Underactuated dynamic positioning of a ship ... - IEEE Xplore

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 5, SEPTEMBER 2000

Underactuated Dynamic Positioning of a Ship—Experimental Results Kristin Y. Pettersen and Thor I. Fossen

Abstract—The paper considers the dynamic positioning problem for a ship having only two independent controls, namely surge force and yaw moment. A time-varying feedback control law including integral action is developed and proved to exponentially stabilize both the North and East positions and the orientation of the ship. Experimental results are presented where the controller is implemented on a model ship, scale 1:70, of an offshore supply vessel. The experiments show that the ship converge to a neighborhood of the desired position and orientation, oscillating about stationary errors, and that with integral action in the control law these stationary errors are reduced. Index Terms—Experimental results, exponential stabilization, marine systems, nonlinear control, time-varying feedback control.

I. INTRODUCTION

D

YNAMIC positioning (DP) of ships is required in many offshore oil field operations, such as drilling, pipe-laying, tanking between ships, and diving support. The dynamic positioning control problem consists in finding a feedback control law that asymptotically stabilizes both the position and orientation of the ship to desired constant values. Critical for the success of a dynamically positioned ship is its capability for accurate and reliable control, subject to environmental disturbances as well as to configuration related changes, such as a reduction in the number of available actuators. Furthermore, robustness to actuator failures can be crucial for the safety, e.g., during diving support and during oil loading and off-loading between two ships. Robustness to actuator failures may be obtained by equipping the vehicle with redundant actuators. It may also be obtained by changing to a control law that controls the vehicle using only the remaining actuators, when an actuator failure is detected. This represents a software solution to fault handling in the event of an actuator failure, and is a cost-reducing alternative to fault handling by actuator duplication or triplication, i.e., by hardware redundancy [1]. Also, ability to control the ship using only a reduced number of actuators allows for equipping the ship with fewer actuators in the design process, and in this way reduce the cost of the ship. In this paper we consider the dynamic positioning problem for a ship having only two independent control inputs, surge force and yaw moment. Typically the ship is equipped with two aft thrusters located at a distance from the center line in order to give both surge force and yaw moment, but has no side thruster to give sway force control. As we want to control both the North and East positions together

with the orientation of the ship, i.e., three degrees of freedom (DOFs), having only two independent control inputs available, we have an underactuated control problem. The underactuated ship control problem is inherently nonlinear. In particular, if we linearize the ship model about the desired constant position and orientation, the resulting linear model is not controllable. Therefore, a nonlinear ship model must be considered. Moreover, the underactuated ship belongs to a class of systems that cannot be asymptotically stabilized . This implies that the dyby a feedback control law namic positioning problem cannot be solved using a linear proportional integral derivative (PID)-controller or any other linear time-invariant feedback control law. Also, the problem is not solvable using classical nonlinear control theory like feedback linearization. To evade this negative result, we introduce explicit , an time-dependence in the feedback control law, approach first used by [2] for the control of mobile robots. Even though there has been many theoretical developments in the area of underactuated systems, see for instance [3]–[10], there has been relatively few experimental results reported that makes use of the developed theory. Experimental results are important for an understanding of the value and the limitations of the theory of underactuated control systems that now exists. In [11] experimental results are presented for the open-loop control of an underactuated autonomous underwater vehicle. In this paper, we present experimental results for feedback control of an underactuated ship. In [12], using averaging theory and homogeneity properties as proposed in [13], a time-varying feedback control law was developed that exponentially stabilizes both the position and orientation of the ship using only the two available control inputs. In this paper we present experimental results for this feedback control law. Moreover, we extend the result of [12] by introducing integral action in the control law and prove that this extended control law exponentially stabilizes both the position and orientation of the underactuated ship. This paper is organized as follows: The possibility for asymptotically stabilizing underactuated marine vehicles is addressed in Section II. In Section III the ship model is presented. A timevarying feedback control law with integral action is developed and proven to exponentially stabilize the ship in Section IV. In Section V the experimental setup is described and experimental results are presented. Finally, the conclusions are given in Section VI.

