CAMS_M4_manoeuvring models.pdf

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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

http://www.marinecybernetics.com/books.htmhttp://www.itk.ntnu.no/ansatte/Fossen_Thor/book/book.html


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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:


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


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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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.



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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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03/09/2007 One day Tutorial CAMS'07 Bol Croatia 33

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.

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CAMS_M4_manoeuvring models.pdf