Higher-order derivatives of rigid body dynamics with application to the dynamic balance of spatial linkages

Higher-order derivatives of rigid body dynamics with application to the dynamic balance of

spatial linkages

de Jong, J. J.; Müller, A.; Herder, J. L.

DOI

10.1016/j.mechmachtheory.2020.104059

Publication date

2021

Document Version

Final published version

Published in

Mechanism and Machine Theory

Citation (APA)

de Jong, J. J., Müller, A., & Herder, J. L. (2021). Higher-order derivatives of rigid body dynamics with

application to the dynamic balance of spatial linkages. Mechanism and Machine Theory, 155, [104059].

https://doi.org/10.1016/j.mechmachtheory.2020.104059

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ContentslistsavailableatScienceDirect

Mechanism

and

Machine

Theory

journalhomepage:www.elsevier.com/locate/mechmachtheory

Higher-order

derivatives

of

rigid

body

dynamics

with

application

to

the

dynamic

balance

of

spatial

linkages

J.J.

de

Jong

a,∗

_,

_A.

_Müller

b

_,

_J.L.

_Herder

c

a Laboratory of Precision Engineering, University of Twente, PO Box 217, AE Enschede 7500, The Netherlands b Johannes Kepler University, Linz, Austria

c Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD DELFT, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 3 July 2020 Revised 8 August 2020 Accepted 9 August 2020 Available online 24 August 2020

Keywords: Dynamic balance Higher-order derivatives Momentum Screw theory Parameter-linear form Multipole representation Rigid body dynamics Parallel mechanisms

a

b

s

t

r

a

c

t

Dynamicbalanceeliminatesthefluctuatingreactionforcesandmomentsinducedby high-speedrobots thatwould otherwisecauseundesiredbase vibrations, noiseand accuracy loss.Manybalancingprocedures,suchastheaddition ofcounter-rotatinginertiawheels, increasethecomplexityandmotortorques.Thereexist,however,asmallsetofclosed-loop linkagesthatcanbebalancedbyaspecificdesignofthelinks’massdistribution, poten-tiallyleadingtosimplerandcost-effectivesolutions.Yet,theintricacyofthebalance con-ditionshindertheextensionofthissetoflinkages.Namely,theseconditionscontain com-plexclosed-formkinematic models toexpress theminminimalcoordinates. Thispaper presentsanalternativeapproachbysatisfyingallhigher-orderderivativesofthebalance conditions,thusavoidingfiniteclosed-formkinematicmodelswhileprovidingafull solu-tionforarbitrarylinkages.Theresultingdynamicbalanceconditionsarelinearinthe iner-tiaparameterssuchthatanullspaceoperation,eithernumericorsymbolic,yieldthefull designspace.Theconceptofinertiatransferprovidesagraphicalinterpretationtoretain intuition.Anoveldynamicallybalanced 3-RSR spatiallymovingmechanism ispresented togetherwithknownexamplestoillustratethemethod.

© 2020TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Fluctuatingreactionforcesandmomentsgeneratedbyfastmovingrobotscauseunwantedbasevibrationsandaccuracy lossattheend-effector[1].Theseshakingforcesandmomentsmaybereducedoreveneliminated byaspecificdesignof therobot’sstructureandinertiaparameters[2].Such mechanisms,that emitneithershakingforcesnorshakingmoments, aretermeddynamicallybalanced,orforce-balancedwhenonlytheshakingforcesarezero.Wedistinguishthreemajor ap-proachestodesignmechanismswiththisfeature.Firstly,onemayaddsupplementarycounter-mechanismstoagiven mecha-nism,suchascounter-rotatingwheels[3,4]oridlerloops[5–7].Secondly,varioussynthesismethodscombineandrecombine elementarydynamicallybalanced modules suchasfour-barlinkages [8,9]orpantograph-like structures [10–12]intoforce balancedordynamicallybalancedmechanismswithmoredegreesoffreedom (DOFs).Thirdly,suchan elementarymodule itselfisobtainedfortheanalysisofitsdynamicbalanceconditions.Byinspectingtheequationsthatdescribeitsmotionand

∗ _{Corresponding author.}

E-mail address: j.j.dejong@utwente.nl (J.J. de Jong). https://doi.org/10.1016/j.mechmachtheory.2020.104059

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

dynamics,arangeofinertiaparameters,i.e.masses,centresofmass(COMs),andmomentsofinertia(MOIs)maybefound thatbalancethisspecificmechanism[8,13,14].

Fortheviabilityofdynamicbalance,itisessentialtofindsimpleandlow-weightmechanismsthatstillfulfilthedesired kinematictask.Inthisview,theadditionofcounter-mechanismsseems undesirablesinceitwillincreasesthemass, com-plexityandtherequiredmotortorque.The synthesisapproaches, ontheother hand,haveprovento beversatileforforce balance,yet,incompleteforfulldynamicbalance[15].Inordertoexpandthescopeofdynamicbalanceandtoenablenew synthesis methods, thefocus ofthispaperlies intheimprovement ofthe thirdapproach, i.e.thegeneralization andthe automationoftheanalysisapproach.

Thenecessaryandsufficientdynamicbalanceconditionsareaconstantlinearandangularmomentum,astheir deriva-tives,theshakingforcesandmoments,thenwillbezero[16].Inpractice,whenthesystemisinitiallyatrest,azerolinear andangularmomentumsuffices. Asetofinertiaparameters that satisfytheseconditionsissaid tobe adynamically bal-anced solution, whereas the full description ofall solutions is termed the design spaceof dynamically balanced inertia parameters,ordesignspaceforsakeofbrevity.Itshouldbenotedthatopen-chainlinkagescannotbedynamicallybalanced withoutadditionalcounter-mechanismsasthey requirezeroornegativemomentsofinertia[4].Closed-chainlinkages,on theother hand,caninsome casesby dynamicallybalancedbya suitablechoiceofinertiaparameters.However, obtaining thecompletedesignspacefortheselinkagesisnottrivialasthedynamicbalanceconditionsaretobeexpressedinminimal coordinates [17].This involveskinematic loop-closuremodels,which maybe intricate,even forrelatively simplelinkages

[8],orunavailableinclosed-formformorecomplex mechanisms[18].The lackofsymbolictransparencyandclosed-form descriptionrenderstheprocessofsolvingthebalanceconditionsanarduoustask.

ThiscomplexityofthebalanceconditionsispartlyovercomebytheLinearIndependentVectorMethod[19]andderived methods[20–22]andtheInertiaFlowMethod[23].Thesemethodseliminateonlyasubsetofdependentcoordinates, lead-ingtosimplerbalancingconditionswhilestillyieldingthecompletedesignspaceingeneral. However,inspecialkinematic cases,suchasparallelograms,theseincompletekinematicmodelsleadtospuriousforceormomentbalanceconditionsand thereforetoan incompletedescriptionofthedesign space[17].Such specialkinematiccasesareofparticularinterest for dynamicbalanceastheypermitmoresolutionsthanthegeneralcase.Forinstance,RicardandGosselin[8]showedthatthe kite-typeandantiparallogram-typeoftheplanar4Rfour-barlinkagemaybefullydynamicallybalancedbyaspecificmass distribution.Thisincontrasttothegeneralfour-barlinkagethatdoesrequireadditionalcounter-rotatingmeasures.

Toprovetheworkof[8]formally,Mooreetal[24].factorizedthebalanceconditionsandloop-closureequationsbymeans oftoricgeometryandLaurentpolynomials.Subsequentlytheyshowedthroughasimilaralgebraicapproachthatthe spher-icalfour-barlinkagecannotbe dynamicallybalancedwithoutadditionalstructures[25].Currently,thesealgebraicmethods still require a tailored inspection per mechanism and are yet to be extended to multi-DOF mechanisms. An alternative methodtodealwiththekinematiccomplexityoftheloop-closureequationswasadoptedin[26].There,screwtheorywas applied to findinstantaneous dynamicbalance, yielding a singleposein whichmechanism accelerationswill not induce shakingforcesandmoments.Sinceoutsidethisposethebalance qualityisnot guaranteed,thismethodyields andsolves onlythenecessarybutnotsufficientconditionsfordynamicbalance.

