The problems of camera measurements in tracking-error fuzzy control of mobile robots

Transcription

1 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca h poblms of cama masumns n ackng-o fu conol of mobl obos SAŠO BLAŽIČ, EL-HADI GUECHI, JIMMY LAUBER, MICHEL DAMBRINE, GREGOR KLANČAR Facul of Elccal Engnng, LAMIH laboao UMR CNRS 8 Unvs of Ljubljana, Unvs of Valncnns and Hanau-Cambéss žaška, Ljubljana, Mon Hou, 9 Valncnns cd SLOVENIA, FRANCE {saso.blac, ggo.klanca}@f.un-lj.s, {l-had.guch, jmm.laub, mchl.dambn}@unvvalncnns.f Absac: hs pap dals h akag-sugno modllng and conol of nonholonomc mobl obos n h cas hn h masumns a gvn b h cama. h masumns n such cas a dffcul o dal h: h a gvn onl n dsc m sampls, usuall h a dlad, a hgh-lvl nos s psn on h masumns. h nonlna ackng o-modl s solvd analcall und h pms of ZOH psn a h ssm npu. h nonlna dsc modl s dvlopd. Sval dscaon ssus a dscussd and h modllng os a analsd. h sco nonlna appoach s usd fo consucng h akag-sugno modl. h conol s dsgnd n h LMI famok. Som pfomanc ssus a dscussd on h smulaon cass. K-Wods: akag-sugno, PDC conol, Mobl obo, Knmac modl, Dscaon, Dla, Dca a Inoducon Mobl, auonomous obos a abou o bcom an mpoan lmn of h faco of h fuu [9]. h flbl and h abl o ac n dffn suaons [] opn up oall n applcaons, lavng no lm o h magnaon. o dv h mobl obo fom s nal pon o h ag pon, h obo mus follo pvousl plannd pah. Man scons a usuall mposd on h pah ha s bng dsgnd. hs ma as fom phscal lmaons [], (movng obsacls c. Sval conolls poposd fo mobl obos h nonholonomc consans, h h o man appoachs o conollng mobl obos a posu sablaon and ajco ackng. h am of posu sablaon s o sabl h obo o a fnc pon, hl h am of ajco ackng s o hav h obo follo a fnc ajco. Fo mobl obos ajco ackng s as o achv han posu sablaon. hs coms fom h assumpon ha h hl maks pfc conac h h gound, sulng n nonholonomc consans, hch mans ha no all h vlocs a possbl a a can momn. An nsv v of nonholonomc conol poblms can b found n []. Accodng o Bock s condon [4] nonholonomc ssms canno b asmpocall sabld aound qulbum usng smooh m-nvaan fdback. Compll nonholonomc, dflss ssms a conollabl n a nonlna sns; hfo, asmpoc sablaon can b oband usng m-vang, dsconnuous o hbd conol las. An ponnall sabl, dsconnuous fdback conoll as poposd b [] and h pon sablaon of mobl obos va sa-spac ac-fdback lnaaon usng poposd coodnas as sudd n []. ajco ackng s mo naual fo mobl obos. Usuall, h fnc ajco s oband b usng a fnc obo; hfo, all h knmac consans a mplcl consdd b h fnc ajco. h conol npus a mosl oband b a combnaon of fdfoad npus, calculad fom fnc ajco, and fdback conol la,.g. n [8]. Lapunov sabl m-vang sa-ackng conol las also usd [4], h h ssm s quaons a lnad h spc o h fnc ajco, and b dfnng h dsd paams of h chaacsc polnomal h conoll paams a calculad. h sablaon o h fnc ajco qus a nono moon condon. Man vaaons and mpovmns of hs smpl and ffcv sa-ackng conoll follod n la sach. ISSN: Issu 4, Volum 8, Apl 9

