1

Chapter 2 - Kinematics

2.1Referenceframes2.2TransformationsbetweenBODYandNED2.3TransformationsbetweenECEFandNED2.4TransformationsbetweenBODYandFLOW

“Thestudyofdynamics canbedividedintotwoparts: kinematics,whichtreatsonlygeometricalaspectsofmotion,andkinetics,whichistheanalysisoftheforcescausingthemotion”

BODY

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

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Overall Goal of Chapters 2 to 8

Thenotationandrepresentationareadoptedfrom:

Fossen,T.I.(1991). NonlinearModelingandControlofUnderwaterVehicles,PhDthesis,DepartmentofEngineeringCybernetics,NTNU,June1991.

Fossen,T.I.(1994).GuidanceandControlofOceanVehicles,JohnWileyandSonsLtd.ISBN:0-471-94113-1.

Representthe6-DOFdynamicsinacompactmatrix-vectorformaccordingto:

!" ! J!!"#M#" " C!#"# " D!#"# " g!!" " g0 ! $ " $wind " $wave

# #

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

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2.1 Reference Frames

ECI{i}: Earthcenteredinertialframe;non-acceleratingframe(fixedinspace)inwhichNewton’slawsofmotionapply.

ECEF{e}: Earth-CenteredEarth-Fixedframe;originisfixedinthecenteroftheEarthbuttheaxesrotaterelativetotheinertialframeECI.

NED{n}: North-East-Downframe;definedrelativetotheEarth’sreferenceellipsoid(WGS84).BODY{b}: Bodyframe;movingcoordinateframefixedtothevessel.

xb- longitudinalaxis(directedfromafttofore)yb- transversalaxis(directedtostarboard)zb-normalaxis(directedfromtoptobottom)

N

ED

e

x

yl

z

e

e

BODY

ECEF/ECI

NED

e t

e

x y

y

x

z

i i

e

e

i z e,

ECEF/ECI

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

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2.1 Reference Frames – Body-Fixed Reference Points

• CG - Centerofgravity• CB- Centerofbuoyancy• CF - Centerofflotation

CFislocatedadistanceLCF fromCOinthex-direction

ThecenterofflotationisthecentroidofthewaterplaneareaAwp incalmwater.Thevesselwillrollandpitchaboutthispoint.

u! " u1nn!1 # u2nn!2 # u3nn!3

un ! !u1n,u2n,u3n"!

Coordinate-freevector

n!i !i " 1,2,3" are the unit vectors that define #n$

Coordinateformof u! in !n"

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

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forces and linear and positions and

DOF moments angular velocities Euler angles

1 motions in the x-direction (surge) X u x2 motions in the y-direction (sway) Y v y3 motions in the z-direction (heave) Z w z4 rotation about the x-axis (roll, heel) K p !

5 rotation about the y-axis (pitch, trim) M q "

6 rotation about the z-axis (yaw) N r #

2.1 Reference frames and 6-DOF motions

xb

yb

zb

u ( )surge

r ( )yaw

v ( )sway

( )heavew

( )rollp

( )pitchqThenotationisadoptedfrom:

SNAME(1950). NomenclatureforTreatingtheMotionofaSubmergedBodyThroughaFluid.TheSocietyofNavalArchitectsandMarineEngineers,TechnicalandResearchBulletinNo.1-5,April1950,pp.1-15.

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

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2.1 Reference Frames - Notation

Generalizedposition,velocityandforce

ECEFposition:

pb/ee !

xyz

! !3Longitude andlatitude

!en !l!

! S2

NEDposition:

pb/nn !

NED

! !3Attitude(Euler angles)

!nb !

"

#

$

! S3

Body-fixedlinearvelocity

vb/nb !

uvw

! !3Body-fixedangularvelocity

"b/nb !

pqr

! !3

Body-fixedforce:

fbb !

XYZ

! !3Body-fixedmoment

mbb !

