Download - 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)
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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
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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
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Stability Analysis via Routh-Hurwitz (3)
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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Stability Analysis via Routh-Hurwitz (4)
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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Stability Analysis via Routh-Hurwitz (5)
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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Stability Analysis via Routh-Hurwitz (6)
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