multi‐objective optimization of a parallel manipulator for the ......prosthetic arm, biomechanics,...

Original Article

LatinAmericanJournalofSolidsandStructures,2018,15 3 ,e26

Multi‐objectiveoptimizationofaparallelmanipulatorforthedesignofaprostheticarmusinggeneticalgorithms

AbstractThis paper presents a synthesis of a spherical parallelmanipulator for ashoulder of a seven‐degrees‐of‐freedom prosthetic human arm using amulti‐objective optimization. Three design objectives are considered,namelytheworkspace,thedexterity,andtheactuatorstorques.Theparallelmanipulatorismodelledconsidering13designparametersinanoptimiza‐tionprocedure.Duetothenon‐linearityofthedesignproblem,geneticalgo‐rithmsareimplemented.Theoutcomesshowthatasuitableperformanceofthemanipulatorisachievedusingtheproposedoptimization.

KeywordsProsthetic arm, Biomechanics,Multi‐objective optimization, Genetic algo‐rithms.

1INTRODUCTION

Thesuccessofaprosthetichumanarmdesigncanbeevaluatedmainly in termsof the functionalityof thedesignanditscost.Forthesereasonsthedesignofthemechanismthatperformsthemovementoftheprosthesisisanissuethatimpactsgreatlyinthefinalproduct.Upperlimbdisarticulationisoneofthecaseswithlessinci‐dences,thereforethepeoplewiththisdisabilityhavefeweroptionsofprosthesesavailable Dillinghametal.,2002 .Oneofthemostadvancedprostheticarmcanbefoundin Johannesetal.,2011 thatisaprostheticarmwith2DOFattheshoulder,ahumeralrotator,elbowandthreeDOFatwrist.Thisdevicecanbeusedfordifferentlevelsofamputation.In Resniketal.,2014 theDekaarmispresented,whichisaprostheticarmwith6DOFatthearmand4athandandaweightof4.45kg.Heetal. 2014 presentedaprostheticarmwith7DOFthatusesunderactuatedmechanismsandweighs4.45kg.Mostofthesesolutionsarebasedonserialkinematicchainconfigurations.Inthiswork,aparallelconfigurationisaddressedforanoveldesign.

Mainadvantagesofparallelmanipulatorsarethatthepayloadissharedbetweentheactuators,theactuatorsarelocatedinthebasesothattheinertiaisreducedandparallelmanipulatorsexhibitstiffbehavior.DisadvantagesoftheparallelmanipulatorsarethesmallworkspaceandlimiteddexterityascomparedtotheserialmanipulatorsCeccarelli,2004 .

For thedesignofaprostheticarm,designcriteriacanbeconsidered inworkspace,stiffness,anddexterityamongothersthatareusedalsoindesignproceduresofmanipulators.

Manyauthorshaveinvestigatedtheoptimizationofmechanismsusingdifferenttechniques.In BoudreauandTurkkan,1996 theforwardkinematicsofthreedifferentplanarparallelmanipulatorsaresolvedusingageneticalgorithm GA .Theoptimizationoftwodifferentparallelplanarmanipulatorsisachievedaswellusingagenetic

José‐AlfredoLeal‐Naranjoa,b*MarcoCeccarellibChristopher‐RenéTorres‐San‐MiguelaLuis‐AntonioAguilar‐PerezaGuillermoUrriolagoitia‐SosaaGuillermoUrriolagoitia‐Calderóna

aInstitutoPolitécnicoNacional,Seccióndeestu‐diosdeposgradodelaEscuelaSuperiordeInge‐nieríaMecánicayEléctricaUnidadZacatenco,MexicoCity,Mexico.E‐mails:lealna‐[email protected],[email protected],lagui‐[email protected],[email protected],ur‐[email protected]:LaboratoriodiRoboticaeMeccatronica,DiCEM,UniversitádiCassinoedelLazioMeridio‐nale,Cassino,Italy.E‐mails:[email protected]

