Observability Criteria for Bearings-Only Target Motion Analysis
01 Mar 1981-IEEE Transactions on Aerospace and Electronic Systems (IEEE)-Vol. 17, Iss: 2, pp 162-166
TL;DR: In this paper, the observability requirements for bearings-only target motion analysis (TMA) were rigorously established by solving a third-order nonlinear differential equation, and closed form expressions were developed and subsequently used to specify necessary and sufficient conditions on own-ship motion that insure a uniquetracking solution.
Abstract: 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.
Citations
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TL;DR: In this paper, the problem of estimating the position and velocity of an object from noise corrupted bearing measurements obtained by a single moving observation platform is considered and a maximum likelihood estimate (MLE) of the target motion analysis solution is developed and its performance analyzed.
Abstract: 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.
495 citations
TL;DR: In this paper, the authors derived exact state equations for the MP filter without imposing any restrictions on own-ship motion; thus, prediction accuracy inherent in the traditional Cartesian formulation is completely preserved.
Abstract: 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.
477 citations
TL;DR: In this article, a modified gain extended Kalman observer (MGEKO) was developed for a special class of systems and a sufficient condition for the estimation errors of the MGEKF to be exponentially bounded in the mean square was obtained.
Abstract: 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.
287 citations
TL;DR: Two types of tracking filters are presented and the closed form expression of the Cramer–Rao lower bound of the estimate is evaluated in a number of operational cases of practical interest.
197 citations
Cites background from "Observability Criteria for Bearings..."
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TL;DR: In this article, the Fisher information matrix (FIM) is used to determine the course of a constant speed observer that minimizes an accuracy criterion deduced from the FIM.
Abstract: In bearings-only tracking, observer maneuver is critical to ensure observability and to obtain an accurate target localization. Here, optimal control theory is applied to the determination of the course of a constant speed observer that minimizes an accuracy criterion deduced from the Fisher information matrix (FIM). Necessary conditions for optimal maneuver (Euler equations) are established and resolved, partly by analytical means and partly by an iterative numerical procedure. Examples of optimal observer maneuvers are presented and discussed.
189 citations
References
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TL;DR: Taylor-series estimation as mentioned in this paper gives a least-sum-squared-error solution to a set of simultaneous linearized algebraic equations and provides the statistical spread of the solution errors.
Abstract: Taylor-series estimation gives a least-sum-squared-error solution to a set of simultaneous linearized algebraic equations. This method is useful in solving multimeasurement mixed-mode position-location problems typical of many navigational applications. While convergence is not proved, examples show that most problems do converge to the correct solution from reasonable initial guesses. The method also provides the statistical spread of the solution errors.
1,273 citations
TL;DR: In this paper, the extended Kalman filter applied to bearings-only target tracking is theoretically analyzed, and 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.
Abstract: 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.
431 citations
TL;DR: In this article, the problem of performing target motion analysis using noisy bearing measurements derived from multiple observation platforms or from a single moving observer is addressed, and the properties of an estimator based on a Cartesian model of the process are detailed.
Abstract: The problem of performing target motion analysis using noisy bearing measurements derived from multiple observation platforms or from a single moving observer is addressed. For the latter case, the properties of an estimator based on a Cartesian model of the process are detailed. Methods of providing estimates both before an observer maneuver, when the process is unobservable, and following an observer maneuver are developed. The results of an experimental study are presented.
278 citations
TL;DR: Two sufficient conditions of global observability of nonlinear systems are presented: the ratio condition which is the generalization of Fujisawa and Kuh's (1971) ratio condition of circuit theory and the strongly positive semidefinite condition.
Abstract: The purpose of this paper is to investigate the problem of observability of nonlinear systems. Two sufficient conditions of global observability of nonlinear systems are presented: (1) the ratio condition which is the generalization of Fujisawa and Kuh's (1971) ratio condition of circuit theory, (2) the strongly positive semidefinite condition. The relationships between these two conditions as well as the condition of positive definiteness of Fitts (1970) are given.
140 citations
TL;DR: In this paper, it was shown that if a set of cofactors of the Jacobian matrix satisfies a "ratio condition," the network has a unique solution, and the class of matrices under consideration is a generalization of the class P recently introduced by Fiedler and Ptak, and includes the familiar uniformly positive-definite matrix as a special case.
Abstract: This paper deals with nonlinear networks which can be characterized by the equation f(x) = y , where f(\cdot) maps the real Euclidean n -space R^{n} into itself and is assumed to be continuously differentiable x is a point in R^{n} and represents a set of chosen network variables, and y is an arbitrary point in R^{n} and represents the input to the network. The authors derive sufficient conditions for the existence of a unique solution of the equation for all y \in R^{n} in terms of the Jacobian matrix \partial f/ \partial x . It is shown that if a set of cofactors of the Jacobian matrix satisfies a "ratio condition," the network has a unique solution. The class of matrices under consideration is a generalization of the class P recently introduced by Fiedler and Ptak, and it includes the familiar uniformly positive-definite matrix as a special case.
63 citations