Robust Formation Control of Marine Craft using Lagrange Multipliers

Transcription

1 Robust Formation Control of Marine Craft using Lagrange Multipliers Ivar-André Flakstad Ihle 1, Jérôme Jouffroy 1, and Thor Inge Fossen 1,2 1 SFF Centre for Ships and Ocean Structures (CeSOS), Norwegian University of Technology and Science (NTNU), NO-7491 Trondheim, Norway 2 Department of Engineering Cybernetics, Norwegian University of Technology and Science, NO-7491 Trondheim, Norway ihle@ntnu.no, jouffroy@ntnu.no, fossen@ieee.org Summary. This paper presents a formation modelling scheme based on a set of inter-body constraint functions and Lagrangian multipliers. Formation control for a fleet of marine craft is achieved by stabilizing the auxiliary constraints such that the desired formation configuration appears. In the proposed framework we develop robust control laws for marine surface vessels to counteract unknown, slowly varying, environmental disturbances and measurement noise. Robustness with respect to time-delays in the communication channels are addressed by linearizing the system. Simulations of tugboats subject to environmental loads, measurement noise, and communication delays verify the theoretical results. Some future research directions and open problems are also discussed. 1 Introduction Ever since man was able to construct ships that were able to cross the open seas have formations been used to improve safety during travel and for tactical advantages in naval battles. Indeed, the ancient Greeks employed a strategy where light and fast vessels would ram their bow into heavier vessels to disable them, thus making them an easy target [4]. This is an old example but still proves how a formation of smaller, but flexible, vessels can outperform larger and more specialized vessels. Even today, this incentive remains valid for vehicle formations, cooperative robotics, and sensor networks. The motivation has, together with rapid developments in computation, communication, control, and miniaturization abilities, led to a major research interest in cooperative control of mobile robots, satellites, and marine vessels during the last decade see for example [17] and references therein. Formation control schemes aim to develop decentralized control laws that yield formation stability, and has been pursued using artificial potentials, virtual structures,

2 2 Ivar-André F. Ihle et al. leader-follower architecture, and behavioral rules [1, 5, 15, 21, 23, 29]. Some related topics of wide interest are networked control systems, synchronization of dynamic systems (see, e.g., [24]), consensus problems in computer science, and computer graphics applications, e.g., [26]. Coordinated control of several independent objects in an unknown environment pose a demanding task for the designer. The controllers should be robust to changes in the environment, unknown disturbances, communication/sensor noise, model uncertainties, and communication constraints between formation members. Previously, stability has been addressed for formations with changing topologies and time-delays see the recent survey paper [25] for an overview but many of the issues mentioned above have not yet been addressed. An interesting application for marine surface vessels where safety towards perturbations is critical is under-way replenishment operations [18]. One of the principal problems in abeam refuelling is the suction effect, caused by the bow waves, which draws the two vessels together. The magnitude of suction forces can increase rapidly, but they are effectively zero when the vessels are in a certain distance from each other. A reliable control design can improve safety of replenishment operations by designing robust control laws that maintain an inter-vessel distance larger than that threshold. This paper develops control laws for marine surface vessels based on a recent formation control scheme [11, 12]. The vessel models are based on new results in nonlinear ship modelling [7]. The resulting model of the formation is on feedback form such that the designer may apply control designs available in the literature. In particular, we develop control laws that are robust with respect to unknown environmental disturbances, and it is proved that the resulting closed-loop system is Input-to-State Stable [28] with respect to these disturbances. Examples of other possible control designs are also given. A linearization of the system provides information about robustness to time delays. The following sections are organized as follows. In Section 2, the formation will be modelled using classical tools from analytical mechanics. Robust control laws for assembling individual ships into a predefined configuration are derived in Section 3. A formation of marine surface vessels exposed to unknown environmental disturbances, noise, and time delays are investigated in Section 4. Finally, concluding remarks are given in Section 5. 2 Auxiliary Constraints, Analytical Mechanics, and Formation Setup The modelling approach in this section is motivated by how Lagrange s method of the undetermined multiplier can treat a collection of independent bodies and a set of auxiliary constraints. The method works quite generally

