Genetic Algorithm Based Optimal Control for a 6-DOF Non Redundant Stewart Manipulator

Transcription

1 ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 Genetc Algothm Based Otmal ontol fo a 6-DOF Non Redndant tewat Manlato A. Oman, G. El-Baym, M. Bayom, and A. Kassem Abstact Alcablty of tnng the contolle gans fo tewat manlato sng genetc algothm as an effcent seach technqe s nvestgated. Knematcs and dynamcs models wee ntodced n detal fo smlaton ose. A PD task sace contol scheme was sed. Fo demonstatng technqe feasblty, a tewat manlato nmecal-model was blt. A genetc algothm was then emloyed to seach fo otmal contolle gans. he contolle was tested onste a genec ccla msson. he smlaton eslts show that the technqe s hghly convegent wth seo efomance oeatng fo dffeent ayloads. Keywods tewat Knematcs, tewat Dynamcs, ask ace ontol, Genetc Algothm. F. NRODUON LGH smlatos mtate the hyscal feelng of lotng an acaft by ovdng gahcal wndows, sond, and moton latfom. One of the most ola flght smlato latfoms s tewat manlato, whee a movng late s connected to a base late by sx legs. Each leg has an e at sldng nsde a lowe at smlatng the thee tanslatonal motons (sge, sway, and heave) and the thee otatonal motons (tch, oll, and yaw) as shown n Fg. 1. task sace contol scheme has been nvestgated by [1-]. n ths scheme, the fame wok s mlt-nts mlt-otts (MMO). hs the fowad knematcs model s mbedded n the contol loo to estmate the task sace dslacements (X) fom the meased jont dslacements (q) as shown n Fg.. he task sace contol s exacebated by the fact that the dect knematcs of tewat manlato has no closed fom solton. Fo examle, Detmae [3] has addessed 4 ossble soltons fo the fowad knematcs. A lot of stdes have ted to smlfy the dect knematcs oblem by dffeent aoaches. Patk [4], and adjadan [5-6] sed the neal netwok aoach. he accacy of ths aoach s vey senstve to the stcte of the neal netwok. Fo examle, adjadan [6] showed that changng the stcte of neal netwoks can lead to dffeent accacy levels n fowad knematcs modelng fo the tewat manlato. lan [7] esented a new closed-fom solton of the oblem bt t sed thee exta sensos. L [8-9] oosed a nmecal algothm based on a fndamental geometc oeaton wth thee nonlnea smltaneos algebac eqatons, whch s mactcal fo the contol ocess. All ths lteate emhaszes the comlexty of alyng task sace contol scheme. Jacoban Matx X & d X d ontolle PLAN q& d q d Fowad Knematcs Fg. 1 tewat Manlato wo schemes ae commonly sed n contol of the tewat manlato: task sace contol and jont sace contol. he A. O; PhD stdent, Old Domnon Unvesty, Aeosace Det., Nofolk, VA, UA (coesondng atho. e-mal: aoma1@od.ed hone (UA): ). G. E; Pofesso, Aeosace Eng. Det., ao Unvesty,.Egyt. M. B; Assocate ofesso, Aeosace Eng. Det., ao Unvesty, Egyt A. K; Assocate ofesso, Aeosace Eng. Det., ao Unvesty, Egyt. 73 Fg. cheme of the task sace contol On the othe hand, the jont sace scheme s develoed by the nfomaton of jont dslacements only, snce each leg of the manlato s contolled as a sngle-nt sngle-ott (O) system. he eo between the actal and desed jont dslacement s sed as a feedback sgnal to the contolle. he nvese knematcs of the tewat manlato has a closed fom and t s easy to be mlemented. n ths way, the sohstcated comtatons of the fowad knematcs ae omtted fom the contol loo. hs scheme has been wdely sed by many eseach eots, esecally fo exemental alcaton. Pasqale [1] sed a obst contol scheme wth acceleaton feedback. L [11] desgned a ootonal gan contolle. Fang [1] mlemented a fzzy contol. [13]

