Nonlinear Formation Control of Marine Craft

Transcription

1 Nonlinear Formation Control of Marine Craft Roger Skjetne, Sonja Moi, and Thor I. Fossen Abstract This paper investigates formation control of a fleet of ships. The control objective for each ship is to maintain its position in the formation while a (virtual) Formation Reference Point (FRP) tracks a predefined path. This is obtained by using vectorial backstepping to solve two subproblems; a geometric task, and a dynamic task. The former guarantees that the FRP, and thus the formation, tracks the path, while the latter ensures accurate speed control along the path. A dynamic guidance system with feedback from the states of all ships ensures that all ships have the same priority (no leader) when moving along the path. Lyapunov stability is proven and robustness to input saturation is demonstrated using computer simulations. Keywords Nonlinear control; Maneuvering; Backstepping; Formation control; Ship control. I. INTRODUCTION The topic of this paper is control of a group or fleet of ships in formation. A design procedure based on maneuvering and not trajectory tracking is applied for this purpose. In maneuvering, the desired behavior of the plant in the output space is separated into two subproblems; 1) converging to and following a desired parametrized path, and 2) satisfying a desired dynamic behavior along the path, here referred to as a speed assignment, see [1], [2]. This is contrary to trajectory tracking where the main goal is for all time t to track a desired output y d (t) which implicitly must contain information of both the desired path and speed. The field of formation control with applications towards mechanical systems, ships, aircraft, satellites, etc., has recently received a lot of attention, see [3], [4], [5], [6], for instance. In [4] a procedure for the design of n 1controllers which ensure that n 1autonomous vehicles follow a path without altering their formation, has been developed. To determine the locations along the path, they used an orthogonal projection from the state of one of the vehicles, being the leader. The time in the n already existing trajectory tracking controllers (for the other vehicles) is then replaced with this projection. If the leader slows down or stops, all the vehicles will slow down and stop. Therefore, the speed and performance of the leader will affect all other members in the formation, but not vice versa. A full-state maneuvering controller was proposed in [7] which ensures that the states converge to a desired path ξ(θ) and then proceed along it. This was done by converting a trajectory tracking controller into a maneuver regulation controller. To determine the path variable θ,theyusedap-orthogonal numerical projection to find the point on the path which minimizes the weighted distance between x(t) and ξ(θ(t)). The methodology of [7] was a hybrid result, and applies to feedback linearizable systems, where the desired path is specified for the full state. More recently, [8] has extended this methodology to output maneuvering by using backstepping. However, the proposed controller cannot be said to have the same advantages as in [8] since the numerical projection only works on the output subset All three authors are with the department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. E- mails: skjetne@ieee.org, moi@stud.ntnu.no and tif@itk.ntnu.no of the states. Extensions