Generalized Labeled Multi-Bernoulli Multi-Target Tracking with Doppler-Only Measurements

Abstract

The paper addresses the problem of tracking multiple targets with Doppler-only measurements in multi-sensor systems. It is well known that the observability of the target state measured using Doppler-only measurements is very poor, which makes it difficult to initialize the tracking target and produce the target trajectory in any tracking algorithm. Within the framework of random finite sets, we propose a novel constrained admissible region (CAR) based birth model that instantiates the birth distribution using Doppler-only measurements. By combining physics-based constraints in the unobservable subspace of the state space, the CAR based birth model can effectively reduce the ambiguity of the initial state. The CAR based birth model combines physics-based constraints in the unobservable subspace of the state space to reduce the ambiguity of the initial state. We implement the CAR based birth model with the generalized labeled multi-Bernoulli tracking filter to demonstrate the effectiveness of our proposed algorithm in Doppler-only tracking. The performance of the proposed approach is tested in two simulation scenarios in terms of the optimal subpattern assignment (OSPA) error, OSPA${}^{\left(2\right)}$ error, and computing efficiency. The simulation results demonstrate the superiority of the proposed approach. Compared to the approach taken by the state-of-the-art methods, the proposed approach can at most reduce the OSPA error by 58.77%, reduce the OSPA${}^{\left(2\right)}$ error by 43.51%, and increase the computing efficiency by 9.56 times in the first scenario. In the second scenario, the OSPA error is reduced by 62.80%, the OSPA${}^{\left(2\right)}$ error is reduced by 43.65%, and the computing efficiency is increased by 2.61 times at most.

1. Introduction

Tracking of multiple moving targets with Doppler measurements in multi-sensor systems has attracted interest from researchers during the past few decades [1,2,3,4,5,6,7,8,9]. Compared with other forms of measurements, Doppler is attractive for a number of reasons. For instance, Doppler sensors are much cheaper, since no hardware array is required. Another reason is that Doppler measurements are relatively accurate. In addition, the amount of information exchanged, or the bandwidth requirement between each sensor and the fusion center and/or between sensors, is significantly lower than that of other measurement types [5]. However, the Doppler measurement cannot provide complete information about the target state. This issue is often referred to as single-sensor unobservability [6]. It is a challenging problem to accurately estimate the number of targets and extract their trajectories or states from Doppler-only measurements under clutter and missed detections.

In general, multiple sensors can be used to alleviate the unobservability problem via multi-sensor data fusion, as well as achieving multi-sensor diversity gain. Chan and Jardine [2] have shown that a unique target state can be obtained using at least four Doppler measurements from different sensors. In [10], Shames et al. algebraically derived the minimum number of Doppler measurements to provide finite solutions of the target state. A dual-stage approach based on observability decomposition for centralized tracking using Doppler-only measurements was proposed in [11]. These papers, however, only study the single target case, while the multi-target case is far more complex. One major issue that comes with the multi-target case is the ghost problem, which is a consequence of the unknown data association. While multiple targets are detected by each sensor, multiple associations of the corresponding measurements from all sensors often result in fictitious or ghost states [12].