II. STABILIZABILITY OF UNDERACTUATED MARINE VEHICLES Manuscript received October 20, 1998; revised November 9, 1999. Recommended by Associate Editor, M. Jankovic. The authors are with the Department of Engineering Cybernetics, Norwegian University of Science and Technology, 7491 Trondheim, Norway (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1063-6536(00)05735-3.

This section reviews properties of the dynamics of underactuated vehicles, and the stabilizability of these vehicles. Underactuated vehicles are vehicles with fewer independent control inputs than degrees of freedom to be controlled. The dynamics

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and kinematics of underactuated marine vehicles can be written [14], [15] (1) (2) , with . The vector of control where , and the vehicle is underactuated as . inputs is The matrices have the following properties: The matrix of inertia including hydrodynamic added inertia, , is constant, symmetric, nonsingular and positive definite. The symmetry property is based on the assumptions of low speed, ideal fluid and no incoming waves. This is a good assumption for dynamically positioned ships [14]. The Coriolis and cen, is tripetal matrix also including added inertia effects, is skew-symmetric. The hydrodynamic damping matrix symmetric and positive definite. The hydrodynamic damping matrix, can be linear in the velocity , giving a constant or it can include terms that are higher-order functions of the velocity. The kinematic transformation matrix has full rank ) except for a pitch angle of deg. The vector (rank is the vector of gravitational and buoyant forces and denotes the linear moments. The vector velocities in surge, sway and heave, and the angular velocities in roll, pitch and yaw, decomposed in the body-fixed frame, see Fig. 1. The vector denotes the position and orientation decomposed in the Earth-fixed frame, and denotes the control forces and moments decomposed in the body-fixed frame. If is a discontinuous feedback law, the Filippov solutions of the system (1), (2) are considered (see, e.g., [16]). Without loss of generality we assume in the following that the position and orientation that we want to stabilize is at the origin of the Earth-fixed coordinate system. The first author together with Egeland in [12] presented the following result, showing that the gravitation and buoyancy , is important for the stabilizability properties of vector, consists of elements underactuated vehicles. The vector , and elements corresponding to the actuated dynamics, , i.e., corresponding to the unactuated dynamics,

(3) Proposition 1: Consider the system (1), (2), and suppose that is an equilibrium point of the system. If has a zero element, then there exists no continuous or discon, that tinuous state feedback law, makes (0, 0) asymptotically stable. This implies that for this class of marine vehicles a constant position and orientation cannot be asymptotically stabilized using a linear PID-controller or any other linear time-invariant feedback control law. Also, the problem is not solvable using classical nonlinear control theory like feedback linearization.

Fig. 1.

Ship variables.

III. SHIP MODEL The underactuated dynamic positioning problem for the ship, is to find a feedback control law that asymptotically stabilizes both the North and East positions together with the orientation of the ship, using only the two available control inputs, surge force and yaw moment. As we seek to control the horizontal motion of the ship, we will restrict the six-dimensional dynamics to the horizontal plane by making the following assumption. Assumption 1: The dynamics associated with the motion in , for heave, roll, and pitch is negligible, that is DP operations. Where is the position variable in heave, is the roll angle, and is the pitch angle, see Fig. 1. This is a well-known assumption [14] which is used in all industrial ship control systems since the magnitude of the heave, roll and pitch variables will be small (second-order damped oscillators), and therefore not significantly influence the horizontal motion that we are interested in controlling. In addition, conventional ships are not equipped with actuators affecting the dynamics in heave, roll denotes the position and pitch. Then the vector and orientation of the ship in the Earth-fixed coordinate system. denotes the linear velocities in surge The vector and sway, and the angular velocity in yaw, decomposed in the debody-fixed coordinate system. The vector notes the control inputs, is the surge force and is the yaw moment. The control input in sway is zero. The gravitation and buoyancy vector field is identically zero since the gravity and buoyancy have no effect on the dynamics in the horizontal plane as this is perpendicular to the direction of gravity. To obtain a model with the appropriate homogeneity properties needed for the development of the control law in Section IV, we make the following assumption. and are diagonal. Assumption 2: The matrices This is true for vehicles having three planes of symmetry, for which the axes of the body-fixed reference frame are chosen to be parallel to the principal axis of the displaced fluid, which are equal to the principal axis of the ship. Most ships have port/starboard symmetry, and moreover bottom/top symmetry is not required for horizontal motion (DOF’s 1, 2, and 6). Nonand symmetry fore/aft of the ship implies that

