A parametric formulation of tracking methods: application to chained systems

A PARAMEIRIC FORMULATION OF ‘I’KACKINC ME‘IHODS: APPLICATION TO CHAlNED SYSTEMS F. DiAZ DEL Rio, G. JIMkNE7- J, I,, SEVILLANO, A. CIVI‘T BAI,CELI,S, Escucla Tbcnica Supc io dc lngcnic ia In onnl ica. Uni c sidad dc Sc illa. AV. Kcin, Mc ccdcs sin. 4 ID1 2 Sc illa. SPAIN, L diaz. pj!, sei i. [email p o ec ed]/.~.c. ABS’IKAC’I Whcn ollowing a pa h, hc c a c sc c al possibili ics a cnding o hc way in which hc ac ual obo s a c can bc cla cd wi h hc wliolc pa h. In his wo k wc o mula c di c cn acking mc hods on hc basc ha a mcmo izcd pa h can bc dcsc ibcd by a singlc dcsc ip o pa amc c ( hc objcc i c poin is cadily gi cn by his pa amc c ). Wc classi y pa h acking acco ding o hc way in which wc impose o design hc p og css o dcsc ip o pa amc c . Bcnc i s and disad an agcs o cach inc hod a c idcn i icd. Wc also sumnia izc how o cons uc cach mc hod. Dcspi c ha his classi ica ion is gcnc ic, applica ion o mobilc obo s and chaincd sys cms is c y aluablc duc o hc impo ancc o acking p oblc n in licm and bccausc hci dcsi cd pa i is usun ly mc no izcd. KEY WORDS: pa amc lc cqualions, di c cn jal gcomc y, mobilc obo s, nonholonomic cons ain s, pa h ollowing, ajcc o y acking, chaincd sys ems. 1. INTROUUCTlON i is wcll know ha hc con c gcncc o a non-omnidi cc ional mobi c obo o a ixcd pos u c ( hc s abiliza ion p oblcm), can no bc achic cd h ough a smoo h ccdback s abiliza ion con ol law duc o B ockc ’s hco em [I]. On hc o hc hand, in mobilc obo s i is usual ha1 hc pa h o ajcc o y is mcmo izcd, and hc poin s abiliza ion p oblcm is c y di lic cn o hc pa h’s con c gencc p oblcm. Fo mos niobilc obo s imc dc c minism is no a cqui cmcn o hc acking, and hc con ol objcc i c can bc hough o as ollowing a pa h wi h a a c ha can bc a iablc. Thc c is no doub ha hc nunibc o mobilc obo s applica ions will bc la gc in hc ncx cw yca s, cspccially in iclds such as in clligcnl anspo a ion sys cms (ITS), cxplo c chiclcs, and pc sonal o assis an obo s. Canscqucn ly, in hc las dccadc hc c has bccn a g ca in e cs in inding con olle s and acking mc hodologics To hcsc obo s. As a csul o hc wo k wc ha c donc du ing hc las yca s in his icld, wc p cscn hc c a acking classi ica ion bascd on a pa amc ic o mula ion. Wc mus undc linc ha wc do no considc ncw con ol cchniqucs in his wo k, nc c hclcss wc conccn a c on a p io subjcc : which mc hods o usc whcn ollowing a pa h. In ac , con ol law applica ion can bc considc cd as a sccond s cp in hc acking p oblcm dcsign. Mo co c , al hough his o mula ion can be applicd Eo c c y sys cm, wc cniphasizc same aspcc s e y aluablc o chaincd sys cms. In hc ncx scc ions wc will in oducc and analyzc hc ncw acking classi ica ion. In scc ion 2 and 3 wc o mula c hc di e cn acking mc hods. In scc ion 4 wc cxplain hc acking classi ica ion bascd on an cxamp c and ina ly wc cxposc hc conclusions. 