 5 13 and

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89 Object 5 3 provide a confirmation of the same through visuals of the deployment pattern in MATLAB and MSC/ADAMS respectively. The time axis in Figure 5 12 is only for representation purposes as we know this analysis is quasi stati c. Figure 5 12. Joint Angles from quasi static analysis MATLAB Configuration 3 . Figure 5 13. Deployment Pattern in MATLAB Configuration 3 .

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90 Configuration 4 Figure 5 14. Stabilizing Boom Length Configuration 4 . Figure 5 15. Stabilizing Joint Angles Configuration 4 . Object 5 4 . ADAMS Simulation Reaching Equilibrium Configuration 4 (.avi file 736 KB) Figures 5 14 and 5 15 demonstrate the stabilizing characteristics of the boom length and the joint angles of the dynamic simulation in MSC/ADAMS. Also, using Eq. 5 1 , it can be seen

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91 that the equilibrium angles predicted by the quasi static theoretical approach (Figure 5 16) and the software simulation (Fi gure 5 15) are within 2 3 degrees of each other. Figure 5 17 and Object 5 4 provide a confirmation of the same through visuals of the deployment pattern in MATLAB and MSC/ADAMS respectively. The time axis in Figure 5 16 is only for representation purposes as we know this analysis is quasi static. Figure 5 16. Joint Angles from quasi static analysis MATLAB Configuration 4 . Figure 5 17. Deployment Pattern in MATLAB Configuration 4 .

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92 Configuration 5 Figure 5 18. Stabilizing Boom Length Configuration 5 . Figure 5 19. Stabilizing Joint Angles Configuration 5 . Object 5 5 . ADAMS Simulation Reaching Equilibrium Configuration 5 (.avi file 924 KB) Figu res 5 18 and 5 19 demonstrate the stabilizing characteristics of the boom length and the joint angles of the dynamic simulation in MSC/ADAMS. Also, using Eq. 5 1 , it can be seen that the equilibrium angles predicted by the quasi static theoretical approach (Figure 5 20) and

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93 the software simulation (Figure 5 19) are within 2 3 degrees of each other. Figure 5 21 and Object 5 5 provide a confirmation of the same through visuals of the deployment pattern in MATLAB and MSC/ADAMS respectively. The time axis in Fi gure 5 20 is only for representation purposes as we know this analysis is quasi static. Figure 5 20. Joint Angles from quasi static analysis MATLAB Configuration 5 . Figure 5 21. Deployment Pattern in MATLAB Configuration 5 .

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94 Configuration 6 Figure 5 22. Stabilizing Boom Lengths Configuration 6 . Figure 5 23. Stabilizing Joint Angles Configuration 6 . Object 5 6 . ADAMS Simulation Reaching Equilibrium Configuration 6 (.avi file 837 KB) Figures 5 22 and 5 23 demonstrate the stabilizing characteristics of the boom lengths and the joint angles of the dynamic simulation in MSC/ADAMS. Also, using Eq. 5 1 , it can be seen

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95 that the equilibrium angles predicted by the quasi static theoretical approach (Figure 5 24) and the software simulation (Figure 5 23) are within 2 3 degrees of each other. Figure 5 25 and Object 5 6 provide a confirmation of the same thro ugh visuals of the deployment pattern in MATLAB and MSC/ADAMS respectively. The time axis in Figure 5 24 is only for representation purposes as we know this analysis is quasi static. Figure 5 24. Joint Angles from quasi static analysis MATLAB Config uration 6 . Figure 5 25. Deployment Pattern in MATLAB Configuration 6 .

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96 In Summary, a ll the simulation results presented above amply demonstrate that: The theoretical framework presented in this is in conformation with the predictions of a well establis hed dynamics solver. The damping introduced into spring models in the software and the round off errors in the theoretical computations maybe the possible contributors to the 1 2 degrees of difference between the computation and simulation. The systems sim ulated in ADAMS reach a steady state 2 seconds of deployment and without high amplitudes of fluctuations. This can therefore be assumed to be a quasi static simulation within reasonable limits. However, as part of the post analysis validations small pertur bations were applied on to the systems to test for equilibrium stability characteristics and although the software modeling technique mentioned in the chapter helped realize correct equilibrium configurations, it was found that the above mentioned modeling technique often produced errant results when tested for stability. M ost equilibrium parameter sets gave a stable end result. However, deeper investigations of the results and modeling technique revealed that due to the dynamic n ature of software simulatio ns, the boom length (Figure 5 27 ) oscillated about the stable length for some time instead of immediately settling to the actual stable length of the boom (as per the theoretical assumption in this work). This in turn allowed the boom to absorb the perturb ations and gave the spring loaded system leverage to adjust itself and reach sta ble configurations again (See Figure 5 2 6 and Object 5 7 ). Although it can be readily seen that by bring ing the lower and the upper trigger val ues in the BISTOP function close r to each other and by raising the stiffness coefficient of the collision damper one may reduce the amplitude of oscillations of the boom length in the simulation , this, as observed, hampered the computation capability of the software, often leading to ter mination of simulations. That is, very close upper and lower limits of the collision damper made the matrices and equations of the dynamics solver often ill conditioned . Note that the

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97 perturbations in the above analysis and other analyses that follow are applied on to the system in the form of 1N impulse forces using the STEP function on Joint 4 (See Figure 1 16) . Figure 5 26 . Stabilizing Joint Angles . (before and after perturbation) Figure 5 2 7 . Stabilizing Boom Length . (before and after perturbatio n) gives leverage to the system Object 5 7. MSC/ADAMS Simulation with perturbation using BISTOP function (.avi file 22 MB) To tackle the above described problem, an alternate method which required slightly mor effort was developed and two cases of which, one unstable (Table 5 2, Figure 5 28 and Object 5 8) and one stable (Table 5 3, Figure 5 29 and Object 5 9) according to the theoretical analysis for Configuration 4 are described below. The analysis was split into two parts manually and for

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98 the second part a fixed length member was inserted into the system to represent the boom while still retaining the same orientations of li nks and the residual preloads that the first part of the analysis had at the point where it was stopped. This was done to eliminate the fluctuating behavior of the BISTOP function. Table 5 2. Parameter Values for Software Simulation of an unstable set C onfiguration 4, with fixed length of boom for Part 2 of simulation Preloads, Stiffness Coefficients and Force Parameters Value Force applied by the boom on the system 100 N Boom Length 0.3 m

