Time-Domain Models of Marine Surface Vessels based on … MCMC2006... · 2014. 8. 3. · Obtaining models of marine vessels Mathematical Models (Simulation,GNC-design HIL-testing,

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Perez & Fossen – MCMC Plenary Lecture Sept 2006

Time-Domain Models of Marine Surface Vessels based onSekeeping Computations

Tristan Perez and Thor Fossen

Centre for Ships and Ocean Structures CeSOSNorwegian University of Sc. and Tehc. NTNU Trondheim, NORWAY

Sept 2006

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Motivation

This presentation aims at answering the following questions:

• How to obtain preliminary models for simulation and control design based on main characteristics of the vessel?

• How can these models be updated when there is data available from experiments?

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Outline

• Part I – Overview of models of marine surface vessels

• Part II – Classical seakeeping models in the frequency domain

• Part III – Time-domain Seakeeping models based on frequency domain data—towards a unified model for manoeuvring in a seaway.

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I- Models of marine surface vessels: an overview

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Obtaining models of marine vessels

Mathematical Models

(Simulation,GNC-designHIL-testing, Diagnosis)

SystemIdentification

Data-base Model testing

Full-scale Experiments

Numerical Hydrodynamics

Scaling System Identification

SystemIdentification

Focus of this presentation

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Manoeuvring and seakeeping theoriesThe study of marine vessel dynamics response has traditionally been

separated into two main areas

Manoeuvring

The aim is to study steering characteristics and response to the command of propulsion systems and control surfaces. This is done in calm water.

Seakeeping

The aim is to study the behaviour of the vessel in waves while keeping a constant speed and course.

Although both areas are concerned with the study of motion, stability and control, the separation allows one making assumptions that simplify the study in each case.

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The model is forced to move and forces, velocities and accelerations are measured.

Then curve fitting is used to obtain a parametric model of the forces.

Manoeuvring models

• Nonlinear parametric models

• Obtained by fitting data from scalledmodel experiments

• Calm water models• Horizontal motion (surge-sway-yaw)• Not commonly available• Restricted to the few sepeeds and

loading conditions of the experiment

u)f(x,x =&

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Seakeeping models

Linear non-parametric models

Obtained from hydrodynamic calculations based on simplifying assumptions:

• Constant course and speed• Linear wave loads• Potential theory• Viscous effects can be added

)( ωjH

For the design of control systems, seakeeping models are very useful.

They provide preliminary models based on little data of the ship.

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Speed–environment envelope

Hull supported by mostly by hydrostatic pressure (forces)

Aero and hydrodynamic forces; strong flow separation

Hydrostatic and hydrodynamic forces; Lift

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Superposition and control problems

Then, motion control problems can have different objectives:

• Control only the non-oscillatory motion• Autopilots, • Dynamic positioning (DP) • Thruster assisted position mooring

• Control only the oscillatory motion • Ride control of high speed vessels (roll and pitch stabilisation)• Heave compensation of offshore structure• Roll stabilisation

• Control both • Dynamic positioning in extreme seas (roll & pitch stabilisation)• Autopilots with rudder roll stabilisation• Unmanned Surface Vehicles USV

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Motion superposition model

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Force superposition model

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II – Classical seakeeping models in the Frequency domain

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Ship motion description

To describe the motion of a ship at sea we use two reference frames:

• North-East-Down (n-frame)• Body-fixed (b-frame)

Generalised position vector

Generalised velocity vector

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Seakeeping frame (h-frame)

A distinctive characteristic of seakeeping theory is the use of an equilibrium frame to formulate the equations of motion, rather thana body-fixed frame.

The h-frame (forward starboard-down). The h-frame (oh, xh, yh, zh) is not fixed to the hull; it moves at the average speed of the vessel following its path.

The xh-yh plane coincides with the mean water free surface.

The origin oh is usually determined such that the zh-axis passes through the time-average position of the centre of gravity.

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Equilibrium

Perturbed

Seakeeping coordinatesThe constant course and speed assumed define a

state of equilibrium of motion

The action of the waves makes the ship oscillatewith respect to this equilibrium—theoscillations may not necessarily be harmonic.

In the absence of wave excitation, the origin ohcoincides with the location of a point s in the ship.

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Kinetics in seakeeping theory

Because the h-frame, moves at a constant speed (including zero), it is inertial. Therefore, the Newton-Euler equations of motion are

where

The mass matrix describred from the h-frame is time varying, but if we make the approximation of small motion we can consider it constant.

