Chapter 7 Models for Ships, Offshore Structures and Underwater … · 2014-08-11 · 4! Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) ! M RB! m 0 0 0 mzg!myg

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

Chapter 7 - Models for Ships, Offshore Structures and Underwater Vehicles

This chapter presents hydrodynamic models for ships, offshore structures and underwater vehicles. The founda9on for the models are the kinema9c equa9ons (Chapter 2), rigid-‐body kine9cs (Chapter 3), hydrosta9cs (Chapter 4), seakeeping theory (Chapter 5) and maneuvering theory (Chapter 6). The models are all expressed in a vectorial seHng for effec9ve computer simula9on and to simplify control design. Focus is made towards preserva9on of matrix proper9es such as symmetry, skew-‐symmetry, posi9ve definiteness and orthogonality which are key elements in nonlinear control and es9ma9on theory.

7.1 Maneuvering Models (3 DOF) 7.2 Autopilot Models for Heading Control (1 DOF) 7.3 DP Models (3 DOF) 7.4 Maneuvering Models including Roll (4 DOF) 7.5 Equa9ons of Mo9on (6 DOF)


The horizontal mo9ons of a ship or semi-‐submersible are described by the mo9on components in surge, sway, and yaw.

This implies that the dynamics associated with the mo9on in heave, roll and pitch are neglected, that is w = p = q = 0.

7.1 Maneuvering Models (3 DOF)

! ! !u, v, r"!

Rigid-‐body dynamics – horizontal plane models for maneuvering

MRB!" ! CRB!!"! " #RB # !RB ! !hyd " !hs " !wind " !wave " ! #

! ! !N,E,!"!

!hyd ! !MA"# ! CA!"r""r ! D!"r""r #

N!!r"!r :! CA!!r"!r " D!!r"!r #


7.1 Maneuvering Models (3 DOF) Consider the 6-‐DOF kinema9c expressions:

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

Rbn!!" !

c!c" !s!c# " c!s"s# s!s# " c!c#s"s!c" c!c# " s#s"s! !c!s# " s"s!c#!s" c"s# c"c#

T!!!" "

1 s!t" c!t"0 c! !s!0 s!/c" c!/c"

J!!" !Rbn!"" 03!303!3 T"!""

J!!" 3 DOF! R!!! !

cos! ! sin! 0

sin! cos! 0

0 0 1

Copyright © Bjarne Stenberg/NTNU


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 !I zx !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 !m!xgr " v"

0 0 mu

m!xgr " v" !mu 0

MA !

!Xu" 0 0

0 !Yv" !Yr"

0 !Yr" !Nr"

CA!!" !

0 0 Yv" v # Yr" r

0 0 !Xu" u

!Yv" v ! Yr" r Xu" u 0

X

X

X

X X X

Assume that the ship has homogeneous mass distribu9on, xz-‐plane symmetry and yg= 0



!" ! R!!"#M "# # CRB!#"# # N!#r"#r ! $ # $wind # $wave

# #

N!!r"!r :! CA!!r"!r " D!!r"!r # J!!" 3 DOF! R!!" !

cos! ! sin! 0

sin! cos! 0

0 0 1

MA !

!Xu" 0 00 !Yv" !Yr"0 !Yr" !Nr"

CA!!" !0 0 Yv" v # Yr"r0 0 !Xu" u

!Yv" v ! Yr"r Xu" u 0

Resul&ng Model

MRB !

m 0 0

0 m mxg

0 mxg Iz

CRB!!" !

0 0 !m!xgr " v"

0 0 mu

m!xgr " v" !mu 0

M !

m ! Xu" 0 0

0 m ! Yv" mxg!Yr"

0 mxg!Yr" Iz!Nr"

6


Dynamic Posi9oning Models •  3-‐D poten9al theory •  2-‐D poten9al theory (strip theory)

0 1.5 m/s (3 knots) …. …..

Speed Sta9onkeeping

Low-‐speed maneuvering

Maneuvering at moderate speed (transit)

Maneuvering Models •  2-‐D poten9al theory (strip theory) up to Froude numbers of 0-‐3-‐0.4

•  2.5-‐D poten9al theory for high-‐speed cra^

LgU 3.0=

Maneuvering at high speed (high-‐speed cra^)

Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Speed Regimes – Hydrodynamic Methods (see Chapters 5 and 6)

7

-8 -6 -4 -2 0 2 4 6 8 -8

-6

-4

-2

0

2

4

6

8 Linear and quadratic damping and their regimes

u (m/s)

linear damping: 0.5u*exp(-0.5u*u) quadratic damping: 0.05|u|u linear + quadratic damping

Dynamic positioning |u| < 2 m/s


The figure illustrates the significance of linear and quadra9c damping for low-‐speed and high-‐speed applica9ons.

Dynamic Posi9oning (sta9onkeeping and low-‐speed maneuvering): Linear damping dominates

Maneuvering (high-‐speed):

Nonlinear damping dominates

7.1 Maneuvering Models (3 DOF) Speed Regimes – Linear and Nonlinear Damping


7.1.1 Nonlinear Maneuvering Models based on Surge Resistance and Cross-Flow Drag Expanding the 3-‐DOF N-‐matrix into: Added mass Coriolis-‐centripetal terms + linear damping + nonlinear damping

N!!r"!r ! CA!!r"!r " D!!r"!r! CA!!r"!r " D!r " d!!r"

# #

CA!!r" !0 0 Yv" vr # Yr"r0 0 Xu" ur

!Yv" vr ! Yr"r !Xu" ur 0

D !

!Xu 0 00 !Yv !Yr0 !Nr !Nr

d!!r" !

12 !S!1 # k"Cfnew!ur"|ur|ur

12 ! "!Lpp /2

Lpp /2 T!x"Cd2D!x"|vr # xr|!vr # xr"dx

12 ! "!Lpp /2

Lpp /2 T!x"Cd2D!x"x|vr # xr|!vr # xr"dx

#

#

#

ITTC surge resistance + cross-‐flow drag

Linear damping

Added mass Coriolis-‐ centripetal matrix

9

N!!r"!r ! CA!!r"!r " D!!r"!r

!

!Yv# vrr " Yr# r2

Xu# urr

!Yv# !Xu# "urvr"Yr#urr

alternatively:Xvrvrr " Xrrr2

Yururr

Nuvurvr"Nururr

"

!X |u|u|ur |ur!Y |v|v |vr|vr!Y |v|r|vr|r ! Yv|r|vr|r|!Y |r|r|r|r|

!N |v|v |vr|vr!N |v|r|vr|r ! Nv|r|vr|r|!N |r|r|r|r|


7.1.2 Nonlinear Maneuvering Models based on 2nd-order Modulus Functions The idea of using 2nd-‐order modulus func9ons to describe the nonlinear terms in the N-‐matrix dates back to Fedyaevsky and Sobolev (1963). This is mo9vated by first principles (quadra9c drag). A simplified form of Norrbin's (1970) nonlinear model which retains the most important terms for steering and propulsion loss assignment has been proposed by Blanke (1981). This model corresponds to fiHng the cross-‐flow drag to 2nd-‐order modulus func9ons.



7.1.2 Nonlinear Maneuvering Models based on 2nd-order Modulus Functions Maneuvering model in matrix form

Recall that D(νr) = D + Dn(νr). No9ce that linear poten9al damping and skin fric9on are neglected in this model (D=0) since the nonlinear quadra9c terms Dn(νr) dominate at higher speeds

N!!r"!r !!X |u|u|ur |ur!Yv" vrr # Yr" r2

Xu" urr ! Y |v|v |vr|vr!Y |v|r|vr|r ! Yv|r|vr|r|!Y |r|r|r|r|

!Yv" !Xu" "urvr#Yr"urr ! N |v|v |vr|vr!N |v|r|vr|r ! Nv|r|vr|r|!N |r|r|r|r|

CA!!r" !0 0 Yv" vr # Yr"r0 0 Xu" ur

!Yv" vr ! Yr"r !Xu" ur 0

D!!r" !!X |u|u|ur | 0 0

0 !Y |v|v |vr |!Y |r|v |r| !Y |v|r|vr |!Y |r|r|r|0 !N |v|v |vr |!N |r|v |r| !N |v|r|vr |!N |r|r|r|

#

#

11

Experimental data can be curve fihed to Taylor series of 1st and 3rd order (odd func9ons). The idea dates back to Abkowitch (1964). Consider the nonlinear rigid-‐body kine9cs:


7.1.3 Nonlinear Maneuvering Models based on Odd Functions

MRB!" ! CRB!!"! " #RB # !RB !

