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Professor Thor I. FossenDepartment of Engineering CyberneticsNTNU Centre for Autonomous Marine Operations and SystemsNorwegian University of Science and Technology (NTNU)

Lecture Notes: TTK 4190 Guidance and Control of Vehicles

Home page: http://www.fossen.biz/

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

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Textbook and Notation

Fossen,T.I.(2011)HandbookofMarineCraftHydrodynamicsandMotionControl.JohnWiley&SonsLtd.

MSSToolbox:www.marinecontrol.org

xb

yb

zb

u ( )surge

r ( )yaw

v ( )sway

( )heavew

( )rollp

( )pitchq

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

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Useful Reference Book

ü Fossen,T.I.(1994). “GuidanceandControlofOceanVehicles”, JohnWiley&SonsLtd.ISBN0-471-94113-1

xb

yb

zb

u ( )surge

r ( )yaw

v ( )sway

( )heavew

( )rollp

( )pitchq

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

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Goals for the Course:1. Mathematicalmodeling

ofvehicles.Thisincludes:§ Kinematics§ Kinetics§ Equationsofmotion

formarinecraftandaircraft

§ Wind,waveandoceancurrentmodels

§ Hydrodynamics:maneuveringandseakeepingtheory

Copyright © Bjarne Stenberg/NTNU

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

2. Designofguidance,navigation andmotioncontrolsystemsforalargenumberofapplications

3. Simulate themotionsofmarinecraftandaircraftinthetime-domainusinghydrodynamic/aerodynamicmodels

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Marine Craft in Operation

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

7 TTK 4190 Guidance and Control (T. I. Fossen) Lecture Notes 2005Pipe-laying vessel

Geologicalsurvey

Heavy lift operations

Positionmooring

Pipe and cable laying

Marine Craft in Operation

ROV operations

Vibrationcontrolof marinerisers

Cable-layingvessel

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Italian supply ship Vesuviorefueling 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: SeaLaunchhttp://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 andMarine Control

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

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NTNU Infrastructure

Copyright © Bjarne Stenberg/NTNU

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

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ROV Minerva

Applied Underwater Robotics Laboratory

Hugin AUV

Remus 100 AUVLight AUV (LAUV)

ThefleetoftheAUR-LabatNTNUTrondheimhttps://www.ntnu.no/aur-lab

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

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Marine Cybernetics Laboratory (MCLab)

MCLab Dimensions:

üThesoftwareisdevelopedbyusingrapidprototypingtechniquesandautomaticcodegenerationunderMatlab/Simulink™andRT-Lab™.

üThetargetPConboardthemodelscalevesselsrunstheQNX™real-timeoperatingsystem,whileexperimentalresultsarepresentedinreal-timeonahostPCusingLabview™.

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 Vessel Gunnerus31meterslongTopspeed13knots http://www.ntnu.edu/marine/gunnerus

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

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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

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

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Penguin equipped with Camera, INS and GPS Sensor Suite for Data Logging

Penguinnavigationpayload:TwoIMUs,opticalcamera,infraredcamera,RTKGPSandembeddedcontrollerfordatalogging

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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

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

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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

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

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NTNU Airfield at AgdenesWealsousetheairportsatEggemoen andØrland

Located94kmNorth-WestofTrondheim

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Offshore UAV Launch and Recovery Systems

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Launching – The Easy Part“Human-Inspired Approach”

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Automatic Launch of the Penguin UAV

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Net Landing onboard a Research Vessel outside the Azores in 2016

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Coordinated Control: Master-Slave Principle: The fixed-wing aircraft acts like an master, while the cooperative multicopters is the slave following the master.

Virtually-Moving Airfield: The airfield/net is moved in an optimal manner vertically and horizontally to improve landing accuracy and reduce impact speed.

K. Klausen, J. B. Moe, J. C. van den Hoorn, A. Gomola, T. I. Fossen and T. A. Johansen. Coordinated Control Concept for Recovery of a Fixed-Wing UAV on a Ship using a Net Carried by Multirotor UAVs. Proc. of the 2016 International Conference on Unmanned Aircraft Systems (ICUAS'16), Arlington, VA, 7-10 June, 2016.