KMN

! !3

! !pb/nn !or pb/ne )

"nb, # !

vb/nb

$b/nb

, % !fbb

mbb

#

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

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2.2 Transformations betweenBODY and NED

Specialorthogonal groupoforder3:

SO!3" ! #R|R ! !3!3, R is orthogonal and detR !1$

Orthogonal matricesoforder3:

O!3" ! #R|R ! !3!3, RR" ! R!R ! I$

RR! ! R!R ! I, detR ! 1

Rotationmatrix:

SinceR isorthogonal, R!1 ! R!

! to ! R fromto ! from

Example:

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

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! ! a :! S!!"a

Cross-productoperatorasmatrix-vectormultiplication:

S!!" ! !S!!!" !

0 !!3 !2!3 0 !!1!!2 !1 0

, ! "

!1!2!3

whereisaskew-symmetricmatrixS ! !S!

2.2 Transformations betweenBODY and NED

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

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2.2 Transformations betweenBODY and NEDEulerstheoremonrotation:

R11 ! !1 ! cos!" "12 " cos!R22 ! !1 ! cos!" "22 " cos!

R33 ! !1 ! cos!" "32 " cos!

R12 ! !1 ! cos!" "1"2 ! "3 sin!R21 ! !1 ! cos!" "2"1 " "3 sin!

R23 ! !1 ! cos!" "2"3 ! "1 sin!

R32 ! !1 ! cos!" "3"2 " "1 sin!R31 ! !1 ! cos!" "3"1 ! "2 sin!

R13 ! !1 ! cos!" "1"3 " "2 sin!

R!,"! I3#3 ! sin" S!"" ! !1 ! cos"" S2!""

where

! ! !!1,!2,!3"!, |!| ! 1

!

vb/nn ! Rbnvb/nb , Rbn :! R!," #

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

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2.2.1 Euler Angle TransformationThreeprincipalrotations:

(2) Rotation over pitch angle about . Note that .

yv =v

2

2 1

x2x3

y3

y2

u3

u2

v2

v3

(1) Rotation over yaw angle about . Note that .

zw =w

3

3 2

x1

x2

z1 z2

u1

u2

w1

w2

U

U

(3) Rotation over roll angle about . Note that .

xu =u

1

1 2

z =z0 b z1

y1

y =y0 bv=v2

v1

w=w0

w1 U

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

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

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

Rx,! !

1 0 00 c! !s!0 s! c!

Ry,! !

c! 0 s!0 1 0!s! 0 c!

Rz,! !

c! !s! 0s! c! 00 0 1

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

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2.2.1 Euler Angle Transformation

Linearvelocitytransformation(zyx-convention):

Smallangleapproximation:

where

Rbn!!nb" !

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#

Rbn!!!nb" ! I3"3 ! S!!!nb" "

1 "!# !$

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

Rbn!!nb"!1 ! Rnb!!nb" ! Rx,!! Ry,"! Rz,#!Rbn!!nb" :! Rz,!Ry,"Rx,#

p! b/nn ! Rbn!"nb"vb/nb #

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

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NEDpositions(continuoustimeanddiscretetime):

2.2.1 Euler Angle Transformation

Component form:

Eulerintegration

p! b/nn ! Rbn!"nb"vb/nb #

N! " u cos!!"cos!"" # v!cos!!"sin!"" sin!#" ! sin!!"cos!#""# w!sin!!" sin!#" # cos!!"cos!#" sin!"""

Ė " u sin!!"cos!"" # v!cos!!"cos!#" # sin!#"sin!"" sin!!""# w!sin!"" sin!!"cos!#" ! cos!!" sin!#""

D! " !u sin!"" # vcos!"" sin!#" # wcos!""cos!#"

#

# #

pb/nn !k ! 1" " pb/n

n !k" ! hRbn!!nb!k""vb/n

b !k" #

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

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Angularvelocitytransformation(zyx-convention):

2.2.1 Euler Angle Transformation

where

1. Singularpointat ! ! " 90o

Smallangleapproximation: Noticethat:

T!!1!!nb" "

1 0 !s!0 c" c!s"0 !s" c!c"

# T!!!nb" "

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

T!!!!nb" !1 0 !"

