FUZZY AUTOPILOT FOR SHIPS EXPERIENCING SHALLOW WATER EFFECT IN MANOEUVERING

Zoran Vuki�*, Edin Omerdi� and Ljubomir Kulja�a

University of Zagreb, Faculty of Electrical Engineering and Computing Department of Control and Computer Engineering in Automation

Unska 3, HR-10000 Zagreb, Croatia

*(e-mail: zoran.vukic@fer.hr)

Abstract: A fuzzy autopilot for ship path control is proposed. Nonlinear model of a ship and a steering subsystem is used. The autopilot uses heading signal and yaw rate signal to produce a command rudder angle. The autopilot does not use lateral offset from the nominal track. Input variable fuzzyfication, fuzzy associative memory rules and output set defuzzyfication are described. The influence of the shallow water effect during larger maneuver is analyzed.

Keywords: Disturbance rejection, Fuzzy control, Nonlinear control systems, Ship control

1. INTRODUCTION

The conventional autopilot for ship's course keeping involves the heading angle feedback. However, by including an additional position feedback, a ship guidance system can be designed (Fossen, 1994). The desired route is most easily specified by way points. Modern sea going vessels have a range of navigation aids including globa positioning system (GPS) receivers, Doppler sonar, gyrocompass etc. These devices provide information required to implement track guidance. Nonlinearity of the ship model and the steering gear subsystem during course-changing and the lack of a simple mathematical model makes it appropriate to design the controller with fuzzy logic instead of the conventional approach. An accurate following of the desired track is of great importance here. Although strictly speaking normal navigation is also a track-keeping problem, this paper particularly discusses an autopilot for accurate track-keeping in manoeuver. The fuzzy autopilot proposed here was designed and tested by simulation in MATLAB using SIMULINK with Fuzzy Logic Toolbox. The autopilot is designed to be used for a wide range of

ship types and ship velocities. Notch filter was used to minimize the effects of wave disturbance. Shallow water effects during manoeuvering were analyzed for the nonlinear model of ESSO 190000 dwt Tanker. This paper is organized as follows: Section 2 introduces mathematical models of the ship, the steering machine and the disturbances. Section 3 describes the course controller designed with fuzzy logic. Section 4 introduces the turning concept and the implementation of course controller in track-keeping systems. Section 5 presents the simulation results. Finally, Section 6 summarizes the concluding remarks.

2. MATHEMATICAL MODELS Although the design of a fuzzy controller does not depend on a mathematical model of the process, such a model is necessary to simulate various motions of ship.

2.1 Ship dynamics In order to verify controller behavior in different conditions, it is useful to simulate the control law against a realistic model of the vessel. In this paper we shall deal with two nonlinear ship models: Mariner Class Vessel (Length between perpendiculars is Lpp=160.93 m) and ESSO 190000 dwt Tanker (Lpp=304.8 m). The nonlinear mathematical models which describe the dynamic between the rudder angle δ and the yaw motion for these ships are given in (Fossen, 1994, Appendix E). Ship's models were transformed to S-functions and adapted for the on-line simulation. 2.2 Steering Gear Subsystem The steering gear subsystem considered is the "two-loop" electrohydraulic steering subsystem common on many ships. The nonlinear steering gear model is shown in Fig. 1. (Vuki�, 1989).

Fig. 1. Nonlinear steering gear model.

Table 1 Parameter values for steering subsystem Telemotor Position Servo

Rudder Servo Actuator

K = 4 [°/s] N = 5 [°/s] 2D = 0.4 [°] PB = 7 [°] H = 0.8 [°] 2.3 Disturbances There are several disturbances with various effects on the system to be taken into account (Amerongen, 1979). Three classes of disturbances can be distinguished: • "disturbances" which affect the dynamics of the

system (e.g. the depth of water), • disturbances which cause additional signals in

the system (e.g. waves), • disturbances that corrupt the measurements (e.g.

noise on the position measurements).

In this paper we consider wave disturbance during course-keeping and change of depth under keel during track-keeping. Wind-Generated Waves We shall deal with the second-order wave transfer function approximation (Fossen, 1994). This model is written as

h sK s

s w s w( ) =

+ +ω

ξ20 0

22 (1)

A linear state-space model can be obtained by transforming this expression to the time-domain by:

d ydt

dydt

y Kddt

2

2 0 022+ + =ξω ω

ωω (2)

Defining dxdt

xhh

12= and x yh h2 = , the state-

space model can be written as: dxdt

dxdt

x

x K

h

h

h

hh

1

2 02

0

1

2

0 12

0�

�

���

�

�

���

=− −�

��

�

���

��

�

�� +

�

��

�

��ω ξω ω

ω

(3)

where ωh is a zero-mean white noise process. Due to its simplicity, this model is useful for control systems design. Depth under keel The shallow water effect for the ESSO Tanker is given by a water depth parameter (Fossen, 1994):

ζ =−T

h T (4)

where T (m) is the ship draft (T=18.46 m) and h > T is the water depth. If states ζ ≥ 08. (i.e.

h m≤ 41535. ), additional terms appear in the nonlinear equations of motion and changes dynamics of the ship.

