A review of flight trajectory optimisations

Abstract The paper reviews the optimisation methods of the flight trajectory for airliners. In contrast to maritime navigation, where the shortest route (the orthodrome) is preferred, in air navigation, the brachistochrone is the optimal flight trajectory on the sphere or on the ellipsoid, considering the wind vector field (maximising the tail wind and minimising the head wind over the duration of the flight). The major impact of the wind on the flight trajectory results from the possible significant velocity at the normal cruise flight levels, which could reach 200 kts, or 40% of the aircraft true airspeed (TAS). Brachistochrone is independent of the flight performance optimisation (range versus speed), as computed by the flight management system. Whichever cost index (CI) is selected (and consequently, the cruise Mach number), the brachistochrone is the minimum time of flight trajectory at that target Mach number. In cruise flight, the minimum time of flight is also equivalent to the minimum fuel consumption. It concerns just the wind velocity field. All these qualify the brachistochrone as the greenest trajectory, the most fuel and emissions efficient solution relative to the atmosphere. The paper classifies the brachistochrone problems (2D, 3D and 4D brachistochrones, with or without flexible time of departure). Some numerical examples are provided. The overall optimal 4D trajectory considers many aspects, including safety, by minimisation of total costs and risks of the 4D trajectory.