Stiffness Analysis of 3-d.o.f. Overconstrained Translational Parallel Manipulators

Transcription

1 008 IEEE Internatonal Conference on Robotcs and Automaton Pasadena, CA, USA, May 19-, 008 Stffness Analyss of -d.o.f. Overconstraned ranslatonal Parallel Manpulators Anatoly Pashevch, Damen Chablat, Phlppe Wenger Abstract he paper presents a new stffness modellng method for overconstraned parallel manpulators, whch s appled to -d.o.f. translatonal mechansms. It s based on a multdmensonal lumped-parameter model that replaces the ln flexblty by localed 6-d.o.f. vrtual sprngs. In contrast to other wors, the method ncludes a EA-based ln stffness evaluaton and employs a new soluton strategy of the netostatc euatons, whch allows computng the stffness matrx for the overconstraned archtectures and for the sngular manpulator postures. he advantages of the developed technue are confrmed by applcaton examples, whch deal wth comparatve stffness analyss of two translatonal parallel manpulators. R I. INRODUCION / REAED WORKS elatve to seral manpulators, parallel manpulators are clamed to offer an mproved stffness-to-mass rato and better accuracy. hs feature maes them attractve for nnovatve machne-tool structures for hgh speed machnng [1,, ]. When a parallel manpulator s used as a Parallel Knematc Machne (PKM), stffness becomes a very mportant ssue n ts desgn [4, 5, 6, 7]. hs paper presents a general method to compute the stffness analyss of -dof overconstraned translatonal parallel manpulators. Generally, the stffness analyss of parallel manpulators s based on a netostatc modelng [8], whch proposes a map of the stffness by tang nto account the complance of the actuated jonts. However, ths method s not approprate for PKM whose legs are subject to bendng [9]. Several methods exst for the computaton of the stffness matrx: the nte Element Analyss (EA) [10], the matrx structural analyss (SMA) [11], and the vrtual jont method (VJM) that s often called the lumped modelng [8]. he EA s proved to be the most accurate and relable, however t s usually appled at the fnal desgn stage because of the hgh computatonal expenses reured for the repeated remeshng of the complcated D structure over the whole worspace. he SMA also ncorporates the man deas of the EA, but operates wth rather large elements D flexble beams descrbng the manpulator structure. hs leads obvously to the reducton of the computatonal expenses, but does not provde clear physcal relatons reured for the parametrc stffness analyss. And fnally, the VJM method s based on the expanson of the A. Pashevch s wth the IRCCyN (UMR CNRS 6597), Nantes, rance and wth the Department of Automatcs and Producton Systems, École des Mnes de Nantes, rance (anatol.pashevch@emn.fr) D. Chablat s wth the IRCCyN (UMR CNRS 6597), Nantes, rance (Damen.Chablat@rccyn-nantes.fr) P. Wenger s wth the IRCCyN (UMR CNRS 6597), Nantes, rance (Phlppe.Wenger@rccyn-nantes.fr). tradtonal rgd model by addng the vrtual jonts (localed sprngs), whch descrbe the elastc deformatons of the lns. he VJM technue s wdely used at the pre-desgn stage. Next secton ntroduces a general methodology to derve the nematc and stffness model. Secton descrbes the manpulator complant elements and the ln stffness evaluaton methods. nally n secton 4, we apply our method on two applcaton examples. II. GENERA MEHODOOGY A. Manpulator Archtecture et us consder a general d.o.f. translatonal parallel manpulator, whch conssts of a moble platform connected to a fxed base by three dentcal nematcs chans (g. 1). Each chan ncludes an actuated jont (prsmatc or rotatonal) followed by a oot and a eg wth a number of passve jonts nsde. Certan geometrcal condtons are assumed to be satsfed wth respect to the passve jonts to elmnate the platform rotatons and to acheve stablty of ts translatonal motons. Base Moble platform g. 1. Schematc dagram of a general -d.o.f. translatonal parallel manpulator ( actuated jont, passve jonts, foot, - eg) ypcal examples of such archtectures are: (a) -PUU translatonal PKM (g a) where each leg conssts of a rod ended by two U-jonts (wth parallel ntermedate and exteror axes), and actve jont s drven by lnear actuator [1] (b) Delta parallel robot (g b) that s based on the -RRPaR archtecture wth parallelogram-type legs and rotatonal actve jonts [14] (c) Orthoglde parallel robot (g c) that mplements the - PRPaR archtecture wth parallelogram-type legs and translatonal actve jonts [10]. Here R, P, U and Pa denote the revolute, prsmatc, unversal and parallelogram jonts, respectvely. It should be noted that examples (b) and (c) llustrate overconstraned mechansms, where some nematc constrans are redundant but do not affect the resultng degrees of freedom. However, most of the past wors deal wth non-overconstraned archtectures, whch motvates the subject of ths paper [8]. B. Basc Assumptons o evaluate the manpulator stffness, let us apply a modfcaton of the vrtual jont method (VJM), whch s based on the lump modelng approach [8, 10]. cordng to ths approach, /08/$ IEEE. 156

2 (a) -PUU translatonal PKM [1] (b) Delta parallel robot [14] (c) Orthoglde parallel robot [10] the orgnal rgd model should be extended by addng the vrtual jonts (localed sprngs), whch descrbe elastc deformatons of the lns. Besdes, vrtual sprngs are ncluded n the actuatng jonts to tae nto account stffness of the control loop. o overcome dffcultes wth parallelogram modelng, let us frst replace the manpulator legs (see g. ) by rgd lns wth confguraton-dependent stffness. hs transforms the general archtecture nto the extended - xuu case allowng treatng all the consdered manpulators n the smlar manner. Under such assumptons, each nematc chan of the manpulator can be descrbed by a seral structure (g. ), whch ncludes seuentally: Base platform (rgd) 1-d.o.f. sprng Rgd oot 6-d.o.f. sprng End-effector (rgd) U Rgd eg U g.. lexble model of a sngle nematc chan 6-d.o.f. sprng (a) a rgd ln between the manpulator base and the th actuatng jont (part of the base platform) descrbed by the constant homogenous transformaton matrx base (b) a 1-d.o.f. actuatng jont wth supplementary vrtual sprng, whch s descrbed by the homogenous matrx functon V a( 0 + 0) where 0 s the actuated coordnate and 0 s the vrtual sprng coordnate (c) a rgd oot lnng the actuatng jont and the leg, whch s descrbed by the constant homogenous transformaton matrx foot (d) a 6-d.o.f. vrtual jont defnng three translatonal and three rotatonal foot-sprngs, whch are descrbed by the homogenous matrx functon V s( 1, 6), where { 1,, } and { 4, 5, 6} correspond to the elementary translatons and rotatons respectvely (e) a -d.o.f. passve U-jont at the begnnng of the leg allowng two ndependent rotatons wth angles { 1, }, whch s descrbed by the homogenous matrx functon V u1( 1, ) (f) a rgd eg lnng the foot to the movable platform, whch s descrbed by the constant homogenous matrx transformaton leg (g) a 6-d.o.f. vrtual jont defnng three translatonal and three rotatonal leg-sprngs, whch are descrbed by the homogenous matrx functon V s( 7, 1), where { 7, 8, 9} and { 10, 1, 1} correspond to the elementary translatons and rotatons, respectvely (h) a -d.o.f. passve U-jont at the end of the leg allowng two ndependent rotatons wth angles {, 4}, whch s descrbed by the homogenous matrx functon V u(, 4) () a rgd ln from the manpulator leg the end-effector (part of the movable platform) descrbed by the constant homogenous matrx transformaton tool g.. ypcal d.o.f. translatonal parallel mechansms 156 he expresson defnng the end-effector locaton subject to varatons of all coordnates of a sngle nematc chan may be wrtten as follows = V ( + ) V (, ) base a 0 0 foot s 1 6 Vu1( 1, ) leg Vs( 7, 1) Vu(, 4) tool where matrx functon V a(.) s ether an elementary rotaton or