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Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions

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A special issue of Sensors (ISSN 1424-8220). This special issue belongs to the section "Physical Sensors".

Deadline for manuscript submissions: closed (20 December 2022) | Viewed by 24277

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Special Issue Editors

Dr. Mahendra Mallick

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Guest Editor

Independent Consultant, Anacortes, WA 98221, USA
Interests: estimation; nonlinear filtering; multitarget tracking; data fusion

Dr. Ratnasingham Tharmarasa

E-Mail Website
Guest Editor

Electrical and Computer Engineering Department, McMaster University, Hamilton, ON L8S 4K1, Canada
Interests: estimation; nonlinear filtering; multitarget tracking; data fusion; sensor resource management; radar signal processing

Special Issue Information

Dear Colleagues,

Bearing-only filtering (BOF) in 2D arises in many real-world tracking problems including underwater tracking using a passive sonar, aircraft surveillance using a passive radar, navigation of a robot using a passive sonar, and undersea exploration of natural resources using sonar. BOF using a single sensor is also a challenging nonlinear filtering problem due to poor observability and the nonlinear measurement model. This filtering problem and associated tracking problem have been studied extensively.

Angle-only filtering (AOF) in 3D is counterpart of BOF in 2D. Real-world AOF problems include passive ranging using an infrared search and track (IRST) sensor, passive sonar, passive radar in the presence of jamming, ballistic missile and satellite tacking using a telescope, satellite to satellite passive tracking, and missile guidance using bearing-only seekers. The number of publications in the AOF and angle-only tracking in 3D is rather limited compared with the corresponding problems in 2D.

We invite original research articles and reviews for this Special Issue that includes advanced filtering and tracking algorithms for bearing-only and angle-only filtering and tracking problems. Related topics include, but are not limited to:

Bearing-only filtering/tracking in 2D
Angle-only filtering/ tracking in 3D
Maneuvering target tracking in 2D and 3D
Filtering/tracking with a von Mises distributions
Filtering/tracking with heavy-tailed noise
Continuous-discrete filtering in angle-only filtering in 3D
Filtering and tracking with unknown noise distributions
Multi-sensor bearings-only/angle-only tracking in the presence of registration errors
Sensor trajectory optimization in single-sensor bearings-only/angle-only tracking
Performance bounds for angle only tracking
Observability of BOF and AOF problems
Centralized/Distributed multisensor fusion

Dr. Mahendra Mallick
Dr. Ratnasingham Tharmarasa
Guest Editors

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Keywords

bearing-only tracking
angle-only tracking
observability
maneuvering target tracking
performance bounds
bias estimation
non-Gaussian and multimodal noise distributions

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Published Papers (12 papers)

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Research

17 pages, 1008 KiB

Open AccessArticle

Optimal Geometries for AOA Localization in the Bayesian Sense

by Kutluyil Dogancay

Sensors 2022, 22(24), 9802; https://doi.org/10.3390/s22249802 - 14 Dec 2022

Cited by 4 | Viewed by 1636

Abstract

This paper considers the optimal sensor placement problem for angle-of-arrival (AOA) target localization in the 2D plane with a Gaussian prior. Optimal sensor locations are analytically determined for a single AOA sensor using the D- and A-optimality criteria and an approximation of the Bayesian Fisher information matrix (BFIM). Optimal sensor placement is shown to align with the minor axis of the prior covariance error ellipse for both optimality criteria. The approximate BFIM is argued to be valid for a sufficiently small prior covariance compared with the target range. Optimal sensor placement results obtained for Bayesian target localization are extended to manoeuvring target tracking. For sensor trajectory optimization subject to turn-rate constraints, numerical search methods based on the D- and A-optimality criteria as well as a new closed-form projection algorithm that aims to achieve alignment with the minor axis of the prior error ellipse are proposed. It is observed that the two optimality criteria generate significantly different optimal sensor trajectories despite having the same optimal sensor placement for the localization of a stationary target. Analysis results and the performance of the sensor trajectory optimization methods are demonstrated with simulation examples. It is observed that the new closed-form projection algorithm achieves superior tracking performance compared with the two numerical search methods. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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21 pages, 8168 KiB

Open AccessArticle

Signal Source Localization with Long-Term Observations in Distributed Angle-Only Sensors

by Shenghua Zhou, Linhai Wang, Ran Liu, Yidi Chen, Xiaojun Peng, Xiaoyang Xie, Jian Yang, Shibo Gao and Xuehui Shao

