ROBUST TRAJECTORY TRACKING CONTROL OF UNDERACTUATED UNDERWATER VEHICLE SUBJECT TO UNCERTAINTIES

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1 Joal of Mai Scic ad chology, Vol 5, No, pp 8-98 (7) 8 DOI: 69/JMS-6-9- OBUS AJECOY ACKING CONOL OF UNDEACUAED UNDEWAE VEHICLE SUBJEC O UNCEAINIES Ya Ch, Jig Li, Kagli Wag, ad Shog Nig Ky wod: dactatd dwat hicl, backtppig cotol modl, paamt ctaiti, tajctoy tackig cotol ABSAC A compoit obt cotol chm i popod by combiig a lidig mod cotoll with a adapti fzzy cotol algoithm to cotol a -DOF dactatd dwat hicl with modl paamt ptbatio ad iomtal ditbac bad o th backtppig cotol mthod ad th Lyapo tability thoy h adapti fzzy cotol algoithm i mployd to compat fo modl paamt ptbatio ad th lidig mod cotoll i adoptd to limiat th ffct of iomtal ditbac ad appoximatio o A hoizotal dyamic modl ad tackig o qatio a tablihd to dcib th tajctoy tackig cotol fo th dactatd dwat hicl h latio btw lidig mod cotol gai ad modl paamt ctaiti i did to dtmi th o limiatig ability of th cotoll h cogc ad tability of th compoit obt cotoll a dmotatd ig th Lyapo dict mthod h popod cotol chm i imlatd fo a -DOF dactatd dwat hicl ad it fficicy i th o limiatio i alidatd i mical imlatio lt cofim that th compoit obt cotol law ca b d to achi a obt ad pfabl cotol pfomac fo th hoizotal tajctoy tackig cotol of th hicl I INODUCION May of today dwat hicl a dactatd hicl d to thi pottial bfit o fll actatd AUV h icld a good cotol chm ad a tamlid hap a wll a a dctio i wat itac ad wight (Ch t al, 6) ackig ad tabilizatio cotol fo dactatd Pap bmittd 4/6/6; id 9/6/6; accptd /9/6 Atho fo copodc: Shog Nig (-mail: ighog@6com) School of Mchaical, Elctical & Ifomatio Egiig, Shadog Uiity at Wihai, Chia Yatai Idty & ad chicia Collg, Yatai, Chia dwat hicl a difficlt bca of th o-poio of dg of fdom byod th cotol (Khalid t al, 4) Moo, th dyamic modl of a dactatd dwat hicl i highly copld ad olia d to th -chagig at of oca ct ad om hydodyamic cofficit (ho ad Fo, ) h dactatd dwat hicl i o agil that th cotioal lia cotol mthod caot flly xploit it maability Bocktt hom (Bocktt, 98) dmotat that ay fdback cotol law of a cotio tim-iaiat ytm cold ot tabiliz th dactatd hicl aymptotically accodig to th ll oltio O th lat fw ya, a lag mb of tdi w codctd i th aa of motio cotol of dactatd dwat hicl Cotol algoithm potd i th litat ca b claifid ito two catgoi: modl-bad cotol ad omodl-bad cotol h o-modl bad cotol appoach i bad o th PID cotoll (Koh t al, 6; Khodayai ad Balochia, 5), th al twok cotoll (Pak, 4) ad th fzzy cotoll (aimodi ad Mllo, ), i which ach popli moto i cotolld idpdtly I gal, thi kid of appoach poid th implt cotol tct bt oft lt i poo tait pfomac omtim ltig i ohoot ad ddampd po h modl-bad appoach, how, qi th dyamic modl of dwat hicl to dfi th cotol law ad ha xcllt pfomac i qick o cogc ad accacy h lidig modl cotol mthod, a wll-kow modl-bad appoach, wa d to ol th tajctoy tackig cotol poblm fo dactatd dwat hicl (X t al, 5; Ya t al, 5) How, th dag foc i th modl wa tablihd ig a lia fctio of th locity ad th wa alid oly at low lociti Moo, thi kid of cotol mthod ltd i a did high fqcy aiatio i th tady tat Aoth wll-kow modl-bad cotol appoach i th backtppig mthod, which ca poid a atifactoy cotol pfomac i th pc of a accat dyamic modl Som backtppig mthod w ptd by om ach (Ghommam ad Saad, ; Li t al, 5; Qi, 5) fo tajctoy tackig cotol of dactatd dwat hicl h taimt of a accat dyamic modl, how, i a ticky tak d to iitabl implificatio ad oth tagibl facto ch a backlah o fictio