X!x"Y!x"N!x"

# x ! !u,v,r,u" ,v" ,r" ,!"! #

xo! !U,0,0,0,0,0,0"! #

X!x"!X!x0"!"i"1

n#X!x"#xi x0

#xi ! 12#2X!x"!#xi"2 x0

#xi2 ! 16#3X!x"!#xi"3 x0

#xi3

Y!x"!Y!x0"!"i"1

n#Y!x"#xi x0

#xi ! 12#2Y!x"!#xi"2 x0

#xi2 ! 16#3Y!x"!#xi"3 x0

#xi3

N!x"!Z!x0"!"i"1

n#N!x"#xi x0

#xi ! 12#2N!x"!#xi"2 x0

#xi2 ! 16#3N!x"!#xi"3 x0

#xi3

3rd-‐order truncated Taylor-‐series expansion about

!x ! x ! x0 ! !"x1,"x2, . . ."xn"!

12

Unfortunately, a 3rd-‐order Taylor series expansion results in a large number of terms. By applying some physical insight, the complexity of these expressions can be reduced. Assump&ons:

1. Most ship maneuvers can be described with a 3rd-‐order truncated Taylor expansion about the steady state condi9on u = u₀. 2. Only 1st-‐order accelera9on terms are considered. 3. Standard port/starboard symmetry simplifica9ons except terms describing the constant force and moment arising from single-‐screw propellers. 4. The coupling between the accelera9on and velocity terms is negligible.


7.1.3 Nonlinear Maneuvering Models based on Odd Functions

X ! X! " Xu#u# " Xu$u " Xuu$u2 " Xuuu$u3 " Xvvv2 " Xrrr2 " X!!!2

" Xrv!rv! " Xr!r! " Xv!v! " Xvvuv2$u " Xrrur2$u " X!!u!2$u" Xrvurvu " Xr!ur!$u " Xv!uv!$u

Y ! Y! " Yu$u " Yuu$u2 " Yrr " Yvv " Yr# r# " Yv# v# " Y!! " Yrrrr3 " Yvvvv3

" Y!!!!3 " Yrr!r2! " Y!!r!2r " Yrrvr2v " Yvvrv2r " Y!!v!2v " Yvv!v2! " Y!vr!vr" Yvuv$u " Yvuuv$u2 " Yrur$u " Yruur$u2 " Y!u!$u " Y!uu!$u2

N ! N! " Nu$u " Nuu$u2 " Nrr " Nvv " Nr# r# " Nv# v# " N!! " Nrrrr3 " Nvvvv3

" N!!!!3 " Nrr!r2! " N!!r!2r " Nrrvr2v " Nvvrv2r " N!!v!2v " Nvv!v2!

" N!vr!vr " Nvuv$u " Nvuuv$u2 " Nrur$u " Nruur$u2 " N!u!$u" N!uu!$u2 #

F! ! F!x0"

Fxi !!F!x"!xi x0

Fxixj ! 12

!2F!x"!xi!xj x0

Fxixjxk ! 16

!3F!x"!xi!xj!xk x0

F ! !X,Y,N"


Nonlinear models of Clarke (2003); see Clarke’s lecture notes

3rrr

2vrr

2vvr

3vvvrv

3rrr

2vrr

2vvr

3vvvrv

rNrvN

rvNvNrNvNN

rYrvY

rvYvYrYvYY

′′+′′′+

′′′+′′+′′+′′=′

′′+′′′+

′′′+′′+′′+′′=′

Taylor Series Cubic Curves

For a symmetric ship hydrodynamic force Y' and moment N' can be modelled by 1st and 3rd order terms using an odd func9on

2nd-‐order Modulus Curves

rrNrvN

rvNvvNrNvNN

rrYrvY

rvYvvYrYvYY

rrrv

rvvvrv

rrrv

rvvvrv

′′′+′′′+

′′′+′′′+′′+′′=′

′′′+′′′+

′′′+′′′+′′+′′=′

v′r′

Y ′

ODD FUNCTION TAYLOR SERIESFIT

v′r′

Y ′

SECOND ORDERMODULUSFIT

gives a smooth surface (physical)

gives a flaher facehed surface (nonphysical)

Taylor Series Cubic Curves or 2nd-order Modulus Curves


Taylor Series Cubic Curves or 2nd-order Modulus Curves

ODD FUNCTIONTAYLOR

SERIES FIT

SECOND ORDERMODULUS FIT

Y ′

( )TAYLORvY ′

( )MODvY ′

v′−

Nonlinear models of Clarke (2003); see Clarke’s lecture notes

This is a big problem when comparing deriva9ves from different experimental facili9es

We obtain different deriva9ve values from cubic and 2nd-‐order modulus curve fit.

15

Forward speed model Starboard-‐port symmetry implies that surge is decoupled from sway and yaw: where t₁ is the sum of control forces in surge.


7.1.4 Linearized Maneuvering Models Linearized Decoupled Models for Forward Speed and Maneuvering

For marine cra^ moving at constant (or at least slowly varying) forward speed the 3-‐DOF maneuvering model can be decoupled in a: ü Forward speed (surge subsystem) ü Sway-‐yaw subsystem for maneuvering

2-‐DOF linear maneuvering model (sway-‐yaw subsystem)

!m ! Xu! "u! ! Xuur ! X |u|u|ur |ur " !1 #

U ! u2 " v2 ! u #

M!" ! N!uo"!r " b! #

δ is the rudder angle is based on the assump9ons that the cruise speed. The current forces are treated as a linear term:

!r ! !vr,r"! u ! uo ! constant #

N!uo"#vc,0$!


7.1.4 Linearized Maneuvering Models

! ! !v, r"!

! ! !!R

M !! " N!u0"! # b!

Model representa9on of Fossen (1994) and Clarke and Horn (1996) The poten9al theory representa9on is obtained by extrac9ng the 2nd and 3rd rows in CRB(ν) and CA(ν) with u = uo, resul9ng in:

Resul9ng model

M !m ! Yv" mxg!Yr"

mxg!Yr" Iz!Nr"b!

!Y!

!N!

N!uo" !!Yv !m ! Xu" "uo!Yr

!Xu" !Yv" "uo!Nv !mxg!Yr" "uo!Nr

#

#

C!!"! !!m " Xu! "uor

!m " Yv! "uov " !mxg"Yr! "uor " !m " Xu! "uov

#0 !m " Xu! "uo

!Xu! "Yv! "uo !mxg"Yr! "uo

v

r #

D !!Yv !Yr!Nv !Nr

#

! ! b! !!Y!

!N!! #

No&ce: C(n) includes the famous destabilizing Munk moment (from aerodynamics) and some other CA-‐terms


7.1.4 Linearized Maneuvering Models

MRB!" ! CRB!u0"! " #RB

! ! !v, r"!

! ! !!R

M !! " N!u0"! # b! M !m ! Yv" mxg ! Yr"mxg ! Nv" Iz ! Nr"

, b !!Y!!N!

N!u0" ! C!u0" # D !!Yv mu0 ! Yr!Nv mxgu0 ! Nr

#

#

Resul9ng model

Comment: Model representa9on of Davidson and Schiff (1946) This model assumes that the hydrodynamic forces τRB are linear in and (linear strip theory) such that:

NB! This representa9on miss the CA-‐terms (see red circles on previous slide)

!RB ! !Y!N!

b

! "Yv# Yr#Nv# Nr#

MA

"# "Yv YrNv Nr

D

"r #

!,!" !