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99 Figure 5 28 . Configuration 4 Unstable remains unstable with fixed length boom model . Object 5 8. MSC/ADAMS simulation of an unstable set of Configuration 4 with fixed boom length and perturbation (.avi file 47 MB) Table 5 3. Parameter Values for Software Simulation of a stable set Configuration 4, with fixed length of boom for Part 2 of simulation Preloads, Stiffness Coefficients and Force Parameters Value Force applied by the boom on the system 100 N Boom Length 0.3 m

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100 Figure 5 29. Configuration 4 Stable remains stable with fixed length boom model . Object 5 9 . MSC/ADAMS simulation of a stable set of Configuration 4 with fixed boom length and perturbation (.avi file 3 MB) With this new modeling tweak, i t was seen that the set of parameters (Table 5 2) that was found to be unstable from the quasi static theoretical analysis was also found to be unstable from the dynamics simulation in MSC/ADA MS. Figure 5 28 shows the unstable joint angles and also only represents the simulation after the perturbation was introduced into the system at equilibrium. Object 5 8 gives a visual representation of the unstable behavior of the system. Similarly, anothe r set of parameters (Table 5 3) which was predicted to be stable by the quasi static theoretical analysis was again found to be stable from the dynamics simulation in MSC/ADAMS. In this case too, Figure 5 29 and Object 5 9 demonstrate the stable behavior o f the system only after the perturbation was introduced into the system and represent joint angles and visuals of the dynamic simulation respectively. Object 5 10 . MSC/ADAMS Static Analysis of the stable set Configuration 4 (.avi file 767 KB) It was also found that the static analysis in MSC/ADAMS of the unstable set (Table 5 2) failed to execute as the software could not find the new equilibrium point whereas the stable set (Table 5 3) did yield a new equilibrium point. Object 5 10 shows the static analysis of the stable set.

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101 CHAPTER 6 DISCUSSIONS AND CONCLUSION Discussions This thesis has demonstrated the usage of tape spring booms in 2D deployable t russes and mechanisms quasi statically. The force fraction is an important element in the calculations as it helps maintain the kinematic integrity of the system. If the applied force at each step were high, then based on the displacement modes of the mech anism, the system would have experienced axial strains and changes in link lengths. For example, a force fraction of 0.005 kept all changes in link lengths to 0.1 mm whereas; a force fraction of 0.05 saw link lengths changing by 2 3 mm. Therefore, very sma ll increments in forces keep the strains very close to zero. However , the force fraction might be designed to be adaptive in nature based on the external load vector at every step to reduce the computation time. Also, since this analysis is a force respons e problem, there is no need to use the corrector algorithms typically used in incremental displacement response problem s . (See Kumar and Pellegrino [ 31 ] ) This method can also be used to ascertain if the given force generated by the boom and the spring prel oads or individually is sufficient for the boom to deploy to its second stable state. (6 1 ) Where f can be , or based on the source of force in focus. This becomes more important when the system might start at a configuration with one or more springs having zero preload and the springs start resisting motion straightaway. Sample sets of total loads were presented in Chapter 4, Case Study 2. As was shown in Chapter 5, even the BISTOP function has its limitati ons in modeling the boom. But b y way of a simpler yet a more tedious method, it was shown that the theoretical

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102 results matched those of the approximated dynamics models. It has been further demonstrated that this method can be used to tackle aspects such a s ternary links, quaternary angulated links, triple revolute joints, fixed joints and double booms. Future Work As part of improvement pointers, Parallel computations, multi core computations can be used to reduce the computation time. Internally within the computation process, computation adaptive force fractions can help improve the computation times. Also, certai n minimization techniques can be used to find the optimal set of parameters or sometimes even to fine tune dimensions and parameters. 2D and 3D Plots between the varying spring stiffness coefficients and varying spring preloads in Chapter 4 can give insigh ts into certain stable regions between the given bounds of para meters as explained in the same chapter. This can also be further explored to quantify relative stability. Furthermore, the quasi static method developed and discussed in this thesis can even b e used , as part of future work, to carry out buckling analysis if there arise situations where the internal forces experienced by the boom is around the buckling limit load of the boom. As a mechanism gets more and more complex in member connections and nu mber, the number of parameters that need to be kept track of, increases significantly. Also, since most joints and links are numbered in order, it is important to keep track of these numbers when the member connections are complex and intertwined. Thirdly, although the initial signs of being able to use this method for 3D mechanisms are promising, it must be mentioned here that the very designing technique of 3D mechanisms is more complex in terms of revolute joint axes alignments etc., hence, the additiona l of at least two more revolute joints via the boom may cause difficulty in retaining the mobility of the system. Fourthly, as the number of variables in the system increases, it gets tougher to capture the effects of all variables on the system efficientl y

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103 through just graphical plots. This is also partially true for 3 variable system, as can be seen from the above analyses. It is also pointed out here that the potential energy equation in this thesis was written in terms of the Cartesian coordinates by us ing the force moment equivalence technique. But in terms of polar coordinates it is a far complex process because the relative rotations between bars have to be taken into consideration . This means that although writing the kinematic constraint equations a re still straightforward in terms of link angles , the first term of the potential energy equation which consists of work done by the forces or the strain energy r eleased by springs will have a combination of functions of This necessitates the nee d to keep track o f the rotation variables. However, on the bright side though, an attempt to solve the potential energy equations in terms of polar coordinates may provide further insights into the behavior of the system in terms of rotation angles of the system and also give a more concise set of insights due to the lesser number of polar variables x y . In the context of contribution to the research world, this thesis is an attempt to inject new ideas into the field of deployable structures. In particular, the integration of these booms offer the potential of enhanced stiffness behavior of the deployed mechanism while simultaneously providing compact stowed configurati ons and a means of actuation.