Cooperative Control: A suspended net is transported away from the ship to a safe distance by using two or more multicopters, which cooperates. They UAVSs are intelligent and share positions in order to control the tension of the net.

The NTNU AMOS Ship-Landing Concept

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The NTNU AMOS Ship-Landing ConceptTwo or more Multicopters Catches the UAV

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Marinecraft:ships,high-speedcraft,semi-submersibles,floatingrigs,submarines,remotelyoperatedandautonomousunderwatervehicles,torpedoesandotherpropelled/poweredstructuresforinstanceafloatingairfield.

Vehicles thatdonottravelonland(oceanandflightvehicles)areusuallycalledcraft.

Vessel: "hollowstructuremadetofloatuponthewaterforpurposesoftransportationandnavigation;especially,onethatislargerthanarowboat".

Thewordsvessel,shipandboatareoftenusedinterchangeably.InEncyclopediaBritannica,ashipandaboataredistinguishedbytheirsizethroughthefollowingdefinition:

Ship:"anylargefloatingvesselcapableofcrossingopenwaters,asopposedtoaboat,whichisgenerallyasmallercraft.Thetermformerlywasappliedtosailingvesselshavingthreeormoremasts;inmoderntimesitusuallydenotesavesselofmorethan500tonsofdisplacement.

Submarine: "anynavalvesselthatiscapableofpropellingitselfbeneaththewateraswellasonthewater'ssurface.

UnderwaterVehicle: "smallvehiclethatiscapableofpropellingitselfbeneaththewatersurfaceaswellasonthewater'ssurface.Thisincludesunmannedunderwatervehicles(UUV),remotelyoperatedvehicles(ROV)andautonomousunderwatervehicles(AUV).

Marine Craft Ref.EncyclopediaBritannica

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

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Marinevesselsarealsoclassifiedaccordingtotheirmaximumoperatingspeed.ForthispurposeitiscommontousetheFroudenumber

Marine Craft

Fn = UgL

Thepressurecarryingthevesselcanbedividedintohydrostatic and hydrodynamicpressure.Thecorrespondingforcesare:

• Buoyancyforceduetothehydrostaticpressures(proportionaltothedisplacementoftheship)

• Hydrodynamicforceduetothehydrodynamicpressure(approximatelyproportionaltothesquareofthespeed)

Thenwecanclassifythevesselsaccordingto(Faltinsen 2005):

• Displacementvessels(Fn1.0-1.2):Thehydrodynamicforcemainlycarriestheweight.

U:shipspeedL:overall(submergedlengthoftheship)G:accelerationofgravity

Inthiscourse,onlydisplacementvesselsarecovered

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

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Classification of ModelsSimulation Model: Thismodelisthemostaccuratedescriptionofasystem,forinstancea6-DOFhigh-fidelitymodelforsimulationofcoupledmotionsinthetimedomain.Itincludesthemarinecraftdynamics,propulsionsystem,measurementsystemandtheenvironmentalforcesduetowind,wavesandoceancurrents.

Themodelshouldbeabletoreconstructthetimeresponsesoftherealsystemanditshouldalsobepossibletotriggerfailuremodessoyoucansimulateaccidentsanderroneoussignalsetc.Simulationmodelswherethefluidmemoryeffectsareincluded(frequency-dependentmodels)typicallyconsistof50-200ODEswhileafrequency-independentmodelcanberepresentedin6DOFwith12ODEsforgeneralizedpositionandvelocity.

Inadditionyouneedsomestatestodescribetheenvironmentalloadsandactuatorsbutstillthenumberofstateswillbelessthan50forafrequency-independentvesselmodel

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

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Classification of ModelsControl Design Model: Thecontrollermodelisareduced-orderorsimplifiedsimulationmodelthatisusedtodesignthemotioncontrolsystem.Initssimplestform,thismodelisusedtocomputeasetofconstantgainsforaPIDcontroller.Moresophisticatedcontrolsystemsuseadynamicmodeltogeneratefeedforward andfeedbacksignals.Thisisreferredtoasmodel-basedcontrol.

