Symmetric solutions for two-body dynamics in a collision prevention model

Abstract

This paper presents the first analytical solutions for the three-dimensional motion of two idealized mobiles controlled by a particular guidance law designed to avoid a collision with minimal path deviation. The mobiles can be regarded as particles, and guidance can be interpreted as complex forces of interaction between the particles. The motion is then a generalized form of two-body Newtonian dynamics. If the mobiles have equal speeds, the relative motion is determined through various transformations of the differential equations. Solvability relies on congruence and symmetries of the paths, which is exploited to reduce the original twelve first-order differential equations to three first- order equations for the relative motion. The resulting state space is partitioned into five invariant subsets, with various symmetries and stabilities. One of these sets describes planar motion, where simple explicit solutions are given. In nonplanar motion, the solution is formally reduced to quadrature. A numerical calculation gives the separation at the closest point of approach, which provides control over minimum separation. The results should be of interest because of their application, which includes, most importantly, the prevention of midair collisions between aircraft, but also potential application to land, water and space vehicles. The solutions should be of interest to mathematical specialists in dynamical systems, because of some novel constants of the motion, novel symmetries, and the associated reducibility of the equations. doi:10.1017/S1446181111000691