est to the vehicle is provided as a reference. The reference posture is formally computed as ( ()), where () =arg min R k ()k P is a projection mappin

Transcription

1 PROCEEDINGS OF MED' - 9 T H MEDITERRANEAN CONFERENCE ON CONTROL AND AUTOMATION, DUBROVNIK, CROATIA, JUNE. Combined trajectory tracking and path following for marine craft P. Encarnac~ao A. Pascoal Institute for Systems and Robotics and Dept. Electrical Eng. Instituto Superior Tecnico Av. Rovisco Pais,, 49- Lisboa, Portugal fpme,antoniogisr.ist.utl.pt Abstract The paper presents a solution to the problem of combined trajectory tracking and path following for marine craft. This problem is motivated by the practical need to develop control systems for marine vehicles that can yield good trajectory tracking performance while keeping some of the desired properties associated with path following. The solution described builds on and extends previous work by Hindman and Hauser on so-called maneuver modied trajectory tracking. The theoretical tools used borrow from Lyapunov stability theory and backstepping techniques. Simulations with a nonlinear model of an underactuated marine craft illustrate the performance of the combined trajectory tracking and path following controller derived. Keywords Trajectory Tracking, Path Following Systems, Nonlinear Control, Backstepping Techniques, Autonomous Marine Crafts. I. Introduction Over the last few years, there has been considerable interest in the development of powerful methods for motion control of autonomous vehicles. The problems of motion control addressed in the literature can be roughly classied in three groups: point stabilization, where the goal is to stabilize a vehicle at a given point, with a desired orientation; trajectory tracking, where the vehicle is required to track a time parameterized reference, and path following, where the vehicle is required to converge to and follow a desired path, without any temporal speci- cations. Point stabilization presents a true challenge to control system designers when the vehicle has nonholonomic constraints since, as pointed out in the celebrated result of Brockett [], there is no smooth (or even continuous) constant state-feedback control law that will do the job. To overcome this diculty two main approaches have been proposed: smooth time-varying control laws [], [3] and discontinuous feedback laws [4], [5], [6], [7]. The trajectory tracking problem for fully actuated systems is now well understood and satisfactory solutions can be found in standard nonlinear control textbooks [8]. However, when it comes to underactuated vehicles, this is, when the vehicle has less actuators then state variables to be This work was supported in part by the Commission of the European Communities under contract no. MAS3-CT97-9 (ASI- MOV) and the Portuguese PRAXIS and PDCTM programs under projects CARAVELA and MAROV. The rst author beneted from a PRAXIS XXI Graduate Student Fellowship. tracked, the problem is still a very active topic of research. Linearization and feedback linearization methods [9], [], as well as Lyapunov based control laws [], [], [] have been proposed. Applications to underactuated surface vessels can be found in [3], [4], [5]. Path following control has received relatively less attention than the other two problems. See for example [], [6] and the references therein. Path following systems for marine vehicles have been reported in [7], [8], [9], []. While on a path following mode, the vehicle forward speed need not be controlled accurately since it is sucient to act on the vehicle orientation to drive it to the path. Typically, smoother convergence to a path is achieved in this case when compared to the performance obtained with trajectory tracking controllers, and the control signals are less likely pushed to saturation []. These three classes of problems have traditionally been addressed separately. In fact, Brockett's condition implies that the point stabilization problem cannot be simply solved by designing a trajectory tracking control law and applying it to a trajectory that degenerates into a single point. Conversely, it is not clear how to extend stabilization methodologies now available to trajectory tracking problems. However, a paper by