The Apollonian paradigm in Cockpit and Ground-based Pilot Display Design

The design of aircraft Detect and Avoid (DAA) systems is critical to enabling Unmanned Aircraft Systems (UAS) greater access to nonsegregated airspace and has the potential to enhance the safety of conventionally piloted aircraft operations. Existing approaches to the “avoid” component of DAA use a sampling approach to determine the range of trajectories (based on a hazard zone) that must be avoided. In at least one existing DAA display, to inform the pilot about the location of the hazard zone the space where DAA Well Clear (DWC) is predicted to be lost is depicted. The abstractness of such a visualization may make it difficult to reconstitute operational reasoning behind the rendering and means the validation of the depicted heading guidance and hazard zone is limited to the cases simulated. Geometric models are part of the suite of fundamental mathematical analyses required to specify cue design and presentation for cockpit displays. In a complementary approach to simulation methods the Apollonian Circles have been shown to have a direct application to the design of DAA systems with the potential to enhance, through this geometric construction, the specification, robustness, and safety of all proximate aircraft operations. The property that, for a constant speed ratio, the locus of all collision points lie on the Apollonius Circle in the 2D conflict plane, has been used to explore an augmentation concept for DAA cockpit displays that are based on conflict probing. The conflict probe underlying the horizontal guidance is derived from all spatial locations where ownship (ownaircraft) is predicted to lose DWC as defined in RTCA DO-365. Color-coded heading bands indicate whether the time until the DWC boundary is crossed has decreased below the Corrective (yellow) or Warning (red) threshold. Should the intruder be maneuvering at the time of a DAA Warning Alert, uncertainty regarding intruder intent may require a pilot to reverse the direction of a horizontal maneuver that was initiated at the moment the Warning Alert was declared. The underlying cause is that the heading guidance bands answer the question “What if ownship were flying in this direction?” assuming the reported intruder track is maintained and that neither intent nor turn-rate is available. To be able to anticipate the result of an intruder maneuver, an obvious follow-on question is “How do the heading guidance bands change as a function of a changing intruder track?”. Whereas with the conflict probe display the possible point of collision (PPC) results from ownship tracks tested against the reported intruder track, the Apollonius Circle allows the PPCs for other intruder directions and the associated ownship track to be analytically determined. In this paper we also extend the Apollonius Circle paradigm to include the Apollonian Circles being a geometric construction comprised of two families (pencils) of coaxal circles each with collinear centers and each with a radical axis. The first family is comprised of Apollonius Circles whose construction is based on a segment. These are Hyperbolic in nature and nested such that there is no intersection between the circles. The second family, the X-Track circles, is Elliptic in nature and each circle in the family intersects the other circles in two specific points being the locations of the aircraft in the proximity pair. A distinguished member of the X-Track circles is the Line of Sight Circle that enables the behaviour of the relative velocity vector and the miss distance vector at closest approach to be readily visualized. The Apollonian Circles provide a rigorous closed-form framework enabling more transparent construction and validation of complex Well Clear Volume surfaces. They lend themselves directly to cue design, the establishment of a more complete and rigorous test regime, and can aid crew and operators to make more informed decisions regarding proximity and conflict resolution. This geometric method is analytic thus results can be calculated in real-time, and provides a more complete and accurate advisory generation capability specific to the imposed encounter condition. Flight crew and ground-based remote pilots are better aided to identify available actions, decide on satisficing actions, nominate an avoidance manoeuver, and physically execute it; all in a safe, defendable, and timely manner.