Trajectory Tracking of a Tri-Rotor Aerial Vehicle Using an MRAC-Based Robust Hybrid Control ...

17 Pages·2017·3.7 MB·English

aerospace
Article
Trajectory Tracking of a Tri-Rotor Aerial Vehicle
Using an MRAC-Based Robust Hybrid
Control Algorithm
ZainAnwarAli*,DaoboWang,MuhammadAamirandSuhaibMasroor
CollegeofAutomationEngineering,NanjingUniversityofAeronauticsandAstronautics,Nanjing210016,
China;[email protected](D.W.);[email protected](M.A.);[email protected](S.M.)
* Correspondence:[email protected];Tel.:[phone]
AcademicEditor:MichaelWing
Received:2November2016;Accepted:20December2016;Published:19January2017
Abstract: Inthispaper,anovelModelReferenceAdaptiveControl(MRAC)-basedhybridcontrol
algorithm is presented for the trajectory tracking of a tri-rotor Unmanned Aerial Vehicle (UAV).
The mathematical model of the tri-rotor is based on the Newton–Euler formula, whereas the
MRAC-basedhybridcontrollerconsistsofFuzzyProportionalIntegralDerivative(F-PID)andFuzzy
ProportionalDerivative(F-PD)controllers.MRACisusedasthemaincontrollerforthedynamics,
whiletheparametersoftheadaptivecontrollerarefine-tunedbytheF-PDcontrollerforthealtitude
controlsubsystemandtheF-PIDcontrollerfortheattitudecontrolsubsystemoftheUAV.Thestability
ofthesystemisensuredandprovenbyLyapunovstabilityanalysis. Theproposedcontrolalgorithm
is tested and verified using computer simulations for the trajectory tracking of the desired path
asaninput. TheeffectivenessofourproposedalgorithmiscomparedwithF-PIDandtheFuzzy
LogicController(FLC).Ourproposedcontrollerexhibitsmuchlesssteadystateerror,quickerror
convergenceinthepresenceofdisturbanceornoise,andmodeluncertainties.
Keywords: ModelReferenceAdaptiveControl;FuzzyLogicController(FLC);trajectorytracking;
tri-rotorUAV
1. Introduction
Currently,researchintheareaofcontrolengineeringisfocusedonthefieldofunmannedflight
bodyaircraft,suchashelicopters,hex-rotor,quad-rotor,andtri-rotorrobots,duetotheirvarietyof
applications,especiallyintheareaofdefense[1–4].Otherareasincludesurveillance,environmental
monitoring,agriculture,andmediacoverage. Foranyunmannedflight,thepositionandaltitudeof
therobotcantakeadvantageofsensorinformation[5,6].Thispaperprovidesatrajectorytracking
controlalgorithmforthetri-rotoraerialvehiclebytakingadvantageofverticaltakeoffandlanding
(VTOL).Theunmannedtri-rotorsystemisusedforimagingofforestfires,accidents,surveillance,
transportation, and the detection of manholes [7–11]. The full weight of the system depends on
controllingtheexternalbars,whichrequireshighenergyconsumption.
Thetri-rotoraerialvehiclehasfourinputcommands,Col,Lat,Lon,andPed,foraltitude,latitude,
longitude,andangularcontrolcommand. Thenineoutputsare(p,q,r),(u,v,w),and(ϕ, θ, ψ),which
aretherotationalvelocity,translationalvelocity,andEulerangles[12]. Torectifytherotorreactionthat
isfoundinyawmoments,aBrushlessDirectCurrent(BLDC)motorisfixedtothetriangularstructure
ofthetri-rotor.
Someofthemainreasonsthatthetri-rotorUAVissuperiortothequad-rotoraerialvehicleareas
follows: (i)OrientationofUnmannedAerialVehicle(UAV):Bycomparingthetri-rotorUAVwiththe
quad-rotorUAVstructure,theorientationofthequad-rotorrapidlydisappearsatlargerdistancesdue
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toitsstructurebeingsymmetrical. Moreover,byusingLight-EmittingDiodes(LED),theoperativesare
notablyconfusedinthedaytime. Comparedtoquad-rotorUAV,tri-rotorUAVorientationisobviousat
farlongerdistances;(ii)NaturalFlyingDynamics: Tri-rotorUAVsareabletoflyinaroutethatclosely
resemblesthefixed-wingUAV.Rapidturningability,forwardflight,andincreasingordecreasingthe
velocityinanintuitivepatternaretheleadingadvantages. Ontheotherhand,quad-rotorUAVsare
notveryintuitive; themainadvantageofthetri-rotorUAVisthatitresemblesfixedwingaircraft
dynamicsduringflight,whileaquad-rotorbehaveslikeasinglerotoraircraft. Thestructureofthe
tri-rotorUAVallowstheswitchingbladetoenjoynotonlythebenefitofUAV’sintuitivestructure,
hoveringofflight, andforwardflightcapability, butalsoVTOL;(iii)Yawanglecontrollingability:
TheyawcontrolisusedfortherotationofaUAVarounditsverticalaxis. Theyawanglepermitsthe
UAVtomakequickerturnsasperagivenreference. Ascomparedtotri-rotorUAVs,thequad-rotor
UAVyawcontrollingisdonebythevariationoftorquebyusingallfourrotors. Thetri-rotorUAV
movesdownwardifanyoftherotorsdecelerate,whereasacceleratingtherotormakesUAVrotate.