2. PARAMETRIC FORMULATlON A memo ized, e c cncc o dcsi cd pa h (o mc cly pn h) can bc dcsc ibcd by a singlc dcsc ip o pa amc c [13], namcly P, and i can bc cxp csscd as a cc o o s a c coo dina cs qacl( ). As a csul , hc acking p ogcss can bc idcn i icd wi h hc p og css o . Al hough hc pa amc c may bc imc, in cascs whc c imc dcpcndcncc is nob clc an , many o hc s a c possiblc. Fo cxamplc, in di c cn ial gcomchy hc na u al a c pa amc c 1131, which makcs he linca spced cqual o I, is gcnc ally p c c cd. Fu hc mo c, whcn s udying con c gcncc o a pa h, i is usual o supposc ha dcsi cd ajcc o y has no cnd. 37 I Whcn s udying hc acking p oblcm, wc can hink o as a ncw s a c coo dina c, which mus bc addcd o hc o he n coo dina cs ha dc inc hc s a e q( l o hc cal sys cm. Thc c o c wc inc casc hc nunibc o coo dinalcs in 1 i wc wan lo cxp css hc sia c o he acking: { . 9). In addi ion, i is usual lo cxp css sys cm s alc h ough c o coo dina cs e&), which a c somc kind o ela ion bc wccn cal nnd dcsi cd s a cs. E u s will easily cxp css how a hc sys em is om i s con c gc icc objcc i c. Thc c o c, wc can classi y hc acking o hc pa h acco ding o Ihe way in which wc imposc o design (whcn possiblc) pa amc c . F om his poin o icw, wc can considc a ncw sys cm inpu : ihc onc ha go cms pa amc c ’s e olu ion, which wc will call a Wc will SCC ha o some acking mc hods, U is ob ious, bu o somc o he s, osclcc ion can in oducc in c cs ing ca u cs on hc acking. To sum up, wc can cxp css acking o whec cd mobilc obo s as: wlic c U is he cc u o inpu a iables, which has a liiica da ion o chained sys cms. Oncc hc acking mc hod has bccn sclcc cd, a sccond s cp is inding a con c gcncc. Dcpcnding on hc sclcc cd acking mc hod, con ol law ex ac ion may bc di e en . 2. TRACKING METHODS’ CLASSIFICATION Many csca chc s ha c s udicd a ious acking me hods whcn he dcsi cd pa h is mcmo izcd o p c iously gcnc a cd. In lic icld o mobilc obo s wo main acking mc hods ha c bccn s udicd. In a i s g oup we ind hose ha considc hc cxplici ly in hc acking [2,3,4j (usually callcd “/ ajecioq, ading” (TT)), and y o app oach he obo o a mo ing objcc i c poin . A usual case o TT is acking in sc osys cms, whc c wc ack a mobilc sys cm o a ge a hc imc i IIIOYC~. The dcsi cd cuo dina cs a c simply qba(l), bccause = , and c o coo dina es may bc dc incd as eq( )q(i)-qded ). In sc osys cms, ime is c i ical and his is hc only possibili y wc ha c. Nonc hclcss, when wc I y o ack a mcmo izcd pa h, hc acking mc hodology can bc dcsigncd opcnly, as wc know a p io i he wholc ajec o y. O cou sc classical sc osys cm acking can bc applicd jus by idcn i ying hc pa amc c associa cd o hc pa h wi h imc, ha is ( )= . Fu hc mo c, IT can bc cx cndcd in a mo c gcnc al assump ion han sc osys cms: Ic he pa amc c I’ bc jhc ion uj’ imc = (l). An asynip olically s ablc con ol law may gua an cc ha hc sys cm will con c gc o a poin qdes( ) in hc dcsi cd pa h in a dc c minis ic imc, cxccp o hc inhe en pc u ba ions ha il may su e . Acco ding o Eq. (I), U will bc simply I o se osys cms, and a unc ion o limc To a mo c gcnc al TT: a=u( ). In a sccond g oup, wc ind lhosc me hods ha do no considc iming cqui cmcn s and y o con c gc o a pa h [4,5,6,7,8], which a c usually callcd “p l h olk~wing”(PF) o mobilc obo s. WE can also ind sc c al cxccllcn compcndia o bo h mc hods in [2,9]. PF Is based on some da ion bc wccn ac ual sys cm’s s alc q(l) and hc