Automation and Robots for Disaster Response

Generalized Labeled Multi-Bernoulli Filter-Based Passive Localization and Tracking of Radiation Sources Carried by Unmanned Aerial Vehicles

By inergency On Mar 13, 2024

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1. Introduction

The goal of passive localization and tracking is to continuously or periodically localize the target or radiation source without the active participation of the target or radiation source, thus obtaining the motion trajectory of the target or radiation source carried by unmanned aerial vehicles (UAVs) [1]. Passive localization and tracking of radiation sources is a critical technology used to determine the position and motion of wireless signal transmitters without active contact, which has a wide range of applications in various fields, such as wireless communications, radio frequency interference (RFI) monitoring, environmental surveillance, and military reconnaissance [2,3,4]. By analyzing the characteristics of received wireless signals, passive localization and tracking provide valuable insights into the spatial distribution of radiation sources, ensuring efficient spectrum management and enhancing situational awareness.

With the rapid proliferation of wireless devices and communication systems, the electromagnetic spectrum is becoming increasingly congested, leading to challenges in efficient spectrum utilization and interference management [5]. In such complex and dynamic electromagnetic environments, it becomes critical to precisely locate and track the radiation sources. Furthermore, passive techniques prove to be essential in situations where the active detection or engagement of the sources is not feasible or desirable. Also, they provide valuable information about the location and behavior of the radiation source, which enables effective monitoring, interference mitigation, and spectrum allocation.

Recently, researchers have proposed different methods for passive localization and tracking of radiation sources. These approaches can be broadly categorized as follows: (1) Time-of-arrival (TOA) techniques [6,7]—These methods estimate the source location based on the time difference between the arrival of the signal at multiple receivers. By calculating the delay time, the position of the transmitter can be deduced. (2) Angle-of-arrival (AOA) method [8]—The AOA method depends on the observation of the direction of arrival (DOA) of the signal at multiple receiving antennas. By triangulating the angle, the location of the transmitter can be determined. (3) Received-signal-strength (RSS, [6]) approaches—RSS-based methods utilize the strength of the received signal to estimate the distance between the source and the receiver. By combining distance measurements from multiple receivers, the location of the transmitter can be inferred. (4) Time-of-flight (TOF) measurements [9,10]—ToF ranging methods are bi-directional ranging techniques, which primarily use the time of flight of a signal traveling back and forth between two asynchronous transceivers (or reflected surfaces) to measure the distance between nodes. In line-of-sight (LOS) environments, ToF-based ranging methods can compensate for the shortcomings of RSSI-based methods.

However, the above methods have some limitations. Firstly, in a complex multipath signal environment or when there are obstacles between the target and the receiver (when the number of targets observed by the receiver is variable), the accuracy of localization and tracking may be compromised. Secondly, the non-ideal signal propagation results in an increased error, which affects the accuracy of localization and tracking. Finally, wireless signal propagation environments are usually non-stationary and impacted by factors such as weather, terrain, and other wireless devices. These variations may lead to a change in parameters such as propagation loss, multipath effect, etc., compromising the localization and tracking results. The random finite-set (RFS) method, as introduced by Mahler in [11], offers a solution to the multi-target tracking problem. Bayesian filtering based on RFS theory is adept at dynamically detecting the number of targets and their states, finding extensive applications in various domains [12,13,14,15,16,17,18].

Because of the multi-target Bayesian filter’s numerical complexity, some alternative approaches have been proposed, including the probability hypothesis density (PHD) [19,20,21], cardinality PHD (CPHD) [22,23], and multi-Bernoulli filter [24,25] methods. Nevertheless, these methods do not function as multi-target trackers since they lack the ability to track target trajectories and, consequently, cannot discern the identity information of the targets. Therefore, Vo et al. proposed the RFS-based generalized labeled multi-Bernoulli (GLMB) [26,27,28] and

δ

-GLMB [29] filters, where the

δ

-GLMB [29] filter yields stronger results compared to the GLMB filter and is more applicable to the multi-target tracking problem [30,31,32,33,34,35].

In this paper, we extend the

δ

-GLMB [29] filter to the passive localization and tracking scenarios. The major contributions of this paper are as follows:

For the complex electromagnetic environment, we model the “scenario with obstacles between the target and the receiver” as an RFS, in which both the state and the number of targets received by the receiver of the base station change during the observation time.
The non-stationary wireless signal propagation environment is usually affected by weather, terrain, other wireless devices, etc. Therefore, we model external factors such as weather and terrain (which may impact the information received by the receiver) as a clutter RFS and identify that each clutter generates a false alarm (a false measurement). Our proposed filter is capable of accurately tracking targets of interest from clutter interference and capturing their trajectory onset remarkably well.
We describe the extended Kalman filter (EKF) and unscented Kalman filter (UKF) implementations of the $δ$ -GLMB filter, which are able to accurately capture the target’s motion state. Moreover, we extend the PHD and CPHD filters to the scenarios of interest in this paper for comparison with the proposed method. Simulation tests verify the effectiveness of the $δ$ -GLMB filter for target number identification and state tracking.