to systems of arbitrary relative degree was made by [1] for systems in vectorial strict feedback form, and in [9], [1] it is also demonstrated that application of a dynamic gradient optimization algorithm recaptures the advantage of minimizing the weighted distance between x(t) and ξ(θ(t)). The main contribution of this paper is to represent and solve the formation control problem by defining a Geometric Task and a Dynamic Task, constituting a Maneuvering Problem, as defined in [1], [2]. With this design procedure, a formation is viewed as a flexible system (as one unit) that maneuvers along a parametrized path. Each individual vessel will have a relative position to a point called the Formation Reference Point (FRP). The Geometric Task ensures that the individual ships converge to their positions in the formation and stay at their respective paths relative to the FRP. The Dynamic Task, which in this case will be a speed assignment for θ, ensures that the FRP (and thus the ships) will move along the path with the specified speed. In this approach there will be no leader of the traditional type. The desired motion for the FRP is equally based on the states of all vessels and gives a centralized guidance system. This implies that it is easy to reconfigure the formation. The formation controller is simulated using three offshore supply vessels. Emphasis is placed on maneuvering performance and input saturation. Since the update law for θ contains an optimization algorithm that minimizes an instantaneous Lyapunov cost function based on the states of all vessels, it is demonstrated that the proposed algorithm handles input saturations effectively by simply modifying the speed of the formation so that the errors in the formation geometric task is unaffected. This is one of the advantages of maneuvering over trajectory tracking. Notation: Abbreviations like GS, LAS, LES, UGAS, UGES, etc., are G for Global, L for Local, S for Stable, U for Uniform, A for Asymptotic, and E for Exponential. Differentiation with respect to θ is indicated by primes: ξ (θ) := ξ θ, ξ (θ) := 2 ξ θ 2. The Euclidian vector norm is x := x > x 1/2. For a matrix P = P > >, let p m := λ min (P ) and p M := λ max (P ). II. FORMATION SETUP A formation with n vessels is created by a set of formation designation vectors l i, i =1,...,n, relative to a Formation Reference Point (FRP), see Figure 1. The idea is for the FRP to follow a given parametrized path ξ(θ) with a desired formation speed along it. Let the FRP be the origin of a moving body frame {b}, and denote the earth fixed frame {e}. The path is in general not a straight line, but a feasible curve in the output space of the vessels. The individual path for vessel i is then ξ i (θ) =ξ(θ)r e b(θ)l i (1) where Rb e is a kinematic rotation matrix from {b} to {e}.

2 y e l 3 l 2 l 4 l 1 l 5 FRP Fig. 1. Illustration of a formation setup. x e ξ(θ) For ships moving on the ocean surface, the output is the 3 DOF vector η = [x, y, ψ], where (x, y) is the position and ψ is the heading. The desired path is then given by ξ(θ) = [x d (θ), y d (θ), ψ d (θ)] >. The tangent vector along the path in the (x, y) directions, T (θ) =[x d (θ), y d (θ)]>, is chosen as the x-axis of the moving frame {b}. The angle of the tangent vector in the {e} frame then gives the desired heading ψ d (θ) = arctan µ Ty (θ) T x (θ) =arctan µ y d (θ) x d (θ). (2) The rotation matrix R(θ) fortheshipsisgivenby