translaton, matrx functons V u1(.) and V u (.) are compostons of two successve rotatons, and the sprng matrx V s(.) s composed of sx elementary transformatons. In the rgd case, the vrtual jont coordnates 0, 1 are eual to ero, whle the remanng ones (both actve 0 and passve 0, 4 ) are obtaned through the nverse nematcs, ensurng that all three matrces, = 1,, are eual to the prescrbed one that characteres the spatal locaton of the movng platform (nematc loop-closure euatons). Partcular expressons for all components of the product (1) may be easly derved usng standard technues for the homogenous transformaton matrces. It should be noted that the nematc model (1) ncludes 18 varables (1 for actve jont, 4 for passve jonts, and 1 for vrtual sprngs). However, some of the vrtual sprngs are redundant, snce they are compensated by correspondng passve jonts wth algnng axes or by combnaton of passve jonts. or computatonal convenence, nevertheless, t s not reasonable to detect and analytcally elmnate redundant varables at ths step, because the developed below technue allows easy and effcent computatonal elmnaton. C. Dfferental Knematc Model o evaluate the manpulator ablty to respond to the external forces and torues, let us frst derve the dfferental euaton descrbng relatons between the end-effector locaton and small varatons of the jont varables. or each th nematc chan, ths euaton can be generaled as follows δ t = J δ + J δ, = 1,,, () where the vector δ t (δ, δ, δ, δ, δ, δ ) = px py p ϕx ϕy ϕ descrbes the translaton δ p (δ, δ, δ ) = px py p and the rotaton δ ϕ (δ, δ, δ ) = ϕx ϕy ϕ of the end-effector wth respect to the Cartesan axes vector δ = ( δ0, δ1 ) collects all vrtual jont coordnates, vector δ = ( δ1, δ4 ) ncludes all passve jont coordnates, symbol '' δ stands for the varaton wth respect ι ι to the rgd case values, and J, J are the matrces of ses 6 1 and 6 4 respectvely. It should be noted that the dervatve for the ι actuated coordnate 0 s not ncluded n J but t s represented n the frst column of ι J through varable 0. he desred ι ι matrces J, J, whch are the only parameters of the dfferental model (), may be computed from (1) analytcally, usng some software support tools, such as Maple, MathCAD or Mathematca. However, a straghtforward dfferentaton usually yelds very awward expressons that are not convenent for further (1)

3 computatons. On the other hand, the fractoned structure of (1), where all varables are separated, allows applyng an effcent sem-analytcal method. o present ths technue, let us assume that for the partcular vrtual jont varable 0 the model (1) s rewrtten as = H V ( ) H, () 1 j j j j where the frst and the thrd multplers are the constant homogenous matrces, and the second multpler s the elementary translaton or rotaton. hen the partal dervatve of the homogenous matrx for the varable j at pont j = 0 may be computed from a smlar product where the nternal term s replaced by V j (.) that admts very smple analytcal presentaton. In partcular, for the elementary translatons and rotatons about the X-axs, these dervatves are: V ran x = V Rot x = (4) urthermore, snce the dervatve of the homogenous matrx 1 ( = Hj V j j) H j may be presented as 0 ϕ ϕ y p x ϕ 0 ϕ x p y = ϕy ϕx 0 p, (5) ι then the desred jth column of J can be extracted from (usng the matrx elements 14, 4, 4,, 1, 1 ). ι he Jacobans J can be computed n a smlar manner, but the dervatves are evaluated n the neghborhood of the nomnal values of the passve jont coordnates j nom correspondng to the rgd case (these values are provded by the nverse nematcs). However, smple transformaton j = j + δ nom j and correspondng factorng of the functon Vj( j) = Vj( j ) V j( δ j) allow applyng the above approach. nom It s also worth mentonng that ths technue may be used n analytcal computatons, allowng one to avod buly transformatons produced by the straghtforward dfferentatng. D. Knetostatc