Virtual Target Algorithm in Cooperative Control of Marine Vessels

Transcription

1 Virtual Targt Algorithm in Cooprativ Control of Marin Vssls Zoran Triska, Nikola Mišković, Đula Nađ and Zoran Vukić Laboratory for Undrwatr Systms and Tchnologis Faculty of Elctrical Enginring and Computing Univrsity of Zagrb Unska 3, Zagrb, Croatia {zoran.triska, nikola.miskovic, dula.nad, Abstract - Marin vssls (surfac and undrwatr) prsnt dynamic systms which ar difficult to control du to thir complxity, hydrodynamic ffcts and xtrnal forcs. Morovr, cooprativ control and guidanc of ths systms prsnts a task which still prsnts an intriguing rsarch topic in th ara of marin robotics. This papr prsnts a guidanc mthod basd on virtual targts (VT) applid to marin vhicls. Th proposd algorithm is drivd using a Lyapunov basd approach which guarants convrgnc and stability undr xtrnal disturbancs. Firstly, th VT algorithm is applid for path following of a singl vhicl. It is dmonstratd that this algorithm is convnint for following of convntional paths (lin, circl arc, splin). Scondly, th VT algorithm is applid for cooprativ control btwn two marin vhicls. Th following formations of marin vssls will b dscribd and analyzd: vhicl following whr on vhicl is th ladr followd by th followr vhicl at a prdfind distanc; and th wingman problm whr two vhicls ar rquird to follow a joint path at a constant prdfind distanc btwn thm. Simulation rsults of th dscribd algorithms will b givn in th papr as wll as th discussion on thir quality. I. INTRODUCTION Marin robotics in gnral prsnts an intrsting and challnging ara whr th application of control thory prsnts an ssntial part. This coms dirctly as a consqunc of harsh nvironmnt in which marin vhicls oprat, charactrizd with unprdictabl disturbancs (wavs, winds and currnts). Exploring th sa dpths is on of th biggst tasks for th futur. Thanks to dvlopmnt of advancd control thory and tchnological improvmnt, nowadays, th main intrsts of marin robotics ar focusd on cooprativ control and guidanc of marin vssls. Dvlopmnt of th multi-vhicl framworks and thrby including mor vhicls in cooprativ missions, lad us to dsign mor complx control systms. This papr dals with virtual targt algorithm in cooprativ control of marin vssls. Th work prsntd in this papr will b laboratd on an ovractuatd marin surfac platform which dvlopmnt has bn publishd at MIPRO confrnc, [3]. Th platform is dvlopd at th Laboratory for Undrwatr Systms and Tchnologis (LABUST) by a group of studnts. This papr addrss simulation rsults of th dscribd algorithms obtaind in MATLAB, so only som important tchnical charactristics of th platform will b mntiond in following part. Th marin platform which mathmatical modl is usd in simulations is actuatd with four thrustrs in an X-shapd configuration which nabl motion in surg, sway and yaw dgr of frdom. This configuration is highly convnint for dynamic positioning as wll as path following. Marin vhicls ar oftn dsignd to b ovractuatd; thy hav mor actuators than strictly ncssary to manuvr. In this cas, four thrustrs ar usd to control thr dgr of frdom: surg, sway and yaw. This proprty guarants rliability in th cas of faults in th actuators. Platform dvlopmnt compriss drivr lctronics for th thrustrs, a GPS unit, a compass, battris and a singl board computr usd for snsor intgration and control. Ths componnts nabl manual control from th ground station as wll as autonomous opration mod. Th communication with th ground station is accomplishd using a two-way wirlss link. A schmatic rprsntation of th hardwar infrastructur implmntd insid th platform can b sn on Fig.. Th control algorithm which nabls autonomous movmnt is implmntd on a singl board computr. This systms configuration has provd to b convnint for laboratory us and tsting of control and guidanc algorithms. Th main motivation for building this typ of a platform was, firstly, to introduc studnts to th ara of marin robotics. As th projct volvd, th task was to build a rathr small, portabl platform which will b usd in th fild for th purpos of assisting divrs in undrwatr navigation and communication with th ground station. Furthr on, th dvlopd platform will b usd in cooprativ guidanc of unmannd undrwatr vhicls. This application would allow mor prcis undrwatr localization and onlin mission rplanning. Th rlvanc of th nvisiond projct is significant from th ducational sid, and th intrdisciplinarity through marin robotics. Fig : Schmatic rprsntation of th hardwar infrastructur

