CAD-based approach for identification of elasto-static parameters of robotic manipulators

Transcription

1 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 1 CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators Alexadr Klmchk a,b, * Aatol Pashkevch a,b, Dame Chablat b a Ecole des Mes de Nates, 4 rue Alfred-Kastler, Nates 4437, Frace b Isttut de Recherches e Commucatos et e Cyberetque de Nates, UMR CNRS 6597, 1 rue de la Noe, 4431 Nates, Frace Abstract - he paper presets a approach for the detfcato of elasto-statc parameters of a robotc mapulator usg the vrtual expermets a CAD evromet. It s based o the umercal processg of the data extracted from the fte elemet aalyss results, whch are obtaed for solated mapulator lks. hs approach allows to obta the desred stffess matrces takg to accout the complex shape of the lks, couplgs betwee rotatoal/traslatoal deflectos ad partculartes of the jots coectg adjacet lks. hese matrces are tegral parts of the mapulator lumped stffess model that are wdely used robotcs due to ts hgh computatoal effcecy. o mprove the detfcato accuracy, recommedatos for optmal settgs of the vrtual expermets are gve, as well as relevat statstcal processg techques are proposed. Effcecy of the developed approach s cofrmed by a smulato study that shows that the accuracy evaluatg the stffess matrx elemets s about.1%. Keywords: Robotc mapulator, Stffess modelg, Elasto-statc parameters, CAD-based Idetfcato, Fte elemet aalyss, Stffess matrx 1 Itroducto Curret treds mechacal desg of robotc mapulators are targeted at essetal reducto of movg masses, order to acheve hgh dyamc performaces wth relatvely small actuators ad low eergy cosumpto. hs motvates usg advaced kematcal archtectures (Orthoglde, Delta, Gatry-au, etc.) ad lght-weght materals, as well as mmzato of the cross-sectos of all mapulator elemets [1-4]. he prmary costrat for such mmzato s the mechacal stffess of the mapulator, whch s drectly related wth the robot accuracy defed by the desg specfcatos. For ths reaso detfcato (or evaluato) of the mapulator elasto-statc parameters becomes oe of the key ssues developmet ad optmzato of moder robotc systems Smlar to geeral structural mechacs, the robot stffess characterzes the mapulator resstace to the deformatos caused by a exteral force or torque appled to the ed-effector [5,6]. Numercally, ths property s usually defed through the stffess matrx K, whch s corporated a lear relato betwee the traslatoal/rotatoal dsplacemet ad statc forces/torques causg ths trasto (assumg that all of them are small eough). he verse of K s usually called the complace matrx ad s deoted as k. As t follows from related works, for coservatve systems K s a 6 6 sem-defte o-egatve symmetrcal matrx but ts structure may be o-dagoal to represet the couplg betwee the traslato ad rotato [7]. I stffess modelg of robotc mapulator, because of some specfcty, there are some partculartes termology. I partcular, the matrx K s usually referred to as the Cartesa Stffess Matrx K C ad t s dstgushed from the Jot-Space Stffess Matrx K θ that descrbes the relatoshp betwee the statc forces/torques ad correspodg deflectos the jots [8]. Both of these stffess matrces ca be mapped to each other usg the Coservatve Cogruecy rasformato [9], whch s trval f the exteral (or teral) loadg s eglgble. I ths case, the trasformato s etrely defed by the correspodg Jacoba matrx. However, f the loadg s essetal, t s descrbed by a more complcated equato that cludes both the Jacoba as well as the Jacoba dervatves ad the loadg vector [1,11]. Other specfc cases, where the above trasformato s o-trval (o-lear or eve sgular), are related to mapulators wth passve jots ad over-costraed parallel archtectures [1]. I the most geeral sese, exstg approaches to the mapulator stffess modelg may be roughly dvded to three ma groups: () the Fte Elemets Aalyses (FEA), () the Matrx Structural Aalyses (MSA), ad () the Vrtual Jot Modelg method (VJM). her advatages ad dsadvatages are brefly preseted below. A evdet advatage of the FEA-modelg s ts hgh accuracy that s lmted by the dscretzato step oly. For robotc applcato t s very attractve, sce the lks/jots are modeled wth ts true dmeso ad shape [13-15]. However, whle creasg of the umber of fte elemets, the problem of lmted computer memory ad the dffculty of the hgh-dmeso matrx verso become more ad more crtcal. Besdes the hgh computatoal efforts, ths matrx verso geerates umerous accumulatve roud-off errors, whch reduce accuracy. I robotcs, ths causes rather hgh computatoal expeses for the repeated re-meshg ad re-computg, so ths area the FEA method s usually appled at the fal desg stage oly [16,17]. * Correspodg author. el ; fax ; E-mal address: alexadr.klmchk@mes-ates.fr (A. Klmchk).