Rb(θ) e =R(θ) := cos ψ d(θ) sin ψ d (θ) sin ψ d (θ) cosψ d (θ) (3) 1 and note that differentiation gives Ṙ(θ) =R (θ) θ = R(θ)S(θ) θ (4) where S(θ) is a skew-symmetric matrix ψ d(θ) S(θ) = ψ d(θ) (5) and ψ d(θ) = x d (θ)y d (θ) x d (θ)y d (θ). (6) (x d (θ))2 (yd (θ))2 III. PLANTS AND PROBLEM STATEMENT An uncertain mechanical system is represented by the vector relative degree two model ẋ 1i = G 1i (x 1i )x 2i f 1i (x 1i )E 1i (x 1i )δ 1i (t) ẋ 2i = G 2i (x i )u i f 2i (x i )E 2i (x i )δ 2i (t) (7) y i = h i (x 1i ) where the subscript i denotes the i th system. x ji R m are the states and x i denotes the vector x i := x > 1i,x> 2i >, y i R m are the system outputs, u i R m are the controls, and δ ji are unknown bounded disturbances. The matrices G ji and h i := hi x 1i are invertible for all x i, the output maps h i (x 1i ) are diffeomorphisms, and all functions are smooth. The uncertainty vectors δ ij represent bounded exogenous disturbances and/or unmodeled dynamics that is uniformly bounded in the state space. The particular bounds do not need to be known. In the proceeding, the mechanical systems described by (7) are referred to as vessels. For a cluster of n vessels, each represented by a position output y i, let a Formation Reference Point (FRP) represent the position of the formation as a whole, and let each individual vessel y i have a designation l i relative to the FRP. Let ξ(θ) be the desired path for the FRP and then ξ i (θ) =ξ(θ) R(θ)l i is the corresponding path for the individual vessels. We are now ready to state the Formation Maneuvering Problem along the lines of [2]: Definition 1: The Formation Maneuvering Problem: Design a set of robust control laws for the individual vessels and a guidance system that solve the tasks: 1. Geometric Task: For each ε GT >, force the output y i to enter an ε GT neighborhood of the desired path ξ i (θ), that is, T such that, y i (t) ξ i (θ(t)) ε GT, t T (8) for any C 1 function θ(t). 2. Dynamic Task: For each ε DT >, force the speed θ to enter an ε DT neighborhood of a desired speed assignment υ s (θ,t), that is, T such that, θ (t) υ s (θ(t),t) ε DT, t T (9) The geometric task ensures that the individual vessels converge to and stay at their designated positions l i in the formation. The speed assignment task ensures that the FRP will move along the path ξ(θ) with a desired velocity υ s (θ,t). IV. CONTROL DESIGN In the procedure that follows, a recursive backstepping design is proposed to solve the formation maneuvering problem for n vessels with the dynamics given in (7). Step 1: Define the error variables z 1i := y i ξ i (θ) =y i ξ(θ) R(θ)l i (1) z 2i := x 2i α 1i (11) ω s := υ s (θ,t) θ (12) where α 1i are virtual controls to be specified later. Differentiating (1) with respect to time results in ż 1i = h i G 1i z 2,i h i G 1i α 1i h i f 1i h i E 1i δ 1i ξ (θ) θ R(θ)S(θ)l i θ. (13) Denote ρ 1i (θ) :=ξ (θ)r(θ)s(θ)l i, and choose Hurwitz design matrices A 1i, so that P 1i = P 1i > > are the solutions to P 1i A 1i A > 1i P 1i = Q 1i where Q 1i = Q 1i >. Define V 1 := z 1iP > 1i z 1i (14)