Hindman and Hauser [] that has not received much attention in the literature shows clearly how to go from a trajectory tracking to a path following controller. In their work, the authors assume that a trajectory tracking controller is available and that a Lyapunov function is known that yields asymptotic stability of the resulting control system about a desired trajectory. The Lyapunov function captures the distance between the vehicle's posture (position and orientation) and the desired posture along the trajectory as measured in some P -metric, that is, V = k (t) (t)k P ; where (t) is the vehicle's posture, (t) is the desired posture, and kxk P = x T P x with P >. The key idea in the work of Hindman and Hauser can now be simply explained as follows. To execute a path following maneuver, the vehicle should look at the "closest point on the path" and adopt the corresponding hypothetical vehicle's posture as a reference to which it should converge. Thus, instead of feeding a desired posture (t) to the tracking controller, the posture corresponding to the path point that is clos-

2 est to the vehicle is provided as a reference. The reference posture is formally computed as ( ()), where () =arg min R k ()k P is a projection mapping that computes the trajectory time that minimizes the distance k ()k P. Under some technical conditions on path shape and path parametrization, it is shown that this strategy actually drives the vehicle to the path. Interestingly enough, the work of Hindman and Hauser goes even one step further. In fact, it shows how to combine in a single control law trajectory tracking and path following behaviors, thus achieving smooth spatial convergence to the trajectory as well as time convergence. This is accomplished by modifying the projection function () through the addition of a time dependent penalty term to obtain (; t) = arg min R h ( ) k ()k P + (t ) i : The constant > weighs the relative importance of convergence in time over spatial convergence to the path. If = is chosen, pure path following is achieved ( (; t) = ()). If =, (; t) = t since = t is always a minimizer of (t ), and the original trajectory tracking controller is recovered. When < <, the system displays attributes of both schemes. Since the Lyapunov function that is used to prove stability of the trajectory tracking control system is also used to dene the distance to the path, the methods proposed by Hindman and Hauser are intrinsically suitable for vehicles with negligible dynamics. This follows from the fact that in practical applications the distance to the path should only take into account kinematic variables such as vehicle position and possibly orientation, thus avoiding the need to feedforward dynamic variables. With the objective of applying similar methods to the control of marine craft, this paper extends the key results of Hindman and Hauser to encompass systems that capture the motion of vehicles with non negligible dynamics. This is done by resorting to backstepping techniques after modication of a standard tracking control law for marine craft. Using the methodology developed, the distance to the path is evaluated by taking into consideration the kinematic variables only, namely the vehicle position. The paper is organized as follows: Section II derives a control law that enables balancing combined requirements of trajectory tracking and path following for vehicles with non negligible dynamics. The application to an autonomous marine craft is described in Section III. Finally, Section IV presents the main conclusions. II. Combined trajectory tracking and path following control Consider the following general nonlinear control system _ = f () + g () () where R n is the state and R m is the control input. The functions f : D! R n and g : D! R n+m are smooth in a domain D R n that contains = and f () =. Let (t) be a reference trajectory for and let (; (t)) a trajectory tracking control law for () such that, with = (; (t)),!. Assume that local asymptotic stability of the trajectory tracking system can be proven using the Lyapunov function V (; t) = k (t)k P ; () where P is a positive denite weighting matrix (notice that V can be simply viewed as the tracking error measured in the P metric). Assume also that _V (; t) = h (t) ; _ _ (t)i P k V (; t) ; where hx; yi P = x T P y is the inner product relative to P and k is a positive scalar. The condition V _ k V can be weakened if one is able to prove stability with V _ negative semidenite by resorting to LaSalle-Yoshizawa or Barbalat's lemmas [8]. To convert the trajectory tracking control law into a combined trajectory tracking and path following control law, let the square of the distance from the path point ( () ; ) to the point (; t) be given by the function U (; t; ) = ( ) V (; ) + (t ) = ( ) k ()k P + (t ) : Given U (; t; ), dene the mapping (; t) =arg min U (; t; ) = arg min R R h ( ) k ()k P + (t ) i ; (3) where is a positive scalar that allows balancing the relative importance of time convergence to the trajectory against spatial convergence to the path. If =, (; t) =arg min R k ()k P gives the trajectory time that minimizes the distance k ()k P and a pure path following controller is obtained. If =, (; t) = t (since = t is always the minimizer of (t ) ), yielding the original trajectory tracking controller. When < <, the mapping gives the trajectory time of the path point ( () ; ) that is closest to (; t) in the metric induced by U (; t; ), and a mixed trajectory tracking and path following behavior is achieved. Under some mild technical conditions on the reference path shape and parameterization (see [] for details), () is unique and continuous. Note that a necessary condition for = (; t) to be a minimizer of U (; t; ) is that U (; t; ) = ( ) ( (; t)) ; d d ( (; t)) (t (; t)) = : (4)

3 Consider now the candidate Lyapunov function Y (; t) =min R U (; t; ) = U (; t; (; t)) (5) = ( ) k ( (; t))k P + (t (; t)) : Dierentiating Y with respect to time gives _Y (; t) = ( ) D ( (; t)) ; _ d d ( (; t)) d E dt (; t) P (; t) (t (; t)) d dt = ( ) D ( (; t)) ; _ d E d ( (; t)) + P d E d d ( (; t)) h dt (; t) ( ) D ( (; t)) ; i + (t (; t)) : From (4), the second term of this expression is zero, hence _Y (; t) = ( ) V _ (; (; t)) ( ) k V (; (; t)) : To conclude asymptotic time convergence to the path one must resort to Barbalat's Lemma [8] since Y _ is not negative denite in t. Assuming that the rst and second derivatives of () (the velocity and the acceleration along the reference path, respectively) are bounded, it is easy to prove that Y is bounded. Thus, Y _ is uniformly continuous. Since Y _ and Y is bounded below by zero, Y (; t)! Y as t!, where Y is positive and nite. Now, Barbalat's Lemma allows for the conclusion that _Y (; t)! as t!. This implies that! ( (; t)) as t!. Using (4), it follows that (; t)! t as t!. Therefore,! (t) as t!, that is, the system tracks the reference path asymptotically. In order to apply the method described to motion control of autonomous vehicle, some modications are required. This stems from the fact that the equations of motion of these vehicles exhibit a clear split between kinematics and dynamics and the specication of a trajectory to be tracked includes the kinematic variables only. For example, in the case of a marine craft a reference trajectory may simply include the desired evolution of the center of mass (and possibly the orientation) of that craft. In this case, however, the methodology proposed by Hindman and Hauser still provides a powerful tool to obtain a path following controller for the craft assuming a trajectory tracking control law exists and its stability has been proved using Lyapunov theory. In fact, the work of Hindman and Hauser provides a clear recipe to suitably modify the Lyapunov function for trajectory tracking so as to yield a new Lyapunov function for path following stability. Suppose for example that () captures the kinematics of a vehicle, and that (after a suitable coordinate transformation) the dynamics can be cast in the form _ = ; (6) where is the actual input vector. Suppose also that a stabilizing trajectory tracking controller = (; (t)) for () and a corresponding Lyapunov function V (; t) have been found. The vehicle dynamics can be included in the control design through a backstepping step [] as follows. Consider the change of variables z = : The variable z can be interpreted as the dierence between the virtual control variable and the virtual control law. Thus, if z tends to zero, the system approaches