2 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators Nevertheless, ths work ths method s appled for the lks stffess matrx detfcato that ca be used further the frame of the VJM ad MSA techques. Such combato allows us to use the advatages of the FEA whle avodg tesve computatos for dfferet mapulator cofguratos. Matrx Structural Aalyss method corporates the ma deas of the FEA but operates wth rather large complat elemets such as beams, arcs, cables, etc. [18]. hs obvously leads to the reducto of the computatoal expeses ad, some cases, allows us to obta a aalytcal stffess matrx for the specfc task. For parallel robots, ths method has bee developed works [19,], where a geeral techque for stffess modelg of the mapulator wth rgd/flexble lks ad passve jots was proposed. It has bee llustrated by stffess aalyss of parallel mapulator of Delta archtecture where the lks were approxmated by regular beams. he latter causes some doubts the model accuracy compared to the combato of the FEA ad VJM techques that are beg developed here he core of Vrtual Jot Modelg method s a exteso of the covetoal rgd-body model of the robotc mapulator, where the lks are treated as rgd bodes but the jots are assumed to be complat ( order to accumulate all types of exstg flexbltes the jots oly). Geometrcally, such approxmato s equvalet to addg to the jots some auxlary vrtual jots (wth embedded vrtual sprgs). It s obvous that such lumped presetato of the mapulator stffess (that realty s dstrbuted) essetally smplfes the model. So, at preset t s the most popular stffess aalyss method robotcs. hs method was frst troduced by Salsbury [1] ad Gossel [], who assumed that the ma flexblty sources were cocetrated the actuator jots. he derved expresso defg relato betwee the jot ad Cartesa stffess matrces (Coservatve Cogruecy rasformato) became a bass for the mapulator stffess aalyss may research works. Later, these results have bee further developed order to take to accout some specfc geometrcal costras [3] ad exteral loadg [1,4]. Nevertheless, exteral loadg s assumed small eough to detect ay o-lear effects dscovered ths work. Due to ts computatoal smplcty, the VJM method has also bee successfully appled to the aalytcal stffess aalyss of a traslatoal parallel mapulator [5]. A key ssue of ths method s how to defe the vrtual sprg parameters. At the begg, t was assumed that each actuated jot s preseted by a sgle oe-dmesoal vrtual sprg [6]. Further, to take to accout the lks flexbltes, the umber of vrtual jots was creased ad each actual actuated or passve jot several traslatoal ad rotatoal vrtual sprgs were cluded [5]. he latest developmets ths area operate wth 6-dmesal vrtual sprgs detfed usg the FEA-based method [7]. hs leads to essetal creasg of the VJM method accuracy that becomes comparable wth the accuracy of the FEA-based techques, but wth much lower computatoal expeses. Hece, both for the MSA ad VJM techques the problem of accurate stffess matrces of the lks wth a complex shape become a crtcal. I the frst works, t was explctly assumed that the ma sources of elastcty are cocetrated actuated jots. Correspodgly, the lks were assumed to be rgd ad the VJM model cluded oe-dmesoal sprgs oly. I other recet works, complace of the lks has bee take to accout by troducg addtoal vrtual jots descrbg ther logtudal elastcty [6] or stffess propertes several drectos [5]. Recet developmet ths area use 6-dmesoal vrtual jots to descrbe elastcty of each lk [8]. I the most of precous works, the stffess parameters of the vrtual jots descrbg the lk elastcty (ad corporated the matrx K θ ) were evaluated rather roughly, usg a smplfed represetato of the lk shape by regular beams. Besdes, t was assumed that all lear ad agular deflectos (compresso/teso, bedg, torso) are decoupled ad are preseted by depedet oe-dmesoal sprgs that produce a dagoal stffess matrx of sze 66 for each lk. Latter, ths elastcty model was ehaced by usg complete 66 o-dagoal stffess matrx of the catlever beam [8]. hs allowed takg to accout all types of the traslatoal/rotatoal complace ad relevat couplg betwee dfferet deflectos. Other ehacemets clude the lk approxmato by several beams, but t gves rather modest mprovemet accuracy. Fgure 1 Itegrato of VJM modelg approach wth FEA-based detfcato techque of the stffess model parameters

3 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 3 Further advace ths drecto (applcable to the lks of complcated shape) led to the FEA-based detfcato techque that volves vrtual loadg expermets CAD evromet ad stffess matrx estmato usg dedcated umercal routes [7]. he latter essetally creased accuracy of the VJM-modelg whle preservg ts hgh computatoal effcecy. It s worth metog that usual hgh computatoal expeses of the FEA s ot a crtcal ssue here, because t s appled oly oce for each lk ( cotrast to the straghtforward the FEA-modelg for the etre mapulator, whch requres complete re-computg for each mapulator posture). As a result, ths approach allowed the authors to tegrate accuracy of the FEA-modelg to the VJM-modelg techque that provdes hgh computatoal effcecy. Geeral methodology of ths hybrd approach s preseted Fgure 1. However, spte of good results obtaed for some case studes, there are stll a umber of ope questos the FEA-based detfcato techque. hey clude optmal settg of the vrtual expermet (.e. defto of the mesh parameters, the jot cotact surfaces to apply forces, etc.) ad ehacemet of umercal algorthms used for computato of the stffess matrx elemets (creasg robustess wth respect to FEA-modelg errors, dstgushg zero elemets from small oes, etc.). hese ssues have ever bee gve proper atteto prevous works (oly some prelmary results have bee preseted our prevous work [9]) ad wll be the focus of ths work. Problem statemet I robotc lterature, there are several ways to obta the stffess matrx of a lk. he smplest of them assumes that a real lk (wth rather complex shape) ca be approxmated by a smple beam, for whch the stffess matrx ca be easly expressed aalytcally. Aother, more accurate, techque deals wth mult-beam approxmato where the lk s preseted as a seral cha of rgd bodes separated by vrtual sprgs. For ths presetato, the stffess matrx of the whole lk s computed usg a commo procedure kow from stffess aalyss of seral mapulators. However, spte of computatoal coveece ad better accuracy, the secod approach ca be hardly appled to may dustral mapulators (where lks may be ohomogeeous, ther shape s qute complex ad cross-secto s o-costat, etc). For stace, for the Orthoglde foot (Fgure ), eve a four-beam approxmato provdes a accuracy of oly 3-5% [7]. Fgure CAD model of the lk wth complex shape o acheve desred accuracy, here t s proposed to apply the FEA-based detfcato methodology that was frstly troduced [7] but eeds some further developmet. he correspodg algorthm s schematcally preseted Fgure 3. At the frst step, a CAD model of the lk s created, whch properly descrbes the lk shape, cross-sectos, dstrbuto ad physcal propertes of the materal (desty, Youg's modulus, Posso's rato, etc.). he, at the secod step, oe of the lk eds s fxed accordace wth cotact surfaces of a adjacet elemet (for example by cyldrcal surface for the revolute jot). At the secod ed of the lk, certa dstrbuted (or localzed) force/torque s appled accordace wth cotact surfaces of a adjacet elemet. For these settgs, the FEA-modelg s performed whch yelds the deflecto feld for a huge umber of pots geerated by the meshg procedure. From ths set, a subset (so called 'deflecto feld') s extracted correspodg to the eghborhood of the referece pot. hs feld cotas desred formato o the lk complace wth respect to the appled force/torque. Such vrtual expermets are repeated several tmes, for dfferet drectos of forces ad torques (ssue of ther magtude s dscussed the followg sectos). Ad fally, at the thrd step, the proposed detfcato procedure that gves the desred stffess matrx of the lk s appled. It s worth metog that the above descrbed algorthm has some essetal advatages, whch are ot achevable whle usg ay aalytcal techque. I partcular, here t s possble to take to accout (straghtforwardly ad explctly) the jot elastcty that usually has a umber of partculartes, such as sgfcat complace of lk/jot areas located closed to the cotact pots or surfaces [3]. For stat, for a case study preseted ths paper, some of the stffess matrx elemets are reduced by the factor of 1-1 f the jot partculartes are modeled properly. It s evdet that exstg aalytcal expressos do ot take to accout these ssues.