2 84 Joal of Mai Scic ad chology, Vol 5, No (7) Som obt backtppig cotoll (Jia t al, ) o adapti backtppig cotol appoach (Ghommam ad Saad, ) w popod to gaat th cotol pfomac i pc of modl paamt ctaiti Althogh th ffct of mall paamt ctaiti o th cotol pfomac w itigatd by tho ach, th latiohip btw th cotol gai ad paamt ctaiti wa ot coidd i thi tdi A hicl d th ol of th cotoll may lo th ability to qickly ad accatly tack th did tajctoi I additio, tho backtppig mthod did ot coid th iflc of iomtal ditbac ad ma ctaiti i thi modl, which may affct thi tait po, cogc, cotol ffot ad obt I thi pap, a compoit obt cotol chm i popod to combi th lidig mod cotoll with th adapti fzzy cotol algoithm hi cotol chm i d to cotol a -DOF dactatd dwat hicl with modl paamt ptbatio ad iomtal ditbac bad o backtppig cotol mthod ad Lyapo tability thoy Adapti fzzy cotol algoithm i mployd to compat fo modl paamt ptbatio ad th lidig mod cotoll i adoptd to limiat th ffct of iomtal ditbac ad appoximatio o h hoizotal dyamic modl ad tackig o qatio a tablihd fo th tajctoy tackig cotol of th dactatd dwat hicl h latio btw th lidig mod cotol gai ad modl paamt ctaiti i did to dtmi th cotoll o limiatig ability h popod cotol chm i imlatd ad it fficicy i ttd ad alidatd ig mical imlatio of a -DOF dactatd dwat hicl I thi pap, dyamic modllig i bifly itodcd i ctio ad a kimatic o modl i ptd i ctio h dig of th compoit obt cotoll ad it paamt lctio ad tability aalyi a poidd i ctio 4 ad 5, pctily Fially, th mical imlatio ad coclio a ptd i ctio 6 ad 7, pctily II DYNAMICAL MODELING AND POBLEM FOMULAION h dyamic modl i tablihd i thi ctio to dcib a dactatd dwat hicl moig i th hoizotal pla h copodig hicl tajctoy tackig cotol poblm i th fomlatd Dyamic Modlig of a Udactatd Udwat Vhicl h dyamic modl of a dactatd dwat hicl i th hoizotal pla i fit itodcd Fig illtat a dactatd dwat hicl ad it fc fam h itial fam {O I X I Y I } i coidd to b fixd, i which axi Z i i th gaitatioal dictio ad th oth two ax (X ad Y) a ppdicla to it I cotat, th body fc fam, alo kow a th moig fam, i t at th gomtic ct of th dwat hicl (amly, th ct of gaity) h (g) x B (yaw) z B τ {B} y B (way) Fig h dactatd dwat hicl ad it fc fam logitdial axi (x B ) poit i th dictio fom th tail to th o whil th hoizotal axi (y B ) poit fom th lft id to th ight id Accodig to Fo (ho ad Fo, ), th olia dyamic modl of th -DOF dactatd dwat hicl ca b witt i th fom of th followig ifom matix: F X ψ J M C D w wh = [x, y, ] dot th diplacmt (x, y) ad th yaw agl of th dwat hicl i th itial fam; = [,, ] pt th g, way, ad yaw lociti of th dwat hicl i th body fc fam whil th cto = [F,, ] icldig th g foc of F ad th yaw toq of h xtal ditbac of th oca ct i Eq () a xpd a w = [w, w, w ] ad matic J(), D(), C(), ad M a dfid a follow: co i J = i co, X X D = Y Y N N C = m m m m, m M= m, m x B, Z {I} Y y B ()

3 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 85 i which m m X, m my, m Iz N I th abo matic, X, X, Y, Y, N, N dot th qadatic ad lia dag cofficit dfid i th fc (Gamh ad Nkoo, 5); m dot th dwat hicl ma ad X, Y, N a th addd ma ad I z i th itia with pct to th tical axi h omial al of dag cofficit X, N, X, Y, Y, N a gally ot accat ogh h latio amog th actal al Q, th omial al Q, ad th ptbd al Q of y dag cofficit ca b xpd a Q = Q Q I pactical applicatio, th ptbd al Q i alway bodd Accodig to th fc (Atp t al, 5), th omial al of a dwat hiclʼ ma ad it addd ma ca b aily obtaid ad th diffc btw th omial al ad th actal al i ally y mall h, it i amd that th hicl ma ad th addd ma atify th followig iqality: m m ad m m, amly m m ad m m, wh -5 < < Wh th actal, th omial, ad th ptbd paamt a icopoatd ito th dyamic modl i Eq (), th dyamic qatio i witt a: wh m m X X w F m m Y Y w m m m N N w w m m w ; w m m w ; w m -m m w ; X X ; Y Y ad N N Oca Ct Ditbac ad Poblm Fomlatio h pfomac of a dwat hicl i gatly affctd by th oca ct d th oca fac How, thi kid of ditbac i y difficlt to dcib accatly i th modl I od to btt xp th oca ct ditbac with a boday, th ditbac w = [w, w, w ] i dfid i th body fc fam i thi pap a how blow (ho ad Fo, ): ii bi ii i bi,mi bi bi, b i ii bi ii g bi () i bi bi,mi o bi bi, () O {I} y y Y locity dictio Y B y x {B} O B Y F x x locity dictio Q {F} X B = locity dictio X F X ψ ψ actal path ψ locity dictio did path Fig h chmatic diagam of hoizotal path followig 7 J b t w t 7 wh w = [w, w, w ] dot th ditbac foc ad momt i th itial fam; xp th Ga whit oi with zo ma al; diag ii diagoal matix ad (4) a a poiti diag a a diagoal matix xp Ga whit oi h cotol poblm of th dactatd dwat hicl with ctaiti ca b xpd a follow: fo a gi did tajctoy [x, y, ], th g cotol foc of F ad th yaw cotol toq of hold b fod ch that th tajctoy tackig o cto [x, y, ] = [x x, y y, ], wh th cto [x, y, ] dot th actal tajctoy, cog to a al a th oigial poit with a y mall ocillatio III KINEMAIC EO MODEL OF AN UNDEACUAED UNDEWAE VEHICLE I thi ctio, a kimatic o modl i tablihd to dcib th tajctoy tackig cotol fo th dactatd dwat hicl ad i d to gid th dig of cotoll I Fig, {O B X B Y B }, {OXY} ad {O F X F Y F } pt th actal body-fixd fam, th itial fam ad th did path cotitt fam pctily, wh Q i a abitay poit o th did tackig path to b followd h coodiat of ital poit Q o th did tackig path of Q = [x, y, ] ca b dcibd a a fctio of tim ad th th poitio of poit Q i th itial fam o th actal path ca b xpd a: ii Q x, y, (5)

