TIME-OPTIMAL PATHS FOR LATERAL NAVIGATION OF AN AUTONOMOUS UNDERACTUATED AIRSHIP

Transcription

1 IME-OPIMAL PAHS FOR LAERAL NAVIGAION OF AN AUONOMOUS UNDERACUAED AIRSHIP Slim Him nd Ysmin Bsoui Looi Sysèms Comls, CNRS-FRE 9, Unisié d Ey Vl d Essonn 8, Ru du Plou, 9 Ey, Fnc {him, soui}@iu.uni-y.f ABSRAC his dls wih chciion of h shos hs fo ll nigion of n uonomous undcud ishi king ino ccoun is dynmics nd cuo limiions. h iniil nd minl osiions gin. W would lik o scify h conol focs h s h unmnnd il hicl o h gin minl osiion quiing h miniml im fo ll nigion. h licion of Ponygin s Miml Pincil, llows us o find fmily of imoiml hs. Bsd on h symmy of ishi dynmics, i.. wih sc o oion nd nslion, i is ossil o consuc glol jcois conncing wo configuions y succssion of fini num of hs im-oiml hs using gomic soning. INRODUCION Unmnnd il hicls nw focus of sch, cus of hi imon licion onil. hy cn diidd ino h diffn ys : ducd scl fid wing hicls (ilns), oy wing icf (hlico) o ligh hn i hicls (ishis). Ligh hn i hicls sui wid ng of licions, nging fom dising, il hooghy nd suy wok sks. hy sf, cosffci, dul, nionmnlly nign nd siml o o. Aishis off h dng of qui ho wih nois lls much low hn hlicos. Unmnnd moly-od ishis h ldy od hmsls s cm nd V lfoms, suillnc nd fo scilid scinific sks such s h monioing nd nionmnl conol. An cul nd is owd uonomous ishis. Wh mks hicl ligh hn i is h fc h i uss lifing gs (i.. hlium o ho i) in od o ligh hn h suounding i. h incil of Achimds lis in h i s wll s und w. Aishis owd nd h som mns of conolling hi dicion. Non igid ishis h mos common fom nowdys. hy siclly lg gs lloons. h mos common fom of diigil is n llisoid. I is highly odynmiclly ofil wih good sisnc o osics ssus. Is sh is minind y is innl ossu. h only solid s h gondol, h s of oll ( i of oll mound h gondol) nd h il fins. h nlo holds h hlium h mks h lim ligh hn i. In ddiion o h lif oidd y hlium, ishis di odynmic lif fom h sh of h nlo s i mos hough h i. h ojci of his is o gn dsid fligh jcoy o followd y h ishi. h jcoy gnion modul gns nominl s jcoy nd nominl conol inu. A mission ss wih k-off fom h lfom wh h ms h holds h mooing dic of h ishi is mound. yiclly, fligh oion mods cn dfind s: k-off, cuis, lnding nd ho. Af h us hs dfind h gol sks, h h gno hn dmins h fo h hicl h is jcoy in sc. In Aonuics, ln fligh conol ofn inols ll nd longiudinl s dcouling. h olm of jcoy gnion fo ll conol is fomuld s n oimiion olm. his moion gnion ks ino ccoun h consins on lociy nd h ound on h udd ngl. h minimum im olm is sold using h mimum incil of Ponygin. Onc his fnc jcoy dmind, h ishi cn follow i wih n oi fdck. h ligh hn i lfom of h 'Looi ds Sysèms Comls' is h AS y Aisd Aishis. I is moly ilod ishi dsignd fo mo snsing. I is non igid 6m long,.m dim nd 8.6m olum ishi quid wih wo col ngins on h sids of h gondol nd conol sufcs h sn. h fou silis nlly cd on h full nd udd momn is oidd y dic linkg o h sos. Enlo ssu is minind y i fd fom h olls ino h wo llons locd insid h cnl oion of h hull. hs llons slf guling nd cn fd fom ih ngin. h ngins sndd modl icf y unis.