InswitchingbladeUAVs,anadvancedpivotbackyawmechanismessentiallypermitsthebackrotorto
rotatelaterallyalongthelongitudinalaxis. RotatingtheUAVbyusingtheseforcesratherthantorque
vectoringrequiresmorepropellerstoactasanalternatefortheapplicationofforceontheverticalaxis.
ThismeansthatthepropellersoperateatfullcapacityandthebackpropellercanstillturntheUAV.
Ifwemaintaintheorientationeveninarough,windyenvironment,itisefficientandeasiertocontrol:
turningthebackpropellerintothewindcounteractsitsforce,andtheswitchingbladeUAVcould
easilybeoperatedinthesetypesofenvironments,whereasaquad-rotorUAVofthesamesizewould
notflywellintheabovementionedcase[13–15].
The latest unmanned tri-rotor systems are easy to use provided that they rely on the Fuzzy
Proportional Integral (F-PI) control algorithm discussed in [16–19]. For the tri-rotor, the operating
conditionsofthesystemandtherotormovementcoupledwiththenonlinearF-PIcontrollerislocated
outside(intheouterloop)ofthesystem(theyawheadingisfastandreliableasitishardtocontrol
the angle). To design a control algorithm for the nonlinear characteristics (noise and disturbance)
ofatri-rotor,thispaperpresentstheconditionsforfuzzyalgorithmswithanexistingPDcontroller.
Forcomparativeanalysis,atri-rotorFuzzyProportionalDerivative(F-PD)computersimulationof
responseperformanceispresentedin[20]. ThemainobjectiveoftheModelReferenceControl(MRC)
orpole-placementmethodistofindtheinputofthesystemanddrivethesystemoutputbytracking
thereferenceprovidedbytheinputmodelascloselyaspossible.
Thebasicthemeistoequalizethesystemoutputprovidedbythereferencesysteminput;wecan
alsosaythatthesystemoutputconvergestothereferenceinputexponentially[21–24]. AFuzzyLogic
Controller(FLC)mainlyconcernsthelinguisticrules. Thebenefitofthefuzzylogiccontrolleristhata
clarificationofaspecificdifficultycanberecognizedwithrespecttohumanbehaviorsothatitcan
berecognizedbyanoperator,andtheirinvolvementcanbeusedtodesignthecontroller’sIF/THEN
rules[25]. ThebiggestadvantageofFLCisclarityastotheenlargement,estimation,andmaintenance
ofthecontrolsystem[26]. Manyhybridcontrollerschemeswereformerlyappliedforthestabilization,
desired path tracking, and trajectory tracking of UAV, like the Model Reference Adaptive Control
(MRAC)base(ProportionalIntegralDerivative,PID)controllerin[27,28]. In[29],Mohammadietal.
useafuzzy-basedPIDcontrollertocontroltheattitudeandaltitudeofaquad-rotorUAV.Regulation,
Pole-Placement,andTracking(RST)basedualcontrollerschemewereproposedforcontrollingthe
stabilizationofatri-rotorUAV[30]. Afuzzy-basedhybridcontrolalgorithmwasdesignedforthe
stabilizationoftri-rotorUAVs[31]. Anadaptivehybridcontrollerschemewasusedtocontrolthe
attitudeandaltitudeofthetri-rotorUAV[32].
Inthisarticle,theperformanceoftheMRAC-hybridcontrolleriscomparedwithafuzzy-based
PID(F-PID)controller[33]andaFuzzylogiccontroller(FLC)[34].Thekeycontributionsofthisarticle
are(1)anovelMRAC-basedfuzzyPDandfuzzyPIDcontrolleraredesignedtocontrolthealtitude
andattitudemovementoftheaerialvehicleandremovethetransientandsteadystateerror;(2)the
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proposedcontrolleruseslinearandangularvelocitycomponents,whicharegivenasaninputtothe
The rest of the manuscript is structured as follows. The tri-rotor system model and preliminaries,
controller;and(3)thestabilityanalysisoftherotationalsystemisprovedbyLyapunovstabilitytheory.
as well as the system main engine, are discussed in Section 2. In Section 3, the design of the control
Therestofthemanuscriptisstructuredasfollows. Thetri-rotorsystemmodelandpreliminaries,
algorithm is provided, such that the position, altitude, and rotational control of the vehicle are
aswellasthesystemmainengine,arediscussedinSection2. InSection3,thedesignofthecontrol
discussed along with the system stability proof. Section 4 presents the simulation results and
algorithmisprovided,suchthattheposition,altitude,androtationalcontrolofthevehiclearediscussed
discussions of the paper. Finally, Section 5 gives the conclusions of the paper.
alongwiththesystemstabilityproof. Section4presentsthesimulationresultsanddiscussionsofthe
paper. Finally,Section5givestheconclusionsofthepaper.
2. The Tri-Rotor System Model and Preliminaries
2. ThTehTer i-mRoodtoerl Soyf sttheem trMi-roodteolr aanedriaPlr evleimhiicnlea riise staken from [35–37]. Tri-rotor UAV dynamics and
parameters are multivariable and nonlinear in nature. The model of the system is shown in Figure 1,
The model of the tri-rotor aerial vehicle is taken from [35–37]. Tri-rotor UAV dynamics and
which depicts the three rotors of the UAV placed in a triangular frame. However, one rotor of the
parametersaremultivariableandnonlinearinnature. ThemodelofthesystemisshowninFigure1,
three is used as a tilt rotor as per the requirements, and this is used to nullify the effects of the torque
whichdepictsthethreerotorsoftheUAVplacedinatriangularframe. However,onerotorofthethree
reaction.
isusedasatiltrotoraspertherequirements,andthisisusedtonullifytheeffectsofthetorquereaction.