7.2 Autopilot Models for Heading Control (1 DOF) 1-‐DOF autopilot model (yaw subsystem)

M!" ! N!uo"!r " b! #

yaw rate as output r ! c!!r, c! ! !0,1" #

Applica9on of the Laplace transforma9on yields (Nomoto, 1957)

r!!s" ! K!1 " T3s"

!1 " T1s"!1 " T2s" # Nomoto's 2nd-‐order model

r!!s" ! K

!1 " Ts" # Nomoto's 1st-‐order model T ! T1 " T2 ! T3

!"!s" ! K

s!1 " Ts" # !! " r

Equivalent 9me constant

The “celebrated” autopilot model

v! !s" ! Kv!1"Tvs"

!1"T1s"!1"T2s"

19

L (m) u0 (m/s) ! (dwt) K (1/s) T1 (s) T2 (s) T3 (s)cargo ship 161 7.7 16622 0.185 118.0 7.8 18.5

Oil tanker 350 8.1 389100 "0.019 "124.1 16.4 46.0

Matlab MSS toolbox T1=118; T2=7.8; T3=18.5; K=0.185; nomoto(T1,T2,T3,K) T1=-‐124.1; T2=16.4; T3=46.0; K=-‐0.019; nomoto(T1,T2,T3,K)

Bode plots of the Nomoto 1st-‐ and 2nd-‐order models

10-4 10-3 10-2 10-1 100 101-100

0

100

Gai

n[d

B]

1st-order2nd-order

10-4 10-3 10-2 10-1 100 101-200

-150

-100

-50

Phas

e[d

eg]

10-4 10-3 10-2 10-1 100 101-200

-100

0

100

Gai

n[d

B]

10-4 10-3 10-2 10-1 100 101-300

-250

-200

-150

Phas

e[d

eg]

Frequency [rad/s]

Mariner class cargo ship course stable ship

course unstable ship Oil tanker


7.2.1-7.2.2 Autopilot Models for Heading Control (1 DOF)

20

7.2.3 Nonlinear Extensions of Nomoto’s Model

The linear Nomoto models can be extended to include nonlinear effects by adding sta9c nonlineari9es referred to as maneuvering characteris9cs.

Nonlinear Extension of Nomoto's 1st-‐Order Model (Norrbin, 1963)

Tr! " HN!r" # K!

HN!r" ! n3r3 " n2r2 " n1r " n0

where HN(r) is a nonlinear func9on describing the maneuvering characteris9cs. For HN(r) = r, the linear model is obtained.

Nonlinear Extension of Nomoto's 2nd-‐order Model (Bech & Wagner-‐Smith, 1969)

T1T2r! " !T1 " T2"r# " KHB!r" $ K!! " T3!# "

HB!r" ! b3r3 " b2r2 " b1r " b0

where HB(r) can be found from Bech's reverse spiral maneuver. The linear equivalent is obtained for HB(r) = r


21

7.2.3 Nonlinear Extensions of Nomoto’s Model

-20 -15 -10 -5 0 5 10 15 20-2

-1

0

1

2course stable ship

linear (K>0)nonlinear

-20 -15 -10 -5 0 5 10 15 20-2

-1

0

1

2course unstable ship

linear (K<0)nonlinear

r (deg/s)

r (deg/s)

δ (deg)

δ (deg)

Bech spiral

Dieudonne spiral (hysteresis)

Bech and Dieudonne spirals

Linear and nonlinear maneuvering characteris&cs

The maneuvering characteris9cs are found by solving for r as a func9on of δ, using the steady-‐state solu9ons of the nonlinear Nomoto models:

HN!r" ! K!HB!r" ! !

# #


22

7.2.4 Pivot Point (Yaw Rotation Point)


Defini&on 7.1 (Pivot Point) A ship's pivot point xp is a point on the centerline measured from the CG about which the ship turns. Consequently, it has no sideways movement (Tzeng 1998)

xp!t" ! !vg/n!t"r!t" , r!t" " 0 #

When turning a ship it is important to know which point the ship turns about. This rota9on point in yaw is referred to as the pivot point.

The pivot point for a turning ship can be computed by measuring the velocity vg/n(t) in CG and the turning rate r(t):

vp/n ! vg/n " xpr ! 0 #

This expression is not defined for a zero yaw rate corresponding to a straight-‐line mo9on. This means that the pivot point is located at infinity when moving on straight line or in a pure sway mo9on. It is well known to the pilots that the pivot point of a turning ship is located at about 1/5∼1/4 ship length a^ of bow X


23

Yv! " !1160 ! 10!5 Nv! " !264 ! 10!5

Yr! ! m! " !499 ! 10!5 Nr! " !166 ! 10!5

Y!! " 278 ! 10!5 N!! " !139 ! 10!5

7.2.4 Pivot Point (Yaw Rotation Point)


xp!t" ! !vg/n!t"r!t" , r!t" " 0 #

Linearized maneuvering equation (steady-state Nomoto model):

AP FP BowCG CO

Lpp

Loa

xp=0.4923Lpp0.023Lpp 0.5Lpp 0.03Lpp

Pivotpoint

vr !

Kv!1 " Tvs"K!1 " T3s"

s!0! KvK # r

!!s" ! K!1 " T3s"

!1 " T1s"!1 " T2s" #

v! !s" ! Kv!1"Tvs"

!1"T1s"!1"T2s"

xp!s,s" ! ! KvK #

xp!s,s" ! ! NrY! ! !Yr ! mu0"N!

YvN! ! NvY! #

Location of the pivot point: Example: The Mariner Class vessel where the nondimensional linear maneuvering coefficients are given as:

xp!s,s" ! 0.4923Lpp #

24

When designing course autopilots it is o^en convenient to normalize the ship steering equa9ons of mo9on such that the model parameters can be treated as constants with respect to the instantaneous speed U defined by:

The most commonly used normaliza9on forms for marine cra^ are:

§  Prime-‐system (SNAME 1950) This system uses the ship's instantaneous speed U, the length L = Lpp (the length between the fore and a^ perpendiculars), the 9me unit L/U, and the mass unit ½ rL³ or ½ rL²T as normaliza9on variables. The prime system cannot be used for low-‐speed applica9ons such as DP, since normaliza9on of the speeds u,v,w implies dividing by the cruise speed U, which can be zero.

§  Bis-‐system (Norrbin 1970) This system can be used for zero speed since division of speed U is avoided. The Bis-‐system is based on the use of the length L = Lpp, the 9me unit √(L/g) such that speed becomes √(Lg) > 0. It also uses the body mass density ra9o m=m/r, where m is the mass unit and is the displacement.

7.2.5 Nondimensional Maneuvering and Autopilot Models

U ! u2 " v2

∇


25

Unit Prime-system I Prime-system II Bis-systemLength L L LMass 1

2 !L3 1

2 !L2T "!!

Inertia moment 12 !L

5 12 !L

4T "!!L2

Time LU

LU L/g

Reference area L2 LT " 2!L

Position L L LAngle 1 1 1Linear velocity U U Lg

Angular velocity UL

UL g/L

Linear acceleration U2L

U2L g

Angular acceleration U2L2

U2L2

gL

Force 12 !U

2L2 12 !U

2LT "!g!

Moment 12 !U

2L3 12 !U

2L2T "!g!L

Body mass density ra9o:

m < 1 : ROVs, AUVs, submarines, etc.

m  = 1 : Floa9ng ships/rigs and neutrally buoyant underwater vehicles, that is:

m > 1 : Heavy torpedoes (typically m = 1.3-‐1.5)

∇=ρ

µ m

Nota9on: x ʹ′ x ʹ′ xʹ′ʹ′

Normaliza&on Table



m ! !!

26

Example 7.3(Normaliza&on of Parameters) The hydrodynamic coefficient Yr can be made nondimensional by using the Prime-‐ and Bis-‐systems.

First, let us determine the dimension of Yr. Consider:

Y ! Yr r

N=kgm/s²

Sway force (N):

rad/s ?

Hence, the unknown dimension must be kgm/s since rad is a nondimensional unit.

The nondimensional values Yrʹ′ and Yrʹ′ʹ′ are found by using kg, m and s in the Conversion Table. Consequently:

Yr! " Yr! 12 !L

3"!L"

!L/U"

" 112 !L

3UYr

Yr!! " Yr!!"!"!L"L/g

" 1!"! Lg

Yr

Prime Bis ! ! 1 and m ! "! Yr!! " 1

m LgYr

floa9ng ship



27

7.2.5 Nondimensional Maneuvering and Autopilot Models Example 7.4 (Normaliza&on of States and Parameters) Consider a linearized maneuvering model. Prime-‐Normaliza9on gives:

M! "!! # N!!u0! "! ! $ b!!!

!! " !v !, r!"!

M! "m ! ! Yv#! m !xg! ! Yr#!

m!xg! ! Nv#! Iz! ! Nr#!, b! "

!Y!!

!N!!

N!!u0! " "!Yv! m!u0! ! Yr!

