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1 TTK 4190 Guidance and Control (T. I. Fossen) Lecture Notes 2005

Chapter 3 - Dynamics of Marine Vessels

3.1 Rigid-Body Dynamics

3.2 Hydrodynamic Forces and Moments

3.3 6 DOF Equations of Motion3.4 Model Transformations Using Matlab

3.5 Standard Models for Marine Vessels

M C

D

g

go w

M - system inertia matrix (including added mass)

C

- Coriolis-centripetal matrix (including added mass)

D

- damping matrix

g

- vector of gravitational/buoyancy forces and moments

- vector of control inputs

go - vector used for pretrimming (ballast control)

w - vector of environmental disturbances (wind, waves and currents)

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A floating or submerged vessel can be pretrimmed by pumping water between the

ballast tanks of the vessel. This implies that the vessel can be trimmed in heave,

pitch and roll:

3.2.4 Ballast Systems

z zd, d, d 3 modes with restoring forces/moment

Steady-state solution:

M C

D

g

go XX X X

gdgo w

d, , ,zd,d,d, where

The ballast vectorgo is computed by using hydrostatic analyses.

main equation for ballast computations

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Consider a marine vessel with n ballast tanks of volumes ViVi,max (i=1,,n).

For each ballast tank the water volume is defined:


Vi hi o

hiAi hdh Ai hi , (Ai hconstant)

A (h )i i

hi Vi

Wi

g

x

z

zoom in

Zballast i1n

Wi g i1n

Vi

The gravitational forces Wi in heave are:

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Restoring moments due to the heave force Zballast:

rib xi ,yi ,zi , i 1, , nBallast tanks location with respect to O:

Kballast gi1

n

y iVi

Mballast gi1

n

x iVi

m r f

x

y

z

0

0

Zballast

yZballast

xZballast

0

go

0

0

Zballast

Kballast

Mballast

0

g

0

0

i1n

Vi

i1

nyi Vi

i1

nx i Vi

0

Resulting

ballast

model:

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Conditions for Manual Pretrimming

Trimming is usually done under the assumptions that and are small such:


Reduced order system (heave, roll, andpitch):

d d

gd

Gd

Gr Zz 0 Z

0 K 0

Mz 0 M

gor g

i1n

Vi

i1

ny i Vi

i1

nx i Vi

dr zd,d,d

wr w3, w4, w5

Grdr go

r wr

Zz 0 Z

0 K 0

Mz 0 M

zd

d

d

g i1

nVi w3

g i1

ny i Vi w4

g i1

nx i Vi w5

Steady-state

condition:

This is a set of linear

equations wherethe volumes Vi

can be found by

assuming that wi=0

(zero disturbances)

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Assume that the disturbances in heave, roll, andpitch have means of zero.Consequently:

and

can be written:


Zz 0 Z

0 K 0

Mz 0 M

zd

d

d

g i1

nVi w3

g i1

nyi Vi w4

g i1

nxi Vi w5

wr w3, w4, w5 0

H y

g

1 1 1

y1 yn1 yn

x1 xn1 xn

V1

V2

Vn

ZzzdZd

Kd

MzzdMd

H y H HH 1y

The water volumes Viis found by using the pseudo-inverse:

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Example (Semi-Submersible Ballast Control) Consider a semi-submersible

with 4 ballast tanks located at

In addition,yz-symmetry implies that


P P

P

P

P P

V1 V2

V4 V3

xb

yb

O

p1

p2

p3

+

+

+

r1b x, y, r2

b x, y, r3b x,y,r4

b x,y

Z Mz 0

H g

1 1 1 1

y y y y

x x x x

y

Zzzd

Kd

Md

gAwp0zd

gGMTd

gGMLd

V1

V2

V3

V4

14g

1 1y1x

1 1y 1x

1 1y 1x

1 1y1x

gAwp

0zd

gGMTd

gGMLd

H y H HH 1y

Inputs: zd,d,d

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An example of a highly sophisticated pretrimming system is theSeaLaunch trim and heel correction system (THCS):

3.2.4 Ballast SystemsSeaLaunch:

This system is designed such

that the platform maintains

constant roll and pitch angles

during changes in weight. Themost critical operation is when

the rocket is transported from

the garage on one side of the

platform to the launch pad.