2 This papr prsnts th nxt stp in rsarch activitis rgarding th guidanc algorithms which includ simulation rsults obtaind using virtual targt algorithm in ordr to achiv cooprativ mission xcution. It should b mntiond that th dvlopd simulation modl assumd that th dynamics of th platform wr uncoupld. Th papr is organizd as follows. Sction II dscribs th mathmatical modl of th basic motion of th marin platform usd in simulation. In sction III, a brif introduction to virtual targt algorithm is givn and in sction IV this algoritm is xpandd to path following approach and multi-ntwork framwork cooprativ control. In sction V, simulation rsults ar prsntd and finally, in sction VI som conlusions ar mad and guidlins for futur work don. paramtr which ar spd dpndant and includ all spds, th drag paramtr can b approximatd with a constant trm, i.. ( u) ( v) = u, [5]. In a similar mannr, yaw modl is givn with (4) whr r is inrtia and () r drag. Th ue, ve and re rprsnt xtrnal disturbancs and unmodlld dynamics of th systm, [7]. x cos sin 0 u y = sin cos 0 v. () 0 0 r = ue, = ve, =. uu u u X () uv v v Y (3) r r r N (4) r re II. MATHEMATICAL MODEL A. Th marin platform modl Th mathmatical modl of a marin platform is dfind using two coordinat frams: an Earth-fixd (inrtial) fram {E} dscribd with axs N (pointing to th north), E (pointing to th ast) and D (pointing down so that th NED fram forms a positivly orintd coordinat systm); a body-fixd coordinat systm {B}, which is usually attachd to th cntr of gravity ( CG ) of th vhicl and is dscribd with thr axis x, y and z pointing rspctivly in th sam dirctions as th NED fram whn x and N ar alignd, [4]. Th mathmatical modl of a marin platform is dscribd with an assumption that th platform is moving only in th horizontal plan, i.. only translation in th N E plan and rotation about th z axis is possibl. Th platform's spds ar dfind in th fixd coordinat fram {B}: surg u and sway v spds ar translation spds in th x and y axis dirctions, rspctivly, and yaw spd r is rotational spd around th z axis. Earth-fixd coordinat fram is usd to dfin positions x and y in th horizontal plan and orintation of th platform. Th motion of th platform is achivd by applying surg ( X ) and sway ( Y ) forc and yaw ( N ) momnt. Th schmatic rprsntation of th mathmatical modl is givn in Fig..,[8]. Th kinmatic modl in th horizontal Fig.. Block-diagram of th mathmatical modl usd to dscrib marin vssls. plan is givn with (). Th dynamic modl is assumd to b uncoupld so ach controllabl dgr of frdom (DOF) can b modlld sparatly. In addition to that, sinc th platform is symmtric in th horizontal plan, th sam modl (and th paramtrs) can b usd to dscrib surg and sway dynamics, as shown with () and (3) whr u is a constant paramtr and ( u) and () v ar drag Th actuator allocation matrix givs rlation btwn th forcs xrtd by thrustrs (,, 3 and 4 ) and th forcs that act on th rigid body ( X, Y, N ). For th cas of th Platform, whos actuator configuration is givn in Fig. 3b) and 45, th actuator matrix is givn with (5) whr D is shown in Fig.3.b). X cos 45 cos 45 cos 45 cos 45 Y = sin 45 sin 45 sin 45 sin 45.(5) 3 N D D D D 4 Sinc four actuators ar usd to control thr dgrs of frdom, this prsnts an ovractuatd systm. This allows for th dsign of fault tolrant control algorithms, [6]. Th invrs actuator allocation matrix cannot b found (sinc th matrix is not squar), but a psudoinvrs can b calculatd instad, [0]. Th actuators can b simply modlld using an affin i modl givn with (6) whr n rprsnts th individual thrustr's control input (commandd rotation spd, input i voltag, tc.), K T is th thrustr cofficint and is thrust xrtd by th i -th thrustr. i = K i i T n n (6) Th nonlinar static thrustr charactristic can asily b compnsatd within th control algorithm, [7], []. D a) b) δ x 3 4 Fig. 3. a) Virtual modl of th platform for dynamic positioning and b) x-shap actuator configuration. y