!Nv! m !xg! u0! ! Nr!

u0! " u0U ! 1, for #u ! 0 and #v ! 0

v ! Uv ", r ! UL r

", ! ! ! "

where

The nondimensional veloci9es and control input can be transformed to its dimensional values by using the Conversion Table:


28

6-‐DOF normaliza&on procedure A systema9c procedure for 6-‐DOF normaliza9on is found by defining the transforma9on matrix:

T!diag!1,1, 1, 1L ,1L ,

1L ", T!1 ! diag!1, 1,1,L,L,L"

Prime-system Bis-systemacceleration !" ! U2

L T!" " !" ! gT!" ""

velocity ! ! U T!" ! ! Lg T!""

position/attitude # $LT#" # $LT# ""

control forces/moments % ! 12 !U

2L2 T!1% " % ! "!g" T!1% ""

Generalized accelera9on, velocity, posi9on and force are then found from:



29

12 !L

3T!2!TM !T!1" !" " 12 !L

3! UL "T!2!TD !T!1! ! " 12 !L

3! UL "2T!2!TG!T!1" # # $

7.2.5 Nondimensional Maneuvering and Autopilot Models Again consider the nondimensional MIMO model:

M!!" ! " D !! ! " G!# ! # $!

When designing vessel simulators and gain-‐scheduled controllers it is convenient to perform the numerical integra9on in real 9me using dimensional 9me.

It is convenient to use the nondimensional hydrodynamic coefficients as input to the simulator/controller, while the states and inputs have their physical dimensions.

M! LU2T!1!" " D! 1

U T!1! " G!! 1L T

!1#" # 112 !U

2L2T$

!TM!T!1"!" " ! UL " !TD!T!1"! " ! UL "2!TG!T!1"# # 1

12!L

3 T2$

M D G

6-‐DOF normaliza9on procedure

!" ! !! ! ! ! !


30

!"!T"2!TM!!T"1" !" " !"! gL T

"2!TD !!T"1" ! " !"! gL T

"2!TG !!T"1" # # $

7.2.5 Nondimensional Maneuvering Autopilot Models 6-‐DOF bis-‐scaling

M!!!" !! " D !!! !! " G!!# !! # $!!

!TM!!T!1"!" " gL !TD !!T!1"! " g

L !TG!!T!1"# # 1!"" T

2$

nondimensional states/inputs nondimensional parameters

dimensional states/inputs nondimensional parameters

Transforming nondimensional states/inputs to dimension:

Resul9ng model:

M D G


31

7.2.5 Nondimensional Maneuvering and Autopilot Models Example 7.5 (Normaliza&on of Parameters while keeping the Actual States) Consider the autopilot model:

M! "!! # N!!u0! "! ! $ b!!!

!TM!T!1" "! # UL !TN

!!u0! "T!1"! $ U2L Tb

!!

T ! diag!1,1/L"

m11! Lm12!1L m21

! m22!v"r"

# UL

n11! Ln12!1L n21

! n22!vr

$ U2L

b1!1L b2

!!

! ! !"

Transforming the states and control input to dimensional quan99es, yields:

Component form:

!! " !v !, r!"!

actual states (dimension) Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

32

7.2.5 Nondimensional Maneuvering and Autopilot Models Example 7.6 (Normaliza&on Procedure for the Nomoto Time and Gain Constants) The gain and 9me constants in Nomoto's 1st-‐ and 2nd-‐order models can be made invariant with respect to L and U by defining:

K! " !L/U"K, T ! " !U/L"T

!L/U"T! r" # r $ !U/L"K ! ! r! " !! UL "1T#r $ ! UL "

2 K#

T#!

!L/U"2 T1! T2! !!3" " !L/U"!T1! " T2! "!# " !$ % !U/L"K! " " K! T3! "$

This representa9on is quite useful since the nondimensional gain and 9me constants will typically be in the range: 0.5 < Kʹ′ < 2 and 0.5 < Tʹ′ < 2 for most ships. An extension to Nomoto's 2nd-‐order model is obtained by wri9ng:

where the nondimensional 9me and gain constants are defined as:

Tiʹ′ = Ti (U/L) (i=1,2,3)

Kʹ′ = (L/U)K



7.3.1 Nonlinear DP Model using Current Coefficients

For low-‐speed applica9ons such as DP, current forces and damping can be described by three area-‐based current coefficients CX,CY and CN. These can be experimentally obtained using scale models in wind tunnels. The resul9ng forces are measured on the model which is restrained from moving (U = 0). The current coefficients can also be related to the surge resistance, cross-‐flow drag and the Munk moment. For a ship moving at forward speed U > 0, quadra9c damping will be included in the current coefficients if rela9ve speed is used.

Xcurrent ! 12 !AFcCX!"c"Vc2

Ycurrent ! 12 !ALcCY!"c"Vc2

Ncurrent ! 12 !ALcLoaCN!"c"Vc2

#

#

#

Zero-‐speed representa9on

Current direc9on (counter clock-‐wise rota9on) Frontal projected current area Lateral projected current area Length over all

!cAFcALcLoa

!c ! " ! #c ! $ #

uc ! !Vc cos!!c"vc ! Vc sin!!c"

# #


7.3.1 Nonlinear DP Model using Current Coefficients Forward speed representa9on

The forward speed representa9on introduces quadra9c damping using the concept of rela9ve velocity. This is NOT the case for the zero-‐speed representa9on.

Rela9ve speed and direc9on Xcurrent ! 12 !AFcCX!"rc"Vrc2

Ycurrent ! 12 !ALcCY!"rc"Vrc2

Ncurrent ! 12 !ALcLoaCN!"rc"Vrc2

#

#

#

Vrc ! urc2 " vrc2 ! !u ! uc"2 " !v ! vc"2

!rc ! !atan2!vrc,urc"

#

#

uc ! Vc cos!!c ! ""

vc ! Vc sin!!c ! ""

# #


7.3.1 Nonlinear DP Model using Current Coefficients Rela9onship between current coefficients and surge resistance/cross-‐flow drag

The current coefficients can be related to the surge resistance and cross-‐flow drag coefficients by assuming low speed such that u ≈ 0 and v ≈ 0. This is a good assump9on for DP.

ur|ur | ! "uc|uc |! Vc2 cos!!c"|cos!!c"|

vr|vr | ! "vc|vc |! "Vc2 sin!!c"|sin!!c"|

urvr ! ucvc! " 1

2 Vc2 sin!2!c"

#

#

#

Xcurrent ! 12 !AFcCX!"c"Vc2 :! X |u|u|ur |ur

Ycurrent ! 12 !ALcCY!"c"Vc2 :! Y|v|v |vr|vr

Ncurrent ! 12 !ALcLoaCN!"c"Vc2 :! N |v|v |vr|vr|"!Yv# ! Xu# "urvr

Munk moment

#

#

#

u ! v ! r ! 0

This gives the following analy9cal expressions for the area based current coefficients similar to Fal9nsen (1990), pp. 187-‐188.

CX!!c" ! !2 !X |u|u"AFc

cos!!c"|cos!!c"|

CY!!c" ! 2 !Y|v|v |"ALc

sin!!c"|sin!!c"|

CN!!c" ! 2"ALcLoa

!!N |v|v sin!!c"|sin!!c"|" 12 !Xu# ! Yv# "

A22!A11

sin!2!c""

#

#

#

36

0 20 40 60 80 100 120 140 160 180 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Angle of attack g c relative bow - counter-clockwise rotation (deg)

C X C X = -C X,max *cos( g )*abs(cos( g )) C Y C Y = C Y,max *sin( g )*abs(sin( g )) C N C N = C N,max *sin(2* g )


7.3.1 Nonlinear DP Model using Current Coefficients Current coefficients

Experimental current coefficients for a tanker compared with analy9cal formulae


7.3.1 Nonlinear DP Model using Current Coefficients Nonlinear DP model based on current coefficients

Op9onal quadra9c damping coefficient used to counteract the destabilizing Munk moment since the current coefficients do not include damping in yaw

!" ! R!!"#M#" " CRB!#"# " Dexp!!"Vrc"#r " d!Vrc,#rc" ! $ " $wind " $waves

# #

D !

!Xu 0 0

0 !Yv !Yr

0 !Nv !Nr

#

The model includes an op9onal linear damping matrix to ensure exponen9al convergence at low rela9ve speed

d!Vrc,!cr" !

! 12 "AFcCX!!rc"Vrc

2

! 12 "ALcCY!!rc"Vrc

2

! 12 "ALcLoaCN!!rc"Vrc

2 ! N |r|rr|r #


7.3.2 Linearized DP Model Vessel-‐parallel coordinates

This defini9on removes the rota9on matrix in yaw under the assump9on that the yaw rate r is small

!p ! R!!!"! #

!" p ! #

M#" " D# ! R!!!"b " $ " $wind " $wavesb" ! 0

#

# #

!" p ! R"!!!"! " R!!!"!" ! R!!!"!"