3 Robust Formation Control of Marine Craft 3 Fig. 1. Imposed constraints transform a group of independent bodies to a formation. for any number of constraints, and has been applied with success, e.g., in computer graphics [2] and multibody dynamics. 2.1 Modelling Consider a system of independent bodies on a 2-dimensional surface as in the left part of Figure 1. Each body has a kinetic energy T i and potential energy U i, and the Lagrangian of the total system is L = T U = i T i U i. (1) Let the generalized position and velocity of body i be q i and v i, respectively. A set of functions can relate the position of two bodies i and j to each other as in the right part of Figure 1 by using a norm q i q j = r ij (2) where r ij is the desired distance between body i and j. To apply the tools from analytical mechanics, we rewrite the functions as a kinematic relation of all positions q := [..., q i,...] C (q) = 0 (3) and denote it the constraint function. A constraint adds potential energy to the system, as preserving forces are necessary to maintain (3), and the modified Lagrangian becomes [20] L = T U + λ C (q) (4) where λ is the Lagrangian multiplier(s). The equations of motion for system i are then obtained by applying the Euler-Lagrange equations with auxiliary conditions to Eq. (4) d L L C (q) + λ = τ i, (5) dt q i q i q i

4 4 Ivar-André F. Ihle et al. where τ i is the known generalized external force associated with the i-th body, and the constraint force is given by τ constraint,i = λ W i, (6) where W i is the i-th column of the Jacobian of (3), W i = C(q) q i. When the set of constraint functions are applied to the independent bodies, a formation emerges, see Figure 1. The configuration is given by the constraint functions, and the equations of motion for all bodies are d L dt q L q C (q) = τ λ. (7) q The left hand side of (7) is the decoupled Euler-Lagrange equations of motion for the independent bodies, and the term for the constraint force interprets how the bodies interact when connected through constraint functions. This is the Lagrangian approach to multi-body dynamics originally described in J.-L. Lagrange s book [19] and appears in many textbooks on analytical mechanics, see for example [20]. The Lagrangian λ-factor has a physical interpretation as a measure of the violation of the constraint function (3), [20], and is found by combining the constraint function with (7). Since we want the position of the bodies to satisfy the constraint functions, neither the velocity nor the acceleration should violate them. To find the kinematic admissible velocities, the constraint function is differentiated with respect to time. Similarly, we differentiate twice to find the acceleration of the constraints. This gives the additional conditions. C (q) = W (q) q = 0.. C (q) = W (q) q + Ẇ (q) q = 0 (8) where W (q) is the Jacobian of the constraint function, i.e., W (q) = C(q) q. An expression for the Lagrangian multiplier is found by obtaining an expression for q in (7), insert it into (8) and solve for λ. Consider a system of point masses with mass equal to identity: We solve for λ to obtain λ = ( W W ) ( ) 1 Ẇ v + W τ, det ( W W ) 0. (9) In order to obtain λ, the product W W must be nonsingular, that is, the Jacobian W must have full row rank. The combination of (3) and (5) yields a Differential-Algebraic Equation (DAE), which we will discuss further in Section Constraint functions The approach above is valid both for systems where the constraints appear in the model, and when functions are imposed to keep the formation together. An

