Longitudinal and Lateral Vibration Analysis of Cables in a Cable Robot using Finite Element Method

Transcription

1 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 Longituina an Latra Vibration Anaysis of Cabs in a Cab Robot using Finit Emnt Mtho H. Tourajizah* Dpartmnt of Mchanica Enginring, Univrsity of Kharazmi, Iran *Corrsponing author, E-mai: Tourajizah@khu.ac.ir M. Yousfzah, M. H. Koraym Dpartmnt of Mchanica Enginring, Iran Univrsity of Scinc an Tchnoogy, Iran E-mai: myousfzah@iust.ac.ir / hkoraym@iust.ac.ir Rciv: 4 Jun 5, Rvis: 5 August 5, Accpt: Sptmbr 5 Abstract: In this papr, vibrationa rspons of a variab-ngth cab in ongituina, atra an torsiona irctions is anays in a cab robot using FE mtho. Th fxibiity of cabs has rmarkab ffct on positioning of th nffctor in cab robots. Aso consiring th fact that th ngth of th cabs ar tim pnnt in a ynamic cab structur ik robocran, th numrica approachs ar prfrab compar to anaytic soutions. To o so, th cab is ivi into finit mnts in which th virtua work quation an Garkin mtho can b impmnt for th quations. Consiring th stiffnss matrix, th charactristic quations an Eign vaus of ach mnt can b fin. A simuation stuy is on in th ASIS on a panar robocran with -DOF an aso for a spatia cas with 6-DOF that is contro by th ai of six variab-ngth fxib cabs in th spac for two iffrnt typs of soi an fxib nffctors. Who th cab robot fxibiity is anayz simutanousy insta of sparation cacuation of ach cab. ot ony a of th 3-D vibrating bhaviour of th who structur is stui in this papr but aso th ngths of th cabs ar consir as variab. Th vibrating rspons of mo shaps, ampitu an frquncis ar xtract an anays, an th rsuts ar compar for two cas of soi an fxib n-ffctor which shows th ffct of th fxibiity in th position of th n-ffctor an th tnsion of th cabs in iffrnt situations. Kywors: Cab, FEM, Garkin mtho, Robocran, Wight rsiua mtho Rfrnc: H. Tourajizah, M. Yousfzah, H. Koraym, Longituina an Latra Vibration Anaysis of Cabs in a Cab Robot Using Finit Emnt Mtho, Int J of Avanc Dsign an Manufacturing Tchnoogy, Vo. /o., 7, pp.. Biographica nots: H. Tourajizah rciv his MSc from Iran Univ. of Scinc an Tch. in 8 in appi mchanica sign. H is currnty a PhD caniat in IUST with a numbr of rsarch pubications. His rsarch intrsts incu robotic, automotiv ng., contro an optimization, an mchatronic systms. M. Yousfzah rciv his BSc in Mchanica Enginring in an his MSc in 3 in th fi of automotiv systm sign. H is currnty a PhD stunt in Iran Univrsity of Scinc an Tchnoogy in th fi of contro. H has yars of xprinc in spcia machinry sign for automotiv inustry an taching xprinc in iffrnt univrsitis. M. Habibnja Koraym rciv his MSc in Mch. Eng. from Amirkabir Univ. of Tch. in 987. H obtain his PhD in Mch. Eng. from Univ. of Woongong, Austraia, in 994. H is a Profssor in Mch. Eng. at IUST.

