MATHEMATICAL MODELLING OF A SINGLE LINK FLEXIBLE MANIPULATOR

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1 MATHEMATICA MOEING OF A SINGE INK FEXIBE MANIPUATOR Malik oudii,ϒ, jamel Boukheala ϒ ad Mohamed Tadjie ϒ,ϒ Isiu Naioal d Iformaique, B.P. 68M, 639 Oued Smar, Algiers, Algeria ϒ Ecole Naioale Polyechique, aboraoire de Commade des Processus,, aveue Hassa Badi, B.P. 8, 6 El-Harrach, Algiers, Algeria Absrac: This paper deals wih he Timosheko beam heory (TBT) based mahemaical modellig of a plaar sigle lik fleible robo maipulaor clamped a is acuaed base ad carryig a payload a is ed-poi. The emphasis has bee, esseially, se o obaiig accurae ad complee equaios of moio ha display he mos releva aspecs of he lighweigh fleible lik srucural properies. Two impora dampig mechaisms: ieral srucural viscoelasiciy effec (Kelvi-Voig dampig) ad eeral viscous air dampig have bee icluded i addiio o he classical effecs of shearig ad roaioal ieria of he elasic lik cross-secio. Those of graviy, orsio, ad logiudial elogaio have bee egleced. The combied agrage-assumed modes mehod i he clamped-free (payload) case wih suiable iiial ad boudary codiios is proposed o derive a dyamic model. Copyrigh 6 USTARTH Keywords: Roboic maipulaors, Fleible arm, PE Model, Timosheko beam heory, agrage-assumed modes mehod.. INTROUCTION Modellig ad corol of fleible maipulaors have bee a acive research field i rece years. Tradiioal robos have bee desiged for "rigidiy" wih shor arms ad a heavy srucure, which sigificaly resric heir rage of applicaios. ighweigh maipulaors wih lower arm cos, higher moio speed, beer eergy efficiecy, safer operaio ad improved mobiliy are highly desirable. A major problem wih such fleible robos, however, is ha he ed-poi accuracy is severely impaired due o srucural deformaio of he fleible liks. This elasiciy leadig o udesirable vibraios is he mos complicaig parameer i he ask of derivig a mahemaical model of such sysems. eails abou modellig ad corol of fleible lik robo maipulaors ca be foud i Caudas de Wi, e al., (996); Beosma, e al., (); wivedy ad Eberhard, (6). I order o fully eploi he poeial advaages offered by hese lighweigh robo arms, oe mus eplicily cosider he effecs of srucural lik fleibiliy ad hus have a complee ad accurae dyamic model a disposal. I his work, we aim o prese he deails of our ivesigaios cocered wih a derivig procedure of a complee ad accurae closed form dyamic model for a fleible sigle lik robo maipulaor clamped a is base ad carryig a payload a is ed-poi. Earlier works, i he same coe of he prese oe, ca be foud i Qi ad Che, (99); Whie ad Heppler, (995); Wag, e al., (996); Sooraksa ad Che, (998); oudii, e al., (6). For our prese ivesigaio, we have adoped he same cosideraios give i oudii, e al. (6). Oly he boudary codiios (B.C) are modified o be i agreeme wih a eperimeal resul esablished i Wag ad Gua (99). These cosideraios are summarized below: To esure corol efficiecy, a high fideliy ad accurae model icludig he mai srucural properies is eeded. Therefore, i addiio o roary ieria ad shear deformaio, we have icluded he dampig effecs, i.e. Kelvi-Voig ieral

2 viscoelasic effec ad eeral viscous air dampig, clearly poied ou as beam promie physical characerisics i Baks, e al. (996). Clamped-mass ype of he cosraied mode shapes raher ha he pied-pied oe. Ideed, eperimes ad aalyical sudies (Hasigs ad Book, 987; Ceiku ad Yu, 99) have prove ha he clamped assumpio is more adequae ad especially whe closig a feedback corol loop aroud he joi where he lik is clamped. Usig such cosraied mode shapes, i is more reasoable o cosider ha he lighweigh lik ieria is small compared o he hub ieria (Bellezza, e al., 99). The robo maipulaor is assumed be acuaed by a high