An efficient finite element formulation of dynamics for a flexible robot with different type of joints

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Mechanism and Machine Theory

journalhomepage:www.elsevier.com/locate/mechmachtheory

An efficient finite element formulation of dynamics for a flexible robot with different type of joints

Chu A My

^a,∗

, Duong X Bien

^a

, Chi Hieu Le

^b

, Michael Packianather

^c

aDepartment of Special Robotics and Mechatronics, Le Quy Don Technical University, 236 Hoang Quoc Viet, Hanoi, Vietnam

bFaculty of Engineering and Science, University of Greenwich, Central Avenue, Chatham Maritime, Kent ME4 4 TB, United Kingdom

cSchool of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, United Kingdom

a r t i c l e i n f o

Article history:

Received 12 September 2018 Revised 11 November 2018 Accepted 22 December 2018

Keywords:

Flexible robots Flexible robotic arms Kinematics Dynamics

Finite elements method

a b s t r a c t

Iftwoadjacentlinksofaflexiblerobotareconnectedviaarevolutejointorafixedpris- maticjoint,therelativemotionofthenextlinkwilldependonboththejointmotionand theelasticdisplacementofthedistalendofthepreviouslink.However,ifthetwoadjacent linksareconnectedviaaslidingprismaticjoint,therelativemotionofthenextlinkwill dependadditionallyontheelasticdeformationdistributedalongthepreviouslink.There- fore,formulationofthemotionequations foramulti-linkflexiblerobotconsistingofthe revolutejoints,thefixedprismaticjointsandtheslidingprismaticjointsischallenging.In thisstudy,thefiniteelementkinematicanddynamicformulationwassuccessfullydevel- opedandvalidatedfortheflexiblerobot,inwhichatransformationmatrixisproposedto describethekinematicsofboththejointmotionandthelinkdeformation.Additionally,a newrecursiveformulationofthedynamicequationsisintroduced.Ascomparedwiththe previousmethods,thetimecomplexityoftheformulationisreducedbyO(2η^),whereη^is thenumberoffiniteelementsonalllinks.Thenumericalexamplesandexperimentswere implementedtovalidatetheproposedkinematicanddynamicmodellingmethod.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

Industrialrobotscanbe categorizedastherigidorflexibleones. Thetraditionalrigidrobotshavebeenwidely usedin various industries.Today, there hasbeen an emerging needto develop the flexible robots tomeet well moreand more sophisticatedrequirements,especiallyinmanufacturingindustries.Theflexiblerobotshaveoutstandingadvantagesoverthe traditionalrigidones,withalower overallmass,smalleractuators,lower energyconsumptions,andagreater payload-to- manipulator-weightratio,takingtheadvantagesofthelatestadvancementsinrobotics,designandmanufacturing, aswell asthematerialengineering.Thecontrolalgorithmsofrigidrobotsareinsufficientwhentheycometoflexibleones;andthe mathematicalmodeling process fortheflexible robotsis muchmorecomplex, especiallywhen workingonthe nonlinear controlissues,andthedynamicmodelingandanalysisofmulti-linkflexiblerobots.Moreover, formulationsofthemotion equationsforamulti-linkflexiblerobotconsistingoftherevolutejoints,thefixedprismaticjointsandtheslidingprismatic jointsarealwayschallenging.

Fig.1showstherevolutejoint(Fig.1a),thefixedprismaticjoint(Fig.1b)andtheslidingprismaticjoint(Fig.1c)thatare usuallyusedinrobotarchitectures.Thefixedprismaticjointimpliesthatitisfixedtothepreviouslinki−1,thecurrentlink

∗ Corresponding author.

E-mail address: mychuanh@yahoo.com (C.A. My).

https://doi.org/10.1016/j.mechmachtheory.2018.12.026 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

Nomenclature

R revolutejoint

Pa slidingprismaticjoint P_b fixedprismaticjoint

n totalnumberoflinks

i indexofalink(i=1÷n) l_i lengthofalinki

n_i totalnumberofelementsofalinki j indexofelementofalinki(j=1÷n_i) l_ie lengthofelementsofalinki

γ

i anglebetweenalinki−1andalinki a_i rearpartofalinki

d_i variableofaprismaticjointiconnectingalinki−1andalinki

θ

i variableofarevolutejointiconnectingalinki−1andalinki O_i≡(O_iX_iY_iZ_i) localcoordinatesystemattachedtoalinki

O₀≡(O₀X₀Y₀Z₀) referencecoordinatesystemfixedtothebase(link0).

u_(i)(2j+1) flexuraldisplacementatthecommonjunctionofelementsjand j+1ofalinki u_(i)(2j+2) flexuralslopeatthecommonjunctionofelementsjandj+1ofalinki

u_{(i−1)(2k+1)} flexuraldisplacementatthedistalnodeofaslidingelementkofalinki−1currentlythroughthe slidingprismaticjointi(P_a)

u_{(i−1)(2k+2)} flexuralslopeatthe distalnode ofaslidingelement kofa linki−1currentlythrough thejointi (Pa)

u_(i−1)(2n

i−1+1) flexuraldisplacementatthedistalendofalinki−1 u_(i−1)(2n

i−1+2) flexuralslopeatthedistalendofalinki−1

u_(i−1)f flexuraldisplacementatthenominalarticulationpointbetweenalinki−1andalinki. u_(i−1)s flexuralslopeatthenominalarticulationpointbetweenalinki−1andalinki. x apointontheelementjofalinki

