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03/09/2007 One-day Tutorial, CAMS'07, Bol, Croatia 1
Manoeuvring Models
(Module 4)
Dr Tristan PerezCentre for Complex DynamicSystems and Control (CDSC)
Professor Thor I FossenDepartment of EngineeringCybernetics
Prepared together with Andrew Ross
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- body velocities:- position and Euler angles:- M, C and D denote the system inertia,Coriolis and damping matrices
- g is a vector of gravitational and buoyancy
forces and moments
- q is a vector of joint angles- is a vector of torque- M and C are the system inertia and Coriolis matrices
Vectorial Representation for Ships
From robotics to ship modeling (Fossen 1991)
Consider the classical robot manipulator model:
Mqq Cq,qq
This model structure can be used as foundation to write the 6 DOF marinevessel equations of motion in a compact vectorial setting (Fossen 1994, 2002):
u, v, w,p, q, rT
x,y,z,, ,T
M C D g
It is here assumed that thehydrodynamic coefficients arefrequency independent.
This will be relaxed later!
J
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Rigid-Body Equations of MotionNewtonian Formulation (Body Frame)
MRB
m 0 0 0 mzg myg
0 m 0 mzg 0 mxg
0 0 m myg mx g 0
0 mz g my g Ix Ixy Ixz
mz g 0 mx g Iyx Iy Iyz
myg mxg 0 Izx Izy Iz
Rigid-body system inertia matrix
MRB CRB RB
where
MRB rigid-body system inertia matrixCRB rigid-body Coriolis/centripetal matrix
The generalized forces on a floating vessel are superpositioned:
RB H wave wind current control
Hydrodynamic radiation-induced forces + viscous damping
See Fossen (1994, 2002) for parameterizations ofCRB
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Forces on the body when the body is forced to oscillate with thewave excitation frequency and there are no incident waves(Faltinsen 1990):
(1) Added mass due to the inertia of the surrounding fluid(2) Radiation-induced (linear) potential damping due to the energy
carried away by generated surface waves
(3) Restoring forces due toArchimedes (weight and buoyancy)
Faltinsen (1990). Sea Loads on Ships and Offshore Structures, Cambridge.
Radiation-Induced Hydrodyn. Forces
R MA CAadded mass
DPpotential damping
g gorestoring forces
hydrodynamic mass-damper-spring
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Fluid Kinetic Energy The concept of fluid kinetic energy:
can be used to derive the addedmass terms.
Any motion of the vessel will inducea motion in the otherwise stationaryfluid. In order to allow the vessel topass through the fluid, it must moveaside and then close behind the
vessel. Consequently, the fluid motion
possesses kinetic energy that itwould lack otherwise (Lamb 1932).
TMRB RB
=1/2
T
Kinetic energy of fluid: T MA A=1/2 T
TA 12
MA
Added Mass and Inertia
MA
Xu Xv Xw Xp Xq Xr
Yu Yv Yw Yp Yq Yr
Zu Zv Zw Zp Zq Zr
Ku Kv Kw Kp Kq Kr
Mu
Mv
Mw
Mp
Mq
Mr
Nu Nv Nw Np Nq Nr
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6 DOF Body-Fixed Representation for Added Mass(Includes Coriolis/Centripetal Terms due to Added Mass)
XA Xu u Xww uq Xqq Zwwq Zqq2
Xv v Xpp Xr r Yvvr Yprp Yr r2
Xv ur Ywwr
Yw vq Zppq Yq Zr qr
YA Xv u Yww Yqq
Yv v Ypp Yr r Xvvr Ywvp Xr r2
Xp Zr rp Zpp2
Xwup wr Xuur Zwwp
Zqpq Xqqr
ZA Xwu wq Zww Zqq Xuuq Xqq2
Yw v Zpp Zr r Yvvp Yr rp Ypp2
Xv up Ywwp
Xv vq X
p Y
q pq X
r qr
KA Xp u Zpw Kqq Xvwu Xr uq Yw w2 Yq Zr wq Mr q2
Yp v Kpp Kr r Ywv2 Yq Zr vr Zp vp Mr r2 Kqrp
Xwuv Yv Zwvw Yr Zqwr Yp wp Xq ur
Yr Zqvq Kr pq Mq Nr qr
MA Xqu wq Zqw uq Mqq Xwu2 w2 Zw Xuwu
Yq v Kqp
Mr r
Ypvr Yr vp Kr p2
r2
Kp Nr rp Yw uv Xvvw Xr Zpup wr Xp Zr wp ur
Mr pq Kqqr
NA Xr u Zr w Mr q Xvu2
Ywwu Xp Yquq Zpwq Kqq2
Yr v Kr p Nr r Xvv2 Xr vr Xp Yqvp Mr rp Kqp
2
Xu Yvuv Xwvw Xq Ypup Yr ur Zq wp
Xq Ypvq Kp Mqpq Kr qr
ddt T1 S2 T1
1
ddt
T2
S2T2
S1T1
2
Kirchhoff's Equations (1869)
TA 12
MA
d
dt
TA
