modelling and simulation of marine craft dynamics...desirable to derive equations of motion about co...

Modelling and Simulation of Marine Craft

DYNAMICSEdin OmerdicSenior Research FellowMobile & Marine Robotics Research CentreUniversity of Limerick

Mobile & Marine Robotics Research Centre

University of Limerick

Outline

DynamicsRigid-Body Equations of Motion

Equations of Motion about CG

Equations of Motion about CO

6 DoF Equations of Motion (ROV)Restoring Forces and Moments

Ocean Current Forces and Moments

Wave Forces and Moments

Propulsion SystemPropeller Thrust and Torque Modelling

Full thruster model

Simulation DiagramsNonlinear 6DoF ROV model (Euler Angles)

Nonlinear 6DoF ROV model (Quaternions)



GNC Signal Flow

GNC Signal Flow

Weather routing

program

Trajectory

Generator

Motion Control

System

Control

Allocation

Fusion Marine Craft INS (GNSS + AHRS)

OA Sensors

(Radar, Cameras, etc.)

Observer

Weather

data

Waypoints

Guidance

System

Control

System

Obstacle

Avoidance

Module

Estimated

positions and

velocities Navigation

System

Waves, wind

and ocean

currents



𝑛

𝑏 CO CG

Ԧ𝑟𝑏/𝑛

Ԧ𝑟𝑔

Ԧ𝑟𝑔/𝑛


𝑚𝑰3×3 𝟎3×3𝟎3×3 𝑰𝑔

ሶ𝒗 Τ𝑔 𝑛𝑏

ሶ𝒘 Τ𝑏 𝑛𝑏

+𝑚𝑺 𝒘 Τ𝑏 𝑛

𝑏 𝟎3×3

𝟎3×3 −𝑺 𝑰𝑔𝒘 Τ𝑏 𝑛𝑏

𝒗 Τ𝑔 𝑛𝑏

𝒘 Τ𝑏 𝑛𝑏

=𝒇𝑔𝑏

𝒎𝑔𝑏

Equations of Motion about CG

Desirable to derive Equations of Motion about CO (to take advantage of geometric properties of craft)

Coordinate Transformation:

𝒗 Τ𝑔 𝑛𝑏


= 𝑯 𝒓𝒈𝑏

𝒗 Τ𝑏 𝑛𝑏


𝑯 𝒓𝑔𝑏 =

𝑰3×3 𝑺𝑻 𝒓𝑔𝑏

𝟎3×3 𝑰3×3

𝑴𝑅𝐵𝐶𝐺

𝑪𝑅𝐵𝐶𝐺




𝑴𝑅𝐵𝐶𝐺𝑯 𝒓𝑔

𝑏ሶ𝒗 Τ𝑏 𝑛𝑏

ሶ𝒘 Τ𝑏 𝑛𝑏 + 𝑪𝑅𝐵

𝐶𝐺𝑯 𝒓𝑔𝑏


𝒘 Τ𝑏 𝑛𝑏 =

𝒇𝑔𝑏

𝒎𝑔𝑏

Equations of Motion about CO

𝑯𝑻 𝒓𝒈𝑏 𝑴𝑅𝐵


ሶ𝒗 Τ𝑏 𝑛𝑏

ሶ𝒘 Τ𝑏 𝑛𝑏 +𝑯𝑻 𝒓𝑔

𝑏 𝑪𝑅𝐵𝐶𝐺𝑯 𝒓𝑔

𝑏𝒗 Τ𝑏 𝑛𝑏

𝒘 Τ𝑏 𝑛𝑏 = 𝑯𝑻 𝒓𝑔

𝑏𝒇𝑔𝑏

𝒎𝑔𝑏 =

𝒇𝑏𝑏

𝒎𝑏𝑏

𝑴𝑅𝐵𝐶𝑂 𝑪𝑅𝐵

𝐶𝑂

𝑴𝑅𝐵𝐶𝑂 =

𝑚𝑰3×3 −𝑚𝑺 𝒓𝑔𝑏

𝑚𝑺 𝒓𝑔𝑏 𝑰𝑏

= 𝑴𝑅𝐵𝐶𝑂𝑇 > 0

Rigid-Body System Inertia Matrix Inertia Matrix (Parallel-Axes Theorem)