Multi-sensor multi-target tracking (MTT) is a difficult problem because of measurement origin uncertainty, the presence of false alarms, missed detections, measurement errors, and birth and death of targets [13]. As the classical MTT approaches, the joint probabilistic data association (JPDA) [14] and multiple hypothesis tracking (MHT) [15,16] algorithms were first proposed in the 1970s. The JPDA and MHT produce tracks using data association and each track has a label or identity. Multi-scan, s-dimensional (sD) assignment, or multi-frame based assignment approaches for JPDA [17] and MHT [18,19] were proposed for improved data association accuracy. Other variants of the algorithms also exist. For example, the exact nearest neighbour JPDA filter [20] was proposed to circumvent the coalescence problem of the JPDA filter, where all data association hypotheses are drastically pruned except the one with the most significant or highest probability. Another variant of JPDA named JPDA* [21] was developed based on the selective pruning, and the best data association hypothesis is chosen to calculate the measurement-to-target probabilities. There are also some methods [22,23,24] that attempt to improve the accuracy of the marginalization by optimizing the posterior density. In recent years, the random finite set (RFS) [25,26] based MTT algorithms have attracted a lot of attention [27]. For the RFS based algorithms, the posterior multi-target density encapsulates multi-target state information based on evidence from the observed data and prior information on the motion of targets [25]. As more data are processed, the posterior density of the set of real tracks will be higher than that for ghost tracks. Earlier RFS based algorithms, including probability hypothesis density (PHD) [28], cardinalized PHD [29], and multi-Bernoulli [30] filters, were not developed to output the trajectory of each target. These algorithms are approximations of the multi-target Bayesian filter but are not real trackers in principle. The first RFS-based tracker was formulated using the labeled RFS, which led to the generalized labeled multi-Bernoulli (GLMB) filter [31,32]. Its multi-scan version [33,34], as well as approximations such as the labeled multi-Bernoulli filter [35], also produce tracks. A unique feature of the GLMB approach is that the approximate error bounds can be computed analytically [32]. Moreover, the GLMB filter can achieve linear complexity in the number of measurements using Gibbs sampling [36], enabling it to handle large-scale MTT problems [37].

A number of RFS-based solutions have been proposed for MTT with Doppler-only measurements. In [5], the PHD filter was applied to Doppler-only measurements in a passive multi-static setting. Tracking of multiple cooperative targets with the multi-Bernoulli filter using Doppler-only measurements was studied in [38]. These methods use the PHD and multi-Bernoulli filters which cannot produce the target trajectories. By assuming that targets are indistinguishable, the PHD and multi-Bernoulli filters model the set of target states as the RFS and estimate the multi-target state at each time step. However, the multi-target state history does not necessarily represent the multi-target trajectory. For more details of these filters, please refer to [25,26,28,30,39,40,41]. To address the trajectory, Papi considered an application of the GLMB filter in [42] for tracking multiple targets using Doppler measurements. In [43], we studied the sensor management problem in Doppler multi-sensor systems for MTT applications using the labeled multi-Bernoulli filter. Do and Nguyen [44] used the GLMB filter for tracking multiple marine vessels in Doppler multi-static systems, and the detection probability is kept as unknown. In [45], an efficient multi-sensor GLMB filter was developed and demonstrated on various sensor types including Doppler. These RFS-based methods all require prior knowledge of the target birth distribution (or initial-state distribution) and therefore can be restrictive in practical applications.

To solve the problem of unknown target birth distribution, some researchers developed measurement-driven birth models [46,47]. The measurement-driven birth model uses the received measurements to generate target births, and hence the prior knowledge of birth distribution is not necessary. However, the notorious poor observability of Doppler measurements makes this approach difficult to apply. In general, samples are used to approximate the birth distribution, and a large number of samples is required to cover the entire surveillance area. Within the RFS filtering framework, the accept-reject (A-R) method is widely used in the state-of-the-art methods to generate birth distributions [48,49,50]. In the A-R method, samples are used to approximate the birth distribution. A large number of samples are drawn from a diffuse prior covering the entire state space, and only those samples whose Doppler shifts are similar to the generated Doppler measurement are accepted. Since a large number of samples are first drawn and then rejected, this method is apparently inefficient. In addition, there could be a large difference between the estimated birth distribution and the true one. In some cases, the under-determined solution produces a birth distribution with sufficient ambiguity and makes the subsequent MTT process difficult.

