Multiobjective Design Optimization of 3-PRR Planar Parallel Manipulators

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1 Multobjectve Desgn Optmzaton of 3-P Planar Parallel Manpulators aza Ur-ehman, Stéphane Caro, Damen Chablat, Phlppe Wenger To cte ths verson: aza Ur-ehman, Stéphane Caro, Damen Chablat, Phlppe Wenger. Multobjectve Desgn Optmzaton of 3-P Planar Parallel Manpulators. 2th CIP Desgn conference, Apr 21, Nantes, France. pp.1-1, 21. <hal-46411> HAL Id: hal Submtted on 16 Mar 21 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Abstract Multobjectve Desgn Optmzaton of 3 P Planar Parallel Manpulators aza U-EHMAN, Stéphane CAO, Damen CHABLAT, Phlppe WENGE Insttut de echerche en Communcatons et Cybernétque de Nantes UM CNS 6597, 1 rue de la Noë, Nantes, France {ur-rehman, caro, chablat, wenger}@rccyn.ec-nantes.fr Ths paper addresses the dmensonal synthess of parallel knematcs machnes. A multobjectve optmzaton problem s proposed n order to determne optmum structural and geometrc parameters of parallel manpulators. The proposed approach s appled to the optmum desgn of a three-degree-of-freedom planar parallel manpulator n order to mnmze the mass of the mechansm n moton and to maxmze ts regular shaped workspace. Keywords: Multobjectve Optmzaton, Parallel Manpulators, Workspace, Desgn. 1 INTODUCTION The desgn of parallel knematcs machnes s a complex subject. The fundamental problem s that ther performance heavly depends on ther geometry [1] and the mutual dependency of almost all the performance measures. Ths makes the problem computatonally complex and yelds the tradtonal soluton approaches neffcent. As reported n [2], snce the performance of a parallel manpulator depends on ts dmensons, the latter depend on the manpulator applcaton(s). Furthermore, numerous desgn aspects contrbute to the Parallel Knematcs Machne (PKM) performance and an effcent desgn wll be one that takes nto account all or most of these desgn aspects. Ths s an teratve process and an effcent desgn requres a lot of computatonal efforts and capabltes for mappng desgn parameters nto desgn crtera, and hence turnng out wth a multobjectve desgn optmzaton problem. Indeed, the optmal geometrc parameters of a PKM can be determned by means of a the resoluton of a multobjectve optmzaton problem. The solutons of such a problem are nondomnated solutons, also called Pareto-optmal solutons. Therefore, desgn optmzaton of parallel mechansms s a key ssue for ther development. Several researchers have focused on the optmzaton problem of parallel mechansms the last few years. They have come up ether wth mono- or mult-objectve desgn optmzaton problems. For nstance, Lou et al. presented a general approach for the optmal desgn of parallel manpulators to maxmze the volume of an effectve regular-shaped workspace whle subject to constrants on ther dexterty [3, 4]. Hay and Snyman [1] consdered the optmal desgn of parallel manpulators to obtan a prescrbed workspace, whereas Ottavano and Ceccarell [5, 6] proposed a formulaton for the optmum desgn of 3-Degree-Of-Freedom (DOF) spatal parallel manpulators for gven poston and orentaton workspaces. They based ther study on the statc analyss and the sngularty loc of a manpulator n orderto optmze the geometrcdesgn of the Tsa manpulator for a gven free from sngularty workspace. Hao and Merlet [7] dscussed a mult-crteron optmal desgn methodology based on nterval analyss to determne the possble geometrc parameters satsfyng two compulsory requrements of the workspace and accuracy. Smlarly, Ceccarell et al. [8] dealt wth the mult-crteron optmum desgn of both parallel and seral manpulators wth the focus on the workspace aspects, sngularty and stffness propertes. Gosseln and Angeles [9, 1] analyzed the desgn of a 3-DOF planar and a 3-DOF sphercal parallel manpulators by maxmzng ther workspace volume whle payng attenton to ther dexterty. Pham and Chen [11] suggested maxmzng the workspace of a parallel flexure mechansm wth the constrants on a global and unformty measure of manpulablty. Stamper et al. [12] used the global condtonng ndex based on the ntegral of the nverse condton number of the knematc Jacoban matrx over the workspace n order to optmze a spatal 3-DOF translatonal parallel manpulator. Stock and Mller [13] formulated a weghted sum mult-crteron optmzaton problem wth manpulablty and workspace as two objectve functons. Menon et al. [14] used the maxmzaton of the frst natural frequency as an objectve functon for the geometrcal optmzaton of the parallel mechansms. Smlarly, L et al. [15] proposed dynamcs and elastodynamcs optmzaton of a 2-DOF planar parallel robot to mprove the dynamc accuracy of the mechansm. They proposed a dynamc ndex to dentfy the range of natural frequency wth dfferent confguratons. Krefft [16] also formulated a mult-crteron elastodynamc optmzaton problem for parallel mechansms whle consderng workspace, velocty transmsson, nerta, stffness and the frst natural frequency as optmzaton objectves. Chablat and Wenger [17] proposed an analytcal approach for the archtectural optmzaton of a 3-DOF transla-

