Kalman Filter Behavior in Bearings-Only Tracking Applications

The extended Kalman filter applied to bearings-only target tracking is theoretically analyzed. Closed-form expressions for the state vector and its associated covariance matrix are introduced, and subsequently used to demonstrate how bearing and range estimation errors can interact to cause filter instability (i.e., premature covariance collapse and divergence). Further investigation reveals that conventional initialization techniques often precipitate such anomalous behavior. These results have important practical implications and are not presently being exploited to full advantage. In particular, they suggest that substantial improvements in filter stability can be realized by employing alternative initialization and relinearization procedures. Some candidate methods are proposed and discussed.     

Observability Criteria for Bearings-Only Target Motion Analysis

The observability requirements for bearings-only target motion analysis (TMA) are rigorously established by solving a third-order nonlinear differential equation. Closed form expressions are developed and subsequently used to specify necessary and sufficient conditions on own-ship motion that insure a uniquetracking solution. It is shown that for certain types of maneuvers the estimation process remains unobservable, even when the associated bearing rate is nonzero. Such maneuvers are frequently overlooked in heuristic discussions of TMA observability, which may account for some common misconceptions regarding the characteristics of acceptable own-ship motion.     

Biased Estimation Properties of the Pseudolinear Tracking Filter

Estimation bias in the pseudolinear filter applied to bearings-only target tracking is discussed. Approximate expressions for the pertinent error terms are developed and subsequently used to predict tracking performance under realistic operating conditions. It is shown that once own-ship executes a maneuver, only the estimated range vector remains biased; the corresponding velocity vector becomes asymptotically unbiased. Further investigation reveals that this range bias is highly dependent upon geometry and can be altered by additional own-ship maneuvers. Experimental data are presented to support these findings.     

Observability Requirements for Three-Dimensional Tracking via Angle Measurements

Observability requirements previously established for bearings-only tracking in two dimensions are extended to a class of three-dimensional estimation algorithms capable of processing any pairwise combination of azimuth bearing, conical bearing, and depth/elevation angle measurements. Although these algorithms are intrinsically nonlinear, it is shown that they can be analyzed in a linear framework without sacrificing mathematical rigor. A simplified observability criterion, applicable to both autonomous and nonautonomous linear systems, is presented and utilized to specify conditions on own-ship motion which are both necessary and sufficient for a unique tracking solution. Further analysis reveals that observability dependence on own-ship maneuvers for the three-dimensional algorithms considered here parallels the concomitant two-dimensional requirements. An interesting difference, however, is that under certain conditions, a unique tracking solution can be obtained in three dimensions for unaccelerated own-ship motion.