!" ! R!!"#

Note that: ü  The kinema9cs is linear ü GPS-‐posi9on measurements h must be transformed to vessel-‐parallel coordinates using

Linearized DP model for controller-‐observer design

!p ! R!!!"! #

39

7.4 Maneuvering Models including Roll (4 DOF) 4-‐DOF models in surge, sway, roll and yaw The speed equa9on can be decoupled from the sway, roll, and yaw modes: The resul9ng linear model takes the form:

M !! " N!uo!! "G" # #

M !

m ! Yv" !mzg ! Yp" mxg ! Yr"!mzg ! Kv" Ix ! Kp" !Ixz ! Np"mxg ! Nv" !Ixz ! Np" Iz ! Nr"

N!uo! !

!Yv !Yp mu0 ! Yr!Kv !Kp !mzgu0 ! Kr!Nv !Np mxgu0 ! Nr

! ! !v,p, r"!

uo !constant

G !diag!0, WGMT, 0"

!! " p"! " cos! r ! r

# #


! ! !E,!,""!

40

7.4 Maneuvering Models including Roll (4 DOF)

3-‐ and 4-‐DOF maneuvering models The maneuvering models presented in Sec9on 7.3 only describe the horizontal mo9ons (surge, sway and yaw) under a zero-‐frequency assump9on. These models are intended for the design and simula9on of DP systems, heading autopilots, trajectory-‐tracking and path-‐following control systems. Many vessels, however, are equipped with actuators that can reduce the rolling mo9on. This could be an9-‐rolling tanks, rudders and fin stabilizers. In order to design a control system for roll damping, it is necessary to augment the roll equa9on to the horizontal plane model. Inclusion of roll means that the restoring moment due to buoyancy and gravity must be included. The resul9ng model is a 4-‐DOF maneuvering model (surge, sway, roll and yaw) that includes the rolling mo9on


41

7.4 Maneuvering Models including Roll (4 DOF) State-‐space model The linear model can be wrihen in state-‐space model according to:

x! !

a11 a12 a13 a14 0

a21 a22 a23 a24 0

a31 a32 a33 a34 0

0 1 0 0 0

0 0 1 0 0

x "

b11 b12 ! b1r

b21 b22 ! b2r

b31 b32 ! b3r

0 0 ! 0

0 0 ! 0

u

a11 a12 a13

a21 a22 a23

a31 a32 a33

! !M!1N!uo!

" a14 "

" a24 "

" a34 "

! !M!1G

#

#

A B where the matrix elements are:

x ! !v,p, r,!,""!

bij are given by B ! M!1TK

! ! TKu

Roll and yaw outputs:

! ! !0, 0, 0, 1,0"x" ! !0, 0, 0, 0,1"x

# #


42

7.4 Maneuvering Models including Roll (4 DOF) Decomposi9ons in roll and sway-‐yaw subsystems The state variables associated with steering and roll can be separated:

v!r!!!

p!

"!

"

a11 a13 0 a12 a14a31 a33 0 a32 a340 1 0 0 0a21 a23 0 a22 a240 0 0 1 0

vr!

p"

#

b11 b12 ! b1rb31 b32 ! b3r0 0 ! 0b21 b22 ! b2r0 0 ! 0

u

!x!!x"

"A!! A!"

A"! A""

x!x"

#B!

B"u

p!

!!"

a22 a24

1 0

p

!#

b21 b22 ! b2r

0 0 ! 0u

v!

r!

!!

"

a11 a13 0

a31 a33 0

0 1 0

v

r

!

#

b11 b12 ! b1r

b31 b32 ! b3r

0 0 ! 0

u

x! ! !v, r,!"!

x! ! !p,!"!

If the coupling matrices are small, we get:

decoupled steering model (sway-‐yaw)

decoupled roll model


43

7.4 Maneuvering Models including Roll (4 DOF) Transfer func9ons for steering and rudder-‐roll damping The transfer func9ons for the decoupled models are:

Plot showing decoupled models and total model

!" !s" !

b2s2"b1s"b0s4"a3s3"a2s2"a1s"a0

! Kroll #roll2 !1"T5s"

!1"T4s"!s2"2$#rolls"#roll

2 "

!" !s" !

c3s3"c2s2"c1s"c0s!s4"a3s3"a2s2"a1s"a0"

! Kyaw !1"T3s"s!1"T1s"!1"T2s"

This is a rough approxima9on since all interac9ons are neglected. In par9cular, the roll mode is inaccurate as seen from the plots.

Matlab MSS toolbox: ExRRD1.m Lcontainer.m

Container ship: Son and Nomoto (1981) Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

44

7.4 Maneuvering Models including Roll (4 DOF)

!"!s" ! 0. 0032!s ! 0. 036"!s " 0. 077"

!s " 0. 026"!s " 0. 116"!s2"0. 136s " 0. 036"

" 0. 083!1 " 49. 1s"!1 " 31. 5s"!s2"0. 134s " 0. 033"

#

!"!s" ! 0. 0024!s " 0. 0436"!s2"0. 162s " 0. 035"

s!s " 0. 0261"!s " 0. 116"!s2"0. 136s " 0. 036"

! 0. 032!1 " 16. 9s"s!1 " 24. 0s"!1 " 9. 2s" #

Length: L = 175 (m) Displacement: 21,222 (m³) Service speed u0 = 7.0 (m/s)

!roll ! 0.189 (rad/s)

! ! 0. 36

right-‐half-‐plane zero: z ! 0. 036 (rad/s)

non-‐minimum phase property

Matlab MSS toolbox: ExRRD3.m


45

7.4.1 The Nonlinear Model of Son and Nomoto High-‐speed container ship given by:

!m ! mx"u" ! !m ! my"vr # X ! !x!m ! my"v" ! !m ! mx"ur ! my"yr" ! myl yp" # Y ! !y

!Ix ! Jx"p" ! mylyv" ! mxl xur # K ! WGMT# ! !k!Iz ! Jz"r" ! my" yv" # N ! xgY ! !n

# # # #

X ! X!u" " !1 ! t"T " Xvrvr " Xvvv2 " Xrrr2 " X!!!2

" X" sin" " Xext

Y ! Yvv " Yrr " Y!! " Ypp " Yvvvv3 " Yrrrr3 " Yvvrv2r " Yvrrvr2

" Yvv!v2! " Yv!!v!2 " Yrr!r2! " Yr!!r!2 " Y" cos" " Yext

K ! Kvv " Krr " K!! " Kpp " Kvvvv3 " Krrrr3 " Kvvrv2r " Kvrrvr2

" Kvv!v2! " Kv!!v!2 " Krr!r2! " Kr!!r!2 " K" cos" " Kext

N ! Nvv " Nrr " N!! " Npp " Nvvvv3 " Nrrrr3 " Nvvrv2r " Nvrrvr2

" Nvv!v2! " Nv!!v!2 " Nrr!r2! " Nr!!r!2 " N" cos" " Next

#

#

#

#

Matlab MSS toolbox: container.m (nonlinear model) Lcontainer.m (linear model)

[xdot,U] = container(x,ui) [xdot,U] = Lcontainer(x,ui,U0)

3rd-‐order Taylor-‐series expansion


46

7.4.2 The Nonlinear Model of Blanke and Christensen An alterna9ve model formula9on describing the steering and roll mo9ons of ships has been proposed by Blanke and Christensen (1986):

m ! Yv! !mzg!Yp! mxg!Yr! 0 0

!mzg!Kv! Ix!Kp! 0 0 0

mxg!Nv! 0 Iz!Nr! 0 0

0 0 0 1 0

0 0 0 0 1

v!

p!

r!

!!

"!

"

Yuv |u| Y|u|p |u| !mu " Yuru Yuu!u2 0

K|u|v |u| Kupu " Kp Kuru WGMT"Kuu!u2 0

Nuvu 0 N|u|r|u|!mxgu N|u|u! |u|u 0

0 1 0 0 0

0 0 1 0 0

v

p

r

!

"

#

Yext

Kext

Next

0

0

" !

#

#

2nd-‐order modulus model


47

7.4.3 Nonlinear Model based on Low- Aspect-Ratio Wing Theory This approach is well suited to derive a physical model structure which can best describe the nonlinear damping forces ac9ng on a marine cra^. The parameters of the model must, however, be found by curve fiHng the simulated response to 9me-‐series for instance by using system iden9fica9on.

Li^ and drag due to low-‐ aspect-‐ra9o wing theory

XLD ! XuuL u2 " XuuuL u3 " XvvL v2 " XrrL r2 " XrvL rv " XuvvL uv2

" XrvuL rvu " XurrL ur2 " Xvv!!L v2!2 " Xvr!!L vr!2 " Xrr!!L r2!2

#XLD

YLD ! YuvL uv " YurL ur " YuurL u2r " YuuvL u2v " YvvvL v3 " YrrrL r3

" YrrvL r2v " YvvrL v2r " Yuv!!L uv!2 " Yur!!L ur!2

#YLD

KLD ! YLDzcp! KuvL uv " KurL ur " KuurL u2r " KuuvL u2v " KvvvL v3

" KrrrL r3 " KrrvL r2v " KvvrL v2r " Kuv!!L uv!2 " Kur!!L ur!2

NLD ! YLDxcp! NuvL uv " NurL ur " NuurL u2r " NuuvL u2v " NvvvL v3

" NrrrL r3 " NrrvL r2v " NvvrL v2r " Nuv!!L uv!2 " Nur!!L ur!2

#

#

#

#


48

D!!" !

!XuuL u ! Xuuu

L u2

!XrvuL rv

!XvvL v ! Xrv

L r

!XuvvL uv ! Xvv!!

L v!2

!Xvr!!L r!2

0 !XrrL r ! Xurr

L ur ! Xrr!!L r!2

!Yuv!!L v!2!Yur!!

L r!2!YuvL u ! Yuuv

L u2!YvvvL v2

!YrrvL r2!Y |v |v |v|!Y |r|v |r|

0!YurL u ! Yuur

L u2!YrrrL r2

!YvvrL v2!Y |v |r|v|!Y |r|r|r|

!Kuv!!L v!2!Kur!!

L r!2!KuvL u ! Kuuv

L u2!KvvvL v2

!KrrvL r2!K|v |v |v|!K|r|v |r|

!Kp!Kpppp2!KurL u ! Kuur

L u2!KrrrL r2

!KvvrL v2!K|v |r|v|!K|r|r|r|

!Nuv!!L v!2!Nur!!

L r!2!NuvL u ! Nuuv

L u2!NvvvL v2

!NrrvL r2!N|v |v |v|!N|r|v |r|

0!NurL u ! Nuur

L u2!NrrrL r2

!NvvrL v2!N|v |r|v|!N|r|r|r|

7.4.3 Nonlinear Model based on Low- Aspect-Ratio Wing Theory In addi9on to the low-‐aspect-‐ra9o li^ and drag components it is necessary to include cross-‐flow drag and viscous roll damping.

Resul9ng nonlinear damping matrix


49

7.4.3 Nonlinear Model based on Low- Aspect-Ratio Wing Theory

Nonlinear 4-‐DOF maneuvering model (surge, sway, roll and yaw)

M!! " CRB!!"! " CA!!"! " D!!"! " G" # # " #wind " #wave #

M !

m ! Xu" 0 0 0

0 m ! Yv" !mzg!Yp" mxg!Yr"

0 !mzg!Kv" Ix!Kp" 0

0 mxg!Nv" 0 Iz!Nr"

,CRB!!" !

0 0 mzgr !m!xgr # v"

0 0 0 mu

!mzgr 0 0 0

m!xgr # v" !mu 0 0

CA!!" !

0 0 0 Yv" v

0 0 0 !Xu" u

0 0 0 Yv" v

!Yv" v Xu" u !Yv" v 0

,G !

0 0 0 0

0 0 0 0

0 0 !K! 0

0 0 0 0

#

#



7.5 Equations of Motion (6 DOF) Ship models are usually reduced order models for control of the horizontal plane mo9ons (surge, sway and yaw) in combina9on with roll if roll damping is an issue. Semi-‐submersible control system are also designed for the stabiliza9on of the horizontal plane mo9ons but for these type of vessels it is also of interest to simulate the heave, roll and pitch mo9ons during cri9cal opera9ons as drilling. In this sec9on we will discuss 6-‐DOF models (surge, sway, heave, roll, pitch and yaw) which are useful for predic9on and simula9on of coupled mo9ons. A 6-‐DOF model can be used in an accurate simula9on model of your system or even in a model-‐based control system if all DOFs are actuated. Some underwater vehicles have actua9on in all DOF.


51

7.5.1 Nonlinear 6-DOF Vector Representations in BODY and NED Equa9ons of mo9on expressed in BODY


M ! MRB "MA

C!!" ! CRB!!" " CA!!"D!!" ! D " Dn!!"

# # #

where different representa9ons for Jk(η) can be used: Euler angles: Unit quaternions:

!! " Jk!!"" #

M !! " C!!"! " D!!"! " g!"" " go # # " #wind " #wave #

J!!!"Jq!!"

! :! !N,E,D,!,",#"!

! :! !N,E,D,!,"1,"1,"1"!

52

7.5.1 Nonlinear 6-DOF Vector Representations in BODY and NED Equa9ons of mo9on expressed in NED using Euler angles Kinema9c transforma9on using: Euler angles are singular at ! ! !"/2


M!!!"!! " C!!",!" #! " D!!",!" #! " g!!!" " go! $ #! " #wind! " #waves! #

J!!"" !Rbn!!nb" 03!303!3 T"!!nb"

!! " J#!!"" # " " J#!1!!" !!!# " J#!!" !" $ !J#!!"" # !" " J#!1!!"$%! ! !J#!!"J#!1!!" !!%

! :! !N,E,D,!,",#"!

M!!!" ! J""!!!"MJ""1!!"

C!!",!" ! J""!!!"#C!"" "MJ""1!!" #J"!!"$J""1!!"

D!!",!" ! J""!!!"D!""J""1!!"

g!!!" $ go!!!" ! J""!!!"#g!!" $ go $

#! $ #wind! $ #wave

! ! J""!!!"!# $ #wind $ #wave" #

53

7.5.1 Nonlinear 6-DOF Vector Representations in BODY and NED Equa9ons of mo9on expressed in NED using quaternions Kinema9c transforma9on using: The unit quaternions are nonsingular Pseudoinverse



Jq!!" !Rbn!q" 03!304!3 Tq!q"

! :! !N,E,D,!,"1,"1,"1"!

!! " Jq!!"" # " " Jq!!!" !!!# " Jq!!" !" # !Jq!!"" # !" " Jq!!!"$$! ! !Jq!!"Jq!!!" !!%

Tq!!q" ! !Tq"!q"Tq!q""!1Tq"!q"

! 4Tq"!q" #

M!!!" ! Jq! !!""MJq! !!"C!!",!" ! Jq! !!""#C!"" "MJq! !!" "Jq!!"$Jq! !!"D!!",!" ! Jq! !!""D!""Jq! !!"

g!!!" # go!!!" ! Jq! !!""#g!!" # go $

#! # #wind! # #wave

! ! Jq! !!""!# # #wind # #wave" #

54

M !

m ! Xu" !Xv" !Xw"

!Xv" m ! Yv" !Yw"

!Xw" !Yw" m ! Zw"

!Xp" !mzg!Yp" myg!Zp"

mzg!Xq" !Yq" !mxg!Zq"

!myg!Xr" mxg!Yr" !Zr"

!Xp" mzg!Xq" !myg!Xr"

!mzg!Yp" !Yq" mxg!Yr"

myg!Zp" !mxg!Zq" !Zr"

Ix!Kp" !Ixy!Kq" !Izx!Kr"

!Ixy!Kq" Iy!Mq" !Iyz!Mr"

!Izx!Kr" !Iyz!Mr" I z!Nr"

Property 7.1 (System Iner&a Matrix) For a rigid body the system iner^a matrix is posi9ve definite

M ! MRB "MA # 0


7.5.1 Nonlinear 6-DOF Vector Representations in BODY and NED

Property 7.2 (Coriolis and Centripetal Matrix): For a rigid body moving through an ideal fluid the Coriolis and centripetal matrix can always be parameterized such that it is skew-‐symmetric:

C!!" ! !C!!!", "! # "6

55


(1)M!!!" ! M!!!"! " 0 " ! # "6(2) s! #M!

!!" $ 2C!!",!" s ! 0 " s # "6, " # "6, ! # "6

(3) D!!",!" " 0 " " # "6, ! # "6

Equa9ons of mo9on expressed in NED



since M = MT > 0 and M = 0. It should be noted that C∗(ν,η) will not be skew-‐symmetrical although C(ν) is skew-‐symmetrical.

.

56

V! " !!!" ! C"!#! ! D"!#! ! g"## # Jk!"##Kp#$ #

V! " !!M!" # #!Kp#"

" !!M!" # #!KpJk!#"!

" !!#M!" # Jk!!#"Kp#$ #

Example 7.8.1 (Lyapunov Analysis exploi&ng MIMO Model Proper&es)



A Lyapunov func9on candidate can be based on kine9c and poten9al energy

V ! 12 !

!M! " 12 "!Kp" #

M! ! 0

M ! M! " 0 Kp ! Kp! " 0

M!" ! C!!"! ! D!!"! ! g!#" " $

Kd ! 0V! " !!!!Kd # D"!#$!" 0 #

GAS follows from Krasowskii-LaSalle's theorem if Jk(η) is nonsingular.

Nonlinear PD-‐controller

!" ! Jk!!"#

! ! g!"" ! Kd# ! Jk!!""Kp" #

!" ! Jk!!"#

M#" " C!#"# " D!#"# " g!!" ! $

# #

57

7.5.2 Symmetry Considerations of the System Inertia Matrix


(iii) yz-‐plane (fore/a^) symmetry)

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 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

(i) xy-‐plane (bohom/top ) symmetry) (ii) xz-‐plane (port/starboard) symmetry

(iv) xz-‐ and yz-‐planes (port/starboard and fore/a^) symmetries

M !

m11 0 0 0 m15 m16

0 m22 m23 m24 0 0

0 m32 m33 m34 0 0

0 m42 m43 m44 0 0

m51 0 0 0 m55 m56

m61 0 0 0 m65 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

58

Assump&on 7.1 (Small Roll and Pitch Angles) The roll and pitch angles:

These are good assump9ons for vessels where the pitch and roll mo9ons are limited-‐that is,

highly metacentric stable cra^ When deriving the linearized equa9ons of mo9on it is convenient to introduce a vessel

parallel coordinate system obtained by rota9ng the body axes an angle ψ about the z-‐axis at each 9me step.

This implies that: where

7.5.3 Linearized Equations of Motion (Vessel-Parallel Coordinates)

!," are small

!" ! J!!!#!!"!0! P"#!#

P!!! !R!!! 03"303"3 I3"3


59

Defini&on 7.2 (Vessel-‐Parallel Coordinate System) The vessel parallel coordinate system is defined as:

where is the NED posi9on/aHtude expressed in BODY coordinates and P is given by

No9ce that PTP = I6×6.

!p ! P!!!"!

!p

P!!! !R!!! 03"303"3 I3"3


NED

BODY !p!

xb

yb xn

yn Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

60

Low-‐speed applica9ons (sta9onkeeping) Vessel-‐parallel (VP) coordinates implies:


!" p ! P" !!!"! # P!!!"!"

! P" !!!"P!!"!p#P!!!"P!!"$

! rS!p#$ #

where and r ! !"

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 applica9ons r ≈ 0 and

!" p ! #

Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen) Copyright © Bjarne Stenberg/NTNU

61

7.5.3 Linearized Equations of Motion (Vessel-Parallel Coordinates) The gravita9onal and buoyancy forces can also be expressed in terms of VP coordinates.

For small roll and pitch angles: No9ce that this formula confirms that the restoring forces of a leveled vessel φ = θ = 0 is

independent of the yaw angle ψ.

g!!"!!"!0! P!!#"G! "P!!#"GP!#"

G!p ! G!p

For a neutrally buoyant submersible (W = B) with xg = xb and yg = yb we have: For a surface vessel G is defined as:

G !diag!0, 0,0, 0, "zg ! zb#W, "zg ! zb#W, 0!

G !

02!2 02!30

0

03!2 Gr0

0

0 0 0 0 0 0

, Gr !!Zz 0 !Z"

0 !K# 0

!Mz 0 !M"

P!!!"GP!!" ! GNo9ce that:


62

Low-‐speed maneuvering and DP: ν ≈ 0 implies that the nonlinear Coriolis-‐centripetal, damping, restoring, and buoyancy forces and moments can be linearized about ν = 0 and φ = θ = 0. Since C(0) = 0 and Dn(0) = 0 it makes sense to approximate:

!" p ! #

M#" $ D# $ G!p ! % $ w

#

#


The resul9ng state-‐space model becomes:

!x " Ax # Bu # Ew

A !0 I

!M!1G !M!1D, B !

0M!1

, E !0M!1

x ! !!p!,"!!!, u # $

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

! " P!!!!p


M!" ! C!!"!0!#D ! Dn!!"$!

D!!%g!#"

G#p

! go " $ ! $wind ! $wavesw

#

63

7.5.3 Linearized Equations of Motion (Vessel-Parallel Coordinates) Vessels in transit (cruise condi9on): For vessels in transit the cruise speed is assumed to sa9sfy: 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 " A!uo"x # Bu # Ew ! F"o

A!uo" !0 I

!M!1G !M!1N!uo!, B !

0M!1

, E !0M!1

, F !I0

x ! !!p!,""!!!


64

7.5.4 Transforming the Equations of Motion to a Different Point


P

COCG

rg

rp

When deriving the nonlinear equa9ons of mo9on it is convenient to transform iner9a, damping, gravita9onal, and buoyancy forces between different points in {b} to exploit structural proper9es of the model. The rigid-‐body transla9onal and rota9onal parts of the system iner9a matrix is decoupled if the coordinate system is located in the CG while it is common to express hydrodynamic added mass and damping in CF alterna9vely CO. System transforma9on matrix -‐ transforms the generalized veloci9es, accelera9ons, and forces between two points in {b}.

vp/nb ! vb/nb " !b/nb ! rp/nb

! vb/nb ! S!rpb"!b/nb

! vb/nb " S!!rpb"!b/nb #

65



P

COCG

rg

rp

System transforma9on matrix

fbb

mbb

!fp

b

rpb ! fp

b " mpb

!I3!3 03!3S!rp

b" I3!3

fpb

mpb

#

! ! H!!rpb"!p

#

#

H!rpb" !

I3!3 S!!rpb"

03!3 I3!3, H!1!rp

b" !I3!3 S!rp

b"

03!3 I3!3 #

vp/nb

!b/nb

! H!rpb"

vb/nb

!b/nb

"

"p ! H!rpb""

#

#

66



Equa9ons of mo9on about CO

M !! " C!!"! " D!!"! " g!"" # # #

Mp ! H!!!rpb"MH!1!rp

b"

Cp!!" ! H!!!rpb"C!!"H!1!rp

b"

Dp!!" ! H!!!rpb"D!!"H!1!rp

b"

gp!"" ! H!!!rp

b"g!""

#

#

#

#

!p ! H!rpb"! ! ! H!1!rp

b"!p

Equa9ons of mo9on about P

M ! H!!rpb"MpH!rp

b"

C!!" ! H!!rpb"Cp!!"H!rp

b"

D!!" ! H!!rpb"Dp!!"H!rp

b"

g!"" ! H!!rpb"gp!""

#

#

#

#

H!!!rpb"MH!1!rpb"Mp

!!p " H!!!rpb"C!!"H!1!rpb"Cp!!"

!p

" H!!!rpb"D!!"H!1!rpb"Dp!!"

!p " H!!!rpb"g!""gp!""

# H!!!rpb"##p

#

67

7.5.5 6-DOF Models for AUVs and ROVs


!! " J!!"" #

M !! " C!!"! " D!!"! " g!"" # # # M !

m ! Xu" 0 !Xw"

0 m ! Yv" 0

!Xw" 0 m ! Zw"

0 !mzg!Yp" 0

mzg!Xq" 0 !mxg!Zq"

0 mxg!Yr" 0

0 mzg!Xq" 0

!mzg!Yp" 0 mxg!Yr"

0 !mxg!Zq" 0

Ix!Kp" 0 !Izx!Kr"

0 Iy!Mq" 0

!Izx!Kr" 0 Iz!Nr"

D!!" ! !diag#Xu,Yv ,Zw,Kp,Mq,Nr$! diag#X |u|u|u|,Y|vv| |v|,Z |w|w|w|,K |p|p|p|,M |q|q|q|,N |r|r|r|$

#

Approximate model for noncoupled mo9ons

d!Vrc,!cr" !

12 "AFcCX!!rc"Vrc

2

12 "ALcCY!!rc"Vrc

2

12 "AFcCZ!!rc"Vrc

2

12 "ALcHLcCK!!rc"Vrc

2

12 "AFcHFcCM!!rc"Vrc

2

12 "ALcLoaCN!!rc"Vrc

2

3DOF!

