lf ship dynamics

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Control Eng. Practice,Vol. 4 , No. 3 , pp. 369-376, 1996Copyr igh t © 1996 E l s ev i e r Sc i ence L td

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I D E N T I F I C A T I O N O F D Y N A M I C A L L Y P O S I T I O N E D S H I P S

T.I. Fossen*, S.I. Sagatun** and A.J. Scrensen**

*Department of E ngineering Cybernetics, Norwegian University of Science an d Technology, No7034 Trondheim, Norw ay([email protected])

**ABB lndustri AS, Hasleveien 50, P.O. B ox 65 40 RodelCkka, N-0501 O slo, Norway

(Rece ived Augus t 1995 ; in f ina l fo rm Novem ber 1995)

A b s t r a c t . Toda y's model-based dy namic positioning (DP) system s require that the ship andthruster dynamics are known with some accuracy in order to use linear quadratic optimal controltheory. However, it is difficult to identify the ma them atical model of a dynam ically positioned (DP)ship, since the ship is not persistently excited un der DP. In addition, the ship param eter-estim ationproblem is nonlinear and multivariable, with only position and thruster state measurements avail-able for param eter estimation. The process and me asurem ent noise mu st also be modeled in order toavoid par am ete r drift due to environmental disturbances and sensor failure. T his article discusses anoff-line parallel extended Ka lm an filter (EKF ) algorithm utilizing two meas urem ent series in parallelto estim ate the par am eters in the DP ship model. Full-scale experiments with a supply vessel areused to demon strate the convergence and robustness of the proposed p aram eter estimator.

K ey W or ds . Dyna mic positioning system s, identification algorithms, full-scale sea trials, Kalm anfiltering, self-tuning control, marine systems, ship control.

1. I N T R O D U C T I O N

M o d e r n d y n a m i c p o s it i o n in g ( D P ) s y s t e m s a r eb a s e d o n m o d e l - b a s e d f e e d b a c k co n tr o l. T h e s t a t ee s t i m a t o r a n d c o n t r o l l a w a r e d e s i g n e d b y a p p l y -i n g a l o w - f r e q u e n c y ( L F ) m a t h e m a t i c a l m o d e l o ft h e s h i p m o t i o n s c a u s e d b y c u r r e n t s , w i n d a n d2 n d - o r d e r w a v e l o a d s , a n d a h i g h - f r e q u e n c y ( H F )m o d e l o f t h e l s t - o r d e r s h i p m o t i o n s c a u s e d b y l s t -o r d e r w a v e d i s t u r b a n c e s ; s e e ( F o ss e n , 1 9 9 4 ).

M o d e l - b a s e d c o n t r o l s y s t e m s u ti l iz i n g s t o c h a s t i co p t i m a l c o n t r o l t h e o r y a n d K a l m a n f i lt e r in g te c h -n i q u e s w e r e f i rs t e m p l o y e d w i t h t h e D P p r o b l e m

b y ( B a l c h e n et a l . , 1 9 7 6) . L a t e r e x t e n s i o n s a n dm o d i f i c a t i o n s o f t h i s w o r k h a v e b e e n r e p o r t e d b yB a l c h e n e t a l . ( 1 9 8 0 a , b ) , G r i m b l eet a l . ( 1 9 8 0 a ,b ) , F u n g a n d G r i m b l e ( 1 9 8 3 ) a n d S e e l i det a l .( 1 9 8 3 ) .

I n o r d e r t o a c h i e v e g o o d p e r f o r m a n c e o f t h e c o n -t r o l s y s t e m i t i s n e c e s s a r y t o h a v e a s u f f i c i e n t l yd e t a il e d m a t h e m a t i c a l m o d e l o f t h e s hi p . A B B I n -d u s t r i A S i n O s l o h a s m a r k e t e d a n e w s e l f - t u n i n gm o d e l - b a s e d D P s y s t e m b a s e d o n t h e r e s u l t s p r e -s e n t e d i n t h i s a r t ic l e . T h e i d e n t i f ie d m o d e l i s u s e da s b a s is f o r t h e c o n t r o l s y s t e m d e s i g n . T h i s s i m -

p l if ie s th e t u n i n g o f t h e c o n t r o l l a w. T h e c o n t r o ls y s t e m d e s i g n i s d i s c u s s e d i n d e t a i l b y ( S c r e n s e net a l . , 1995) .

2. S H I P A N D T H R U S T E R M O D E L S

T h i s s e c ti o n d e s c ri b e s th e m a t h e m a t i c a l m o d e l o ft h e t h r u s t e r s a n d t h e L F m o t i o n o f t h e s h i p .

2.1. T h r u s t e r M o d e l

M o s t D P s h i p s u s e t h r u s t e r s a n d m a i n p r o p e l l e r st o m a i n t a i n t h e i r p o s i t i o n a n d h e a d i n g . T h et h r u s t f o r c e o f a p i t c h - c o n t r o ll e d t h r u s t e r c a n b ea p p r o x i m a t e d b y :

F(n,p)=

K(n) IP- P01 ( P - P0)(1 )

w h e r e t h e f o r c e c o e f f i c i e n tK ( n ) i s a s s u m e d t o b ec o n s t a n t f o r c o n s t a n t p r o p e l l e r r e v o l u t i o n n , P i st h e " t r a v e l e d d i s t a n c e p e r r e v o l u t i o n " , D i s t h ep r o p e l le r d i a m e t e r a n d :

p = P / D (2 )

i s t h e p i t c h r a t i o . P 0 is p i t c h r a t i o o f f - s e t d e f i n e ds u c h t h a t p = P0 y ie l d s z e r o t h r u s t , t h a t i s :

F ( n , p 0 ) = 0 . ( 3 )

T h r u s t F o r c e s a n d M o m e n t . T h e t h r u s t f o r c e sa n d m o m e n t v e c t o r ~" E R 3 ( s u rg e , s w a y a n d y a w )

36 9



370 T.I . Fossenet al.

~ . = : ~ L I ~ _ _ ~ _ ' _ - . - 7 < - S - . -z - w - . . . . J ~ ~ " . . . . ' -

F i g . 1 . P i c t u r e s h o w i n g t h e s u p p l y v e s s e l w h i c h w a s u s e d d u r i n g t h e s e a t r i a l s i n t h e N o r t h S e a ( L = 7 6 . 2 m ) .

m a i n p r o p e l l e r ( k N )

. o o . . . . i . . . . . . . . ! . . . . . . . . i . . . . . . . .

2 0 0 -- -i . . . .. . . . i . .. . . .. .. . . - . - ~ . . . ..

) Y ~ e

- 1 - 0 . 5 0 0 . 5 1P / D

2 0 0t u n n e l t h r u s t e r ( k N )

1 5 0 . - : . . . .

1 O 0i i i i ~ i

5 0 . . . . ::. . . . . :,. . . . . . . . . . . . . ~ . . . . . . i

0

- 5 0

- 1 0 0

- 1 5 0

- 2 0 0- 1 - 0 . 5 0 0 . 5

P / D

F i g . 2 . E x p e r i m e n t a l l y m e a s u r e d t h r u s t ( a s t e r is k s ) a n d t h r u s t e r m o d e l a p p r o x . E q . ( 1 ) v e r s u sp = P / D . L e f tp l o t: F ( 1 2 2 , p ) - - 3 7 0 p i p [ a n d F ( 1 6 0 , p ) = 6 5 5 p [ p [. R ig h t p l o t: F ( 2 3 6 , p ) = 1 3 7 p[ p [. P r o p e l le r r e v o lu t i o ni s in rpm.

f o r t h e s u p p l y v e s se l in F i g u r e 1 c a n b e w r i t t e n :

r = T K u (4)

w h e r e u E l~ r i s a c o n t r o l v a r i a b l e d e f i n e d a s :

u = [[Pl - P lo[ (P l - PlO), [P2 - P2o[(P2 - P2 o),

• . , [P r - P ro l (Pr - P ro) ] T (5 )

w h e r e Pio ( i = 1 . .. r ) a r e t h e p i t c h r a t i o o f f - s e t s f o rt h r u s t e r N o . i a n d r is t h e m a x i m u m n u m b e r o ft h r u s t e r s .

Thru s t Fo rce Coe ff i c ien t Ma t r ix .T h e t h r u s t f o rc ec o e f fi c ie n t m a t r i x K i s a d i a g o n a l m a t r i x o f t h r u s tf o r c e c o e f f i c i e n ts d e f i n e d a s :

K = d i a g { K l ( n l ) , K 2 ( n 2 ) , ...,K r ( n r ) } (6 )

w h e r e n i ( i = 1 . .. .r ) i s t h e p r o p e l l e r r e v o l u t i o n o fp r o p e l l e r n u m b e r i. T h e th r u s t f o r c e sK i ( n i ) u ia r e d i st r i b u te d t o t h e s u rg e , s w a y a n d y a w m o d e sb y a 3 x r t h r u s t e r c o n f i g u r a t i o n m a t r i x T .

Thrus t e r Conf igu ra t i on Ma t r ix .C o n s i d e r t h e s h i pi n F i g u r e 1 , w h i c h i s e q u i p p e d w i t h t w o m a i n p r o -p e ll e rs , t h r e e t u n n e l t h r u s t e r s a n d o n e a z i m u t ht h r u s t e r w h i c h c a n b e r o t a t e d t o a n a r b i t r a r y a n -g l e a . T h e c o n t r o l v a r i a b le s a r e a s s ig n e d a c c o r d -i n g t o :

U l =

U 2 - -

U 3

U 4 =

U 5

U 6 =

p o r t m a i n p r o p e l l e rs t a r b o a r d m a i n p r o p e l le ra f t t u n n e l t h r u s t e r Ia f t t u n n e l t h r u s t e r I Ib o w t u n n e l t h r u s t e rb o w a z i m u t h t h r u s t er .

T h e f o l lo w i n g t h r u s t e r c o n f i g u r a t i o n m a t r i x i s o b -t a i n e d :

T =1 1 0 0 0 cos c~ l0 0 1 1 1 s in~ ) (7 )

/ 1 - - / 2 - - 1 3 - - / 4 / 5 /6 s i n a

w h e r e l i ( i = 1 . .. 6 ) a r e t h e m o m e n t a r m s i n y a w.I t i s a l s o s e e n t h a t 12 = 11 ( s y m m e t r i c a l l o c a t i o n o f



Identification o f Dyna mically Positioned Ships 371

m a i n p r o p e l le r s ). T h e t h r u s t d e m a n d s a r e d e fi n e ds u c h t h a t p o s i t i v e t h r u s t f o r c e / m o m e n t r e s u l ts inp o s i t i v e m o t i o n a c c o r d i n g t o t h e v e s s e l ' s p a r a l l e la x i s s y s t e m , d e f in e d s u c h t h a t p o s i t i v e x - d i r e c t i o ni s f o r w a r d s , p o s i t i v e y - d i r e c t i o n i s s t a r b o a r d a n dp o s i t i v e z - d i r e c t i o n i s d o w n w a r d s . T h e o r ig i n isl o c a t e d i n t h e c e n t e r o f b u o y a n c y.

f i n e d a c c o r d i n g t o ( 9 ) s u c h t h a t :

r o l l = m - X a m 2 3 =m x G - - Y ÷m22 = m - Y9 m33 = I~ - N÷.

(12 )

2.3. K i n e m a t i c s

T h e k i n e m a t i c e q u a t i o n o f m o t i o n f o r a s h i p is :

2 .2 . L F S h ip D y n a m i c s

T h e L F s h ip m o d e l i n s u rg e , s w a y a n d y a w c a n b ed e s c r i b e d b y ( F o s s e n 1 9 9 4 ) :

/7 = J ( w ) v ( 1 3 )

w h e r e W = [ x, y, ¢ ] T a n d J ( w ) i s a r o t a t i o n m a t r i xd e f i n e d a s :

M I ) + C ( v ) v + D ( u - u c ) = ~" + w ( 8 )

w h e r e v = [ u ,v, r] T d e n o t e s t h e L F v e l o c i t y v e c -t o r , v c = [uc,vc,rc] T i s a v e c t o r o f c u r r e n t v e l o c -i ti e s, 7 - i s a v e c t o r o f c o n t r o l f o r c e s a n d m o m e n t sa n d w = [Wl ,W2,W3] Ti s a v e c t o r o f z e r o - m e a n

G a n s s i a n w h i t e n o i s e p ro c e s s e s d e sc r i b in g u n m o d -e ll ed d y n a m i c s a n d d i s t u r b a n c e s . N o t i c e t h a t r ,d o e s n o t r e p r e s e n t a p h y s i c a l c u r r e n t v e l o c i t y, b u tc a n b e i n t e r p r e t e d a s t h e e f f ec t o f c u r r e n t s i n y a w.T h e c u r r e n t s t a t e s a r e u s e fu l in t h e p a r a m e t e r e s -t i m a t o r s in c e t h e y r e p r e s e n t s l o w l y - v a ry i n g n o n-z e r o b i a s t e r m s .

c o s ¢ - s i n e 0 ]d ( r / ) = s i n e c o s ¢ 0 .

0 0 1(14)

3 . O F F - L IN E P A R A M E T E R E S T I M A T O R

T h e o f f- li ne p a r a m e t e r e s t i m a t o r i s b a s e d o n t h es t a t e a u g m e n t e d e x t e n d e d K a l m a n f i l t e r ( E K F ) .

3 .1 . S t a t e A u g m e n t e d E x t e n d e d K a l m a n F i l t e r

C o n s i d e r t h e f o l lo w i n g n o n l i n e a r s y s t e m :

T h e i n e r t i a m a t r i x i n c l u d i n g h y d r o d y n a m i c a d d e dm a s s t e r m s i s a s s u m e d t o b e p o s i t i v e d e f i n i t eM = M T > 0 f o r a d y n a m i c a l l y p o s i t i o n e d s h i p ,w h e r e a s D > 0 is a s t r i c t l y p o s i t i v e m a t r i x r e p r e -s e n t in g li n e a r h y d r o d y n a m i c d a m p i n g . N o n l i n e a rd a m p i n g i s a s s u m e d t o b e n e g l i g ib l e f o r s t a t i o n -k e e p i n g o f s h i ps , w h e r e a s t h e a s s u m p t i o n o f s ta r -b o a r d a n d p o r t s y m m e t r y i m p li es t h a t M a n d Dc a n b e w r i t t e n :

x ( k + 1) = / ( x ( k ) , 'o , ( k ) , 0 ( k ) ) + W l ( k ) (1 5)0 ( k + 1 ) = 0 ( k ) + r / ( k ) ( 16 )

w h e r e x E l ~ n is t h e s t a t e v e c t o r , u E R r i s t h ei n p u t v e c t o r, 8 E l ~ p i s t h eu n k n o w n p a r a m e t e r

v e c t o r t o b e e s t i m a t e d a n d w l , ~7 E R n a r e z e r o -m e a n G a n s s i a n w h i t e n o i se p r o ce s s e s. T h i s m o d e lc a n b e e x p r e s s e d in a u g m e n t e d s t a t e - s p a c e f o r ma s :

m - X u 0 0 ]M = 0 m - Y ~ m x G - Y ÷ (9 )

0 r n x a - - Y ÷ I z - N ÷

0 ]D = o -Yo -Y~ . (10)

0 - - N v - N r

T h e C o r io l is a n d c e n t r i f u g a l m a t r i x C ( v ) i s i n-

c l u d ed i n t h e m o d e l t o i m p r o v e t h e c o n v e r g e n c eo f t h e p a r a m e t e r e s t i m a t o r . M o r e o v e r , t h is m a -t r i x m a y b e s i g n i fi c a n t fo r a s h i p o p e r a t i n g a ts o m e sp e e d , w h e r e a s C ( v ) = 0 f o r a sh i p a t re s t .I t s h o u ld b e n o t e d t h a t i n c lu s io n o f C ( v ) i n t h em o d e l w i ll n o t i n c r ea s e t h e n u m b e r o f p a r a m e t e r st o b e e s t i m a t e d , s i n c e C ( v ) i s o n l y a f u n c t i o n o ft h e e l e m e n t s m i j o f t h e i n e r t i a m a t r i x ; s e e T h e -o r e m 2 . 2. o n p a g e 2 7 i n (F o s s e n , 1 9 9 4 ) . I n f a c tM = ( m ~ j } y i e l d s :

C ( v ) -=

[ 0 00 0 m l l U ( 11 )- ~ 2 v + - ~ 3 r - m l l u 0

w h e r e t h e n o n - z e r o e le m e n t s m i j =- m j i a r e d e -

~ ( k + 1 ) = ~ : ( ~ ( k ) , u ( k ) ) + w ( k ) ( 1 7 )

w h e r e ~ = [x T, o T ] T i s t h e a u g m e n t e d s t a t e v e c -to r, w = [Wl ,~T]Tand:

~ : ( ~ ( k ) , u ( k ) ) = [ S ( x ( k l ' u ( k l ' 0 ( k l )0 ( k ) ( 1 8 )

F u r t h e r m o r e , i t i s a s s u m e d t h a t t h e m e a s u r e m e n te q u a t i o n c a n b e w r i t t en :

z ( k ) = n ( ~ ( k ) ) + v ( k ) (19)

w h e r e z E R m a n d m i s t h e n u m b e r o f s e n-s o rs . T h e d i s c r e t e - ti m e e x t e n d e d K a l m a n f i lt e ra l g o r i t h m i n Ta b l e 1 c a n t h e n b e a p p l i e d t o e s t i-m a t e ~ = [x T, o T ] T i n ( 17 ) b y m e a n s o f th e m e a -s u r e m e n t ( 1 9 ) . F o r d e t a i l s o n t h e i m p l e m e n t a t i o ni s s u e s , s e e ( G e l b et al . , 1 9 8 8 ) .

3 .2 . O ff - L i n e E K F f o r P a r a ll e l P ro c e s si n g

I n o r d e r t o i m p r o v e t h e c o n v e r g e n c e a n d p e r f o r -m a n c e o f th e p a r a m e t e r e s t i m a t o r, t h e s a m e q u a n -



37 2 T.I . Fossen e t al .

Ta b l e 1 S u m m a r y o f d i s c r e te - t im e e x te n d e d K a l m a n f il te r ( E K F ) .

S y s t e m m o d e lM e a s u r e m e n t

I n i t i a l c o n d i t i o n s

S t a t e e s t i m a t e p r o p a g a t i o n

E r r o r c o v a r i an c e p r o p a g a t i o nG a i n m a t r i x

S t a t e e s t i m a t e u p d a t eE r r o r c o v a r ia n c e u p d a t e

Def i n i t i on s

{ ( k + 1 )= . ~ ( { ( k ) , u ( k ) ) + I 'w ( k ) ;z ( k ) = n ( ~ ( k ) ) + v ( k ) ;

w ( k ) , , ~ N ( O , Q ( k ) )v ( k ) , , ~ N ( O , R ( k ) )

~ ( 0 ) = ~ 0 ; p ( 0 )= P o~(k + 1) = ~ ( ~ ( k ) , u ( t ) )

P ( k + 1 ) = 4 ~ (k ) P ( k )~ T ( k ) + I ' ( k ) Q ( k )r r ( k )K ( k ) = - P ( k ) H T ( k ) [ H ( k ) - P ( k ) H T ( k ) +R ( k ) ] - '

~(k) = -~(k) + K (k )[z (k ) - 7 - / (~(k) ) ]P ( k ) = [ I - K ( k ) H ( k ) ] - P ( k ) [ I - K ( k ) H ( k ) ] T

+ K ( k ) R ( k ) K T ( k )o . ~ • o T - I .

t i t y c a n b e m e a s u r e d N > 2 t i m e s f o r d i ff e r-e n t e x c i t a t i o n s e q u e n c e s . M o r e o v e r , l e t th e i n p u tu i E l ~ r c o r r e s p o n d t o t h e s t a t e v e c t o r x i E 1~n

a n d m e a s u r e m e n t v e c t o r z i e R r n f o r (i = 1 . . .N ) .U n d e r t h e a s s u m p t i o n o f c o n s t a n t p a r a m e t e r s , t h ep a r a m e t e r v e c t o r 0 E IRp w i l l b e t he s am e fo r a l lt h e s e s u b s y s t e m s . T h i s c a n b e e x p r e s s e d m a t h e -m a t i c a l l y a s :

X l ( k + l ) =

x 2 ( k + l ) =

X N ( k + l ) =0 ( k + l ) =

f (X l(k) , U l (k) , 0 (k) ) -b W l(k)f(x 2( k) , u2(k ) , 0(k)) -Jr-?M2(k)

f(XN(k),UN(k),O(k)) +W)N(k)O ( k ) + n ( k ) ( 2 0 )

w i t h m e a s u r e m e n t s :

z , ( k ) =

z 2 ( k ) =

Z N ( k ) =

h l ( X l ( k ) , 0 ( k ) ) -4 - V l (] e)h2(x2(k) , O(k) ) + v2(k)

hN(X N(k), O (k)) q- VN(k).(21)

H e n c e , t h i s s y s t e m c a n b e w r i t t e n i n a u g m e n t e ds t a t e - s p a c e f o r m a c c o r d i n g t o :

~ ( k + l ) = . ~ ' ( ~ ( k ) , u ( k ) ) + w ( k ) (22)

z ( k ) = 7 " ~ ( ~ ( k ) ) + v ( k ) (23)

w h e r e u = [ u T , . . . , u ~ ] T, z = [ z T , . . . , z ~ ] T, ~ =[ X ~ , . . . , X ~ , o T ] Ta n d :

~ ' ( ~ ( k ) , u ( k ) ) =

n ( ~ ( k ) ) =

f(Xl(k) ,~1 (k), 0(k ))f ( x 2 ( k ) , u 2 ( k ) ,O(k))

y(xN(k),uN (k), O(k))o(~)

h(xl(k),O(~))

h(~2(k),O(k))

h(XN(k),O(k))

( 2 4 )

( 2 5 )

I t is o b s e r v e d t h a t d i m x =N n + p ,d i m u = N ra n d d i m z = N m . I t i s t h e n e v i d e n t t h a t m o r ei n f o r m a t i o n a b o u t t h e s y s t e m i s o b t a i n e d b y u s i n g

m u l t i p l e m e a s u r e m e n t s e q u e n c e s . I n c r e a s e d in f o r -m a t i o n i m p r o v e s p a r a m e t e r i d e n t i f i a b i l i t y a n d r e -d u c e s t h e p o s s ib i l i ty f o r p a r a m e t e r d r i f t . H o w e v e r ,i t s h o u l d b e n o t e d t h a t p a r a l l e l p r o c e s si n g im p l i e st h a t t h e p a r a m e t e r e s t im a t i o n m u s t b e p e r f o r m e doff - l ine .

F o r s h i p a p p l i c a t i o n s s i g n i f ic a n t p e r f o r m a n c e i m -p r o v e m e n t h a s a l r e a d y b e e n r e p o r t e d f o r N = 2 ;s ee (Abk owi t z , 1975 , 1980 ; Hw ang , 1 980 ).

i 1 1" Yl

p2 Eq.3 u 2 SHIP - = v2

M i5k

Fig . 3 . Para l le l conf igura t ion of EK F for N = 2 .

4 . I D E N T I F I C A T I O N O F A S U P P LY V E S S E L

F u l l - s c a l e e x p e r i m e n t s w i t h t h e s u p p l y v e s s e l i nF i g u r e 1 w i ll b e u s e d t o d e m o n s t r a t e t h e c o n v e r -g e n c e o f t h e p r o p o s e d p a r a m e t e r - e s t i m a t i o n a lg o -r i t h m .

4 .1 . Sys tem Iden t i f i ca t ion Mode l

T h e s y s t e m i d e n t i f i c a t i o n m o d e l i s b a s e d o n t h em a t h e m a t i c a l m o d e l p r e s e n t e d i n S e c t i o n 2 . A s -s u m i n g n o e n v i r o n m e n t a l d i s tu r b a n c es , a d y n a m -i c a l l y p o s i t i o n e d s h i p c a n b e d e s c r i b e d b y t h ef o l lo w i n g n o n - d i m e n s i o n a l m o d e l ( B i s - s y s te m ) i nsu rge , sw ay and yaw ( s ee p age 178 i n (Fo s sen ,



Identification of Dynamically Positioned Ships 37 3

1994)) :

M " i / ' + C " ( t / ' ) v " + D " v " = T " K " u "(26)

w h e r e

M "

D " [1 - X ~ 0 0 ]

0 1 - Y! ' x" Y" (27)" y y ~ - - , ,

- - k z - - I V ÷x G r

= o - v ~ ' , - W ' • ( 2 8 )0 -- N v" - N r"

T h e t h r u s t e r c o n f i g u r a t i o n m a t r i x w a s c o m p u t e dto b e :

u 3 , u 4 a n d u s . Tw o m a n e u v e r s s h o u l d b e p e r -f o r m e d ; s e e F i g u r e 6 .

( 3 ) a z i m u t h t e s t : t h e l a st t e s t i nv o lv e s r u n n i n gt h e a z i m u t h t h r u s t e r u 6 a l o n e. T w o m e a s u r e -men t s e r i e s a r e r equ i r ed ; s ee F igu re 7 .

4 .3 . I m p l e m e n t a t i o n I s s u e s

Th i s imp l i e s t ha t a t l e a s t 6 s ea tr i a l s mus t be pe r-fo rm ed fo r N - - 2 . T he f i r s t t wo s ea tr i a l s a r e u sedt o i d e n t if y t h e p a r a m e t e r s K ~ ' = K ~ ' a n d X " i nt h e d e c o u p l e d s u rg e e q u a t i o n :

1.0000 1.0000 0

T" = 0 0 1.0000

0 . 0 4 7 2 - 0 . 0 4 7 2 - 0 . 4 1 0 8

0 0 0 ]1.oooo 1 .oo oo 1 .oooo (29)

-0 .3858 0 .455 4 0 .3373

wh e rea s K " = d i ag {K l ' , K~ ' , K~ ' , K~ ' , K~ ' , K~ ' } i st h e u n k n o w nm a t r i x t o b e e st i m a t e d . I n a d d i ti o nt o t h i s i t w i ll b e a s s u m e d t h a t D " i su n k n o w n .An a p r i o r i e s t ima t e o f M ' 1 i s c a l cu l a t ed by ap -p l y i n g s e m i - e m p i r i c a l m e t h o d s . F o r m o r e d e t a i lsa b o u t t h e c o m p u t a t i o n o f t h e a d d e d m a s s d e r iv a -t ives i , , , (Fa l t i nsen , 1990) .t~ , Y~ , N r an d Y~', seeT h e i n e r t i a m a t r i x M " w a s c o m p u t e d t o b e :

1 . 1 2 7 4 0 0 ]M " = 0 1 .890 2 -0 .0744 . ( 30 )

0 -0.0744 0.127 8

H e n c e K '1 a n d D " a r e t h e o n l y r e m a i n i n gu n-k n o w n m a t r i c e s i n t h e D P m o d e l ( 2 6 ) .

4 .2 . Sea Tr ia ls

I n o r d e r t o i m p r o v e t h e c o n v e rg e n c e o f t h e p a -r a m e t e r e s t i m a t o r i t i s p r o p o s e d t o u s e s e v er a l of f-l i n e m e a s u r e m e n t s e r i e s g e n e r a t e d b y a n u m b e r o fc a r e f u l l y p r e d e f i n e d m a n e u v e r s . F o r i n s ta n c e , i ti s a d v a n t a g e o u s t o d e c o u p l e t h e s u rg e m o d e f r o mt h e s w a y a n d y a w m o d e s i n o r d e r t o i m p r o v e t h ec o n v e r g e n ce o f t h e p a r a m e t e r e s t i m a t o r . T h i s i sm o t i v a t e d b y t h e b l o c k d ia g o n a l s t r u c t u r e o f M "a n d D ' .

Decoup led Sh ip Maneuve r s . T h e f o l l o w i n g t h r e ed e c o u p l e d s h i p m a n e u v e r s a r e p r o p o s e d :

( 1 ) u n c o u p l e d s u r g e : t h e sh i p i s o n l y a l lo w e dt o m o v e i n s u rg e ( c o n s t a n t h e a d i n g ) b y m e a n so f t h e m a i n p r o p e l l e r s u l a n d u 2 . T h e h e a d -i n g i s c o n t r o l l e d b y m e a n s o f o n e o f t h e b o wt h r u s t e r s . A t l e a s t t w o m a n e u v e r s s h o u l d b e

p e r f o r m e d ; s e e F i g u r e 5 .( 2 ) c o u p l e d s w a y a n d y a w : t h e s hi p s h ou l dp e r f o r m t w o c o u p l e d m a n e u v e r s i n s w a y a n dy a w b y m e a n s o f t h e t h r e e t u n n e l t h r u s t e r s ,

1 1 1 " I f 1 1 1 1 I I I 1 I I I I

- X~) u - = +K1 u l K2 u2,,u (31)

} " ---- u " ( 3 2 )

w h e r e X ~ / i s c o m p u t e d b y u s i n g s t r i p t h e o r y( F a l t i n se n , 1 99 0 ) . T h e p a r a m e t e r v e c t o r c o r -

r e s p o n d i n g t o t h i s s y s t e m i s d e n o t e d a s 0 ~ ' =^ II

t-*lrr"", LuJY'0T" T he es t im at ed pa ra m et er ve ct or 01i n su rge i s f rozen a n d u s e d a s i n p u t f o r t h e s e c -o n d s y s t e m i d e n t i f i c a t io n s c h e m e ( S I 2 ), t h a t i s t h ecoup l ed sway and yaw iden t i f i c a t i on . S imi l a r l y,t h e o u t p u t f r o m t h e s e co n d p a r a m e t e r - e s t i m a t i o n

^ I I

s cheme 02 i s f rozen a n d u s e d a s in p u t f o r t h el a s t p a r a m e t e r - e s t i m a t i o n s c h e m e ( S I 3 ) , t h a t i s^ II

0 3 . T h e l a s t s c h e m e i s u s e d t o e s t i m a t e o n l y o n ep a r a m e t e r , t h a t i s 0 N = K ~ ' w h e r e a s t h e s e c o n dp a r a m e t e r - e s t i m a t i o n s c h e m e i s u s e d t o e s t i m a t et h e c o u p l i n g t e r m s i n s w a y a n d y a w ; s e e F i g u r e4 . I n t h e la s t t w o p a r a m e t e r - e s t i m a t i o n s c h e m e s

U l u 2 u 3

I Yl I Y2 I Y3

Fig . 4 . Decoupled parameter es t imat ion in te rms ofthree sys tem ident i f ica t ion schemes SI1-SI3 .

t h e s h i p i s c o m m a n d e d t o c h a n g e h e a d i n g d u r i n gt h e m a n e u v e r s , w h i c h i m p l i e s t h a t t h e n o n l i n e a rk i n e m a t i c e q u a t i o n :

/ / " = J " ( v " ) # ' ( 33 )

whe re ~ /" = Ix " , y " , ¢ , , ]T, shou ld be u sed t o ge th e rw i t h t h e d y n a m i c m o d e l ( 26 ) . H e n c e t h eu n k n o w np a r a m e t e r v e c t o r c o r r e s p o n d i n g t o s e a t r ia l 2 is0' ' = [y" , y " , , , , , , , K~,]T~ ,, N ' , K 3 , . I n t h i s e x a m p l et h e t u n n e l t h r u s t e r s a t t h e s t e r n a r e o f s a m e t y p e

(K~ ' = K~ ' ) . I t is co nv en i en t t o r ewr i t e ( 26 ) and(33 ) i n t e rms o f t he ve s se lm o m e n t u m :

h " - - M " v " ( 34 )



374 T.I . Fossene t a l .

1 5 0 0s u r g e p o s i t i o n [ m ]

1 0 0 0 .

5 0 0

0 ~ . . . . . . . . . . . . . . . . . . . . . .

- 50 ~ On 2 0 0 4 0 0 6 0 0

1 3 5

s a m p l e s

y a w a n g l e [ d e g ]

1 3 4

1 3 3

1 3 2

1 3 1

1 3 C2 0 0 4 0 0 6 0 0

s a m p l e s

4s u r g e v e l o c i t y [ m / s ]

3

2

1

0

-10

1 0 0

5 0

0

- 5 0

- 1 0 ( ~

2 0 0 4 0 0s a m p l e s

P / D m a i n p r o p e l l e r s [% ]

6 0 0

2 0 0 4 0 0 6 0 0s a m p l e s

Fig . 5 . Sea Tr ia l 1 : Fu l l - sca le exper iment wi th a supp ly vesse l (uncoupled surge) .

x y - p l o t [ m ]5 0 2 5 0

0

- 5 0

- 1 0 ( ~ ' 0 0 5 0 1 0 0 1 5 0

1 0 0P / D s t e r n t h r u s t e r s [ % ]

5 0

0

- 5 0

- 1 0 C 2 0 0 4 0 0 6 0 0s a m p l e s

y a w a n g l e [ d e g ]

2 0 0

1 5 0

1 0 U0 -- 2 0 0 4 0 0 6 0 0

1 5 0

s a m p l e s

P / D b o w t h r u s t e r [% ]

1 0 0 [ . . . .

: : : o 2 0 0 4 0 0 6 0 0s a m p l e s

F i g . 6 . S e a Tr ia l 2 : F u l l - sc a l e e x p e r i m e n t w i t h a s u p p l y v e s se l ( c o u p l e d s w a y a n d y a w ) .

x y - p l o t [ m ] y a w a n g l e [ d e g ]1 0 2 0 0

0

- 1 0

- 2 0

-~_~0 0 5 0 1 0 0

y a w a n g l e r a t e [d e g / s ]

0 . 5

0

- 0 . 5

- 1 . . . .

- 1 . ~4 0 0 6 0 0

s a m p l e s

1 5 0

1 O 0

5 0

%5 0 2 0 0 4 0 0 6 0 0s a m p l e s

~ l m u t h t h r u s t e r [% ]100''" 1- 5 0 f

- 1 O 0 "

- t % i i 6 0 0s a m p l e s

Fig . 7 . Sea Tr ia l 3 : Fu l l - sca le exper iment wi th a supp ly vesse l ( az imuth t es t ) .



Identification of Dynam ically Positioned Ships

A h e l e m e n t s

o . 4 ! ! , !

o .2 . . . . . . . . . . . . . , . . . . . . . . . . . . .

• i i ~

0

- 0 . 2 -

i : :

- 0 . 4 I , , I1 O 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

s a m p l e s

x 10 .3 K e l e m e n t s2 0 T " T 1 I I"

1 5

1 0

5

O

i i- 5 . i

0 1 O 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0

s a m p l e s

37 5

tt t,Fig . 8 . Parameter es t imates 01-09 versus t ime for the supply vesse l .

( 2 3 ) b e c o m e s :

z l ( k ) = H 1 x l ( k ) + v l ( k )

z 2( k) - - H 2 x 2 ( k ) + v 2 ( k ) .

a n d a m o m e n t u m b i as te r mbh w h i c h y i e l d s t h ef o l l o w i n g m o d e l :

( 4 4 )• H

h + C h ( h " ) h " = (45)

A ~ ( O " ) h " + T " K " ( O " ) u " + bh + W h(35 )

• = ' " - h "f ' J ' ( r f ' ) M 1 ( 36 )

bh = wb (37 )

/ ¢ ' = w 0 ( 3 8 )

H e r e W h , W b a n d w e a r e z e r o - m e a n G a u s s i a n

w h i t e n o i s e p r o c e s s e s ,bh i s a s l o w l y - v a r y i n g p a -r a m e t e r r e p r e s e n t i n g u n m o d e l l e d d y n a m i c s a n d [O'x' 0 0 ~ ]e n v i r o n m e n t a l d i s t u r b a n c e s , a n d0" i s t he pa ra m - A~ = 0 0~' ( 46 )e t e r v e c t o r t o b e e s t i m a t e d . T h e n e w m a t r i c e s i n 0 0~' 0"5the m ode l a r e de f ined acco rd ing t o : g " = d iag{O~ ', 0~ ' , 0~ ' , 0~ ' , 0~ ' , 0~ '} . ( 47 )

A h = - D M (39 )

C h (h ) = C ( M - l h ) . (40 )

S i n c e M i s a s s u m e d t o b e k n o w n w i t h s u f f ic i e n ta c c u r a c y, t h e o n l y u n k n o w n q u a n t i t i e s i n ( 3 5 ) -(38 ) a r e A h a n d K . T h e m a i n m o t i v a t i o n f o r

u s i n g t h e m o m e n t u m e q u a t i o n i n s t e a d o f t h e s t an -d a r d d y n a m i c e q u a t i o n s o f m o t i o n i s t h e i m p r o v e d ^ ,, [ - 0 . 0 3 1 8 0 0 ]p e r f o r m a n c e o f t h e s t a t e a n d p a r a m e t e r e s t i m a t o r . A h = [ 0 -0 .06 02 0 .0618 J (48)M o r e o v e r e s t i m a t i o n o f t h e s t a t e s h = M v, y a n d 0 - 0 .0 0 7 5 - 0 .2 4 5 4bh , t o g e t h e r w i t h t h e p a r a m e t e r v e c t o r 8 , i s e a s -i e r t o p e r f o r m t h a n e s t i m a t i o n o f v, ~?,bh a n d 0 .H e n c e , t h e r e s u l t i n g m o d e l c a n b e w r i t t e n :

x ~ ( k + 1) = f ( X l ( k ) , ' U l ( k ) , O ( k ) ) +W l ( k ) ( 41 )

x e ( k + 1) = f ( x 2 ( k ) , u 2 ( k ) , O ( k ) ) + w 2 ( k )( 42 ) D " a " a z " ' - I= - - h - - - ( 5 0 )

O(k + 1) = O(k) + w o(k) (43)

"h T T bT ITwh ere x~ = [ i , r / i , h iJ , W~ =[Wm,wbT]T( i = 1 , 2 ) a n d w i t h t h e o b v i o u s d e f i n i t io n o f f . ^ [ 0 .0 35 8 0 0 ]I f p o s i t io n ( x , y ) a n d h e a d i n g ( ¢ ) a r e m e a s u r e d , D " = 0 0 .1183 -0 .01 24 . (51)

0 -0 .00 41 0 .0308

4.4 . Experimental Results

S i x m a n e u v e r s w i t h t h e s u p p l y v e s s e l w e r e u s e d t oe s t im a t e t h e D P m o d e l• T h e u n k n o w n p a r a m e t e r

= 0 tt] T i s o rg a n i z e d a c c o r d i n gec to r 0" [0~', .. ., 9J

to :

T h e p a r a m e t e r e s t i m a t e s f o r t h e o ff - li n e p a r a l l elc o n f i g u r at i o n o f t h e E K F a l g o r i t h m a r e s h o w n i nF i g u r e 8 , w h i l e t h e s t e a d y - s t a t e n u m e r i c a l v a l u e sa r e g i v e n b e l o w.

Identified Momentum Equation.