Tosummarize;inliteratureseveralsystematicanalysismethods werepresentedthat solvethedynamicbalance condi-tionsforgivenlinkages.Yet,nomethodyieldsthecompletedynamicallybalanceddesignspaceofarbitrarylinkageswithout atailoredmanipulationoftheloop-closureequations.Such amethodisdesiredtoadvanceourunderstandingofdynamic balanceandtofindnew,simpleandlightweightbalancesolutions.

Inthispaper,thislong-standingproblemistackledbyextendingtheinstantaneousapproachof[26]overthecomplete workspacebyincludingandsolvingasufficientnumberofhigher-orderderivativesofthedynamicbalanceconditions.The higher-orderkinematicanddynamicmodelsarereadilyavailablethroughrecursiveapplicationoftheimplicitfunction theo-rem[27],thusavoidingtheuseofclosed-formkinematicmodels.Thismethodleadstothenecessaryandsufficientdynamic balanceconditions,andan automaticandcompletecharacterizationofalldynamicallybalanced designsofanygiven non-singularmechanismconsistingoflowerkinematicpairs.Tothat end,thispaperforthefirsttimepresentsanalgorithmto compute thederivatives ofthebodies’ massmatricesandmomentum equationsinopen- andclosed-loop linkagesup to arbitraryorder.

Thispaperstartswithasynopsisofthemethodtoguidethereaderthroughthefollowingsections(Section2).Thereafter thehigher-orderderivativesofkinematicsisoutlined(Section3),followedbyarecapitulationoftherigidbodydynamicsin thescrew theoryframework (Section4).Thisleadstoa recursivealgorithmthat yieldsthehigher-orderderivativesofthe linearandangularmomentumequations(thedynamicbalancingconditions)ofopenandclosed-chainlinkages(Section5). Theresultinghigher-ordermomentumequationsarethenrecastintotheparameter-linearform[28,29]toprovidedynamic balanceconditionsthat arelinearintheinertia parametersandsolvablebynullspacealgorithms (Section6).Generalnull spacealgorithms,i.e.singularvaluedecompositionorGaussianelimination,yieldacompletedescriptionofalldynamically balanced mass distributions. This description however, is strongly mixedin the inertia parameters, causing a loss of in-terpretation anddesignintuition. Therefore,an alternative,meaningfuldescription ofthedesign spaceofopen-chain and closed-chainlinkagesispresented(Section7).Thisdescriptionisderivedfromtheconceptofinertiatransferandthe

multi-polerepresentationoftheinertiaparameters,asusedintheparameteridentificationofrobots[30].Thisinterpretation,here

termed themultipole-rodrepresentation,is showntoaidthe feasibilitystudyofdynamicbalancedlinkages. Itshould be noted that open chains receivequite some attention inthiswork. Although they cannot be dynamically balanced them-selves,theyprovideinsightintothesolutionspaceofclosed-looplinkages.Morespecifically, itwillbeshownthatalarge

portionof thedesign spaceof a closed-loop linkageis build up fromthe open-chain equivalentsinto whichthe linkage maybe decomposed.Casestudiesofa6-DOF serialrobot,a4R planarfour-barlinkage,anda3-RSRmechanismillustrate thehigher-orderdynamicbalance method(Section8).Thisresultsinanovel3-RSRmechanismdesignthat isdynamically balancedforthe 2-DOFthat lie onthree planesofmirror symmetry.Referto TableA.1 foralist ofsymbolsusedinthis paper.

2. Synopsisofthehigher-orderdynamicbalancemethod

Ground-basedopen-chain linkages are dynamicallybalanced ifthe momentumh iszero forall n_b jointcoordinates q andalljointvelocitiesq˙.Notethatthishisacombinationofthelinearandangularmomentumandthusa6-dimensional vector.Sincethemomentummustbezeroforalljointvelocitiesandsincethesejointvelocitiesarelinearinthemomentum weobtainthefollowingbalancingcondition

¯h

(

q,¯z

)

≡ 0 (1)

inwhich ¯h₌

{

∂

_/

∂

q˙₁

(

h

)

_,...,

∂

_/

∂

q˙_n

b

(

h

)

}

denote the collection of thebasis vectorsof h with respect to q˙. ¯z denotes the

collectionofthe inertia parameters of thelinkage,i.e. masses, centresof mass,andmoments of inertia.The aim ofthis papernowliesinthederivationofthecompletesetofinertiaparameters ¯zthatguaranteedynamicbalanceforanygiven open-orclosed-chainlinkage.

Forclosed-looplinkagesthejointcoordinatesqarenolongerindependentsinceasetofloopclosureconstraintequations holdforallmotion

f

(

q

)

≡ 0 (2)

Thisleadstodependenciesinqandq˙,andconsequently, inareducedsetofbalancingconditions.Conventionally,this de-pendencywouldbeincorporatedintoEq.1byselectingasetofminimalcoordinatesuandsolvingEq.2forthedependent coordinatesq=c

(

u

)

.However,thisapproachisnotalwaysapplicableasthereisingeneralnoclosedformsolutiontothe loop-closureequation,i.e.cisnotalways knownexplicitly.Furthermore,ifthesolutionisfoundnevertheless,itistypically asetofinvolvedequationsthatarehardtouseinthebalancingprocedure.

Inthis paperwe take a different,Taylor-based approach.It relies on three features. Firstly we leverage the fact that, althoughcmightnotbeavailableinclosedform,itshigher-orderderivativesDk_u

(

c

)

areavailableinthereference configura-tionu0througharecursiveapplicationoftheimplicitfunctiontheorem([27]andrecapitulatedinSection3).Theresulting

higher-orderderivativesenableaTaylorexpansionofthedynamicbalanceconditionssuchthatwithaslightabuseof nota-tiontheseread

¯h

(

u,¯z

)

=¯h

(

u0,¯z

)

+Du



¯h

(

u0,¯z

)



(

u− u0

)

+ 1 2!D 2 u



¯h

(

u0,¯z

)



(

u− u0

)

2+· · · = ∞  k=0 1 k!D k u



¯h

(

u0,¯z

)



(

u− u0

)

k≡ 0 (3)

Asthismustholdforallmotionu,all Taylorcoefficients(the higherpartialderivativesDku



_¯h



)are requiredtobezerofor dynamicbalance.Sincehandfareanalyticfunctionsinanon-singularconfiguration,weobtainthefollowingnecessaryand sufficientconditions Dku



¯h

(

u0,¯z

)



≡ 0 for k=0...kmax. (4)

Notethat thisdoesnot requirean explicit solutionto the loop-closureequations(Section 5). Furthermoresince the bal-anceconditionsareanalytic,onlyafinite(butunknown)numberkmaxofpartialderivativesissufficienttoensuredynamic

balance,enablinganalgorithmictreatmentofthisproblem.

The second feature isthat these Taylorcoefficients (Eq.4) are linear in inertia parameters ¯z(Section 4). The balance conditionscantherefore,withthehelpoftheregressionmatrixXk,bewrittenintheform

Xk

(

u0

)

¯z≡ 0 for k=0...kmax (5)

andconsecutivelysolvedbynullspacealgorithms(Section6),leadingtoafulldescriptionofthedynamicallybalancedmass distributions ¯z_∈ker

(

Xk

)

,providedthatasufficientnumberofderivativesisused.

Thirdly,in thispaperwe presentasystematicpartitioningandinterpretation oftheresultingdesignspacein orderto retainsome insightinofthedesigndependenciesandthefeasibilityofthe solution(Section7). Weillustratethemethod onknownandnewexamples(Section8).