3 III. VIRTUAL TARGET ALGORITHM IN PATH-FOLLOWING This sction dscribs only fundamntal concpts and problm formulation of th us of virtual targt algorithm for singl-vhicl guidanc. For dtails and whol kinmatic rror modlling, intrstd radr can rfr to [], [3]. Bfor w dal with th ssnc of th problm, th kinmatic modl () is rprsntd with x U cos whr U x y U sin (7) r y y arctg (8) x With alrady dfind Earth-fixd (inrtial) fram {E}and a body-fixd coordinat systm {B}, a Srrt- Frnt fram is dfind,[5]. Virtual targt attachd to {F} movs along prdfind path. Whol problm can b asily xplaind with rfrnc to Fig.4. Fig.4. Vhicl paramtrs and fram dfinitions Aftr som mathmatical calculations, th following kinmatic rror modl is obtaind and xprssd with rspct to th fram {F} s s ( cc y) U cos y ccss U sin (9) r ccs whr s is spd of virtual targt, attachd to th fram {F}, is path curvatur, U is vhicl s total vlocity and is diffrnc btwn vhicl s dirction of motion and orintation of virtual targt which is th sam as local path tangnt at point P. Control dmands can b formulatd as follows. Aftr som transint tim, platform must b in origin of {F} fram, which mans that and convrg to zro. Scond task is to rduc β towards zro. Whn ths rquirmnts ar fulfilld, path following is achivd. In ordr to dsign control law which nsur convrgncs both th vhicl and virtual targt, on of th possibl Lyapunov s function candidat is chosn. V E s y (0) Computing its tim drivativ and aftr substitutions ar mad, with propr choic of, bcoms ngativ dfinit. Choosing s U cos ks () as control signal which guarants that, a nw additional dgr of frdom is introducd in th control structur. Th spd of virtual targt latr taks a part in dfining yaw rat rfrnc. Considring th candidat Lyapunov function: V ( ) () and choosing control law: r k( ) cc ( s) s (3) is scurd whr is nonngativ controllr gain and φ is angl of approach which is modlld as function of latral coordinat. Th approach angl is instrumntal in shaping transint manuvrs during th path approach phas. Th following hyprbolic tangnt shapd function, paramtrizd by and has dsirabl proprtis du to inhrnt saturation. ) tanh( k ) (4) ( y a y Th advantags of proposd kinmatic control laws ar in following. Asymptotically convrgnc to zro of th variabls β, and is achivd, classic singularity of th path following problm is rmovd and vhicl can prform following of various kinds of th paths. Th abov kinmatic control law applis to th kinmatic modl of marin vhicls only. Using th backstpping tchniqus, this control law can b xtndd to dal with th vhicl dynamics. Howvr, aftr that yaw rat rfrnc is dtrmind, a classic I-P controllr is dsignd for yaw rat control,[9]. Backstpping tchniqu supposs that procss paramtrs ar vry accuratly dtrmind. If w hav som uncrtainty in mathmatical modl control signal can xprincd chattring which is not accptabl bhaviour, []. Th complt control structur usd for simulation can b sn on Fig. 5. Low lvl control is an I-P controllr dsignd according to scond ordr binomial modl transfr function as dsird bhaviour. High lvl control is basd on th virtual targt algorithm. Fig.5. Virtual targt basd control structur IV. COOPERATIVE CONTROL OF MARINE VESSELS In this sction, intntion is to xpand control algorithm and show hav to fficintly us thm to control svral vhicls in ordr to achiv coopration btwn thm. Guidanc and control of cooprativ missions is dmanding

4 task and still prsnts a challng in th ara of marin robotics. Two or mor marin vhicls oprating togthr form a multi-vhicl framwork, [], []. Usually, on vhicl is th ladr which xcuts som mission objctivs and th rst of th vhicls (followrs) follow his movmnts in som way. In th continuing part of this sction, two coordinativ tasks will b dscribd. Each of thm us singl vhicl path following algorithm basd on virtual targt. With th virtual targt basd approach, coordination task is compltly uncoupld from singl - vhicl path following task. A. Vhicl following In vhicl following scnario, th Ladr has to prform som motion. Through th communication link, th Ladr snds basic navigation information to th Followr and thn Followr vhicl rconstructs onlin th path to b followd. For instanc, Ladr follows a prdfind path and Followr has to follow a Ladr, maintaining a dsird distanc btwn thm. Also, Ladr vhicl can b opnloop controlld by human oprator, [4]. As alrady mntiond, Followr vhicl follows th path shapd by Ladr vhicl at fixd or variabl distanc, masurd in curvilinar abscissa of th rconstructd path. Furthr on, if w add mor vhicls in multi-framwork systm, a rlation Ladr Followr can b gnralizd. On vhicl will b ultimat ladr which only xcuts its own mission. Nxt