I£ " = lO -3d ia g{9 .3 , 9 .3 , 2 .0 , 2 .0 , 2 .8 , 2 .6} . ( 49 )

F r o m ( 3 9 ) i t is s ee n t h a t :

S u b s t i t u t i n g ( 3 0 ) a n d ( 4 8 ) i n to ( 5 0 ) , y i e l d s:



376 T.I. Fossenet al.

Ident i f ied Sta te-Space Model . T h e e s t i m a t e dmod e l (35 ) can be r e la t ed to the D P con t ro l mod e lb y a s s u m i n g t h a tC h ( h ) = O . Hence :

/ / ' = A " u 't + B r " ( 5 2)

w h e r e A " " 1 ̂ u u B u M " - I .M - A h M a n d = T h e

numerical values are :

^, , [ -0.031 8 0 0 ]A --- 0 -0.06 28 -0.003 0 (53)

0 -0 .0045 -0 .2428

A,, [B =

0 . 0 0 8 2 0 . 0 0 8 2 00 .0001 -0 .0001 0 .00080.0035 -0 .0035 -0 .0059

0 0 00 .0008 0 .002 0 0 .0017

-0 .0055 0 .011 3 0 .0079.(54)

The non-d imens iona l e igenva lues o f A " a re :

(Yaw) ~ = -0 .2 42 9(Sway) AN = - 0 . 0 6 2 7(Surge ) ~ = -0 .0 31 8 .

( 5 5 )

The d imens iona l t ime cons tan t s a re g iven by( Ti = - ( 1 / ) ~ ' ) v/-L/g) :

(Ya w) T~' -- 11.5 (s)( S w a y ) = 4 4 . 5 ( s )(Surg e) T~' = 87.8 (s).

( 5 6 )

F o r m o r e d e t a il s a b o u t t h e D P c o n t r o l s y s t e m d e -s ign, see (S0rensenet al., 1995).

5 . C O N C L U S I O N S

In th i s pape r a new approach fo r t he iden t i f i -ca t ion o f dynam ica l ly pos i t ioned sh ips has beenproposed . Three d i f f eren t sh ip maneu ver s wereused in a decoup led iden t i f i ca tion scheme ba sed onan o ff- li ne pa ra l le l conf igu ra t ion o f the e x tende dKalm an f i lt e r a lgo r i thm. S imula t ion s tud ie ss h o w e d t h a t t h e p r o p o s e d p a r a m e t e r - e s t i m a t i o nscheme was r emarkab ly accura t e fo r sh ip mode l stha t were coup led in su rge , sway and yaw. Thep a r a m e t e r - e s t im a t i o n a l g o r it h m h a s b e en i m p le -men ted and t e s t ed on a supp ly vesse l . The e s -t ima te d mo de l o f t he sup p ly vessel has been im-p lemen ted , and used fo r mode l -based DP con t ro lsys t em des ign . The e s t ima ted va lues o f t h i s sh ipshowed good ag reem en t wi th expe r im en ta l r e su l t sf rom mode l t e s t s .

6 . R E F E R E N C E S

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and Na vy App l i ca t ions.Monte rey, CA. pp . 337-393.

Abkow i tz , M. A . (1980). Measu remen t o f Hyd ro -dynamic Charac te r i s t i c s f rom Sh ip Maneuvr ingTria ls by System Ident i f ica t ion. In :Transac-t i ons on SN AM E, 88 :283-318 .

Balchen, J . G. , N. A. Jenssen and S. Stol id (1976) .

Dynamic Pos i t ion ing Us ing Ka lman F i l t e r ingand Opt ima l Con t ro l Theory. In :I FA C / / I F I PSym pos iu m on Au to ma t ion in Offsho re O i lFie ld Opera t ion .H o l l an d , A m s t e r d a m . p p . 1 8 3 -186.

Balchen, J . G. , N. A. Jenssen and S. S~el id(1980a) . Dynam ic Pos i t ion ing o f F loa t ing Ves -se l s Based on Ka lman F i l t e r ing and Opt i -mal Con trol . In : Proceedings of the 19thIE EE Conference on Decis ion a nd Control.New York , NY. pp . 852-864 .

Ba lchen , J . G . , N . A . Jenssen , E . Math i sen and

S . S~e l id (1980b). D ynam ic P os i t ion ing Sys -t e m B a s e d o n K a l m a n F i l t e r i n g a n d O p t i m a lCon t ro l . Model ing, Ident i f ica t ion and ControlM I C - I ( 3 ) , 1 3 5 - 1 6 3 .

Fal t insen, O. M. (1990) .Sea Loads on Sh ipsand Offshore St ructures .C a m b r i d g e U n i v e r s it yPress .

Fossen, T. I . (1994) . Guidance and Con t ro l o fOcean Vehicles. J o h n Wi l e y a n d S o n s L t d .

Fung , P. T-K. and M. J . Gr imble (1983) . DynamicSh ip Pos i t ion ing Us ing a Se l f Tun ing Ka lm anFi l ter. I E E E Tr a n sa c t io n s o n A u t o m a t i c C o n -trol A C - 2 8 ( 3 ) , 3 3 9 - 3 4 9 .

Gelb , A. , J . F. Kasper, J r. , R. A. Nash, J r. , C. F.P r i ce and A . A . Su the r l and , J r. (1988) .Appl iedOpt ima l Es t ima t ion .M I T P r e s s . B o s t o n , M a s -sachuset ts .

Gr imble , M. J . , R . J . Pa t ton and D . A . Wise(1980a) . The D es ign o f Dy nam ic Pos i t ion -ing Con t ro l Sys tems Us ing S tochas t i c Op t ima lCon t ro l Theory. Opt ima l Con t ro l App l i ca t ionsand Methods 1 , 1 6 7 - 2 0 2 .

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