3. Kinematics

Inthissectionthe groundworkofthemethodislaid by describingthe notationandkinematics,andby recapitulating thehigher-orderderivativesofkinematics.

3.1. Kinematicsofopen-chainandclosed-chainlinkagesusingliegroupandscrewtheory

Screwtheory isusedthroughoutthisworkasitgivesa conciserepresentationofthe higher-orderderivativesof kine-maticsanddynamics.Thisscrewtheoryframeworkinterpretsmotionofabodyasacombinationofanangularvelocity

ω

aroundanaxisinspace,passingthroughpointrt,andavelocityalongthataxis,termedthepitch

λ

t

t=



ω

v



=



ω

rt×

ω



+

λ

t



0

ω



(6)

The twist t is a function of the angular velocity

ω

and the velocity v of the body’s particles that instantaneously pass throughtheoriginofthereferenceframe.Twospecialcasesexist:1)apurerotation,i.e.theangularvelocityisorthogonal tothevelocity, resultinginazeropitch(

λ

t=0),and2)apuretranslation,whentheangularvelocityiszeroandthepitch

isinfinite(

λ

t=∞).

Thetwistofthenb bodiesinaopenchainislinearlydependentonthejointvelocitiesq˙ ofthejointsinthechain.The

twistbasisvectors_jassociatedtoeachjointjistermedunittwistsorinstantaneousscrewaxis(ISA).Inthecurrentcontext sjisalwaystakeninthereferenceconfigurationq0.Thejointsinasinglechainarenumbered1tonb,fromthebasetothe

end-effector.Therefore,theJacobianJiofbodyiisformedbytheISAslowerinthechain ti=Jiq˙ =



s1· · · si0



˙ q,s=



n rs× n



+

λ

s



0 n



,s_∞=



0 n



(7)

TheseISAarepuregeometricquantitiesthatsolelydependentonthejointlocationrs,theorientationofthejoint,encoded

byunitvectorn,andthepitchofthejoint

λ

s.InthiscasewetreatthethreesingleDOFlowerkinematicpairs:revoluteR,

helicalH,orprismaticP.ForanR-jointthepitchiszero

λ

s=0,whileforaP-jointthepitchisinfinite

λ

s=∞,resultingina

limitcases_∞andisthereforetreatedseparately.MultiDOFjoints,suchasball-socketjoints,aretreatedasinstantaneously identicaltoasetofseriallyconnectedsingleDOFjoints.

Inclosed-chainlinkages,thebodytwistsarerestrictedbyasetofloopclosureconditionsf.Theresultingtwistsmaybe foundby regardingeachloop asa connectionofmultipleopenchains.Asingleloopforexampleisopenedby cuttingan arbitrarybody,resultingintwoopenchainsofwhichthelast‘virtual’bodieshavethesametwists.Theseloop-closure con-ditionsconstrainthetwistsofthebodies,asencodedbymatrixK.Byselectinganindependentsetofndinputcoordinates

u,thissystemissolvedandalldependentjointvelocitiesdetermined

Dq

(

f

)

q˙ =KJ¯q˙ ≡ 0,q˙ ∈ker

(

KJ¯

)

,q˙ =Cu˙ (8)

ThelinkageJacobianJ¯=[J₁· · · J

n_b]isthecollectionofallnbbodyJacobians.Thenb × nd C-matrixdenotesthefirst-order

solutiontotheloop-closureequations.

Toexpressfinitemotion,areferenceframe

ψ

_iisassociated toeachbody i.A homogeneoustransformationmatrixH_i, consistingofa rotationmatrix Rianda translationvector oi,expressa pointa fromabody-fixed frame intothe inertial

frame ofreferencea0_=H

iai.Inthisconvention theai vectorconsist offourvalues; 3Cartesiancoordinates anda1.The

superscriptsdenotetheframesofexpressionofthevector.ThesetransformationmatricesrelateagaintotheISAsinaopen chainthroughaproductofmatrixexponentials,leadingtothegeneralforwardkinematicsofopen-chainlinkages[31]

Hi= i j=1 exp

(

qj



sj×



)

,H=



R o0 1



,



s×



=



n×



rs× n+

λ

sn0 0



(9) inwhichexp

(

qj



sj×



)

denotesthematrixexponentialofthe4 × 4matrixoftheISAinthereference(initial)configuration (q=q0=0

)

,and



n×



the3 × 3skewsymmetricmatrixofn.

The ISAare expressed inanother coordinateframe by the adjointtransformation matrixAd



H



.The ISA isexpressed fromthebodyfixedreferenceframesi

iintheinertialframeofreferences0i accordingto s0_i =Ad



Hi



si_i,Ad



H



=



R 0



o×



R R



(10)

The time derivative of the transformationmatrix relatesto the body twist through thematrix formof theadjoint twist transformation,heretermedadjointtwistmatrixad(ti)

d dt

(

Ad



Hi



)

=ad

(

ti

)

Ad



Hi



,ad

(

t

)

=



ω

×



0



v

×

 

ω

×





(11)

3.2. Higher-orderderivativesofkinematics

Forparallelmechanismaclosed-formsolutiontothekinematicloop-closureequationsdoesnot existingeneral.Yet,a higher-orderapproximation ofthemotionisavailable by treating theclosedloop asa connectionofseveralopen chains. For such a connection,the higher-order derivativesof the loop-closureequations are found andsolved yielding a Taylor

approximationoffinitemotion[27].Inthatapproach,thehigher-orderpartialderivativesofthebodytwistsarefoundfrom theadjointtwistmatricescorrespondingtotheISAthatarelowerintheopen-chainequivalentlinkage[32].SinceeachISA isconstantwhenexpressedinalocalbody-fixedframe,allthesederivativesfollowfromarepetitiveapplicationofEq.11to

Eq.10,suchthat

Dαq

(

si

)

= i−1 j=1 ad



sj



αj si (12)

InhereDαq

(

A

)

=

∂

k/

(

∂

qα11· ...·

∂

qαnn

)

(

A

)

denotesthehigher-orderpartialderivativeswithrespecttotheelementsofq.

Vec-tor

α

=

(

α

1,...,

α

n

)

comprisestheorderofthederivativescorresponding toq,runningfromthe baseto theend-effector.

Henceweassumeanorderedsequence,i.e.

α

icorrespondstothejointqi.Thek=

α

1+...+

α

n=

|

α|

isthetotalorder,see AppendixA.Thejointshigherinthechainhavenocontributiontothemotionofthelowerjoints,suchthatthisderivative (Eq.12)issettozero,i.e.if

α

_j=0forj_{≥ i}.Bythis,allthehigher-orderpartialderivativesofthebodyJacobiansDα_q

(

J_i

)

are available.

Thisprocedureisused forthesolutionofthe higher-orderclosed-loopconstraints [27]by recastingitinto thematrix derivativeframeworkofVetterforbookkeeping[33]andAppendixA.Inthisnotationthecollectionofallfirst-orderpartial derivativesofmatrixA=[a1 · · · am]aresortedaccordingto1

Dq

(

A

)

=



Dq

(

a1

)

· · · Dq

(

am

)



,Dq

(

ai

)

=



∂

/

∂

q₁

(

ai

)

· · ·

∂

/

∂

qn

(

ai

)



(13)

Withthis, thederivatives oftheloop-closure solutionD_u

(

C

)

arefound through applicationofthe chainrule andproduct rule(AppendixA)toEq.8.Thecollectionofsecond-orderloop-closureconstraintsread

Dq



K¯J



(

CC

)

+K¯JDu

(

C

)

≡ 0 (14)

InhereABdenotestheKroneckerproductoftwomatrices(AppendixA).FromthisequationDu

(

C

)

isdetermined.A

recur-siveapplicationleadstothek-thorderconstraints

Dku



K¯JC



= nb  i=1



D_qk



K¯J



· · · K¯J



C¯k≡ 0,

(

C¯k

)

=



(

Ck

)

 · · · Dku

(

C

)





(15)

fromwhichDk_u

(

C

)

maybesolvedthroughthealgorithmpresentedin[27].TheKroneckerpowerisdenotedbyak super-script.Theexact compositionoftheC¯k collectionmatrixisfound throughrepetitiveapplicationofthe chainandproduct

rules,butisomittedhereduetospacelimitation.

