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Thor I. Fossen Professor of Guidance, Navigation and Control
Centre for Autonomous Marine Operations and Systems (AMOS)
Department of Engineering Cybernetics Norwegian University of
Science and Technology (NTNU)
NO-7491 Trondheim, Norway
Lecture Notes: TTK 4190 Guidance and Control of Vehicles
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
WIKI page: http://www.fossen.biz/TTK4190
Home page: http://www.fossen.biz/
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Textbook Fossen, T. I. (2011) Handbook of Marine Cra/
Hydrodynamics and Mo4on Control. John Wiley & Sons Ltd.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Useful References
Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles,
John Wiley & Sons Ltd. ISBN 0-471-94113-1
xb
yb
zb
u ( )surge
r ( )yaw
v ( )sway
( )heavew
( )rollp
( )pitchq
SNAME (1950). Nomenclature for TreaEng the MoEon of a Submerged
Body Through a Fluid. The Society of Naval Architects and Marine
Engineers, Technical and Research Bulle4n No. 1-5, April 1950, pp.
1-15.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Goals for the Course: 1. MathemaEcal modeling
of vehicles. This includes: KinemaEcs KineEcs EquaEons of
moEon
for marine craO and aircraO
Wind, wave and ocean current models
Hydrodynamics: maneuvering and seakeeping theory
Copyright Bjarne Stenberg/NTNU
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
2. Design of guidance, navigaEon and moEon control systems for a
large number of applicaEons
3. Simulate the moEons of marine craO and aircraO in the
Eme-domain using hydrodynamic/aerodynamic models
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TTK 4190 Guidance and Control (T. I. Fossen)
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TTK 4190 Guidance and Control (T. I. Fossen) Lecture Notes
2005
Pipe-laying vessel
Geological survey
Heavy lift operations
Position mooring
Pipe and cable laying
Marine Craft in Operation
ROV operations Vibration control of marine risers
Cable-laying vessel
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Marine Craft in Operation
Copyright Bjarne Stenberg/NTNU Lecture Notes TTK 4190 Guidance
and Control of Vehicles (T. I. Fossen)
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Italian supply ship Vesuvio refueling two ships at sea.
Courtesy: Hepburn Eng. Inc.
Path Following and Trajectory Tracking
Underactuated container ship in transit. Fully actuated supply
ship cruising at low speed.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Formation Control/Underway Replenishment
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Interdisciplinary: Rocket Launch / DP system / THCS
SMMarine Segment
Courtesy: SeaLaunch
http://www.sea-launch.com
Courtesy: SeaLaunch LLC
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Trim & Heel Correction System (THCS)
Process and Marine Control
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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NTNU Infrastructure
Copyright Bjarne Stenberg/NTNU
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Optimisation of hull resistance with and without waves
Identification of hydrodynamic parameters by:
Planar Motion Mechanism (PMM) tests
Vertical Motion Mechanism (VMM) tests
Towing Tests of Ships and Floating Structures
Model testing in Peerlesspool in London
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Marine Cybernetics Laboratory (MCLab)
MCLab Dimensions:
The soOware is developed by using rapid prototyping techniques
and automaEc code generaEon under Matlab/Simulink and RT-Lab.
The target PC onboard the model scale vessels runs the QNX
real-Eme operaEng system, while experimental results are presented
in real-Eme on a host PC using Labview.