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. These terms will, however, be small compared and ( ) for most ships. to the diagonal elements Control design under a relaxation of this assumption to the general case is trivial to solve for fully actuated ships, while it is a topic for future research for underactuated ships. The three DOF ship model is then

systems, see [20]. To obtain a model having the appropriate homogeneity properties, we use the following global coordinate transformation :

(10) (4)

The state equations of the ship are then

(5) where diag

(6) (7)

diag

(11)

(8)

In order to introduce integral action in the feedback law, the variables are defined by

(9) (12)

IV. CONTROL LAW The control objective is to find a feedback control law that makes the ship converge to a desired constant position and orientation, and to stabilize this configuration, i.e., to asymptotically stabilize a desired equilibrium point. Without any loss of generality we have assumed the desired equilibrium point to be the origin. From Proposition 1 and the ship model (4), (5) we see that the ship belongs to the class of systems that cannot be asymptotically stabilized by a static feedback control . However, it is shown [12] that the ship still law is small time locally controllable from any equilibrium point. By [17] this implies that it is possible to asymptotically stabilize the ship using a periodic time-varying feedback control law . In this section we develop a periodic time-varying feedback control law, and prove that it asymptotically stabilizes both the position and orientation of the ship using only the two available control inputs. Moreover, the convergence to the desired equilibrium point is exponential. The feedback control law is derived using averaging theory and homogeneity properties as proposed by [13]. Furthermore, we here introduce integral control in the control law, in order to remove stationary errors due to constant disturbances. For the definitions of dilations, homogeneity, and exponential stability with respect to a dilation, the reader is referred to [18], [19]. For the extension of these definitions to time-varying

, . We propose the following continuous and periodic time-varying feedback law:

(13) where

(14) and and where the polynomial

and

have roots strictly in the left half-plane. such Proposition 2: There exist constants , and , the that for any feedback control law (13) asymptotically stabilizes the origin of the system (11), (12). Moreover, the origin is exponentially , , , , , stabilized with respect to the dilation , , , , , , , . This implies that there such that along any solution of exists two constants the system the following inequality is satisfied: (15)

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Fig. 2. Experimental setup.

Proof: The closed-loop system (11)–(13) can be written (16) where

Fig. 3. Trajectory in the xy -plane.

As the system (16) satisfies the conditions of [13, Prop. 2], can be viewed as a higher order perturbation of the system (18) and if the system (18) is asymptotically stable, then so is the system (16). Moreover, it is then exponentially stable with re. spect to We reduce the system (18) by defining the velocities and as our new control variables

(19) The vector fields and are homogeneous of degree 0 and 2, respectively, with respect to the dilation

(17)

Let (14) define the control inputs to (19). Due to the periodic time-variant controls, the resulting system is a periodic time-

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Fig. 4. The time evolution of the position variables x and y [m], the orientation [deg/s].

