Minimization of Cross-track and Along-track Errors for Path Tracking of Marine Underactuated Vehicles

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1 Minimization of Cross-track and Along-track Errors for Path Tracking of Marine Underactuated Vehicles Anastasios M. Lekkas and Thor I. Fossen Abstract This paper deals with developing a guidance scheme for minimizing the position error of a marine underactuated vehicle during a path-tracking scenario. The desired position is determined by a virtual vehicle, which is assumed to navigate on the desired path, and the position error is analyzed in two components: a) the along-track error, and b) the cross-track error. Initially, perfect heading tracking is assumed and the well-known Line-of-Sight (LOS) guidance is used to minimize the cross-track error. Moreover, by using the vehicle kinematics and assuming perfect velocity tracking, a surge velocity guidance law for minimizing the along-track error is proposed. Then, the perfect velocity and heading tracking assumptions are relaxed and the stability of the total system, including the heading and velocity controllers, is studied and the system is shown to be globally κ-exponentially stable. The results are supported by computer simulations. I. INTRODUCTION The performance of unmanned vehicles (UVs) is largely dependent on the guidance system, which is assigned to generate appropriate reference trajectories to be given as inputs to the control system. These trajectories are generated by taking into account the vehicle s velocity and position, as well as the desired trajectory to be followed or tracked. Depending on the motion control objectives involved, the following motion control scenarios are often considered for marine vehicles: a) target tracking, b) path following, c) path tracking, and d) path maneuvering. For more details on the motion control scenarios the reader is referred to []. Path tracking refers to the case where the vehicle is assigned to track an object that moves along a predefined path. This implies that the mission involves both temporal and spatial constraints that have to be satisfied in order for the mission to be accomplished. Naturally, in this scenario an underactuated vehicle should be able to control both its heading angle (or course angle, in the case where environmental forces are present) and its surge velocity in order to satisfy the corresponding constraints. This is different compared to a path following scenario (see for instance [], [3]) where it can be assumed that the vehicle has a constant total speed and it is necessary to control only its heading angle in order to achieve the desired result. However, the path tracking *This work was supported by the Norwegian Research Council through the Centers for Ships and Ocean Structures, and Autonomous Marine Operations and Systems at NTNU. A. M. Lekkas is with the Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, NO-749, Trondheim, Norway anastasios.lekkas at itk.ntnu.no T. I. Fossen is with the Centre for Autonomous Marine Operations and Systems, Department of Engineering Cybernetics, Norwegian University of Science and Technology, NO-749, Trondheim, Norway fossen at ieee.org scenario is not so far from the target tracking case and it can be implemented as such []. Therefore, for an underactuated vehicle the guidance system should generate reference trajectories for both the heading angle and the surge velocity such that the vehicle will manage to track an object. For marine vehicles, the most popular guidance techniques in the literature are the following: a) Pure Pursuit (PP), b) Line-of Sight (LOS) and c) Constant Bearing (CB). These are presented in detail in [4]. The combination of the guidance system (which generates the reference trajectories) with the velocity and heading autopilots (which are assigned to track these reference trajectories) form a system which can often be difficult to analyze and infer upon its stability. An integrated approach for accurate trajectory tracking was presented in [5]. In [6] the authors reported