4 86 Joal of Mai Scic ad chology, Vol 5, No (7) h did hoizotal locity of th dwat hicl i fa mall tha it did logitdial locity wh o xtmly lag xtal ditbac xit Moo, th iflc of th hoizotal locity o th tackig pfomac i alo ot igificat I od to th mooth of tackig c, x ad y a qid to b cotioly difftiabl h, th did locity at ital poit Q i th itial fam ca b dfid a: x y (6) Q S Q Q Q Q S Q xco yi S xi yco () h itctio agl of btw th did locity cto of Q ad th hoizotal axi i th itial fam i dfid a th otatioal agl fom fam {O B X B Y B } to fam {OXY} ad ca b xpd a: y acta x x = y acta x x By dfiig Q = [x, y, ] a th actal poitio cto of th dwat hicl i th itial fam ad Q = [x, y, ] a th did poitio cto of th poit, th tackig o cto, = [x, y, ], ca b xpd i th actal bodyfixd fam by th followig qatio: (7) Q Q (8) wh () i th otatio tafomatio matix fom th itial fam to th body-fixd fam ad i paamtizd locally by agl with th followig latiohip: co i = i co By takig th diati of ɛ, Eq (8) bcom (9) Q Q Q Q () ad wh S S Eq () ca b witt a S = Symbol i Fig dot th itctio agl btw th did locity Q ad th logitdial axi i th actal body-fixd fam ad ca b calclatd fom th diffc btw th actal yaw agl () ad th did yaw agl ( ) a = h cod tm of x co y i, xi y co, i Eq () pt th locity compot i th actal body-fixd fam fo th did - locity cto x, y, i th itial fam Wh two locity compot x ad y a ythizd ito th d- id locity Q a dfid by x y, th two tm Q x co y i ad xi y co i Eq () ca b th xpd a Q co ad Q i pctily With a fw maiplatio, Eq () bcom x Q co y Qi y Q co x Qi IV CONOLLE DESIGN () I thi ctio, g foc F ad yaw momt a did accodig to th backtppig cotol mthod ad Lyapo thoy Bad o th o modl i Eq () i th body-fixd fam, a olia backtppig mthod i adoptd to dig th cotoll followig a did path h cotoll dig poc i otlid i th followig tp: Stp : Dfi th Lyapo fctio a V = x y ad tak th tim diati of V Sbtittig th fit two tm i th o modl of Eq () ito th tim diati qatio of V lt i th followig tim diati of V : V x co +y i () Q Q

5 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 87 Now, lt itodc two ital cotol fctio ad with thi did al ad digd a kxx Q co ad kyy Q i, wh k x ad k y a two poiti cotat Sbtittig th two did al of ad ito Eq () yild V = x y kx ky Sic ad a ot th actal cotol fctio of th dwat hicl, th two o aiabl ad ca b xpd a = ad = Coqtly, th tim diati qatio of V bcom V k x k y x y (4) = x y Stp : Fo th coidatio of th tabilizatio of th o aiabl ad, th followig Lyapo fctio i d: V= V+ (5) h tim diati of V bcom V k x k y y x (6) x y h actal cotol foc F i th obtaid a follow: F X X m g m x k ˆ (7) wh k ad a two poiti cotat ad i th did g acclatio alog th logitdial dictio h followig adapti fzzy cotol igal ˆ ad it adapti cotol law a digd to limi- at th ctaity X X of i Eq () Lt tah b a clo appoximatio of X X of i Eq () Sbtittig Eq (7) ito th fit tm of Eq () yild ˆ wg ˆ tah x k m tah h, Eq (6) ca b witt a V k x k y y k wh m w g x y ˆ m ˆ tah m tah w m m g m m x k m w () () Stp : o tabiliz th o aiabl ad, th followig Lyapo fctio i mployd: V = V + m () ad it copodig tim diati i th xpd a ˆ ˆ ˆ tah ˆ (8) (9) wh = [,, ] coit of th locity o cto of th g, way, ad yaw lociti i th body fc fam; ( ) pt th adapti fzzy ba fctio; ˆ i th adapti cotol paamt; i a poiti cotat ad i a abitay mall poiti cotat V = k x k y y x y g k m w ˆ m ˆ tah m tah m ()

