Motion Control and Trajectory Tracking Control for a Mobile Robot Via Disturbance Observer

Transcription

1 WSEAS RANSACIONS on SYSEMS Haan Wang G L Moon Conol an ajcoy ackng Conol fo a Mobl Robo Va Dubanc Obv Haan Wang 3 G L Dpan of Elccal Engnng Shangha Jaoong Unvy Cla A739 8 Dongchuan Roa Mn Hang Shangha 4 PR Chna Inu of Plaa phyc Chn Acay of Scnc PO Box 6 Hf Anhu 33 PR Chna 3 Dpan of Elconc an Infoaon Engnng Jayng Collg Mong Roa Mhou Guangong 545 PR Chna Abac: - h pap nvga h ackng conol of a whl obl obo n h unknown nvonn A ubanc obv vlop wh ulaon h ngal fl h popo conol ch ploy h ubanc obv conol appoach o gn an auxlay whl vlocy conoll o ak h ackng o whch nclu h vlocy ackng o h angula vlocy ackng an h ajcoy ackng o vco a all a pobl n conaon of unknown boun ubanc n h knac of h obl obo an ak u of h ubanc conoll o jc h unknown boun ubanc h appoxaon o an h unknown boun ubanc can b ffcnly jc by ployng h ngal fl A ajo avanag of h popo ho ha h poon o vlocy/angula vlocy an h ajcoy o vlocy/angula vlocy a no long ncay h bcau h obv conoll ackng boh h obl knac an h unknown boun ubanc Mo poanly all gnal n h clo-loop y can b au o b unfoly ulaly boun h y ably an convgnc of h oon conol an h ajcoy ackng o a pov ung h Lyapunov ably hoy Sulaon ul a pov o vfy h popo conol agy I hown ha h conol agy fabl Ky-Wo: - Dubanc obv Vlocy ackng ajcoy ackng -vayng ubanc Mobl obo Moon conol Lyapunov ably Inoucon Moon conol of whl obl obo WMR hav aac h anon of any ach [ ] In n uch y paly fo h WMR wh loang capacy whch ncay n nuy Howv WMR ha nonholonoc nau an on Bock conon whch h ncay conon o ak a ooh -nvaan conol law Kanayaa al [] popo an aypoc oon conoll whch u connuou fback conol o Howv h conoll aop local lnaaon ung Lyapunov nc ably ho an canno b globally abl Fo an Lw [3] gn a conoll fo boh oon conol an pon ablaon ung backppng o Bak an Jan [7] popo global aypoc oon conoll ung backppng Coulau al [8] an AG Lonc al [9] alo popo a globally aypocally abl conoll ung ag-pocng algoh An-ubanc aapv conol wa u fo h obl obo ung ual aapv nual conol wh unknown nwok paa a a n al [] h ackng pobl of h obl obo ha alo aac h anon of any ach []-[5] Ung Babal la o h backppng ho o conoll hav bn popo uch ha h obl obo coul globally follow h pcal pah uch a ccl an agh ln W Dong ISSN: Iu Volu 9 Januay

