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 and Piero GiliAngelo LerroPolytechnic University of Turin, 10129 Turin, Italy*Assistant Professor, Department of Mechanical and Aerospace Engineering, C.so Duca degli Abruzzi 24; .Search for more papers by this author, Alberto BrandlPolytechnic University of Turin, 10129 Turin, Italy†Research Assistant, Department of Mechanical and Aerospace Engineering, C.so Duca degli Abruzzi 24; .Search for more papers by this author and Piero GiliPolytechnic University of Turin, 10129 Turin, Italy‡Associate Professor, Department of Mechanical and Aerospace Engineering, C.so Duca degli Abruzzi 24; .Search for more papers by this authorPublished Online:1 Dec 2020https://doi.org/10.2514/1.G005591SectionsPDFPDF Plus ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionThe recent perspective to improve aviation safety, by means of using synthetic air data on board commercial aviation, opened a new scenario in avionics [1]. In fact, air data sensing is still based on different probes and vanes (used as direct sources of measure) protruding externally from the aircraft fuselage. On the other hand, integrated digital avionics offer the opportunity for air data estimation with data fusion techniques and without using physical (or mechanical) sensors. This approach is also correlated to analytical redundancy [2,3] and can be used as part of a redundant flight control system (FCS) architecture, to monitor physical sensors or to replace failed sensors [4,5], for example.An air data synthetic sensor, therefore, enables the replacement of a physical sensor with consequent benefits in terms of weight, power consumption, reliability, maintainability and emissions. Air data synthetic sensors can be split into three main categories: 1) pitot-free aircraft speed estimators [6]; 2) vane/sensor-free aerodynamic angle estimators [7,8]; 3) pitot and vane/sensor-free for both airspeed and aerodynamic angle estimators [9]. Using synthetic air data estimation in addition, or in place, of physical air data sensors would also be beneficial for next-generation air vehicles, e.g., unmanned aerial vehicles (UAVs) and urban air mobility (UAM) aircraft, to overcome some issues toward certification [10]. In fact, a redundant air data system (ADS) with limited use of synthetic sensors [11] can lead to a compact solution able to overcome some issues related to common failure modes or incorrect failure diagnosis of modern ADS [12–14].Although recently re-emerged, the idea of synthetic aerodynamic angle estimators can be dated back to 1949 thanks to the U.S. Air Force (USAF) technical report [15]. In [15] several methods are analyzed to estimate the aerodynamic angles, angle-of-attack (AoA) and angle-of-sideslip (AoS), and one of them was implemented and discussed in 1973 in [16], usually referred to as the Freeman’s method. These are considered the first solutions to the problem of the estimation of the aerodynamic angles without using physical sensors. Since then, several other synthetic solutions were conceived [17–24]. The state-of-the-art presented here highlights that synthetic sensors for AoA and AoS estimation can be grouped in two main categories: model based (e.g., Kalman filter) and data driven (e.g., neural networks).This Note presents a scheme to estimate AoA and AoS. With respect to the state-of-the-art, the proposed scheme is model-free as it does not need any aircraft model or flight test database. Moreover, the latter aspect makes the proposed scheme independent from the aircraft configuration and the flight regime that, instead, it highly affects the design of model-based or data-driven synthetic sensors.In this work the classical flight mechanic equations are rearranged in order to obtain a scheme that can be solved for AoA and AoS estimation. The proposed scheme is basically a system of nonlinear equations governing the aerodynamic angles based on aircraft dynamics, airspeed, and wind data. To solve the proposed scheme for a preliminary numerical validation, an iterative method has been applied, but several other possibilities exist (e.g., Kalman filters).In Sec. II notations used in this work are