2 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca h appoach poposd n ou pap s basd on dsc akag-sugno fu modl, noducd n [7], oband fom h dsc nonlna modl of h knmac o. hn, a classcal Paalll Dsbud Compnsaon (PDC la [8] s compud usng LMI chnqus []. h poposd achcu s vald fo all h ajcos hn h lna and angula vlocs a boundd. Moov, h hs appoach conol la can b mplmnd asl n al m snc s possbl o fnd sablng gans ha can opa fo sval ajcos hn h lna an angula vlocs a boundd and h sabl pop s povn fo an nal condon n a p-spcfd compac s of h sa spac. h man pon n hs pap s o dscuss dscaon ssus of h -S modl and also o dscuss h pfomanc of h PDC conol. hs pap s ogand as follos. In Scon h ackng-o modl of h mobl obo n connuous cas s shon. In Scon sval dsc vsons of h o modl a dscussd. h -S modl s dvlopd n Scon 4. Scon dals h h PDC conol of h mobl obo. Scon 6 dscusss h poblm of dlad masumns, scon 7 gvs som smulaon suls, hl h conclusons a sad n Scon 8. Connuous knmac o-modl of ajco ackng B akng no accoun h non slppng condon, h knmac modl of h mobl obo n h X-Y plan can b n as follos: vcos vsn ( h h consdd conol npus of h mobl obo v and a h lna and h angula spd of h obo, spcvl. h oupu vaabls a and (h obo gavcn poson and (h angl bn h spd vco and h X-as,.. h obo onaon. Fg. llusas h dfnon of h posu o pssd n fam of h al obo and dmnd usng h acual posu q [ ] of h al obo and h fnc posu q [ ] of an vual fnc obo b h quaon: Y ( ( ( ( ( cos sn sn cos q q (, Fg.. Posu o ( Fom ( and ( and assumng ha h vual obo has a knmac modl smla o (, h posu o modl can b n as follos [7]: cos v sn u ( h v s h lna fnc vloc and s h angula fnc vloc. h conol la s hn dfnd as u [ v ]. V ofn (.g. [9] h conol s dcomposd as: v v cos( vb u (4 b Insng h conol (4 no (, h sulng modl s gvn b: sn v u B( h u [ v ] dfnd la. s h fdback sgnal o b b b b (, Dscaon of h knmac modl h connuous modl s no suabl fo h mplmnaon. In ou cas h nfomaon abou X ISSN: Issu 4, Volum 8, Apl 9

3 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca h posu of h obo s oband ach ms and also h conol s sn o h obos h h sam fqunc.. Eul dscaon h smpls mhod of dscaon s h Eul ngaon fomula h h dvav of h funcon s placd h h dffnc quon, and h follong dsc modl s oband fom (: ( k sn ( k v ( k ub (6 h all h sgnals on h gh hand-sd of h quaon a funcons of k.. h ac ZOH-dscaon h ac dsc nonlna knmac modl ll b dvd n hs scon. I s oband on h pms of consan npu sgnals bn samplng nsans (k of h oupu,.. ZOH s bng usd a h ssm npu: v( v( k cons., < ( k (7 ( ( k cons., < ( k L us no anals h hd quaon n ( hn < ( k : d ( k d ( k( ( ( k ( k( (8 No ha h funconal dpndnc on k o s no usd v fomall n h,.. ( dos no hav h manng. h onaon of h obo changs pc-s lnal and s possbl o analcall dmn h poson of h obo n h n samplng nsan basd on h posu and h conol npu n h cun samplng nsan. h quaons fo h poson can b ngad: ( k ( k d v( k cos( ( d ( k ( k ( k ( k ( vk ( cos ( k k ( ( d d v( k sn( ( d ( vk ( sn ( k k ( ( d Analcal soluon fo (k s: vk ( k ( ( vk ( ( k ( k sn( ( k ( k sn( ( k k ( vk ( k ( ( cos( ( sn( ( k k ( sn( ( cos( ( k Smlal h modl fo (k can b oband: k ( k ( ( cos( ( v( k k ( vk ( k ( cos( ( k sn( ( k sn( ( k (9 ( ( h dsc modl of h mobl obo s compld b h modl fo ( k oband fom (8: ( k ( k ( k ( h fnc modl fo h mobl obo s h sam as h modl (, (, (, onl h npu sgnals a v ( k and ( k : ( k ( k ( cos( ( k sn( ( k v ( k ( k v ( k sn( ( cos( ( ( k k k v ( k ( k v ( k sn( ( sn( ( ( k k k ( k ( k ( cos( ( k cos( ( k ( ( k ( k ( k akng no accoun (, (, (, (, and ( h follong nonlna dsc modl s oband: ( ( sn( v sn( sn ( ( ( cos( cos( ( cos( ( k cos sn v v ( k sn cos v v ( k ( k ( k ( k (4 ISSN: Issu 4, Volum 8, Apl 9