4. Rigidbodydynamics

Therigidbodydynamicsofspatially movingobjectsandmechanismsisconciselywrittenwiththeuseofscrewtheory

[34,35]. This section briefly introduces the use of screw andLie group theory in rigid body dynamics, followed by the presentationofthemultipole-rodrepresentationoftheinertiaparameters asusedintheinterpretationofthedynamically balancedsolutionlateron.

4.1. Momentumwrenchandmassmatrix

The momentum of a body is the product of the body’s spatial mass matrix M and the twist t associated to it. The momentumisaco-screworawrench-likeentityandthereforetermedmomentumwrenchhereafter

h=



ξ

p



=Mt. (16)

Themassmatrixofabodyisformedbytheintegraloverthebodyvolume

M= V



−



r×



2



r×



−



r×



I3



dm=



E



mc×



−



mc×



mI3



. (17)

Thisgivesrisetotheclassicaldescriptionwithamassm,acentreofmasscandinertiamatrixEwithrespecttotheinertial frameofreference.TheinertiamatrixEcontains3inertiamomentsand3products ofinertia,respectivelyonitsdiagonal e_d =[e1 e2 e3]andits off diagonaleo =[e4 e5 e6].The matrixI3 denotes a3× 3identity matrix.Dueto theframe

invarianceofkinetic energyK₌1_/2tMt_,themassmatrixtransformswithanadjointtransformationmatrixontheright anditstransposedontheleft.Bychoosingaframethatislocatedatthecentreofmassandalignedwiththeprincipalaxis

1 Please note the two distinct uses of the differentiation operator. When the superscript is a vector, i.e. D α

q , it denotes a repeated partial derivatives, but

when the superscript is a scalar, i.e. D k

Fig. 1. Three representations of the inertia parameters of a body. (a) The conventional representation with a mass m , a centre of mass c and an inertia matrix G around c . (b) The multipole representation [30] with parameters that are linear in the mass matrix; a monopole m at r , a dipole δin the direction

a , and a quadripole ηin the direction of b . One monopole, three dipoles and six quadripoles are sufficient to describe arbitrary bodies. (c) The multipole- rod representation reduces the number of graphical elements by interpreting the quadripole as an infinitely long, slender rod, termed ‘pure-inertia rod’ and depicted as a striped bar. The monopole is termed ‘point mass’, whereas the dipole is treated as a ‘displacement’ of the point mass with negative pure-inertia rod in the same direction.

ofinertia,anymassmatrixcanbediagonalized.Thecorrespondingtransformationmatrixfromthisprincipalaxesframeto thecurrentframeisAd



H_p



.ThisgivesrisetothreeprincipalMOIsg1,g2,andg3

M=Ad



Hp



− diag

(

g1,g2,g3,m,m,m

)

Ad



Hp



−1 . (18)

Inthisbody-fixedframe,themassmatrixisconstant,i.e.m˙ andg˙i=0,duetotherigidbodyassumption.Sincethemass

matrixisformedbyacollectionofpositivemassparticles,themassmatrixitselfissymmetricpositivedefinite,leadingto 7inequalityconditionsonthemassandtheprincipalMOIs

m>0,gi>0,gi+gj>gk (19)

4.2. Momentumwrenchbasis

Similartothetwistbasis,wedefinealinkage’smomentumbasisthatspansallpossiblemomentumwrenchesatagiven pose. Thebasis vectors, termedthe instantaneousmomentum wrenches(IMW) anddenotedwithhˆi,are themomentum

wrenchesgeneratedbyunitactuationofeachjoint.Thetotalmomentumwrenchofaopenchainisthereforegivenby

h=Miti=M¯¯Jq˙=



_ˆ h1 · · · hˆn



˙ q≡ 0,_hˆ_i= nb  j=i Mjsi≡ 0 (20)

In hereM¯ =[M1 · · · Mn] denotes the collection ofall mass matricesin the chain.For dynamicbalance all the IMWs

mustbezeroforarbitrarymotion.Forclosed-chainlinkagesthemomentumwrenchbasisiscomputedbyapplyingthefirst orderloop-closuresolutionC

h=M¯¯JCu˙ ≡ 0. (21)

4.3. Multipole-rodinterpretationofthemassmatrix

Inthecurrentdynamicbalancingprocedurewewillusethefactthatthebalancingconditionsarelinearintheelements ofthemassmatrixsuchthattheycanbesolvedthroughasetoflinearoperations.Theconventionalmassmatrix parametri-sation, consistingofmassesm,COMscandprincipalMOIsg,is notsuitable fortheinterpretationoftheresultingdesign space, sinceitisnot linearintheelementsofthemassmatrix.Thereforewe willuseaslightadaptationofthemultipole conceptofRosetal[30].,termedthemultipole-rodrepresentation(Fig.1).Thisinterpretationreliesonthefactthatamass matrix can be decomposed into threeprimitive elements; 1) a singlepoint mass at r, denoted with a subscript m, 2) a

displacementofthepointmassinthedirectionofaunit vectoracombinedwithapure-inertiarodofoppositemagnitude,

denotedwithasubscript

δ

,and3)apure-inertiarodinthedirectionofaunitvector b,denotedwithasubscript

η

.These pure-inertiarodsareinterpretedasinfinitelylongslenderrodsinthedirectionoftheirunitvector. Theirmassisassumed zerosuchthatonlytherotationalvelocitycomponentinaperpendiculardirectiongeneratesangularmomentum.Arotation aroundtheirlongitudinalaxisgeneratesnoangularmomentum.Thesoledifferencewiththemethodof[30]isthegraphical representation.Thisreducesthelargernumberofpointmasses(poles),whichotherwisemightcongestthefigures.

Now,anymassmatrixcan berepresentedbychoiceof10oftheseprimitiveelements,onepoint mass,three displace-ments,andsixpure-inertiarods,aslongastheunitvectorsaiandbiareunique

M=mMm

(

r

)

+  i=1···3

δ

iM_δ

(

ai,r

)

+  i=1···6

η

iMη

(

bi

)

. (22)

mm=1,

(

mc

)

m=r,Em=−



r×



2,m_δ=0,

(

mc

)

_δ=a,E_δ=1/2



r− a×



2− 1/2



r+a×



2,

mη=0,

(

mc

)

η=0,Eη=−



b×



2. (23)

Fortheplanarcase,thisrepresentationrequiresonepointmass,twodisplacementsandonepure-inertiarod,ofwhichthe elementsreduceto

mm=1,

(

mc

)

m=r,em=

r

2,m_δ=0,

(

mc

)

_δ=a,e_δ=2ar,mη=0,

(

mc

)

η=0,eη=1. (24)

Forfeasibilityof eachbody,they must consist ofatleastone positive point mass,andthreenon-coplanar positive pure-inertiarods(Eq.19),sincetwo pure-inertiarods representan infinitelyflatobject.Anegativepure-inertiarodrequiresat least3arbitrarilyorientedpositivepure-inertiarods(ortwopositivecoplanarpure-inertiarods)ofsufficientmagnitudeto representa feasible body.A closed-formfeasibilitydescription of an arbitrarycollection oftheseelements canbe found througheigendecompositionoftheresultingmassmatrix,butliesoutsidethescopeofthispaper.

