Bearings-only Tracking 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.

Continuous-discrete extended Kalman filtering based on the neural ordinary differential equations method

In bearings-only tracking (BOT), the accuracy of state estimation significantly diminishes when confronted with an unknown target motion model. Therefore, this study proposed a novel continuous-discrete filtering algorithm, the neural ordinary differential equations-based continuous-discrete extended Kalman filtering (NODE-CD-EKF). The approach utilizes the neural ordinary differential equation (NODE) to build the target's system model, which is then used in the time update to precisely predict the target state at the next time point within the continuous-discrete Kalman framework. The NODE concisely and accurately describes the motion model in the form of ordinary differential equations in a data-driven way. This approach addresses the limitations of traditional Kalman filters, which suffer from degraded performance due to a lack of a precise mathematical model of target states. Our method is evaluated and compared to existing multiple-model and other data-driven algorithms in simulations of BOT and another scenario using the Accumulated Root Mean Square Error (ARMSE) of position. The ARMSE of our method is only 16.30 % of the multiple-model methods at least. Overall, this approach outperforms other multiple-model and data-driven algorithms and possesses the potential for solving other nonlinear tasks in real-world applications.

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.

Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements.

Handling the challenge of missing measurements in nonlinear systems is a difficult problem in various scientific and engineering fields. Missing measurements, which can arise from technical faults during observation, diffusion channel shrinking, or the loss of specific metrics, can bring many challenges when estimating the state of nonlinear systems. To tackle this issue, this paper proposes a technique that utilizes a robust cubature Kalman filter (RCKF) by integrating Huber's M-estimation theory with the standard conventional cubature Kalman filter (CKF). Although a CKF is often used for solving nonlinear filtering problems, its effectiveness might be limited due to a lack of knowledge regarding the nonlinear model of the state and noise-related statistical information. In contrast, the RCKF demonstrates an ability to mitigate performance degradation and discretization issues related to track curves by leveraging covariance matrix predictions for state estimation and output control amidst dynamic disruption errors-even when noise statistics deviate from prior assumptions. The performance of extended Kalman filters (EKFs), unscented Kalman filters (UKFs), CKFs, and RCKFs was compared and evaluated using two numerical examples involving the Univariate Non-stationary Growth Model (UNGM) and bearing-only tracking (BOT). The numerical experiments demonstrated that the RCKF outperformed the EKF, EnKF, and CKF in effectively handling anomaly errors. Specifically, in the UNGM example, the RCKF achieved a significantly lower ARMSE (4.83) and ANCI (3.27)-similar outcomes were observed in the BOT example.

Sensor Management with Dynamic Clustering for Bearings-Only Multi-Target Tracking via Swarm Intelligence Optimization

Sensor management is a crucial research subject for multi-sensor multi-target tracking in wireless sensor networks (WSNs) with limited resources. Bearings-only tracking produces further challenges related to high nonlinearity and poor observability. Moreover, energy efficiency and energy balancing should be considered for sensor management in WSNs, which involves networking and transmission. This paper formulates the sensor management problem in the partially observable Markov decision process (POMDP) framework and uses the cardinality-balanced multi-target multi-Bernoulli (CBMeMBer) filter for tracking. A threshold control method is presented to reduce the impact on tracking accuracy when using bearings-only measurements for sequential update. Moreover, a Cauchy–Schwarz divergence center is defined to construct a new objective function for efficiently finding the optimal sensor subset via swarm intelligence optimization. This is also conducive to dynamic clustering for the energy efficiency and energy balancing of the network. The simulation results illustrate that the proposed solution can achieve good tracking performance with less energy, and especially that it can effectively balance network energy consumption and prolong network lifetime.

3D generalized bias compensated pseudolinear Kalman filter for colored noisy bearings-only measurements

Bias compensated pseudolinear Kalman filter (BC-PLKF) degrades for bearings-only tracking if measurements are corrupted with time-correlated noises. Generalized bias compensated pseudolinear Kalman filter (GBC-PLKF), on the other hand, outperforms BC-PLKF and many other comparison filters for two-dimensional (2D) cases under colored noise. However, GBC-PLKF is not practicable for three-dimensional (3D) real-life problems and due to the coupling between lateral and longitudinal planes, its 3D extension is not trivial. In this paper, bias analysis of 3D pseudolinear Kalman filter (PLKF) with azimuth and elevation measurements including colored noise is performed. Then, the low-cost recursive filter, 3D-GBC-PLKF is proposed for 3D real-life applications. The performance and effectiveness of the proposed algorithm is demonstrated with an air-to-surface guided missile pursuing a typical target.