5 Robust Formation Control of Marine Craft 5 example of systems with model constraints can be molecules consisting of multiple atoms connected by chemical bonds. The configuration of the molecule depends on the properties of the interacting atoms. When the constraints are not physically present in the model, it can be denoted a virtual constraint. Two types of constraint functions for formation control purposes are considered in this paper, but others can be found in [13], the position relative constraint C l (q, t) = (q i q j ) (q i q j ) r ij (t) = 0, r ij (t) R, (10) which is equivalent to the Euclidean norm of the distance between two bodies, and the position offset constraint C l (q, t) = q i q j o ij (t) = 0, o ij (t) R n, (11) where q i and q j are the position of body i and j. The scalar r ij (t) is the desired distance between q i and q j while the column vector o ij (t) describes the offset between q i and q j in each degree of freedom (DOF). We say that two bodies are neighbors if they appear in the same index of the constraint function, and in which case they can access each others information. 2.3 Formation Topology As mentioned above, the Lagrangian multipliers exist if the Jacobian W has full row-rank. This limits the number of constraint functions that can be imposed on the formation. In addition, each imposed function reduces the degrees of freedom of the formation, thus the total number of constraints must not exceed the total degree of freedom. The full-row rank condition of W implies that the given constraint functions cannot be contradictory nor redundant. Suppose the formation has r members, each with an n DOF system. For constraints on the form (11), this means that there must be p < r such constraints for W to have full row rank since a new constraint would be a linear combination of the previous. An example of a feasible formation topology is a line. A formation can be subject to p nr 3 position relative constraints as long as they neither contradict the existing constraints nor is a linear combination of other constraints. The above conditions yield a Jacobian with full row rank and we can solve (9) to obtain the Lagrangian multiplier. The results can be extended to a time-varying formation topology as long as W has full row rank for all t 0. Illustrations of constraint functions that lead to singularities are shown in Figure Control Plant Ship Model We consider a fully actuated surface vessel i in three degrees of freedom where the surge mode is decoupled from the sway and yaw mode due to

6 6 Ivar-André F. Ihle et al. Fig. 2. Examples of redundant auxiliary constraints. Consider the position offset constraints (11) on the left: when two constraints are given, the third is simply a linear combination of those. For the position relative constraints (10) on the right two constraints are either contradictory or redundant. In both cases, the Jacobian W has less than full row rank. port/starboard symmetry. Let an inertial frame be be approximated by the Earth-fixed reference frame called NED (North-East-Down) and let another coordinate frame be attached to the ship as in Figure 3. The states of the vessel are then η i = [x i, y i, ψ i ] and ν i = [u i, v i, r i ] where (x i, y i ) is the position on the ocean surface, ψ i is the heading (yaw) angle, (u i, v i ) are the body-fixed linear velocities (surge and sway), and r is the yaw rate. The Earthfixed frame is related to the body-fixed frame through a rotation ψ i about the z i -axis, expressed in the rotation matrix R (ψ i ) = cos ψ i sin ψ i 0 sin ψ i cos ψ i 0 SO (3) The equations of motion for vessel i are given as (see [6] for details) η i = R (ψ i ) ν i M i ν i + C i (ν i ) ν i + D i (ν i ) ν i + g (η i ) = τ i. (12a) (12b) The model matrices M i, C i, and D i denote inertia, Coriolis plus centrifugal, and damping, respectively, g i is a vector of gravitational and buoyancy forces and moments, and τ i is a vector of generalized control forces and moments. For notational convenience system (12) is converted to the Earth-fixed frame using a kinematic transformation [6, Ch ]. The following relationships M ηi (η i ) = R (ψ i ) M i R (ψ i ) [ ] C ηi (ν i, η i ) = R (ψ i ) C i (ν i ) M i R (ψ i ) Ṙ (ψ i ) R (ψ i ) D ηi (ν i, η i ) = R (ψ i ) D i (ν i ) R (ψ i ) g ηi (η i ) = R (ψ i ) g (η i ) n i (ν i, η i, η i ) = C ηi (ν i, η i ) η i + D ηi (ν i, η i ) η i + g ηi (η i )

7 Robust Formation Control of Marine Craft 7 Fig. 3. Inertial Earth-fixed frame and body-fixed frame for a ship. give the new equations of motion for vessel i subject to the following assumptions: M ηi (η i ) η i + n i (ν i, η i, η i ) = R (ψ i ) τ i Assumption A1: The mass matrix M i is positive definite, i.e. M i = M i > 0; hence M ηi (η i ) = M ηi (η i ) > 0 by [6] and [8]. Assumption A2: The damping matrix can be decomposed in a linear and a nonlinear part: D i (ν) = D lin,i + D nonlin,i (ν) where D nonlin,i is a matrix of nonlinear viscous damping terms for instance quadratic drag. We consider a group of r vessels subject to constraint functions as in (10) or (11) such that the vector of control forces and moments is the force due to the violation of the constraint function C, τ i = τ constraint,i. Furthermore, we collect the vectors into new vectors, and the matrices into new, block-diagonal, matrices by defining η = [η 1,..., η r ], M η = diag {M η1,..., M ηr }, and so on. For i = 1,..., r, this results in the model M η (η) η + n (ν, η, η) = R (ψ) τ constraint (13) where the constraint force τ constraint is given by τ constraint = W λ and the Lagrangian multiplier is found to be λ = ( W Mη 1 RW ) ( ) 1 W Mη 1 n + Ẇ η. (14)

