Path Following Control of Mobile Robot Using Lyapunov Techniques and PID Cntroller

Transcription

1 Innaonal Jounal of Fuzz ogc and Inllgn Ssms, vol. 11, no. 1, Mach 11, DOI : 1.591/IJFIS Pah Followng Conol of Mobl obo Usng aunov chnqus and PID Cnoll asok Jn * and Han-Ho ack ** * D. of Mchaoncs Engnng, Dongso Unvs, Busan, , Koa **D. of Elconcs Engnng, Jnju Naonal Unvs, Jnju, , Koa Absac Pah followng of h mobl obo s on sach ho fo h mobl obo navgaon. Fo h conol ssm of h whld mobl obo(wm) bng n nonhonolomc ssm and h comlx laons among h conol aams, s dffcul o solv h oblm basd on adonal mahmacs modl. In hs a, w sns a sml and ffcv wa of mlmnng an adav followng conoll basd on h PID fo mobl obo ah followng. h mhod uss a non-lna modl of mobl obo knmacs and hus allows an accua dcon of h fuu ajcos. h oosd conoll has a aalll sucu ha consss of PID conoll wh a fxd gan. h conol law s consucd on h bass of aunov sabl ho. Comu smulaon fo a dffnall dvn nonholonomc mobl obo s cad ou n h vloc and onaon ackng conol of h nonholonomc WM. h smulaon suls of whl mobl obo lafom a gvn o show h ffcvnss of h oosd algohm. K Wods : Pah followng, whld mobl obo, aunov, Nonholonomc, Knmacs. 1. Inoducon In h las dcad as, mobl obo has bn dvlod as ndusal Auomad Gudd Vhcls (AGV) wh hgh nonlnas ha a ofn unknown and m-vang. hfo, f w wan o dsgn a conoll fo mobl obo, w should consd ha h xac ah-followng(pf) fomanc, whch s concnd wh h abl o dv a mobl obo auonomousl as clos as ossbl o a vousl dfnd fnc ah[1]. Whld mobl obos hav man alcaon flds n ndusal and svc obocs, aculal whn flxbl moon caabls a qud on asonabl smooh gounds and sufacs []. h a scall ncssa fo asks ha a dffcul and dangous fo mn o fom. Man sachs hav shown ns n mobl obos. Mos of hm hav focusd on ajco ackng o conol h obo o follow a dsd ajco, and on sablzaon o sablz a obo a a dsd on. [-3]. Among h man conol sags ha hav bn oosd fo vaous nonholonomc ssms, sach suls can gnall b classfd no wo classs. h fs class s knmac conol, whch ovds h soluons onl a h u knmac lvl, wh h ssms a snd b h knmac modls and vloc acs as h conol nu. Basd on xac ssm knmac, dffn conol sags hav Manusc cvd Jan. 4, 11; vsd Ma. 1, 11; Accd Ma., 11. hs sach was suod b Basc Scnc sach Pogam hough h Naonal sach Foundaon of Koa(NF) fundd b h Mns of Educaon, Scnc and chnolog(1-154) bn oosd [4,5,7]. cnl, a fw sach woks hav bn cad ou o dsgn conolls agans ossbl xsnc of modlng uncans and xnal dsubanc[8-1]. obus xonnal gulaon s oosd n [8] b assumng known bounds of h nonlna dfs. I s also qud ha h x -subssm s schz. o lax hs condon, adav sa fdback conol s oosd n [9] fo ssm wh song nonlna dfs. Howv, h conol nu of h conoll fo h knmac modl s gnall vloc, bu s mo alsc ha h nu s oqu. In [6], a knmac conoll s dsgnd fs so ha h ackng o bwn a al obo and a fnc obo convgs o zo, and scondl a oqu conoll s dsgnd b usng backsng so ha h vlocs of a mobl obo convg o h dsd vlocs, whch a gvn b h knmac conoll dsgnd a h fs s. h convnonal fdback conolls, ncludng ooonal ngal dvav (PID) conoll, a commonl usd n h fld of nduss bcaus h conol achcus a v sml and as o mlmn. Bu whn hs convnonal fdback conolls a dcl ald o nonlna ssms, h suff fom h oo fomanc and obusnss du o h unknown nonlnas and h xnal dsubancs. Dung dcads, vaous conol sags o dal wh h unknown nonlnas and h xnal dsubancs a oosd such as auomac unng of PID conol, vaabl sucu conol known as nonlna obus conol, fdback lnazaon, modl fnc adav conol, dc adav conol, and nllgn conol aoachs, c [11-1]. An adav conol sag has found man alcaons n such aas as obo manulaos, sh sng, acaf conol, and ocss conol, bcaus can connuousl adjus aams of a conoll o accommoda changs n ssm 49