Adaptive Kernel Kalman Filter

Sequential Bayesian filters in non-linear dynamic systems require the recursive estimation of the predictive and posterior distributions. This paper introduces a Bayesian filter called the adaptive kernel Kalman filter (AKKF). With this filter, the arbitrary predictive and posterior distributions of hidden states are approximated using the empirical kernel mean embeddings (KMEs) in reproducing kernel Hilbert spaces (RKHSs). In parallel with the KMEs, some particles, in the data space, are used to capture the properties of the dynamical system model. Specifically, particles are generated and updated in the data space, while the corresponding kernel weight mean vector and covariance matrix associated with the feature mappings of the particles are predicted and updated in the RKHSs based on the kernel Kalman rule (KKR). Simulation results are presented to confirm the improved performance of our approach with significantly reduced particle numbers, by comparing with the unscented Kalman filter (UKF), particle filter (PF) and Gaussian particle filter (GPF). For example, compared with the GPF, the proposed approach provides around 5% logarithmic mean square error (LMSE) tracking performance improvement in the bearing-only tracking (BOT) system when using 50 particles.

Robust Estimation in Continuous–Discrete Cubature Kalman Filters for Bearings-Only Tracking

The model of bearings-only tracking is generally described by discrete–discrete filtering systems. Discrete robust methods are also frequently used to address measurement uncertainty problems in bearings-only tracking. The recently popular continuous–discrete filtering system considers the state model of the target to be continuous in time, and is more suitable for bearings-only tracking because of its higher mathematical solution accuracy. However, the sufficient evaluation of robust methods in continuous–discrete systems is not available. In addition, in the different continuous–discrete measurement environments, the choice of a robust algorithm also needs to be discussed. To fill this gap, this paper firstly establishes the continuous–discrete target tracking model, and then evaluates the performance of proposed robust square-root continuous–discrete cubature Kalman filter algorithms in the measurement of uncertainty problems. From the simulation results, the robust square-root continuous–discrete maximum correntropy cubature Kalman filter algorithm and the variational Bayesian square-root continuous–discrete cubature Kalman filter algorithm have better environmental adaptability, which provides a promising means for solving continuous–discrete robust problems.

Implementation of ensemble Kalman filter algorithm for underwater target tracking

ABSTRACT Surveillance of underwater for maritime warfare is traditionally being carried out by bearings-only tracking from many decades. The measurements used for state estimation here are nonlinear. Also the noise in the measurements and the process cannot be always Gaussian. The traditional nonlinear filtering algorithms like extended Kalman filter and modified gain extended Kalman filter use the linearisation of the system. The unscented Kalman filter (UKF) uses the sigma point approach based on Gaussian distribution to deal with nonlinearity. The particle filter (PF) uses the randomly generated particles based on the pdf of the state. PF is highly complex to implement and it also suffers from sample impoverishment. Hence, ensemble Kalman filter (EnKF) which is a simplified form of PF will be tried out for bearings-only tracking in this research work. The performance of EnKF is compared with PF and UKF and the results obtained using these filters in Matlab are presented.

Underwater Bearing Only Tracking Using Optimal Observer Maneuver Strategies

This paper considers the problem of tracking a uniform moving source using noisy bearing measurements obtained from a distant observer. Observer trajectory optimization plays a central role in this problem, with the objective to minimize the estimation error of the target state. The Bearing Only Tracking (BOT) of passive targets is mainly focused on the observer maneuver with known trajectories and rarely focused on the future prediction of observer states using adaptive optimization strategies. In this paper, observer paths using one-step ahead optimization based on a performance index are devised which are potentially useful for longer horizon observer trajectory planning in passive tracking. This performance index is the function of source parameters termed as the determinant of Error Covariance Matrix (ECM) which is numerically more efficient than the determinant of Fisher Information Matrix (FIM). The determinant of the FIM requires the calculation of future values for target states and measurements rather than the current values, which is not feasible for Kalman like filters. Therefore, in this paper, the optimization technique is implemented using the state error covariance which is readily available through Kalman filter equations and does require separate numerical calculations. Due to optimal observer maneuver, the performance of the proposed algorithm does not depend on the initial conditions as compared to the conventional tracking methods. The efficiency of the evolutionary algorithm is calculated in terms of range, position and velocity errors and simulation results show 4% fewer estimation errors for ECM based optimization than the determinant of the FIM method.