3. FUZZY LOGIC COURSE AUTOPILOT The fuzzy autopilot for course keeping uses two control inputs: heading error e d= −ψ ψ and yaw

rate r d dt= ψ / . The control action generated by

the autopilot is the command rudder angle δC . Fig. 2. shows a simple block diagram of the autopilot (Tovornik, 1995).

Conditioning

Conditioning

e

r

error

errordot Conditioningy δC

Fuzzy LogicCourse Controller

Fig. 2. Block diagram of the fuzzy logic course

autopilot. Block "Conditioning" serves for scaling and conditioning controller input and output variables.

The membership functions of the fuzzy sets are labeled as follows:

Table 2 Labels for the membership functions NB negative big PS positive small NM negative medium PM positive medium NS negative small PB positive big ZE zero

Fig. 3. Membership functions of fuzzy sets for error and errordot.

Fig. 4. Membership functions of fuzzy sets for y. Fig. 3. gives the membership functions of fuzzy sets used for input variables error and errordot (�onlagi�, 1996). In Fig. 4. membership functions of fuzzy sets for output variable y are given. Different shapes of membership functions were analyzed, but forms of membership functions shown in Fig. 3. and Fig. 4. with Mamdani inference mechanism gave the best results (Polkinghorne, 1995).

Table 3 Rulebase of the fuzzy course autopilot error e NB NM NS ZE PS PM PB r NB NB NB NB NB NM NS ZE r NM NB NB NB NM NS ZE PS o NS NB NB NM NS ZE PS PM r ZE NB NM NS ZE PS PM PB d PS NM NS ZE PS PM PB PB o PM NS ZE PS PM PB PB PB t PB ZE PS PM PB PB PB PB Each control input has seven fuzzy sets so that there is a maximum of 49 fuzzy rules. Table 3 shows the complete rulebase for the controller.

In Fig. 5. the block diagram of the control system for the course-keeping with fuzzy autopilot is given.

Fuzzy Logic CourseController Steering Subsystem Ship

Ψd

Ψ

e

rδC δ

Ψ

Disturbances

Notch Filter

Fig. 5. Block diagram of the control system for the

course-keeping.

4. FUZZY LOGIC TRACK-KEEPING AUTOPILOT

A track-keeping autopilot (given in Fig. 6.) can be obtained by introducing an additional position feedback in the control system shown in Fig. 6. A ship position (X(t), Y(t)) is calculated from kinematics equations. In a real system it can be obtained from GPS (Global Positioning System).

Fuzzy Logic CourseController Steering Subsystem Ship

Ψd

Ψ

e

rδC δ

Ψ

Disturbances

Calculating desired yawangle

X

Y

Way points (Xd, Yd)

X Y

Xd Yd

Fig. 6. Block diagram of control system for track-keeping.

The desired route is most easily specified using way points (P1, P2, ..., Pn) with coordinates Pi=(Xi, Yi). We shall use a turning concept shown in Fig. 7., where it is supposed that ship moves in a straight line between way points. A track changing maneuver is performed in such a way that the ship moves in a circle arc.

Pi

Pi+1

ϕ

Ship

Pi-1

WOPWOP* ( )ρ0 = d WOP Pi,

R

ρ0

Fig. 7. Turning concept for track-keeping. The wheel over point (WOP*) is the point where a ship leaves a straight line motion and enters the circle arc and vice versa (Holzhüter, 1995). The WOP* will not be a starting point of the turning manoeuver, because it is impossible to change the turn rate r of the ship instantaneously. The model-

based wheel-over point WOP which indicates the start of the manoeuver lies about one ship length before WOP*. The position of WOP and value of radius R depend on the angle ϕ=∠Pi-1PiPi+1. Fig. 8. shows dependence of the position of the WOP ( ρ0 ) on the angle of the ship's course change (ϕ ) for a Mariner Class Vessel. This dependence is built in a simulation algorithm and used for automatic calculation of WOP. Dependence of the radius of the circle (R) upon the angle ϕ can be found in a similar way.

( )ρ ρ ϕ0 0= Mariner Class Vessel

Fig. 8. Dependence of the position of the WOP

upon the angle of the ship's course change for the Mariner Class Vessel.

