A generalization of path following for mobile robots

P oceedings o he 1999 IEEE In e na ional Con e ence on Robo ics & Au oma ion De oi , Michigan May 1999 A Gene aliza ion o Pa h Following o Mobile Robo s F. Diaz del Rio, G. Jimknez, J. L. Se illano, S. Vicen e, A. Ci i Balcells. Facul ad de In o mh ica. Uni . Se illa. T : (+34) 54552 779. E-mail: [email p o ec ed]. us. es Add ess: A da. Reina Me cedes s/n. 41012 Se illa. SPAIN Abs ac Se e al au ho s ha e p oposed some me hoh o applying pa h ollowing in specijic cases o mobile obo s ([3J [9], [Ill, e c.). When we hy o ex end he pa h ollowing app oach o he gene al p oblem se e al dsy7cul ies a ise. In his pape we p esen a gene alized echnique o apply pa h ollowing o a mobile obo wi h nonholonomic cons ain s. As an applica ion example, we expose he case o mobile obo s wi h a highe deg ee o maneu e abili y han he ypical ca -like obo s. In pa icula we conside a obo ha can u n a ound i sel making a ze o- adius u n; a case s ill no esol ed as a as we know. Finally we p opose a sui able con ol law o his example ha ensu es asymp o ical con e gence. 1 In oduc ion The mos ex ended sys ems in au oma ic con ol heo y a e se osys ems. He e we ack a mobile sys em a he ime i mo es; i.e. posi ion, eloci y o , in gene al, any magni ude in which we a e in e es ed, is he ins an aneous e e ence ha ou sys em mus ollow. In igu e ig. la we show his case o he s a e coo dina es q( ). he desi ed coo dina es qdLF( ), and he e o coo dina es de ined as e,,( )=q( )-qdes( ). In se osys ems his is he only possibili y we ha e, because he e e ence ajec o y is collec ed as we do he acking. On he o he hand, in mobile obo s i is usual ha he ajec o y is memo ized o p e iously gene a ed by a pa h gene a o module [ diazl]. Fo ou pu poses bo h cases a e he same, and he e m memo izedpa h o me ely pa h is used o bo h o hem. A e e ence o desi ed pa h o be ollowed is desc ibed by a single pa ame e , namely , and i can be exp essed as a ec o o s a e coo dina es qk( ). Fu he mo e, we mus emphasize he impo ance ha con e gence o a pa h acqui es in mobile obo s, as con e gence o a ixed poin qo, can no be achie ed h ough a smoo h eedback s abiliza ion con ol law (a di ec esul o B ocke ’s heo em [ 11). When we y o ack a memo ized pa h, he acking me hodology can be e y di e en , as we know a p io i he whole ajec o y. Thus we can ind se e al possibili ies o do he acking. O cou se he classical se osys em acking can be done jus by iden i ying he pa ame e ( ) associa ed o he pa h wi h ime, ha is ( )= . Ano he simila possibili y called ajec oly acking (U) is based in a mo e gene al assump ion han simple se osys ems (see ig. Ib): he pa ame e ( ) is a gene ic unc ion o ime. The e o e he e o coo dina es a e e,( )=q( )-qd-( ( )). Then we can go h ough he s o ed ajec o y wi h he mos app op ia e scale o ( ), o example =a . Using an asymp o ically s able con ol law (e.g. [SI), i is gua an eed ha he sys em will con e ge o he desi ed ajec o y in a de e minis ic ime (excep o he inhe en pe u ba ions ha i may su e ). ,*’ Ac uol desi ed ,’ mjec o y ,‘ Memo ized mjec oy, =a ; a=scale ,‘ -- Fig. la: T acking in a se osys em Fig. Ib: T ajec o y T acking Memo ized ob ained h ough some ela ion b Fig. IC: Pa h Following Al hough TT is s aigh o wa d, i is no he only me hod (no he mos sui able) o ollowing memo ized pa hs. Du ing he las yea s se e al al e na i es ha e been p oposed. The bes es ablished in li e a u e and mos sui able o many si ua ions in which ime is no a c i ical pa ame e ( his is he case o mos