6 88 Joal of Mai Scic ad chology, Vol 5, No (7) o limiat th ctaity of Y Y o i Eq (), th adapti fzzy cotol igal ˆ ad it adapti cotol law a adoptd ad xpd a ˆ ˆ ˆ tah ad ˆ, i which ˆ i th adapti cotol paamt; ( ) i th adapti fzzy ba fctio; i a poiti cotat ad i a abitay mall poiti cotat Sbtittig th adapti fzzy cotol igal of ad w m m w ito th cod tm of Eq () with a fw maiplatio yild ˆ ˆ tah m mw YY m tah h ital cotol ca b xpd i th fom of k hfo, Eq () ca b -witt ig th abo xpio fo ad with V = k x k y k k k m m x y m w g ˆ m ˆ tah m m tah ˆ m ˆ tah wh m tah m m y YY m w m (4) Bca aiabl i ot a actal cotol paamt, it ca b placd with a o xpio of = h, btittig th adapti cotol law of ˆ i Eq (9) ito Eq (4) yild V = k x k y k k k m m m w g m m m x y ˆ m ˆ tah m tah ˆ m ˆ tah m tah (5) Stp 4: I thi tp, a cotoll fo o cto ad tabilizatio i itodcd ig th followig Lyapo fctio: 4 V = V + m (6) 4 By takig th tim diati of V 4 ad btittig th tm of ad th adapti cotol law of ˆ ito th tim diati of V, th tim diati of V 4 ca b xpd 4 a g V = k x k y k k m m x y k m m m w ˆ m ˆ tah m m tah m ˆ m ˆ tah m tah (7) h actal cotol law of th yaw toq of i obtaid a follow:

7 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 89 4 g NN m m m k ˆ (8) o limiat th ctaity of N N o i Eq (), th adapti fzzy cotol igal ˆ i Eq (8) ad it adapti cotol law a dfid a ˆ ˆ ˆ tah ad ˆ, i which ˆ i th adapti cotol paamt; ( ) i th adapti fzzy ba fctio; i a poiti cotat ad i a abitay mall poiti cotat Sbtittig i Eq (8) ito th thid tm of Eq () with a fw maiplatio lt i w m mk g m ˆ ˆ tah tah h, Eq (7) bcom V = k x k y k k k m m x y m w g m w g ˆ k ˆ mm m tah m m tah ˆ m ˆ m tah ˆ m tah ˆ m tah m tah (9) wh m m w mm m m m kw () mm g Stp 5: o tabiliz th o aiabl of, th followig Lyapo fctio i lctd: 5 4 V = V + m () 5 h copodig tim diati of V 5 bcom by btittig th adapti cotol law of ˆ ito Eq (): V = k x k y k k k m m x y m w g m w g k m m ˆ ˆ tah m m m tah m ˆ m ˆ tah m tah ˆ m ˆ tah m tah m V CONOL PAAMEE SELECION AND SABILIY ANALYSIS () h latio btw lidig mod cotol gai ad modl paamt ctaiti i obtaid i thi ctio to a th o limiatig ability of th cotoll I th matim, th aymptotic tability of th oall ytm i alo dmotatd bad o Lyapo tability thoy Fit, th followig Lmma i itodcd o a bodd popty: Lmma h followig iqality hold fo ay > ad ay (Polycapo ad Ioao, 996): tah () wh i a poiti cotat that atifi, i, = 785 h two obt cotol tm of g ( ) i th cotol foc of F i Eq (7) ad g ( ) i th yaw cotol toq of i Eq (8) a d to limiat th two ctaiti of w ad w i Eq () ad () pctily h two poiti cotat of ad i th two obt cotol tm dictly dtmi th o limiatig ability fo th cotoll h two ctaiti i Eq () ad () ca b witt a g w m m m m (4) g w m m m m (5)