2 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 ITRODUCTIO Cab robots ar on of th nw gnrations of para robots in which th n-ffctor is contro by th ai of svra cabs that just can xrt tnsiona forc to th n-ffctor. Th appi cabs which ar us as th actuator of th n-ffctor shou b fxib nough to provi th possibiity of its rotation aroun a rum without forming a rotationa spring on th rum which is a ngativ rawback on th cacuat kinmatic an kintic of th robot. That s why a itt fxibiity of th cabs is unavoiab. Incrasing th inrtia forcs of th En-ffctor an cabs as to consirab formation in th cabs an consqunty gnrats significant vibrationa rror on th position of th n-ffctor which infuncs th accuracy of th robot. Hnc, vibration of th cabs is th most changing probm in controing of such robots which can caus normous viation in th position of th n-ffctor [, ]. Both of axia an transvrsa fxibiitis appar in ths cabs. Cab robots ar manufactur unr two main catgoris: Unr-constrain an Fuy-constrain. Accoring to an anaytic rsarch on a fuyconstrain cab robot, th transvrsa fxibiity is ignorab compar to th axia fxibiity [3]. Howvr this assumption is not compty vai for unr-constrain robots. Cab suspn robots which ar unr-constrain ar so popuar sinc thy o not hav imit work spac [4, 5]. Th most important chang in orr to anays th vibrationa rspons of such cabs is thir variab ngth uring th ynamic procss of th robot which maks it ifficut to sov thir PDE using orinary soutions. In prvious rsarchs continuum an muti ynamic mos of cabs wr us for vibrationa moing. Authors in [6] prsnt a procur for stuying th ynamics of a sing variab ngth cab systm. Th cab is mo as a chain an is trat as a mutiboy systm. Th chain inks in turn ar mo as ump masss. Hr th ynamic is xtract for a sing cab. Som vibrating anaysis of astic cab robots can b foun in [7]. This papr iscusss a fback contro mtho for incompty rstrain wir-suspn mchanisms an anti-sway contro mtho with xact inarization using invrs ynamics is sign for incompty rstrain typ mchanism. Hr sinc th cab is incompty rstrain, th vibration an swing is unavoiab which is nutraiz using th mntion controing stratgy. Again th moing an simuation is on for a sing rop. Dynamic an contro of a compt robocran actuat by svn cabs is stui in [8] howvr th important probm of fxibiity of th cabs is ignor hr. Workspac stuy of ths kins of robocrans is on in [9] in which again th ffct of fxibiity is not consir in th obtain workspac. Anothr rsarch of soi robocran is on in [] in which a iffrnt mtho of controing of th cab robots is prsnt using activ bounary contro. Sinc th vibrating anaysis of cabs using anaytic mthos, spciay for th tim pnnt ngth vrsions is xtrmy ifficut, numrica agorithms ar prfrr in som itraturs. Vibration anaysis of a sing cab with a constant ngth is on in [] using th FE mtho, an it is xtn in [] for th variab-ngth on. FEM an FEA ar us togthr in orr to anayz th ynamic an panar vibration of a cab in [3]. Vibration anaysis of th cabs us in a simp structur is on in [4]. For th mntion rsarchs again th numrica mthos ar mpoy for a sing cab which os not show th ffct of fxibiity of a compt ynamic cab structur ik robocrans on viation of thir n-ffctors. Thrfor, consiring th mntion shortag in th itraturs, in this papr fxibiity anaysis of a ynamic cab structur ik cab robots is xtract for tim-variab ngth cabs using numrica finit mnt approach by which th ffct of th fxibiitis can b asiy