cases in indus ial mobile obo s) is pa h ollowing (PF). This is based in some ela ion be ween ac ual sys em’s s a e q( ) and he memo ized pa h. This ela ion will gi e us he desi ed poin qda( ) o memo ized pa h o be acked. The eal sys em should y o ollow his poin ins ead o he one gi en by he o he app oach (see ig. IC). The e o coo dina es a e also e,( )=q( )-qda( ). Using his app oach, i is no gua an eed ha he sys em will each a poin o he desi ed ajec o y in a de e minis ic ime. The i ues o PF can be unde s ood conside ing his example: i big pe u ba ions o ce he sys em o be a es , o TT he desi ed poin will mo e una oidably. This means ha e o s will g ow up o some alue ha may in oduce ins abili y. On he o he hand, i PF is used, he desi ed poin will be he same in spi e o hese pe u ba ions. This allows he sys em o o e come la ge pe u ba ions a oiding possible uns able s a es. Mo eo e , he ex ac ion o an asymp o ically s able law using PF is no mo e di icul han using TT, as we can see in he 0-7803-51 80-0-5/99 $10.00 0 1999 IEEE 7 men ioned li e a u e and also in he example o sec ion IV. Thus in e es in PF is g owing apidly. The e ha e been se e al ials o apply PF (we desc ibe hem below) in speci ic cases, bu when we y o gene alize PF se e al p oblems a ise. In he nex sec ions we will y o sol e hese p oblems, cla i ying hem wi h a comple e example. 2 De ini ions and Robo Model Le 's conside a gene al mobile obo as shown in ig. 2 and le q=(X, Y, 4' be i s s a e coo dina es, which ep esen he Ca esian posi ion (x, VE@ o a ce ain poin PO ( ypically he midpoin be ween he ea wheels) wi h espec o a ixed ex insic coo dina e sys em %(O, i, j) and he o ien a ion #E(- z l o he obo wi h espec o he X axis. We will choose u=( , U)'€@ as he pai o con ol a iables o ou sys em which ep esen he linea eloci y o poin PO and he angula eloci y o he obo (o he pai o a iables such as o ques o ol ages supplied o he mo o s, a e analogue o he acking s udy, as showed by [2]). Fo hese ec o a iables he s a e equa ion o he mobile obo a e he well-known equa ions ( ha a e non-linea in q and linea in U): q=B(q)u ; B= i4 sen4 0 11 ; U= (I); C ] To s udy he acking o a memo ized e e ence pa h qk( ) =(Xda( ), Yd,( ), 4,,,( ))' le us de ine ano he in insic coo dina e sys em 2?{qdes( ( )). , n] linked o he pa h. is he uni a y ec o angen o he plana pa h in he desi ed poin qks( ( )) and n he no mal o i '. Le (em e,,)€@ be he posi ion e o s o poin Po ela i e o hese axis and e&(-zn] he obo o ien a ion e o , so e,( )=(e, e,,, e$' will be ou ela i e e o ec o 2. Le udes=( des( ), wdes( ))' be he desi ed con ol s a e exp essed as a unc ion o he desc ip o pa ame e . A his poin i is impo an o de ine exac ly which pa hs qd,( )=(Xd,( ). Yk( ), q5d,( )) a e alid. We can no choose an a bi a y unc ion on and assign i o he h ee desi ed s a e coo dina es. These desi ed coo dina es mus ha e some p ope ies o gua an ee ha he acking is possible. Fi s he domain o mus be in ini e ( o example he posi i e eal line R') in o de o gua an ee ha he e e ence ajec o y does no end. This mus be done o ensu e he possibili y o con e gence, because, as i is well known o a mobile obo , B ocke 's heo em [ 11 p e en s eedback s abiliza ion3 o he obo o a ixed poin . Second, and o he same eason, he e e ence ajec o y can no con ain singula poin s whe e he inpu s a e null, i.e. udes=O. Finally, he pa h can be made by a ' The ec o s i and n exis s only when he linea eloci y hs a his poin o he i ual obo ha wen h ough he e e ence pa h was no ze o; i i we e null, can be chosen pa allel o he i ual obo o ien a ion, as he nonholonomic cons ain equi es he wo ec o s o be pa allel. * An analogous coo dina e sys em was used in [Kanay90]. The e he sys em was linked o he obo i sel . We mean smoo h con ol laws, so ha we elude non-smoo h laws o casons o con inui y on he con ol a iables. 