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104 APPENDIX A OVERALL FULL SIZED EQUILIBRIUM MATRIX 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0

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105 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

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106 APPENDIX B T RANFORMATION (ROTATION JACOBIAN) MATRIX The Rotation Jacobian Matrix is multiplied by a column vector,

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107 APPENDIX C SA MPLE CODE IN MATLAB FOR CONFIGURATION 4 Main File clear all clc;clf num=0; % Sweeping Loops for th01=20:5:40 for th02=110:5:130 for th03=20:5:40 n_Links=6; n_Joints_min=5; % moving joints n_found=2; % foundation joints n_onlytern=1; % ternary only links n_doubjoi=0; % triple revolute joints n_actjoi=n_Joints_min+n_found+n_doubjoi; n_Joints=n_actjoi+n_Joints_min n_onlytern; n_Springs=7; n_tern=1; L=[0.3 0.3 0.05 0.05 0.3 0.3]; % Link Lengths Lfix=0.1 ; % Fixed Link Length L_boom=0.06; % Initial boom length wi_L=2e 2;br_L=5e 2; ArBoom=1e 4; Ar=[1e 4 1e 4 1e 4 1e 4 1e 4 1e 4 ArBoom]; % Cross sectional area E=[70e9 70e9 70e9 70e9 70e9 70e9 63e9]; I=wi_L*br_L^3/1 2; %MI of beams d_iter=[0 0 0 0 0 0 0 0 0 0]'; r_iter=[0 0 0 0 0 0 0]; r_iter_te=[0 0 0 0 0 0 0]'; xy=[0 0 0 0 0 0 0 0 0 0]; th=([5.7392+r_iter(1) 180 5.7392+r_iter(2) 180+r_iter(3) 180+r_iter(4) ... 180+5.7392+r_iter(5) 5.7392+r_iter(6) 90]); %incremental alf_spr=[5.7392 180 2*5.7392 5.7392 0 5.7392 180 2*5.7392 5.7392]; num=num+1; if num==1 data_store=zeros(length(th01)*length(th02)*length(th03),13); end k1=40;k2=40;k3=40; k6=k1; k5=k2; k4=k3; theta0=[th01 th02 th03 0 30 120 30]; %preloads % Diagonal Matrix of Spring Stiffnesses K=[k1 0 0 0 0 0 0; ... 0 k2 0 0 0 0 0; ...

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108 0 0 k3 0 0 0 0; ... 0 0 0 0 0 0 0; ... 0 0 0 0 40 0 0; ... 0 0 0 0 0 40 0; ... 0 0 0 0 0 0 40]; Loadm ag_boom=100;Lo_app=0; %Boom force magnitude and sum of boom force at each step % Joint Connectivity Matrix (Rows = Movable Joints, Columns = Links) Joi_Con=[1 0 0 0 0 0;... % JCM 2 0 0 0 0 0;... 0 1 0 0 0 0;... 0 2 0 0 0 0;... 0 0 1 0 0 0;... 0 0 2 1 0 0;... 0 0 0 2 0 0;... 0 0 0 0 1 0;... 0 0 0 0 2 0;... 0 0 0 0 0 1;... 0 0 0 0 0 2]; Jo_Co=[1 0 0 0 0 0;... % Modified JCM to find the reactions matrix 2 1 0 0 0 0;... % binary joints collapsed to one row 0 2 1 0 0 0;... 0 0 0 2 1 0;... 0 0 0 0 2 1;... 0 0 0 0 0 2]; J_C=[2 1 0 0 0 0;... % Further Modified JCM to find the Rotation % Transformation Jacobian fixed joint s rows removed 0 2 1 0 0 0;... 0 0 2 1 0 0;... 0 0 0 2 1 0;... 0 0 0 0 2 1]; % calling function for part 1 of computation delta=0.005; [x_store,y_store,L_app,alf,alf_spr,q,L_f,L_boom,x,y,d_iter_a,Lo_tot,sigma_ten]=feb11_conf1_p 1(n_Link s,n_Joints,n_Joints_min,n_actjoi,n_onlytern,n_Springs,n_tern,L, ... Lfix,Ar,E,I,r_iter_te,L_boom,th,alf_spr,K,Joi_Con,Jo_Co,J_C,theta0,delta,Loadmag_boom,Lo_a pp); % Initializing and calling part 2 of computation Load_tot_p1=Lo_tot; th=alf; n_Links=n_Li nks+1; n_found=3; n_onlytern=0; % the ternary only joint is now a binary joint with the boom n_doubjoi=0; n_actjoi=n_Joints_min+n_found+n_doubjoi;

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109 n_Joints=n_actjoi+n_Joints_min n_onlytern; r_iter_te=[0 0 0 0 0 0 0]'; L=L_f; % Joint Connectivity Matrix ( Rows = Movable Joints, Columns = Links) Joi_Con=[1 0 0 0 0 0 0;... 2 0 0 0 0 0 0;... 0 1 0 0 0 0 0;... 0 2 0 0 0 0 0;... 0 0 1 0 0 0 0;... 0 0 2 1 0 0 0;... 0 0 0 0 0 0 2;... 0 0 0 2 0 0 0;... 0 0 0 0 1 0 0;... 0 0 0 0 2 0 0;... 0 0 0 0 0 1 0;... 0 0 0 0 0 2 0;... 0 0 0 0 0 0 1]; Jo_Co=[1 0 0 0 0 0 0;... 2 1 0 0 0 0 0;... 0 2 1 0 0 0 0;... 0 0 2 1 0 0 2;... 0 0 0 2 1 0 0;.. . 0 0 0 0 2 1 0;... 0 0 0 0 0 2 0;... 0 0 0 0 0 0 1]; J_C=[2 1 0 0 0 0 0;... 0 2 1 0 0 0 0;... 0 0 2 1 0 0 2;... 0 0 0 2 1 0 0;... 0 0 0 0 2 1 0]; delta=0.5; d_iter_a=[1e 5 1e 5 1e 5 1e 5 1e 5 1e 5 1e 5 1e 5 1e 5 1e 5]'; zero_load=[1 1]'; % work done at equilibrium [x_store_2,y_store_2,Load_0_store,L_app,alf,alf_spr,q,L_f,L_boom,x,y,d_iter_a,sigma_ten,U_ m,Lo_tot,C_mat,A,B,R_rows,zero_load]=feb11_conf1_p2( n_Links,n_Joints,n_Joints_min,n_act joi,n_onlytern,n_Springs,n_tern,L, ... Lfix,Ar,E,I,r_iter_te,th,alf_spr,K,Joi_Con,Jo_Co,J_C,theta0,delta,d_iter_a,zero_load); Load_tot_p2=Lo_tot; % Calculating the Jacobian and the Hessian Matrices of the Potential En ergy % of the System [Jac_mat,Jac_rank,Hess_mat]=Jac_Hess_feb11_conf1(n_Joints_min,L,Lfix); % Evaluating the Hessian Matrix at specific numerical values % L_app is the total force applied Lax1=L_app(1);Lay1=L_app(2);Lax2=L_a pp(3);Lay2=L_app(4);Lax3=L_app(5); Lay3=L_app(6);Lax4=L_app(7);Lay4=L_app(8);Lax5=L_app(9);Lay5=L_app(10);

