Bearings-only Tracking Problem Research Articles

Passive underwater tracking with unknown measurement noise statistics using variational Bayesian approximation

This paper considers a bearings-only tracking problem with unknown measurement noise statistics. It is assumed that the measurement noise follows a Gaussian probability density function where the mean and the covariance of the noise are unknown. Here, an adaptive nonlinear filtering technique is proposed, where the joint distribution of the measurement noise mean and its covariance are considered to follow a normal inverse Wishart (NIW) distribution. Using the variational Bayesian (VB) approximation, joint distribution of the target state, the measurement noise mean and covariance is factorized as the product of their individual probability density function (pdf). Minimizing the Kullback-Leibler divergence (KLD) between the factorized and true joint pdfs, probability distributions of the noise mean, covariance and the target states are evaluated. The estimation of states with the proposed VB based method is compared with the maximum a posteriori (MAP) and the maximum likelihood estimation (MLE) based adaptive filtering. Deterministic sigma points are used to realize the filtering algorithms. The proposed adaptive filter with VB approximation is found to be more accurate compared to their corresponding MAP-MLE based counterparts.

A Monte Carlo-Based Iterative Extended Kalman Filter for Bearings-Only Tracking of Sea Targets.

In this paper, a Monte Carlo (MC)-based extended Kalman filter is proposed for a two-dimensional bearings-only tracking problem (BOT). This problem addresses the processing of noise-corrupted bearing measurements from a sea acoustic source and estimates state vectors including position and velocity. Due to the nonlinearity and complex observability properties in the BOT problem, a wide area of research has been focused on improving its state estimation accuracy. The objective of this research is to present an accurate approach to estimate the relative position and velocity of the source with respect to the maneuvering observer. This approach is implemented using the iterated extended Kalman filter (IEKF) in an MC-based iterative structure (MC-IEKF). Re-linearizing dynamic and measurement equations using the IEKF along with the MC campaign applied to the initial conditions result in significantly improved accuracy in the estimation process. Furthermore, an observability analysis is conducted to show the effectiveness of the designed maneuver of the observer. A comparison with the widely used UKF algorithm is carried out to demonstrate the performance of the proposed method.

Multi-Sensor Data Fusion for Target Tracking Using Machine Learning Techniques

Target detection using multi fusion data is one of the common techniques used in military as well as defence units. The usage of a wide variety of sensors is now possible due to modern data fusion technology. The major problem is the existing multi-sensor fusion technique is loss of data and delay is message transfer. To overcome the existing problems, proposed work includes optimization, machine learning, and soft computing techniques. Multi Sensor Data Fusion (MSDF) is becoming an increasingly significant field of study and is being explored by a broad range of individuals. Data defects, outliers, misleading data, conflicting data, and data association are some data fusion concerns. In addition to the statistical advantages of more independent observations, the precision of an observation may be improved by using a variety of different types of sensors. Target tracking has earned a lot of attention in recent years in the realm of surveillance and measurement systems, particularly those in which the state of a target is approximated based on measurements. Academics as well as implementers in the fields of radar, sonar, and satellite surveillance are interested in the bearings-only tracking (BOT) problem. The BOT is the sole option available in many surveillance systems, such as those found aboard submarines. Significant difficulties arise because of the constrained observability of target states based only on bearing measurements. The work that is suggested tackles the limitations of EKF and its derivatives in controlling MSDF within the context of BOT. Specifically, the study identifies divergence as a primary challenge and works to devise solutions for it. It is recommended that two key methods of fusion, data level and feature level (or state level), be investigated in depth. This is in recognition of the fact that the MSDF may increase observability, thereby reducing the tendency of the tracking algorithm to diverge and realizing a better estimate of the states. The Information Filter, which is a casting of the Kalman Filter, and its expansions are employed via extensive simulation to lessen the influence of initial assumptions on the convergence of MSDF tracking algorithms. This is accomplished by using the Kalman Filter.

Gaussian Sum Shifted Rayleigh Filter for Underwater Bearings-Only Target Tracking Problems

In this paper, a new Gaussian sum (GS) filter with shifted Rayleigh filter (SRF) as proposal has been developed for the bearings-only tracking (BOT) problem. The proposed filter estimates the prior and posterior probability density functions as weighted sum of several Gaussian densities, where the individual Gaussian densities are generated from the SRF. Performance of the proposed filter on the BOT problem is compared with the SRF, the GS sparse-grid Gauss-Hermite filter, and other available GS and quadrature-based filters. Performance of filters is compared in terms of a root-mean-square error (RMSE), track divergence, and computational time. The effect of initial uncertainty, measurement noise covariance, and sampling time on filtering accuracy are also studied. Finally, RMSEs of all the filters are evaluated in comparison with the Cramer-Rao lower bound. From simulation results, it is observed that the performance of the proposed GS-SRF is superior to all other nonlinear filters considered.