Sensors 2022, 22(24), 9655; https://doi.org/10.3390/s22249655 - 9 Dec 2022

Cited by 1 | Viewed by 1292

Abstract

Angle-only sensors cannot provide range information of targets and in order to determine accurate position of a signal source, one can connect distributed passive sensors with communication links and implement a fusion algorithm to estimate target position. To measure moving targets with sensors on moving platforms, most of existing algorithms resort to the filtering method. In this paper, we present two fusion algorithms to estimate both the position and velocity of moving target with distributed angle-only sensors in motion. The first algorithm is termed as the gross least square (LS) algorithm, which takes all observations from distributed sensors together to form an estimate of the position and velocity and thus needs a huge communication cost and a huge computation cost. The second algorithm is termed as the linear LS algorithm, which approximates locations of sensors, locations of targets, and angle-only measures for each sensor by linear models and thus does not need each local sensors to transmit raw data of angle-only observations, resulting in a lower communication cost between sensors and then a lower computation cost at the fusion center. Based on the second algorithm, a truncated LS algorithm, which estimates the target velocity through an average operation, is also presented. Numerical results indicate that the gross LS algorithm, without linear approximation operation, often benefits from more observations, whereas the linear LS algorithm and the truncated LS algorithm, both bear lower communication and computation costs, may endure performance loss if the observations are collected in a long period such that the linear approximation model becomes mismatch. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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21 pages, 453 KiB

Open AccessArticle

2D and 3D Angles-Only Target Tracking Based on Maximum Correntropy Kalman Filters

by Asfia Urooj, Aastha Dak, Branko Ristic and Rahul Radhakrishnan

Sensors 2022, 22(15), 5625; https://doi.org/10.3390/s22155625 - 27 Jul 2022

Cited by 11 | Viewed by 2156

Abstract

In this paper, angles-only target tracking (AoT) problem is investigated in the non Gaussian environment. Since the conventional minimum mean square error criterion based estimators tend to give poor accuracy in the presence of large outliers or impulsive noises in measurement, a maximum correntropy criterion (MCC) based framework is presented. Accordingly, three new estimation algorithms are developed for AoT problems based on the conventional sigma point filters, termed as MC-UKF-CK, MC-NSKF-GK and MC-NSKF-CK. Here MC-NSKF-GK represents the maximum correntropy new sigma point Kalman filter realized using Gaussian kernel and MC-NSKF-CK represents realization using Cauchy kernel. Similarly, based on the unscented Kalman filter, MC-UKF-CK has been developed. The performance of all these estimators is evaluated in terms of root-mean-square error (RMSE) in position and % track loss. The simulations were carried out for 2D as well as 3D AoT scenarios and it was inferred that, the developed algorithms performed with improved estimation accuracy than the conventional ones, in the presence of non Gaussian measurement noise. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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21 pages, 1801 KiB

Open AccessArticle

A New Coarse Gating Strategy Driven Multidimensional Assignment for Two-Stage MHT of Bearings-Only Multisensor-Multitarget Tracking

by Zheng Wei, Zhansheng Duan, Yina Han and Mahendra Mallick

Sensors 2022, 22(5), 1802; https://doi.org/10.3390/s22051802 - 24 Feb 2022

Cited by 4 | Viewed by 1899

Abstract

The problem of two-dimensional bearings-only multisensor-multitarget tracking is addressed in this work. For this type of target tracking problem, the multidimensional assignment (MDA) is crucial for identifying measurements originating from the same targets. However, the computation of the assignment cost of all possible associations is extremely high. To reduce the computational complexity of MDA, a new coarse gating strategy is proposed. This is realized by comparing the Mahalanobis distance between the current estimate and initial estimate in an iterative process for the maximum likelihood estimation of the target position with a certain threshold to eliminate potential infeasible associations. When the Mahalanobis distance is less than the threshold, the iteration will exit in advance so as to avoid the expensive computational costs caused by invalid iteration. Furthermore, the proposed strategy is combined with the two-stage multiple hypothesis tracking framework for bearings-only multisensor-multitarget tracking. Numerical experimental results verify its effectiveness. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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18 pages, 1878 KiB

Open AccessArticle

Signal Source Positioning Based on Angle-Only Measurements in Passive Sensor Networks

by Yidi Chen, Linhai Wang, Shenghua Zhou and Renwen Chen

Sensors 2022, 22(4), 1554; https://doi.org/10.3390/s22041554 - 17 Feb 2022

Cited by 3 | Viewed by 1846

Abstract

Some passive sensors can measure only directions of arrival of signals, but the real positions of signal sources are often desirable, which can be estimated by combining distributed passive sensors as a network. However, passive observations should be correctly associated first. This paper studies the multi-target data association and signal localization problem in distributed passive sensor networks. With angle-only measurements from distributed passive sensors, multiple lines in a 3-dimensional (3D) scenario can be built and then those that will intersect in a small volume in 3D are classified into the same source. The center of the small volume is taken as an estimate of the signal source position, whose statistical distributions are formulated. If the minimum distance is less than an association threshold, then two lines are considered to be from the same signal source. In numerical results, the impacts of angle measurement accuracy and platform self-positioning accuracy are analyzed, indicating that this method can achieve a prescribed data association rate and a high positioning performance with a low computation cost. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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22 pages, 757 KiB