u r
TA
vq
TA
wXA
ddt
TAv
pTAw
rTAu
YA
ddt
TAw
qTAu
pTAv
ZA
ddt
TAp
wTAv
vTAw
rTAq
qTAr
e KA
ddt
TAq
uTAw
wTAu
pTAr
rTAp
MA
ddt
TAr
vTAu
uTAv
qTAp
pTAq
NA
kinetic energy dueto the fluid
MA CA()
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In addition to potential damping we have to include otherdissipative viscous terms like skin friction, wave drift damping etc:
Total hydrodynamic damping matrix:
The hydrodynamic forces and moments can be now bewritten as the sum of :
Viscous Hydrodynamic Damping
D DSskin
friction
DWwave drift
damping
DMdamping due to
vortex shedding
D : DP DS DW DM
H MA CA D g go
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M C D g wave wind current control
Equations of Motion
M MRB MA
C CRB CA
The resulting model is (frequency-independent coefficients):
M
mXu Xv Xw
Xv m Yv Yw
Xw Yw mZw
Xp mzgYp mygZp
mzgXq Yq mxgZq
mygXr mxgYr Zr
Xp mzgXq mygXr
mzgYp Yq mxgYr
mygZp mxgZq Zr
IxKp IxyKq IzxKr
IxyKq IyMq IyzMr
IzxKr IyzMr IzNr
System inertia matrix including added mass
Linear mass-damper-spring
(frequency-independent)
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In classical manoeuvring theory, the forces are modelled at ageneral non-linear function:
A particular affine parameterization is then used, and thecoefficients are estimated linear regression from the data.
The disadvantage of this model representation to a energy-based(Lagrangian) approach is that model reduction,symmetry/skew-symmetry properties, positive matrices, etc.are difficult to exploit in simulation and control design.
This model can, however, be related to the Lagrangian model: asshown by Ross et al. 2007:
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Manoeuvring Hydrodynamics
g(D(C(M =+++ )))&
f(M += ),,&&
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Parameterisations
Two types of parameterisations for the hydrodynamic forces aregenerally used in classical manoeuvring theory:
Truncated Taylor-series expansions:
Davison and Shiff (1946): 1st-order (linear) terms.
Abkowitz (1964): odd terms up to 3rd
order.
2nd -order modulus Fedyaevsky and Sobolev (1963)
Norrbin (1970)
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Parameterisations
2nd -order modulus Taylor-series
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rrNrvN
rvNvvNrNvNN
rrYrvY
rvYvvYrYvYY
rrrv
rvvvrv
rrrv
rvvvrv
++
+++=
++
+++=
3rrr
2vrr
2vvr
3vvvrv
3rrr
2vrr
2vvr
3vvvrv
rNrvN
rvNvNrNvNN
rYrvY
rvYvYrYvYY
++
+++=
++
+++=
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Parameterisations
As commented by Clarke (2003),
Taylor expansions give rise to a smoothrepresentation of the forces, but have nophysical meaning.
2nd-order modulus expansions represent well thehydrodynamic forces at angles of incidence:
cross-flow drag.
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Taylor-Series Expansions
Where the partial derivatives are taken at anequilibrium:
...)()()( 22
2
+
+
+= xx
x
fxx
x
fxf
hydhyd
hydhyd
[ ]TU 00000=
[ ]Tx &=
[ ]T00x =
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Model of Abkowitz (1964)
The coefficients arecalled hydrodynamic
derivatives.
Many terms are set tozero by exploiting
physically properties.If not, there willthousands ofcoefficients.
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Model of Norrbin (1970)
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2nd-Order ModulusFrom Blanke and Christiansen (1986):
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Measurement of Hydrodynamic Derivatives
Experiments with model tests. Full scale sea trials and system identification. Theoretical prediction methods. Regression analysis results from similar designs.
Model tests that can be performed
Straight line in a towing tank,
Rotating arm, Planar motion mechanism PMM, Oscillator tests, Free running (radio controlled).