the manifold corresponding to the "kinematic system". This motivates the choice of V = Y (; t) + zt z as a candidate Lyapunov function. Dierentiating V with respect to time yields Setting _V = Y Y () [f () + g () ] + () g () z + zt _ ( ) k V (; (; t)) + Y () g () z + zt _ = _ g T () Y T () K dyn z; where K dyn is a positive denite matrix, it follows that _V ( ) k V (; (; t)) z T K dyn z : Using the same arguments as before, convergence to the path can be concluded. In practice, the modications required to deal with more realistic cases may warrant further complexity of the resulting control laws, as the next section will show. However, the key recipe provided by Hindman and Hauser still applies. III. Application to a surface craft The methodology described in Section II is now applied to the design of a combined trajectory tracking and path following controller for the DELFIM autonomous surface craft developed at the Instituto Superior Tecnico of Lisbon, Portugal. See [3], [4] for details on the construction, modeling, and control of the DELFIM vehicle. A. Vehicle model Following standard practice, the equations of motion of a marine craft can be developed using a global coordinate frame fug and a body-xed coordinate frame fbg that moves with the vehicle. The following notation is required [5], [6]: U p Borg := (x; y) T - position of the origin of fbg measured in fug; B v Borg := (u; v) T - velocity of the origin of fbg relative to fug, expressed in fbg (i.e., body-xed linear velocity); B! B := r - angular velocity of fbg relative to fu g, expressed in fbg (i.e., body-xed angular velocity). The symbol U BR ( ) denotes the rotation matrix from : 3

4 fbg to fug, parameterized by the yaw angle. With this notation, the dynamics and kinematics of the marine craft can be cast in the form Dynamics: M RB B _v Borg + C RB B v Borg B v Borg = RB (7) Kinematics: I z _r = N (8) U _p Borg = U BR ( ) B v Borg (9) _ = r; () where RB denotes the vector of external forces and N is the external yawing torque. The symbols M RB and I z denote the rigid body mass matrix and the moment of inertia about the z B axis respectively, and C RB represents the matrix of Coriolis and centripetal terms. The vector RB can further be decomposed as RB = add ( _; ) + body () + ; () T where add is the added mass term and = B vb T org ; r. The term body consists of the hydrodynamic forces acting on the vehicle's body. The vector = (T c ; ) T denotes the forces applied to the vehicle by the propulsion system, where T c is the force exerted on the vehicle's body by the two back propellers working in common mode. A simplied model for the yawing torque N is [4], [7] N = N v uv + N r ur + T d ; () where T d represents the torque generated by the two back propellers in dierential mode. The symbols N v and N r denote hydrodynamic torque coecients. Combining equations (7)-(8) and ()-(), the body- xed dynamic model of the AUV can be written as [4], [7] M B _v Borg + rjm B v Borg + D B v Borg ; r B v Borg = Tc (3) I z _r N v uv N r ur = T d ; (4) where M includes the rigid body mass matrix and added mass terms, and J is the skew symmetric matrix J = The term rjm B v Borg captures Coriolis and centripetal, as well as added mass terms, and D B v Borg ; r includes the hydrodynamic damping terms. For the DELFIM vehicle, a simplied model of the hydrodynaminc damping terms is Xu Xuuu D BvBorg ; r X = uuuu X vvv Y rr Y vu : ; where X () and Y () denote hydrodynamic force coecients. The D () matrix can be further decomposed as D B v Borg ; r = D B v Borg + D r, where D BvBorg = D r = Xu Xuuu X uuuu X vvv Y vu Y r Observe that the DELFIM model was written in such a way as to separate the surge and sway dynamics from the yaw dynamics. This is because control design will resort to two backstepping-like steps, the rst to include the surge and sway dynamics, and the second to include the yaw dynamics. B. Trajectory tracking controller As explained in Section II, the starting point for the derivation of a combined trajectory tracking and path following controller for the surface craft is the availability of a trajectory tracking controller for the same vehicle. In what follows, the general structure of the trajectory tracking controller for underactuated vehicles