0 1 "!#0 !# 1

T!!1!!nb" " T!

! !!nb"

!" nb ! T"!!nb"#b/nb # !b/nb !

!"

00

# Rx,!!0""

0# Rx,!! Ry,"!

00#"

:! T$!1!"nb""# nb #

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

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ODEforEulerangles:ODEforrotationmatrix

2.2.1 Euler Angle Transformation

Componentform:

!! " p # qsin" tan ! # rcos" tan !"! " qcos" ! rsin"

#! " q sin"cos! # r cos"

cos! , ! " $90o

# #

# + algorithmforcomputationofEuleranglesfromtherotationmatrix

where

Eulerangleattituderepresentations:

Rbn!!nb"!nb! !!,",#"!

!" nb ! T"!!nb"#b/nb # R! b

n ! RbnS!"b/n

b " #

S!!b/nb " !

0 !r qr 0 !p!q p 0

#

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

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Summary:6-DOFkinematicequations:

2.2.1 Euler Angle Transformation

Componentform:

3-parameterrepresentation

withsingularityat ! ! " 90o

N! " u cos!cos" # v!cos!sin"sin# ! sin!cos#"# w!sin!sin# # cos!cos#sin""

Ė " u sin!cos" # v!cos!cos# # sin#sin"sin!"# w!sin"sin!cos# ! cos!sin#"

D! " !u sin" # vcos"sin# # wcos"cos#

#

# #

!! " p # q sin! tan" # rcos! tan""! " q cos! ! rsin!

#! " q sin!cos" # r cos!

cos" , " " $90o

# #

#

!nb! !!,",#"!!! " J!!""

#

p# b/nn

$# nb"

Rbn!$nb" 03!3

03!3 T$!$nb"

vb/nb

%b/nb

#

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

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2.2.2 Unit Quaternions4-parameterrepresentation:

-avoidstherepresentationsingularityoftheEulerangles-numericaleffective(notrigonometricfunctions)

Q ! !q|q!q !1,q ! !!,"!"!, " ! "3 and ! ! "" ! " !!1,!2,!3!!

R!,! ! I3"3 " sin! S!!" " !1 ! cos!" S2!!"

Unitquaternion(Eulerparameter)rotationmatrix(Chou1992):

! ! cos "2

! ! !!1,!2,!3"! ! " sin "2

q !

!

"1"2"3

!cos #

2

! sin #2

! Q

Rbn!q! : ! R!,! ! I3"3 " 2!S!!! " 2S2!!"

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

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2.2.2 Unit QuaternionsLinearvelocitytransformation

where

Rbn!q! !

1 ! 2!!22 " !32" 2!!1!2 ! !3"" 2!!1!3 " !2""

2!!1!2 " !3"" 1 ! 2!!12 " !32" 2!!2!3 ! !1""

2!!1!3 ! !2"" 2!!2!3 " !1"" 1 ! 2!!12 " !22"

Componentform(NEDpositions):

Rbn!q"!1 ! Rbn!q"!

q!q ! 1NB! mustbeintegratedundertheconstraintor!2 ! "1

2 ! "22 ! "3

2 " 1

N! " u!1 ! 2!22 ! 2!3

2" # 2v!!1!2 ! !3"" # 2w!!1!3 # !2""

Ė " 2u!!1!2 # !3"" # v!1 ! 2!12 ! 2!3

2" # 2w!!2!3 ! !1""

D! " 2u!!1!3 ! !2"" # 2v!!2!3 # !1"" # w!1 ! 2!12 ! 2!2

2"

# # #

p! b/nn ! Rbn!q"vb/nb #

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

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2.2.2 Unit QuaternionsAngularvelocitytransformation

Tq!q" ! 12

!!1 !!2 !!3" !!3 !2!3 " !!1!!2 !1 "

, Tq!!q"Tq!q" ! 14 I3#3

where

!! " ! 12 !"1p # "2q # "3r"

"! 1 " 12 !!p ! "3q # "2r"

"! 2 " 12 !"3p # !q ! "1r"

"! 3 " 12 !!"2p # "1q # !r"

#

#

#

#

Componentform:

NB! nonsingulartothepriceofonemoreparameter

Alternativerepresentation(Kane1983)

Theequationsarederivedusing

q! ! Tq!q""b/nb # q! !!"