and Stffness Models or the manpulator netostatc model, whch descrbes the force-and-moton relaton, t s necessary to ntroduce addtonal euatons that defne the vrtual jont reactons to the correspondng sprng deformatons. In accordance wth the adopted stffness model, three types of vrtual sprngs are ncluded n each nematc chan: 1-d.o.f. vrtual sprng descrbng the actuator complance 6-d.o.f. vrtual sprng descrbng complance of the foot 6-d.o.f. vrtual sprng descrbng complance of the leg. Assumng that the sprng deformatons are small enough, the reured relatons may be expressed by lnear euatons τ = K 0 act = K oot τ 6 6 τ τ 7 7 = K eg, (6) τ 1 1 where τ j s the generaled force for the jth vrtual jont of the th nematc chan, K act s the actuator stffness (scalar), and, K oot, K eg are 6 6 stffness matrces for the foot and leg respectvely. It should be stressed that, n contrast to other wors, these matrces are assumed to be non-dagonal. hs allows tang nto account complcated couplng between rotatonal and translatonal deformatons, whle usual lump-based approach does consder ths phenomena [8]. or analytcal convenence, expressons (6) may be collected n a sngle matrx euaton τ = K δ, = 1,, (7) where τ ( τ 0, τ 1 ) = s the aggregated vector of the vrtual jont reactons, and K = dag ( K act, Koot, K eg) s the aggregated sprng stffness matrx of the se 1 1. Smlarly, one can defne the aggregated vector of the passve jont reactons τ = ( τ, τ ) but all ts components must be eual to ero: 1 4 τ = 0, = 1,, (8) o fnd the statc euatons correspondng to the end-effector moton δt, let us apply the prncple of vrtual wor assumng that the jonts are gven small, arbtrary vrtual dsplacements (, ) n the eulbrum neghborhood. hen the vrtual wor of the external force f appled to the end-effector along the correspondng dsplacement t = J + J s eual to the sum ( f J) + ( f J). or the nternal forces, the vrtual wor s τ snce the passve jonts do not produce the force/torue reactons (the mnus sgn taes nto account the adopted drectons for the vrtual sprng forces/torues). herefore, because n the statc eulbrum the total vrtual wor s eual to ero for any vrtual dsplacement, the eulbrum condtons may be wrtten as J f = τ J f = 0. (9) hs gves addtonal expressons descrbng the force/torue propagaton from the jonts to the end-effector. Hence, the complete netostatc model conssts of fve matrx euatons (), (7) (9) where ether f or δt are treated as nown, and the remanng varables are consdered as unnowns. Obvously, snce separate nematc chans posses some degrees-of-freedom, ths system cannot be unuely solved for gven f. However, vce versa, for gven end-effector dsplacement δt, t s possble to compute both the correspondng external force f and the nternal varables, δ, τ, δ (.e. vrtual sprng reactons and dsplacements n passve jonts, whch may also provde useful nformaton for the desgner). Snce matrx K s non-sngular (t descrbes the stffness of the vrtual sprgs), the varable δ can be expressed va f usng euatons τ = K δ and J f = τ. 1 hs yelds substtuton δ ( = K J ) f allowng reducng the netostatc model to system of two matrx euatons 1 + δ =δ ( J K J ) f J t wth unnowns f and matrx form S J f δ = t δ J 0 0 J f = 0 (10). hs system can be also rewrtten n a (11) where the sub-matrx S = J K J descrbes the sprng complance relatve to the end-effector, and the sub-matrx J taes nto account the passve jont nfluence on the end-effector motons. herefore, for a separate nematc chan, the desred stffness matrx K defnng the moton-to-force mappng f = K δt, can be computed by drect nverson of relevant matrx n the left-hand sde of (11) and extractng from t the 6 6 sub-matrx wth ndces correspondng to S. It s also worth mentonng that computng S reures 6 6 nversons only, snce