AAssssuummppttiioonn 11.. TThhee ffoorrmmaattiioonn oorr aarrrraannggeemmeenntt ooff tthhee ttrrii--rroottoorr aaeerriiaall vveehhiiccllee iiss wweellll aaddjjuusstteedd iinn tteerrmmss ooff tthhee
oorriieennttaattiioonn ooff tthhee xx,, yy,,a annddz za axxisisw witithhr reessppeeccttt toor roototorr1 1,,r roototorr3 3,,a annddr orototorr2 2,,r ersepspecetcitviveleyly..

Figure1.Thestructureofthetri-rotoraerialvehicle.
Figure 1. The structure of the tri-rotor aerial vehicle.
22..11.. SSyysstteemm DDyynnaammiiccss
TThhee mmoototorrb blaladdeer ardadiuiussis ids edneontoetdedby bRy T(cid:1844)a(cid:3021)n danδTd is(cid:2012)(cid:3021)th eist othtael taontgalu alanrgvuellaorc ivteyloocfitthye osfy tshteem sy;astiesmth; ea
bisla dtheea brelaad,ρea aisretah,e (cid:2025)a(cid:3028)ir dise nthsiet ya,iqr sdheonuslidtyb, eqt hsehoroutlodr bshea tfhteto rroqtuoer, tsrhiasftth teotroqtuael,t h(cid:1872)r(cid:3045)u sits ptrhoed tuocteadl itnhrtuhset
spyrsotedmuc,ethde inae trhoed ysnyastmemic,c otheeffi aceiernodtoyfntahmruics tciosedffeincioetnetd obfy thArcut,stth ies adeernodotyenda mbyic (cid:1827)ro(cid:3030)(cid:3047)to, rthseh aafetrcoodeyffincaimenitc
irsoAtocrq ,sahnadft Acocepfifsictiheenta iesr o(cid:1827)d(cid:3030)y(cid:3044)n, aamndic (cid:1827)co(cid:3030)(cid:3043)e ffiisc tiehnet aoefropdoywnearm. ic coefficient of power.
A(cid:1827)(cid:3030)(cid:3047)==t(cid:1872)(cid:3045)/⁄ρ(cid:2025)(cid:3028)a(cid:1853)(δ((cid:2012)(cid:3021)R(cid:1844)(cid:3021)))2(cid:2870) ((11))
ct r a T T
(cid:1827) =(cid:1869)⁄(cid:2025) (cid:1853)((cid:2012) (cid:1844) )(cid:2870)(cid:1844) (2)
(cid:3030)(cid:3044) (cid:3028) (cid:3021) (cid:3021) (cid:3021)
A = q/ρ a(δ R )2R (2)
cq a T T T
(cid:1827) =(cid:1829) ⁄(cid:2025) (cid:1853)((cid:2012) (cid:1844) )(cid:2871) (3)
(cid:3030)(cid:3043) (cid:3043) (cid:3028) (cid:3021) (cid:3021)
A =C /ρ a(δ R )3 (3)
cp p a T T
The coefficient of power can be written as (cid:1829) =(cid:1869)(cid:2012) .
(cid:3043) (cid:3021)
THheenccoee, fficientofpowercanbewrittenasCp = qδT.
Hence,
(cid:1827) =(cid:1827) =(cid:1869)(cid:2012) ⁄(cid:2025) (cid:1853)((cid:2012) (cid:1844) )(cid:2871). (4)
A(cid:3030)(cid:3043)= A(cid:3030)(cid:3044)= qδ(cid:3021)/ρ(cid:3028)a(δ (cid:3021)R (cid:3021))3. (4)
cp cq T a T T
The total external forces and torques acting in the frame of the vehicle are given by:
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Thetotalexternalforcesandtorquesactingintheframeofthevehiclearegivenby:
(cid:40)
f = f + f + f
ext. x y z
(5)
f = 0+t δ |δ |sinα−t (δ |δ |cosα+δ |δ |+δ |δ |).
ext. r 1 1 r 1 1 2 2 3 3
Thetiltangleisdefinedbyαandcanbecontrolledbyusingu ,theyawcontroller.
4

 τfext. = (cid:18)(3)20.5 ∗ltr(δ2|δ2|−−δ3(|lδt3r|δ)1(cid:19)|δτ1+|fesx(cid:16)itn.12α=L−tτrx(kδτ+2δ|1δτ|2yδ|1++|cτoδzs3α|δδ32||)δ2−|+ltrδδ31||δδ31||)cosα+kτδ1|δ1|sinα(cid:17)  (6)
where f , f , and f arethetotalforcesthatareexertedonabody,andτ , τ , τ arethetotaltorque
x y z x y z
responsesalongthex,y,orzdirectionoftheEarthcoordinatesystem.
Assumption 2. The three-rotor UAV is rigid. Then, the nonlinear dynamics can be derived using the
Newton–Eulerformulas.
Fortheidealhoveringandtrajectorytrackingcoefficientofprofile,dragistakentobeconstant
andsupposedtobe0.015. Thevalueistakenfromthemomentumtheory[38],inwhichthesolidity
ratioisdenotedby“σ”. ToavoidalltheexternaleffectsexertedontheUAVbodymoments,rotational
andtranslationaldynamicsareusedtogovernthetri-rotorUAV,andarewrittenasfollows.
TranslationalDynamics: Wehavetoneglectalltheeffectsofframemovementsonthetranslational
velocitycomponentsleadingthetri-rotoraerialvehicle,whicharegivenby:
..