In this paper, the problem of tracking multiple distinguishable and non-cooperative targets using Doppler measurements is studied. While this is an important practical problem, to the best of our knowledge, only a few publications exist on this topic. For real-world applications, we need a realistic measurement model that accounts for the measurement noise, clutter, and missed detections. Moreover, we need a tracker that can fuse Doppler-only measurements from multiple radars to alleviate the poor observability. The tracker must operate without the prior knowledge of the birth distribution, since it is not available in practice. To address these challenges, we propose a constrained admissible region (CAR) based adaptive birth model in the GLMB filter. Recent studies have presented various CAR based approaches (e.g., see [51,52]) for the generation of prior state distributions. However, existing approaches are all designed for the space-object tracking problem in astrodynamics. For the first time, we combine a generated Doppler measurement with a set of physics-based constraints to instantiate the birth distribution. This is achieved by first analytically determining the set of samples representing possible initial states for each single Doppler measurement based on the CAR, and then eliminating the infeasible samples using the multi-sensor GLMB update. We present the resulting GLMB tracking approach in detail in the sequential Monte Carlo (SMC) implementation and demonstrate its capability.

The organization of the paper is as follows: Section 2 and Section 3 present a description of the tracking model and a brief review of the GLMB recursion. In Section 4, the proposed GLMB tracking with the CAR based adaptive birth model is described in detail, outlining the general framework and the step-by-step implementation. Numerical simulations and results are presented in Section 5. Section 6 draws conclusions.

2. Target and Sensor Models

In the 2D surveillance area, the kinematic state of a moving target at time k is described by a vector [53,54]

$$x_k={\left[p_{k,x}{\dot{p}}_{k,x}{p}_{k,y}{\dot{p}}_{k,y}\right]}^{\mathrm{T}},$$

(1)

where the target position and velocity are given as $p_k={\left[p_{k,x}{p}_{k,y}\right]}^{\mathrm{T}}$ and ${\dot{p}}_k={\left[{\dot{p}}_{k,x}{\dot{p}}_{k,y}\right]}^{\mathrm{T}}$, respectively. The measurement time interval T is assumed constant and targets move with the nearly constant velocity (NCV) motion. The direct discrete-time kinematic model [55] is used, and the dynamic equation is as follows [55]

$$x_k=F{x}_{k-1}+w_{k-1},$$

(2)

where

$$F=I_2\otimes \left[\begin{array}{cc}1& T\\ 0& 1\end{array}\right],$$

(3)

with ⊗ denoting the Kronecker product [56], and $w_{k-1}$ is the zero-mean white Gaussian process noise with covariance Q [55]

$$Q=I_2\otimes {\sigma }_w^2\left[\begin{array}{cc}\frac{T^4}{4}& \frac{T^3}{2}\\ \frac{T^3}{2}& T^2\end{array}\right],$$

(4)

where ${\sigma }_w$ is the standard deviation (SD) of the process noise. Note that, if the discretized continuous-time kinematic model is used, the covariance Q of the process noise has a different expression. For more details of the kinematic model, please refer to [55].

The targets are observed by a Doppler multi-sensor system with $N_s$ sensors. In the multi-sensor system, target $x_k$ is illuminated by the sinusoidal waveform from sensor j located at $s^{\left(j\right)}={[s_x^{\left(j\right)},s_y^{\left(j\right)}]}^{\mathrm{T}}$. If the signal is scattered by the target, sensor j will receive it and report a Doppler measurement as [57]

$$z_k^{\left(j\right)}=h_k^{\left(j\right)}\left(x_k\right)+{\epsilon }_k^{\left(j\right)},$$

(5)

where

$$h_k^{\left(j\right)}\left(x_k\right)=-2{\dot{p}}_k^{\mathrm{T}}\frac{p_k-s^{\left(j\right)}}{∥p_k-s^{\left(j\right)}∥}\frac{f_{c,j}}{c}$$

(6)

is the true Doppler shift, $f_{c,j}$ is the carrier frequency for sensor j, c is the speed of light, and ${\epsilon }_k^{\left(j\right)}$ is the Gaussian measurement noise with zero mean and SD ${\sigma }_{\epsilon }$.

