Adaptive Learning Approach of Fuzzy Logic Controller with Evolution for Pursuit Evasion Games

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1 datv Larnng roach of Fuzzy Logc Controllr wth Evoluton for Pursut Evason Gams Hung-Chn Chung and Jng-Sn Lu Insttut of Informaton Scnc, cadma Snca Nangang, Ta, Tawan 59 bstract. Ths ar studs a smlfd ursut-vason roblm. W assum that th vadr movs wth constant sd along a trajctory that s wll-dfnd and known a ror. Th objctv of strng control of th ursur modld as a nonholonomc uncycl-ty mobl robot s to ntrct th movng vadr. n adatv larnng aroach of fuzzy logc controllr s dvlod as an nvrs knmatcs solvr of uncycl to nabl a mobl robot to us th vadr trajctory to adat ts control actons to ursut-vason gam. In ths roosd aroach, G volvs th aramtr valus of th fuzzy logc control systm amng to aroxmat th nvrs knmatcs of ursur so as to gnrat a trajctory caturng th vadr. Smulaton rsults of ursut-vason gam llustrat th rformanc of th roosd aroach. Kywords: Fuzzy logc control, Pursut-vason, Invrs knmatcs, Gntc algorthm. Introducton Pursut-vason gams hav bcom th ncrasngly mortant ssus n robotcs scurty and survllanc rcntly. Thr ar many tys of ursut-vason gams, such as rul-basd fuzzy systm for ursut-vason gams [], ursut-vason gams wth sarchng th nods of a grah [], vsblty basd ursut-vason gams [3]. [] rsntd a tm-otmal control stratgy of ursur for th ursutvason gam. Th authors [4] roosd a nw mthod basd on hrarchcal rnforcmnt larnng to study mult-agnt ursut-vason roblm, and th xrmntal rsult has also showd that basd on th oton algorthm of hrarchcal rnforcmnt larnng, th algorthm ffcncy can rduc th comlxty of th ursut-vason task, avodng tradtonal rnforcmnt larnng curs of dmnsonalty. Fuzzy systm osssss charactrstcs of lngustc nformaton and logc control, whl nural ntworks hav charactrstcs of aralllsm, fault tolranc and assocaton. Thrfor, fuzzy systms and nural ntworks ar ald to svral control roblms [-] wth satsfactory rsults and ossss th charactrstcs of unvrsal aroxmaton [3-4]. Tradtonally, th fuzzy systms and nural ntworks ar trand by usng th gradnt dscnt mthod. Howvr, such tchnqus may lad to local otmum. Som rsarchrs hav bn tryng to us volutonary algorthms, such as gntc algorthms (Gs), to ovrcom such dffcults [5-7]. J.-S. Pan, S.-M. Chn, and N.T. Nguyn (Eds.): ICCCI, Part I, LNI 64, ,. Srngr-Vrlag Brln Hdlbrg

2 datv Larnng roach of Fuzzy Logc Controllr 483 In ths ar, our objctv s to dvlo an adatv larnng aroach of fuzzy logc controllr wth voluton to nabl a uncycl-ty whld mobl robot adat ts actons to ursut-vason gam, assumng th vadr trajctory s wll-dfnd and known. Th moton control of ursur s to ntrct th vadr. Th wghtng aramtrs and mmbrsh functons of th fuzzy logc controllr ar tund va volutonary mthod such as Gs. In addton, xrmntal rsults of ursut-vason gams llustrat th rformanc of th roosd aroach. Ths ar s organzd as follows. In Sc., statmnt of th ursut-vason gam that ths ar dals wth s formulatd as an nvrs knmatcs roblm. fuzzy logc control algorthm of vadr trajctory trackng s roosd n Sc. 3. In Sc. 4, th fuzzy logc controllr aramtrs ar tund by gntc algorthm. In Sc. 5, smulaton rsults of ursut-vason ar shown to dmonstrat th ffctvnss of our roosd aroach. In Sc. 6, w conclud ths ar. Problm Statmnt sml structur of two crcular-sha mobl robots n ursut-vason gam s shown n Fgur.Th knmatc quatons of th ursur robot modld as a nonholonomc uncycl mobl robot [] ar wrttn as: x y θ = = v cosθ = vsnθ v R tan u whr ( x, y ) dnots th oston of th ursur robot, θ s th orntaton, v s th vlocty, u s th strng angl and R s th whl bas. W assum that th vadr movs wth constant sd along a trajctory ( x, y ) that s wll-dfnd and known a ror. W consdr a smlfd control roblm: v s constant and th only control s th strng angl u.ltrnatvly, for control uros, dfn th rfrnc sgnal to b th vsblty ln angl tan y y θ d = ( ), and th rlatv dstanc btwn ursur and vadr x x d = ( x x ) + ( y y ) as th vadr nformaton. Th control objctv s to dsgn an ntllgnt controllr, scfcally an adatv fuzzy controllr for () such that by controllng th orntaton angl θ of ursur to guarant that th ursur can follow th drcton of th vadr,.. θ = θ d. Snc th nonholonomc ursur () can t chang ts orntaton θ wthout changng ts oston ( x, y ), th catur occurs whn th ursur can touch (or ntrct) th ()