4 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 4 Real shape Materal propertes Jots partculartes CAD model FEA-based vrtual expermets Couplgs betwee traslatoal\rotatoal deflectos Deflectos feld Stffess matrx detfcato Fgure 3 Stffess matrx Algorthm for stffess matrx detfcato procedure Hece, the goal of ths paper s the accuracy mprovemet of the stffess matrx detfcato for mapulator elemets. As t follows from the related aalyss, partcular problems that should be cosdered here are the followg: () () () (v) (v) Further developmet of the FEA-based methodology for the lk stffess matrx detfcato, whch allows us to take to accout the lk shape, couplg betwee rotatoal/traslatoal deflectos ad jot partculartes; Developmet of umercal techque, whch computes the stffess matrx from the set of pots (dsplacemet feld) extracted from the FEA-based vrtual expermets; Evaluato of ths techque accuracy wth respect to vrtual expermet settgs (meshg type\sze, deflecto rage, fxg ad force applcato method) ad sze\shape of vrtual sesor; Mmzato of the detfcato errors by statstcal processg of the expermetal data (outles elmato, determato of the cofdece tervals ad detectg "zero" elemets of the stffess matrx); Valdato of the developed techque o applcato examples related to typcal parallel mapulators ad ther comparso wth covetoal regular-shape approxmato models. o address these problems, the remader of ths chapter s orgazed as follows. Secto 3 troduces the FEA-based methodology for detfcato of the lk stffess matrx. Secto 4 proposes a umercal techque for evaluatg the stffess matrx elemets from the feld of pots. Secto 5 focuses o the accuracy estmato. Secto 6 deals wth mmzato of the detfcato errors. I Secto 7, the developed techque s appled to the lks of Orthoglde mapulator. Ad fally, Secto 8 summarzes ma results ad cotrbutos of ths paper. 3 FEA-based approach for detfcato of lk stffess matrx Let us start from a detaled descrpto of the FEA-based methodology for the lk stffess matrx detfcato. It s based o a umber of the vrtual expermets that are coducted the CAD-evromet (CAIA wth ANSYS, for stace). Each of these expermets gves some formato o the resstace of a elastc body or mechasm to deformatos caused by a exteral force or torque. he desred stffess model s obtaed by tegratg data obtaed from several dfferet expermets, whch dffer the drecto of appled forces/torques. For relatvely small deformatos, the stffess propertes are defed through the so-called stffess matrx K, whch defes the lear relato F p K M δφ betwee the three-dmesoal traslatoal/rotatoal dsplacemets p ( px, py, pz ) ; δφ ( δx, δ y, δz ) ad the statc forces/torques F ( Fx, Fy, Fz ), M ( M x, M y, M z ) causg ths trasto. As kow from mechacs, K s a 66 symmetrcal sem-defte o-egatve matrx, whch may clude o-dagoal elemets to represet the couplg betwee the traslatos ad rotatos [7]. he verse of K s usually called the complace matrx ad s deoted as k. (1)

5 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 5 For robotc mapulators, the matrx K ca be computed sem-aalytcally provded that the stffess matrces of all separate compoets (lks, actuators, etc.) are kow wth desred precso [7]. However, explct expressos for the lk stffess matrces ca be obtaed smple cases oly (truss, beam, etc.). For more sophstcated shapes that are commoly used robotcs, the stffess matrx s usually estmated va the shape approxmato, usg relatvely small set of prmtves [5]. However, as t follows from the studes [7], the accuracy of ths approach s rather low. Hece, a geeral case, t s prudet to apply to each lk the FEA-based techques, whch hypothetcally produce rather accurate result. Fgure 4 Idetfcato expermet for the elemet wth a complex shape Usg the FEA, the stffess matrx K (or ts verse k ) s evaluated from several umercal expermets, each of whch produces the vectors of lear ad agular deflectos (p, ) correspodg to the appled force ad torque (F, M) (see Fgure 4). he, the desred matrx s computed from the lear system 1 F... F p... p k 1 m 1 m M1... M m δφ1... δφ m where m s the umber of expermets ( m 6 ) ad the matrx verse s replaced by the pseudo-verse the case of m 6. It s obvous that the case of m 6 wth specal arragemet of the forces ad torques s umercally attractve F,,,, ;,, M,, 1 x z F M F M, Fy,,, ;,,, M y,,, Fy,, ;,,,, Mz F M F M 5 5 F M F M (). (3) correspodg to the dagoal structure of the matrx to be verted. I ths case, each FEA-expermet produces exactly oe colum of the complace matrx p1 / Fx p / Fy p3 / Fz p4 / M x p5 / M y p6 / M z k δ 1 / Fx δ / Fy δ 3 / Fz δ 4 / M z δ 5 / M y δ 6 / M φ φ φ φ φ φ z ad the values (p, ) may be clearly physcally terpreted. O the other had, by creasg the umber of expermets ( m 6) t s possble to reduce the estmato error. Besdes, t s worth metog that the classcal methodology of the optmal desg of a expermet caot be appled here drectly, because t s ot possble to clude equato (4) the measuremet errors duced by vrtual expermets as addtve compoets. Some aspects of ths problem are studed secto 5.4. Hece, to obta the desred stffess (complace) matrx, t s requred to estmate frst the deflectos (p, ) correspodg to each vrtual expermet. hs ssue s the focus of the followg secto. 4 Numercal techque for evaluatg the stffess matrx elemets Usually, FEA-based expermets, the values (p, ) are computed from the spatal locato of a sgle fte elemet eclosg the referece pot. I cotrast to ths approach, here t s proposed to evaluate (p, ) from the set of pots (dsplacemet feld) descrbg trastos of a rather large umber of odes located the eghborhood of referece pot. It s reasoable to assume that such modfcato wll yeld postve mpact o the accuracy, sce the FEA-modelg errors usually dffer from ode to ode, exposg almost quas-stochastc ature. 4.1 Related optmzato problem o formulate ths problem strctly, let us deote the dsplacemet feld by a set of vector couples { p, p 1, } (see Fgure 5) where the frst compoet p defes the ode's tal locato (before applyg the force/torque), p refers to the (4)

