Nonlinear Formation Control of Marine Craft

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P r o m d i n g s o f th e 41- IEEEConkrence an Derision and Cootroi

1.85 Vegas, Newda USA, December2001 WeM04-3

Nonlinear Formation Control ofMarine CraftRoger Skjetne, Sonja Moi, and Thor I. Fossen

A b sm c l -Th b paper investigate$ formation control of B B e t of ships.Th e wonbol objecllveforesrh ship is to maintain ia position io th e forma-

tion while P rirtu.1) Formation Reference Point (FRP) back s a predefinedp t h . This is obtsined by using vectorial backstepping to solve hvo rub-

problems; B gwrnetrlr task, and a dynamic tsrk. Th e fo rmer guarantppl

that the FRP, and thus the formatiom, backs the path, whUe the lauer en-

sures accurate s p e d contml along th e path. A dynamic guidmce systemwith feedback fmm th e s t a t e o f dl ships ensures that SU ships haw the

same priority (no leader) when m oving along the path. Lyapunov rtrhiiity10 pmven and mbuitnns to input saturation io demanrtrsted using com-

of the states. Extensions to system s of arbitrary relative degree

was made by [ l ] for system s in vectorial strict feedbac k form,

and in [9], [IO] t is also demonstrated that application of a dy-namic gradient optimization algorithm recaptures the advantage

of minimizing the weighted distance between z ( t ) nd ((@(t)) .

The main contribution of this paper is to represen t and solvethe formation control problem by defining a Geometric Task

and a D ynamic Task, constituting a Maneuvering Problem, asdefined in [l], [2]. With this design procedure, a formation is

viewed as a flexible system (asone ,,,,it) that along

a param etrized path. Each individu al vessel will have a relative

puler simulations.

control; Ship control .Kc7"od-NonUn-r C o n W Maneuvering; Backstepping; Formation

position to a po int called the Formation Reference Point (FRP).

TheGeom etric Task ensures that the individual ships conve rge

or fleet ofships in to their positions in the formation and stay at their respective

formation. A design procedure based on manewering and not paths relative to the FRP. The Dynam ic Task, which in this casetrajectory tracking is applied for this purpo se. In maneuvering, will be a speed assignment for 8, ensures that the FRP (and thus

the desired behavior of the plant in the output space is separated the ships) W i l l move along the Path With the specified s p e d . Ininto two subproblems; 1) converging to and following a desired this approa ch there will be no leader ofthe traditional type. The

param etrized path, and 2) satisfying a desired dynam ic behavio r desired motion for the FRP is equally based on the states of allalong the path, here referred to as a speed assignment, see [I], vessels and gives a centralized guidance system. This implies[2]. This is contrary to trajectory tracking where the main goal that it iS easy to reconfigu re he form ation.

is for all tirne t to track a desired ? l d ( t ) hich implicitly The formation controller is simulated using three offshoremust con tain information of both the desi red path and speed. SUPPlY vess els. Emphasis is placed on maneuvering perfor-

ne ield offo rmation control with applications towards me- mance and input saturation. Since the update law for 8 con-

satellites, e@.,has recentlyre- tains an o ptimization algorithm that minimizes an instantaneousceived a lot of atten tion, see [3], [4], [5], [6], for instance. In LYaPunov cost function based on the States of ll vessels, it is

[4] a procedu re for the design of n+1 controllers which ensure dm~ onstrat ed hat the proposed algorithm handles input saturn-that n+1 autonom ous vehicles follow a path without altering tionseffectively by simply modifying the speed of the formation

their form ation, has been deve loped, TO determin e the locatio ns SO that the errors in the form ation geom etric task is unaffec ted.

along the path, they used an orthog onal projection from the state This is one of the advantages of maneuvering Over trajectoryofo ne o fth e vehicles, being the leader. The time in the nalready tracking.existing trajectory tracking controllers (for the other vehicles) is Notation: Abbreviations ike GS, LAS, LES, UGAS, UGES,

then replaced with this projection, If the leader slows down or etc., are G or Global, L for Local, S for Stable, U for Uniform,

stops, all the vehicles will slow down and stop. Therefore, the A for Asym ptotic, and E for Expon ential. Differentiation with

speed and performance of the leader will affect all other mem- respect to 8 is indicated by primes: ( ' ( 8 ) :=$, 6" ( 8 ) :=3 .