This paper is structured as follows. In Section 2, we provide an overview of the relevant background, encompassing the dynamic physics model of a single target, measurement model, multi-target random finite-set (RFS) system model, Bayesian multi-target recursion, and

δ

-GLMB filter. Section 3 presents a concise overview of the nonlinear recursive implementation of the

δ

-GLMB filter for passive localization and tracking. Section 4 and Section 5 offer some numerical examples and conclusions, respectively. Additionally, Appendices Appendix A and Appendix B present some fundamental theories. We use upper-case (lower-case) bold characters to indicate matrices (vectors). Some symbols and their implications are given in Abbreviations.

4. Numerical Example

The EKF and UKF recursion implementations of the $δ$ -GLMB filter possess their respective advantages and disadvantages. It is worth noting that both the EKF and UKF approximations are computationally more efficient than the SMC approximation when dealing with nonlinearities. Furthermore, extracting state estimates remains straightforward due to the Gaussian mixture implementation at the core of these methods. In particular, the EKF recursion requires the computation of the Jacobi matrices, limiting its applicability to scenarios where both the state and measurement models are differentiable. In contrast, the UKF recursion avoids the requirement of differentiation altogether and can be applied to the nonlinear model. Therefore, in the subsequent simulation results, we only provide the UKF implementation.

In this section, we conduct two simulation scenarios: stationary and moving BSs. Unfortunately, there is a lack of existing multi-target RFS tracking methods tailored to passive localization and tracking contexts, making direct performance comparisons challenging. Then, we adapt the following established algorithms to the passive localization and tracking context:

We first extend the $δ$ -GLMB filter to the passive localization and tracking scenarios and provide closed-form solutions for both the prediction and update. Compared to the PHD and CPHD filters, the $δ$ -GLMB can provide a multi-target Bayesian filtering solution for target trajectory estimation (i.e., it can recognize the birth information of the tracked target).

4.1. A Stationary BS for Passive Localization and Tracking

The two-dimensional workspace has a size of

[- 2000, 1500]

\times [- 200, 1800]

m, as illustrated in Figure 3. The number of targets changes over time due to the occurrence of births and deaths. The specific time instances for the births and deaths of the true radiation sources are outlined in Table 1. The state kinematics equation for the radiation source is

$x_{k} = F (ϖ) x_{k - 1} + Γ_{k} v_{k - 1},$

(75)

and

$F (ϖ) = [\begin{matrix} 1 & \frac{sin (ϖ Δ T)}{ϖ} & 0 & - \frac{1 - cos (ϖ Δ T)}{ϖ} \\ 0 & cos (ϖ Δ T) & 1 & - sin (ϖ Δ T) \\ 0 & \frac{1 - cos (ϖ Δ T)}{ϖ} & 0 & \frac{sin (ϖ Δ T)}{ϖ} \\ 0 & sin (ϖ Δ T) & 0 & cos (ϖ Δ T) \end{matrix}],$

(76)

$Γ_{k} = [\begin{matrix} Δ T^{2} https://www.mdpi.com/ 2 \end{matrix}] 0 Δ T & 0 0 & Δ T^{2} https://www.mdpi.com/ 2 0 & T,$

(77)

where $Δ T = 1 s$ and $ϖ = 0.06$ rad/s. $v_{k - 1} \sim N (\cdot; 0, Q_{k - 1})$ , where $Q_{k - 1} = d i a g () [1, 1]$ . We assume that the spontaneous newborn model is the LMB RFS with the parameters $π_{B} = {\{r_{B}^{(i)}, p_{B}^{(i)}\}}_{i = 1}^{4}$ , where $r_{B}^{(1)} = r_{B}^{(2)} = 0.02$ , $r_{B}^{(3)} = r_{B}^{(4)} =, 0.03$ , and $p_{B}^{(i)} (x) = N (x; m_{B}^{(i)}, P_{B})$ , where $m_{B}^{(1)} = {[- 1500, 0, 250, 0]}^{T}$ , $m_{B}^{(2)} = {[- 250, 0, 1000, 0]}^{T}$ , $m_{B}^{(3)} = {[250, 0, 750, 0]}^{T}$ , $m_{B}^{(4)} = {[1000, 0, 1500, 0]}^{T}$ , and $P_{B} = d i a g {({[10, 5, 10, 5]}^{T})}^{2}$ . $P_{S, k} = 0.99$ , $P_{D, k} = 0.90$ , $J_{m a x} = 3000$ , and $K_{B} = 5$ .

