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Detection, Estimation, and Modulation Theory

This paper considers the problem of estimating the position and velocity of an object from noise corrupted bearing measurements obtained by a single moving observation platform. The process is inherently nonlinear and exhibits unusual observability properties that are geometry-dependent. A maximum likelihood estimate (MLE) of the target motion analysis solution is developed and its performance analyzed. A comparison is drawn between the MLE and two previously reported methods, a nonlinear modified-instrumental variable estimate (MIV) and the pseudo-linear estimate (PLE). Both the MIV and PLE are shown to derive from approximations to the nonlinear measurement equation and therefore share some common properties with the MLE. The limits on performance that can be expected from processing bearing data are detailed. Specifically, for long range-to-baseline geometries, approximate expressions for the Cramer-Rao bound are derived. Extension of the results to the practical filters approximately predicts numerically observed behavior. For less restrictive geometries, bounds are presented. Incorporation of a target speed constraint on the MLE results in a transition to a lower dimensional problem as noise level and range increases. Monte Carlo experimental results are presented and the improvements realized by the MLE techniques are evident.

Fundamental properties and performance of conventional bearings-only target motion analysis

Previous studies have shown that the Cartesian coordinate extended Kalman filter exhibits unstable behavior characteristics when utilized for bearings-only target motion analysis (TMA). In contrast, formulating the TMA estimation problem in modified polar (MP) coordinates leads to an extended Kalman filter which is both stable and asymptotically unbiased. Exact state equations for the MP filter are derived without imposing any restrictions on own-ship motion; thus, prediction accuracy inherent in the traditional Cartesian formulation is completely preserved. In addition, these equations reveal that MP coordinates are well-suited for bearings-only TMA because they automatically decouple observable and unobservable components of the estimated state vector. Such decoupling is shown to prevent covariance matrix ill-conditioning, which is the primary cause of filter instability. Further investigation also confirms that the MP state estimates are asymptotically unbiased. Realistic simulation data are presented to support these findings and to compare algorithm performance with respect to the Cramer-Rao lower bound (ideal) as well as the Cartesian and pseudolinear filters.

/pdf/utilization-of-modified-polar-coordinates-for-bearings-only-3y9q7a1m2s.pdf

Utilization of modified polar coordinates for bearings-only tracking

A new globally convergent nonlinear observer, called the modified gain extended Kalman observer (MGEKO), is developed for a special class of systems. This observer structure forms the basis of a new stochastic filter mechanization called the modified gain extended Kalman filter (MGEKF). A sufficient condition for the estimation errors of the MGEKF to be exponentially bounded in the mean square is obtained. Finally, the MGEKO and the MGEKF are applied to the three-dimensional bearings-only measurement problem where the extended Kalman filter often shows erratic behavior.

A stochastic analysis of a modified gain extended Kalman filter with applications to estimation with bearings only measurements

This is the third part of a series of papers that provide a comprehensive survey of the techniques for tracking maneuvering targets without addressing the so-called measurement-origin uncertainty. Part I [1] and Part II [2] deal with general target motion models and ballistic target motion models, respectively. This part surveys measurement models, including measurement model-based techniques, used in target tracking. Models in Cartesian, sensor measurement, their mixed, and other coordinates are covered. The stress is on more recent advances - topics that have received more attention recently are discussed in greater details.

A Survey of Maneuvering Target Tracking—Part III: Measurement Models

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.

Observability Criteria for Bearings-Only Target Motion Analysis

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.

Biased Estimation Properties of the Pseudolinear Tracking Filter

The local behavior of a maximum likelihood estimator that adaptively weights data of uncertain origin to make a probabilistic measurement-to-track assignment is examined. The results are placed in the framework of the classic estimation theory of Fisher and Cramer. The Cramer-Rao bound is derived and the MLS error level is compared with this lower bound and the level achieved with known data association. Deterioration in performance was found to depend solely on the false-detection to valid-contact ratio, and the actual (measured) covariance matrix is a scalar multiple of the covariance matrix computed for associated data.

Trajectory Estimation with Uncertain and Nonassociated Data

: 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 ensure a unique tracking solution. It is shown that for certain types of maneuvers the estimation process remains unobservable, even when the associated bearing rate in non-zero. Such maneuvers are frequently overlooked in heuristic discussions of TMA observability, which may account for some common misconceptions regarding the characteristics of accepatable own ship motion

/pdf/necessary-and-sufficient-observability-conditions-for-3odjcwn2bb.pdf

Steven C. Nardone

Papers

Observability Criteria for Bearings-Only Target Motion Analysis

Biased Estimation Properties of the Pseudolinear Tracking Filter

Trajectory Estimation with Uncertain and Nonassociated Data

Necessary and Sufficient Observability Conditions for Bearings-Only Target Motion Analysis