Aircraft Trajectory Optimization. Aircraft Trajectory Optimization. Motivation. Motivation

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1 Universidad Rey Juan Carlos Departamento de Estadística e Investigación Operativa October 13, 29 Outline Motivation Aircraft equations of motion a Optimal control Acknowledgments References a See references [1], [2], [3], [4], [5], [6] Motivation Motivation Air transport key data a a SESAR ATM Master Plan: million of commercial flights in 25 only in Europe. 1.5% of the European Gross Domestic Product. Responsible for the 2% of CO 2 global emissions.

2 Motivation Motivation ATM: current situation Air Traffic Management (ATM), as defined by the International Civil Aviation Organization (ICAO), is the dynamic, integrated management of air traffic and airspace safely, economically and efficiently through the provision of facilities and seamless services in collaboration with all parties Air Traffic Control Air Traffic Flow Management Air Space Management ATM: new concept SESAR a NextGen b a b SESAR main goals The future ATM system must be able to cope with the expected market growth and meet the societal requirements. 3-fold increase in air traffic movements whilst reducing delays. Improve the safety performance by a factor of 1. 1% reduction in the effects aircraft have on the environment. Provide ATM services at a cost to airspace users which at least 5% less. Motivation Aircraft Equations of Motion Research scope within this framework To plan more efficient flight profiles seeking: flight efficiency cost-effectiveness sustainability Aircraft definition An aircraft is a vehicle which is able to fly by being supported by the air, or in general, the atmosphere of a planet. Fixed-wing aircraft definition A fixed-wing aircraft, usually called an airplane or aeroplane, is a heavier-than-air aircraft capable of flight whose lift is generated not by wing motion relative to the aircraft, but by forward motion through the air.

3 Aircraft Equations of Motion Aircraft Equations of Motion Figure: forces acting on an aircraft Figure: Aircraft controls and moments Aircraft Equations of Motion Aircraft Equations of Motion The problem is approached under the following hypothesis: 1 Flat earth model. (The earth is flat, nonrotating, and approximate inertial reference frame). 2 Atmosphere is at rest relative to the earth and atmospheric properties are functions of altitude only. 3 The ariplane is a conventional jet airplane modeled as a variable-mass particle. 4 For trajectory analisys, the translational equations (F = ma) are uncoupled from the rotational equations (M = Iα) by assuming that the airplane rotational rates are small and that control surface deflections do not affect forces. 5 The forces acting on an airplane are thurst, aerodinamic forces and gravity. 6 We assume a parabolic drag polar, i.e, C D = C D + KC 2 L. The main problem is defined by the following differential-algebraic equation (DAE) system: m V = T cos ǫcos ν D mg sinγ mv ( χcos γ cos µ γ) = T cos ǫsinν Q + mg sinµcos γ mv ( χcos γ sinµ + γ) = L + T sinǫ mg cos µ cos γ X e = V cos γ cos χ Y e = V cos γ sinχ Z e = V sinγ ṁ = CT g

4 Aircraft Equations of Motion Aircraft Equations of Motion Symmetric Flight in a Vertical Plane The problem is approached under the following hypothesis: 1 Flight in a vertical plane ( Y e = ). It implies sinχ = (χ = ). 2 Symmetric flight, which means there is no thrust and speed sideslip (β = ; ν = ). Due to this symmetry there are no forces along Y axis, Q =. Figure: Euler angles Aircraft Equations of Motion Aircraft Equations of Motion Symmetric Flight in a Vertical Plane The main problem is defined by the following differential-algebraic equation (DAE) system: m V = T cos ǫ D mg sinγ mv γ = L + T sinǫ mg cos γ X e = V cos γ Z e = V sinγ = ḣ ṁ = CT g Figure: Coordinate systems for flight in a vertical plane

5 Aircraft Equations of Motion Optimal Control Problem We consider a simple optimal control problem with path constraints in which one has to determine the control inputs u(t) that minimize J = φ[x(t f ),t f ] subject to the differential constraint ẋ = f [x(t), u(t), t] the path constraints φ L φ[x(t),u(t),t] φ U and the boundary conditions ψ[x(t f ),u(t f ),t f ] = Figure: Forces acting on an airplane in flight where the initial conditions x(t ) = x at the fixed initial time t are assigned and the final time t f is free. Problem We studied the distance maximization problem for a commercial airplane with a direct method approach using nonlinear programming. We considered a symmetric flight in a vertical plane, considering the descent phase, modeling thus the aircraft as a glider. Symmetric Flight in a Vertical Plane The main problem is defined by the following differential-algebraic equation (DAE) system: m V = T cos ǫ D mg sinγ mv γ = L + T sinǫ mg cos γ X e = V cos γ Z e = V sinγ = ḣ ṁ = γ

6 Glider 1 There is no thrust, T =, so the mass stays constant, ṁ =. Additional simplification 1 There is no variation of the temperature with the altitude, so the atmospheric variables are constant, ρ = constant and p = constant. It is asumed as well that g = constant. The resulting DAE system is: m V = D mg sinγ mv γ = L mg cos γ X e = V cos γ ḣ = V sinγ dimensionless variables The dimensionless velocity, ˆV : ˆV = V The lift coefficient, C L : C L = The drag coefficient, C D : C D = V B L 1 2 ρv 2 S D The load factor, n: n = L mg = ˆV 2 C L C Lopt The dimensionless drag, DEmax mg = C 1 2 ρv 2 S D + KCL 2 = 1 2 (ˆV 2 + n2 ˆV 2) dimensionless DAE system E max V B g ˆV = ˆV 2 2 (1 + C2 L C 2 Lopt V B g ˆV γ = ˆV 2 C L C Lopt cos γ X e V B = ˆV cos γ ḣ V B = ˆV sinγ ) E max sinγ