3 tonal parallel mechansm, named Orthoglde 3-axs, based on prescrbed knetostatc performance to be satsfed n a gven Cartesan workspace. Most of the foregong research works amed to mprove the mechansms performance throughout ther whole workspace. In ths paper, the mechansms performance are mproved over a regular shaped workspace that s defned based on the specfcatons. As a result, we propose a methodology to deal wth the multobjectve desgn optmzaton of PKMs. The sze of the regular shaped workspace and the mass n moton of the mechansm are the objectve functons of the optmzaton problem. Its constrants are determned based on the mechansm accuracy, assembly and the condtonng number of ts knematc Jacoban matrx. The proposed approach s hghlghted wth the optmal desgn of a 3-DOF Planar Parallel Manpulator (PPM). The non-domnated solutons, also called Pareto-optmal solutons, are obtaned by means of a genetc algorthm. 2 3-P PLANA PAALLEL MANIPU- LATOS Planar parallel manpulators are dstngushed by the fact that all ther components and correspondng motons, ncludng ther end-effector, generate planar motons. Ther archtecture s smple and they are usually smple to control. They can fnd many applcatons n planar motons that requre hgh dynamcs. Ther weakness s ther dffculty to carry out a large payload whose the weght s normal to the plane of moton. 3-DOF PPMs are classfed n famles, namely, the 3, 3 P, 3 P and 3 PPP PPMswhereandP standforrevoluteandprsmatc jonts, respectvely. Those famles are descrbed n[2]. In the scope of ths paper, we focus on the optmum desgn of 3 PPP PPMs, where P denotes an actuated prsmatc jont. However, the proposed approach can be appled to any type of PPM. 2.1 Manpulator archtecture A 3-DOF PPM wth three dentcal chans s shown n Fg. 1. Each knematc chan s of P-type and conssts of one actuated prsmatc jont, P; two revolute jonts, ; and two lnks. Ths 3-P manpulator s ntended to poston and orent the equlateral trangle-shaped platform C 1 C 2 C 3 n the plane of moton. The geometrc center of platform C 1 C 2 C 3, denoted by P, s the operaton pont of the manpulator. The dsplacements of the three prsmatc jonts,.e., ρ 1, ρ 2 and ρ 3, are the nput varables whereas the Cartesan coordnates of pont P,.e., x p and y p, and the orentaton φ of the platform are the output varables. The base-platform of the manpulator s also an equlateraltrangle wth vertces A 1, A 2 and A 3, pont O s ts geometrc center and the orgn of the reference frame. The prsmatc actuators are algned to ts sdes and are attached to ponts A (=1,2,3) wth orentaton angles α 1, α 2 and α 3 beng equal to, 12 and 24, respectvely. Here are the parameters descrbng the manpulator geometry: : radus of the crcumscrbed crcle of trangle A 1 A 2 A 3 of center O,.e., =OA ; r: radus of the crcumscrbed crcle of trangle C 1 C 2 C 3 of center P,.e., r=pc ; L b : the length of the ntermedate lnks,.e., L b = B C ; r j : the cross-secton radus of the ntermedate lnk; r p : the cross-secton radus of the movng platform lnk. B 3 r a1 a 1 A 3 P p O A 1 A 2 ρ 1 ρ 3 b 3 a 3 C 3 r a3 C 2 e 2 e 3 B 1 e 1 b 1 b 2 C 1 ra2 Fgure 1: 3 P planar parallel manpulator 2.2 Knematc modelng of the 3-P PPM Knowng the geometrc parameters of the mechansm,.e.,, r and L b, ts Inverse Knematcs Model (IKM) gves the relaton between the actuators dsplacements ρ and the movng platform pose,.e., x p, y p and φ: ρ = f (x p ) (1) wth ρ = [ ρ 1 ρ 2 ρ 3 ] T (2a) x p = [ x p y p φ ] T φ B 2 a 2 ρ 2 (2b) Eq. (1) can be expressed as a quadratc equaton [18]. The latter may have eght solutons correspondng to the eght workng modes of the mechansm [19]. The choce of the workng mode can also be used as a desgn parameter of the mechansm as t modfes the locaton of ts sngular confguratons. The Drect Knematcs Model(DKM) of the manpulator characterzes the movngplatform posen termsofthe prsmatc actuators dsplacements: x p = f (ρ) (3) The DKM of the 3 P PPM may have sx solutons correspondng to the sx assembly modes of the mechansm [2].