The Doppler measurement (5) is detected by sensor j with the detection probability $p_D^{\left(j\right)}\left(x_k\right)\le 1$. The false detections in the surveillance area are assumed independent of the target states [31]. Specifically, for sensor j, the false detections are modeled as the Poisson RFS, and the intensity function is given as ${\kappa }^{\left(j\right)}$ [31]. Hence, the number of false detections received at sensor j is Poisson distributed, with mean ${\lambda }^{\left(j\right)}$. The measurement set collected by sensor j at time k is denoted as $Z_k^{\left(j\right)}$. We assume that multiple sensors can simultaneously observe the targets at time k. This set of sensors is called the set of activated sensors, which is denoted by $A_k\subseteq \{1,\dots ,P\}$. The set of measurements from all activated sensors is denoted as $Z_k=\{Z_k^{\left(j\right)},\dots ,Z_k^{\left(P\right)}\}$.

3. Generalized Labeled Multi-Bernoulli Recursion

Since the number of targets is not known, the set of targets, called the multi-target state, can be modeled as an RFS. By introducing the label issue, the labeled RFS has been proposed to incorporate target identity and perform tracking [31,32]. In this paper, we use a labeled RFS named GLMB, which is conjugate with respect to the multi-target likelihood function and is closed under the multi-target Chapman–Kolmogorov equation with respect to the multi-target transition kernel [31,32]. These properties make it possible to provide analytic solutions to the multi-target filtering problems.

Each target is augmented with a label $\ell \in \mathbb{L}$, where $\mathbb{L}$ denotes a discrete countable space. For clarity, lowercase letters are used to represent single-target states, e.g., x, $\mathit{x}$, while uppercase letters are used to represent multi-target states, e.g., X, $\mathit{X}$. To distinguish the labeled states from unlabeled ones, symbols for the former are bolded, e.g., $\mathit{x}$, $\mathit{X}$, etc. The multi-target state $\mathit{X}$ is a labeled RFS given as

$$\mathit{X}=\{{\mathit{x}}_1,\dots ,{\mathit{x}}_N\}=\{(x_1,{\ell }_1),\dots ,(x_N,{\ell }_N)\}\in \mathcal{F}(\mathbb{X}\times \mathbb{L}),$$

(7)

where $\mathbb{X}$ and N denote the state space and the number of single-target states, respectively.

As a labeled RFS on $\mathbb{X}\in \mathbb{L}$, the GLMB [31,32] is important that it is closed under the Chapman–Kolmogorov equation and is also a conjugate prior [31]. The GLMB distributed according to [31,32]

$$\mathit{\pi }\left(\mathit{X}\right)=△\left(\mathit{X}\right)\sum _{c\in \mathbb{C}}{\omega }^{\left(c\right)}\left(\mathcal{L}\left(\mathit{X}\right)\right){[p^{\left(c\right)}]}^{\mathit{X}},$$

(8)

where $▵\left(\mathit{X}\right){\delta }_{\left|\mathit{X}\right|}\left(\right|\mathcal{L}\left(\mathit{X}\right)\left|\right)$ denote the distinct label indicator, $\mathbb{C}$ is a discrete index set, $p^{\left(c\right)}$ is a probability density that $\int p^{\left(c\right)}(x,\ell )dx=1$, each ${\omega }^{\left(c\right)}$ satisfies ${\sum }_{L\in \mathbb{L}}{\sum }_{c\in \mathbb{C}}{\omega }^{\left(c\right)}\left(L\right)=1$, and $\mathcal{L}:\mathbb{X}\times \mathbb{L}\to \mathbb{L}$ denotes the projection $\mathcal{L}\left(\right(x,\ell \left)\right)=\ell$. For numerical implementation of multi-target trackers, it is more convenient to use the so-called $\delta$-GLMB form [32], which is a special case of GLMB with

$$\mathbb{C}=F\left(\mathbb{L}\right)\times \mathsf{\Xi },$$

(9)

$${\omega }^{\left(c\right)}={\omega }^{(I,\xi )}\left(L\right)={\omega }^{(I,\xi )}{\delta }_I\left(L\right),$$