X = (u /m)(cosϕcosψsinθ+sinψsinϕ) (7)
1
..
Y = (u /m)(sinθsinψcosϕ−sinϕcosψ) (8)
1
..
Z = −g+(u /m)(cosθcosϕ). (9)
1
where“m”isthemass,u theverticalorcollectiveisforce.
1
RotationalDynamics: Wehavetoneglecttheframemovementsthatareexertedonthebodyof
UAV,similartothetranslationalvelocityeffects:
.. (cid:0) (cid:1)
ϕ = qr I −I /I +(L/I )u (10)
y z x x 2
.. (cid:0) (cid:1) (cid:0) (cid:1)
θ = pr I −I /I + L/I u (11)
z x y y 3
.. (cid:0) (cid:1)
ψ = pq I −I /I +(L/I )u (12)
x y z z 4
where“L”isthelengthfromthecenterofUAV,u , u , u aretheroll,pitch,andyawcontrolcommands,
2 3 4
respectively,generatedbythepropellersofthetri-rotoraerialvehicle. Theinertialmomentsofthe
tri-rotorare I , I , and I .
x y z
AngularRatesorEulerAngles:
. .
p = ϕ−ψsinθ (13)
. .
q = θcosϕ+ψcosθsinϕ (14)
. .
r = ψcosθcosϕ−θsinϕ (15)
The position from the center of mass in terms of the inertial frame is defined by x, y, and
z, respectively. The Euler angles, ϕ, θ, and ψ, are used to define the position of UAV. For the
trajectory tracking, neglect the higher order dynamics and let u = u + mg. Consequently,
1 T
.. .. ..
X = −gϕ, Y = gθ, Z = u /m.
T
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2.2. MainEngine(ElectricMotors)
ABrushlessDirectCurrent(BLDC)motorhasamagnetontherotorsideandwindingonthe
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statorsideisdrivenbyapresetserialarrangementoftheDirectCurrent(DC)powersourcecalled
caocmommmutuattoatro. rI.nI nstsattaotro wrwinidnidnign,g b,abcakc kEElelcetcrtor oMMaaggnneteitci cFFieieldld (E(EMMFF) )isis ggeenneerraatteedd wwhheenn tthhee rroottaattiinngg
mmaaggnneett iinntteerraaccttss wwiitthh tthhee ssttaattoorr ppoollee.. TThhee mmooddeell ooff BBLLDDCC mmoottoorr iiss ttaakkeenn ffrroomm [[3399]]..
EΕT(cid:3021)==ZΖT(cid:3021)IΙT(cid:3021)++ϕφd(cid:1856)d(cid:1856)IΙtT(cid:1872)(cid:3021)++(cid:2027)ς(cid:3021)T (1(41)6 )
ς(cid:2027)(cid:3021)== ζ(cid:2014)(cid:3045)..ΓΓ(cid:3045)(cid:1858)f((ΘΘ(cid:3045))),, (1(51)7 )
T r r r
where (cid:2014) , Θ , Γ are the angular rotor velocity, the rotor position, and the rotor magnetic flux
whereζ (cid:3045), Θ(cid:3045), Γ(cid:3045)aretheangularrotorvelocity,therotorposition,andtherotormagneticfluxconstant,
croesnpsetacntitrv, erleys,rpaencdrtivtheelyg, eannedr attheed gteonrqeruaeteisd gtiovreqnuea sisT gei=venς TaζrsIT (cid:2286).(cid:3032) = (cid:3098)(cid:3269)(cid:3085) (cid:3293)(cid:2945)(cid:3269) .
IInn tteerrmmss ooff mmaacchhiinnee ppaarraammeetteerrss,, ttoorrqquuee iiss ddeefifinneedd aass::
T(cid:2286)(cid:3032)==T(cid:2286)(cid:3013)++(cid:1836)J d(cid:3031)(cid:3031)(cid:3085)ζ(cid:3047)(cid:3293)r++(cid:2022)ξ(cid:2014)ζ(cid:3045). . (1(61)8 )
e L r
dt
33.. DDeessiiggnn ooff tthhee CCoonnttrrooll AAllggoorriitthhmm
TThhee ccoommpplleettee ssyysstteemm ccoonnttrrooll mmooddeell iiss ddiissccuusssseedd iinn FFiigguurree 22..

Figure 2. The block diagram of the reference trajectory control system.
Figure2.Theblockdiagramofthereferencetrajectorycontrolsystem.
3.1. The Control Objective and Its Approach
3.1. TheControlObjectiveandItsApproach
The dynamics of the UAV includes altitude, roll, and pitch and yaw control command;
The dynamics of the UAV includes altitude, roll, and pitch and yaw control command;
(cid:2012) , (cid:2012) ,and (cid:2012) are the angular velocity of all three rotors, respectively.
δ(cid:2869), δ(cid:2870), andδ(cid:2871) aretheangularvelocityofallthreerotors,respectively.
1 2 3
RRoollll CCoonnttrrooll:: TTwwoo ccoonnddiittiioonnss ccoonnttrrooll tthhee rroollll mmoommeenntt ooff tthhee vveehhiiccllee..
I. Clockwise (cid:2012) <(cid:2012) <(cid:2012)
(cid:2870) (cid:2869) (cid:2871)
I. Clockwiseδ < δ < δ
II. Anticlo2ckwis1e (cid:2012) 3<(cid:2012) <(cid:2012)
(cid:2871) (cid:2869) (cid:2870)
II. Anticlockwiseδ < δ < δ
3 1 2
Pitch Control: There are also two conditions for pitch controlling.
