Identification of Systems with Friction via Distributions using the Modified Friction LuGre Model

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1 Idetfato of Syte wth rto a Dtrbto g the Modfed rto LGre Model RADU ZGLIMBEA, VIRGINIA INCA, EMILIAN GREABAN, MARIN CONSANIN Deartet of Atoato Uerty of Craoa Bd. Deebal, o.5, Craoa ROMANIA rad@atoato..ro Abtrat: - h aer reet the detfato roedre baed o dtrbto theory to oto te yte wth frto g the odfed LGre frto odel. Maly t ale the relt of the detfato roedre baed o dtrbto theory to oto te yte wth frto. here are defed the o alled geeralzed frto dya yte (GDS a a loed loo trtre arod a ooth yte wth doto feedba loo rereetg frto reato etor. Both GDS wth t frto odel (SM ad dya frto odel (DM, alo the odfed frto LGre odel aalyzed. he detfato roble forlated a a odto of ahg the extee relato of the yte. he, th relato rereeted by ftoal g tehqe fro dtrbto theory baed o tetg fto fro a fte deoal fdael ae. he adage of rereetg forato by dtrbto are oted ot whe eal eolto a ldg ode, or lt yle a aear.he rooed ethod a bath o-le detfato beae detfato relt are obed drg the yte eolto after oe te teral bt ot ay te oet. h ethod doe ot reqre the derate of eared gal for t leeto. Soe exerel relt are reeted to llate t adage ad ratal e. Keyword: - Idetfato; Dtrbto theory; rto; the odfed frto LGre odel. 1 Itrodto here are ay ehaal yte, fro ahe tool otog to trag agato, where the o-alled frto fore flee the oto. rto odel o oe ef olearty h a tto, hyteret, Strbe effet, t-l, deedg o eloty [1], [], [3], [4]. he ale of odel araeter a hage drg the yte eolto or are fleed by oe other ae a exteral teeratre, qalty of ateral et. A large arety of frto odel, a Colob frto odel, Dahl odel [5], [6], LGre odel [19], exoel odel [7], brtle odel [8], te arable odel [9], there are aeted lteratre [1]. Igorg frto otrollg h yte a lead to trag error, lt yle, dered t-l oto []. Sh adate tratege ole detfato roedre of the otrolled yte, ldg detfato of the odel frto araeter. Ufortely frto odel are olear, olg a doto deedee wth reet to eloty. Beae of th, ay tehqe, a detfato baed o te-dretzed odel fal to offer good relt. A rey of odel, aaly tool ad oeato ethod for the otrol of ahe wth frto reeted [9]. A dtrbto-baed aroah of ehaal yte wth frto detfato deeloed [1]. h exted the relt o oto te yte detfato baed o dtrbto theory, reorted [11], for lear yte, or [1], for olear yte. o erfor oto te doa detfato the yte dfferel eqato trafored to a algebra yte that reeal the ow araeter, [13]. h a be doe alo by g oe odlatg fto to geerate ftoal to aod the dret oto of the t-ott da derate [14], [15], [16]. hrogh th, teded to deotrate the effetee of the dtrbto baed detfato tehqe fro [1], for frto ehaal yte. A the detfato relt are obed drg the yte eolto after oe te teral bt ot ay te oet, rooed elf-tg ethod a bath o-le. he aer orgazed a follow: After trodto the frt eto, Seto dedated to the geeralzed frto dya yte GDS, a reeted [1]. Seto 3, reet the a te ISSN: ISBN:

2 the of oto te yte detfato baed o dtrbto. Seto 4 lltrate alato for a frto ehaal yte detfato g the odfed frto LGre odel. Seto 5 reet exerel relt, aga olo are reed Seto 6. Geeralzed rto Dya Syte A geeraled dya frto yte (GDS a yte haratered by the te eqato of the for r x f( x,, r,.., r,.., r (1 x f( x,, r 1 where x( t X R, t t, the te etor ad q t ( U R, tt the t etor. he etor r ( t R R, t t, 1: ( are alled frto reato etor. hey deed o x ad throgh a ef oerator {}, alled frto oerator, r {,}, x 1: (3 here are two ategore of frto odel: t frto odel (SM ad dya frto odel.(dm. or SM, the oerator (3 a o dya ag r ( x, : X UR R, 1: (4 wth a ef trtre a follow. or ay 1:, there are two fto ( x, : X UV R, 1:, (5 whh detere the o alled geeralzed eloty etor, ad a ( x, : X UA R, 1:, (6 exreg the o alled ate ooet of the eloty etor. I SM, the o dya ag (4 a be exreed a a fto of, ad a oly, that ea, r (, a ( x,, 1: (7 he trtre of a GDS wth SM lltrated g.1. r ( x x (, a a ( x, r ( x,, 1: g.1, he feedba trtre of a GDS wth SM x hx (, y or dya frto odel, (DM the oerator (3 a dya yte haraterzed by a addtoal te etor z, exreg teral hage oe rfae of relate oeet. he frto reato etor r the ott of a dya yte r (,, h z x (8 z (,, f z x (9 here are ay tye of DM bt we hall reet a lfed Dhal odel [], [5], for 1, haraterzed by a frt order olear dya yte z a z (1 r z (11 where the eloty ( R x. (1 h yte a be exreed a a gle dfferel eqato wth reet to the frto reato arable r [1 ( a / g( r ] (13 3 Coto e Syte Idetfato Baed o Dtrbto h eto reet the a relt o oto te yte detfato baed o dtrbto, a hae bee reeted [11]. Let be the fdael ae fro dtrbto theory [17] of the real tetg fto, :, t ( t, hag oto derate at leat to the order, wth oat ort for ay of the aboe derate. he lear ae orgazed a a toologal ae oderg a ef or [17]. A dtrbto a lear, oto real ftoal o, :, (. Let q:, t q( t be a fto that adt a Rea tegral o ay oat teral fro. Ug th fto, a qe dtrbto q :, q( a be bld by the relato q( q( t ( t dt,. Coderg, at leat, q C(, the followg ort eqalee e lae [18], q(, q( t, t (14 he -order derate of a dtrbto a ew ( / dtrbto, q qely defed by the relato, ISSN: ISBN:

3 ( ( 1 (, (15 ( ( q q Whe qc (, the ( ( q ( q ( ( q ( t ( t dt that ea the -order derate of a dtrbto geerated by a fto q C ( eqal to the dtrbto geerated by q (, the -order te derate of the fto q. If q C (, fro (13, (16 oe a wrte, ( ( ( ( q ( t ( t dt( 1 q( t ( t dt(16 q Let oder a dyaal oto te yte exreed by a dfferel oerator, q/( y, C(, y, (17 whoe exreo deed o a etor of araeter [ ]. (18 It rereet a faly of odel wth a ge trtre ot araeter. A eal ae the odel (17 exreg a lear relato the araeter q /(, ( y C, y, w w 1, (19 where w ad rereet a of the derate of oe ow, oble olear, fto,, wth reet to the t ad ott arable, ( w [ (, ], 1: 1 y, ( [ (, ] ( 1 y. (1 where araeter,,, are ge teger ber. I [11] the extee ad qee odto for a roble of dtrbto baed oto te yte detfato are reeted. Soe that t oble to reord the fto (, y a the te teral, alled oberato te teral or t te wdow. he retrto of the fto ( y, to the te teral deoted by (, y reetely. If o ofo wold aear, the we ay dro the brt. A detfato roble ea to detere the araeter, ge the ror forato o the odel trtre C, (17, ad a et of obered tott ar (, y, (, y, C a h a way that, q ( t, t ( /(, y h odto ole, q ( t, t, (, y (3 /( y, for ay t-ott ar ( y, obered to that yte. Let oder two fale of reglar dtrbto, w, 1:, ad ( reated baed o the fto (, (1, ( w ( [ ( ] ( 1 R t t dt (4 ( ( 1 [ ( t] ( t dt 1 R whh detere the row etor, ( [ (,..., (,..., ( ]. (5 w w1 w w ( ( [ 1 R ( ] ( t t dt ( 1 R t ( 1 [ ( ] ( t