4 K = dag( K act, K oot, K eg ). Solvablty of system (11) n general case,.e. for any gven J and J, cannot be proved. Moreover, f the matrx J s sngular, the passve jont coordnates can not be found unuely. rom a physcal pont of vew, t means that f the nematc chan s located n a sngular posture, then certan dsplacements δt can be generated by nfnte combnatons of the passve jonts. But for the varable f the correspondng soluton s unue (snce the matrx J s obvously non-sngular f at least one 6 d.o.f. sprng s ncluded n a seral nematc chan). On the other hand, the sngularty may produce an nfnte number of stffness matrces for the same spatal locaton of the end-effector and for dfferent values provded by the nverse nematcs. A specal technue to tacle ths case, based on the sngular value decomposton, has been also developed. After the stffness matrces K for all nematc chans are computed, the stffness of the entre manpulator can be found by smple addton Km = = 1K. hs follows from the superposton prncple, because the total external force correspondng to the end-effector dsplacement δt (the same for all nematc chans) can be expressed as f = = 1f where f = K δt. It should be stressed that the resultng matrx K s not nvertble, snce some motons of the end-effector do not produce the vrtual sprng reactons (because of passve jonts nfluence). However, for the entre manpulator, the stffness matrx K m s s postve defnte and nvertble for all non-sngular (for the rgd model) postures. E. Comparson wth Other Results he man advantage of the proposed methodology s ts applcablty to overconstraned mechansms. o descrbe t n more detals, let us brefly revew an alternatve technue [8]. he latter s orgnated from the same prncpal euatons but the soluton strategy ncludes straghtforward elmnaton of the passve jont varables usng the dfferental nematc euatons () only. Obvously, the feasblty of ths step depends on the solvablty of the euvalent matrx system 1 1 I J δt J δ1 δ1 I J = J δ δ δ I J δ J (1) where δt and δ are treated as unnowns. In the nonconstraned case (for the -PUU archtecture, for nstance) the matrx n the left-hand sde of (14) s suare, of se 18 18, so t can be nverted usually. However, for overconstraned manpulators, ths matrx s non-suare, so the system cannot be solved unuely. or example, for manpulators wth the parallelogram-type legs (Orthoglde, Delta, etc.) the matrx se s So, n [10] three addtonal (vrtual) passve jonts were ntroduced to solve the problem. But, obvously, such a modfcaton changes the manpulator archtecture and ts stffness matrx, doubtng valdty of the correspondng model. Besdes, the developed technue allows computng the stffness matrx even for the sngular manpulator postures and does not ncorporate the least-suare pseudo-nversons appled by other authors. hs s acheved by applyng another soluton strategy, whch consders smultaneously the nematc and statc-eulbrum euatons for each nematc chan separately. Some hdden convenences are ncluded n the modelng stage. In partcular, the nematc models of the chans may nclude several redundant sprngs that are totally compensated by relevant passve jonts. However, there s no need to elmnate these sprngs from the model manually, snce they do not ncrease the matrx ses n system (11). hs allows ncludng n the model 6-d.o.f. vrtual sprngs of general type, wthout any modfcatons. Another advantage of the proposed technue s that t can be generaled easly. Wthn ths paper, t s appled to the stffness modelng of -d.o.f. translatonal manpulators wth actuators located between the base and the foot. However, t can be easly modfed to cover other actuator locatons, whch may be ncluded n the foot or n the leg. A further generalaton s related to a number of nematc chans and ther smlarty. hey are also not crucal assumptons and nfluence on the Jacoban computng only. But after the Jacobans are determned, the stffness matrces for separate chans may be computed n the same manner and then aggregated. III. PARAMEERS O HE COMPIAN EEMENS he adopted