2 ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 oosed a new technqe of obst ato dstbance ejecton contolle (ADR). Howeve all these stdes ovelook the manlato dynamcs, becase the contolle s desgned based on actato model sch as hydalc system [11, 13], o sevo motos [1, 1], whch estcts the ablty of desgnng a contolle wth hgh efomance tackng. he contbton of ths ae s to show the met of sng genetc algothm n tnng the contolle gans fo tewat manlato based on manlato dynamcs nstead of actato dynamcs as n the evos stdes [1-13]. Fo smlcty, a PD contolle n jont sace s consdeed. An otmzaton oblem s assgned seekng fo mnmm settlng tme and mnmm eo when a ste nt s consdeed as a efeence nt. he eo s defned hee as the dffeence between the obseved and the desed jont dslacements. hs ae s oganzed as follows: n secton two, the descton of nvese knematcs enables one to detemne the lnk lengths n tems of desed/secfed e latfom oston and angla oentatons. ecton thee ncldes a dscsson abot the dynamcs model. he se of a genetc algothm to seach fo contolle gans s esented n secton fo. ecton fve offes the eslts of the smlaton sed to examne the oosed technqe. Fnally, secton sx s the conclson.. NVERE KNEMA MODEL hee ae two fames descbng the moton of the movng late: an neta fame (X, Y, Z) located at the cente of the base late and a body fame (x B, y B, z B ) located at the cente of the movng late wth the z B -axs ontng otwad. he angle between the local x B -axs of the movng late and the lne of the jont J s denoted by β as shown n Fg. 3. B B B B [ X Y Z ] PJ = = 1,,..., 6 = [ R cos ] ( β ) R sn( β ) n the same manne, an angle α s defned between the neta X-axs and the lne of the jont J l. he oston of the jont J l n the neta fame s defned as: [ X Y Z ] PJ l = l l l = 1, = [ Rl cos l ( α ) R sn( α ) ],..., 6 he e late has a caablty fo 6-DOF moton (thee otatonal motons and thee tanslatonal motons). he otatonal motons of the late ae defned by Ele angles n seqence hs the tansfomaton fom the body fame (x B, y B, z B ) to the neta fame (X, Y, Z) s gven by the Matx R : R late late θ = θ θ θ ϕ + ϕ ϕ ϕ + θ ϕ ϕ whee efes to angle cosne and efes to angle sne. he angles, θ, and φ ae Ele angles. he absolte angla velocty of the movable late n body fame s gven by θ 1 & ω = φθ φ& (4) φθ θ θ& n addton to the otaton, one shold consde the tanslaton vecto as: late (1) () (3) x(t) late = y(t) (5) z(t) + h whee h s the ntal heght of the e late s cente. he tajectoy of the e late s cente s defned by x(t), y(t), and z(t). he oston of the jont J n netal fame (X, Y, Z) s then calclated as: J P B B B B [ X Y Z ] = R P + = (6) late J late Fg. 3 Jont oston on the movng late he oston of the jont J n the late body fame s he length vecto of the th leg () and (6) as: J L = P -P J l L can then be comted fom = 1,,..., 6 By sbstttng fom (3) and (5) nto (6), and consdeng the sqae vale of vecto L n (7), the elatonsh between the jont sace vaables and task sace vaables can be smmazed as: (7) 74

3 ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 whee θ X = θ R cos + x l Y = θ R cos + y Z = L = X + Y + Z ( β ) + ( ) R sn( β ) () t R cos( α ) ( β ) + ( + ) R sn( β ) () t Rl sn( α ) R cos( β ) + R sn( β ) + z( t) + h ϕ ϕ and = 1,,, 6. Based on (8), the nvese knematcs has a closed fom. On the othe hand, t s mossble to develo any closed fom fo the fowad knematcs. Each leg has thee degees of feedom: two otatonal and one tanslatonal moton. he leg can otate aond the nvesal jont, whle the e at of the leg s sldng nsde the lowe at by an actatng foce F as shown n Fg. 4. hs a shecal jont s emloyed to connect the e at of each leg by the movable late whle the lowe at s connected to the base late by a nvesal jont. (8) Γ = tan 1-1 ε = tan Y X Y X Z ( X X ) + ( Y Y ) l l whee =1,,,6. he tansfomaton matx (x bl, y bl, z bl ) fame to (X leg, Y leg, Z leg ) fame s gven as R Leg ε Γ = ε Γ ε Z l Γ Γ l ε ε ε l Γ Γ RLeg (9) fom (1) he angla velocty ω of the th leg wth esect to the leg body fame s defned by Γ& sn() ε () ω = ε& (11) Γ& cos ε. DYNAM MODEL Fg. 4 Leg mechansm of tewat manlato he moton of the each leg s consdeed by two fames: a leg fxed fame (X leg, Y leg, Z leg ) located at the jont J l aallel to the neta fame and the leg body fame (x bl, y bl, z bl ) located at the same ont wth x bl -axs ontng n wad. he otaton seqence of the leg stats fom otatng aond Z Leg -axs wth an angle Γ, followed by a otaton abot the y bl -axs wth an angle ε. he otatonal angle of the leg can be secfed by the oston of the e jont P J and the oston of the lowe jont P J l as he dynamcs model of tewat manlato has been addessed by many methods sch as the Lagange eqaton [1, 11], Newton-Ele eqaton [13-14], and the ncle of vtal wok [16-17]. Based on the eslts that have been shown by Khall [18], Newton-Ele method emeged as the most effectve way to model tewat manlato dynamcs. Howeve ths method has been hghlghted by some common eos n evos eseach eots. hese eos wee lsted by haowen [19], and coected n the cent eseach. n Newton Ele method, the dynamcs model of tewat manlato s descbed thogh 4 govenng eqatons, sx eqatons fo the movable late, and the othes fo the legs. Fo comleteness, the dynamc eqatons wll be lst hee. Moe detals of the devaton ae gven by many efeences [13, 14, and 18]. hs the moment eqaton aond the nvesal jont s ml l + ω ( g + al ) + m ( g + a ) + ([ ] + [ l ]) ([ ] + [ ]) ω L f = l α (1) whee ml s the lowe leg mass, m s the e leg mass, [ ] s the nvaant neta matx of the lowe leg, l [ ] s the vaant neta matx [17] of the e sldng leg, a l s the acceleaton of the lowe leg, a s the acceleaton of the e leg, g s the gavtatonal acceleaton, ω s the angla velocty of the leg, α s the angla acceleaton, L s the length of the whole smatc 75

4 leg, l s the oston vecto of the lowe leg fom nvesal jont, s the oston vecto of the e leg meased fom the nvesal jont, and f s the eacton foce between the shecal jont and the e late. he eacton foce f s decomosed nto thee comonents n the leg s body fame as shown n Fg. 4. he foce eqaton n x bl -decton of the sldng mechansm s gven as m g + a = F f x (13) ( ) he dynamcs of the e late have thee moment eqatons and thee foce eqatons as 6 = + R m a m g f late R leg = 1 + = 6 R α ω ω f b late Rleg = 1 (14) whee =1,,,6, m s the mass of late ls the extenal ayload, [ ] s the neta matx of the e late, a s the acceleaton fo the e late s cente of mass, ω and α ae the angla velocty and acceleaton of late, and b the oston vecto fom the cente of late to the jont J l. olvng the dynamc eqatons of tewat manlato has two models. he fst model s the nvese dynamcs comtng the eqlbm foces. he second model s the fowad dynamcs bldng the smlaton tool. he nt of the nvese dynamcs s the desed tajectoy of the movable latfom as a fncton of the tme and the otts ae the actato foces. On the othe hand, the nts of the fowad dynamcs models ae the actato foces aled at cylndcal jonts, and the otts ae the movable e late ostons and oentatons. he algothm of the nvese dynamcs model can be smmazed n the followng stes: te1: ecfy the desed task sace dslacements as [φ(t), θ(t), (t), x(t), y(t), z(t)] and the devatves wth tme. te: Obtan the tansfomaton matx R late of the movng late fom (3). te4: Obtan the angla velocty of the movng late ω fom (4). ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 te5: Use nmecal dffeentaton to comte the angla acceleaton of the movng late α. te6: omte the ostons of jonts J and J l fom () and (6). te7: Obtan the vale of angles ε and Γ fo each leg fom (9). te8: Obtan the tansfomaton matx fo each leg R leg fom (1). te9: Use nmecal dffeentaton to evalate the tme devatves of angles ε and Γ. te1: Obtan the angla velocty ω of each leg fom (11). 