asymptotic trajectory tracking by employing a cascade control strategy consisting of a kinematic and a dynamic control law. A solution where the primary task is to steer the vehicle on the desired path and the secondary task is to assign the speed so as to track the target was given in [7]. A robust methodology for the maneuvering problem, which consists of a geometric task and a dynamic task, in the presence of bounded disturbances was developed in [8]. A generalization of the maneuvering problem was presented in [9] where the LOS guidance law was employed in order to demonstrate the stabilization of more general manifolds. Following the same distinction between the tasks, the authors in [] implemented a LOS guidance scheme for the geometric task and an appropriate speed assignment for the dynamic task, assuming the vehicle is a particle. Many other solutions have appeared in the literature, including the combined problem of path planning and trajectory tracking (see [], []), cases where there is large modeling parametric uncertainty [3], and cases which account for the presence of ocean currents, see [4], [5], [6]. This paper builds upon and extends the methodology presented in []. One major difference is that, instead of a particle, an underactuated vehicle model is used for the analysis. Moreover, contrary to [], we include a more complete proof regarding the minimization of both the along-track and cross-track errors. First, assuming perfect heading angle and surge velocity tracking, we employ the LOS guidance for minimizing the cross-track error and develop a methodology for obtaining a velocity assignment that minimizes the alongtrack error. Next, we consider stability of the total system in cascade by taking into account the convergence of the heading angle and surge velocity controllers. By using well-

2 known results from nonlinear cascade systems theory we show that the total system is globally κ-exponentially stable, a stability concept which was introduced by [7]. The rest of the paper is organized as follows: Section II presents the vehicle model and the controllers as well as the virtual vehicle model that is used for tracking. Section III gives a brief overview of the LOS guidance law for crosstrack error minimization. In Section IV, a new surge velocity guidance technique is developed in order to minimize the along-track error. Section V deals with the stability of the total system. In Section VI the theoretical analysis is supported by simulation results and, finally, Section VII concludes the paper. II. VEHICLE AND VIRTUAL VEHICLE MODELS A. Vehicle Model The ship equations of motion are usually represented in three DOFs by neglecting heave, roll and pitch [8]: η = R(ψ)ν () M ν + C(ν)ν + D(ν)ν = τ + τ wind + τ wave () where η := [x, y, ψ], ν := [u, v, r] and cos(ψ) sin(ψ) R(ψ) = sin(ψ) cos(ψ) SO() (3) is the rotation matrix in yaw. It is assumed that wind and wave-induced forces τ wind and τ wave can be linearly superpositioned. The system matrices M = M RB +M A and C(ν) = C RB (ν) + C A (ν) are given by [8]: M = m X u m Y v mx g Yṙ mx g N v I z Nṙ C RB (ν) = C A (ν) = mr mx g r mr mx g r Y v v + Yṙr X u u Y v v Yṙr X u u, (4), (5), (6) Hydrodynamic damping will in its simplest form be linear: X u D = Y v Y r (7) N v N r while a nonlinear expression based on second-order modulus functions describing quadratic drag and cross-flow drag is: X u u u D(ν) = Y v v v Y r v r N v v v N r v r Y v r v Y r r r N v r v N r r r (8) An exhaustive definition of all the model variables and parameters, as well as other nonlinear representations, can be found in [8]. B. Surge Velocity and Heading Angle Controllers The maneuvering model used in guidance and control systems only needs to capture the most important hydrodynamic effects. Such a model can be based on the following assumptions [8]: A: Surge can be decoupled from the sway and yaw motions. A: The yaw dynamics can be accurately described by a nonlinear Nomoto model (see [8] for more details) and stabilized by a heading autopilot. This implies that the sway