*Correspondingauthor

http://dx.doi.org/10.1590/1679-78254044

Received:May19,2017InRevisedForm:November26,2017Accepted:November26,2017Availableonline:February05,2018

José‐AlfredoLeal‐Naranjoetal.Multi‐objectiveoptimizationofaparallelmanipulatorforthedesignofaprostheticarmusinggeneticalgorithms

LatinAmericanJournalofSolidsandStructures,2018,15 3 ,e26 2/15

algorithmin BoudrearandGosselin,1999 wheretheobjectivefunctionisformulatedtogetaworkspaceascloseaspossibletoapredefinedone.InCabreraetal. 2002 thepathsynthesisofafour‐barmechanismisworkedoutusinggeneticalgorithms.CeccarelliandLanni 2004 workedouttheoptimizationoftheworkspaceandthedi‐mensionsofgeneralthreerevolutejointsmanipulatorusingsequentialquadraticprogramming.In Hernandezetal.,2015 theoptimizationofacable‐basedparallelmanipulatorisworkedoutusingevolutionaryalgorithms.Theworkspaceofasphericalparallelmechanismforlaparoscopicsurgeryisoptimizedusingageneticalgorithmin LiandPayandeh,2002 .In Castejónetal.,2010 amulti‐objectiveoptimizationwasperformedinordertodesignaroboticarmforservicetasks. InZhenetal. 2010 theoptimizationof thestiffnessanddexterityofasixDOFsparallelmanipulatorispresentedusinggeneticalgorithmsandneuralnetwork.Chakeretal. 2012 performedthesynthesisofasphericalmanipulatorforsurgeriesbyoptimizingtheworkspaceanddexterityofthemechanismusingGA.In Essombaetal.,2016 itispresentedthesynthesisofasphericalmanipulatorusedasaprobeholderconsidering amulti‐objective optimization and GA. The synthesis of a 3‐RRR spherical parallelmanipulator isworkedoutin Wu,2012 usingamulti‐objectiveoptimizationconsideringasobjectivefunctionthedynamicdex‐terityandtheisotropy.InRamanaetal. 2016 itisworkedouttheoptimizationofa3DOFparallelmanipulatorusingasobjectivesthedexterity,workspaceandstiffness.

Anarisingprobleminthedesignofaparallelmechanismcomesfromthefactthattheirperformanceishighlydependentontheirgeometricparametersandconfigurationsaswellasthedifferentperformancemeasures work‐space,dexterity,etc aremutuallydependent Wu,2012 .Inthiswork,thesynthesisofaparallelsphericalmanip‐ulatoriscarriedoutusingamulti‐objectiveoptimizationcombinedwithGA.Theproposedsphericalmanipulatorisusedasamechanismtoreplicatetheshouldermovementinaprosthetichumanarm.Theshoulderofthepros‐theticarmisanalyzedasoneofthemostcomplexpartoftheprostheticdevicesinceitrequiresagreatmobilityandvolumerestrictions.Threeobjectivefunctionsaretakenintoaccountintheproposedoptimization.Forthefirstobjective,thespecifiedtrajectorymustbeinsidetheavailableworkspace;thesecondobjectiveisthemaximizationofthemechanismaccuracy measuredbythedexterity ;whilethethirdobjectiveistheminimizationofthetorquesmeasuredherebythemaximumvalues ofthethreeactuators.Thisisdonesinceareductioninthetorquecouldyieldtoareductionintheactuatorsandthusreductioninweightandpowerconsumption.Withtheobtainedre‐sults,aparallelmanipulatorisdesignedandasetofdynamicsimulationsareperformedtovalidatetheresultsoftheoptimizationandtocharacterizethedesignsolution.

