Modeling,IdenticationandControl,Vol. 33,No. 3,2012,pp. 111122,ISSN18901328MultiobjectiveOptimumDesignofa3-RRRSpherical Parallel ManipulatorwithKinematicandDynamicDexteritiesGuangleiWu11DepartmentofMechanical andManufacturingEngineering,AalborgUniversity,9220Aalborg,Denmark.E-mail: gwu@m-tech.aau.dkAbstractThispaperdealswiththekinematicsynthesisproblemofa3-RRRsphericalparallelmanipulator,basedon the evaluation criteria of the kinematic,kinetostatic and dynamic performances of the manipulator. Amultiobjectiveoptimizationproblemisformulatedtooptimizethestructural andgeometricparametersof thespherical parallel manipulator. Theproposedapproachisillustratedwiththeoptimumdesignofaspecial spherical parallel manipulatorwithunlimitedrollingmotion. Thecorrespondingoptimizationproblemaimstomaximizethekinematicanddynamicdexteritiesoveritsregularshapedworkspace.Keywords: Sphericalparallelmanipulator, multiobjectiveoptimization, Cartesianstinessmatrix, dex-terity,GeneralizedInertiaEllipsoid1 IntroductionAthreeDegreesofFreedom(3-DOF)spherical paral-lel manipulator (SPM) is generallycomposedof twopyramid-shapedplatforms, namely, amobileplatform(MP)andaxedbasethatareconnectedtogetherbythreeidentical legs, eachoneconsistingoftwocurvedlinksandthreerevolutejoints. Theaxesof all jointsintersectatacommonpoint,namely,thecenterofro-tation. Suchasphericalparallelmanipulatorprovidesathree degrees of freedomrotational motion. Mostof the SPMs ndtheir applications as orientingde-vices, such as camera orienting and medical instrumentalignment(GosselinandHamel, 1994; Li andPayan-deh, 2002; CavalloandMichelini, 2004; Chakeretal.,2012). Besides, theycanalsobeusedtodevelopac-tive spherical manipulators, i.e., wrist joint (Asada andGranito,1985).Indesigning parallel manipulators, a fundamentalproblemisthattheirperformanceheavilydependsontheirgeometry(HayandSnyman, 2004)andthemu-tual dependencyof the performance measures. Themanipulator performance depends onits dimensionswhile the mutual dependency among the performancesis related to manipulator applications (Merlet, 2006b).Theevaluationcriteriafordesignoptimizationcanbeclassied into two groups: one relates to the kinematicperformance of the manipulator while the other relatestothe kinetostatic/dynamic performance of the ma-nipulator (Caroet al., 2011). Inthekinematiccon-siderations,acommonconcernistheworkspace(Mer-let, 2006a; KongandGosselin, 2004; Liuetal., 2000;Bonev andGosselin, 2006). The size andshape oftheworkspaceareofprimaryimportance. Workspacebaseddesignoptimizationcanusuallybesolvedwithtwodierent formulations, therst formulationaim-ingtodesignamanipulatorwhoseworkspacecontainsaprescribedworkspace(HayandSnyman, 2004)andthe secondapproachbeing to designa manipulatorwhose workspace is as large as possible (Louet al.,2005). InRef.(Bai,2010),theSPMdexteritywasop-timizedwithinaprescribedworkspacebyidentifyingdoi:10.4173/2012.3.3 c2012NorwegianSocietyofAutomaticControlModeling,IdenticationandControl(a) (b)Figure1:3-RRRunlimited-rollSPM:(a)CADmodel,(b)applicationassphericallyactuatedjoint.thedesignspace. Itisknownfrom(GosselinandAn-geles, 1989)thattheorientationworkspaceof aSPMisamaximumwhenthegeometricanglesofthelinksareequalto90o. However,maximizingtheworkspacemayleadtoapoordesignwithregardtothemanip-ulator dexterityandmanipulability(Stamper et al.,1997;DurandandReboulet,1997). Thisproblemcanbesolvedbyproperlydeningtheconstraintsondex-terity(Merlet, 2006a; Huang et al., 2003). For theoptimumdesignof SPMs, anumber of works focus-ingonthekinematicperformance, mainlythedexter-ityandworkspace, havebeenreported, whereas, thekinetostatic/dynamic aspects receive relatively less at-tention. Ingeneral,thedesignprocesssimultaneouslydealswiththetwopreviouslymentionedgroups,bothof whichinclude anumber of performance measuresthat essentially vary throughout the workspace. On thekinetostaticaspect,theSPMstinessisanimportantconsideration (Liu et al., 2000) to characterize its elas-tostaticperformance. Whentheyareusedtodevelopsphericallyactuatedjoint, not