L ! B ! D " 40 m! 6.5 m! 1. 5 m
Cybership II
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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NTNU Research Groups: Department of Marine Technology Department
of Biology incl. UNIS and CalPoly Department for Archaeology and
Religious Studies Department of Engineering CyberneEcs Centre for
Autonomous Marine OperaEons and Systems (AMOS)
Scientific Focus Areas: Development of technology for guidance,
navigaEon and control of underwater vehicles (ROVs and AUVs)
Environmental monitoring and mapping at sea surface, water
column, and sea bed
OperaEons under ice in the arcEc Study of any object of interest
(bio-geo-chemical objects) InspecEon/surveillance for environmental
agencies, oil industry, ecotoxicology
Deepwater archaeology Deepwater ecology research Complex
deepwater underwater operaEons including inspecEon and
intervenEon
Deepwater mineral extracEon
ROV Minerva
AUV REMUS 100 at Svalbard
Applied Underwater Robotics Laboratory AUR-Lab
RV Gunnerus
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NTNU Research Vessel Gunnerus 31 meters long Top speed 13
knots
http://www.ntnu.edu/marine/gunnerus
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UAV Factory Penguin B w/ Piccolo SL
28 m/s cruise speed
Gasoline, 8 hr
endurance
MTOW 21 kg
2-5 kg payload
capacity
Large payload bay
80W generator
Avionics system
integration made with Maritime Robotics based on Cloudcap
technology
Telemetry on 2.4 GHz radio, GPRS (and VHF)
Catapult launch
Custom payload
system integration with avionics interface
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Skywalker X8 w/Ardupilot 18 m/s cruise speed
Catapult launch
Belly or net landing
Electric, 1hr
endurance
Large payload bay
>1 kg payload
capacity
Inexpensive
Flexible avionics and
payload system integration with ArduPilot open source autopilot
and mission planning SW
Currently telemetry on 433 MHz or 5.8 GHz radio for VLOS
Can be set up for BLOS with GPRS and VHF radio links
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3DRobotics hexa-copter w/Ardupilot
Electric, 5-10 min
endurance
1 kg payload capacity
Inexpensive
Flexible system
integration with Ardupilot open source autopilot and mission
planning SW
Telemetry on 433 MHz og 5.8 GHz radio
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Microdrone Quad-copter
Turn-key solution
Various camera, video
and radio systems
Electric, 45 min
endurance
2-3 kg payload
capacity
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Marine craD: ships, high-speed craO, semi-submersibles, oaEng
rigs, submarines, remotely operated and autonomous underwater
vehicles, torpedoes and other propelled/powered structures for
instance a oaEng air eld. Vehicles that do not travel on land
(ocean and ight vehicles) are usually called craO. Vessel: "hollow
structure made to oat upon the water for purposes of transportaEon
and navigaEon; especially, one that is larger than a rowboat ". The
words vessel, ship and boat are oOen used interchangeably. In
Encyclopedia Britannica, a ship and a boat are disEnguished by
their size through the following deniEon: Ship: "any large oaEng
vessel capable of crossing open waters, as opposed to a boat, which
is generally a smaller craO. The term formerly was applied to
sailing vessels having three or more masts; in modern Emes it
usually denotes a vessel of more than 500 tons of displacement.
Submarine: "any naval vessel that is capable of propelling itself
beneath the water as well as on the water's surface. This is a
unique capability among warships, and submarines are quite dierent
in design and appearance from surface ships Underwater Vehicle:
"small vehicle that is capable of propelling itself beneath the
water surface as well as on the water's surface. This includes
unmanned underwater vehicles (UUV), remotely operated vehicles
(ROV) and autonomous underwater vehicles (AUV).
Marine Craft Ref. Encyclopedia Britannica
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Marine vessels are also classied according to their maximum
operaEng speed. For this purpose it is common to use the Froude
number
Marine Craft
Fn ! UgL
The pressure carrying the vessel can be divided into hydrostaEc
and hydrodynamic pressure. The corresponding forces are:
Buoyancy force due to the hydrostaEc pressures (proporEonal to
the displacement of the ship) Hydrodynamic force due to the
hydrodynamic pressure (approximately proporEonal to the square of
the speed)
Then we can classify the vessels according to (FalEnsen
2005):
Displacement vessels (Fn1.0-1.2): The hydrodynamic force mainly
carries the weight.
U: ship speed L: overall (submerged length of the ship) G:
acceleraEon of gravity
In this course, only displacement vessels are covered
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Degrees-of-Freedom (DOF)
xb
yb
zb
surge
yaw
sway
heave
roll
pitch
In maneuvering, a marine craO experiences moEon in 6 DOF. The
moEon in the horizontal plane is referred to as surge (longitudinal
moEon, usually superimposed on the steady propulsive moEon) and
sway (sideways moEon). Heading, or yaw (rotaEon about the verEcal
axis) describes the course of the vessel. The remaining three DOFs
are roll (rotaEon about the longitudinal axis), pitch (rotaEon
about the transverse axis), and heave (verEcal moEon). Roll is
probably the most troublesome DOF, since it produces the highest
acceleraEons and, hence, is the principal villain in seasickness.