ThenumberofODEsusedinconventionalmodel-basedshipcontrolsystemsisusuallylessthan20.APIDcontrollertypicallyrequirestwostates:onefortheintegratorandoneforthelow-passfilterusedtolimitnoiseamplification.Consequently,setpoint regulationin6DOFcanbeimplementedbyusing12ODEs.However,trajectory-trackingcontrollersrequireadditionalstatesforfeedforwardaswellasfilteringsohigher-ordercontrollawsarenotuncommon.

Observer Design Model: Theobservermodelwillingeneralbedifferentfromthemodelusedinthecontrollersincethepurposeistocapturetheadditionaldynamicsassociatedwiththesensorsandnavigationsystemsaswellasdisturbances.Itisasimplifiedversionofthesimulationmodelwhereattentionisgiventoaccuratemodelingofmeasurementnoise,failuresituationsincludingdead-reckoningcapabilities,filteringandmotionprediction.

Formarinecraft,themodel-basedobserveroftenincludesadisturbancemodelwherethegoalistoestimatewave,windandoceancurrentforcesbytreatingtheseascolorednoise.FormarinecraftthenumberofODEsinthestateestimatorwilltypicallybe20foraDPsystemwhileabasicheadingautopilotisimplementedwithlessthan5states.

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

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The Classical Models in Naval Architecture

mu̇ − vr + wq − xgq2 + r2 + ygpq − ṙ + zgpr + q̇ = Xmv̇ − wp + ur − ygr2 + p2 + zgqr − ṗ + xgqp + ṙ = Ymẇ − uq + vp − zgp2 + q2 + xgrp − q̇ + ygrq + ṗ = ZIxṗ + Iz − Iyqr − ṙ + pqIxz + r2 − q2Iyz + pr − q̇Ixy

+ mygẇ − uq + vp − zgv̇ − wp + ur = KIyq̇ + Ix − Izrp − ṗ + qrIxy + p2 − r2Izx + qp − ṙIyz

+ mzgu̇ − vr + wq − xgẇ − uq + vp = MIzṙ + Iy − Ixpq − q̇ + rpIyz + q2 − p2Ixy + rq − ṗIzx

+ mxgv̇ − wp + ur − ygu̇ − vr + wq = N

Themotionsofamarinecraftexposedtowind,wavesandoceancurrentsareusuallymodeledin6DOFbyapplyingNewton's2ndlaw:

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

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The Classical Models in Naval ArchitectureTheexternalforcesandmomentsX,Y,Z,K,M andN actingonamarinecraftareusuallymodeledbyusing:

ManeuveringTheory: ThestudyofashipmovingatconstantpositivespeedUincalmwater withintheframeworkofmaneuveringtheoryisbasedontheassumptionthatthehydrodynamiccoefficientsarefrequencyindependent(nowaveexcitation).Themaneuveringmodelwillinitssimplestrepresentationbelinearwhilenonlinearrepresentationscanbederivedusingmethodslikecross-flowdrag,quadraticdampingorTaylor-seriesexpansions.

SeakeepingTheory:Themotionsofshipsatzeroorconstantspeed inwavescanbeanalyzedusingseakeepingtheorywherethehydrodynamiccoefficientsandwaveforcesarecomputedasafunctionofthewaveexcitationfrequencyusingthehullgeometry.Thefrequency-dependentmodelsareusuallyderivedwithinalinearframeworkwhileextensionstononlineartheoryisanimportantfieldofresearch.

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

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Maneuvering TheoryAssumesthattheshipismovinginrestrictedcalmwater.Hence,themaneuveringmodelisderivedforashipmovingatpositivespeedUunderazero-frequencywaveexcitationassumptionsuchthataddedmassanddampingcanberepresentedbyusinghydrodynamicderivatives(constantparameters).