2Geneticalgorithmsandmulti‐objectiveoptimization

Geneticalgorithms GA ,developedinHolland 1992 ,areheuristicmethodsthatconsistinoptimizationpro‐ceduresasinspiredinnaturalevolution.InGA,aninitialrandompopulationneedstobecreated.Thecharacteristicsofeachsolutionareusedinequivalentchromosomes.Eachsolutionisevaluatedandclassifiedaccordingtohowwellitsatisfiestheobjectivefunctionandthenitisassignedaprobabilityofreproduction.Thefittestindividualsaremorelikelytobereproduced selection ,andthustheyinheritthosecharacteristics.Thecombination crosso‐ver oftheparentgenesyieldstoaconsecutivegeneration replacement .Mutationcanoccurinthechromosomesofsomeindividuals.Itisprojectedthatsomeindividualsofthisnewgenerationwillhaveinheritedthebestchar‐acteristicsoftheirparentsandwillbeabettersolutiontotheproblem.Thisnewpopulationgoesthroughthesameprocessandthiscycleisrepeateduntilallthememberssharethesamegeneticinformation.Thislastgenerationisthebestsolutiontotheoptimizationproblem Fogel,1994 .Ageneticalgorithmisstoppedwhenthepopulationconvergestoanoptimalsolutionorwhenamaximumnumberofgenerationsisreached,Figure1.

Geneticalgorithmsrequireobjectiveandfitnessfunctions.Theobjectivefunctiondefinestheoptimalcondi‐tionandthefitnessfunctionassesseshowwellaspecificsolutionsatisfiestheobjectivefunctionandassignsarealvaluetothatsolution Coelloetal.,2007 .



Figure1:Aflowchartofageneticalgorithmoptimization.

Formerly,theinformationoftheindividualswasencodedinbitstringscalledBinary‐coded,butnowadaystheindividualsarecodedusingrealnumbers DebandKumar,1995 .Real‐codedGAusuallyachievesbetterresultsthanBinary‐codedGA Davis,1991 .

Theselectionprocessconsistsintakingtwoparentstocreateoffspring.Theobjectiveistoprovidetothefitterindividualsagreaterchanceofreproductionexpectingthattheiroffspringwillhavehigherfitness.Typicaltypesofselectionschemearetheproportionateselectionandtheordinal‐basedselection.Intheproportionatebasedselec‐tion,theindividualsarepickedupbasedontheirfitnessvaluesthatarerelativetothefitnessoftheotherindivid‐uals.Theordinal‐basedselectionselectsindividualsbyconsideringtheirrankwithinthepopulation SivanandamandDeepa,2007 .

Thecrossovercombinestwodifferentindividualstogeneratenewoffspring.Manycrossovermethodsexist,andthemostcommonissingle‐pointcrossover Blumetal.,2012 .Inthiswork,alinearcrossoverisused.Twoparentsareselectedasp1andp2andthreeoffspringsaregeneratedas0.5p1 0.5p2,1.5p1‐0.5p2,‐0.5p1 1.5p2respectively Herreraetal.,1998 .Then,thethreeoffspringareevaluatedandthebesttwoareselectedforthenextgeneration.

Mutationconsistsinarandommodificationofageneduringreproduction Cabreraetal.,2002 .Ingeneral,themutationpreventsconvergencetoalocaloptimum LiandPayandeh,2002 .Themutationoperatorisappliedwithasmallprobability Wangetal.,2004 .Inthiswork,anon‐uniformmutationisused Michalewicz,1996 .

Multi‐objectiveoptimizationcommonlyinvolvesmultipleconflictingobjectivesthatmustbeconsideredsim‐ultaneouslyandthereisasetofmathematicallyequallygoodsolutions Miettinen,2008 .ThesesetofsolutionsareknownasnondominatedorParetooptimalset.ThesetofsolutionsformstheParetooptimalfront.Thecharac‐teristicofaParetooptimalsolutionis thatanychange inthevaluesof thesolutionwillnot improveanyof theobjectivefunctions.

Theconceptofdominanceordominationisusedinmostoftheoptimizationalgorithms.IfthereareNobjectivefunctions,asolutionx 1 dominatesanothersolutionx 2 ifthetwofollowingconditionsaretrue Deb,2014 :1. The solution x(1) is no worse than x(2) in all objectives. 2. The solution x(1) is better than x(2) in at least one objective.