5. Higher-orderderivativesofthemomentumequationsandofthedynamicbalanceconditions

Thepreviouslypresentedhigher-orderanalysisofthekinematicsisextendedtorigidbodydynamicsinthissection.The aimistofindandsolvethenecessaryandsufficientdynamicbalanceconditionsofarbitrarylinkageswithoutinvokingthe closed-formsolutiontotheloop-closureequations.Fordynamicbalancingpurposesthisstudyisconfinedtothechangeof rigidbody momentum.Othereffectssuch asgravity, elasticity,orexternalforcesarenottakenintoaccount.Thedynamic balanceconditionsareobtainedfromthepartialderivativesofthemassmatricesandmomentumequationsofopen-chain linkages,whichareextendedthereaftertoclosed-chainlinkagesbyincludingthehigher-orderderivativesoftheloop-closure solution.Itshouldbenotedthatalthoughopen-chainlinkagescannotbedynamicallybalancedwithoutadditional counter-mechanisms,their descriptionis importantfordynamicbalance sinceclosed-loop linkages canbe regarded asconnected openchains.

5.1. Derivativesofthemassmatrixinaopenchain

Themassmatrixof abody iin aopen chaindependsontheposeofthejoints thatare lower inthekinematicchain accordingtoEq.9andEq.18.Therefore,itspartialderivativewithrespecttoajointj,lowerinthechain(j≤ i),isfoundby applyingEq.11toEq.18

∂

∂

q_j

(

Mi

)

=−ad



sj



 Mi− Miad



sj



(25)

Herewehaveusedthefactthatthemassmatrixisconstantinthebody-fixedframe.Forallpartialderivativeswithrespect tojointshigherinthechain(j>i)thisderivativeiszero.

Asecond (non-zero)partialderivativeiseitherwithrespecttoajointhigher(j≤ l≤ i)orjointlower (l≤ j≤ i)inthe chain.Inthefirstcase(j_{≤ l≤ i})thepartialderivativebecomes

∂

∂

ql

∂

∂

q_j

(

Mi

)

=ad



sl







ad



sj



 Mi+Miad



sj



+



ad



sj



 Mi+Miad



sj



ad



sl



. (26)

HeretheJacobiidentity

∂

/

∂

q_l

(

ad



sj



)

=ad

(

ad



sl



sj

)

=ad



sl



ad



sj



− ad



sj



ad



sl



isused.Forthesecondcase(l ≤ j≤ i) only the derivative of the mass matrix has to be taken into account as a higher joint does not influence a lower ISA (

∂

/

∂

q_l

(

ad



sj



)

=0).ThisresultsinasimilarequationasEq.26,withthesoledifferencethattheindicesjandlareswapped. Thisalsofollowsfromthesymmetryofpartialderivatives.Thisnestedstructure,i.e.thepre-andpostmultiplicationof ad-jointtwistmatrices,ispreservedforthehigherorders,leadingtoarecursiveformulaforallpartialderivativesofthemass matrix

∂

∂

qj

(

Dαq

(

Mi

)

)

=−ad



sj



 Dαq

(

Mi

)

− Dαq

(

Mi

)

ad



sj



(27)

inherejisthelowestjointtowhichapartialderivative istaken, i.e.

α

_l=0forall l_<j.Incase

α

_{l =}0foranyl_> i,this equationissettozero.

5.2.Derivativesofthemomentumwrenchinaopenchain

Now thatthe derivativesofthemass matrixup toarbitraryorder areavailable, we considerthepartial derivativesof themomentum wrench withtheaim ofobtaining all higher-orderdynamicbalance conditions.Consider themomentum wrench generated by the jthbody dueto unit actuation ofjoint i,which is lower inthe chain. Two typesof non-zero partialderivativesappear.Eitherjointl— withrespecttowhichthederivativeistaken— isbelowthejointi,orbetween

thejointiandthejthbody.Inthefirstcase(l_{≤ i≤ j}),thepartialderivativeofboththemassmatrixandtheISAhaveto betakenintoaccount,partiallycancelingout

∂

∂

q_l



Mjsi



=

∂

∂

q_l



Mj



si+Mj

_∂

∂

_q l

(

si

)

=−ad



sl



 Mjsi. (28)

Inthesecondcase(i<l≤ j),thepartialderivativeoftheISAvanishes

∂

/

∂

q_l

(

si

)

=0.Therefore,thepartialderivativeofthe

momentumwrenchbecomes

∂

∂

q_l



Mjsi



=

∂

∂

q_l



Mj



si=−

(

ad



sl



 Mj+Mjad



sl



)

si. (29)

Thehigher-orderpartialderivativesarefoundsimilarlybymakingasplitbetweenthepartialderivativesrelatedtojoints lower thanthemomentum generatingISA,andtheonesrelatedtothejointsbetweentheISAandthebody.Therefore.a secondmulti-index isintroduced forwhichholds

β

_l=

α

l foralli<l≤ j and

β

l=0foralll≤ i.Thepartialderivativesof

themomentumwrencharefoundfromEq.27accordingto

Dα_q



Mjsi



= i l=1



−ad



sl







αl Dβq



Mj



si. (30)

Again thisequation iszeroif

α

_{l =}0 foranyl_> j.Thesepartial derivativesmaybe summedto obtainthe derivativesof thetotalmomentumofthelinkage.Noticethatinthisequationthemomentumderivativesareformulatedasasequenceof matrixoperations,whicharelinearinthemassmatrix.

5.3. Derivativesofthedynamicbalanceconditionsofaopen-chainlinkages

Thedynamicbalanceconditionsdictatethatthemomentumwrenchofalinkageiszeroforallmotion.Thereforealsoall higher-orderderivativesofthemomentumwrenchmustbezero.Withalargeenoughnumberofderivativeskmaxtheseare

not onlythenecessarybutalsothesufficientdynamicbalanceconditionsfornonsingularlinkages.Infact, hereit willbe shownthatforopen-chainlinkagesonlyderivativesuptothesecondorderareneeded(kmax≤ 2).Whenthesearesatisfied,

allthehigher-orderdynamicbalanceconditionssatisfied,andthelinkageisdynamicallybalancedforfinitemotion. Forzeroth-orderdynamicbalance,thecondition(Eq.20)imposedoneachIMWis

ˆ hi= nb  j=i Mjsi=M˜isi≡ 0,M˜i= nb  j=i Mj=



_˜ Ei



mici×



−



m ici×



˜ miI3



(31)

TheaggregatedmassmatrixM˜_iisthesumofthemassmatricesbelongingtobodieshigherinthechainthans_i.Consider nowthefollowingmomentumderivativesofhˆjandhˆl,involvinganytripletsl,sj,andsiofzeroorfinitepitchISA,which

arearrangedinascendingorder(l≤ j≤ i)

∂

∂

q_i



_ˆ hl



=

∂

∂

q_i



_˜ Mi



sl≡ 0,

_∂

∂

_q i



_ˆ hj



=

∂

∂

q_i



_˜ Mi



sj≡ 0 (32)

∂

∂

ql

∂

∂

q_i



_ˆ hj



=−ad



sl





∂

∂

q_i



_ˆ hj



−

∂

∂

q_i



_˜ Mi



ad



sl



sj≡ 0 (33)

NoticethatthesedynamicbalancingconditionsimposeconstraintsonthesameaggregatedmassmatrixM˜isinceqiishigher

inthechainthanqlandqjsuchthat

∂

/

∂

qi

(

Mj

)

=0forj≤ i.Asthefirst-orderbalancingconditions(Eq.32.b)ensurethat

∂

/

∂

q_i

(

hˆj

)

=0,thesecond-orderdynamicbalanceconditions(Eq.33)reduceto

∂

∂

q_i



_˜ Mi



ad



sl



sj≡ 0 (34)

Arecursiveapplicationshowsthatthisextendstothehigherorders,suchthatallbalanceconditionsareoftheform

∂

∂

q_i



_˜ Mi



i l=j



ad



sl







αl sj≡ 0 (35)