2 Innaonal Jounal of Fuzz ogc and Inllgn Ssms, vol. 11, no. 1, Mach 11 dnamcs and dsubancs [1]. In hs alcaons, an onln adaaon law s usuall usd o sma h unknown aams of h ssm and hn, an aoa conoll s dsgnd o conol h lan o sasf a dsd fomanc. o al an adav conol mhod o obo manulaos as a man conoll, h a o knowldg abou h obo manulao s qud: h lna abou unknown aams and skw-smm faus, c. Addonall aks a long m o calcula h gsson max usd n h algohm. In hs a, h adav followng conol s usd as a followng conoll of mobl obo [9-11]. And also, a classcal PID aoach s ald fo h ah-followng conoll, whch s h conol sag mos fqunl usd n h ndus. A v sml modl of h mobl obo knmacs s usd and hus a obus PID unng s ncssa. PIDs advanags nclud smlc, obusnss and h famla n h conol commun. Bcaus of hs, a ga dal of ffo has bn sn o fnd h bs choc of PID aams fo dffn ocss modls. h sucu of h nwl oosd conol algohm of nonlna PID-basd adav conol s shown n Fg. 3. hs conol algohm has h chaacscs such as sml sucu, ll comuaon m, and connuous auo-unng mhod of h PID conoll. h a s oganzd as follows: In Scon, w dscuss h modl and modl ansfomaon of h ssms ncludng h knmacs of h mobl obo. Scon 3 dals wh h oblm of dsgnng conoll basd on aunov chnqus and PID, and Scon 4 dals wh sval suls o show ha h oosd mhod s ffcv. Fnall, som conclusons a dawn n Scon 5.. Knmacs Analss of Mobl obo A wo-whl dffnal dv mobl obo was chosn as h objc n hs a. Is whl oaon s lmd o on axs. hfo, h navgaon s conolld b h sd a chang on h sd of h obo. hs knd of obo has nonholonomc consans. h knmacs schm of a wowhl dffnal dv mobl obo s as shown n Fg. 1, wh {O, X, Y} s h global coodna, v m s h vloc of h obo cnod, s h angula vloc of h obo cnod, v l s h vloc of h lf dvng whl, v s h vloc of h lf dvng whl, s h dsanc bwn wo dvng whls, s h adus of ach dvng whl, x and a h oson of h obo, θ s h onaon of h obo, and m s h angula vloc. Accodng o h moon ncl of gd bod knmacs, h moon of a wo-whl dffnal dv mobl obo can b dscbd usng quaons (1) and (), wh ml, and m, a angula vlocs of h lf and gh dvng whls scvl. Y Cx (, ) v l m v v m X O Fg. 1 Knmacs schm of dffnal dv mobl obo. vl ml, v m, vm ( v + vl)/ ( v v )/ m l h nonholonomc consan quaon of h obo s as followng: x snθ cosθ (3) Dvng (4) fom (1) and (): vm m m, + ml, m, ml, Moov, w can dfn h dnamc funcon of h obo as fomula (5). θ x vmcos θ ( m, + m, l)cosθ vmsn θ ( m, + m, l)snθ θ m ( m, m, l) wng h quaon (5), w can xss h Jacoban max as (6) cosθ cosθ, snθ snθ ml, θ m All vaabls a nlad n quaon (6), whch causs h conoll dsgn o b mo comlx. hfo, quaon (6) should b dcould. Fo θ s onl lad o m, x and a onl lad o v m. (1) () (4) (5) (6) 5