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110 lam1=sigma_ten(1);lam2=sigma_ten(2);lam3=sigma_ten(3);lam4=sigma_ten(4); lam5=sigma_ten(5);lam6=sigma_ten(6);lam7=sigma_ten(7);lam8=sigma_ten(8); % H ess_mat=simplify(subs(second_part_diff)) Jac_mat_sub=subs(Jac_mat) Jac_rank_sub=rank(Jac_mat_sub) Hess_mat_sub=subs(Hess_mat) Hess_rank_sub=rank(Hess_mat_sub) % Checking the Stability of the Structure stability_mat=U_m'*Hess_mat_sub*U_m; pos_vect=ones(1,si ze(U_m,2)); stability=pos_vect*stability_mat*pos_vect' % % Overall Stiffness Matrix and Modal Analysis K_mat=A*(C_mat \ B); K_spr=R_rows*K*R_rows'; K_overall=K_mat+K_spr % data storage data_store(num,1)=k1; data_store(num,2)=k2; data_store(num,3)=k3; data_ store(num,4)=k4; data_store(num,5)=k5; data_store(num,6)=k6; data_store(num,8)=theta0(1); data_store(num,9)=theta0(2); data_store(num,10)=theta0(3); data_store(num,11)=theta0(5); data_store(num,12)=theta0(6); data_store(num,13)=theta0(7); data_store(num,14 )=Loadmag_boom; if stability>0 data_store(num,7)=1; else if stability==0 data_store(num,7)=0; else data_store(num,7)= 1; end end end end end Part 1 Computation Function File %% MATLAB Code to generate Equilibrium, Compatability and Flexibility Matrices %% and to find the displacement response with figures % Static Analysis Configuration 4 Part 1 Deployment

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111 function [x_store,y_store,L_app,alf,alf_spr,q,L_f,L_boom,x,y,d_iter_a,Lo_tot,sigma_ten]=feb11_conf1_p 1(n_Links,n_Joint s,n_Joints_min,n_actjoi,n_onlytern,n_Springs,n_tern,L, ... Lfix,Ar,E,I,r_iter_te,L_boom,th,alf_spr,K,Joi_Con,Jo_Co,J_C,theta0,delta,Loadmag_boom,Lo_a pp) i_store=0;Lo_tot=0; % total load over all steps gets added up while L_boom<=0.3000 % loop until the boom reaches 0.3000 i_store=i_store+1; alf=subs(th); % Spring Initial and Current Angles alf_spr=alf_spr+r_iter_te'; alf_spr=subs(alf_spr); x_prev=Lfix;y_prev=0; for i=1:n_Joints_min x(i)=x_prev+L(i)*cosd(alf(i)); y(i)=y_prev+L(i)*sind(alf(i)); x_prev=x(i); y_prev=y(i); end Equi_mat=cell(size(Joi_Con,1),size(Joi_Con,2)); C_mat=zeros(n_Links+n_tern,n_Links+n_tern); Zero_mat=zeros(3,3); for i=1:n_Links*n_Joints_min Equi_mat{i}=zeros(3,3); end reacns_mat=zeros(n_Joints*3,(n_actjoi n_only tern)*2); for i=1:n_Links Rot_mat=[cosd(alf(i)) sind(alf(i)) 0;... sind(alf(i)) cosd(alf(i)) 0;0 0 1]; % Element Equilibrium Matrix A_mat{i}=[ 1 0 0;0 1/L(i) 1/L(i);... 0 1 0;1 0 0;... 0 1/L(i) 1/L(i);0 0 1]; % Eleme nt Flexibility Matrix C_mat(i,i)=L(i)/(Ar(i)*E(i)); % Combining the Elemental Equilibrium Matrices elem_eq{i}=[Rot_mat Zero_mat;Zero_mat Rot_mat]*A_mat{i}; % Combining the Elemental Equilibrium Matrices for j=1:n_Joints if Joi_Con(j,i)==1 j1=j; Equi_mat{j1,i}=elem_eq{i}(1:3,1:3); end if Joi_Con(j,i)==2 j2=j;

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112 Equi_mat{j2,i}=elem_eq{i}(4:6,1:3); end end % forming the reactions matrix within the equilibrium matrix for jk=1:n_Joints if nnz(Joi_Con (jk,:))==1 && nonzeros(Joi_Con(jk,:))==2 reacns=[ 1 0;0 1;0 0]; else if nnz(Joi_Con(jk,:))==1 && nonzeros(Joi_Con(jk,:))==1 reacns=[1 0;0 1;0 0]; else if nnz(Joi_Con(jk,:))>1 reacns=[0 0;0 0;0 0]; end end end for kl=1:(n_actjoi n_onlytern) if Jo_Co(kl,i)~=0 && Jo_Co(kl,i)==Joi_Con(jk,i) reacns_mat(3*(jk 1)+1:3*(jk 1)+3,2*(kl 1)+1:2*(kl 1)+2)=reacns; end end end end for i=1:size(Equi_mat,1) for j=1:size(Equi_mat,2) for m=1:size(Equi_mat{i,j},1) for n=1:size(Equi_mat{i,j},2) A((i 1)*3+m,(j 1)*3+n)=Equi_mat{i,j}(m,n); end end end end A=[A reacns_mat]; % condensing the rows and columns A=[A(:,1);A(:,2);A(:,3);A(:,4);A(:,5);A(:,6);A(:,7);A(:,8);A(:,9);A(:,10);A(:,11);A(:,12);A(:,13); A(:,14);A(:,15);A(:,16);A(:,17);A(:,18)]; A=[A(4:5,:)+A(7:8,:);A(10:11,:)+A(13:14,:);A(16:17,:);A(19:20,:)+A(22:23,:);A(25: 26,:)+A(28: 29,:)]; A=[A(:,1) A(:,4) A(:,7) A(:,9)+A(:,11) A(:,10) A(:,13) A(:,16);]; % Checking the Rank of the Overall Equilibrium Matrix "A" Arank=rank(A) % Kinematic (or Compatibility) Matrix B=A'; % Developing the Matrix "R" whose each column contains a set of spring % rotation angles for the corresponding mechanism in matrix M. R_temp=cell(size(J_C,1),size(J_C,2)); for i=1:n_Links r{i}=[sind(alf(i))/L(i); cosd(alf(i))/L(i); sind(alf(i))/L(i);cosd(alf(i))/L(i)];