wholc mcmo izcd pa h. This cla ion o p ojeclic n will gi c us hc dcsi cd poin qdJ ), i.c. hc dcsc ip o pa amc c Y as a unc ion o hc ac ual posi ion and pa h: Nq, )=O, whc c IT is somc kind o p ojcc ion o hc ac ual posi ion o lhc pa h. Thcn hc eal sys cm mus y o ollow his poin qdn( ) ins cad o hc one gi en by he o hc app oach. Fo cxamplc, he dcsi cd poin is usually sc cc cd o bc hc ”closes poin on hc pa h” o he ac ual obo ’s posi ion [13]. Thc c m “closes ” mcans hc pa h’s poin ha makcs ce ain dis ancc c ilc ion minimum. Tlic c o coo dina es may csul also as eq(pq(+qdm( ). O cou sc using his app oach, i is no gua an ccd ha hc sys cm will each a poin o he dcsi cd ajcc o y in a dc c minis ic imc. Bu hc main p oblcm wi h PF is ha p ojec ion uniqucncss has no bccn gua an ied yc , hcncc no PF ha c bccn dcsigncd ha can be applicable o all possiblc pa hs ( o hc au ho s’ knowlcdgc). Wc mus no c ha PF p ojcc ion can bc considc cd as an imposed holonomic cons ain on hc whole s a e {q, }, and his mcans lha hc sys cm will immcdia cly oosc onc o i s s a e coo dina cs. Inpu D can bc ob aincd by di l’c cn ia ing hc p ojcc ion wi h cspcc o imc, subs i u ing s a c cqua ion (I), and hen sol ing o IT= i. : 372 Thc c o c PF in oduces a dcpendencc likc U yu(q, U) = /(q) U. No e ha i denomina o is null in Eq. (2), a ia ion o is undc incd. We ha c shown in 141 ha his caw is cqi i alcn o hc non-uniqucncss o hc choscn p ojcc ion. In o hc wo ds, PF is no app icablc o a11 kind o pa hs. Thc key ad an age o PF is ha i is mo c sui ablc o many si ua ions in which imc is no a c i ical pa amc c [5,6,10]. This can bc unde s ood iY wc considc he ollowing cxamplc: i pc u ba ions o cc hc mobile obo o bc a cs , dcsi cd poin o TT will mo c una oidably, This mcans ha c o s will g ow up o somc aluc ha may in oducc ins abili y. On hc o hc hand, i PF wc c uscd, dcsi cd poin will be he samc in spi c o hcsc pc u ba ions, bccausc pa h’s shapc and cal obo sh c cmain hc samc (c is linca o inpu s U To chaincd sys ems). A di c cn acking mc hod was p oposcd in a p c ious papc o ou s [I I]. Wc namcd i e o adop i e ocking (EAT), bccausc acking adap s o sys cm c o s. Wc showcd ha i c ains hc cxposcd ad an agcs o PF and i can bc applicd o all sc s o pa hs. I s dcsign is simila o ha o TT bu i is in cndcd o pa h cco c ing wi hou s ic iming cqui cmcn s. No e ha o mos mobi c obo s (including ad anced whcclchai s [ 121) imc dc cnninism is no a cqui cmcn o he acking. Whcn imc is no c i ical, wc can “dcsign” he a ia ion o I’, and wc can cga d obo s a c, i.c. aking c o s in o accoun . Thus, con as ing hc IT‘S igid a ia ion o , ha is o=o(/,), wc p oposcd o=g(e,J, whc c g(eJ is a “con cnicn ” unc ion o hc c o s. Hc c “con cnicn ” is c c cd o hc dcsignc objcc i cs, bu also i should mcan ha acking is done co ec ly, and he p c iously cxplaincd ad an agcs o PF a e p csc cd. Tha is, unc ion g(eJ should ul ill: I c o s a c small, g{eJ should cnd o I (o nio c cxac ly o lul/1udea( )l i wc wan o mo c along hc pa h a ano hc pacc), so he acking escmb cs IT. c o s a c la gc, hc c c cncc obo should “wai o ” hc ac ual obo . Tha is, g(e.J should bc small un il a good can c gcncc