5. SIMULATION RESULTS Fig. 9. shows simulation results for the fuzzy course keeping autopilot. It presents time responses of heading, yaw rate, command rudder angle and rudder angle and controller inputs and output during a course changing maneuver without disturbances. Heading time response is without overshoot and oscillation during transient response. Fig. 10. and 11. shows the influence of wave disturbance on the course changing performance. Fig. 10 shows the time history and power spectral density of wave disturbance. Since most of the energy in the wave spectrum is located around the modal frequency of the wave spectrum, notch filter with the natural frequency equal to the encounter frequency is used for wave filtering. A small overshoot appears in the heading time response due to an additional phase lag introduced by a notch filter.

Fig. 9. Course changing manoeuver without wave

disturbance.

Fig. 10. Time history and power spectral density of wave disturbance.

Fig. 11. Course changing manoeuver with wave

disturbance. Fig. 12. presents way points guidance with the fuzzy autopilot for a Mariner ship. The desired route is given with way points P1, P2, ..., P6. Fuzzy autopilot uses only two inputs (heading and yaw rate) and has a good tracking perfomance.

dy

P1

P2P3 P4

P5

P6

Fig. 12. Way point guidance by Line of Sight

(LOS). Way points are denoted with P1, P2, ..., P6. Ship moves in a straight line between way points.

break point (h=41.535 m, zeta=0.8)

shallow water (h=30 m, zeta=1.5997)introduces additional hydrodynamic terms

(h=250 m, zeta=0.0797)

A

B

P

Fig. 13. Dependence of a water depth parameter ς

upon water depth h (see (4)). In a shallow water (ς≥0.8, h≤41.535 m) additional hydrodynamic terms are introduced into mathematical model of ESSO Tanker.

The dependence of a water depth parameter ζ upon water depth h is given in Fig. 13. Two points are considered on this graph: point A (deep water, h=250 m, ζ=0.0797) and point B (shallow water, h=30 m, ζ=1.5997). Changes in ship dynamics for different depths of water can be seen from Fig. 14., where it is supposed that manoeuver begins at the same point. The only difference between two simulations was water depth: in the first case it was 250 m and in the second 30 m. Now we consider the influence of the shallow water effect in manoeuvering. A ship moves in a straight line with forward speed of aprox. 16 knots. Water depth is 250 m. The ship begins manoeuver at the point (415, 0) and moves in a circle arc (approximately, see Fig. 15.). At t=130 (s) water depth suddenly changes to 30 m.

Beginning of the manoeuver(415, 0)

------- Trajectory of the ship for water depthh=30 m

Trajectory of the ship for water depthh=250 m

Fig. 14. Manoeuver in a different depths of water.

Dynamics of the ESSO Tanker differs at deep water (h=250 m) and shallow water (h=30 m).

(580, 420)R=420 m

Deep waterh=250 m

Shallow waterh=30 m

Shallow watereffect

Pstart

Pfinal

Fig. 15. Shallow water effect in manoeuvering1. At

t=130 s depth of water has changed suddenly from 250 m to 30 m.

Fig. 16. Time responses of variables during manoeuvering with shallow water effect at t=130 s.

1 The circle in this picture looks like an ellipse because aspect ratios for x and y axes are not equal

Dynamics of the ship have changed, as can be seen from Fig. 16. A small picture gives an enlarged part of time responses for forces and moment around point where depth of water was changed abruptly. Fuzzy autopilot brings a control action in such a way that ship stops moving in a circle arc and enters the desired direction easily. A fuzzy autopilot reacts appropriately to the shallow water effect by forcing the ship to enter the desired heading without unacceptable errors. However, the tracking error is higher, as would be expected because of the change in ship dynamics.

6. CONCLUSION

This paper presents the development of a fuzzy autopilot for course and track-keeping of ships. The research was conducted in two parts: course keeping and track keeping. The initial research concentrated on the design of a simple course keeping system. The next stage of the research used this system as a starting point for building accurate track keeping system. The results of the simulation show that it is possible to design an autopilot for track-keeping of ships using fuzzy logic course keeping autopilot. In a large manoeuver (where nonlinear dynamics of ship and steering gear play an important role) with shallow water effect producing change in ship dynamics, the fuzzy autopilot performs well. There are two possibilities for improving the perfomance in manoeuvering: 1. The fuzzy autopilot stays unchanged, but scaling

factors on controller inputs and output are dynamically modified. This leads to an adaptive fuzzy autopilot.

2. The fuzzy autopilot is augmented with an additional input - lateral offset from the nominal path.

Research in both of these directions is continuing. The effective use of the same fuzzy autopilot design with two different ships (Mariner and ESSO 190000 dwt Tanker) with a ratio of lengths near 2 illustrates the versatility of this type of control. The simplicity of the proposed autopilot and the possibility of using it for different ships without retuning its parameters was one of our goals from the beginning. This autopilot achieved the goal.

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