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113 for j1=1:size(J_C,1) if J _C(j1,i)==1 R_temp{j1,i}=r{i}(1:2,1); else if J_C(j1,i)==2 R_temp{j1,i}=r{i}(3:4,1); end end end end for i=1:size(R_temp,1) for j=1:size(R_temp,2) for m=1:size(R_temp{i,j},1) for n=1:size(R_temp{i,j},2) R_mat((i 1)*2+m,(j 1)*1+n)=R_temp{i,j}(m,n); end end end end R_0=zeros(size(R_mat,1),1);R_I=R_mat(:,1);R_II=R_mat(:,2);R_III=R_mat(:,3); R_IV=R_mat(:,4);R_V=R_mat(:,5); R_VI=R_mat(:,6); % R_rows is the Rotation Transformation Jacobian R_rows(:,1)=R_I R_0;R_rows(:,2)=R_II R_I;R_rows(:,3)=R_III R_II; R_rows(:,4)=R_IV R_III;R_rows(:,5)=R_V R_IV;R_rows(:,6)=R_VI R_V; R_rows(:,7)= R_VI+R_0; % SVD Decomposition of the Equilibri um Matrix [U,S,V] = svd(A); U_m=U(:,Arank+1:2*n_Joints_min); % displacement modes M=U_m; M_store{i_store}=M; R=R_rows'*(M); R_store{i_store}=R; for i=1:n_Springs r_ini(i,1)=(alf_spr(i) theta0(i))*pi/180; end % Calculating the initial "Load" to keep the mechanism in the stowed % position q= K*r_ini; q_store{i_store}=q; r_ini_store{i_store}=r_ini; R_rows_store{i_store}=R_rows; Load_0=R_rows*q; % force vector equivalent to moments Load_0_store{i_store}=Load_0; % L_app is th e force applied by the Boom Load_boom=[0 0 0 0 Loadmag_boom*cosd(th(7)) Loadmag_boom*sind(th(7)) 0 0 0 0]'; % total force applied per step multiplied by delta L_app=delta*(Load_boom+Load_0);

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114 psi=M'*L_app; S_mat=R'*K*R; phi=S_mat \ psi; % generalized displa cements % Displacement and Rotation Response d_iter=vpa(M*phi,5); r_iter_te=vpa(R*phi*180/pi,5); Lo_app=Lo_app+delta*Load_boom; % adding the force in this step to the total Lo_tot=Lo_tot+L_app; d_iter_a=d_iter; % Updated Coordinates of the Joints i2=0; fo r i=1:n_Joints_min x(i)=x(i)+d_iter_a(2*i2+1); y(i)=y(i)+d_iter_a(2*i2+2); x_store(i_store,i)=0; y_store(i_store,i)=0; x_store(i_store,i+1)=Lfix; y_store(i_store,i+1)=0; x_store(i_store,2+i)=x(i); y_store( i_store,2+i)=y(i); i2=i2+1; end % calculation of angles and link lengths and confirming the kinematic constraints are maintained [th,L_f]=angles_boomlength_conf1(x,y,Lfix) th_store{i_store}=th; r_iter=th alf; r_iter_store{i_store}=r_iter; L_boom=L_f(7) ; % plotting the deployment pattern if i_store==1 x_plot=[0 Lfix x(1) x(2) x(3) x(4) x(5) Lfix/2 Lfix x(1) x(2) x(3) x(4) x(5) 0 x(3)]; y_plot=[0 0 y(1) y(2) y(3) y(4) y(5) 0 0 y(1) y(2) y(3) y(4) y(5) 0 y(3)]; line(x_plot,y_plo t) axis([ 0.6 0.6 0.2 0.4]) else if rem(i_store,10)==0 x_plot=[0 Lfix x(1) x(2) x(3) x(4) x(5) Lfix/2 Lfix x(1) x(2) x(3) x(4) x(5) 0 x(3)]; y_plot=[0 0 y(1) y(2) y(3) y(4) y(5) 0 0 y(1) y(2) y(3) y(4) y(5) 0 y(3)]; lin e(x_plot,y_plot) axis([ 0.6 0.6 0.2 0.4]) end end

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115 end end Part 2 Computation Function File %% MATLAB Code to generate Equilibrium, Compatability and Flexibility Matrices %% and to find the displacement response with figures % Static Analysis Configuration 4 Part 2 Reaching Equilibrium function [x_store_2,y_store_2,Load_0_store,L_app,alf,alf_spr,q,L_f,L_boom,x,y,d_iter_a,sigma_ten,U_ m,Lo_tot,C_mat,A,B,R_rows,zero_load]=feb11_conf1_p2(n_Links,n_Joints,n_Joints_min,n_act joi,n_onlytern,n_Sprin gs,n_tern,L, ... Lfix,Ar,E,I,r_iter_te,th,alf_spr,K,Joi_Con,Jo_Co,J_C,theta0,delta,d_iter_a,zero_load) i_store=0;Lo_tot=0; while norm(zero_load,2)>=1e 6 % loop until work done at equilibrium goes below 1e 6 i_store=i_store+1; alf=subs(th); % Spring In itial and Current Angles alf_spr=alf_spr+r_iter_te'; alf_spr=subs(alf_spr); x_prev=Lfix;y_prev=0; for i=1:n_Joints_min x(i)=x_prev+L(i)*cosd(alf(i)); y(i)=y_prev+L(i)*sind(alf(i)); x_prev=x(i); y_prev=y(i); end Equi_mat=cell(size(Joi_Con,1) ,size(Joi_Con,2)); C_mat=zeros(n_Links+n_tern,n_Links+n_tern); Zero_mat=zeros(3,3); for i=1:n_Links*n_Joints_min Equi_mat{i}=zeros(3,3); end reacns_mat=zeros(n_Joints*3,(n_actjoi n_onlytern)*2); for i=1:n_Links Rot_mat=[cosd(alf(i)) sind(alf(i)) 0;... sind(alf(i)) cosd(alf(i)) 0;0 0 1]; % Element Equilibrium Matrix A_mat{i}=[ 1 0 0;0 1/L(i) 1/L(i);... 0 1 0;1 0 0;... 0 1/L(i) 1/L(i);0 0 1]; % Element Flexibility Matrix C_mat(i,i)=L(i)/(Ar(i)*E(i)); % Combinin g the Elemental Equilibrium Matrices elem_eq{i}=[Rot_mat Zero_mat;Zero_mat Rot_mat]*A_mat{i}; % Combining the Elemental Equilibrium Matrices for j=1:n_Joints