(10)

$$p^{\left(c\right)}=p^{(I,\xi )}=p^{\left(\xi \right)},$$

(11)

where $\mathsf{\Xi }$ is a discrete space. Therefore, the distribution of the $\delta$-GLMB is as follows:

$$\mathit{\pi }\left(\mathit{X}\right)=△\left(\mathit{X}\right)\sum _{(I,\xi )\in \mathcal{F}\left(\mathbb{L}\right)\times \mathsf{\Xi }}{\omega }^{(I,\xi )}\left(\mathcal{L}\left(\mathit{X}\right)\right){[p^{\left(\xi \right)}]}^{\mathit{X}}.$$

(12)

If the multi-target prior is a $\delta$-GLMB, its prediction ${\mathit{\pi }}_{+}\left({\mathit{X}}_{+}\right)$ is also a $\delta$-GLMB with the form

$${\mathit{\pi }}_{+}\left({\mathit{X}}_{+}\right)=△\left({\mathit{X}}_{+}\right)\sum _{(I_{+},\xi )\in \mathcal{F}\left({\mathbb{L}}_{+}\right)\times \mathsf{\Xi }}{\omega }_{+}^{(I,\xi )}\times {\delta }_{I_{+}}\left(\mathcal{L}\left({\mathit{X}}_{+}\right)\right){[p_{+}^{\left(\xi \right)}]}^{{\mathit{X}}_{+}},$$

(13)

where

$${\omega }_{+}^{(I_{+},\xi )}={\omega }_B\left(I_{+}\cap \mathbb{B}\right){\omega }_S^{\left(\xi \right)}\left(I_{+}\cap \mathbb{L}\right),$$

(14)

$$p_{+}^{\left(\xi \right)}\left(x,\ell \right)=1_{\mathbb{L}}\left(\ell \right)p_S^{\left(\xi \right)}\left(x,\ell \right)+\left(1-1_{\mathbb{L}}\left(\ell \right)\right)p_B\left(x,\ell \right),$$

(15)

$$p_S^{\left(\xi \right)}\left(x,\ell \right)=\frac{〈p_S\left(·,\ell \right)f\left(x|·,\ell \right),p^{\left(\xi \right)}\left(·,\ell \right)〉}{\eta {}_S^{\left(\xi \right)}\left(\ell \right)},$$

(16)

$$\eta {}_S^{\left(\xi \right)}\left(\ell \right)=\int 〈p_S\left(·,\ell \right)f\left(x|·,\ell \right),p^{\left(\xi \right)}\left(·,\ell \right)〉dx,$$

(17)

$${\omega }_S^{\left(\xi \right)}\left(L\right)={\left[\eta {}_S^{\left(\xi \right)}\right]}^L\sum _{I\subseteq \mathbb{L}}{1}_I\left(L\right){\left[q_S^{\left(\xi \right)}\right]}^{I-L}{\omega }^{\left(I,\xi \right)},$$

(18)

$$q_S^{\left(\xi \right)}(\ell )=〈q_S\left(·,\ell \right),p^{\left(\xi \right)}\left(·,\ell \right)〉.$$

(19)

In the above equations, $p_S\left(·,\ell \right)$ is the survival probability, $q_S\left(x,\ell \right)=1-p_S\left(x,\ell \right)$ denotes the probability of death, $p_S^{\left(\xi \right)}\left(·,\ell \right)$ is the density of a surviving target, $p_B\left(·,\ell \right)$ is the density for the new birth target, $f\left(·|·,\ell \right)$ is the transition density, ${\omega }_B$ is the weight of birth labels, ${\omega }_B^{\left(\xi \right)}$ is the weight of surviving labels, $〈f,g〉=\int f\left(x\right)g\left(x\right)dx$ is the standard inner product of functions f and g, and $1_S\left(X\right)$ is a inclusion function, which is equal to 1, if $X\subseteq S$ and 0, otherwise. The multi-target posterior is also a $\delta$-GLMB with the form

