Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012
Adaptive Formation Control of Multiple Autonomous Underwater Vehicles Bidyadhar Subudhi, Senior Member IEEE
Basant Kumar Sahu
Centre for Industrial Electronics and Robotics Dept. of Electrical Engineering National Institute of Technology, Rourkela-769008, India
Centre for Industrial Electronics and Robotics Dept. of Electrical Engineering National Institute of Technology, Rourkela-769008, India
Abstract: This paper presents an adaptive formation control for a multiple Autonomous underwater vehicles (AUVs) using Lyapunov stability criterian. This controller considers the hydrodynamic parameter uncertainties of the AUVs. Simulations are performed considering three AUVs in a group and the results obtained demonstrate effective formation control of these AUVs while tracking a circular trajectory.
way points. In this method the leader navigates the waypoint using acoustic long base line measurements of the position. A leader-follower formation method based on line-of-sight methods has been used in [8] in which the controller is developed using feedback linearization methods. The formation control of marine surface crafts using the Lagrangian mechanics has been proposed in [12] in which the distance between members are used as constraints. The formation control of multiple unmanned vehicles has been investigated in [13], in which two types of controllers i.e. LSi and L-L has been used. The virtual leader concept whose trajectory is known to all the members belong to formation is introduced in this paper. Formation control of AUVs and obstacle avoidance, which is based on the potential function approach, has been proposed in [14]. In this paper the total maneuvering area has been divided into three different parts such as safety area, avoidance area and danger area. Control laws for different areas have been developed successively based on the potential functions. A formation control law of multiple AUVs using fixed interaction topology criteria has been developed in [15], where an adaptive sliding variable structure control law has been designed for controlling this formation. A nonlinear formation keeping and mooring control of multiple AUVs in chained form has been proposed in [16], where the leader follower method of formation control employing integrator back stepping has been used. The above investigations have not considered the uncertainties due to hydrodynamic damping factors. But it is important to consider these uncertainties to achieve good formation performance. Hence the hydrodynamic effects are considered and a new adaptive control law is developed for the formation control of the AUVs in this paper using Lyapunov stability criteria.
Key Words: AUV, Formation Control, Adaptive control law
I. Introduction Formation control of multiple AUVs is an important control problem in recent years due to several advantages as follows. AUVs in formations instead of a single AUV increase the robustness and efficiency of the system while providing redundancy, reconfiguration ability and structure flexibility [1]. The formation control of AUVs have applications also in commercial purposes such as in gas and oil industries, in precise surveys or areas where traditional bathymetric surveys are less effective, post-lay pipe surveys and in military field to map mines area [17]. Controlling a single AUV is difficult due to its nonlinear and uncertain dynamics thus formation control of multiple AUVs is more challenging. In [2], control of a group of ships in formation has been investigated, where the objective is to maintain positions of ships in formation along with a virtual formation reference point (FRP). The above problem considers two tasks named geometric task and dynamic task. The FRP methods have been used in [3], alongwith some inter vessel communications tasks taking into account. A decentralized controller method has been proposed in [5] where the inter vessel communication problems and FRP methods is used to solve the formation control of a group of vessels. The problems of formation control of multiple AUVs, such as geometric task, dynamic task and synchronization tasks have been solved by using Lyapunov stability based back stepping controller in [6]. A leader follower algorithm has been developed for multiple AUVs formation in [4, 7]. These algorithms maintain a fixed geometrical formation which navigating the mission
The organization of the paper is as follows. Section II describes the problem formulation. AUV kinematics and dynamics were reviewed in brief and an adaptive control law for formations of multiple AUVs is developed in section III. In this section the stability of the developed controller using Lyapunov method has been proved. To verify the efficacy of the control law for the formation AUVs, simulation results are presented in section IV. The conclusions are made in section V.
Corresponding authors: Bidyadhar Subudhi, Centre for Industrial Electronics and Robotics, Dept. of Electrical Engineering, National Institute of Technology, Rourkela-769008, India, E:
[email protected]. Basant Kumar Sahu, Centre for Industrial Electronics and Robotics, Dept. of Electrical Engineering, National Institute of Technology, Rourkela769008, India, E:
[email protected].