76 te11: Use nmecal dffeentaton to comte the angla acceleaton α fo each leg. te1: olve (1) to comte f y and f z fo each leg. te13: olve (14) as a set of lnea homogenos eqatons n f x te14: alclate the actatng foce F fom (13). eqentally, the fowad dynamcs model s develoed n evese decton of the nvese knematcs algothm. V. OPMZAON PROEDURE UNG GA Genetc algothm s now consdeed as one of the most ola otmzaton and seach technqes. he fst obvos alcaton fo the algothm taced back to 196 when Holland ntodced the algothm n hs wok stdyng adatve systems []. he algothm then eceved an enomos exloaton by Goldbeg [1].he man advantages of GA ae ts global otmzaton efomance and the ease of dstbtng ts calclatons among seveal ocessos o comtes as t oeates on the olaton of soltons that can be evalated concently. t s a vey smle method, geneally alcable, not nclned to local otmzaton oblems that ase n a mltmodal seach sace, and no needs fo secal mathematcal teatment. Moeove the algothm s moe alcable fo the dscontnos oblem nlke the conventonal gadent-based seachng algothms. Genetc algothm bascally woks based on the mechansm of natal selecton and evoltonay genetcs. he algothm stats by codng the vaables to bnay stngs (chomosomes). Evey chomosome has n genes. he gene s a bnay bt by vale zeo o one. hee man oeatons contol the ocede of the GA: eodcton, cossove, and mtaton. Reodcton s ocessng to select the aent fom a geneaton. he ocess s based on svval of the fttest (hghest efomance ndex). n ths way, the eodcton ocess gdes the seach fo the best ndvdals (hgh efomance ndex). Afte the ndvdals ae selected, the cossove ocess s then sed to swa between two chomosomes by secfc obablstc decson. he cossove ocess geneates offsng cayng mxed nfomaton fom swaed aents (chomosomes). Mtaton s the mechansm to event the algothm fom local otmal onts by addng some degee of andomness. he ocess s efomed by altenaton of the gene fom zeo to one o fom one to zeo wth the mtaton ont detemned nfomly at andom. he mtaton ate shold be consde caeflly snce the hghe mtaton ate means moe nmbe of geneatons ae eqed fo algothm convegence and a low mtaton ate may lead to a convegence fo a local mnmm. he algothm mantans a constant sze of geneaton by selectng the fttest chomosomes fom aents and offsngs. he algothm teatvely oeates to convege fo schema matches by some toleance. Roghly, a genetc algothm woks as shown n Fg. 5. Fthe descton of genetc algothms can be fond n Goldbeg [1-]. Fg. 6 shows a jont sace PD contolle scheme. n ths scheme, the nvese knematcs s emloyed to comte the

5 ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 desed jont dslacements (L 1, L,, L 6 ) fom the desed task sace dslacements (x d, y d, z d, θ d, φ d, d ), the desed and meased jont dslacements ae then comaed feedng the contol logc. actcal oblems; the model s a nonlnea mathematcal one. hs [6] has aled an otmzaton technqe to contol a obot n tackng oblem as a hghly nonlnea oblem. he eslts showed the owe of sng otmzaton technqes n tnng contolle aametes. hs encoages sng otmzaton technqes n ths eseach to tne contolle gans of tewat manlato. he efomance ndex gven n (16) s selected hee to mnmze the absolte aea nde the eo cve (dffeence between desed and meased jont dslacements) wth tme. n addton, anothe tem ootonal to the settlng tme s added avodng the flatness of the eo cve. A % ctea s sed to defne the settlng tme. ettlng tme s also consdeed as the tme san fo the ntegaton he weghtng factos w 1 and w ae selected sch that the two tems n the cost fncton beng n the same level of the magntde. F cos t e (t) = L = w1 d 6 t= tsan = 1 t= () t L () t e (t)dt + w t san (16) GA n Fg. 5 oagates seachng fo otmal contolle gans n (15) to mnmze the cost fncton n (16) fo nt ste nts as a efeence. V. MULAON REUL AND DUON Fg. 5 Flow chat of genetc algothm he oosed otmzaton technqe