Model-Free Scheme for Angle-of-Attack and Angle-of-Sideslip Estimation

Open AccessEngineering NotesModel-Free Scheme for Angle-of-Attack and Angle-of-Sideslip EstimationAngelo Lerro, Alberto Brandl Piero GiliAngelo LerroPolytechnic University of Turin, 10129 Italy*Assistant Professor, Department Mechanical Aerospace Engineering, C.so Duca degli Abruzzi 24; .Search more papers by this author, BrandlPolytechnic Italy†Research Assistant, author GiliPolytechnic Italy‡Associate authorPublished Online:1 Dec 2020https://doi.org/10.2514/1.G005591SectionsPDFPDF Plus ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionThe recent perspective improve aviation safety, means using synthetic air data on board commercial aviation, opened a new scenario in avionics [1]. In fact, sensing is still based different probes vanes (used as direct sources measure) protruding externally from the aircraft fuselage. On other hand, integrated digital offer opportunity estimation with fusion techniques without physical (or mechanical) sensors. This approach also correlated analytical redundancy [2,3] can be used part redundant flight control system (FCS) architecture, monitor sensors or replace failed [4,5], example.An sensor, therefore, enables replacement sensor consequent benefits terms weight, power consumption, reliability, maintainability emissions. Air split into three main categories: 1) pitot-free speed estimators [6]; 2) vane/sensor-free aerodynamic angle [7,8]; 3) pitot both airspeed [9]. Using addition, place, would beneficial next-generation vehicles, e.g., unmanned aerial vehicles (UAVs) urban mobility (UAM) aircraft, overcome some issues toward certification [10]. (ADS) limited use [11] lead compact solution able related common failure modes incorrect diagnosis modern ADS [12–14].Although recently re-emerged, idea dated back 1949 thanks U.S. Force (USAF) technical report [15]. [15] several methods are analyzed estimate angles, angle-of-attack (AoA) angle-of-sideslip (AoS), one them was implemented discussed 1973 [16], usually referred Freeman’s method. These considered first solutions problem angles Since then, were conceived [17–24]. The state-of-the-art presented here highlights that AoA AoS grouped two model (e.g., Kalman filter) driven neural networks).This Note presents scheme AoS. With respect state-of-the-art, proposed model-free it does not need any test database. Moreover, latter aspect makes independent configuration regime that, instead, highly affects design model-based data-driven sensors.In work classical mechanic equations rearranged order obtain solved estimation. basically nonlinear governing dynamics, airspeed, wind data. To solve preliminary numerical validation, an iterative method has been applied, but possibilities exist filters).In Sec. II notations presented. A rearrangement introduced III, formulation IV. derived V, verification VI before concluding work.II. Notations Reference FramesIn work, vectors indicated bold-italic lower case letters v), v) vector components, whereas matrices capital A). An inertial reference frame FI={XI,YI,ZI} considered, noninertial frames centered center gravity (CG): body [25]. FB={XB,YB,ZB} axes oriented along fixed directions onboard, Fig. 1b. FW={XW,YW,ZW} X axis aligned freestream velocity vector; Z intersection plane normal trajectory (XB,ZB) directed downward (i.e., upper surface). surrounded mass enclosed virtual volume moves together its own FCV={XCV,YCV,ZCV}.Fig. 1 Representation a) frames, b) frames.From 1a relative distance r between expressed r=rB+rW, where rW rB are, respectively, FCV flying object measured origin FI. angular FB FI ω=pi^B+qj^B+rk^Bwhere i^B, j^B, k^B unit FB. Recalling time-derivative properties [26], FI, relationship velocities written