8 obo like he s udied one, and his implies ha (X&( ), Yd,( ), mus espec he nonholonomic cons ain s o ou mobile obo . e' J d=0.25m. R=0.16m e*= 4- 4de, X W jig. 2: ex insic and in insic obo coo dina es. Using he abo e ela i e a iables and coo dina es linked o he pa h, and by simple calcula ions, he ollowing s a e equa ions can be easily ound [6]: 0 w,(~ 0 e, CO&,,) In i s mo e gene al o m i we de ine e o s in a na u al ashion, ha is e, = R(q,,)(q - qa,) , hen he ma ix o m o he abo e equa ion can be exp essed as: = Bdcs(eq)Ude ( )+B(e,)u (3) 3 The Gene al Me hod o Pa h Following P e ious s udies. Du ing he las decade he e has been a g ea esea ch e o o de elope a acking based in a PF. This has lead o se e al good app oaches ha ha e made emphasis in di e se aspec s o PF acco ding o he pa icula cha ac e is ics o he analyzed sys em o he e e ence pa hs o be acked. The mos impo an can be summa ized in he ollowing ca ego ies: 1. In [3] and [9] he desi ed poin in he pa h is ob ained h ough a no mal p ojec ion along he ec o ha we ha e called n. The e o e his p ojec ion chooses he poin o he e e ence pa h ha has null e, coo dina e (see ig. 2). They ha e o p ohibi pa hs con aining ci cles wi h small adius (we will call u ns wi h null adius and in ini e cu a u e "ze o- adius u ns") o ensu e ha he no mal p ojec ion exis s and is unique. A simila pa h ollowing was used in Na lab [12]. As Na lab is a ca -like obo , i canno make ze o- adius u ns, so hese pa hs we e no conside ed. In [ll] he p ojec ion poin chosen by he au ho s is he one ha minimizes he euclidian dis ance be ween he eal and he e e ence poin s PI (see ig. 2). Using poin PI hey a oid pa hs wi h cu a u e ending o in ini e. Bu his s a egy ails when he e e ence pa h is a u n a ound poin Po. In his case any ac ual con igu a ion (ha ing di e en o ien a ions) whose poin PI is on he desi ed posi ion o PI will ha e ze o dis ance. Tha is, he couple (XI, YI) does no 2. ep esen he whole s a e o a mobile obo , al hough hese coo dina es always change o e e y ajec o y. This example shows us ha he whole s a e o he obo mus be conside ed o cons uc a gene ic acking. I is impo an o men ion ha p e ious s udies ha e no gi en an exac de ini ion o cons uc ion o PF as a as we know; hey ha e speci ied a pa icula me hod, sui able o hei equi emen s, ha can be called PF. Con e sely, as ou goal in his pape is o cons uc a gene al app oach o PF, and conside ing he expe ience ex ac ed om p e ious wo ks, we mus ha e in mind he ollowing wo s a emen s. Fi s we mus conside he whole obo ’s s a e ( ep esen ed he e by (X, K 4) o simplici y, and usually called he obo pos u e). Second, we mus con empla e all he possible e e ence pa hs ha can be ollowed by he obo , including ze o- adius u ns. Gene al Da h ollowinP cha ac e is ics. As a i s s ep and in o de o ge a gene al cons uc ion o PF, we a e going o ex ac he gene ic cha ac e is ics o pa h ollowing. Acco ding o hese s udies and he in ui i e beha io men ioned a he In oduc ion, he main cha ac e is ics ha mus ule PF and ha di e en ia e i om TT, can be summa ized as ollows: 1. We only mus conside he global shape o he pa h o do he ollowing. The desi ed ajec o y e olu ion (go e ned by ) mus no play any ole in he ack as i does in TT. In