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Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012 II. Problems Formulation
III. Deevelopment of Adaptive Forrmation controol law
Starting frrom arbitrary points p in space, the AUVs arre to track the desiired circular trrajectory mainttaining formattion. i.e. the errorrs found betw ween the positions of desired trajectory and positions of AUVs A should bee zero as follow ws.
To develop an addaptive control for achieving successful formatiion, a brief revview on AUV kinematics k andd dynamics were prroved in this seection. (i) AUV Kinemaatics and Dynaamics
(1)
lim η - η r = 0
Thee AUV kinem matics and dynnamics in six degrees of freedom m i.e. the vehhicle is movinng in three diimensional spaces are briefly reeviewed here. There are twoo types of frames of references are taken intoo account i.e. body b fixed frame of o reference {B B} and anotheer is earth fixed frame of referennce which is known k inertial frame of refeerence {I}. The oriigin of B is coinciding with the t centre of mass m of the vehiclee.
t→ ∞
where η = position and a orientation n vector of AU UV
ηr = desired trajectory t and orientation possition vector
Fig. 1. Schemattic representation of three AUVss moving in circcular trajectory keepingg formation.
During foormation it iss necessary thhat, the distaance between AUV V1 and AUV2 is equal to thhat of the distaance between AUV V2 and AUV3 i.e. i (2) lim η - η = η - η t→∞
where
1
2
2
Fig. 3. Schematic S represenntation of AUV showing s forces annd torques in earth fixeed frame of referennce
Thee motion of an AUV in six deegrees of freeddom (DOF) can be described by thhe following vectors v [9]:
3
η = [ x, y,z,φ ,θ,ψ ]
T
η1 , η2 , η3 are positioon vectors of AUV1, A AUV2 and
AUV3 respecttively (Fig. 2).
v = [u,v,w, p,q,,r ]
T
(3)
τ = [ X,Y,Z,K,M M,N ] where η is the position and orientatiion vector in the t inertial T
frame. x, y, z are the coordinates off position and φ, θ, ψ are orientattion along lonngitudinal, trannsversal and veertical axes respecttively. v is the velocity vectoor with coordinnates in the body-fiixed frame. u, v, v w denote linear velocitties p, q, r are anggular velocitiess. X, Y, Z aree forces, K, M, M N denote momennts. τ is the forcces and momennts acting on thhe AUV in the boddy-fixed frame.. Thee nonlinear dyynamical and kinematical k eqquations of motionn can be expressed as [9]: & & + g(η) = τ && + C(η, η)η & η& + D(η, η)η M (η)η η& = J ( η ) v Fig. 2. Schhematic represen ntation of positioons of three AUVs A maintaining samee distance among th hem during keepinng formation.
(4)
where M(η) is the inertia matrixx including addded mass,
C(η, η) η& is the matrrix of Coriolis and centrippetal terms 2
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Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012 & denottes hydrodynaamic including addded mass. D(η, D η) damping andd lift matrix and g(η) is the vector of gravitational forces and moments. J(η) is veloocity e fixed fram mes. transformationn matrix betweeen robot and earth This transform mation matrix can c be presenteed as ( 03×3 ⎤ ⎡ J (η) J(η) = ⎢ 1 ⎥ ⎣ 03×3 J 2 (η) ⎦
where
s = e v + Λe p e p = η - ηr ev = v - vr m Λ = Possitive definite matrix
(5)
It will w proved shhortly that by choosing the parameter adaptattion law as givven below in eq.(11), the closed loop formatiion control off AUVs achievve the desiredd trajectory trackingg and ensure thhe stability of the t system. T & (11) αˆ = -ΓY Y s where Γ is the positive definite symm metric matrix.
s ⎤ ⎡cψcθ -ssψcφ +cψsθsφ sψsφ +cψcφ sθ ⎢ J1 (η η) = ⎢sψcθ cψ ψcφ +sφ sθcψ -cψsφ +sθsψc ψcφ ⎥⎥ ⎥⎦ ⎢⎣ -sθ cθcφ cθsφ (6)
⎡1 and J (η)) = ⎢ 0 2 ⎢ ⎢⎣ 0
sφ tθ
cφ tθ ⎤ -sφ ⎥⎥ cφ / cθ ⎥⎦
cφ sφ / cθ c
(10)
Thee structure of the t proposed adaptation a conntrol law is shown below.