4 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca h h plc dpndnc on k s omd n h gh hand-sd of h quaons o mak hm cla. Inoducng h conol (4 no (4 a v compl modl s oband. h quaons of h sulng modl h v b and b dpnd lnal on ( k and ( k, so s v smpl o pu o h ma fom: vsn( ( cos( v sn( sn( ( sn ( ( ( cos k ( sn co s k ( ( h ( k ( k ( k ( k. I nds o b sssd h ha ( as no oband b lnaaon of h nonlna modl. Acuall, all h nonlna dpndncs a kp, hfo h modl s ac. h soluon fo h npu pa s much mo nonlna and onl appoma soluon fo h B ma can b dvd: ( sn b cos( B b ( cos( v cos( ( cos( v cos( v cos( sn( ( sn ( v cos( sn( v sn( v sn( v cos( ( cos( b sn b cos (6 h o modl oband fom ( and (6 k ( Ak ( Bu( k (7 b h u ( k [ v ( k ( k ] s v compl fo b b b h alaon of h akag-sugno modl oband b sco nonlna appoach [8]. hs s du o h hgh numb of nonlnas n macs A (4 and B (4. Egh nonlnas mans ha h -S modl ould hav 8 6 fu uls and h numb of h LMIs ould b n h ang of 6.. h smplfd ZOH-dscaon In od o smplf h nonlna modl fom h pvous subscon and duc h numb of nonlnas h follong smplfcaons noducd no ( and (6: ( ( cos sn (8 h ma A of h sulng modl s h sam as n h Eul modl (6, hl h ma B bcoms: v sn B v cos (9 h smplfd modl posssss smla compl as h Eul on snc h numb of nonlnas fo h -S modl s 4 ( n h ma A and n h ma B..4 Compason of ndvdual modls h modls psnd n subscons.,. and. a compad n hs subscon. h compason mans h ha h lmns of macs A and B a compad b calculang h absolu dffnc. I s v as o s ha h dffnc bn A macs s h hghs hn and a h hghs hl. Fo h ma B h hghs dffnc as oband b usng numcal mhods. In hs analss h sach s lmd o h π s:.,.,, v,. h suls a shon n abl. h o alas mans mamum absolu dffnc bn h Eul modl (scon. o h smplfd modl (scon. and h ac nonlna modl (scon.. Each o shos h valus h h o s h bggs. hs mans ha B Eul and B smplfd a no analsd n h sam pon of h spac (,,, v,. Onl h cass h h absolu o s h bggs a shon. h analss shos ha h onl o bg (lav dffncs occu n h cas of h lmns B and B f h Eul appomaon s usd (n boh cass h appomaons a mo han h faco oo bg hl n h cas of h smplfd modl h o s lo. ISSN: Issu 4, Volum 8, Apl 9