stffness model of each nematc chan ncludes three complant components, whch are descrbed by one 1-d.o.f. sprng and two 6-d.o.f. sprngs correspondng to the actuator, and to the foot/leg lns (see g. ). et us descrbe partcular technues for ther evaluaton. A. tuator Complance he actuator complance, descrbed by the scalar parameter 1 act = K act, depends on both the servomechansm mechancs and the control algorthms. Snce most of modern actuators mplement the dgtal PID control, the man contrbuton to act s done by the mechancal transmssons. he latter are usually located outsde the feedbac-control loop and consst of screws, gears, shafts, belts, etc., whose flexblty s comparable wth the flexblty of the manpulator lns. Because of the complcated mechancal structure of the servomechansms, the parameter act s usually evaluated from statc load experments, by applyng the lnear regresson to the expermental data. B. n Complance ollowng a general methodology, the complance of a manpulator ln (foots and legs) s descrbed by 6 6 symmetrcal 1 1 postve defnte matrces Kleg, K foot correspondng to 6-d.o.f. sprngs wth relevant couplng between translatonal and rotatonal deformatons. hs dstngushes our approach from other lumped modelng technues, where the couplng s neglected and only a subset of deformatons s taen nto account (presented by a set of 1-d.o.f. sprngs). he smplest way to obtan these matrces s to approxmate the ln by a beam element for whch the non-ero elements of the complance matrx may be expressed analytcally: = EA = EI = 5 EI = EI y = 44 = 55 = EI GJ EI y 6 = EI y (1) Here s the ln length, A s ts cross-secton area, I y, I, and J are the uadratc and polar moments of nerta of the cross-secton, and E and G are the Young s and Coulomb s modules respectvely. However, for certan ln geometres, the accuracy of a snglebeam approxmaton can be nsuffcent. In ths case the ln can be approxmated by a seral chan of the beams, whose complance s evaluated by applyng the same method (.e. consderng the nematc chan wth 6-d.o.f. vrtual sprngs, but wthout passve 1565

5 jonts). hs leads to the resultng complance matrx Kn = Jb Kb J b, where J b and K b ncorporate the Jacoban and the complance matrces for all vrtual sprngs. C. EA-based evaluaton of stffness or complex ln geometres, the most relable results can be obtaned from the EA modelng. o apply ths approach, the CAD model of each ln should be extended by ntroducng an auxlary D object, a pseudo-rgd body, whch s used as a reference for the complance evaluaton. Besdes, the ln orgn must be fxed relatve to the global coordnate system. hen, seuentally and separately applyng forces x, y, and torues M x, M y, M to the reference object, t s possble to evaluate correspondng lnear and angular dsplacements, whch allow computng the stffness matrx columns. he man dffculty here s to obtan accurate dsplacement values by usng proper EA-dscretaton ( mesh se ). Besdes, to ncrease accuracy, the dsplacements must be evaluated usng redundant data set descrbng the reference body moton. or ths reason, t s worth applyng a dedcated SVDbased algorthm. As follows from our study, the sngle-beam approxmaton of the Orthoglde foot gves accuracy of about 50%, and the four-beam approxmaton mproves t up to 0% only. Whle the EA-based method s the most accurate one, t s also the most tme consumng. However, n contrast to the straghtforward EA-modelng of the entre manpulator, whch reures recomputng for each manpulator posture, the proposed technue nvolves a sngle evaluaton of ln stffness. IV. APPICAION EXAMPES o demonstrate effcency of the proposed methodology, let us apply t to the comparatve stffness analyss of two -d.o.f. translatonal mechansm, whch employ Orthoglde archtecture. CAD models of these mechansms are presented n g. 4. A. Stffness of U-Jont Based Manpulator rst, let us derve the stffness model for the smplfed Orthoglde