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RD-Al69 897 AN AUTOPILOT FOR COURSE KEEPING AND TRACK FOLLOUING(U) In-NAVAL POSTGRADUATE SCHOOL MONTEREY CA S VINUKTANANDA

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NAVAL POSTGRADUATE SCHOOLMonterey, California

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4: A

DTIC( : rE CT E

THESISEAN AUTOPILOT FOR COURSE KEEPING

ANDTRACY FOLLOT'Ir

by

p',... oiT[mart Virnu;tananda0.... SeDternber 103 '

LUJ

L-Thesis Advisor: C-3J. Thd1er

~ Approved for puhlic release; distribution is unlimitec'

85 .K. u~ 10

SECURITY CLASSIFICATION OF THIS PAGE (b D1 2i,, ee00

REPORT DO MENTATION! PAGE EFORE COMPLETING FORM-t. REPORT NUMER. GOVT ACCESSION NO. 3.-RECIPIENT*S CATALOG NUMBER

4. TITLE (and Subtitle) S. TYPE ;F REPORT & PERIOD COVERED

An Autopilot for Course Keeping Master's Thesisand rackFollwingSeptember 1985

and Track =ollowing 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(e) 6. CONTRACT Oi GRANT NUMBER(s)

Sommart Vimuktananda

S." PERFORMING ORGANIZATION NAME AND ADDRESS -10. PROGRAM ELEMENT. PROJECT, TASK. EAREA A WORK UNIT NUMBERS

Naval Postgraduate SchoolMontereyCalifornia 93943

1,. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

September, 1985Naval Postgraduate School 13. NUMUER OF PAGES

Monterey,California 93943 89

* 14. MONITORING AGENCY NAME A AOORESS( I different from Controlling Office) Is. SECURITY CLASS. (of tisle report)

Unclassified

ID. OECLASSIFICATION, DOWNGRADINGSCHEDULE

I. OISTRIUUT!ON STATEMENT (of this Report)

Approved for public release; distribution is unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20. it differenlt from Report)

Is. SUPPLEMENTARY NOTES

19. KEY WOROS (Continue on reverse side if necessary and Identify by block number)

Course Keeping,Track Following,0 arole ProgramStabilizing,Compensation.Trajectory

20. ABSTRACT (Continue on reverse side If necesary a'd Identify by block number)

Computer control of ship steering provides track followingas well as course keeping.The desired track is stored in thecomputer,and the position of the ship(as provided by a satellite

navigation system) is compared with track coordinates.A heading

correction is calculated continuously and used to un date the

course command.

D O 1473 EDITION OF I Nov $s is OBSOLETES. N OiO2-tL. OIJ-.6OI 1SSECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

%"%

Approved for public release; distribution is unlimited.

An Autopilot for Course Keeping and Track Following

by

Sommart VimuktanandaLieutenantRoyal Thai Navy

B.S., Royal Thai Naval Academy, 1974

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLSeptember 1985

Author:_ ______

Sopmffarr V3imu a

Approved by:

Alex Gerba Jr., econ eaer

Harrletr B. k1gas Lnairman,Department of Electrical ano Computer Engineering

. onn i. uyer,Dean of Science and Engineering

2

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AB STRACT

Computer control of ship steering provides track

following as well as course keeping. The desired track is

stored in the computer,and the position of the ship(as

provided by a satellite navigation system) is compared with

track coordinates.A heading correction is calculated contin-

uously and used to up date the course command.

Acoe~-z1.-m For

I"T

i+s

.3

I- omue coto=fsipsern roie2rc

. flloingas wll s curs keeing Th desredtrak i

TABLE OF CONTENTS

I. INTRODUCTION....................9

Ii. COURSE-KEEPING...................11

*A. THE PRIMARY TRANSFER FUNCTION OF THE

SYSTEM.....................11

B. STABILIZING THE SYSTEM ............. 12

C. IMPROVEMENT OF THE SYSTEM...........32

1. Cascade Lead Compensation ......... 33

III. TRACK FOLLOWING...................50

A. FINDING. THE COURSE CORRECTION..........52

B. FINDING'THE DESIRED COURSE...........57

C. FIRST ATTEMPT TO DO THE TRACK FOLLOWING . . . 58

IV. CONCLUSION.....................78

APPENDIX A: COMPUTER PROGRAM...............79

LIST OF REFERENCES....................87

BIBLIOGRAPHY........................88

INITIAL DISTRIBUTION LIST.................89

4

LIST OF FIGURES

2.1 Block Diagram Of The System With An Autopilot .13

2.2 Ship Heading(1) and Heading Command(2) with

G=20. ....................... 162.3 Rudder Angle~i) and Ship Heading(2) with G=20 ... 172.4 Ship Heading(1) and Heading Command(2) with

G=24.2. ...................... 18

2.5 Rudder Angle(1) and Ship Heading(2) with G=24.2 .19


G=30. ....................... 20

2.7 Rudder Angle(1) and Ship Heading(2) with G=30 ... 21

2.8 Ship Heading(l) and Heading Command(2) with

G=36. ....................... 22

2.9 Rudder Angle~i) and Ship Heading(2) with G=36 ... 232.10 Ship Heading(1) and Heading Command(2)-with

Disturbance and G=20O................24

2.11 Rudder Angle(l) and Ship Heading(2) with

Disturbance and G=20 ................ 25


Disturbance and G=24.2 ............... 26

2.13 Rudder Angle(1) and Ship Heading(2) with

Disturbance and G=24.2 ............... 27


Disturbance and G=30................28




Disturbance and G=36................30



5

2.18 Bode Plot of the System with Steering Servo . . .. 32

2.19 The System with the Compensator. .......... 33

2.20 Family of the Root Loci. .............. 35

2.21 Rudder Angle(1) and Ship Heading(2) without

Limiter and G=3.5 .. ................ 37


Limiter and G=10. ................. 38


Limiter and G=15. ................. 39


Limiter and G=24.2 ................. 40

2.25 Block Diagram of the Steering Gear ......... 41

2.26 The Complete Block Diagram for Course-Keeping . .. 41


Limiter and K1=24.2'. ................ 42


Limiter and K1=15. ................. 43

2.29 Rudder Angle~l) and Ship Heading(2) with

Limiter and K1=10. ................. 44


Limiter and K1=3.5 ................. 45


Limiter and K1=4.6 ................. 46


Limiter and K1=5.3 ................. 47


Limiter and K1=6..................48

2.34 Bode Plot of the System with Steering Servo

and Filter ..................... 49

3.1 Orientation of the Space Axis(XO,YO) and the

Moving Axis (X,Y). ................. 50

3.2 Course-Keeping and Coordinate Calculation. ......52

3.3 Ship Heading(l) and Desired Heading(2) ........53

6

. . -r . _ .- r . T-. - . 7. - . -- -r -

3.4 Distance in X(1) and Y(2) ............. 54

3.5 The Trajectory Followed by the Ship ........... .55

3.6 The Coordinate of The Desired Trajectory ...... .56

3.7 The Algorithm to Compute The Course Correction . 56

3.8 Algorithm to Compute The Desired Course ........ .57

3.9 The Condition such that the Algorithm Fails . ... 58

3.10 Block Diagram of An Autopilot With The

Integrator ......... .................... .59

3.11 Desired Trajectory and X when K4=.0l .. ....... .60

3.12 Desired Trajectory and Y when K4=.OI .. ....... .61

3.13 Rudder Angle when K4=.01. ..... ............. .62

3.14 Ship Heading and Heading Command when K4=.Ol . 63

3.15 Desired Trajectory and X with Disturbance when

K4=.0l ...................... 64

3.16 Desired Trajectory and Y with Disturbance when

K4=.Ol. ...................... 65

3.17 Rudder Angle with Disturbance when K4=.01 ..... .. 66

3.18 Ship Heading and Heading Command with

Disturbance when K4 =.01 ..... ............. .67

3.19 Desired Trajectory and X with Initial Value of

X~l .......... ........................ .693.20 Desired Trajectory and Y with Initial Value of

x=l. .......... ....................... ... 70

3.21 Rudder Angle with Initial Value of x=l ........ 71

3.22 Ship Heading and Heading Command with Initial

Value of x= 1 ........ ................... .723.23 The Complete Block Diagram for Course-Keeping

and Track-Following ....... ................ .73

3.24 Desired Trajectory and Actual Trajectory(y)

with K5=.75 ........ .................... .743.25 Desired Trajectory and Actual Trajectory(x)

with K5=.75 ........ .................... .753.26 Rudder Angle with K5=.75 ..... ............. .76

7

3.27 Ship Heading and Desired Heading with K=.75 .... 77

8

I. INTRODUCTION

The basic reasons for using an autopilot on a ship are

to reduce the number of operator controls,minimize rudder

orders for Course-keeping,and reduce the travelizg time

between destinations.