(7)
with s( • )= sin( s • ),c( • )= = cos( • ),t( • ) = tan( • ) Assumptions Taken Due to pressence of asym mmetrical com mplexities in body b structure of AUV, A deriving control c law is very difficult. For sake of conveeniences the fo ollowing assum mptions are takken. These are as follows. f CM (ccenter of masss) and CB (Center of buoyancy) coincides each other. Masss distributionn all over the bodyy is homogeneo ous. The hydroodynamic term ms of higher order as a well as pitch and role motioons are negligiible.
Fig. 4. Schemaatic representation of control structurre
Considdering four DOF of AUV moddel for simpliccity, eq. (8) can be simplified as follows f
(ii) Formatiion control law w In AUV syystems, the parrameter uncertainties are present initially and which w may cau use inaccuracyy or instabilityy for the control systems s if th hese uncertainnties will not be compensated properly. Adaaptive control law is thereffore needed used too maintain con nsistent perform mance of a sysstem in presence of above unccertainties by estimating thhese parameters. Thhe adaptation mechanism is used to adjustt the parameters inn the control laaw. Managingg the uncertainnties present in thhe equation (4), ( an adapttive controllerr is designed so that t it will fo orce the AUV Vs to track in the desired trajecctories. For th his, define a regressor maatrix hat & η& r , && Y = Y(η, η, η r ) such th
⎡ m11 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0
]
T
r
is
the
position
Similarly v r = [ur , vr , wr , pr , qr , rr ]
and
0 ⎤ ⎡ u&r ⎤ ⎡ d11 0 ⎥⎥ ⎢⎢ v&r ⎥⎥ ⎢⎢ 0 + 0 ⎥ ⎢ w& r ⎥ ⎢ 0 ⎥⎢ ⎥ ⎢ m44 ⎦ ⎣ r&r ⎦ ⎣ 0
0 d 22 0 0
0 0 d 33 0
0 ⎤ ⎡ ur ⎢ 0 ⎥⎥ ⎢ vr 0 ⎥ ⎢ wr ⎥⎢ d 44 ⎦ ⎣⎢ rr
ur vr wr rr
⎤ ⎡τx ⎤ ⎥ ⎢ ⎥ ⎥ = ⎢τ y ⎥ ⎥ ⎢τz ⎥ ⎥ ⎢ ⎥ ⎦⎥ ⎣ τ r ⎦
m11u&r + d11 ur ur = τ x m22 v&r + d 22 vr vr = τ y
iss the velocity and
(15)
m33 w& r + d 33 wr wr = τ z
angular veelocity vector of o the referencee trajectory. Let the control law has the following fo structture (9) τ = Y αˆ - K D s simple P.D teerm. where Yαˆ = Feed F forward term, t = a K Ds ve definite gain n matrix and s = error vectorr. KD is a positiv
m44 r&r + d 44 rr rr = τ r Equatioon (15) can be expressed in matrix m form as
Y(v r , v& r )α = τ
(16)
where v r = [ u r ,vr ,w wr ,rr ]
T
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0 0 m33 0
Equatioon (13) can bee presented im mplicitly in thee following form
orientationn vector of the referencee trajectory with w coordinatees in the inertiaal frame T
0 m22 0 0
(13) Vs to follow bee defined Let the reference trajeectory the AUV as folloow x&r = ur , y& r = vr , z&r = wr ,ψ& r = rr (14)
&& r + C(η, η& )η& r + D(η, η)η & & r + g(η) = Y(η , η, & η& r , && Mη η r )α (88)
η r = [ x r , y r , z r , ϕ r , θ r ,ψ
(12)
M && η r + D ( η , η& )η& r = τ or
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(17)
Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012 ⎡u&r ur ur 0 0 0 ⎢ 0 0 v&r vr vr 0 Y(vr , v& r ) = ⎢ ⎢0 0 0 0 w& r ⎢ 0 0 0 0 ⎣⎢ 0 Y(vr , v& r ) is the regressor matrix.