5 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca abl. Wos cas lmns of h macs Ma Elmn Eac valu Appo. ma. o A, A A, -A A A B B B Eul B smplfd B Eul B smplfd Hbd ad-hoc modl h Eul modl (spcall s ma B s no v suabl fo h conol dsgn. I s pfabl o usd h smplfd o modl. I s also possbl o us som hbd modl hl sll kpng h numb of nonlna funcons n h - S modl qual o 4. Obvousl, h nonlna funcons (fo A, B, and B can b usd dcl fom h ac modl, h fouh on can b sn nsad of hch s usd n h smplfd modl. No ha h appomaon of h cos h s os ha h appomaon of h sn h, so ould b b o us h appomaon fo h cosn funcon, bu hs ould add anoh nonlna no h modl. I s also possbl o appoma h lmn B hou ncasng h numb of nonlnas: cos( sn... ( 4 In h smplfd modl (n Scon. s usd as an appomaon fo B,.. h o of h appomaon s O ( hl n h cas of ( h o s of h s O (. 4 akag-sugno modl of h obo h dsc S modl s psnd hough h follong polopc fom [7]: ( ( ( k ( h k ( Ak ( Buk ( In od o consuc h modl h appoach h sco nonlna ll b usd [8]. hs mans ha h nonlnas hav o b akn fom h nonlna modl and usd n h pms vco (k. h vco (k n ou cas s: h macs sn vsn( sn( k ( b b A and B a: A B 4 ( ( No hav o fnd mnmum and mamum valus of h 4 nonlna funcons: lmn < l < lma l,,, 4 (4 h agan lm ou sach o h follong s of h spac:.,.,, v, ( π 4 h numb of uls s 6. h macs of h modl a: h ε ε 4 A ε ε B ε ε 4 4mn k k 4ma ls (6 mn fo 8 ε ma ls mn fo 4 and 9 ε ma ls (7 mn fo {,,, 6, 9,,, 4} ε ma ls fo,,, 8 ε ISSN: Issu 4, Volum 8, Apl 9

6 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca PDC conol of h obo In od o sabl h dsc S fu modl (, a PDC (Paalll Dsbud Compnsaon conol la s usd [8]: B ( (8 u ( k h ( k F( k F( k Sval suls concnng h sabl of h -S modl h h PDC conolls s. h poblm s ofn solvd hn h LMI famok. H h soluon ha s o opms h dca a of h ssm ll b usd [8]: mnm β XYM,,... M subjc o X>, Y, β X ( Y XA M B >, AX BM X β X Y, ( AX AjX BM j BM j, ( AX A jx BM j BM j X < j subjc o h h j (9 h abov gnald gnvalu poblm can b solvd b usng numcal algohms. h suls a opmal dca a ha has o sasf β < and F MX ( h poblm s ha h LMI oolbo n Malab canno solv h LMIs h h. In hs cas h > laon can b usd nsad n h dfnon of h nquals. h oh poblm ha s mo sous s ha h soluon fo β dos no ca much nfomaon abou h acual dca a. V ofn h dca a β pdcs nsabl alhough h ssm s sabl. hs poblm can b pal ovcom f a n vaabl s noducd: γ β ( A slghl modfd gnald gnvalu poblm s hn oband: mnmγ XYM,,... M subjc o X>, Y, γ X X ( Y XA M B >, AX BM X γ X X Y, ( AX AjX BM j BM j, ( AX A jx BM j BM j X < j subjc o h h j ( h n algohm fnds h soluon fo γ < as han h ognal algohm fo β <. h oband dca a sll dos no sho h acual dca a. hs s of cous du o a v consvav appoach hch s a gnal chaacsc of hs appoach. h opmal suls a unusabl bcaus h conol ssm s oo quck. hs poblm can b solvd b usng h consan on h npu [8]: h consan v ( ( < μ s nfocd a all ms f h follong LMIs a addd o h s of LMIs (: X M, M μ I X φ I ( h ( φ. h abov addon dos lm h conol sgnal succssfull, bu s also v consvav snc oks fo an nal condon. If ( s o b usd, h paams φ and μ nd som unng. A lo of laaons of can condons s n h lau. h ffcn soluon of LMIs as no h goal of hs pap. 6 h poblm of dla Bsds havng h oupu dfnd onl n dsc momns, h masumns a also dlad. In od o hav good pfomanc conol, hs dla has o b compnsad. hs s usuall achvd b unnng h obo modl n paalll o h acual obo. h conol schm s shon n Fg.. ISSN: Issu 4, Volum 8, Apl 9