Bearings-Only Tracking Using Augmented Ensemble Kalman Filter

Tracking targets with bearings-only measurement is a great challenge caused by poor observability and highly nonlinear estimation. In this brief, a novel augmented ensemble Kalman filter (AEnKF) is presented to address this bearings-only tracking problem. Different from the conventional ensemble Kalman filter (EnKF), the AEnKF overcomes the limitation of the linear measurement update rule in the linear minimum mean-square error (LMMSE) framework. The AEnKF utilizes a nonlinear transform of the measurement, called uncorrelated conversion (UC), to augment the measurement space. This conversion serves as a pseudomeasurement and is uncorrelated with the original measurement statistically. Unlike other UC filters based on the Gaussian assumption in the existing literature, the AEnKF does not impose any assumption on the probability density of the measurement by using generalized orthogonal polynomials to construct the UCs in a systematic way. The simulation results show that the AEnKF outperforms the conventional EnKF and other UC filters in the bearings-only tracking problem.

Spherical Simplex Unscented Kalman Filter-Based Jumping and Static Interacting Multiple Model

A modified interacting multiple model (IMM) method called spherical simplex unscented Kalman filter-based jumping and static IMM (SSUKF-JSIMM) is proposed to solve the problem of nonlinear filtering with unknown continuous system parameter. SSUKF-JSIMM regards the continuous system parameter space as a union of disjoint regions, and each region is assigned to a model. For each model, under the assumption that the parameter belongs to the corresponding region, one sub-filter is used to estimate the parameter and the state when the parameter is presumed to be jumping, and another sub-filter is used to estimate the parameter and the state when the parameter is presumed to be static. Considering that spherical simplex unscented Kalman filter (SSUKF) is more suitable for a real-time system than the unscented Kalman filter (UKF), SSUKFs are adopted as the sub-filters of SSUKF-JSIMM. Results of the two SSUKFs are fused as the estimation output of the model. Experimental results show that SSUKF-JSIMM achieves higher performance than IMM, SIR, and UKF in bearings-only tracking problem.

Continuous Singularity Free Approach to the Three-Dimensional Bearings-Only Tracking Problem

This work presents a new description of the three-dimensional bearings-only tracking problem. A review of established approaches reveals that the common use of elevation and azimuth angles in the state or the measurement model can cause unrealistic error distributions and is not well suited to Kalman filtering under certain relevant circumstances. It is furthermore shown that such methods immanently corrupt rotational invariance and thus let estimation performance rely on a fortunate choice of the inertial reference system. Based on these considerations, an alternative representation is derived that aims at eliminating the prior identified shortcomings. A corresponding tracking filter is designed and demonstrated to give equivalent or superior performance compared with usual approaches.

Optimal design of the Own Ship maneuver in the bearing-only target motion analysis problem using a heuristically supervised Extended Kalman Filter

The state estimation algorithm and the Own Ship maneuvering specifications have great effects on the accuracy of the bearing-only tracking (BOT) method. In the BOT problem, the EKF algorithm is widely used as the nonlinear state estimation algorithm while it suffers from its sensitivity to the initial values of covariance matrixes. This paper aims at improving the accuracy of the BOT problem by using the metaheuristic evolutionary optimization algorithms as supervising algorithms. Three different optimization algorithms, Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Cuckoo Search (CS) are used in finding the optimal initial values of the dynamic and the measurement process noise covariance matrixes of the Extended Kalman Filter (EKF). Moreover, the optimal path (leg) planning of the Own Ship maneuver is also done by minimizing the Fisher Information Matrix (FIM). The Monte Carlo analysis of the simulation results demonstrates the effectiveness of the evolutionary algorithms in improving the performance of the EKF in a BOT problem.