mechancs, where the legs are comprsed of euvalent lmbs wth U-jonts at the ends. cordngly, to retan major complance propertes, the lmb geometry corresponds to the parallelogram bars wth doubled cross-secton area. et us assume that the world coordnate system s located at the end-effector reference pont correspondng to the sotropc manpulator posture (when the legs are mutually perpendcular and parallel to relevant actuator axes). or ths assumpton, the geometrcal models of separate nematc chans can be descrbed by the expresson (1) Because for the rgd manpulator the end-effector moves wth only translatonal motons, the nomnal values of the passve jont coordnates are subject to the specfc constrants = 4 = 1, whch are mplctly ncorporated n the drect/nverse nematcs [10]. However, the flexble model allows varatons for all passve jonts. Usng the ln stffness parameters obtaned by the EA-modelng and applyng the proposed methodology, we computed the complance matrces for three typcal manpulator postures, the prncpal components of whch are presented n able 1. Below, they are compared wth the complance of the parallelogram-based manpulator. B. Stffness of Parallelogram Based Manpulator Before evaluaton the complance of the entre manpulator, let us derve the stffness matrx of the parallelogram. Usng the adopted notatons, the parallelogram euvalent model may be wrtten as = R ( ) ( ) R ( ) V (, ) (14) Plg y x y s 7 1 where, compared to the above case, the thrd passve jont s elmnated (t s mplctly assumed that = ). On the other hand, the orgnal parallelogram may be splt nto two seral nematc chans (the upper and lower ones) = ( d/) R ( + ) ( ) (, ) ( ) ( d/) up up y 1 x up up up Vs 1 6 Ry + = ( d/) R ( + ) ( ) (, ) ( + ) ( d/) dn dn y 1 x dn dn dn Vs 1 6 Ry (15) (16) where, d are the parallelogram geometrcal parameters, 1,, { up, dn} are the varatons of the passve jont coordnates and the sub/superscrpts up and dn correspond to the upper and lower chan respectvely. Hence, the parallelogram complance matrx may be also derved usng the proposed technue that yelds an analytcal expresson K K K K dck dsk K44+ 0 (17) = 4 8 dck dsk dsk 0 K6 0 0 K Plg where C = cos( ) S = sn( ). Usng ths model and applyng the proposed technue, we computed the complance matrces for three typcal manpulator postures (see table able 1). As follows from the comparson wth the U-jont case, the parallelograms allow ncreasng the rotatonal stffness roughly n 10 tmes. hs justfes applcaton of ths archtecture n the Orthoglde prototype desgn [15]. V. CONCUSIONS he paper proposes a new systematc method for computng the stffness matrx of overconstraned parallel manpulators. It s based on multdmensonal lumped model of the flexble lns, whose parameters are evaluated va the EA modelng and descrbe both the translatonal/rotatonal complances and the couplng between them. In contrast to prevous wors, the method employs a new soluton strategy of the netostatc euatons, whch consders smultaneously the nematc and statc relatons for each separate nematc chan and then aggregates the partal solutons n a total one. hs allows computng the stffness matrces for overconstraned mechansms for any gven manpulator posture, ncludng sngular confguratons and ther neghborhood. Another advantage s computatonal smplcty that reures low-dmensonal matrx nverson compared to other technues. Besdes, the method does not reure manual elmnaton of the redundant sprng correspondng to the passve jonts, snce ths operaton s nherently ncluded n the numercal algorthm. he effcency of the proposed method was demonstrated through applcaton examples, whch deal wth comparatve stffness analyss of two parallel manpulators of the 1566