$$\mathit{\pi }\left(\mathit{X}\right|Z)=▵\left(\mathit{X}\right)\sum _{(I_{+},\xi )\in \mathcal{F}\left({\mathbb{L}}_{+}\right)\times \mathsf{\Xi }}\sum _{\theta \in \mathsf{\Theta }}{\omega }_{+}^{(I,\xi )}\times {\delta }_I\left(\mathcal{L}\left(\mathit{X}\right)\right){\left[p^{(\xi ,\theta )}(·|Z)\right]}^{\mathit{X}},$$

(20)

where $\mathsf{\Theta }$ is the space of mappings $\theta :\mathbb{L}\to \{0,1,\dots ,|Z\left|\right\}$ such that $\theta \left(i\right)=\theta \left(i^{{}^{\prime }}\right)$ implies $i=i^{{}^{\prime }}$, and

$${\omega }^{(I,\xi ,\theta )}\left(Z\right)\propto {\omega }^{(I,\xi )}{[{\eta }_Z^{(\xi ,\theta )}]}^I,$$

(21)

$$p^{(\xi ,\theta )}(x,\ell |Z)=\frac{p^{\left(\xi \right)}(x,\ell ){\psi }_Z(x,\ell ;\theta )}{{\eta }_Z^{(\xi ,\theta )}\left(\ell \right)},$$

(22)

$$\eta {}_Z^{(\xi ,\theta )}\left(\ell \right)=〈p^{\left(\xi \right)}(·,\ell ),{\psi }_Z(·,\ell ;\theta )〉,$$

(23)

$${\psi }_Z(x,\ell ;\theta )={\delta }_0\left(\theta (\ell )\right)q_D(x,\ell )+(1-{\delta }_0\left(\theta (\ell )\right))\frac{p_D(x,\ell )g\left(z_{\theta (\ell )}\right|x,\ell )}{\kappa \left(z_{\theta (\ell )}\right)},$$

(24)

where $p_D(·,\ell )$ is the detection probability, $q_D(·,\ell )=1-p_D(·,\ell )$ denotes the probability of miss detection, $\kappa (·)$ is the clutter intensity, $g(·|·,\ell )$ is the likelihood, and ${\delta }_S\left(X\right)$ is a Kronecker delta function, which is equal to 1, if $X=S$ and 0, otherwise.

The aforementioned recursion considers the propagation of GLMB for a single sensor. For the multi-sensor case, a feasible fusion approach is the iterated-corrector strategy [28] which approximates the multi-sensor update through iterated single sensor updates. Compared with the single sensor case, the multi-sensor case provides more observations about the tracked target, resulting in better tracking performance. Although the iterated-corrector strategy has no rigorous mathematical derivation, it is still an attractive multi-sensor fusion approach because of its low computational complexity and simple implementation.

4. Multi-Sensor GLMB Filter with CAR Based Birth Model

4.1. Problem Formulation

The implementation of the standard GLMB filter is introduced in Section 3. Obviously, the filter relies on the prior knowledge of birth distributions which limits its practical applications. In Section 1, it is analyzed that measurement driven methods provide reasonable solutions for estimating birth distributions, since measurements contain observable information about targets. However, Doppler measurements are significantly less informative, which makes the problem much more challenging.

The Doppler measurement (5) is a nonlinear function of target position and velocity. We have a scalar measurement and four unknowns in the components of the target state. Therefore, many 4-dimensional target states actually exist $x={[p_x,{\dot{p}}_x,p_y,{\dot{p}}_y]}^{\mathrm{T}}$ that are compatible with a single Doppler measurement, even without considering the noise [12]. To visualize the high sensitivity of the Doppler measurement (5) regarding the target position, Figure 1 shows the contour lines for the Doppler measurement in the $(p_x,p_y)$ plane, for constant velocity ${\dot{p}}_x={\dot{p}}_y=1$ m/s of a target, and the sensor position is $s_x=s_y=0$ m with the transmitting frequency $f_c=900$ MHz. Figure 1 shows that a large number of different target positions can produce similar Doppler shift values.