in th Ladr hirarchy will b scond vhicl which is at sam tim th Followr of ultimat Ladr and also a Ladr to th vhicl blow in hirarchy systm. At th bottom is simpl Followr as a tail of such a formation systm. Whol framwork act as a giant snak and som collision avoidanc algorithm must b applid in ordr to prvnt ultimat ladr crashing with th followrs whil prforming sharp manouvrs. Collision avoidanc is byond th scop of this papr and an assumption of smooth path with small curvatur is mad. Th vhicl following approach proposd in this work is basd on th singl vhicl path following guidanc tchniqu for ach vhicl. In addition, to achiv dsird distanc btwn formation mmbrs, th vhicl s surg spd must b controlld according to th rror from dsird distanc from local Ladr. So, th curvilinar distancs btwn th Ladrs and Followrs ar dfind as th diffrnc btwn rspctiv curvilinar abscissas. Fig.6. Hirarchy systm and dfind curvilinar distancs Thus, th task is to dsign a control law in ordr to rach a dsird distanc. Dfining th distanc rror as, th implmntd solution to mak is in th form of tangnt hyprbolic function addd with fd-forward Ladr surg forc signal. Th final form of th surg control law is X X L ku tanh( ks ) (5) whr and ar tunabl controllr gains. With th proprly calculatd surg forc Followr vhicl producs rquird surg spd to maintain dsird distanc. B. Wingman problm On of th most common formation tasks is holding paralll formation whil vhicls mov along diffrnt paths. As in vhicl-following, Ladr xcuts manouvrs whil Wingmn vhicls must maintain paralll formation with rspct to th Ladr vhicl. Fig.7. Exampl of paralll vhicl formation Obviously, spd of th wingmn vhicls must b controlld to accomplish paralll formation. To achiv coopration, ach vhicl is controlld with virtual targt basd approach. Thir virtual targts ar usd for th coopration purpos. Fig.8. Gnral wingman problm ntwork Th ky ida is to add a nw virtual targt, Coordination Virtual Targt (CVT) for ach wingmn vhicl. Controlling th spd of CVT, dsird position of CVT is dtrmind. Control law which nsurs propr CVT curvilinar spd is U cos k (6) s CVT s whr is total vlocity of Vhicl-, is th diffrnc btwn th local path tangnt givn by position of th CVT and th virtual targt of Vhicl-, is projctd longitudinal rror, and is a tunabl gain. CVT bhavs lik th virtual targt for Ladr, moving on th path of Wingman vhicl. Whn CVT is st on th path of wingman vhicl, curvilinar distanc can b calculatd. With rfrnc to Fig. 8., Vhicl - is a wingman vhicl and formation position rror is. To obtain compltly paralll formation is st to zro.

5 V. SIMULATION RESULTS In this sction, simulation rsults ar prsntd in ordr to illustrat th prformanc of th prviously proposd guidanc algorithm. Th simulation cas study which is prsntd hr consists of two coopration scnarios and this papr is focusd only on thm. Vido and rsults of all othr scnarios with diffrnt mission paths can b found onlin at A. Vhicl-following simulation rsults Th first scnario includs vhicl following for thr vhicls whr path to b followd is a circl with and constant path curvatur which is in this cas. At th momnt th Ladr is positiond at th origin of th NE coordinat fram and it is commandd to follow th circl. Th first Followr is positiond at point and th scond at point. Throughout th whol simulation rfrnc distanc btwn vhicls was kpt at. Rcordd rsponss of vhicl distancs in th trm of curvilinar abscissa can b sn in Fig. 0. Followr- gnrats path for Followr-, so distanc rror must b obsrvd rlativ to th vhicl- l which is in front of him. Fig.9. givs th path for all thr vssls during th simulatd mission. Th control rrors ar shown in Fig.. and it is clar that aftr initial ovrshoots th vssls rach zro stady stat rror. An ovrshoot is dirct consqunc of fforts to gt closr to th vssl as soon as possibl. Fig.. Rspons of distanc rror Fig.. Surg spds of th vhicls B.Wingman Problm Simulativ Rsults For th Wingman framwork simulation, a motion coordination task btwn thr vhicls is prsntd. Ladr vhicl xcuts sam path following as in vhicl following (circl with ) whil othr two vhicls hav to follow concntric circls with and, rspctivly. During th whol simulation, dsird paralll formation position is succssfully achivd. Fig. 3. shows th convrgnc and maintnanc of th dsird formation whil prforming th path following tasks. It is obvious that vhicl which has to cross th longst path must hav th largst surg spd. In Fig. 4. surg spds of th formation mmbrs can b sn. Ladr spd has smooth asymptotically rspons whil Followr spds ar rsults of controlling surg forcs in ordr to achiv th coordination task. Fig.9.Th paths of th vssls during simulation Fig.0.Distanc btwn th vhicls Fig.3. Th paths of th vssls Th wingman problm

