Bearings-Only Tracking: Observer Maneuver Recommendation

Abstract: Doppler-Bearing Tracking (DBT) is commonly used in target tracking applications for the underwater environment using the Hull-Mounted Sensor (HMS). It is an important and challenging problem in an underwater environment.,The system nonlinearity in an underwater environment increases due to several reasons such as the type of measurements taken, the speeds of target and observer, environmental conditions, number of sensors considered for measurements and so on. Degrees of nonlinearity (DoNL) for these problems are analyzed using a proposed measure of nonlinearity (MoNL) for state estimation.,In this research, the authors analyzed MoNL for state estimation and computed the conditional MoNL (normalized) using different filtering algorithms where measurements are obtained from a single sensor array (i.e. HMS). MoNL is implemented to find out the system nonlinearity for different filtering algorithms and identified how much nonlinear the system is, that is, to measure nonlinearity of a problem.,Algorithms are evaluated for various scenarios with different angles on the target bow (ATB) in Monte-Carlo simulation. Computation of root mean squared (RMS) errors in position and velocity is carried out to assess the state estimation accuracy using MATLAB.

Abstract: In passive target tracking, target motion parameters (TMP) i.e. range, course, and speed are estimated using bearing measurements. In the simulation mode, the accuracy of the estimated solution can...

Abstract: Using the recently proposed measure of nonlinearity (MoN), the authors try to find the magnitude of nonlinearity for passive target tracking with bearings-only measurements in underwater environment. The method derived to measure the nonlinearity is completely based on the state covariance matrices of the filters. It is tried to find the allowable magnitude of nonlinearity in terms of MoN with which a filter can perform to estimate the target motion parameters with required accuracy. In this paper, MoN values for different filters are computed for different scenarios. Results obtained in the Monte Carlo simulation are presented.

Abstract: Two sensor arrays, hull-mounted array, and towed array sensors are considered for bearings-only tracking. An algorithm is designed to combine the information obtained as bearing (angle) measurements from both sensor arrays to give a better solution. Using data from two different sensor arrays reduces the problem of observability and the observer need not follow the S-maneuver to attain observability of the process. The performance of the fusion algorithm is comparable to that of theoretical Cramer–Rao lower bound and with that of the algorithm when bearing measurements from a single sensor array are considered. Different filters are used for analyzing both algorithms. Monte Carlo runs need to be done to evaluate the performance of algorithms more accurately. Also, the performance of the fusion algorithm is evaluated in terms of solution convergence time.

Abstract: The Kalman Filter (KF) is one of the most widely used methods for tracking and estimation due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the Extended Kalman Filter (EKF) which simply linearizes all nonlinear models so that the traditional linear Kalman filter can be applied. Although the EKF (in its many forms) is a widely used filtering strategy, over thirty years of experience with it has led to a general consensus within the tracking and control community that it is difficult to implement, difficult to tune, and only reliable for systems which are almost linear on the time scale of the update intervals. In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely sampled points can be used to parameterize mean and covariance, the estimator yields performance equivalent to the KF for linear systems yet generalizes elegantly to nonlinear systems without the linearization steps required by the EKF. We show analytically that the expected performance of the new approach is superior to that of the EKF and, in fact, is directly comparable to that of the second order Gauss filter. The method is not restricted to assuming that the distributions of noise sources are Gaussian. We argue that the ease of implementation and more accurate estimation features of the new filter recommend its use over the EKF in virtually all applications.

Abstract: This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples.

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.

Abstract: The problem of bearings-only target localization is to estimate the location of a fixed target from a sequence of noisy bearing measurements. Although, in theory, this process is observable even without an observer maneuver, estimation performance (i.e., accuracy, stability and convergence rate) can be greatly enhanced by properly exploiting observer motion to increase observability. This work addresses the optimization of observer trajectories for bearings-only fixed-target localization. The approach presented herein is based on maximizing the determinant of the Fisher information matrix (FIM), subject to state constraints imposed on the observer trajectory (e.g., by the target defense system). Direct optimal control numerical schemes, including the recently introduced differential inclusion (DI) method, are used to solve the resulting optimal control problem. Computer simulations, utilizing the familiar Stansfield and maximum likelihood (ML) estimators, demonstrate the enhancement to target position estimability using the optimal observer trajectories.

Abstract: The nature of observability of nonlinear systems encountered in passive target motion analysis (TMA) is carefully examined. The approach proposed here is based upon a well-chosen criterion which allows us to answer the major observability questions.