φ

^m(^x),m=1÷4 shapefunctions

S vectorofshapefunctions

S_ij elementalvectorofshapefunctions

w_ij(x) flexuraldisplacementatthepointxontheelementjofalinki r_ij(x) vectorfromO_itothepointxontheelementjofalinki r_0ij(x) vectorfromO₀ tothepointxontheelementjofalinki

(i−1)i transformationmatrixdescribingthemotionofaflexiblelinkirelativetoaflexiblelinki−1 H_(i−1)i transformationmatrixdescribingthemotionofarigidlinkirelativetoarigidlinki−1 m_i uniformdensity(masspermeter)ofalinki

E Young’smodulus

I_i momentofinertiaofalinki M globalmassmatrixofthesystem

M_ij elementalmassmatrixfortheelementjofalinki K globalstiffnessmatrixofthesystem

K_ij elementalstiffnessmatrixfortheelementjofalinki p_i variableofajointi

p_ij vectorofelasticdisplacementsandslopesoftheelementjofalinki

κ

i vectorofelasticdisplacementsandslopesofalinki q vectorofgeneralizedcoordinatesofthesystem F vectorofappliedtorques/forcesofthesystem

i slidesthroughthejoint,andthelengthofthecurrentlinkivarieswithrespecttotime.Incontrast,theslidingprismatic jointisfixedtothecurrentlinki,boththejointandthelinktranslaterelatively totheprevious linki−1,andthelength ofthepreviouslinki−1varies.

Generally,thekinematicsofaflexiblelinkidependsmuchonwhichkindofjointsconnectingthelinkiwiththeprevious linki−1.Forthecaseinwhichthetwolinksareconnectedbyarevoluteorafixedprismaticjoint(Fig.1a,b),therelative motionofthelinki dependsonthemotionofthejointiandtheelasticdeformationatthedistalendofthepreviouslink i−1.Nevertheless,inthecaseofslidingprismatic joint(Fig.1c),themotionofthelink idoesnot dependontheelastic deformationatthedistalendofthelinki−1,itdependsontheelasticdeformationoftheslidingelementonthelinki−1. Thiselementvariesalongthelengthofthelinki−1,withrespecttotime.Forthecasesinwhichtherevolutejointandthe slidingprismaticjoint(Fig.1a,c),itisusuallyassumedthat theelasticdisplacementsatthefirstnodeofthefirstelement

Fig. 1. Three types of joints for the multi-link flexible robots.

onthelink iequal tozero.However, forthecaseinwhichthe fixedprismatic joint(Fig.1b), theelement ofzeroelastic deformationis theslidingelement ofthelink ithrough thefixed prismaticjoint.Obviously,the elasticityeffectsoflinks associatedwiththeuseofthethreejointtypesshouldbetakenintoaccountswhenworkingonthekinematicanddynamic modelingforageneralflexiblerobotthatconsistsofallthreejointtypes.

Decades ago, the dynamic modeling and analysis of multi-link flexible robots have been well documented [47,71,9,12,65,32,53,79,22,6,19,75,37,85,44].There have also beenattempts to generalize the kinematic anddynamic mod- eling problemsforthe multi-link flexible robots,andmostof theseinvestigations focus on themulti-link flexible robots consistingofall revolutejoints.Therewerealsostudies consideringtheuseoftheprismaticjointsfortheflexible robots;

however,thesestudies mainly focused on the robotwithone or two linksonly. Wang andWei[81] andAl-Bedoorand Khulief[5]investigatedthedynamicsofaslidingflexiblelinkthroughafixedprismaticjoint.Juetal.[33],YuhandYoung [84] andPanetal.[67] developedthe modelingfora two-link flexible robotwitha fixed prismaticjoint.Korayemetal.

[47]addressedthe dynamicmodellingof a multi-linkflexible robot, ofwhich each link rotatesandreciprocates through afixed prismaticjoint.WangandMills [82] studieda parallelrobotthat hasthe flexiblelinkswiththe slidingprismatic joints.KhademandPirmohammadi[37]formulated thedynamicequationsfora multi-linkflexible robotconsistingofthe slidingprismaticjointsaswell.

Although the dynamicmodelling of the flexible robot that consists of the sliding prismatic joints (Fig. 1c) has been considered in the researches by Khadem and Pirmohammadi[37], Ju et al. [33], and Wang and Mills [82], the flexible deformationoftheslidinglink,alongwhichtheprismaticjointslides,hasbeenoverlooked.Inparticular,littleattentionhas beenpaidtothekinematicanddynamicmodellingofthemulti-linkflexiblerobotsthathavesimultaneouslytherevolute joints,thefixedprismaticjointsandtheslidingprismaticjoints.Noticethatcombinationsofdifferenttypesofjointsprovide theflexibilitiesin designandadditionalfunctionsforthemulti-link robots,tohelp meetingwell the additionaltechnical requirementsandexpandingthescopeofapplicationsforthemulti-linkrobots.

Basically,theflexiblerobotsarethecontinuousdynamicalsystemswhichareoftencharacterizedbyaninfinitenumber ofdegrees offreedom and are governed by the nonlinear coupleddifferential equations; andthe exact solution ofsuch systemsisnot practical. Inorderto formulatethe dynamicequations,thecontinuousdynamical systemsare usually dis-

Table 1

A summary of the related work and key references about the dynamic modeling and analysis of the flexible robots.