Bearings-only Tracking Based on Distributed Multisensor Pseudolinear Kalman Filter

For bearings-only tracking (BOT), there are mainly two problems of nonlinear filtering and poor range observability. In the paper, a new distributed multisensor pseudolinear Kalman filter (PLKF) algorithm is proposed. The sensors use an instrumental vector PLKF (IV-PLKF) to process the measurements of the target independently, which can tackle the bias arising from the correlation between the measurement vector and pseudolinear noise by the bias compensation PLKF (BC-PLKF). The IV-PLKF embeds the recursive instrumental vector estimation method into the BC-PLKF, uses it to construct the instrumental vector, and applies the method of selective angle measurement to modify the local target state estimation and covariance. In the fusion center, the target state can be estimated by using the multisensor optimal information fusion criterion. Then the Cramer-Rao lower bound (CRLB) of multisensor BOT is derived. Simulation results show the effectiveness of the algorithm.

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 state estimators for three dimensional angles-only underwater target tracking problems

Gaussian sum filters are considered to be more accurate in terms of estimation accuracy when compared to the conventional algorithms. In this work, Gaussian sum state estimation algorithms are implemented for three dimensional angles-only target tracking problem. Shifted Rayleigh filter (SRF) has been considered as the most accurate estimation algorithms for bearings-only tracking, with moderate computational load. Therefore, SRF formulated in the Gaussian sum framework is developed for solving three dimensional angles-only target tracking problems. The estimation accuracy of the developed algorithm, and other Gaussian as well as Gaussian sum algorithms is validated in terms of percentage track-loss and root mean square error (RMSE).

Bearing‐only underwater uncooperative target tracking for non‐Gaussian environment using fast particle filter

Bearing-only tracking of an underwater uncooperative target is essential to defend the sea territory. Considering the influences by uncertain underwater environment, the purpose of this work is to estimate 2-D locations and velocities of an interested underwater target for non-Gaussian environment. In this work, a fast particle filter (FPF) based on the traditional particle filter (PF) with novel jet transport (JT) technique is proposed to deal with this problem. Aiming to overcome the heavy computation burden of the traditional PF that limits most of its practical applications, the JT technique can dramatically reduce the computation time and complexity in the particle evolution process, which contributes huge computational complexities to the traditional PF. Then, the proposed FPF is tested through simulations in the 2-D underwater uncooperative target tracking scenario. Finally, the Monte Carlo simulation results demonstrate that the proposed FPF can track the underwater uncooperative target with the similar accuracies as the traditional PF but only occupies less than 20% of the computational resources.

Variational Bayesian Kalman filter using natural gradient

We propose a technique based on the natural gradient method for variational lower bound maximization for a variational Bayesian Kalman filter. The natural gradient approach is applied to the Kullback-Leibler divergence between the parameterized variational distribution and the posterior density of interest. Using a Gaussian assumption for the parametrized variational distribution, we obtain a closed-form iterative procedure for the Kullback-Leibler divergence minimization, producing estimates of the variational hyper-parameters of state estimation and the associated error covariance. Simulation results in both a Doppler radar tracking scenario and a bearing-only tracking scenario are presented, showing that the proposed natural gradient method outperforms existing methods which are based on other linearization techniques in terms of tracking accuracy.

Prevalence of shifted Rayleigh filter for passive surveillance in underwater

PurposeFrom many decades, bearings-only tracking (BOT) is the interested problem for researchers. This utilises nonlinear filtering methods for state estimation as there is only information about the target, i.e. bearing is a nonlinear measurement. The measurement bearing is tangentially related to the target state vector. There are many nonlinear filtering algorithms developed so far in the literature.Design/methodology/approachIn this research work, the recently developed nonlinear filtering algorithm, i.e. shifted Rayleigh filter (SRF), is applied to BOT.FindingsThe SRF is tested for two-dimensional BOT against various scenarios. The simulation results emphasise that the SRF performs well when compared to the standard nonlinear filtering algorithm, unscented Kalman filter (UKF).Originality/valueSRF utilises the nonlinearities present in the bearing measurement through the use of moment matching. The SRF is able to produce the solution in highly noisy environment, long ranges and high dimension tracking.