PitchIC. ontNrools:eT-Uhepr e(cid:2012)ar<ea(cid:2012)lso=tw(cid:2012)o conditionsforpitchcontrolling.
(cid:2869) (cid:2871) (cid:2870)
II. Nose-Down (cid:2012) =(cid:2012) <(cid:2012)
(cid:2870) (cid:2871) (cid:2869)
I. Nose-Upδ < δ = δ
1 3 2
II. YNaows eC-Donotwronl:δ Ya=wδ co<ntrδol has only one condition along with tilt angle.
2 3 1
• (cid:2012) =(cid:2012) =(cid:2012) (cid:1875)(cid:1861)(cid:1872)(cid:1860) (cid:2009) =0
(cid:2871) (cid:2870) (cid:2869)
YawControl: Yawcontrolhasonlyoneconditionalongwithtiltangle.
Altitude Control: To achieve the preferred altitude of the UAV, the RPM of all rotors of the UAV
m•ustδ be= thδe s=amδe:w ithα =0
3 2 1
• (cid:2012) =(cid:2012) =(cid:2028) .
(cid:2869) (cid:2870) (cid:3031)(cid:2869)
AltitudeControl: ToachievethepreferredaltitudeoftheUAV,theRPMofallrotorsoftheUAVmustbe
From translation and the rotational dynamics of UAV, the parameters (cid:1873) , (cid:1873) , (cid:1873) , and (cid:1873) are
(cid:2869) (cid:2870) (cid:2871) (cid:2872)
thesame:
given as
(cid:1873) = (cid:1864) (cid:2028) (δ(cid:2870)−δ(cid:2870)) (17)
(cid:2869) (cid:2869) (cid:3030)(cid:2869) (cid:2869) (cid:2870)
(cid:1873) = (cid:1864) (cid:2028) (δ(cid:2870)−δ(cid:2870))−(cid:1864) (cid:2028) δ(cid:2870)(cid:1855)(cid:1867)(cid:1871)(cid:2009) (18)
(cid:2870) (cid:2870) (cid:3030)(cid:2869) (cid:2869) (cid:2870) (cid:2871) (cid:3030)(cid:2870) (cid:2871)
(cid:1873) = (cid:1864) (cid:2028) (−δ(cid:2870)−δ(cid:2870))+(cid:1864) (cid:2028) δ(cid:2870)+(cid:1864) (cid:2028) δ(cid:2870)(cid:1871)(cid:1861)(cid:1866)(cid:2009) (19)
(cid:2871) (cid:2872) (cid:3031)(cid:2869) (cid:2869) (cid:2870) (cid:2871) (cid:3031)(cid:2870) (cid:2871) (cid:2871) (cid:3031)(cid:2870) (cid:2871)
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• δ = δ = τ .
1 2 d1
From translation and the rotational dynamics of UAV, the parameters u , u , u , and u are
1 2 3 4
givenas
(cid:16) (cid:17)
u = l τ δ2−δ2 (19)
1 1 c1 1 2
(cid:16) (cid:17)
u = l τ δ2−δ2 −l τ δ2 cosα (20)
2 2 c1 1 2 3 c2 3
(cid:16) (cid:17)
u = l τ −δ2−δ2 +l τ δ2+l τ δ2sinα (21)
3 4 d1 1 2 3 d2 3 3 d2 3
(cid:16) (cid:17)
u = τ δ2+δ2 +τ δ2 cosα (22)
4 c1 1 2 c2 3
wherethethrustcoefficientisτ , τ ,andthedragcoefficientisτ , τ .
c1 c2 d1 d2
3.2. ModelReferenceAdaptiveControl
The MRAC algorithm is used to insure the stability of proposed system dynamics as well as
the noise and time delay in the system [40]. MRAC warrants serious consideration as a means of
controllingthenonlinearandadaptivesystem. Theparametersofacontrolleraretunedbyusingthe
errorbetweenthereferencemodelandcloseloopdynamics.
Assumption3. Thesystemerroriswrittenase = T −T ,whereTisthex,y,orzaxis,respectively.
T a d
Assumption4. BycomparingF-PIDandtheFLCcontroller,theproposedMRAC-Hybridcontrolleraccurately
followsthereferencetrajectoryproofoftheirrotationalsystemstability,asshowninTheorem1.
3.3. PositionalControl
Proposition1. Thepositioncontrolofthesystemiswrittenusingtheadaptivelawfortheorientationofthe
systembyusingEquations(7)and(8).
. . . . . .
Lete , e ,ande betheerrorsofspeed;x ,y ,andz arethedesiredspeeds,wherex , y ,andz
x y z d d d a a a
aretheactualspeedsinthex,y,andzdirections,respectively.
. .
e = x − x (23)
x d a
. .
e = y −y (24)
y d a
. .
e = z −z (25)
z d a
Thus,thedesiredroll,pitch,andyawcanbecalculatedbyusingtheseequations:
. .
ϕ = p+ψsinθ (26)
d
. .
θ = q−ψcosθsinϕ/cosϕ (27)
d
. .
ψ =r+θsinϕ/cosθcosϕ. (28)
d
3.4. AltitudeControl
FromEquation(9),thealtitudeofUAVcontainstheverticalforceofinputu andcanbewrittenas:
1
..
u = Zm+gm/cosθcosϕ, (29)
1
wherecosθandcosϕ=0andthefuzzy-basedproportionalderivativecontrollerusedtocontrolthe
altitudeofUAVcanbeexpressedas:
Aerospace2017,4,3 7of17
.