8 9 Joal of Mai Scic ad chology, Vol 5, No (7) wh mwm x k ad m m w m m m k It i kow that ab a b wh a ad b a two al mb, h, takig th abolt al o both id of th qatio fo Eq (4) ad (5) lt i th followig two iqaliti w m m m m (6) w m m m m (7) It ca b fom iqality (6) that wh m m, th ctaity caot b gaatd to b limiatd by th g cotol foc of F Similaly, a fom Eq (7), wh m m, th ctaity caot b gaatd to b limiatd by th yaw cotol toq of I oth wod, th ctaiti of w ad w may b gat tha th two obt poiti cotat of ad h, th two ctaity ma of m ad m mt b amd to atify m m ad m m, amly m m ad m m fo -5 < < Lt m m m ad m m m h followig iqaliti hold t: m m m (8) m m m (9) Sbtittig iqaliti (8) ad (9) ito iqaliti (6) ad (7) pctily yild w ad w h, th ctaiti ca b limiatd by th g cotol foc of F ad th yaw cotol toq of if th ma ad obt cotol paamt atify coditio of m m ad m m ; m m m ad Wh m ad m m m m, m m m ad m m m Wh m ad m al of both m m m ad m m m, th imm i qal to + h, th boday coditio fo ad bcom o + Fo a gi al of ad ctaity al of m, th followig coditio mt hold: th cotol paamt of ad d to wh atify th followig iqality o m ad m a poiti al ad to mt th boday coditio of o + wh m ad m a gati al h boday al fo gati al of m ad m a lag tha that fo poiti al of m ad m I oth wod, th cotol cot fo m ad m i mo tha that fo poiti al of m ad m ˆ i Sic i tah, th tah i i ˆ Lt ˆ, ˆ ad ˆ i h th followig iqaliti ca b obtaid bad o Lmma : i i i tah i Additioally, by ig th two iqaliti g m w ad m g w, Eq () ca b witt a: V k x k y k k k m m 5 x y k m m (4) akig th tim diati of kyy Qi ad th btittig th cod tm of Eq () ito th diati qatio yild i co Q Q k i x y Q Sbtittig Eq (4) ito i Eq (4) gt (4) Qi Q co my Y Y m m ky Qi x w m (4) Ad th btittig Eq (4) ito Eq (4) yild

9 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 9 V k x k k m m y 5 y Q Q k x k y k k k m m k x y (4) wh th bcipt dot th imm al ad th al of ca b xpd m m Q m Y mky Y w Bad o Yog iqality, th fit th ctaity tm i Eq (4) a l tha th followig fctio fo poiti cotat i k x, (i =,, ): y Q k yq k m m ad y Lt xami th lat th ctaiti tm i Eq (4) ig a wot ca aalyi by amig all of thm a poiti I thi ca, Eq (4) bcom V y (44) 5 x x y wh k k ; y ky 5 ; k k ; x x 5 y ad k 5 Q k Q k m m 5k 5 5 k k m m Q y Q Ud aio difft coditio, th appopiat al of k x, k y, k, k,, ad ca b lctd to a poiti al fo x, y ad By takig x(t) = [x, y,,,, ] ad = mi { x, y,,,, }, iqality (44) bcom V 5 - V5+, which lt i t V5 t V5 +5 fo t t ig th Compaio Lmma (Khalil, 996), fial h, th followig coclio ca b daw: t,, fial x t x t t (45) Eq (45) idicat that th tat o mai i a bodd ttig a zo ad ca b dcd by icaig th gai i Eq (44) Sic kxx Q co, kyy Q i, ad k, th locity tackig o of, ad ca alo b kpt i a bodd ttig a zo wh th tat o of x t cog to a al with a mall ighbohood a zo, which wold th global tability of th cotol ytm VI NUMEAL SIMULAION ESULS AND DISCUSSIONS Nmical imlatio w codctd to alat th popod cotoll pfomac i modl mooth tait po, qick cogc, low cotol ffot, ad obt o ify th alidity of th popod cotoll, a dactatd dwat hicl wa tak a a xampl to alat it ability i th tajctoy tackig cotol Fo th hicl coidd i thi xampl, th followig cto a d = [,, ], = [x, y,] ad = [F,, ] I th followig imlatio, th am cotoll tct i applid fo th g foc of F ad th yaw toq of ig th am gai al i th cotoll qatio I that way, th cotoll dig wold b idpdt of th tackd tajctoy h ma (m) of th dactatd dwat hicl i 85 kg ad it otatioal itia aod z axi (I z ) i 5 kgm h addd ma i th dictio of ad ad th addd momt of itia i th dictio of a gi a X kg, Y 8 kg ad N kgm, pctily h g, way ad yaw lia dag cofficit ha al of X = 7 kg/, Y = kg/ ad N = 5 kgm / h g, way, ad yaw qadatic dag cofficit a at X = kg/m, Y = kg/m ad N = kgm Alo, th cotat of m, m, m ptig th addd ma ad combid itia a gi a m m X 5 kg, m = my 65 kg, ad m Iz N 8 kg m All al mtiod abo w d a th omial al i th cotoll dyamic modl I th followig imlatio ca, th modl paamt i th plat dyamic modl icldig phyical paamt of th dwat hicl ad hydodyamic cofficit a dtmid bad o th followig two amptio: () th actal al a ot kow; () th actal al ay withi % fom th omial al Fo xampl, th omial al of th fi paamt i th cotoll dyamic modl mˆ, Iˆ, ˆ, ˆ z X X a qal to {85 kg, 5 kgm, - kg, kg/m} ad thi copodig actal al i th plat dyamic modl ca b lctd withi th followig ag: m [665 kg, 5 kg], I z [495 kgm, 55 kgm ], X [- kg, -97 kg], X [9 kg/m, kg/m] Nmical imlatio w caid ot ig th foth-od g-ktta fomla with a cotat tim tp at h iitial coditio ad cotol paamt w gi a follow: k x = 5, k y = 65, k =, k = k = 5, = 4, = 8, = = 8 h adapti fzzy tm i th fom of Eq (8) w d to appoximat th th ctaiti of X X, Y Y, ad N N i