"!! 1

2!"!

!I3"3 # S!""#b/nb #

R! bn ! RbnS!"b/nb "

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

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4-parameterrepresentation

NonsingularbutonemoreODEisneeded

Summary:6-DOFkinematicequations(7ODEs):

Componentform:

q ! !!,"1, "2, "3!!

!! " ! 12 !"1p # "2q # "3r"

"! 1 " 12 !!p ! "3q # "2r"

"! 2 " 12 !"3p # !q ! "1r"

"! 3 " 12 !!"2p # "1q # !r"

#

#

#

#

2.2.2 Unit Quaternions

N! " u!1 ! 2!22 ! 2!3

2" # 2v!!1!2 ! !3"" # 2w!!1!3 # !2""

Ė " 2u!!1!2 # !3"" # v!1 ! 2!12 ! 2!3

2" # 2w!!2!3 ! !1""

D! " 2u!!1!3 ! !2"" # 2v!!2!3 # !1"" # w!1 ! 2!12 ! 2!2

2"

# # #

!! " J!!""

#

p# b/nn

q#"

Rbn!q" 03!3

04!3 Tq!q"vb/nb

$b/nb

#

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

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Discrete-timealgorithmforunitquaternionnormalization

q!q ! !12 " !2

2 " !32 " "2 ! 1

2.2.2 Unit Quaternions

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

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Continuous-timealgorithmforunitquaternionnormalization:

Ifq isinitializedasaunitvector,thenitwillremainaunitvector.

However,integrationofthequaternionvectorq fromthedifferentialequationwillintroducenumericalerrorsthatwillcausethelengthofq todeviatefromunity.

InSimulink thisisavoidedbyintroducingfeedback:

!q " Tq!q"!nbb # !2 !1 ! q

!q"q

ddt !q

!q" ! 2q!Tq!q"!nbb " !!1 ! q!q"q!q ! !!1 ! q!q"q!q0ifqisinitializedasaunitvector

! ! 0 (typically 100!

x ! 1 ! q!qChangeofcoordinates(x=0gives)

x! " !!x!1 ! x" x! " !!xlinearizationaboutx=0gives

q!q ! 1

2.2.2 Unit Quaternions

ddt !q

!q" ! 2q!Tq!q"!b/nb ! 0 #

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

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2.2.3 Quaternions from Euler AnglesRef.Shepperd (1978)

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

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2.2.3 Quaternions from Euler Angles

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

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2.2.4 Euler Angles from QuaternionsRequirethattherotationmatricesofthetwokinematicrepresentationsareequal:

q ! !!,"1, "2, "3!!

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#

"

R11 R12 R13R21 R22 R23R31 R32 R33

Algorithm: Onesolutionis:

! ! atan2!R32,R33"

" ! !sin!1!R31" ! ! tan!1 R311 ! R312

; " " "90o

# ! atan2!R21,R11"

#

#

#

whereatan2(y,x) isthe4-quadrantarctangentoftherealpartsoftheelementsofx andy,satisfying:

!! " atan2!y,x" " !

Rbn!!nb" :! Rbn!q" !nb! !!,",#"!

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

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N

ED

e

x

yl

z

e

e

2.3 Transformation betweenECEF and NED

ECEF{e}-frame

NED{n}-frame

Longitude:l (deg)Latitude:µ (deg)Ellipsoidalheight:h (m)

Apointon orabove theEarth’ssurfaceisuniquelydeterminedby:

h

NEDaxesdefinitions:N - NorthaxisispointingNorthE - EastaxisispointingEastD – DownaxisispointingdowninthenormaldirectiontotheEarth’ssurface

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

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2.3.1 Longitude and Latitude TransformationsThetransformationbetweentheECEFandNEDvelocityvectorsis:

Twoprincipalrotations:1. arotationlaboutthez-axis2. arotation()aboutthey-axis.!! ! "/2

!en! !l,!"! ! S2

Rne!!en" ! Rz,lRy,!!! "2!

cos l ! sin l 0

sin l cos l 0

0 0 1

cos !!! ! "2 " 0 sin!!! ! "

2 "

0 1 0

! sin!!! ! "2 " 0 cos !!! ! "

2 "

Rne!!en" !

! cos l sin! ! sin l ! cos lcos!! sin l sin! cos l ! sin lcos!cos! 0 ! sin!

p! b/ee ! Rne!"en"p! b/en ! Rne!"en"Rbn!"nb"vb/eb #

p! b/nn ! Rbn!"nb"vb/nb #

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

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SatellitenavigationsystemmeasurementsaregivenintheECEFframe:Nottousefulfortheoperator.

Presentationofterrestrialpositiondata isthereforemadeintermsoftheellipsoidalparameterslongitudel,latitudeµ andheighth.

Transformation:

2.3.2 Longitude/Latitude from ECEF Coordinates

andheighth

!en! !l,!"!pb/ee ! !x,y,z"!

pb/ee ! !x,y,z"!

pb/ee ! !x,y,z"!

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

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N ! re2

re2 cos2!"rp2 sin2!

2.3.2 Longitude/Latitude from ECEF Coordinates

l ! atan! yexe "

tan! ! zep 1 ! e2 N

N " h!1

h ! pcos! ! N

#

#

whilelatitudeµ andheighth areimplicitlycomputedby:

pb/ee ! !x,y,z"!Parameters Commentsre ! 6 378 137 m Equatorial radius of ellipsoid (semimajor axis)rp ! 6 356 752 m Polar axis radius of ellipsoid (semiminor axis)!e ! 7.292115 ! 10!5 rad/s Angular velocity of the Earthe ! 0.0818 Eccentricity of ellipsoid

e ! 1 ! !rpre "

2

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

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2.3.2 Longitude/Latitude from ECEF Coordinates

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

Ref. Hofman-Wllenhof et al. (2004)

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2.3.3 ECEF Coordinates from Longitude/Latitude

Ref.Heiskanen (1967)

Thetransformationfromforgivenheightsh toisgivenby!en! !l,!"!

xyz

!

!N " h"cos!cos l!N " h"cos!sin lrp2

re2N " h sin!

pb/ee ! !x,y,z"!

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

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2.4 Transformation between BODY and FLOW FLOWaxesareoftenusedtoexpresshydrodynamicdata.TheFLOWaxesarefoundbyrotatingtheBODYaxissystemsuchthatresultingx-axisisparalleltothefreestreamflow.

InFLOWaxes,thex-axisdirectlypointsintotherelativeflowwhilethez-axisremainsinthereferenceplane,butrotatessothatitremainsperpendiculartothex-axis.They-axiscompletestheright-handedsystem.

xb-

xstabzb

yb

Uxflow

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

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Ry,! !

cos! 0 sin!0 1 0

!sin! 0 cos!, Rz,!" ! Rz,"! !

cos" sin" 0!sin" cos" 00 0 1

uvw

! Ry,!! Rz,!"!