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Hydrodynamic forces on marine vessels

Viscous

(function of the vesselvelocity)

Forces due to flowseparation and skin

friction

Restoring(proportional to the

vessel position)

Gravity and buoyancy

Radiation forces(proportional to

velocity and acceleration)

Added madd and potential damping

Waveexcitation

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Radiation forces for sinusoidal motion

If the vessel is moving sinusoidally in each DOF,the radiation forces can be expressed as

Added mass (matrix): due to the change in momentum of the fluid.

Potential damping (matrix): due to the energy carried away by the radiated waves.

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Restoring forces (linear)

The resotring forces are usually linearized:

=restτ

(Awp waterplane area)

These are usually computed for calm water—Calm water stability

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Frequency-domain model

Putting all the terms together:

These are not true equations of motion, but a different way of writing the frequency response of the system

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Wave exitation forces—Force RAO

Sea surface elevation 1st order wave force

Force RAO

Linear assumption

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RAOS (frequency response functions)

Sea surface elevation

Force RAO

combined

Motion RAO

motion

hwexj τωξ ~)(

~ G=

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Force to motion frequency response

The relationship between the force and motion RAO is

Wave to forceForce to motionWave to motion

Where the force to motion frequency response is

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Hydrodynamic Computations

There are several hydrodynamic programs that based on different potential theories (2D, 2D+t, 3D) compute the hydrodynamic coefficients and the frequency response functions:

• Added mass, damping and restoring coefficients.• Force RAO• Motion RAO

These are computed for a set of discrete frequencies.

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Potential theory—summary

Inviscid fluid and irrotational flow

(t,x,y,z) Potential function

Flow velocity Pressure

02 =Φ∇

For most problems related ship motion in waves, potential theory is sufficient to obtain results with appropriate accuracyfor engineering purposes. Viscous effects are added to the models using experimental data.

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Example Austal Ships H260

Added mass in Sway Potential damping in Sway

0 1 2 3 4 5 6 70.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

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frequency (rad/s)

A22 (U=18.0056 m/s)

0 1 2 3 4 5 6 70.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 10

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frequency (rad/s)

B22 (U=18.0056 m/s)

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Example Austal Ships H260

Force RAO Sway Motion RAO Sway

0 1 2 3 4 5 6 70

500

1000

1500

2000

wave frequency (rad/s)

Ampl

itude

(kN

/m)

Force RAO amplitude and phase: SWAY

120deg, 18.0056m/s

0 1 2 3 4 5 6 7-32

-30

-28

-26

-24

-22


Phas

e (d

eg)

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8


Ampl

itude

(m/m

)

Motion RAO amplitude and phase: SWAY

120deg, 18.0056m/s

0 1 2 3 4 5 6 7-54

-52

-50

-48

-46


Phas

e (d

eg)

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III – Time-domain seakeeping models based on frequency-domain data

(Towards and unified model for manoeuvringand seakeeping)

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A linear time-domain model (Cummins Eq.)

Cummins (1962) considered the behaviour of the fluid and the vessel ab initio in the time domain.

Under the assumption of linearity he found the time-domain equation of motion in the h-frame:

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A linear time-domain model (Cummins Eq.)

- The added mass matrix is constant; frequency and speed independent.

- There is a constant damping term which appears only if the vessel has forward speed.

- The convolution term accounts for fluid memory effects.

- The kernel of the convolution is a matrix of retardation functions or impulse responses, which depend on the forward speed.

- If the vessel has forward speed, restoring forces appear due to hydrodynamic pressure. (usually ignored for Fn

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Example Retardation Functions

0 5 10 15 20-1

-0.5

0

0.5

1

1.5

2

2.5x 10

6

Time (s)

K22

Retardation function

K22(t)

0 5 10 15 20-2

-1

0

1

2

3

4

5

6x 10

7 Retardation function

Time (s)

K44

0 5 10 15 20-12

-10

-8

-6

-4

-2

0

2

4x 10

6 Retardation function

Time (s)

K24

K44(t)

K24(t)

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Ogilvie relationships

Ogilvie (1964), provided the link between the Cummins equation and its frequency-domain counterpart:

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Properties of the convolution terms

• Time-domain properties

(Relative degree 1)

(BIBO stable)

• Frequency-domain properties

(zero at zero frequency)

(corroborate relative degree)

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Convenient representations

The convolution terms are not convenient for analysis, control design and simulation. Thus we can chose a more convenient representation: state-space or transfer function—this leads to different identification problems.

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Identification problems

1. State-space

2. Transfer functions

Force to motion TF 3.

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From Impulse response to SS

Propose a canonical realization:

Obtain the parameters via NL LS:

• This method was proposed by Yu and Falnes (1995).

• It is not easy to chose the order of the system

• Convergence depends on the initial value of the parameters and the weighting function.