IS cachcd (D should cnd o ze o). O cou sc in his si ua ion, no dc c minis ic ollowing is cxpcc cd. O cou sc many passible unc ions g(eJ can bc dcsigncd a cnding o hc cha ac e is ics and pu poses o cach sys cm, bu in [I I] wc showcd ha a c y sui ablc unc ion is U = g(eJ = ap(- K#i) ; whc c K,>O is a scalc ac o . Wi h hcsc condi ions hc acking canno bc dc c minis ic and cminds ha o PF. Mo co c , a sccond kind o EAT can bc dcsigncd. I wc nccd somc aspcc s o imc dc c minism, a ia ion o can bc cx cndcd o include he “inaccu acy in hc dc c minis ic acking”, ha is, hc di c cncc bc wccn hc dcsc ip o pa amc c ( ha indica cs hc a gc o ou con ol dcsign qdl ( )) and imc 1 ( ha indica cs hc a gc o hc iming cqui cmcn s qde,(i)). Hcncc awou d bc g( .e,J: a combined unc ion o c o s eg and hc di c cncc bc wccn pa amc c and imc . Oncc again he e a c many possiblc unc ions g(eJ, bu an in c cs ing s a egy [I I] is in cndcd o gc a dc c minis ic acking a long las ( ha is, a “ claxcd” dc c minis ic acking). Pa amc c will cmain “a es ” whcn he obo is a om hc pa h (in spi c o hc inc casc o di c cncc - ). Whcn obo “ ccupc a cs” and app oachcs hc pa h, wc conccn a c on cducing hc di c cncc -/. Thc c o c g( ,e,J can no bc boundcd by I, bccausc in hc sccond casc I- mus app oach /. Mo co c a hc o igin (/- =O. e,+), g mus bc I, Thus wc showcd ha a con cnicn choice o his mo e complcx EAT can bc: g( ,eJ- exp(-K,eqz) (I+K,p c/on(/- ;)), ; whc c KpO, K,,W a c scalc ac o s ha indica cs how as hc con c gcncc o o is. Hc c g( ,eJ is uppc boundcd by I + K,~ .and lowc boundcd by I - K,~ 2 2 o 0. Thc o mula ion ad an age o EAT is ha i can ob iously bc applicd o all sc s o pa hs, and implies a clnlion To hc inpu c Iikc U =c (q) o c =u( , 9). Wc can scc his cla ion as a (in gcnc al) non-in cg ablc da ion bc wccn lic whole s a c gi cn by {q, }. To sum up, wc can ind in hc cu cn li c a u c he ollowing p oposals o , which supposc a di cc cla ion cons ain bc wccn {q, }: 373 1. 7T: = ( ). * u=a( ). 2. PF: n(q, )=O u=o(q,u). In IT, wc “loose” uoo dina c , and in PF WE usually loosc onc o hc e o coo dina cs e,, [4], Mo co c hc wo p oposals o EAT a c non-in cg ablc ela ions bc wccn coo dina cs {q, ): 3. EAT: a--i=o(q) 4. EAT: = :=a( ,q). 5. Q= a(q, U). 7. u=cT( , U). s. u=a(u). I wc cambinc all possiblc dcpcndcnccs o q wc no icc ha hc c a e ou cases IcA: 6. a=a( , u,q). Wc mus poin ou ha inc hod 5 is no hc samc as PF, bccausc wc suppose ha cla ion 5 is no in cg ablc (In PF i is clca ly in cg ablc, bccausc dq, )=Ocxis s). Mo co c we will no conside hc c hc las wo acking mc hods (7 and 8). Thcy do no sccm o bc in c cs ing o acking pu poscs, bccausc a ia ion o pa amc c docs no dcpcnd on hc s a c q. On hc con a y, cases 5 and 6 a c a mo c in ui i c and c cc i c cx cnsion o hcm, and wc will p cscn hci cha ac c islics and applica ions in his pape o he i s ime. Nc c hclcss hcsc mc hods 7 and 8 will be s udicd in ulu c wo k in o dc o dc c minc i hcy can bc wo hwhile o somc pa icula applica ions. In o dc o ix idcas and cxplain he wo new acking mc hods, wc will analyzc a simplc sys cm in hc ncx sec ion. Dcspi e ha his examplc is c y simplc, ou classi ica ion is gcnc ic o c c y sys cm; so ex ac ion o con ol laws o chained sys cins can akc ad anlagc o using somc o hcsc me hods. Somc cxamplcs o hcsc bcnc i s hs c bccn cxainincd in somc o -ou p c ious wo ks [4,1 I]. 