8 8 Ivar-André F. Ihle et al. When the Jacobian W has full row rank W Mη 1 RW exists since M η is positive definite, hence Mη 1 exists and W Mη 1 RW is nonsingular. Furthermore, the constraint force arise pairwise between neighbors and is always present when C 0. 3 Robust Formation Assembling If the vessels are not already in a formation, they must be assembled into the specific configuration at the start of an operation. Furthermore, the members of the formation should stay assembled in the presence of unknown environmental disturbances, noise, and time delays in the communication channels. This section presents a robust control law that assembles vessels into a desired formation shape and maintain the configuration in the presence of unknown environmental perturbations. 3.1 Formation Assembling We want the formation to be configured as defined by (3), that is, we want to stabilize (3) as an equilibrium of (13). This is a higher-index DAE, which is known to be inherently unstable. Indeed, the procedure in the previous section gives d 2 dt 2 C = 0 (15) which is unstable when (3) is a scalar function the Laplace-transform of (15) yields a transfer function with two poles in the origin. Hence, if the initial conditions of (13) do not satisfy the imposed constraint function, or there are measurement noise or external disturbances acting on the system, the trajectories of (15) will go to infinity. We stabilize the system by replacing the right hand side of (15) with u and consider the stabilization of the constraint function as an ordinary control design problem. By applying negative feedback from the constraint and its derivative, we get a Proportional-Derivative type control law d 2 dt 2 C = u = K pc K d C (16) which corresponds to the Baumgarte technique for stabilization of constraints in dynamic systems [3]. The resulting constraint forces are still given by τ constraint = W λ, but the Lagrangian multiplier from (14) has been replaced with the stabilizing Lagrangian multiplier λ = ( W Mη 1 RW ) ( ) 1 W Mη 1 n + Ẇ η u. (17) The control law (6) is locally implementable as each λ i in (17) only requires information about neighboring system and the ith column of the Jacobian, W i,.

9 Robust Formation Control of Marine Craft Constraint Forces Position Offset Position Relative Forces Relative Distance Fig. 4. The resulting constraint forces between two point masses as a function of their relative distance. Position relative constraint function ( ) as in (10) and position offset constraint functions (- -) as in (11). depends on the position of neighbors. The stabilizing Lagrangian multiplier is then substituted in the constraint force in (13) and the formation configuration is stabilized around C = 0. The resulting forces from imposing a constraint between two bodies are found in Figure 4. The formation assembling problem is now transformed into a linear control system which can be analyzed with the large number of methods in the literature. An extension of this framework to underactuated vessels is given in [13]. 3.2 Extension to Other Control Schemes Coordinated control laws for several independent models in an unknown environment poses challenges to the designer. When the designer has some a priori knowledge of the environmental effects, they can be incorporated into the design of the control law. Due to the structure of (15), we are able to use many of the designs in the control literature. Assume that the constraints are of the form (10) or (11) and satisfy the conditions in Section 2.3.