Observer control for bearings-only tracking using possibility functions

Bearings-only tracking using passive sensors is important for covert surveillance of moving targets. This paper adopts a mathematical formulation of bearings-only tracking in the framework of possibility theory, where uncertainties are represented using possibility functions, rather than usual probability distributions. Possibility functions have the capacity to deal robustly with partial (incomplete) specification of mathematical models and have been found particularly useful in model mismatch situations. The paper explores the design of reward functions which provide information gain in the context of observer motion control, in the framework of possibilistic recursive filter for bearings-only tracking. Numerical results demonstrate that in the presence of a model mismatch, the proposed framework performs better than the Bayesian probabilistic framework for stochastic filtering and control.

A New Pseudolinear Filter for Bearings-Only Tracking without Requirement of Bias Compensation.

In bearings-only tracking systems, the pseudolinear Kalman filter (PLKF) has advantages in stability and computational complexity, but suffers from correlation problems. Existing solutions require bias compensation to reduce the correlation between the pseudomeasurement matrix and pseudolinear noise, but incomplete compensation may cause a loss of estimation accuracy. In this paper, a new pseudolinear filter is proposed under the minimum mean square error (MMSE) framework without requirement of bias compensation. The pseudolinear state-space model of bearings-only tracking is first developed. The correlation between the pseudomeasurement matrix and pseudolinear noise is thoroughly analyzed. By splitting the bearing noise term from the pseudomeasurement matrix and performing some algebraic manipulations, their cross-covariance can be calculated and incorporated into the filtering process to account for their effects on estimation. The target state estimation and its associated covariance can then be updated according to the MMSE update equation. The new pseudolinear filter has a stable performance and low computational complexity and handles the correlation problem implicitly under a unified MMSE framework, thus avoiding the severe bias problem of the PLKF. The posterior Cramer–Rao Lower Bound (PCRLB) for target state estimation is presented. Simulations are conducted to demonstrate the effectiveness of the proposed method.

Application of Cubature Information Filter for Underwater Target Path Estimation

Bearings-only tracking plays a pivotal role in passive underwater surveillance. Using noisy sonar bearing measurements, the target motion parameters (TMP) are extensively estimated using the extended Kalman filter (EKF) because of its simplicity and low computational load. The EKF utilizes the first order approximation of the nonlinear system in estimation of the TMP that degrades the accuracy of estimation due to the elimination of the higher order terms. In this paper, the cubature Kalman filter (CKF) that captures the system nonlinearity upto third order is proposed to estimate the TMP. The CKF is further extended using the information filter (IF) to provide decentralized data fusion, hence the filter is termed as cubature information filter (CIF). The results are generated using Matlab with Gaussian assumption of noise in measurements. Monte-Carlo simulation is done and the results demonstrate that the CIF accuracy is same as that of UKF and this indicates the usefulness of the algorithm for state estimation in underwater with the required accuracy.

Adaptive Two-Step Bearing-Only Underwater Uncooperative Target Tracking with Uncertain Underwater Disturbances.

The bearing-only tracking of an underwater uncooperative target can protect maritime territories and allows for the utilization of sea resources. Considering the influences of an unknown underwater environment, this work aimed to estimate 2-D locations and velocities of an underwater target with uncertain underwater disturbances. In this paper, an adaptive two-step bearing-only underwater uncooperative target tracking filter (ATSF) for uncertain underwater disturbances is proposed. Considering the nonlinearities of the target’s kinematics and the bearing-only measurements, in addition to the uncertain noise caused by an unknown underwater environment, the proposed ATSF consists of two major components, namely, an online noise estimator and a robust extended two-step filter. First, using a modified Sage-Husa online noise estimator, the uncertain process and measurement noise are estimated at each tracking step. Then, by adopting an extended state and by using a robust negative matrix-correcting method in conjunction with a regularized Newton-Gauss iteration scheme, the current state of the underwater uncooperative target is estimated. Finally, the proposed ATSF was tested via simulations of a 2-D underwater uncooperative target tracking scenario. The Monte Carlo simulation results demonstrated the reliability and accuracy of the proposed ATSF in bearing-only underwater uncooperative tracking missions.