Moreover,thezeroth-orderbalanceconditions(Eq.31)satisfiesallhigher-orderforcebalancingconditionssince

∂

/

∂

q_i

(

M˜i

)

isafunctionofthelinearmomentumandthemassisassumedtobeconstant

∂

∂

q_i

(

m ici

)

=pˆi≡ 0,

∂

∂

q_i

(

m˜i

)

=0 (36)

Therefore,onlythefollowingfirst-andsecond-ordermomentbalanceconditionsremain:

∂

∂

q_i



_˜ Ei



nl≡ 0,

_∂

∂

q_i



_˜ Ei



nj≡ 0,

_∂

∂

q_i



_˜ Ei



nl×



nj≡ 0 (37)

Inthegeneralcase,whenn_jࢲn_l,thisimposes9independentconstraintsonthederivativeoftheinertiamatrix,requiring

∂

/

∂

q_i

(

E˜i

)

=0,thus directlysatisfyingall higher-orderpartial derivatives(Eq.35). Thisshowsthat derivativesof ahigher

orderthankmax=2imposenonewdynamicbalanceconditionsforopen-chainlinkages.When,inthespecialcase,all

non-infinitepitchISAlowerinthe chainareparallel, i.e.n_j

n_iforall j_< i,the momentbalanceconditions(Eq.37)vanishor become equivalent. Then, only threehigher-order constraints are imposed onthe aggregated inertia matrixE˜i. Prismatic

joints(infinitepitchISA)lower inthechainimposenohigher-ordermomentbalanceconditionsastheirangularvelocities n_jarezero.

Tosummarize:foropen-chainlinkagesthezero-orderforce andmomentbalanceconditions(Eq.31) andthefirst-and second-ordermomentbalanceconditions(Eq.37)arenecessaryandsufficient,leadingtoakmax=2.

5.4.Derivativesofthedynamicbalanceconditionsofclosed-chainlinkages

Thedynamicbalanceconditionsofclosed-chainlinkagesdictateazeromomentumwrench(Eq.21)forallindependent velocitiesu˙.Thereforethezeroth-orderbalancingconditionsread

¯

M¯JC≡ 0 (38)

Alsoall higher-orderpartialderivativeswithrespect toushould bezerofordynamicbalance.Theseconditionsarefound byrepetitiveapplicationofthechainrule,theproductruleandderivativesoftheKroneckerproduct.SimilartoEq.15,the first-orderdynamicbalancingconditionsbecome

D_u



M¯¯JC



=D_q



M¯¯J



₍

CC

)

+M¯¯JD_u

(

C

)

≡ 0 (39)

Thisgeneralizestohigher-ordersbyarepetitiveapplicationofthechainandproductrules

Dk u



_¯ M¯JC



=



Dkq



_¯ M¯J



· · · _M¯¯J



C¯k≡ 0 (40)

Fromtheanalyticityofthemomentum equationsitmaybededucedthatthereisfinitekmax whichrendersthese

con-ditions not only necessary but also sufficient for the dynamic balance for closed chains in nonsingular poses. Refer to

Section9foradiscussiononthenecessityandsufficiencyoftheseconditions.

Itshouldbe notedthatthesehigher-orderdynamicbalance conditionsarelinearinthemassmatricesandcan be ob-tainedthroughaseriesofmatrixmultiplicationsandlinearoperations.Thismethodisthereforeabletotreatsymbolicor numericalinput.

6. Dynamicbalancesolutionusingtheparameter-linearform

Now,to solvethesehigher-orderdynamicbalanceconditions,we recastEq.40intheparameter-linearform[28,29]as usedintheparameteridentification.Thisenablesnullspaceprocedurestoextractthedynamicallybalancedmass distribu-tions.

6.1. Parameter-linearform

Sincethem,mcandE(Eq.17)arelinearinthemomentumequation,thefollowingparameter-linearformholds

h=Mt=



t∗



z,z=



m mc e_d e_o



(41)

inwhichthez-vectorisformedby theinertiaparameters ofthebody.Thetwistdependent‘regression’ matrixisgivenby



t∗



=



0 −



v

×



diag

(

ω

)



ω

∗



v



ω

×



0 0



,



ω

∗



=



_ω

5

ω

6 0

ω

4 0

ω

6 0

ω

4

ω

5



. (42)

Notice thatthe ordering oftheinertia parameter slightlydiffersfrom [29].The parameter-linear formofthe momentum basisofaopen-chainlinkagesisdirectlycomputedfromEq.20

¯h=vec

(

M¯¯J

₎

=

⎡

⎢

⎣

ˆ h1 . . . ˆ hn

⎤

⎥

⎦

=

⎡

⎢

⎣



s1∗



· · ·



s1∗



. . . ... ... 0 · · ·



sn∗



⎤

⎥

⎦

⎡

⎣

z1 . . . zn

⎤

⎦

=W¯z (43)

inhere ¯hand¯zdenotetheconcatenationofallIWMandallinertiaparametersinthechain,respectively.

Toobtaintheparameter-linearformofclosed-chainlinkages,thevectorizationofmatrixproducts(AppendixA)isapplied toEq.38

¯h=vec

(

M¯¯JC

)

=

(

CI6

)

W¯z=X¯z (44)

6.2. Higher-orderdynamicbalanceconditionsintheparameter-linearform

Theparameter-linearformalsoappliestohigher-orderderivativesofthebalanceconditionsastheyareformedthrough asequenceofmatrixoperationsthatarelineartheinertiaparameters.Thehigher-orderopenchainregressionmatricesW_k canbefoundaccordingly,i.e.bytheapplicationofEq.41toEq.27andEq.30,resultinginthefollowingcondition

vec



Dk_q



¯h



=Wk¯z≡ 0 (45)

Forclosedchainstheparameter-linearformisfoundbyapplyingthematrixvectorizationtoEq.40,suchthat

vec



Dkq



¯h



=

(

C¯kI6

)

W¯k¯z=Xk¯z≡ 0 (46)

inwhichW¯k =



W₁ · · · W_k



.Nowwehavearrivedattheparameter-linearformofthehigher-orderderivativesofthe momentum equationsofopen- andclosed-chainlinkages.Itshould beobservedthat allthesesteps solelyrelyonmatrix operationssuitableforalgorithmictreatment.

6.3. Solvingthedynamicbalancecondition

Dynamicbalancerequiresinertiaparameters ¯zthatareontheintersectionofthenullspacesofalltheXimatrices

¯z∈ker

(

Xi

)

,¯z∈ker

(

X¯kmax

)

,¯z=Ny (47)

inwhichX¯kmax=



X1 · · · Xkmax



isthecollectionofall regressionmatricesuptoorderkmax.Itshouldbe emphasised

that thereisa finitekmax,which makesthe approachpractically feasible.The columnsoftheNmatrix formabasis that

span thisnullspaceandtherewith describe thefull designspace ofthedynamically balanced inertia parameters.This N matrix istermed the designspace matrix andmaybe found through numeric orsymbolic nullspacealgorithms such as Gauss-Jordan eliminationor singularvalue decomposition. The corresponding designparameters are collectediny.With thisthecompletesetofdynamicallybalancedinertiaparametersofanygivennonsingularlinkagemaybefound.

7. Partitioningandinterpretationofthedynamicbalancesolution

Theapplicationofnullspacealgorithmstothedynamicbalanceproblem(Eq.47)mayresultinadesignspace descrip-tion thatis stronglymixedintheinertia parameters, compromisingstructure anddesignintuition. Toaidthedesigner, a partitioningof thedesign spacewithrespect to thejoint topology ispresented alongsidea multipole-rodrepresentation (Fig.1) ofthese partitions.We shallshow that 6types ofinertia transfer matricescompletely describethe design space ofopen-chainlinkages.Theseinertiatransfermatricescontainallinertiaparametersthat maybetransferredbetweentwo hingedbodies, i.e.subtracted fromone body andadded tothe other,withoutchangingthe momentumgenerated bythe linkage.Thispartitioningwillleadtoageneraldescriptionofthedesignspaceofopen-chainlinkagesthat,moreimportantly, alsocovers alarge partofthedesign spaceofclosed-looplinkages.Closed-looplinkages maybeseen asaconnection of multipleopenchains.Abalancingsolutionthatisvalidforopen-chainlinkagesisthereforealsovalidforclosed-chain link-ages.Althoughtheopen-chaindesignspaceitselfisalways unfeasible,incombinationwitha closed-chaindesignspaceit allowsformorefeasiblesolutionsasshownlaterintheexamples.