Asolutionx 1 isPareto‐optimalifisnondominatedwithrespecttothesearchspace orfeasibleregion Coello,2015 .

Manyevolutionaryalgorithmshavebeenproposedovertheyearstosolvemulti‐objectiveoptimizationprob‐lems.Inthiswork, it isusedacontrolledelitistnon‐dominatedsortingGA CENSGA DebandGoel,2001 .ThealgorithmofCENSGAisbasedinNSGA‐II Debetal.,2002 ,butthereisadifferenceintheselectionapproachthatimprovestheconvergencetotheParetofront.



TheCENSGAalgorithmworksasfollows.First,arandomPtpopulationofsizeNisformed.Fromthispopula‐tion,QoffspringsarecreatedusingthecommonGAoperators crossover,mutation .Then,acombinedRtpopula‐tion PtÈQt ofsize2Nis formed.ThepopulationRt issortedaccordingtonon‐dominationanddifferentnon‐dominatedfrontsarecreated.Thefirstnon‐dominatedfrontisformedonlybyelementsnon‐dominatedwithre‐specttoRt.Thesecondnon‐dominatedfrontisformedbyelementsdominatedbyjustonesolution.AlltheelementsofRtareassignedafrontinasimilarfashion Elarbietal.,2017 .ThecrowdingdistanceiscalculatedforeveryelementofRt Tanetal.,2006 .ThenNelementsareselectedfromRttoformthenextgenerationPt 1.Themaxi‐mumnumberofelementsselectedfromi‐thnon‐dominatedfrontisdefinedbyageometricdistribution

11

1i

i K

rn N r

r--

=-

1

whereristhereductionratio r 1 andKisthenumberofnon‐dominatedfronts.nidenotesthemaximumallowablenumberofindividualstakenfromthei‐thfront.Ifniislargerthanthenum‐

berofelementsinthei‐thfront,thenalltheelementsofthisfrontarechosenandtheremainingslotsareaddedtoni 1.Butifniissmallerthanthenumberofelementsinthefront,nitheelementsarechosenusingacrowdedbinarytournamentselection.Thecrowdedbinarytournamenttakestwoelementsandreturnstheonewiththebiggercrowdingdistance.GAoperatorsareappliedtothenewpopulationofPt 1toformQt 1andarecombinedtoformRt 1andthedescribedprocessisrepeatedforaspecificnumberofgenerations.TheresultofthisprocessistheParetofront,thismeansasetofoptimalsolutions.CENSGAalgorithmissummarizedinFigure2.

Figure2:AflowdiagramoftheCENSGAalgorithm.

3Kinematicdesignanditsanalysis

Theprostheticdeviceconsideredinthisworkisbasedinthedesignin Leal‐Naranjoetal.,2016 ,Figure3 a .Thisdevicehassevendegreesoffreedom,withthreeDOFsthatareintheshoulder,oneintheelbowandthreeinthewrist.Theshoulderofthisprototypeisdrivenbya3DOFparallelsphericalmanipulator,Figure3 b .Themech‐anismoftheelbowisa4barmechanism.Thewristmechanismisalsoaparallelsphericalmanipulator.Theweightofthisprototypeis1250gincludingthehand.Themainissueofprostheticarmsisthefunctionalityandweightofthedevice.Animprovementintheprosthetickinematicdesigncouldpotentiallyyieldinimprovementsinthefunc‐tionalityandalsointheweightduetolowerpowerrequirements.



Figure3:3Dprintedprototype:a aprosthetichumanarm;b shouldermechanism.

TheshoulderoftheprosthesisismodelledasathreeDOFssphericalmanipulator,Figure4 a .Themanipula‐torconsistsofthreelegswiththreerevolutejointseach,whoseaxesconvergeatonepointthat isthecenterofrotation.Theaxesoftheserevolutejointsaredefinedbytheunitvectorsui,viandwi.Thelinksai attachedtothe

lowerplatform arecharacterizedbytheangle 1_i ,whichrepresentstheanglebetweenthejointsofthelink.The

linksbi attachedtothemobileplatform arecharacterizedbytheangle 2_i ,whichrepresenttheanglebetween

thejointsofthelink,Figure4 b .