invstigat on th viation of th robot n-ffctor. Garkin mtho is us hr in both ongituina an atra irctions. This cacuation is on for who of th cabs an thir structur simutanousy insta of anayzing ach cab sparaty. FEM is chosn hr sinc a fast cacuation with an accptab accuracy cou b provi for a variab ngth cas. Sinc th stui cab robot is unr-constrain, both of atra an ongituina vibrations n to b anayz simutanousy for th robot with tim variab cabs ngth. This stuy is first on for a thr-cab panar robot with two grs of from an it is thn xtn for a 6-cab spatia robot with six grs of from. Dynamic formuation of th panar structur can b foun in [5, 6] whi th spatia cas is prsnt in [7]. First of a, ynamic moing of a sing cab rop is xtract using Lagrangian mtho which rsuts in ongituina, atra an torsiona vibration quations of th cab. Aftrwars, rsutant iffrntia quations ar sov using wight rsiua functions an th Garkin mtho. Using th finit-mnt mtho, th rop is ivi into finit mnts which giv us th shap functions, stiffnss matrix, charactristic quation of th systm an finay Eign vaus or natura frquncis of th vibration of th systm. This procss is thn xtn for a six-cab robot with variab ngth cabs. Corrctnss of th prsnt

3 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 3 thortica formuations is invstigat using a simuation stuy for both of panar an spatia samps of cab robots. A 3-cab robot with a massiv nffctor an two grs of from is first mo in th ASYS an thn it is xtn for a six cab trianguar n-ffctor with six grs of from. Th first simuation anayss th panar vibrations of th cab whi th scon on covrs th vibration of th cabs in thr irctions. Aso an intrsting comparison is prform for a robot structur in which th n-ffctor s vibration is ignorab compar to th cab s vibration an th structur in which th nffctor is aso fxib. atura frquncis, mo shaps an maximum strss of th cabs for a of th mntion systms ar gain an anayz. Rsuts show that cab vibrations can affct th position of th n-ffctor in th systms which ar not quipp with a suitab contror spciay for th spatia cas. DYAMIC FORMULATIO First th formuation of a sing cab is rprsnt. Consiring a cab compos of svra twist rops ik Fig. rsuts in ongituina ongation [8], Hr is rotationa fxibiity of th cab, U is its ongituina fxibiity aong Z axis, f an q ar forc an torqu of th cab rspctivy, T is th tnsion of th cab at position S an C is torsiona strss of th cab. Impmntation of wight rsiua function an Garkin mtho rsuts in th foowing iffrntia quations: L U W U( m U k U K )x L W ( I k3u k 4 )x Diviing th cab into finit mnts an using partia intgra, w hav: L k k L 4 k4 k () ku W U U( m U ) [ U U ] U (3) k W U( I ) [ ] U Shap functions can b fin as: U( ) U aa U ( ) tt (4) By supposing th shap functions as bow: U( ) C sin( ) C cos( ) L m / k ( ) D sin( ) D cos( ) L I / k 4 (5) Th fina rsutant shap functions ar: Fig. Schmatic of a sing rop [8] Using Lagrangian mtho, th foowing ynamic quations can b achiv which incu two paramtrs of anguar an ongituina ispacmnt: m U ku K I k3u k4 () whr m an I ar th mass an inrtia of th cab rspctivy, K i is fxibiity cofficint of th rop, is tim rivation of rotationa fxibiity an th inx x inicats rivation rspct to S. sin( L( )) ( ) a sin( L ) sin( L ) ( ) a sin( L ) m / k sin( rl( )) ( ) r sin( rl ) sin( rl ) ( ) r sin( rl ) r I / k 4 (6) By substituting th mntion shap functions into th wight rsiua function quations, w hav:

4 4 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 K u U K W U U W whr: K K 4 u u W un k( ) u 4* 4 n 4* k( ) k( ) k( ) uncoup u couping (7) (8) Garkin rsuts in th foowing gnra charactristic quation: k W W Un K( ) Un mnts (9) atura frquncis of th systm can b obtain by th ai of mntion charactristic quation. Th sam procur can b on for a six-cab robocran of th Fig.. us anaytica mthos for anaysing thir fxibiitis. First of a ynamic quation of th systm is xtract using Hamitonian mtho. Kintic an potntia nrgis of th systm ar cacuat without consiring any xtrna forcs: u v T / (( ) ( ) ) / I(( ) ) t t t u v V / AE(( ) ( ) ) / k(( ) ) x x x () whr u, v an ar ongituina, atra an torsiona ispacmnt of th cabs rspctivy. Aso is th nsity, E is th asticity mou an A is th cross sction ara of th cabs. Using Hamitonian formuation, th foowing ynamic quations can b prsnt: ( T V ) t u( x,t ) u( x,t ) k t x v( x,t ) ( x,t ) k k 3 x x v( x,t ) u( x,t ) k t x v( x,t ) ( x,t ) k k 3 x x ( x,t ) u( x,t ) k 3 t x v( x,t ) ( x,t ) k3 k 33 x x () In orr to us th wight rsiua mtho, th foowing functions ar chosn: Fig. Schm of a spatia cab robot [9] Hr six DOFs of th owr trianguar n-ffctor pat ar contro using six fxib cabs connct to th uppr trianguar fix pat. Th ngths of th cabs ar tim pnnt which maks it ifficut to u( x, t) U ( x)sin t v( x, t) V ( x)sin t ( x, t) ( x)sint () Substituting th abov functions in th ynamic quations, vibrating formuations can b fin as bow: m U ku m V ku m U k3u k V k k 3 V V k 3 k 3 33 k (3) Using th wight rsiua mtho, th foowing virtua work formuation can b gain, which fins th vibrating ampitu:

5 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 5 W W W U V L( t) U ( m U k U L( t) L( t) 3 V ( m V k U ( m k U k V 3 k V k V k k k 3 STIFFESS MATRIX CALCULATIO ) x ) x ) x (4) ow w can ivi th cab into finit ongituina mnts with two nos an thr grs of from. Consiring bounary conition, th foowing virtua work formuation can b obtain: W W W U V U ( ( k U V ( ( k U ( ( k 3U k k k V k ) Um U ) V k V k 3 3 ) Vm V ) 3 ) m ) 33 (5) Accoring to th mntion quations, Garkin quations prouc th foowing wight functions which ar th vibrating ampitu of th systm: U( ) Usin( ) U cos( ) L( t ) m / k V( ) V sin( ) V cos( ) L( t ) m / k ( ) sin( ) cos( ) L( t ) m / k33 (6) Vibrating ampitus ar finab accoring to th shap function as bow: U U ( ) U U U V V ( ) V V V ( ) (7) In orr to hav mor accurat rsuts, shap functions ar suppos to b harmonic: sin( ( )) U sin sin( ) U sin sin( ( )) V sin sin( ) V sin sin( ( )) sin sin( ) sin (8) Substituting th abov shap functions in th virtua work formuations rsuts in: k U, U, U WU U U ( U, U, U k U V U V V U V U V V k 3 U U U U k V, V, V WV V V ( V, V, V k V U V U U V U V U U k 3 V V V V k 33,, W (,, k 3 U U U U U U k 3 V V V V V V (9) An so by fining th movmnt vctor of th mnts as: a U U V V () Stiffnss matrix of th who systm can b cacuat as: T

6 6 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 K K K 3 KT K K K3 K3 K3 K 33 k U, U, k U V U V K K U, U, U V U V () k 3 U U k V U V U K3 K U U V U V U k V, V, k 3 V V K K3 V, V, V V k 3 U U k 3 V V K3 K3 U U V V k 33,, K33,, 4 ASYS MODELIG Anaysis Typ Emnt typ Outsi iamtr Wa thicknss Pip wa mass TABLE PARAMETERS OF PLAAR MODELIG Structura Moa Sub-spac PIPE6.4 unit. unit.3 unit 6 Pip axia stiffnss 4.5 unit Linar Eastic Isotrop Matria proprtis 6 E=4.5 unit,υ=.3 Msh typ Lin Msh Ky Point (,,) Ky Point (-,,) Ky Point3 (,,) A. MODELIG