In the past autopilots were used extensively to maintain

a given course. They work well but most autopilots are for

course control only,the effect of an ocean current and/or

wind during travel are not considered. When a disturbance is

applied,the ship follows the desired course but does not

follow the given track without correction. This problem has

been studied [Ref. 1]. The correction needed to compensate

for errors due to the effect of a current and/or wind were

found.

The purpose of this thesis is to develop a procedure to

compensate for the error in position and reduce it as much

as possible, then the autopilot can be used for both

Course-keeping and Track following.

Today,many ships have computers on board. The computer

can be used to solve the problem of Track following.

First,the desired trajectories are stored in the computer.

When the computer receives measured X and Y positions the

computer will provide the desired position with respect to X

and Y and compare with the actual position.An error signal

is obtained from trajectory information,and can be used to

drive the ship close to the desired trajectory.

Actual position must be provided by the navigation

system. The navigation system used in the Track following

autopilot must be accurate,must give continuous information

about the position of the ship,and the system must be

useable everywhere in the world. It is felt that the

NAVSTAR/GLOBAL POSITIONING SYSTEM can be used.

9

Z7.............................

The NAVSTAR/GLOBAL POSITIONING SYSTEM(GPS),currently

being developed by the Department of Defense,is a satellite

based navigation system that will provide the user with

extremely accurate three-dimensional position,velocity and

time information on a 24-hour basis and in all weather

conditions at any point on the earth.The user position is

determined by measuring its range to four satellites.For

more details refer to [Refs. 2,3].

10

. . . .. . . . . . . . . . . . . . . .. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

II. COURSE-KEEPING

A. THE PRIMARY TRANSFER FUNCTION OF THE SYSTEM

The equations of motion by Davidson [Ref. 4] are:

Q + c mc - c5

where Q(s)= (Z/V)o

e = turning angular velocity

V = ship speed;

= drift angle ;

6 = helm angle ;

Z = ship length ;

denotes d/ds = (Z/V)d/dt s = (V/)tm = (ml-cf)

m),m2,n = coefficient of inertia

c9 .,cc,Cm,Cf = coefficient of resistance

" c = coefficient of rudder force.

The former equation is the equation of lateral transla-

tion and the latter is the equation of turning angular

motion.

Then rewriting the equations of motion in terms of time

and taking the Laplace transform,we obtain the turning rate

of the ship in steering [Ref. 5] is:

6(s) K(I+Tls) S(s) + rTT2s+(Ti+T2)16(0-)+TiT 26(0-)(1+Tls)(l+T 2s) (1+T~s)(1+T2s)

The first term corresponds to the ship motion excited by

the steering,and the second corresponds to the memory of her

motion at the beginning of the steering. Therefore,a rela-

tional function is:

P.1

....... II.

_"(SX = K(I+T-s)6(s) (l+Tis)(I+T2 s)

-...(1)

Equation(l) describes the response character of the ship

to steering, which may be called the transfer function of

the ship in steering.

In this study of automatic Course-keeping we will be

working with a 200,000 DWT super-tanker of the following

characteristics [Ref. 6],

length = 310.00 meters

breadth 47.16 meters

Draft 18.90 meters

Steering Quality indices.

Tl -269.3 seconds

T2 : 9.3 seconds

T3 : 20.0 seconds

K : -0.0434 red/sec

Maximum Rudder Deflection = 30 degree

Maximum Rudder rate = 2.32 degrees/second

Substitute Tl,T2,T3 and K in equation (1), we get:

O(s) -0.0434(1+20s)6(s) (1-269.3s)(1+9.3s)

--------(2)

B. STABILIZING THE SYSTEM

From equation(2),we see that one pole of the steering

transfer function is in the right half plane,so the system

is unstable. We have to stabilize the system by using an

autopilot.

12

.................................. ...... °.... *."°i......... ... ... '. '.... ......... .. ,................. ...... .... "..............., .,....,.............-.,

Assuming the autopilot has a gain G, Figure 2.1 is the

block diagram of the system with an autopilot.

j d I _ _ _ _ _ _ _ 0 I e

Figure 2.1 Block Diagram Of The System With An Autopilot.

The open loop transfer function for the system is

H(s) GK(1+Tss)

s(I+Tls)(I+T 2s)(I+TE S)

Where 1 the rudder servo transfer function.

(1+TE s)

From experience the value of T1 is 1 to 2 seconds and a

good choice is 1.7 seconds,so the only parameter available

to make the system stable is G.

To get the values of G for stability,we refer to the

Routh criterion:

GK(1+T3s) - -l

s(l+Tgs)(I+Tls)(l+T2s)

s(l+Tgs)(ITls)(1+T2s)+GK(l+T3s) = 0

s(1+1.7s)(1-269.3s)(l+9.3s)-0.0434G(1+20s) = 0s +0.692s3+0.06s2-(2.35XlY-2.04XlOG)s+l.02X OG: 0

s 1 0.06 1.02XO'G

13

°.I

s 3 0.692 -(2.35X10%-2.04XlM0G) 0

s~ A 1.02)'10'sG

s' B 0

so 1.02X10'G

Where

A =0.692 O.06+(2.35xltT"-2.04Xl0'G)

0.692

B =-A(2.35X10f*-2.04XlO~'G)-0.692 1.02xl10sG

A

To find the limits of G for stability,the values of A,B

and 1.02x10'IG must be greater than zero.

In the s2 row

0.692 0.06+(2.35Xl1-2.04Xl1G) > 0

4.1755 X102 -2.04X10MG > 0

G < 204.68

In the s' row

B >0

-A(2.35Xl0-4-2.04Xl0'kG)-0.692Xl.02Xl0'sG > 0

After manipulation.

G 2 - 85G + 240 < 0

3 < G < 82

In the s' row

l.02xl0'G > 0

G >0

So the condition of G for stability is

3 < G < 82

By using DSL/36O,the system was simulated. The system

was represented by the block diagram of Figure 2.1. The

computer program for this system is contained in Appendix A.

14

Values of G were selected between 12 and 36,which are the

ones with better time constants.

Figure 2.2 to Figure 2.9 are the computer outputs for

different values of G. By comparing,we found that when

G=24.2 (Figure 2.4,2.5) and G=30 (Figure 2.6,2.7) the

settling time is almost the same and shorter, with less

oscillation than for other values of G. When we compare the

rudder angle for both values,we can see that when G=30 the

rudder angle is larger so in this case G=24.2 is the best

value.

Next we simulate a disturbance that produces the rate of

turn caused by waves. For a large super tanker the rate of

turn can easily reach as much as 0.2 or 0.3

deg./sec.(0.00349 or 0.00524 rad./sec.).

In this thesis the rate of turn 0.2 deg./sec. was used

in the program. Figure 2.10 to Figure 2.17 are the computer

outputs of this program. Figure 2.12,2.13 show that when

G=24.2 the settling time and rudder angle are better than

other values. Figure 2.18 is the Bode plot of this system

which phase margin = 11.17 degrees and gain margin =-18.46

DB that are lower than normally used in practise.

-..

'2 15

% *% a * .. **. . . . . . . . . . . . . . . . . . . . . .

hErOD RNO HEROAF VS TIME0HE;D0AF=I0 IN2=0.0 G=20

C,-r0

CD

"10

0SAE 0.0 UISIC U O

YSRE .0 UI,/NHPO O

Fiue22 Si0edn adHaigCmad20i§Gl)0

01

RUDOER RNGLE RND HERO VS TIMEOHERDRF=10 IN2=C.0 G=20

CDJ)

I

o-o 1 0 2b00 3 . , 00 ,.0 6 .o b 0 o.0

|~ o,0

0

C,0 ,

I

0o0 10.00 20.00 30.o0 40.00 50.o0 60.00 70.00 80.00

XSCALE= 100.00 UNITS/INCH RUN NO. I

TSCRLE= 50.00 UNITS/INCH PLOT NO. 2

Figure 2.3 Rudder An~le( and Ship Heading(2)

17

.. ...°* * ** . .% ** . . . . .... . . .........*- ~ . ... . .

HERD P'\,D HEROAF VS TIME~HEROflF=iO0 IN2=O.O.G=2L4.2

010

Xc'E 1C 0 NIS IC R N N .

