Estimation of thrusting trajectories in 3D from a single fixed passive sensor

The problem of estimating the state of thrusting/ballistic endo-atmospheric projectiles moving in three-dimensional space for the purpose of impact point prediction (IPP) using two-dimensional measurements from a single passive sensor (stationary or moving with constant velocity) is investigated. The location of a projectile's launch point (LP) is generally unavailable, and this could significantly affect the performance of the estimation and the IPP. However, if the altitude of the LP is known, the launch position can be obtained with negligible error from the first line of sight measurement intersected with the terrain map. The estimability is analyzed based on the Fisher Information Matrix (FIM) of the target parameter vector that determines its trajectory: the initial launch (azimuth and elevation) angles, drag coefficient, and thrust. Lack of knowledge about the LP altitude makes the problem substantially more difficult, since this altitude is then an additional unknown target parameter and must be included into the target parameter vector that needs estimability analysis. The full rank of the FIM, with/without the LP altitude, ensures that one has estimable target parameters. The corresponding Craḿer-Rao lower bound quantifies the estimation performance of the estimator that is statistically efficient and can be used for the IPP accuracy evaluation. In view of the inherent nonlinearity of the problem, the maximum likelihood estimate of the target parameter vector can be found by using a suitable numerical approach. A search strategy with two stages-a mixed (partially grid-based) search followed by a continuous search-is proposed. For even a coarse grid, this approach is shown to have reliable estimation performance and leads to an IPP of good accuracy. Due to its parallelizable nature, the mixed search allows the two-stage strategy to be implementable in real time.