velocity v(t) is bounded for all t. A3: The drift forces due to ocean currents, wind and waves can be neglected in the model since the guidance and control systems are designed to include integral action. Based on Assumptions 3 an underactuated ship with two controls, thrust T and rudder angle δ, can be modeled as: ẋ = u cos(ψ) + σ (9) ẏ = u sin(ψ) + σ () ψ = r () where σ = sin(ψ)v and σ = cos(ψ)v are known timevarying signals that can be measured, while the surge and yaw dynamics are modeled as: (m X u ) u X u u u u = ( t T )T () T n ṙ + n r + n r 3 = K n δ (3) where t T > is the thrust deduction number, X u > is the added mass in surge, and X u u > is the resistance or quadratic damping in surge. The Nomoto gain and time constants are recognized as K n > and T n >, respectively, while according to Norrbin n = for coursestable ships, n = for course-unstable ships, and n > [9]. The model of [9] is a first-order model, which can be used to describe the yaw dynamics of most commercial ships. Consequently, it is trivial to design a feedback linearizing controller that will result in GES surge velocity error dynamics: t T = [(m X u )( u d K pu ũ K iu ũdτ) t T X u u u u], (4) with ( ) = ( ) ( ) d, where ( ) denotes the true value and ( ) d denotes the desired value of the corresponding variable. Regarding the real controller gains, we have K pu, K iu >. Similarly for the heading angle error dynamics: δ = K n (n r 3 + n r K pr ψ Kir t ψdτ K dr r). (5) where the real controller gains satisfy K pr, K ir, K dr >. The controllers (4) (5) result in GES equilibrium points at ũ = r =. In reality this is hardly the case due to the saturation of the actuators. However, this result is useful in order to obtain a proof of concept in Section V where the

3 x n x b the path but also helps avoid singularities when computing the cross-track error, for more details the reader is referred to [], []. p x e x b y b y e x e y b III. CROSS-TRACK ERROR MINIMIZATION In order to minimize the cross-track error we employ the LOS guidance law. This problem has been studied extensively in the literature and in this paper we will use the formulation presented for the horizontal plane in [], briefly revised here. The time-derivative of () can be written in phase-amplitude form as []: y b x b ẏ e = U sin(ψ γ p + β), () where U = u + v is the total speed of the vehicle and the sideslip angle is defined as: β = atan(v, u). (3) Fig.. Illustration of the path-tracking problem addressed in this paper. stability of the overall path-tracking system is studied and the convergence time of the controllers can affect performance. C. Virtual Vehicle Kinematics and Tracking Error We consider a -D continuous straight path that connects two successive waypoints (x k, y k ) for k =,,..., N. In this case, the path-tangential angle is constant between the waypoints and can be computed as: γ p = atan(y k+ y k, x k+ x k ). (6) For the path-tracking scenario it is reasonable to assume that a virtual particle is navigating with a total speed U t on the desired path, therefore its position p n t = (x t, y t ) is computed by integrating the inertial velocities: ẋ t = U t cos (γ p ), (7) ẏ t = U t sin (γ p ). (8) Then the position error for a given vehicle position (x, y) is given by: [ ] [ ] xe = R x xt (γ y p ), (9) e y y t therefore, the along-track and the cross-track error can be rewritten: x e = (x x t ) cos(γ p ) + (y y t ) sin(γ p ), () y e = (x x t ) sin(γ p ) + (y y t ) cos(γ p ). () The objective of the vehicle in this case is to track the virtual particle, that is p p t. For this study we will assume that the virtual vehicle moves on a straight line and that no unknown external disturbances act on the vessel. An illustration of the problem can be seen in Fig.. The virtualvehicle approach not only determines the desired position on y n Remark : It should be noted that in a tracking scenario () holds only when the virtual vehicle moves on a straight line. On the other hand, for a path-following scenario () holds for curved paths as well. Assuming perfect heading tracking (ψ = ψ d ) the LOS guidance generates the following heading angle reference trajectories: ( ) ye ψ d = γ p + atan β, (4) where > is the lookahead distance. Combining () and (4) yields: y e ẏ e = U. (5) + ye Proposition : For U, > the system (5) has a globally κ-exponentially stable equilibrium point at y e =. Proof: The proof can be found in []. This was first proven in []. IV. ALONG-TRACK ERROR MINIMIZATION Contrary to the path-following task, the path-tracking scenario requires the minimization of both () and (). As it was shown in Section III, the cross-track error can be minimized by generating heading commands according to the LOS guidance law. The along-track error, on the other hand, will be minimized by generating appropriate surge velocity reference trajectories. Before proceeding we assume the following: A4: The heading commands are perfectly tracked (ψ = ψ d ). A5: The surge velocity commands are perfectly tracked (u = u d ). The time-derivative of () in combination with () and taking into account Assumption A5 gives: ẋ e =(ẋ ẋ t ) cos (γ p ) + (ẏ ẏ t ) sin (γ p ), =(u d cos(ψ d ) v sin(ψ d ) U t cos (γ p )) cos (γ p )+ + (u d sin(ψ d ) + v cos(ψ d ) U t sin (γ p )) sin (γ p ), =u d cos(γ p ψ d ) + v sin (γ p ψ d ) U t. (6)

4 Vehicle (x, y,, r), ũ Vehicle (x, y,, r) x e,y e Controllers (T, ) (u d, d ) {z } desired commands Guidance ( d p 6= / ± k, U d > ) {z } constraints Fig.. The heading and surge velocity controllers along with the vehicle form the driving system of the cascade. Fig. 3. The guidance system along with the vehicle form the driven system. Combining (6) with the heading reference trajectories (4) yields: ẋ e =u d cos(β + atan(y e / ))+ }{{} ζ + v sin(β + atan(y e / )) U t. (7) }{{} ζ In order to transform (7) in a more practical form, we exploit the following property: ( ) a ± b atan(a) ± atan(b) = atan (8) ab Combining (3), (7) and (8) gives: a = v, b = y e u d ( ) v + ud y e ζ = atan u d vy e (9) (3) where ζ (π/) ± κπ for κ =,,..., n, for more details on this see Remark at the end of this section. For the sake of simplicity we define: From (7), (3), (3), we have: ξ := v + u dy e u d vy e (3) ẋ e = u d cos (atan(ξ)) + v sin (atan(ξ)) U t, u d = + vξ U t. (3) + ξ + ξ Consequently, we choose the desired surge velocity as: ( ) u d = vξ + + ξ ( k x x e + U t ), (33) with k x >. Proposition : Assuming perfect heading and surge velocity tracking, the system (3) has a GES equilibrium point at x e = if the surge velocity assignment is given by (33). Proof: This follows from (3) (33) which gives ẋ e = k x x e. Remark : Regarding Eq. (6), for underactuated vehicles only the surge velocity and the heading angle are available for control. This means that it is not possible to find surge velocity commands capable of minimizing the along-track error (6) if the vehicle is moving at a direction normal to the straight line, that is cos(γ p ψ d ) = γ p ψ d = (π/) ± κπ for κ =,,..., n. However, such a value for the heading angle will be generated by the LOS guidance law only if the cross-track error is infinite, since in that case the most effective way to approach the path is to move on a direction normal to the path: y e ψ d γ p (π/). However, due to the stability result established in Section III, the cross-track error will indeed be bounded. V. STABILITY OF THE TOTAL SYSTEM The stability analysis presented in Sections III IV assumed perfect heading and surge velocity tracking. In reality, however, this is never the case since the heading and speed controllers will always need time before converging to the desired values. Consequently, the stability of the overall system should be considered where the heading and surge velocity controllers act as the driving systems and the guidance system, which minimizes the along-track error and the cross-track error, is the driven system. This is depicted in Figs 3. For the sake of clarity, we rewrite first the along-track error/speed controller subsystem in cascade form, followed by the cross-track error/speed controller/heading controller subsystem. A. Formulating the