[deg], the surge and sway velocities u and v [cm/s], and the yaw velocity r

varying system. The averaged system, [21], of this is the autonomous system

(20)

is asympAs the linearization of (20) about totically stable, the origin is a locally asymptotically stable equilibrium of the averaged system (20). The system satisfies the conditions of [21, Th. 4.1], and we can thus conclude that there such that for any the origin of exists an the time-varying system (19) is asymptotically stable. The equations for and in (18) are (21)

(22) The system (19) together with (21), (22) satisfy the conditions of [22, Prop. 1]. There thus exists lower bounds for the parameters and , such that for parameters chosen above these bounds, the last terms of equations (21), (22) dominate the first terms, and define inner control loops that make the actual velocities and track the desired velocities and , respectively. Consequently, by [22, Prop. 1], for positive and sufficiently large of the system values of and the origin (18) is asymptotically stable. is an exponentially stable equilibThus rium of (16), with respect to the dilation [13, Prop. 2]. V. EXPERIMENTAL RESULTS The experiments were performed at the Guidance, Navigation, and Control Laboratory, at the Department of Engineering Cybernetics, NTNU. The Laboratory includes a model ship, scale 1:70, which is a model of an offshore supply vessel, see Fig. 2. The model ship has a mass of 17.6 kg, and a length of m aft 1.19 m. The center of gravity is located at

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and moment disturbance to the ship. The sampling frequency used in the experiments was 50 Hz. A. Experiments Without Integral Action In the experiments, the ship was first controlled by the control law (13) without integral action, using the control parameters

The ship was initially steered to the point

1 1 1),  (0)

Fig. 5. Time evolution of the commanded forces and moment  ( [N],  ( ) [Nm].

01

by a dynamic positioning control law using three control inputs, and then the controller was switched to the underactuated control law using only two control inputs. The switching implies an initial step disturbance to the system. The desired equilibrium point was

The results are shown in the Figs. 3–5. We see that the ship converges to a neighborhood of the desired equilibrium point, and eventually oscillates about constant errors. The mean error in the position variable is 0.05 m, in the course angle the mean error is 7 deg, and in there is a mean error of 0.53 m. This motivated us to introduce integral action into the time-varying control law, in order to diminish or remove the stationary errors, particularly in . B. Experiments with Integral Action

Fig. 6. Trajectory in the xy -plane, with integral control.

of midships. This is also the origin of the body-fixed coordinate system. We have the following relationship between the speed of the ship and the model ship: m/s

(23)

denote the ship and the model, where the subscripts and respectively. The position and orientation of the ship are monitored by two infrared cameras, Fig. 2. The camera coordinates are transmitted to a PC-386 computer where the position measurements are transformed to Earth-fixed coordinates, and these coordinates are transmitted to a dspace signal processor. A nonlinear passive observer [23] is used to reconstruct the unmeasured states, that is low-frequency (LF) positions and velocities, first-order waveinduced disturbances, and biases. Hence a full-state feedback controller can be implemented using the LF position and velocity estimates for feedback. The dspace signal processor communicates over a dspace bus with a Pentium 166 MHz computer where the feedback control law is implemented. The proposed controller can be implemented in real time using state-of-the-art computers since the requirement to CPU is quite limited. The thruster commands are sent through the signal processor to the ship. In future, the communication from the signal processor to the ship thruster servos will be performed by a radio transmitter. Today, there is a wire connection between the signal processor and the ship servos, and this wire results in a nonconstant force

Integral action was then included by changing the control pato rameters

The ’s were chosen such that the inverse of the integral time was approximately one decade below the bandwidth of the corresponding equation in the linearization of (20). The experimental results are shown in Figs. 6–8. We see from the figures that introducing integral action, the mean stationary errors were reduced. The position variable here oscillates about a mean error of 0.01 m, the course angle oscillates about a mean error of 0.4 deg and the mean error in is reduced to 0.25 m. (The amplitude of some of the variables in Figs. 6–8 are still increasing because they reach their stationary amplitudes, which are only slightly larger than the amplitudes at 1000 s, after approximately 3000 s. We have however chosen to show the first 1000 s in order to better see the transient behavior of the ship.) To understand the reasons for the stationary errors and oscillatory behavior we performed some simulations. Simulations under the ideal conditions of no modeling errors, actuator saturation, measurement noise or environmental disturbances like currents, waves or the wire disturbance, showed that the ship did converge exponentially to the desired equilibrium point, both with and without integral action. Simulations in which the ship inertia and damping matrices had small off-diagonal terms, and and , simulations with quite large errors in the parameters , showed that despite these modeling errors the ship

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Fig. 7. Time evolution of the position variables x and y [m], the orientation with integral control.