1. INTRODUCTION

n e opic ofth is paper is controlof a

systems, ships,

I" . . "". .

bers in the formation, but not vice versa.

A fLN-state maneuvering controller was proposed in [7]

which ensures that the states converge to a desired path c(8)and

The Euclidian vector norm is Iz/= (zTz)l'*. For a matrix

p =PT,, let p , :=xrnin(p)ndP n, :=X__(P) .~ ~ . .

then proceed along it. This was don e by converting a trajectory I I . FORMATIONSE T UPtracking controller into a maneuver regulation controller. To

ical projection to find the point on the path w hich minimizes the

of [7 ] was a hybrid resulf and applies to feedback linearizablesystems, where the desired path is specified for the full state.

More =cently, P I has ex tended th is m e t h o d o b ~o ourput

manewering by using backstepping. However, the proposedcontroller cannot be said to have the same advantages as in [8]

since the numerical projection only works on the output subset

A formation with n vessels is created by a set of formation

The idea is for th e FRP to

determine thepath variable 8, they used a P-orthogonal t m ~ e r -designa tion vectors l ,, =1, ,,,, , elative to a ~~~~~i~ ~ Ref.erence point (FRP), see Figure

speed along it ,

weighted distance z ( t ) nd Theme tho do lo^f o ~ ~ o wgiven parametrized path ( ( 8 ) with a desired formation

Let the FRP be the origin of a moving body frame {b}, and

line, but a feasible curve in the output space of the

denote the fixedhe

The individual path for vessel is then

ne ath is in general no t

a

All three authom are with the depamnent of EngineeringCybernehcs, Nor- ~ ~ ( 0 )< ( e ) +R ; w i (1)wegian Universily of Science and Technology, N-7491 Tm dh eim , Now ay. E-mails: [email protected], [email protected] [email protected] where R; is a kinematic rotation matrix from { b } to {e}.

0-7803-7516-5/02/$17.00 02002 IEEE 1699

mailto:[email protected]

mailto:[email protected]



.6,; are unknown'bounded disturbances. The matrices Gi i and

V h ; :=f re invertible for all z;, the output maps h;(zli)

are diffeomorphisms,and all functions are smooth.The uncertainty vectors 6; j represent bounded exogenous

disturbances and/or unmodeled dynamics that is uniformly

bounded in the state space. The particular bounds d o not needto be known.

In the proceeding, the m echanical systems described by (7)are referred to as vessels. For a cluster of n vessels, each repre-

sented by a position output y;, let a Form ation Reference Point

(FRP) represent the position of the formation as a whole, and

let each individual vessel y; have a designation l i relative to

the FW. Let E ( @ ) he the desired path for the FRP and then":~, .~,:\>-%z<y

Ye 1, , E ; ( @ ) = E ( @ ) +R(8) l i is the corr&ponding path for the indi-

vidual vessels. We are now ready to state the For mation Maneu-

vering Problem along the lines of [ 2 ] :Dejnition I: The Formation Maneu vering Problem: De-

Fig. I. llustrationa f a f o r n t h sehlp.

s i 5 set of robust control laws for the individual vessels and a

@idance system that solve the tasks:

1. Geometric Task: For each EGT >0, force the output yi to

enter an -neighborhood of the desired uath E ; (8) hat is.

For ships moving on the ocean surface, the output is the 3

DOF vector 9 = [Z, , 1L1 9 where (s,Y) is the Position and

$ is the heading. The desired Path is then given- . ..,[ X d ( 8 ) , yd(O), 7 j d ( O ) ] ' .The tangent vector along the path in

the (z,y) directions, T(8)=[z&(8),y&(S)lT, is chosen as the

in the {e} frame then gives the desired heading

3~- 0 that, -

x-axis o fthe moving frame {b}. The angle ofthe tangent vector ( t ) E , ( s ( t ) ) l 5 EG TI vt2 T (8)

for any C' function 8 ( t ) .2. Dynam ic Task: For each ED T >0, force the speed 8 to enter

an EDT -neighborhood of a desired speed assignment u s (& ) ,that is, 3T 2 0 such that,

Ild(') = (""')dO) = (@);(e) '

(')

The rotation matrix R ( 8 ) or the ships is given by

- . Iirt) - m.t)I< vt >T (9)~ , - . .