2 Figu LSC ishi lfom AS dscid li o h inil fnc fm whil h lin nd ngul lociis of h hicl should ssd in h ody-fid coodin sysm. his fomulion hs n fis usd fo undw hicls. In his, h oigin C of coincids wih h R m cn of olum of h hicl. Is s h incil s of symmy whn ill. hy mus fom igh hndd ohogonl nomd fm. h is of h onuic fm follows h dicion of h ishi li lociy V wih sc o h wind. α is h ngl of ck wihin h m m ln, nd β h skid ngl wihin h m y m ln. o dsci h osiion nd h oinion of h ishi AIRSHIP DYNAMIC MODELING Kinmic modling A gnl sil dislcmn of igid ody consiss of fini oion ou sil is nd fini nslion long som co. h oionl nd nslionl s in gnl nd no ld o ch oh. I is ofn sis o dsci sil dislcmn s cominion of oion nd nslion moions, wh h wo s no ld. How, h comind ffc of h wo il nsfomions (i.. oion, nslion ou hi sci s) cn ssd s n quiln uniqu scw dislcmn, wh h oionl nd nslionl s in fc coincid. h conc of scw hus sns n idl mhmicl ool o nly sil nsfomion. h fini oion of igid ody dos no oy o h lws of co ddiion (in icul commuiiy) nd s sul h ngul lociy of h ody cnno ingd o gi h iud of h ody. h mny wys o dsci fini oions. Dicion cosins, Rodigu - Hmilon's (qunions) ils, Eul ms, Eul ngls, cn s s mls. Som of hs gous of ils y clos o ch oh in hi nu. h usul miniml snion of oinion is gin y s of h Eul ngls, ssmld wih h h osiion coodins llow h dsciion of h siuion of igid ody. A dicion cosin mi (of Eul oions) is usd o dsci h oinion of h ody (chid y succssi oions) wih sc o som fid fm fnc. h fnc fms considd, figu, in h diion of h kinmics nd dynmics quions of moion. hs h Eh fid fm considd s Gliln, nd wo locl fms chd o ishi, h ody fid fm R nd onuic fm R. h m osiion nd oinion of h hicl should R f Figu Gnl configuion of fms wih sc o h inil fnc fm, h Eulin miion is usd. h h oinion ngls : h Roll φ, h Pich θ nd h Yw ψ. h cun configuion is hn dducd fom h lmny oions. h osiion η nd h oinion η of h hicl in cn scily dscid y: R f ( y ) nd η ( φ θ ψ η ) hn h oinion mi fm R nd fnc R is gin y m f H R f () wn h ody fid, :

