On the Convergence of Iterative Methods for Bearings-Only Tracking

Bearings-only tracking (BOT) is the determination of the trajectory of a target solely from bearing measurements. The BOT area has been widely investigated [1—4] and numerous solutions for this problem have been proposed [5]. However the measurement equation relating the target bearing to its state is highly nonlinear. Classical least square methods as the Kalman filter cannot be directly applied. This advocates for the use of extended Kalman filter for which convergence is improved by using modified polar coordinates [4]. Another approach is the pseudolinear estimator (PLE) formulation proposed in [1] which lumps the nonlinearities into the noise term, resulting in a linear measurement equation. However, the measurement matrix contains elements that are functions of noisy bearings and, overall, are correlated with the noise terms of the measurement equation. As a result, the PLE exhibits a bias which can be severe [1, 3]. To overcome this bias, maximum likelihood and instrumental variables have been developed [1, 3, 6] and give satisfactory results. These are gradient search based on a batch processing of all the available measurements. The latter feature may be quite useful if there are missing data (track interruption). So batch methods are generally considered as reliable. However, their proper convergence are reputed to be sensitive to initial conditions and step sizes. So the objective of this work is to present sufficient conditions ensuring convergence of iterative (i.e., gradient-like, instrumental variable) search methods. A common point of all these algorithms is that they are of batch type. First, we give a general presentation of a basic result of Iltis and Anderson [7]. In fact, a general analysis based on linear and multilinear algebra reveals the fundamental nature of the problem and (almost) completely allows us to avoid boring calculations. It is then possible to consider a unique framework for target-observer scenarios of increasing complexity, common to all the methods. For instance, we consider successively the cases of a maneuvering observer, a maneuvering source (with known maneuvering instants) and, finally, unknown maneuvering instants. Thus, we can obtain sufficient convergence conditions for maneuvering source and observer. Besides their own theoretical interests, these conditions yield feasible methods for the general BOT problems. All these conditions come from a unique result which is strongly related to the quasiconcavity property (see Section III). Thus, even if the tribute to the seminal work of Iltis and Anderson is clear, different interpretations and extensions are developed here. The paper is organized as follows. Section II deals with the general formulation of the BOT problem. The case of nonmaneuvering target and