U00

u ! Ucos!cos"v ! Usin"w ! Usin!cos"

# # #

or

vstab ! Ry,!vb

v flow ! Rz,!"vstab

# #

xb-

xstabzb

yb

Uxflow

2.4 Transformation between BODY and FLOW

U ! u2 " v2 #

Principalrotations:

Velocitytransformation:

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

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2.4 Transformation between BODY and FLOW ExtensiontoOceanCurrents

Foramarinecraftexposedtooceancurrents,theconceptofrelativevelocitiesisintroduced.Therelativevelocitiesare:

ur ! u ! ucvr ! v ! vcwr ! w ! wc

# # #

Ur ! ur2 " vr2 " wr2 #

ur ! Ur cos!!r"cos!"r"vr ! Ur sin!"r"wr ! Ur sin!!r"cos!"r"

# # #

!r ! tan!1 wrur

"r ! sin!1 vrUr

#

#

Hence,

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

Sideslipangle(SSA)andangleofattack(AOA)

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2.4.1 Definitions of Course, Heading and Sideslip Angles

Sideslip angle:

Crab angle:

Course angle:

Heading (yaw) angle

Speed over ground:

Relative speed:

Current speed:

Current direction:

GNSS measures course angle 𝜒and speed over ground UCompass measures heading angle 𝜓Currents can be measured by an Acoustic Doppler Current Profiler (ADCP)

North

East

Ocean Current Triangle: Horizontal Plane

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2.4.1 Definitions of Course, Heading and Sideslip AnglesTherelationshipbetweentheangularvariablescourse,heading,andsideslipisimportantformaneuveringofavehicleinthehorizontalplane(3DOF).

Thetermscourseandheadingareusedinterchangeablyinmuchoftheliteratureonguidance,navigationandcontrolofmarinecraft,andthisleadstoconfusion.

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

Definition(Courseangleχ): Theanglefrom thex-axisoftheNEDframeto thevelocityvectorofthevehicle,positiverotationaboutthez-axisoftheNEDframebytheright-handscrewconvention.MeasuredusingGNSS(orHPRunderwater).

Definition:Heading(yaw)angleψ:Theanglefrom theNEDx-axisto theBODYx-axis,positiverotationaboutthez-axisoftheNEDframebytheright-handscrewconvention.Measuredusingacompass.

Differencebetweencourseandheadingangles:

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2.4.1 Definitions of Course, Heading and Sideslip Angles

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

Definition(Crabangleβ): Theanglefrom theBODYx-axisto thevelocityvectorofthevehicle,positiverotationabouttheBODYz-axisframebytheright-handscrewconvention:

Thecrabangleisafunctionoftheswayvelocityandspeedoverground:

Definition(Sideslipangleβr): Thesideslipangleisdefinedintermsofrelativevelocities:

North

East

Remark: InSNAME(1950)andLewis(1989)thesideslipangleformarinecraftisdefined as:

βSNAME=-βr

Weusethesignconventionbytheaircraftcommunitye.g.Nelson(1998)andStevens(1992).Thisdefinitionismoreintuitivefromaguidancepoint-of-viewthanSNAME(1950).

37

Someinterestingobservationsregardingsideslip:

1) Avehiclemovingonastraightlineincalmwater(U > 0 andv =0)willhaveazerocrabangle

2) Assoonasyoustarttoturn,theswayvelocitywillbenon-zeroandconsequently,.Thecrabanglecorrespondstotheamountofcorrectionavehiclemustbeturnedinordertomaintainthedesiredcourse.

3) Avehicleisalsoexposedtoenvironmentalforces,whichinducesaflowvelocity(wind/current).Thisforcesthevehicleto“sideslip”.Moreover,

2.4.1 Definitions of Course, Heading and Sideslip Angles

� = 0

� 6= 0

! ! arcsin vU

! small" ! ! v

U

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

!r ! tan!1 wrur

"r ! sin!1 vrUr

#

#

38

2.4.1 Definitions of Course, Heading and Sideslip Angles

Relative flight path angle:

Flight path angle:

Angle of attack:

Pitch angle:

Speed over ground:

Relative speed:

Current speed:

GNSS measures flight path γand speed over ground UAHRS or INS measure pitch angle 𝜃Currents can be measured by an Acoustic Doppler Current Profiler (ADCP)

North

Down

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

Ocean Current Triangle: Vertical Plane