3 CψCθ SψCφ + CψSθSφ H SψCθ CψCφ + SψSθSφ Sθ SφCθ () SψSφ + CψSθCφ CψSφ + SψSθCφ CφCθ nd h nsfom mi wn h ody fid H fm Rm nd h onuic fm R cn win s: CαCβ CαSβ Sα H Sβ Cβ SαCβ SαSβ Cα () () () wh C cos nd S sin H nd H SO() dnos h oion mi h scifis h oinion of h ishi fm li o h inil fnc fm in inil fnc fm coodins. SO() is h scil ohogonl gou of od which is snd y h s of ll ohogonl oion mics h chcisics : RR I nd d( R) I sns h idniy mi. π π his dsciion is lid in h gion θ. A singuliy of his nsfomion iss fo: π θ ± k π, k Ζ L's now inoduc V ( u w) s h lin lociy of h oigin C ssd in R nd ( ) Ω q s h ngul lociy ssd in h fm. h kinmics of h ishi cn ssd in h following wy: wh η R η η ( η ) V ( η ) Ω m () Sφ nθ Cφ Cφ Sφ () Sφ Cφ Cθ Cθ Dynmic modling In his scion, nlyic ssions fo h focs nd momns cing on h ishi did. I is dngous o fomul h quions of moion in ody fid fm o k dng of h hicl's gomicl ois. Alying Nwon's lws of moion ling h lid focs nd momns o h suling nslionl nd oionl cclions ssmls h quions of moion fo h 6 dgs of fdom. h focs nd momns fd o sysm of ody-fid s, cnd h ishi cn of olum. W will mk in h squl som simlifying ssumions: h h fid fnc fm is inil, h giionl fild is consn, h ishi is suosd o wll infld, h olsic ffcs ignod, h dnsiy of i is suosd o unifom, nd h influnc of gus is considd s coninuous disunc, ignoing is sochsic chc. h dfomions considd o ngligil. Glol dynmics quion cn win s : M η τ + τ (η) d wh M d nd τ m (η ) scily h ini mi nd h dynmicl (Coiolis nd cnifugl) nsos which du o h mss of ishi, nd τ is h sum of h diffn nl nso, which inol: Glol odynmic nso du o h ddd mss hnomnon lus focs nd momns gnd y h ishi ody (hull, fins nd gondol). Aosic nso dsciing h focs nd momns du o h giy nd uoyncy Poulsion nso du o h cod hus. m (6) Ll dynmics of h ishi h ishi modl consiss of ss, comlicing h conol dsign. In onuics, y nul simlificion consiss of dcomosing h fligh mods ino: k off, cuis nd lnding. hs sks cn diidd ino wo min mods: longiudinl mod nd ll on. In his w focus on ll mod in consn liud. h cod huss nd los ssocid o h longiudinl nigion conoll o hold h liud nd li lociy nigion consn. h udds llow h ishi o nig in h hoionl ln. h mhmicl modl of h immd ll dynmics in h locl fm is gin y 6 : my m + Y + Y + Y + m Y m W m U + Y + Y + + Y ( F G F B ) φ cos( θ ) (7)

4 + N m U + N + m G + N + F φ cos G + m U + L + F φ cos m ( θ ) L N mw + + N ( θ ) L + L + + N mw L (8) (9) hs quions cosond o h Ll, Yw nd oll dynmics. U nd h il nd ll lociy comonns in locl fm. wh Y cofficins. m i, Y, N U, N V V, L cos sin ( β ) ( β ) nd L () h odynmic is h n mss in h i h dicion. i ini mi lmns. m is h ishi mss. θ is h quiliium ich ngl. In gnl, h ishi mos wih low sd. h quiliium wn h cnifugl momn ound is cusd y h udd dflcion nd giionl momn is h cus of n insignificnly smll Roll ngl nd which cn omid. king hs considions ino ccoun, h modl cn simlifid s: m y + m + m N Y mu + Y + Y + N + Y mu + + N + N nd h kinmic quions gin y: ψ V cos y V sin ( ψ + β ) ( ψ + β ) m () () () Rciuling, h modl of h ll dynmics of h ishi cn win s: Figu Ll configuion of ishi (o iw) i fm h coodins of h cn of mss in h locl Rm 6. β is h skid ngl wn h li lociy V nd m is ino m ym ln. In h snc of wind, his ngl s whn h ishi follows h wih non o cuu. Fo fid udd dflcion, i.. cosonding o h cicl h, his ngl ks consn lu whn n quiliium wn odynmic momn, cusd y h ishi ody (hull, h icl fins nd conol sufcs) moion wih sc o h suounding i nd h cnifugl on is slishd. his ngl ks on smll lus. wh: m y β β + + ψ y V V β + + cos sin ( ψ + β ) ( ψ + β ) Y + m + Y N m Y m N Y mu + m + Y N mu my m Y m N () () (6) (7) (8)