0 SRE .0 UIS/NHPO O

Fiue24 ShpHaig n edng.mad2

wih0142

01

RUDDER ANGLE RND HERO VS TIME9HERDAF=1O 1N2=O.0 G=24.2

CD

4. ru

0z

Lo

0=

0.0 0 00 00 00 .0 1 0 00 0 00

x0 R

0 SAE 0 .0 U IS IC U O

0SRE 00 NT/NHPO O

Fiue25RddrAge adShpHaig2

wih0= 2

01

-0

HERO RND HERORF VS TIMEgHERDRF=IO 1N2=0.O G=30

01

02

. .. . . . .

RUDDER RNC-;LE RN' HERD VS TIMEqoHEROF=10 IN2=O. :=30V

CD

0

C=r

01

XSAE 10.0 0NT/NH U O

CSRE 00 NT/NHPO O

Fiue27Rde0nie n hpHaig2

wi =

A,.1

HERO RND HERORF VS TIME-HERORF=lo 1N2=O.O G=36

0

0 0 1 . 0 2 . 0 0 0 0 0 b o 6b 0 7 .0I0 0 0

0SAE 0 .0 U IS ICTSRE .0 U ISIC U O

PLT0o

0 i u e2 8 S i ed nad H ai g C m ad20iRGl)6

(-22

[ -'-r- .5 - W - w-L *-... - . + - d. . . .J ,,. . , ,, ,

RUODER RNGLE RND HERD VS TIMEOHEROFF=1O IN2=o.o 0=36

° !00010

i-i

%" 0

-..

-WI 0

XSCPLE= 100.00 UNITS/INCH RUN NO. ITSCRLE= 80.00 UNITS/INCH PLOT NO. 2

' Figure 2.9 Rudder Anfle(l and Ship Heading(2)

23

°. • . . . • o . , . . - - - . • °

HERD RNO HEL-'DRF VS TIMEgHEROPF=i'O IN2=O.00349 G=20.0

0

0D

CD

CO

C

Co

0

C

C 0 0 00 b 0 30 0 4.0 5.0 600 b 0 ac.0

0o

XSAE 0 .0 U ISIC U O

0 SRE .0 U IS/NHPO O

Fiue21 hpHaig0)adHaigCmad2

wihDsubnc0n =0

02

RUDDER RNGLE RND HERD VS TIME'HERDRF=V,- 1N2=0.00349 0=20.00

C

CD

0

C

100 'Cc 1.0 00 00 00 00


0SRE 00 NT/NHPO O

0 iue21 ud nlel n hpHaig2

wihDsubnc n =0

02

HERO RNO HERODF VS TIME.HEROI9F=1O IN2=0.00349 G=24.2

9

.0

to-

e'J

"-2i

°0

XSC.LE 10 .0 U ISIC U O

.'.

YSCRLE= 4.00 UNITS/INCH PLOT NO. 1

Figure 2.12 *Ship Heading(1) and Heading Conimand(2)With Disturbance and G=24.2.

26

.." .°* .

*"-~ ° *,1. * ' 4 4 -

*.b .. e *.. **°..* ..-:: -*

RUDDER RNGLE RNO HERO VS TIMEgHEROFRF=1O [email protected] G=214.20

010

0SAE 0 .0 U IS IC U O

0SRE 00 NT/NHPO O

Fiue210udrAgel n=MHaig2

wihDsubnc0n 4

02

5%

HERO RNO HEROAF VS TIME-HERDOiF=10 1N2=0.003'49 G=30

CD,

0

C0

C.'

lb.oo 100 1.0 b o 4 .0 ,.0 6 .0 b 0 80004

o


0SRE .0 UIS/NHPO O

Fiue21 hpHaigl)adHaigCmad2

0ihDsubnc n =0

02

RUDDER RNGLE AND HERD VS TIME09HEROFRF=1O IN2=O.00349 G=30

iiz

{0

Li

Cr.

[0F0

?0

N.

010

RUDDLER 100.00 AN /N HEA RUN NO.METSCE = O I0 N N2.00OT9NO=30

Fa

0

12129

0--. 0

00o0w

'oo0bo bo 6o 6oo ~o ao oo bo

0 SAE 0.0 UISIC U O

- SCIL =8000 U IT/ICHPOTN0

oN

£iur .1 RderAgl~l ndSipHedng2- wthDitubacean G30

: 20dN

w0

w0

m"0o0

0.00- - 10.00 20.0 • i5 0 4 .00. 00 . .0. 7.00 B -.0

i

HERO RNO HERDAF VS TIMEoHERORF=1O IN2=0.00349 G=36

(D-00

CDrv

00 110. 002'0 3 .0 b o s . u 6 .D ,.0 e .

010

TscL•

, .0 .NT/IC PL INO

03

%

0

0

010

ThSoo iLE: 200 .3.00 TS/INCHb~o RUNo NO.00

TSCRLE= 4.00 UNITS/INCH PLOT NO.

Figure 2.16 Ship Heading(1) and Heading Command(2)

with Disturbance and G=36.

30

RUDDER RNCLE RND HERD VS TIMEOHERORF=1O 1N2=O.00349 0=36-

en

0

0.0 1.0 20.0 3.o b o b o 6 .oo 7.o 00

0SAE 0.0 UIS IC U O

0SRE 00 NT/NHPO O

% igr 21 RddrAnl~l ndShpHed0g2

%: wth Dsturanceand =3-

03

(SOP) 3SV1IdWool-- a0l=-

.. .. .... .....

... . ......... ....... ...

. *.... .. . .. .~7 .

..........

................

C. IMPROVEMENT OF THE SYSTEM

Now the system is stable but the rudder angle is too

large which is not practical at all. It is necessary to

improve the system performance by using a compensator to

obtain acceptable transient and rudder angle performances.

1. Cascade Lead Compensation

Lead filter compensation has a transfer function as

follows:

_,____6) l+1s1 (s+Z) =z If -___

ai (s+P) Z p C1+s) 1+Js

Wherec=Z/P and P>Z. The system is shown in Figure

2.19.

Figure 2.19 The System with the Compensator.

The characteristic equation is:

It (G/oc)K(l+T3s)(s+z)

s(s+p)(l+Tls)(l+T2s)(l+Tgs)0

-O.O434(C/D()(s+ p)(l+2Os)

s(s+p) (l-269.3s)(1+9.3s)(l+l.7s) 0

33

-q. . . . . . . .

. M

Dm.s (s~p) (1-269.3s) (19.3s) (11.7s)-O. 0434(G ) sop (120s)

"0

4 2 5 7 .6 3 3 s'+( 2 94 6.4 9 +4 257. 6 33p)s4+(258.3+2946.49p)sI +

(-1+258.3p+.868(G/i))s2 +((-l+.868G)p.O434(G/ )}s+.0434Gp

=0

From this equation,the NPS PAROLE program was used

to find the best values of P and Z. The family of root loci

are given on Figure 2.20.

From experience a good value of Oc was 0.1 and we

select the damping ratio about 0.42. -We obtain:

P = 0.4

C = 0. 1

Z = XP = 0.04

Hence the transfer function of the lead filter

compensator is:

+ s 1+25s

Figure 2.21-2.24 are the computer outputs of Figure

2.19 with different values of G. It can be observed that to

have a quick response a high gain is necessary but this

could demand an excessive operation of the rudder which can

be bad for the following reasons:

(1) To get a quick response,faster rudder operation and

greater rudder angle are needed.

(2) This faster response may cause accelerations that

are too sharp for personnel aboard ship.

(3) Too much rudder action would cause the rudder

machine with all the hardware to have its life

reduced.

In Figure 2.22 - 2.24 the system reaches steady

state in very short time but the rudder angle is too high.

34

• . .- .. -.................................-....-. ,..-......... .-............ ....... ,.. ,. . . .-

REAL RXIS(UNITS PER INCH) = 0.2500IMAG AXIS(UNITS PER INCH) = 0.2500

RLPHR=.O01,.O1,. 1VIMUKTRNRNDR G=24.2

0

b0

I~Mal

_-

Cc

L3 +

ci +

ci +

Mo +

-0-,

+

+

10 0

-1.2 1.0 -0 7, -0 s G.2

d . 5

in o +cl {Lf 0 +

gm r 0 +

-1.5 -. 00 -0.5 -3. 0 -0. + 0 20 5

00 ++

p.Figure 2.20 Family of the Root Loci.

: 35

cic

. - . . . .. . • •. - ° . ° . .. , - , , o •. ° . • • - . . ° - , - .- ,

In Figure 2.21 the rudder angle is at the limit but the

system takes a long time to reach steady state.