10 10 Ivar-André F. Ihle et al. Equation (15) is basically a double integrator which can be put into an upper-triangular form. Indeed, with φ 1 (t) := C (η (t)) and φ 2 (t) := Ċ (η (t)) we obtain φ 1 = φ 2 φ 2 = u. (18a) (18b) The constraint stabilization problem is then to design a controller u that renders (φ 1, φ 2 ) = (0, 0) stable. The structure of (18) allows us to take advantage of existing control designs. For example, with a quadratic cost function, the controller can be designed using LQR-techniques. In the presence of unknown model parameters, an adaptive control scheme can be used [14]. The system (18) has an upper triangular structure and falls into the class of strict feedback systems. This class of systems has been thoroughly investigated and is frequently used as a basis for systematic and constructive control design as it encompasses a large group of systems. If disturbances or unknown model parameters appear as nonlinearities designs for uncertain systems or adaptive control, for example, in [16], can be applied. Consider the formation of r vessels perturbed by unknown bounded disturbances δ (t) M η (η) η + n (ν, η, η) = R (ψ) τ constraint + W d δ (t) (19) where Wd is a smooth, possibly nonlinear, function. These disturbances may represent slowly-varying environmental loads due to second-order waveinduced disturbances (wave drift), currents and mean wind forces. The method in Sections 2.1 and 2.4 transforms (19) to the form of (18) φ 1 = φ 2 φ 2 = u + W d (φ 1, φ 2 ) δ (t) (20a) (20b) where W d = W M 1 η Wd. The goal will now be to render the closed-loop system Input-to-State Stable [28] from the disturbances with respect to the origin of the closed-loop system (20). This can be achieved by using a design procedure from [16]: Control Design: We define the error variable as z (t) = φ 2 α (21) where α is a virtual control to be specified later. The time derivative of φ 1 is φ 1 = φ 2 = z + α We choose Hurwitz design matrices A i, i = 1, 2, such that P i = Pi solution of P i A i + A i P i = Q i where Q i = Q i > 0. > 0 is the

11 Let the first control Lyapunov function be Robust Formation Control of Marine Craft 11 V 1 (φ 1, t) = φ 1 P 1 φ 1 The time derivative V 1 becomes, with the choice α = A 1 φ 1, V 1 = φ 1 Q 1 φ 1 + 2φ 1 P 1 z. Differentiating (21) with respect to time gives ż = φ 2 α = u + W d (φ 1, φ 2 ) δ (t) A 1 φ 2. We define the second control Lyapunov function with the following time derivative V 2 (φ, t) = V 1 + z P 2 z V 2 = φ 1 Q 1 φ 1 + 2z P 2 ( u + P 1 2 P 1 φ 1 + W d δ A 1 φ 2 ) and the control law is chosen as u (φ, t) = A 2 z P 1 2 P 1 φ 1 + A 1 φ 2 + α 0 (22) where α 0 is a damping term to be determined. Young s inequality yields and we obtain 2z P 2 W d δ 2κz P 2 W d W d P 2 z + 1 2κ δ δ, κ > 0 V 2 φ 1 Q 1 φ 1 z Q 2 z + 1 2κ δ δ + 2z P 2 ( α0 + κw d W d P 2 z ). The choice α 0 = κw d W d P 2z yields V 2 φ 1 Q 1 φ 1 z Q 2 z + 1 2κ δ δ q min y κ δ 2 < 0, y > 1 2κq min δ where q min = min (λ min (Q 1 ), λ min (Q 2 )) and y := [φ 1, z ]. Hence, the control law (22) renders the closed-loop system ISS from δ (t) to z. In φ-coordinates the control law (22) is written as u = K p φ 1 K d φ 2 κw d W d P 2 (φ 2 A 1 φ 1 ) where K d = (A 1 + A 2 ) and K p = A 2 A 1 P 1 2 P 1 so the robust backstepping design encompasses the Baumgarte stabilization technique (16).