opposi ion o TT (whe e he desi ed pos u e is exac ly de e mined by a igid law like = ( )), in PF we mus choose some ela ionship o de e mine he desi ed pos u e. We will call his ela ionship “p ojec ing unc ion” as i p ojec s he ac ual pos u e o he e e ence pa h. We will deno e i as I he obo s ops, he e e ence o goal poin mus also s ops, as he pa ame e does no g ow by i sel . The p og ess o mus no be independen (as in TTj bu dependen on he eal obo mo emen , ha is i equa ion mus be d i less. The exis ence o he igid law = ( ) in TT implies he e e ence e olu ion o be qdes( ( )), and consequen ly “pulling” o “d agging” he obo o each he e e ence. On he o he hand, in PF he e e ence pa h can no “pull” (o “d ag”) he obo : he obo mus mo e independen ly by some condi ion (o cou se, meanwhile a con ol law mus ensu e con e gence o he pa h). We mus impose a mo ion in he eal sys em o gua an ee i mo es o p og esses. We will call his condi ion “mo ion exigency” and we will deno e i as mo eX g(u)=O. A di ec esul o wha is explained be o e is ha he e is no ime exigency in he ollowing. This means ha we can no ensu e ha he obo will each a e e ence poin in a p edic able pe iod o ime. Pa h ollowinsJ cons uc ion. TT’s cons uc ion is elemen a y. I equi es only choosing he mos sui able ela ion = ([j, o a de e minis ic acking in he sys em. On he con a y PF cons uc ion is no so ba e because i implies a special ela ionship be ween he ac ual poin and he global pa h. Pa adoxically, due o he mo e labo ious cha ac e o PF, i pe mi s mo e lexibili y han TT. 2. ,*,O=O. 3. 4. 5. 9 [STEP 1: p ojec ion une io? . p o, (e,, )=o ........................................................... ..J 2 e o coo dina es , ............................................................. 1 I STEP 2: mo ion exigency I 1 ................ *:!::!!.?!!:o .................. .i , 1 2 e o;cm&ina es 1 ig. 3. Gene al pa h ollowing cons uc ion scheme. Some me hods o PF cons uc ion ha e been de eloped in he e e ences men ioned in his sec ion, bu hey only apply o speci ic cases. Con e sely, and based on he p e ious cha ac e is ics, we can s aigh o wa dly ind a PF cons uc ion based in wo gene ic s eps (see Jig. 3). In his scheme we begin wi h a mobile obo ha has h ee s a e coo dina es ( h ee e o coo dina es eq exp essed ela i e o he e e ence pa h) and wo deg ees o eedom U (DOF). Al hough we p esen he case o a mobile obo ; he case o a gene ic nonholonomic sys em can be easily deduced. Pi s s ep: “p ojec ing unc ion”. This is de i ed as a consequence o cha ac e is ics 1 and 2 o PF. Once a dis ance c i e ion is chosen, a “p ojec ing unc ion” ,,i0 =O ela es eal pos u e wi h global pa h. This gi es us a p ojec ing poin on he desi ed pa h: i is he desi ed pos u e qh( ( )) a his ins an o ime. A he same ime he p ojec ion is an holonomic cons ain be ween e o coo dina es; so i supposes he elimina ion o one e o coo dina e. As he p ojec ion mus be s a ed be ween e o coo dina es and he desi ed pa h, i depends on ac ual e o s eq and on he memo ized pa h shape (in gene al, pa ame e ); ha is p uj(eq, )=O. As we a e alking abou a geome ic p ojec ion, ec o U can no play a ole in his s ep, because he eal obo eloci y does no in luence on a geome ic p ojec ion. Consequen ly he p ojec ion in oduces he ollowing coo dina e ans o ma ion: Poin s {eq} ha obey &,(eq, )=O de ine a su ace ( wo- dimensional in ou case) whe e he obo is placed. Hence obo pos u e is now gi en by only wo e o coo dina es ep=(el, e2) ins ead o he h ee eq=(ex, e e@) on a TT. A classical example o p ojec ing unc ion