2 WSEAS RANSACIONS on SYSEMS Haan Wang G L al [6] popo conoll nu h n a of h ynac y aypocally ack h ajcoy conng unknown na paa Whn h npu of h obl obo appa auaon ZP Jang al [7] pn a conol agy o al wh h pobl of global ablaon an global ackng conol fo h obl obo All h pap au ha ubanc a o b conan o low -vayng o vn whou conaon Whn h ubanc a fa -vayng h pfoanc of ho conol o a unafacoy An bcau of h ffculy n ynac olng afcal nllgnc conol ung nual nwok an fuy logc can b con a an ffcv ool fo nonlna conoll gn In [8] an [9] h obv ung ullay nual-nwok wa vlop fo h obl obo ackng conol bu h conoll ucu an h nual-nwok lanng algoh a coplca an copuaonally xpnv In [] h obv ung fuy ho wa pn o copna h loa ubanc ha ak h ackng naccuacy fuho h ho con only h ackng o an h al xnal ubanc n h vlocy an h angula vlocy of h obl obo w no con In h pa a novl ubanc obv [] whch u n h obl applcaon wh abaly fa -vayng ubanc ha bn uccfully u fo ha k v Ba on h ubanc obv w con h uaon wh ackng conol ung ubanc obv a o oban vlocy an ajcoy n unknown nvonn a hown n Fg On conol pupo of h obl obo ha h acual vlocy qual o h vlocy an akng u h angula vlocy of h obl obo on Anoh conol pupo ha h al ajcoy of h obl obo can b loca quckly o h ajcoy o h ffc acuao ynac cobn wh h obl obo an h npu oqu of wo vng whl W popo a nw conol ho ung ubanc obv fo h obl obo whch jc boun ubanc h popo ch a unknown paa an conol h obl obo wh pou whl havng h chaacc of global ably B n pn ch can jc xnal abay fa -vayng ubanc h an conbuon of h pap a l a follow: Dcoupl ackng an onaon conol ag a popo fo h WMB whou pong any con on h y ynac; Conoll gn fo h WMB wh anubanc; 3 Dubanc obv gn fo abaly fa -vayng ubanc n h WMB y; an 4 ajcoy ackng ba h conol gn vlop n unknown nvonn Sulaon ul a cb n al o how h ffcvn of h popo conol h an of h pap ogan a follow h ol of a nonholonoc obl obo nouc n Scon h an pobl of h foulaon o poon an onaon conol a cu n Scon 3 h nonlna obv an h conoll gn a pn n con 4 Sulaon u a how n con 5 Conclung ak a gvn n Scon 6 Mol of a Nonholonoc Mobl Robo h obl obo hown n Fgu a ypcal xapl of a nonholonoc chancal y I con of a vhcl wh wo vng whl oun on h a ax an a fon pav whl h poon an h onaon a achv by npnn acuao povng h ncay oqu o h a whl h wo vng whl hav h a au no by an a paa by R Pon C loca n cn of a of h obl obo; pon P loca n h ncon of h ln of h obl ba an h ax of h vng whl h anc bwn pon C an pon P no by h poon an h onaon of h obo n an nal Caan fa {O X Y} coplly pcf by h vco q [ x c y c θ ] wh x c yc a h coona of h cn of a of h vhcl an θ h onaon of obl plafo {C X c Y c } au fo X ax ISSN: Iu Volu 9 Januay

3 WSEAS RANSACIONS on SYSEMS Haan Wang G L O Y Dvng whl θ R P YC Fg A nonholonoc obl plafo C XC Pav whl A nonholonoc obl obo y havng an n-nonal confguaon pac L wh gnal coona q q n an ubjc o conan can b cb by [] M q q C q q q D B q τ A qλ n n wh M q R a yc pov n n fn na ax C R h Cnpal an Cool ax D R n h xnal n ubanc vco B q R h npu anfoaon ax τ R no h npu n vco A q R h ax aoca wh h conan an λ R h vco of conan foc All knac conan of h obl plafo a npnn of an can b xp a A q q Wh pc o h ynac of obl obo h followng pop a known [3] Popy : M q a yc an povfn ax; Popy : h ax M C kw- yc [4] ha x M C x n x R Au ha S q b a full ank ax n - fo by a of ooh an lnaly npnn vco fl pannng h null pac of A q S q A q 3 Ung an 3 pobl o fn an n auxlay vco funcon v R uch ha fo all [3] [5] q S q v 4 Knac an ynac of a obl plafo X h pu ollng an nonlppng nonholonoc conan a ha h obo can only ov n h con noal o h ax of h vng whl an h obl ba af h nonholonoc conan [6 7] An h vlocy coponn of h conac pon wh h goun ppncula o plan of h whl o naly y co θ nθ c xc θ 5 Fo 3 an 5 h conan ax of h obl plafo xp a A q [ nθ coθ ] 6 hu ax S q can now b xp a coθ nθ S q nθ coθ 7 hfo ay o vfy ha h knac quaon of ackng 4 can b xp a x c coθ nθ v q y c nθ coθ 8 ω θ v v ω v an ω a boun lna an angula vloc of h obl obo pcvly Eq 8 call h ng y of h vhcl h Lagang foal u o fn h ynac quaon of h obl obo h ynacal quaon of h obl plafo can b xp n h ax fo wh nθ M q coθ nθ coθ I θ coθ C q q θ nθ coθ nθ coθ B q coθ nθ D nθ R R nθ τ τ A q τ coθ 9 l wh an a h ubanc of lna an angula vloc of h obl obo pcvly; wh [ ] ISSN: Iu Volu 9 Januay