4 2.3 Knematc Jacoban matrx of the 3-P PPM The knematc Jacoban matrx defnes the relatonshp between the actuators and moble platform velocty vectors. For the th knematc chan, a closed loop vector equaton can be wrtten as: OP = OA + A B + B C + C P (4) Equaton (4) can be expressed algebracally as: p = r a +ρ a +L b b re (5) 2.4 Stffness matrx The stffness model of the 3-P PPM s obtaned by means of the refned lumped mass modelng presented n [21]. Let us consder a general schematc of the 3-P PPM that s composed of a moble platform connected to a fxed base by means of three dentcal knematcs chans, as shown n Fg. 2. Each knematc contans an actuated prsmatc jont and two passve revolute jonts. Base wth a, b, e and r a beng the unt vectors depcted n Fg. 1. Upon dfferentaton of Eq. (5) wth respect to tme we get, P P P ṗ = ρ a +L b ḃ rė (6) ḃ and ė beng wrtten as: ḃ = β Eb ė = φee (7) β s the angular velocty of the th ntermedate lnk and E s the rght angle rotaton matrx gven by, [ ] 1 E = (8) 1 Accordngly, Eq. (6) becomes ṗ = ρ a +L b β Eb r φee (9) Upon multplcaton of Eq. (9) by b T, we obtan the matrx form: Aẋ p = B ρ (1) wth A = and B = bt 1 b T 2 rb T 1 Ee 1 rb T 2 Ee 2 b T 3 rb T 3Ee 3 bt 1 a 1 b T 2 a 2 b T 3a 3 (11) (12) Therefore, the prsmatc jonts rates are expressed n terms of the movng platform twst as follows: ρ = B 1 Aẋ p = Jẋ p (13) where J s the knematc Jacoban matrx of the manpulator. J = B 1 A = 1 b 1 rk(b 1 e 1 ) b 2 rk(b 2 e 2 ) (14) a.b b 3 rk(b 3 e 3 ) The sngular confguratons of the 3 P PPM can be obtaned by means of a sngularty analyss of J as explaned n [18]. Moble Platform Fgure 2: Schematc dagram of a 3-P Accordng to the flexble model descrbed n [21], each knematc chan of the 3-P manpulator can be consdered as a seral archtecture as shown n Fg. 3 that contans sequentally: Base Platform (gd) A c L gd Body r gd Body 1-dof 6-dof 6-dof Sprng Sprng Sprng Fgure 3: Flexble model of the sngle knematc chans of the 3-P PPM, A c stands for actuatng jont and for revolute jont a rgd lnk between the manpulator base and the th actuated jont (part of the base platform) descrbed by the constant homogeneous transformaton matrx T Base ; a 1-dof actuated jont, defned by the homogeneous matrx functon V a (q ) where q s the actuated coordnate; a 1-dof vrtual sprng descrbng the actuator mechancal stffness, whch s defned by the homogeneous matrx functon V s1 ( θ ) where θ s the vrtual sprng coordnate correspondng to the translatonal sprng; a 1-dof passve -jont at the begnnng of the leg allowng one rotaton angle q2, whch s descrbed by the homogeneous matrx functon V r1 (q2 ) a rgd leg of length L lnkng the foot and the movable platform, whch s descrbed by the constant homogeneous transformaton matrx T L ;

5 a 6-dof vrtual sprng descrbng the leg stffness, whch s defned by the homogeneous matrx functon V s2 ( θ 1 θ 6), wth θ 1, θ 2, θ 3 and θ 4, θ 5, θ 6 beng the vrtual sprng coordnates correspondng to the sprng translatonal and rotatonal deflectons; a 1-dof passve -jont between the leg and the platform, allowng one rotaton angle q 3, whch s descrbed by the homogeneous matrx functon V r2 (q 3 ); a rgd lnk of length r from the manpulator leg to the geometrc center of the moble platform, whch s descrbed by the constant homogeneous transformaton matrx T r ; a 6-dof vrtual sprng descrbng the stffness of the movng platform, whch s defned by the homogeneous matrx functon V s3 ( θ 7 θ 12), θ 7, θ 8, θ 9 and θ 1, θ 11, θ 12 beng the vrtual sprng coordnates correspondng to translatonal and rotatonal deflectons of lnk C P; a homogeneous transformaton matrx T End characterzng the rotaton from the 6-dof sprng assocated wth lnk C P and the manpulator base frame; The correspondng mathematcal expresson defnng the