Similarly, pitching and heaving feel uncomfortable to humans. When
designing ship autopilots, yaw is the primary mode for feedback
control. StaEonkeeping of a marine craO implies stabilizaEon of the
surge, sway and yaw modes.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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When designing feedback control systems for marine vessels,
reduced order models are oOen used since most vehicles do not have
actuaEon in all DOF. This is usually done by decoupling the moEons
of the vessel according to: 1-DOF models can be used to design
forward speed controllers (surge), heading autopilots (yaw) and
roll damping systems (roll). 3-DOF models are usually: Horizontal
plane models (surge, sway and yaw) for ships, semi-submersibles and
underwater vehicles that are used in dynamic posiEoning systems,
trajectory-tracking control systems and path-following systems. For
slender bodies like submarines, it is also common to assume that
the moEons can be decoupled into longitudinal and lateral moEons.
Longitudinal models (surge, heave and pitch) for forward speed,
diving and pitch control. Lateral model (sway, roll and yaw) for
turning and heading control. 4-DOF models (surge, sway, roll and
yaw) are usually formed by adding the roll equaEon to the 3 DOF
horizontal plane model. These models are used in maneuvering
situaEons where it is important to include the rolling moEon
usually with the purpose of reducing roll by acEve control of ns,
rudders or stabilizing liquid tanks. 6-DOF models (surge, sway,
heave, roll, pitch and yaw) are fully coupled equaEons of moEon
used for simulaEon and predicEon of coupled vessel moEons. These
models can also be used in advanced control systems for underwater
vehicles which can be actuated in all DOF.
Degrees of Freedom (DOF)
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Classification of Models Simulation Model: This model is the
most accurate descripEon of a system, for instance a 6-DOF
high-delity model for simulaEon of coupled moEons in the Eme
domain. It includes the marine craO dynamics, propulsion system,
measurement system and the environmental forces due to wind, waves
and ocean currents. The model should be able to reconstruct the Eme
responses of the real system and it should also be possible to
trigger failure modes so you can simulate accidents and erroneous
signals etc. SimulaEon models where the uid memory eects are
included (frequency-dependent models) typically consist of 50-200
ODEs while a frequency-independent model can be represented in 6
DOF with 12 ODEs for generalized posiEon and velocity. In addiEon
you need some states to describe the environmental loads and
actuators but sEll the number of states will be less than 50 for a
frequency-independent vessel model
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Classification of Models Control Design Model: The controller
model is a reduced-order or simplied version of the simulaEon model
that is used to design the moEon control system. In its simplest
form, this model is used to compute a set of constant gains for a
PID controller. More sophisEcated control systems use a dynamic
model to generate feedforward and feedback signals. This is
referred to as model-based control. The number of ODEs used in
convenEonal model-based ship control systems is usually less than
20. A PID controller typically requires two states: one for the
integrator and one for the low-pass lter used to limit noise
amplicaEon. Consequently, setpoint regulaEon in 6 DOF can be
implemented by using 12 ODEs. However, trajectory-tracking
controllers require addiEonal states for feedforward as well as
ltering so higher-order control laws are not uncommon. Observer
Design Model: The observer model will in general be dierent from
the model used in the controller since the purpose is to capture
the addiEonal dynamics associated with the sensors and navigaEon
systems as well as disturbances. It is a simplied version of the
simulaEon model where asenEon is given to accurate modeling of
measurement noise, failure situaEons including dead-reckoning
capabiliEes, ltering and moEon predicEon. For marine craO, the
model-based observer oOen includes a disturbance model where the
goal is to esEmate wave, wind and ocean current forces by treaEng
these as colored noise. For marine craO the number of ODEs in the
state esEmator will typically be 20 for a DP system while a basic
heading autopilot is implemented with less than 5 states.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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The Classical Models in Naval Architecture
m!u! ! vr " wq ! xg"q2 " r2# " yg"pq ! r!# " zg"pr " q! #$ #
Xm!v! ! wp " ur ! yg"r2 " p2# " zg"qr ! p! # " xg"qp " r!#$ # Ym!w!