4 WSEAS RANSACIONS on SYSEMS Haan Wang G L τ an l τ no oqu of gh an lf whl pcvly Sucual pop of a obl plafo h y can b anfo no a o appopa fo fo conol pupo Ung 4 ubung an hn ulplyng by S h conan ax A q λ can b lna hu h appopa fo of y can b oban h copl quaon of ackng of h nonholonoc obl plafo a gvn by qs q v Mv C v D Bτ wh M S MS C S MS C S D S D B S B Eq cb h bhavo of h nonholonoc y n a nw of local coona S q a Jacoban ax ha anfo vloc n obl ba coona v o vloc n Caan coona q hfo h pop of h ognal ynac hol fo h nw of coona Popy 3: h ax M q yc pov fn; Popy 4: h ax M C kwyc; an Popy 5: h D a boun U of 9 n yl M I B R R v C D ω Fnally h coupl y can b xp a v τ ω τ 3 wh I τ τ τ l / τ τ τ l R / Fuho an 3 can b wn a v τ 4 ω τ 5 3 Pobl Foulaon o Moon Conol h ovvw of a obl obo hown n Fg Snc h obl obo wok n an unknown nvonn w houl focu on h accuacy of h ackng an onaon conol whch acc va coponng no h poon o gnal PES nca h placn of h cn of a of h obl ba fo h ackng conol locaon an h onaon o gnal OES pn h X C ax vaon fo h onaon conol of X C ax Ung 4 an 5 w con h oubl ngao ol pnng h obl obo a follow: wh y y 6 7 τ y τ v y ω y y y a h poon vlocy an acclaon of h obl obo pcvly; y y y a angl angula vlocy an angula acclaon of h obl obo pcvly Rak 3: In a ana vo conol y gnal pacc an unanng ha h poonal gnal nclung poon y vlocy y an o acclaon y a avalabl fo fback conol gn Fo ackng conol pupo h ajcoy f an con vav an y y y a alo known boun an connuou gnal Fuho h ackng conol o y y aly copuabl A uch h nvolv angula vaabl y y y an a avalabl Howv n a obl obo y h hng wll b chang W can no g boh h poon of h cn of a of h obl obo y an h ajcoy A uch h acual an y onaon a unavalabl h an objcv of h ack followng vo o anan h cn of a on h ack a a o anan h acual onaon of h obl n qu on Snc w o no know h xac hap of h vo ack w can only oula a gnal ung a no o ll u h lav anc bwn h cn of a an ack cn whch PES A PES can b au qu accualy vav can b a qu wll an ISSN: Iu Volu 9 Januay

5 WSEAS RANSACIONS on SYSEMS Haan Wang G L gnally au o b known fo conol y gn h ca of coponng angula vaabl la h poonal on hn PES an OES a xp a y y 8 y y 9 wh an no PES an OES pcvly hough y y y an y a unavalabl fo h obl obo Aupon 3: h xnal ubanc an a boun wh an unknown boun Aupon 3: In a obl obo y boh h PES gnal an f vav a avalabl; h OES gnal an a avalabl oo Aupon 33: h ack poon ack onaon y y an an h f an con vav a boun an connuou gnal hough hy a no avalabl fo fback conol n h obl obo h conol objc o pn a conol agy o h xnal abaly fa vayng ubanc an o ak h PES an h OES ally a o n h ackng conol 4 Dubanc Obv Dgn an Sulaon 4 Dubanc obv gn Fo 6 7 an 8 h followng ackng conol o ynac can now b xp a τ y wh An h ubcp no o n h followng x Rak 4: In h ana vo ng wh gnal y y an y a avalabl w can gn h followng al cany quvaln conol τ k k y p wh k p an k a pov conan h ubanc obv fo A non al y no avalabl n h y Copa wh ana vo conol ng con h followng conol ba on avalabl gnal: τ k k p wh k >/ Subung no yl a clo-loop y k k 3 p wh v v v y v w w ω y ω Fo h clo-loop ynac 3 f w can gn a ubanc obv uch ha α β wh α an β a pov conan hn h ably of h y 3 wll b achv aly Con h followng ffnal quaon: y 4 wh a pov conan I oluon λ I how ha convg o h u valu xponnally Howv bcau an a no avalabl canno b oban fo 4 cly La 4: Accong o h l blow ngal fl ν 5 ν 6 wh a pov conan an a nal valu h followng concluon can b oban: h gnal can convg o u valu xponnally α β wh α pov conan an β pov gn paa; h gnal can b oban fo h followng ngal quaon ψ ψ 7 wh ISSN: Iu Volu 9 Januay

