Abstract
The paper presents methods to determine the time, positions, and distance of closest approach for two vehicles following arbitrary trajectories in two or three dimensions. The distance of closest approach of two vehicles following arbitrary curved trajectories is determined by two conditions: (i) the relative velocity must be orthogonal to the relative position in order for the distance to be a nonzero extremum; (ii) the radial acceleration including centripetal terms must have a direction that increases the separation for the extremum to be a minimum. This theorem on the distance of closest approach simplifies in the case of uniform motion along rectilinear trajectories. Three examples are given: (i) the two-dimensional motion of surface vehicles changing the velocity of one of them so as to enforce a given minimum separation distance; (ii) the three-dimensional motion of two aircraft, one flying horizontally and the other climbing, changing the vertical velocity of the latter to ensure a minimum separation distance set “a priori”; (iii) the case of an aircraft flying with constant velocity in a straight line so that its closest approach to another aircraft flying in a circular holding pattern in the same plane occurs at a given time chosen “a priori”.
Highlights
In the traffic of vehicles safety is identified with the absence of collisions or conflicts
The conflict detection and resolution (CDR) required that the trajectories of two vehicles lead to a relative distance not less than the safe separation distance (SSD) at all times
This can be ensured if the distance of closest approach (DCA) is not less than the SSD
Summary
In the traffic of vehicles safety is identified with the absence of collisions or conflicts. A conflict occurs when the distance between the centroids of two vehicles is less than a safe separation distance (SSD) determined by their size. (i) the absence of conflicts and (ii) the confirmation that a conflict has been resolved depend on determining the distance of closest approach (DCA) that is not less than the SSD. The paper presents methods to determine the time, positions, and distance of closest approach for two vehicles following arbitrary trajectories in two or three dimensions. The three-dimensional cases include all types of flying vehicles, like airplanes, helicopters, drones, rockets and satellites, and submerged submarines. The differences in conflict detection and resolution (CDR) between all these types of vehicles concern the speed, size, and distances that enter as parameters in the same methods of calculation of distance and time of closest approach
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