A New Approach for Ship Motion Control Based on Experiment

Youngbok Kim Department of Mechanical System Engineering,

Pukyong National University, Busan, Korea Republic kpjiwoo@pknu.ac.kr

Bui Van Phuoc Department of Control & Mechanical Engineering,

Pukyong National University, Busan, Korea Republic

phuocpknu@gmail.com

Abstract—In this paper, the authors propose a new approach to control problem of the marine vessels which are moored or controlled by actuators. The vessel control system is basically based on the Dynamic Positioning System(DPS) technology. And, using DPS and sensing system, the actuators are installed in the vessel to be positioned in the specified area. There are many types of actuator system in the real applications. The useful and widely applicable systems are thruster and mooring winch based systems. It may be difficult to compare the control performances of two types. If we consider this problem in point of usefulness including cost, we can easily find out that the mooring winch system is more useful and popular to the real field than the thrust system except the special purpose. Considering these facts, in this paper we consider a DPS design problem which can be extended to the many application fields including two types of actuator system. The main object of this paper is to obtain more useful control design method for DPS. In this problem, a complicate fact is the control allocation. For this problem, many results have been given and verified by other researchers with a process followed from two individual steps. It means that the controller and control allocation design process is carried out individually. In this paper, the authors give more sophisticated design solution for this issue. The authors propose a new design method in which the controller design and control allocation problem is considered simultaneously. In other word, the system stability, control performance and allocation problem are unified by a LMI(linear matrix inequality) based on control theory. Because, as well known, LMI expression gives us a solution so efficiently. The usefulness of proposed approach is verified by experiment with a supply vessel model and found work well.

Keywords-control allocation; redistributed pseudo-inverse algorithm; ship berthing; sliding mode control; slow speed maneuvering.

I. INTRODUCTION Automatic berthing control of ships has been of interest in

the 1990s. In the marine context, the berthing maneuvering is a complicated procedure, which relies on both human experience and control strategy. At the berthing stage, the ship has to follow the given path with a slow speed to prevent collisions. However, one has to cope with challenges associated with the following: i) the significant reduction in controllability of actuators due to the low speed operations, ii) the influence on ship handling due to changes in hydrodynamic coefficients when the ship moves from open seas to confined water, and

iii) the relatively large effect of environmental disturbance. For these reasons, it is not surprising that the recent research efforts have concentrated on developing nonlinear and intelligent control schemes. These schemes include adaptive, backstepping schemes [1, 2] and neural network based control schemes or others [3-7]. However, the limitation on actuator controllability during dead-slow speed operations has not yet been solved. Hence, it is not safe to apply these methods to actual ship berthing. Thus, currently, large ship maneuvering in the harbor area is still conducted manually with assistance from tugboats.

To overcome the abovementioned challenges and drawbacks and achieve a complete automated solution, the authors propose a new approach to ship berthing by using autonomous tugboats. In this study, to cope with the uncertainties such as the environmental disturbance forces and moments, modeling errors and changes in hydrodynamic coefficients, a robust control design approach is introduced. Exactly, ∞H control based controller is designed to suppress the difference between the control and actuator signal, and occupy some control performance.

II. SYSTEM MODEL The low frequency motion of a surface ship can be

described by the following model [8]

,.

c ϕϕ

+ TMv + Dv = τ R ( )b η = R( )v (1)

The inertia matrix 3x3R∈M which includes hydrodynamic added inertia can be written as

0 00 ,0

u

v r

v z r

m Xm Y Y

N I N

−⎡ ⎤⎢ ⎥= − −⎢ ⎥− −⎣ ⎦

M

(2)

where m is the vessel mass and zI is the moment of inertia about the vessel fixed z-axis. For control application, the motion of the ship is restricted to low frequency. The wave

978-1-4577-2091-8/12/$26.00 ©2011 IEEE

frequency is assumed to be independent from added inertia, which implies that M = 0 .