6 WSEAS RANSACIONS on SYSEMS Haan Wang G L ψ ϑ [ E [ ϑ E ] a copuabl gnal wh V k k ] p 8 ϑ an conan > gn paa Poof: S Appnx A h ably of h y 3 gvn n h followng ho ho 4: Au h quaon 3 conng of y 4 an 5 afyng Aupon 4 43 h conoll an h obv 7 h xnal ubanc can b jc xponnally h PES an h OES can convg o o h a valu of an globally xponnally convg o h u valu pcvly h ackng conol o an a all convg o o a nfny Poof: Ung h La 4 w hav β α A h a valu n 7 globally xponnally convg o h u valu pcvly Con h Lyapunov funcon cana V k 9 p hn ffnang V wh h an ngang 3 an w hav V kp k k / 3 k k / k k / > wh k wh k > an > k Whn / k w hav V hfo w know / k Nocng ha a a can b oban hu w hav a I follow ha a 4 Sulaon h popo conoll an obv n h con a vf wh copu ulaon ung MALAB h paa valu of h obl obo a akn a 9 kg I 5 kg R 36 5 h paa of h conoll a chon a k k 3 5 ; k k 6 h p p paa of h obv a chon a 6; 4 ; 8 Inal vlocy an angula vlocy a akn a / an pcvly; h ag pou akn a vlocy an angula vlocy a akn a v / an ω a/ pcvly h pak fqunc of wo ubanc a oponal Fo h pupo of ulaon h paa of wo ubanc a akn a: n4π an ω 5coπ h ulaon ul fo vlocy ackng an angula vlocy ackng a hown n Fg an Fg 3 Vloc /a/ oqu N Fg 3 h npu oqu Fg how h ackng an h onaon conol pfoanc pcvly Fg 3 how v vlocy ackng angula vlocy ackng Fg Vloc Rgh whl Lf whl ISSN: Iu Volu 9 Januay

7 WSEAS RANSACIONS on SYSEMS Haan Wang G L h npu oqu of wo vng whl h ulaon ul how ha h ackng an h onaon of conol n o h valu whch vala h ffcvn of h ubanc obv n ho 4 Un h popo conol o ackng conol of h ajcoy an onaon achv an h anly u o h ubanc obv chan h ulaon ul ona h ffcvn of h popo ubanc obv n h pnc fa -vayng xnal ubanc Alhough fa -vayng xnal ubanc a nouc no h ulaon ol h ackng/onaon conol pfoanc of y un h popo conol no ga Dffn ackng/onaon conol pfoanc can b oban by ajung h valu of gn paa Fuho h kn of ubanc obv alo can ack vlocy wh -vayng h nal vlocy an angula vlocy a h a a abov ca h ag vloc a akn a v / an ω 5 a/ pcvly h ulaon ul fo vlocy ackng an angula vlocy ackng a hown n Fg 4 an Fg 5 Vlocy / Fg 4 h vlocy 5 ackng an Conoll Dgn 5 ackng o In h con un h vlocy h ackng pobl fo obl obo pn o vala h ackng au ha h fnc ajcoy x y θ can b xp x v coθ q y v nθ θ ω 3 wh v an ω a fnc vlocy an angula vlocy A n [8] an [9] h ackng o xp a E p x coθ y nθ θ nθ x x coθ y y θ θ 3 x θ f an Claly fo any valu of θ y only f x y θ θ of Angula Vlocy a/ Fg 5 Angula vlocy x y h f vav E p can b wn a x ωy v v co E p y ωx v n θ ω ω θ θ 33 5 Popo conol law o olv h ackng pobl h conol law a popo a follow v v coθ ρ acan x π 34 ISSN: Iu Volu 9 Januay