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116 if Joi_Con(j,i)==1 j1=j; Equi_mat{j1,i}=elem_eq{i}(1:3,1:3); end if Jo i_Con(j,i)==2 j2=j; Equi_mat{j2,i}=elem_eq{i}(4:6,1:3); end end for jk=1:n_Joints if nnz(Joi_Con(jk,:))==1 && nonzeros(Joi_Con(jk,:))==2 reacns=[ 1 0;0 1;0 0]; else if nnz(Joi_Con(jk,:))==1 && nonzeros(Joi_Con(jk,:))==1 reacns=[1 0;0 1;0 0]; else if nnz(Joi_Con(jk,:))>1 reacns=[0 0;0 0;0 0]; end end end for kl=1:n_actjoi if Jo_Co(kl,i)~=0 && Jo_Co(kl,i)==Joi_Con(jk,i) reacns_mat( 3*(jk 1)+1:3*(jk 1)+3,2*(kl 1)+1:2*(kl 1)+2)=reacns; end end end end for i=1:size(Equi_mat,1) for j=1:size(Equi_mat,2) for m=1:size(Equi_mat{i,j},1) for n=1:size(Equi_mat{i,j},2) A((i 1)*3+m,(j 1)*3+n)=Equi_mat{i,j}(m,n); end end end end A=[A reacns_mat]; % Condensing is slightly different in Part 2 because of the extra boom member A=[A(4:5,:)+A(7:8,:);A(10:11,:)+A(13:14,:);A(16:17,:)+A(19:20,:); A(22:23,:)+A(25:26,:);A(28: 29,:)+A(31:32,:)]; A=[A(:,1) A(:,4) A(:,7) A(:,9)+A(:,11) A(:,10) A(:,13) A(:,16) A(:,19)]; % Checking the Rank of the Overall Equilibrium Matrix "A" Arank=rank(A) % Kinematic (or Compatibility) Matrix B=A'; % Developing the Matrix "R" whose each column contains a set of spring % rotation angles for the corresponding mechanism in matrix M.

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117 R_temp=cell(size(J_C,1),size(J_C,2)); for i=1:n_Links r{i}=[sind(alf(i))/L(i); cosd(alf(i))/L(i); sind(alf(i))/L(i);cosd(alf(i))/L (i)]; for j1=1:size(J_C,1) if J_C(j1,i)==1 R_temp{j1,i}=r{i}(1:2,1); else if J_C(j1,i)==2 R_temp{j1,i}=r{i}(3:4,1); end end end end for i=1:size(R_temp,1) for j=1:size(R_temp,2) for m=1:size(R_temp{i,j},1) for n=1:size(R_temp{i,j},2) R_mat((i 1)*2+m,(j 1)*1+n)=R_temp{i,j}(m,n); end end end end R_0=zeros(size(R_mat,1),1);R_I=R_mat(:,1);R_II=R_mat(:,2);R_III=R_mat(:,3) ; R_IV=R_mat(:,4);R_V=R_mat(:,5);R_VI=R_mat(:,6); R_rows(:,1)=R_I R_0;R_rows(:,2)=R_II R_I;R_rows(:,3)=R_III R_II; R_rows(:,4)=R_IV R_III;R_rows(:,5)=R_V R_IV;R_rows(:,6)=R_VI R_V; R_rows(:,7)= R_VI+R_0; % SVD Decomposition of the Equilibrium Matrix [U,S,V ] = svd(A); U_m=U(:,Arank+1:2*n_Joints_min); M=U_m; M_store{i_store}=M; R=R_rows'*(M); R_store{i_store}=R; for i=1:n_Springs r_ini(i,1)=(alf_spr(i) theta0(i))*pi/180; end % Calculating the initial "Load" to keep the mechanism in the stowed % position q = K*r_ini; q_store{i_store}=q; r_ini_store{i_store}=r_ini; R_rows_store{i_store}=R_rows; Load_0=R_rows*q; Load_0_store{i_store}=Load_0; % L_app is the force applied by the Boom % No force applied by the boom in Part 2

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118 L_app=delta*Load_0; psi=M'*L_app; S_ma t=R'*K*R; phi=S_mat \ psi; % Displacement and Rotation Response d_iter=vpa(M*phi,5); r_iter_te=vpa(R*phi*180/pi,5); Lo_tot=Lo_tot+L_app; % Calculating internal forces at equilibrium sigma1=0; for cou=1:Arank sigma1=sigma1+((U(:,cou)'*(Load_0))/S(cou,cou ))*V(:,cou); end d_iter_a=d_iter; % Checking if the columns of U corresponding to mobility give zero when % multiplied by the Applied Load Vector. zero_load=U_m'*(L_app/delta); sigma_ten=sigma1; % Updated Coordinates of the Joints i2=0; for i=1:n_Joints_mi n x(i)=x(i)+d_iter_a(2*i2+1); y(i)=y(i)+d_iter_a(2*i2+2); x_store_2(i_store,1)=0; y_store_2(i_store,1)=0; x_store_2(i_store,2)=Lfix; y_store_2(i_store,2)=0; x_store_2(i_store,2+i)=x(i); y_store_2(i_store,2+i)=y(i); i2=i2 +1; end [th,L_f]=angles_boomlength_conf1(x,y,Lfix) th_store{i_store}=th; r_iter=th alf; r_iter_store{i_store}=r_iter; L_boom=L_f(7); if i_store==1 x_plot=[0 Lfix x(1) x(2) x(3) x(4) x(5) Lfix/2 Lfix x(1) x(2) x(3) x(4) x(5) 0 x(3)]; y_plot =[0 0 y(1) y(2) y(3) y(4) y(5) 0 0 y(1) y(2) y(3) y(4) y(5) 0 y(3)]; line(x_plot,y_plot) axis([ 0.6 0.6 0.2 0.4]) else if rem(i_store,20)==0 x_plot=[0 Lfix x(1) x(2) x(3) x(4) x(5) Lfix/2 Lfix x(1) x(2) x(3) x(4) x(5) 0 x(3)];