3. A SIMPLE EXAMPLE Lc us conside a simplc sys cm wi h wo s a c coo dina cs x={xl, x }, whose goal is o ollow a c c cncc pa h xd,,=/. l,jJ i, .k-ldw( )} made by a i ual obo . S a c cqua ions a c simply: . , =U* (3) . , = 11, ; x:.,& = l ,,&(i-) Xi,,&.” = u2,,A, ) whc c (‘) holds o dc i a i c wi h cspcc o , Applying chain’s law: u,,k3(Q= i i+,*>( ), As wc a c in c cslcd in hc acking wc dclinc hc syslcm’s c o s as hc dij’c ence: .j=1,2. And hcn wc ha c .ii,,&, = lj,8h( ) ,/=1,2. ed j =.VI(!) -. lcle$i (i)l ; L:, (I)= 1 d ) -?141des( 1 ; (4) .?(I) = a( ) - . ~~~( ( )) : L;: I ) = u?( ) - uj&J ): WC ha c cxp csscd explici ly hc depcndcncc o hcsc a iables bccausc hc dcsign In a scwosys ciii J.={, a=l and a c y siniplc con e gcn con ol law can be: I TT is choscn, dcpcndencc = ( ) will only add a scalc on hc clcc ioii o -, supposing ha i>O H. NOIC ha in bo h cascs, hc condi ion iW Vi implics ha hc c c cncc poin {xI&( )). x2dc,( (g)/ will una oidably ad ance in spi c o eal obo mo ion. Bu i wc could apply PF mc hod, hc si ua ion would bc c y di c cn . Fi s we mus look o he mos sui ablc p ojcc jon @x, ). in o dc o sclcc hc desc ip o pa amc c and hcncc, hc c c cncc poin /. lk,( ), x?,kJ /] a any ins an . The mos ob ious choicc is hc p ojcc ion ha sclcc s hc closcs poin on lic pa h o hc obo ’s posi ion, i.c. ha makes C:_,.,! minimum. Fo cxamplc, i hc dcsi cd pa h is hc [in, {xJkS( )=s , xjb,( )=O, s>O/ hcn hc closcs poin wi 1 bc xd6=j.xI. O}, and bc p ojcc ion: F x,/s. Di c cn ia Ion o p ojcc ion gi c us hc a c o ( J D =uI/s. Thcn hc acking will p og ess only i inc cascs, ix. i U,># (o uj<O i hc acking is o mclhodology mus now choose bc wccn di k cn o ms o ( ) o a. ~#) = i?,,,J k&e, i ~,~J )-K,c,, j=1.2, whc c K/ >O. 374 bc donc in c c sc o dc ). Thcn wc mus impose some condi ion o “mo ion cxigcncy”[4] o cnsu c ha ( cal and hcncc i ual) obo s ad ancc, and hcncc o gua an cc ha hc acking is being donc. The simples mo ion cxigcncy is o cou sc u,=com on >O, bu o hc mo c sophis ica cd can bc p oposcd. I hc i ial mo ion cxigcncy is p c c cd (and cnongh o ou con ol pu poses) hcn hc p c ious con ol law s ill succccds (o cou sc only o inpu 2): i 2 ( ) = u2 d&) - Kj e? = - K2 el. This is ob iously a pa icula casc. I a gcnc ic ajec o y had o bc ackcd, hcn a mo c gcnc al mo ion cxigcncy is p c c ablc, o cxamplc U/’+ u?’=c in.~ioni>O [4]. This has hc addi ional ad an agc o main aining inpu s wi hin cc ain alucs, a oiding an cxccssi c inc casc o inpu s (which may in oduce ins abili y). Howc c , p oblcms a ise whcn wc canno gua an cc p ojcc ion uniqucncss. Fo cxamplc, i hc pa h wc c a ci clc and hc p c ious p ojec ion wc c o bc applicd, hcn p ojec ion uniqucncss is b okcn whcn hc obo app oachcs hc ci clc ccn c . So i is no ully applicablc o all kind o pa hs, and p ojcc ion uniqucncss should bc ca c ully analyzed (SCC [4]). In EAT wc can design he mos app op ia c acking alc, i.e. an cqua ion o uas a unc ion o hc c o s. As