3 whose time derivative then becomes V 1 = 2z 1iP > 1i h i G 1i z 2i 2z 1iP > 1i ρ 1i ω s 2z 1i > P 1i [ h i G 1i α 1i h i f 1i ρ 1i υ s h i E 1i δ 1i ]. The first virtual controls α 1i are chosen as α 1i = G 1 1i ( h i) 1 [A 1i z 1i h i f 1i ρ 1i υ s α i ] (15) where α i are damping terms to be picked. Define the first tuning functions, τ 1i R, as τ 1i := 2z > 1iP 1i ρ 1i. (16) To handle the perturbations we use nonlinear damping and apply Young s inequality V 1 = z 1iQ > 1i z 1i τ 1i ω s 2z 1iP > 1i h i G 1i z 2i 2z 1iP > 1i h i E 1i δ 1i 2z 1iP > 1i α i z 1iQ > 1i z 1i τ 1i ω s 2z 1iP > 1i h i G 1i z 2i 2z > 1iP 1i α i 1 2 κ 1i ( h i ) E 1i E > 1i ( h i ) > P 1i z 1i 1 κ 1i δ > 1iδ 1i and the nonlinear damping terms are picked as α i = 1 2 κ 1i ( h i ) E 1i E > 1i ( h i ) > P 1i z 1i, κ 1i > (17) which gives V 1 z 1iQ > 1i z 1i τ 1i ω s 2z 1i > P 1 1i h i G 1i z 2i δ > 1i κ δ 1i. (18) 1i In aid of the next step, we differentiate α 1i to get where α 1i = σ 1i ρ 2i θ $1i δ 1i (19) σ 1i := α 1i [G 1i x 2i f 1i ] α 1i x 1i t (2) ρ 2i := α 1i θ (21) $ 1i := α 1i E 1i. x 1i (22) Step 2: Differentiating (11) with respect to time gives ż 2i = G 2i u i f 2i E 2i δ 2i σ 1i ρ 2i θ $1i δ 1i. (23) Choose Hurwitz design matrices A 2i so that P 2i = P > 2i > are the solutions to P 2i A 2i A > 2i P 2i = Q 2i <, and define V 2 := V 1 whosetimederivativebecomes z 2iP > 2i z 2i (24) V 2 z 1iQ > 1i z 1i τ 1i ω s 2z 2iP > 2i ρ 2i ω s 2z 2iG > > 1i ( h i ) > 1 P 1i z 1i δ > κ 1iδ 1i 1i 2z 2iP > 2i [G 2i u i f 2i E 2i δ 2i σ 1i ρ 2i υ s $ 1i δ 1i ]. The control laws are then chosen as u i = α 2 (x i, θ,t) = G 1 1 2i [P2i G> 1i ( h i ) > P 1i z 1i A 2i z 2i f 2i σ 1i ρ 2i υ s u i ] (25) where u i are nonlinear damping terms to be designed. Define z i := z 1i > > 2i z>,qi := diag (Q 1i,Q 2i ), and the final tuning functions as τ 2i := τ 1i 2z 2iP > 2i ρ 2i. (26) Using Young s inequality again, the derivative V 2 is bounded by V 2 z i > Q i z i τ 2i ω s 2z 2iP > 2i ½u i 1 2 κ 2i 1 κ 1i δ > 1iδ 1i ¾ E2i E 2i > $ 1i $ > 1i P2i z 2i 1 h i δ > κ 2iδ 2i δ > 1iδ 1i 2i and the final nonlinear damping terms u i are assigned as u i = 1 2 κ 2i E2i E 2i > $ 1i $ > 1i P2i z 2i, κ 2i >. (27) ³ Define i := [δ > 1i, δ > 2i] > 1 and K i := diag κ 1i 1 1 κ 2i, κ 2i. The result is then V 2 z i > Q i z i ω s n X τ 2i > i K i i. (28) If we disregard the sign indefinite tuning function terms, each system in the z i -coordinates is an ISS system from the disturbances i to z i. It follows that for any ε GT >, the output error z 1i (t) = y i (t) ξ i (θ(t)) ε GT, for some time t T, can

4 be guaranteed by choosing the nonlinear damping coefficients κ ji large enough, and this solves the Geometric Task. Next, we must deal with the tuning functions. Choosing ω s to solve the dynamic task is equivalent to a trajectory tracking design with θ = υ s (θ,t). A better choice is to design an update law for θ or ω s that uses feedback from the states of the vessels. In [9], [1] it was demonstrated that τ(x, θ,t):= τ 2i (x i, θ,t)= V 2 (x, θ,t), (29) θ that is, the total tuning function is the gradient of V 2 with respect to θ. We therefore consider the Direct Gradient Update Law and the Filtered Gradient Update Law next. Direct Gradient Update Law: From (28) and according to [9], [1], let ω s = µ τ 2i, µ > (3) which gives the new bound for (28) Ã n! 2 X V 2 z i > Q i z i µ τ 2i > i K i i, (31) and by choosing the gains K i large enough, we can guarantee any residual bound for z i (t). In particular, this means that as t, z 1i (t) ε GT, that is, y i (t) ξ i (θ(t)) ε GT, and each individual vessel enters its designation. The controller realization becomes θ = υ s (θ,t)µ τ 2i (x i, θ,t) (32) and since the states z i (t) are made small, the tuning functions τ 2i (t) are made small. Hence, as