Along-track Error/ Speed Controller/ Heading Controller Cascade System The next step is to rewrite the along-track error subsystem as a function of ũ, u, ψ and ψ. The assumptions of perfect heading angle and surge velocity tracking do not hold anymore and since ψ = ψ ψ d, ũ = u u d, we rewrite (6) as: ẋ e =(ũ + u d ) cos(γ p ψ d ψ) + v sin (γ p ψ d ψ) U t, (34) which, after application of several trigonometric properties and adding/subtracting terms, can be written as: ẋ e =u d cos (γ p ψ d ) + v sin (γ p ψ d )+ + (cos ( ψ) )(u d cos (γ p ψ d ) + v sin (γ p ψ d ))+ + sin ( ψ)(u d sin (γ p ψ d ) v cos (γ p ψ d ))+ ( cos ( + ũ ψ) ) ξ sin ( ψ) + U t. (35) + ξ + ξ

5 Inspired by [3], we transform (35) in the form: ẋ e = u d cos (γ p ψ d ) + v sin (γ p ψ d ) U t + ω T x, (36) where x := [ũ ψ] T (x will be defined in Section V-C) and ω T = [ω ω ] with: ω = sin ( ψ + φ ξ ), φ ξ = atan(, ξ), (37) ω = cos ( ψ) (u d sin (γ p ψ d ) + v cos (γ p ψ d ))+ ψ sin ( ψ) + ψ (u d sin (γ p ψ d ) v cos (γ p ψ d )). (38) Combining (33) and (36) yields: ẋ e = k x x e + ω T x. (39) B. Formulating the Cross-track Error/ Speed Controller/ Heading Controller Cascade System Since ũ = u u d and ψ = ψ ψ d, we rewrite the crosstrack error system () as: ẏ e = (ũ + u d ) sin ( ψ + ψ d γ p ) + v cos ( ψ + ψ d γ p ), (4) which, again, after several trigonometric transformations and addition/subtraction of terms, can be written as: ẏ e =u d sin (ψ d γ p ) + v cos (ψ d γ p )+ + (cos ( ψ) )(u d sin (ψ d γ p ) + v cos (ψ d γ p ))+ + sin ( ψ)(u d sin (ψ d γ p ) + v cos (ψ d γ p ))+ ( sin ( + ũ ψ) ) ξ cos ( ψ) +. (4) + ξ + ξ Inspired by [3], we transform (4) in the form: ẏ e = u d sin (ψ d γ p ) + v cos (ψ d γ p ) + χ T x, (4) where x = [ũ ψ] T and χ T = [χ χ ] with: χ = sin ( ψ + φ ξ ), φ ξ = atan(ξ), (43) χ = cos ( ψ) (u d sin (ψ d γ p ) + v cos (ψ d γ p ))+ ψ sin ( ψ) + ψ (u d sin (ψ d γ p ) + v cos (ψ d γ p )). (44) Combining (4) and (4) yields: ẏ e = U dy e + χ T x. (45) + ye C. Stability of the Cascaded System Equations (39) and (45) indicate that the total system has the same structure as (54) (55). We define: x := [x e y e ] T, x := [ũ ψ] T, (46) and compare with (54) (55), hence getting: [ f (t, x ) = kx e U dy e + y e ] T, (47) g(t, x)x = [ ω T x χ T ] T x. (48) [ ( ) ( ) t f (t, x ) = K puũ + K iu m X ũdτ u ] T (K T t d r r + K pr ψ + ψdτ) Kir (49) Therefore we can proceed with the following theorem: Theorem : The total cascade system (47) (49) has a globally κ-exponentially stable equilibrium point at x = if the control laws are given by (4)-(5) and the desired yaw angle and surge velocity are given by (4) and (33), respectively. Proof: This follows from satisfying Assumptions A6- A8 in Appendix A. Assumption A6: From Propositions and we already know that the equilibrium point y e = x e = x = is globally κ-globally exponentially stable when the heading angle and the surge velocity are perfectly tracked. By choosing the LFC V T = (/)(x e + ye), and: V T x x = x x V T x x c V T (x ) for c and x. (5) The condition V T x c x µ (5) is also satisfied x µ, µ >. Assumption A7: This condition is related to the interconnecting terms, namely ω T x and χ T x. It can be shown that this constraint is satisfied. Regarding (43), we have that χ ũ. Eq. (44) is bounded since the terms (cos ( ψ) )/ ψ and sin ( ψ)/ ψ are bounded and well-defined at ψ =. Similarly, from (37) we have that ω ũ and ω can be shown to be bounded in a similar way as χ. Assumption A8: It has already been proved that the equilibrium points ũ, ψ =, consequently x = is a GES equilibrium point. This means that the solutions satisfy: ũ(t) λ uo ũ(t o ) e (t to) (5) ψ(t) λ ψo ψ(t o ) e (t to) (53) and therefore by choosing ν ( ũ(t o ) ) = (λ uo ) ũ(t o ) and ν ( ψ(t o ) ) = (λ ψo ) ψ(t o ) the integrability condition is satisfied. Since all three assumptions are satisfied and, in addition to this, the nominal system Σ has a globally κ- exponentially stable equilibrium and the system Σ has a GES equilibrium, we conclude that the cascade system has a globally κ-exponentially stable equilibrium at x =.