𝑰𝑏 = 𝑰𝑔 −𝑚𝑺𝟐 𝒓𝑔𝑏

𝑰𝑔 =

𝐼𝑥 −𝐼𝑥𝑦 −𝐼𝑥𝑧−𝐼𝑦𝑥 𝐼𝑥 −𝐼𝑦𝑧−𝐼𝑧𝑥 −𝐼𝑧𝑦 𝐼𝑥

= 𝑰𝑔𝑇 > 0Unique representation!

𝝉𝑅𝐵𝐶𝑂𝝂𝑅𝐵

𝐶𝑂ሶ𝝂𝑅𝐵𝐶𝑂




𝑪𝑅𝐵𝐶𝑂 𝒗 =

𝟎3×3 −𝑚𝑺 𝒗 Τ𝑏 𝑛𝑏 −𝑚𝑺 𝑺 𝒘 Τ𝑏 𝑛

𝑏 𝒓𝒈𝑏

−𝑚𝑺 𝒗 Τ𝑏 𝑛𝑏 −𝑚𝑺 𝑺 𝒘 Τ𝑏 𝑛

𝑏 𝒓𝒈𝑏 𝑚𝑺 𝑺 𝒗 Τ𝑏 𝑛

𝑏 𝒓𝒈𝑏 − 𝑺 𝑰𝑏𝒘 Τ𝑏 𝑛

𝑏 = −𝑪𝑅𝐵𝐶𝑂𝑇> 0

Rigid-Body Coriolis and Centripetal MatrixMultiple representations!