OF PLAAR ROBOT Bas on th mntion formuations, two cass of th cab robots ar simuat in ASYS softwar an thir natura frquncis an mo shaps ar gain. Th first structur is a panar cab robot with thr cabs an two grs of from ik Fig. 3. Hr two DOFs of th ump mass of m (X, Y) ar contro using thr fxib cabs with tim pnnt ngth of L i which ar connct to thr fix rums of inrtia J i an rotationa amping cofficint of c i at th position of A i with th ang of. Th istanc of th rums is L B. i Fig. 4 ASYS mo of th panar cab robot Fig. 3 Schmatic of a panar cab robot [6] This structur is mo in ASYS by making fu constraint in th trianguar shap fram an aso making z irction constraint for th cabs movmnt in orr to stuy thir panar vibration (Fig. 4), whr th rat paramtrs ar prsnt in Tab. Fig. 5 Schmatic of a spatia cab robot [9] B. MODELIG OF SPATIAL ROBOT In th scon cas thr is a spatia cab robot with six cabs an six grs of from for th trianguar shap n-ffctor as iustrat in Fig. (5). Hr F n is

7 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 7 th fix goba coorinat attach to th fix uppr pat of th robocran, F b is oca coorinat of th moving n-ffctor, A,B,C ar th cornrs of th fix pat an E,D,F ar th cornrs of th moving nffctor an ths points ar connct using six fxib cabs of tim pnnt ngth. Th bas triang is constrain in a of its grs of from an th n ffctor is fr to vibrat through th cabs (Fig. 6): 5 SIMULATIO STUDY A. SIMULATIO OF PLAAR ROBOT Using th mntion paramtrs rsuts in th foowing noa mo shap functions of Fig. 7 an frquncis of Tab 3: Fig. 6 ASYS mo of a spatia cab robot Rat paramtrs can b foun in Tab : Anaysis Typ TABLE : PARAMETERS OF SPATIAL MODELIG Structura Moa Sub-spac Emnt typ PIPE6 Outsi iamtr.4 unit Wa thicknss Pip wa mass. unit.3 unit Pip axia stiffnss unit Matria proprtis Linar Eastic Isotrop 6 E=4.5 unit,υ=.3 Msh typ cabs: Lin msh pats: Trianguar Distanc btwn bas an n-ffctor En-ffctor Bas units Ky-point:(,,) Ky-point:(-,,) Ky-point3:(,,) Ky-point:(,3,) Ky-point:(-,3,) Ky-point3:(,-,)

8 8 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 for a ot of bounary conitions in th cas of panar robot. Tab 4: First tn natura frquncis of th fxib n-ffctor spatia systm Stp Tim (sc)/frquncy(hz) B. SIMULATIO OF SPATIAL ROBOT Spatia simuation is on using two iffrnt conitions. First th n-ffctor is suppos to b astic that its vibration is not ignorab compar to th cabs vibrations. In this conition th strss of th cabs is: Fig. 7 First fiv mo shaps of th panar systm TABLE 3: FIRST TE ATURAL FREQUECIES OF Stp THE PLAAR ROBOT Tim (sc)/frquncy(hz) It can b sn that th critica vibrations ar occurr at th mi of th cabs an as a rsut, it os not criticay affct th position of th n-ffctor. Th critica vibration causs significant ispacmnt at th cabs with ow frquncy an can b occurr asiy MIIMUM VALUES ODE VALUE MAXIMUM VALUES ODE VALUE Th sam no numbrs ar rat to iffrnt cabs. Aso th natura frquncy of th who systm is prsnt in Tab 4. First fiv mos of th systm in noa ispacmnt contour ar shown in Fig. 8. In this cas two catgoris of fxibiity ar obsrvab in th mo shaps. First two mos ar mosty affct by ongituina fxibiitis of th cabs whi th ast thr mos ar mosty affct by atra vibrations. It can b sn that for th cass in which th ongituina fxibiitis of th cabs ar ignorab compar to thir atra vibrations, th position of th n-ffctor is not consiraby viat whi th rror is not ignorab for th cass in which th ongituina vibrations ar ominant.

