Unobservable Problem of Target Tracking with Bearing-only Measurements in 3-dimensional Space

The bearings-only tracking (BOT) system is said to be observability if and only if the target motion parameters can be uniquely determined by noise-free bearing measurements. By utilizing the method of orthogonal vectors and characteristic of linear matrix equation, the problem of observability for BOT in noise-free bearings measurements from single observer is discussed based on the target and observer traveling in the 3-dimensional space. A proposition that BOT for target and observer traveling in the 3-dimensional space with constant acceleration remains unsolvable is presented and proved. By proving the proposition, it is also shown that some motion parameter ratios of target can be estimated under certain condition satisfied by measurements and time samples. The proposition is extended to arbitrary rank of manoeuvre for the observer and the target, which BOT remains unobservable property while the rank of target manoeuvre is higher than that of the observer manoeuvre. The theoretical analysis of this paper provides the guidelines for how the observer trajectory should be formulated to avoid unobservable state for BOT in practice application.