12 12 Ivar-André F. Ihle et al. Hence, by exploiting existing design methodologies the formation scheme in Section 2 is rendered robust against unknown disturbances. The control scheme can also be extended to include parameter adaptation and to find constant unknown biases: Let ϕ R x be a vector of constant unknown parameters M η (η) η + n (ν, η, η) = R (ψ) τ constraint + W a ϕ where Wa is a smooth function. Recall that the Lagrangian multiplier λ is still as in (17). The transformed model becomes φ 1 = φ 2 (23) φ 2 = u + W a (φ 1, φ 2 ) ϕ (24) where W a is smooth. By adopting the adaptive control design procedure from [16] or [14] we can find a control law that renders the equilibrium points C = Ċ = 0 and ϕ = ϕ ˆϕ uniformly globally convergent and guarantees that C, Ċ, ϕ 0 in the limit as t. 4 Case Study We investigate a formation of three vessels where one vessel track a desired path while the others follow according to the formation constraint function. All vessels are subject to unknown environmental perturbations, measurement noise and the communication channels are affected by time delays. We will use a time-varying constraint function to allow a time-varying configuration. Consider the following functions (x 1 x 2 ) 2 + (y 1 y 2 ) 2 r 2 12 C fc (η, t) = (x 2 x 3 ) 2 + (y 2 y 3 ) 2 r23 2 (t), C tt (η, t) = η := η 1 η d (t) (x 3 x 1 ) 2 + (y 3 y 1 ) 2 r31 2 where r 23 (t) and η d (t) are three times differentiable. The first functions are on the constraint function form (10), while the last is a constraint function that yields a control law for trajectory tracking [13]. Since the two functions are not conflicting we can collect them into the following constraint function [ ] Cfc (η, t) C (η, t) = = 0. (25) C tt (η, t) Together with the ship model (19), where W d = W Mη 1 R (ψ), the backstepping design in Section 3 yields robust control laws for formation control and trajectory tracking with φ = C V (φ, t) = φ P φ where P = P > 0.

13 Robust Formation Control of Marine Craft 13 The closed-loop equations of motion for the three vessels are M η (η) η + n (ν, η, η) = R (ψ) W fcλ fc τ tt + R (ψ) δ (t) where the formation control laws are given as R (ψ) Wfcλ fc = ( W fc Mη 1 Wfc) ( Wfc n + Ẇ η (26) ) + K p C fc + K d Ċ fc + P fc2 (Ċfc A fc1 C fc where K p, K d R 3 3 are positive definite. The trajectory tracking control law is τ tt = [R (ψ 1 ) λ tt, 0, 0] where R (ψ 1 ) λ tt is the control law for the first vessel to track the desired path η d R (ψ 1 ) λ tt = n 1 (ν 1, η 1, η 1 ) M η1 ( η d k tp η (27) k td η P tt2 ( η A tt η ) ) where k tp, k td R 3 3 are positive definite. 4.1 Linearized Analysis of Robustness to Time-delays We know that the delay robustness for a single-input-single-output linear system is given by [10] T max = P M ω gc (28) where T max is the maximum delay in the feedback loop that does not destabilize the system, P M is the phase margin, and ω gc is the gain crossover frequency. Thus, increasing the phase margin and/or decreasing the bandwidth improves delay robustness. We linearize by assuming small variations in the constraint functions and heading angle. The loop-gain of the linearized system (about the heading angle ψ 0) from the disturbance δ to the constraint C fc1 is found to be G cδ (s) = κ/2 s 2 + k d s + k p (29) where k p and k d are the (1, 1)-elements of K p and K d, respectively. Using tools from linear systems theory we can adjust the gains to maximize the delay that does not destabilize the system. This has to be done in a tradeoff relation with other performance properties. A critically damped system is desired since it implies no overshoot, and this is achieved for k d = 2 k p. This analysis is no guarantee for stability in the presence of delays, but it gives us an indication.

14 14 Ivar-André F. Ihle et al. 4.2 Simulation Results The control plant model of a fully actuated tugboat in three degrees of freedom (DOF), surge, sway, and yaw, is used in the case study.the model has been developed using Octopus SEAWAY for Windows [27] and the Marine Systems Simulator [22]. SEAWAY is a frequency-domain ship motions PC program based on the linear strip theory to calculate ship motions. The Marine Systems Simulator (MSS) is a Matlab/Simulink library and simulator for marine systems. A nonlinear speed dependent formulation for station-keeping (u = v = r = 0) and maneuvering up to u = 0.35 gl pp = 6.3 m / s (Froude number 0.35) is derived in [7] where L pp is the length between the perpendiculars. Based on [7], output from SEAWAY were used in MSS to generate a 3 DOF horizontal plane vessel model linearized for cruise speeds around u = 5 m / s with nonlinear viscous quadratic damping in surge. Furthermore, the surge mode is decoupled from the sway and yaw mode due to port/starboard symmetry. The model is valid for cruise speeds in the neighborhood of u = 5 m / s. The model in body-fixed reference frame is then M i ν i + D i (ν i ) ν i + C i ν i = τ i (30) with the following model matrices M i = C i = D i (ν i ) = u i The desired path for vessel 1 is η d (t) = x d (t) t y d (t) = A sin( ωt ψ d (t) atan2 ẏd where A = 200 and ω = 0.005, and the unknown environmental disturbances are δ i (t) = sin (0.1t) sin (0.1t) + white noise (31) sin (0.1t) acting the same on all vessels. ẋ d )