Open AccessArticle

Angle-Only Filtering of a Maneuvering Target in 3D

by Mahendra Mallick, Xiaoqing Tian, Yun Zhu and Mark Morelande

Sensors 2022, 22(4), 1422; https://doi.org/10.3390/s22041422 - 12 Feb 2022

Cited by 5 | Viewed by 2068

Abstract

We consider the state estimation of a maneuvering target in 3D using bearing and elevation measurements from a passive infrared search and track (IRST) sensor. Since the range is not observable, the sensor must perform a maneuver to observe the state of the [...] Read more.

X Y

-plane and nearly constant velocity (NCV) along the Z-axis. The natural choice for the NCT motion is to allow perturbations in speed and angular rate in the stochastic differential equation, as has been pointed out previously for a 2D scenario using range and bearing measurements. The NCT motion in the

X Y

-plane cannot be discretized exactly, whereas the NCV motion along the Z-axis is discretized exactly. We discretize the continuous-time NCT model using the first and second-order Taylor approximations to obtain discrete-time NCT models, and we consider the polar velocity and Cartesian velocity-based states for the NCT model. The dynamic and measurement models are nonlinear in the target state. We use the cubature Kalman filter to estimate the target state. Accuracies of the first and second-order Taylor approximations are compared using the polar velocity-based and Cartesian velocity-based models using Monte Carlo simulations. Numerical results for realistic scenarios considered show that the second-order Taylor approximation provides the best accuracy using the polar velocity or Cartesian velocity-based models. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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26 pages, 1239 KiB

Open AccessArticle

Ground Target Tracking Using an Airborne Angle-Only Sensor with Terrain Uncertainty and Sensor Biases

by Dipayan Mitra, Aranee Balachandran and Ratnasingham Tharmarasa

Sensors 2022, 22(2), 509; https://doi.org/10.3390/s22020509 - 10 Jan 2022

Cited by 1 | Viewed by 1911

Abstract

Airborne angle-only sensors can be used to track stationary or mobile ground targets. In order to make the problem observable in 3-dimensions (3-D), the height of the target (i.e., the height of the terrain) from the sea-level is needed to be known. In most of the existing works, the terrain height is assumed to be known accurately. However, the terrain height is usually obtained from Digital Terrain Elevation Data (DTED), which has different resolution levels. Ignoring the terrain height uncertainty in a tracking algorithm will lead to a bias in the estimated states. In addition to the terrain uncertainty, another common source of uncertainty in angle-only sensors is the sensor biases. Both these uncertainties must be handled properly to obtain better tracking accuracy. In this paper, we propose algorithms to estimate the sensor biases with the target(s) of opportunity and algorithms to track targets with terrain and sensor bias uncertainties. Sensor bias uncertainties can be reduced by estimating the biases using the measurements from the target(s) of opportunity with known horizontal positions. This step can be an optional step in an angle-only tracking problem. In this work, we have proposed algorithms to pick optimal targets of opportunity to obtain better bias estimation and algorithms to estimate the biases with the selected target(s) of opportunity. Finally, we provide a filtering framework to track the targets with terrain and bias uncertainties. The Posterior Cramer–Rao Lower Bound (PCRLB), which provides the lower bound on achievable estimation error, is derived for the single target filtering with an angle-only sensor with terrain uncertainty and measurement biases. The effectiveness of the proposed algorithms is verified by Monte Carlo simulations. The simulation results show that sensor biases can be estimated accurately using the target(s) of opportunity and the tracking accuracies of the targets can be improved significantly using the proposed algorithms when the terrain and bias uncertainties are present. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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23 pages, 6099 KiB

Open AccessArticle

Intelligence-Aware Batch Processing for TMA with Bearings-Only Measurements

by Gabriele Oliva, Alfonso Farina and Roberto Setola

Sensors 2021, 21(21), 7211; https://doi.org/10.3390/s21217211 - 29 Oct 2021

Cited by 2 | Viewed by 1929

Abstract

This paper develops a framework to track the trajectory of a target in 2D by considering a moving ownship able to measure bearing measurements. Notably, the framework allows one to incorporate additional information (e.g., obtained via intelligence) such as knowledge on the fact the target’s trajectory is contained in the intersection of some sets or the fact it lies outside the union of other sets. The approach is formally characterized by providing a constrained maximum likelihood estimation (MLE) formulation and by extending the definition of the Cramér–Rao lower bound (CRLB) matrix to the case of MLE problems with inequality constraints, relying on the concept of generalized Jacobian matrix. Moreover, based on the additional information, the ownship motion is chosen by mimicking the Artificial Potential Fields technique that is typically used by mobile robots to aim at a goal (in this case, the region where the target is assumed to be) while avoiding obstacles (i.e., the region that is assumed not to intersect the target’s trajectory). In order to show the effectiveness of the proposed approach, the paper is complemented by a simulation campaign where the MLE computations are carried out via an evolutionary ant colony optimization software, namely, mixed-integer distributed ant colony optimization solver (MIDACO-SOLVER). As a result, the proposed framework exhibits remarkably better performance, and in particular, we observe that the solution is less likely to remain stuck in unsatisfactory local minima during the MLE computation. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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20 pages, 622 KiB