The aforementioned problem makes target state initialization a critical issue. Successful Doppler-only MTT has either used the A-R approach [48,49,50] or assumed an accurate prior knowledge of the birth distributions [5,38,42,44]. To refine the state estimates of birth targets while tracking multiple targets simultaneously, we go a step further and propose a novel multi-sensor GLMB tracking solution. Such an approach relies on: (i) an efficient measurement classification; (ii) a smart birth distribution estimation technique; and (iii) an iterated-corrector scheme for multi-sensor information fusion.

4.2. Measurement Classification

This paper attempts to generate birth distributions adaptively using the received measurements. In order to reduce errors of the generated birth distributions caused by measurement uncertainties, the birth distribution cannot be generated by measurements originating from persistent tracks. Hence, it is necessary to divide the received measurements into two sets, i.e., the set of persistent measurements and its complementary set. This is similar to the approach proposed in [58], which developed adaptive birth models within SMC-PHD and SMC-CPHD filters. However, since there is more information available in the GLMB filter, the classification of measurements is more straightforward.

As analyzed in Section 3, the discrete space $\mathsf{\Xi }$ is used to estimate the multi-target posterior $\delta$-GLMB. $\mathsf{\Xi }$ is actually the space of association map histories, i.e., ${\mathsf{\Theta }}_{0:k}={\mathsf{\Theta }}_0\times \dots \times {\mathsf{\Theta }}_k$, where ${\mathsf{\Theta }}_t$ is the association map space at time t. At each time step, the association map records the relation between measurements and targets, i.e., undetected targets are assigned 0 and a target with label ℓ associated with a measurement $z_{\theta (\ell )}$ is assigned $\theta (\ell )$. Hence, using the association map, it is easily to obtain the measurement set corresponding to the persistent tracks. At time k, given that the entire measurement set is $Z_k$ and the persistent measurement set is $Z_{s,k}$, the complementary set ${\tilde{Z}}_{s,k}$ is obtained by eliminating $Z_{s,k}$ from Z as

$${\tilde{Z}}_{s,k}=Z_k/Z_{s,k}.$$

(25)

Note that a measurement associated with an existing target can not originate from a new birth target based on the assumption that a measurement is produced by at most one target. Therefore, the complementary set ${\tilde{Z}}_{s,k}$ at current time has not been associated with any of persistent tracks, and hence it may from new targets. The elements of ${\tilde{Z}}_{s,k}$ are used to estimate the birth distribution ${\mathit{\pi }}_B\left(X\right)$, which is then propagated to the next time step.

The birth distribution ${\mathit{\pi }}_B\left(X\right)$ is a $\delta$-GLMB RFS [31,32]:

$${\mathit{\pi }}_B\left(X\right)=▵\left(\mathit{X}\right)\sum _{L\in \mathcal{F}\left(\mathbb{B}\right)}{\omega }_B\left(L\right)\times {\delta }_L\left(L\left(\mathit{X}\right)\right){\left[p_B\right]}^{\mathit{X}},$$

(26)

where L is the birth label set with weight [31,32]

$${\omega }_B\left(L\right)=\prod _{\ell \in \mathbb{B}}(1-r_B^{(\ell )})\prod _{\ell \in L}\frac{1_{\mathbb{B}}(\ell )r_B^{(\ell )}}{1-r_B^{(\ell )}}.$$

(27)

The birth $\delta$-GLMB can be completely characterized by

$${\left\{r_B^{(\ell )}\left(z\right),p{}_B^{(\ell )}(·;z):\ell ={\ell }_B^{\left(j\right)}\left(z\right)\right\}}_{z\in {\tilde{Z}}_{s,k},j\in J},$$