10 9 Joal of Mai Scic ad chology, Vol 5, No (7) 5 y (m) -5 Solid: did tajctoy Dottd: actal tajctoy x (m) Fig Cicla tackig cotol compaio btw th actal ad did tajctoi ig omial modl paamt Eq () h ipt cto of th th adapti fzzy tm i dfid a = [,, ] Accodig to th o cad by ctaiti, th i of dico of ach fzzy ipt cto i diidd ito fi fzzy labl ad thi copodig mmbhip fctio a dfid a A xp i i i ci i, wh c i ha th al of -, -5,, 5 ad ad i qal to 4 h fzzy bai fctio ( ) i Eq (8) wa lctd a ( ) = [ ( ),, M ( )], wh th l-th fzzy bai fc- M tio wa digd a l l i l i A i Ai i l i with l Ai i a th mmbhip fctio of th i-th ipt cto i th l-th fzzy l ad M a th mb of l Bad o th abo appoximatio, w ha ˆ i ˆ i ˆ i tah h cotat of i th paa- mt adapti law (9) wa lctd a = o alat th tajctoy tackig cotol pfomac of th popod cotoll, two difft po w obtaid d th followig two coditio: (a) h actal modl paamt a kow ad (b) h actal modl paamt ha al withi % of it copodig omial al h popod cotoll wa focd to tack a did cicla path of x (t) = i (t) m, y (t) = co (t) m I thi ca th cod diati of th path a qid ic th cicla tackig of th hicl i achid by a cotat agla locity of ad lia lociti of ad It i fod that Q = m/ ad Q fom Eq (6) ad - ad / fom Eq (7), which idicat that th hicl tal alog a cotat clockwi path Ca : h ca foc o th tajctoy tackig cotol fo a cicla path bad o th omial modl paamt I thi ca, th dactatd dwat hicl mo fom th iitial poitio = [-5 m, 5 m, ad] at a iitial locity of = [ m/, m/, ad/], which ha iitial poitio ad oitatio o at x = 5 m, y = - m ad = ad iitial locity o at = 9 m/, = m/ ad = - ad/ h total imlatio tim wa t a 64 Fo th ppo of btt obatio of th tait ad tady tat po, th tajctoy tackig o w diplayd fo th fit cod ad fo th ti datio of 64 cod h tajctoy of th dactatd dwat hicl i diplayd i th itial fam pla, a how i Fig h olid li i th did path whil th dahd li pt th actal path calclatd by th popod cotoll It ca b fom Fig that a lag diffc xit btw th did path ad th imlatio datm dig th tait tat po h g cotol foc of F ad th yaw cotol momt of a illtatd i Fig 4 It i obd fom Fig 4 that th g cotol foc of F ad yaw cotol momt of a y high at th iitial ad gadally cog to thi did al aft a hot of piod of tim Dig th tady tat po aft th cogc, both th g foc ad th yaw momt xhibit a faoabl tady pfomac withot ay igificat big ohoot

11 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 9 F (N) τ (N m) (a) Sg foc F fo th fit cod (b) Oiw of g foc F fo th ti datio (c) Yaw momt τ fo th fit cod Fig 4 po of th g cotol foc ad th yaw cotol momt (d) Oiw of yaw momt τ fo th ti datio 5 (m/) (a) Lia locity tackig o fo th fit cod (b) Lia locity tackig o fo th ti datio 5 (m/) (c) Lia locity tackig o fo th fit cod (d) Lia locity tackig o fo th ti datio 5 (ad/) () Agla locity tackig o fo th fit cod (f) Agla locity tackig o fo th fit cod Fig 5 Vlocity tackig o i th body-fixd fam