7.1. Partitioningthedesignspaceofopen-chainlinkages

The dynamic balancing conditions of open-chain linkages (Eq. 31 and Eq.37) are formulated in terms of aggregated massmatricesM˜_i.Beforepresentingthegeneralsolutionitmayalreadybeobservedthat solutiontotheseequationswill alsobeintermstheaggregatedmassmatrices.Fromtheseaggregatedsolutionseachindividual massmatrixcanbefound accordingly

Mi=M˜i− ˜Mi+1,zi=Niyi− Ni+1yi+1. (48)

Therefore,thecompletedesignspacematrixNofanopenchain(Eq.47)maybepartitionedasabanddiagonalmatrix

⎡

⎢

⎢

⎢

⎢

⎣

z1 z2 . . . zn−1 zn

⎤

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

N1 −N2 · · · 0 N2 −N3 . . . ._. . ... . . . Nn−1 −Nn 0 · · · Nn

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎣

y1 y2 y3 . . . yn−1 yn

⎤

⎥

⎥

⎥

⎥

⎥

⎦

(49)

inherethesubmatrixN_idescribesallinertiaparametersthatcanbeexchangedbetweenthetwobodiesconnectedbyjoint

iwithoutchangingthedynamicbehaviorofthechain.TheseNisubmatricesarethereforetermedinertiatransfermatrices.

InSection 7.3it isshownthat thereexistactually6 typesofinertia transfermatricesdependingonthetype ofjointand parallelismwiththejointaxeslowerinthechain.

Table 1

The dimensions of the 6 inertia transfer matrices. Each joint i in a chain extends the design space depending on the type of joint; revolute ( R ), helical ( H ), or prismatic ( P ) and the alignment with all non-prismatic joints j < i lower in the chain; a) skew or b) parallel. ∗_{With a prismatic joint, the prismatic joint direction applies n i =}

ni,∞ .

Joint type R ( λi = 0 ) H ( λi = finite ) P∗( λi = ∞ )

Skew nj ࢲ n i 3 1 6

Parallel nj n i 5 4 7

Itshouldbenotedthatasimilarconceptisusedinthecontextofparameteridentificationtodescribethesetof unidenti-fiableinertiaparameters[30,36].Broadlyspeaking,inertiaparametersaresaidtobeunidentifiableiftheydonotcontribute tothekineticenergyofthelinkage.Thedynamicallybalanceddesignspaceofopen-chainlinkages,asfoundhere,isformed byunidentifiableinertiaparametersaszeromomentuminthiscasealsoimplieszerokineticenergy.Theinverseisnottrue ingeneral.Thisalsoshowsthat theinertiaparameters inthisdesignspacedonotaffecttherequiredmotoreffortofthe linkage.

7.2.Partitioningthedesignspaceofclosed-chainlinkages

Wehave alreadyseen that closed-looplinkages can be convertedinto an open-chain equivalentby openingthe loop. Therefore,thedynamicbalanceconditions,andhencethesolutions,foropenchainsarealsovalidforclosed-chainlinkages. Yet,thisisnotnecessarilythecompletedesignspace,sincetheloop-closureequationsallowfordynamicallybalancedmass distributions thatlie outside thedesign spaceofopen-chain linages, i.e.rank

(

X¯

)

≤ rank

(

W¯

)

.The design spaceof closed-chain linkages can therefore be partitioned into NO, dealingwith the equivalentopen-chains, termed open-chain design

spacematrix,andintoaremainderNC associatedtotheloopclosure,termedclosed-chaindesignspacematrix

N=



NO NC



,NO=



NI · · · NN



. (50)

Theopen-chainequivalentdesignspacematrixNO isfoundbycuttingtheloopsofaclosed-looplinkagesuchthatasetof

Nchainsarefound.Theopen-chaindesignspacematrixNIassociatedtochainIhastheband-diagonalformofEq.49.The

completeopen-chaindesignspaceistheunionoftheopen-chain designspacesofthechainsintowhichthelinkagemay bedecomposed.Theopen-chain designspacesoftheindividual chainsarenot necessarilydisjoint,e.g.twodesign spaces basesNIandNII ofasingleloopmaypartlycoverthesamedesignspace.Thismeansthattherankoftheopen-chaindesign

spaceisequalto,orsmallerthan,thesumoftherankoftheindividualopen-chaindesignspaces.Furthermoreitshouldbe notedthattheopen-chaindesignspaceisinvarianttowherealoopisopened,althoughthebasismightbedifferent.

Ameaningfulclosed-chaindesignspacematrixisfoundbyintroducingasuitabletestmatrixT,whoseinertiaparameters arenot inthespanofthe open-chaindesign space.The nullspacebasis

(

X¯T

)

⊥ oftheresulting higher-ordermomentum wrenchesX¯T yieldsaninterpretabledesignspacematrixNC

NC=T

(

X¯T

)

⊥. (51)

7.3.Interpretationofthedesignspaceviatheconceptofinertiatransfer

InSection 5.3, itwasshownthat dynamicbalanceimposes two conditionson theaggregatedmassmatricesof open-chainlinkages:Firstly,eachaggregatedmassmatrixM˜_ishould bechosensuch thatitsIMW vanishes(M˜_is_{i≡ 0}).Secondly, theactuationofthecorrespondingjointqishouldnotchangetheangularmomentumgeneratedbyaanyjointlowerinthe

chain(

∂

/

∂

q_i

(

E˜i

)

nj≡ 0forallj<i).Fromthefirstconditionthreecasesarise;an ISAofzero,finiteorinfinitepitch,while

forthesecond conditiontwo casesexist; eitherall axesup to n_i are parallel (n_j

n_iforall j _< i) orat leastone isskew (njࢲniforj <i).Thisgivesriseto6typesofdesignspacefor1-DOFlowerkinematicpairs,and, consequently,6typesof

inertiatransfermatricesNi(Eq.49).Thesearediscussednow. Forhigher-DOFjointsandjointsinplanarlinkagesasimilar

representationexistasshownsubsequently.

7.3.1. Sixinertiatransfermatrices

Here,themultipole-rodrepresentationofthesesixinertiatransfermatricesaregiven(Fig.2).Inthisnotationthepoint mass,displacementandpure-inertiarodelementsofthemultipole-rodrepresentation(Eq.23)arerespectivelydenotedby zm(r), z_δ(r, n), andzη(n). The dimensionsoftheseinertia transfermatricesare in Table1.Starting froma revolute joint,

whoseaxishasnoparticularalignment,thesixcasesarediscussedandinterpreted.