presented. A rearrangement of some flight mechanic equations is introduced in Sec. III, and the problem formulation is presented in Sec. IV. The proposed scheme for aerodynamic angle estimation is derived in Sec. V, and a preliminary numerical verification is presented in Sec. VI before concluding the work.II. Notations and Reference FramesIn this work, vectors are indicated with bold-italic lower case letters (e.g., v), and lower case letters (e.g., v) are used for vector components, whereas the matrices are in bold-italic capital letters (e.g., A). An inertial reference frame FI={XI,YI,ZI} is considered, and two noninertial frames are considered centered in the aircraft center of gravity (CG): the body and wind reference frames [25]. The body reference frame FB={XB,YB,ZB} has axes oriented along fixed directions onboard, as in Fig. 1b. The wind reference frame FW={XW,YW,ZW} has the X axis aligned to the freestream velocity vector; the Z axis is the intersection of the plane normal to the trajectory and the (XB,ZB) plane of the aircraft and directed downward (i.e., from the upper to the lower wind surface). The aircraft is considered surrounded by an air mass enclosed in a virtual control volume that moves together with its own reference system FCV={XCV,YCV,ZCV}.Fig. 1 Representation of a) inertial and control volume reference frames, and b) body and wind reference frames.From Fig. 1a the relative distance r between the aircraft and the inertial reference frame can be expressed as r=rB+rW, where rW and rB are, respectively, the distance of the control volume reference system FCV and the flying object both measured from the origin of the inertial one FI. In addition, the angular velocity of the frame FB with respect to the inertial frame FI is ω=pi^B+qj^B+rk^Bwhere i^B, j^B, and k^B are the unit vectors in FB. Recalling the time-derivative properties [26], in the inertial reference frame FI, the relationship between velocities can be written as r˙=vI=r˙B+w(1)where vI is the inertial velocity, r˙B is the relative velocity between the aircraft and the surrounding air, and w=r˙W is the velocity of the control volume, or wind speed. Therefore, the generic flying vehicle is considered to fly with the velocity r˙B, or true air speed, relative to a moving control volume animated by the wind speed w, i.e., control volume velocity of the surrounding air in the inertial frame FI. Finally, the inertial velocity of the aircraft is the vectorial sum of the r˙B and w as represented in Fig. 1a.The vector transformation from the inertial reference frame FI to the body frame FB is obtained considering the ordered sequence 3–2–1 of Euler angles: heading angle ψ, elevation angle θ, and bank angle ϕ. Henceforth, in order to ease the notation, the cosine and sine functions will be denoted as C and S, respectively, whose arguments are indicated as subscript. The full rotation matrix from FI to FB is composed as follows: CI2B=[CθCψCθSψ−SθSϕSθCψ−CϕSψSϕSθSψ+CϕCψSϕCθCϕSθCψ+SϕSψCϕSθSψ−SϕCψCϕCθ](2)The full rotation matrix from FW to FB is composed as follows: CW2B =[CαCβ−CαSβ−SαSβCβ0SαCβ−SαSβCα](3)Among all transformation properties [27], it is worth underlying that CI2BC˙B2I=ΩB(4)where ΩB=[0−rqr0−p−qp0](5)III. Rearrangement of Flight Mechanic EquationsRecalling velocity definitions [28], Eq. (1) can be rewritten as vI=CB2IvB+w(6)and the relative velocity vB can be obtained from the wind reference frame using Eq. (3) as vB=V∞i^WB(7)where V∞ is the magnitude of the relative velocity vector, V∞=|vB|=u2+v2+w2, and i^WB=(CβCα)i^B+(Sβ)j^B+(CβSα)k^B, i.e., the unit vector of the relative velocity in the body reference frame.Recalling Eqs. (6) and (4), the inertial acceleration aI=v˙I projected on the body reference frame can be written as aB=CI2BaI=v˙B+ΩBvB+CI2Bw˙(8)Equation (8) highlights the ambiguity coming from inertial acceleration measured onboard. In fact, it can be generated from aircraft maneuver (v˙B+ΩBvB) and/or change in the external wind (CI2Bw˙). Typically, aB can be derived onboard from the proper acceleration nB, measured by accelerometers from the Inertial Measurement Unit (IMU), Attitude and Heading Reference System (AHRS), or Inertial Navigation Systems (INS). In this case, the inertial acceleration aB is calculated as aB=nB−CI2B[0,0,g0]T, where g0≃9.81 is the gravitational acceleration.From Eq. (8), the acceleration v˙B can be written as v˙B=aB−ΩBvB−CI2Bw˙(9)From Eq. (7), the time derivative of the relative velocity’s magnitude is V˙∞=(vBTv˙B/V∞), and substituting v˙B with its expression of Eq. (9), the following equation is obtained: V˙∞V∞=vBTv˙B=vBT(aB−ΩBvB−CI2Bw˙)=vBT(aB−CI2B)w˙(10)where vBTΩBvB is null, and all terms refer to the same time instant.IV. Problem FormulationThe proposed scheme is based on the hypothesis that the relative velocity vB in Eq. (10), and hence the aerodynamic angles, at a certain time instant t can be modeled using information from the past. Therefore, by means of the integral definition, the relative velocity vector vB at time t can be expressed starting from vB at a generic time τ, with t≥τ, as vB(t)= vB(τ)+∫τtv˙B(T) dT (11)Henceforth, the subscript notations vB,t or (vB)t are used in place of vB(t), and the independent variable of the integrand function is omitted in order to ease the notation. Recalling Eq. (9), Eq. (11) can be rewritten as 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 with Eq. (13), Eq. (10) can be written at time τ as V∞,τV˙∞,τ=[vB,t−∫τtaB dT+∫τtΩBvB dT+∫τtCI2Bw˙ 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 all terms depending on vB, and hence the aerodynamic angles, are collected on the right-hand side.V. Proposed SchemeThe proposed scheme, named “Angle of Attack and Sideslip Estimator” (ASSE), is based on making dependencies from the relative body velocity vB explicit. In this case, the integral term ∫τtΩBvB dT of Eq. (14) must be explicated in terms of vB. Several levels of approximations can be assumed. In this work, the proposed formulation is based on the assumption that the integrand function is constant in the generic time interval [τ,t]. The latter hypothesis is identified as the zero-order approximation.A. Zero-Order ASSE ApproximationIn a generic time window, from τ to t, the integrand function of ∫τtΩBvB dT in Eq. (14) can be approximated constant; therefore ∫τtΩBvB dT=(ΩBvB)tΔt(15)where Δt=t−τ. Substituting the latter expression into Eq. (14) and recalling matrix properties, Eq. (14) can be rewritten as V∞,τV˙∞,τ+[∫τtaB dT−∫τtCI2Bw˙ dT]T(aB−CI2Bw˙)τ=V∞,ti^WB,tT(I−ΩB,tΔt)(aB−CI2Bw˙)τ(16)Equation (16) is the basic expression of the zero-order scheme referred to the generic time τ where the aerodynamic angles α(t) and β(t) are the only unknowns and all other terms are supposed to be measured. Therefore, the aerodynamic angle estimation proposed here is based on direct measure of 1) true airspeed V∞ and its time derivative V˙∞, 2) the inertial body acceleration aB (described in Sec. III), 3) angular rates, and 4) the wind field. As far as the wind field is concerned, the wind velocity is assumed to be known in order to be able to measure the wind acceleration term w˙ in Eq. (16). Even though this assumption is not practicable, it is used here to demonstrate the feasibility of the proposed ASSE scheme. For the sake of clarity, conclusion of this work can always be applicable in the case of null, steady wind field or discrete wind change.B. Zero-Order ASSE SchemeTo simplify the proposed scheme notations, the measurable quantities of Eq. (16) are grouped and denoted as follows: nτ=V∞,τV˙∞,τ+[∫τtaB dT−∫τtCI2Bw˙ dT]T(aB−CI2Bw˙)τ(17)and mτ=V∞,t(I−ΩB,tΔt)(aB−CI2Bw˙)τ=hτi^B+lτj^B+mτk^B(18)Therefore, Eq. (16) can be rewritten in a more compact form: nτ=i^WB,tTmτ=hτCβCα+lτSβ+mτCβSα(19)Equation (19) represents a generic nonlinear scalar equation in two variables α(t) and β(t). For the latter reason, the aerodynamic angles are represented without subscripts related to time. Equation (19) can be expanded back in time starting from t to n-th generic τi with i∈[0,1,…,n], where τ0≡t. Therefore, the following system of n+1 nonlinear equations is obtained: {nt=i^WB,tTmt=htCβCα+ltSβ+mtCβSαnτ1=i^WB,tTmτ1=hτ1CβCα+lτ1Sβ+mτ1CβSα⋮nτn=i^WB,tTmτn=hτnCβCα+lτnSβ+mτnCβSα(20)Equation (20) is the generic form of the proposed zero-order ASSE scheme based on n+1 equations. In this work, an expansion in the past is considered (τi+1