t, θ(t) υ s (θ(t),t) which satisfies the speed assignment. Since we can rewrite (32) as θ = υ s (θ,t) µ V 2 (x, θ,t) (33) θ it follows by the analysis in [9], [1] that choosing µ large induces a separation of time scales between the vessel dynamics and θ. In the fast time scale, (33) becomes a dynamic gradient optimization algorithm that selects the point on the path for the FRP which minimizes V 2 with respect to θ. Therefore, errors in the geometry of the formation are rapidly minimized with respect to the instantaneous cost function V 2 (x(t),,t). Filtered Gradient Update Law: In [1], [2], the update law was constructed as θ = υ s (θ,t) ω s P ω s = λω s µ n (34) τ 2i µ, λ > by extending the Lyapunov function to V = V 2 1 2µ ω2 s, which yields V z i > Q i z i λ µ ω2 s > i K i i. (35) 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 has the same gradient properties as discussed above if λ and µ are chosen large. Experience has shown, however, that the filtered version gives an improved numerical response for θ(t). The caveat is higher order in the controller. Remark 2: In the control laws (25) it is seen that each vessel only needs information about its own states, time, and the path variable θ. The guidance system, on the other hand, incorporate the dynamic update law, (33) or (34), which needs information of all states in the formation. Therefore, each vessel must communicate its state information to the guidance system which processes this and returns the path variable θ. Remark 3: In this setup, no precautionary measures are included to avoid inter-vessel collisions. Using maneuvering with gradient optimization, however, will minimize transients, improve performance, and in that way reduce the risk for collisions. A further idea would be to include potential functions in the overall Lyapunov function. V. CASE STUDY: SHIPS IN FORMATION For maneuvering of ships in formation we will use a model for which there is no coupling between the surge and the sway-yaw subsystems, see [11], [12]. Let η i =[x i,y i, ψ i ] > be the position vector in the {e} frame, where (x i,y i ) is the position on the ocean surface and ψ i is the yaw angle. Let ν i =[u i,v i,r i ] > be the {b} frame velocity vector. The subscript i denotes the i th ship. The equations of motion in surge, sway, and yaw for each ship is written η i = R i (ψ i )ν i ν i = Mi 1 D i ν i Mi 1 T i R i > (ψ (36) i)w where R i (ψ i ) is the rotation matrix (3), M i = M i > > is the system inertia matrix including the hydrodynamic added inertia, D i is the hydrodynamic damping matrix, T i =[T 1i,T 2i,T 3i ] > is the fully actuated vector of control forces and moments, and w is vector of environmental disturbances decomposed in the {e} frame. Remark 4: In the underactuated case, the maneuvering methodology is the same. We refer the reader to the final ship case in [9], and references therein, on this topic. The dynamical system (36) is in the form of (7), where η i is the output and T i is the control. Let the desired path for the FRP be µ y > η d (θ) = x d (θ) y d (θ) arctan d (θ) x d (θ) (37) where x d (θ) and y d (θ) are three times differentiable with respect to θ, and ψ d (θ) is given by (2). The individual paths for each ship are then η di (θ) =η d (θ)r(ψ d (θ))l i where l i =[l xi,l yi, ] >. Let u d be the desired surge speed for the FRP along the path. Then υ s (θ,t) is given by υ s (θ,t)= u d (t) p x d (θ) 2 y d (θ)2.