described in [4] is adopted. In his work, Godhavn presents both a nonadaptive and an adaptive controller where the latter aims at estimating and rejecting constant ocean currents. Both schemes are applicable to surface vehicles that are only actuated in surge and yaw. In this paper, only the nonadaptive version will be used since, to derive the adaption laws for the adaptive case, the estimation errors are included in the Lyapunov functions, thus preventing the use of the Hauser and Hindman methods. The controllers developed by Godhavn are applicable to low speed slender ships for which decoupling the surge from the steering (sway and yaw) dynamics is natural. Since that is not the case for the DELFIM vehicle, a slight modication in the design method is necessary. The presentation that follows borrows considerably from the work in [4]. Let (t) = (x ref (t) ; y ref (t)) T be a reference for the position U p Borg and let ~p = U p Borg (t) be the corresponding position error. Notice how in this case the reference trajectory only includes the desired position of the vehicle, thus avoiding specifying its orientation explicitly. Consider the Lyapunov function : V = ~pt ~p: (5) Dierentiating V with respect to time yields Let _V = ~p T U R B B v Borg _ (t) : U R B B v Borg where k is a positive scalar. Then Following backstepping tech- is negative denite in ~p. niques, dene = _ (t) k ~p; _V = k ~p T ~p z = U R B B v Borg ( _ (t) k ~p) 4

5 as the dierence between the virtual control variable U R B B v Borg and the virtual control law _ (t) k ~p, and let V be the Lyapunov function V = V + zt z : (6) Finally, consider the Lyapunov function with time derivative V 3 = V + z The time derivative of V is given by where k >, and with _V = k ~p T ~p k z T z + z T U R B M Tc a = h r + h a ; _V 3 = k ~p T ~p k z T z z T U R B M + a z h I z (T d + N v uv + N r ur) + h = k ~p T ~p k z T z + z T U R B M z z h I z (T d + N v uv + N r ur) + h : The control signal + Choosing yields h = (JM MJ + D ) B v Borg h = D B v Borg B v Borg k M B v Borg M U R T B (~p (t) k _ (t) + k z ) : T c = a (7) _V = k ~p T ~p k z T z U R B M z T To include the yaw dynamics, consider now z = a; : a with >. The time derivative of a can be computed to give _a = h _r + h ; where h = (JM MJ + D ) B _v Borg r+ _D B v B Borg v Borg + D B v B Borg _v Borg k M B _v Borg U _R B T (~p (t) k (t) + k z ) M U RB T ~p _ (3) (t) k (t) + k _z : Therefore, h _z = h : I z (T d + N v uv + N r ur) In order to simplify the expressions, dene With these denitions, h h = h = h h = h : _z = h I z (T d + N v uv + N r ur) + h : T d = N v uv N r ur I z h h + z T U R B M where k 3 is a positive scalar, renders _V 3 = k ~p T ~p k z T z k 3 z + k 3 z ; (8) negative semidenite. In [4] it is shown, using LaSalle arguments, that the control law (7)-(8) provides exponential tracking of z as long as saturation of the inputs is avoided, and the vehicle surge velocity u is kept nonzero. It is now straight forward to derive a combined trajectory tracking and path following controller for the surface craft. To do this, one starts by identifying the kinematic equation (9) with the general model () by making = (x; y) T and as the vehicle inertial velocity, that is, = U R B B v Borg. Hence, f () = and g () = I, where I denotes the identity matrix. The Lyapunov function (5) should then be modied according to (5). The last step consists of redening the Lyapunov function (6) as V = Y (; t) + zt z ; where Y is dened in (5). The details are omitted. C. Simulation results Simulations with the model of the autonomous surface craft DELFIM were performed choosing k =, k = k 3 = :, and = :7, as in [4]. In the simulations, the initial posture of the vehicle was set to (x; y; )j t= = 35 m; ; and the desired velocity along the reference path was chosen as ms. The performance achieved with pure trajectory tracking ( = ), pure path following ( = ), and for combined trajectory tracking and path following ( = :) was assessed. Figure shows the vehicle following a path consisting of two straight line segments joined together by a cubic spiral, for the case when =. Figure depicts the actuator signals, that is, the force T c and the torque T d generated by the common and dierential thruster modes, respectively, and Figure 3 shows how the tracking errors x (t) x ref (t) and y (t) y ref (t) converge to zero. The 5