𝟎3×3 −𝑚𝑺 𝒗 Τ𝑏 𝑛𝑏 −𝑚𝑺 𝒘 Τ𝑏 𝑛

𝑏 𝑺 𝒓𝒈𝑏

−𝑚𝑺 𝒗 Τ𝑏 𝑛𝑏 +𝑚𝑺 𝒓𝒈

𝑏 𝑺 𝒘 Τ𝑏 𝑛𝑏 −𝑺 𝑰𝑏𝒘 Τ𝑏 𝑛

𝑏 = −𝑪𝑅𝐵𝐶𝑂𝑇> 0


𝑚𝑺 𝒘 Τ𝑏 𝑛𝑏 −𝑚𝑺 𝒘 Τ𝑏 𝑛

𝑏 𝑺 𝒓𝒈𝑏

𝑚𝑺 𝒓𝒈𝑏 𝑺 𝒘 Τ𝑏 𝑛

𝑏 −𝑺 𝑰𝑏𝒘 Τ𝑏 𝑛𝑏

Generalized vector of External forces and moments

𝝉𝑅𝐵𝐶𝑂 =

𝒇𝑏𝑏

𝒎𝑏𝑏

𝑯𝑻 𝒓𝒈𝑏 𝑴𝑅𝐵


ሶ𝒗 Τ𝑏 𝑛𝑏

ሶ𝒘 Τ𝑏 𝑛𝑏 +𝑯𝑻 𝒓𝑔

𝑏 𝑪𝑅𝐵𝐶𝐺𝑯 𝒓𝑔

𝑏𝒗 Τ𝑏 𝑛𝑏

𝒘 Τ𝑏 𝑛𝑏 = 𝑯𝑻 𝒓𝑔

𝑏𝒇𝑔𝑏

𝒎𝑔𝑏 =

𝒇𝑏𝑏

𝒎𝑏𝑏

𝑴𝑅𝐵𝐶𝑂 𝑪𝑅𝐵

𝐶𝑂𝝉𝑅𝐵𝐶𝑂𝝂𝑅𝐵

𝐶𝑂ሶ𝝂𝑅𝐵𝐶𝑂




6 DoF Rigid-Body Simulation ModelAttitude representation: Euler Angles

r

q

p

w

v

u

b

nb

b

nb

/

/

w

vv

nb

n

nb

θ

pη

/

Y

P

R

z

y

x

n

n

n

nb

n

nb

θ

pη /

Gain Integrator

0

00 /

nb

n

nb

θ

pη

0

00

/

/

b

nb

b

nb

w

vv

ሶ𝒑 Τ𝑏 𝑛𝑛

ሶ𝜽𝑛𝑏=

𝑹𝑏𝑛 𝜽𝑛𝑏 𝟎3×3𝟎3×3 𝑻𝜃 𝜽𝒏𝒃



ሶ𝜼 = 𝑱𝜣 𝜼 𝝂

b

nb

b

nb

/

/

w

vv

𝑴𝑅𝐵 ሶ𝝂 + 𝑪𝑅𝐵𝒗 = 𝝉𝑅𝐵

ሶ𝝂 = 𝑴𝑅𝐵−1 𝝉𝑅𝐵 − 𝑪𝑅𝐵𝒗

Gain

𝑪𝑅𝐵KINEMATICS

DYNAMICS

+

−

𝝉𝑅𝐵𝑴𝑅𝐵

−1

GainIntegrator




6 DoF Rigid-Body Simulation ModelAttitude representation: Quaternions

r

q

p

w

v

u

b

nb

b

nb

/

/

w

vv

Gain Integrator

0

00

/

/

b

nb

b

nb

w

vv

b

nb

b

nb

/

/

w

vv

𝑴𝑅𝐵 ሶ𝝂 + 𝑪𝑅𝐵𝒗 = 𝝉𝑅𝐵

ሶ𝝂 = 𝑴𝑅𝐵−1 𝝉𝑅𝐵 − 𝑪𝑅𝐵𝒗

Gain

𝑪𝑅𝐵KINEMATICS

DYNAMICS

+

−

𝝉𝑅𝐵𝑴𝑅𝐵

−1

GainIntegrator

q

pη

n

nb /

3

2

1

0

/

q

q

q

q

z

y

x

n

n

n

n

nb

q

pη

ሶ𝒑 Τ𝑏 𝑛𝑛

ሶ𝒒=

𝑹𝑏𝑛 𝒒 𝟎3×3𝟎4×3 𝑻𝒒 𝒒



ሶ𝜼 = 𝑱𝒒 𝜼 𝝂

0

00 /

q

pη

n

nb



Dynamics6 DoF Equations of Motion (ROV)