The geometric task ensures that the individual vessels con-I (3) verge to and stay at their designated positions l i in the forms-

cosrjd(B) -sin+d(8) 0

R E ( 8 ) =' := I s i n t d ( 8 ) cos*(8)"" 1 ) tion. The s peed assignm ent task ensur& that the FR P will move

along the path E(8) with a desired velocity v s 8, )and note that differentiationgives

IV. C O N T R O L E S I G N

In the procedur e that follows, a recursive backstepping designis proposed to solve the formation maneuvering problem for n

vessels with the dyna mics given in (7).

Step 1: Define the error variables

k(B)=R'(8)b=R(O)S(8)b(4 )

where S(8) s a skew-symmetricmatrix

zli :=y; - e i ( @ )=y; - ( 6 )- R(6)li

111. P L A N T S N D PRO BLEM TATEMENT

An uncertain mechanical system is represented by the vector

relative degree two model

Sii =G i i ( ~ i ; ) z z i+ i;(zi;) +Eii(zii)6ii(t)i z i =G z i ( ~ i ) u i+ z ; ( ~ i )+E z i ( z i ) b z i ( t )

Yi =h i ( Z l i )

(7)

where the subscript i denotes the i'th system. x j i E Wm

are the states and 5 ; denotes the vector zi := [ x L , z & ] ' ,

y; E Wm re the system outputs,ui E W m re the contro ls, and

y i :=xzi - 1 ;

w s :=U,(& t ) -

where a'; re virtual controls to he specified later. Differentiat-in g (10) with respect to time results in

i i i =VhiGiirz,; +VhiGiioii+Vh,fi;

+VhiEii61i - ' (8 )b-R(S)S(S)liS. (13)

Denote p , ; (8 ) :=c ' ( 8 )+R(@)S(O)l;,nd choose Hunvitz de -

sign matrices A l i , so that P1i =PL >0 are the solutions to

Pl;Al; +A:,Pl; =- Q l i whereQl; =Ql; >0.Define

n

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whose time derivative then beco mes

Step 2: Differentiating (1 1) with respect to time givesn n

i z i =Gziui+2 i +Ezi6zi - uli- 2,e -wii61;. (23)VI = 2zAPiiVh,GlizZi+ 2zAP1;pliw,

i=l i s 1

The first virtual controlsa l i are chosen asv, :=K+Cz,:P2;z2; (24)

i= l

ai;=GL' (Vhi)-' [Aii~ii Vhifi i+pl iv . +aoi] (15)

where aoi are damping terms to be picke d. D efine the first tun-

ing functions, T I ; E W s

T I ; :=2~;Plipli. (16)

To handle the pelturbations we use nonlinear dam ping and applyYoung's inequality n

i= l

whose time derivative becomes

n n n

V z 5 -C~;Qiizii=I +CTI~W.=l + =1 2zLPzip2,~s

"" 1+E 2 z L G Z (VhdTPl;zl,+ --SASlil i

i=l=1

+E z2 2i [Gziw+ z i +E2i62i - u ~ i pZivs- w1i61il.

The control laws are then chosen as

VI =- 2 z L Q i i z 1 ; + e r l i w . + ~ ~ Z ~ P ~ ; V ~ , G ~ ; Z ~ ~i = l i=1 i=l

" "+~ 2 z ~ P i i V h i E i i 6 l i~ 2 z ~ P i i a o i

U,= ( 2 ( 5 ; , 8, )i= l i= 1

" " " =-G,'[P,;'G; (V h; )T l; tl ;+Azizzi

5 -c A Q i i z i i +CTI;~.E 2z~PiiVhiGiizzi - i i + 1; +P2iU. +UOi] (25)

where uoi are nonlinear damping terms to be designed. Define

zi :=[A = L I T , i :=diag Q l ; , Qz;) nd the final tuningfunctions as

Using Young's ineq uality again , the derivative Vz is bound ed by

:=I i= 1 i d

"

Tzi :=T I ; +22LP2;pzi. ( 2 6 )