For a stable ship following a straight line, 3x3R∈D is a strictly positive damping matrix because of the linear wave drift damping and laminar flow. The linear damping matrix is defined as

0 00 .0

u

v r

v r

XY YN N

−⎡ ⎤⎢ ⎥= − −⎢ ⎥− −⎣ ⎦

D

(3)

3[ , , ]Tx y Rϕ= ∈η represent the inertial position (x, y) and the heading angle ϕ in the earth fixed coordinate frame and

3[ , , ]Tu v r R= ∈v describes the surge, sway, and yaw rate of ship motion in the body fixed coordinate frame. The rotation matrix in yaw ( )ϕR is used to describe the kinematic equation of motion; that is

cos sin 0( ) sin cos 0 .

0 0 1

ϕ ϕϕ ϕ ϕ

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

R

(4)

The slow varying external forces and moment caused by wind, current, and waves are lumped together into a bias term

3R∈b . 3R∈cτ is the control input composed of forces and moments provided by the propulsion system, which includes the main propellers of the ship, bow thrusters, and stern thrusters. In this effort, the propulsion system is replaced by tugboats to prevent collisions of the ship. The vector cτ is the result of combined efforts of four tugboats, as shown in Fig. 1; this vector is defined as follows:

( ) ,c α=τ B f (5)

where the vector 1 2 3 4[ ]Tf f f f F= ∈f presents the thrusts produced by tugboats.

Furthermore, max0 , (1,..., 4)if f i≤ ≤ ∀ ∈ . The geometric

configuration matrix 3x4( ) Rα ∈B captures the relationship between all four tugboats and the ship. The i-th column of matrix ( )αB is defined as

cos( )( ) sin( ) ,

cos( ) sin( )

i

i

yi i xi il l

αα α

α α

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥− +⎣ ⎦

iB

(6)

where the angle iα defines the force direction of the i-th tugboat. Measured clockwise, it is relative to the x-axis of the body fixed coordinate frame. The location of the i-th contact point in the body fixed coordinate system is at ( , )xi yil l .

1f

2f

3f

4f

1α

2α

3α

4α

11

( ,)

xyl l

33

(,

)x

yll

44

(,

)x

yll

22

(,

)x

yll

ϕ

Figure 1. Control of ship motions using four tugboats.

Thus the control input vector cτ can be expressed in the form of the geometric configuration matrix ( )αB and thrust vector f as

1 1 1 1 1 1 1

2 2 2 2 2 2 2

3 3 3 3 3 3 3

4 4 4 4 4 4 4

,

Ty x

y xc

y x

y x

c s l c l s fc s l c l s fc s l c l s fc s l c l s f

α α α αα α α αα α α αα α α α

− +⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− +⎢ ⎥ ⎢ ⎥=⎢ ⎥− + ⎢ ⎥⎢ ⎥ ⎢ ⎥− + ⎣ ⎦⎣ ⎦

τ

(7)

where sin( ) and cos( ).i i i is cα α α α= = In this article, the contact positions between the ship and the tugboats are assumed to be fixed. The adequate set ( , )i ifα is solved by using the control allocation approach.

III. CONTROL ALLOCATION AND CONTROLLER DESIGN In this section, the robust control design based on ∞H is

developed for ship trajectory tracking purpose. This design has the following potential and significant advantages: i) insensitivity to plant nonlinearities and parameter variations and ii) remarkable stability and performance robustness with respect to disturbances. With these advantages, this method satisfies the trajectory tracking requirements in the harbor area. Here, at first we think about the control allocation problem. A mentioned above, it is a kind of control signal distribution skill. Because there exist number differences between control signals calculated from controller and actuators. A solution is to introduce Moore Penrose pseudo-inverse matrix [9]. Then a vector on Eq. (5) is computed by using the Moore Penrose pseudo-inverse matrix which is a special case of pseudo-inverse matrix and the relation can be given as following:

,′ ′ ′ ′= ⇒ = *c cτ B f f B τ (8)

where 1( )−′ ′ ′ ′=* T TB B B B . If we use an optimization algorithm, the actual control force input to the ship is given by

( ( ) ).α= − + +*cf c B τ B c (9)

Where 111* })()({)( −−−= TT ααα BWBBWB . Here let us check what is unreasonable in the previous results given by other researches. In general, if we consider control allocation problem, at first we design a controller and next control allocation process to be considered. It means that the two process (controller design and control allocation) is operated respectively. In other word, total control performance and system stability problems are not handled simultaneously.