UGHF for acoustic tracking with state-dependent propagation delay

Acoustic tracking with propagation delay is a challenging problem for the following reasons: 1) It is difficult to perform an accurate state prediction, as the time interval between the current state and the previous state is varying and unknown due to the propagation delay; and 2) the target time (signal emission time) needs to be estimated in addition to the target position and velocity. With the state augmented with the target time, the state transition cannot be described by the commonly used explicit Gauss–Markov model. In this paper, we propose a new approach to solving this difficult problem by using the Gauss–Helmert model for the state transition, which consists of an implicit equation between two consecutive states. An unscented Gauss–Helmert filter is then developed based on this formulation. The new approach is applied to the bearings-only tracking problem with state-dependent propagation delay. Simulation tests are conducted to demonstrate the performance of the unscented Gauss–Helmert filter, which is shown to outperform other approaches in terms of estimation accuracy, especially when a target is moving at high speed.

Observation noise modeling based particle filter: An efficient algorithm for target tracking in glint noise environment

In this paper, a novel particle filtering algorithm for target tracking in the presence of glint noise based on observation noise modeling is proposed. The algorithm samples particles using the observation likelihood function, the construction of which is converted to a modeling problem of observation noise. Additionally, the Gaussian mixture model is incorporated to approximate the distribution of observation noise at each time instant. In order to derive a recursive form update for the parameters of the Gaussian components, the maximum likelihood estimation method is employed, enabling noise to be effectively tracked by fusing the latest observations. The algorithm is then used in simulations of bearings-only tracking problems in a glint noise environment with two types of targets: non-maneuvering and maneuvering. The results of the proposed algorithm are evaluated and compared to several existing filtering algorithms through a series of Monte Carlo simulations. The simulation results demonstrate that the proposed algorithm is more precise, robust, and even has a faster convergence rate than the comparative filters. Lastly, the performance of the proposed filter in situations with different numbers of particles and Gaussian components is explored using the simulation results.

Flight path optimization for an electronic intelligence unmanned aerial vehicle

The bearings-only tracking problem with angle-of-arrival (AoA) information received from unmanned aerial vehicle (UAV) is considered. The Kalman filter and joint integrated probabilistic data association filters (JIPDAF) are utilized for single or multiple targets tracking, respectively. An optimal UAV trajectory is constructed for both single and multiple target tracking scenarios. The special case of stationary radar stations localization is numerically simulated.

Gaussian-Sum Cubature Kalman Filter with Improved Robustness for Bearings-only Tracking

This letter presents a Gaussian-sum cubature Kalman filter with improved robustness compared to the original algorithm proposed by the authors in , which demonstrated excellent accuracy and efficiency for the bearings-only tracking problem. Modifications are made in the splitting and merging procedure of the Gaussian components in the algorithm. Simulation results confirm the improved robustness of the modified algorithm against the choice of threshold level for the splitting procedure.

The Recursive Bayesian Estimation Problem via Orthogonal Expansions: an Error Bound

When solving the non linear non Gaussian filtering problem via orthognal series expansions the involved probability density functions are approximated with truncated series expansions. Inevitable the truncation introduces an error. In this paper an upper bound on the 1-norm of the approximation error in the probability density function of the state vector conditional on the system output measurements, due to the truncation, is derived and numerically evaluated in a simulation example. The bound quantifies the proximity of the obtained approximate solution to the true one. To explore the choice of orthonormal basis as a degree of freedom in the proposed method, a comparison between the Fourier and Legendre bases in a bearings-only tracking problem is performed.

A variable neighborhood search particle filter for bearings-only target tracking

In this paper a novel filtering procedure that uses a variant of the variable neighborhood search (VNS) algorithm for solving nonlinear global optimization problems is presented. The base of the new estimator is a particle filter enhanced by the VNS algorithm in resampling step. The VNS is used to mitigate degeneracy by iteratively moving weighted samples from starting positions into the parts of the state space where peaks and ridges of a posterior distribution are situated. For testing purposes, bearings-only tracking problem is used, with two static observers and two types of targets: non-maneuvering and maneuvering. Through numerous Monte Carlo simulations, we compared performance of the proposed filtering procedure with the performance of several standard estimation algorithms. The simulation results show that the algorithm mostly performed better than the other estimators used for comparison; it is robust and has fast initial convergence rate. Robustness to modeling errors of this filtering procedure is demonstrated through tracking of the maneuvering target. Moreover, in the paper it is shown that it is possible to combine the proposed algorithm with an interacted multiple model framework.

