Using Ellipsoids to Construct Dynami c Four -Dimensional Trajectories For Airport Arrivals

An operational concept is presented wherein four -dimensional trajectories ( 4DTs ) are negotiated between the air traffic control (ATC) responsible for the airspace surrounding an airport of interest, and aircraft arriving to that airport . The co ncept involves several steps including the arrival of aircraft into the airspace; the specification of an arrival time to an initial approach fix (IAF) by ATC ; the determination of many 4DTs from the aircraft ’s current location to th e IAF within the time a llotted; and, the issuance of a specific 4DT for execution. Subsequently, a n approach is presented for identifying and evaluating an entire family of 4DTs that connect two points in space along with a specified transit time between them . The approach invol ves placing the two points of interest at the foci of an ellipsoid in space whose eccentricity is determined by the transit time and an assumed average transit speed . 4DTs are then identified by three points in space and time: (1) an initial point at the aircraft ’s initial location, (2) a turning point at the ellipsoid surface, and (3) a final point at the initial approach fix . The surface of the ellipsoids represents a family of turning points between the two foci . The ellipsoid is then trimmed to account for such things as exclusion zones (e.g. noise abatement areas; high terrain; tall structures; etc ), aircraft performance (e.g. turn angles; climb and descent angles; winds), and avoidance of other aircraft ’s wake vortices .