9 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 9 Fig. 8 First fiv mo shaps of th fxib n ffctor spatia cas TABLE 5: FIRST TE ATURAL FREQUECIES OF THE RIGID ED-EFFECTOR SPATIAL SYSTEM Stp Tim (sc)/ Frquncy(Hz) So it can b concu that th structiv fxibiity of a spatia cab robot which ns to b inhibit by a propr contror is its ongituina vibrations. Hr, in contrary to panar cas, th ampitu of vibrations is suprpos at th point of th n-ffctor which as to a big viation. In th scon approach th vibration of th n-ffctor is consir ignorab compar to th vibration of th cabs. Thrfor, just th vibration of th cabs can b anays hr, whr th natura frquncis ar shown in Tab 5. Th amounts of strss in th cabs ar cacuat as: MIIMUM VALUES ODE VALUE E+ MAXIMUM VALUES ODE VALUE.3E+.7E+.4E+.39E+ It can b sn that in this cas both of th strss of th cabs an ampitu of atra vibrations ar incras whi its frquncy is cras rspct to prvious

10 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 stuy which shows that most of th vibrating nrgy hr is consum to vibrat th cabs whi for th formr cas, fxibiity of th n-ffctor issipats a sction of th nrgy which as to owr vibrating rspons of th cabs. Th first fiv mos of this systm in th noa ispacmnt contour ar shown in Fig. 9. As it was xpct th n-ffctor is rmain soi hr. It can b obsrv that hr, in contrary to fxib n-ffctor cas most vibration of th cabs ar atra which o not xtrmy affct th position of th n-ffctor. Fig. 9 First fiv mo shaps of th rigi n-ffctor spatia cas Athough th ampitu of ths vibrations is highr than prvious cas, thy o not isturb th accuracy sinc no ongituina fxibiity is prouc. So it can b concu that proviing a soi n-ffctor for th spatia cab robot hps its accuracy whi th rmain ongituina vibrations can aso b amp using a contror. 7 COCLUSIO In this papr vibrating formuation of th cabs in ongituina, atra an torsiona irctions was rprsnt for a variab-ngth cab using FEM. A simuation stuy was on for two samps of cab structurs ik robocran in which th ngth of th cabs is tim-variab. FEM was prform as a strong vibrating anaysr too for who of th cab robot structur insta of sparation anaysis of ach cab. ot ony th 3-D vibrations of th cabs wr stui but aso th ngths of th cabs wr consir variab. Rsuts ar prsnt for a cas of panar robot with thr cabs an a spatia robot with six cabs.

11 Int J Avanc Dsign an Manufacturing Tchnoogy, Vo. / o. / March 7 It was sn that th critica vibration by which th maximum ispacmnt an minimum frquncy is prouc an can b occurr asiy for a ot of bounary conitions in th cas of panar robot ar occurr at th mi of th cabs an as a rsut, it os not criticay ffct th position of th nffctor. But th rsut is opposit in th cas of spatia robot. In this cas bcaus of spatia structur of th robot in which th ampitu of vibrations is suprpos at th point of th n-ffctor, an important rror at th position can b obsrv which shows th ncssity of signing a propr vibration contror. On th othr han comparing th rsuts of soi an fxib n-ffctor for spatia cas show that in th cas of th soi n-ffctor most vibrations ar atra vibration which os not caus a major ispacmnt rror in th n-ffctor but th ampitu an strss of th vibrations is highr an frquncy of th vibrations is owr. This is contribut to th fact that in this cas a of th vibrating nrgy is xrt on th cabs. Howvr th vibrations in th fxib n-ffctor systm can b issipat by transmitting a sction of vibrating nrgy to th n-ffctor an thus proucs a owr ampitu an strss with