Bound on the Estimation of a 3-D Trajectory from a Stationary Passive Sensor and its Attainability

It has been shown in previous works that the trajectory of a thrusting/ballistic object in three-dimensional space is observable with two-dimensional measurements from a stationary passive sensor. The measurements can either start from the launch point or start in flight, i.e., with delayed acquisition. The observability of the target trajectory was investigated by testing the invertibility of the Fisher Information Matrix (FIM) numerically. This work discusses the observability of the trajectory via the uniqueness of the target state vector for a certain sequence of 2-d angle-only measurements (azimuth and elevation angles) from a single fixed passive sensor. The discussion starts with polynomial motion from which the results are extended to nonlinear thrusting/ballistic motion. Two cases: (i) known thrust and drag coefficient, (ii) unknown thrust and drag coefficient are considered. The gravity acceleration is shown to be the crucial part that guarantees the observability in all the cases.