dt (6 Ay t-ott ar ( y, obered fro the yte (7 derbed by a ar of reglar dtrbto ( w, for ay, [11]. he roble of the yte (17 araeter detfato a be rereeted ow by dtrbto. or exale, the reglar dtrbto geerated by the oto fto q /( y, fro (17, related to the araeter etor, a (7 ( ( ( ( qθ qθ /(, y 1 w w ( y( If a trle ( *, y*, * a realzato of the odel (17, the the detty (39 e lae, q ( θ* qθ* /( *, y* (, (8 ad e era, f a t-ott ar (*, y * of the faly of odel (7, wth ow araeter, geerate a dtrbto q ( ( ( ( θ qθ /( *, y * 1 w (9 whh atfe q ( θ qθ /( *, y* (, *, (3 A ha ooet t eogh a hoe (tlze a fte ber N of fdael fto, 1: N ad to bld a algebra eqato, w (31 where w a ( N atrx of real ber [ ( 1;...; ( ;...; ( ] w w w w N (3 where -th row w ( ge by (5. he ybol deote a N -ol real etor blt fro (6, [ ( 1,..., (,..., ( ] N. (33 Whe oly the retrto (, y of the ar ( y, o the te teral aalable, ay t hae ISSN: ISBN:

4 ( for t derate (, t 1: the ae oat ort, ( { ( t} [, ], 1:, 1: N (34 Below there are oe le tetg fto, ( t ( t, t (, t t, t (35 a b a b [ ( / ( t ],, (, t, (36, t (, ] [, where a alg fator ad oralze the area t b (, 1/ (,,, t. (37 If r ra( w, the a qe olto obed. 1 ( * (38 w w w 4 Alato for a rto Mehaal Syte Idetfato Ug the Modfed of rto LGre Model o obe the eleet of GS wth the theory of detfato baed o dtrbto, a reeted [1], let oder a le ehaal yte otg of a a, og o a rfae wth brtle. he yte exted by a fore, whle the (earable frto fore f ret the oto (g.. he odel for th yte : (39 g., A le ehaal yte wth frto he frto fore ge of the odfed LGre odel: z (4 the zoe of frto where z z (41 g( ad g ( ( (4 1 wth ow a dya frto araeter ad a tffe araeter, reetely, ad ow a t frto araeter, where the Colob frto, the t frto ad a o frto oeffet, aga z the teral te of the odfed frto odel. he te x of (1 ha three ooet, x 1 x ad x3 z, o x 1 x 1 x x3 x x x x x where x1 x oto, x aga x3 3 3 x ( x 1 x, (43 rereet eloty, z rereet the dlaeet of brtle whh eareable. he yte a be derbed by the followg dfferel eqato (44: exet a et of ot of a zero eare. Deotg 1 ; ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ad the araeter that hae to be detfed, 1 (44 exreed a the oerator (19, where 1. he dtrbto age (9 of th dfferel oerator, ealated for a tetg fto o the te teral, t b, o the eleet ge by (4, (5, (6 of the for w( t ( t dt where, w ISSN: ISBN:

5 t ( ( tdt w1 w3 t ( ( tdt t ( ( tdt w t ( t ( t dt t ( t dt w4 t ( t ( ( tdt w5 t ( t ( ( tdt t ( t ( t ( ( tdt w6 w7 t ( ( t ( tdt t ( t ( ( tdt ( t ( t dt w8 t ( t ( ( tdt t ( t ( ( tdt w9 t ( t ( ( tdt t ( t ( t ( ( tdt w1 t ( ( tdt. or the ealato of thee tegral oly t-ott ar (,, reetely (, ad the odle of are eeary. Otherwe alo the eed ad the aelerato wll be eared, for to eare aelerato wll e a aelerator. Itegral are tlzed to bld the yte (31, (3, (33, whoe olto (38. 5 Exerel Relt o leet dtrbto baed detfato ethod, a exerel latfor (DBI ha bee deeloed. It allow reatg tetg fto wth ele araeter, atoatally to reate ad ole the yte (31. he t-ott da for detfato are obed fro a exteral ore (a da fle or terally by lato. May exale ad tye of frto yte hae bee leeted for detfato bt, beae of lted ae th aer, oly oe exale aalyed, baed o the alato reeted eto 4. I th exale, the eared gal, a dated g.3., are geerated by a te t t ( 1( t aed.1 throgh fto of trafer wth l te x( [ ] ad oly,,,, a araeter eeary for detfato. e tetg fto o, a (35, wth =1 ad 1 =[1,1.4]; =[1.4,1.8]; 3 =[1.8,.]; 4 =[.,.6]; 5 =[.6,3]; 6 =[3,3.4]; 7 =[3.4,3.8]; 8 =[3.8,4.]; 9 =[4.,4.6]; 1 =[4.6,5] are tlzed. 