Topics Related work and key references

One link flexible robot [52,26,27,34,81,80,50,5,59,78,33,69]

Two link flexible robot [65,55,67,84,2,8,11,29,35,38,63,7]

Multi-link flexible robot [12,79,32,47,9,22,6,53,19,75,37,85,44,71,49,45,46]

Newton-Euler’s equations – based approach [1,14,65,32,81,84,13,9,16,38]

Lagrange’s equations - based approach [15,12,65,53,79,34,55,18,53,67,7,22,80,5,78,19,75,85,2,8,71,31,64,20,76,21,59,29]

Kane’s equations - based approach [6,63]

Gibbs-Appell’s equations - based approach [37,47,44–46]

Hamilton’s equations - based approach [69,33]

Finite element method - based formulation [23,68,32,65,78,4,58,35,36,5,67,82,79,28,30]

Assumed mode method – based formulation [31,64,20,12,22,53,19,75,85,71,80,76,55,2,8,21,59,29,37,47,33,74,51]

Implementation of flexible robots and experiments

[61,47,57,73,66,83,43]

cretized by using two main methods, including (i) the Assumed Modes Method(AMM) and (ii) FiniteElements Method (FEM).TheodoreandGhosal[77]andDwivedyandEberhard[24]showedthatthemaindrawbackofAMMisthedifficulty in findingthe modesforlinkswiththe non-regularcross sectionsandthe multi-linkrobots,andFEM normallyrequires fewer computations; therefore, FEM lendsitself ideally suited forthe modeling of multi-links flexible robots. Practically, FEMhasbeenutilizedbymanyauthorsforthedynamicformulationofflexiblerobots[23,68,32,65,78,4,58,35,36,5,67,82,79]. Inthisapproach,eachlinkofarobotisdivided intothefinitebeamelementsofthesamelength.The kineticenergyand thepotentialenergyofeachelementarecalculatedtoformulatetheelementalmassandstiffnessmatricescorrespondingly.

Unfortunately, theelementalmass andstiffnessmatrix areusually calculatedin termsofthe nodalcoordinates.Withre- specttotheorderofentriesinsome globalvectorsofthegeneralizedcoordinates,theindexofthelastfourrowsandthe lastfourcolumnsofeachelementalmatrixisdifferentfromthatoftheother elementalmatrices. Moreover,theelemental matricescorrespondingtoelementsondifferentlinkshavedifferentsizes.Therefore,constructionsoftheglobalmassand stiffnessmatricesofgoverningequationsusuallyrequirethecomplexcomputationalproceduresandtransformationsforas- semblingtheelementalmassandstiffnessmatricesrespectively.Furthermore,inthecasethattheflexiblerobotconsistsof theslidingandfixed prismaticjoints,aconstructionoftheglobalmassandstiffnessmatricesisevenmorecomplexsince theboundaryconditionsvarywithrespecttotime.

Theaboveraisedcriticalissuesandproblemsleadtothemotivationsofdevelopinganewkinematicanddynamicformu- lationforthemulti-linkflexiblerobots.Inthispaper,anewkinematicmodellingmethodforaplanarn-linkflexible robot that consistsofthe revolute,fixedprismatic andsliding prismaticjointsispresentedanddiscussed.Anewhomogeneous transformationmatrixisaddressedtodescribethekinematicsofboththejointmotionandthelinkdeformation.Sincethe proposedmatrixisgiveninageneralform,itisthereforeapplicabletomodellingofanyplanarflexiblerobots.Inaddition, anewrecursiveformulationofdynamicsfortheflexiblerobotsisalsopresented.Basedonthefiniteelement—Lagrangian approach,alltheelementalmassandstiffnessmatricesaretransformedandexpressedexplicitlyandgloballywithrespect totheentiregeneralizedcoordinates.Consequently,theglobalmassandstiffnessmatricesareformulatedbysummingover alltheelementalmassandstiffnessmatricesofthesamesizerespectively.Byusingthesymboliccomputingsyntaxesavail- able inthesymboliccomputingenvironments, theformulationcan be implementedin asimplified andefficientmanner.

Theproposedmethodinthisstudyismoreefficientanduseful,especiallywhencomparedtothecommonmethodswhich usethecomplexschemesforassemblingtheglobalmassandstiffnessmatrices.Inordertopresentvalidationsofthenewly developedandproposeddynamicmodellingmethodformulti-linkflexiblerobots,thenumericalexamplesandexperiments areimplemented.

The restof thepaper isorganizedas follows.Section 2presents a briefliterature review,especially the relatedwork asfoundationsforthestudy. Sections3and4presentthekinematics, dynamicsandnumericalexamplesofthe dynamic analysisofamulti-linkflexiblerobotrespectively.Section5presentstheexperimentsforvalidationoftheforwarddynamic analysisresultsbasedontheproposeddynamicmodellingmethod.Finally,thesummary andconclusionsare presentedin Section6.