onlythe MPangulardisplacement but also the translational displacement ofthe rotation center should be evaluated from the Carte-sianstinessmatrixofthemanipulatorandshouldbeminimized. Moreover, the dynamic performance of themanipulatorshouldbeashighaspossible.Amongtheevaluationcriteriaforoptimumgeomet-ricparametersdesign,anecientapproachistosolvea multiobjective optimization problem, which takes allormostoftheevaluationcriteriaintoaccount. Astheobjective functions are usually conicting, no single so-lutioncanbeachievedinthisprocess. Thesolutionsof suchaproblemarenon-dominatedsolutions, alsocalledPareto-optimal solutions. Somemultiobjectiveoptimizationproblemsofparallelmanipulators(PMs)have been reported in the last few years. Hao and Mer-letproposedamethoddierentfromtheclassical ap-proaches to obtain all the possible design solutions thatsatisfyasetofcompulsorydesignrequirements,wherethe designspace is identiedviathe interval analy-sisbasedapproach(HaoandMerlet,2005). Ceccarellietal. focusedontheworkspace, singularityandsti-ness properties to formulate a multi-criterion optimumdesignprocedureforbothparallel andserial manipu-lators(Ceccarelli etal., 2005). StockandMillerfor-mulatedaweightedsummulti-criterionoptimizationproblem with manipulability and workspace as two ob-jectivefunctions(StockandMiller, 2003). KretandHesselbach formulated a multi-criterion elastodynamicoptimization problemfor parallel mechanisms whileconsideringworkspace, velocitytransmission, inertia,stiness and the rst natural frequency as optimizationobjectives(KretandHesselbach,2005). Altuzarraetal. dealtwiththemultiobjectiveoptimumdesignofaparallelSchoniesmotiongenerator,inwhichthema-nipulator workspace volume and dexterity were consid-eredasobjectivefunctions(Altuzarraetal.,2009).Inthis work, amultiobjective designoptimizationproblemisformulated. Thedesignoptimizationprob-lemof the3-DOFspherical parallel manipulatorcon-siders the kinematic performance, the accuracy and thedynamic dexterity of the mechanism under design. Theperformances of the mechanism are also optimized overaregular shapedworkspace. Themultiobjectivede-112G.Wu,Multiobjectiveoptimizationofsphericalparallelmanipulator(a) (b)Figure2:ArchitectureofageneralSPM:(a)overview,(b)parameterizationoftheithleg.signoptimizationproblemisillustratedwitha3-RRRSPMshowninFigure1, whichcanreplacetheserialchains basedwrist mechanisms. Thenon-dominatedsolutions, alsocalledPareto-optimal solutions, of themultiobjective optimization problem are obtained withageneticalgorithm.2 ManipulatorArchitectureThe spherical parallel manipulator under studyis anovel robotic wrist with an unlimited roll motion (Bai,2010; Bai et al., 2009), whichonlyconsists of threecurved links connected to a mobile platform (MP). Themobile platform is supposed to be quite stier than thelinks, whichisconsideredasarigidbody. Thethreelinks are drivenbythree actuators movingindepen-dentlyonacircularrailofmodelHCR150fromTHKviapinionandgear-ringtransmissions. Thankstothecircular guide, the overall stiness of the mechanism isincreased. Moreover, suchadesignenablestheSPMto generate an unlimited rolling motion,in addition tolimitedpitchandyawrotations.AgeneralsphericalparallelmanipulatorisshowninFigure2(a)(Liuetal., 2000). Figure2(b)representsthe parameters associated with the ith leg of the SPM,i=1,2,3. TheSPMiscomposedof threelegsthatconnectthemobile-platformtothebase. Eachlegiscomposed of three revolute joints. The axes of the revo-lutejointsintersectandtheirunitvectorsaredenotedbyui, wiandvi, i =1,2,3. Thearcanglesof thethreeproximalcurvedlinksarethesameandequalto1. Likewise,thearcanglesofthethreedistalcurvedlinksarethesameandequal to2. Theradii of thelink midcurves are the same and equal to R. Geometricangles and dene the geometry of the two pyrami-dalbaseandmobileplatforms. ThepresentedSPMinFigure1(a)isaspecialcasewith= 0. TheoriginOof the reference coordinate system Fais located at thecenterofrotation.3 KinematicandKinetostaticModelingoftheSPMThe kinematics of