American Control Conference June -. Portland OR USA FrB. Nonlinear Tracking Control of Underactuated Surface Vessel Wenjie Dong and Yi Guo Abstract We consider in this paper the tracking control problem of an underactuated surface vessel. Based on the cascaded structure of tracking error dynamics we present two new tracking controllers using Barbalat s lemma and backstepping techniques under mild assumptions on the reference trajectory. The closed-loop systems achieve global asymptotical and exponential tracking respectively. Simulation results show that the proposed control laws are effective. I. INTRODUCTION Control of underactuated surface vessels has attracted much attention recently. The state of the underactuated surface vessel cannot be stabilized to the origin by any smooth pure state feedback control law due to Brockett s necessary condition ([3]). Furthermore the model of the underactuated surface vessel is not drift-less and the control methods developed for stabilizing nonholonomic systems cannot be directly used to solve the stabilization problem of this system. With the effort of researchers several stabilizing control laws have been developed. Interested readers may refer to [][7][7][9][][][]. The tracking control problem of underactuated surface vessel has also been studied. In [] and [9] output tracking control is discussed based on feedback linearization and Lyapunov theory. In [] and [] global practical tracking controllers are presented where the tracking errors are made to converge to a neighborhood of the origin. In [] a K-exponential tracking controller is proposed based on the cascaded structures of the tracking error dynamics. In [] an asymptotical tracking controller and an exponential tracking controller are proposed. The asymptotical tracking controller is derived with the passivity-based L g V -type control strategy and the exponential tracking controller is proposed by combining the cascade structure of the tracking error dynamics and backstepping technique. In this paper we consider the tracking control problem of an underactuated surface vessel. Under mild assumptions we propose new tracking controllers with the aid of the cascade structure of the closed-loop system. In order to propose the controllers we first apply a nonlinear coordinate transformation to the system and the tracking error dynamics is derived with a cascaded structure. Then an asymptotical tracking controller is designed using Barbalat s lemma and backstepping. Furthermore in order to improve the convergence rate of the tracking error an exponential tracking controller is proposed by fully exploiting the Department of Electrical and Computer Engineering University of Central Florida -73-99-9//$. AACC 3 cascaded structure of the tracking errors which ensures that the tracking error globally exponentially converges to zero. Compared with existing controllers our controllers make the tracking errors converge to zero asymptotically or exponentially. Good tracking performance is guaranteed under mild assumptions made on the reference trajectories. The rest of the paper is organized as follows. In section the problem under study is stated. In Section 3 an asymptotical tracking controller is designed. Then an exponential tracking control law is presented in Section. In Section simulation results are shown. Finally we conclude the paper in Section. II. PROBLEM STATEMENT Consider the tracking controller problem of an underactuated surface vessel. The vessel has two propellers which are the force in surge and the control torque in yaw. Following the results in [] the kinematics of the system can be written as ẋ cos ψ sin ψ u ẏ = sin ψ cos ψ v () ψ r where (x y) denotes the coordinate of the center of mass of the surface vessel in the earth-fixed frame ψ is the orientation of the vessel and u v and r are the velocities in surge sway and yaw respectively. Assume that ) the environment forces due to wind currents and waves can be neglected in the model ) the inertia added mass and damping matrices are diagonal. The dynamics of the surface vessel is described as ([]) u = m vr d u + τ m m m v = m ur v m m ṙ = m m uv d 33 r + τ where m ii > are given by the vessel inertia and the added mass effects d ii > are given by the hydrodynamic damping m ii and d ii are assumed to be constant. τ and τ are the surge control force and the yaw control moment respectively. Given a bounded feasible reference trajectory (x d y d θ d u d v d r d ) with reference input (τ d τ d ) ()