2. Abriefliteraturereviewandrelatedwork

Inrecentdecades,agreatdeal ofwork hasbeendevotedtothedynamicmodeling andanalysisofflexiblerobots.The comprehensiveliterature reviewsoftheflexiblerobotdynamicsandcontrol werepresentedby TheodoreandGhosa[77], DwivedyandEberhard[25],BenosmanandLeVey[10],RahimiandNazemizadeh[70],Kiangetal.[40],Sayahkarajyetal.

[72],LochanandSubudhi[54],andAlandolietal.[3].Table1presentsthesummaryoftherelatedworkandkeyreferences aboutthedynamicmodelingandanalysisoftheflexiblerobots.

Thecontentsandanumberoftheresearchabouttheone-linkflexiblerobotdynamicsarehuge.TheLagrange’sequations wereusedbyKalkerandOlsder[34]andLiandSankar[51]toderivetheequationsofmotionofflexible robotanddriven mechanismswithflexible links.Thedynamicequationswerederived byWangandVidyasagar[80]foraclassofmultilink manipulatorswiththelastlink flexible.KwonandBook[50] reportedamethodto solvetheinversedynamicproblemin

time domain. Al-Bedoorand Khulief[5]and Tokhi etal. [78] emphasized on the finite element formulation ofdynamic equations,andJuetal.[33]andMarghituandDiaconescu[59]utilizedAMMforthedynamicmodelling.

Therehasbeenalotofeffortsworkingonthedynamicmodelingandanalysisoftwo-linkflexiblerobots.However,most ofthesestudies focused on the flexible robotswith two revolute joints.Pan etal.[67] investigated the modelingof the flexiblerobotwiththefixedprismaticjoints,inwhichtheaxialshorteningeffectofthelinkwastakenintoaccounts.Yuh andYoung[84]studiedabeamthathasarotationalandtranslationalmotion,inwhichthetime-varyingpartialdifferential equationandtheboundaryconditionswerederivedtodescribethelateraldeflectionofthebeam.

Recently,therehasbeenanumberofstudiesthatfocusonthedynamicmodelingandanalysisofthemulti-linkflexible robotsand the related robotic mechanisms.However, mostof these studies emphasised on the flexible robots with the useof the revolute joints. As discussed and mentioned in Section 1, there wasa few studies [48,37] that proposed the methods formodeling ofa multi-linkflexible robot that usesthe prismatic joints.However, the studydone by Korayem etal.[47]onlyconsideredthecaseinwhicheacharmlinkofthegivenlengthrotatesandreciprocatesthrougharigidjoint.

Thework done byKhademandPirmohammadi[37]focused onthe slidingprismaticjoint;however,the elasticityeffects that involve inthe time-varying value of a link length, associated withthe use ofall typesof the prismatic joint,have notbeeninvestigated.Basically,thegeneralizedmethodsandtechniquesforthekinematicanddynamicalmodellingofthe flexiblerobotthatconsistsoftherevolute,fixedprismaticandslidingprismaticjointshavenotbeenwellstudied.

Apartfromtheworksinvestigatingdifferentarchitecturesofflexiblerobot,therehasalsobeenawidespectrumofstud- iesfocusingonvariousanalyticalapproaches,incorporatedwithFEMorAMM,forthedynamicmodellingofthemulti-link flexiblerobotasshowninTable1.Aspreviouslymentioned,FEMhasbeenappliedfordecadestomodelandanalysecom- plexstructuralengineeringproblems.Using FEM,Jonker [32]addressedtheNewton Eulerdynamicmodelfortheflexible robotusing therotary joins, withthe use of separate rigidbody displacements andactual deformations to describe the currentstate ofafiniteelement.InFEM,thebehavioroftheelasticityofaflexible robotarm isobservedonarigidbody motioninwhichtheelasticdeformationisthensuperimposed.FEMwasalsousedforthedynamicmodellingapplications.

Duetal.[23] presentedthe generalnonlineardynamics modelwhich wasdeveloped forthree-dimensional(3D) flexible robots.Eachlink isdiscretized by the finiteelementswith its inertia lumped atthenodes ofeach element. Naganathan andSoni[65]investigatedthecouplingeffectsofkinematicsandflexibilityintheflexiblerobots.AugustynekandAdamiec- Wójcik [9]studiedthe mechanismwiththeflexible beam-likelinkswhichuse therevolute joints.Usoro etal.[79] used theLagrangianapproachforthemathematicalmodelingofthelightweightflexible robots.Panetal.[67]developedagen- eraldynamicmodelfordesignandcontroloftheflexible robotswhichusetheprismaticjoints.Tokhietal.[78]presented thetheoretical andexperimental investigations intothe dynamicmodelling andcharacterisation ofaflexible robotic sys- tem. Al-Bedoor and Khulief[5] formulated the dynamicmodel of a sliding link with a prismatic joint. Wang and Mills [82]presenteda finiteelement– Lagrangianformualtionofdynamicequationsforaflexible-link planarparallel platform.