5 m/s y [meters] The design procedure in the previous section gives the following signals: 1 5 y e Response on the ocean surface z 1i := η i η d (θ) R(ψ d (θ))l i z 2i := ν i α 1i ρ 1i = η d (θ)r(ψ d(θ))s(θ)l i α 1i = R i > (ψ i)[a 1i z 1i ρ 1i υ s ] σ 1i = Ṙ> i (ψ i)r i (ψ i )α 1i R i > (ψ i)[a 1i R i (ψ i )ν i ρ 1i υ s ] ρ 2i = R i > (ψ i)[ A 1i ρ 1i η d (θ)υ s R(ψ d (θ))s 2 (θ)l i υ s R(ψ d (θ))s (θ)l i υ s ] x [meters] Fig. 2. Simulation of 3 offshore supply vessels in a line formation following a desired sinusoidal path. x e τ 2i =2z > 1i P 1iρ 1i 2z > 2i P 2iρ 2i Tuning functions T i = M i [ P 1 2i R> i (ψ i)p 1i z 1i Control laws A 2i z 2i Mi 1 D i ν i σ 1i ρ 2i υ s u i ] u i = 1 2 κ 2iP 2i z 2i where T i is the control law for Ship i. The controller realization, using a Filtered Gradient Update Law,is θ = υ s (θ,t) ω s ω s = λω s µ n P τ 2i (η i, ν i, θ,t). The following two simulations are performed for a formation of 3 ships. The numerical values of the M i and D i matrices, taken from [13], represent true data of supply ships that operate in the North Sea. In both simulations, the output path is given by (37), where x d (θ) =θ and y d (θ) = 5 sin 2π 4θ. The desired surge speed of the FRP starts out with the set-point u d =4m/s. At time t =5sthe formation chief sets the new formation speed to u d =1m/s. A. Simulation 1: Maneuvering with ocean disturbances The aim of this simulation is to show that with the formation maneuvering design we can robustly perform the path following maneuver for a formation of ships influenced by environmental disturbances. Starting off the path, we want the vessels to converge 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 l 1 = [,, ] >, l 2 =[, 15, ] > and l 3 =[, 15, ] >. This means that the FRP coincide with Ship 1, and the ships will travel in a transversal line formation as one unit. The environmental disturbances are w = sin (.1t), (38) 2 acting the same on all the vessels. To attenuate these disturbances, the nonlinear damping gains are set to κ 2i = 2. The other controller parameters are set as: A 1i = diag(.2,.2,.5), A 2i = diag(2, 2, 2), P 1i = diag(.2,.2, 1), P 2i = diag(1, 1, 4) and µ = λ =2. The initial conditions were η 1 () = >,, 2, π 4 η 2 () = π >, 5,, 3 η3 () = [, 5, ] >, ν 1 () = ν 2 () = ν 3 () = [1,, ] > and θ() = ω s () =. Figure 2 shows how the ships in the formation converge smoothly to their designated path and accurately track it. With the substantial environmental disturbances (38), the position error was attenuated to less than 1 m in x and y, and less than 1 in heading Surge speeds of the 3 vessels Time (Seconds) Fig. 3. Time-plot of the surge speeds, u 1 (t),u 2 (t),u 3 (t), for the three ships. In Figure 3 the surge speed of the ships are shown. Since the center ship are chosen to coincide with the FRP, this ship is seen to obtain the desired speed u d as assigned by the formation chief. The two side ships obtain a periodic path speed according to their individual positions, necessary to keep the formation. B. Simulation 2: Thrust saturation failure 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 θ is projected from the state of the leader vessel. Hence, only if the leader experiences a problem will the formation as a whole 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 ξ(θ(t)) along the path is equally influenced by the states of all the vessels through the tuning function (29) and the update law. Therefore, if one vessel experiences a problem, all the vessels will act upon it. We continue the experiment by forcing a saturation constraint on Ship 2, so that it will maximally be able to go with surge speed of 8 m/s. The surge speed assignment will be the same as in the previous simulation, that is, 4 m/s for t<5 s and 1 m/s for t 5 s, which now is infeasible for Ship 2. The environment in this simulation is disturbance free, w =, so that no nonlinear damping is required. The other controller parameters are set to: A 1i = diag(.5,.5,.5), A 2i = diag(2, 2, 2), P 1i = diag(.6,.6,.6), P 2i = u 1 u 2 u 3