8 WSEAS RANSACIONS on SYSEMS Haan Wang G L ρv y nθ ω ω ρ3 acan θ x y θ π 35 wh ρ ρ an ρ 3 a pov gn paa ho 5: Au ha ω an v a boun an unfoly connuou ov [ If h ω o v o no convg o o hn h o qulbu of h clo-loop y globally aypocally abl Poof: Con h Lyapunov funcon cana V3 ρ log x y θ 36 hn ffnang V 3 wh h conng an w hav 4ρx acan x 4ρθ 3 acan θ V 3 π x y π 37 hfo h ajco x y θ a unfoly boun on [ Accong o [3] l [ x acan x θ acan θ ] 38 whch n un w know l x θ 39 I an o pov ha y a h can b ablh by ho of agun u n h poof of [3] 53 Sulaon h popo conoll n h con a vf wh copu ulaon ung MALAB conng h popo conoll an h obv n con 4 h paa valu of h obl obo an h ubanc of vloc a h a a on n con 4 h ajcoy ackng ba on h vlocy ackng an h angula ackng h paa of h conoll a chon a k ; k 6 h paa of p h obv a chon a 4; 55 ; 5 h ajcoy ha bn gvn o b v / ω a/ a ccl ackng of h obl obo wh nal o vco E [ x y θ ] [ 5] h p ulaon ul of ackng a hown n Fg 6 8 Fg 6 how h poon ackng o n X an Y coona Fg 7 how h angula ackng o Fg 8 how h npu oqu of wo vng whl h ulaon ul how ha h ackng o n o h valu whch vala h ffcvn of h ubanc obv n ho 5 Un h popo ubanc obv an h conoll law ackng pobl can b achv an h ackng o vco xponnally convg o o vco ackng o akng o a Fg 6 Poon ackng o Fg 7 Angl ackng x y θ ISSN: Iu Volu 9 Januay

9 6 Concluon In h pap ffcv ubanc obv ba on h of ngal fl ha bn pn yacally o vlocy/angula vlocy ackng an ajcoy ackng fo h obl obo wh unknown nvonn Fo h conoll h ably an o bounn pov ung Lyapunov ably hoy h popo obv qu no nfoaon on h y Sulaon u hav vf h ffcvn of h popo obv Appnx A Poof fo La 4 Poof Con h followng ffnal quaon y 4 I oluon I can b foun ha convg o xponnally Fo 5 an 6 on ha [ ] 4 [ ] 4 Subung 4 an 4 no 4 ul n 43 wh [ ] u 44 Dfn h followng vaabl χ 45 Dffnang boh of 45 an cobnng 43 yl χ χ I oluon χ 46 Dffnang boh of 46 wc yl χ 47 wh [ ] 48 Dffnang boh of 47 wc yl χ 49 Copang 47 an 49 on ha 5 Subung 44 an 48 no 5 yl c c c 3 wh c c an 3 c a conan Obvouly h wll x pov conan α an β uch ha α β wh n β Fo 3 on ha V 5 Dfn E hn on ha E Fuho w o no n gnal Subung 5 no 4 yl ψ 5 wh Fg 8 h npu oqu oqu N Rgh whl Lf whl WSEAS RANSACIONS on SYSEMS Haan Wang G L ISSN: Iu Volu 9 Januay

