Transcript TMR 4240 Marine Control Systems

Mathematical Modelling of
Dynamically Positioned
Marine Vessels
Professor Asgeir J. Sørensen,
Department of Marine Technology,
Norwegian University of Science and Technology,
Otto Nielsens Vei 10, NO-7491 Trondheim, Norway
E-mail: [email protected]
1
Outline
• Kinematics
• Vessel dynamics
– Nonlinear low-frequency vessel model
– Linear wave-frequency model
• Environmental loads
– Wind load model
– Wave load model
• Mooring system
• Full-scale tests
2
Dynamic Positioning and Position Mooring
Pipe and cable laying
Vibration control
of marine risers
Position mooring
ROV operations
Geological survey
Heavy lift
operations
3
Cable laying vessel
Pipe laying vessel
Functionality: Control Modes
•
•
•
•
•
•
•
•
Station keeping models
Marine operation models
Slender structures
Multibody operations
Manoeuvring models
Linearized about some Uo
Sea keeping
Motion damping
High speed tracking/Transit
Low speed tracking
Marked position
Station keeping
0
4
1
2
Speed [knots]
3
4
5
6
7 …..
Modelling
The mathematical models may be formulated in two complexity levels:
 Control plant model: Simplified mathematical description containing only the
main physical properties of the process. This model may constitute a part of the
controller. Examples of model based output controllers are e.g. LQG, H₂/H∞,
nonlinear feedback linearization controllers, back-stepping controllers, etc. The
control plant model is also used in analytical stability analysis, e.q Lyapunov
Stabilty.
 Process plant model: Comprehensive description of the actual process. The
main purpose of this model is to simulate the real plant dynamics including
process disturbance, sensor outputs and control inputs. The process plant
model may be used in numerical performance and robustness analysis of the
control systems.
Maneuvering Model
Low-Speed Model
Station Keeping Model
U0
-3 m/s < U < 3 m/s
U  Uo
5
Kinematics - Reference Frames
Vc
c
w V
 Earth-fixed XEYEZE - frame
w
 The hydrodynamic XhYhZh - frame is

moving along the path of the vessel.
The XhYh-plane is assumed fixed and
(x , y )
parallel to the mean water surface. In
XE
sea keeping analysis the
(x , y )
hydrodynamic frame is moving
YR
forward with constant vessel speed
U. In station keeping operations
about the coordinates xd, yd, and ψd
YE
the hydrodynamic frame is Earthfixed and denoted as reference Body-fixed XYZ - frame is fixed to
parallel XRYRZR - frame
vessel body with origin located at
mean oscillatory position in average
water plane, (xG, 0, zG). Submerged
part of vessel is assumed to be
symmetric about xz-plane
(port/starboard)

d
d
6
d
X
Kinematics Relations
Linear and angular velocity of vessel in body-fixed frame
relative to earth-fixed frame for 6 DOF - surge, sway,
heave, roll, pitch and yaw:
 1  J 1 2  0 33   1 

  

 J 




2   0 33 J 2 2    2 
v (sway)
q (pitch)
Y
r (yaw)
Earth-fixed position and orientation vectors are:
1  x y z , 2    
T
T
u (surge)
p (roll)
w (heave)
X
Z
Linear and angular vessel velocity vectors in
body-fixed frame are defined:
1  u v wT ,  2  p q r T
7
Kinematics Relations
 1  J 1 2  0 33   1 

  

 J 




2   0 33 J 2 2    2 
Where J 1( ) and J2( ) are Euler rotation matrices.
v (sway)
cfcS ?sfcd+cfsSsd sfsd+cfcdsS
J1Ý
R2 Þ=
q (pitch)
Y
sfcS cfcd+sdsSsf ?cfsd+sSsfcd
?sS
cSsd
cScd
r (yaw)
d
%
X2 =
+Cx,d
0
0
0
S%
0
+Cx,dCy,S
0
1
=J ?
R2 ÞR
%2
2 Ý
u (surge)
Z
1 sdtS cdtS
J 2 ÝR2 Þ=
0
cd
?sd
cS 0
0 sd/cS cd/cS
8
w (heave)
X
f
%
0
p (roll)
1
J?
R2 Þ=JT1 ÝR2 Þ
1 Ý
1
J?
R2 ÞJT2 ÝR2 Þ
2 Ý
Vessel motion
Low-frequency Motion
LF