3 Pah Followng Conol of Mobl obo Usng aunov chnqus and PID Cnoll 3. Dsgnng Conoll Cona o h usual suaon, ackng s as han gulaon fo a nonholonomc WM. An nuv xlanaon of hs can b gvn n ms of a comason bwn h numb of conolld vaabls (ouus) and h numb of conol nus. Fo h wo-whl dffnal dv mobl obo of Scon, wo nu commands a avalabl whl h vaabls ( x, and h onaon θ ) a ndd o dmn s confguaon. hus, gulaon of h WM osu o a dsd confguaon mls zong h ndndn confguaon os. Whn ackng a ajco, nsad, h ouu has h sam dmnson as h nu and h conol oblm s squa. us dno h cun oson of h mobl obo as m, h vloc as m. h Casan vloc m, s snd n ms of jon vaabls as (7). So, h knmacs modl of h obo s as followng: J( ) q (7) m m m cosθ vm snθ J( m) q m (8) θ 1 m hfo, n od o oban h al-m oson of h obo, w can conol h conol law as qm [ v m, m]. h followng s h wa o g h dsd nus of h mobl obo ssm mahmacall. h ssu of mobl obo ah followng can gnall b ansfomd no followng on fnc mobl obo. o assum h obo cun oson o b,,θ m x, and h sd o b qm [ v m, m], h fnc mobl obo oson s,,,θ m x, and h sd o b q, [ v, ], as shown n Fg.. Y m o od [ x,, θ] [ x x,, θ θ], and h mobl obo oson o quaon (9) s oband fom h gomc laonsh shown n Fg.. h fnc sa n h obo coodna s xssd as follow b h ansfomaon fom h wold coodna o h obo coodna. x cosθ snθ snθ cosθ θ 1 (, ) (, ) m m m m hn h oson o dffnal quaon s as followng: (9) m v+ vcosθ xm + v snθ (1) θ m Pah followng of mobl obo basd on knmacs modl s o sach h boundd nu as q [ v m, m], ha causs oson o vco, [ x,, θ ] o b boundd, and lm [ x,, θ], wh aba nal o and h quaon (1) conolld b h conol law. B usng h o vco ( x,, θ), h followng cuv n h obo coodna can b calculad as follow; m v cosθ (7) v θ + snθ m m v θ snθ m (8) (9) A aunov candda funcon s dfnd as n quaon (3). 1 λ 1 θ θ V V1 + V + ( + h ) (3) θ m, x θ wh V 1 mans h o ng o h dsanc and V mans h o ng n h dcon. Af dffnang boh sds n quaon (3) n ms of m, w can acqu h sul as n quaon (31). V V 1 + V λ + ( θθ + θθ h ) (31) O m x θ m ackng cuv Dsd ah Fg. Poson o of mobl obo und Casan coodna. X us subsu quaon (31) no h cosondng a n quaon (31), suls n quaon (3). V λv mcosθ v sn θ ( θ + θ ) + θ [ + m h m ] θ (3) No ha V < s qud fo a gvn V o b a sabl ssm. On hs bass, w can dsgn h nonlna conoll of h mobl obo as n quaon (33),(34). 51

4 Innaonal Jounal of Fuzz ogc and Inllgn Ssms, vol. 11, no. 1, Mach 11 v γ ( cos θ ), ( γ > ) (33) m cosθ snθ θ + γ m k ( θ + h θ), ( kh, > ) (34) θ hfo, usng hs conoll fo h mobl obo, V aoachs o zo as ; and θ also aoach almos o zo as shown n (35). x θ x θ V λγ ( cos θ) k θ (35) v m m Fg. 3 Conol ssm dagam. x θ 4. Exmns In od o s h mhod of hs a, h conoll s smulad wh h ccula ack and h ln ack as fnc ajcos, scvl. h smulaon: h fnc ack s as x + 1.5, h fnc vloc s as v., and h fnc angula vloc s as.. Whl h nal vloc of h mobl obo s as v m.4, and h nal angula vloc of h mobl obo s as m.3, h nal oson fo h mobl obo s as m ( 1, 1., π /3). h smulaon suls a shown n Fg. 6. I s moan o no h good fomanc of h conoll n s of h v sml non-lna modl usd fo h comuaon of h conol law. hs of modl smlfcaon s v moan whn a sml and low-m consumng conol law s ndd bcaus of ocsso lmaons. In hs a, w us a PID algohm fo conollng oson and vloc of h mobl obo lafom. Usng h PID algohm, w can conol h moan ssm chaacscs, l sa us, sng m, sad sa o, ssm sabl, c. Each m n h conol algohm has a dffn ffc on h ssm chaacscs [15]. In h PID conol, h nu fo conol of a sandad PID conoll n connuous m s sad as n quaon (36). 1 d( ) m () K () + ( τ) dτ+ d d (36) d () K () + K ( τ) dτ+ K d Fg. 5 Pah followng fo h mobl obo. wh () sns o sgnal ha s h dffnc bwn dsd nu and ouu sgnal. Fo h dgal conol PID quaon n dsc m s xssd as quaon (37). Δ mk () K Δ k () + Kk () + K Δ k () (37) d Also h abov quaon (37) can b wn as quaon (38). Δ mk () vk () + vk ( 1) + v k ( ) (38) 1 wh v K + K + K d, v1 K K, and v K d. d Fg. 6 Followng o: x,, and θ. K () Δm() K ( τ ) dτ m () d () K d Fg. 4 PID conoll dagam. Fg. 7 ajco os of mobl obo. 5