r˙=vI=r˙B+w(1)where vI velocity, r˙B surrounding air, w=r˙W volume, speed. Therefore, generic vehicle fly r˙B, true speed, moving animated w, i.e., Finally, vectorial sum w represented 1a.The transformation obtained considering ordered sequence 3–2–1 Euler angles: heading ψ, elevation θ, bank ϕ. Henceforth, ease notation, cosine sine functions will denoted C S, whose arguments subscript. full rotation matrix composed follows: CI2B=[CθCψCθSψ−SθSϕSθCψ−CϕSψSϕSθSψ+CϕCψSϕCθCϕSθCψ+SϕSψCϕSθSψ−SϕCψCϕCθ](2)The FW CW2B =[CαCβ−CαSβ−SαSβCβ0SαCβ−SαSβCα](3)Among all [27], worth underlying CI2BC˙B2I=ΩB(4)where ΩB=[0−rqr0−p−qp0](5)III. Rearrangement Flight Mechanic EquationsRecalling definitions [28], Eq. (1) rewritten vI=CB2IvB+w(6)and vB (3) vB=V∞i^WB(7)where V∞ magnitude vector, V∞=|vB|=u2+v2+w2, i^WB=(CβCα)i^B+(Sβ)j^B+(CβSα)k^B, frame.Recalling Eqs. (6) (4), acceleration aI=v˙I projected aB=CI2BaI=v˙B+ΩBvB+CI2Bw˙(8)Equation (8) ambiguity coming onboard. generated maneuver (v˙B+ΩBvB) and/or change external (CI2Bw˙). Typically, aB onboard proper nB, accelerometers Inertial Measurement Unit (IMU), Attitude Heading System (AHRS), Navigation Systems (INS). case, calculated aB=nB−CI2B[0,0,g0]T, g0≃9.81 gravitational acceleration.From (8), v˙B v˙B=aB−ΩBvB−CI2Bw˙(9)From (7), time derivative velocity’s V˙∞=(vBTv˙B/V∞), substituting expression (9), following equation obtained: V˙∞V∞=vBTv˙B=vBT(aB−ΩBvB−CI2Bw˙)=vBT(aB−CI2B)w˙(10)where vBTΩBvB null, refer same instant.IV. Problem FormulationThe hypothesis (10), hence at certain instant t modeled information past. integral definition, starting τ, t≥τ, vB(t)= vB(τ)+∫τtv˙B(T) dT (11)Henceforth, subscript vB,t (vB)t place vB(t), variable integrand function omitted notation. (11) vB,t=vB,τ+∫τt(aB−ΩBvB−CI2Bw˙) dT(12)and vB,τ=vB,t−∫τtaB dT+∫τtΩBvB dT+∫τtCI2Bw˙ dT(13)Replacing vB,τ (13), (10) τ V∞,τV˙∞,τ=[vB,t−∫τtaB dT]T(aB−CI2Bw˙)τ⇒⇒V∞,τV˙∞,τ+[∫τtaB dT−∫τtCI2Bw˙ dT]T(aB−CI2Bw˙)τ=[vB,t+∫τtΩBvB dT]T(aB−CI2Bw˙)τ(14)where depending vB, collected right-hand side.V. Proposed SchemeThe scheme, named “Angle Attack Sideslip Estimator” (ASSE), making dependencies explicit. term ∫τtΩBvB (14) must explicated vB. Several levels approximations assumed. assumption constant interval [τ,t]. identified zero-order approximation.A. Zero-Order ASSE ApproximationIn window, t, approximated constant; therefore dT=(ΩBvB)tΔt(15)where Δt=t−τ. Substituting recalling properties, V∞,τV˙∞,τ+[∫τtaB dT]T(aB−CI2Bw˙)τ=V∞,ti^WB,tT(I−ΩB,tΔt)(aB−CI2Bw˙)τ(16)Equation (16) basic α(t) β(t) only unknowns supposed measured. measure V˙∞, (described III), rates, 4) field. As far field concerned, assumed known w˙ (16). Even though practicable, demonstrate feasibility scheme. For sake clarity, conclusion always applicable steady discrete change.B. SchemeTo simplify notations, measurable quantities nτ=V∞,τV˙∞,τ+[∫τtaB dT]T(aB−CI2Bw˙)τ(17)and mτ=V∞,t(I−ΩB,tΔt)(aB−CI2Bw˙)τ=hτi^B+lτj^B+mτk^B(18)Therefore, form: nτ=i^WB,tTmτ=hτCβCα+lτSβ+mτCβSα(19)Equation (19) represents scalar variables β(t). reason, subscripts time. Equation expanded n-th τi i∈[0,1,…,n], τ0≡t. n+1 {nt=i^WB,tTmt=htCβCα+ltSβ+mtCβSαnτ1=i^WB,tTmτ1=hτ1CβCα+lτ1Sβ+mτ1CβSα⋮nτn=i^WB,tTmτn=hτnCβCα+lτnSβ+mτnCβSα(20)Equation (20) form equations. expansion past (τi+1<τi) forward equally feasible leading conclusions present work. highlighting no spacing steps here. even very uncommon, nonuniform subsequent nonadjacent steps. useful condition number (20).The nn=(i^WB,tTMnT)T=Mni^WB,t(21)where nn=[nt,nτ1,…,nτn]T Mn=[mtT,mτ1T,…,mτnT]T. components independent, extra given constraint i^WB,tTi^WB,t=1. {1=i^WB,tTi^WB,tnn=Mni^WB,t(22)The most suitable solver adopted (22) estimation.C. Solution Existence ConditionsUnder could linear, nn*=Mn*i^WB,t(23)where nn*=[1,nnT]T Mn*=[i^WB,tT,Mn]T. If Mn* invertible, n=1 have