6 Fig.4. Surg spds of th vhicls It should b noticd, th Ladr commandd surg forc was kpt constant throughout th whol simulation which rsultd in a constant stady-stat cruising spd. Th Wingman vhicl surg forcs ar calculatd according to (5). Aftr transint tim, vn surg spds of th Wingman vhicls ar in stady-stat. Considring that circls ar concntric and diffrncs in a radius btwn Ladr vhicl and Wingman vhicls ar sam, th absolut diffrnc btwn surg spds would also b sam. VI. CONCLUSION AND FUTURE WORK In this papr, a rlativly nw control approach in marin applications has bn dscribd. Firstly, singl vhicl path algorithm is dvlopd with th aim of xpanding on cooprativ tasks. Us of virtual targt as control tchniqu provids highly dcoupling btwn th coordination systm and ach individual vhicls control systm. Th systm stability and th prformanc of control systm ar shown in simulation rsults. Th simulations prov th functionality of th proposd control law. Th futur work will includ implmnting th dvlopd virtual targt algorithm on a ral vssl to b usd in fild conditions. Aftr succssfully applid singl vhicl path following algorithm, our final goal would b to achiv coopration btwn two or mor vhicls in fild conditions. ACKNOWLEDGMENT Th work was carrid out in th framwork of a Coordination and Support Action typ of projct supportd by Europan Commission undr th Svnth Framwork Programm "CURE - Dvloping Croatian Undrwatr Robotics Rsarch Potntial" SP-4 Capacitis (call FP7-REGPOT-008-) undr Grant Agrmnt Numbr: Th authors would lik to thank all th studnts who participatd in th dvlopmnt of th laboratory marin platform. REFERENCES [] M. Brivik and T.I. Fossn. Guidanc Laws for Planar Motion Control. Proc. of 47th IEEE Confrnc on Dcision and Control, pags , Cancun, Mxico, 008. [] M. Brivik and T.I. Fossn. Path following for marin surfac vssls. Proc. of Ocans 004, pags 8-89, Kob, Japan, 004. [3] N. Miskovic, Z. Triska, D. Nad and Z. Vukic. Guidanc of Small Scal Ovractuatd Marin Platform Exprimntal Rsults. Procdings of th 34rd Intrnational convntion on information and communication tchnology, lctronics ad mircrolctronics MIPRO 0, CTS, 0. [4] T.I. Fossn. Guidanc and Control of Ocan Vhicls. John Wily & Sons, Nw York, NY, USA, 994. [5] L. Morira and T. I. Fossn and C. Guds Soars. Path following control systm for a tankr ship modl. 34: , 007. Ocan Enginring, [6] E. Omrdic. Thrustr fault diagnosis and accommodation for ovractuatd opn--fram undrwatr vhicls. PhD thsis, Univrsity of Wals Collg, Nwport, Unitd Kingdom, 00. [7] N. Miskovic and Z. Vukic and M. Barisic and P.P. Soucacos. AUV idntification by us of slf--oscillations. Proc. of th CAMS'07 Confrnc, 007. [8] N. Mišković and M. Bibuli and G. Bruzzon and M. Caccia and Z. Vukić. Tuning Marin Vhicls' Guidanc Controllrs through Slf--Oscillation Exprimnts. Confrnc, 009. Procdings of th MCMC'09 [9] Jong-Woo Choi and Sang-Chol L. Antiwindup Stratgy for PI-Typ Spd Controllr. Transactions on, 56(6): , 009. Industrial Elctronics, IEEE [0] N. Miskovic. Us of slf-oscillations in guidanc and control of marin vssls. PhD thsis, Univrsity of Zagrb, Faculty of Elctrical Enginring and Computing, Zagrb, Croatia, 00. [] Z. Vukic and Lj. Kuljaca. Automatic Control -- Analysis of Linar Systms. Kign, Zagrb, 005. (in Croatian). [] M. Bibuli. Advancd Guidanc Tchniqus and mission control architctur dvlopmnt towards marin robotic systms nhancmnt. PhD thsis, Univrsity of Gnova, Faculty of Enginring, Gnova, Italy, 00. [3] M. Bibuli, G. Bruzzon, M. Caccia and L. Lapirr.Path-Following Algorithms and Exprimnts for an Unmannd Surfac Vhicl. Journal of Fild Robotics -0(009), Wily Priodicals,Inc,009. [4] M. Bibuli, M.Caccia and L. Lapirr.Virtual targt Basd Coordinatd Path-following for Multi-Vhicl Systms. Procdings of th... [5] L.Lapirr, D.Sotanto and A.Pascoal. Nonlinar Path Following with Applications to th Control of Autonomous Undrwatr. Procdings of th 4 nd Control, Hawaii USA, Dcmbr,003. IEEE confrnc on Dcision and