If detected, each target generates a noisy azimuth, angular velocity, and Doppler frequency change-rate measurement $z = {[θ_{k}, {\dot{θ}}_{k}, {\dot{f}}_{k}]}^{T}$ . The other parameters are $δ_{θ}^{} = 2 (π / 180) rad$ , $δ_{\dot{θ}}^{} = 10^{- 2} rad / s$ , and $δ_{\dot{f}}^{} = 10^{- 3} Hz / s$ . The clutter measurements are expected to appear within the range of $[- π, π] rad \times [- π / 10, π / 10] rad / s \times [- 5, 0] Hz / s$ , and the number of clutter obeys a Poisson RFS with a mean value of 5 and an intensity of $κ_{k} (z) = λ_{c} U (Z)$ , where $λ_{c} = 5.07 \times 10^{- 2} {({rad}^{2} {Hz / s}^{2})}^{- 1}$ , and $U (Z)$ denotes the uniform distribution density over the observation region.

The output of the

δ

-GLMB-UKF filter for one MC run is presented in Figure 4. Notably, although our form of clutter arises from

[- π, π] rad \times [- π / 10, π / 10] rad / s \times [- 5, 0] Hz / s

, the corresponding

x - y

coordinates can be derived from Equations (8) and (9). The presence of several clutter interferences at each time moment can be seen in Figure 4, but the

δ

-GLMB-UKF filter can launch and terminate the tracks with a small delay. We also notice that the estimated trajectory of the third target shows an orbital switch at the 22nd moment, but the newborn labeled target is quickly captured.

We further validate our results by comparing the performance of the

δ

-GLMB, PHD, and CPHD filters via the UKF implementation. Figure 5 compares the OSPA, [39] distance of the three filters (

p = 1

c = 100

m) and its localization and cardinality components. As can be seen from Figure 5, the

δ

-GLMB-UKF filter generally outperforms the other two filters for this scenario, despite the overestimation of the target number at the 61st moment. The cardinality statistics of the three filters are shown in Figure 6 over 500 MC trials. It can be seen from Figure 6 that the

δ

-GLMB-UKF is superior to the PHD-UKF and CPHD-UKF methods in terms of target number detection.

4.2. A Moving BS for Passive Localization and Tracking

Unlike Section 4.1, in this section, the base station is in motion (as depicted in Figure 7), and the duration of this scenario is 60 s. The specific time points of the true radiation sources are provided in Table 2. The other experimental parameters are consistent with those in Section 4.1.

Figure 8 gives the results of the true and estimated trajectories from the

δ

-GLMB-UKF filter outputs in the x and y coordinates from one MC trial. Figure 8 shows that the proposed algorithm can still accurately capture the start of the target’s trajectory even in the presence of clutter interference. We also observe that, as expected, although there is no trajectory switching, there is a slight occurrence of lost or spurious trajectories, ensuring that the estimated track identity remains consistent throughout the scenario. Moreover, compared to the trajectory results in Figure 4, the output of the tracked trajectories in the case of the moving base station scenario is more accurate. This is due to the fact that when the base station is moving, we only need to extract the measurement information for time k, which is independent of the base station’s position at the previous

k - 1

moments, thus reducing the data correlation requirements.

Similarly, the cardinality statistics of the three filters over 500 MC trials are shown in Figure 9. Compared to Figure 6, their target number estimation performance is significantly improved. Although

δ

-GLMB-UKF and PHD-UKF still think that the targets have not disappeared at the 41st moment, the overestimation error is immediately corrected at the 42nd moment. Figure 10 compares the OSPA distances with the localization and base components of the three filters. Figure 10a shows that the

δ

-GLMB-UKF filter significantly outperforms the PHD-UKF and CPHD-UKF filters. As can be seen from the OSPA cardinality component, the error of the

δ

-GLMB-UKF method is generally smaller than that of the other two methods for target cardinality estimation (Figure 9 provides a more intuitive representation). The OSPA results for both the localization and cardinality components show that the

δ

-GLMB-UKF filter outperforms the PHD-UKF and CPHD-UKF filters.

5. Conclusions

This paper explores the application of the $δ$ -GLMB filter in the field of passive localization and tracking of radiation sources. By extending the $δ$ -GLMB filter to the scenario of passive localization and tracking of radiation sources, this paper presents EKF and UKF implementations based on this filter. It provides two simulation cases (stationary base station and moving base station) and tests the UKF implementation of the proposed $δ$ -GLMB, and the simulation results demonstrate the effectiveness of this approach. Moreover, our proposed $δ$ -GLMB-UKF filter accurately tracks the target and captures its trajectory, as demonstrated in simulation examples alongside other filters, thereby verifying the effectiveness of the $δ$ -GLMB filter in target identification and state tracking. Furthermore, the $δ$ -GLMB filter can achieve accurate localization and tracking in complex and dynamic wireless environments, which provides significant theoretical and technical support for enhancing the performance of passive localization and tracking systems for radiation sources and for further research in application areas. In the future, the parameters of the $δ$ -GLMB filter can be further optimized to achieve higher positioning and tracking accuracy. Also, we will consider adding new measurements to the existing trajectories and validating the proposed algorithm in engineering experimental tests.

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