end-effector locaton subject to varatons n all above defned coordnates of the th knematc chan can be wrtten as follows: T =T BaseVa( ) ( ) ( ) q Vs1 θ Vr1 q 1 T L V s2 (15) ( θ 1 θ6 ) Vr2 (q2)t ( rv s3 θ 7 θ12 ) T Base From [21], the knetostatc model of the th leg of the 3-PPPMcanbereducedtoasystemoftwomatrx equatons, namely, [ ][ ] [ ] S θ J q f δt = 2 2 δq 2 J q (16) where the sub-matrx S θ = J θ K θ 1 J θ T descrbes the sprng complance relatve to the geometrc center of the movngplatform, and the sub-matrxj q takesnto account the passve jont nfluence on the movng platform motons. K θ 1 matrx, of sze 13 13, descrbes the complance of the vrtual sprngs and takes the form: K θ 1 = K 1 act 6 1 K leg K pf (17) where K act s the 1 1 stffness matrx of the th actuator, K leg s the 6 6 stffness matrx of the th ntermedate leg and K pf s the 6 6 the stffness matrx of the th lnk of the movng platform. The complance matrces of the ntermedate legs and the th lnk of the movng platform are calculated by means of the stffness model of a cantlever beam, namely, L EA L 3 L 3EI z 2 2EI z L 3 K L 1 = 3EI y L2 2EI y L GI x L2 L 2EI y EI y L 2 2EI z L EI z (18) L s the length of the beam,.e., L = L b for the ntermedate legs and L = r for the movng platform lnks. A s the cross-sectonal area of the beam,.e., A Lb = πrj 2 and A r = πrp 2. I z=i y s the polar moment of nerta about y and z axes,.e., for the ntermedate legs and the movng platform lnks, ther expressons are πrj 4/4 and πr4 p/4, respectvely. I x =I z +I y s the polar moment of nerta about the longtudnal axs of the beam. E s the Young modulus of the materal and G ts shear modulus J θ of sze 6 13 s the Jacoban matrx related to the vrtual sprngs and J q of sze 6 2, the one related to the passve jonts. f s the wrench exerted on the th leg of the 3-P PPM at the geometrc center of the movng platform and δt s the correspondng translatonal and rotatonal dsplacements vector. Therefore, the Cartesan stffness matrx K of the th leg defnng the moton-to-force mappng s obtaned from Eq. (16). f = K δt (19) Fnally, thecartesanstffnessmatrxkofthe3-p PPM s found wth a smple addton of K matrces, namely, K = 3 K (2) =1 3 MULTIOBJECTIVE DESIGN OPTI- MIZATION In general, the desgn process of PKMs smultaneously deals wth two groups of crtera, one relates to the knematc propertes whle the other relates to the knetostatc/dynamc propertes of the mechansm. Bothofthesegroupsncludeanumberofperformance measures that essentally vary throughout the workspace but reman wthn the prescrbed bounds. Knematc aspects are comparatvely less complex and are usually based on the concept of crtcal ponts whereas knetostatc aspects work wth a detaled descrpton of the structure and ther evaluaton s usually tme consumng. Indeed, one of the major desgn ssues n knetostatc desgn s the computaton of the stffness matrx[22]. Accordngly, a multobjectve desgn optmzaton approach s proposed based on performance measures/crtera from both knematc and knetostatc domans. On the one hand, ths approach deals wth the geometrc/knematc desgn n order to determne the PKM geometry ncludng the lnk

6 lengths and the jont lmts. On the other hand, t consders the knetostatc desgn to determne the sze and the mass propertes of the lnks. 3.1 Optmzaton objectves The multobjectve optmzaton problem ams to determne the optmum geometrc parameters of a PKM n order to maxmze ts workspace as well as to mnmze the mass of the mechansm n moton. Here, the workspace of the mechansm s dscretzed and the consdered performance measures and constrants are evaluated and verfed for each pont Mass n moton of the mechansm The mass n moton of the mechansm s consdered to be the frst objectve functon of the optmzaton problem. Mass and nerta are functons