12 94 Joal of Mai Scic ad chology, Vol 5, No (7) 5 x (m/) (a) Poitio o x fo th fit cod (b) Poitio o x fo th ti datio 6 y (m) (c) Poitio o y fo th fit cod 4 5 (d) Poitio o y fo th ti datio ψ (dg) () Oitatio tackig o ψ fo th fit cod (f) Oitatio tackig o ψ fo th ti datio Fig 6 Poitio ad oitatio tackig o i th itial fam h locity tackig o i th body-fixd fam a how i Fig 5, i which th lft fig diplay th o fo th fit of th imlatio tim to ob th tait po ad th ight fig diplay th o fo th ti datio of 64 to poid a oall iw of th o It ca b fom Fig 5 that th locity tackig o cog to a al a zo with a ocillatio th od of - m/ o ad/ withot ay lag ohoot ad th lowly cog to a tady po towad zo aft Oc th locity tackig o ach th tady tat, th aiatio i o i y limitd h imm abolt locity o of,, calclatd fom th data how i Fig 5 a 5 (m/), (m/), 6 (ad/), pctily ad thi copodig tadad diatio a (m/), (m/) ad (ad/) h poitio ad oitatio tackig o a dpictd i Fig 6, i which th lft fig how data fo th fit imlatio tim ad th ight fig how th data fo th ti imlatio datio of 64 It i obd fom th lft fig that aft a hot piod of tim, th tackig o cog to a zo with a mall ocillatio i th od of - m o dg h poitio tackig o of x ad y calclatd fom th data i Fig 6 a withi th ag fom -8 m to 46 m ad th oitatio tackig o ai withi th ag fom

13 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 95 5 y (m) -5 Solid: did tajctoy Dottd: actal tajctoy x (m) Fig 7 Cicla tajctoy compaio btw th did path ad actal path ig th modl paamt withi ±% of th omial al F (N) τ (N m) (a) Sg foc F fo th fit cod (b) Sg foc F fo th ti datio (c) Yaw toq τ fo th fit cod 4 5 (d) Yaw toq τ fo th ti datio Fig 8 h g cotol foc ad yaw cotol toq ig th modl paamt withi ±% of th omial al 6-7 ad to 6 ad h lt dmotat that th popod cotoll pfom xtmly wll i tajctoy tackig cotol ad qick cogc Ca : I thi ca, it i amd that th actal modl paamt a ot kow bt thy a withi a imal ctaity of % of th omial al fo th pio tajctoy tackig cotol aod a cicla path h iitial coditio a th am a tho i Ca h tajctoy of th dactatd dwat hicl i th itial fam i diplayd i Fig 7 It ca b obd fom Fig 7 that imila o xit btw th did path ad th imlatio datm Wh compad to th lt fom Ca, th popod cotoll i thi ca ca alo ffctily limiat ctaiti althogh it ha th kow modl paamt Fig 8 dmotat th g cotol foc of F ad th yaw cotol toq of ltd fom th popod cotoll It i otwothy that th timatd cotol foc ad toq x- hibit latily mall ocillatio thogh th kow modl paamt a amd h locity tackig o i th body-fixd fam a diplayd i Fig 9, i which th lft fig diplay th lt fo th fit of th imlatio tim ad th ight fig illtat th lt fo th ti datio of 64 h locity tackig o qickly cog to a a zo al with a y mall ocillatio i th od of - m/ o ad/ ad ach to th tady tat aft a hot piod of tim Oc th locity tackig o ach to th tady tat, th ocillatio ag i ly bodd with th imm abolt locity o of,, at (m/), 5458 (m/), 74 (ad/), pctily ad th copodig tadad diatio at (m/), 95 (m/) ad 469 (ad/) Wh compad to th lt fom Ca, th kow dyamic modl paamt i thi ca ltd i th imila locity tackig o to tho fom Ca ho lt cofim that th popod cotoll i ffcti i ctaity limiatio h poitio ad oitatio tackig o a dpictd i

14 96 Joal of Mai Scic ad chology, Vol 5, No (7) 5 (m/) (a) Lia locity o fo th fit cod (b) Lia locity o fo th ti datio 5 (m/) (c) Lia locity o fo th fit cod (d) Lia locity o fo th ti datio 5 (ad/) () Agla locity o fo th fit cod (f) Agla locity o fo th ti datio Fig 9 Vlocity tackig o i th body-fixd fam ig th modl paamt withi ±% of th omial al Fig hy both qickly cog to a a zo al ad tabiliz at thi al with a ocillatio i th od of - m o dg h poitio tackig o of x ad y ay withi th ag fom -55 m to 4 m th oitatio tackig o ai fom -45 ad to 6 ad h lt how that althogh om kow modl paamt a amd i th cod ca, th popod cotoll pfom xtmly wll ad alo how it capability of cogig qickly VII CONCLUSIONS I thi pap, th compoit obt cotol chm copld with th lidig mod cotoll ad th adapti fzzy cotol algoithm wa itodcd to tackl th hoizotal tajctoy