square matrix, i^WB,t=M1*−1n1*(24)In words, if (23) solvable, (τ0≡t τ1). sets minimum considered. nonzero determinant required unique solution. consideration translated conditions. Firstly, (17) (18), clear i-th step uniform conditions introduces null M1*. Hence, observed [29], (based approach) cannot performed trim) Secondly, M1* should rank, or, each shall add guarantee linearly rows. condition, important chance base grids.Therefore, assuming may linearized, general existence least (τ0 τ1) available, conditions, equations.VI. Numerical VerificationIn section, verified simulated presence simulation intended provide exhaustive performance evaluation demonstration Maneuver DefinitionThe validation simulator, inspired two-seat light motorized aircraft. simulator coupled six-degree-of-freedom equipped thrust models designed accordingly results engine datasheet. run explicit Δt=0.1 milli/second.Two maneuvers excite longitudinal lateral-directional beyond their linearity: stall maneuver, described 2; sideslip sweep 3. Both begin trim Aileron commands maintained positions though, due gyroscopic effects, perfectly symmetric. aileron during slightly coupled.Fig. 2 Input–output maneuver: relevant input output data.Fig. 3 data.After short dive, acting sole elevator command, 2a, producing initially increase then smooth deceleration high attack, seen 2b, changes sideslip.The rudder 3a, exciting large range, attack almost 3b.B. ResultsAs V.A, estimated simultaneously (22). measures derivative, acceleration, (implicitly nB IV), 5) 6) acceleration. implement noise delay, signals noise-free synchronized.Even numbers here: current single τ1=t−Δt; times τ1=t−Δt τ2=t−2Δt; four τ1=t−Δt, τ2=t−2Δt, τ3=t−3Δt.In Levenberg–Marquardt algorithm [30]. Because importance guess iteration methods, applied strategy: AoA/AoS values previously initial condition) initialized imposing values.Using estimations reported 4 5 maneuver. good agreement estimations. noted although maximum absolute error smaller than 0.6° maneuvers, there few peaks mainly approximation shown later VI.C. confirm validity maneuvered equations.Fig. Estimation two, three, sideslip.Fig. sweep: sideslip.By extending equations, accuracy increased significantly sometimes higher errors 5a. contrary, 4b, four-equation-scheme improved compared two-equation-scheme. increasing side, VI.C, two-equation two-equation-scheme seems preferable because mathematical complexity three- four-equation schemes supported significant improvement.C. Approximation ErrorTo evaluate (22), exact value (15) approximation. under field, (0-OAE) 0-OAE=−((ΩBvB)tΔt−∫τtΩBvB dT)aB(∫τtΩBvB dT)aB(25)Although range before, 6 shows (shorter 0.25 s) largest increased. straightforward longer 0-OAE bounded within ±6% reduced ±2% highest beginning transition state dynamic conditions.Fig. Zero-order maneuvers.VII. ConclusionsIn estimators, concurrent After simplification set accelerations, attitudes, vector. Each refers instant. when maneuvering, applicable. introduce corresponding increases From demonstrates Note. analysis results, best tradeoff References [1] Flottau J., “Boeing 737 MAX Return Decision January,” Aviation Week & Space Technology, 2019, https://aviationweek.com/air-transport/easas-director-expects-boeing-737-max-return-decision-january. 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TopicsAeronautical EngineeringGuidance, SystemsCivil AviationAviation SafetyAirspeedMilitary AviationAeronauticsAir ForcesAviationAerodynamic PerformanceSlip (Aerodynamics)Commercial AviationFlight TestAvionicsAerodynamics KeywordsAerodynamic AngleAngle SideslipAircraft SpeedBody FrameAttitude SystemFlight MechanicsKalman FilterFlight TestingAileronsFlight Received7 August 2020Accepted25 October 2020Published online1 December