biggr frquncis but bcaus of its ongituina natur it has mor structiv ffct on th position an accuracy of th systm. This phnomnon shows that a soi n-ffctor can a to a mor accurat motion of th robot. ACKOWLEDGMETS Th authors wou ik to thank Rsarch an Tchnoogy Vic Chancor of Kharazmi Univrsity for his financia support. REFERECES [] Zhang, X., Mis, J. K., an Cghorn, W. L., Couping Charactristics of Rigi Boy Motion an Eastic Dformation of a 3-PRR Para Manipuator with Fxib Links, Mutiboy Systm Dynamics, Vo., o., 9, pp [] Caraccioo, R., Richii, D. an Trvisani, A., Exprimnta Vaiation of a Mo-Bas Robust Contror for Muti-Boy Mchanisms with Fxib Links, Mutiboy Systm Dynamics, Vo., 8, pp [3] Diao, X. an Ma, O., Vibration Anaysis of Cab- Drivn Para Manipuators, Mutiboy Systm Dynamics, Vo., o. 4, 9, pp [4] Oh, S.R., Mankaa, K.K., Agrawa, S.K. an Abus, J.S., Dynamic Moing an Robust Contror Dsign of a Two-Stag Para Cab Robot, Mutiboy Systm Dynamics, Vo. 3, o. 4, 5, pp [5] Hyn, T. an Worn, Ch., Dynamics an Fatnss- Bas Contro of a Kinmaticay Untrmin Cab Suspnsion Manipuator, Mutiboy Systm Dynamics, Vo. 6, o., 6, pp [6] Kamman, J. W. an Huston, R. L., Mutiboy Dynamics Moing of Variab Lngth Cab Systms, Mutiboy Systm Dynamics, Vo. 5, o. 3,, pp. -. [7] Yanai,., Yamamoto, M. an Mohri, A., Fback Contro for Wir-Suspn Mchanism with Exact Linarization, Intignt Robots an Systm, IEEE/RSJ Intrnationa Confrnc, 3,, pp [8] Fang, Sh., Franitza, D., Toro, M., Bks, F., an Hir, M., Motion Contro of a Tnon-Bas Para Manipuator Using Optima Tnsion Distribution, IEEE/ASME, Transaction on Mchatronics, Vo. 9, o. 3, 4, pp [9] Pappas, G. J., Lygros, J. an Gobo, D.., Stabiization an Tracking of Fback Linarization Systms unr Input Constarints, Intignt Machins an Robotic Laboratory Dpartmnt of Ectrica Enginring an Computr Scinc, Univrsity of Caifornia at Brky, Brky CA-947. [] Oh, S. R. an Agrawa, S. K., Cab-Suspn Panar Para Robots with Runant Cabs, Contrors with Positiv Cab Tnsions, Mchanica Systms Laboratory, Dpartmnt of Mchanica Enginring, Univrsity of Dawar, wark, DE-976, U.S.A, 5. [] Hashmi, S. M. an Roach, A., A Dynamic Finit Emnt for Vibration Anaysis of a Cab an Wir Rop, Asian Journa of Civi Enginring, Vo. 7, o. 5, 6, pp [] Wang, P. H., Fung, R. F. an L, M. J., Finit Emnt Anaysis of a Thr Dimnsiona Unrwatr Cab with Tim Dpnnt Lngth, Journa of Soun an Vibration, Vo. 9, o., pp. 3-49, 998. [3] Gorgakis, C. T. an Tayor, C.A., oninar Dynamics of Cab Stays, Dpartmnt of Civi Enginring, Earthquak Enginring Rsarch Cntr, Univrsity of Bristo, UK, Journa of Soun an Vibration, o. 8, 5, pp [4] Fung, R. F., Lu, Y. an Huang, S. C., Dynamic Moing an Vibration Anaysis of a Fxib Cab- Stay Bam Structur, Journa of Soun an Vibration,Vo. 54, o. 4,, pp [5] Wiiams, R. L. an Gaina, P., Rossi A., Panar Cab Dirct Drivn Robots Part, Kinmatics an Statics, ASME Dsign Tchnica Confrncs,. [6] Wiiams, R. L., Gaina, P. an Rossi A., Panar Cab Dirct Drivn Robots Part, Dynamics an contro, ASME Dsign Tchnica Confrncs,. [7] Ap, A. B. an Agrawa, S. K., Cab Suspn Robots, Dsign, Panning an Contro, Dpartmnt of Mchanica Enginring, Univrsity of Dawar, wark, DE- 976,USA,. [8] Samaras, R.K., Skop, R.A. an Miburn, D.A., An Anaysis of Coup Extnsiona-Torsiona Osciations in Wir-Rop, Transaction of th ASME, Journa of Enginring for Inustry Vo. 96, o. 4, 974, pp [9] Zhang Y., Agrawa S.K. an Piovoso M.J., Coup Dynamics of Fxib Cabs an Rigi En-ffctor for a Cab Suspn Robot, Amrican Contro Confrnc, Vo. 4, o. 6, 6.