6 - x 1 Meared gal Ste: x=[x1 x x3] x(=[ ] x x 3 aelerato e (e g.3, Meared arable for the yte wth the odfed frto LGre odel ad l te x(=[ ] he real ad detfed araeter ale are reetely: : : : : : or the ae t bt wth x(=[1 1 1] ad 1 =[1,1.3]; =[1.3,1.6]; 3 =[1.6,1.9]; 4 =[1.9,.]; 5 =[.,.5]; 6 =[,5..8]; 7 =[.8,3.1]; 8 =[3.1,3.4]; 9 =[3.4,3.7]; 1 =[3.7,4], the detfato relt are: : : : : : he eared arable for th ae are lltrated g.4. ISSN: ISBN:

6 6 - x 1 Meared gal Ste: x=[x1 x x3] x(=[1 1 1] x x 3 aelerato e (e g.4, Meared arable for the yte wth the odfed frto LGre odel ad l te x(=[1 1 1] 6 Colo he aboe relt lltrate the adage of dtrbto baed detfato for yte wth dotte o the rght de. Derto by ftoal allow to elarge the area of yte to whh detfato roedre a be aled. h aer deeloet of a aer Idetfato of yte wth frto a dtrbto. Referee: [1] B. Artrog-Helory, Cotrol of ahe wth frto, Boto MA, Klwer, 1991 [] C. Cada de Wt, H. Olo, K. J. Åtrö, P. Lhy, "A New Model for Cotrol of Syte wth rto", IEEE ra. Atoat. Cotr., ol.4, , Marh [3] H. D ad N. Sath, "Idetfato of rto at Low Velote Ug Waelet Ba to Networ", Pro. of Aera Cotrol Coferee, Phladelha, Je 1998, [4] S.J. K. ad I.J. Ha, A reqey-doa Aroah to Idetfato of Mehaal Syte wth rto, IEEE ra. Atoat. Cotr., ol.46, , Je. 1. [5] P. Dahl, A old frto odel, Aeroae Cor., El Segdo, CA,eh. Re. OR- 158( , [6] B. Artrog-Helory, P. Dot, ad C. Cada de wt,, "A Srey of Model, Aaly ool, ad Coeato Method for the Cotrol of Mahe wth rto," Atoata, Vol. 3, No. 7, , [7] Cada de wt, C., Noel, P., Ab, A., ad Broglato, B., "Adate rto Coeato Robot Malator: Low Velote," It. Joral of Robot Reearh, Vol. 1, No. 3, , [8] Haeg, D. A. ad redlad, B., "O the Modelg ad Slato of rto," ASME Joal of Qya Syte, Meareet, ad Cotrol, Vol. 113, , Set [9] B. Helory, P. Dot, ad C. de Wt, A rey of odel, aaly tool ad oeato ethod for the otrol of ahe wth frto, Atoata, ol. 3, o. 7, , [1] S. J. K ad I. J. Ha, O the extee of Carathèodory Solto ehaal yte wth frto, IEEE ra. Atoat. Cotr., ol. 44, , No [11] C. Mar, "Syte Idetfato Baed o Dtrbto heory". Proeedg of the IASED Iteratoal Coferee Aled Slato ad Modellg (ASM, Crete, Je, [1] C. Mar, E. Petre, D. Seltea, D. Sedre, "Idetfato of Nolear Plat a Dtrbto. Alato for Watewater Bodegradato Proe," he 15th It. Coferee o Cotrol Syte ad Coter See-CSCS15, May 5-7,5, Bharet. [13] N. K.Sh, G. P. Rao, Idetfato of oto te yte. Dordreht: Klwer Aade Pre. [14] C. Mar, Syte Idetfato by the ethod of weghted oet, 1th It. Coferee o Cotrol Syte ad Coter See, CSCS 1, Bharet. [15]. Batoge, H. Garer, P. Sblle, "PM bae ethod for oto-te odel detfato detert tdy," Pro. Of IAC Sy.SYSID'97, oa, Jly, [16] A. E. Pearo, A. E. Lee, "O the detfato of olyoal t-ott dfferel yte," IEEE ra.at.cotr.,vol. AC3, No8, [17] W. Ke, P. eodore, Itrodto to dtrbto theory wth tehal alato, Edtra eha, Bharet [18] V. Barb, Dfferel eqato. Edtra Jea, Ia [19] D. P. L, "Reearh o Paraeter Idetfato of rto Model for Sero Syte Baed o Geet Algorth", Proeedg of the orth Iteratoal Coferee o Mahe Learg ad Cyberet, Gagzho, 18-1 Agt 5, ISSN: ISBN: