Passive trajectography using angle measurement

Passive sensors play a vital role in surveillance, measurement and observation systems. Firstly, they have the advantage of being discreet, unlike radar or active sonar, which emit pulses. What's more, they are energy-efficient, which is an advantage for on-board satellite equipment, for example. Last but not least, these sensors cover a very broad spectrum, from low-frequency acoustics (underwater passive listening) through electromagnetics (electronic warfare interceptors) to optics (infrared cameras).

On the other hand, the major disadvantage of passive sensors is that they don't directly measure the distance to the detected sources. They generally only provide angular measurements. So how do you go from angular measurements to distance?

This problem is referred to as passive localization if the source is fixed, and passive trajectography (TP or TMA, "target motion analysis") if the source's kinematics need to be involved to obtain a solution.

If several sensors are available, widely separated geographically and simultaneously delivering angle measurements of the target, then localization can be carried out by triangulation. This estimate is instantaneous.

On the other hand, if you only have one moving sensor facing a moving source, then you need to use a trajectography method. A series of angles measured over a sufficiently long period of time must be exploited. This angular track is not enough: we also need to know the sensor's navigation, i.e. its position over time. Finally, we need to make an assumption about the kinematics of the source, the simplest being uniform rectilinear motion (URM).

If all these elements are present, using one of the passive angle trajectography (PAL) methods, an attempt can be made to estimate the source's trajectory. But there's still one major difficulty: for a unique solution to be obtained, the sensor must manoeuvre, and not in just any way. This problem of uniqueness of solution is known as observability. The trajectory followed by the observer also plays a part in the accuracy of the localization: this is the problem of optimal control.

The basic problem of TPA in a plane is therefore not just a simple estimation problem. . It is in itself a complete field that has been and still is the subject of much ink. Theoretically, TPA belongs both to the field of automatic control (observability, optimal control) and to the field of mathematical statistics, especially estimation theory. In the context of the latter, we introduce an important notion, the Cramèr-Rao bound (CRB), which is the lower limit of uncertainty on the estimation of a quantity, in this case distance. In other words, under certain conditions, no...