Observability of a Thrusting/Ballistic Trajectory in 3-D From a Single Fixed Passive Sensor

Abstract

In previous works, it has been shown that the motion of a thrusting/ballistic object in a three-dimensional (3-D) space is observable with 2-D measurements from a stationary passive sensor. The measurements can either start from the launch point or can be delayed. The observability was investigated by testing the invertibility of the Fisher Information Matrix (FIM). This paper discusses the observability via the uniqueness of the target state vector for a certain sequence of 2-d angle-only measurements from a single fixed passive sensor. The discussion starts with polynomial motion from which the results are extended to nonlinear thrusting/ballistic motion. To illustrate the observability of a thrusting/ballistic target, the estimation problem of such a target with delayed acquisition after burn-out time is solved with a 7-d parameter vector (velocity vector azimuth angle and elevation angle, drag coefficient, 3-d acquisition position, and target speed at the acquisition time). A maximum likelihood (ML) estimator is used for the motion parameter estimation at acquisition time. The impact point prediction is then carried out with the ML estimates. The FIM is investigated via simulations to prove the observability numerically and the maximum likelihood estimator is shown to be efficient.