s aled to the tewat latfom wth aametes gven n able. Fg. 6 Jont sace PD contolle he contol law of ths PD s gven as F = K e + K e& = K (L L) + K (L& L) & (15) d ef he PD contolle s commonly desgned by analytcal methods sch as oot locs, o state sace model [3-5]. f the nonlneaty s sgnfcant, then t s dffclt to se sch analytcal methods and wegh the nflence of each gan on the esonse. n ths case, othe methods wee oosed. Hahsa [3] has develoed a gan tnng technqe based on tme devatves. Faa-Jeng [4] sed a ecent fzzy-nealnetwok (RFNN) to tne an P contolle. he alcaton of otmal tnng technqe to the contolle gans has been extensvely exloed fo ocesses that ae dffclt to be tned analytcally. Baogang [5] oosed a new methodology fo a nonlnea PD contol. hs contolle s based on the theoetcal fzzy analyss and genetc-based otmzaton. he contolles gave bette eslts than the conventonal PD. he contolle has not been aled to d ef 77 ABLE MULAON PARAMEER OF EWAR MANPULAOR Vaable Descton Vale Unt L Length of e leg.95 m L l Length of lowe leg.95 m R Rads of e late 1 m R l Rads of base late 1 m α Jont angles of base late [-5, 5, 7, 17,-17, -7] deg β [-,, 118, 1, -1-18] deg Jont angles of e late m l Mass of each e leg kg m ll Mass of each lowe leg kg m Mass of each e late kg K L Lowe vale of K 1 4 N/m K Ue vale of K 1 6 N/m K dl Lowe vale of K d 1 3 Ns/m K d Ue vale of K d 1 5 Ns/m Moment of neta of the lowe leg.63,.8536,.8536 kg Moment of neta of the e leg.94, 1.591, kg xl yl zl x y z Fo GA otmzaton, the mtaton ate s 1%. Each geneaton has a fxed olaton sze 1 o no geneaton ovela. he algothm s hghly convegent. he nmbe of

6 geneatons fo convegence s 3. he otmzaton algothm conveges at the vales of K and K d as { } x1 5 N/m and { } x1 4 N.sec/m esectvely. he efomance of the contolle s tested agan by a genec msson. hs msson s a hozontal ccla tack wth ads.1 m. he aametc eqaton of ths msson s defned as π πt ζ = t sn π xt =. 1sn( ζ ) yt =. 1( 1 cos(ζζ) z = 1 ζ π and t (17) whee the dmmy vaable ζ s mlemented to ledge that all fnctons n (17) have zeo veloctes and acceleatons at the begnnng and end of the msson. Also the msson has been assgned to be nsde the geometc woksace gven n Fg. 7. he nvese knematcs was emloyed to comte the efeence jont sace dslacements (L d ) shown n Fg. 8. Now the contol model s tested onste ths msson, when (L d ) s consdeed as nts (see Fg. 6). Fg. 9 shows the geneated actato foces based on the contol law gven n (15). he tme ecods of the actato foces look vey smooth. hs mles that the contolle has the caablty to cate the elaton between the aled foces and the meased jont dslacements. Fg. 1 mentons the eo between the desed and meased task sace dslacements. n Fg. 1, the ode of maxmm eo s 1-5 m, whle the ode of the desed task dslacement s 1 - m, whch s qte adeqate fo the flght smlato alcatons. n addton two dffeent ayloads ae added to the e latfom. Fg. 1 demonstates the caablty of the contolle to efom at dffeent oeatng condtons wth accetable accacy levels. y ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 z= z=.1 z=. z=.4 z=.6 z= x Fg. 7 he geometc woksace at zeo oentaton L 1 L 3 L 5 F 1 F 3 F L L ack eo L tme (sec) tme (sec) Fg. 8 Actve Jont Dslacements fo Desed ack tme (sec) 6 x F F 4 F tme (sec) Fg. 9 Actato foces 67 tme (sec) eo x eo y tme (sec) Fg. 1 Eo ove the ack 78

