Adaptive Fuzzy Model-Based Predictive Control of Nonholonomic Wheeled Mobile Robots Including Actuator Dynamics Z. Sinaeefar, M.

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1 Iteatoa Joua of Scetfc & Egeeg Reseach, Voume 3, Issue 9, Septembe- ISS Adaptve Fuzzy Mode-Based edctve Coto of ohooomc Wheeed Mobe Robots Icudg Actuato Dyamcs. Saeefa, M. Faokh Abstact hs pape pesets a adaptve oea mode-based pedcto coto (C) fo tajectoy tackg of heeed mobe obots (WMRs). Robot dyamcs ae subject to vaous ucetates cudg paamete vaatos, uko oeates of the obot ad toque dstubaces fom the evomet. I ths pape, a dscete-tme fuzzy mode combato th C s descbed to ao appoxmato of the uko dyamcs of the obot, cudg the actuato dyamc. Moeove, by tug the eghtg paametes the cost fucto of the C, the tackg eo of a gve tajectoy ca be mmzed. Fay, the paametes of the fuzzy mode may be adjusted o-e by the use of a gadet descet agothm cosdeato of the ucetates. he smuato esuts of a WMR exampe sho the effectveess of the poposed method. Idex ems Gadet descet agothm, Fuzzy system, oea mode pedctve coto, Adaptve coto, Mobe obots, ajectoy tackg. IRODUCIO oadays, obots ae beg seted moe ad moe to dyamc evomets such as obotc socce, maufactug pats, etc. ajectoy tackg coto fo mobe obots s a fudameta pobem, hch has bee tesvey vestgated the obotcs commuty. he desg of coto as fo mobe obots th a dyamc mode s cosdeed sevea papes, fo stace tajectoy tackg [], [], [3]. Oe of the eay studes of ths pobem used a Lyapuov fucto to desg a oca asymptotc tackg cotoe. Goba tackg as expoed by dyamc feedback eazato techques [4], [5], [6], backsteppg techques [7], [8], [9], ad sdg mode techques []. hese cotoes eque that ea ad agua veoctes must ot covege to zeo, so they ca ot be used fo the eguato pobem of ohooomc mobe obots. Aso, these cotoes do ot take to accout the estctos the coto sgas due to the dffcuty of mpemetato. Mode pedctve coto (MC), aso ko as ecedg hozo coto (RHC), has become oe of the most successfu coto stateges deveoped dug the ast fe decades, ad uke may othe advaced coto techques, t has desabe featues sutabe fo dusta appcatos [], ad ts appcatos ae aso expadg to obot coto. I MC, a pocess pat s used to pedct futue outputs ove a pescbed peod. opetes that set MC apat fom othe coto as ae ts o-e optmzato ad costats. Recet eves of MC agothms ad techooges ca be foud [], [3]. Hoeve, the possbe appcatos of MC ae mted to Autho ame s cuety pusug mastes degee pogam eectc coto egeeg Ia Uvesty of Scece ad echoogy, Ia. E- ma: saeefa@gma.com Co-Autho ame s cuety pusug Assocate ofesso eectc coto egeeg Ia Uvesty of Scece ad echoogy, Ia. E- ma: faokh@ust.ac. ea systems. Whee ea modes ae ot suffcety accuate, the detfcato of o-ea modes fo coto becomes absoutey ecessay. heefoe, the moe chaegg task of deveopg a oea MC (C) has aso bee attempted [4]. A eactve tajectoy tackg cotoe based o oea mode pedctve coto has bee peseted [5]. A oea mode pedctve coto scheme th obstace avodace fo tajectoy tackg of a mobe obot has bee poposed [6]. Moe exampes ca be foud [7], [8], [9], []. he success of ay MC mpemetato depeds o the effectveess of the souto method used. Oe possbe ad vey pomsg appoach to dyamc optmzato s to appy teget agothms such as eua etoks () ad fuzzy systems, hch have bee used cotoe desgs to dea th vaous ucetaty pobems the system. A path tackg scheme fo a mobe obot based o fuzzy ogc pedctve coto s peseted [3], hee pedctve coto s used to pedct the posto ad the oetato of the obot, he the fuzzy coto s used to dea th the oea chaactestcs of the system. he ma cotbuto of ths pape s the deveopmet of a C fo tackg coto of WMRs. I the poposed cotoe, a fuzzy mode th paametes adapted o-e s used to estmate the dyamcs of the obot. It s assumed that thee ae ucetates both kematc ad dyamc paametes ad actuato paametes. o dea th the ucetates, a adaptve cotoe s desged usg a gadet descet agothm. he poposed method s apped to a type (, ) WMR. he est of ths pape s ogazed as foos: I Secto, the WMR dyamcs ad the C stategy ae peseted. Secto 3 descbes the adaptve C desg. Secto 4 shos smuato esuts, ad fay, cocusos ae gve Secto 5. IJSER