1 1 1),  (0)

Fig. 8. Time evolution of the commanded forces and moment  ( [N],  ( ) [Nm], with integral control.

01

still converged exponentially to the desired equilibrium point. We then simulated the behavior of the ship without modeling errors for the case where measurement noise of zero mean value and standard deviation 5 cm (corresponding to 3.5 m for the offshore supply vessel) was added to the position values, and zero mean value noise with a standard deviation of 1 deg was added to the orientation variable. This corresponds to the maximum measurement noise a supply vessel typically could experience. This of course gave some stationary errors. Also, it gave some

[deg], the surge and sway velocities u and v [cm/s], and the yaw velocity r [deg/s],

stationary oscillations, but these were of smaller magnitude than those observed in the experiments. Moreover, the oscillations were mainly in the position variable while negligible in , and thus did not resemble the oscillations in the experiments. We furthermore, in addition to the measurement noise, included actuator saturation of approximately 1 N for the surge control force and 1 Nm for the yaw control moment, but the results remained the same. Then we performed simulations without measurement noise and actuator saturation, in which the ship was subjected to a constant earth-fixed environmental disturbance. A constant earthfixed force can be used as a model for the bias of environmental forces like wind, currents and waves that affect the ship [14]. At the lab there was no wind, but significant currents and also waves. Moreover the transmission wire introduced a force on the ship. When exposing the ship to a force of 0.05 N in both the earth-fixed - and -direction, the simulated behavior showed to be very similar to the behavior in the experiments, both with and without integral action. To conclude, the stationary errors and oscillations of the ship shown in the experiments may be due to a combination of several factors, but the simulations indicated that the main reason

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was constant environmental forces that acted on the ship, due to the bias of currents and waves (and the force of the transmission wire). The integral action reduced the stationary errors, but they were not eliminated and the stationary ship behavior was still oscillatory. The experimental results showed that the control law did not give the desired local behavior that it was designed for. As the simulations indicated that the main reason for this was the environmental disturbance, it will be interesting in future work to include a constant environmental force in the analysis and control design, possibly including adaption in the control scheme, in order to reduce or eliminate the stationary errors and oscillations. VI. CONCLUSIONS In the paper the dynamic positioning problem was considered for a ship having only two independent control inputs. The ship belonged to a class of underactuated marine vehicles that cannot be asymptotically stabilized using a time-invariant . To evade this negative result, feedback control law a time-varying feedback control law including integral action was proposed and proven to asymptotically stabilize the ship, providing exponential convergence to the desired position and orientation. Then experiments were presented showing that the controlled ship converged to a neighborhood of the desired position and orientation, oscillating about stationary errors. The stationary errors and oscillation could be due to a combination of several factors, but simulations indicated that the main reason was constant environmental forces that acted on the ship, due to the bias of currents, waves, and the force of the transmission wire. The experiments showed that with integral action in the control law, the stationary errors were reduced, but they were not eliminated. In future it would be interesting to include a constant environmental force due to wind, waves, and currents in the analysis and control design, possibly including adaption in the control scheme, in order to reduce or eliminate the stationary errors and oscillations. REFERENCES [1] M. Bodson, “Emerging technologies in control engineering,” IEEE Contr. Syst. Mag., vol. 15, no. 6, 1995. [2] C. Samson, “Velocity and torque feedback control of a nonholonomic cart,” in Proc. Int. Workshop Nonlinear Adaptive Contr.: Issues Robot., C. Canudas de Wit, Ed. Berlin, Germany: Springer-Verlag, 1991, pp. 125–151. [3] G. Oriolo and Y. Nakamura, “Control of mechanical systems with second-order nonholonomic constraints: Underactuated manipulators,” in Proc. 30th IEEE Conf. Decision Contr., Brighton, U.K., Dec. 1991, pp. 2398–2403. [4] K. Y. Wichlund, O. J. Sørdalen, and O. Egeland, “Control of vehicles with second-order nonholonomic constraints: Underactuated vehicles,” in Proc. 3rd Eur. Contr. Conf., Rome, Italy, 1995, pp. 3086–3091.

