Differential Geometric Guidance Based onthe Involute of the Target's Trajectory

Abstract

This paper presents a novel approach to missile guidance using the differential geometry of curves and not relying on the line of sight information. The target’s trajectory is treated as a smooth curve of known curvature and the new algorithm is based on the involute of the target’s curve. The missile’s trajectory uses the concept of virtual target to generate the correct involute trace. It is shown that the missile is either on the trace immediately or may be able to reach it through an alignment procedure. In general, following the trace requires a three-dimensional maneuver in which the missile flies above the target’s tangent plane. The projection of the three-dimensional trajectory onto the tangent plane coincides with the involute trace, but is traversed in the time-to-go, thus resulting in the intercept. Two air-to-air scenarios of point masses are considered for a maneuvering target of the F-16 fighter class: 1) a two-dimensional engagement with target executing a constant g turn; 2) a three-dimensional engagement with target executing a barrel-roll maneuver. Perfect target information is assumed in simulations. In the first case, intercepts occur both for the involute law and proportional-navigation (PN) guidance; PN based intercepts occur quicker, but the involute-based trajectories are more difficult to evade and always result in a side impact. In the second case, PN fails to intercept the target, while the involute law is successful.