ByusingFEMandKane’sequations,AmiroucheandXie[6]addressedamatrixformulationofthedynamicalequationsfor theflexible multibodysystems. Heidarietal.[30]investigated anonlinear finiteelement modelforthedynamicanalysis ofthree-dimensionalflexiblelinkmanipulators,inwhichboththegeometricelasticnonlinearityandtheforeshorteningef- fectsare considered.TheuseofFEMapproachforthedynamicmodellingofflexiblemultibodysystems wasdiscussedby GeradinandCardona[28].Lochanetal.[54]discussedadvantagesofFEMindynamicmodellingofaflexiblerobot,andit wasshowedthatthe mainadvantage ofFEMoverthe other methodsisthat theconnections inFEM aresupposed to be clampedfree,withatleasttwomodesshapeperlink.AnotheradvantageofFEMoveranalyticalsolutionmethodsisthatit iscapableofhandlingnonlinearconditionseasily.FEMcanalsohandleirregularitiesinthestructureandmixedboundary conditions.Meanwhile,AMMhaslongbeenappliedaswellforthedynamicmodellingandanalysisoftheflexiblerobots,in whichthelinkelasticityisusuallymodelledbyatruncatedfinitemodalseries,intermsofthespatialmodeeigenfunctions andtime-varyingmode amplitudes.In particular, ShafeiandShafei [74]have employed AMM approach forthe dynamic modelingoftheobliqueimpactofamulti-flexible-link roboticmanipulator.Itisnotedthatthe dynamicmodellingofthe flexiblerobotsalsoneedstobeconsideredwhenworkingonthecontrollawdesign.Matsunoetal.[62]analysedtheAM– Lagrangiandynamicmodeltodeveloptheclosed-loopfeedbackcontrollawsforasinglelinkflexiblerobot.BolandiandEs- maeilzadeh[11]presentedanexacttiptrajectorytrackingcontrolofaflexiblerobot,basedonthetippositionandthestrain gaugemeasurementsthroughthenonlinearcontroldesign.BasedonHamilton’seqautions,Zhangetal.[86]formulatedthe dynamicequationstocontrolvibrationsoftheone-linkflexiblerobot.

Forvalidationofthedynamicmodellingandthecontroldesign,therehasbeenanumberofresearchesconductingexper- imentsonactualflexiblerobots.Massoudetal.[61],Korayemetal.[47],CannonandSchmitz[17]andLuo[57]designedand conductedexperimentsforonelink flexiblerobot. Experimentsformulti-linkflexible robotswereimplemented byShafei andKorayem[73],Nagarajetal.[66],Yimetal.[83],andKorayemandDehkordi[43].Inparticular,KorayemandDehkordi [43]presentedexperiments fora mobilemanipulator,with two flexiblelinks andrevolute–prismatic jointsto demonstrate theperformanceofadynamicmodelformulatedbythemself.

3. Kinematicsofaplanarflexiblerobot

Inthissection,thekinematicmodelofamulti-linkflexiblerobotisformulatedandanalysed.Specifically,letusconsider thecaseofaplanarflexible robotconsistingofnlinksandnjoints.Itisassumedthattheelasticdeformationofalinkis small.Eachlinkistreatedasanassemblageofafinitenumberofbeamelements.

Fig. 2. A generalized schematic of an arbitrary pair of flexible links: a nominal & actual configuration.

Table 2

Joint variables and parameters of a link.

Joint Type χi ϑi u ₍i−1)s u ₍i−1)f a i Note

R l i−1 θi^∗ u (i−1)(2ni−1+2) u (i−1)(2ni−1+1) 0 ( ^∗): joint variable P a d _i^∗ γi u ₍i−1)(2k+2) u ₍i−1)(2k+1) 0 k ∈ {^{1 , 2 , 3 , ..., n} i−1} P b l i−1 γi u ₍i−1)(2ni−1+2) u ₍i−1)(2ni−1+1) d _i^∗−l i

Fig.2presentsageneralizedschematicofanarbitrarypairofflexiblelinksofaplanarflexible robot,withan arbitrary couple oflinks, namely link i−1 andlink i. A linki−1 connects witha link i by a joint i which can be the following threejointtypes:Revolutejoints (R), Slidingprismaticjoints(Pa) andFixedprismatic joints(P_b). Thelink iwithalength l_i is divided inton_i elementsof theequal length l_ie.Each element j ofthe link i (elementij) hastwo nodes. Eachnode hasa flexuraldisplacementanda flexuralslope.Forthe elementij,p_{i j}=[ u_(i)2j−1 u_(i)2j u_(i)2j+1 u_(i)2j+2 ]^T isthe vectoroftheflexuraldisplacementandflexuralslope.In Fig.2,u_(i−1)f andu_(i−1)sare theflexuraldisplacement andslope atthe nominalarticulationpoint betweentwo links. Ifthejoint iis arevolute joint orafixed prismatic joint,thepoint isat thedistalend oflinki−1,andu_(i−1)f=u_(i−1)(2n

i−1+1) andu_(i−1)s=u_(i−1)(2n

i−1+2). Ifthejointi isa slidingprismatic joint,thepointisatthedistalendoftheslidingelementkofthelinki−1throughthejoint,andu_(i−1)f=u_{(i−1)(2k+1)}and u_(i−1)s=u_{(i−1)(2k+2)}.

Let usdefine O_i≡(O_iX_iY_iZ_i) asthe local coordinate systemattached to the link i, where the originO_i isfixed to the proximalendofthelinkiandtheaxisX_ipointsinthedirectionofthelinki.Similarly,O_i−1≡(^Oi−1X_i−1Y_i−1Z_i−1)^isdefined forthelinki−1.O₀≡(O₀X₀Y₀Z₀)isthereferencecoordinatesystemfixedtothebase.