which satisfies ẋ d = u d cos ψ d v d sin ψ d ẏ d = u d sin ψ d + v d cos ψ d ψ d = r d u d = m v d r d d u d + τ d m m m v d = m u d r d v d m m ṙ d = m m u d v d d 33 r d + τ d the tracking control problem under study is to design a control law for system ()-() such that lim t (x x d )= lim t (y y d )= lim t (ψ ψ d )= lim t (u u d )= lim t (v v d )= lim t (r r d )=. For the tracking problem there are several controllers proposed in [9] [] [] [] [] []. In the following sections we propose two new controllers: an asymptotical tracking controller and an exponential tracking controller. Due to space limit proofs are omitted. III. ASYMPTOTICAL TRACKING CONTROLLER To facilitate the controller design we transform ()-() into a suitable form. Applying the following state and input transformations ([]): z = x cos ψ + y sin ψ z = v z 3 = x sin ψ + y cos ψ + m v z = ψ z = r z = m u z w = m m uv d 33 r + τ w =( d )u z 3 z τ we have ż = z z + z 3 z m z z m m ż = z + z (z + z ) m m ż 3 = z z ż = z ż = w ż = w. (3) () () () 3 Similarly by the transformation z d = x d cos ψ d + y d sin ψ d z d = v d z 3d = x d sin ψ d + y d cos ψ d + m v d z d = ψ d z d = r d z d = m u d z d w d = m m u d v d d 33 r d + τ d w d =( d )u d z 3d z d τ d we have ż d = z d z d + z 3d z d m z d z d m m ż d = z d + z d (z d + z d ) m m ż 3d = z d z d ż d = z d ż d = w d ż d = w d. () Define the tracking error to be (7) e = [e e e 3 e e e ] T =[z z d z z d z 3 z 3d z z d z z d z z d ] T. (9) We have ė = e e + e 3 e + z 3d e m m +z d e 3 m (e e + z d e + e z d ) ė = e + d () (e e + z d e m m +e z d + e e + e z d + z d e ) ė 3 = e e + e z d + z d e ė = e () ė = w w d ė = w w d. Lemma : Using the transformations () () and (7) lim t e i = ( i ) implies that (x y ψ u v r) asymptotically converges to (x d y d ψ d u d v d r d ) System ()-() is a cascaded system. An important issue in controlling cascaded systems is to design a control law to avoid peaking phenomenon []. However for system ()-() the peaking phenomenon never appears if the sub-system () is asymptotically stable. Lemma : For the cascaded system ()-() if (x d y d u d v d r d ) is bounded and e i (3 i ) globally converge to zero then e and e globally converge to zero. Based on Lemma we propose a global tracking controller in two steps using backstepping []. In the first step we consider the stabilization problem of the subsystem (e 3 e e ). Assuming e is a virtual control we design e and w such that e 3 e and e globally asymptotically
converge to zero. The result of the first step is stated in the following lemma. Lemma 3: In () if z d and z d are bounded and lim t z d (t) let Lemma : For system () if (x d y d ψ d u d v d r d ) is bounded and r d satisfies t r d(s)ds δt ( t ) () e = χ () where constant δ> the control inputs w = k (e α) z d e 3 + w d + α (3) w = w d (k + k )e k k e (9) where w = w d k 3 ż d e 3 k 3 z d ė 3 k e (k 3 k +)z d e 3 e 3 e () α = k 3 z d e 3 () ensure e χ = k e k e 3 (e + z d ) () i (3 i ) globally exponentially converge to zero where control parameters k i > ( i ) and constants k i > ( i ) then e 3 e and e k k. asymptotically converge to zero respectively. By Lemma and Lemma we have the following In the second step we design an actual control w so theorem. that e 3 e e and e asymptotically converge to zero with Theorem : For system () if (x d y d ψ d u d v d r d ) is the aid of the result in Lemma 3. We have the following bounded and r d satisfies () control law (9)-() ensure result. e i ( i ) globally exponentially converge to zero Lemma : For system () if z d and z d are bounded where control parameters k i > ( i ) and k k. and lim t z d (t) the control law In the control laws (9) and () the control parameters are k i ( i ). Ifk i > ( i ) and k k the w = k (e χ) e 3 (e + z d ) e + w d + χ() states of the closed-loop system () globally exponentially w = k (e α) z d e 3 + w d + α (7) converge to zero. The exponential convergence rate of e i (3 i ) can be adjusted by the control parameters globally asymptotically stabilize e 3 e e and e to the k i ( i ). The exponential convergence rate of e origin where constants k i > ( i ) α and χ are and e depends on m m and the exponential defined in () and (). With the aid of the results in Lemma and Lemma we have the following theorem. Theorem : For system () if (x d y d ψ d u d v d r d ) is bounded and lim t r d (t) then the control law ()- (7) ensures that (x y ψ u v r) globally asymptotically converges to (x d y d ψ d u d v d r d ). In Theorem a global tracking controller is proposed convergence rate of e 3 e e and e. It should be noted that the assumption on r d is not strict. In this paper we propose two tracking controllers. The first one in Theorem makes the tracking errors asymptotically converge to zero while the second one in Theorem makes the tracking errors exponentially converge to zero. However the assumptions made on the reference trajectories in Theorem are less conservative than that in Theorem. with the assumption that (x d y d ψ d u d v d r d ) is bounded Several tracking controllers are proposed in literature. and lim t r d (t). In the controller there are only four In [] and [] global practical tracking controllers are control parameters k k k 3 and k. In order to make the presented. The tracking errors are made to converge to a state of the closed-loop system asymptotically converge to neighborhood of the origin. While our exponential controller makes the tracking errors exponentially converge the given trajectory they are only needed to be positive. Generally large values of k i ( i ) make the tracking to zero. In [] a K-exponential tracking controller is errors converge to zero fast. However controller ()-(7) proposed based on the cascaded structures of the tracking does not guarantee that the convergence rate of the tracking error dynamics. In [] an exponential tracking controller error is exponential. In order to make the tracking errors is proposed. The exponential tracking controller is proposed converge to zero fast an exponential tracking controller is by combining the cascade structure of the dynamics of proposed in the next section. the tracking errors and the backstepping technique. The assumptions made on the reference trajectories are the IV. EXPONENTIAL TRACKING CONTROLLER same as those in Theorem. Advantages of our proposed controller over some previous results in literature include: Based on the transformation ()-() we have the following lemma. zero; ) The assumptions made on the reference trajectories ) It can make the tracking errors exponentially converge to Lemma : For the cascaded system ()-() if are less conservative and easily verified; 3) The controller (x d y d ψ d u d v d r d ) is bounded and () is globally exponentially stable then ()-() is globally exponentially is simple and the control parameters are easily chosen. stable. V. SIMULATION By Lemma it only needs to consider the system (). For () we have the following result. In this section we study the effectiveness of the proposed control laws by simulation. Consider an underactuated sur- 33
face vessel with the model parameters []: m = kg m = kg =kg d =7kg/s = kg/s d 33 =kg/s. Assume the initial condition of the system is (x()y()ψ()u()v()r()) = (3. ). The reference trajectory to be tracked is similar to that in []. Assume the initial reference state are x d () = m y d () = m ψ d () = rad u d () = m/s v d () =.m/s r d () =.rad/s and u d (t) =m/s r d (t) =.rad/s the reference trajectory (x d y d ψ d u d v d r d ) can be generated by (3) with proper τ d and τ d. We first use the control law ()-(7) to system ()-(). The control parameters are chosen as k = k =. k 3 = and k =. Figs - are simulation results. Figs - show the tracking errors of the closed-loop system converge to zero. Fig. 7 shows the desired trajectory and the actual trajectory of the vessel in X Y plane. Fig. shows that the two control inputs are bounded. The simulation results verify the result in Theorem. In the second simulation the exponential stabilizing law (9) and () is applied to system () with the model parameters and initial condition stated above. In the control law we choose k =. k = k 3 = k =.. Figs 9-7 are simulation results. Figs 9- show that the given desired trajectories and the response of each state. They show that the states of the closed-loop system converge to the desired trajectories. Furthermore Fig. shows that the tracking errors exponentially converge to zero because logarithm of absolute value of each tracking error decrease linearly. Fig. shows the desired trajectory and the actual trajectory of the vessel in X Y plane. Fig. 7 shows that the control inputs are bounded. From the simulation the two proposed control laws all make the tracking error converge to zero. Since the first control law does not guarantee an exponential convergence rate of the tracking error of the closed-loop system the tracking performance may be not good as one desires. The second control law guarantees that the tracking error exponentially converge to zero. Therefore the performance of the tracking error system is satisfactory. VI. CONCLUSION In this paper the tracking control problem of an underactuated surface vessel is considered. New asymptotical tracking controller and exponential tracking controller are proposed with the aid of the cascaded structure of the error dynamics Barbalat s lemma and the backstepping technique. They can make the tracking errors converge to zero asymptotically or exponentially. Simulation results show the proposed controllers are effective. The ideas developed in this paper can be applied to design controllers of other underactuated systems. REFERENCES [] A. Behal D. M. Dawson W. E. Dixon and Y. Fang Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics IEEE Trans. on Automatic Control vol.7 no.3 pp.9-. 