•
•
•
•
Wind loads
Current loads
Wave loads; 2. order
Thruster action
Wave-frequency Motion
WF
• Wave loads; 1. order
time
Superposition may be assumed:
9
tot   WF
Nonlinear Low-frequency Vessel Model
Nonlinear 6 DOF low-frequency model - surge, sway,
heave, roll, pitch and yaw :
MX
DÝ
%+CRB Ý
XÞ
X+CA Ý
Xr Þ
Xr +
Xr Þ+GÝ
RÞ=benv +b
b
thr
moor +
Relative velocity vector is defined:
 r  u  u c
10
v  vc
w p q r T
u c =Vc cosÝ
Kc ?fÞ
, v c =Vc sinÝ
Kc ?fÞ
benv =bwind +bwave2
Environmental loads: Wind and 2. Order wave loads
b
moor
Generalised mooring forces
bthr
Generalised thruster forces
Nonlinear Low-frequency Vessel Model
System inertia matrix:
M=
m ?Xu%
0
?Xw%
0
mz G ?Xq%
0
0
m ?Y v%
0
?mz G ?Y p%
0
mx G ?Y r%
?Z w%
0
m ?Z w%
0
?mx G ?Z q%
0
0
?mz G ?K v%
0
I x ?K p%
0
?I xz ?K r%
mz G ?M u%
0
?mx G ?Z q%
0
I y ?M q%
0
0
mx G ?N v%
0
?I zx ?N p%
0
I z ?N r%
M =MT >0
11
Nonlinear Low-frequency Vessel Model
Generalized Coriolis and centripetal forces:
CRB ÝXÞ=
0
0
0
c41
?c51 ?c61
0
0
0
?c42
c52
?c62
0
0
0
?c43 ?c53
c63
?c41
c42
c43
0
c51
?c52
c53
c54
0
?c65
c61
c62
?c63
c64
c65
0
CRB ÝXÞX
?c54 ?c64
c41 =mzG r
c42 =mw
c43 =mÝ
z G p ?v Þ
c51 =mÝ
x G q ?wÞ c52 =mÝ
z G r +x G pÞ c53 =mÝ
z G q +u Þ c54 =Ixz p ?Iz r
c61 =mÝ
v +x G rÞ c62 =?mu
c65 =Ix p +Ixz r.
12
c63 =mxG p
c64 =Iy q
Nonlinear Low-frequency Vessel Model
Generalized Coriolis and centripetal forces:
CA ÝXr Þ=
?ca51 ?ca61
0
0
0
0
0
0
0
?ca42
0
0
0
?ca43 ?ca53
0
ca51
ca42 ca43
0
ca61 ca62
0
CA ÝXr ÞXr
0
?ca62
0
?ca54 ?ca64
ca53
ca54
0
?ca65
0
ca64
ca65
0
ca42 =?Zw%w ?Xw%u r ?Zq%q ca43 =Yp%p +Yv%v r +Yr%r
ca51 =Zq%q +Zw%w +Xw%u r ca53 =Xq%q ?Xu%u r ?Xw%w ca54 =Yr%v r +Kr%p +Nr%r
ca61 =Yv%v r ?Yp%p ?Yr%r
ca62 =Xu%u r +Xw%w +Xq%q ca64 =Xq%u r +Zq%w +Mq%q
ca65 =Yp%v r +Kp%p +Kr%r
13
Nonlinear Low-frequency Vessel Model
Generalized damping and current forces:
DÝ
dNL Ý
Xr Þ
=DL X+
Xr , r Þ
where:
r =atan2Ý?v r , ?u r Þ
d NL ÝXr , r Þ= 0. 5w L pp
14
U cr = u 2r +v 2r
DC cx Ýr Þ|U cr |U cr
Xu
0
Xw
0
Xq
0
DC cy Ýr Þ|U cr |U cr
0
Yv
0
Yp
0
Yr
Zu
0
Zw
0
Zq
0
0
Kv
0
Kp
0
Kr
L pp BC cSÝr Þ|q|q ?z px DC cx Ýr Þ|U cr |U cr
Mu
0
Mw
0
Mq
0
L pp DC cfÝr Þ|U cr |U cr
0
Nv
0
Np
0
Nr
BC cz Ýr Þ|w|w
B 2 C cdÝr Þ|p|p +z py DC cy Ýr Þ|U cr |U cr
DL = ?
Examples of current coefficients surge, sway and yaw for supply ship:
Current coefficient in surge [Ns 2 /m2 ]
4
6
x 10
Current coefficient in sway [Ns 2 /m2 ]
5
5
x 10
4.5
4
4
3.5
2
3
0
2.5
2
-2
1.5
1
-4
0.5
-6
0
20
40
80
100
Degrees [deg]
120
140
160
140
160
180
Current coefficient in yaw [Ns 2 /m2 ]
6
6
60
x 10
4
2
0
-2
-4
-6
-8
15
0
20
40
60
80
100
Degrees [deg]
120
180
0
0
20
40
60
80
100
Degrees [deg]
120
140
160
180
Examples of current coefficients heave, roll and pitch for supply ship:
Current coefficient in heave [Ns 2 /m2 ]
Current coefficient in roll [Nms 2 /m2 ]
6
0
1
x 10
0.8
-0.5
0.6
0.4
0.2
-1
0
-0.2
-1.5
-0.4
-0.6
-2
-0.8
-1
0
20
40
80
100
Degrees [deg]
120
140
160
180
Current coefficient in pitch [Nms 2 /m2 ]
5
2.5
60
x 10
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
16
0
20
40
60
80
100
Degrees [deg]
120
140
160
180
-2.5
0
20
40
60
80
100
Degrees [deg]
120
140
160
180
Damping properties
Dominating
High sea state
Low sea state
damping
Linear wave drift.
Surge
Nonlinear turbulent
skin friction.
Nonlinear turbulent skin friction,
when |u r | >0.
Linear laminar skin friction,
for low KC number and u r ¸ 0.
Nonlinear turbulent skin friction,
Sway
Nonlinear eddy-making.
when |u r | >0.
Linear wave drift.
Linear laminar skin friction,
for low KC number and u r ¸ 0.
Nonlinear turbulent skin friction,
Yaw
Nonlinear eddy-making.
when |u r | >0.
Linear wave drift.
Linear laminar skin friction,
for low KC number and u r ¸ 0.
17
Nonlinear Low-frequency Vessel Model
Generalized restoring forces: GÝRÞ
G =?
0 0
0
0
0
0
0 0
0
0
0
0
0 0 Zz
0
ZS 0
0 0
Kd 0
0
0 0 Mz
0
0 0
0
0
0
MS 0
0
0
Zz ª?w gAWP
ZS =Mz ªw g XXxdA
AW P
Kd ª?w g4Ý
z B ?z G Þ?w g XXy 2 dA =?w gVGMT
AW P
MS ª?w g4Ýz B ?z G Þ?w g XXx 2 dA =?w gVGML
AW P
18
Nonlinear Low-frequency Vessel Model
Wind load :
Ax C wx Ý
w Þ
|U wr |U wr
Ay C wy Ý
w Þ
|U wr |U wr
bwind =0. 5a
0
u w =Vw cosÝ
Kw ?fÞ
, v w =Vw sinÝ
Kw ?fÞ
Ay L yz C wy Ý
w Þ
|U wr |U wr
?Ax L xz C wx Ý
w Þ
|U wr |U wr
Ay L oa C wfÝ
w Þ
|U wr |U wr
Xrw =
u ?u w v ?v w w p q r
U wr = u 2rw +v 2rw
w =atan2Ý
?v rw , ?u rw Þ
19
T
Examples of wind coefficients surge, sway and yaw for supply ship:
Wind coefficient in surge [Ns 2 /m2 ]
Wind coefficient in sway [Ns 2 /m2 ]
400
2000
300
1500
200
1000
100
500
0
0
-100
-500
-200
-1000
-300
-1500
-400
0
50
100
200
Degrees [deg]
250
300
350
400
Wind coefficient in yaw [Ns 2 /m2 ]
4
2
150
x 10
1.5
1
0.5
0
-0.5
-1
-1.5
-2
20
0
50
100
150
200
Degrees [deg]
250
300
350
400
-2000
0
50
100
150
200
Degrees [deg]
250
300
350
400
Examples of wind coefficients heave, roll and pitch for supply ship:
Wind coefficient in roll [Nms 2 /m2 ]
4
Wind coefficient in heave [Ns 2 /m2 ]
2
x 10
1
1.5
0.8
0.6
1
0.4
0.5
0.2
0
0
-0.5
-0.2
-0.4
-1
-0.6
-1.5
-0.8
-1
-2
0
50
100
150
200
Degrees [deg]
250
300
350
400
300
350
400
Wind coefficient in pitch [Nms 2 /m2 ]
5000
4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
-5000
21
0
50
100
150
200
Degrees [deg]
250
0
50
100
150
200
Degrees [deg]
250
300
350
400
Nonlinear Low-frequency Vessel Model
2. Order Wave loads :
i
biwave2 =b
biwsv , i =1..6
#
wm +
=2 >
22
N
A
j=1 j
Tijj Ý
gj , Kwave
?fÞ
1/2
cosÝ
gj t +P
jÞ
2
Mooring System
Overview