1+~ Z Z ~ P i ioi + K i i (Vhi)E liE; (Vh i)T P~i z~ ii- 1 1 2

" 1

;=I K l i

+c 6z61i

and the nonlinear damp ing terms are picked as " n

1 T V z 5 - ~ T Q i z i+ T Z ~ W .

2 ;= Ioi = -&I; (Vhi)EliE; (Vhi) Piitl;, ~ l i 0 (17) i= l

Iwhich gives [&;E&+W i i P ; ] Pzit2ii= l

"+E 2 z ; p l i ~ h i ~ l i z z i" 1,5&. (18) and the final nonlinear damping terms uoi are assigned as

K l i

1

2

;= I := I

=- - ~ z ; [ E ~ ~ E , :w, ;w; ] P ~ ; z Z ; , nzi >0. (27)In aid of the next step, we differentiateali to ge t

irii =UI; + 2$ +w1i61i (19) Define Ai :=[S;, S L I T and Ki := diug (&+&, 12,

The result is thenwhere

n " n

(21)

(22)

If we d isregard the sign indefinite tuning func tion terms, eachsystem in the 2,-coordinates is an ISS system from the distur-

bances A, to 2. It follows that forany EGT >0, the output error

Izi.(t)l = I y , ( t ) - , (8 ( t ) ) l 5 EGT , for some time t t , can

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be guaranteed by choosing the nonlinear damping coefficients

f i j ; large enough, and this solves the G eome hic Task.

Next, we must deal with the tuning fun ctions. Choosing w s E

0 to solv e the dynamic task is equiv alent to a trajectory tracking

design with B =u,(9, t ) .A better choice is to design an update

law for 0 or 1, hat uses fee dback from the states of the vessels.

In [9], [IO] it was demons trated that

that is, the total tuning function is the gradient of V2with respect

to 9. We therefo re consider the Direct Gradient Up date Law andthe Filtered Gradient UpdateLmo ext.

Direct Gradient Update Law: From (28) and according to

PI, [lo], k t

which gives the new bound for (28)

and by choosing the gains K; large enough, we can guarantee

any residual bound for l z ; ( t ) l . In particular , this mean s that as

t +CO; lzii(t)/ 5 EGT, that is, I y ; ( t ) - ( O ( t ) ) I 5 EGT, an deach individual vessel enters its designation. The c ontroller re-

alization becomes

and since the states z ; ( t ) are made small, the tuning functions

r*;(t)re made small. H ence, as t +CO , O(t) c;: u. (9 ( t ) , t )

which satisfies the speed assignment. Since we can rewrite (32)

it follows by the ana lysis in [9], [IO] that choosing p large in-duces a separation of time scales between the vessel dynam ics

an d 9. In the fast time scale, (33) becomes a dynamic gradient

optimization algorithm that selects the point on the path for the

FRP which minimizesV2with respec t to B . Therefore, errors in

the geometry of the formation are rapidly minimized with re-

spect to the instantaneous cost function v*(.(& ., ).Filtered Gradient Update Law: In [I], [2], the update law

was constructed as

s = t ) - w s

aso=u,(e,t) - P - ( x , s , t )v,

as (33)

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It is clear that this solves the Formation Maneuvering Problem

for the same reasons as above. In [9] it was demonstrated that

(34) is just a filtered version of (33). It bas the same gradientproperties as discussed above if X an d p are chosen large. Ex-

per ienc e has show n, however, tha! the filtered version gives animproved numerical response for B ( t ) . The caveat is higher or-der in the controller.

Remurk2: In the control laws (25 ) it is seen that each vesselonly needs info rmation about its own states, time, and the path

variable 8. The guidance system, on the other hand, incorpo-rate the dyna mic update law, (33)or (34), which needs informa-

tion of all states in the formation. Therefore, each vessel must

communicate its state information to the guidance system whichprocesses this and returns the path variable 8.

Remark 3: In this setup, no precautionary measures are in-

cluded to avoid inter-vessel collisions.Using maneuvering with

gradient optimization, however, will minimize transients, im-

prove performance, and in that way reduce the risk for colli-sions. A further idea would be to include potential functions inthe overall Lyapnnov function.