6 m/s y [meters] m/s diag(1, 1, 4) and µ = λ =2. The initial conditions were η 1 () =,, π 5 >, η2 () = 1, 1, π 5 >, η3 () = [, 25, ] >, ν 1 () = ν 2 () = ν 3 () = [4,, ] > and θ() = ω s () = Speed Assignment υ s θ Dot 6 1 Response on the ocean surface 4 5 y e 2 x e Time [seconds] x [meters] Fig. 4. Resulting response of the formation when Ship 2 saturates. Interestingly, Figure 4 shows that the formation follows the path as desired in spite of the failure in Ship 2. Figure 5 reveals that the speed of the formation is considerable slower than the assigned speed of 1 m/s. In fact, the speed of the slowest vessel converges to its maximum speed of 8 m/s while the two other vessels follow at what speed necessary to keep the formation assembled. The formation is as fast as its slowest member. The important part is that the vessels keep following the path and therefore do not cause any accidents. This feature is due to the inherent gradient optimization algorithm that tries to minimize the Lyapunov cost function which incorporates the states of all the vessels, see [9], [1] Surge speeds of the 3 vessels Time [seconds] Fig. 5. Surge speeds of the ships, where Ship 2 maximally makes 8 m/s. Commanded speed for the formation was 1 m/s. Figure 6 shows a time-plot of the assigned speed υ s (θ(t),t) and the resulting response of θ(t). Clearly, θ(t) is slower than the assigned speed. VI. CONCLUSIONS A robust nonlinear control design method has been proposed that solves the Formation Maneuvering Problem as defined. Individual decentralized control laws were developed for each vessel in the formation. These control laws only uses information of each vessel s own states and, in addition, information about the desired path and speed comming from a centralized guidance system. The central guidance system incorporates information from all states to select the desired position ξ(θ) for the FRP, and thus the individual designations ξ i (θ). It ensures that the FRP moves along the path ξ(θ) with desired speed υ s (θ,t). u 1 u 2 u 3 Fig. 6. Time-plot of the speed assignment υ s(θ(t),t) for the FRP and the resulting response of θ(t). Notice that θ(t) is slower than the assigned speed. The formation acts as one unit where all vessels have the same priority. This means that there is no leader vessel which takes priority over the others. In practice, however, in for example a fleet of ships, the central guidance system should be processed in the computer onboard one ship where then also the formation chief sets the desired path and speed along it. Simulation of three supply ships in a transversal line formation demonstrated the performance and robustness of the controller. The first simulation illustrated the robustness towards environmental disturbances, while the second simulation analyzed a scenario of thrust saturation failure in one ship. REFERENCES [1] R. Skjetne, T. I. Fossen, and P. Kokotović, Output maneuvering for a class of nonlinear systems, in Proc. 15th IFAC World Congress Automatic Control, (Barcelona, Spain), July 22. [2] R. Skjetne, T. I. Fossen, and P. Kokotović, Robust output maneuvering for a class of nonlinear systems, Automatica. Submitted May 7th, 22. [3] W. Kang and N. Xi, Non-time referenced tracking control with application in unmanned vehicle, in Proc. 14th IFAC World Congress Automatic Control, (Beijing, China), IFAC, July [4] W. Kang, N. Xi, and A. Sparks, Theory and applications of formation control in a perceptive reference frame, in Proc. 39th IEEE Conf. Decision & Control, (Sydney, Australia), pp , IEEE, Dec. 2. [5] C. R. McInnes, Ring formation for a planar constellation of satellites, Guidance, Control, and Dynamics, vol. 18, no. 5, pp , [6] T. J. Tarn, A. K. Bejczy, and N. Xi, Intelligent planning and control for robot arms, in Proc. 12th IFAC World Congress Automatic Control,(Sydney, Australia), IFAC, July [7] J. Hauser and R. Hindman, Maneuver regulation from trajectory tracking: Feedback linearizable systems, in Proc. IFAC Symposium on Nonlinear Control Systems Design, (Lake Tahoe, CA, USA), pp , IFAC, June [8] P. Encarnação and A. Pascoal, Combined trajectory tracking and path following: An application to the coordinated control of autonomous marine craft, in Proc. 4th IEEE Conf. Decision & Control, (Orlando, Florida, USA), IEEE, Dec. 21. [9] R. Skjetne, A. R. Teel, andp. V. Kokotović, Stabilization of sets parametrized by a single variable: Application to ship maneuvering, in 15th Int. Symp. Mathematical Theory of Networks and Systems, (Notre Dame, Indiana, USA), Aug [1] R. Skjetne, A. R. 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