3 484 H.-C. Chung and J.-S. Lu vadr such that th dstanc d s lss than or qual to th sum of radus of ursur and vadr: d <= r + r () ( x, y ) r θ ( x, y ) r Fg.. Two mobl robots n ursut-vason Formally, th ursut-vason roblm that w dal wth n ths ar can b statd as follows. Gvn a wll-dfnd vadr trajctory ( x ( t), y ( t)), an ntal locaton ( y (, y x, ) of ursur, fnd a u that gnrats a trajctory x, y ) ( of ursur assng through x ), such that th rror = θ d θ along ths trajctory aroachs zro and d aroachs a valu wthn an accuracy tolranc dfnd n () vntually. Th soluton to ths roblm rqurs that th nvrs of th vlocty knmatcs () of ursur must b solvd aroxmatly to fnd control u that strs th ursur n th drcton algnng to th rfrnc sgnal and catch th vadr. Ths s achvd by th fuzzy logc control n ths ar. 3 Fuzzy Logc Controllr for Pursur Moton 3. Dscrton of Fuzzy Logc Systm Th basc confguraton of fuzzy logc systms conssts of fuzzy IF-THEN ruls and a fuzzy nfrnc ngn. Th fuzzy nfrnc ngn uss th fuzzy IF-THEN ruls to rform a mang from nut lngustc varabls to outut lngustc varabls. Gvn th nut data xq, q =,,, n, and th outut data y, =,,, m, th th fuzzy rul has th followng form:

4 datv Larnng roach of Fuzzy Logc Controllr 485 R : IF THEN x y s s w and x n and y m s s n w m (3) whr s a rul numbr, q s ar th fuzzy sts of th antcdnt art, and ral numbrs of th consqunt art. Whn th nuts [ ] T th outut x n w ar x = x x ar gvn, y of th fuzzy nfrnc can b drvd from th followng quatons: y ( x w ) = h n w ( = q= h n ( = q= μ ( x )) q q q q μ ( x )) (4) whr μ ( x q ) s th mmbrsh functon of q h T ruls. w [ ww w ] y (x). q, h s th numbr of th fuzzy = s a wghtng vctor rlatd to th th outut 3. Fuzzy Control as Invrs Knmatcs Solvr Now w rocd to fuzzy control dsgn of ursur moton. Lt th outut trackng T T rror = θ d θ. Th fuzzy logc controllr has two nuts = (, ) = (, ) = θ θ (5) = d v = = tanu (6) y y R + ( ) x x By usng cntrod of ara for dfuzzfr, th strng angl u s th outut, whr h u Fuzzy = w j j= h j= whr h s th numbr of ruls (3) to guarant a rch nough sac of controls, and B j () s a mmbrsh functon dfnd as Gaussan bass functon B B j j ( ) ( ) (7)