of manpulator dmensons,.e., lnk lengths, cross-sectonal area, thckness. Hence, n general, the mass n moton m t of the mechansm s composed of the mass of the platform, m pf, the mass of the three ntermedate bars, m b, and the mass n moton of the three prsmatc actuators, m s : optmzaton problem of PKMs s based on the formulaton of workspace maxmzaton,.e, to determne the optmum geometry of PKM n order to maxmze a regular-shaped workspace. Workspace sze can be defned by ts geometrc shape parameters lke the radus of a cylndrcal/sphercal workspace or the sdes of the cube for a cubc workspace. In the scope of the paper, a cylndrcal workspace defned wth ts radus w s consdered. Furthermore, at each pont of the workspace, an angular rotaton range φ=2 of the platform about the Z-axs can be acheved. A 3-dmensonal schematc of the regular shaped workspace s shown n Fg. 4, where x c, y c are the coordnates of the center of the regular dextrous workspace and φ c s the orentaton of the platform at ts home-posture (see Fg. 1). w (x c, y c, φ c ) φ m t = m pf +3m b +3m s (21) Snce the actuators are fxed, ther mass s consdered to be constant whle the mass of the other two components can easly be calculated by usng the geometry of the components and the densty d of ther materal, gven as, m pf = πr 2 prd, m b = πr 2 jl b d (22) Consequently, the frst objectve functon of the optmzaton problem s wrtten as: f 1 (x) = m t mn (23) x beng the vector of the geometrc desgn parameters of the mechansm egular workspace sze Workspace s one of the most mportant desgn ssues as t defnes the workng volume of the robot/manpulator and determnes the area that can be reached by a reference frame located on the movng platform or end-effector [23, 12]. The sze and shape of the workspace are of prmary mportance for the global geometrc performance evaluatons of the manpulators [24]. The qualty of the workspace that reflects the shape, sze, presence of sngulartes s of prme mportance n PKM desgn. Workspace based desgn optmzaton of parallel mechansms can usually be solved wth two dfferent formulatons. The frst formulaton ams to desgn a manpulator whose workspace contans a prescrbed workspace and the second approach ams to desgn a manpulator, of whch the workspace s as large as possble. However, maxmzng the workspace may result n poor desgn wth regard to the manpulator dexterty and manpulablty [12]. Ths problem can be solved by properly defnng the constrants of the optmzaton problem. Here, the multobjectve Fgure 4: 3-P workspace Consequently, n order to maxmze the manpulator workspace, the second objectve of the optmzaton problem can be wrtten as: f 2 (x) = w max (24) 3.2 Optmzaton constrants Besdes, the geometrc and actuator constrants of the PKM, condtonng of the knematc Jacoban matrx and accuracy obtaned from the stffness characterstcs of the mechansm are consdered. Constranng the condtonng of the Jacoban matrx guarantees sngularty free workspace whereas lmts on accuracy consderaton ensure suffcent mechansm stffness Geometrc Constrants The frst constrant s related to the mechansm assembly, namely, L b +r /2 (25) In order to avod ntersectons between prsmatc jonts, the lower and upper bounds of the prsmatc lengths are defned as follows: < ρ < 3 (26) Condton number of the knematc Jacoban matrx The condton number κ F (M) of a m n matrx M, wth m n, based on the Frobenus norm s defned as follows, κ F (M) = 1 m tr(m T M)tr[(M T M) 1 ] (27)

