Observability conditions for target states with bearing-only measurements in three-dimensional case

In target tracking with a passive sensor such as infrared seekers, angle-only information is determined. In this case no information about the range of the target is provided and unobservable system is resulted. This paper studies the observability of discrete time three-dimensional bearing-only target tracking. The target is assumed to be moving in a straight line while a single moving observer (own-ship) measures its relative (elevation and azimuth) bearing angles. By transforming the inherently nonlinear bearing measurements into a pseudo-linear form, a linear least squares (LS) estimator is formulated. Observability is then analyzed by studying the solvability of the associated LS problem. This approach has the advantage of providing simple unobservability conditions as opposed to other approaches in the continuous time setup. In this paper, most of motivation and problem formulation is taken from (P. Shar and X.R. Li, 1999). It is shown that for no maneuvering target, at least three independent measured bearings for system observability is required