5 486 H.-C. Chung and J.-S. Lu m + j Bj ( ) = x( ( ) ), j =,, h (8) σ = + j 4 Tunng of th Fuzzy Logc Controllr Paramtrs va Evolutonary Mthod 4. Gntc algorthm G s a stochastc sarch and otmzaton tchnqu that mtats natural voluton wth Darwnan survval. Tradtonal Gs rforms on th codng of th aramtrs, thrfor, th codng mthod allows Gs to handl multaramtrs or multmodl ty of otmzaton roblms asly. Th mchansm of a G, shown n Fg., can b dvdd nto four arts: frst, k a oulaton of solutons codd as artfcal chromosoms. Scond, choos th bttr solutons (n trms of ftnss) for rcombnaton. Thrd, rform crossovr and mutaton on th chromosoms. Fourth, us ths offsrng to rlac orgnal chromosoms and obtan a nw gnraton. Much work has shown that Gs lad to narly global otmum solutons n many roblms, such as control systm, mag rcognton, ath lannng, and robot larnng, tc. For otmal dsgn of fuzzy systms, thortcal and mrcal rsults hav dmonstratd that Gs ar good canddats for slctng and gnratng fuzzy ruls [8-9]. Fg.. Th rocss of th adatv larnng aroach shows th mchansm of th offln tunng m σ w m σ w m3 σ 3 w m4 σ 4 3 w m5 σ 5 4 w m6 σ 6 5 w 6 w 7 w 8 w 9 Fg. 3. Th st of aramtrs of control, ncodd as th chromosom, conssts of wghtng aramtrs and mmbrsh functons of th fuzzy logc systm

6 datv Larnng roach of Fuzzy Logc Controllr Controllr Paramtrs +Tunng va G n adatv larnng aroach for controllng ursur moton s to fnd such a st of aramtrs of control that str th ursur moton () n th dsrd drcton. Frst, th wghtng aramtrs and mmbrsh functons of th fuzzy logc controllr (7) shown n Fg. 3 ar ncodd as a chromosom to b volvd by G. W comut th outut acton of ach chromosom, and rform th acton. Nxt, a ftnss functon dfnd by = L ftnss d k (9) k= whr d = ( x x ) + ( y y ) and L s th st numbr. Fnally, basd on th ftnss functon, th oratons of G ar rformd accordng to Fg., ncludng rroducton, crossovr and mutaton. Ths rocss s ratd untl th trmnaton crtra ar mt. Th ovrall schm of th roosd controllr s shown n Fgur 4. tan y y θd = ( ) x x u x& = V cosθ y& = V snθ V θ& = tan u R θ d dt Fg. 4. Block dagram of th adatv larnng control systm for ursur, whr th aramtrs of fuzzy logc controllr ar tund va Gs. Th vadr nformaton s n trms of th rang and drcton btwn th ursur and vadr. 5 Smulaton Rsults Consdr th knmatcs quatons of th ursur modl n (). Th radus of th ursur and vadr ar assumd as r =. 6m and r =. 4m, rsctvly. Smulaton ar rformd wth vadr trajctory rrsntd by a crcl x = 5cos( t), y = 5sn( t), whr t [, π ] s a aramtr. Th ntal oston of th ursur and vadr ar assumd as ( x, ) = (, ) and y ( x, ) = (5,), rsctvly. Th ntal orntaton θ = rad and th trcycl y

7 488 H.-C. Chung and J.-S. Lu whl bas R =. m. Th sd of th ursur and vadr s qual V = V =.6m / s. For th fuzzy mmbrsh functons, w choos thr gaussan mmbrsh functons. Each mmbrsh functon has two aramtrs m and σ, rsctvly. Th fuzzy controllr has nuts, so w hav 9 ruls. Thus, w obtan a chromosom wth lngth gns, as shown n Fgur 3. Th oulaton conssts of 3 chromosoms whch ar all randomzd ntally. Th crossovr and mutaton robablty ar assumd as 65% and %, rsctvly. Th trackng rsults of th ursur and vadr ar shown n Fgur 5. On can obsrvs that th roosd controllr can str th ursur robot to succssfully catur th vadr. Onc th ursur caturd th vadr, th ursur wll follow th vadr aftrwards. Fgur 6 shows th control nut rrsnts th strng angl of th ursur, and th catur tm s 7 sconds. Fgur 7 s th ftnss functon of th voluton, showng th dcras of dstanc btwn ursur and vadr, thus convrgnc. Th catur tm of qual sd s tabulatd n Tabl. Th catur tm s monotoncally dcrasd at low sd, but s not ncssarly dcrasd as th sd s hghr. Tabl. sd ( V = V ) Catur tm.m/s.m/s.3m/s.4m/s.5m/s.6m/s.7m/s.8m/s.9m/s m/s 9 s 88 s 79 s 79 s 7 s 7 s 8 s 68 s 53 s 8 s y[m] angl x[m] 5 5 t[s] Fg. 5. Th trackng rsults n xy lan shows that th ursur can catur th vadr succssfully Fg. 6. Th control nut u shows that th catur tm s 7 sconds