7 Here, the condton number s computed based on the Frobenus norm as the latter produces a condton number that s analytc n terms of the posture parameters whereas the 2-norm does not. Besdes, t s much costler to compute sngular values than to compute matrx nverses. The terms of the drect Jacoban matrx of the 3- P PPM are not homogeneous as they do not have same unts. Accordngly, ts condton number s meanngless. Indeed, ts sngular values cannot be arranged n order as they are of dfferent nature. However, from [25] and [26], the Jacoban can be normalzed by means of a normalzng length. Later on, the concept of characterstc length was ntroduced n [27] n order to avodthe random choceof the normalzng length. For nstance, the prevous concept was used n [18] to analyze the knetostatc performance of manpulators wth multple nverse knematc solutons, and therefore to select ther best workng mode. Accordngly, for the desgn optmzaton of 3- P PPM, the mnmum of the nverse condton number of the knematc Jacoban matrx, κ 1 (J), s supposed to be hgher than a prescrbed value, say.1, throughout the manpulator workspace, for any rotaton of ts end-effector,.e., mn ( κ 1 (J) ).1 (28) Accuracy constrants The poston and orentaton accuracy s assessed by usng the stffness parameters of the mechansm. Let (δx, δy, δz) and (δφ x, δφ y, δφ z ) be the poston and orentaton errors of the end-effector subjected to external forces (F x, F y, F z ) and torques (τ z, τ y, τ z ). The constrants related to the accuracy of the manpulator are defned as follows: δx δx max δy δy max δz δz max δφ x δφ max x δφ y δφ max y δφ z δφ max z (29) (δx max, δy max, δz max ) beng the maxmum allowable poston errors and ( ) δφ max x, δφ max y, δφ max z the maxmum allowable orentaton errors of the end-effector. These accuracy constrants can be expressed n terms of the components of the mechansm stffness matrx and the wrench appled to the end-effector. Let us assume that the accuracy requrements are: δx2 +δy 2.1m (3a) δz.1 m (3b) δφ z 1deg (3c) If the end-effector s subjected to a wrench, whose components are F x,y =F z =1N and τ z =1Nm, then the accuracy constrants can be expressed as: kxy mn F x,y / δx 2 +δy 2 = 1 6 N.m -1 (31a) kz mn F z /δz = 1 5 N.m -1 (31b) kφ mn z τ z /δφ z = 1 π/18 N.m.rad-1 (31c) 3.3 Desgn Varables Along wth the above mentoned geometrc parameters (, r, L b ) of the 3-P PPM, the dmenson of the crcular-cross-secton of the ntermedate bars defned wth radus r j and the crcular-cross-secton of the platform bars defned wth r p are consdered as desgn varables, also called decson varables. The platform s assumed to be made up of three crcular bars, each of length r. Hence, the desgn parameters vector x s gven by: x = [ r L b r j r p ] T (32) 3.4 Multobjectve optmzaton problem statement The Multobjectve Optmzaton Problem (MOO) for a 3 P PPM can be stated as: Fnd the optmum desgn parameters x of a 3 P PPM n order to mnmze the mass n moton of the mechansm and to maxmze ts regular shaped workspace subject to some desgn constrants,.e., the nverse condton number of the knematc Jacoban matrx and accuracy are to be hgher than prescrbed values throughout the manpulator workspace. Mathematcally, the problem can be wrtten as: mnmze f 1 (x) = m t (33) maxmze f 2 (x) = w over x = [ r L b r j subject to : g 1 : L b +r 2 g 2 : < ρ < 3 g 3 : κ 1 (J).1 g 4 : k mn xy g 5 : k mn z g 6 : k mn φ z x lb x x ub r p ] T F x,y δx2 +δy 2 = 16 F z δz = 15 τ z δφ z = 1 π/18 where x lb and x ub are the lower and upper bounds of x, respectvely. 4 ESULTS AND DISCUSSIONS The multobjectve optmzaton problem (33) s solved by means of modefontie[28] and by usng ts bult-n multobjectve optmzaton algorthms. MATLAB code s ncorporated n order to analyze the system and to get the numercal values for the objectve functons and constrants that are analyzed n modefontie for ther optmalty and feasblty. The lower and upper bounds of the desgn varables are gven n Table 1. The manpulator s supposed to be bult nsteel wth adensty equalto d=785kg/m 3 and a Young modulus equal to E= N/m 2.