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119 y_plot=[0 0 y(1) y(2) y(3) y(4) y(5) 0 0 y(1) y(2) y(3) y(4) y(5) 0 y(3)]; line(x_plot,y_plot) axis([ 0.6 0.6 0.2 0.4]) end end end end Function to calculate the angles and link lengths after computations function[th,L_f]=angles_bo omlength_conf1(x,y,Lfix) %Link I hyp1=sqrt((y(1) 0)^2+(x(1) Lfix)^2); sin_th1=(y(1) 0)/hyp1; cos_th1=(x(1) Lfix)/hyp1; th(1)=atan2(sin_th1,cos_th1)*180/pi; L_f(1)=hyp1; %Link II hyp2=sqrt((y(2) y(1))^2+(x(2) x(1))^2); sin_th2=(y(2) y(1))/hyp2; cos_th2=(x(2 ) x(1))/hyp2; th(2)=atan2(sin_th2,cos_th2)*180/pi; L_f(2)=hyp2; %Link III hyp3=sqrt((y(3) y(2))^2+(x(3) x(2))^2); sin_th3=(y(3) y(2))/hyp3; cos_th3=(x(3) x(2))/hyp3; th(3)=atan2(sin_th3,cos_th3)*180/pi; L_f(3)=hyp3; %Link IV hyp4=sqrt( (y(4) y(3))^2+(x(4) x(3))^2); sin_th4=(y(4) y(3))/hyp4; cos_th4=(x(4) x(3))/hyp4; th(4)=atan2(sin_th4,cos_th4)*180/pi; L_f(4)=hyp4; %Link V hyp5=sqrt((y(5) y(4))^2+(x(5) x(4))^2); sin_th5=(y(5) y(4))/hyp5; cos_th5=(x(5) x(4))/hyp5; th(5)=360+atan2(sin_th5, cos_th5)*180/pi; L_f(5)=hyp5; %Link VI hyp6=sqrt((0 y(5))^2+(0 x(5))^2); sin_th6=(0 y(5))/hyp6; cos_th6=(0 x(5))/hyp6; th(6)=atan2(sin_th6,cos_th6)*180/pi; L_f(6)=hyp6;

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120 %Link VII Boom hyp7=sqrt((y(3) 0)^2+(x(3) Lfix/2)^2); sin_th7=(y(3) 0)/hyp7; cos_th7=(x(3) Lfix/2)/hyp7; th(7)=atan2(sin_th7,cos_th7)*180/pi; L_f(7)=hyp7; e nd Jacobian and Hessian Matrices Function File function[Jac_mat,Jac_rank,Hess_mat]=Jac_Hess_feb11_conf1(n_Joints_min,L,Lfix) % Forming the Kinematic Constraint Equations % In Ca rtesian Coordinates syms x1 x2 x3 x4 x5 y1 y2 y3 y4 y5 real syms x01 x02 x03 x04 x05 y01 y02 y03 y04 y05 real syms alf1 alf2 alf3 alf4 alf5 alf6 real syms Lax1 Lay1 Lax2 Lay2 Lax3 Lay3 Lax4 Lay4 Lax5 Lay5 real xy=[x1 y1 x2 y2 x3 y3 x4 y4 x5 y5]; xy0=[x01 y 01 x02 y02 x03 y03 x04 y04 x05 y05]; alf=[alf1 alf2 alf3 alf4 alf5 alf6]; L_app=[Lax1 Lay1 Lax2 Lay2 Lax3 Lay3 Lax4 Lay4 Lax5 Lay5]'; % Constraint Equations F(1)=0.5*((x1 Lfix)^2+(y1 0)^2 1)/L(1); F(2)=0.5*((x2 x1)^2+(y2 y1)^2 1)/L(2); F(3)=0.5*((x3 x2)^2+ (y3 y2)^2 1)/L(3); F(4)=0.5*((x4 x3)^2+(y4 y3)^2 1)/L(4); F(5)=0.5*((x5 x4)^2+(y5 y4)^2 1)/L(5); F(6)=0.5*((0 x5)^2+(0 y5)^2 1)/L(6); F(7)=0.5*((x3 Lfix/2)^2+(y3 0)^2 1)/L(7); F(8)=(((x3 x2)*(y4 y3) (x4 x3)*(y3 y2))/(L(3)*L(4))) sin(alf(3) alf(4)); % bendi ng constraint syms lam1 lam2 lam3 lam4 lam5 lam6 lam7 lam8 real sigma_ten=[lam1 lam2 lam3 lam4 lam5 lam6 lam7 lam8]'; % internal forces % Symbolic Potential Energy Equation of the System SE=0; % Strain Energy of the Torsion Springs for i4=1:2:2* n_Joints_min SE=SE+L_app(i4)*(xy(i4) xy0(i4))+L_app(i4+1)*(xy(i4+1) xy0(i4+1)); end % Lagrangian Multipliers or the (Internal) Axial Forces in the Links PE= S E+sigma_ten(1)*F(1)+sigma_ten(2)*F(2)+sigma_t en(3)*F(3)+sigma_ten(4)*F(4) +sigma_ten(5)*F( 5)+sigma_ten(6)*F(6)+sigma_ten(7)*F(7)+sigma_ten(8)*F(8); % Calculating the Jacobian Matrix of the System and the second differential % of the Potential Energy for j5=1:size(sigma_ten,1) for i5=1:size(xy,2) Jac_mat(j5,i5)=diff(F(j5),xy(i5)); fi rst_part_diff{j5,i5}=diff(PE,xy(i5)); for k5=1:size(xy,2) % second_part_diff{k5,i5}=diff(first_part_diff{j5,i5},theta(k5));

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121 Hess_mat1{j5,1}(k5,i5)=diff(Jac_mat(j5,i5),xy(k5)); pd_SE(k5,i5)=diff(SE,xy(k5)); end end end Jac_mat=simplify(Jac_mat); Jac_rank=rank(Jac_mat) Hess_mat2=pd_SE; for i30=1:size(sigma_ten,1) Hess_mat2=Hess_mat2+sigma_ten(i30)*Hess_mat1{i30}; end Hess_mat=Hess_mat2;