8 datv Larnng roach of Fuzzy Logc Controllr 489 x Fg. 7. Th ftnss functon shows th dcras of rlatv dstanc btwn ursur and vadr 6 Concluson In ths ar, an adatv larnng aroach of fuzzy logc controllr usng voluton s dvlod as an nvrs knmatcs solvr of uncycl for strng constant sd ursur moton to catur an vadr wth wll-dfnd, a ror known trajctory for ursut-vason gams. G s mloyd to tun th st of aramtrs of control consstng of wghtng aramtrs and mmbrsh functons of th fuzzy logc controllr usng vadr trajctory. Th alcablty and fasblty of th roosd tchnqu s dmonstratd by th smulaton rsults. Rfrncs. Wang, C.H., Wang, W.Y., L, T.T., Tsng, P.S.: Fuzzy B-sln mmbrsh functon (BMF) and ts alcatons n fuzzy-nural control. IEEE Trans. Syst. Man, Cybr. 5, (995). Wang, L.X.: datv fuzzy systms and control: dsgn and stablty analyss. Prntc- Hall, Englwood Clffs (994) 3. Hornk, K., Stnchcomb, M., Wht, H.: Multlayr fdforward ntworks ar unvrsal aroxmators. Nural Ntworks, (989) 4. Wang, L.X., Mndl, J.M.: Fuzzy bass functons, unvrsal aroxmaton, and orthogonal last squars larnng. IEEE Trans. Nural Ntworks 3, (99) 5. Wang, C.H., Lu, H.L., Ln, C.T.: Dynamc otmal larnng rats of a crtan class of fuzzy nural ntworks and ts alcatons wth gntc algorthm. IEEE Transactons on Systms, Man and Cybrntcs. 3, () 6. Wang, W.Y., L, T.T., Hsu, C.C., L, Y.H.: G-basd larnng of bmf fuzzy-nural ntwork. In: Procdngs of th IEEE Intrnatonal Confrnc on Fuzzy Systms (FUZZ-IEEE ), vol., () 7. Farag, W.., Quntana, V.H., Lambrttorrs, G.: Gntc-Basd Nuro-Fuzzy aroach for modlng and control of dynamcal systms. IEEE Trans. on nural ntworks 9 (998) 8. Yuan, Y., Zhuang, H.: gntc algorthm for gnratng fuzzy classfcaton ruls 84, 9 (996)

9 49 H.-C. Chung and J.-S. Lu 9. Sng, T.L., Khald, M.B., Yusof, R.: Tunng of a nuro-fuzzy controllr by gntc algorthm. IEEE Trans. Syst. Man, Cybr. Part B 9, 6 36 (999). Hladk, D., Vascak, J., Sncak, P.: Hrarchcal fuzzy nfrnc systm for robotc ursut vason task. In: 6th Intrnatonal Symosum on ld Machn Intllgnc, SMI 8, (8). Khagas,., Hollngr, G., Sngh, S.: grah sarch algorthm for ndoor ursut/vason. Mathmatcal and Comutr Modllng 5, (9). Lm, S.., Furukawa, T., Dssanayak, G., D-Whyt, H.: Tm-Otmal Control Stratgy for Pursut-Evason Gams Problms. In: Procdngs of th 4 IEEE Intrnatonal Confrnc on Robotcs and utomaton, vol. 4, (4) 3. Islr, V., Kannan, S., Khanna, S.: Randomzd ursut-vason n a olygonal nvronmnt. IEEE Transactons on Robotcs, (5) 4. Lu, J., Lu, S., Wu, H., Zhang, Y.: ursut-vason algorthm basd on hrarchcal rnforcmnt larnng. In: 9 Intrnatonal Confrnc on Masurng Tchnology, vol., (9)