8 Desgn Varable r L b r j r p Lower Bound (lb) [m] Upper Bound (ub) [m] Table 1: Lower and upper bounds of the desgn varables The Pareto fronter s shown n Fg. 5 whereas the desgn varables and the correspondng objectve functons for two extreme and one ntermedate Pareto optmal solutons, as shown n Fg. 5, are depcted n Table 3. For each desgn teraton, workspace lmts are calculated based on the set of desgn parameters of the mechansm. Then, workspace dscretzaton s performed wth respect to ts x, y coordnates and wth respect to the orentaton angle φ of the movng platform. The constrants of the problem are evaluated at each grd pont of the workspace., r, Lb[m] r L b w [m] Scheduler MOGA-II Number of teratons 2 Drectonal cross-over probablty.5 Selecton probablty.5 Mutaton probablty.1 DNA (DeoxyrboNuclec Acd) strng mutaton rato DOE algorthm.5 Sobol DOE number of desgns 3 Total number of teratons 3 2 = 6 Table 2: modefontie algorthm parameters A multobjectve genetc algorthm (MOGA) s used to obtan the Pareto fronter based on the mechansm mass and the workspace radus. modefon- TIE scheduler and Desgn Of Experments (DOE) parameters are gven n Table 2. MATLAB s used to process and analyze the system for any ndvdual of the current populaton (generated by the modefon- TIE scheduler). Correspondng to each populaton set, MATLAB returns the output varables that are analyzed by modefontie for the feasble solutons accordng to the gven constrants. At the end, the Pareto-optmal solutons are obtaned from the generated feasble solutons. Workspace adus w [m] ID-I ID-II ID-III Mass m t [kg] Fgure 5: Pareto fronter for 3-P optmzaton problem rj, rp[m] r j r p w [m] Fgure 6: Desgn varables as a functon of w for the Pareto-optmal solutons The desgns assocated wth the three foregong solutons are shown n Fg. 7. Fgure 8 llustrates the varatonal trends as well as the nter-dependency between the objectve functons and desgn varables by means of a scatter matrx. The lower trangular part of the matrx represents the correlaton coeffcents, ξ, whereas the upper one shows the correspondng scatter plots. The dagonal elements represent the probablty densty charts of each varable. The correlaton coeffcents vary from -1 to 1. Two varables are strongly dependent when ther correlaton coeffcent s close to 1 or -1 and ndependent when the latter s null. Fgure 8 shows that: Both objectves functons m t and w are strongly dependent as ther correlaton coeffcent s equal to.97; Both objectves functons m t and w are strongly dependent on all desgn varables as all of the correspondng correlaton coeffcents are greater than.7; w (ξ.83) s slghtly more dependent than m t (.711 ξ.981) of the desgn varables; Fgure 6 llustrates the desgn varables, r, L b, r j and r p as a functon of w for the Pareto-optmal solutons. It s noteworthy that the hgher w, the hgher the desgn varables. It s apparent that the varatons n varables, r, L b and r j wth respect to (w.r.t.) w are almost lnear whereas the varatons n r p w.r.t. w s rather quadratc. As a matter of fact, t should be due to the fact that the hgher the sze of the mechansm the hgher the bendng of the movng platform lnks whereas the ntermedate lnks are manly subjected to tenson and compresson. Fnally, the three sets of desgn varables correspondng

9 Desgn ID Desgn Varables Objectves [m] r [m] L b [m] r j [m] r p [m] m t [kg] w [m] I II III Table 3: Three Pareto optmal solutons egular Workspace egular Workspace egular Workspace (a) ID I (b) ID II (c) ID III Fgure 7: CAD Desgns of three Pareto-optmal solutons r L b r j r p m t w r L b r j r p m t w Fgure 8: Scatter matrx llustratng the correlatons between the objectve functons and the desgn varables to the Pareto-optmal solutons depcted n Fg. 5 are shown n Fg. 6 by means of the green, pnk and red symbols. 5 CONCLUSIONS In ths paper, the problem of dmensonal synthess of parallel knematcs machnes was addressed. A multobjectve desgn optmzaton problem was formulated n order to determne optmum structural and geometrc parameters of any parallel knematcs machne. The proposed approach s smlar to that used n [29] but we took nto account the mass and the regular workspace nstead of consderng the entre volume of the manpulator. The proposed approach was appled