Figure4:Asphericalparallelmanipulatorfortheshoulderjoint:a aCADdesign;b aschemewithdesignparameters.

Themobileandthefixedplatformsarecommonlytriangularpyramidsdefinedbyi and

i ,withanequilat‐

eraltriangleasabase.Duetotheshapeofthebaseofthepyramids,theattachmentpoints vertexesofthebase arespacedevery120°butinthisworkinsteadofregularpyramids,theattachmentpointsinthefixedandmobileplatformsareconsideredasaparameterthatcantakeanyvalue.Thismeansthatthebaseisascalenetriangleandtheattachmentpointsaredefinedbytheangle

i ireferstothei‐thleg;i 1,2,3 whichisconsideredasanother

parameterofthemechanism,Figure5.Anglei istheanglebetweentheYaxisandtheaxisoftheattachment

point.



Figure5:Distributionoftheattachmentpointsatthefixedandmobileplatforms.

Forthisparallelmanipulator,theinversekinematicanalysisisprettystraightforwardincomparisontothedirectkinematic.Theinverseanalysisisasfollows.Vectoruiisdefinedas

( ) ( / 2)[0 1 0] , 1,2,3 Ti z i xu R R id g p= - = 2

whereRa b representsarotationof“b”degreesaroundthe“a”axis.Unitvectorviisgivenby

( ) ( ) ( )v 1_ 0 1 02

T

i z i x y i z iR R R Rp

d g q aæ ö÷ç é ù= - ÷ç ê ú÷ ë ûç ÷è ø

3

unitvectorwiisafunctionoftheorientationofthemobileplatform,thusthisvectorisdefinedas

w Q w 'i i= 4

wherewi'istheinitialorientationofthevectorswhenthemanipulatorisinitshomeconfigurationandisgivenby

( ) ( ) ( )w ' 0 sin cosi

T

i zR d b bé ù= ê úë ûandQistheorientationmatrixofthemobileplatform.

Forthesecondlinkthisrelationfollows

( )v w 2_cosi i ia=⋅ 5

substitutingwiandviinEquation5,andmakingalgebraicsimplificationsityieldsto

2 0AX BX C+ + = 6

Where

( ) ( ) ( ) ( )_ 1 3 4 5 7 2_cosy zi x i i iA w a a w a a w a a= - + + - + - 7

( ) ( ) ( )_ 2 6 82 2 2y zi x i iB w a w a w a= + + 8

( ) ( ) ( ) ( )_ 1 3 4 5 7 2_cosy zi x i i iC w a a w a a w a a= + + + + - 9

( )tan /2iX q= 10

( ) ( )1 1_sin cosi ia a d=- 11

( ) ( ) ( )2 1_sin sin cosi i ia a d g=- 12

( ) ( ) ( )3 1_cos sin sini i ia a d g=- 13



( ) ( ) ( )4 1_cos cos sini i ia a d g= 14

( ) ( )5 1_sin sini ia a d=- 15

( ) ( ) ( )6 1_sin cos cosi i ia a d g= 16

( ) ( )7 1_cos cosi ia a g=- 17

( ) ( )8 1_sin sini ia a g= 18

Equation6isaquadraticequationthatprovidestheangleoftheinputlinksofthesphericalmanipulator.Itcanbeseenfromequation6thatforeachleg,twosolutionsexist.Thus,foreverypositionintheworkspaceofthismanipulatorcanbedesignedforeightpossibleassemblysolutions,Figure6.

Figure6:Representationofthetwopossiblesolutionsforeachlegofthemechanism.

4Modellinganddesignoptimizationprocedure

Oneoftheaimsoftheoptimizationofthisparallelmanipulatoristomaximizetheworkspaceinawaythattheprostheticarmshouldbeable todescribeahumeralextensionmovementof20degreesandahumeral flexionmovementof90degrees,Figure7 a .