Thezero-frequencyassumptionisonlyvalidfor surge,sway andyaw sincethenaturalperiodofaPDcontrolledshipwillbeintherangeof100-150s.For150s thisresultin:

ωn = 2πT≈ 0.04 rad/s #

Thenaturalfrequenciesinheave,rollandpitcharemuchhighersoitisnotstraightforwardtoremovethefrequencydependenceinthesechannels.

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

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Maneuvering TheoryItiscommontoformulatetheshipmaneuveringmodelasacoupledsurge-sway-yawmodelandthusneglectheave,rollandpitchmotions:

• Hydrodynamicaddedmasspotentialdampingduetowaveradiationandviscousdamping• Hydrostaticforces(springstiffness)• Windforces• Waveforces(1st- and2nd-order)• Controlandpropulsionforces

mu̇ − vr − xgr2 − ygṙ = Xmv̇ + ur − ygr2 + xgṙ = YIzṙ + mxgv̇ + ur − ygu̇ − vr = N

#

MRB ν̇ + CRBνν = τRB #

τRB = τhyd + τhs hydrodynamic andhydrostatic forces

+ τwind + τwaveenvironmental forces

+ τcontrol #

τi = 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 ModelsInthelinear6-DOF casetherewillbeatotalof36massand36dampingelements proportionaltovelocityandacceleration.Inadditiontothis,therewillberestoringforces,propulsionforcesandenvironmentalloads.Ifthegeneralizedforce iswrittenincomponent,thelinearaddedmassanddampingforcesbecome:

where Xu,Xv , . . . ,Nr are the linear damping coefficientsand Xu̇,Xv̇ , . . . ,Nṙ represent hydrodynamic added mass.

τhyd

X1 = Xuu + Xvv + Xww + Xpp + Xqq + Xrr+ Xu̇u̇ + Xv̇ v̇ + Xẇẇ + Xṗṗ + Xq̇q̇ + Xṙṙ⋮

N1 = Nuu + Nvv + Nww + Npp + Nqq + Nrr+ Nu̇u̇ + Nv̇v̇ + Nẇẇ + Nṗṗ + Nq̇q̇ + Nṙṙ

#

#

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

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Nonlinear Maneuvering ModelsApplicationofnonlineartheoryimpliesthatmanyelementsmustbeincludedinadditiontothe36linearelements.

TruncatedTaylor-seriesexpansionsusingoddterms(1st- and3rd-order)whicharefittedtoexperimentaldata(Abkowitch 1964)or2nd-ordermodulusterms(Fedyaevsky 1963)

Theequationsbecomerelativelycomplicatedduetothelargenumberofhydrodynamiccoefficientsontheright-handsideneededtorepresentthehydrodynamicforces.

Taylor-seriesexpansionsarefrequentlyusedincommercialplanarmotionmechanism(PMM)testswherethepurposeistoderivethemaneuveringcoeffs.experimentally.

X1 = Xu̇u̇ + Xuu + Xuuuu3 + Xv̇v̇ + Xvv + Xvvvv3 + ⋯⋮

N1 = Nu̇u̇ + Nuu + Nuuuu3 + Nv̇v̇ + Nvv + Nvvvv3 + ⋯

#

#

X1 = Xu̇u̇ + Xuu + X |u|u|u|u + Xv̇v̇ + Xvv + X |v|v |v|v + ⋯⋮

N1 = Nu̇u̇ + Nuu + N |u|u|u|u + Nv̇v̇ + Nvv + N |v|v |v|v + ⋯

#

#

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

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Seakeeping,isthestudyofmotionwhenthereiswaveexcitationandthecraftkeepsitscourseanditsspeedconstant(whichincludesthecaseofzerospeed).Thisintroducesadissipativeforceknownasfluidmemoryeffects(Cummins1962).

