marine vehicle motion

8/12/2019 Marine Vehicle Motion

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Marine Vehicle Dynamics and Control

Mar i ne vessel mot i on i s mor e compl i cat edt han spacecr af t mot i on due t o t he dependenceof f or ces and moment s on at t i t ude andvel oci t y

Mar i ne vessel mot i on i s i n some ways mor ecompl i cat ed t han ai r craf t

mot i on

Typi cal f i r st i nvest i gat i onsof shi p dynami cs and cont r ol

f ocuses on t he hor i zont almot i on of symmet r i c( l ef t - t o- r i ght ) shi ps( sur ge, sway, and yaw)


2/15

Reference Frames for Ship D&CInertial Frame,Fi. Typically assume flat Earth, but rotatingEarth is used for some problems. The inertial frame has 3-axisdown, though some sources have 3-axis up

For maneuvering problems, an Earth-fixed Fi is used, and forseakeeping problems, a frame moving with the vehicles nominalvelocity is used

Body Frame, Fb. A 3-2-1 rotation from Fi through yaw (),pitch (), and roll () angles: Rbi =R1()R2()R3()


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Reference Framesfor Ship D&C (2)

Illustration (from T. Fos-sensGuidance and Control ofOcean Vehicles) shows rela-tionship between intermediate

axis systems and roll, pitch,and yaw, angles


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Motion Equations for a Rigid Ship

Fb : v = v +

1

m

ffluid+ fthrust

+ Rbiagrav

Fb : = I1

I+ g

fluid+ gthrustFi : r = R

ib

v = S1()

where agrav= [0 0 g]T

Note that v and are expressed in Fb, and ris expressedin Fi

Much of the ship d & c literature follows aircraft notation:

v = [u v w]T, = [p q r]T

Assuming left-right symmetry (but not up-down symme-

try) implies that Ixy =Iyz = 0, but in general Ixz 6= 0


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Motion Equations for a Rigid Ship

Equations of motion are typically written in this form

M+ C()+ D()+ g() = E+

where = [vT

T

]T

, = [rT

T

]T

, M is the modifiedmass matrix, Cis a matrix involving the gyroscopic terms,Dis the damping matrix, gis the vector of restoring forcesand moments,E is the vector of environmental forces and

moments, and is the vector of control forces and moments

Determining the various matrices, forces and moments, israther complicated and is the subject of texts such as

T. I. Fossen, Guidance and Control of Ocean Vehicles, Wi-ley, 1994

M. S. Triantafyllou and F. S. Hover, Maneuvering and

Control of Marine Vessels, Department of Ocean Engineer-ing, MIT, 2003 (available online)


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Modeling Fluid Forces

Assumption: Forces and moments acting on a rigid body are a

linear combination of

1. Radiation-induced forces: body is forced to oscillate withwave excitation frequency and there are no incident wavesF Added mass: MACA()F Potential damping: DP()

F Weight and buoyancy:f

R()F Other damping effects, including skin friction, wave drift

damping, and vortex shedding damping

2. Froude-Kriloffand Diffraction Forces

3. Environmental forces: currents, waves, wind

4. Propulsion forces: thruster/propellor, control surfaces


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Horizontal Motion

The state vector is

x =h

vT T rT TiT

= [u v w p q r x y z ]

T

For horizontal motion, the ship has zero heave, roll, and pitch;the motion is restricted to surge, sway, and yaw

The control for horizontal motion typically involves thrustersand rudders; linearized equations are:

(m Xu)u = Xuu +X0

(m Yv)v+ (mxG Yr)r = Yvv+ (Yr mU)r+Y0

(mxG Nv)v+ (Izz Nr)r = Nvv (Nr mxGU)r+N0

The various Xu etc. terms are constants; the X0

etc. terms arethe control forces and moments


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Horizontal Motion (2)

Linearized equations for horizontal motion are:

(m Xu)u = Xuu +X0

(m Yv)v+ (mxG Yr)r = Yvv+ (Yr mU)r+ Y0

(mxG Nv)v+ (Izz Nr)r = Nvv (Nr mxGU)r+ N0

Clearly surge (u) is decoupled from sway (v) and yaw()

The coupled sway-yaw system is

m Yv mxG YrmxG Nv Izz Nr

x =

Yv Yr mUNv Nr mxGU

x+

f

Mx = A0x + f

x = M1A0x + M1f

x = Ax + Bu


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Horizontal Motion (3)

The coupled sway-yaw system is m Yv mxG YrmxG Nv Izz Nr

x =

Yv Yr mU

Nv Nr mxGU

x + f

Mx = A0x + f

x = M1A0x + M1f

x = Ax + Bu

Stability analysis leads to a simplified stability condition:

C=YvNr+Nv(mU Yr)>0

The term Cis called the vessel stability parameter

Ship design criteria are influenced by making C > 0 true; forexample, adding more aft surface area drives Nv positive, in-

creasing stablity


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Stability Analysis via Routh-Hurwitz

Suppose we know A in terms of system parameters; i.e., A =A(m, I, x, u, )

How can we determine stability using system parameters?

Routh-Hurwitz stability criteria

Develop the characteristic polynomial for A:

p() =n +an1n1 + +a0

Develop the Hurwitz matrix

H=

an1 an3 01 an2 00 an1 0...

......

...0 0 a0


11/15

Stability Analysis via Routh-Hurwitz (2)

The Hurwitz matrix is

H=

an1 an3 01 an2 0

0 an1 0......

......

0 0 a0

Define the principal minors ofH by

1 = an1

2 = det an1 an3

1 an2

...

n = det H


12/15


Necessary and sufficient conditions for the asymptotic stabilityofx=Ax are

i > 0 i= 1, , n

Since A depends on the parameters, p, the coefficients ofp()depend on p, and the principal minors depend on p

Thus these conditions defi

ne regions in parameter space wherethe linearized system is asymptotically stable

Stable and unstable regions are separated by stability bound-aries, which correspond to one or more eigenvalues crossing theimaginary axis

If a real eigenvalue crosses the imaginary axis, then at that pointin parameter space a0(p) = 0 (exercise: convince yourself that

this statement must be true)


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If a real eigenvalue crosses the imaginary axis, then at that pointin parameter space a0(p) = 0

If a c.c. pair crosses the imaginary axis, then

n1(p) = 0

Thus the stability boundaries in parameter space are defined byone of the two conditions :

a0(p) = 0

n1(p) = 0


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Suppose A =

A11 A12A21 A22

. Then the characteristic polyno-

mial is p() =2 (A11+A22) +A11A22 A12A21

The principal minors ofH are

1 = a1 = (A11+A22)>0 A11+ A22 0

(A11+A22)(A11A22

A12A21)0

These conditions were used in the sway-yaw example. Details

are in 4.2 of Triantafyllou and Hover


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Suppose A=

1 A122 4

. Then the conditions are

1 > 0 1 4 0 (4 + 2A12)>0

2A12 > 4 A12 > 2

Exercise: verify numerically that the appropriate stabilityboundary condition is satisfied when A12 = 2 and that thestability condition developed here is valid

marine vehicle motion

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