Estimability of thrusting trajectories in 3D from a single passive sensor

Abstract

The problem of estimating the state of thrusting/ballistic endoatmospheric projectiles moving in 3-dimensional (3-D) space using 2-dimensional (2-D) measurements from a single passive sensor (stationary or moving with constant velocity) is investigated. The estimability is analyzed based on the Fisher Information Matrix (FIM) of the target parameter vector, comprising the initial launch (azimuth and elevation) angles, drag coefficient and thrust, which determine its trajectory according to a nonlinear motion equation. The initial position is assumed to be obtained from the first line of sight (LoS) measurements intersected with a known-altitude plane. The full-rank FIM ensures that this is an estimable system. The corresponding Cram´er-Rao lower bound (CRLB) quantifies the estimation performance of the estimator that is statistically efficient and can be used for impact point prediction (IPP). Due to the inherent nonlinearity of the problem, the maximum likelihood estimate of the target parameter vector is found by using iterated least squares (ILS) numerical approach. A combined grid and ILS approach searches over the launch angles space is proposed. The drag coefficient-thrust grid-based ILS approach is shown to converge to the global maximum and has reliable estimation performance. This is then used for IPP.