7 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca fnc vlocs dsgn v fdfoad conol v cos coodna ansfomaon fdback conol vb b v acual obo h dlad masumns d d d knmac modl m m m dla md md md Fg.. h conol schm n h cas of dlad masumns Alhough s possbl o solv h poblm b dffn obsvs, n ou cas a smpl Smh pdco s usd. W hav o oban h cun posu of h obo (,, hou havng masud dcl. Insad can calcula fom h acual masumns ( d, d, d, h oupu of h modl ( m, m, m, and h oupu of h modl ha s dlad fo h sam m as h masud sgnals a ( md, md, md. h da s o sma h undlad oupu h h pms ha h dffnc bn h undlad and h dlad oupus of h modl s h sam as h dffnc bn h undlad and h dlad oupus of h acual obo. W hfo fd back h undlad sgnals: ˆ ( d m md ˆ ( d m md ˆ ( d m md (4 7 Smulaon pmns h o pmns conducd n hs pap. h fs shos h ajco ackng h h dsc PDC conoll poposd n hs pap. h suls a shon n Fgs., 4, and. V good suls a oband n hs cas. I can b obsvd ha h ackng os convg o fnc acual Fg.. h fnc ajco (dashd and h acual on (sold v fnc acual fnc acual Fg. 4. h lna (upp fgu and h angula (lo fgu vlocs ISSN: Issu 4, Volum 8, Apl 9

8 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca Fg.. Convgnc of h os In h scond pmn h dla as also psn on h masud oupus. In h fs cas (Fgs. 6, 7, 8 h dla s on sampl m (, ms. h masud sgnal s usd dcl as h oupu of h ssm. h dla causs ha h ssm sas oscllang hl no oscllaons a found n h dla-f cas (Fgs., 4,. In h scond cas (Fgs. 9,, h ffc of dla s ducd b fdng back h sgnals accodng o (4. Whn sng h dla of sampl ms (66,7 ms nsabl occud n h cas hn masud sgnals ad as h oupus of h ssm. V good suls a sll oband n hs cas f usng Smh pdco. I nds o b sssd ha h modl of h ssm as no nalsd h h sam valus as h acual ssm. h nal o of h modl as. m n h -dcon,. m n h -dcon, and 4 o n h oaon. v fnc acual fnc acual Fg 7. h pmn h h dla of sampl m h dla no akn no accoun: h lna (upp and h angula (lo fgu vlocs Fg 8. h pmn h h dla of sampl m h dla no akn no accoun: h os fnc acual Fg 6. h pmn h h dla of sampl m h dla no akn no accoun: h fnc ajco (dashd and h obo pah (sold fnc acual Fg 9. h pmn h h dla of sampl m h dla compnsad: h fnc ajco (dashd and h obo pah (sold ISSN: Issu 4, Volum 8, Apl 9

9 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca v fnc acual fnc acual Fg. h pmn h h dla of sampl m h dla compnsad: h lna (upp and h angula (lo vlocs Fg. h pmn h h dla of sampl m h dla compnsad: h os fnc acual Fg. h pmn h h dla of sampl ms h dla compnsad: h fnc ajco (dashd and h obo pah (sold v fnc acual fnc acual Fg. h pmn h h dla of sampl ms h dla compnsad: h lna (upp fgu and h angula (lo fgu vlocs Fg 4. h pmn h h dla of sampl ms h dla compnsad: convgnc of h os 8 Concluson hs pap dals h akag-sugno modllng and conol of nonholonomc mobl obos. h nonlna ackng o-modl s solvd analcall und h pms of ZOH psn a h ssm npu. h nonlna dsc modl s dvlopd. Sval dscaon ssus a dscussd and h modllng os a analsd. I s shon ha h modl oband b h Eul ngaon mhod posssss bg os n som cass and should b no usd. Modls of smla compl sul n much b pcson. h sco nonlna appoach s usd fo consucng h akag-sugno modl. h ffc of dla s also sudd. h conol s dsgnd n h LMI famok. Som ISSN: Issu 4, Volum 8, Apl 9