Lemma:Thegeneralhomogeneoustransformationmatrix₍i−1)ⁱdescribingthekinematicsofboththemotionofajoint iandthedeformationofalinki−1canbewritteninthefollowingform:

(ⁱ−1)ⁱ=

⎡

⎢ ⎣

cos

ϑ

^i+u(ⁱ−1)^s

−sin

ϑ

^i+u(ⁱ−1)^s

0

χ

^i+aicos

ϑ

^i+u(ⁱ−1)^s

sin

ϑ

i+u_(i−1)s

cos

ϑ

i+u_(i−1)s

0 u_(i−1)f+a_isin

ϑ

i+u_(i−1)s

0 0 1 0

0 0 0 1

⎤

⎥ ⎦

^{, (1)}

where

χ

i,ϑi,u_(i−1)s,u_(i−1)f,anda_iaresymbolicnotationsrepresentingthejointvariablesandparameters accordingtothe jointtypesasshowninTable2.

Proof: InordertoobtainthecoordinatesystemO_iX_iY_iZ_i,fromO_i−1X_i−1Y_i−1Z_i−1,thefollowingtranslationsandrotations areperformed.

(i) A puretranslation ofthe coordinatesystem O_i−1 along

χ

i inthe directionX_i−1. Thetranslating distance

χ

iwill be the jointvariable d_i,ifthe jointi isa sliding prismaticjoint.Otherwise,

χ

i isthe length l_i−1ofthe link i−1.The homogeneoustransformationmatrixcharacterizingthistransformationisdenotedasT(

χ

i).

(ii) AtranslationofO_i−1,attheprevious location,along u_(i−1)finthedirectionY_i−1.Ifthejointiisa slidingprismatic joint,the translatingdistance u_(i−1)f will be the flexuraldisplacement u_{(i−1)(2k+1)} calculated at the distal node of an element k ofthe link i−1. At a time instance,this element currentlyslides inside the joint i. Otherwise,it is equaltotheflexuraldisplacementatthetipofthelinki−1,u_(i−1)(2n

i−1+1).Thehomogeneoustransformationmatrix describingthistranslationisdenotedasT(^u(i−1)f)^.

Itisnotedthat, sincethesmallelastic deformationsareassumed, thetranslation(ii)isalsoassumedtobe carriedout beforethenextrotation(iii).

(i) ArotationofO_i−1,attheprevious location,around Z_i−1.Ifthejointiisa slidingprismaticjoint,therotationangle u_(i−1)swillbetheflexuralslopeu_{(i−1)(2k+2)}calculatedatthedistalnodeoftheelementkofthelinki−1.Atatime instance,thiselementcurrentlyslidesinsidethejointi.Otherwise,itisequaltotheflexuralslopeatthetipofthe linki−1,u_(i−1)(2n

i−1+2)ThehomogeneoustransformationmatrixrepresentingthisrotationisR(^u(i−1)s)^.

(ii) ArotationofO_i−1,atthepreviouslocation,aroundZ_i−1.Ifthejointiisarevolutejoint,therotationangleϑiwillbe thejointvariable

θ

i.Otherwise,itisthegeometricangle

γ

ibetweenthelinki−1andthelinki.The homogeneous transformationmatrixisR(ϑi).

(iii) AtranslationofO_i−1,atthepreviouslocation,backtotheproximalendofthelinki.Ifthejointiisafixedprismatic joint,a_i=d_i−l_i,whered_iisthejointvariable.Otherwise,itisassumedthata_i=0.Thehomogeneoustransformation matrixisT(a_i).

Consequently,thecumulativetransformationmatrixiscalculatedasfollows:

(ⁱ−1)ⁱ=T

( χ

i

)

^T

u_(i−1)f

R

u_(i−1)s

R

( ϑ

i

)

^T

(

^ai

)

(ⁱ−1)ⁱ=

⎡

⎢ ⎣

cos

ϑ

i+u_(i−1)s

−sin

ϑ

i+u_(i−1)s

0

χ

i+a_icos

ϑ

i+u_(i−1)s

sin

ϑ

i+u₍i−1)^s

cos

ϑ

i+u₍i−1)^s

0 u₍i−1)^f+aisin

ϑ

i+u₍i−1)^s

0 0 1 0

0 0 0 1

⎤

⎥ ⎦

^{. (2)}

Byusingthematrices(i−1)i,thekinematics oftheflexiblearm canbe systematicallydetermined.Givena valueofx, x∈[0,l_ie],ontheelementjofthelinki,inthelocalframeO_i,thepointr_ij(x)canbedeterminedasfollows:

r_{i j}

(

^x

)

⁼

⎡

⎢ ⎣

(

^j−1

)

^lie+x w_{i j}

(

^x

)

0 1

⎤

⎥ ⎦

^{, (3)}

wheretheflexuraldisplacementatthepointxisdeterminedasfollows:

w_{i j}

(

^x

)

⁼⁴_m=1

φ

^m

(

^x

)

^u(ⁱ)(²j−2+m). (4)

Byusingthetransformationmatrices(i−1)i(i=1÷n),inthereferenceframeO₀,thevectorr_0ij(x)canbecalculatedas follows:

r_{0i j}

(

^x

)

⁼

0ir_{i j}

(

^x

)

^{, (5)}

where

0i=

01

12...