6 DoF Equations of Motion including Ocean Currents (ROV)

𝑴𝑅𝐵 ሶ𝝂 + 𝑪𝑅𝐵 𝒗 𝒗 + 𝒈 𝜼 +𝑴𝐴 ሶ𝝂𝒓 + 𝑪𝐴 𝒗𝑟 𝒗𝑟 + 𝑫 𝒗𝑟 𝒗𝑟 = 𝝉𝑤𝑎𝑣𝑒 + 𝝉𝑝𝑟𝑜𝑝𝑢𝑙𝑠𝑖𝑜𝑛

rigid-body and hydrostatic terms hydrodynamic terms

Restoring Forces and Moments 𝑊 = 𝑚𝑔 𝐵 = 𝜌𝑔𝛻

𝒈 𝜼 = −𝒇𝑔𝑏 + 𝒇𝑏

𝑏

𝒓𝑔𝑏 × 𝒇𝑔

𝑏 + 𝒓𝑏𝑏 × 𝒇𝑏

𝑏 𝒇𝑔𝑛 =

00𝑊

𝒇𝑏𝑛 = −

00𝐵

𝒇𝑔𝑛

𝒇𝑏𝑛

𝐶𝐵

Buoyancy force

Weight force

𝐶𝐺

𝑧𝑛

Euler Angles Quaternions

𝒇𝑔𝑏 = 𝑹𝑏

𝑛 𝜽𝑛𝑏−1𝒇𝑔

𝑛

𝒇𝑏𝑏 = 𝑹𝑏

𝑛 𝜽𝑛𝑏−1𝒇𝑏

𝑛

𝒇𝑔𝑏 = 𝑹𝑏

𝑛 𝒒 −1𝒇𝑔𝑛

𝒇𝑏𝑏 = 𝑹𝑏

𝑛 𝒒 −1𝒇𝑏𝑛





𝑴𝑅𝐵 ሶ𝝂 + 𝑪𝑅𝐵 𝒗 𝒗 + 𝒈 𝜼 +𝑴𝐴 ሶ𝝂𝒓 + 𝑪𝐴 𝒗𝑟 𝒗𝑟 + 𝑫 𝒗𝑟 𝒗𝑟 = 𝝉𝑤𝑎𝑣𝑒 + 𝝉𝑝𝑟𝑜𝑝𝑢𝑙𝑠𝑖𝑜𝑛

rigid-body and hydrostatic terms hydrodynamic terms

Ocean Current Velocity Vector(Irrotational Fluid)

n

cnb

n

b

n

cnb

b

n

b

c

b

c vθRvθRv0

vv

13

c

1,

Relative Velocity Vector

𝒗𝑟 = 𝒗 − 𝒗𝑐 =𝒗 Τ𝑏 𝑛𝑏

𝒘 Τ𝑏 𝑛𝑏 −

𝒗𝒄𝑏

𝟎3×1=

𝒗 Τ𝑏 𝑛𝑏 − 𝒗𝒄

𝑏


Assumption: ሶ𝝂𝒄 ≈ 𝟎

𝑴𝑅𝐵 +𝑴𝐴 ሶ𝝂 + 𝑪𝑅𝐵 𝒗𝑟 + 𝑪𝐴 𝒗𝑟 𝒗𝑟 +𝑫 𝒗𝑟 𝒗𝑟 + 𝒈 𝜼 = 𝝉𝑤𝑎𝑣𝑒 + 𝝉𝑝𝑟𝑜𝑝𝑢𝑙𝑠𝑖𝑜𝑛

𝑴 𝑪 𝒗𝑟





Hydrodynamic System Inertia (Added-Mass) Matrix

Hydrodynamic Coriolis-Centripetal (Added-Mass) Matrix


𝑴 𝑪 𝒗𝑟

𝑴𝐴 = 𝑴𝐴𝑇 ≥ 𝟎

r

q

p

w

v

u

N

M

K

Z

Y

X

00000

00000

00000

00000

00000

00000

AM

𝑴𝐴 =𝑴11 𝑴12

𝑴21 𝑴22𝑪𝐴 𝒗𝑟 =

𝟎3×3 −𝑺 𝑴11𝒗1𝑟 +𝑴12𝒗2𝑟−𝑺 𝑴11𝒗1𝑟 +𝑴12𝒗2𝑟 −𝑺 𝑴21𝒗1𝑟 +𝑴22𝒗2𝑟

𝑪𝐴 𝒗𝑟 = −𝑪𝐴𝑇 𝒗𝑟

Hydrodynamic Damping Matrix

rrrr

rqqq

rppp

rwww

rvvv

ruuu

rNN

qMM

pKK

wZZ

vYY

uXX

00000

00000

00000

00000

00000

00000

rvD




Wave Spectra



𝑴 𝑪 𝒗𝑟




Modelling of Ocean Waves

Reference:T. Perez. Ship Motion Control Course Keeping and Roll Stabilisation using Rudders and Fins. Springer, 2005.