2 Iteatoa Joua of Scetfc & Egeeg Reseach Voume 3, Issue 9, Septembe- ISS ROBLEM FORMULAIO. Dyamcs of WMR Usg the Eue-Lagage fomuato, the dyamcs of WMRs ca be descbed by [4], [5], [6]: M( q) q C( q, q) q F( q) G( q) τ B( q) τ A ( q) λ () hee Mq ( ) s a symmetc, postve defte eta matx, C( q, q ) s the cetpeta ad Coos matx, Fq ( ) s the vecto of suface fcto, Gq ( ) s the gavtatoa vecto, τ d deotes bouded uko dstubaces cudg ustuctued umodeed dyamcs, Bq ( ) s the put tasfomato matx, τ s the m put vecto, Aq ( ) s the matx assocated th the m costats, ad λ s the vecto of costat foces. Suface fcto s as foos: f ( q) Fvq Fd sg( q ) () hee F v s the coeffcet of the vscous fcto ad F d s the coeffcet of the dyamc fcto. he dyamcs of the DC sevomotos hch dve the hees of the obot ca be expessed as foos: τs Ka (3) La Ra Kee u hee τ e s the vecto of toque geeated by the moto, K s the postve defte costat dagoa matx of the moto toque costat, a s the vecto of amatue cuets; L, R, ad K e ae the dagoa matx of amatue ductace, amatue esstace ad back eectomotve foce costat of the motos, espectvey; e s the agua veoctes of the actuato motos. he moto toque τ s ad the hee toque τ ae eated by gea ato as: τ τ (4) s hee s a postve defte costat dagoa matx, ad the agua veoctes of the actuatos e ae eated to the hee agua veoctes v as: e v (5) By gog the amatue ductace ad cosdeg eatos (4)-(5), Eq. (3) ca be defed as foos: τ Ku Kv (6) hee K K R a, K K ek. he eato betee the hee agua veoctes v ad the veocty vecto v s: b v v v (7) b Substtutg (6) ad (7) (), the equato of WMR, cudg actuato dyamcs, ca be obtaed as: M( q) q C( q, q) q F( q) G( q) τd (8) B( q) ( Ku Kv) A ( q) λ he kematc mode of WMR ca be expessed as foos: q S( q) v (9) d By takg the tme devatve of the kematc mode (8), the obot dyamcs (8) ca be tasfomed to: Mv Cv F τd K B u () hee M S MS, C S MS S CS K B () B S B, F S F, τd S τ d Accodg to (), the hee actuato put votages ae cosdeg as the coto puts.. Mode edctve Coto Agothm he MC s a optma coto hch uses pedctos of the system output to cacuate the coto a [7]. At each sampg stat, the mode of the system s used to pedct the output of the system ove a pedcto hozo p, ad by mmzg a pedefed objectve fucto, the futue sequece of coto puts ae computed. By use of the ecedg hozo stategy, oy the fst coto acto the sequece s apped to the system ut the ext sampg tme s eached [8]. he hozos ae the moved oe sampe peod toads the futue, ad optmzato s epeated. Cosde the foog oea state-space mode: x t f ( xt, ut ) () m hee xt ad ut ae the system state ad coto put, espectvey. I ths pape, t s assumed that fucto m f () s cotuous ove. By defg eo vectos x x x ad u u u, e ca fomuate the cost fucto as foos: p J ( k) x ( k j k) Q x( k j k) j c u ( k j ) R u( k j ) j (3) hee p, c ae the pedcto hozo ad coto hozo espectvey, ad Q, R ae eghtg matces fo the eo vectos of state ad coto vaabes espectvey. We cosde aso a costat apped to the amptude of the coto vaabe: u u( k j k) u (4) m max Hece, the oea optmzato pobem ca be expessed as foos: xu, u agm Jk ( ) (5) Such that: X( k k) x m X( k j k) f ( X( k j k), u( k j k) u u( k j K) u d max (6) At each sampg tme k, the optmzato pobem (5) ca be soved, yedg a sequece of optma coto sgas * * { u ( k K),, u ( k c K)}. he the fst eemet of the sequece of optma cotos, u ( k K), ca be apped to the optmzato pobem as the actua coto acto. hs poce- IJSER

3 Iteatoa Joua of Scetfc & Egeeg Reseach Voume 3, Issue 9, Septembe- 3 ISS due epeats at tme k. Fg.. Bock dagam of the poposed C 3 ADAIVE C DESIG he pupose of tajectoy tackg of WMRs s to obta a coto a based o a adaptve fuzzy C techque. he ovea coto stuctue s sho Fg.. Fuzzy systems ae appopate caddates fo modeg ad coto of oea systems. A adaptve fuzzy system s defed as a fuzzy ogc system hose ues ae deveoped though a tag pocess. he poposed fuzzy mode s used to appoxmate the mode of a mobe obot, cudg actuato dyamcs ode to pedct the futue output. Aso the gadet descet agothm s empoyed to adapt the paamete ucetates. he goveg equato of a mobe obot, cudg actuato dyamcs, ca be descbed geeay as a oea dscete system: x( k ) f( x( k), u ( k)) (7) hee x [ v, v ] s the vecto of system states ad u( k) [ u ( k), u ( k)] s the vecto of put votages. I the poposed coto scheme, a fuzzy mode s used to estmate the mode of a mobe obot th actuato dyamcs (7) fo pedctg the obot behavo. he fuzzy mode cossts of to paae fuzzy systems as sho Fg.. Each fuzzy system has thee puts ad oe output. he vectos v( k ), ( k ), u ( ) ( ) k u k, ad ( k ), v( k ), u ( ) ( ) k u k ae the fst ad secod fuzzy system put vaabes, espectvey. he paametes u ( k ) ad u ( k ) deote the ght hee votage ad eft hee votage of the obot. he outputs ae ea veocty, ad agua veocty ( k) at tme stat k. é ù u ( k) + u ( k) ê ( k) ú é ( k) ù u ( k) - u ( k) ê ú ë û Fuzzy System Fuzzy System Fg.. Bock dagam of Fuzzy mode v ( k ) ( k ) Fom the obot dyamcs (), (), v ad ca be obtaed as: K K ( m- m ) d v ( u u ) v q & & = m m m (8) K b m d K b C & = ( u - u )- v - hee = ( I + ( m - m ) d ). c he puts u( k ) u( k ) ad u( k ) u( k ) ae seected fo fuzzy systems because (8), the coeffcets of u ad u the equato used to cacuate v ad the coeffcets of u ad u the equato used to cacuate ae detca. he membeshp fuctos of puts ad outputs ae sho Fg. 3. he ext step s the ceato of the fuzzy ues based o sampe data obtaed fom the appoxmate obot dyamcs (7). he fuzzy ue-base hch s sho abes ad, cotas ues coveg a combatos of membeshp fuctos of the 3 put vaabes, gvg a tota of 45 ues. I ths pape e use the set of fuzzy system hch cudes a sgeto fuzzfe, a poduct feece ege, ad the cete-aveage defuzzfe. he set of fuzzy system ca be expessed as foos: M x x y exp a f( x) (9) M b x x exp M M () (((7 a y z, b z x x z exp () hee M ad ae costats hch deote the umbe of fuzzy ues ad puts espectvey. he fuzzy membeshp fucto that s used ths pape s a Gaussa-shaped fom th a cetod x ad a dth. Aso, y s the cetod of the fuzzy membeshp fucto coespodg to the th ue. he pupose of adjustg the paametes of the fuzzy mode s to mmze the adaptve eo hch s defed as foos: e( k) ( f ( x( k)) y( k)) () hee f( x ) s the fuzzy output, ad yk ( ) s the ea output of the pat at tme k, ad ek ( ) s the eo at tme k. he paametes of the fuzzy mode ae updated o-e va the foog gadet descet method: e( k) e( k) ( f ( x( k)) y( k)) y ( k ) y ( k), (3) y ( k) y ( k) b IJSER

4 MF of Output v(k+) o (k+) MF of Output v(k+) o (k+) MF of (k) [ FS] o v(k) [ FS] MF of U+U(k)[ FS] o U-U(k)[ FS] MF of v(k)[ FS], (k)[ FS] Iteatoa Joua of Scetfc & Egeeg Reseach Voume 3, Issue 9, Septembe- 4 ISS V[m/s] o W[ad/s] U+U[Vot] o U-U[vot] [ad/s].o.4 v[m/s].6.8 v[m/s] o [ad/s] (c) ABLE FUY RULE-BASE OF FUY SYSEM FOR HE IUS: (A) u ( k) + u ( k) Î, (B) u ( k) + u ( k) Î, AD (C) u ( k) + u ( k) Î. ek ( ) (C) x ( k ) x ( k), x ( k) e( k) ([ f ( x( k)) y( k)][ y f ( x( k))] z [ x x ]) x ( k) b( ) ek ( ) ( k ) ( k), ( k) 3 e( k) ([ f ( x( k)) y( k)][ y f ( x( k))] z [ x x ] ) ( k) b( ) hee, s the eag ate of fuzzy system. (A) (B) (4) (5) v[m/s] o [ad/s] (d) 4 SIMULAIO RESULS I ths secto, some compute smuatos ae pefomed to evauate the pefomace of the poposed cotoe. I these smuatos, the ea physca paametes of the WMR ad coto paametes ae summazed abe 3. he paamete m c s the mass of the patfom thout the dvg hees ad the otos of the DC motos, m deotes the mass of each dvg hee pus the oto of ts moto, I c deotes the momet of eta of the patfom thout the dvg hees ad the otos of the motos ad I m deotes the momet of eta of each hee ad the moto oto about a hee damete. Fg. 3. Membeshp fuctos of fst put, secod put, (c) thd puts, ad (d) the output, of each fuzzy system IJSER