gear moor raher ha a direc drive oe. The clamped assumpio is eve eforced whe such a acuaor is used (Krisha, 988). The oulie of his paper is as fellows. I secio, he differe sages followed i obaiig he fleible maipulaor moio goverig equaio are deailed afer preseig he plaar robo arm wih is mai parameers. Secio 3 is devoed o he proposed derivig procedure o obai a dyamic model of he sudied robo o he basis of he combied agrageassumed modes approach. Secio repors simulaio resuls. The paper is cocluded wih secio 5.. MATHEMATICA MOEING. efiiios ad variables. The fleible robo sysem uder cosideraio is show i Figure. I cosiss of a clamped-free wih ip payload plaar movig fleible arm which ca bed freely i he horizoal plae. The deflecio which is he rasverse displaceme of he lik from he X -ais is deoed by w. This displaceme a ime is perpedicular o he X -ais. This figure is a op view of he maipulaor i deflecio ad he ais of roaio of he hub is perpedicular o he maipulaor evoluio plae. As usual, he fleible lik ca be cosidered as a beam. The heigh of he beam cross-secio is assumed o be larger ha he base. The, deflecios are cosraied o occur oly i he horizoal plae. The ierial frame of referece is idicaed by X Y coordiae frame. The X Y coordiae frame is a frame of referece ha roaes wih he overall srucure. The X -ais is age o he beam a he base (Bellezza, e al., 99).. Timosheko Beam Theory. A rigorous mahemaical model widely used for describig he rasverse vibraio of beams is based o he TBT (or hick beam heory) (Timosheko, 97) developed by Timosheko i he 9s. This PE based model is chose because i is more accurae i predicig he beam s respose ha he oe based o he Euler Beroulli (EB) beam heory (Ima, 99). I has bee show i he lieraure ha he predicios of he Timosheko beam model are i ecelle agreeme wih he resuls obaied from he eac elasiciy equaios ad eperimeal resuls (Trail-Nash ad Collar, 953; Huag, 96; Ha, e al., 999). The Timosheko heory accous for boh he effec of roary ieria ad shear deformaio, which are egleced whe applied o Euler-Beroulli beam heory (or hi beam heory). The rasverse vibraio of he beam depeds o is geomerical ad maerial properies as well as he eeral applied orque. Oe of he mai geomerical parameers is he Timosheko s shear coefficie k which is a modifyig facor ( k < ) o accou for he disribuio of shearig sress such ha effecive shear area is equal o ka. The maerial properies refer, for eample, o is desiy i mass per ui volume ρ, Youg s modulus E (see he lis of symbols a he ed of he paper). Ispec a eleme of he defleced lik wih widh d, a posiio (Fig.). Y τ y y Y J o efleced lik Tip payload Fig.. Cofiguraio of he sigle lik fleible maipulaor arm. Y Z X Beam eleme (Figure ) M α Rigid hub V Y θ d ρ Iβ d parallel wih eural ais perpedicular o face parallel wih X -ais Fig.. Kiemaics of deformaio of a bedig lik (beam) eleme. ue o he effec of shear, he origial recagular eleme chages is shape o somewha like a parallelogram wih is sides slighly curved. M p, J p w ρ Aw d l, E, I, ρ M dm V dv β X X X w

3 This eleme is subjeced o a shearig force V, ad a bedig mome M. O he opposie side of his segme, which correspods o a posiio d, he shearig force ( V dv ) is V V ( d, = V d () ikewise he mome force ( M dm ) a he posiio d is M M ( d, = M d () Noe ha he oal deflecio is due o boh bedig ad shear forces, so ha he shear agle σ (or loss of slope) is equal o he slope of cerelie (eural ais) w(, or, simply, w less slope of bedig β (, i he form: σ = w β (3) The shear force V is give by V = kagσ = kag[ w β ] () By cosiderig he "sadard liear solid" of Zeer (Zeer, 98), wih a sress-srai law give by ν K & ν = Eε C & ε (5) ad assumig liear variaios of srai ad sress across he beam deph, he oal mome obaied by iegraig firs mome of sress across he beam cross secio is (Baker, e al., 967) M = K M = I E C β (6) As i Baks ad Ima, (99); Baks, e al., 99), he oal ieral mome (bedig ad dampig) M is he give by M = EIβ C Iβ (7) where C is he Kelvi-Voig dampig coefficie. The equaio of moio of he sudied sigle lik fleible robo arm ca be derived by cosiderig boh he equilibrium of he momes ad he forces. Takig momes as posiive i he couer-clockwise direcio, heir summaio wih disregardig he secod order erm of d, yields he relaio bewee he spaial chage i he bedig mome ad he shear force: M = V ρiβ (8) where he erm ρ Iβ sads for he disribued roaioal ieria give by he produc of he mass mome of ieria of he cross secio ad he agular acceleraio. The relaio ha fellows balacig forces is V γ w = ρaw (9) where he erms γ w, ρ Aw represe, respecively, he air resisace force ad he disribued rasverse ierial force. Subsiuio of Eq.() ad Eq.(7) io Eq.(8) ad likewise Eq.() io Eq.(9) yields he wo coupled equaios of he damped Timosheko beam moio: ( C I ) ( EIβ ) kag( w β ) ρiβ = β () [ kag( w )] ρaw γw = β () If he dampig effecs erms are suppressed, he classical se of wo coupled PE developed by Timosheko i he 9s arises, see (Timosheko, 9; Timosheko, 9): ( EI ) kag( w β ) ρiβ = β () [ kag( w )] ρaw = β (3) The modelled beam cross-secioal area ad desiy beig uiform, Eq.() ad Eq.() ca be easily decoupled as follows: C Iw C Iρ w EIw E C γ ρ I ρi w w ρkag EIγ ρiγ w w ρaw γw kag kag = () C Iρ C Iβ β EIβ E C γ ρ I ρi β β (5) ρkag EIγ ρiγ β β ρaβ γβ = kag kag Similar o he oe esablished by Sooraksa ad Che (996), equaio () is he damped TB homogeeous liear fifh order PE epressig he deflecio w. ue o he crucial imporace of he BC, we affec o his equaio he followig clamped-mass oes, used i oudii, e al. (6), bu wih akig io accou he resul esablished by Wag, ad Gua (99) abou he very small ifluece of he ip payload ieria o he fleible maipulaor dyamics: w) = w, w ) = w& ; w(, = w (, = ; M p w ( l, = M ( l,, J p w ( l, = M ( l, =. (6) The classical fourh order Timosheko beam PE is rerieved if he dampig effecs erms are suppressed: 3

4 EIw ρi E w ρ I w ρaw = (7) If he shear effec is egleced we are led o he Rayleigh beam equaio (ord Rayleigh, 95): EIw ρ Iw ρaw = (8) Moreover, if boh he roary ieria ad shear deformaio are egleced, he he goverig equaio of moio reduces o ha based o he classical Euler- Beroulli heory give by EIw ρ Aw = (9) If he cosidered dampig effecs are associaed o he Euler-Beroulli Beam, he correspodig PE is C Iw ρ Aw γw = () EIw To solve he PE wih mied derivaive erms (Eq.()), w ca ake he followig epaded separaed form wih he chose deflecio mode shapes W () ad he modal ampliudes δ ( : w(, where m =. = W ( ) ( = = = mπ δ cos. δ( () By subsiuig Eq.() io Eq.(), we fid his fourh order emporal OE: () c δ ( ( cm c )&&& 3 δ ( ( cm c5 )&& δ( ( c m c m c )& δ ( c m δ ( = () ad cosiderig, for eample, a alumium beam, we foud, afer umerical calculaios, ha he coefficies () of δ ( ad && δ & ( ) are very small compared o hose of he lower order erms of Eq.() (oudii, e al., 6). This laer is, he, approimaely reduced o he followig secod order OE: cf && δ ( cf & δ ( cf3δ ( = ρiπ E Cγ where cf = m ρa ; ρkag C Iπ EIγπ cf = m m KAG γ ; cf EIπ 3 m (3) =. Eq.(3) has he geeral form of a secod order OE characerizig a liear elasic sysem wih dampig. Cosiderig he uderdampig mode case, wih a se of iiial codiios, δ () ad & δ (), is soluio is ξω e ( ωd δ ( = C cos ψ ) () where c9m m cf3 ω = = ; cf c c cf c6m c7 m = cf ω ( cm c 5 5 c8 ξ = ; ωd = ω ξ ; ) ω () () & δ δ ξ ω C = [ δ ()] ; ωd & δ () δ () ξ ω ψ = arca ; =,,. ωdδ () The robo lik rasverse displaceme, approimaely obaied, as a soluio of he damped Timosheko equaio (Eq.) is fially epressed by w(, = = ξω C e cos mπ ( ω ψ ) cos 3. ROBOT YNAMIC MOE d (5) I order o obai a se of OE of moio o adequaely describe he dyamics of he fleible lik maipulaor, he agrage's approach is used. A dyamic sysem compleely locaed by geeralized coordiaes qi mus saisfy differeial equaios of he form: where d d q & i qi q& i = Q, i =,,, (6) is he so called agragia which is give by i = T U (7) T represes he kieic eergy of he sysem, U is poeial eergy. ad is he Rayleigh's dissipaio fucio. is he geeralized eeral force acig o Q i he correspodig coordiae q i. Theoreically here are ifiie umber of OE, bu for pracical cosideraios such as boudedess of acuaig eergy ad limiaio of he acuaors ad he sesors workig frequecy rage, i is more reasoable o rucae his umber a a fiie oe (Cao ad Schmiz, 98; Qi ad Che, 99). The oal kieic eergy of he robo lik is give by: T = ρ Aw d ρiβ d (8) The poeial eergy of he arm due o he ieral bedig mome ad he shear force ca be wrie as U = EIβ d KAGσ d (9)

5 The dissipaed eergy due o he dampig effecs is: = γ w d C Iw d (3) Subsiuig hese eergies epressios io Eq.(6) accordigly ad usig he rasverse deflecio separaed form (Eq.()) we ca, afer edious maipulaios, derive he desired dyamic equaios of moio i he mass ( B ), dampig ( H ) ad siffess ( K ) mari form: B. q& ( H. q& ( K. q( = Q( (3) T ( [ δ ] wih q = θ ( δ ( δ ( ( ), [ τ ] T Q( =. 5. CONCUSION I his paper, he developme of a fleible lik robo maipulaor mahemaical model usig Timosheko beam heory has bee repored. The emphasis has bee, esseially, se o obaiig accurae ad complee equaios of moio ha display he mos releva aspecs of srucural properies ihere o he modelled lighweigh fleible lik. I paricular, wo impora dampig mechaisms: ieral srucural viscoelasiciy effec (Kelvi-Voig dampig) ad eeral viscous air dampig have bee icluded i addiio o he classical effecs of shearig ad roaioal ieria of he elasic lik cross-secio. To derive a closed-form fiiedimesioal dyamic model for he plaar sigle lik lighweigh robo, a eergeic derivig procedure based o he agragia approach combied wih he assumed modes mehod has bee proposed. The mari differeial equaio i Eq.(3) ca be easily represeed i a sae-space form as z& ( = Az z( Bzu( y( = C z z( (3) mode mode mode3 mode [ ] T wih u( = τ, [ θ ( δ ( δ ( & θ ( & δ ( & ( ] T z = ). ( δ Solvig he sae-space marices gives he vecor of saes z(, ha is, he agular displaceme, he modal ampliudes ad heir velociies. eails of he obaied dyamic model (Eq.(3) or Eq.(3)) are o icluded here due o he space limi. W Fig. 3. The firs four mode shape fucios SIMUATION To simulae he vibraioal behaviour of he modelled elasic lik robo, is physical parameers umerical values are he same of hose used i oudii, e al. (6). To formulae a simple, physically correc fiie dimesioal dyamic model for behaviour aalysis, he geeral goverig equaio is rucaed o oly he wo lower (domia modes of vibraio ( = ). The correspodig wo firs mode shape fucios W ( ) ad W ( ) are show i Figure 3 where W 3 ( ) ad W ( ) are also illusraed. The umerical simulaio has bee performed o show he free evoluio behaviour characerized by he ip rasverse deflecio impulse resposes wihou a payload (Fig.) ad wih a payload (Fig.5). Comparig hese resposes, i is evide ha he resul of addig a payload has reduced he frequecy of vibraio ad icreased he vibraios magiude. Furhermore, he obaied resposes are ideical o hose obaied i oudii, e al. (6). Thus, we ca sae ha our resuls are i ecelle agreeme wih hose foud by Wag ad Gua (99) abou he egligible effec of he payload ieria o he fleible maipulaor dyamics. 5 Tip Trasverse eflecio [m] Time [s] Fig.. Tip deflecio respose wihou payload. Tip Trasverse eflecio [m] Time [s] Fig. 5. Tip deflecio respose wih payload.