X
t
Anchor lines
(x,y)
Seabed
Y
XE
XE
YE
COT
YE
23
Mooring System

Z
Single Line Modelling
H
X
TP
3 types of excitation:
• Large amplitude LF motions
• Medium amplitude WF motions
• Very high frequency vortex-induced
vibrations
D
Ltot
Seabed
T
~
T T T
~
HHH
24
xh
Xh
TP
H
Ls
Mooring System

Z
Line Characteristics
H
X
TP
T  fT ( Xh )
H  f H ( Xh )
D
Ltot
Ls
7000
Seabed
xh
Xh
5000
T
3000
H
1000
0
1370
1380
1390
1400
1410
1420
Horizontal distance to anchor [m]
25
1430
Mooring System
Forces and moment on moored structure
• Additional damping term
d mo
XE
( xiA ,yiA )
g mo
• Restoring term
Anchor
i
Hi cosi




g mo   
Hi sin i

i 1 
Hi x i sin i  Hi yi cosi 
n
xih
hi
YE
TP
xiTPE  x  cos(   t ) xiTP 0  sin(   t ) yiTP 0
yiTPE  y  sin(   t ) xiTP 0  cos(   t ) yiTP 0
x ih  x iA  x iTPE
hi 
y ih  y iA  y iTPE
 i  atan2( y ih , x ih )
xi  x  xiTPE
yi  y  yiTPE
26
( xiTPE, yiTPE )
yi h
( xih ) 2  ( yih ) 2
Turret
Mooring System
Generalized mooring forces in LF model
mo  dmo (.)  JT ()gmo (.)
Quasi-static mooring model:
- use the line characteristics
for each line i in
Hi  f Hi (hi )
Hi cosi




g mo   
Hi sin i

i 1 
Hi x i sin i  Hi yi cosi 
n
27
Mooring System
Linearized Mooring Model
Hi  H0i  ci hi
g mo  g mo (0 ) 
d mo  D M 
28
df Hi
ci 
(hi  h0i )
dhi
g mo
|  0 (  0 )  g mo (0 )  G (  0 )

Linear Wave-frequency Vessel Model
Potential theory is assumed, neglecting viscous effects.
Two sub-problems:
•
•
29
Wave Reaction: Forces and moments on the vessel when the vessel
is forced to oscillate with the wave excitation frequency. The
hydrodynamic loads are identified as added mass and wave radiation
damping terms.
Wave Excitation: Forces and moments on the vessel when the
vessel is restrained from oscillating and there are incident waves. This
gives the wave excitation loads which are composed of so-called
Froude-Kriloff (forces and moments due to the undisturbed pressure
field as if the vessel was not present) and diffraction forces and
moments (forces and moments because the presence of the vessel
changes the pressure field).
Linear Wave-frequency Vessel Model
Linear 6 DOF Wave-frequency model - surge, sway, heave, roll,
pitch and yaw :
MÝ
gÞ
R
Rw +D p Ý
gÞ
R
%Rw +GRRw =bwave1
R
%w =JÝ
R
R
%Rw ,
#
2Þ
30
T

R
=
0
0
f
#
d
2
Motion vector in hydrodynamic frame:
RRw  R 6
Earth-fixed motion vector:
Rw  R 6
1. Order wave loads
bwave1  R 6
Verification tests on Varg FPSO
Turret
Anchor lines
Seabed
31
Full-scale results, Varg FPSO
Measured wind direction [deg]
Measured tension, line 1-5 [kN]
-20
1150
1100
-25
1050
1000
-30
950
-35
900
-40
850
800
-45
750
-50
700
0
100
200
300
400
500
0
600
100
200
300
400
500
600
500
600
Measured wind velocity [m/s]
Measured tension, line 6-10 [kN]
1150
27
1100
26
1050
25
1000
24
950
23
900
22
850
21
800
20
750
19
700
650
0
18
100
200
300
time [sec]
32
400
500
600
0
100
200
300
time [sec]
400
Full-scale results, Varg FPSO
Estimated (blue) and measured (red) North position [m]
Measured (red) RPM, thruster 1
2
1000
1.5
800
1
600
0.5
400
0
200
-0.5
0
-1
-200
-1.5
-400
-2
-2.5
-600
0
100
200
300
400
500
0
600
100
Estimated (blue) and measured (red) East position [m]
200
300
400
500
600
500
600
500
600
Measured (red) RPM, thruster 2
1000
4
800
3
600
2
400
200
1
0
0
-200
-1
-400
-600
-2
0
100
200
300
400
500
0
600
100
Estimated (blue) and measured (red) heading [deg]
200
300
400
Measured (red) RPM, thruster 3
1000
57
800
56.5
600
56
400
55.5
200
55
0
-200
54.5
-400
54
33
-600
53.5
0
100
200
300
time [sec]
400
500
600
0
100
200
300
time [sec]
400