5 Iteatoa Joua of Scetfc & Egeeg Reseach Voume 3, Issue 9, Septembe- 5 ISS ABLE FUY RULE-BASE OF FUY SYSEM FOR HE IUS: (A) u ( k) - u ( k) Î, (B) u ( k) - u ( k) Î, AD (C) u ( k) - u ( k) Î (A) (B) (C) he kematc ad dyamc matces () ae expessed as: cos m S( q) s, M I K (6) md c C bk md c hee m mc m, I Ic Im mcd mb. he moto votage bouds ae sted as [- v, v]. he cotoe paametes ae seected as p 5, c. he sampg tme s. sec. he eghtg matces ae assumed as Q dag5,3,, R dag.5,.5. As dscussed befoe, the mode of the mobe obot th moto dyamcs s estmated by fuzzy systems dug the oe optmzato. he eag ate of each fuzzy system uses.9. I ode to sho the pefomace of the poposed cotoe, the fuzzy C as apped to the obot fo to cases: I the fst smuato, the adaptve tackg cotoe s tested oy fo uceta paametes. It s assumed that thee s o koedge about the WMR paametes, ad thee s o dstubace ths case. he desed tajectoy fo ths case s a ccua path hch s chose as foos: x ( t) 7.5cos( t), y ( t) 5 7.5s( t) hee ( )., ad v ( t).5. t ABLE 3 WMR ARAMEERS aamete Smuato vaue (m).5 b (m).75 d(m).3 L(m). m (m) m (m) c 36 I (Kg.m ) m.5 I (Kg.m ) c 5.65 I (Kg.m ).5 (s). K K (7) he ta posto of the WMR s seected as q ( t) 9,4, p /. Smuato esuts of the poposed fuzzy C ae sho fgues 4 to 6. As these fgues sho, the WMR ca foo the desed path ad at the desed veocty th good accuacy. Moeove, the moto votages ae th the pedefed bouds. Mea squae posto ad veocty eos fo oadaptve Fuzzy C ad adaptve Fuzzy C afte eachg the tajectoy ae gve abe. 4. As sho the tabe, the coto behavo of the adaptve Fuzzy mode base pedctve cotoe s see to be eatvey dea fo tackg a ccua path. I the secod case, a extea dstubace τ d s apped to the WMR at t 5 sec. he desed tajectoy fo ths case s a cce ad the othe coto paametes ae the same as case oe. he smuato esuts of the o-adaptve Fuzzy C ad the adaptve Fuzzy C ae sho Fg. 7 espectvey. As sho ths fgue, the o-adaptve Fuzzy C foos the efeece tajectoy th a szeabe eo vesus the adaptve Fuzzy C. As these fgues sho, the adaptato capabty of the fuzzy system ca cope th ths dstubace vey qucky ad etu the mobe obot to ts desed path, yedg a adaptve ad obust coto method. ABLE 3 MEA SQUARE OSIIO AD VELOCIY ERRORS FOR O- ADAIVE FUY C AD ADAIVE FUY C MSE Mode osto.3445 o-adaptve Veocty.453 Mode osto.99 Adaptve Veocty.9 IJSER

6 (ad/s) y (m) v (m/s) y (m) y (m) U (Vot) U (Vot) Iteatoa Joua of Scetfc & Egeeg Reseach Voume 3, Issue 9, Septembe- 6 ISS COCLUSIO o acheve bette path tackg fo WMRs, a adaptve fuzzy C coto method as desged ths pape. he poposed cotoe soves the tegated kematc ad dyamc tackg pobem the pesece of both paametc ad opaametc ucetates. Futhemoe, a fuzzy system hose paametes ae updated o-e by a gadet descet agothm has bee empoyed. Whe ths fuzzy system ca povde a appopate mode of the obot, t ca aso dea th ay chages obot paametes. he smuato esuts o a type (, ) WMR ustate the effectveess of the poposed coto scheme. Futue ok shoud focus o the stabty aayss of the poposed method Refece ajectoy x (m) Fg. 4. Desed ad actua tajectoes fo WMR V Ref Fg. 5. Desed ad actua veoctes of WMR V W Ref W IJSER Refece ajectoy x (m) Fg. 6. Coto votages of the WMR fo: the ght hee, the eft hee Refece ajectoy x (m) Fg. 7. Desed ad actua tajectoes fo WMR pesece of extea dstubaces fo: o-adaptve Fuzzy C, ad Adaptve Fuzzy C

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