Figure7:Motionrequirements:a Flexion‐Extensionmovementofthehumerus;b referenceframes.

Tofulfillthisrequirement,itisnecessarythemanipulatorwilltobeabletoperformarotationfrom‐20°to90°aroundtheaxisX”,whichisperpendiculartothesagittalplaneofthebody,Figure7 b .TheaxisX”,wheretherotationoccurs,notnecessarilycoincideswiththehomepositionofthemanipulator.Forthisreason,itwasconsid‐eredthatthemanipulatorisrotatedanangleϕ1aroundtheZaxisfromitsreferenceposition,andthemanipulator



rotates doingtheflexion‐extension aroundtheX”axiswhichisrotatedfromX’anangleϕ2,sotheorientationoftheend‐effectorisgivenby

( ) ( )Q 1Z xR Rj e¢¢= 19

with

( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( ) ( )

( ) ( ) ( )''

1 cos cos 1 cos sin

1 cos 1 cos cos sin

sin sin cos

x x x y y

x x y y y x

y x

k k k k k

R k k k k k

k k

e e e ee e e e

e e e

é ù- + -ê úê ú= - - + -ê úê ú-ê úë û

20

wherekx ‐sin ϕ2 ,ky cos ϕ2 andεisavariablewithvaluesfrom‐20°to90°.Therequiredmovementdescribesanarcof110°.Thisarcwasdiscretizedinsegmentsof5°.Theobjective

functionwiththeworkspacecanbedefinedas

Q 1

1 . 6 ( ),

0 .6

if X inEq isrealg h h

if X inEq isnotreal

ìïï= å = íïïî 21

whereg1isevaluatedinthementionedarc.Thus,duetotheadopteddiscretization,themaximumvalueforthisfunctionis23.Forsimplicity,thisfunctionismultipliedby‐1anditistriedtobeminimized.

Thesecondobjectiveoftheoptimizationisrelatedtothekinematicaccuracyofthemanipulator.ThekinematicaccuracyisassociatedwiththeconditioningnumberoftheJacobianmatrix alsorelatedtodexterity .Thedexterityisdefinedas GosselinandAngeles,1987

K J J K 1 1 1 q x x qk k- -= - - £ < ¥ 22

with

( )( )( )

v w

J v w

v w

1 1

2 2

3 3

T

Tx

T

é ùê úê úê úê úê úê úê úë û

´

= ´

´

23

( )K v u w v u w v u w1 1 1 2 2 2 3 3 3q diag= ´ ´⋅ ⋅ ⋅´ 24

where||||denotestheEuclideannormofitsmatrixargumentand denotesthevectorproduct orcrossproduct .Thevalueofk 1correspondstoaconfigurationwithgoodkinematicaccuracy,anddoesnothaveanupper

limit.Thesecondobjectivefunctionisexpressedas

2g k= 25

Thethirdobjectiveisrelatedtothetorquethatisrequiredtoperformtheflexion‐extensionmovement.Duetothelowaccelerationsthatarerequiredduringthemovementandthelowmassoftheprostheticdevice,theinertialeffectscanbeignoredandastaticanalysiscanbeperformedforeachpositioninthedefinedtrajectorywithlimitedcomputationalcosts.

Usingthevirtualworkapproach,theend‐effectoroutputforces forcesandtorques isgivenby Tsai,1999 TF J 26

thus,themotortorquesiscalculatedas

1TJ F

27

whereJisthejacobiananditsgivenby

1q xJ K J 28



andFarethetorquesduetotheweightoftheprostheticdevice 15N ,thehand 5N andtheloadcarried 5N andconsideringitscorrespondingleverarm.