10 Saso Blac, El-Had Guch, Jmm Laub, Mchl Dambn, Ggo Klanca pfomanc ssus a dscussd on h smulaon cass. Rfncs: [] M. Acad, Coodnaon and conol of a am of mobl obos, WSEAS ansacons on Ssms, Vol. 6, No. 6, 7, pp. 6-. [] S. Blažč, I. Škjanc, D. Mako, Smpl fu adapv conol fo a class of nonlna plans, WSEAS ansacons on Ssms, Vol., No., 4, pp [] S. Bod al., Lna ma nqual n ssm and conol ho, Suds n Appld Mahmacs, Phladlpha, 994. [4] R. W. Bock, Asmpoc sabl and fdback sablaon, n: R.W. Bock, R.S. Mllman, H.J. Sussmann (Eds., Dffnal Gomc Conol ho, Bkhaus, Boson, MA, 98, pp [] C. Canudas d W, O.J. Sodaln, Eponnal Sablaon of Mobl Robos h Nonholonomc Consans, IEEE ansacons on Auomac Conol, Vol. 7, No., 99, pp [6] K. N. Fass, M.. El Hag, A. A. El Kos, ajco ackng conol fo a hld mobl obo usng fu logc conoll, WSEAS ansacons on Ssms, Vol. 4, No. 7,, pp. 7-. [7] El H. Guch, J. Laub, M. Dambn, S. Blažč, G. Klanča, ackng-o modl-basd PDC conol fo mobl obos h acclaon lms, submd o IEEE Innaonal Confnc on Fu Ssms FUZZ-IEEE, 9. [8] D.-H. Km, J.-H. Oh, ackng conol of a o-hld mobl obo usng npu oupu lnaaon, Conol Engnng Pacc, Vol. 7, No., 999, pp [9] G. Klanča, I. Škjanc, ackng-o modlbasd pdcv conol fo mobl obos n al m, Robocs and Auonomous Ssms, Vol., No. 6, 7, pp [] I. Kolmanovsk, N. H. McClamoch, Dvlopmns n Nonholonomc Conol Poblms, IEEE Conol Ssms, Vol., No. 6, 99, pp. 6. [] M. Lpč, G. Klanča, I. Škjanc, D. Mako, B. Poočnk, m opmal plannng consdng acclaon lms, Robocs and Auonomous Ssms, Vol. 4,, pp [] K. Pak, H. Chung, J.G. L, Pon sablaon of mobl obos va saspac ac fdback lnaaon, Robocs and Compu Ingad Manufacung, Vol. 6,, pp. 6. [] L. Podsdkosk, J. Noakosk, M. Idkosk, I. Vva, A n soluon fo pah plannng n paall knon o unknon nvonmn fo nonholonomc mobl obos, Robocs and Auo-nomous Ssms, Vol. 4,, pp. 4. [4] C. Samson, m-vang fdback sablaon of ca lk hld mobl obo, Innaonal Jounal of Robocs Rsach, Vol., No., Vol. 99, 64. [] I. Škjanc, M. Lpć, J. L. Fguoa, S. Blažč, Fu modl-basd pdcv conol fo a CSR h mulpl sad sa: A smulaon sud and a compason h oh nonlna MBPC conol algohms, WSEAS ansacons on Ssms, Vol., No., 4, pp [6] I. Škjanc, Monong of as-a amn plan basd on fu modl, WSEAS ansacons on Ssms, Vol. 6, No., 7, pp [7]. akag, M. Sugno, Fu dnfcaon of ssms and s applcaon o modlng and conol, IEEE ans. On Ssm, Man and Cbncs, Vol., No., 98, pp. 6-. [8] K. anaka, H. O. Wang, Fu Conol Ssns Dsgn and Analss: A Lna Ma Inqual Appoach, John Wl & Sons, Inc.,. [9] Y. ng, W. I. L, H. C. Ja, A Pah Plannng Algohm fo Indusal Robos, Compus & Indusal Engnng, Vol. 4,, pp ISSN: Issu 4, Volum 8, Apl 9