15 Robust Formation Control of Marine Craft Fig. 5. Position response of vessels during simulation. Vessel 1 follows the desired (dashed) path, and the desired configuration changes from a triangular shape to a line about halfway through the simulation. The unknown environmental disturbances are seen to be slowly-varying while the first-order wave-induced forces (oscillatory wave motion) are assumed to be filtered out of the measurements by using a wave filter see [9]. This is a good assumption since a ship control system is only supposed to counteract the slowly-varying motion components of the environmental disturbances to reduce wear and tear of actuators and propulsion system. The desired formation configuration is given by r 12 = 70, r 31 = 70, and r 23 is initially 65 and changes smoothly to 130 at about 1000s. The control gains are k tp = 4I, k td = 2I, K p = diag (k pi ), k pi = 3.24, K d = diag (k di ), k di = 6 and κ = 20. The initial values are η 1 (0) = [ 0, 0, π 2 ], η2 (0) = [ 45, 25, π 2 ], η 3 (0) = [ 40, 10, π 2 ] and ν1 (0) = ν 2 (0) = ν 3 (0) = 0. Figure 5 shows the resulting position trajectories and five snapshots of the vessels during the simulation: the vessels assemble into the desired configuration and vessel 1 tracks the desired path. The position tracking and formation constraints errors due to the disturbances (31) were attenuated to less than 1 m, and 5 m, respectively. The time-varying configuration is seen in the third and fourth snapshot as the formation changes from a triangle to a line.

16 16 Ivar-André F. Ihle et al. For the linearized relation (29) the values give a bandwidth of 0.59 rad / s and a phase margin of 85. This corresponds to a maximum time delay of 2.5 s. In the simulation all communication channels are affected by 2.5 s time delays, and simulations show that delays larger than 3 s cause instabilities in the closed-loop system. Thus, the transfer function provides a good estimate of robustness towards time delays. 5 Concluding Remarks and Future Directions This paper has shown how a group of individual vessels can be controlled by imposing formation constraint functions and applying tools from analytical mechanics. Stabilization of these constraints is interpreted as a control design problem and a robust control law that maintains formation configuration in the presence of unknown disturbances has been developed. The theoretical results have been verified by a simulation where a group of marine surface vessels moves while maintaining the desired configuration. The results demonstrated robustness with respect to unknown disturbances affecting the vessels and time delays in the communication channels. In the current literature, cooperative control has become an umbrella term for feedback applications in sensor networks, vehicle formations, cooperative robotics, and consensus problems. The applications have all something in common: multiple, dynamic, entities share information to accomplish a common task. The design of formation control schemes poses challenges to the engineer, and many are not yet addressed [25]. A real-life implementation for formation control will depend heavily on the choice of communication protocols, available hardware, sensors, instrumentation and actuators, onboard computing possibilities, and power consumption. Sharing information is a vital component for cooperation and communication issues such as inconsistent delays, noise, signal dropouts, and possible asynchronous updates should be taken into account. In environments with limited bandwidth a formation control design with a minimal amount of information exchange is desired. Most formation control problems are studied in the context of homogenous groups with double integrator dynamics. A vehicle s dynamics might change with time, or as the formation moves through its environment. It would be interesting to study formations with more complex or uncertain dynamics, or both, and where a single vehicle drop out of formation or its actuators break down. Future research on formation control should facilitate implementation of theoretical results to be verified experimentally. An important step would be to address these topics and mathematically guarantee that a group of systems, subject to a wide range of practical challenges, can cooperate to solve a problem that would be out of reach for a single system.

