Aircraft Proximity: Towards Systematic Test Strategies via the Apollonian Proximity Circle paradigm

Conceptual mathematical models play a crucial role in the development of engineering specifications required in the detailed design process and in the test and evaluation of avionics equipment. Specifying aircraft proximity is a precursor to the development of aircraft Detect and Avoid (DAA) systems that are, in turn, critical to enabling Unmanned Aircraft Systems (UAS) greater access to non-segregated airspace. Such specifications can also significantly enhance the safety of conventionally piloted aircraft operations. A more rigorous mathematical approach to specifying aircraft proximity functions is now possible based on the Apollonian Proximity Circle (APC) paradigm. In turn, this paradigm is based on the Cylindrical Bipolar Co-ordinate system that can be used to formally establish a framework of situational Quadrant Equivalence Sets and Test Regimes that exhibit differing proximity and geometrical behaviour. Some Test Regimes only require a sparse allocation of test points while others, because of functional characteristics such as asymptotic behaviour and contour spacing, require a far more intense allocation of test points. Having addressed the static specification of proximity the dynamic aspects related to short-term intent and Conflict Probing also need to be addressed and this leads to the identification of various anomalous behaviours. In this paper the examples identified are: False Positives; Turn Policy Reversals; the role of Decision Delays; and the nature of operational guidance within Well Clear Volumes. Selected examples show how this behaviour can be better characterized by reference to the APC paradigm and, in particular, with reference to the role of the Line of Sight vector between the aircraft in proximity. The APC paradigm presented in this paper is analytic thus enabling a far more complete and accurate specification of proximity than has previously been reported in the literature. The systematic approach provides a foundational basis for resolving the differing and contrary anecdotal viewpoints presently arising in the international literature.