5 m N Y + my N my m Y m N m N Y mu + myn mu my m Y m N Y Y m N m y m Y m N m N Y + my N my m Y m N h ll dynmics of h ishi h n ffin sucu. In h comc fom h dynmics cn gin y: X f ( X ) + g (9) wh : β + β + f ( X ) nd g V cos( β + ψ ) V sin( β + ψ ) Som difficulis is wih his modl: h fis on is h undcuion of his sysm, i.. ss sd y singl inu conol. h scond on is h nonholonomic chc: non ingl lionshi wn lociis: sin( β + ψ ) + y cos( β + ψ ) () his kind of consins clld Pfffin Consins 8. im Oiml Emls In his gh, l s inoduc fnc imoiml hs fo h sysm und sudy, king ino ccoun h sysm dynmics nd cuo ciliis. Hnc, his olm cn fomuld s follows: min f d f () X f ( X ) + g( X ) u X ( u min ) X nd X ( f ) u u m X f () h olm is o find h dmissil conol u h minimi h im fo which h sysm chs h finl s fom h iniil on X. Wihou loss of X f gnliy, nd y siml nomliion nd shifing (if h wo ounds of h conol domin no symmic), w cn consin h conol o long o uni inl, i.. u. o sol his olm, w ly h Ponygin s Mimum Pincil (PMP) o oin ncssy condiions fo fnc jcoy of sysm o im-oiml. h PMP ss h: if X () is imoiml jcoy dfind on [, ], nd u() is h cosonding im-oiml fnc conol, hn h iss n soluly coninuous co funcion clld h djoin co, :[, ] R, such h h following condiions sisfid,7,9 :. ( ) fo ll [, ]. H ( X ( ), ( ), u( )) min H ( X ( ), ( ), ( )), wih > fo y fid X ( ), ( ) nd, such h H ( X ( ), ( ), u( )) + ( ), X ( ), wh, is h co sc inn oduc.. h djoin co () sisfis h following H quion: X A il ( X,, u) ifying h ncssy condiions is clld n ml. Fis, consid h Hmilonin H, funcionl fo h oiml conol olm wh mulilis h djoin h consins. h Hmilonin funcion of h sysm is gin y: H( X( ), ( ), ( )) + f ( X) + g( X) + + V + ( β + + ) ( β + + ) + sin( ψ + β) + V nd h co-s dynmics gin y: cos( ψ + β) () Sujc o

6 V V cos( ψ + β) + V sin( ψ + β) V sin( ψ + β) cos( ψ + β) () hfo, s nd consn on [, ] h iss µ nd γ [,π ] such h, [, ] hn : V µ sin( ψ + β γ ) µ cos( γ ) µ sin( γ ) + V µ sin( ψ + β γ ) nd h Hmilonin coms: H( X + + µ V,, ) + ( β + ) ( β + ) + cos( ψ + β γ ) + ( + ) () (6) h minimiion of h Hmilonin wih sc o h conol is oind y minimiing g( X ). h conol longs o uni conol domin, hn his minimiion cn chid y king fo h oosi sign of g(x ), hn: if g < if g > (7) h funcion φ ( ) g( X ), dfind long n ml ( X,, ) is clld h swiching funcion ssocid o h sysm. Clly, h os of his funcion imon fo h sudy of oiml synhsis. If h iss nonmy inl such h φ () is idniclly o, h ml is singul on h inl. Assum now h ml o ng, i.. ks is lus in {, } fo lmos im s such h is no lmos ywh consn on ny inl of h fom ε, + ε, ε >. is clld swiching im fo ] [ s s s nd cosonding s is clld swiching s. h jcois cosonding o h conols ±, figu cicls. h fss wy o un is uning wih h smlls dius, mns h mking h udds dflcion in on of is limis. i.. o un lf o igh. figu nd 6 shows, β nd cus whn h ishi ss fom sigh lin o cicl. hy k consn lus whn h cicl mnn h is ind. β() () Y X Figu Cicl h fo 6 Figu skid ngl β sons fo Figu6 Yw sons fo Singul mls As mniond o, h singul conols chcid y h fc h φ () is idniclly o in nonmy inl. How, h PMP loss is disciminion nu, i.. y conols in U sisfy h ncssy condiions. In his cs, w nd som ddiionl condiions. h nulliy of φ () in nonmy 6