! uq " vp ! zg"p2 " q2# " xg"rp ! q! # " yg"rq " p! #$ # ZIxp! "
"Iz ! Iy#qr ! "r! " pq#Ixz " "r2 ! q2#Iyz " "pr ! q! #Ixy
" m!yg"w! ! uq " vp# ! zg"v! ! wp " ur#$ # KIyq! " "Ix ! Iz#rp !
"p! " qr#Ixy " "p2 ! r2#Izx " "qp ! r!#Iyz
" m!zg"u! ! vr " wq# ! xg"w! ! uq " vp#$ # MIzr! " "Iy ! Ix#pq !
"q! " rp#Iyz " "q2 ! p2#Ixy " "rq ! p! #Izx
" m!xg"v! ! wp " ur# ! yg"u! ! vr " wq#$ # N
The moEons of a marine craO exposed to wind, waves and ocean
currents are usually modeled in 6 DOF by applying Newton's 2nd
law:
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
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The Classical Models in Naval Architecture
The external forces and moments X,Y,Z,K,M and N acEng on a
marine craO are usually modeled by using: Maneuvering Theory: The
study of a ship moving at constant posiEve speed U in calm water
within the framework of maneuvering theory is based on the
assumpEon that the maneuvering (hydrodynamic) coecients are
frequency independent (no wave excitaEon). The maneuvering model
will in its simplest representaEon be linear while nonlinear
representaEons can be derived using methods like cross-ow drag,
quadraEc damping or Taylor-series expansions. Seakeeping Theory:
The moEons of ships at zero or constant speed in waves can be
analyzed using seakeeping theory where the hydrodynamic coecients
and wave forces are computed as a funcEon of the wave excitaEon
frequency using the hull geometry. The frequency-dependent models
are usually derived within a linear framework while extensions to
nonlinear theory is an important eld of research.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Maneuvering Theory
Assumes that the ship is moving in restricted calm water. Hence,
the maneuvering model is derived for a ship moving at posiEve speed
U under a zero-frequency wave excitaEon assumpEon such that added
mass and damping can be represented by using hydrodynamic
derivaEves (constant parameters). The zero-frequency assumpEon is
only valid for surge, sway and yaw since the natural period of a PD
controlled ship will be in the range of 100-150 s. For 150 s this
result in:
!n ! 2"T! 0.04 rad/s #
The natural frequencies in heave, roll and pitch are much higher
so it is not straighworward to remove the frequency dependence in
these channels.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Maneuvering Theory It is common to formulate the ship
maneuvering model as a coupled surge-sway-yaw model and thus
neglect heave, roll and pitch moEons:
Hydrodynamic added mass potenEal damping due to wave radiaEon
and viscous damping HydrostaEc forces (spring sEness) Wind forces
Wave forces (1st- and 2nd-order) Control and propulsion forces
m!u! ! vr ! xgr2 ! ygr! " " Xm!v! # ur ! ygr2 # xgr! " " YIzr! #
m!xg#v! # ur$ ! yg#u! ! vr$" " N
#
MRB!" ! CRB!!"! " #RB #
!RB ! !hyd " !hs hydrodynamic andhydrostatic forces
" !wind " !waveenvironmental forces
" !control #
!i ! !Xi,Yi,Zi,Ki,Mi,Ni"!, i ! #hyd, hs, wind, wave, control$
#
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Linearized Maneuvering Models
In the linear 6-DOF case there will be a total of 36 mass and 36
damping elements proporEonal to velocity and acceleraEon. In
addiEon to this, there will be restoring forces, propulsion forces
and environmental loads. If the generalized force is wrisen in
component, the linear added mass and damping forces become:
where Xu,Xv , . . . ,Nr are the linear damping coefficientsand
Xu! ,Xv! , . . . ,Nr! represent hydrodynamic added mass.
!hyd
X1 ! Xuu " Xvv " Xww " Xpp " Xqq " Xrr" Xu# u# " Xv# v# " Xw# w#
" Xp# p# " Xq# q# " Xr#r#
!