6 ye (m) 3 v (m/sec) xe (m) β (deg) Fig. 4. The cross-track and the along-track errors converge to zero. Fig. 6. Plots of the sideslip angle β and the sway v. u vs ud (m/sec) ψ vs ψd (deg) Fig. 5. u u d ψ ψ d Plots of the desired and true surge velocity and heading angle. T δ Fig. 7. Plots of the control inputs T and δ. VI. SIMULATIONS The simulations were implemented with the model for Cybership II, a : 7 replica of a supply ship. The model parameters and more details regarding the vessel can be found in [4]. The virtual vehicle is moving on a straight line connecting the waypoints wpt = (, ) and wpt = (, ) and its total speed is U t = 3 m/sec. The controllers gains were chosen as: K pr =.34, K dr = 44.6, K ir =.38, K pu = 47.47, K iu = 34.. Moreover, = 9 m and k x =.3. When the simulation starts the vessel has an initial cross-track error of approximately. m and the along-track error starts to increase because the vessel is not moving on the desired path, where the virtual vehicle moves, yet, see Fig. 4. As a result, the desired surge velocity increases fast in order for the vessel to catch up with the virtual vehicle, Fig. 5. The total position error converges to zero when both x e = y e =, this occurs after 4 sec approximately. From then on, the vessel keeps moving at a constant speed of u d = U t = 3 m/sec and the steady state heading angle is ψ = γ p = 45. Plots of the sway speed and the sideslip angle can be seen in Fig. 6, while Fig. 7 shows plots of the control inputs T and δ. VII. CONCLUSIONS This paper dealt with the development of a guidance technique that can generate appropriate reference trajectories for the surge velocity of an underactuated system in order to minimize the along-track error in a path-tracking scenario. These velocity reference trajectories are fed into the surge velocity controller. This solution works in cooperation with the, already well-known, LOS guidance which generates the corresponding reference trajectories for the heading angle so as to minimize the cross-track error in the same scenario. The heading angle reference trajectories are fed into the heading autopilot. The total stability of the cascade system consisting

7 of the vehicle, the guidance system and the controllers was studied and it was shown to be globally κ-exponentially stable, a result which was supported by computer simulations. Future work includes solving the 3-D path-tracking problem in the presence of constant unknown environmental forces. APPENDIX In this section we present the theorem which was employed in Section V to show stability of the cascade. The proof can be found in [5] and in our case we include a reformulated version, like the one which was used in [6]. Consider the cascade system: Σ : ẋ = f (t, x ) + g(t, x)x (54) Σ : ẋ = f (t, x ), (55) where x R n, x R m, x [x, x ] T. The function f (t, x ) is continuously differentiable in (t, x ) and f (t, x ), g(t, x) are continuous in their arguments and locally Lipschitz. Theorem A: The cascaded system (54)-(55) is uniformly globally asymptotically stable if the following three assumptions are satisfied: A6) The system f (t, x ) is uniformly globally asymptotically stable with a Lyapunov function V (t, x ), V : R R n : R positive definite (that is V (t, ) = and V (t, x ) > x ) and proper (that is, radially unbounded) which satisfies: V x x c V (t, x ) x µ where c, µ >. (56) We also assume that ( V/ x )(t, x ) is bounded uniformly in t for all x µ, that is, there exists a constant c > such that for all t t o V x c x µ. (57) A7)The function g(t, x) satisfies g(t, x) θ ( x ) + θ ( x ) x, (58) where θ, θ : R R are continuous. A8) Equation ẋ = f (t, x ) is uniformly globally asymptotically stable and for all t o, t o x (t, t o, x (t o )) dt ν( x (t o ) ). (59) REFERENCES [] M. Breivik, V. E. Hovstein, and T. I. Fossen, Straight-line target tracking for unmanned surface vehicles, Modeling, Identification and Control, vol. 9, no. 4, pp. 3 49, 8. [] A. M. Lekkas and T. I. Fossen, Line-of-Sight Guidance for Path Following of Marine Vehicles. LAP LAMBERT Academic Publishing (O. Gal, Ed.), 3, ch. 5, In Advanced in Marine Robotics, pp [3] M. Bibuli, G. Bruzzone, M. Caccia, and