7 inl imlis h ll is im diis null in h inl, i.. φ( ) φ( ) L φ m ( ) h ocss of diion is sod whn h conol in h ssion of hs diion. Fo n ffin sysm, k φ ( ) ( X, ) + ( X, ) (8) k is clld h od of singul conol. Hnc, h singul conol cn ssd s: ( X, ) (9) ( X, ) Poosiion: h singul conols of ou sysm of h fis od nd n noml. Poof: L s di h swiching funcion φ () : + oosiion: h sysm of diffnil quions soluion. Poof: Fom h condiion w find, imlying ( + β γ ) (), (6) (6), hs s sin ψ (7) ψ + β γ kπ (8) wih k Ζ his sul sns h ncssy condiion fo h isnc of h singul conol. Fom h quion (8), w cn s h, h conol cofficin φ( ) g + φ( ) g ( f f g q + + g ) g () () q g is n nishing. So h Hmilonin coms: fom his, ( V cos( ψ + β γ ) H X,, ) + µ V cos( ψ + β γ ) (9) () µ ± () V cos( ψ + β γ ) V nd φ( ) q + g q + ( f f ) q () wh dnos h jcoin mi of h co V. No h V g. Nullifying hs quions, w find: () () lcing in h fis nd h scond quions of h sysm of quions (), w find: + () w cn conclud fom quion ( ) h: µ is n qul o o, cus h o lu of µ imlis null djoin co, his condics PMP smns. o o h minimliy of h singul conols, w mus s h gnlid coniy condiion ofn clld, snghnd Lgnd -Clsh condiion gin y, : d u d k φ k ( ) k k is h od of singul conol. Fo ou sysm: d φ d ( ) µ V cos( ψ + β γ ) fom h quion (): µ V cos( ψ + β γ ) which is u, i.. is osii y dfiniion. () () () 7

8 Hnc, h singul conol, nd fom quion () cn gin y: ( q f f q) ( + ) ( β + ( + ) ) q + g n( ψ + β γ ) () fom h isnc condiion of h singul conol, i.. ψ + β γ kπ, h conol cn ducd o: such h < ( β + ( + ) ) (6) Figu7 singul conol Discussion of h singul conol Onc h singul conol is dmind, w illus h gomic sh of h fnc jcoy of h ishi und his singul conol. Fom h singul conol ncssy condiion, ψ + β γ is consn, nd γ is consn oo, hn ψ + β is consn. his imlis h h ngl wn h li lociy V nd h fnc is is consn. hus h singul jcoy is sigh lin. W cn find h sm sul y lying h conol dfind low in h dynmics of β : β + β + ψ (7) imlying non iion of β + ψ ngl, h fnc jcoy is hus lin. nsiion fom cud h o h lin quis h licion of h singul conol which gows monoonously, figu7. his losion of conol is du o h non ccnc of h disconinuiis in h cuu y h ishi dynmics. Fo his son w wn o smooh his cuu y oimiing h nsiion im wn non o cuu jcoy nd sigh lin. Whn w y o connc wo lignd configuion h oiml h is oiously sigh lin, i.. β, which cosond o (6)..,his conol is mddd in β() () Figu 8 skid ngl sons o 8 - Figu 9 Yw sons o 8

9 wh is mi fomd y h igncos cosonding o h sysm ignlus. h suln digonlid dynmics gin y: Figu ishi configuion on cicl lin nsiion. Oiml nsiion of non null cuu h o sigh lin W now consid h issus ining o h swiching wn h non o cuu h nd sigh lin. W h ldy chcid h foms of h oiml s nd conol jcois in ch mod sly. hfo, w nd o fuh scify h im inl cosonding o h mod swiching nd h ms h dmin whn nd fo how long h singul conol lss. W ocd o ddss oh issus using coninuiy gumn.w look fo h fss wy fo h ishi o mo fom non null cuu hs o sigh lin in oiml im. h lin is chcid y o lus of β nd, nd h non null cuu hs chcid y non o lus of β nd. L s us h dynmics of β nd fo chiing his ojci. In oiml conol liu, h following hom is dmonsd. hom: Fo ny lin noml sysm h oiml conol is of ng-ng y. h nomliy condiion mns h h sysm is conolll wih sc o ch of is conol inus. h dminn of h conolliliy mi is: + ) ( (8) h sysm und considion is noml. L s find h swiching sufc llowing h sysm o insc h oigin, sing fom ny iniil condiion wihin his sufc, nd und scific conol, i.. ±. h dynmics of h Yw nd h skid ngl symoiclly sl. o simlify h clculus, sion of dynmics is fomd y s mi digonliion, y mns of lin s sc il nsfomion. L + + () h cosonding soluions of his sysm gin y: ( ) ( ) + + ( + ) ( + ) () wh nd h iniil condiions. o find h swiching sufc w look fo h iniil condiions fom which h sysm dynmic jcoy coss h oigin und h conol ±. h g oin is ( c ) ( ) c c. How, h coss im wih h oigin is sily clculd fom h sysm (): nd ln + c () ln + c () quliing ims nd soling h suling quion fo w find: + + his quion dfins h swiching sufc in nd () coodins cosonding o ±, figu nd, in figu, w show h whol swiching sufc. β (9) 9