V. CASE TUDY: HIPSN FORMATION

For maneuvering of ships in formation we will use a model forwhich there is no coupling between the surge and the sway-yaw

subsystems, see [ I l l , [12]. Let qi =[ x ; , y ; , $ , lT be the "posi-

tion" vecto r in the {e} frame, where (x;,;) is the position on

the ocean surface and Ili is the yaw angle. Let U;=[U;, ;, r; ]be the {b} liam e velocity vector. The subscript i denote s the i'th

ship. The equations of motion in surge, sway, and yaw for eachship is written

T

where RZ(*~)s the rotation matrix (3), M, =M z 0 is th esystem inertia matrix including the hydrodynamic added inertia,

D; is the hydrodynamicdamp ing matrix, T;=[Tl;,T.;,3JT

is the fully actuated vector of control forces and moments, and

w is vector of environmental disturbances decomposed in the{e} frame.

R e m a r k 4 In the underactuated case, the maneuveringmethodology is the same. We refer the reader to the final ship

case in [9], and references therein, on this topic.

The dynam ical system (36) is in the form of (7), where q , is

the output and T; is the control. Let the desired path for the FRPbe

where q ( 8 ) an d yd(9) are three times differentiable with re-

spect to %, and $ b d ( B ) is given by (2). The individual paths for

each ship are then

where l ; =[Iz;, l , ; , OIT . Let u d be the desired surge speed for

the FRP along the path. Then U"(%,) is given by

by extending the Lyapunov function to V =

yields+&w: , which



The design procedure in the previou s section gives the following

7 2 , =24PliP,,+2z2:P2,pz; Tuning functions

Figure 2 shows how the ships in the formation convergesmoothly to their designated path and accurately track it. Withthe substantial environmental disturbances (38), the position er-ror was attenuated to less than 1 m in x and y, and less than 1

in heading.

T, =f i f ; [ - P Z ; l R T ( $ ; ) P 1 , ~ l i Control laws

+ A z i ~ z+ ifF'Qvi +01;+p z i ~ ~mil

_ I

whereT; is the control law for Ship a. The controller realization,_-,"I-

using a Filtered Gradient Update Law, is

0=U , ( O , t ) - w s

W a = X # . - ~ ~ T z ; ( q ; , u ; , O , t ) ."i= l

i

I..-<-z---==2z:5

IJ

,. II a = - - - - - 0 I

phe following two simulations are performed for a formationof 3 ships. The numerical values of the and D , matrices,

taken from [13], represent true data of supply ships that operatein the North Sea. Inboth simu lations, the output path isgiven by(37), where xd(6')=0 and y d ( 0 ) =500sin &,O. The desiredsurge sp ed of the FRP Starts Out with the set-Po'nt U d =4m/s.

At time t = 500 s th e formation chief sets the new formation

speed to U d =IOmis.

Fig. 3. Time-plot ofthc surge speeds,u l ( t ) ,uz( t ) , ( t ) ,or the three ships.

In Figure 3 the sume sued of the ships are shown. Since the

A. Simularion I : Maneuvering wi f h ocean disturbances

The aim o f this simulation is to show that with the formationmaneuvering design we can robustly perform the path following

maneuver for a formation of ships influenced by environmentaldisturbances. Starting off the path, we want the vessels to con-verge smoothly to their designated locations in the formation

and eventually move along the path with the desired'speed.The formation designation vectors are chosen as 11 =

[O,O, 0IT, 12 =[O, 150,OIT and l3 =[O, 50,OIT.This meansthat the FRP coincide with Ship 1, and the ships will travel in

a transversal line formation as one unit. The environmental dis-turbances are

w = [ ]+[ ] sin ( O . l t ) , (38)

acting the same on all the vessels. To attenuate these dis-

turbances, the nonlinear damping gains are set to nz i =20 . The other controller parameters are set as: A I , =

-diag(0.02,0.02,0.5) , Az; = - d i a g ( 2 , 2 , 2 0 ) , P I , =

d i a g ( 0 . 2 , 0 . 2 , 1 ) , P z i d z a g ( l O , 1 0 , 4 0 ) a n d p = X= 2 0 .

qz(0) = [500, 0 , 51'. q 3 ( 0 ) = [0 , 500, 0IT, ul(0) =

u z ( 0 )=~ ~ ( 0 )[ l , 0, 01 and O(0) =w.(O) =0.