During this operation the

water pumps operate at theirmaximum capacity to

counteract the shift in weight.

A feedback system controls the pumps to maintain the

correct water level in each of the legs during

transportation of the rocket

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Automatic Pretrimming using Feedback from

In the manual pretrimming case it was assumed that wr=0. This assumption canbe removed by using feedback.

The closed-loop dynamics of a PID-controlled water pump can be described by a

1st-order model with amplitude saturation:

Tj (s) is a positive time constant

pj (m/s) is the volumetric flow rate pumpj

pdj is the pump set-point.

The water pump capacity is different for

positive and negative flow directions:

zd,d,d

Tjpj pj satpdj

satpdj

pj,max pj pj,max

pdj pj,max pdj pj,max

pj,max pdj pj,max

0.63 ,

,

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Example (Semi-Submersible Ballast Control, Continues): The water flow

model corresponding to the figure is:


P P

P

P

P P

V1 V2

V4 V3

xb

yb

O

p1

p2

p3

+

+

+

V1 p1

V2 p3

V3 p2 p3

V4 p1 p2

Tp p satpd

Lp

V1

V2V3

V4

, p

p1

p2

p3

, L

1 0 0

0 0 10 1 1

1 1 0

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-T-1 Lsat( ).

ppd

go ( )

r

-

ballastcontroller

Gr

d

Closed-loop pump dynamics with water volume as output

r

r

( )Gr

-1

Steady-state relationship forwater volume and trim


Gr r gor wr

Feedback control system:

pd HpidsGr

dr

r

Hpidsdiagh1,pids, h2,pids, . . . , hm,pids

Tp p satpd

Lp

Equilibrium equation:Dynamics:

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SeaLaunch Trim and Heel Correction System (THCS)

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S

Marine Segment


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0 187.5 375 562.5 750 937.5 1125 1312.5 1500

21.5

10.5

00.5

11.5

22.5

33.5

44.5

55.5

6

Roll and pitch during launch time (secs)

rollandp

itch(

deg)

420 430 440 450 460 4702

0

2

4

6

Measured pitch during launch

time (secs)

Pitch

angle(deg)

4.21

0.95

A1

jp

470420 jp20 10 0 10 20 30

2

0

2

4

6

Calculated pitch motions

time (secs)

pitcha

ngle

(deg)

4.326

0.202

Z4

l180

29.77515Z

1 l

roll

pitch

3.2.4 Ballast SystemsRoll and pitch angles during lift-off

CNN

10th October 1999

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3.3 6 DOF Equations of Motion

Body-Fixed Vector Representation

M C D g go w

J

M MRB MA

CCRBCA

DDPDSDWDM

NED Vector Representation

Kinematic transformation (assuming that exists-i.e., ):J1

/2

J J1

JJ J1JJ1

MJM J1

C, JCMJ1JJ1

D, JDJ1

gJg

MC, D, gJ go w

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3.3.1 Nonlinear Equations of Motion

(1) MM 0 6

(2) s

M

2C

, s 0 s 6

, 6

, 6

(3) D, 0 6, 6

Properties of the NED Vector Representation

ifM M 0 and M 0.

It should be noted that C, will not be skew-symmetrical although Cis skew-symmetrical.