N1 ! Nuu " Nvv " Nww " Npp " Nqq " Nrr" Nu# u# " Nv# v# " Nw# w#
" Np# p# " Nq# q# " Nr#r#
#
#
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Nonlinear Maneuvering Models ApplicaEon of nonlinear theory
implies that many elements must be included in addiEon to the 36
linear elements. Truncated Taylor-series expansions using odd terms
(1st- and 3rd-order) which are sed to experimental data (Abkowitch
1964) or 2nd-order modulus terms (Fedyaevsky 1963)
The equaEons become relaEvely complicated due to the large
number of hydrodynamic coecients on the right-hand side needed to
represent the hydrodynamic forces. Taylor-series expansions are
frequently used in commercial planar moEon mechanism (PMM) tests
where the purpose is to derive the maneuvering coes.
experimentally.
X1 ! Xu" u" # Xuu # Xuuuu3 # Xv" v" # Xvv # Xvvvv3 #!"
N1 ! Nu" u" # Nuu # Nuuuu3 # Nv" v" # Nvv # Nvvvv3 #!
#
#
X1 ! Xu" u" # Xuu # X |u|u|u|u # Xv" v" # Xvv # X |v|v |v|v
#!"
N1 ! Nu" u" # Nuu # N |u|u|u|u # Nv" v" # Nvv # N |v|v |v|v
#!
#
#
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Seakeeping, is the study of moEon when there is wave excitaEon
and the craO keeps its course and its speed constant (which
includes the case of zero speed). This introduces a dissipaEve
force known as uid memory eects (Cummins 1962). The governing model
is formulated in the Eme domain (Cummins equaEon):
Seakeeping Theory
!MRB ! A"!#$!" ! B total"!#!# ! "0
tK"t # !#!#"!#d! ! C! " $wind ! $wave ! "$ #
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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Unified Theory: Seakeeping & Maneuvering
A unied theory for maneuvering and seakeeping is useful since it
allows for Eme-domain simulaEon of a ship in a seaway. This is
usually done by solving the linear seakeeping equaEons of moEon in
the Eme domain (includes uid memory eects). The next step is to
assume linear superposiEon such that wave forces can be added for
dierent speeds U and sea states. A similar assumpEon is used to add
nonlinear damping and restoring forces such that the resulEng model
is a unied nonlinear model combining the most important terms from
both the maneuvering and seakeeping theories. Care with respect to
"double counEng" must be taken. This refers to the problem that
hydrodynamic eects can be modeled twice when merging the results
from two theories. For instance, since both models have linear
damping, the resulEng damping matrix must be chosen such that it
represents the total damping.
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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- body-xed velociEes: - posiEon and Euler angles: - M, C and D
denote the system inerEa, Coriolis and damping matrices - g is a
vector of gravitaEonal and buoyancy forces and moments
- q is a vector of joint angels - is a vector of torque - M and
C are the system inerEa and Coriolis matrices
In Fossen (1991) the robot model:
M!q"q! " C!q,q# "q $ %
was used as foundaEon to write the 6 DOF marine craO equaEons of
moEon in a compact vectorial sexng. Vectorial RepresentaEon The
robot model was modied to describe marine craOs in a vectorial
sexng
!
! ! !u,v,w,p, q, r"T! ! !x,y, z,!, ",#"T
Fossen's Robot-Like Vectorial Model for Marine Craft
M!" ! C!!"! ! D!!"! ! g!#" ! g0 " $ ! $wind ! $wave
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I.
Fossen)
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It is advantageous to use the model representaEon Fossen (1991)
since nonlinear system properEes like:
symmetry of matrices skew-symmetry of matrices posiEveness of
matrices
can be exploited in the stability analysis. These properEes
relates to passivity of the model. These properEes also represents
physical properEes of the system which should be exploited
when designing nonlinear controllers and observers for marine
vessels.
Model Representations for Marine Craft
M!" ! C!!"! ! D!!"! ! g!#" ! g0 " $ ! $wind ! $wave
Lecture Notes TTK 4190 Guidance and Control (T. I. Fossen)
Copyright Bjarne Stenberg/NTNU