6 Orthoglde famly (wth U-jont based and parallelogram based lns). Relevant smulaton results have confrmed essental advantages of the parallelogram based archtecture and valdated adopted desgn of the Orthoglde prototype. Another contrbuton s the analytcal stffness model of the parallelogram, whch was derved usng the same methodology. Whle appled to the -d.o.f. translatonal mechansms, the method can be extended to other parallel archtectures composed of several nematc chans wth rotatonal/prsmatc jonts and lmb- or parallelogram-based lns. So, future wor wll focus on the stffness modelng of more complcated parallel mechansm wth another actuator locaton (such as the Verne machne [16]) and also on the expermental verfcaton of the stffness models for the Orthoglde robot. REERENCES [1] J. lusty, J. Zegert and S. Rdgeway, undamental Comparson of the Use of Seral and Parallel Knematcs for Machne ools, In: Annals of the CIRP, vol. 48(1), [] P. Wenger, C. M. Gosseln and B. Mallé, A Comparatve Study of Seral and Parallel, In: Mechansm opologes for Machne ools, PKM 99, pp. -, Mlano, []. Majou, P. Wenger and D. Chablat, he desgn of Parallel Knematc Machne ools usng Knetostatc Performance Crtera, In: rd Internatonal Conference on Metal Cuttng and Hgh Speed Machnng, Met, rance, June 001. [4] G. Prtschow and K.-H. Wurst, Systematc Desgn of Hexapods and Other Parallel n Systems, In: Annals of the CIRP, vol. 46(1), pp 91 95, [5] O. Company and. Perrot, Modellng and Desgn Issues of a -axs Parallel Machne-ool, Mechansm and Machne heory, vol. 7, pp , 00. [6]. Brogardh, PKM Research - Important Issues, as seen from a Product Development Perspectve at ABB Robotcs, In: Worshop on undamental Issues and uture Research Drectons for Parallel Mechansms and Manpulators, Quebec, Canada, October 00. [7] A. Pashevch, P. Wenger and D. Chablat, Knematc and stffness analyss of the Orthoglde, a PKM wth smple, regular worspace and homogeneous performances, In: IEEE Internatonal Conference On Robotcs And Automaton, Rome, Italy, Aprl 007 [8] C.M. Gosseln, Stffness mappng for parallel manpulators, IEEE ransactons on Robotcs and Automaton, vol. 6, pp. 77 8, [9] X. Kong and C. M. Gosseln, Knematcs and Sngularty Analyss of a Novel ype of -CRR -DO ranslatonal Parallel Manpulator, he Internatonal Journal of Robotcs Research, vol. 1(9), pp , September 00. [10]. Majou, C. Gosseln, P. Wenger and D. Chablat. Parametrc stffness analyss of the Orthoglde, Mechansm and Machne heory, vol. 4(), pp , March 007. [11] B.C. Bougarrou, J.C. auroux, G. Gogu and Y. Heerah, Rgdty analyss of R1 parallel robot uncoupled nematcs, In: Proc. of the 5th Internatonal Symposum on Robotcs (ISR), Pars, rance, March 004. [1] D. Deblase, X. Hernot and P. Maurne, A Systematc Analytcal Method for PKM Stffness Matrx Calculaton, In: IEEE Internatonal Conference on Robotcs and Automaton (ICRA), pp , Orlando, lorda, May 006. [1] Y. and Q. Xu, "Stffness Analyss for a -PUU Parallel Knematc Machne", Mechansm and Machne heory, (In press: Avalable onlne 6 Aprl 007) [14] R. Clavel, DEA, a fast robot wth parallel geometry, Proceedngs, of the 18th Internatonal Symposum of Robotc Manpulators, IR Publcaton, pp , [15] D. Chablat and Ph. Wenger, Archtecture Optmaton of a -DO Parallel Mechansm for Machnng Applcatons, the Orthoglde, IEEE ransactons On Robotcs and Automaton, vol. 19(), pp , 00. [16] D. Kanaan, P. Wenger and D. Chablat, Knematcs analyss of the parallel module of the VERNE machne, In: 1th World Congress n Mechansm and Machne Scence, IoMM, Besançon, June 007. A A A A 1 B y 1 x C 1 B C P C A A 1 B 1 y x 1 j 1 C 1 B C P C B A A 1 B 1 y x (A) U-JOIN BASED ARCHIECURE (B) PARAEOGRAM BASED ARCHIECURE (C) WORKSPACE AND CRIICA POINS Q 1 AND Q IG. 4. KINEMAICS O WO -DO RANSAIONA MECHANISMS EMPOYING HE ORHOGIDE ARCHIECURE 1 j 1 C 1 B C P C B A ABE I: RANSAIONA AND ROAIONA SINESS O HE -PUU AND -PRPAR MANIPUAORS MANIPUAOR ARCHIECURE Pont Q 0 x, y, = 0.00 mm Pont Q 1 x, y, = 7.65mm Pont Q x, y, = mm tran [N/mm] rot [N mm/rad] tran [N/mm] rot [N mm/rad] tran [N/mm] rot [N mm/rad] -PUU manpulator PRPaR manpulator