Thegoverningmodelisformulatedinthetimedomain(Cumminsequation):

Seakeeping Theory

MRB + A∞ξ̈ + B total∞ξ̇ + ∫0

tKt − τξ̇τdτ + Cξ = τwind + τwave + δτ #

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

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- body-fixedvelocities:- positionandEulerangles:-M,CandDdenotethesysteminertia,Coriolisanddampingmatrices- gisavectorofgravitationalandbuoyancyforcesandmoments

- q isavectorofjointangels- isavectoroftorque-M andC arethesysteminertiaandCoriolismatrices

InFossen(1991)therobotmodel:

Mqq̈ + Cq, q̇q = τ

wasusedasfoundationtowritethe6-DOFmarinecraftequationsofmotioninacompactvectorial setting.

Matrix-VectorRepresentation

Therobotmodelwasmodifiedtodescribemarinecraftaccordingto:

τ

ν = u,v,w,p, q, rTη = x,y, z,φ, θ,ψT

Fossen's Robot-Like Vectorial Model for Marine Craft

Mν̇ + Cνν + Dνν + gη + g0 = τ + τwind + τwave

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

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Itisadvantageoustoexpresstheequationsofmotioninmatrix-vectorform:

sincenonlinearsystempropertiessuchas:

§ symmetry ofmatrices§ skew-symmetryofmatrices§ positiveness ofmatrices

canbeexploitedinthepassivity/stabilityanalysis.Thesepropertiesrelatestopassivity ofthemodel.NoticethThese propertiesaredestroyedbylinearization.

Thesepropertiesalsorepresentsphysicalpropertiesofthesystemwhichshouldbeexploitedwhendesigningnonlinearcontrollersandobservers formarinecraft.

Thematrix-vectornotationmakesitveryeasytosimulatethemodel(inMatlab andC++)

Why use Matrix-Vector Form?

Mν̇ + Cνν + Dνν + gη + g0 = τ + τwind + τwave

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 φ5 rotation about the y-axis (pitch, trim) M q θ6 rotation about the z-axis (yaw) N r ψ

xb

yb

zb

u ( )surge

r ( )yaw

v ( )sway

( )heavew

( )rollp

( )pitchq

ThenotationisadoptedfromSNAME(1950).

Foramarinecraft,DOFisthesetofindependentdisplacementsandrotationsthatcompletelyspecifythedisplacedpositionandorientationofthecraft.Acraftthatcanmovefreelyinthe3-Dspacehasmaximum6DOFs—threetranslationalandthreerotationalcomponents.

Consequently,afullyactuatedmarinecraftoperatingin6DOFmustbeequippedwithactuatorsthatcanproduceindependentforcesandmomentsinalldirections.

Degrees of Freedom (DOF)

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Whendesigningfeedbackcontrolsystemsformarinecraft,reduced-ordermodelsareoftenusedsincemostvehiclesdonothaveactuationinallDOF.Thisisusuallydonebydecouplingthemotionsofthevesselaccordingto:

1-DOFmodelscanbeusedtodesignforwardspeedcontrollers(surge),headingautopilots(yaw)androlldampingsystems(roll).

3-DOFmodelsareusuallyhorizontal-planemodels(surge,swayandyaw)forships,semi-submersiblesandunderwatervehiclesthatareusedinDPsystems,trajectory-trackingcontrolsystemsandpath-followingsystems.Forslenderbodiessuchastorpedo-shapedAUVsandsubmarines,itisalsocommontoassumethatthemotionscanbedecoupledintolongitudinalandlateralmotions.

·Longitudinalmodels(surge,heaveandpitch)forforwardspeed,divingandpitchcontrol.·Lateralmodel(sway,rollandyaw)forturningandheadingcontrol.

4-DOFmodels (surge,sway,rollandyaw)areusuallyformedbyaddingtherollequationtothe3-DOFhorizontal-planemodel.Thesemodelsareusedinmaneuveringsituationswherethepurposeistoreducerollbyactivecontroloffins,ruddersorstabilizingliquidtanks.

6-DOFmodels (surge,sway,heave,roll,pitchandyaw)arefullycoupledequationsofmotionusedforsimulationandpredictionofcoupledvesselmotions.Thesemodelscanalsobeusedinadvancedcontrolsystemsforunderwatervehicles,whichareactuatedinallDOF.

Degrees of Freedom (DOF)