So,it was necessary to reduce the rudder angle and

simultaneously reduce the settling time of the system. Inorder to introduce a limiting value of rudder rate,the

steering gear transfer function must be replaced in the

system. The block diagram for an equivalent circuit was

given on Figure 2.25.

Where K3 =1/Tg = 0.588

With this transformationwe have the system on

Figure 2.26

Figure 2.27 - 2.30 are the computer outputs of this

block diagram with the gain selected before. We can see that

when the value of the gain(Kl) is greater than 10,the system

reaches steady state in a short time (about 3 minutes for

K1=10) but the rudder angle reaches the limit on one side.

In Figure 2.30,Kl=3.5,the rudder angle is .within the.limits

but the settling time is still too long. Then we must try to

find a value of gain(Kl) between 3.5 and 10 to get the best

results for rudder angle and settling time.

Figure 2.31 - 2.33 are the outputs of this trial

with gain:4.6,5.3 and 6. We can see that:

For gain = 6 the settling time is about 450

sec.(7.5min.) the maximum rudder rudder angle is about 28

deg. and the heading error about 0.77 deg..

For gain 5.3 the settling time is about 470

sec.(7.8min.) the maximum rudder angle is about 26 deg. and

the heading error about 0.9 deg..

For gain = 4.6 the settling time is about

510sec.(8.5min) the maximum rudder angle is about 22 deg.

and the heading error about 1.1 deg..

36

RUDDER RNGLE RNO HERD VS TIME-SOMMRBT G=3.5 1N2=.00349

0

-10

0SAE 0 .0 U IT/NHRN N .

YSCLE 5.0/IS ICHPO O

0 iue22 udrAgel n hpHaig2

0ihu iieradG35

03

RUDDER RNGLE RND HEAD VS TIME.S0MMRRT G=10 IN2=.003149ri-

0o

0

0n

0

0CD

0 0 b 0 20 0 1.0 0 0 51 0 ,.0 70 0 11

14 1 OR

0 SAE 0 -0 U IS IC U O

0 SRE .0 U IS/NHPO O

Fiue220udrAgel n hpHaig2

~4JJ

PJ-IDER PNGLEF RND HERD VS TIMEO5OM RTG=1'5 1N2=.003149

00

-10

XSCALE= 100.00 UNITS/INCHRU NO3TSCRLE= 10.00 UNITS/INCH PC O

Figure 2.23 Rudder Angle~i) and Ship Heading(2)without Limiter and G=15.

39

RUDDER RNGLE RND HERD VR TIME-, g 8SOMMRRT G=24.2 IN2:.00349

CD

-,

-

,%"D.a. 0

v

00

C -o!

'"" 0

0

010

a'-

04

..-• 0

, ' ,0:.'. 0

i;,:0, 0

172 0

0.00L10.00 20.00 3.00 4000/0.0NCHo PL.00 N.0

-'o.

"("Figure 2.24 Rudder Angle 1) and Ship Hteading(2)"-" ' without Limiter and G=24.2.

": 40

d.4

Figure 2.25 Block Diagram of the Steering Gear.

*ISSTuMRSF4 L

ev -- L I*3S

Figure 2.26 The Complete Block Diagram for Course-Keeping.

So we can conclude that the best value forCourse-keeping with the limiter is 4.6. Figure 2.34 is theBode plot of this system with phase margin 42.2 degreesand gain margin = -15.97 DB that are better than the Bodeplot of Figure 2.18. Then the system of Course-keeping is

satisfactory.

41

..................-

RlUDDER RNGLE HERO VS TIME-IN2=O.00319 KI=24.2 HEROD9F=3

0

0

C3

c a

t~42

RUDOEP RNGLE HERO VS TIMEgIN 2=O.OO 3 19 K1=15 HERORF=3

Lo'

0 ' 0 1'000 200 3'000 4 .0 4k 00 56000 64.0

0 SAE 00 NTSIC U O

0 SRE .0 UIS/NHPO O

/Figr .8 Rde nl> n hpHaig2

wihLmtr0aK=5

'13-z0

R UDDER ANGLE &HERO VS TIME01N2=O.00349 K1=10 HERORF=3

LAl

0.0 b o 10 0 200 h 0 40.0) 40 0 6.0 64.0

0SAE CD NTSIC U O

0SRE .0 UIS/NHPO O

Fiue220udrAge~ n hpHaig2

wihLmtr0AK=0

04

*2 WT** * * EU ~ U *

RUDDER RNGLE &HERD VS TIME-IN2O0.O349 K1=3.5 HERDPRF=3

CD0

CD

CD

. 0 8 -0 100 4 .0 3000 400 b~ o s .0 6oo

0SAE 00 NTSIC U O0SAE .0 UIS/NHPO O

0 iue23 udrAge~ n hpHaig2wihLmtr0n 135

04

0:

RUDDER RNGLE HERO VS TIME-IN2=O.OO349 K1=L.6 HERDf9F=3cu

rCD

,

o0 0 8100 11000 2000 32.0 b.o e.o h o 6o o

0SAE 00 NTSIC U O

0SRE .0 UIS/NHPO O

Fiue230udrAgel n hpHaig20ihLmtr n 146

i46

RUD'DER RNGLE HERD VS TIME-IN2=O.OO349 K1=5.3 HEF1DRF=3

0

00

Ln

c,

0.0 8 .0 1 . 0 2 0 0 320. 0 b .0 0.0 5'C 0 6 0 0

XSAE 00 NTSIC U O

0SRE .0 U IS/NHPO O

Fiue230udrAgel n hpHaig2

wihLmtrn l53

04

RUDDER RNGLE HERO VS TIME

-IN2=O.OO3L49 K1=6 HERDf3F=3

C,-0

LI,

0

c0

0

CD

0n

CD

0C)

G0 0 800 11000 20.0 3'000 bO00 '8.0 560 0 64 .0

0SAE 00 NTSIC U O

0SAE .0 UIS/4HPO O

Fiue230udrAl n=hpHaig2wih0m r~l~ l6

04

(NOP) aSVH

.. .. . . .

.... .. ...

.. . . . .. .. . .. . . . . . .

.. . . . . . .. . . . . . . . .. . . . . . .

.. .. .... . .. ....

.... .... .... .... .... ... ... ..

... .. .. .... .... . ... ....... .... ... .... ... ... . ..... . . .... .

.... .... .. . ... . .

... .. .. ... .. .. .. .... ..

.... .. .. . ..

.. .. ... . . . . . . . .

. .. .. . . .... ...

.... .. t .... ...... .9 ..... L, 0

................

. ................

~ ~. . . . %9~%.. ........ .. . ........ . . . . . .

'7z

III. TRACK FOLLOWING

Using the ship heading controller obtained previ-

ously,and assuming that the velocity of the ship is

constant(at 14 knots) during underway on the open sea,the

position of the ship can be found at any time.

In order to find the position of the ship,a right hand

rectangular coordinate system is established, the origin of

which is chosen to be in the body itself,as shown in Figure

3.1.

XO

.0C •

YOoo

oy

Figure 3.1 Orientation of the Space Axis(XO,Y0)and the Moving Axis(XY).

The origin and the axes are fixed with respect to the

body but movable with respect to other systems of coordinate

50

axes fixed in space.It is assumed that the two systems coin-

cide at t=O.

The transformation from the ship to space coordinate

system is 4.Zfined by the following relations,obtained from

Figure 3.1

S= VcosG + Using

Y = Vsine - UcosO

and

X XO+JXdt

Y = YO+/Ydt

where

X = Velocity in X-direction

= Velocity in Y-direction

XO = Initial position of X

YO = Initial position of Y

V = Ship velocity

U = Lateral velocity

Assuming constant velocity and no lateral force,the

equations become:

X Vcose

Y VsinE

and

X= dx/dt

S= dy/dt

then

X = XO + JdxY = YO + dy

Knowing initial values of X and Y,the coordinates of the

ship are calculated. Figure 3.2 is the block diagram of this

procedure.

51

.Z '

S.,

'.. . . . . - -. . . . . . . .. . . .

IJ

DI US~ ~VUt Sim

Figure 3.2 Course-Keeping and Coordinate Calculation.

These equations were included in the program of the

Course-keeping autopilot in order to find the position of

the ship.

Figure 3.3-3.4 are the output of this program.

The purpose of the Track following is to keep the ship

following the desired trajectory from the beginning to the

destination. When the ship is not on the track a course

correction must be calculated to return the ship to the

track. The corrected course is a function of the distance to

the track.

A. FINDING THE COURSE CORRECTION

Initially the navigator must design the trajectory from

the beginning point to the destination on the map. He has to

know the course and speed. When the ship is underway,it is

necessary to know(measure) the position of the ship.

Although the course of the ship may be the same as the

desired course,it may not be on the desired trajectory due

to the effect of sea current and wind. Therefore the navi-

gator has to calculate a new course to keep the ship

52

HERD RND ' ERDRF VS TIMEHERORF=5 7N2=0

CDi

CD ;

0D

cc

o o 8C. oo 1,60.00 240.00 30. 00 'ibO000 480. 00 S' 6 G. 00

XSCALE= 80.00 UNITS/INCH RUN NO.1ISCRLE= 0.80 UNITS/INCH PLOT NO.I

Figure 3.3 Ship Heading(1) and Desired Heading(2).

53

[. -

[ .

X RND Y VS TMEHERO91F=5 7.N2=0

'.

TSC-0.Z= 0.0 UISICHPO O""1

:':"

F 354

"1t.00 " r0.00 iE ,.003 240. 00320.00 1400.0O0 LI90. 0 560.00 640 .0

IXSCRLE= 80.00 UNITS/INCH RUN NO. 1TSCPLF_ 0.40 UNITS/INCH PLOT NO. 2

Figure 3.4 Distance in X(1) and Y(2).

v 54

a

following the desired trajectory until the ship reaches the

destination. This situation is depicted in Figure 3.5.

A

/-i A

x

Figure 3.5 The Trajectory Followed by the Ship.

The procedure of an autopilot for track-following is the

same as that used by a navigator. First the coordinates of

the desired trajectory are stored in the computer as shown

in Figure 3.6 The position of the ship is measured by

NAVSTAR/GPS.The computer calculates the errors in X and Y

positions and calculates the course correction. These errors

should be zero in order to keep the ship always on the

desired trajectory.

The procedure to calculate the course correction(8c) is

shown in Figure 3.7

55

bL'.-

-1 O to 0 4, 0 ( .0 70 feqo je It* *

Figure 3.6 The Coordinate of The Desired Trajectory.

xee

Figure 3.7 The Algorithm to Compute The Course Correction.

DS/Xe =SIN 9

DS = Xe SIN 8

SIN ec DS/M

*ec SIN (DS/M)

56

B. FINDING THE DESIRED COURSE

The desired trajectory is stored in the computer,the

desired course can be found in a simple way,Figure 3.8 shows

the method. The X axis is divided into constant small inter-

vals. For each point of X,the corresponding Y coordinate is

found. The desired course is obtained by taking the arctan-

gent of the result of the division between increment in Y

and increment in X.

- Y

Y4

Y3

-: ,,I ,I I

I I

X1 XZ X3 X4 X 5 X

Figure 3.8 Algorithm to Compute The Desired Course.

TAN(X1.)

-,

Ga TAN ( Y Y2-I

e3 = TAN'Y3-Y,,. \(X3-X?./

E)4 = TA'Y4-Y3" \x4-x3)

E5 = TA,'Y -Y,"- X5 - X41

57

.-'.% .;.- ." .. . . .,.".:* ... -" "-'-'.-*." ,. .". -.:. .' *-',.". . .- ' -' .".. ., . .' ,",; - ....-" ',<- -;-' -" ' ,. " "

For selecting the value of the constant small interval

of X,we consider Figure 3.9

Y

B

x

Figure 3.9 The Condition such that the Algorithm Fails.

If the value of X is big,the straight line AB is not on

the curve,so the desired course is not true. To get the true

desired heading the straight line AB has to be on the curve

all the time. To accomplish this we have to select the value

of X small enough,so the straight line AB is part of the

curve.

C. FIRST ATTEMPT TO DO THE TRACK FOLLOWING

From the Figure 3.2, X and Y positions are obtained.

With these values, the desired heading and trajectory are

obtained. By using an integrator with the gain K4(as shown

in the block diagram of figure 3.2 ) we will obtain a new

block diagram which is shown in Figure 3.10. The purpose of

the integrator is to eliminate steady state position errors.

A suitable value of integrator gain is K4=.01.The value

was found by repeated simulation runs. Figure 3.11-3.14 are

58

. . . . . . .

-'Figure 3.10 Block Dia ram of An Autopilot-" With The Integrator.

the results obtained without disturbances. From Figure,'.."3.11-3.12 the actual trajectory and the desired trajectory

" are coincident. In Figure 3.14 the time for the output toi reach steady state is about 800 seconds(13.3 minutes).

~The system was also simulated with disturbances. Figure-]. 3.15-3.18 are the results with K4-.01. In Figure 3.15-3.16

the actual trajectory and the desired trajectory are not~coincident but the distance between them gradually increases

"5. with time. In Figure 3.18 the time for the output to reach".' steady state is about 1600 seconds(26.67 minutes).

"-. 59

--

OC

K

Co

• ..< .. : ' .-..: ' .° ..., , .. -, ., ' ... .; ..... .... , % . -...o ...: .., ... '.I. ..N. -. .> ..:

7r 72

4.4

S& 56

48 46

40 49

32

2 24

I

Si' 2 S 2 1 6 8 2 24TIME 1109 SECI

DESIRED TRA..ECTORY RICACTUL. TR&ECTORY I VS TIME

Figure 3.11 Desired Trajectory and X when K4=.01.

60

.I.. .

54

4.

4 0 4 842- 42

N

2 4 24

1 12 ,

IF

i "hiFIME (101 S

221 24

IDESIRED TRAICTORY AME)

ACTUAL TRAECTORY (Y) VS TIlE

Figure 3.12 Desired Trajectory and Y when K4=.O0.

61

6.0 S.. 3 ~ us 3.3 30 I

Figure 3.13 Rudder Angle when K4=.O.

62

536 53

4W 43

47 47

4 45

43 43

O.6 6.5 1.6 1.5 zLo 2s 3.6 3.5 4.6 4.5 5.6

TIME (101 SEC)

SHiP IECOING MWD OESIREO IEFIOING VS TIlEIN2-O.O X-O K4-.01

Figure 3.14 Ship Heading and Heading Command when K4=.O1.

63J

. . . . . . .

4s-I 4S

39

4 IL

TIME 110' SEC)

DESIRED TRALJECTORY FMACTUft. TR&JECTORY (XI VS, TIME

Figure 3.15 Desired Trajectory and X with Disturbancewhen 14=0.

64

4& 4?

24 24

I LL12.

9 4 5 1 1 14 is i' is 22 24TIME (10, SEC)

CESIRED TRRJCTORY AM0ACTUlAL TRP..CTORY (Y) VS TIME

Figure 3.16 Desired Trajectory and Y with Disturbancewhen K4=.61.

65

* . -# -. A I .* -,. - ..3. -

h26.

0 6 .3 0.6 0.2 1.2 1.5 1. 25 2.4 Z7 3.0 3.3 3.6TMtf 411 SECi

RLU2ER M4GLE VS TI ME

Figure 3.17 Rudder Angle with Disturbancewhen K4=.Q1.

66

55

SS. S3

49. 49

47

45 4S

4- 43

41 4

37.

Tiff (10' 5ECI

SHIP HEADING AND3 DESIRED HEADING VS TIME112-0.00349 X-0 94-.01

Figure 3.18 Ship Heading and Heading Commandwith Disturbance when K4 =.Q1.

67

Additional simulation runs included disturbances,but it

was also assumed that the ship was not on the desired

trajectory at the beginning. Figure 3.19-3.22 are the

results. In Figure 3.19-3.20,we can notice that the actual

trajectory stays away from the desired trajectory all the

time,there are no control actions to put the ship back to

desired trajectory.

The next step was to include the course correction(O )in the control loop, as shown in Figure 3.23.

Simulations were run with course correction and gain

K5,a best value of K5=.75 was obtained. Figure 3.24-3.27

are the results using this value. In Figure 3.24-3.25 the

ship comes back to the desired trajectory and follows it all

the time. The time from the initial position to the desired

trjectory is about 1400 seconds(23.3 minutes).

68

....................

~48

4 0 - 4 0

33-

24- 24

ILI

0- 7 24

Till flog SECICESIRED TRPAJCTORY fAO

FCTUft TRA.ICTORY IX) VS TIME

Figure 3.19 Desired Trajectory and X withInitial Value of X=1.

69

- .S * *.o.~~* t. , -

42 4z

24-/ 24

II

1 2 '4 is is k~ 2 14 I 5 Z 24TIME 110' SECI

DES IRED TRREC TORY ANDACTUAL TRAX.CTORY (Y) VS TIME

Figure 3.20 *Desired Trajectory and Y withInitial Value of ,c=l.

70

%37

*.

M

rL

g~e .3 .6 . 1 . S . 1.0 .1 .4 27 5.0. .TIME (10' SEC)

RWO(R ANIGLE VS TIME

Figure 3.21 Rudder Angle with Initial Value of x~l.!71

,o..

51 71

4 4

41L 47

45- 45

43- 43

0.0 0.5 1.0 I.5 0. 8. sU 0. 4. Is5 .Time lice SECI

SHIP MEItNG AMOCESIRED HEROING VS TIME112-0.00349 X-1 K4-.01

Figure 3.22 Ship H eading and Heading Command withInitial Value of x= 1.

72

and inck Folwzg

FILTE73

t-

.... .........v.-............

*2~.. Y

sr 54

4& 48

30-.

24 24

1 125

ACTUAL TRAJECTORY (Y) V TIME

Figure 3.24 Desired Trajectory andActual Trajectory(y) with K5=. 15.

74

eA V-V- 7.77 V. K- 17 -

7r 72

G+ 64

5.

46 48

40 43

24 24

si's *24

14M 110 SEC) 2

DSRDTRAJECTORY F0ACTUL TRAJECTORY WX VS TIME

Figure 3.25 Desired Trajectr andActual Trajectory x) with 1K5=7.

75

67

b'

.5

-1.

°10

. 4.U,,,U *3. .5 0. 6. ,,.2 1's.5 .L 2.42.7 5.3 5.5 5.6• .- TIME (105 S 1

"-' RUOCER ANGLE VS TIME

"- Figure 3.26 Rudder Angle with K5=.75.

.,, 76

75

52

31- 51

i::s2+ 2

07 17 6 12 1 S 1

Tilt (102 SECI

SHIP lEROING (IC OESIREO MEWOING VS TIMEK5-.75 X-1 IN2-0.OO349

Figure 3.27 Ship Heading and Desired Headingwith K. 75.

77

4.5 'A . . . . . - % .A *5**

A.. . . . '... . . 38... .--. A'~-

IV. CONCLUSION

Use of the computer to control the autopilot for both

Course-keeping and Track following achieves an accuracy for

steady state conditions as high as may be desired. An

optimal value of gain for the autopilot was obtained *by

using the PAROLE program to study families of Root Loci.

Both negative and positive feedback can be used,but only

negative feedback is recommended.

In Chapter 2 for Course-keeping,the Bode plot is used to

show that the filter POLE ZERO and gain were the best for

the system. The actual heading in steady state was exactly

the desired heading with zero error.

In Chapter 3 the problem of acquiring the track was

studied. If the Track following mode is turned on when the

ship is many ship lengths off track,the amount of rudder

activity is determined by the distance off-track,the nature

of the Tracking algorithm,and the shape of track itself,as

well as the initial value of the ship heading.. It was found

that the initial ship heading was important,and rudderactivity could be minimized by proper choice of the initial

heading angle. This suggests that further study of the

effect of initial heading angle is desirable,since manual

adjustment of ship heading prior to activating the autopilot

could greatly reduce rudder motions. Once the ship reached

the track,the system was able to follow the desired trajec-

tory exactly in calm and rough sea.

Further studies may be conducted to include the cost

function into these programs to minimize the fuel consump-

tion and to apply optimal control theory to find the optimal

track for ship maneuvering at sea.

78

-. * -.

APPENDIX A

COMPUTER PROGRAM

*THIS PROGRAM REFERS TO FIGURES 2.2-2.17//SOMM047 JOB (2259.1435), 'PROJECT',CLASS=B//*MiAIN ORG=NPCVM1 .22S9P//*FORMAT PR, DDNAN4E=PLOTX. SYSVECTR. DEST--LOCAL// EXEC DSL//DSL. INPUT DDTITLE AUTOPILOT DESIGN WITHOUT COMPENSATIONINTOER NPLOTCONST NPLOT-1,IC1=O. , C2=0. ,C3=0. , C4=O. ,C1=3 .1415927PAEAN HEADRE = 10.PARAN rN2 = 0.0PARAN G = 20PARAM K = -. 0434PAEAN TE = 1.7PAEAN TI = -269.3PAEAN T2 = 9.3PARAN T3 = 20.DERIVATIVE

IN1 = HEADRE*C1/1SO.ERROR = INI-THETADELTAR = G*ERRORDELTA = REALPL(1C1,TEDELTAR)AMPLI - K*DELTACOMP = LEDLAG(IC2,T3,T1,AMPLI)CORREC = REALPL(IC3,T2,COMP)THETAl = IN2+CORRECTHETA = INTGRL(14,THETA1)HEAD = THETA*180./Cl

-. RUDANG = DELTA*180./C1ERRORD = HEADRE-HEAD

SAMPLECALL DRWG(1,iTIME,HEAD)CALL DRWG(1,2,TIME,HEADRF)CALL DRWG(2,1,TIME,RUDANG)CALL DRWG(212,TIME,HEAD)

TERMINALCALL ENDRW(NPLOT)

PRINT 8. ,HEADRF, HEAD, ERRORD, RUDANGCONTRL FINTIM=800. ,DELT=.8,DELS=8.ENDSTOP//PLOT.PLOTPARM DD*&PLOT SCALE=.65 &END

//PLOT.SYSIN DD*HEAD AND HEADRF VS TIMEHEADRF=10 1N2=0.Q G=20RUDDER ANGLE AND HEAD VS TINEREADRF=1O 1N2=0. G=20

79

FILE: EE44181 DSL Al

*THIS PROGRAM REFERS TO FIGURES 2.21-2.24//SOMMAR3 JOB (2259,1435), 'EE44181 ,CLASS=B//*MAIN ORG=NPGVM1 .2259P, LINES=( 10)//*FOMAT PR, DDNAME=PLOTX. SYSVECTR, DEST-LOCAL// EXEC DSL//DSL. INPUT DD*TITLE AUTOPILOT DESIGN WITH COMPENSATIONINTGER NP LOTCONST NPLOT=-4, 1C10., 1C2=O. , C3=O. , C4-O. ,C1=3 .1415927PAEAN HEADRF=O.PAEAN rN2=O.00349PARAM G=3.SPAEAN K-.0434PAEAN TE=1.7PAEAN T1=-269.3PAEAN T2=9.3PAEAN T3=20.DERIVATIVE

INi = HEADRF*Cl/18O.ERROR = INI-THETAFILTER = LEDLAG(O.,2S.,2.S,ERROR)DELTAR = G*FILTERDELTA = REALPL(IC1,TE,DELTAR)AMPLI = K*DELTACOMP = LEDLAG(1C2,T3,T1,AMPLI)CORREC = REALPL(IC3,T2,COMP)THETAl = IN24-CORRECTHETA = INTGRL(I4,THETA1)HEAD = TEETA*18O./ClRUDANG = DELTA*180./ClERRORD = HEADRF-HEAD

SAMPLECALL DRWG(1,1,TIME,RUDANG)CALL DRWG(1,2,TIME,HEAD)


PRINT 3. ,HEAD,RUDANGCONTRIL FINTIM=SOO. ,DELT=.8,DELS=8.ENDPARAM G=10ENDPARAM G=l5ENDPAEAN G=24.2ENDSTOP//PLOT.PLOTPARN DD

&PLOT SCALE=.65 &END//PLOT.SYSIN DD *RUDDER ANGLE AND HEAD VS TIMESOMM~ART G=3.5 IN2=.00349RUDDER ANGLE AND HEAD VS TIMESOMMART G=10 IN2=.00349RUDDER ANGLE AND HEAD VS TIMESOMMART G-15 1N2=.00349

L-

FILE: EE44185 DSL Al

*THIS PROGRAM REFERS TO FIGURES 2.27-2.33//SOM!4AR7 JOB (2259,1435),'EE44184',CLASS=B//*MAIN ORG=NPGVM1.2259P, LINES=( 10)//*FORMAT PR,DDNAME=PLOTX. SYSVECTR,DEST=LOCAL1/EXEC DSL

//DSL. INPUT DDTITLE AUTOPILOT DESIGN WITH COMPENSATION AND LIMITERPARAM IN2=0.00349PARAM K1=24.2PARAM K2=-.0434,K3=.588,...