cxplaincd bc o c a c y simp c and in c cs ing possibili y would bc: CT =g(e,J=exp (-K (e 2+ e:)) ; K,>O is hc scalc ac o . This p oposal ics ha c o s do no inc casc g ca ly. Thc c o c, i consc cs hc PF ad au agcs, whilc a oiding hc p ojcc ion di licul ics. Bcsidcs, o his i ial sys cm hc p c ious TT con ol law s il wo ks: ui( )=up(-KJe,’+ e )) u,,,jJ )-K,e, : j=1.2, K, >O. Thc main di c cncc is ha whcn c o s a c Ia gc cnough, hc con c gcncc cscmblcs ha o hc s abiliza ion p oblcm ( i ual obo is a cs ). Going u hc , we p oposc he c hc wo ncw acking inc hods 5 and 6. Mc hod 5 has hc samc kind o cla ion c = u(q, U) han PF, which cll us ha i will bcha c in a simila ashion o PF. Bu now we a c ce o “design” his cla ion, a oiding hc p oblcms o hc p ojcc ion di licul ics. Fo cxamplc a o m o c ha cminds us o PF can bc: u.u, n=- 2 (1 + K.WF a c an( ( ,x)) ” dcs whc c Kn F>O is a scale ac o ha indica es haw as he con c gcncc o o hc p ojcc ing poin x,&pb) will bc, and Nx, )=O is a sui ablc p ojcc ion. Thc p ojcc ing poin xde,( p+j is hc c hc poin o hc pa h ha ul ils n x, ,-)=O. In gcnc al o poin s xaeS( ) ha a c nca o xdes( pp), hc sign and magni udc o Ns, ) gi c us an idca o how a is xdes( ) om i (in 121 some cxamplcs can bc consul ed). Thcn hc sccond ac o o c will bc I when PF objcc i c has bccn achic cd ( cquals pF), and will bc lcss o biggc o hc wisc, ying o app oxima c I’ o pF~ Func ion a da has bccn choscn o simila casons o ha o EAT mc hod. Thc i s ac o has bccn choscn simila o hc dcpcndcncc ha can bc ound in hc c7 o PF. O cou sc hc con ol law has o dccidc hc cxac o m o U, bu we mus calizc ha i obo app oachcs sd,( }, U will bc pa allcl o uaS. In his casc pa ame e will mo c a hc pace ha con ol law dccidcs. Finally no c also ha whcn obo is s oppcd (inpu s U a c null), Y docs no p og csa. Likc in PF wi h his mc hod i is no p cdic ablc whcn hc obo will cach a pa h’s poin in hc gcnc al casc, and i may bc mo c s ablc han TT whcn c o s a c big. On ic o hc hand, p ojcc ion uniqucncss is no a p oblcm: i i wc c sa is icd o a wholc pa h, hc only conscqocncc is ha hc sccond ac o o U is 1. In addi ion, ano lw o m o c ha cminds us o EAT may bc: whc c KA+O is a scalc Fac o ha indica cs how as hc con c gcncc o o objcc i c poin xd s( ) is. The p c ious analysis is alid again (considc ing EAT ca u cs). As his mc hod can bc considc cd as a PF wi h adap a ion o c o s, wc can namc i “Adup ii~ Pa h Fdollowing” (APF). 375 Fo hc sainc dcsi cd pa h dcsc ibcd o PF (a s aigh linc) wc can usc Lc samc p ojec ion: n(x, )= - x,Ls -U. Thcn wc can usc bc samc con ol law o p c ious PF: imposc somc "mo ion CXigC~lCy"[4] o cnsii c ha eal and i ual obo s ad ance, and imposc hc con o law o inpu 2. Hc c again a ino c sophis i a cd mo ion cxigcncy likc U,'+ u2'=conSs an >0 would ha c hc addilional ad an agc o main aining inpu s wi hin ce ain alues. Mo co c hc c may bc o hc possibili ics o dcsigning con ol laws ha can bc considc cd in u u c wo ks Finally wc can dcsign a acking wi h hc cla ion: a =U (1, U, q). Following hc samc mc hodology p cscn cd o hc EAT wi h U= g( ,eJ, and o hc wo APF a cla ions, a pai o in c cs ing cascs a c gi cn by: Thc p c