the SPMs has been well docu-mented(GosselinandAngeles,1989),whichisnotre-peatedindetailhere. Hereafter,theorientationofthemobileplatformisdescribedbytheorientationrepre-sentationof azimuth-tilt-torsion( ) (Bonev,2008),forwhichtherotationmatrixisexpressedasQ =__ccc( ) + ss( ) ccs( ) sc( ) csscc( ) cs( ) scs( ) + cc( ) sssc( ) ss( ) c__(1)113Modeling,IdenticationandControlwhere (,], [0,), (,],ands() =sin(),c() = cos().Under the prescribed coordinate system, unit vectoruiisexpressedinthebaseframe Fabelow:ui=_sin i sin cos i sin cos T(2)wherei= 2(i 1)/3,i = 1,2,3.Unit vector wi of the intermediate revolute joint axisintheithlegisexpressedin Faas:wi=__sisc1 + (cisisicci)s1cisc1 + (sisi + cicci)s1cc1 + scis1__(3)Theunitvectorviof thelastrevolutejointaxisintheithleg,isafunctionofthemobile-platformorien-tation,namely,vi= Qvi(4)where vicorresponds to the unit vector of the last rev-olute joint axis in the ith leg when the mobile platformisinitshomeconguration:vi=_sin i sin cos i sin cos T(5)3.1 KinematicJacobianmatrixLet denote the angular velocity of the mobile-platform, thescrewsvelocityequationviatheithlegcanbestatedas$=_0_ =i$iA +i$iB +i$iC(6)with the screws for the revolute joints at points Ai, BiandCiexpressedas$iA=_ui0_,$iB=_wi0_,$iC=_vi0_Since the axes of the two passive revolute joints in eachleg lie in the plane BiOCi, the following screw is recip-rocaltoalltherevolutejointscrewsoftheithleganddoesnotlieinitsconstraintwrenchsystem:$ir=_0wivi_(7)Applyingtheorthogonal product () (Tsai, 1998) tobothsidesofEqn.(6)yields$ir $= (wivi)T= (uiwi) vi i(8)As aconsequence, the expressionmappingfromthemobile platform twist to the input angular velocities isstatedas:A= B (9)withA =_a1a2a3,ai= wivi(10a)B = diag_b1b2b3,bi= (uiwi) vi(10b)where=_123T. MatricesAandBaretheforwardandinverseJacobianmatricesofthemanipu-lator, respectively. If Bisnonsingular, thekinematicJacobianmatrixJisobtainedasJ = B1A (11)3.2 CartesianstinessmatrixThestinessmodel oftheSPMunderstudyisestab-lishedwithvirtualspringapproach(Pashkevichetal.,2009), byconsideringtheactuationstiness, linkde-formationandtheinuenceofthepassivejoints. TheexiblemodeloftheithlegisrepresentedinFigure3.Figure3(b) illustrates thelinkdeections andvaria-tionsinpassiverevolutejointangles.Letthecenterof rotationbethereferencepointofthe mobile platform. AnalogtoEqn. (6), the smalldisplacementscrewofthemobile-platformcanbeex-pressedas:$iO=_p_ = i$iA + i$iB + i$iC(12)wherep=[x,y,z]Tislineardisplacementofthe rotationcenter and=[x,y,z]Tisthe MP orientation error. Note that this equation onlyincludesthejointvariations,whilefortherealmanip-ulator,linkdeectionsshouldbeconsideredaswell.Thescrewsassociatedwiththelinkdeectionsareformulatedasfollows:$iu1=_ririC ri_,$iu2=$iC,$iu3=_niriC ni_(13)$iu4=_0ri_,$iu5=_0vi_,$iu6=_0ni_where ni=wi viis the normal vectors of planeBiOCi,ri= wi ni,andriCisthepositionvectorofpointCifromO. Thedirectionsofthevectorsriandniareidenticaltoui4andui6,respectively.Byconsideringthe linkdeections ui1...ui6andvariations inpassive joint angles andaddingall thedeectionfreedomstoEqn. (12), themobileplatformdeectionintheithlegisstatedas$iO=i$iA + i$iB + i$iC+ ui1$iu1 + ui2$iu2+ ui3$iu3 + ui4$iu4 + ui5$iu5 + ui6$iu6(14)Thepreviousequationcanbewritteninacompactformbyseparatingthetermsrelatedtothevariations114G.Wu,Multiobjectiveoptimizationofsphericalparallelmanipulator(a)(b)Figure3:Flexible model of a single leg: (a) virtualspring model, where Acstands for the actua-tor, R for revolute joints and MP for the mo-bileplatform, (b) linkdeections andjointvariationsintheithleg.inthepassiverevolutejointanglesandthoserelatedtotheactuatorandlinkdeections,namely,$iO= Jiui +Jiqqi(15)withJi=_$iA$iu1$iu2$iu3$iu4$iu5$iu6_(16a)Jiq=_$iB$iC_(16b)ui=_iui1ui2ui3ui4ui5ui6T(16c)qi=_iiT(16d)Let the external wrench applied to the end of the ithleg be denoted by fi, the constitutive law of the ith legcanbeexpressedasfi=_KrrKrtKTrtKtt_i_p_ fi= Ki$iO(17)Ontheotherhand, thewrenchappliedtothearticu-lated joints in the ith leg being denoted by a vector i,theequilibriumconditionforthesystemiswrittenas,JiTfi= i, JiTqfi= 0, ui= Ki1i(18)CombiningEqns.