Therefore in this paper we introduce a new control strategy to handle the two or multi objects problem by one step process. This idea comes out from robust control approach which is described in figure 1 and 2. The basic concept of control allocation can be depicted as Figure 1. In Figure 1, if there is no need to consider the control allocation, then the block ‘Control Allocation + Actuator’ can be deleted. It means that if we consider the control allocation, then we describe this problem by Figure 1 and it can be transformed as Figure 2. The final object of this study is summarized as suppressing the error between the control signal and actual control input to the ship. Also the control performance can be guaranteed.

Then this control problem is given by following theorem.

Figure 2. A new control design problem with control allocation

Figure 3. A transformed control and control allocation

[Theorem] The closed-loop system is stable and the RMS gain from w to z does not exceed γ if and only if there exists a symmetric matrix ∞X such that

.0

,02

>

<⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−

+

∞

∞

∞∞∞

X

IDXCDIB

CXBAXXA

clcl

Tcl

Tcl

Tclcl

Tclcl

γ (10)

IV. EXPERIMENT RESULTS Experiment has been executed to evaluate the performance

and robustness of the controlled system. The ship and experiment apparatus are set up and it is

shown in Figure 4. In our system, the controller is installed on the mother ship and four tug boats are attached to the ship. Especially, each tug boat can produce only pushing force, because we should consider the real situation. And the mother ship position is measured by image processing system using one camera installed on the ceiling.

In the result, the experiment result is shown in Figure 5. From this result, it is verified that a good control performance is obtained.

Figure 4. Control system with four actuators

Figure 5. Ship positioning control result

V. CONCLUDING REMARKS In this paper, the authors have proposed a new approach for

ship berthing, which is based on the assistance of autonomous tugboats. Especially, the author proposes a new control design and control allocation strategy. In this paper, the authors give more sophisticated design solution for this issue. The authors propose a new design method in which the controller design and control allocation problem is considered simultaneously. In other word, the system stability, control performance and allocation problem are unified by a LMI(linear matrix inequality) based on control theory. Because, as well known, LMI expression gives us a solution so efficiently. The usefulness of proposed approach is verified by experiment with a supply vessel model and found work well.

ACKNOWLEDGEMENT This research was a part of the project titled “The

Development of Mooring Position Control System for Offshore Accommodation Barge” funded by the Ministry of Land, Transport and Maritime Affairs, Korea.

REFERENCES [1] R. Skjetne, T. I. Fossen, and P. V. Kokotovic, Adaptive output

maneuvering, with experiment, for a model ship in a marine control laboratory, Automatica, 41(2), 289-298, 2005.

[2] C. Jammazi, Backstepping and partial asymptotic stabilization: application to partial attitude control, International Journal of Control, Automation, and System, 6(6), 859-872, 2008.

[3] Y. Zhang, G. E. Hearn, and P. Sen, A Multivariable Neural Controller for Automatic Ship Berthing, Journal of IEEE Control System, 17(4), 31-44, 1997.

[4] I. M. Namkyun and K. Hasegawa, A Study on Automatic Ship Berthing Using Parallel Neural Controller, The Journal of Kansai Society of Naval Architects of Japan, 237(3), 127-132, 2002.

[5] R. Zhang, Y. Chen, Z. Sun, F. Sun and H. Xu, Path Control of a Surface Ship in Restricted Waters Using Sliding Mode, Transactions on Control System Technology, 8(4), 722-732, 2000.

[6] A. J. Healey and D. Lienard, Multivariable Sliding Mode Control for Autonomous Diving and Steering of Unmanned Underwater Vehicles, Journal of Ocean Engineering, 18(3), 327-339, 1993.

[7] K. A. Bordingnon, Constrained Control Allocation for Systems with Redundant Control Effectors, Phd. Thesis, Dept. Aerospace & Ocean Eng., Virginia Polytechnic Institute & State Univ., Blacksburg, VA, 1996.

[8] T. I. Fossen, Marine Control System- Guidance, Navigation, Rigs and Underwater Vehicle, (Trondheim, Norway, Norwegian University of Science and Technology, 2002).

[9] G. Strang, Introduction to Linear Algebra (Wellesley Cambridge Press, 2003).

[10] J. P. Strand, Nonlinear Position Control System Design for Marine Vessels, PhD. Thesis, Dept of Engineering Cybernetics, Norwegian University of Science and Technology, 1999.