17 Robust Formation Control of Marine Craft 17 Acknowledgments This project is sponsored by The Norwegian Research Council through the Centre for Ships and Ocean Structures (CeSOS), Norwegian Centre of Excellence at the Norwegian University of Science and Technology (NTNU). References 1. R. Arkin. Behavior-based robotics. MIT Press, Cambridge, MA, USA, D. Baraff. Linear-time dynamics using Lagrange multipliers. In Computer Graphics Proceedings, pages SIGGRAPH, J. Baumgarte. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanical Engineering, 1:1 16, L. Casson. The Ancient Mariners: Seafarers and Sea Fighters of the Mediterranean in Ancient Times. Princeton University Press, Princeton, NJ, USA, 2nd edition, M. Egerstedt and X. Hu. Formation constrained multi-agent control. IEEE Transactions on Robotics and Automation, 17(6): , T. I. Fossen. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway, T. I. Fossen. A nonlinear unified state-space model for ship maneuvering and control in a seaway. International Journal of Bifurcation and Chaos, 15(9), T. I. Fossen and Ø. N. Smogeli. Nonlinear time-domain strip theory formulation for low-speed maneuvering and station-keeping. Modelling, Identification and Control, 25(4): , T. I. Fossen and J. P. Strand. Passive Nonlinear Observer Design for Ships Using Lyuapunov Methods: Full Scale Experiments with a Supply Vessel. Automatica, 35(1):3 16, G. F. Franklin, J. D. Powell, and A. Emami-Naeini. Feedback Control of Dynamic Systems. Addison-Wesley, Boston, MA, USA, 4th edition, I.-A. F. Ihle, J. Jouffroy, and T. I. Fossen. Formation control of marine craft using constraint functions. In IEEE Marine Technology and Ocean Science Conference Oceans05, Washington D.C., USA, I.-A. F. Ihle, J. Jouffroy, and T. I. Fossen. Formation control of marine surface craft using lagrange multipliers. In Proc. 44rd IEEE Conference on Decision & Control and 5th European Control Conference, pages , Seville, Spain, I.-A. F. Ihle, J. Jouffroy, and T. I. Fossen. Formation control of marine surface craft: A Lagrangian approach Submitted. 14. P. A. Ioannou and J. Sun. Robust Adaptive Control. Prentice Hall, Inc., (Out of print in 2003), Electronic copy at ioannou/robust Adaptive Control.htm. 15. A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6): , M. Krstić, I. Kanellakopoulos, and P. V. Kokotović. Nonlinear and Adaptive Control Design. John Wiley & Sons Ltd, New York, 1995.

18 18 Ivar-André F. Ihle et al. 17. V. Kumar, N. Leonard, and A. S. Morse (Eds.). Cooperative Control. Lecture Notes in Control and Information Sciences. Springer-Verlag, Heidelberg, Germany, E. Kyrkjebø and K. Y. Pettersen. Ship replenishment using synchronization control. In Proc. 6th IFAC Conference on Manoeuvring and Control of Marine Crafts, pages , Girona, Spain, J. L. Lagrange. Mécanique analytique, nouvelle édition. Académie des Sciences, Translated version: Analytical Mechanics, Kluwer Academic Publishers, C. Lanczos. The Variational Principles of Mechanics. Dover Publications, New York, 4th edition, N. E. Leonard and E. Fiorelli. Virtual leaders, artificial potentials and coordinated control of groups. In Proc. 40th IEEE Conference on Decision and Control, pages , Orlando, FL, USA, MSS. Marine systems simulator, Norwegian University of Science and Technology, Trondheim, Norway R. Olfati-Saber and R. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9): , A. Pogromsky, G. Santoboni, and H. Nijmeijer. Partial synchronization: From symmetry towards stability. Physica D Nonlinear Phenomena, 172:65 87, W. Ren, R. W. Beard, and E. M. Atkins. A survey of consensus problems in multi-agent coordination. In Proc. American Control Conference, pages , Portland, OR, USA, C. W. Reynolds. Flocks, herds, and schools: A distributed behavioral model. Computer Graphics Proceedings, 21(4):25 34, SEAWAY. Octopus seaway, Amarcon B.V., The Netherlands E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 34(4): , P. K. C. Wang. Navigation strategies for multiple autonomous robots moving in formation. Journal of Robotic Systems, 8(2): , 1991.