ByusingEquation 27 ,thetorquethatisrequiredfortheactuatorscanbeevaluatedasthethirdobjectivefunctionintheform

3 g M ax T orque= 29

whereMaxTorqueisthemaximumvalueofthetorqueofthemotorsintheprescribedtrajectorytobeminimized.Sincetheinversekinematicofthismanipulatorithaseightdifferentsolutions twodifferentsolutionsforeach

leg ,soitisnecessarytostablishanequationtolimitthemechanismtobeconstrainedduringtheentireoptimiza‐tionprocessforthesamebranch.Furthermore,fromFigure4itcanbeseenthatinordertochangefromonebranchtoanotheritcanonlybepossibleifthelinksofalegarealigned,thusthemechanismpassesthroughasingularposition.Toconstrainthemechanisminthesamebranch,thesignofthecosinebetweentheplaneformedbytheunitvectorsvianduiandtheunitvectorwineedstoremainconstant.Suchaconditioncanbeexpressedas

( )v u w( )i i isign ´ ⋅ 30

Inordertoassurethatthepositionofthemotorsisnotoverlapped,arestrictionbetween1 ,

2 and3 are

stablishedas:

1 1 2 40h d d= - ³ 31

2 2 3 40h d d= - ³ 32

3 1 3 40h d d= - ³ 33

Aminimumangleof40degreesbetweentheattachmentpointswasstablishedinordertobepossibletofittheactuators.

Thedesignparametersareϕ1,ϕ2,γ,β,α1_1,α1_2,α1_3,α2_1,α2_2,α2_3,δ1,δ2andδ3.Theoptimizationproblemisdefinedas

1 2 3 , , m in g g g 34

1 2 3 4 0 , 40 , 4 0 sub ject to h h h 35

CENSGAalgorithmwasappliedusingarandompopulationof100individualsandareductionratioof0.5.Themaximumnumberofiterationswas200.Forthegeneticoperators,itwasusedalinearcrossoverandanon‐uniformmutationwithamutationprobabilityof0.5%.

5Results

Afterfinishingtheoptimization,theParetofrontfortheeight‐differentassemblysolutionswasobtained.Thefunctiong2 Maximumtorqueofthemotors wasplottedvsg1 amplitudeofthedefinedtrajectory ,Series1 inFigure8.g2vsg3 dexterity ,Series2inFigure8.Series1and2areinthesameplotwithtwoverticalaxesinordertovisualizethethreeobjectivevaluesofapossiblesolution.Fortheg2vsg3plot,onlysolutionsthatcompletelysatisfythemotionwereplotted g1 24 inordertoreducethedataplottedandasthemaininterest isonlyinsolutionsthatwouldsatisfythestablishedtrajectory.Thescaleinalltheplotswasthesameinordertosimplifythecomparison.InFigure8athevaluesoftheobjectivefunctionsforthesolutionsofthefirstpossibleassemblyareplotted.Itcanbeseenthatforthisconfigurationonethemostsuitablesolutionhasanobjectivevalueofg1 24,g2 3.5andg3 1.4.Thereareothersolutionsthatrequirealowertorquebutthevalueofthedexterityincreasestog1 24,g2 3.3andg3 2.9;orsomeothersolutionswithabetterdexteritybutthetorqueincreasestog1 24,g2 3.9andg3 1.2.Figure8bshowsobjectivefunctionsforthesecondconfiguration.Inthisconfiguration,themostsuitablesolutionhasanobjectivevalueofg1 24,g2 5.9andg3 2.5,soclearlyisnotabetteroneascomparedwiththatforconfiguration1.



Figure8:Valuesofthethreeobjectivefunctionsforthedifferentsolutions:a Assemblysolution1;b Assemblysolu‐tion2;c Assemblysolution3;d Assemblysolution4;e Assemblysolution5;f Assemblysolution6;g Assemblyso‐

lution7;h Assemblysolution8.



Forconfiguration3 Figure8c ,thesolutionwiththesmallesttorquehasvaluesofg1 24,g2 3.2andg3 4.9andasthedexteritydecreases,thetorqueincreasessoitisnotbetterasthesolutionofconfiguration1.InFigure8ditcanbeseenthatthemostsuitablesolutionhasthevaluesoftheobjectivefunctiong1 24,g2 3.4andg3 1.4.Thissolutionisslightlybetterthanconfiguration1.Forconfiguration5and6and8 Figure8e,Figure8fandFigure8h ,thereisnosolutionintheusedscale,soitcanbeconcludedthatthereisnotabettersolutionthantheoneproposedintheotherconfiguration.InFigure8gitcanbeseenasetofsuitablesolutionsliketheonewithvaluesforobjectivefunctiong1 24,g2 3.18andg3 2.5.andg1 24,g2 3.5andg3 1.2.