0
0
0
0
wr wr
0 r&r
0
α = [m11 , d11 , m22 , d 22 , m33 , d33 , m44 , d 44 ] and τ = ⎡τ x ,τ y ,τ z ,τ r ⎤ ⎣ ⎦
& - 2(C + D) is the But for an AUV it can easily verify that M skew-symmetric matrix. Using system dynamics eq.(25)will reduced to
0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ rr rr ⎦⎥
& (t) = sT ( τ - Mη &&r - Cη& r - Dr η& - g ) V 1 Now second term of eq.(20) is rewritten as
(18)
1 V2 (t) = α% T Γ-1α% , α% = αˆ - α 2 α&% = α&ˆ , as α = constant definite vector
T
The first derivative of eq.(27) is & (t) = α&% T Γ -1α% V 2 Therefore
The regressor matrix Y(vr , v& r ) helps to represent the AUV model in a linear parametric form. The derivation of the regressor matrix of a high-DOF is very tedious. For that reason here only for DOF is taken into account. In real time realization the regressor matrix is to be computed in every control cycle. Each parameter of the matrix is computed in each iteration [18].
& =V & (t) + V & (t) = V(t) 1 2 &&r - Cη& r - D r η& - g ) + α&ˆ T Γ -1α% s T ( τ - Mη
(28)
(29)
& = s T ( τ - Yα% ) + α&ˆ T Γ -1α% V(t)
(30) Taking the controller input τ = Yαˆ - K Ds and solving eq.(30), one obtains, & = s T Yα% - s T K s + α&ˆ T Γ -1α% (31) V(t) D
Proof of Stability of the Proposed control Law
Let V(t) be the Lyapunov candidate function given by
Substituting α&ˆ = -ΓYTs ,
(19)
& V(t) becomes
& = -s T K s ≤ 0 V(t) D
(32)
Eq.(32) satisfies Lyapunov stability criteria for AUV systems to be stable. Hence the proposed adaptive controller given by eq.(32) is stable.
Taking derivative of eq. (19) gives
IV. Results and Discussions
& =V & (t) + V & (t) V(t) 1 2
(20)
& (t) = s Ms& + 1 sT Ms & V 1 2 T
Let a reference circular path in space for first AUV described as follows
(21)
Using equation (10) referring [10],
x r1 = 10sin(0.01t)
s = η& − η& r
y r1 = 10cos(0.01t)
one gets
z r1 = 10,ψ r1 =
1 & & (t) = sT M(η && - && V ηr ) + s T Ms 1 2
(22)
& (t) it obtains as, and solving for V 1 1 & & (t) = s T ( τ - Cη& - Dη& - g - Mη &&r ) + sT Ms V 1 2 & & But η = s + ηr
(23)
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x&r1 , y& r1 , z&r1 ,ψ& r1 , && xr1 , && yr1 , && zr1 ,ψ&&r1 , x&r 2 , y& r 2 , z&r 2 ,ψ& r 2 , && xr 2 , && yr 2 , && zr 2 ,ψ&&r 2 , x&r 3 , y& r 3 , z&r 3 ,ψ& r 3 , && xr 3 , && yr 3 , && zr 3 ,ψ&&r 3
(24)
(
π
x r2 = 20sin(0.01t) x r3 = 30sin(0.01t) (34) y r2 = 20cos(0.01t) y r3 = 30cos(0.01t) π π z r2 = 10,ψ r2 = z r3 = 10,ψ r3 = 3 3 The first and second derivatives of position and velocity terms used in simulation are
1 & & (t) = sT ( τ - Cη& - Dη& - g - Mη &&r - (C + D)s ) + sT Ms V 1 r r 2 1 2
(33)
Similarly the reference paths for second and third AUVs are described as
Substituting the value of M(η)η && from eq. (4) into eq.(22)
)
& (t) = sT ( τ - Mη & - 2(C + D) s or V &&r - Cη& r - Dr η& - g ) + sT M 1
. The other parameters which are necessary for simulation taken from reference [11] are given in table 1. Table 1. Parameters of AUV used for simulation
(25)
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(27)
using eq.(8) in eq.(29) one gets
The stability of the proposed adaptive controller is proved by using Lyapunov stability craterian.