Open AccessArticle

Closed-Form Pseudolinear Estimators for DRSS-AOA Localization

by Jun Li, Kutluyil Dogancay and Hatem Hmam

Sensors 2021, 21(21), 7159; https://doi.org/10.3390/s21217159 - 28 Oct 2021

Viewed by 1714

Abstract

This paper investigates the hybrid source localization problem using differential received signal strength (DRSS) and angle of arrival (AOA) measurements. The main advantage of hybrid measurements is to improve the localization accuracy with respect to a single sensor modality. For sufficiently short wavelengths, AOA sensors can be constructed with size, weight, power and cost (SWAP-C) requirements in mind, making the proposed hybrid DRSS-AOA sensing feasible at a low cost. Firstly the maximum likelihood estimation solution is derived, which is computationally expensive and likely to become unstable for large noise levels. Then a novel closed-form pseudolinear estimation method is developed by incorporating the AOA measurements into a linearized form of DRSS equations. This method eliminates the nuisance parameter associated with linearized DRSS equations, hence improving the estimation performance. The estimation bias arising from the injection of measurement noise into the pseudolinear data matrix is examined. The method of instrumental variables is employed to reduce this bias. As the performance of the resulting weighted instrumental variable (WIV) estimator depends on the correlation between the IV matrix and data matrix, a selected-hybrid-measurement WIV (SHM-WIV) estimator is proposed to maintain a strong correlation. The superior bias and mean-squared error performance of the new SHM-WIV estimator is illustrated with simulation examples. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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20 pages, 473 KiB

Open AccessArticle

A Total Lp-Norm Optimization for Bearing-Only Source Localization in Impulsive Noise with SαS Distribution

by Ji-An Luo, Chang-Cheng Xue, Ying-Jiao Rong and Shen-Tu Han

Sensors 2021, 21(19), 6471; https://doi.org/10.3390/s21196471 - 28 Sep 2021

Cited by 2 | Viewed by 1706

Abstract

This paper considers the problem of robust bearing-only source localization in impulsive noise with symmetric

α

This paper considers the problem of robust bearing-only source localization in impulsive noise with symmetric

α

-stable distribution based on the Lp-norm minimization criterion. The existing Iteratively Reweighted Pseudolinear Least-Squares (IRPLS) method can be used to solve the least LP-norm optimization problem. However, the IRPLS algorithm cannot reduce the bias attributed to the correlation between system matrices and noise vectors. To reduce this kind of bias, a Total Lp-norm Optimization (TLPO) method is proposed by minimizing the errors in all elements of system matrix and data vector based on the minimum dispersion criterion. Subsequently, an equivalent form of TLPO is obtained, and two algorithms are developed to solve the TLPO problem by using Iterative Generalized Eigenvalue Decomposition (IGED) and Generalized Lagrange Multiplier (GLM), respectively. Numerical examples demonstrate the performance advantage of the IGED and GLM algorithms over the IRPLS algorithm. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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19 pages, 411 KiB

Open AccessArticle

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

by Shizhe Bu, Aiqiang Meng and Gongjian Zhou

Sensors 2021, 21(16), 5444; https://doi.org/10.3390/s21165444 - 12 Aug 2021

Cited by 8 | Viewed by 2042

Abstract

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. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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20 pages, 5518 KiB

Open AccessArticle

TMA from Cosines of Conical Angles Acquired by a Towed Array

by Antoine Lebon, Annie-Claude Perez, Claude Jauffret and Dann Laneuville

Sensors 2021, 21(14), 4797; https://doi.org/10.3390/s21144797 - 14 Jul 2021

Cited by 5 | Viewed by 2201

Abstract

This paper deals with the estimation of the trajectory of a target in constant velocity motion at an unknown constant depth, from measurements of conical angles supplied by a linear array. Sound emitted by the target does not necessarily navigate along a direct path toward the antenna, but can bounce off the sea bottom and/or off the surface. Observability is thoroughly analyzed to identify the ghost targets before proposing an efficient way to estimate the trajectory of the target of interest and of the ghost targets when they exist. Full article

(This article belongs to the Special Issue Advances in Angle-Only Filtering and Tracking in Two and Three Dimensions)

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