ious considc slions shown o EAT and APF will apply now combining bo h ac o s. CONCLUSIONS Wc ha c o inula cd di 'c cn acking mc hods o mcmo izcd pa hs acco ding o hc way in which we hpme 01- design hc p og ess o dcsc ip o pa amc c . Whilc hc so-called pa h ollowing and ajcc o y acking ha c bccn classical mc hods, wc ha c in oduccd o hc mc hods, cach ha ing sc c al ad an agcs: somc may consc c mos o hc ad an agcs o pa h ollowing, whilc a oiding i s p oblcins, o hc s a c alid o all possiblc ajcc o ics, somc o hc s may p csc c dc cm nis ic (non-s ic ) acking a hc samc mc ha acili a c ubus ncss (i s beha io undc la gc c o b o dclaycd csponsc is much bc c han ha o a ajcc o y acking), c c. Thc gcnc ic o muh ion p cscn cd hc c can opcn a lo o possibili ies u many applica ions. REFERENCES . B ockeii. R. W. "Asyinp oiic s abili y and eedback siabili a ion". Di e en ial Geome ic Coniml Theo y, Bi khause .pp. 181-208, 19R3. 2. Llluch, M., M. Reyhmoglu and N. H. McClain och. Con ol and s abiliza ion o nonholonomic dynamic syslcms. IEEF.T ilns,onAu . Conl ol,37(1ll,pp 1746-1757, 1992. 3. Lsmi ~ux. F.. S. Sckha al. J.P. Lauinond, Moiion planning and con ol o Hila e pulling a aile , IEEE T ans on Roboiics and Auiomiiun. Vol 15 No 4. Augus 1999, 4. Dim del Rio. F., G. Jimenez, J. L. Se illano. S. Vicenle. A. Ci il Balcells. A paih ollowing conl ol o unicycle obo s. Joumal o Robolic Sysleins 18(7), 329.342 (2001). 5. Dis del Rio. F., G. Jiinlne , J. L. Se illana, S. Vicenle. A. Ci il Balcells. A gene aliza ion o pa h oll wing o inobik obols. P oc. o lhc IEEE Inl. Con . on Robo ics and Aul. ICRA'99. Del oil. 1999. 6. Sa ka . N.. X. Yun. V. Ku na . Con ol o mechanical syslems wilh olling consi ainls: appliealion 10 dynamic conlml o mobile obo s. The hi. J. Rob. Resea ch. 13(1) 1994. 7. So dalen. O.J. and C. Canudas de Wii. Expc ncnliai eon~ ol aw o a mobile obo : ex ension lo pa h ollowing. ICEE T ans. on Roboiic And Aulom., V. 9. N 6, Dec. 1993. 8. Samson. C. Con ol a chain d sys ems: applicaiion o pa h ollowing and lime- a ying poin -s abiliwciun o mobile obols, IEEETmns. on Au oinSonl ol. 40 (19YS). pp. 61-77. 9. De Luco. A., G. O iolo, and C. Sainson. '"Feedback con ol o B nonh lonoinic ca -like obo ". Robo Mo ion Plannning and Coii~ ol, pages I71 -253. Sp inge -Vz lag. Be lin. I Y98. I0,Casudas de Wi . KhennouT, C. Samson. So dalen. Nonlinea con ol design o mobile obo s. Recenl ends in in bile obolu. Ed. Y.F. Zheng. Wo ld Scien i ic Se ies in Robolics and Aulom. Syalema. 1943. I I.Dia del Rio, F.. Jimenez, G., Se iliano, J.L., Ainaya, C.A., Ci il-Ba cells. A. E o Adapli e T acking o Mobile Robois. P oceedings o he 28 h Annual Con e ence. IEEE Indus ial Elec onics Sociely. Swilla (SPAIN). pp. 2415-2.121. 2002. ISBN 0-7803-7474-6 12.Diaz del Rio. F. Analysis an3 e alua ion o inobile obo conl ol: appliealion lo elecl ic wheelchai s (in Spanish). Ph. D. 'I'hesis. Uni e si y o Se ille (Spain), 1997. 13.Mae zdo do Canno, 1'. Di c enlial geoinc y o cu cs and su aces. P eniice-Hail. 1976. 14.lsido i. A.. Noiilin a Coniml Sys eins: An Inl oduclion. Lec u e Noles in Con ol and In o m. Science, 72. Sp inge -Ve lag. 1985 376