6 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 6 ode's dsplacemet due to the appled force/torque, ad s the umber of cosdered odes. he, assumg that all the odes are close eough to the referece pot, ths set ca be approxmated by a rgd trasformato p p R( δφ) p p, 1,, (5) that cludes as the parameters the lear dsplacemet vector p ad the orthogoal 33 matrx R that depeds o the rotatoal dsplacemet. he, the problem of the deflecto estmato ca be preseted as the best ft of the cosdered vector feld by equato (5) wth respect to the sx scalar varables ( px, py, pz, δ x, δ y, δ z ) corporated the dsplacemet vector p ad the rotato matrx R. Fgure 5 Feld of pots from FEA modelg I practce, the FEA-modelg output provdes the deflecto vector felds for all odes referrg to all compoets of the mechasm. So, t s requred to select a relevat subset correspodg to the eghborhood of the referece pot p. Besdes, the ode locatos p must be expressed relatve to ths pot,.e. the org of the coordate system must be shfted to p. he latter s specfed by the physcal meag of the deflectos the stffess aalyss. o estmate the desred deflectos (p, ), let us apply the least square techque that leads to mmzato of the sum of squared resduals f 1 p p R( δ φ) p p m (6) Rp, wth respect to the vector t ad the orthogoal matrx R represetg the rotatoal deflectos. he specfctes of ths problem (that does ot allow drect applcato of the stadard methods) are the orthogoal costrat R R I ad o-trval relato betwee elemets of the matrx R ad the vector. he followg sub-sectos preset two methods for computg p,, as well as ther comparso study. 4. SVD-based soluto For the comparso purposes, frst let us brefly preset the exact soluto of the optmzato problem (6). It reles o some results from the matrx algebra referred to as Procrustes problem [31]. he correspodg estmato procedure s decomposed two steps, whch sequetally produce the rotato matrx R ad the traslato vector p. he, the desred vector of rotato agles s extracted from R. Let us trasform the orgal optmzato problem (6) to the stadard form. Frst, for the o-costraed varable p, straghtforward dfferetato ad equatg to zero gves a expresso p 1 ( ) p R I p 1 1 (7) he, after relevat substtuto ad deotg 1 pˆ p p ; 1 1 gˆ p p ( p p ), (8) 1 the orgal optmzato problem s reduced to the orthogoal Procrustes formulato f 1 gˆ Rp ˆ m. (9) R wth the costrat R R I. he latter yelds the soluto [31] R VU (1) that s expressed va the sgular value decomposto (SVD) of the matrx

7 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 7 pˆ g U ΣV, (11) 1 whch requres some computatoal efforts. Hece, the above expressos (7), (1) allow to solve the optmzato problem (6) terms of varables R ad p. Further, to evaluate the vector of agular dsplacemets, the orthogoal matrx R must be decomposed to a product of elemetary rotatos aroud the Cartesa axes x, y, z. It s obvous that, a geeral case, ths decomposto s ot uque ad depeds o the rotato order. However, for a small (that s mplctly assumed for FEA-expermets) ths matrx may be uquely preseted dfferetal form 1 δz δy R δz 1 δ x (1) δy δx 1 able 1 Evaluato of the rotato agles from matrx R Method x SVD+ r 3 r 13 r 1 SVD- r3 y r31 r1 SVD± ( r3 r3 ) / ( r13 r31 ) / r1 r1 z ( ) / SVD+as as r 3 asr 13 asr 1 SVD-as asr 3 asr 31 asr 1 SVD±as as ( r r ) / as ( r r ) / as ( r r ) / Usg ths expresso, the desred parameters δx, δ y, δ z may be extracted from R [ r j ] several ways (able 1), whch are formally equvalet but do ot ecessarly possess smlar robustess wth respect to roud-off errors. A relevat comparso study wll be preseted Secto LIN-based soluto o reduce the computatoal efforts ad to avod the SVD, let us troduce learzato of the rotatoal matrx R at the early stage, usg explct parameterzato gve by expresso (1). hs allows rewrteg the equato of the rgd trasformato (5) the form p pδ φ p ; 1, (13) that ca be further trasformed to a lear system of the followg form p δφ I P p ; 1, where P s a skew-symmetrc matrx correspodg to the vector p : pz py P pz p x (15) py px he, applyg the stadard least-square techque wth the objectve (14) f 1 p P δφ p m (16) φt, oe ca get the soluto

8 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 8 1 I P p p 1 1 δφ P P P P p that employs the 66 matrx verso. hs soluto ca be smplfed by shftg the org of the coordate system to the pot 1 p p leadg to expresso c 1 1 I p p 1 1 δ ˆ ˆ φ ˆ P P 1 P p 1 that requres verso of the matrx of sze 33. Here, P ˆ deotes a skew-symmetrc matrx correspodg to the vector p p p. ˆ c It s evdet that the obtaed expresso (18) s more computatoally effcet compared to (8) ad (1). Besdes, for some typcal cases correspodg to regular meshg patters of the FEA, expresso (18) ca be further smplfed as show the followg subsecto. (17) (18) 4.4 Aalytcal expressos for typcal case studes I practce, the pots p are dstrbuted the space a regular way, accordace wth the meshg optos chose for the FEA-modelg. hs allows us to verse the matrx ˆ P P ˆ 1 aalytcally ad to obta very smple expressos for p,. Let us cosder several patters that are useful for practce ad are preseted Fgure 6. Fgure 6 ypcal patters of the deflecto feld Case 1: Symmetrcal feld (Fgure 6a). If the feld s symmetrcal wth respect to ts cetre preseted a compact aalytcal form as p c, the soluto (18) ca be 1 p p ; 1 δ 1 ˆ φ D P p (19) 1 where the matrx D dag ( pˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ y pz px pz px py ) () s dagoal ad easly verted. Case : Cubc feld (Fgure 6b). If the feld s symmetrcal ad, addto, t s produced by uform meshg of the cubc 3 3 subspace aa a, the matrx D s expressed as Dd I where d a ( 1) 6( 1). Case 3: Plaar square feld (Fgure 6c). For the feld produced by uform meshg of the square a a located D dag d d / d / perpedcular to the x-axs, the expresso for the matrx Summarzg Secto 4, t should be oted that the proposed umercal techque s computatoally attractve ad allow smultaeous estmato both traslatoal ad rotatoal deflectos p, from the FEA-produced feld. Below t s evaluated from the pot of vew of the precso ad robustess.