6 NOMENCATURE A : lik cross-secio area; B : ieria mari; C : Kelvi-Voig dampig coefficie; : dissipaed eergy; E : lik Youg s modulus of elasiciy; G : shear modulus; H : dampig mari; I : lik mome of ieria; J : hub ad roor (acuaor) oal ieria; o J : payload ieria; k : shear correcio facor; p K : siffess mari; K : viscoelasic maerial cosa; l : lik legh; : agragia; M : bedig mome; M : payload mass; : mode umber; p q : vecor of geeralized coordiaes; Q : vecor of eeral forces; : ime; T : kieic eergy; U : sored poeial eergy; V : shear force; w : rasverse deflecio; W : rasverse deflecio ormal mode shape; : coordiae alog he beam; α : agular posiio of a poi of he defleced lik; β : roaio of cross-secio abou eural ais; δ : h modal ampliude; ε : srai; ν : ormal sress; θ ( : agular posiio of he hub; ρ : lik uiform liear mass desiy; σ : shear agle; τ : acuaor orque applied a he base of he lik; ω : h aural frequecy of vibraio; ξ : h dampig raio. REFERENCES Baker, W. E., W. E. Woolam ad. Youg (967). Air ad ieral dampig of hi cailever beams, I. J. Mech. Sci., 9, Baks, H. T. ad. J. Ima (99). O dampig mechaisms i beams, J. of Applied Mechaics, 58, Baks, H. T., Y. Wag, ad. J. Ima (99). Bedig ad shear dampig i beams: frequecy domai echiques, J. of Applied Mechaics, 6, Baks, H. T., R. C. Smih ad Y. Wag (996). Smar maerial srucures: modelig, esimaio ad corol. Wiley-Masso, New York. Bellezza, F.,. aari, ad G. Ulivi (99). Eac modelig of he slewig fleible lik, Proc. IEEE I. Cof. Roboics Auoma., pp , Ciciai, OH, USA. Beosma, M., F. Boyer, G..Vey ad. Primau (). Fleible liks maipulaors: from modellig o corol, J. of Iellige ad Roboic Sysems, 3, 38. Cao, R. H., Jr. ad E. Schmiz (98). Iiial eperimes o he ed-poi corol of a fleible oe-lik robo, I. J. Roboics Res., 3, 3, Caudas de Wi, C., B. Siciliao, ad G. Basi (996). Theory of robo corol. Spriger-Verlag, odo. Ceiku, S. ad W.. Yu (99). Closed loop behaviour of a feedback corolled fleible arm: A comparaive sudy, I. J. Roboics Res.,, 3, wivedy S. K. ad P. Eberhard. (6) yamic aalysis of fleible maipulaors, a lieraure review, Mechaism ad Machie Theory,, Ha, S. M., H. Bearoya ad T. Wei (999). yamics of rasversely vibraig beams usig four egieerig heories, J. of soud ad vibraio, 5, 5, Hasigs, G. G. ad W. J. Book (987). A liear dyamic model for fleible roboic maipulaors, IEEE Cor. Sys. Mag., 7, 6-6. Huag, T. C. (96). The effec of roary ieria ad of shear deformaio o he frequecy ad ormal mode equaios of uiform beams wih simple ed Codiios, J. of Applied Mechaics, 8, Ima,. J. (99). Egieerig Vibraio. Preice- Hall, Eglewood Cliffs, NJ. Krisha, H. ad M. Vidyasagar (988). Corol of a sigle-lik fleible beam usig a Hakel-ormbased reduced order model, Proc. IEEE Cof. o Rob. ad Auomaio, vol., pp. 9-. oudii, M.,. Boukheala, M. Tadjie ad M. A. Boumehdi (6). Applicaio of Timosheko beam heory for derivig moio equaios of a lighweigh elasic lik robo maipulaor, I. J. of Auomaio, Roboics ad Auoomous Sysems, 5,, 6. Qi, X. ad G. Che (99). Mahemaical modelig of kiemaics ad dyamics of cerai sigle Fleiblelik robo arms, Proc. IEEE Cof. o Corol Applicaios,, pp ord Rayleigh (95). Theory of soud, over Publicaios, Secod ediio, New York. Sooraksa, P. ad G. Che (998). Mahemaical modelig ad fuzzy corol of a fleible-lik robo. Mahl. Compu. Modellig, 7, Timosheko, S. P. (9). O he correcio for shear of he differeial equaio for rasverse vibraios of prismaic bars, Philosophical Magazie,, Timosheko, S. P. (9). O he rasverse vibraios of bars of uiform cross secio, Philosophical Magazie, 3, 5-3. Timosheko, S.,. H. Youg, W. Jr. Weaver (97). Vibraio problems i egieerig. Wiley, New York. Trail-Nash, P. W. ad A. R. Collar (953). The effecs of shear fleibiliy ad roary ieria o he bedig vibraios of beams, Quarerly J. of Mechaics ad Applied Mahs.,6,86-. Wag F. Y. ad G. Gua. (99) Iflueces of roary ieria, shear ad loadig o vibraios of fleible maipulaors, J. of Soud ad Vibraio, 7, Wag, F. -Y., P. Zhou ad P. ever (996). yamic effecs of roary ieria ad shear deformaio o fleible maipulaors, Proc. IEEE I. Cof. o Sys., Ma, ad Cyber., 3, pp Whie, M. W.. ad G. R. Heppler (995). Timosheko model of a fleible slewig lik, Proc. America Cor. Cof.,, pp Zeer, C. (98). Elasiciy ad ielasiciy of meals. Uiversiy of Chicago Press, Chicago. 6