Modelling of Ocean Waves




Waves in short-crested seas

Wave spectrum (mean direction 45°)

Elevation of the surface (200 components)

Potential (z = 0m) Pressure (z = 0m)

Waves in long-crested seas

Wave spectrum (direction 45°)

Elevation of the surface (20 components)

Potential (z = 0m) Pressure (z = 0m)

Reference:O. Smogeli. Marine Systems Simulator (MSS), Norwegian University of Science and Technology, Trondheim, 2006.

N

E



KinematicsTransformations BODY-NEDQuaternions

Wave

Spectrum

1st order

Force RAO

Sea

State

𝑆 𝜔

𝐻𝑠, 𝑇𝑧

Wave

Amplitude

2nd order

Force RAO

1st order Wave

Induced Force

𝐴𝑘

2nd order Wave

Drift Force

𝝉𝑤𝑎𝑣𝑒1

𝝉𝑤𝑎𝑣𝑒2

Force RAO: Wave Amplitude ↔ Wave-Induced Force



DynamicsPropulsion System

Propulsion System



𝑴 𝑪 𝒗𝑟



Thruster Modelling

Propeller Thrust:

Propeller Torque:

Propeller Efficiency:

Advance Ratio:

→

Bilinear Thruster Model:Linear approximation:

→

Affine Thruster Model:

Reference:Fossen, T.I. and Sagatun, S.I. Adaptive Control of Nonlinear Underwater Robotic Systems. Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, CA, 1991, pp. 1687-1694.




Bilinear Thruster Model




Affine Thruster Model





Full Thruster Model



DynamicsSimulation Diagrams

Nonlinear 6DoF ROV model (Euler Angles)

ሶ𝝂 = 𝑴𝑅𝐵 +𝑴𝐴−1 𝝉𝑤𝑎𝑣𝑒 + 𝝉𝑝𝑟𝑜𝑝𝑢𝑙𝑠𝑖𝑜𝑛 − 𝑪𝑅𝐵 𝒗𝑟 + 𝑪𝐴 𝒗𝑟 𝒗𝑟 − 𝑫 𝒗𝑟 𝒗𝑟 − 𝒈 𝜼

v η η

Gain Integrator

0η 0v

v

Gain

KINEMATICSDYNAMICS

+𝝉𝑝𝑟𝑜𝑝

GainIntegrator

n

cnb

n

b

n

cnb

b

n

b

c

b

c vθRvθRv0

vv

13

c

1

v

+

−

𝑴𝑅𝐵 +𝑴𝐴−1

𝝉𝑤𝑎𝑣𝑒

Gaincr vvv

𝑪𝑅𝐵 + 𝑪𝐴

−

𝑫

−−

𝒈

b

nb

b

nb

/

/

w

vv

nb

n

nb

θ

pη /

+



DynamicsSimulation Diagrams

Nonlinear 6DoF ROV model (Quaternions)

ሶ𝝂 = 𝑴𝑅𝐵 +𝑴𝐴−1 𝝉𝑤𝑎𝑣𝑒 + 𝝉𝑝𝑟𝑜𝑝𝑢𝑙𝑠𝑖𝑜𝑛 − 𝑪𝑅𝐵 𝒗𝑟 + 𝑪𝐴 𝒗𝑟 𝒗𝑟 − 𝑫 𝒗𝑟 𝒗𝑟 − 𝒈 𝜼

v η η

Gain Integrator

0η 0v

v

Gain

KINEMATICSDYNAMICS

+𝝉𝑝𝑟𝑜𝑝

GainIntegrator

n

c

n

b

n

c

b

n

b

c

b

c vqRvqRv0

vv

13

c

1

v

+

−

𝑴𝑅𝐵 +𝑴𝐴−1

𝝉𝑤𝑎𝑣𝑒

Gaincr vvv

𝑪𝑅𝐵 + 𝑪𝐴

−

𝑫

−−

𝒈

b

nb

b

nb

/

/

w

vv

q

pη

n

nb /

+

modelling and simulation of marine craft dynamics...desirable to derive equations of motion about co...

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