7 Model data The airplane is a Subsonic Business Jet powered by two GE CJ61-6 turbojets. The ariplane resembles an early model of the Lear Jet. The characteristics of this model used in the calculations are the following: m = 499 Kg S = 21.5 m 2 C D =.23 K =.73 Atmospheric data It has been assumed that the values are the ISO standard at sea level: ρ = Kg/m 3 g = 9.81 m/s 2 Path constraints h 11 x V stall V V divergencia CL CL max Boundary values t i = s h i = 11 m h f = m x i = m V i = 18 m/s γ i = rad Figure: Optimal path

8 12 h [m] 1 The flight time spent in completing the trajectory is t final = [s] The maximum flight distance is x = 27.8 [km] Figure: Altitude x [km] Figure: Distance V [m/s] Figure: True Airspeed

9 γ [deg] C L Figure: flight path angle Figure: CL Problem We studied the Minimum-Fuel trajectory problem for a commercial airplane with a direct method approach using nonlinear programming. We considered a symmetric flight in a vertical plane, considering the complete flight of a commercial flight modelling the aircraft as a propelled Jet. A standard atmosphere is defined and the BADA database model is considered. Jet in a Symmetric Flight in a Vertical Plane The problem is approached under the following hypothesis: 1 A spherical earth model is considered. 2 There is thrust, T, so the mass is variable, m cte. Consumption and trhust are considered functions only of altitude. 3 The Direction of Thrust is considered to be along the x wind axis, (ǫ = ). 4 The wind is not considered. 5 Flight on a vertical plane ( Y e = ). It implies sinχ = (χ = ). 6 Symmetric flight, which means there is no thrust and speed sideslip (β = ; ν = ). Due to this symmetry there are no forces along Y wind axis, Q =.

10 Jet equations The main problem is defined by the following differential-algebraic equation (DAE) system: m V = T D mg sinγ mv γ = L mg cos γ + m V 2 θ = (R e + h) cos γ X e (R e + h) = V (R e + h) cos γ ḣ = V sinγ dimensionless variables The dimensionless velocity, ˆV : ˆV = V V B ; The lift coeficient, C L : C L = L ; 1 2 ρv 2 S The drag coefficient, C D : C D = = C 1 2 ρv 2 S D + KCL 2; The load factor, n: n = L mg = ˆV 2 C L C Lopt ; The dimensionless drag, DEmax mg D = 1 2 (ˆV 2 + n2 ˆV 2); The dimensionless thrust, ˆT: ˆT = T T B, where T B = mg E max. ṁ = CT g dimensionless DAE system E max V B g ˆV = ˆT ˆV 2 2 (1 + C2 L C 2 Lopt ) E max sinγ V B g ˆV γ = ˆV 2 C L C Lopt cos γ + (V B ˆV) 2 (R e + h)g cos γ θ V B = X e (R e + h)v B = ḣ V B = ˆV sinγ ṁ = C ˆTm E max V (R e + h) cos γ Aircfraft data The performances of a typical commercial aircraft will be analyzed: Airbus A with two CFM56 5 A3 Jet engines. The characteristics of this aircraft are the ones specified in BADA database.

11 Standard Atmospheric Model h tropopause = T ISA 6.5 T = T ISA + T ISA { T (1 + αh) if h tropopause, T = if h tropopause. P = P ISA (1 + αh T ) g Rα T ISA ρ = ρ ISA T ρ = ρ (1 + αh T ) g Rα 1 g = µ r[i] 2 Path constraints h min[h M,h M2 ] x C Vmin V stall V V Mo Mach M M m min m m max C L C Lmax T T maxclimb ṁ min ṁ Boundary values h i = 1 m h f = 1 m x f = 2. Km V i = 18 m/s γ i =.6 rad m i = MTOW Kg Atmospheric values T ISA = T ISA = K P ISA = Pa ρ ISA = Kg m 3 Figure: Optimal path

12 h [m] Objective Function Value A 32: Consumption = Kg Flight Time Values A 32: t final = s Figure: Altitude x [km] 2 θ [deg] Figure: Distance Figure: Longitude

13 V [m/s] 3 V CAS [m/s] Figure: True Airspeed Figure: Calibrated Airspeed Mach 1. γ [deg] Figure: Mach number Figure: Flight path angle

14 1.4 C L 14 T [N] Figure: C L Figure: Thrust m [kg] Consumption [kg] Figure: Aircraft mass Figure: Accumulate consumption

15 simlation Acknowledgments This research is being carried out together with the members of the optimal control group: Alberto Olivares; Ana G. Bouso; Ernesto Staffetti; ; Miguel Angel Maldonado; in the Department of Statistics and Operations Research of the Universidad Rey Juan Carlos 1. The project is being supported by the funds of ATLANTIDA project References JJ Martõnez-Garcõa and MA Gomez-Tierno. Curso de Mecanica del Vuelo. Publicaciones de la Escuela Tecnica Superior de Ingenieros Aeronauticos, Madrid, D.G. Hull. Fundamentals of airplane flight mechanics. Springer, 27. B. Etkin and L.D. Reid. Dynamics of flight. John Wiley & Sons, A Miele. Flight Mechanics, vol. 1. Theory of Flight Paths. Reading, Massachusetts: Addison-Wesly Publishing Company, Inc, T.R. Yechout, S.L. Morris, D.E. Bossert, and W.F. Hallgren. Introduction to aircraft flight mechanics. Aiaa, 23.

16 GE Townsend, AS Abbott, and RR Palmer. Guidance, flight mechanics and trajectory optimization. NASA CR-17, 1968.