15 Y Ch t al: obt Cotol of AUV Sbjct to Uctaiti 97 5 x (m/) (a) Poitio tackig o x fo th fit cod (b) Poitio o x fo th ti datio 6 5 y (m/) (c) Poitio o y fo th fit cod (d) Poitio o y fo th ti datio ψ (dg) () Oitatio tackig o ψ fo th fit cod (f) Oitatio tackig o ψ fo th ti datio Fig Poitio tackig o i th itial fam ig th cotol paamt withi ±% of th omial al tackig cotol poblm fo dactatd dwat hicl with modl paamt ptbatio ad iomtal ditbac A adapti fzzy cotol algoithm wa mployd to compat fo modl paamt ptbatio whil a lidig mod cotoll wa adoptd to limiat th ffct of iomtal ditbac ad appoximatio o h aymptotic tability of th oall ytm wa dmotatd bad o Lyapo tability thoy h latio btw lidig mod cotol gai ad modl paamt ctaiti wa did to dtmi th o limiatig ability of th cotoll h hoizotal dyamic modl ad tackig o qatio fo th dactatd dwat hicl w tablihd fo th tajctoy tackig cotol of th hicl h popod cotol chm wa alo imlatd fo a cicla

16 98 Joal of Mai Scic ad chology, Vol 5, No (7) path ig th omial ad o-omial paamt ad it fficicy i limiatig th ctaiti wa dmotatd ad alidatd ig mical imlatio of a -DOF dactatd dwat hicl Nmical imlatio lt dmotatd that th popod cotoll diplayd xcllt pfomac i it mooth tait po, qick cogc, low cotol ffot, ad obt h popod cotoll ca b d to aaly th cotol pfomac of th cotoll d aio opatioal coditio ACKNOWLEDGEMENS Gatfl ackowldgmt i gi to th fiacial ppot fom th Natioal Natal Scic Fodatio of Chia with Gat No ad Scic ad chology Majo Pojct of Shadog Poic with Gat No 5JMH8 hi wok wa alo patially ppotd by Stat Ky Laboatoy of obotic ad Sytm (HI) with Gat No SKLS-5-MS-6, ad Chia Pot-doctoal Scic Fodatio with Gat No 476 ad M58, ad ach Awad Fd fo Excllt Yog ad Middl-agd Scitit of Shadog Poic with Gat No BSZZ8 EFEENCES Atp, S, Z Q Log, D amthgala, A L Fot ad J Dffy (5) Nmical itigatio of th hydodyamic itactio btw two dwat bodi i lati motio Applid Oca ach 5, 4-4 Bocktt, W (98) Aymptotic tability ad fdback tabilizatio Difftial Gomtic Cotol hoy 8-9 Ch Y, M Zhag, X Y Zhao ad J Gao(6) Adapti fzzy i tajctoy tackig cotol of dactatd dwat hicl with ctaiti Oca Egiig, - Gamh, B ad S Nkoo (5) Nolia boptimal cotol of flly copld o-affi ix-dof atoomo dwat hicl ig th tatdpdt iccati qatio Oca Egiig 96, Ghommam, J ad M Saad () Backtppig-bad coopati ad adapti tackig cotol dig fo a gop of dactatd AUV i hoizotal pla Itatioal Joal of Cotol 87, 76-9 Jia, H M, W L Sog ad Z L Ch () Nolia backtppig cotol of dactatd AUV i diig pla Adac i ifomatio Scic ad Sic Scic 4, 4- Khalid, I, M Ahad ad I Syafizal (4) A hybid-di dwat glid modl, hydodyamic timatio, ad a aalyi of th motio cotol Oca Egiig 8, -9 Khalil, H (996) Nolia Sytm Ptic-Hall, Eglwood Cliff Khodayai, M ad S Balochia (5) Modlig ad cotol of atoomo dwat hicl (AUV) i hadig ad dpth attitd ia lf-adapti fzzy PID cotoll Joal of Mai Scic ad chology, Koh, H, M W S La, G St ad E Low (6) A cotol modl chm fo a dactatd dwat obotic hicl Jomal of Itlligt ad obot Sytm 46, 4-58 Li, Y, C Wi, Q W, P Y Ch, Y Q Jiag ad Y M Li (5) Stdy of dimio tajctoy tackig of dactatd atoomo dwat hicl Oca Egiig 5, 7-74 Pak, B S (4) Nal twok-bad tackig cotol of dactatd atoomo dwat hicl with modl ctaiti Joal of Dyamic Sytm Mamt & Cotol 7, -7 Polycapo, M M ad P A Ioao (996) A obt adapti olia cotol dig Atomatica, 4-47 Qi, X (5) Spatial tagt path followig cotol bad o Nbam gai mthod fo dactatd dwat hicl Oca Egiig 4, aimodi, F M ad M Mllo () Fzzy/kalma hiachical hoizotal motio cotol of dactatd OV Itatioal Joal of Adacd obotic Sytm 7, 9-54 ho, I ad I Fo () Mai Cotol Sytm: Gidac, Naigatio ad Cotol of hip, ig ad Udwat Vhicl odhim, Noway X, J, M Wag ad L Qiao (5) Dyamical lidig mod cotol fo th tajctoy tackig of dactatd mad dwat hicl Oca Egiig 5, 54-6 Ya, Z P, H M Y, W Zhag, B Y Li ad J J Zho (5) Globally fiittim tabl tackig cotol of dactatd UUV Oca Egiig 7, -46