6 x y vehicle trajectory for = (pure path following case) is presented in Figure 4. Figure 5 shows the actuator signals. The trajectory tracking errors x (t) x ref (t) and y (t) y ref (t), and the modied errors x (t) x ref ( (; t)) and y (t) y ref ( (; t)) are plotted in Figures 6 and 7, respectively. From Figure 7 it is possible to see that the vehicle actually converges to the path. However, from Figure 6 it is clear that it is always about : m ahead of the "reference" vehicle. Finally, Figures 8 to refer to the combined trajectory tracking and path following controller that is obtained by setting = :. Figure 8 shows the x y vehicle trajectory, while Figure 9 displays the actuator signals. From the tracking errors in Figure and from the modied errors in Figure, the combined behavior can be interpreted as follows: the vehicle converges quickly to the desired path and only then will it react to achieve zero trajectory tracking error. Notice that in this particular example the overall spatial approach to the desired path does not show drastic dierences in the three dierent simulations. However, the control activity is considerably smoother in the cases where 6=, thus showing the benets of path following. y y ref x Fig. 3. Tracking errors (t) (t) for =. x y Fig. 4. Pure path following: =. T c [N] y Fig.. Pure trajectory tracking: = modied trajectory tracking and allows for its application to the control of vehicles with non negligible dynamics. The methodology described has been used to develop a combined trajectory tracking and path following controller for a fully actuated underwater vehicle and also applied to the coordinated control of autonomous marine craft [8], [9]. Further work will address the problem of controlling surface crafts in the presence of parameter uncertainty and external disturbances, as well as reduced motion sensor information. References [] R. W. Brockett, \Asymptotic stability and feedback stabilization," in Dierential Geometric Control Theory, R. W. Brock- T d [Nm] Fig.. Actuator signals for = : force T c and torque T d. IV. Conclusions The paper presented a solution to the problem of combined trajectory tracking and path following for marine craft. The solution described builds on and extends previous work by Hindman and Hauser on so-called maneuver T c [N] T d [Nm] Fig. 5. Actuator signals for = : force T c and torque T d. 6

7 4 T c [N] 3 y y ref Fig. 6. Tracking errors (t) (t) for =. T d [Nm] Fig. 9. Actuator signals for = :: force T c and torque T d. (π λ ) y y ref (π λ ).5.5 y y ref Fig. 7. Modied error (t) ( (; t)) for = Fig.. Tracking errors (t) (t) for = :. ett, R. S. Millman, and H. J. Sussman, Eds., Boston, 983, pp. 8{9, Birkhauser. [] C. Canudas de Wit, H. Khennouf, C. Samson, and O. J. Srdalen, \Nonlinear control design for mobile robots," in Recent Trends in Mobile Robotics, Yuan F. Zheng, Ed. 993, vol., pp. {56, World Scientic Series in Robotics and Automated Systems. [3] J. M. Godhavn and O. Egeland, \A Lyapunov approach to exponential stabilization of nonholonomic systems in power form," IEEE Transactions on Automatic Control, vol. 4, no. 7, pp. 8{3, July 997. [4] A. Aguiar and A. Pascoal, \Stabilization of the extended nonholonomic double integrator via a logic-based hybrid control," in Proceedings of SYROCO', 6 th IFAC Symposium on Robot Control, Vienna, Austria, Sept.. [5] A. Astol, \Exponential stabilization of a wheeled mobile ro- x y Fig. 8. Combined trajectory tracking and path following: = :. bot via discontinuous control," Journal of Dynamic Systems, Measurement, and Control, vol., pp. {6, Mar [6] J. P. Hespanha, \Stabilization of nonholonomic integrators via logic-based switching," in Proceedings of IFAC'96, 3 th World Congress of IFAC, S. Francisco, CA, USA, July 996, vol. E, pp. 467{47. [7] C. Canudas de Wit and O. J. Srdalen, \Exponential stabilization of mobile robots with nonholonomic constraints," IEEE Transactions on Automatic Control, vol. 37, no., pp. 79{ 797, Nov. 99. [8] H. K. Khalil, Nonlinear Systems, Prentice-Hall, New Jersey, Second edition, 996. [9] G. Walsh, D. Tilbury, S. Sastry, R. Murray, and J. P. Laumond, \Stabilization of trajectories for