Multirotor-Aerial-Vehicles-Modeling-Estimation-and

•ByRobertMahony,VijayKumar,andPeterCorke20�IEEEROBOTICS&AUTOMATIONMAGAZINE�SEPTEMBER20121070-9932/12/$31.00ª2012IEEEDigitalObjectIdentifier10.1109/MRA.2012.2206474Dateofpublication:27August2012Thisarticleprovidesatutorialintroductiontomodeling,es-timation,andcontrolformulti-rotoraerialvehiclesthatincludesthecommonfour-rotororquad-rotorcase.Aerialroboticsisafast-growingfieldofroboticsandmultirotorair-craft,suchasthequadrotor(Fig-ure1),arerapidlygrowinginpopularity.Infact,quadrotoraerialroboticvehicleshavebecomeastandardplatformforroboticsresearchworldwide.Theyalreadyhavesufficientpayloadandflightendurancetosupportanumberofindoorandoutdoorapplications,andtheimprovementsofbatteryandothertechnologyisrapidlyincreasingthescopeforcommercialopportunities.Theyarehighlyma-neuverableandenablesafeandlow-costexperimentationinmapping,navigation,andcontrolstrategiesforrobotsthatmoveinthree-dimensional(3-D)space.Thisabilitytomovein3-Dspacebringsnewresearchchallengescom-paredwiththewheeledmobilerobotsthathavedrivenmobileroboticsresearchoverthelastdecade.Smallquadrotorshavebeendemon-stratedforexploringandmapping3-Denviron-ments;transporting,manipulating,andassemblingobjects;andacrobatictrickssuchasjuggling,balancing,andflips.Additionalrotorscanbeadded,leadingtogeneral-izedN-rotorvehicles,toimprovepayloadandreliability.Modeling,Estimation,andControlofQuadrotor©ISTOCKPHOTO.COM/©ANDREJSZAVADSKISThistutorialdescribesthefundamentalsofthedynamics,estimation,andcontrolforthisclassofvehicle,withabiastowardelectricallypoweredmicro(lessthan1kg)-scalevehicles.ThewordhelicopterisderivedfromtheGreekwordsforspiral(screw)andwing.Fromalinguisticperspec-tive,sincetheprefixquadisLatin,thetermquadrotorismorecorrectthanquadcopterandmorecommonthantet-racopter;hence,weusethetermquadrotorthroughout.ModelingofMultirotorVehiclesThemostcommonmultirotoraerialplatform,thequadro-torvehicle,isaverysimplemachine.Itconsistsoffourindividualrotorsattachedtoarigidcrossairframe,asshowninFigure1.Controlofaquadrotorisachievedbydifferentialcontrolofthethrustgeneratedbyeachrotor.Pitch,roll,andheave(totalthrust)controlisstraightfor-wardtoconceptualize.AsshowninFigure2,rotorirotatesanticlockwise(positiveaboutthezaxis)ifiisevenandclockwiseifiisodd.Yawcontrolisobtainedbyadjustingtheaveragespeedoftheclockwiseandanticlockwiserotat-ingrotors.Thesystemisunderactuated,andtheremainingdegreesoffreedom(DoF)correspondingtothetransla-tionalvelocityinthex-yplanemustbecontrolledthroughthesystemdynamics.Rigid-BodyDynamicsoftheAirframeLetf~x,~y,~zgbethethreecoordinateaxisunitvectorswithoutaframeofreference.Let{A}denotearight-handinertialframewithunitvectorsalongtheaxesdenotedbyf~a1,~a2,~a3gexpressedin{A}.Onehasalgebraicallythat~a1¼~x,~a2¼~y,~a3¼~zin{A}.Thevectorr¼(x,y,z)2fAgdenotesthepositionofthecenterofmassofthevehicle.Let{B}bea(right-hand)bodyfixedframefortheairframewithunitvectorsf~b1,~b2,~b3g,wherethesevectorsaretheaxesofframe{B}withrespecttoframe{A}.TheorientationoftherigidbodyisgivenbyarotationmatrixARB¼R¼½~b1,~b2,~b32SO(3)inthespecialorthogonalgroup.Onehas~b1¼R~x,~b2¼R~y,~b3¼R~zbyconstruction.WewilluseZ-X-YEuleranglestomodelthisrotation,asshowninFigure3.Togetfrom{A}to{B},wefirstrotateabouta3bythetheyawangle,w,andwewillcallthisinter-mediaryframe{E}withabasisf~e1,~e2,~e3gwhere~eiisexpressedwithrespecttoframe{A}.Thisisfollowedbyarotationaboutthexaxisintherotatedframethroughtherollangle,/,followedbyathirdpitchrotationaboutthenewyaxisthroughthepitchanglehthatresultsinthebody-fixedtriadf~b1,~b2,~b3gR¼cwchs/swshc/swcwshþchs/swchswþcws/shc/cwswshcwchs/c/shs/c/ch0@1A,wherecandsareshorthandformsforcosineandsine,respectively.Letv2fAgdenotethelinearvelocityof{B}withrespectto{A}expressedin{A}.LetX2fBgdenotetheangularvelocityof{B}withrespectto{A};thistimeexpressedin{B}.Letmdenotethemassoftherigidobject,andI2R333denotetheconstantinertiamatrix(expressedinthebodyfixedframe{B}).Therigidbodyequationsofmotionoftheairframeare[2]and[3]_n¼v,(1a)m_v¼mg~a3þRF,(1b)_R¼RX3,(1c)I_X¼X3IXþs:(1d)ThenotationX3denotestheskew-symmetricmatrix,suchthatX3v¼X3vforthevectorcrossproduct3andanyvectorv2R3.ThevectorsF,s2fBgcombinetheprinci-palnonconservativeforcesandmomentsappliedtothequadrotorairframebytheaerodynamicsoftherotors.DominantAerodynamicsTheaerodynamicsofrotorswasextensivelystudiedduringthemid1900swiththedevelopmentofmannedhelicop-ters,anddetailedmodelsofrotoraerodynamicsareavail-ableintheliterature[4],[5].Muchofthedetailabouttheseaerodynamicmodelsisusefulforthedesignofrotorsystems,wherethewholerangeofparameters(rotorxyz{B}T1T2T3T4dFrontΦiFigure2.Notationforquadrotorequationsofmotion.N¼4;Uiisamultipleofp=4(adaptedwithpermissionfrom[1]).Figure1.AquadrotormadebyAscendingTechnologieswithVICONmarkersforstateestimation.MICHAELSHOMIN,CMUSEPTEMBER2012�IEEEROBOTICS&AUTOMATIONMAGAZINE�21geometry,profile,hingemechanism,andmuchmore)arefundamentaltothedesignproblem.Foratypicalroboticquadrotorvehicle,therotordesignisaquestionforchoos-ingoneamongfiveorsixavailablerotorsfromthehobbyshop,andmostofthecomplexityofaer