TThe initial conditions were q , (O) =[O, 2 0 0 , 4 ] ,

T

- - .center ship are chosen to coincide with the FRP, this ship i s

seen to obtain the desired speed Ud as assigned by the formationchief. The two side ships obtain a periodic path speed accordingto their individual positions, necessary to keep th e formation.

B. Simulation 2: Thrust saturafion ailure in one ship

It is of interest to see how the formation behave as a whole,if the thrust of one ship saturates. In [4] the path variable 0

is projected from the state of the leader vessel. Hence, only if

the leader experiences a problem will the formation as a w hole

act robustly on it. A failure in one of the other vessels will not

influence the others and can therefore easily lead to an accident.The design procedure proposed in this paper, is not based on

any leader vessel. The time evolution of E(O(t)) along the pathis equally inRuenced by the states of all the vessels through the

tuning function (29)and the update law. Therefore, if on e vesselexperiences a problem, all the vessels will act upon it.

We continue the experim ent by forcing a saturation co nstraint

on Ship 2, so that it will maximally be able to go with surgespeed of 8 m i s . The surge speed assignment will be the same as

in the previous simulation, that is, 4 m /s for t <500 s and 10

m i s f o r t 2 500s,which now is infeasible for S hip 2.

The environment in this simulation is disturbance free, w =

0, so that no nonlinear damping is required. The other con-troller parameters are set to: A l , = - d i a g ( 0 . 5 , 0 . 5 , 0 . 5 ) ,A z; = -diag(2,2,20), 1i = dZag(0.6,0.6, 0.6), Pz , =

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d ia g (10,10,40) and ~r.=X =20. The initial conditions were

q,(O) = [O , O I lz(0) = [100, -100, 51 ,%(o) =

[0, 50, 0IT, ~ ( 0 ) ~ ( 0 ) u3(0) = [4, 0, OlT and

T T

O(0) =w * ( 0 )=0.

. - - I D , A m A dm m m Fig. 6. Time-plot o f the speed assignment v,(ll(t) , ) or the FRP nd the re-

sulting response o f b ( t ) . o t i ~ ehat b ( t ) s slower than the assigned speed.

Fig. 4. Resulting ~ ~ p o n s ef the formation when Ship 2 s am te s.

Interestingly, Figure 4 shows that the formation follows thepath a s desired in spite of the ‘failure’ n Ship 2. Figure 5 reveals

that the speed of the formation is considerable slower than the

assigned speed of 10d s . In facf the speed of the slowest vesselconverges to its maximum speed of 8 m /s while the two othervessels follow at what speed necessary to keep the formationassembled. The formation is as fast as its slowest member. Theimportant part is that the vessels keep following the path andtherefore do not cause any accidents. Th is feature is due to the

inherent gradient optimization algorithm that tries to minimizethe Lyapunov cost function which incorporates the states of allthe vessels, see [9], [lo].

d n m m .m a-m,

Fig. 5. Surge spceds of the ships, where Ship 2 maxima lly makes 8m i s . Com-manded speed for the formation was 10m i s .

Figure 6 shows a time-plot of the assigned speed u.(O(t), )

and the resulting response of B ( t ) . Clearly, B ( t ) is slower thanthe assigned speed.

V I. CO N CL U SI O N S

A robust nonlinear control design method has been proposedthat solves the Formation Maneuvering Problem as defined. In-

dividual decentralized control laws were developed foreach ves-sel in the formation. These control laws only uses informationof each vessel’s own states and, in addition, information aboutthe desired path and speed comming from a centralized guid-

ance system. The central guidance system incorporates infor-mation from all states io select the desired position € ( e ) or theFRP, and thus the individual designations &(e ) . It ensures thatthe FRP moves along the path c(8 ) with desired speed u.(O, t) .

The formation acts as “one unit” where all vessels have the samepriority. This means that there i s no leader vessel which takespriority over the others.

In practice, however, in for example a fleet of ships, the cen-tral guidance system should be processed in the computer on-board one ship where then also the “formation chief’ sets the

desired path and speed along it.Simulation of three supply sh ips in a transversal line forma-

tion demonstrated the performance and robustness of the con-troller. The first simulation illustrated the robustness towardsenvironmental disturbances, while the second simulation ana-lyzed a scenario of thrust saturation failure in one ship.

RE FE RE N CE S

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Nonlinear Formation Control of Marine Craft