systems with nonholonomic constraints," IEEE Transactions on Automatic Control, vol. 39, no., pp. 6{, Jan [] E. Freund and R. Mayr, \Nonlinear path control in automated vehicle guidance," IEEE Transactions on Robotics and Automation, vol. 3, no., pp. 49{6, Feb [] D. H. Kim and J. H. Oh, \Globally aymptotically stable tracking control of mobile robots," in Proceedings of CCA'98, 998 IEEE International Conference on Control Applications, Trieste, Italy, Sept [] R. Fierro and F. L. Lewis, \Control of a nonholonomic mobile robot: Backstepping kinematics into dynamics," in Proceedings of CDC'95, 34 th IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec. 995, pp. 385{38. [3] J. M. Godhavn, \Nonlinear tracking of underactuated surface vessels," in Proceedings of CDC'96, 35 th IEEE Conference on Decision and Control, Kobe, Japan, Dec. 996, pp. 987{99. [4] J. M. Godhavn, Topics in Nonlinear Motion Control: Non- 7

8 y y ref (π λ ) (π λ ) holonomic, Underactuated, and Hybrid Systems, Ph.D. thesis, Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway, 997. [5] K. Y. Pettersen and H. Nijmeijer, \Tracking control of an underactuated surface vessel," in Proceedings of CDC'98, 37 th IEEE Conference on Decision and Control, Tampa, Florida, USA, Dec. 998, pp. 456{4566. [6] Z. Jiang and H. Nijmeijer, \A recursive technique for tracking control of nonholonomic systems in chained form," IEEE Transactions on Robotics and Automation, vol. 44, no., pp. 65{79, Feb [7] P. Encarnac~ao, A. Pascoal, and M. Arcak, \Path following for marine vehicles in the presence of unknown currents," in Proceedings of SYROCO', 6 th IFAC Symposium on Robot Control, Vienna, Austria, Sept.. [8] G. Indiveri, M. Aicardi, and G. Casalino, \Nonlinear timeinvariant feedback control of an underactuated marine vehicle along a straight course," in Proceedings of MCMC', IFAC Conference on Manoeuvring and Control of Marine Craft, Aalborg, Denmark, Aug.. [9] P. Encarnac~ao and A. Pascoal, \3D path following for autonomous underwater vehicles," in Proceedings of CDC', 39 th IEEE Conference on Decision and Control, Sydney, Australia, Dec.. [] M. Aicardi, G. Casalino, G. Indiveri, A. Aguiar, P. Encarnac~ao, and A. Pascoal, \A planar path following controller for underactuated marine vehicles," in Proceedings of MED', 9 th Mediterranean Conference on Control and Automation, Dubrovnik, Croatia, June. [] R. Hindman and J. Hauser, \Maneuver modied trajectory tracking," in Proceedings of MTNS'96, International Symposium on the Mathematical Theory of Networks and Systems, St. Louis, MO, USA, June 99. [] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc, New York, 995. [3] A. Pascoal, P. Oliveira, C. Silvestre, L. Sebasti~ao, M. Runo, V. Barroso, J. Gomes, G. Ayela, P. Coince, M. Cardew, A. Ryan, H. Braithwaite, N. Cardew, J. Trepte, N. Seube, J. Champeau, P. Dhaussy, V. Sauce, R. Moitie, R. Santos, F. Cardigos, M. Brussieux, and P. Dando, \Robotic ocean vehicles for marine science applications: the european ASIMOV project," in Proceedings of OCEANS' MTS/IEEE, Rhode Island, Providence, USA, Sept.. [4] M. Prado, \Modeling and control of an autonomous ocean vehicle," M.S. thesis, Department of Electrical Engineering, Instituto Superior Tecnico, Lisboa, Portugal,. [5] D. Fryxell, P. Oliveira, A. Pascoal, C. Silvestre, and I. Kaminer, \Navigation, guidance, and control of AUVs: An application to the MARIUS vehicle," IFAC Journal of Control Engineering Practice, vol. 4, no. 3, pp. 4{49, Mar [6] C. Silvestre and A. Pascoal, \Control of an AUV in the vertical and horizontal planes: System design and tests at sea," Transactions of the Institute of Measurement and Control, vol. 9, no. 3, pp. 6{38, 997. [7] T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley & Sons, Chichester, England, 994. [8] P. Encarnac~ao and A. Pascoal, \Combined trajectory tracking and path following for underwater vehicles," Accepted for presentation at CAMS', IFAC Conference on Control Applications in Marine Systems, Apr.. [9] P. Encarnac~ao and A. Pascoal, \Combined trajectory tracking and path following control: an application to the coordinated control of autonomous marine craft," Submitted to CDC', 4 th IEEE Conference on Decision and Control, Mar Fig.. Modied error (t) ( (; t)) for = :. 8