1 V(t) = ⎡⎣sT Ms + α% T Γ-1α% ⎤⎦ = V1 (t) + V2 (t) 2 1 1 where V1 (t) = s T Ms and V2 (t) = α% T Γ-1α% 2 2
(26)
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Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012 Mass (kg) m11=99.00 m22=108.50 m33=126.50 m44=29.10
Damping coefficients (kg/s) d11 = 10+227.18|u| d22 = 405.42|v| d33 = 10+227.18|w| d44=1.603+12.937|r
Others
KD = diag{10,20,50,10} Λ = diag{50,10, 20, 20} Γ = diag{200,10, 20,0,0, 0,10,10}
The simulation results for these required trajectories are given below. Formation Control of Three AUVs in a Circular path in space AUV1 AUV2 AUV3 Desired of AUV1
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Fig. 7. Reference velocity and actual velocity of AUV1
Fig.7 shows that of desired and actual linear velocities as well as angular velocities. From both fig.6 and fig.7, it is clear that actual and reference positions and velocities of AUV1 are coinciding with each other.
Position(z) (m)
10 8 6 4 2 0 40 20
40 20
0
0
-20 Position(y) (m)
-20 -40
-40
Position(x) (m)
Fig. 5. Circular trajectory tracking and formation of a group of three AUVs.
Fig.5 shows the result of group of three AUVs tracking circular path. From this figure it is seen that the desired trajectory position coincides with the actual trajectories. Fig. 8 Position errors of AUV1
Fig.8 shows the errors found comparing the reference and actual positions as well as orientation of AUV1 in the earth fixed inertial frame of reference. From this figure it found that the position errors converge to zero. Hence the AUV1 is tracking in the reference path.
Fig. 6. Reference position and actual position of circular path of AUV1
Fig.6 shows comparison of the desired and actual positions as well as orientation of AUV1 in the earth fixed inertial frame of reference.
Fig. 9 Velocity errors of AUV1
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Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012 Fig.9 shows the errors found comparing the desired and actual linear velocities as well as angular velocity of the AUV1 in the body fixed frame of reference. From this figure it found that the velocity errors converge to zero.
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Distance between AUV2 and AUV3 (m)
10
8
6
4
2
0
-2 0
100
200
300
400
500 600 Time-t (s)
700
800
900
1000
Fig. 12. Distance between AUV2 and AUV3 during formation 1.5
Error of distance among AUVs (m)
1
Fig. 10. Forces and torque of AUV1
Fig.10 shows the forces and torque applied to the AUV1 during moving in formation control.
Distance between AUVs during of formation To avoid collision and uniforma motion in the desired trajectories, the vehicles should maintain equal distance of separation to accomplish the given task in a group. As the gooup consist of only three AUVs, the distance between AUV1 and AUV2 is equal to that of the distance between AUV2 and AUV3.
Distance between AUV1 and AUV2 (m)
4 2 0 -2
300
400
500 600 Time (s)
700
800
900
-1.5 -2 -2.5
0
100
200
300
400
500 600 Time-t (s)
700
800
900
1000
From Fig.11 it is clear that the distance between AUV1 and AUV2 remains constant at 10 m. during maintenance of formation moving in circular path. Fig.12 shows the distance between AUV2 and AUV3 also remains constant at 10 m. during maintenance of formation moving in circular path. Fig.13 shows the difference of distance between AUV1 and AUV2 and that of AUV2 and AUV3 during maintenance of formation which is zero, during moving in circular path which is the error of distance of formation control. From figures 5-13, it is clear that, the trajectory tracked by AUV1 coincides with the desired path and the errors goes to zero. It is also clear that the distances among AUVs remain constant during travelling in the desired trajectories in formation. Similarly for other AUVs in the group the results can found out.