7 y ntenatonal Jonal of Mechancal, ndstal and Aeosace Engneeng : 8 m ayload = m ayload = kg m ayload = 5 kg x Fg. 11 hange n tackng accacy wth the ayload V. ONLUON hs ae esents the modelng and contol algothm of a non-edndant 6-DOF tewat manlato. t shows that the PD contol scheme, sng actve jonts degees of feedom feedback and otmzed wth GA, s comtatonally effcent and easy to mlement as fa as the accacy of the nvese knematcs model s gaanteed. he contol scheme s tested on a thee-dmensonal ccla msson. he eslts show the effcency of the algothm and the obstness of the esltng contolle wth vaable load. REFERENE [1] Nag,., and hong, W., Hgh eed ackng ontol of tewat Platfom Manlato va Enhanced ldng Mode ontol, EEE ntenatonal onfeence on Robotcs & Atomaton, Leven, Belgm, May [] Yng,., Y-hn,., and Ho-hn J, Modelng and ontol fo a Gogh tewat Platfom N Machne, Jonal of Robotc ystems, No., Vol. 11, Jne 4. [3] Detmae, P., he tewat-gogh Platfom of Geneal Geomety an Have 4 Real Postes, Advances n Robot Knematcs: Analyss and ontol, Klwe Academc Pblshes, [4] Patk, J., and aah, Y., A Hybd tategy to olve the Fowad Knematcs Poblem n Paallel Manlatos, EEE ans. Robot and Atomat. Vol. 1, Febay 5. [5] adjadan, H., and aghad, H., omason of Dffeent Methods fo omtng the Fowad Knematcs of a Redndant Paallel Manlato, Jonal of ntellgent and Robotc ystems, 5. [6] adjadan, H., aghad, H., and Fateh, A., Neal Netwoks Aoaches fo omtng the Fowad Knematcs of a Redndant Paallel Manlato, ntenatonal Jonal of omtatonal ntellgence Vol., No. 1, [7] Lan, B., Jeha, R., ng-gan, K., and n-ky, L., A losed-fom olton to the Dect Knematcs of Nealy Geneal Paallel Manlatos wth Otmally Located hee Lnea Exta ensos, EEE ans. Robot and Atomat. Vol. 17, Al 1. [8] L, K., Ftzgeald, M., and Lews, F., "Knematc Analyss of a tewat Platfom Manlato," EEE ans. ndstal Electoncs, Vol. 4, No., [9] L, K., Lews, F., and Ftzgeald, M., "olton of Nonlnea Knematcs of a Paallel-Lnk onstaned tewat Platfom Manlato," cts, ystems, and gnal Poc., ecal sse on "mlct and Robst ystems," Vol. 13, No. -3, [1] Pasqale,., Fancos, P., Loenzo,., and Bno,., Robst Desgn of ndeendent Jont ontolles wth Exementaton on a Hgh-eed Paallel Robot, EEE ans on ndstal Electoncs, Vol. 4, Agst, [11] L, D., and alcdean,., Modelng, mlaton, and ontol of a Hydalc tewat Platfom, EEE nt. onf on Robotcs and Atomaton, Albqeqe, New Mexco, Al [1] Fang,., Hng-Hsang,., and hn-eng, L., Fzzy ontol of a xdegee Moton Platfom wth tablty Analyss, EEE M onfeence, Vol. l 1, Octobe, [13], Y., Dan, Y., Zheng,., Zhang, Y., hen, G., and M, J., Dstbance-Rejecton Hgh-Pecson Moton ontol of a tewat Platfom, EEE ans. on contol systems technology, Vol. 1, , May 4. [14] cavcco, L., and clano, B., Modelng and ontol of Robot Manlatos, nge, econd Edton. Al, 1. [15] M.-J. L,.-X. L, and.-n. L, Dynamcs Analyss of the Gogh tewat Platfom Manlato, EEE ans. Robot and Atomat, Vol. 16, Febay,. [16] Dasgta, M., and Mthynjaya,., A Newton Ele Fomlaton fo the nvese Dynamcs of the tewat Platfom Manlato, Mech. Mach. heoy, Vol. 33, No. 8, Novembe, [17] sa, L., olvng the nvese dynamcs of a tewat Gogh Manlato by he Pncle of Vtal Wok, J. Mech. Des., Vol. 1, Mach,. [18] Khall, W., and Gegan,., nvese and Dect Dynamc Modelng of Gogh tewat Robots, EEE ans. Robot and Atomat, Vol., Agst, 4. [19] F,., and Yao, Y., omments on A Newton-Ele Fomlaton fo the nvese Dynamcs of the tewat Platfom Manlato, Mech. Mach. heoy, Vol. 8, Jan, 6. [] Holland, J., Adataton n Natal and Atfcal ystems, he Unvesty of Mchgan Pess [1] Goldbeg, E., he Desgn of nnovaton: Lessons fom and fo ometent Genetc Algothms, Boston, Klwe Academc Pblshes,. [] John, J., Otmzaton of ontol Paametes fo Genetc Algothms," EEE ans on ystem, Man, and ybenetcs, Vol. 16, No. 1, [3] Hahsa K, and Geng L, Gan nng n Dscete-me Adatve ontol fo Robots, E Annal onfeence n Fk. Agst, 3. [4] Faa-Jeng, L, and hh-hong, L., On-lne Gan nng Usng RFNN fo Lnea ynchonos Moto, EEE, PE, Vol., Jne, 1. [5] Baogang, H., eno, M., Geoge, K., and Raymond, G., New Methodology fo Analytcal and Otmal Desgn of Fzzy PD ontolles, EEE an on Fzzy ystem, Vol. 7, Octobe, [6] hs, M., Genetc Algothms fo Ato-nng Moble Robot Moton ontol, Res. Lett. nf. Math. c, Vol. 3,