Aircraft proximity termination conditions in the planar turn centric modes

Closed-form analytic solutions for proximity management strategies are of great importance as a design benchmark when validating both automated systems and procedures associated with the design of air traffic rules. Merz (1973) first presented a solution for a set of optimal strategies for resolving co-planar co-operative encounters between two aircraft (or ships) with identical linear and rotational speeds. This paper extends the solution domain for turning aircraft beyond that of identical aircraft by presenting a rigorous analysis of the problem through a generalised optimisation approach. This analysis provides a dependable method for determining the location of the point of closest approach. This is achieved by using a vector form of Fermat's equation for stationary points. A characteristic of this solution is the identification of a fixed reference point lying on the vector between the aircraft turn centres or on one of its extensions. This point is then used to determine where the location of the minima in the relative range between the aircraft will occur. Bounds for the domain of the solution are constructed in terms of the rotational angles of the aircraft on their turn circles. Four distinct topologies are required to characterise the types of minima that can occur. The methodology has applications in an operational context permitting a more detailed and precise specification of proximity management functions when developing algorithms for aircraft avionics and air traffic management systems.