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122 LIST OF REFERENCES [1]. M erchan , C. H. H. , 1987 Deployable Structures hesis , M assachusetts I nstitute of T echnology, Cambridge, MA, USA . [2]. Xu , Y., Guan , F., Chen, J., Zheng , Y., 2012, Structural design and static analysis of a double ring deployable truss for Acta Astronautica , 81(2) , pp. 545 554 . [3]. Kassabian, P. E., Pellegrino , S. , 1999, Retractable Roof S tructures , Proc. Instn Civ. Engrs Structs & Bldgs , 134 , pp. 45 56 . [ 4]. Mikulas , M. M. , Murphey , T. , Jones , T. C. , Tension Aligned Deployable S tructures for Large 1 D and 2 D Array Applications 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL, USA, pp. AIAA 2008 2243 . [5]. Escrig F., 1985, Expandable Space Structures , Int. J . Space Structures , 1985, 1 ( 2 ) , pp. 79 91. [6]. G uest, S., 1994 Deployable Str Ph . D . Dissertation , University of Cambridge, England. [7]. Knight, B. F., 2000 Deployable Antenna Kinematics Us , Ph . D . Diss ertation, University of Florida, Gainesville FL, USA. [8]. Tibert , G., 2002 tr Dissertation, KTH, Sweden. [9]. Murphey, T., 2008 ), Keynote Address, AIAA Structures TC . [10]. Pellegrino, S., (Ed.), ( 2001 ), D eployable Structures, CISM Courses and Lectures No. 412 , Springer, Wien, New York , USA . [11]. Pellegrino , S. , Kukatha san, S. Small S atellite Deployable Technical Report, CUED/D STRUCT/TR190, Defence Evaluation Research Agency and the University of Cambridge , England. [12]. Barrett, R., 2007 , Deployable Reflectors for Small Satellites 21 st AIAA/USU Conference on Small Satellites , Logan, UT, USA, pp. 109. [13]. Rimrott, F. J., 1966 , Storable Tubular Extendible Members , Engineering Digest. etractable Appendages in Spacecraft Journal of Spacecraft and Rockets , 32(6), pp. 1006 1014.

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123 [15]. Fernandez, J. M ., Lappas, V. J., Daton Lovett, A. J., 2011 C oncept using b i Acta Astronautica, Vol. 69 (1 2) , pp. 78 85. [16]. Banik, J., 2008, Baseline TRAC boom Design SRS/NASA NanoSail D , Air Force Research Laboratory, Albuquerque, NM , USA. [17]. Roybal, F. A., Banik, J. A. , Murphey, T. W. , 2007 Deployable Boom 48th AIAA Structures, Structural Dynamics, and Materials Confer ence, Honolulu, Hawaii, pp. AIAA 2007 1838. [18]. Thomas, G. M., 2010, Prototype Development and Dynamic Characterizat ion of USA. [19]. Beavers, F . L., Munshi, N. A., Lake, M. S. , Maji, A., Qassi m, K., Carpenter, B. F., 2002 , Design and Testing o f an Elastic Memory Composite D eployment Hinge for 43 rd S tructures, Structural Dyn amics, and Materials Conference, Denver, Colorado, USA, pp. 1 5 . [20]. S ickinger , C. , Herbeck, L., Strohlein, T., Torrez Torres , J., 2004 Booms: Design, Manufacture, Verification 55 th International Astronautical Congress, Vancouver Canada. [21]. Jeon, S. K. , Murphey, T. W. , 2011 esign and A nalysis of a Meter class CubeSat Boom with a Motor less Deployment by B i stable Tape Springs, 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy namics and Materials Conference, Denver, Colorado , pp. 1731 1741. [22]. Mallikarachchi, C., 2011 , Thin Walled Composite Deployable Booms with Tape Spring Hinges Ph . D . Dissertation, University of Cambridge, England. [23]. Parera, P. M., 2013 , Deployment Simulations of a Composite Boom for Small Satellites Licentiate Thesis in Engineering Mechanics , Technical Reports from Royal Institute of Technology, Department of Mechanics, KTH University, Stockholm, Sweden. uctures, 22(4), pp. 409 428. elements for 254. [26]. McGuire, W., Gallagher, R. H., 197 9, Matrix Structural Analysis , John Wiley, Chichester, England. [27]. Kanchi, M., 1992, Matrix Methods of Structural Analysis , Wiley Eastern, New Delhi, India .

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124 [28]. Lu, J., Li, N., Luo, Y., 2011, Kinematic Analysis of Planar Deployable Structures with Angulated Be Advances in Structural Engineering, 14(6) , pp. 1005 1015. Internation al Journal for Numeric al Methods in Engineering, 35, pp. 1219 1236. [30]. Pellegrino, S., Calladine, C. R., 199 Structural Computation of an Assembly of Rigid Links, Frictionl J. Appl . Mech . , Vol. 58 (3) , pp. 749 753. [31] Computation of Kinemati Int. J. Solids Structures, Vol. 37 (46 47) , pp. 7003 7027. [32]. Lengyel, A., 2002 , Analogy between Equilibrium of Structures a nd Compatibility of Ph . D . Dis sertation, University of Oxford, England. [33]. Nagaraj, A Constra int Jacobian based approach for S tat Computers and Structures, 88 (1 2) , pp. 95 104. [34]. P ellegrino, S., 1993 , Structural Computations with the Singular Value Decompos ition of Int . J . Solids Structures , 30 ( 21 ) , pp. 3025 3035. [35]. Lu, J., Li, N., Luo, Y., 2007 , Mobility and Equili brium Stability Analysis of Pin jointed Mechani sms with E Journal of Zhejiang University SCIENCE A , 8( 7 ) , pp. 1091 1100.

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125 BIOGRAPHICAL SKETCH Mr. Amrith N . Hansoge pursued his undergraduate studies at Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat , India between 2006 and 2010, where he received the title of Mechanical Engineer. He worked as a research assistant for a year at the Indian Institute of Science , Bangalore, Karnataka, India between 2011 and 2012 . He was a part s research group in the Department of Mechanical Engineering where he w orked on a technique called as static balancing of m echanisms. He enrolled as a graduate student at the Universit y of Florida in the fall of 2012 to pursue a m aster in mechanic al e ngineering.