MC, D, gJ go w

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M

m Xu Xv Xw

Xv m Yv Yw

Xw Yw m Zw

Xp mz gYp mygZp

mz gXq Yq mxgZq

mygXr mxgYr Zr

Xp mz gXq mygXr

mzgYp Yq mxgYr

mygZp mxgZq Zr

IxKp IxyKq IzxKr

IxyKq IyMq IyzMr

IzxKr IyzMr IzNr

Property (System Inertia Matrix) Fora rigid body thesystem inertia matrix is

strictly positive if and only ifMA>0, that is:

If the body is at rest (or at most is moving at low speed) under the assumption of

an ideal fluid, the zero-frequency system inertia matrix is always positive definite,

that is

where:

M MRB MA 0

M M 0

M M

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Property (Coriolis and Centripetal Matrix): For a rigid body moving

through an ideal fluid the Coriolis and centripetal matrix can always beparameterized such that it is skew-symmetric, that is

IfM is nonsymmetric, we write M as the sum of asymmetric andskew-

symmetric matrix:

where

This implies that we can compute C from

CC , 6

12

M M 0M 0 T 12

M 12

M 0

M 12

M M 12

M M

M M 12

M M0

M M 0


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Assumption (Small Roll and Pitch Angles) The roll and pitch angles:

These are good assumptions for vessels where the pitch and roll motions arelimited-i.e., highly metacentric stable vessels

This assumption implies that:

where

3.3.2 Linearized Equations of Motion

,are small

J0

P

P R 033

033 I33

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Definition (Vessel Parallel Coordinate System) The vessel parallel coordinate

system is defined as:

where is the NED position/attitude decomposed in body coordinates and P

is given by

Notice that PTP = I66.

pP

p

P R 033

033 I33


NED

BODYp

xb

ybxn

yn

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Low Speed Applications (Station-Keeping)

Vessel parallel (VP) coordinates implies:


pP

P

P

Pp

PP

rSp

where andr

S

0 1 0 0 0 0

1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

For low speed applications r 0. This gives a linear model:

p

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The gravitational and buoyancy forces can also be expressed in terms ofVP

coordinates. For small roll and pitch angles:

Notice that this formula confirms that the restoring forces of a leveled vessel( ) is independent of the yaw angle .

g0

PG PGP G

pG

p

0

For a neutrally buoyant submersible (W=B) withxg=xb andyg=yb we have:

For a surface vessel G is defined as:

G diag0,0,0,0, zg zbW, zg zbW, 0

G

022 0230

0

032 Gr

0

0

0 0 0 0 0 0

, Gr

Zz 0 Z

0 K 0

Mz 0 M

PGP G

Notice that:

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Low-Speed Maneuvering and DP: implies that the nonlinear Coriolis,

centripetal, damping, restoring, and buoyancy forces and moments can belinearized about and . Since C(0)=0 and Dn(0)=0 it makes

sense to: approximate:

M C0

D DnD

g

Gp

go

p

M D Gp

w

0

0 0


The resulting state-space model becomes:

x Ax Bu E

A0 I

M1G M1D, B

0

M1, E

0

M1

x p, , u

which is the linear time invariant(LTI) state-space model used in DP.

P p

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Vessels in Transit (Cruise Condition):

For vessels in transit the cruise speed is assumed to satisfy:

This suggests that

where

u uo

N uo C D | o

o uo,0,0,0,0,0

p o

M N uo G p w

o

Linear parameter varying(LPV) model:

x Auox Bu Ew F o

A uo 0 I

M1G M1N uo, B

0

M1, E

0

M1, F

I

0

x p,

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Models for ships, semi-submersibles, and underwater vehicles are usually

represented as one of the following subsystemes:

Surge model: velocity u

Maneuvering model (sway and yaw): velocities v and r

Horizontal motion (surge, sway, and yaw): velocities u,v, and r

Longitudinal motion (surge, heave, and pitch): velocities u,w, and qLateral motion: (sway, roll, and yaw): velocities v,p, and r

or:

Horizontal plane models: DOFs 1, 2, 6

Longitudinal motion: DOFs 1, 3, 5

Lateral motion: DOFs 2, 4, 6

3.5 Standard Models for Marine Vessels

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The horizontal motion of a ship or

semi-submersible is described bythe motion components insurge,

sway, andyaw.

This implies that the dynamics

associated with the motion in heave,

roll, andpitch are neglected, that is

w=p=q=0.

3.5.1 3 DOF Horizontal Motion

u, v, r n, e,

Low-speed applications-i.e., dynamically positioned ships where U0,

and maneuvering at high speed will now be treated separately.

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Low-Speed Model for Dynamically Positioned Ship

Consider the 6 DOF kinematic expressions:

For small roll and pitch angles and no heave this reduces to:

Rb

n

cc sccss ssccs

s

c

c

c

s

ss

c

s

ss

c

s cs cc

T

1 st ct

0 c s

0 s/c c/c

JRb

n 033

033 T

J3 DOF

R

cos sin 0

sin cos 0

0 0 1

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Assume that the ship has homogeneous mass distribution,xz-plane symmetry andyg=0:

MRB

m 0 0 0 mzg myg

0 m 0 mzg 0 mxg

0 0 m myg mxg 0

0 mzg myg Ix Ixy Ixz

mzg 0 mxg Iyx Iy Iyz

myg mxg 0 Izx Izy Iz


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

MRB

m 0 0

0 m mxg

0 mxg Iz

CRB

0 0 mxgr v

0 0 mu

mxgr v mu 0

MA

Xu 0 0

0 Yv Yr

0 Yr Nr

CA

0 0 Yvv Yrr

0 0 Xuu

Yvv Yrr Xuu 0

X

X

X

XX

X

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3.5.1 3 DOF Horizontal MotionFor the 3 DOF low speed model, M = MT and C = -CT, that is:

As for the system inertia matrix, linear damping in surge is decoupled from swayand yaw. This implies that:

Linear damping is a good assumption for low-speed applications. Similarly thequadratic velocity terms given by are negligible in DP

M m Xu 0 0

0 m Yv mxgYr

0 mxgYr IzNr

C

0 0 m Yvv mxgYrr

0 0 m Xuu

m Yvv mxgYrr m Xuu 0

D

Xu 0 0

0 Yv Yr

0 Nv Nr

C

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Resulting Low-Speed (DP) Model:

where

B is the control matrix describing the thruster configuration and u is the control input.

Nonlinear Maneuvering Model:

At higher speeds the assumptions that

This suggests the following 3 DOF nonlinear maneuvering model:

R

M D

Bu M = MT>0 and D = DT>0

D D Dn D and C 0 are violated

R

M C D

l d d l f d

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3.5.2 Decoupled Models for Forward

Speed/Maneuvering

For vessels moving at constant (or at least slowly-varying) forward speed:

the 3 DOF maneuvering model can be decoupled in a:

Forward speed (surge subsystem)

Sway-yaw subsystem for maneuvering

Forward Speed Model

Starboard-port symmetry implies that surge is decoupled from sway and yaw:

where is the sum of control forces in surge. Notice that both linearand quadratic

dampinghave been included in order to cover low- and high-speed applications.

U u2 v2 u

m XuuXuu X|u|u|u|u 1

1

3 2 l d d l f d

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Speed/Maneuvering

2 DOF Linear Maneuvering Model (Sway-Yaw Subsystem)

A linear maneuvering model is based on the assumption that the cruise speed:

while v and rare assumed to be small.

Representation 1 (see also lecture notes by Professor David Clark)

The 2nd and 3rd rows in the DP model

with u=uo, yields:

u uo constant

C

0 0 m Yvv mxgYrr

0 0 m Xuu

m Yvv mx gYrr m Xuu 0

C m Xuuor

m Yvuov mxgYruor m Xuuov

0 m Xuuo

XuYvuo mxgYruo

v

r

C C

Notice that

3 5 2 D l d M d l f F d

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Speed/Maneuvering

Assume that the ship is controlled by a single rudder:

and that linear damping dominates:

then:

where

bY

N

D D DnD

M N uo b

M m Yv mxgYr

mxgYr IzNr

NuoYv m XuuoYr

XuYvuoNv mxgYruoNr

bY

N

v, r

This is the linear maneuvering model

as used by Clark, Fossen and others.

Developed from MRB, CRB, MA, CA

Notice:N includes the famous

Munk moment and some otherCA-terms

3 5 2 D l d M d l f F d

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Speed/Maneuvering2 DOF Linear Maneuvering Model (Sway-Yaw Subsystem)

Representation 2 (Davidson and Schiff 1946). Starts with Newtons law:

where linear terms in acceleration, velocity and rudderare added according to:

Notice: This approach does not included the CA-matrix. The resulting model is:

In this model theMunk momentis missing in the yaw equation. This is a destabilizing moment

known from aerodynamics which tries to turn the vessel. Also notice that two other less

important CA-terms are removed from N(uo) when compared to Representation 1.

MRB CRB RB

RB YN

Yv YrNv Nr

Yv YrNv Nr

M N uo b

M m Yv mxgYr

mxgYr IzNr, N uo

Yv muoYr

Nv mxguoNr, b

Y

N

3 5 2 D l d M d l f F d

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Speed/Maneuvering

1 DOF Autopilot Model (Yaw Subsystem)

A linear autopilot model for course control can be derived from the maneuvering model

by defining the yaw rate ras output:

Hence, application of theLaplace transformation yields (Nomoto 1957):

The 1st-order Nomoto modelis obtained by defining the equivalent time constant as:

M N uo b

r c , c 0, 1

r

s

K1T3s

1T1s1T2s2nd-order Nomoto model

T T1 T2 T3

r

s K

1Ts

s Ks1Tsr

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3.5.3 Longitudinal and Lateral Models

The 6 DOF equations of motion can in many cases be divided into two non-

interacting (or lightly interacting) subsystems:

Longitudinal subsystem: states u,w,q, and

Lateral subsystem: states v,p,r, and

This decomposition is good forslender bodies (large length/width ratio). Typical

applications are aircraft, missiles, andsubmarines.

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yz-plane of symmetry

(fore/aft symmetry)


M

m11 m12 0 0 0 m16

m21 m22 0 0 0 m26

0 0 m33 m34 m35 0

0 0 m43 m44 m45 0

0 0 m53 m54 m55 0

m61 m62 0 0 0 m66

M

m11 0 m13 0 m15 0

0 m22 0 m24 0 m26

m31 0 m33 0 m35 0

0 m42 0 m44 0 m46

m51 0 m53 0 m55 0

0 m62 0 m64 0 m66

M

m11 0 0 0 m15 0

0 m22 0 m24 0 0

0 0 m33 0 0 0

0 m42 0 m44 0 0

m51 0 0 0 m55 0

0 0 0 0 0 m66

xy-plane of symmetry(bottom/top symmetry):

xz-plane of symmetry

(port/starboard symmetry)

M diagm11, m22, m33,m44 ,m55, m66

xz-, yz- andxy-planes of symmetry

(port/starboard, fore/aft and bottom/top

symmetries).

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Starboard-port symmetry implies the following zero elements:

The longitudinal and lateral submatrices are:

M

m11 0 m13 0 m15 0

0 m22 0 m24 0 m26

m31 0 m33 0 m35 0

0 m42 0 m44 0 m46

m51 0 m53 0 m55 0

0 m62 0 m64 0 m66


Mlong

m11 m13 m15

m31 m33 m35

m51 m53 m55

, Mlat

m22 m24 m26

m42 m44 m46

m62 m64 m66

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Longitudinal Subsystem (DOFs 1, 3, 5)


Rbn

cc sccss ssccs

sc ccsss csssc

s cs cc

T

1 st ct

0 c s

0 s/c c/c

JRbn 033

033 T

d

cos 0

0 1

w

q

sin

0u

v,p, r,are small

not controlling the N-position

using speed control instead

Resulting kinematic equation:

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For simplicity, it is assumed that higher order damping can be neglected, that is. Coriolis is, however, modelled by assuming that and that

2nd-order terms in v,w,p,q, and rare small. Hence, DOFs 1, 3, 5 gives:

Assuming a diagonal MA gives:


Dn0 u 0

CRB

0 0 0

0 0 mu

0 0 mxgu

u

w

q

CRB

mygq zgrp mxgq wq mxgr vr

mzgp vp mz gq uq mxgp ygqr

mxgq wu mz gr xgpv mz gq uw Iyzq Ixzp Izrp Ixz r Ixyq Ixpr

Collecting terms in u,w, and q, gives:

CRBCRB

The skew-symmetric property is

destroyed for the decoupled model:

CA

Zwwq Yvvr

Yvvp Xuuq

ZwXuuw NrKppr

0 0 0

0 0 Xuu

0 ZwXuu 0

u

w

q

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The restoring forces with W=B andxg=xb:


g

WBsin

WBcossin

WBcoscos

ygWybBcoscos zgWzbBcossin

zg

WzbBsin xgWxbBcoscos

xg

WxbBcossin ygWybBsin

g 00

W BGz sin

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3.5.3 Longitudinal and Lateral ModelsLongitudinal Subsystem (DOFs 1, 3, 5)

m Xu Xw mzg XqXw m Zw mxg Zq

mzg Xq mxg Zq Iy Mq

uw

q

Xu Xw Xq

Zu Zw Zq

Mu Mw Mq

u

w

q

0 0 0

0 0 m Xuu0 ZwXuu mxgu

u

w

q

0

0

W BGz sin

1

3

5

m Zw mxgZq

mxgZq IyMq

w

q Zw Zq

Mw Mq

w

q

0 m Xuuo

ZwXuuo mxguo

w

q

0

BGzWsin

3

5

u uo constant

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3.5.3 Longitudinal and Lateral ModelsLongitudinal Subsystem (DOFs 1, 3, 5)

Linear pitch dynamics (decoupled):

where the natural frequency is:

Iy MqMqBGzW 5

BGz W

IyMq

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Lateral Subsystem (DOFs 2, 4, 6)


Rbn

cc sccss ssccs

sc ccsss csssc

s cs cc

T

1 st ct

0 c s

0 s/c c/c

JRbn 033

033 T

not controlling the E-position

using heading control instead

Resulting kinematic equation:

p

r

u, w,p, r,and are small

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Again it is assumed that higher order velocity terms can be neglected so that. Hence:

Assuming a diagonal MA gives:


Dn0

Collecting terms in v,p, and r, gives:

CRBCRB

The skew-symmetric property is

destroyed for the decoupled model:

CRB

mygp wp mzgr xgpq mygr ur

mygq z gru mygp wv mz gp vw Iyzq Ixzp Izrq Iyzr Ixyp Iyqr

mxgr vu mygr uv mxgp ygqw Iyzr Ixyp Iyqp Ixzr Ixyq Ixpq

CRB

0 0 muo

0 0 0

0 0 mxguo

v

p

r

CA

Zwwp Xuur

YvZwvw MqNrqr

XuYvuv KpMqpq

0 0 Xuu

0 0 0

XuYvu 0 0

v

p

r

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The restoring forces with W=B, xg=xb andyg=zg:

g

WBsin

WBcossin

WBcoscos

ygWybBcoscos zgWzbBcossin

zg

WzbBsin xgWxbBcoscos

xg

WxbBcossin ygWybBsin


g

0

W BGz sin

0

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3.5.3 Longitudinal and Lateral ModelsLateral Subsystem (DOFs 2, 4, 6)

u uo constant

m Yv mzg Yp mxg Yrmzg Yp Ix Kp Izx Kr

mxg Yr Izx Kr Iz Nr

vp

r

Yv Yp YrMv Mp Mr

Nv Np Nr

vp

r

0 0 m Xuu

0 0 0

XuYvu 0 mxgu

v

p

r

0

W BGz

sin

0

2

4

6

m Yv mxgYr

mxgYr IzNr

vr

Yv YrNv Nr

v

r

0 m Xuuo

XuYvuo mxguo

v

r

2

6

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3.5.3 Longitudinal and Lateral ModelsLateral Subsystem (DOFs 2, 4, 6)

Linear roll dynamics (decoupled):

where the natural frequency is:

Ix KpKpWBGz 4

BGz W

IxKp

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CH3 Part2