10 to the optmum desgn of a three-degree-of-freedom planar parallel manpulator wth the am to mnmze the massn moton ofthe mechansmand to maxmze ts regular shaped workspace. It s apparent that other performance ndces can be used as constrants. However, they cannot necessarly be used as objectve functons as the latter are usually formulated as a sum of an ndex over all the manpulator workspace. As another constrant, we could use the collsons between the legs of the manpulator as llustrated n [4]. Our future works wll deal wth the multobjectve desgn optmzaton and the comparson of 3-DOF planar parallel manpulators of dfferent archtectures as well as the optmzaton of the cross-secton type of ther lnks. EFEENCES [1] A. M. Hay and J. A. Snyman. Methodologes for the optmal desgn of parallel manpulators. Internatonal Journal for Numercal Methods n Engneerng, 59(11): , 24. [2] J. P. Merlet. Parallel obots. Kluwer Academc Publshers, Norwell, MA, USA, 26. [3] Y. Lou, G. Lu, N. Chen, and Z. L. Optmal desgn of parallel manpulators for maxmum effectve regular workspace. In Proceedngs of the IEEE/SJ Internatonal Conference on Intellgent obots and Systems, pages 795 8, Alberta, 25. [4] Y. Lou, G. Lu, and Z. L. andomzed optmal desgn of parallel manpulators. IEEE Transactons on Automaton Scence and Engneerng, 5(2): , 28. [5] E. Ottavano and M. Ceccarell. Workspace and optmal desgn of a pure translaton parallel manpulator-tsa manpulator. Meccanca, 35(3):23 214, May 2. [6] E. Ottavano and M. Ceccarell. Optmal desgn of capaman (cassno parallel manpulator) wth prescrbed workspace. In 2 nd Workshop on Computatonal Knematcs KC21, pages 35 43, Seoul, South Korea, 21. [7] F. Hao and J.-P. Merlet. Mult-crtera optmal desgn of parallel manpulators based on nterval analyss. Mechansm and Machne Theory, 4(2): , 25. [8] M. Ceccarell, G. Carbone, and E. Ottavano. Mult crtera optmum desgn of manpulators. In Bulletn of the Polsh Academy of Scences Techncal Scences, volume 53, 25. [9] C. M. Gosseln and J. Angeles. The optmum knematc desgn of a planar three-degree-offreedom parallel manpulator. ASME Journal of Mechansms, Transmsson and Automaton n Desgn, 11:35 41, [1] C. M. Gosseln and J. Angeles. The optmum knematc desgn of a sphercal three-degree-offreedom parallel manpulator. Journal of Mechansms, Transmssons and Automaton n Desgn, 111(2):22 27, [11] H. H. Pham and I-M. Chen. Optmal synthess for workspace and manpulablty of parallel flexure mechansm. In Proceedng of the 11 th World Congress n Mechansm and Machne Scence, pages , Tanjn, Chna, Apr [12].E. Stamper, L.-W.Tsa, andg. C.Walsh. Optmzaton of a three-dof translatonal platform for well-condtoned workspace. In Proceedngs of the IEEE Internatonal Conference on obotcs and Automaton, pages , New Mexco, [13] M. Stock and K. Mller. Optmal knematc desgn of spatal parallel manpulators: Applcaton of lnear delta robot. Transactons of the ASME, Journal of Mechancal Desgn, 125(2):292 31, 23. [14] C. Menon,. Vertechy, M.C. Markot, and V. Parent-Castell. Geometrcal optmzaton of parallel mechansms based on natural frequency evaluaton: applcaton to a sphercal mechansm for future space applcatons. IEEE Transactons on obotcs, 25(1):12 24, Feb 29. [15] H. L, Z. Yang, and T. Huang. Dynamcs and elasto-dynamcs optmzaton of a 2-dof planar parallel pck and place robot wth flexble lnks. Journal of Structural and Multdscplnary Optmzaton, 38(2):195 24, 29. [16] M. Krefft and J. Hesselbach. Elastodynamc optmzaton of parallel knematcs. In Proceedngs of the IEEE Internatonal Conference on Automaton Scence and Engneerng, Edmonton, Canada, Aug [17] D. Chablat and P. Wenger. Archtecture optmzaton of a 3-dof parallel mechansm for machnng applcatons, the orthoglde. IEEE Transactons On obotcs and Automaton, 19(3):43 41, 23. [18] D. Chablat, Ph. Wenger, S. Caro, and J. Angeles. The socondtonng loc of planar 3-dof parallel manpulator. In Proceedngs of DETC 22, ASME Desgn Engneerng Techncal Conference, Montreal, Quebec, Canada, 29 Sep 2 Oct 22. [19] D. Chablat and P. Wenger. Workng modes and aspects n fully parallel manpulators. In Proceedngs of the IEEE Internatonal Conference on obotcs and Automaton, pages , May [2] C. M. Gosseln and J. P. Merlet. The drect knematcs of planar parallel manpulators: specal archtectures and number of solutons. Mechansm and Machne Theory, 29(8): , [21] A. Pashkevch, D. Chablat, and P. Wenger. Stffness analyss of overconstraned parallel manpulators. Mechansm and Machne Theory, 44(5): , 29.

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