N0,ࢲTheinertiatransfermatrixassociatedtoarevolutejoint(

λ

=0)— whosejointaxisisskew(ࢲ)withrespecttoone

ormoreprecedingrevolute orhelicaljoints— compriseofthreeinertiaparameters.Thesethree parameterscanbe freelyexchanged(addedtooneandsubtractedfromtheother)betweenthetwobodieshingedbythisjointwithout

Fig. 2. The interpretation of the six sets of inertia parameters that can be exchanged between the two (grey) bodies attached to joint i (subtracted from one and added to the other) without changing the dynamic behaviour of the chain as a whole. These six cases arise from the three types of 1-DOF lower pairs, and parallelism with all preceding revolute and helical joints. The orientation of the preceding prismatic joints have no influence. It should be noted that for clarity sake the effect of the displacement δon the MOIs is not shown, as it can be compensated by, or absorbed in η1 . Since the pure-inertia rods have no application point, they are displayed at an arbitrary location.

affectingthemomentumgeneratedbythechainasawhole.Theseparametersare: 1)apointmasszm onthejoint

axisrs,2)adisplacementofthispointz_δinthedirectionofthejointaxisn,3)apure-inertiarod

η

inthedirection

ofthejointaxisn.Thecorrespondinginertiatransfermatrixthereforereads

N0,∦=



zm

(

rs

)

z_δ

(

n,rs

)

zη

(

n

)



. (52)

Thereasonforthesethreeinertiatransfersisthattheactuationofajointwithapointmassmanywhereonitsaxis rsdoesnotinduceanylinearorangularmomentum,nordoesitchangetheIWMoflowerjoints(Eq.37),sinceand

equalandopposite point massis attachedtotheconnecting body.Thisyields thedesign freedomszmandz_δ.The

thirddesignfreedom,apure-inertiarod

η

,generatesnoangularmomentumasitisalignedwiththejointaxis.This alignmentalsomakessurethattherotationofthispure-inertiarodbythejointwillnotcauseachangeintheinertia matrixfeltbythelowerjoints.

Any other exchangeofmass orinertia betweenthe two bodiesconnectedby thisjointwill eitherchangethe mo-mentumgeneratedbythisjointorbythejointslowerinthechain.

N_0, Whentherevolute joint(

λ

₌0)isparallelwithrespecttoallprecedingrevolute andhelicaljoints,twoadditional parameters areobtained, incomparisontoN0,ࢲ. Theseparametersare twoperpendicular pairsof pure-inertiarods.

Alltheseallfourrodsareonasingleplaneperpendicularton.Thesepure-inertiarodareofoppositemagnitudeina pairwisemanner(Fig.2).

Thesefouradditionalpure-inertiarodsallowforamodificationoftheinertiatensorwithoutchangingthedynamics of the chain.The first of the pure-inertiarod pairs

η

2 is inthe direction ofb2, which is perpendicular to n. The

angularmomentuminducedbyb2iscancelledbyanequalandnegativepure-inertiarodinadirectionperpendicular

tobothnandb2.Thisalsoholdsforasecond pair

η

3 withcorresponding b3.Thisadditionalpure-inertiarodsarise

sincetheircommonplanewhichisperpendiculartonisnotchangingbyactuationofthejointslowerinthechain. Theinertiatransfermatrixisthereforeparametrizedaccordingto

N0,

(

s

)

=



zm

(

rs

)

z_δ

(

n,rs

)

zη

(

n

)

zη

(

b2

)

− zη

(

n× b2

)

zη

(

b3

)

− zη

(

n× b3

)



N_f,ࢲ Fora helical(

λ

₌finite), non-parallel jointanypoint masswill generatealinearmomentum through itspitching motionsuch that its inertia transfer matrixonly containsa pure-inertiarodin the directionofthe jointaxis. The displacement(z_δ)wouldcauseaposedependentinertiamatrixandnon-constantIWMsassociatedtothelowerjoints. Thesoleinertiachangeistherefore

Nf,∦=zη

(

n

)

(54)

N_f, Whenahelicaljointisparallel toall precedingrevoluteandparalleljointsithasasimilarinertiatransferspaceas N0, (Eq.53)withthesoledifferencethatthemassshouldthereforeequatetozeroasthepitchingmotionwould

gen-eratealinearmomentum.Thedisplacement(z_δ)ontheotherhanddoesnotinducelinearmomentumandtherefore remains

Nf, =



zδ

(

n,rs

)

zη

(

n

)

zη

(

b2

)

− zη

(

n× b2

)

zη

(

b3

)

− zη

(

n× b3

)



(55) N_∞,ࢲ The inertia transfer of a prismatic joint (

λ

_=∞) whose joint axisis not alignedwith all preceding revolute or helicaljointaxeshasasize6,since itsEcanbe selectedfreely. Asthesemomentsandproductsofinertiawill not induceangularmomentum(andareconstant)theycanbeselectedasdesired.Herethischoiceisparameterizedby6 pure-inertiarods

N_∞,∦=



z_η

(

b1

)

· · · zη

(

b6

)



. (56)

N_∞, When theprismaticjoint(

λ

=∞) isalignedwithprecedingzeroandfinitepitchjointsitgainsadisplacementzδ

inthedirectionofthejointaxis,leadingtoaninertiatransfermatrixwithsize7

N_∞, =



z_δ

(

n_∞,n_∞

)

z_η

(

b1

)

· · · zη

(

b6

)



. (57)

7.3.2. Multi-DOFjoints

Thisapproachalsoholdsformulti-DOF jointsthatcan locallybemodelled asa serialconnectionof1-DOF joints,e.g., cylindrical,planar,universalorsphericaljoints.Thesemulti-DOFjointscanthetransmitinertiaparametersthatarecommon inthe lower kinematicpair analogue.For example,a cylindricaljoint canbe modelled asprismatic anda revolute joint inseriessuch thatits inertia transfermatrixis theintersectionofthe image ofN0 andN∞,which istheinertia transfer

associatedtoahelicaljointN_f.Dependingontheparallelismwiththejointslowerinthechain,eithertypeistobeselected. AplanarjointisserialconnectionoftwoprismaticjointssuchthatitsinertiatransferisN_∞.Universalandsphericaljoints locallybehaveasaserialconnectionofmultiplenon-parallel,intersecting revolutejoints.Theassociatedinertiatransferis thereforeapointmasszm(r)ontheintersectionpointroftheseaxes.Weassumeherethatthejointitselfdoesnotcontain

intermediatebodieswithmassorinertia.

7.3.3. Jointsinplanarlinkages

IntheplanarcaseonlyzeroandinfinitepitchISAexist.Therefore,threetypesofinertiatransfermatricesappear,1)the ISAisrevolute andthereforeautomatically parallelto allother revolute joints,2) theISA andalllower ISAare prismatic jointsor3)theISAisprismaticbutatleastoneofthelowerjointsisarevolutejoint

N0=zm

(

rs

)

,N_∞, =



z_δ

(

n∞

)

zη



,N_∞,∦=zη (58)

Inthefirstcase,thepointmassshouldbeontherevolutejointandtheMOIaroundthatpointshouldbezero.Inthesecond case,thebodysolelytranslates, thereforethemassshouldbezeroandthefirstandsecond momentsofmassarefree,as parameterizedbyadisplacementandapure-inertiarod.Inthethirdcase,whentheISAunderinspectionisprismaticand oneormorelowerjointsarerevolute,thisdisplacementwillcausesachangingMOIassociatedtotherotationofthelower joints.Thisthedisplacementshouldthereforebezero.

WiththisdescriptionoftheinertiatransfermatricesNiofcommonjoints,thedynamicallybalanceddesignspaceofany

open-chainlinkage maybeobtained. Alsoforclosed-chainlinkages, theopen-chain equivalentdesignspacematrix NO is

completelydetermined,generalizing[23]to spatiallinkages. Itshouldbe notedthatopen linkagescannot bedynamically balancedwithoutadditionofcounter-mechanisms. Thiscanalsobeestablished fromtheinertia transfermatricesasnone ofthempermitbothapositivemassandpositiveMOIs.Therefore,aspecificNC isrequiredtorenderafeasibledynamically

balanceddesignspace.Theexistenceofthisadditionaldesignspaceisfoundonacase-by-casebasisinthenextsection.

8. Casestudies

The higher-orderdynamicbalance approach is illustrated herewith casestudies ofa serial 6-DOF robot, a planar 4R

four-barlinkage,anda3-RSRmechanism. Inallcases aninterpretation oftheclosed-chaindesignspacebasesNC will be