FPD = −G z−G (z −z ), (30)
D P a d
wherez istheactualandz isthedesiredaltitude. Theinputrulesforerror,thederivativeoferror,
a d
andoutputrulesforthefuzzy-basedPDcontrollerareshowninFigure3,suchthatG andG are
D P
theproportionalandderivativegainsofthefuzzycontroller,respectively. Theoutputsurfaceofthe
fuzzy-basedproportionalandfuzzy-basedderivativecontrollerareshowninFigure4.
Aerospace 2017, 4, 3 7 of 17
Aerospace 2017, 4, 3 7 of 17
(a) (b)
(a) (b)
Figure 3. (a) Input rules for error and the derivative error of the fuzzy-based Proportional Derivative
Figure3.(a)Inputrulesforerrorandthederivativeerrorofthefuzzy-basedProportionalDerivative
(FPiDg)u croe n3t.r (oal)l eIrn;p (but) rthuele os ufotpr uetr rrourl easn fdo rt hthe ed feurzivzayt-ibvaes eerdr oPrD o fc othnetr foulzlezry. -based Proportional Derivative
(PD)controller;(b)theoutputrulesforthefuzzy-basedPDcontroller.
(PD) controller; (b) the output rules for the fuzzy-based PD controller.
(a)
(a)
(b)
(b)
Figure 4. (a) The output surface of the fuzzy-based proportional controller; (b) the output surface of
FigtFhuierg efuu4rz.ez (4ya.-) b(Taa)hs eTedho edu eotrpuiuvtpatutsitvu seru fcarofcanectreoo foltlfeh treh. ef ufuzzzyzy-b-baasseeddp prrooppoorrttiioonnaall ccoonnttrroolllleerr; ;((bb) )ththee oouutptpuut stusrufrafcaec oefo f
the fuzzy-based derivative controller.
thefuzzy-basedderivativecontroller.
3.5. Attitude Control
3.5. Attitude Control
The fuzzy-based proportional integral derivative control method for the rotational dynamics of
UAV Tahree afsuszizgyn-ebda saesd (cid:1873) pr,o(cid:1873)p,o artniodn (cid:1873)al. in tegral derivative control method for the rotational dynamics of
(cid:2870) (cid:2871) (cid:2872)
UAV are assigned as (cid:1873) ,(cid:1873) , and (cid:1873) .
(cid:2870) (cid:2871) (cid:2872)
Proposition 2. The roll tracking error of UAV can be written as:
Proposition 2. The roll tracking error of UAV can be written as:
Aerospace2017,4,3 8of17
3.5. AttitudeControl
Thefuzzy-basedproportionalintegralderivativecontrolmethodfortherotationaldynamicsof
UAVareassignedasu , u ,andu .
2 3 4
Proposition2. TherolltrackingerrorofUAVcanbewrittenas:
e = ϕ −ϕ . (31)
ϕ a d
TheLyapunovfunctioninthesystemcanbeexpressedas:
(cid:90)
S = G e +G e dt, (32)
1 1 ϕ 2 ϕ
(cid:82)
inwhichG andG arethegainsforthefine-tuningofthesystemand e istherollintegralerror.
1 2 ϕ
TheLyapunovstabilitytheoryisusedwhileusingtheLyapunovcandidatefunctionS asapositive
1
definiteanditstimederivativeasanegativesemi-definite,whichisshowninEquation(33).
Assumption5.Bycomparingwiththepreviousworks,ourproposedmethodprovidesrobustnessinthepresence
ofnoiseandmodeldisturbanceaswellasfastconvergenceandazerosteadystateerror.
Theorem 1. The rotational controller in the equation that is applied for the tri-rotor aerial vehicle will
asymptotically converge on the rotational velocity component and tracking errors along with the model
uncertaintyandnoisetozero.
Proof. ConsidertheLyapunovfunction
V = (1/2)S2. (33)
1 1
Itsderivativeis
. . (cid:0) . . (cid:1)
V = S S = S G ϕ −G ϕ +G e . (34)
1 1 1 1 1 a 1 d 2 ϕ
Note: NocontrolinputispresentinEquation(34).
. (cid:0).(cid:1)
Ifwesuppose ϕisthevirtualcontrol,thedesiredvirtualcontrol ϕ iswrittenas
a
(cid:0).(cid:1) . (cid:0) (cid:1)
ϕ = ϕ − G e /G −(S /G ). (35)
a d 2 ϕ 1 1 1
.
Now,thevirtualcontrol ϕistherollcontrolrateofatri-rotoralongwithitsownerror:
. . .
S = ϕ −ϕ = (1/G )(S +S ). (36)
2 a d 1 1 1
NowtheamplifiedLyapunovfunctionforthesecondstagestrategyiswrittenas:
V = (1/2)S2+(1/2)S2. (37)
2 1 2
Nextwetakethederivativeofsecondstepstrategy:
. . .
V = S S +S S . (38)
2 1 1 2 2
. .
Afterthat,weputthevaluesofS andS inthesecondstepstrategyintothederivativeequation:
1 2
V.2=S2(cid:104)eϕ(cid:16)G12+GG21(cid:17)+(cid:82) eϕdt(G1G2)+e.ϕ(cid:16)GG21(cid:17)+qr(cid:16)Iy−IxIz(cid:17)+ LIux2 −ϕ..d(cid:105)−S1(cid:2)G1eϕ+G2(cid:82) eϕdt(cid:3). (39)
Aerospace2017,4,3 9of17
Thedesireddynamicsofthesystemcanbewrittenas
. .
V = −(1/G )(S +S ). (40)
2 1 1 1
.