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[5] P. Morin, C. Samson, J.-B. Pomet, and Z.-P. Jiang, “Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls,” Syst. Contr. Lett., vol. 25, pp. 375–385, Aug. 1995. [6] P. Morin and C. Samson, “Time-varying exponential stabilization of a rigid spacecraft with two control torques,” IEEE Trans. Automat. Contr., vol. 42, pp. 528–534, Apr. 1997. [7] N. E. Leonard, “Periodic forcing, dynamics and control of underactuated spacecraft and underwater vehicles,” in Proc. 34th IEEE Conf. Decision Contr., New Orleans, LA, Dec. 1995, pp. 3980–3985. [8] M. Reyhanoglu, “Control and stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 2371–2376. [9] J.-M. Godhavn, “Nonlinear tracking of underactuated surface vessels,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 987–991. [10] K. Y. Pettersen, “Eponential Stabilization of Underactuated Vehicles,” Ph.D. dissertation, Norwegian Univ. Sci. Technol., Trondheim, 1996. [11] N. E. Leonard, “Control synthesis and adaptation for an underactuated autonomous underwater vehicle,” IEEE J. Oceanic Eng., vol. 20, pp. 211–220, July 1995. [12] K. Y. Pettersen and O. Egeland, “Exponential stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 967–971. [13] P. Morin and C. Samson, “Time-varying exponential stabilization of the attitude of a rigid spacecraft with two controls,” in Proc. 34th IEEE Conf. Decision Contr., New Orelans, LA, Dec. 1995, pp. 3988–3993. [14] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 1994. [15] T. I. Fossen and O.-E. Fjellstad, “Nonlinear modeling of marine vehicles in six degrees of freedom,” Int. J. Math. Model. Syst., vol. 1, no. 1, pp. 17–27, 1995. [16] J.-M. Coron and L. Rosier, “A relation between continuous time-varying and discontinuous feedback stabilization,” J. Math. Syst. Estim., Contr., vol. 4, no. 1, pp. 67–84, 1994. [17] J.-M. Coron, “On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law,” SIAM J. Control Optim., vol. 33, pp. 804–833, May 1995. [18] M. Kawski, “Homogeneous stabilizing feedback laws,” Contr.-Theory Advanced Technol., vol. 6, no. 4, pp. 497–516, 1990. [19] H. Hermes, “Nilpotent and high-order approximation of vector field systems,” SIAM Rev., vol. 33, no. 2, pp. 238–264, 1991. [20] J. B. Pomet and C. Samson, “Time-varying exponential stabilization of nonholonomic systems in power form,” INRIA, Tech. Rep. 2126, Dec. 1993. [21] R. T. M’Closkey and R. M. Murray, “Nonholonomic systems and exponential convergence: Some analysis tools,” in Proc. 32nd IEEE Conf. Decision Contr., San Antonio, TX, Dec. 1993, pp. 943–948. [22] K. Y. Pettersen and O. Egeland, “Robust control of an underactuated surface vessel with thruster dynamics,” in Proc. 1997 Amer. Contr. Conf., Albuquerque, NM, June 1997. [23] T. I. Fossen and J.-P. Strand, “Passive nonlinear observer design for ships using Lyapunov methods: Experimental results with a supply vessel,” Automatica, vol. 35, no. 1, 1999.