5 Pah Followng Conol of Mobl obo Usng aunov chnqus and PID Cnoll Fgs. 7 shows h ouus and conol nus ha mak h mobl obo follow dsd ah. I should b nod ha h hozonal and vcal dslacmns a bough o zo n 5 [sc] and 6 [sc] n Fg. 7, scvl. 5. Conclusons hs a sns an adav followng conoll fo mobl obo basd on h PID. h oosd conoll has a aalll sucu ha consss of PID conoll wh a fxd gan. h conol law s consucd on h bass of aunov sabl ho. And also, hs a shows ha h followng os a boundd unfoml and ulmal und h xsnc of h dsubancs and modlng o, mahmacall. h whld mobl obo s mlod as a s-bd o al h oosd conoll. h xmnal suls show ha h oosd conoll s adaabl o h nvonmn changs and s mlmnd o h al ssm and h xmnal suls dmonsa h ffcvnss of h mhods. Fo h fuu wok, obus unng and oson smaon should b consdd o alz mo accua moons of mobl obos. fncs [1] Fo, ws F, Conol of a nonholonomc mobl obo: backsng knmacs no dnamcs, In Pocdngs of h IEEE Confnc on Dcson and Conol (CDC 95), 1995, []. X. Wang, Adav fuzz ssms and conol dsgn and sabl analss, Pnc hall, [3] W. E. Dxon, D. M. Dawson, E. Zgoglu and A. Bhal, Nonlna Conol of Whld Mobl obos. Sng, 1. [4] C. Samson, m-vang fdback sablzaon of a nonholonomc whld mobl obo, Innaonal Jounal of obocs sach, vol. 1, , [5] I. Kolmanovsk and N. McClamoch, Dvlomn n nonholonomc conol oblm, IEEE Conol Ssm Magazn, vol.15,.-36, [6] PEI Xnzh, sach on ajco ackng and sablzaon of nonholonomc mobl obos, Phd. dssaon, D. conol ho and conol ngnng, Habn Insu of chnolog, Habn, Chna, 3. [7] A. Asolf, Dsconnuous conol of nonholonomc ssms, Ssm and Conol s, vol. 7, , [8] Z. P. Jang, obus xonnal gulaon of nonholonomc ssms wh uncans, Auomaca, vol. 36, ,. [9] S. S. G, Z. P. Wang, and. H., Adav sablzaon of uncan nonholonomc ssms b sa and ouu fdback, Auomaca, vol. 39, , 3. [1] Slon JJ, W. Ald nonlna conol. Englwood Clffs, NJ: Pnc-Hall; [11] F. Poubogha, M.P. Kalsson, Adav conol of dnamc mobl obos wh nonholonomc consans, Comu.Elc.Eng. 8, ,. [1].D.C. hanh, K.K. Ahn, Nonlna PID conol o mov h conol fomanc of axs numac afcal muscl manulao usng nual nwok, Mchaoncs, vol.16, , 6. a-sok Jn H cvd h Ph.D. dgs fom Pusan Naonal Unvs, Busan, Koa, n 3, n lconcs ngnng. H s cunl an Asssan Pofsso a Dongso Unvs. Fom 4 o 5, h was a Posdocoal sach a h Insu of Indusal Scnc, h Unvs of oko, Jaan. Hs sach nss nclud nwok snsos fuson, mobl obos, comu vson, and nllgn conol. D. Jn s a Mmb of h KIIS, ICOS, and SJ. Han-Ho ack H cvd h B.S. dg n Damn of Elconc Engnng fom Bukung Naonal Unvs, Busan Koa, n H cvd h M..S. dg n Damn of Elconc Engnng fom Dong-A Unvs, Busan, Koa, n 199. H cvd Ph. D. dg n Damn of Elconc & Communcaon Engnng fom h Koa Mam Unvs, Busan, Koa, n Snc 1991, h has bn a facul mmb of h Elconc Engnng a h Jnju Naonal Unvs, wh h s cunl a Pofsso. Hs sach nss a Nual Nwok, Fuzz Ssm, obocs, Faco Auomaon, Mchancal Vbaon, ansoaon, and Mulmda Ssm c. H s a mmb of IEEE, KIMISC, KMS, KIEE, and KFIS. 53