(15), (17)and(18), thekinetostaticmodel of the ith leg can be reduced to a system of twomatrixequations,namely,_ SiJiqJiqT022_ _fiqi_ =_$iO021_(19)wherethesub-matrixSi=JiKi1JiTdescribes thespringcompliance relative tothe center of rotation,andthesub-matrixJiqtakesintoaccountthepassivejointinuenceonthemobileplatformmotions.Ki1isa7 7matrix,describingthecomplianceofthevirtualspringsandtakingtheform:Ki1=_Ki1act016061Ki1L_(20)where Kiact corresponds to the stiness of the ith actua-tor. KiL of size 66 is the stiness matrix of the curvedlinkintheithleg,whichiscalculatedbymeansoftheEuler-Bernoullistinessmodelofacantilever. InFig-ure3(b), u1, u2andu3showthethreemomentdirections while u4, u5andu6showthe threeforce directions, thus, using Castiglianos theorem (Hi-bbeler, 1997), the compliance matrix of the curved linktakestheform:Ki1L=__C11C120 0 0 C16C12C220 0 0 C260 0 C33C34C3500 0 C34C44C4500 0 C35C45C550C16C260 0 0 C66__(21)where the corresponding elements are given in Ap-pendixA.ThematrixJiof size6 7istheJacobianmatrixrelatedtothevirtualspringsandJiqof6 2,theonerelatedtorevolutejoints intheithleg. TheCarte-sianstinessmatrixKioftheithlegisobtainedfromEqn.(19),fi= Ki$iO(22)where Kiis a 66 sub-matrix, which is extracted fromtheinverseofthe8 8matrixontheleft-handsideofEqn.(19). Fromf=

3i=1fi,$O= $iOandf= K$O,the Cartesian stiness matrix K of the system is foundbysimpleaddition,namely,K =3

i=1Ki(23)115Modeling,IdenticationandControl3.3 MassmatrixThemassinmotionof themechanisminuencesthedynamic performance, such as inertia, acceleration,etc.,hence,formulatingthemassmatrixisoneimpor-tantprocedureinthedynamicanalysis. Massmatrixis the function of manipulator dimensions and materialproperties,i.e.,linklengths,cross-sectionalarea,massdensity. Generally, the manipulator mass matrix (iner-tiamatrix)canbeobtainedonthebasisofitskineticenergy. The total kinetic energy Tincludes the energyTpof themobileplatform, Tlof thecurvedlinksandTsoftheslideunits: ThekineticenergyofthemobileplatformisTp=12mpvTpvp +12TIp (24)withvp= Rcos p ,Ip= diag [IxxIyyIzz] (25)wherempisthemassofthemobile-platformandIxx, Iyy, Izzare the mass moments of inertia of themobile-platform about x-,y-,z-axes,respectively. ThekineticenergyofthecurvedlinksisTl=123

i=1_mlviTlvil+Il2i_(26)withvil=12R_iwiui +vi_(27a)Il=12mlR2_1 sin 2 cos 22_(27b)i= (uivi) (uiwi) vi= ji (27c)wheremlisthelinkmassandIlisitsmassmo-mentofinertiaaboutwi. ThekineticenergyoftheslideunitsisTs=12_Ign2g +msR2s_T (28)wheremsisthemassof theslideunitandRsisthedistancefromitsmasscentertoz-axis. Igisthemassmomentof inertiaof thepinionandngisthegearratio.Consequently, theSPMkineticenergycanbewritteninthefollowingformT= Tp +Tl +Ts=12TM (29)Figure4:TherepresentationoftheregularworkspacefortheSPMwithapointingcone.withthemassmatrixMofthesystemisexpressedasM =_msR2s +Ign2g +14mlR2sin21_13+JT_Ip +mpR2cos2[p]T[p]+14mlR23

i=1[vi]T[vi] +Il3

i=1jijTi_J (30)where [] stands for the skew-symmetric matrix whoseelementsarefromthecorrespondingvectorand13istheIdentitymatrix.4 DesignOptimizationoftheSpherical Parallel ManipulatorTheinversekinematicproblemof theSPMcanhaveuptoeight solutions, i.e., the SPMcanhave uptoeightworkingmodes. Here, thediagonal termsbioftheinverseJacobianmatrixBaresupposedtobeallnegative for the SPM to stay in a given working mode.Intheoptimizationprocedure,criteriainvolvingkine-matic and kinetostatic/dynamic performances are con-sidered to determine the mechanism conguration andthedimensionandmasspropertiesofthelinks. More-over, the performances are evaluatedover a regularshapedworkspace free of singularity, whichis speci-edasaminimumpointingconeof 90oopeningwith116G.Wu,MultiobjectiveoptimizationofsphericalparallelmanipulatorFigure5:Designvariablesofthe3-RRRSPM.360ofullrotation,i.e., 45oand (180o,180o],seeFigure4.4.1 DesignvariablesVariables 1, 2, and arepart of thegeometricparametersof a3-RRRSPMand=0forthema-nipulatorunderstudy. Moreover, theradiusRofthelinkmidcurveisanotherdesignvariableandthecrosssectionofthelinksissupposedtobeasquareofsidelengtha. ThesevariablesareshowninFigure5. Asaconsequence,the design variable vector is expressed asfollows:x = [1,2,,a,R] (31)4.2 ObjectivefunctionsThekinematicperformanceis oneof themajor con-cernsinthemanipulatordesign, of whichacriterionis the evaluationof the dexterityof SPMs. Acom-monlyusedcriteriontoevaluate this kinematic per-formanceistheglobalconditioningindex(GCI)(Gos-selinandAngeles,1991),whichdescribestheisotropyof the kinematic performance. The GCI is dened overaworkspaceasGCI=_1(J)dW_ dW(32)where(J)istheconditionnumberof thekinematicJacobian matrix (11). In practice, the GCI of a roboticmanipulatoriscalculatedthroughadiscreteapproachasGCI=1nn

i=11i(J)(33)where n is the number of the discrete workspace points.As a result, the rst objective function of the optimiza-tionproblemiswrittenas:f1(x) = GCI max (34)Referringtothekinematicdexterity, animportantcriteriontoevaluatethedynamicperformanceisdy-namic dexterity, whichis made onthe basis of theconcept of Generalized Inertia Ellipsoid (GIE) (Asada,1983). Inordertoenhancethedynamicperformanceand to make acceleration isotropic, the mass ma-trix(30) shouldbeoptimizedtoobtainabetter dy-namicdexterity. SimilartoGCI,aglobaldynamicin-dex(GDI)isusedtoevaluatethedynamicdexterity,namely,GDI=1nn

i=11i(M)(35)wherei(M)istheconditionnumberofthemassma-trixoftheithworkspacepoint. Thus, thesecondob-jectivefunctionoftheoptimizationproblemiswrittenas:f2(x) = GDI max (36)4.3 OptimizationconstraintsInthis section, thekinematicconstraints, condition-ingof thekinematicJacobianmatrixandaccuraciesduetotheelasticdeformationareconsidered. Con-straining the conditioning of the Jacobian matrix aimstoguaranteedexterous workspacefreeof singularity,whereaslimitsonaccuracyconsiderationensuresthatthemechanismissucientlysti.4.3.1 KinematicconstraintsAccording to the determination of design space re-portedin(Bai, 2010), the bounds of the parameter1, 2andsubjecttotheprescribedworkspacearestatedas:45o 90o, 45o 1,2 135o(37)Thesequenceoftherst,secondandthirdslideunitsappearing on the circular guide counterclockwise isconstant. Inorder to avoidcollision,the angles ijbe-tweentheprojectionsofvectorswiandwjinthexyquadrant, i,j =1,2,3, i =j, asshowninFigure6,havetheminimumvalue, say10o. Toavoidcollision117Modeling,IdenticationandControlFigure6:Slideunitcongurationofthe3-DOFSPM.andmakethemechanismcompact, thefollowingcon-straintsshouldbesatised:12,23,31 = 10o(38)R0= 0.120 m Rsin 1 Rs= 0.200 mMoreover, the SPM should not reach any singularityinitsorientationworkspace. Therefore, thefollowingconditionsshouldbesatised.det(A) , bi= (uiwi) vi (39)whereAistheforwardJacobianmatrixofthemanip-ulator denedinEqn. (9) and >0is apreviouslyspeciedtolerancesetto0.001.4.3.2 ConditioningnumberofthekinematicJacobianmatrixMaximizing the GCI and constraining the kine-maticJacobianmatrixcannotpreventtheprescribedworkspace away fromill-conditioned congurations.Forthedesignoptimizationinordertoachieveadex-terousworkspace, theminimumof theinversecondi-tion number of the kinematic Jacobian matrix 1(J),basedon2-norm, shouldbehigherthanaprescribedvaluethroughouttheworkspace,say0.1,namely,min(1(J)) 0.1 (40)4.3.3 AccuracyconstraintsThe accuracy constraints of the optimization prob-lemfor the SPMare related to the dimensions ofTable1:The lower andupper bounds of the designvariablesx.1 [deg] 2 [deg] [deg] a [m] R[m]xlb45 45 45 0.005 0.120xub135 135 90 0.030 0.300the curvedlinkandthe maximumpositional deec-tionof the rotationcenter andangular deectionofthe moving-platform subject to a given wrench appliedonthe latter. The control loopstiness is Kiact=106Nm/rad. Letthestaticwrenchcapabilitybespec-iedas the eight possible combinations of momentsm=[10, 10, 10] Nm, whiletheallowablemaxi-mum positional and rotational errors for the workspacepoints are 1 mmand 2o=0.0349 rad, respectively,thus,theaccuracyconstraintscanbewrittenas:pn=_x2n + y2n + z2n p(41)n=_2x, n + 2y, n + 2z, n rwherethelinearandangulardisplacementsarecom-putedfrom$O=K1f withtheCartesianstinessmatrix(23)andp= 1 mm,r= 0.0349 rad.4.4 FormulationofthemultiobjectiveoptimizationproblemMathematically, themulti-objectivedesignoptimiza-tion problem for the spherical parallel manipulator canbeformulatedas:maximize f1(x) = GCI (42)maximize f2(x) = GDIover x = [1,2,,a,R]subject to g1: 45og2: R0 Rsin 1 Rsg3: 12,23,31 = 10og4: det(A) , (uiwi) vi g5: min(1(J)) 0.1g6:_x2n + y2n + z2n pg7:_2x, n + 2y, n + 2z, n rxlb x xubi = 1,2,3wherexlbandxub,respectively,arethelowerandup-perboundsofthevariablesxgivenbyTable1.118G.Wu,MultiobjectiveoptimizationofsphericalparallelmanipulatorTable2:AlgorithmparametersoftheimplementedNSGA-IIPopulation Generation Directionalcrossover Crossover Mutation Distributionsize probability probability probability index40 200 0.5 0.9 0.1 20Table3:ThreePareto-optimalsolutionsDesign Variables ObjectivesID 1 [deg] 2 [deg] [deg] a [m] R[m] GCI GDII 56.2 81.0 89.8 0.0128 0.1445 0.366 0.711II 51.6 84.3 89.9 0.0133 0.1533 0.453 0.665III 47.2 90.8 89.2 0.0127 0.1641 0.536 0.6254.5 Pareto-optimal solutionsFor the proposed SPM, the actuation transmissionmechanismis a combination of actuator of modelRE35 GB and gearhead of model GP42 C fromMaxon(Maxon, 2012) andaset of gear ring-pinionwithrationg= 8. Moreover,thecomponentsaresup-posed to be made of steel, thus, E= 210 Gpa, = 0.3.Moreover, the moving platform is supposed to be a reg-ulartriangle, thus, theMPandlinkmassesaregivenbymp=334hR2sin2,ml= a2R2(43)whereis themass densityandh=0.006 mis thethicknessofthemovingplatform. Thetotal massmsofeachslideunit, includingthemassoftheactuator,gearhead, pinion and the manufactured components, isequaltoms= 2.1 kg.Thepreviousformulatedoptimizationproblem(42)is solved by the genetic algorithm NSGA-II (Deb et al.,2002)withMatlab,ofwhichthealgorithmparametersaregiveninTable2.The Pareto front of the formulated optimizationproblemfortheSPMisshowninFigure7andthreeoptimal solutions, i.e., two extreme and one intermedi-ate,arelistedinTable3.Figure8illustratesthevariationaltrendsaswellastheinter-dependencybetweentheobjectivefunctionsand design variables by means of a scatter matrix. Thelowertriangularpartofthematrixrepresentsthecor-relationcoecientswhereastheupperoneshowsthecorresponding scatter plots. The diagonal elementsrepresent theprobabilitydensitycharts of eachvari-able. Thecorrelationcoecientsvaryfrom 1to1.Twovariablesarestronglydependentwhentheircor-relation coecient is close to 1 or 1 and independentwhenthelatterisnull. Figure8shows: both objectives functions GCI and GDI are0.35 0.4 0.45 0.5 0.550.620.630.640.650.660.670.680.690.70.710.72ID IID IIID IIIKinematic dexterityDynamic dexterityFigure7:TheParetofrontofthemultiobjectiveopti-mizationproblemfortheSPM.stronglydependentastheircorrelationcoecientisequalto 0.975; both objectives functions GCI and GDI arestronglydependent onall designvariables as allof the corresponding correlation coecients aregreaterthan0.6; GCIisslightlymoredependentthanGDIofthedesignvariablesasall thecorrespondingcorrela-tioncoecientsof formeraregreaterthanthoseoflatter; GDI islessdependentonthedesignvariablesandathantheothervariablesalthoughthetwoformervariablesinuencetheSPMmass, thisisduetothelargeportionoftheslideunitmassinthetotalmechanismmass.119Modeling,IdenticationandControlFigure8:Scattermatrixfortheobjectivefunctionsandthedesignvariables.0.35 0.4 0.45 0.5 0.55406080100Variables [deg]Kinematic dexterityGCI 120.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72406080100Variables [deg]Dynamic dexterityGDI 120.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.720.10.120.140.160.18Variables [m]Dynamic dexterityGDI 10*aRFigure9:Design variables as functions of objectives forthePareto-optimalsolutions.Figure9displays thedesignvariables as functionsof the objectives. It is noteworthy that the higherGCI, thelower 1, conversely, thehigher GDI, thehigher1. Thisphenomenonisoppositewithrespecttovariable2. Thedesignvariableconvergesto90oapproximately, which indicates that = 90ois the pre-ferredgeometricparameterfortheSPMunderstudy.The lower link midcurve R and higher a lead to higherGDI. The three sets of of design variables correspond-ingtothethreePareto-optimal solutionsdepictedinTable3areshowninFigure9withsolidmarkers.5 ConclusionsIn this paper, the geometric synthesis of spherical par-allel manipulators is discussed. Amultiobjectivede-signoptimizationproblembasedonthegeneticalgo-rithmwasformulatedinordertodeterminethemech-anismoptimumstructural andgeometricparameters.The objective functions were dened on the basis of thecriteria of bothkinematic andkinetostatic/dynamicperformances. This approachis illustratedwiththeoptimumdesignofanunlimited-rollspherical parallelmanipulator,aimingatmaximizingthekinematicanddynamicdexterities toachieverelativelybetter kine-maticanddynamicperformancessimultaneously. Itisfound that the parameter being equal to 90ois a pre-ferredstructurefortheSPMunderstudy. Finally,thePareto-frontwasobtainedtoshowtheapproximationof theoptimal solutionsbetweenthevarious(antago-nistic)criteria, subjecttothedependencyof theper-formance. Thefutureworkwill aimtomaximizetheorientationworkspaceandoptimizethecross-sectiontypeofthecurvedlinks.ReferencesAltuzarra, O., Salgado, O., Hernandez, A., andAngeles, J. Multiobjective optimum design ofa symmetric parallel schonies-motion generator.120G.Wu,MultiobjectiveoptimizationofsphericalparallelmanipulatorASMEJ. Mechanical Design, 2009. 131(3):031002.doi:10.1115/1.3066659.Asada, H. Ageometrical representationof manipu-lator dynamics andits applicationtoarmdesign.ASME J. Dynamic Systems, Measurement and Con-trol,1983. 105(3):131142. doi:10.1115/1.3140644.Asada, H. and Granito, J. Kinematic and staticcharacterization of wrist joints and their opti-mal design. In IEEE International ConferenceonRoboticsandAutomation. pages244250, 1985.doi:10.1109/ROBOT.1985.1087324.Bai, S. Optimumdesign of spherical parallel ma-nipulator for a prescribed workspace. Mecha-nismand Machine Theory, 2010. 45(2):200211.doi:10.1016/j.mechmachtheory.2009.06.007.Bai, S., Hansen, M. R., and Andersen, T. O.Modellingof aspecial classof spherical parallel manipulatorswithEulerparameters. Robotica, 2009. 27(2):161170. doi:10.1017/S0263574708004402.Bonev, I. A. Direct kinematics of zero-torsionparallel mechanisms. In IEEE InternationalConference on Robotics and Automation.Pasadena, California, USA, pages 38513856,2008. doi:10.1109/ROBOT.2008.4543802.Bonev, I. A. and Gosselin, C. M. Analyti-cal determination of the workspace of symmetri-cal spherical parallel mechanisms. IEEE Trans-actions on Robotics, 2006. 22(5):10111017.doi:10.1109/TRO.2006.878983.Caro, S., Chablat, D., Ur-Rehman, R., and Wenger, P.Multiobjective design optimization of 3-PRR planarparallel manipulators. InGlobal Product Develop-ment,pages 373383. Springer-Verlag Berlin Heidel-berg,2011. doi:10.1007/978-3-642-15973-237.Cavallo,E.andMichelini,R.C. Aroboticequipmentfortheguidanceofavectoredthrustor. In35thIn-ternational SymposiumonRobotics. Paris, France,2004.Ceccarelli, M., Carbone, G., andOttaviano, E. Multicriteria optimum design of manipulators.In Bulletinof thePolishAcademyof Technical Sciences, 2005.53(1):918.Chaker, A., Mlika, A., Laribi, M. A., Romdhane,L., andZeghloul, S. Synthesis of spherical paral-lel manipulator for dexterous medical task. Fron-tiers of Mechanical Engineering, 2012.7(2):150162.doi:10.1007/s11465-012-0325-4.Deb, K., Pratap, A., Agarwal, S., andMeyarivan, T.Afastandelitistmultiobjectivegeneticalgorithm:NSGA-II. IEEETrans.EvolutionaryComputation,2002. 6(2):182197. doi:10.1109/4235.996017.Durand, S. L. and Reboulet, C. Optimaldesign of a redundant spherical parallel ma-nipulator. Robotica, 1997. 15(4):399405.doi:10.1017/S0263574797000490.Gosselin, C. M. andAngeles, J. Theoptimumkine-maticdesignof aspherical three-degree-of-freedomparallel manipulator. ASMEJ.Mechanisms,Trans-missions, and Automation in Design, 1989.111:202207. doi:10.1115/1.3258984.Gosselin,C.M.andAngeles,J. Aglobalperformanceindexforthekinematicoptimizationofroboticma-nipulators. ASME J. Mechanical Design, 1991.113(3):220226. doi:10.1115/1.2912772.Gosselin, C. M. andHamel, J. F. The Agile Eye:a high-performance three-degree-of-freedom camera-orientingdevice. InIEEEInternational ConferenceonRoboticsandAutomation.SanDiego,CA,pages781786,1994. doi:10.1109/ROBOT.1994.351393.Hao, F. and Merlet, J.-P.Multi-criteria optimal designof parallel manipulatorsbasedoninterval analysis.MechanismandMachineTheory, 2005. 40(2):157171. doi:10.1016/j.mechmachtheory.2004.07.002.Hay, A. M. and Snyman, J. A. Methodologies forthe optimal design of parallel manipulators. In-ter. J. Numerical Methods in Engineering, 2004.59(11):131152. doi:10.1002/nme.871.Hibbeler, R. C. MechanicsofMaterials. Prentice Hall,1997.Huang, T., Gosselin, C. M., Whitehouse, D. J., andChetwynd, D. G. Analytic approachfor optimaldesign of a type of spherical parallel manipula-torsusingdexterousperformanceindices. IMechE.J. Mechan. Eng. Sci., 2003. 217(4):447455.doi:10.1243/095440603321509720.Kong, K. and Gosselin, C. M. Type synthesis ofthree-degree-of-freedomsphericalparallelmanipula-tors. Inter. J. RoboticsResearch, 2004. 23(3):237245. doi:10.1177/0278364904041562.Kret, M. and Hesselbach, J. Elastodynamicoptimization of parallel kinematics. In Pro-ceedings of the IEEE International Conferenceon Automation Science and Engineering. Ed-monton, AB, Canada, pages 357362, 2005.doi:10.1109/COASE.2005.1506795.121Modeling,IdenticationandControlLi, T. and Payandeh, S. Design of sphericalparallel mechanisms for application to laparo-scopic surgery. Robotica, 2002. 20(2):133138.doi:10.1017/S0263574701003873.Liu, X. J., Jin, Z. L., andGao, F. Optimumdesignof 3-dof spherical parallel manipulators with respectto the conditioning andstiness indices. Mecha-nismandMachineTheory, 2000. 35(9):12571267.doi:10.1016/S0094-114X(99)00072-5.Lou, Y., Liu, G., Chen, N., and Li, Z. Optimaldesign of parallel manipulators for maximumef-fective regular workspace. In Proceedings of theIEEE/RSJInternational Conference onIntelligentRobotsandSystems. Alberta, pages795800, 2005.doi:10.1109/IROS.2005.1545144.Maxon. MaxonMotorandGearheadproductscatalog.2012. URLhttp://www.maxonmotor.com/maxon/view/catalog/.Merlet, J.-P. Jacobian, manipulability, conditionnumber, and accuracy of parallel robots. ASMEJ. Mechanical Design, 2006a. 128(1):199206.doi:10.1115/1.2121740.Merlet,J.-P. Parallel Robots. Kluwer,Norwell,2006b.Pashkevich,A.,Chablat,D.,andWenger,P. Stinessanalysis of overconstrained parallel manipulators.MechanismandMachineTheory, 2009. 44(5):966982. doi:10.1016/j.mechmachtheory.2008.05.017.Stamper, R. E., Tsai, L.-W., andWalsh, G. C. Op-timizationof athree-dof translational platformforwell-conditionedworkspace. InProceedings of theIEEE International Conference on Robotics and Au-tomation. Albuquerque, NM, pages 32503255, 1997.doi:10.1109/ROBOT.1997.606784.Stock, M. andMiller, K. Optimal kinematic designofspatial parallel manipulators: Applicationof lin-eardeltarobot. ASMEJ.Mechanical Design,2003.125(2):292301. doi:10.1115/1.1563632.Tsai, L.-W. TheJacobiananalysisofparallel manip-ulatorsusingreciprocal screws. InJ. LenarcicandM. L. Husty, editors, Advances inRobot Kinemat-ics: Analysis andControl, pages 327336. KluwerAcademicPublishers,1998.AppendixAThe elements of the compliance matrix(21) for thecurvedbeamC11=R2_s1GIx+s2EIy_(A-1a)C12=s8R2_1GIx1EIy_(A-1b)C16=R22_s2EIys7GIx_(A-1c)C22=R2_s2GIx+s1EIy_(A-1d)C26=R22_s4GIxs2EIy_(A-1e)C33=R2EIz(A-1f)C34=s5R2EIz(A-1g)C35=s6R2EIz(A-1h)C44=R2A_s1E+s2G_+s3R32EIz(A-1i)C45=s8R2A_ 1E 1G_+s4R32EIz(A-1j)C55=R2A_s1G+s2E_+s2R32EIz(A-1k)C66=R2GA+R32_s3GIx+s2EIy_(A-1l)withs1= 2 + sin 2 cos 2(A-2a)s2= 2sin 2 cos 2(A-2b)s3= 32 + sin 2 cos 2/2 4 sin 2(A-2c)s4= 1 cos 2sin22/2 (A-2d)s5= sin 22(A-2e)s6= cos 21 (A-2f)s7= 2 sin 22sin 2 cos 2(A-2g)s8= sin22(A-2h)whereEistheYoungsmodulusandG = E/2(1 + )istheshearmoduluswiththePoissonsratio. Ix,IyandIzarethemomentsof inertia, respectively. Aistheareaofthecross-section.122