is he no mal p ojec ion desc ibed in [3] and [9], equi alen o making e, null. Tha is, he i s e o coo dina e e, is elimina ed and he obo pos u e is exp essed by only wo: ep=(ep e@) (e o coo dina es a e called @, @ in hese e e ences). The wo- dimensional su ace is he e,, axis ex ended o all he possible obo o ien a ions. This simple me hod o elimina ing one o he e o coo dina es can no always be used, as we show in he example o he nex sec ion. I is impo an o ema k ha pa ame e is a his ime he hi d s a e coo dina e4, and we should men ion i o speci y he /o, 91, 929 931 /qdes( ), , eh e2/ ; ep=(el, ‘ In ‘IT gi es us no s a e because is de e mined only by ime h ough he unc ion = ( ). whole obo pos u e, now gi en by ( , el, e2). Bu is no an e o coo dina e, ha is, i does no play any ole in he con ol o in he s abiliza ion p oblem, ha is cen e ed only in making e,( )-+O, ega dless o . A his s ep we can say ha we ha e isola ed pa ame e om he pa h con e gence p oblem, and ha we ha e a sys em wi h wo deg ees o eedom and wo s a e a iables ( o ge ing ), whe e smoo h s abiliza ion is possible. This asse ion does no con adic he abo e men ioned impossibili y o eedback s abiliza ion o a ixed pos u e in nonholonomic sys ems, because we only s abilize wo coo dina es wi h his PF con e gence, neglec ing he hi d coo dina e . In o he wo ds, we can s abilize e,,( )+O bu no e,( )+O. The addi ion o a p ojec ing unc ion ,,i(e, )=O gi es us he way in which pa ame e a ies. Di e en ia ion o his unc ion lead us o5: Now s a e equa ion (3) can be subs i u ed, and using he “chain-law” = ‘ , we sol e o i and ha e inally: i L’ ‘ p’ B(e, , )u d e- Now we ha e sol ed he p oblem o inding a closed exp ession o he a ia ion o pa ame e in an elegan way. This is clea ly a PF, because a ia ion o does no depend implici ly on ime. In he equa ion (4) he e may be some ope a ion es ic ions; o example i denomina o is null, a ia ion o is unde ined. This case mus be analyzed o each applica ion and we will s udy i in dep h o he example o sec ion IV. The op imum p ojec ion depends on he mobile obo s uc u e and e en on he applica ion, bu we can s a e some gene al condi ions o a “good” p ojec ing unc ion o be cohe en : 1. A p ojec ing poin can always be ound o any sys em’s s a e q and o any alid e e ence pa h, ha is: V(X,Y,Q)) E @x(-z Uniqueness o he goin on he pa h q&( ) mus be ensu ed (a leas locally ). I he ac ual obo s a e is he same as a pos u e o he pa h: q( )=qdes( J, hen he p ojec ed mus be ,. Tha is p oy has a ze o o eq=O: pmy(O, )=O V . I would be desi able ha he analy ic equa ions could ha e a closed o m, o help he inding o a con ol law whose s abili y is analy ically demons able. Second s eD: “mo ion exieenc ”. Finally, as we desc ibed in PF cha ac e is ic 4, we need o imposed a “mo ion exigency” mo eX,g(u)=O o gua an ee ha he obo mo es. Al hough he o m o his unc ion depends on he applica ion, we mus ul ill he ollowing condi ions: 3 ,~ 31/ p (q-qdes( J, J =O 2. 3. 4. 1. 2. No solu ion a he o igin u=O so U is ne e null; I is desi able ha mo eXig(u)=O is an e en unc ion on i s componen s7. This ensu es ha he obo will app oach he pa h h ough he mos sui able inpu s (nega i e o posi i e), mindless o he di ec ion i mus ake on he pa h (inc easing o dec easing ). This is pa icula ly impo an when e o s a e big. To help he con ol law o con e ge o he pa h, i would be desi able ha co n onen s o U beha e symme ically, ha is he o al mo ion [U I should spli iden ically be ween and w. In he cu en mobile obo li e a u e mos mo ion exigencies (no called wi h his e m) a e applied o ca -like obo s, so i is usual o ha e =c e, which is in ui i e o ca s. Fo obo s wi h highe maneu e abili y o he s au ho s ha e p e e ed he exigency gi en by I F,,,, I =de, ha is I w I I I =C ee, o a oid slippage. As he las one has an inde e mina ion o null w o , and ou (2,O) obo does no ha e mo ion es ic ions, we will use he adequa e “mo ion exigency“ gi en by: 3. (5) i=l whe e Km is he whole mo ion applied o he sys em and bi is he scale ac o o each inpu . 