9 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 9 5 Accuracy of the stffess matrx detfcato o be appled practce, the accuracy of the developed umercal techque should be evaluated wth respect to the vrtual expermet settgs (meshg type\sze, deflecto rage, fxg ad force applcato method) ad sze\shape of vrtual sesor. Let us cosder several case studes correspodg to typcal dustral applcatos. 5.1 Error sources ad ther mpact It s obvous that the ma source of estmato errors s related to the FEA-modelg that hghly depeds o the sze ad type of the fte elemets, meshg optos, corporated umercal algorthms, computer word legth ad the roud-off prcple. Hypothetcally, the accuracy ca be essetally mproved by reducg the mesh sze ad creasg the umber of dgts presetato of all varables. But there are some evdet techcal costrats that do ot allow gorg the FEA lmtatos. Aother type of errors arses from umercal dfferetato corporated the cosdered techque. Strctly speakg, the lear relato (1) s vald for rather small deflectos that may be udetectable agast the FEA-modelg defects. O the other sde, large deflectos may be out of the elastcty rage. Hece, t s prudet to fd a compromse for the appled forces/torques takg to accout both factors. 5. Ifluece of learzato ad roud-offs Sce both of the cosdered algorthms (SVD-based ad the proposed oe, see Secto 4) volve umerous matrx multplcatos, they may accumulate the roud-off errors. Besdes, they employ the frst-order approxmato of the matrx R that may create aother source of accuracy. Hece, t s prudet to obta umercal assessmets correspodg to a typcal case study. For these assessmets, there were examed data sets correspodg to the cubc deflecto feld of sze 111 mm 3 wth the mesh step of 1 mm (1331 pots). he deflectos have bee geerated va the rgd trasformato (5) wth the parameters p ( aaa,, ) ad δ φ ( bbb,, ) preseted able, able 3. All calculatos have bee performed usg the double precso floatg-pot arthmetc. able Idetfcato errors for the rotato [deg] able 3 Method 6 SVD+ 1 6 SVD- 1 7 SVD± SVD+as 1 6 SVD-as 1 7 SVD±a LIN LIN as 9 1 Rotato ampltude b Idetfcato errors for the traslato [mm] Method SVD-based techque LIN-based techque raslato ampltude a mm.1 mm 1. mm 1 mm SVD LIN As t follows from the aalyss, the fluece of the learzato ad roud-offs s eglgble for the traslato (the duced errors are less tha 1-14 mm). I cotrast, for the rotato, practcally acceptable results may be acheved for rather small agular deflectos that are less tha 1. (the errors are up to.1 ). he latter mposes a essetal costrat o the ampltude of the forces/torques the FEA-modelg that must esure reasoable deflectos. Aother cocluso cocers comparso of the SVD-based ad LIN-based methods. It justfes advatages of the proposed LIN-based techque that provdes the best robustess ad lower computatoal complexty.

10 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators Ifluece of FEA-modelg errors By ts geeral prcple, the FEA-modelg s a approxmate method that produces some errors caused by the dscretzato. Besde, eve for the perfect modelg, the deflectos the eghborhood of the referece pot do ot exactly obey the equato (5). Hece, t s reasoable to assume that the rgd trasformato (5) corporates some addtve radom errors p p R( δφ) p p ε ; 1, (1) that are supposed to be depedet ad detcally dstrbuted Gaussa radom varables wth zero-mea ad stadard devato. I the frame of ths assumpto, the expresso for the deflectos (18) ca be rewrtte as 1 o 1 ο ; δ δ ˆ ˆ ˆ p p ε φ φ P P P ε () where the superscrpt o correspods to the true parameter value. hs justfes usual propertes of the adopted pot-type estmator (18), whch s obvously ubased ad cosstet. Furthermore, the varace-covarace matrces for t, may be expressed as 1 ˆ ˆ 1 cov p I; cov δφ P P (3) allowg to evaluate the estmato accuracy usg commo cofdece terval techques. As t follows from (1), for the traslatoal deflecto p the detfcato accuracy s defed by the stadard devato ad depeds o the umber of the pots oly. I cotrast, for the rotatoal deflecto, the spatal locato of the pots s a very mportat ssue. I partcular, for the cubc feld of the sze aa a, the stadard devato of the rotato agles may be approxmately expressed as a /6. o evaluate the stadard devato descrbg the radom errors ε, oe may use the resdual-based estmator obtaed from the expresso E ( ) (3 6) p p R δφ p p. (4) 1 where E (.) deotes the expectato of the radom varable. he latter may be easly derved takg to accout that, for each expermet, the deflecto feld cossts of three-dmesoal vectors that are approxmated by the model cotag 6 scalar parameters. Moreover, to crease accuracy, t s prudet to aggregate the squared resduals for all FEA-expermets ad to make relevat estmato usg the coeffcet (3 6) m, where m s the expermets umber. I addto, to crease accuracy ad robustess, t s reasoable to elmate outlers the expermetal data. hey may appear the FEA-feld due to some aomalous causes, such as suffcet meshg of some elemets, volato of the boudary codtos some areas of the mechacal jots, etc. he smplest ad relable method that s adopted ths research s based o the data flterg wth respect to the resduals (.e. elmatg certa percetage of the pots wth the hghest resdual values). Aother practcal questo s related to detectg zero elemets the complace matrx or, other words, evaluatg the statstcal sgfcace of the computed values compared to zero. hese ssues wll be studed Secto Optmal settgs for vrtual expermets Fally, to evaluate the combed fluece of varous error sources, let us focus o the deflecto detfcato from the cubc feld of sze 111 mm 3 (1331 pots, mesh step 1 mm). I partcular, let us cotamate all deflectos usg the Gaussa ose wth the s.t.d mm that s a typcal value dscovered from the examed FEA data sets (able 4). Smlar to the prevous case, all calculatos were performed usg the double precso floatg-pot arthmetc (16 decmal dgts). he obtaed results cofrmed the ma theoretcal dervatos of the prevous subsectos. he detfcato errors obey the ormal dstrbuto (Fgure 7) but ther s.t.d. should be evaluated takg to accout some addtoal ssues. hus, the s.t.d. of the traslatoal error s about mm ad depeds oly o the FEA-duced compoet that s evaluated as 6 / mm. he fluece of the learzato ad roud-offs s eglgble here (ths compoet s less tha 1-14 mm). Also, ths type of the error does ot deped o the traslato ampltude.