Aftercomparingthesolutions,itwaschosentheonewithobjectivefunctiong1 24,g2 3.18andg3 2.5de‐spiteithasaslightlyworstdexteritythanthesolutionofconfiguration1becauseitistheonewiththelowesttorque,andalowertorquerepresentssmalleractuatorsandloweroverallweight.ThevalueoftheparametersassociatedtothissolutionareshowninTable1.

Table1:Parametersusedforthesynthesisofthemechanism.

Parameter Value ° Parameter Value ° ϕ1 48 α2_1 76ϕ2 224 α2_2 52Γ 77 α2_3 53Β 76 δ1 0α1_1 39 δ2 76α1_2 39 δ3 200α1_3 47

Withthecalculatedvalues,aCADdesignof themanipulatorwasdone.Twodynamicsimulationwereper‐

formedinordertoevaluatetheperformanceofthemanipulator.Thefirstsimulationaimedtoperformaflexion‐extensionmovementwitha loadof5Natthe locationofthehand,Figure9.Themovementbeginswith20°ofextensionandfinishesat90°offlexion.Thetimerequiredtoperformthemovementwasstablishedtobe2s.ThematerialoftheprostheticdevicewasABSexceptforthelinksoftheparallelmanipulatorthatweremodelasalu‐minum.

Figure9:Representationofthemovementperformedduringthedynamicsimulation.

Theresultsofthesimulationshowthatduringthemovement,theactuatorsperformsmoothmovements.Themanipulatordoesnotapproachanysingularity,andthisisbecausetheoptimizationofthedexterity.Themaximumtorquethattheactuatorsrequireis2.98Nm,Figure10.



Figure10:Resultsofthehumeralflexion‐extensionsimulation:a angulardisplacementoftheactuators;b torqueof

theactuators.

Thesecondsimulationconsistedinthesamemovementasinthepreviousanalysisbutwiththeelbowinaflexedposition,Figure11.

Figure11:Representationofthemovementperformedduringtheseconddynamicsimulation.

Theresultsshowthatthemaximumrequiredtorquetoperformthismovementis2.9Nm,Figure12.

Figure12:Resultsofthehumeralflexion‐extensionsimulation:a angulardisplacementoftheactuators;b torqueof

theactuators.

Athirdsimulationwasperformedinordertocomparewiththepreviousdesign.Thecharacteristicsofthesimulationarethesameassimulationonebutthepayloadwasincreasedto10Nandshowedthatwhenthepayloadisincreasedto1kg,themaximumrequiredtorqueis4Nm.

6Conclusions

Inthiswork,amulti‐objectiveoptimizationisusedtodesignofasphericalparallelmanipulatorastheshoulderofaprostheticdevice.Theresultsshowthattherequiredtorquetoperformthedefinedmovementwasreducedin



comparisonwiththepreviousdesign 4Nmvs6Nm .Incomparisontothepreviousconfigurationofthemecha‐nismused,therangeofmotionwasincreasedandanimprovementinthecontrollabilityofthemechanismwasachievedintermsofthedexterity.Fromtheresults,itisevidentthattheoptimizationofthethreeobjectivesneedstobecarriedout inasimultaneouswaybecausetheyaredependenttoeachotherandan improvement inonecharacteristiccouldyieldtoanundesiredrepercussionoftheothers.

Acknowledge

ThefirstauthoracknowledgesConsejoNacionaldeCienciayTecnología CONACyT forsupportinghisPhDstudyandresearchwithaperiodofstudyin2016atLARMinCassinowithinadoublePhDdegreeprogram.

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multi‐objective optimization of a parallel manipulator for the ......prosthetic arm, biomechanics,...

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