Enhanced Modified Polar Coordinates Filter Applied to Bearing Only Tracking

Two new algorithms are presented in this paper to solve the bearing only tracking problem, by using the recently proposed algorithms for nonlinear filtering, which are the Unscented Kalman Filter (UKF) and the Cubature Kalman Filter (CKF). These filters are used to solve the high intrinsic nonlinearity in the modified polar coordinates dynamic model. The resulting filters, named the Modified Polar Unscented Kalman Filter (MPUKF) and the Modified Polar Cubature Kalman Filter (MPCKF) are applied to the two dimensional bearing-only tracking problem. The estimation performance of these nonlinear filters is assessed by means of the root mean square error (RMSE). The simulation results show that these new filters are more efficient than other similar filters, such as the Modified Polar Extended Kalman Filter (MPEKF). Compared to this filter, the MPUKF and the MPCKF are, in particular, non-sensitive to the initial range and more stable, by using a dedicated initialization procedure, proposed in this paper.

A Gaussian-Sum Based Cubature Kalman Filter for Bearings-Only Tracking

Herein is presented an efficient nonlinear filtering algorithm called the Gaussian-sum cubature Kalman filter (GSCKF) for the bearings-only tracking problem. It is developed based on the recently proposed cubature Kalman filter and is built within a Gaussian-sum framework. The new algorithm consists of a splitting and merging procedure when a high degree of nonlinearity is detected. Simulation results show that the proposed algorithm demonstrates comparable performance to the particle filter (PF) with significantly reduced computational cost.

Hierarchical Resampling Algorithm and Architecture for Distributed Particle Filters

In this paper, we introduce a hierarchical resampling (HR) algorithm and architecture for distributed particle filters (PFs). While maintaining the same accuracy as centralized resampling in statistics, the proposed HR algorithm decomposes the resampling step into two hierarchies including intermediate resampling (IR) and unitary resampling (UR), which suits PFs for distributed hardware implementation. Also presented includes a residual cumulative resampling (RCR) method that pipelines and accelerates the UR step. The corresponding architecture, when compared with traditional distributed architectures, eliminates the particle redistribution step, and has such advantages as short execution time and high memory efficiency. The prototype containing 8 PEs has been developed in Xilinx Virtex IV FPGA (XC4VFX100-12FF1152) for the bearings-only tracking (BOT) problem, and the result shows that the input observations can be processed at 37.21 KHz with 8 K particles and a clock speed of 80 MHz.

Distributed multi-sensor particle filter for bearings-only tracking

In this article, the classical bearings-only tracking (BOT) problem for a single target is addressed, which belongs to the general class of non-linear filtering problems. Due to the fact that the radial distance observability of the target is poor, the algorithm-based sequential Monte-Carlo (particle filtering, PF) methods generally show instability and filter divergence. A new stable distributed multi-sensor PF method is proposed for BOT. The sensors process their measurements at their sites using a hierarchical PF approach, which transforms the BOT problem from Cartesian coordinate to the logarithmic polar coordinate and separates the observable components from the unobservable components of the target. In the fusion centre, the target state can be estimated by utilising the multi-sensor optimal information fusion rule. Furthermore, the computation of a theoretical Cramer–Rao lower bound is given for the multi-sensor BOT problem. Simulation results illustrate that the proposed tracking method can provide better performances than the traditional PF method.

Particle Filter with Fine Resampling for Bearings-Only Tracking

To solve the nonlinear filtering problem in the Bearings-Only Tracking (BOT), an improved particle filter with fine resampling is proposed, called as PF-FR (Particle Filter with Fine Resampling). By introducing distance-comparing process and generating new particle based on optimized combination scheme, PF-FR filter performs better than generic PF-SIR filter both in terms of effectiveness and diversity of the particle system, hence, evidently enhancing tracking accuracy in bearings-only tracking problem. Simulations indicate that the proposed PF-FR algorithm is effective.

Object Tracking in 3-D Space with Passive Acoustic Sensors using Particle Filter

This paper considers the object tracking problem in three dimensional (3-D) space when the azimuth and elevation of the object are available from the passive acoustic sensor. The particle filtering technique can be directly applied to estimate the 3-D object location, but we propose to decompose the 3-D particle filter into the three planes particle filters, which are individually designed for the 2-D bearings-only tracking problems. 2-D bearing information is derived from the azimuth and elevation of the object to be used for the 2-D particle filter. Two estimates of three planes particle filters are selected based on the characterization of the acoustic sensor operation in a noisy environment. The Cramer-Rao Lower Bound of the proposed 2-D particle filter-based algorithm is derived and compared against the algorithm that is based on the direct 3-D particle filter.