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From impulse response to SS

Discrete-time approach:

Key result (Ho and Kalman, 1966):

The order of the system, and the matrices are obtained from factorizations of the SVD decomposition

• This method was proposed by Kistiansen and Egeland (2003)

• Requires conversion to continuous time, which must be handled carefully to keep the properties of the convolution.

• Model order reduction is usually necessary.

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From frequency response to TFRational TF:

Estimate parameters by fitting the frequency response:

• The problem can be linearised resulting in linear LS problem, which can be used to give initial parameter estimate for nonlinear LS

• The frequency-domain properties of the constraint the relative degree, and the TF has a zero at s=0.

• We start with minimum order that satisfy the constraints and increase the order until we get a good fit.

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Comparison

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Converting to body-fixed coordinates

Small angles

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Converting to body-fixed coordinates

State-space representation of the convolution

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Model attributes and limitations

• The model derived is linear and was obtained from seakeeping calculations based on potential theory.

• Potential theory does not account for viscous effects, and lift in the case high-speed vessels.

• The limitations of the seakeeping program used to obtain the data should be taken into account.

• The model is speed dependant, this can be relaxed for slow variations of the forward speed (beyond the scope of this paper)

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Adding other effects

Hull as a Lifting surface (Blanke, 1981):

Viscous damping (Norrbin, 1970, Blanke, 1981,):

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Adding other effects

After choosing a structure for the added effects, we can use data from experiments to estimate the parameters:

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Summary and Discussion

• We have discussed the use of seakeeping data computed from standard hydrodynamic codes to obtain time-domain models for simulation and control design.

• We have revisited the classical frequency-domain approach used in hydrodynamics together with the resulting linear time-domain model.

• By using system identification, one can approximate the convolution terms by the response of linear systems. By combining these linear systems with the rest of the model we obtain a model in body-fixed coordinates.

• This provides an attractive alternative for obtaining models for control design, since seakeeping codes are nowadays standard tool, and they require little information of the vessel: hull form and loading condition.

• The models obtained can be updated if experimental data of the vessel is available via system identification.

• The model presented, is valid for any excitation provided the linearity assumption is valid, and since it incorporates fluid memory effects, it is a unified model for manoeuvring in a seaway.

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AcknowedgementsThis work has been supported by the Centre for Ships and Ocean Structures CeSOS and the Research Council of Norway.

The main motivation for the work comes from

Bailey, P.A., W.G. Price and P. Tamarel (1997). A unified mathematical model describing the manoeuvring of a ship in s seaway. Transactions The Royal Institution of Naval Architects–RINA 140, 131–149.

Kristansen, E. and O. Egeland (2003). Frequency dependent added mass in models for controller design for wave motion ship damping. MCMC’03, Girona, Spain.

The results of many discussions and collaborative work with the following people has significantly affected the paper (in a good way!):

Prof. O.M. FaltinsenProf. A.J. SørensenProf. M. Blanke

A. RossØ. Smogeli

E. KristiansenK. Unneland

Finally, the authors are also grateful to T. Armstrong and T. Mak, from Austalships, Australia for sharing the data of a modern vessel and for the on-going collaborative work on model validation and system identification.

Time-Domain Models of Marine Surface Vessels based on Sekeeping ComputationsMotivationOutlineI- Models of marine surface vessels: an overviewObtaining models of marine vesselsManoeuvring and seakeeping theoriesManoeuvring modelsSeakeeping modelsSpeed–environment envelopeSuperposition and control problemsMotion superposition modelForce superposition modelII – Classical seakeeping models in the Frequency domainShip motion descriptionSeakeeping frame (h-frame)Seakeeping coordinatesKinetics in seakeeping theoryHydrodynamic forces on marine vesselsRadiation forces for sinusoidal motionRestoring forces (linear)Frequency-domain modelWave exitation forces—Force RAORAOS (frequency response functions)Force to motion frequency responseHydrodynamic ComputationsPotential theory—summaryExampleExampleIII – Time-domain seakeeping models based on frequency-domain data(Towards and unified model for manoeuvring and seakeeping)A linear time-domain model (Cummins Eq.)A linear time-domain model (Cummins Eq.)Example Retardation FunctionsOgilvie relationshipsProperties of the convolution termsConvenient representationsIdentification problemsFrom Impulse response to SSFrom impulse response to SSFrom frequency response to TFComparisonConverting to body-fixed coordinatesConverting to body-fixed coordinatesModel attributes and limitationsAdding other effectsAdding other effectsSummary and DiscussionAcknowedgements

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Time-Domain Models of Marine Surface Vessels based on … MCMC2006... · 2014. 8. 3. · Obtaining models of marine vessels Mathematical Models (Simulation,GNC-design HIL-testing,