4 An Example Applica ion: The Case O A (2,O) Mobile Robo . One o he mos ex ended mobile obo con igu a ions is ha wi h deg ee o maneu e abili y 2 and 0 s ee ing wheels (a (2,0)- obo acco ding o he de ini ion o [2]). The ypical opology o hese (2,O)- obo include wo d i e mo o s a each ea wheel, ha can u n independen ly o wa d o backwa d. Fu he mo e i is one o he obo s in which ajec o ies a e mo e complica ed, as i can no ha e comple e maneu e abili y ( ha is, i is no omnidi ec ional) bu i can make ze o- adius u ns. So i is a e y in e es ing p oblem o apply ou pa h ollowing cons uc ion o hese obo s. As ou g oup has been in e es ed du ing he las yea s in he imp o emen o elec ical wheelchai s ([4][5][7]) ha inco po a e his con igu a ion, we ha e s udied he complica ions ha his opology in oduces. Fi s s eD: “woiec ing unc ion”. Maneu e abili y in hese obo s is e y high, and hey ha e no addi ional mo emen es ic ions (excep o he inhe en nonholonomic cons ain ). Thus we can choose he p ojec ing poin as ha in he pa h ha is nea es o he obo , i.e. he one which dis ance is minimal. As he h ee e o coo dina es mus play a ole in he dis ance, a “good” elec ion o he dis ance dq can be: i=l whe e Ki a e he scale ac o s be ween he di e en e o s o gua an ee dimensional homogenei y. In he case o ou obo his leads o: ’ He e de i a ion espec o a ec o holds o a summa ion. since i can each he same pos u e as many imes as i wishes. Global uniqueness is impossible o a mobile obo ha is ully con ollable, ’ We a e assuming he e ha he obo mo o s ha e no a special s uc u e o he applica ion does no need o ma ch in a unique di ec ion. A ca would be o example his case, because i s ea di ec ion is limi ed. 10 d:( , ) = e: +e; + Kie To choose he minimal dis ance poin we “ eeze” ac ual obo pos u e ( ha is u=O) and “mo e” along he desi ed pa h ( ha is we a y ) looking o he poin wi h a local minimum: lu=O whe e we omi simple calcula ions (using (2)) and use he “chain-law” j- = ’i . He e (‘) holds o di e en ia ion espec o and (‘) espec o 1. So. (7) is ou ,,(e,( ), )=O, whe e des( ), wdes( ) a e he inpu con ol p o iles o desi ed pa h. This p ojec ion is ully in ui i e, because i always chooses he nea es poin o he ac ual obo pos u e. As we desc ibed in sec ion 111, di e en ia ion o his p ojec ing unc ion gi es us he new equa ion o he a ia ion o pa ame e : = w&s( >cose+i + Kiwqgs( ) (8) igs( ) -w&j< ) dgs< )ey + Ki&s( ) - Kie#wies( ) - ex ies( > As his p ojec ion has been ob ained h ough a gene ic PF cons uc ion, we can ge some o he classical p ojec ions as pa icula cases. Fo example he no mal p ojec ion used in [3][9], is ound by doing K,=O (you should emind ha no mal p ojec ion o a plane cu e coincides wi h minimal Euclidean dis ance, gi en by he dis ance dq when K,=O [lo]). In e ec he nulli y o K, and he use o he same pa ame e o hese au ho s (cha ac e ized by des( )=I), leads o: Once we ha e chosen his p ojec ion we should con i m i i e i ies he condi ions o he abo e sec ion o be conside ed a “good” p ojec ing unc ion. All o hese condi ions excep