6
200
-1
Fig. 13. Error of distance among AUVs during formation
8
100
-0.5
-3.5
10
0
0
-3
12
-4
0.5
1000
Fig. 11. Distance between AUV1 and AUV2 during formation
V. Conclusions
An adaptive formation control law for a group of has been developed which considers the uncertain parameters associated with the hydrodynamic effects. This law is used to guide the AUVs to track in the desired trajectories. The stability of the proposed control law has been verified by using the Lyapunov stability criterion. Simulation results
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Proceedings of Advances in Control and Optimization of Dynamic Systems ACODS-2012 2009 IEEE International Conference on Mechatronics and Automation, Changchun, China, p. 4863-4867 August 9–12, 2009. [9] T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley & Sons, 1994 [10] Slotine, J.J.E., and Li, W., Applied Nonlinear Control, Prentice-Hall, 1991. [11] J. Garus, "Nonlinear Adaptive Control of Underwater Robot in. Horizontal Motion," in Proceedings of the 6th WSEAS Int. Conf. on systems theory and scientific computation. [12] I. A. F. Ihle, J. Jouffroy, and T. I. Fossen, "Formation Control of Marine Surface Craft: A Lagrangian Approach," IEEE Journal of Oceanic Engineering, vol. 31, no. 4, pp. 922-934, 2006. [13] F. Fahimi, Sliding mode formation control for under-actuated surface vessels, IEEE Transactions on Robotics, 23 (3), p 617–622, 2007. [14] Jia, Q. and Li, G. “Formation control and obstacle avoidance algorithm of multiple autonomous underwater vehicles (AUVs) based on potential function and behaviour rules”, IEEE International Conference on Automation and Logistics, Jinan, China, p.569-573, August 18 -20, 2007. [15] Rongxin Cui, "Formation control of autonomous underwater Vehicles under fixed topology," IEEE International Conference on Control and Automation, IEEE Press, p. 293-2918, May 2007. [16] E. Yang and D. Gu, “Nonlinear formation-keeping and mooring control of multiple autonomous underwater vehicles,” IEEE/ASME Transactions on Mechatronics, vol. 12, no. 2, p. 164-178,2007. [17] Blidberg, D Richard. “The Development of Autonomous Underwater Vehicles (AUV); A Brief Summary”. Autonomous Undersea Systems Institute, Lee New Hampshire, USA. [18] An-Chyau Huang; Ming-Chih Chien, ” Adaptive control of robot manipulators : a unified regressor-free approach” published by World Scientific Publishing Co. 2010.
show the efficacy of the control law developed when applied to track the desired trajectories. References [1] D. J. Stilwell and B. E. Bishop, “Platoons of underwater vehicles,” IEEE Control Systems Magazine, vol. 20, p. 45 –52, Dec. 2000. [2] R. Skjetne, S. Moi, and T. I. Fossen, “Nonlinear formation control of marine craft,” in Proc. 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA, p. 1699–1704, Dec 2002. [3] R. Skjetne, I.-A. F. Ihle, and T. I. Fossen, “Formation Control by Synchronizing Multiple Maneuvering Systems,” in Proc. 6th IFAC Conference on Manoeuvring and Control of Marine Crafts, Girona, Spain, p. 280–285, Sep. 17-19 2003. [4] D. B. Edwards, T. A. Bean, D. L. Odell, and M. J. Anderson, "A leaderfollower algorithm for multiple AUV formations," IEEE/OES on Autonomous Underwater Vehicles, vol. 1, p. 40-46, 2004. [5] I. A. F. Ihle, Roger Skjetne1, and T. I. Fossen, “Nonlinear formation control of marine craft with experimental results,” in Proc. 43rd IEEE Conf. on Decision & Control, Paradise Island, The Bahamas, p. 680–685, 2004. [6] Weisheng Van, "Formation control of underactuated autonomous underwater vehicles in horizontal plane," Proceedings of the IEEE International Conference on Automation and Logistics, IEEE Press, p. 822827, Sep. 2008. [7] Rongxin Cui , Shuzhi Sam Ge , Bernard Voon Ee How , Yoo Sang Choo, “Leader-follower formation control of underactuated AUVs with leader position measurement”, Proceedings of the IEEE international conference on Robotics and Automation, May 12-17, 2009, Kobe, Japan, , p.967-984, 2009. [8] Y. Wang, W. Yan, W. Yan, “A Leader-Follower Formation Control Strategy for AUVs Based on Line-of-Sight Guidance,” Proceedings ofthe
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