We study the link between observability and non-singularity of the Fisher information matrix (FIM) such as they appear in the problems of tracking: most of the time, the sensors are mounted on a structure having a symmetrical configuration. Therefore, in this case, the parameter of interest is observable up to a symmetry. We prove that the FIM is singular in the subspace of symmetry. Still, there are several examples of observable parameters belonging to the symmetry subspaces. We explain this surprising paradox, and we illustrate it by several examples coming from the target motion analysis (TMA) literature.

Estimating a parameter with biased measurements is the topic of this paper. To do this, we propose three different models when the available data are corrupted by an additive constant bias. For each model, we detail the computation of the CramérRao lower bound (CRLB). For each model, examples using the traditional bearing-only target motion analysis (BOTMA) are given, for which we propose an estimator whose empirical performance is compared to the CRLB.

The problem of biased measurements in bearings-only target motion analysis (BOTMA) is the topic of this article. More precisely, we thoroughly establish the observability condition of the trajectory of the target together with the bias (assumed constant during the whole scenarios).

We address in this paper the problem of observability in target motion analyses (TMA), when the measurements are a sum or a difference of ranges between a target and two motionless observers. Necessary and sufficient conditions of observability are given in time difference of arrival (TDOA) situation and in a bistatic situation.

In this paper, we present a pseudolinear estimate for bearings and frequencies target motion analysis. We propose two ways to reduce its inherent bias and evaluate its performance with Monte Carlo simulations, with reference to the Cramér-Ran lower bound.

Observability in bearings-only target analysis (BOTMA) is studied when the target is in constant-velocity motion and the observer maneuvers gently (a constant turn motion or a constant acceleration motion). During our study, we proved that the rendezvous routes of the observer and the target play a key role in this analysis. We establish necessary and sufficient conditions of observability and we identify the virtual (or ghost) targets giving the same bearings when the system is not observable.

This paper is the companion-paper of another paper presented in FUSION'17 concerning bearings-only target motion analysis (BOTMA). In this one, bearing data are replaced by range data: we study observability in range-only target motion analysis (ROTMA). When the observer is in constant turn motion, the target's trajectory is observable, as in BOTMA. If the observer is in constant acceleration motion, necessary and sufficient conditions of observability in ROTMA are proven to be not the same than in BOTMA, although the equality of the ranks of the respective Fisher information matrices. As in BOTMA, the rendezvous routes of the observer and the target play a crucial role. When the system is not observable, ghost-targets (the ones at the same range than the true target) are identified.

Acoustic propagation delay has not been investigated for a continuous wave multistatic sonar tracking system except for the recent study conducted by Jauffret et al. [4], which estimates the trajectory of a constant velocity target. The results showed that the estimate bias caused by the propagation delay is not negligible, especially for a bistatic system. This paper develops an interacting multiple model unscented Gauss-Helmert filter with numerical Jacobian (IMM-UGHF-NJ) to track a maneuvering target with propagation delay using a bistatic sonar system. The IMM-UGHF-NJ can overcome the two tracking challenges introduced by the delay, namely, implicit state transition model and lack of analytical expression of the Doppler shifted frequency in the measurement model. Simulation tests have been conducted, and the results show that the IMM-UGHF-NJ can reduce the estimation error significantly, especially for fast moving targets.