numbe 2 a e s aigh o wa dly sa is ied. Uniqueness is equi alen in ou case o he non-nulli y o denomina o o equa ion (4) [6]. A deep s udy is made in [6] and i s inal esul is ha local uniqueness is eached unde ce ain non-se e e condi ion?. These condi ions a e wo bounds o K, and cu a u e de i a i e o he desi ed pa h ~;~~( ), ha can be easily sa is ied i e o s a e no unbea able and desi ed pa hs a e no ab up . In opposi ion o hese non-se e e condi ions, no mal p ojec ion mus a oid pa hs con aining ci cles wi h small adius o ensu e ha i exis s and is unique, ha is, i is a mo e es ic i e o he easible e e ence pa hs. Second s eD: “mo ion exieenc ”. In he case o he (2, 0)- obo , we ha e only wo deg ees o eedom; so a e y sui able mo ion exigency is he one men ioned be o e: (9) whe e K,,,,, is he whole eloci y applied o he sys em and b, is he scale o he angula speed, ha gi e us how much he sys em can u n. In conclusion we ha e educed he sys em jus o wo s a e a iables (no explici ly de ined bu ob ained h ough he applica ion o a “p ojec ing” cons ain gi en by equa ion (7) 2( ) + bid( ) = K:,, > 0 o he h ee e o s e,,), and one deg ee o eedom ( esul ing om he use o (9) o ec o inpu U). In hese a iables we ha e condensed wha we need o con e ge o a gene ic pa h h ough a pa h ollowing. Con ol law. Al hough con ol law selec ion is a om ou objec i es, i is con enien o show he beha io o ou sys em and he way o ge o an asymp o ically s able con ol law. We will use Lyapuno ’s second me hod me hod wi h he quad a ic e o unc ion as he Lyapuno unc ion (which ma ches wi h he semidis ance): (10) Di e en ia ing wi h espec o ime and using s a e equa ions (2) we ha e: V = (ex cos(e,) + ey sen(e,))+ Kie+,w Now we impose (as ou con ol law) his de i a i e V o be nega i e semi-de ini e9 o ensu e con e gence, ha ing: (1 1) = (ex cos(e,) + ey sen(e+,)).I + (K,e, )Kp = = -k;(ex cos(e,) + ey sen(e,)p + :Kiei] Equa ion (1 1) ep esen s a line in he plane o he no malized con ol inpu s ( , Kp). I we choose, o con enience, b,=K, in he mo ion exigency (9), his equa ion will ep esen a ci cle. The in e sec ion o ci cle (9) and line (11) will gi e us he eques ed alues o and w (i i does no exis , ci cle’s nea es poin o he line is chosen). Asymp o ic con e gence o he p oposed law is demons a ed in [6] (i can be ob ained, as usual o hese kind o laws in mobile obo s, h ough Ba bala ’s lemma [9]). The in e sec ion o ci cle and line ( ha is, he con ol law) educes o a i s o de di e en ial equa ion when e,+O o ex4. In hese cases, he pa ame e s Z,, Z, play he ole o ime cons an . Pa h ollowine e alua ion. E en when asymp o ic con e gence is ensu ed, simula ion is always a good way o e i y and obse e he con ol beha io . The p oposed e e ence pa hs whe e a smoo h PF con ol law mus be e alua ed ha e o be alid (see sec ion 2). Hence a ca -like obo o ou (2,O) obo can no go h ough a piecewise pa h including cu a u e discon inui ies (e.g. a s aigh line plus a ci cle); ne e heless he pieces should be linked by he pa h planning o ensu e cu a u e con inui y a leas ( o example h ough he addi ion o clo hoids o simila cu es). As we ha e shown analy ically he gene ali y o ou p ojec ion unc ion, hen e e y pa h complying wi h he cu a u e con inui y is iden ical o e alua ing ou PF. We ha e selec ed wo e y in e es ing examples: 1) app oaching o a s aigh line, whe e ypical con e gence is showed, 2) con e ging o a ze o- adius u n, whe e ou PF shows i s gene ali y. The alues o cons an pa ame e s ha e been chosen o ensu e a smoo h con e gence, as 5=0.5s, 2,=0.5s. The