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Estimability of thrusting trajectories in 3-D from a single passive sensor with unknown launch point

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 is investigated. The location of projectile’s launch point (LP) is unavailable and this could significantly affect the performance of the estimation and the IPP. The LP altitude is then an unknown target parameter. 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, thrust and the LP altitude, which determine the trajectory according to a nonlinear motion equation. The full rank of the FIM ensures that one has an estimable target parameters. The corresponding Cram´er-Rao lower bound (CRLB) quantifies the estimation performance of the estimator that is statistically efficient and can be used for IPP. In view of the inherent nonlinearity of the problem, the maximum likelihood (ML) estimate of the target parameter vector is found by using a mixed (partially grid-based) search approach. For a selected grid in the drag-coefficient-thrust-altitude subspace, the proposed parallelizable approach is shown to have reliable estimation performance and further leads to the final IPP of high accuracy.