NowbyputtingS andS inEquation(39),thedesirabledynamicsaregivenby
1 1
. . (cid:90)
V = −e −e (G /G )− e dt(G /G ). (41)
2 ϕ ϕ 2 1 ϕ 2 1
Thedesiredsystemdynamicsverifiesthenegativedefiniteoftrackingerror,itsvelocitytracking
error,andintegration. NowEquation(39)willbenegative,ifu isgivenby
2
u = −(cid:20)e (cid:18)G2 +G2(cid:19)+(cid:90) e dt(cid:18)G2 +G G (cid:19)+e. (cid:18)G2(cid:19)−ϕ.. +qr(cid:18)Iy−Iz(cid:19)(cid:18)Ix(cid:19)(cid:21). (42)
2 ϕ G 1 ϕ G 1 2 ϕ G d I L
1 1 1 x
Similarly,thepitchcontrolequationcanbewrittenas
u = −(cid:20)e (cid:18)G4 +G2(cid:19)+(cid:90) e dt(cid:18)G4 +G G (cid:19)+e. (cid:18)G4(cid:19)−θ.. +pr(cid:18)Iz−Iy(cid:19)(cid:18)Iy(cid:19)(cid:21). (43)
3 θ G 3 θ G 3 4 θ G d I L
3 3 3 y
Lastly,theyawcontrolequationcanbewrittenas
u = −(cid:20)e (cid:18)G6 +G2(cid:19)+(cid:90) e dt(cid:18)G6 +G G (cid:19)+e. (cid:18)G6(cid:19)−ψ.. +pq(cid:18)Ix−Iy(cid:19)(cid:18)Iz(cid:19)(cid:21). (44)
4 ψ G 5 ψ G 5 6 ψ G d I L
5 5 5 z
Nowthefuzzy-basedPIDgainsandroll,pitch,andyawmodulescanbewrittenas
G
FP = 2 +G2 (45)
G 1
1
G
FI = 2 +G G (46)
1 2
G
1
G
FD = 2, (47)
G
1
whereG andG arethegainsforrollcontrolandfuzzyruleforthePIDcontroller,asdefinedafter
1 2
thestabilityperformance. Theinputrulesforerror,itsderivativeerror,andtheoutputrulesforthe
fuzzy-basedPIDcontrollerareshowninFigure5. Theoutputsurfaceofthefuzzy-basedproportional,
fuzzy-basedderivative,andfuzzy-basedintegralcontrollerareshowninFigure6.
Therootsofacharacteristicequationcanbeobtainedbyplacingthepolesofthesystematthe
desiredlocation,i.e.,k , k , k ,byusingthepole-placementmethodofAlietal.[31]. Tuningofthe
a b c
system can be made faster such that the derivative of the Lyapunov candidate function becomes
morenegative.
G
2 = k +k +k (48)
G a b c
1
G
2 +G2 = k k +k k +k k (49)
G 1 a b a c b c
1
G
2 +G G = k k k (50)
G 2 1 a b c
1
ProofofStabilityPerformance. TheLyapunovcandidatefunctionisusedtoinsurethestabilityofour
.
system. ConsidertheLyapunovcandidatefunctiontobeV ≤0,where∀(e )ensuresthetracking
T T
error. Thedesiredpositions ϕ , θ , ψ areboundedbythesuppositionsuchthatthetrackingerroris
d d d
alsobounded. Moreover,thestabilityisalsoprovedbyLyapunovcandidatefunction“V,”whichis
Aerospace 2017, 4, 3 9 of 17
(cid:1873) =−(cid:3428)(cid:1857) (cid:4672)(cid:3008)(cid:3120)+(cid:1833)(cid:2870)(cid:4673)+(cid:1516)(cid:1857) (cid:1856)(cid:1872)(cid:4672)(cid:3008)(cid:3120)+(cid:1833) (cid:1833) (cid:4673)+(cid:1857)(cid:4662) (cid:4672)(cid:3008)(cid:3120)(cid:4673)−(cid:2016)(cid:4663) +(cid:1868)(cid:1870)(cid:3436)(cid:3010)(cid:3301)(cid:2879)(cid:3010)(cid:3300)(cid:3440)(cid:4672)(cid:3010)(cid:3300)(cid:4673)(cid:3432). (41)
(cid:2871) (cid:3087) (cid:3008)(cid:3119) (cid:2871) (cid:3087) (cid:3008)(cid:3119) (cid:2871) (cid:2872) (cid:3087) (cid:3008)(cid:3119) (cid:3031) (cid:3010)(cid:3300) (cid:3013)
Lastly, the yaw control equation can be written as
(cid:1873)(cid:2872) =−(cid:4674)(cid:1857)(cid:3103)(cid:4672)(cid:3008)(cid:3008)(cid:3122)(cid:3121)+(cid:1833)(cid:2873)(cid:2870)(cid:4673)+(cid:1516)(cid:1857)(cid:3103)(cid:1856)(cid:1872)(cid:4672)(cid:3008)(cid:3008)(cid:3122)(cid:3121)+(cid:1833)(cid:2873)(cid:1833)(cid:2874)(cid:4673)+(cid:1857)(cid:4662)(cid:3103)(cid:4672)(cid:3008)(cid:3008)(cid:3122)(cid:3121)(cid:4673)−(cid:2032)(cid:4663)(cid:3031)+(cid:1868)(cid:1869)(cid:4672)(cid:3010)(cid:3299)(cid:3010)(cid:2879)(cid:3301)(cid:3010)(cid:3300)(cid:4673)(cid:4672)(cid:3010)(cid:3013)(cid:3301)(cid:4673)(cid:4675). (42)
Now the fuzzy-based PID gains and roll, pitch, and yaw modules can be written as
(cid:1833)
(cid:1832)(cid:1842)= (cid:2870)+(cid:1833)(cid:2870) (43)
(cid:1833) (cid:2869)
(cid:2869)
(cid:1833)
(cid:2870)
(cid:1832)(cid:1835)= +(cid:1833) (cid:1833) (44)
(cid:1833) (cid:2869) (cid:2870)
(cid:2869)
(cid:1832)(cid:1830)=(cid:3008)(cid:3118), (45)
(cid:3008)(cid:3117)
where (cid:1833) and (cid:1833) are the gains for roll control and fuzzy rule for the PID controller, as defined after
(cid:2869) (cid:2870)
the stability performance. The input rules for error, its derivative error, and the output rules for the
fuzzy-based PID controller are shown in Figure 5. The output surface of the fuzzy-based
proportional, fuzzy-based derivative, and fuzzy-based integral controller are shown in Figure 6.