10 Figu swiching sufc fo h condiion in quion () nd swiching whn h < ε is hold. In his cs h conol swich o h oh ng. h sm lgoihm is lid o dc h inscion wih h oigin, whn is h cs, h conol is hold o o. Figu show h suling im-oiml nsiion conol fom cicl o sigh lin. Figu 6 illus comison of β sons und licion of nd. W s h h β sons o chs h o y fs. In figu 7 w show h configuion of h ishi on h suln jcoy y licion of Figu Swiching sufc fo Figu S sc oiml nsiion fom cicl o sigh lin Figu ol swiching sufc figu shows h nsiion jcois whn lying ng-ng conol. h suln jcois h h hyssis cycl fom. h lgoihm of h oiml nsiion is sd on h dcion of h coss oins of h ng-ng jcois wih h swiching sufc, h id is o com h hoionl disnc of h sysm s oin (, ) fom h swiching sufc y lcing Figu im-oiml cicl o lin nsiion conol

11 Figu6 skid ngl β sons o nd Y Byson, A.E., HO, Y.C., Alid Oiml Conol, Oimiion, Esimion nd Conol, OHN WILEY & SONS, 97. Fossn,., Guidnc nd Conol of Ocn Vhicls,. WILEY PRESS, 996. Hygounnc, E., Souès, P., Lcoi, S., Modélision d un Diigl Eud d l Cinémiqu d l Dynmiqu, LAAS-CNRS, 6, Oco. 6 Khouy, G.A., Gill,.D. Edios, Aishi chnology, CAMBRIDGE AEROSPACE, SERIE, Lumond,.P, Roo Moion Plnning nd Conol, Lcu Nos in Conol nd Infomion Scincs 9, SPRINGER, Muy, R.M., Li, Z., Ssy, S., S., Mhmicl Inoducion o Rooic Mniulion, CRC Pss, Ponigin, L., Bolinski, V., Gmkélidé, R., Michchnko, E., h Mhmicl hoy of Oiml Pocsss, Inscinc Pulishs, 96. Roins, H. M., A Gnlid Lgnd-Clsch Condiion fo h Singul Css of Oiml Conol, IBM ounl of Rsch nd Dlomn, Vol.,. 6-7, uly 967. Zfn, M., Kum, V., Cok, C., Mics nd Conncions fo Rigid-Body Kinmics, Innionl ounl of Rooics Rsch, Vol. 8, #,. -8. f X Figu7 Gomic configuion of ishi sons o Conclusion nd fuu wok In his, chciion of h im-oiml fnc hs fo ll nigion of h ishi is slishd, h Mimum Pincil of Ponygin gis locl infomion of h oimliy of h hs. In h fuu wok, his sudy should comld y gomic soning fo oiding wy o slc insid of his fmily, h oiml h o link ny wo configuions in ll ln. Acknowldgmn h uhos wish o hnks P. Souès nd E. Hygounnc, fom LAAS-CNRS (oulous, Fnc). Rfncs Ahns, M., Fl, P. L., Oiml Conol, An inoducion o h hoy nd is Alicions, McGRAW-HILL BOOK COMPANY, 966. Bsoui, Y., Him, S., Som Insighs in Ph Plnning of Smll Auonomous Aishis, ounl of Achis of Conol Scincs, Polish Acdmy of Scincs, ol.,, ; 9-66.