L. Lapierre, Path-following algorithms and experiments for an unmanned surface vehicle, Journal of Field Robotics, vol. 6, no. 8, pp , 9. [4] M. Breivik and T. I. Fossen, Guidance laws for autonomous underwater vehicles. INTECH Education and Publishing, 9, ch. 4, pp [5] I. Kaminer, A. Pascoal, E. Hallberg, and C. Silvestre, Trajectory tracking for autonomous vehicles: An integrated approach to guidance and control, Journal of Guidance, Control, and Dynamics, vol., no., pp. 9 38, 998. [6] F. Alonge, F. D Ippolito, and F. Raimondi, Trajectory tracking of underactuated underwater vehicles, in Proceedings of the 4th IEEE Conference on Decision and Control, vol. 5,, pp [7] P. Encarnaçao and A. Pascoal, Combined trajectory tracking and path following: an application to the coordinated control of autonomous marine craft, in Proceedings of the 4th IEEE Conference on Decision and Control,, pp [8] R. Skjetne, T. I. Fossen, and P. V. Kokotović, Robust output maneuvering for a class of nonlinear systems, Automatica, vol. 4, no. 3, pp , 4. [9] R. Skjetne, U. Jørgensen, and A. Teel, Line-of-sight path-following along regularly parametrized curves solved as a generic maneuvering problem, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference,, pp [] M. Breivik and T. I. Fossen, Path following for marine surface vessels, in Proceedings of the OTO 4, Kobe, Japan, 4. [] F. Repoulias and E. Papadopoulos, Planar trajectory planning and tracking control design for underactuated AUVs, Ocean Engineering, vol. 34, no., pp , 7. [] G. Ambrosino, M. Ariola, U. Ciniglio, F. Corraro, A. Pironti, and M. Virgilio, Algorithms for 3D UAV path generation and tracking, in 45th IEEE Conference on Decision and Control. IEEE, 6, pp [3] A. P. Aguiar and J. P. Hespanha, Trajectory-tracking and pathfollowing of underactuated autonomous vehicles with parametric modeling uncertainty, IEEE Transactions on Automatic Control, vol. 5, no. 8, pp , 7. [4] M. Aicardi, G. Casalino, G. Indiveri, A. Aguiar, P. Encarnaçao, and A. Pascoal, A planar path following controller for underactuated marine vehicles, in 9th Mediterranean Conference on Control and Automation, Dubrovnik, Croatia,. [5] G. Indiveri, S. Cretí, and A. A. Zizzari, A proof of concept for the guidance of 3D underactuated vehicles subject to constant unknown disturbances, in 9th IFAC Conference on Manoeuvring and Control of Marine Craft, Arenzano, Italy,. [6] A. P. Aguiar and A. M. Pascoal, Dynamic positioning and way-point tracking of underactuated AUVs in the presence of ocean currents, International Journal of Control, vol. 8, no. 7, pp. 9 8, 7. [7] O. J. Sørdalen and O. Egeland, Exponential stabilization of nonholonomic chained systems, IEEE Transactions on Automatic Control, vol. 4, no., pp , 995. [8] T. I. Fossen, Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley and Sons Ltd.,. [9] N. Norrbin, On the design and analysis of the zig-zag test on base of quasi-linear frequency response, The Swedish State Shipbuilding Experimental Tank (SSPA), Gothenburg, Sweden, Tech. Rep. B 4-3, 963. [] A. Micaelli and C. Samson, Trajectory tracking for unicycle-type and two-steering-wheels mobile robots, INRIA, Sophia-Antipolis, Tech. Rep. 97, Nov [] L. Lapierre, D. Soetanto, and A. Pascoal, Nonlinear path following with applications to the control of autonomous underwater vehicles, in 4nd IEEE Conference on Decision and Control, vol., 3, pp [] K. Y. Pettersen and E. Lefeber, Way-point tracking control of ships, in Proceedings of the 4th IEEE Conference on Decision and Control, vol.. IEEE,, pp [3] E. Børhaug and K. Y. Pettersen, Cross-track control for underactuated autonomous vehicles, in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, 5, pp [4] R. Skjetne, The maneuvering problem, Ph.D. dissertation, Norwegian University of Science and Technology, 5. [5] E. Panteley and A. Loría, On global uniform asymptotic stability of nonlinear time-varying systems in cascade, Systems & Control Letters, vol. 33, no., pp. 3 38, 998. [6] A. Loría, T. I. Fossen, and E. Panteley, A separation principle for dynamic positioning of ships: Theoretical and experimental results, IEEE Transactions on Control Systems Technology, vol. 8, no., pp ,.