The roots of a characteristic equation can be obtained by placing the poles of the system at the
desired location, i.e., (cid:1863) ,(cid:1863) ,(cid:1863) , by using the pole-placement method of Ali et al. [31]. Tuning of the
(cid:3028) (cid:3029) (cid:3030)
system can be made faster such that the derivative of the Lyapunov candidate function becomes more
negative.
(cid:1833)
(cid:2870)
=(cid:1863) +(cid:1863) +(cid:1863) (46)
(cid:1833) (cid:3028) (cid:3029) (cid:3030)
(cid:2869)
(cid:1833)
Aerospace2017,4,3 (cid:1833)(cid:2870)+(cid:1833)(cid:2869)(cid:2870) =(cid:1863)(cid:3028)(cid:1863)(cid:3029)+(cid:1863)(cid:3028)(cid:1863)(cid:3030)+(cid:1863)(cid:3029)(cid:1863)(cid:3030) (1407)o f17
(cid:2869)
(cid:1833)
(cid:2870)
positiveanddefinite,andtheirderivative(cid:1833)i(cid:2869)s+ne(cid:1833)g(cid:2870)a(cid:1833)t(cid:2869)iv=e.(cid:1863)F(cid:3028)(cid:1863)o(cid:3029)r(cid:1863)e(cid:3030)x ample,V. (eT) <0, ∀(eT) (cid:54)=0&V(4(80)) =
0,(T = ϕ, θ, ψ).

Aerospace 2017, 4, 3 10 of 17
((aa)) ((bb))
FigFFuiigrgeuurr5ee. 55(.. a ()(aa)I) n IIpnnupptuuttr urruluellseessf offororre ererrrroroorrr a aannnddd dddeeerrriiivvvaaatttiiivvveee eeerrrrrrooorrr ooofff ttthhheee fffuuuzzzzzzyyy--b-bbaaasseseeddd PPPrroorpoppoororttriiotoinnoaanll a IIlnnIttneegtgerragalrl al
DeDrDieverraiitvviavatteiivv(ePe I((DPPII)DDc))o ccnootnrnottrrloloelllrlee;rr(;;b (()bbt))h ttheheeo ououutpttppuuutttr ruruulelleesss f fofoorrr t tthhheee fffuuuzzzzzzyyy---bbbaaassseeeddd PPPIIIDDD cccooonnnttrtrrooollllleelerr.r. .
(a) (b)
(c)
Figure 6. (a) The output surface of the fuzzy-based proportional controller; (b) the output surface of
Figure6.(a)Theoutputsurfaceofthefuzzy-basedproportionalcontroller;(b)theoutputsurfaceof
the fuzzy-based derivative controller; (c) the output surface of the fuzzy-based integral controller.
thefuzzy-basedderivativecontroller;(c)theoutputsurfaceofthefuzzy-basedintegralcontroller.
4. Simulation Results and Discussion
4. SimulationResultsandDiscussion
This section proves the effectiveness and stability of our proposed controller by using two
Thissectionprovestheeffectivenessandstabilityofourproposedcontrollerbyusingtwodifferent
different simulations of trajectory tracking of tri-rotor UAV. We compare our proposed MRAC-
simHuylabtriiodn csoonftrtroalljeecr tworiytht rFa-cPkIiDn ganodf tFriL-rCo tcoornUtrAolVle.rWs ebyc oumtiplizairnego tuhrep droatpao osef d[3M1],R wAhCi-cHh yabreri dshcoownntr oinll er
witThaFbl-eP I1D. andFLCcontrollersbyutilizingthedataof[31],whichareshowninTable1.
Table 1. Tri-rotor aerial vehicle system parameters.
Parameters Mass, m g L Ix Iy Iz
Values 0.785 9.81 0.3050 0.3105 0.3105 0.3212
SI Units kg m/s2 m Kg·m2 Kg·m2 Kg·m2
The block diagram of reference trajectories and block diagram of the desired values for the
control system are shown in Figure 2. Selected trajectories using input as a reference (which
constantly changes its position by using the horizontal and vertical errors of UAV) are exposed as
being real applications. That will also help us to analyze the effectiveness and robustness of our
proposed control algorithm in the absence of noise or disturbance in Simulation I and in the presence
of noise or disturbance in Simulation Case II.
The attitude and altitude control equations are used to control the orientation of UAV, in which
initial errors are controlled by the desired position of the UAV at the required altitude. The overall
system performance depends on MRAC and the stability of the system is proven by Lyapunov
stability.
4.1. Simulation Case I