(ⁱ−1)ⁱ. (6)

Todemonstratethekinematicmodellingasproposed,allnine(3²)feasibleconfigurationsofageneralizedtwo-linkflex- ible robotthat consistsof all threejoint typesare presentedin details inAppendix. Forall configurations, theproposed transformationmatrixisappliedtodescribethekinematicsoftherobots.Thepositionofthedistalendofthefirstlinkand thearmtiparecalculatedforboththeflexiblerobotandtherigidrobotofthesamegeometric parameters.Thefollowing are themain summaries fromthe above calculationsand analyses:(a) ifthe elasticdeformation of the linksis ignored, thekinematicbehavior ofaflexible robotandthecorresponding rigidrobotwill bethe same;(b)applyingtheproposed matrix,akinematicmodellingprocessforaflexiblerobotisquitefamiliarandcommoninthefieldofrobotics;and(c)the proposedmatrixisapplicabletoanyplanarflexiblerobots.

4. Dynamicsofaplanarflexiblerobot

Inthis section, the dynamicmodellingof the generalplanar flexible robot ispresented. The equationsof motion are writtenby applyingthe Lagrangianformulation.Particularly, two newrecursive formulationsare introduced toformulate theglobalmassandstiffnessmatricesrespectively.

Intheformulationoftheglobalmassmatrix,thevelocityofamasspointonanarbitraryelementofalinkiiscalculated viaJacobianmatrixandthegeneralizedvelocities.The elementalkineticenergyisthuscalculatedandexpressedinterms

oftheentiregeneralizedvelocities.Hence,thenumberofrowsandthenumberofcolumnsofeveryelementalmassmatrix equalstothenumberofthegeneralizedcoordinates.

Intheformulationoftheglobalstiffnessmatrix,avectorofextendedshapefunctionsisdefined. Thesizeofthisvector equalstothesizeofthevectorofgeneralizedcoordinates.Theflexuraldisplacementofapointonanarbitraryelement of a linki canbe calculatedvia thedefinedvector. Thus,the elementalpotentialenergyis expressedintermsofthe entire generalizedcoordinates,andthenumberofrowsandthenumberofcolumnsofeveryelementalstiffnessmatrixequalsto thenumberofthegeneralizedcoordinates.

Consequently, theglobalmass andstiffnessmatricescan beformulated by summing overall the elementalmass and stiffnessmatricesofthesamesize,respectively.

BasedontheFiniteElements– Lagrangianapproach,theequationofmotionsfortheflexiblerobotcanbeexpressedas follows:

M

(

^q

)

^¨q+C

q,q˙

˙

q+Kq+G

(

^q

)

^{=F. (7)}

The equation is written with respect to the global vector of the generalized coordinates, q= [tp₁

κ

1^T p₂

κ

2^T ... pn

κ

n^T]^T,where p_i isvariable ofajoint i,and

κ

i=[ u_(i)1 u_(i)2 ... u_(i)(2n

i+1) u_(i)(2n

i+2) ]^T is thevectoroftheelasticdisplacementsofallelementsonalinki.

Thevectorqincludesnjointvariablesp₁,p₂,...,pnandn

i=1(²ⁿi+2)^{arethetotalnodal}displacementsandslopesofall elementsonalllinks.Thus,thesizeofqis(ⁿ+n

i=1(²ⁿi+2))×1. F=[

τ

1 0 ... 0

2n₁+2

...

τ

^{n 0} ... 0

2nn+2

]^T isthevectoroftheappliedforces/torquesimposingonnjoints.

ThesizeoftheglobalmatricesM,CandKis(ⁿ+n

i=1(²ⁿi+2))×(ⁿ+n

i=1(²ⁿi+2))^. 4.1. Recursiveformulationoftheglobalmassmatrix

Consideranelementjofalinki,thekineticenergyoftheelementiscomputedasfollows:

Ti j= 1 2

lie 0

mir˙^T_{0i j}r˙0i jdx. (8)

Inthereferenceframe,thevelocityofthemasspointlocatedatr_0ij(x)canbecomputedasfollows:

˙

r0i j=

∂

^r0i j

∂

^{q q˙. (9)}

Thus,

T_{i j}= 1 2q˙^T

⎛

⎝

lie

0

m_iJ^T_{i j}J_{i j}dx

⎞

⎠

q˙ =1

2q˙^TM_{i j}q˙, (10)

whereJacobianmatrixispresentedasfollows:

J_{i j}=

∂

^r0i j

∂

^{q . (11)}

Consequently,theelementalmassmatrixforanelementjofalinkicanbewrittenasfollows:

Mi j=[msp]_{i j}=

lie

0

miJ^T_{i j}Ji jdx. (12)

ThesizeofmatrixM_ij is(ⁿ+n

i=1(²ⁿi+2))×(ⁿ+n

i=1(²ⁿi+2))^. Theindexesarepresentedasfollows:s,p=1÷(ⁿ+n

i=1(²ⁿi+2))^. Itisnotedthatforallsandpfrom(ⁱ+2j−1+i−1

u=1(²ⁿu+2))^to(ⁱ+2j+1+i−1

u=1(²ⁿu+2))^,

msp=m_il_ie 420

⎡

⎢ ⎣

156 22l_ie 54 −13l_ie 22l_ie 4l_ie² 13l_ie −3l_ie²

54 13lie 156 −22lie

−13lie −3l_ie² −22lie 4l_ie²

⎤

⎥ ⎦

⁽¹³⁾

Thetotalkineticenergyoftherobotsystemispresentedasfollows:

T=n i=1

ni j=1T_{i j}=1

2q˙^TMq˙^T. (14)

Finally,theglobalmassmatrixisobtainedasfollows:

M=n i=1

ni

j=1Mi j. (15)

Fig. 3. A flowchart of the algorithm for formulation of the global mass and stiffness matrices.