10 WSEAS RANSACIONS on SYSEMS Haan Wang G L ψ [ ] [ ϑ V ] V ϑ o plfy 5 w fn h followng vaabl ς 53 I f vav ς 54 Subung 53 an 54 no 5 yl ς ς ψ 55 I oluon ς ψ 56 Subung 56 o 54 yl ψ ψ 57 h copl h poof Rfnc: [] I Kolanovky an N H McClaoch Dvlopn n Nonholonoc Conol Pobl IEEE Conol Sy Mag Vol 5 pp 36 Dcb 995 [] YKanayaa YKua FMyaak an Noguch A abl ackng Conol Mho fo an Auonoou Mobl Robo Innaonal Confnc on Roboc an Auoaon pp [3] RFo an FLw Conol of a Nonholonoc Mobl Robo: Backppng Knac no Dynac Jounal of Roboc Sy Vol4 No3 pp [4] D Wang an G Xu Full-Sa ackng an Innal Dynac of Nonholonoc Whl Mobl Robo IEEE/ASME anacon on Mchaonc Vol 8 No pp 3-4 Jun 3 [5] MP Paulaj R B Aha CR Ha H Falah Eaon of Mobl Robo Onaon Ung Nual Nwok 9 5h Innaonal Colloquu on Sgnal Pocng & I Applcaon CSPA pp 4-46 [6] A Koo A Coan L Bana C Voloncu Apc Rgang h Objc Followng Conol Pocu fo Whl Mobl Robo WSEAS anacon on Sy an Conol Iu 6 Vol 3 pp Jun 8 [7] L Bak V Jan Sabl Nonlna Poon Conol Law fo Mobl Robo Ung Gnc Algoh an Nual Nwok Wol Auoaon Cong WAC '6 pp 7 6 [8] JB Coulau G Capon G Ban an MD Wan Sably Analy of a Von- Ba Conol Dgn fo an Auonoou Mobl Robo IEEE anacon on Roboc Vol No 5 pp 6-69 Ocob 6 [9] AG Lonc MPG Gaffa JAS lo Río Mobl Robo Global Localaon Ung Ju a Vual Lanak Pocng of h 5h WSEAS Innaonal Confnc on Sgnal Pocng Roboc an Auoaon Ma Span 6 ISBN79-59 ISSN [] MK Bugja SG Fab L Call Dual Aapv Dynac Conol of Mobl Robo Ung Nual Nwok IEEE anacon on Sy Man an Cybnc Pa B: Cybnc Vol 39 No pp 9-4 Fbuay 9 [] C W Canua B Sclano an G Ban hoy of Robo Conol Spng-Vlag Lonon 996 [] Z P Jang an H Njj ackng Conol of Mobl Robo: a Ca Suy n Backppng n Auoaca Jounal of IFAC Vol 33 pp [3] R M Muay G Walh an S S Say Sablaon an ackng fo Nonholonoc Conol Sy Ung -vayng Sa Fback n IFAC Nonlna Conol Sy Dgn Boaux M Fl pp [4] C Saon an K A-Abah Fback Conol of a Nonholonoc Whl Ca n Caan pac Pocng IEEE In Confnc on Roboc an Auoaon Sacano CA pp [5] W Oln an J van Aongn Robu ackng Conol of wo-g-fo Mobl Robo Conol Engnng Pacc pp [6] W Dong W Huo S K o an W L Xu ackng Conol of Uncan Dynac Nonholonoc Sy an Applcaon o Whl Mobl Robo IEEE anacon on Roboc an Auoaon Vol 6 No 6 pp Dcb [7] ZP Jang E Lfb H Njj Saua ablaon an ackng of a nonholonoc obl obo Sy & Conol L Vol 4 pp [8] YH K an FL Lw Nual Nwok Oupu Fback Conol of Robo Manpulao Roboc an Auoaon IEEE ISSN: Iu Volu 9 Januay

11 WSEAS RANSACIONS on SYSEMS Haan Wang G L anacon on Vol 5 Iu pp 3-39 Apl 999 [9] F Sun Z Sun an PY Woo Nual Nwokba Aapv Conoll Dgn of Roboc Manpulao wh an Obv Nual Nwok IEEE anacon on Vol Iu pp Januay [] UY Huh JH Pak an JH K Fuy Obv Dgn fo Mobl Robo Conol Applcaon Pocng of 3 IEEE Confnc on Conol Applcaon Vol pp 45 5 Jun 3 [] SS G BB Rn an H L Ha Dk Dv Conol n Mobl Applcaon Jl Sy Sc & Coplxy Vol No pp 5-4 Jun 7 [] N Saka X Yun an V Kua Conol of Mchancal Sy wh Rollng Conan: Applcaon o Dynac Conol of Mobl Robo In J Robo R Vol 3 No pp [3] J E Slon an W L On h Aapv Conol of Robo Manpulao In J Robo R Vol 6 No 3 pp [4] S Aoo an F Myaak Sably an Robun of PID Fback Conol fo Robo Manpulao of Snoy Capably n Roboc Rach M Bay an R P Paul E Cabg MA: MI P 984 pp [5] A D Luca an G Oolo Molng an Conol of Nonholonoc Mchancal Sy n Knac an Dynac of Mul-Boy Sy J Angl an A Kckhy E Nw Yok: Spng-Vlag 995 Vol 36 ch 7 pp [6] R McCloky an R Muay Exponnal Sablaon of Dfl Nonlna Conol Sy Ung Hoognou Fback IEEE anacon on Auoaon Conol Vol 4 pp May 997 [7] Y Yaaoo an X Yun Coonang Locooon an Manpulaon of a Mobl Manpulao n Rcn n n Mobl Robo Y F Zhng E Wol Scnfc 993 pp 57 8 [8] Da IN Ka Dgn an Iplnaon of an Aapv Fuy Logc-ba Conoll fo Whl Mobl Robo IEEE anacon on Conol Sy chnology Vol 4 No 3 pp 5-5 May 6 [9] Fukao H Nakagawa an N Aach Aapv ackng Conol of a Nonholonoc Mobl Robo IEEE anacon on Roboc an Auoaon Vol 6 No 5 pp Ocob [3] Z-P Jang H Njj ackng Conol of Mobl Robo: a Ca Suy n Backppng Auoaca Jounal of IFAC ISSN: Iu Volu 9 Januay