12 "AFcCX!!rc"Vrc

2

12 "ALcCY!!rc"Vrc

2

000

12 "ALcLoaCN!!rc"Vrc

2

#

Dn!!"! !

|!|!Dn1!|!|!Dn2!|!|!Dn3!|!|!Dn4!|!|!Dn5!|!|!Dn6!

# Alterna9ve representa9on using current coefficients

68

7.5.6 Longitudinal and Lateral Models for Submarines The 6-‐DOF equa9ons of mo9on can in many cases be divided into two noninterac9ng (or lightly interac9ng) subsystems:

ü  Longitudinal subsystem: states u,w,q and θ ü  Lateral subsystem: states v,p,r and ψ

This decomposi9on is good for slender bodies (large length/width ra9o). Typical applica9ons are aircrac, missiles, and submarines.


69

yz-‐plane of symmetry (fore/a^ symmetry)

7.5.6 Longitudinal and Lateral Models for Submarines

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 (boeom/top symmetry):

xz-‐plane of symmetry (port/starboard symmetry)

M ! diag!m11,m22,m33,m44,m55,m66"

xz-‐, yz-‐ and xy-‐planes of symmetry (port/starboard, fore/a^ and bohom/top symmetries).


70

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

Starboard-‐port symmetry implies the following zero elements: The longitudinal and lateral submatrices are:


Mlong !

m11 m13 m15

m31 m33 m35

m51 m53 m55

, M lat !

m22 m24 m26

m42 m44 m46

m62 m64 m66


71

Longitudinal subsystem (DOFs 1, 3, 5)


Rbn!!" !


T!!!" "

1 s!t" c!t"0 c! !s!0 s!/c" c!/c"

J!!" !Rbn!"" 03!303!3 T"!""

d!

!!"

cos! 00 1

wq

#! sin!0

u

v,p, r,! are small

not controlling the N-‐posi9on using speed control instead

Resul9ng kinema9c equa9on:


72

Longitudinal subsystem (DOFs 1, 3, 5) For simplicity, it is assumed that higher order damping can be neglected-‐that is, . Coriolis is, however, modeled by assuming that and that 2nd-‐order terms in v,w,p,q and r are small. Hence, DOFs 1, 3, 5 gives: Assuming a diagonal MA gives:


Dn!!" ! 0 u ! 0

CRB!!"! !0 0 0

0 0 "mu

0 0 mxgu

u

w

q

CRB!!"! !

m!ygq " zgr"p ! m!xgq ! w"q ! m!xgr " v"r

!m!zgp ! v"p ! m!zgq " u"q " m!xgp " ygq"r

m!xgq ! w"u ! m!zgr " xgp"v " m!zgq " u"w "!Iyzq " Ixzp ! Izr"p " !!Ixzr ! Ixyq " Ixp"r

Collec9ng terms in u,w, and q, gives:

CRB!!" ! "CRB! !!"

The skew-‐symmetric property is destroyed for the decoupled model:

CA!!"! !

!Zw" wq # Yv" vr

!Yv" vp # Xu" uq

!Zw" !Xu" "uw # !Nr"!Kp" "pr

"0 0 0

0 0 Xu" u

0 !Zw" !Xu" "u 0

u

w

q


73

Longitudinal subsystem (DOFs 1, 3, 5) The restoring forces with W = B and xg = xb:


g!!" !

!W ! B" sin!! !W ! B" cos! sin"! !W ! B" cos! cos"! !ygW ! ybB" cos ! cos" " !zgW ! zbB" cos! sin"

!zgW ! zbB" sin ! " !xgW ! xbB" cos! cos"

! !xgW ! xbB" cos ! sin" ! !ygW ! ybB" sin!

g!!! !

00

WBGz sin !


74

7.5.6 Longitudinal and Lateral Models for Submarines Longitudinal subsystem (DOFs 1, 3, 5)

m ! Xu! !Xw! mzg ! Xq!!Xw! m ! Zw! !mxg ! Zq!

mzg ! Xq! !mxg ! Zq! Iy ! Mq!

u!w!q!

"

!Xu !Xw !Xq!Zu !Zw !Zq!Mu !Mw !Mq

uwq

"

0 0 00 0 !!m ! Xu! "u0 !Zw! ! Xu! "u mxgu

uwq

"

00

WBGz sin!#

"1"3"5

m ! Zw! !mxg!Zq!

!mxg!Zq! Iy!Mq!

w!

q!"

!Zw !Zq

!Mw !Mq

w

q

"0 !!m ! Xu! "uo

!Zw! !Xu! "uo mxguo

w

q"

0

BGzWsin !#

"3

"5 #

u ! uo ! constant


75

7.5.6 Longitudinal and Lateral Models for Submarines Longitudinal subsystem (DOFs 1, 3, 5)

Linear pitch dynamics (decoupled): where the natural frequency and period are:

!Iy !Mq! "!" !Mq!! # BGzW! $ "5


!pitch !WBGzIy ! Mq"

, Tpitch ! 2"!pitch

#

76

Lateral subsystem (DOFs 2, 4, 6)


Rbn!!" !


T!!!" "

1 s!t" c!t"0 c! !s!0 s!/c" c!/c"

J!!" !Rbn!"" 03!303!3 T"!""

not controlling the E-‐posi9on using heading control instead

Resul9ng kinema9c equa9on:

!! " p"! " r

# #

u,w, p, r, ! and " are small


77

Lateral subsystem (DOFs 2, 4, 6) Again it is assumed that higher order velocity terms can be neglected so that . Hence: Assuming a diagonal MA gives:


Dn!!" ! 0

Collec9ng terms in v,p and r, gives:

CRB!!" ! "CRB! !!"

The skew-‐symmetric property is destroyed for the decoupled model:

CRB!!"! !

!m!ygp " w"p " m!zgr " xgp"q ! m!ygr ! u"r

!m!ygq " zgr"u " m!ygp " w"v " m!zgp ! v"w " !!Iyzq ! Ixzp " Izr"q " !Iyzr " Ixyp ! Iyq"r

m!xgr " v"u " m!ygr ! u"v ! m!xgp " ygq"w " !!Iyzr ! Ixyp " Iyq"p " !Ixzr " Ixyq ! Ixp"q

CRB!!"! !0 0 muo

0 0 0

0 0 mxguo

v

p

r

CA!!"! !

Zw" wp ! Xu" ur

!Yv"!Zw" "vw # !Mq" !Nr" "qr

!Xu"!Yv" "uv # !Kp" !Mq" "pq

"0 0 !Xu" u

0 0 0

!Xu" !Yv" "u 0 0

v

p

r


78

Lateral subsystem (DOFs 2, 4, 6) The restoring forces with W = B, xg = xb and yg = zg:

g!!" !

!W ! B" sin!! !W ! B" cos! sin"! !W ! B" cos! cos"! !ygW ! ybB" cos ! cos" " !zgW ! zbB" cos! sin"

!zgW ! zbB" sin ! " !xgW ! xbB" cos! cos"

! !xgW ! xbB" cos ! sin" ! !ygW ! ybB" sin!


g!!" "

0WBGz sin!

0


79

7.5.6 Longitudinal and Lateral Models for Submarines Lateral subsystem (DOFs 2, 4, 6)

u ! uo ! constant

m ! Yv! !mzg ! Yp! mxg ! Yr!!mzg ! Yp! Ix ! Kp! !Izx ! Kr!mxg ! Yr! !Izx ! Kr! Iz ! Nr!

v!p!r!

"

!Yv !Yp !Yr!Mv !Mp !Mr

!Nv !Np !Nr

vpr

"

0 0 !m ! Xu! "u0 0 0

!Xu! ! Yv! "u 0 mxgu

vpr

"

0WBGz sin!

0#

"2"4"6

m ! Yv! mxg!Yr!

mxg!Yr! Iz!Nr!

v!

r!"

!Yv !Yr!Nv !Nr

v

r

"0 !m ! Xu! "uo

!Xu! !Yv! "uo mxguo

v

r#

!2

!6 #



7.5.6 Longitudinal and Lateral Models for Submarines Lateral subsystem (DOFs 2, 4, 6)

Linear roll dynamics (decoupled): where the natural frequency and period are:

!Ix ! Kp! "!" ! Kp!! #WBGz! $ "4

!roll !WBGzIx ! Kp" , Troll !

2"!roll

#

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