11 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 11 able 4 Parameters of the FEA-modelg ose for dfferet mesh type Meshg optos ype of the mesh Abbrevato Sze of the fte elemet σ, mm 3L 3 mm Lear mesh L mm L 1 mm P 5 mm Parabolc mesh 3P 3 mm P mm Fgure 7 Hstograms for the detfcato errors ( a = 1. mm, b=.1, = mm) I cotrast, for the rotatoal deflectos, there exsts strog depedece o the ampltude (Fgure 8). I partcular, for the agular deflecto.1, the s.t.d. of the detfcato error s about deg, whle the FEA-duced compoet s 5 evaluated as / a / deg ad the learzato compoet s about deg (see able ). Moreover, the smulato results allow to defe preferable values of the agular deflecto that may be extracted from the FEA-data wth the hghest accuracy. hey show that the deflecto agles should be the rage.1. to esure the detfcato accuracy of about.% Fgure 8 Idetfcato errors for dfferet ampltudes of the rotatoal deflectos hus, the case studes cosdered Secto 5 cofrm that the proposed LIN-based algorthm esures the same accuracy as the SVD-based oe, whle possessg lower computatoal complexty. Besdes, these results gve some practcal recommedatos for settg the FEA-based expermets ad evaluatg the detfcato accuracy. 6 Mmzato of the detfcato errors o demostrate effcecy of the developed techque ad to evaluate ts applcablty to real-lfe stuatos, let us cosder a example for whch the desred complace matrx ca be obtaed both umercally ad aalytcally. A comparso of these two solutos provdes coveet bechmarks for dfferet FEA-modelg optos ad also gves some practcal recommedatos for achevg the requred accuracy.

12 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators Bechmark example As a example, let us cosder a catlever beam of sze 111 mm 3 (see Fgure 9) wth the Youg's Modulus 5 E 1 N / mm ad the Posso's Rato.66. hese data correspod to geometry ad materal propertes of a typcal robot lk studed ths work. Fgure 9 Examed catlever beam For ths elemet, a aalytcal expresso for the complace matrx ca be preseted as k11 k k 6 k k k53 k55 k6 k k (5) k where o-zero elemets are: k35 k53 L /EIy, k11 L / E A, k L /3EIz, k33 L /3EIy, k44 L / G J, k55 L /3EIy, k66 L /3EIz, k6 k6 L /EIz. Here L s the legth of the beam, A s ts cross-secto area, I y, I z are the secod momets, J s the cross-secto property. Durg modelg, the loads have bee appled at oe ed of the beam wth the other ed fully clamped. he force/torque ampltudes were determed usg expresso (3) ad the optmal accuracy settgs for the deflectos.1 1. mm ad.1., whch yelded the followg values: Fx 1 N, Fy 1 N, Fz 1 N, Mx 1 N m, M y 1 N m, Mz 1 N m. hese loads were appled sequetally, provdg 6 elemetary FEA-expermets, each of whch produced a sgle colum of the complace matrx k, accordace wth (4). 6. Optmal selecto of FEA-modelg optos Sce the detfcato errors, whch are studed here, essetally deped o the dscretzato of the FEA-based model ad defto of the deflecto feld, let us focus o the fluece of the meshg parameters of the FEA-model ad o dmesos of the vrtual sesor that specfes ths feld. Meshg optos. he adopted software provdes two basc optos for the automatc mesh geerato: lear ad parabolc oes. It s kow that, geerally, the lear meshg s faster computatoally but less accurate. O the other had, the parabolc meshg requres more computatoal resources, but leads to more accurate results. For the cosdered case study, both meshg optos have bee examed ad compared wth respect to the accuracy of the obtaed complace matrx. he mesh sze was gradually reduced from 5 to 1 mm, utl achevg the lower lmt mposed by the computer memory sze. he obtaed results (able 5 ad Fgure 1) clearly demostrate advatages of the parabolc mesh, whch allow achevg approprate accuracy of.1% for the mesh step of mm usg stadard computg capactes. I cotrast, the best result for the lear mesh s 1% ad correspods to the step of 1 mm. able 5 Maxmum errors estmato of complace matrx elemets Meshg optos ype of the mesh Abbrevato Sze of the fte elemet Lear mesh Parabolc mesh Maxmum errors elemets of detfed matrx k j 3L 3 mm 7% L mm % 1L 1 mm 1% 5P 5 mm 3.3% 3P 3 mm.19% P mm.1%