PMM
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Experimental Methods
Model testing in Peerlesspool in London
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Measurement of Hydrodynamic Derivatives
Rotating arm
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Typical Tests
Pure Sway:
Pure yaw:
Drift and yaw:Different tests are used tofit different parts of the
model.
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During the model tests, themodel is forces to moveand forces velocities andaccelerations are recorded.
Then the hydrodynamicderivatives are estimated
from regression analysis.
Measurement of Hydrodynamic Derivatives
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A Novel 4 DOF Manoeuvring Model
Ross et. al. (2007) has reassessed the manoeuvring models in theliterature, and formulated a novel 4 DOF (surge, sway, roll, yaw)Lagrangian model using first principles and superposition of:
Potential (added mass)
Circulation effects: lift and drag
Effect of roll on circulation effects
Cross-flow drag.
The advantage of the Lagrangian model is its vector representationwhich is tailor made for energy-based control design (Lyapunov).
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Added Mass and Coriollis
The 4 DOF solution of Kirchhoffs equations can beexpressed as (Fossen, 2002)
Added mass Added mass Coriollis andCentripetal terms
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Model of Ross et al. (2007)Circulation effects (lift and drag), effect of roll on circulation effects and cross-flow drag (modulus representation) are derived in Ross et al. (2007):
where the components are:
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Manoeuvring Model
Combining all the terms in a matrix for, we obtainthe manoeuvring equations in Lagrangian form
(Fossen 1994, 2002).
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Model Validation with PMM Data
To validate the model, Ross et al. (2007) used
data of several PMM tests, and perform aregression based on the model structurederived.
Then compared the fit with that of a model fitted
by a tank testing facility to the same dataset.
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Fitting Using PMM Data @ 30kt
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Validation in Full Scale (Perez et al.,2007)
Perez et al. (2007) fitted a simplified model to datarecorded on full scale manoeuvres of Austals
Trimaran Hull 260.
The parameters were fitted with data of a 20-20 zig-zag test, and then the model validated with data of a10-10 zig-zag test.
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Simplified Model
The model was simplified according to the that of Blanke (1981).This was done because the excitation signal was not richenough to estimate all the parametersthe zig-zag test is not
designed for system identification!
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Model Fitting (20-20 ZZ)
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Model Validation (10-10 ZZ)
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Effects of Currents
The current has to effects, which are represented with thevelocity of the vessel relative to the current velocity:
Potential: The Munk moment is incorporated in the added
mass Coriollis-Centripetal terms.
Viscous: eddy making and skin friction. These areincorporated in the cross-flow drag.
cr
rrArrARBARB
GDCCMM
J
=
=+++++
=
)()()()()(
)(
&
&
In some applications, where positioning is important,the effects of current must be considered:
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References Davidson, K. S. M. and L. I. Schiff (1946). Turning and Course Keeping Qualities. Transactions of SNAME.
Abkowitz, M. A. (1964). Lectures on Ship Hydrodynamics - Steering and Manoeuvrability. Technical ReportHy-5. Hydro- and Aerodynamic Laboratory. Lyngby, Denmark.
Fedayevsky, K.K. and G.V. Sobolev (1963). Control and Stability in Ship Design. State Union ShipbuildingPublishing House. Leningrad, USSR.
Norrbin, N. (1971). Theory and observations on the use of a mathematical model for ship manoeuvring indeep and conned water. Technical Report 63.Swedish State Shipbuilding Experimental Tank. Gothenburg.
Clarke, D. (2003). The foundations of steering and manoeuvring. In: Proceedings of the IFAC Conferenceon Control Applications. P lenary talk.
Ross, A., T. Perez, and T. Fossen (2007) "A Novel Manoeuvring Model based on Low-aspect-ratio LiftTheory and Lagrangian Mechanics." IFAC Conference on Control Applications in Marine Systems (CAMS).Bol, Croatia, Sept.
Blanke, M. (1981). Ship Propulsion Losses Related to Automated Steering and Prime Mover Control. PhDthesis. The Technical University of Denmark, Lyngby.
Christensen, A. and M. Blanke (1986). A Linearized State-Space Model in Steering and Roll of a High-SpeedContainer Ship. Technical Report 86-D-574.Servolaboratoriet, Technical University of Denmark. Denmark.
Perez,T., T, Mak, T. Armstrong, A.Ross, T. I. Fossen (2007) Validation of a 4DOF Manoeuvring Model of aHigh-speed Vehicle-Passenger Trimaran." In Proc. 9th International conference on Fast Transportation.Shanghai, China Sept.