3 [] S. Berge K. Ohtsu and T. Fossen Nonlinear control of ships minimizing the position tracking errors Modeling Identification Control vol. pp.-7 999. [3] R. Brockett Asymptotic stability and feedback stabilization in Differential Geometric Control Theory R. Brockett R. Millman and H. Sussmann Eds. Boston MA: Birkhauser 93. [] F. Bullo Stabilization of relative equilibria for underactuated systems on Riemannian manifolds Automatica vol.3 no. pp.9-3. [] F. Bullo N. E. Leonard and A. D. Lewis Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups IEEE Trans. Automatic Control vol. pp.37-. [] W.E. Dixon D.M. Dawson F. Zhang and E. Zergeroglu Global exponential tracking control of a mobile robot system via a PE condition IEEE Transactions on Systems Man and Cybernetics vol.3 pp.9-. [7] I. Fantoni R. Lozano F. Mazenc and K.Y. Pettersen Stabilization of a nonlinear underactuated hovercraft Int. J. Robust Nonlinear Control vol. no. pp.-. [] T.I. Fossen Guidance and Control of Ocean Vehicles New York: Wiley 99. [9] J.-M. Godhavn Nonlinear tracking of underactuated surface vessels Proc. 3th Conf. Decision Control Kobe Japan December 99 pp.97-9. [] Z.P. Jiang Global tracking control of underactuated ships by Lyapunov s direct method Automatica vol.3 pp.3-39. [] M. Krstic I. Kanellakopoulos and P. Kokotovic Nonlinear and Adaptive Control Design. New York NY: Wiley 99. [] H.K. Khalil Nonlinear Systems. nd Edition Prentice-Hall Englewood Cliffs NJ 99. [3] I. Kolmanovsky and N.H. McClamroch Developments in nonholonomic control systems IEEE Control System Magazine vol. no. 99 pp.-3. [] E. Lefeber K. Y. Pettersen and H. Nijmeijer Tracking control of an underactuated ship IEEE Trans. on Control systems Technology vol. no. 3 pp.-. [] F. Mazenc K.Y. Pettersen and H. Nijmeijer Global uniform asymptotic stabilization of an underactuated surface vessel IEEE Trans. on Automatic Control vol.7 no. pp.79-7. [] K.Y. Pettersen Exponential stabilization of underactuated vehicles Ph.D. thesis Norwegian University of Science and Technology 99. [7] K.Y. Pettersen and O. Egeland Exponential stabilization of an underactuated surface vessel Proc. 3th IEEE Conf. Decision and Control Kobe Japan 99 pp.97-97. [] K.Y. Petterson and H. Nijmeijer Global practical stabilization and tracking for an underactuated ship A combined averaging and backstepping approach Proc. IFAC Conf. System Structure Control Nantes France July 99. pp.9-. [9] K.Y. Pettersen and O. Egeland Time-varying exponential stabilization of the position and attitude of an underactuated autonomous underwater vehicle IEEE Trans. on Automatic Control vol. no. 999 pp.-. [] M. Reyhanoglu Control and stabilization of an underactuated surface vessel Proc. 3th IEEE Conf. Decision and Control Kobe Japan 99 pp.37-37. [] H.J. Sussmann and P.V. Kokotovic The peaking phenomenen and the global stabilization of nonlinear systems IEEE Trans. on Automatic Control vol.3 no. 99 pp.-. x: dashdot x d 3 3 Fig.. Response of x and x d
y: dashdot y d r: dashdot r d : solid (rad/sec).9..7....3.. Fig.. Response of y and y d Fig.. Response of r and r d reference ψ: dashdot ψ d : solid (rad) Y (m) 3 3 X (m) Fig. 3. Response of ψ and ψ d Fig. 7. Trajectory in X Y plane 9 x u: dashdot u d τ (N): solid τ (N.m): dashdot 7 3 Fig.. Response of u and u d Fig.. Control inputs v: dashdot v d 3 3 x: dashdot x d 3 3 Fig.. Response of v and v d 3 Fig. 9. Response of x and x d
. y: dashdot y d r: dashdot r d : solid (rad/sec)...... Fig.. Response of y and y d Fig.. Response of r and r d ψ: dashdot ψ d : solid (rad) 7 3 logarithm of absolute value of tracking errors 3 7 9 Fig.. Response of ψ and ψ d Fig.. Logarithm of absolute value of tracking errors. u: dashdot u d.... 3 Y (m) reference 3. 3 3 X (m) Fig.. Response of u and u d Fig.. Trajectory in X Y plane v: dashdot v d....... τ (N): dashdot τ (N.m): solid 3 3 7 9 3 Fig. 3. Response of v and v d Fig. 7. Control inputs 3