13 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 13 Accuracy of these results ca also be evaluated by the dfferece betwee the deflectos computed usg aalytcal expresso (5) ad the detfed complace matrx (assumg that the lk s uder smlar loadgs as vrtual expermets). hs allows us to smplfy physcal meag of the detfcato errors ad also to obta a weghted performace measure that eglects some detfcato falures, whose mpact o the referece pot deflecto s ot essetal. Relevat results are preseted Fgure 11, where there are preseted the worst values selected from sx vrtual expermets. hey cofrm that the mesh P (parabolc wth step mm) esures accuracy of.1% ad s preferable for practce. O the other had, as t follows from separate study, further reducto of the mesh step does ot lead to essetal reducto of the detfcato errors. Obvously, ths cocluso s vald for ths case study oly, but t ca be appled to other cases usg proper scalg of dmesos. Fgure 1 Errors (%) estmatos of o-zero elemets of the complace matrx (3L,L,1L are lear mesh wth steps 3,,1 mm; 5P,3P,P are parabolc mesh wth steps 5,3, mm) Fgure 11 Ifluece of the stffess matrx errors o the posto accuracy Defg the vrtual sesor. he developed techque operates wth the deflecto feld bouded by the vrtual sesor (see Fgure 4), whch s located the eghborhood of the referece pot (RP). As stated above, ths eghborhood should be large eough to eutralze the fluece of the FEA-duced errors, but ts ureasoable crease may lead to volato of some essetal assumptos ad, cosequetly, to a reducto the accuracy. able 6 Evoluto of the detfcato errors for dfferet vrtual sesors h-sesor k 11 k k 33 k 44 k 55 k 66 k 6 k 53 1-layer.1%.5%.5%.3%.1%.1%.4%.4%.1%.11%.13%.13%.1%.8%.9%.9%.9%.%.9%.%.%.5%.4%.4%.14%.14%.3%.4%.8%.8%.1%.%.%.19%.19%.5%.6%.4%.4%.%.13%.13%.9%.9% 1%.33%.8%.8%.46%.41%.41%.54%.54% %.86% 1.55% 1.55%.98%.91%.9% 1.3% 1.3% o get a realstc ferece cocerg ths ssue, a umber of expermets have bee carred out, for dfferet deftos of the RP-eghborhood (.e., the vrtual sesor sze). he obtaed results (able 6) show that the hghest accuracy.1% s acheved for the oe-layer cofgurato of the deflecto feld, whch s composed of the odes located o the rare edge of the examed beam. hs patter s very close to the square-type feld 11 mm studed above (see Secto 4.4). I cotrast, creasg the

14 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 14 sesor sze up to the cubc-type of 111 mm 3 leads to the detfcato error of about.8%. Hece, practce, t s reasoable to estmate the deflecto values from the feld of pots correspodg to the square-type vrtual sesor. It s worth metog that, practce, the referece pot of the lk may be located outsde the the lk materal (see Fgure ). Cosequetly, the RP-eghborhood does ot clude ay fte elemets that may be used for creatg the deflecto feld requred by the detfcato procedure. I ths case, the lk CAD-model should be complemeted wth a addtoal sold body cetered the referece pot ad restraed by the surfaces of the correspodg jot (ths body should be rgd eough to sure correctess of FEA-based smulatos). After such modfcatos, the vrtual sesor ca be defed the usual way. As t follows from our experece, proper defto of the vrtual sesor (ad addtoal sold body) as well as appled loadg (dstrbuted o the jot surfaces) play a crucal role coductg of vrtual expermets. 6.3 Statstcal processg of FEA-based data hough the fte elemet aalyss s based o strctly determstc assumptos, t cludes tedous computatos that may geerate some errors, whch may be treated statstcally. hs dea s appled below order to mprove the detfcato accuracy ad to detect the stffess matrx elemets that may be set to zero ( practce, the stffess matrces wth strctly zero elemets are rather commo, see eq. (5)). Elmatg outlers. As otced above, the FEA-modelg data may clude some aomalous samples that do ot obey the assumed statstcal propertes. hs pheomeo has bee detected of 6 expermets, (see Fgure 1) where the hstograms demostrated obvous presece of the outlers chagg the regular dstrbuto shape (local maxmums aroud the tals). For ths reaso, a straghtforward flterg techque was appled that elmated 1% the odes correspodg to the hghest resdual values (see Fgure 13). hs techque essetally mproved the detfcato accuracy, the maxmum error for the complace matrx elemets reduced from.1% to.5%. Fgure 1 Resduals for stffess model detfcato wth parabolc mesh of mm It s worth metog that here, because of the 3-dmesoal ature of the problem, each ode has bee evaluated by three resduals ad t was elmated f ay of these resduals was treated as a outler. I addto, the detaled aalyss showed that the outlers were cocetrated at the beam edges, whch cofrms prevous assumptos cocerg the FEA-duced errors. Elmatg o-sgfcat elemets. Accordg to (3), the desred complace matrx cludes a umber of zero elemets (6 of 36), but the proposed detfcato procedure may produce some small o-zero values. o evaluate ther statstcal sgfcace, for each elemet k j the cofdece terval was computed. Relevat computatos were performed usg expressos for the varaces of the deflectos (1) ad the s.t.d. value of the FEA-modelg ose, whch was estmated as (4), to be adopted to correspodg elemets of the matrx k (ths procedure volves smple dvso by the magtude of the appled force or torque) mm (by averagg for all 6 expermets). he, the computed cofdece tervals were scaled accordace wth Usg ths approach, the complace matrx was revsed by assgg to zero all o-sgfcat elemets. he employed "decso support" algorthm treated a elemet as a o-sgfcat oe f ts cofdece terval cluded zero. As show able 7, ths techque allowed us to detect all 6 zero elemets metoed above. It should be also oted that all o-zero elemets were evaluated as sgfcat oes, wth essetal safety factor of 1 ad hgher.

15 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 15 able 7 Elmatg o-sgfcat elemets Fgure 13 Flterg of the deflecto feld outlers Parameter Estmated value CI Exact value k ± k. 1 ± k ± k ± k ± k ± k ± k ± k ± k ± k ± k ± k ± k ± k ± k ± Remarks ad commets. he preseted llustratve example that deals wth a classcal elemet (catlever beam) cofrmed valdty of the developed method but also demostrated some lmtatos of the FEA-modelg wth respect to the stffess aalyss. I partcular, some (ot very essetal but o eglgble) dsagreemet betwee umercal values of the appled forces/torques ad ther values extracted from the modelg protocol were detected. Besdes, there are a umber of o-trval ssues defg modelg optos that are ormally set by default. All these factors cotrbute to the accuracy, but a practcally acceptable level.1% ca be acheved rather easly, usg stadard computg facltes. Some addtoal ehacemet ca be acheved symmetrzg the obtaed matrx k : ( k k ) / that s motvated by the physcal reasos. 7 Applcato example: stffess matrces for Orthoglde lks Let us apply the proposed methodology to the detfcato of the lk stffess matrces for a parallel mapulator of Orthoglde famly [3]. he prcpal compoets of the mapulator are preseted Fgure 14, where the elemets (b, e, f) are treated as flexble oes ad the elemet (c) s assumed to be rgd. I prevous works, relevat stffess matrces have bee obtaed ether va sgle-beam approxmato [5] or by usg FEA-based detfcato procedure wth lear meshg opto [7]. Besdes, jots partculartes have ot bee take to accout ad ther fluece o the elemets of the stffess matrx was ot studed. Hece, t s qute possble that some stffess matrx elemets have bee detfed wth essetal errors.