Pl=-.1222,P2=.1222,P3=-.52,P4=.52,K4=0.OPARAM HEADRF=3INTGER NP LOTCONST NPLOT=1,C=3 .1415927DERIVATIVE

IN1 = HEADRF*C1/180.ERROR1 IN1-THETAFILTER = LEDLAG(O.,2S.,2.5,ERRORl)DELREF = FILTER*KlDELDES = LIMIT(P3,P4,DELREF)

-DELTAl = DELDES-DELTACDELDOT = LIMIT(Pl,P2,DELTAl)DELDOC = K3*DELDOTDELTAC = INTGRL(O...DELDOC)AMPLI = K2*DELTACCOMP = LEDLAG(O.,20.,-269.3,AMPLI)CORREC =REALPL(O.,9.3,COMP)THETAl = IN2+COR.ECTHETA = INTGRL(O.,THETA1)RUDANG = DELTAC*180./ClHEAD =THETA*180./ClERRORD = HEADRF-HEAD

SAMPLECALL DRWG(1,1,TIME,RUDANG)CALL DRWG(1,2,TIMEHEAD)


PRINT S. ,RUDANG, HEAD, ERRORDCONTRL FINTIM=700. ,DELT-.7,DELS=7.INTEG RKSFXENDSTOP//PLOT.PLOTPAJ4 DD*&PLOT SCALE=.65 &END

//PLOT.SYSIN DD*RUDDER ANGLE & HEAD VS TIME1N2=0.00349 K1=24.2 HEADRF=3

FILE: T2 DSL Al

*THIS PROGRAM REFERS TO FIGURES 3.3-3.4//SOMMA21 JOB (2259,1435), 'PROJECT' ,CLASS=B//*MAIN ORG=NPGVMl.2259P,LINES=( 10)//*FORMAT PR, DDNAME=PLOTX. SYSVECTR, DEST=-LOCAL

F. XEC DSL//DSL. INPUT DD*TITLE SIMULATION OF SHIP DYNAMICS (TEST OF FOLLOWING)PARAM K1=4.6,K2=-.0434,K3=.588,...

P1=-.1222,P2=.1222,P3=-.S2,P4=.S2PARAM V-0.0038889*SPEED IN MILES/SECOND WHICH IS EQUIVALENT TO 14 KNOTS

PARAM IN2=0.0 ,HEADRF=5CONST NPLOT=1, Cl=3 .1415927INTGER NP LOTDERIVATIVE

IN1 = HEADRF*Cl/180.El = IN~1-THETAFILTER= LEDLAG(O.,25.,2.S,EI)E2 = FILTER*KlE3 = E1*K4E4 = INTGRL(O.,E3)DELREF= E2+E4DELDES= LIMIT(P3 ,P4,DELREP)DELTAl= DELDES-DELTACDELDOT=- LIMIT (P1, P2, DELTAl)DELDOC= K3*DELDOTDELTAC= INTGRL(0. ,DELDOC).AMPLI = K2*DELTACCOMP = LEDLAG(O.,20.,-269.3,AMPLI)CORREC= REALPL(O. ,9.3,COMP)THETAl=. 1N2+CORRECTHETA = INTGRL(O.,THETAl)HEAD = THETA *180./ClRUDANG= DELTAC* 180 ./ClDX = V*COS(THETA)DY = V*SIN(THETA)X = INTGRL(O.,DX)Y = INTGRL(O.,DY)

SAMPLECALL DRWG(1,1,TIME,HEAD)CALL DRWG(1,2,TIME,HEADRF)CALL DRWG(2,1,TIME,X)CALL DRWG(2,2,TIME,Y)


PRINT 8. ,HEAD,HEADRF,X,YCONTRL FINTIM=650. ,DELT=.4,DELS=.6SINTEG RKSFXENDSTOP//PLOT.PLOTPARM DD&PLOT SCALE=.65 & END

//PLOT.SYSIN DD*HEAD AND HEADRE VS TIMEHEADRF=5 1142=0

FILE: SOM2 DSL Al

*THIS PROGRAM REFERS TO FIGURES 3.11-3.22TITLE SIMULATION OF SHIP DYNAMICS (TEST OF FOLLOWING)PAPAM Kl=4.6,K2=-.O434,K3=:.588,K4=O.O1,...

P1=-. 1222,P2=.1222 ,P3=-.52,P4=.52PARAM V=-O.0038889*SPEED IN MILES/SECOND WHICH IS EQUIVALENT TO 14 KNOTS

PARAM EPSILX=O0.125,G=-.074PARAN 1N2=0.00349CONST C1=3. 1415927NLEGEN YPATH=-20.,-30. ,-1O.,-12.5,O.,,O.,1O.,1O.,20.,...18.,30.,25.,40.,31.,5O.,36.,60., 40.,70. ,43.,SO.,...45. ,90. ,46.5,100. ,48.

NLFGEN XPATH=--30.,-20.,-12.5,-10.,O.,O.,1O.,1O.,18.,...20.,2S.,30.,31.,40.,36.,50..,40.,60.,43.,,70.,45 .....SO. .46.5,90. ,48.,100.

DYNAMICTRAJE1 = NLFGEN(YPATH,X)TRAJE2 = NLEGEN (XPATH, Y)DELTAX = (X+EPSILX)-XDELTAY = NLFGEN(YPATH,X4EPSILX) -NLFGEN(YPATH,X)Z = DELTAY/DELTAX .-

ZR = ATAN(Z)THECOM = ZR*180./Cl

DERIVATIVEINi = THECOM*Cl/180.E = INI-THETAEl = 10.*EFILTER = ZEROPL(0.,.04,0.4,El)E2 = FILTER*KlE3 = E*K4E4 - INTORL(0.,E3)DELREF = E2+E4DELDES = LIMIT(P3,?4,DELREF)DELTAl = DELDES-DELTACDELDOT = LIMIT(P1,P2,DELTAI)DELDOC = K3*DELDOTDELTAC = INTGRL(0.,DELDOC)AMPLI1 = K2*DELTACAMPLI = G*AMPLI1COMP = ZEROPL(0.,.05,-.0037,AMPLI)CORREC = REALPL(0.,9.3,COMP)THETAl = IN2+CORRECTHETA2 = INTGRL(0.7853982,THETAl)

PROCED THETA = COR(THETA2,Cl)THETA3 = THETA2

3 IF(ABS(THETA3).LT.Cl) GO TO 5IF(TEETA3.LT.0) GO TO 4THETA3 = THETA2-ClGO TO 3

4 THETA3 = THETA2#ClGO TO 3

5 THETA = THETA2ENDPRO

HEAD = THETA *180./ClRUDANG = DELTAC*180./Cl

83

7........... . . . . . . . . .

FILE: S0M2 DSL Al

DX = V*COS(THETA)DY = V*SIN(THETA)X = INTGRL(O.,DX)Y = INTGRL(O.,DY)

PRINT 125 ., RUDANG. HEAD, THECOM,X, TRAJE1,Y, TRAJE2CONTRL FINTIM=24000.,DLT ,DELS=2O.SAVE (Gl)12.,HEAD,THECOMSAVE (G2)12.,TRAJEI,YSAVE (G3)12.,TRAJE2,X

* -SAVE (G4)12.,RUDANGGRAPH (Gl/Gl,DE-TEK61S,PO=O,.5) TIME(LE=8.O,SC=500 .....

NI=1O,UN&-SEC'); .EAD(LE=9,NI=9,LO=37,SC=2.,UN='DEGREES') ..THECOM(LE=9,NI=9,Po=8. DLO=37,SC=2.,UN='DEGREES')

GRAPH (G2/G2.,DE=-TEK6l8,Po=o, .5) TIME(LE=8. ,SC=2000.,..NI=12,tJN='SEC) ...TRAJE1(LE=9,NI=9,LO=O,SC=6.,UN=-'MILES'),..Y(LE=9,NI=9,PO=8. ,LO=O,SC=6. ,UN='MILES')

GRA2H (G3/G3,DE=-TEK61S,PO=O, .5) TIME(LE=8.,SC=2000 ..... ..NI=12,UN& SEC') ....TRAJE2(LE=9,NI=9,LO=O,SC=g. ,UN='MILES') ..X(LE=9,NI=9,PO=8. ,LO=O,SC=-9. ,UN='MILES')

GRAPH (G4/G4,DE=-TEK618) TIME(LE=8. ,SC=300.,...N1=12,UN=-'SEC'),..RUDANG(LE=9,NI=9,Lo=-30,SC=7. ,UN-&DEGREES')

LABEL (01) SHIP HEADING AND DESIRED HEADING VS TIMELABEL (Gi) IN2-O.OO349 X=O K4=.OlLABEL (02) DESIRED TRAJECTORY ANDLABEL (G2) ACTUAL TRAJECTORY (Y) VS TIMELABEL (G3) DESIRED TRAJECTORY ANDLABEL (G3) ACTUAL TRAJECTORY (X) VS TIMELABEL (G4) RUDDER ANGLE VS TIMEEND

* . STOP

8~4

FILE: SOMi DSL Al

*THIS PROGRAM REFERS TO FIGURE 3.24 TO 3.27TITLE SIM4ULATION OF SHIP DYNAMICS (TEST OF FOLLOWING)PARAM Kl=4.6,K2=-.0434,K3-.5S88,K4=O.O1,K5=O.75,....

Pl=-.1222,P2=.1222,P3=-.52,P4-.52PARAN V--0.0038889*SPEED IN MILES/SECOND WH~ICH IS EQUIVALENT TO 14 KNOTS

PARAM EPSILX7=O.125,G=-.074PARAM L=2.5,1N2=O.00349CONST C1=3 .1415927NLEGEN YPATRH=-2O.,-30.,-1O.,-12.5,O.,Ol.,1O.,10.,20.,...18.,30.,2S.,40.,31.,SO.,36.,60.,40.,70.,43.,...80. ,45. ,90.,.46.5,100. ,48.NLFGEN XPATH--30. ,-20. ,-12.S,-1O. ,O. ,0. 10.,40.,18.,...20. .25. ,30. .31. ,40. .36.,50. ,40. .60.,43. .70.,..45.,80. ,46.S,90.,.48.,100.

DYNAMICTRAJE1 = NLEGEN(YPATN,X)TRAJE2 = NLEGEN(XPATS,Y)DELTAX = (X+EPSILX)-XDELTAY = NLFGEN(YPATH, X+EPSILX) -NLFGEN(YPATH, K)z = DELTAY/DELTAXZR = ATAN(Z)THECOM = ZR*180./Cl

DERIVATIVEINI = TEECOM*C1/180.E = IN1-THETA+TKETACEl = 10.*EFILTER = ZEROPL(0.,.04,O.4,E1)E2 = FILTER*K1 l-~E3 = E*K4E4 = INTGRL(O.,E3)DELREF = E2+E4DELDES = LIMIT(P3,P4,DELREF)DELTAl = DELDES-DELTACDELDOT = LIMIT(Pl,P2,DELTAl)DELDOC = K3*DELDOTDELTAC = INTGRL(0.,DELDOC)AMPLI1 = 2DTAAMPLI = G*AMPLI1COMP = ZEROPL(O.,.05,-.0037,AMPLI)CORREC = REALPL(O.,9.3,COMP)THETAl = lN2+CORRECTHETA2 = INTGRL(O.9599311.,THETAl)

PROCED THETA = COR(THETA2,C1)THETA3 = THETA2

3 IF(ABS(THETA3).LT.Cl) GO TO 5IF(THETA3.LT.0) GO TO 4THETA3 = THETA2-ClGO TO 3

4 THETA3 = THETA2+C1GO TO 3

S THETA = THETA2ENDPRO

HEAD = THETA *180./Cl'RUDANG = DELTAC*180./Cl

FILE: SOMi DSL Al

DX -V*COS(THETA)DY - V*SIN(THETA)x = INTGRL(l.,DX)Y = INTGRL(O.,DY)YERROR = TRAJEl-YXERROR = X-TRAJE2DS - XERROR*SIN(INl)N - DS**2+L**2P = SQRT(N)M = DS/PTEETC1 = ASIN(M)THETAC = KS*THETClT = THETAC*1SO./Cl

PRINT 125. ,RUDANG, HEAD, THECOM,X, TRAJE1,Y, TRAJE2 .DSCONTRL FINTIM=-20000. ,DELT=6. ,DELS=20.SAVE (Gl)12. ,HEAD,THECOMSAVE (G2)12.,TRAJE1,YSAVE (G3)12.,TRAJE2,XSAVE (G4)12.,RUDANGGRAPH (Gl/Gl,DE=-TEK6l8,PO=O, .5) TIME(LE=8.O,SC=2000.,..

NI=1O,UN=' SEC') ..EAD(LE=9,NI=9,LO=1O,SC=7. ,N'DEGREES') ..THECOM(LE=9,NI=9,PO=8. ,L010,SC-7. ,TN='DEGREES')

GRAPH (02/G2,DE=TEK618,PO=O, .5) TIME(LE=8. ,SC=2000....NI=12,UN-'SEC') ..TRAJEl(LE=9,NI=9,LO=OSC=6.,UN=-'MILES'),....Y(LE=9,NI=9,PO=8. ,LO=O,SC=6. ,UN-'MILES')

GRAPH (G3/G3,DE=-TEK6l8,PO=O, .5) TIME(LE=8. ,SC=2000.,..NI=12,UN='SEC'),..TRAJE2(LE=9,NI=9,LO=O,SC=8. ,UN='MILES'),..X(LE=9,NI=9,PO=8.,LO=O,SC=8.,UN=-'MILES')

GRAPH (G4/G4,DE=-TEK618,PO=O, .5) TIME(LE=8. ,SC=300.,..NI=12,tJN='SEC'),...RUDANG( LE=9 ,NI19, LO=-2S, SC=6. ,UN='DEGREES')

LABEL (Gi) SHIP HEADING AND DESIRED HEADING VS TIMELABEL (01) KS=.75 X1- IN2=0.00349LABEL (G2) DESIRED TRAJECTORY ANDLABEL (G2) ACTUAL TRAJECTORY (Y) VS TIMELABEL (G3) DESIRED TRAJECTORY ANDLABEL (03) ACTUAL TRAJECTORY (X) VS TIMELABEL (04) RUDDER ANGLE VS TIMEENDSTOP

86

LIST OF REFERENCES

1. Miguel Alvaro Marques Policarpc, Course and StationKe ngwt Automatic ship Control M.S.~ Thesisr'aa

Flotgrd-UAre bcnoolc R-1-90 eey~talifornia,December1980.

2. Beser Jacques and Parkinson Bradford W., itTheApplication of N4VSTAR Differential GPS in theCivilian Communit VJounal of the Institute ofNavigation, p. 107 ,Voi.29,N6.2,l9n2.

3. Jor ensen P.S, "Navstar/ Global Positioning System18- Satellite Constellations" I Journal of the Instituteof Navigation, p.89,Vol.27,No.zI980T

4. Davidson Kenneth S.M.,and Sch~ff Leonard I., "unnadCourse-Keeping Qualities , Transaction NAEp.179,Vol.54, 1946.

5. Nomoto K. ,Taguchi T. Honda K. and Hirano S., "On theSteerig Qu~alitiesof ship" Internation ShipbuildingProgress, pp.355-360,Vol .4,No.35,July £7.

6. Koyama T., "Inprovement of Coursg Stability by theSubsidi*ary Automatic control' nt ernationSjhipbuilding Progress, pp.126-l33,Voi.19,No.22,1,IL.

87

BIBLIOGRAPHY

Comstock,J.P.. Principles of Naval Architecture, The Societyof Naval Aciesaa Mrne n-gineers,t967.

David~on Kenneth S.M " On the Turning and Steering ofShips', Transactions 9*AME, pp.287-307,VoI.57,1949.

Schiff Leonard I.and Gimprich Marvin, "Automatic Steering ofShips lby proportion control", Transactions SNAME,pp.94- 18,Vol.57,1949.

88

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No. Copies1. Library,Code 0142 2

Naval Postgraduate SchoolMonterey, California 93943-5100

2. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22304-6145

3. Professor Geor e J. Thaler, Code 62Tr 5Department of lectrical andComputer Engineering,Naval Postgraduate SchoolMonterey, California 93943-5100

4. Professor Alex Gerba Jr., Code 62GzDepartment of Electrical andComputer Engineering,Naval Postgraduate SchoolMonterey, California 93943-5100

5. Department Chairman, Code 62RrDepartment of Electrical andComputer Engineering,Naval Postgraduate SchoolMonterey, California 93943-5100

6. Mr George CurryHENSCHEL9 Hoyt Drive, P.O. Box 30Newburyport, Mass 01950

7. Mr John CarterHENSCHEL9 Hoyt Drive, P.O.Box 30Newburyport, Mass 01950

8. LT. Sommart Vimuktananda27/40 Soi Suan luang,Klong-TunBangkok 10250Thailana

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