whole mo ion is K,,,,,=5Ocm/s, K,,,,,=0.5 ds. The cons an s bo=K, a e 0.25m. 1 1 2 2 . V = - di ( , ) = -(ex2 + e: + Kie:) No e ha Vcan ne e be nega i e de ini e because he exis ence o he * Mainly he non-se e e condi ion is due o nonholomic na u e o he sys em. nonholonomic cons ain , see [Diazgla] o a demons a ion. 11 In he i s case we ha e selec ed big ini ial e o s o p o e he good con e gence o he me hod in p esence o ex eme condi ions (see ig. 4) and o compa e i wi h TT. In PF he e o e, mus always be ze o acco ding o he p ojec ing unc ion (7) and he solid line o he e o ey gi es us he eal obo ajec o y. We ha e used o TT he con ol law o [8] uning i s cons an s so i beha es in a simila ashion o s nall e o s (KX=20s-’; Ky=6.Ocm-‘; KB=3.2cm-’) (i s eal ajec o y is he dashed line). No e ha in PF he obo begins he acking in e e se di ec ion in he i s ansien , in o de o educe he e o s as e , and pa ame e dec eases oo. In he end his will imply ha PF me hod will each less dis ance in he desi ed pa h. This case will ne e happen in a pu e TT, whe e = , and he e e ence obo s “pulls” he eal one and i will ad ance he same as he e e ence ajec o y. This is a well-known ad an age o PF equen ly commen ed in he li e a u e [11][3], ha educes oscilla ions in he end. Mo eo e inpu commands in PF a e limi ed by ou mo ion exigency while in TT hey a e no . I hey we e in TT, i s esponse would be e en poo e . I Ini ial .... ig. 4: PF o a line (X axis) unde big ini ial e o s. -*..... -. ”..... e-. * qc ual 1.5 W ,(TT) -0.1 --0.1 ig. 5: PF o a ze o- adius u n(e,(O) =-0.03m, e,(O) =-0. Im). In he second example he desi ed ajec o y is he ‘ e ical axis, because e e ence obo u ns a ound i s poin Po (see ig. 5). The eal obo mo emen would be gi en by he p ojec ion on plane AY. As in he p e ious case, al hough ini ial e o s we e big, PF chooses he nea es poin on he desi ed ajec o y, i.e. ha wi h ze o e) (acco ding o p ojec ing unc ion (7)). No e ha in PF inpu con ols , w a e spli in he mos con enien o m o ge a as con e gence. In TT con e gence is slowe because cons an s we e uned o he acking o a line. I cons an s we e uned o his las case hen con e gence o a line would be slowe . 5 Conclusions. We p esen a echnique o cons uc pa h ollowing in mobile obo s ( ha has been shown by se e al s udies o be mo e ad an ageous han ajec o y acking). I consis s o wo s eps: choosing a “p ojec ing unc ion” o ela e ac ual pos u e o desi ed pa h as a unc ion o e o s and in insic desc ip o pa ame e , and imposing a “mo ion exigency” o ensu e obo ad ances on he pa h. These s eps ha e been ob ained based on he gene al pa h ollowing cha ac e is ics ha we ha e p e iously ex ac ed. We also s a ed he condi ions ha bo h s eps mus sa is y o be cohe en . The ea e we co obo a e ou p oposi ion wi h a (2,0)- obo (acco ding o he de ini ion o [2]) ha can make ze o- adius u ns, a case ha has ne e been sol ed, as a as we know, using pa h ollowing. Finally we p esen simula ion esul s unde big ini ial e o s o exhibi he good and as con e gence o ou pa h ollowing app oach. Acknowledgemen s This wo k has been suppo ed by IASS ag eemen “Segundo Con enio de Colab. IASS-Uni . de Se illa” and by CICYT p ojec TER96-2056-C02-01. Au ho s will also hank P o . C. Samson (INRIA, F ance) o his kind mailing o esea ch epo s and he e iewe s o hei wo hwhile commen s. Re e ences. R.W. B ocke . “Asymp o ic S abili y and Feedback S abiliza ion”. in Di e en ial Geom. Con . Theo y, Bi khause , pp. 181-208,1983. G. Campion, G. 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