16 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 16 I accordace wth Secto 6.1, the FEA-based vrtual expermets were performed usg the parabolc mesh of sze mm (P Fgure 1) ad the test forces/torques of 1 N ad 1 N m respectvely, whch, for cosdered lk sze ad materal propertes, correspod to the area of lear force-deflecto relato. As t follows from able 5, such settgs esure accuracy of about.1%. Idetfcato results are preseted able 8, where the obtaed matrces clude a umber of zero elemets that were detected usg the techque preseted Secto 6.. It s also worth metog that these matrces are symmetrcal, whch cofrms valdty of the developed method. Fgure 14 CAD model of Orthoglde ad ts prcpal compoets: Orthoglde (a), foot (b), ed-effector (c), parallelogram (d), parallelogram axs (e) ad bar (f) For comparso purposes, smlar matrces have bee computed usg other methods (sgle- ad mult-beam approxmatos, the FEA-modelg wth lear meshg). hey are preseted able 9 ad show essetal dssmlarty the evaluato of some matrx elemets by dfferet methods. For stace, for the parallelogram bar (.e., lk (f) Fgure 14), the torsoal complace defed by the elemet k 44 dffers by a factor of 13. he ma reaso for ths s that the developed techque takes to accout the jot partculartes that defe the force dstrbuto rule for the appled loadgs (prevous results assumed that the loadg was localzed the referece pot). However, the fal cocluso cocerg accuracy of the obtaed stffess matrces ca be obtaed after tegrato of these matrces the stffess model of the etre mapulator ad comparg t wth a straghtforward FEA-modelg of the mapulator assembly. Hece, preseted examples cofrm advatages of the developed stffess matrx detfcato techque ad demostrate ts ablty to take to accout complex lk shapes as well as jot partculartes related to the force/torque dstrbuto. I addto, t produces symmetrcal matrces wth some zero elemets accordace wth physcal propertes of the cosdered mapulator compoets. I the followg chapters, these matrces wll be used for the VJM-based stffess modelg of parallel mapulators. able 8 Complace matrces of Orhoglde lks Lk Complace matrx foot k Foot axs k Axs bar k Bar

17 A. Klmchk, A. Pashkevch, D. Chablat, CAD-based approach for detfcato of elasto-statc parameters of robotc mapulators 17 able 9 Comparso of the complace matrx elemets obtaed from dfferet methods Method k 11 [mm/n] k [mm/n] Lk (b): foot k 33 [mm/n] Complace matrx elemets k 44 [rad/n mm] k 55 [rad/n mm] k 66 [rad/n mm] Sgle-beam approxmato Four-beam approxmato FEA-based evaluato (lear mesh) FEA-based evaluato (parabolc mesh) Lk (e): parallelogram axs Sgle beam approxmato FEA-based evaluato (lear mesh) FEA-based evaluato (parabolc mesh) Lk (f): parallelogram bar Sgle-beam approxmato FEA-based evaluato (lear mesh) FEA-based evaluato (parabolc mesh) Coclusos he paper proposes a CAD-based approach for detfcato of the elasto-statc parameters of the robotc mapulators. he ma cotrbutos are the areas of vrtual expermet plag ad algorthmc data processg, whch allows to obta the stffess matrx wth requred accuracy. I cotrast to prevous works, the developed techque operates wth the deflecto feld produced by vrtual expermets a CAD evromet. he proposed approach provdes hgh detfcato accuracy (about.1% for the stffess matrx elemet) ad s able to take to accout the real shape of the lk, couplg betwee rotatoal/traslatoal deflectos ad jot partculartes. o compute the stffess matrx, the umercal techque has bee developed, ad some recommedatos for optmal settgs of the vrtual expermets are gve. I order to mmze the detfcato errors, the statstcal data processg techque was appled. he advatages of the developed approach have bee cofrmed by case studes dealg wth the lks of parallel mapulator of the Orthoglde famly, for whch the detfcato errors have bee reduced to.1%. Further developmet ths area wll focus o automato of some operatos that curretly are performed the teractve mode ad full tegrato of the developed techque exstg CAD evromet. I partcular, t s reasoable to develop a dedcated routes that smplfy data exchage betwee the CAD ad the computatoal procedures as well as defto of the smulato parameters whle performg the vrtual expermets. 9 Ackowledgmets he work preseted ths paper was partally fuded by the Rego Pays de la Lore, Frace, by the project ANR COROUSSO, Frace ad FEDER ROBOEX, Frace. 1 Refereces [1] R. Avles, J. Vallejo, J. Agurrebeta, I.F. de Bustos, G. Ajura, Optmzato of lkages for geeralzed rgd-body gudace sythess based o fte elemet models, Fte Elemets Aalyss ad Desg 46 (1) [] B. Sclao, ad O. Khatb, Sprger Hadbook of Robotcs, ISBN: , Berl, 8 [3] J.-P. Merlet, Parallel Robots, Kluwer Academc Publshers, Dordrecht, 6. [4] I. yap, G. Hovlad, Kematc ad elastostatc desg optmzato of the 3-DOF Gatry-au parallel kamatc mapulator, Modellg, Idetfcato ad Cotrol, 3() (9) [5] J. Duffy, Statcs ad Kematcs wth Applcatos to Robotcs, Cambrdge Uversty Press, New York, [6] J. Ageles, Fudametals of Robotc Mechacal Systems: heory, Methods, ad Algorthms, Sprger, New York, 7. [7] J. Kövecses, J. Ageles, he stffess matrx elastcally artculated rgd-body systems, Multbody System Dyamcs 18() (7) [8] N. Cblak, H. Lpk, Sythess of Cartesa Stffess for Robotc Applcatos, I: IEEE Iteratoal Coferece o Robotcs ad Automato (ICRA), Detrot, MI, 1999(3), pp