TherecursivealgorithmforthesymbolicformulationoftheglobalmassmatrixMis showninFig.3.Theinputsofthe formulationarethenumberoflinksn,thenumberoffiniteelementsonthelinksn₁,n₂,...nn,andtheotherparametersof therobotsystem.Theglobalmatricesareinitiatedaszeromatrices.Foreveryelementjofalinki,thepositionvectorr_0ij(x) isdeterminedwithEq(5),the JacobianJ_{i j} isimplemented withEq(11),andthe elementalmassmatrixM_ij iscalculated withEq(12).Forallelementsonalllinks,theabovestepsarerecursivelyloopedtoarchiveallM_ij.Finally,theglobalmatrix MisobtainedwithEq(15).

NoticethatthesymbolicformulationofM_ijandMcanbe effectivelyimplementedbyusingafewsyntaxesinthesym- boliccomputingenvironments.Forexample,inMaple,asoftwaredevelopedbyWaterlooMaple,twosyntaxesJacobianand intareneededfortheJacobianmatrixformulation,J_{i j}= ^∂_∂^r0_q^{i j},andtheintegrationM_{i j}=

l_ie

0

m_iJ^T_{i j}J_{i j}dx,respectively.

4.2. Recursiveformulationoftheglobalstiffnessmatrix

Theelasticpotentialenergyofelementjoflinkiiscalculatedas:

P_{i j}= 1 2

lie 0

EI_i

∂

^2wi j

(

^x

)

∂

^x2

2

dx (16)

Basedon thevectorofshapefunctionsS=[

φ

1(^x)

φ

2(^x)

φ

3(^x)

φ

4(^x) ^{]T,thefollowingnotationS}ij (the vector ofextendedshapefunctions)isdefinedasanelementalvectorofshapefunctions,whichcorresponds toanelementjona linki.ThesizeofS_ijequalstothesizeofq:

S_{i j}=

0 ... 0

(

^{i+2j−2+iu−1=1(2nu+2)}

)

S^T 0 ... 0

T

(

^{n+ni=1(2ni+2)}

)

^×1

(17)

Theflexuraldisplacementw_ij(x)expressedinEq(4)isnowrecalculatedwiththefollowingequation:

w_{i j}

(

^x

)

^=STi jq (18)

SubstitutingEq(18)intoEq(16)yieldsthefollowingresult:

P_{i j}= 1 2

_lie

0

EI_i

∂

^2STi j

∂

^{x2 q}

²

dx, (19)

Ui j=

∂

^2STi j

∂

^x2

(20) Thus,

P_{i j}= 1 2q^T

EI_{i lie}

0

U^T_{i j}U_{i j}dx

q=1

2q^TK_{i j}q (21)

Consequently,theelementalstiffnessmatrixforelementjoflinkicanbewrittenas:

K_{i j}=[ksp]_{i j}=EI_i lie

0

U^T_{i j}U_{i j}dx (22)

ThesizeofmatrixK_ijis(ⁿ+n

i=1(²ⁿi+2))×(ⁿ+n

i=1(²ⁿi+2))^{,andtheindexess},p=1÷(ⁿ+n

i=1(²ⁿi+2))^. Thetotalelasticpotentialenergyoftherobotiscalculatedasfollows:

P=n i=1

ni j=1P_{i j}=1

2q^TKq (23)

Finally,theglobalstiffnessmatrixcanbecalculatedasfollows:

K= n

i=1 ni

j=1

K_{i j} (24)

ThematricesKandK_ij havethesamesizeof(ⁿ+n

i=1(²ⁿi+2))×(ⁿ+n

i=1(²ⁿi+2))^. Itisnotedthat,forall

sorp∈/

i+2j−1+

i−1

u=1

(

^2nu+2

)

,

i+2j+1+

i−1

u=1

(

^2nu+2

)

, (25)

thecorrespondingentriesinmatrixK_ij areequaltozero,ksp=0.Thisisbecausew_ij(x)calculatedviaEq(18)dependsonly onfournodalelasticdisplacementsatthetwonodesofanelementjonalinki.

Forallsandp∈

i+2j−1+

i−1

u=1

(

^2nu+2

)

,

i+2j+1+

i−1

u=1

(

^2nu+2

)

, (26)

thecorrespondingentriesksparethefollowingconstants:

ksp=EIi

l_ie³

⎡

⎢ ⎣

12 6l_ie −12 6l_ie 6lie 4l_ie² −6lie 2l_ie²

−12 −6lie 12 −6lie

6l_ie 2l_ie² −6l_ie 4l_ie²

⎤

⎥ ⎦

⁽²⁷⁾

Similartothesymbolicformulationoftheglobalmassmatrix,therecursivealgorithmfortheformulationofK_ijandK isshowninFig.3,anditcanbeimplementedeffectivelyinthesymboliccomputingenvironments.