(28)

where $r_B^{(\ell )}\left(z\right)$ is the existence probability, $p_B^{(\ell )}(·;z)$ is the probability density, ${\ell }_B^{\left(j\right)}\left(z\right)$ is the label assigned for the birth target initiated by z from sensor j, and J is the set of sensor numbers. To distinguish the birth distributions generated by different sensors, the label is augmented as ${\ell }_B^{\left(j\right)}=\left(k_B^{\left(j\right)},i{}_B^{\left(j\right)},j\right)$, where $k_B^{\left(j\right)}$ is the birth time and $i_B^{\left(j\right)}$ is the index that distinguishes targets born at the same time.

For a measurement $z\in {\tilde{Z}}_{s,k}$, the new-born likelihood is given by [47]

$$r_U\left(z\right)=1-\sum _{(I,\xi )\in \mathcal{F}\left(\mathbb{L}\right)\times \mathsf{\Xi }}\sum _{\theta \in \mathsf{\Theta }\left(I\right)}{1}_{z_{\theta }}\left(z\right){\omega }^{(I,\xi ,\theta )},$$

(29)

where the inclusion function ensures that the sum is performed only on updated hypotheses that assign the measurement to a target and the weight ${\omega }^{(I,\xi ,\theta )}$ is given in (21). The probability of existence for the next time for the birth distribution initiated by measurement $z\in {\tilde{Z}}_{s,k}$ depends on the new-born likelihood obtained at current time [47]:

$$r_B^{(\ell )}\left(z\right)=min\left(r_{B\max },\frac{r_U\left(z\right)}{{\sum }_{\zeta \in {\tilde{Z}}_{s,k}}{r}_U\left(\zeta \right)}·{\lambda }_B\right),$$

(30)

where $r_{Bmax}\in [0,1]$ is the maximum probability of existence for the new birth target, and ${\lambda }_B$ denotes the expected number of target births the next time.

4.3. CAR Based Birth Distribution Estimation

In the SMC formulation of the GLMB filter, the birth probability density $p_B^{(\ell )}(·;z)$ generated by the measurement $z\in {\tilde{Z}}_{s,k}$ is approximated as

$$p_B^{(\ell )}(·;z)=\sum _{i=1}^{M_b}\frac{1}{M_b}{\delta }_{x_z^{\left(i\right)}}\left(x\right),z\in {\tilde{Z}}_{s,k}$$

(31)

where $M_b$ is the number of samples for the birth target, and $x_z^{\left(i\right)}$ is the state of the ith sample. To obtain $x_z^{\left(i\right)}={[p_x,{\dot{p}}_x,p_y,{\dot{p}}_y]}^{\mathrm{T}}$ in (31), it is assumed that the measurement noise is irrelevant since Doppler measurements are typically accurate (small ${\sigma }_{\epsilon }$).

Referring to (6) and dropping the time index k and sensor index j, the unit vector along the radar line-of-sight (RLOS) is defined by

$$u:=\frac{p-s}{∥p-s∥}=\frac{p-s}{\rho },$$

(32)

where $\rho$ is the range of the target from the sensor. We define the relative position coordinates $(x_{\mathrm{r}},y_{\mathrm{r}})$

$$x_{\mathrm{r}}:=p_x-s_x,y_{\mathrm{r}}:=p_y-s_y.$$

(33)

Then, the range of the target from sensor is given by

$$\rho =\sqrt{{x_{\mathrm{r}}}^2+y_{\mathrm{r}}^2}.$$

(34)

The radial velocity is defined as the projection of the target velocity along the RLOS [59] (see Figure 2) and is given by

$$v_r:={\dot{p}}^{\mathrm{T}}u=vcos\alpha ,0\le \alpha \le 2\pi$$

(35)

where $v=\parallel \dot{p}\parallel =\sqrt{{\dot{p}}_x^2+{\dot{p}}_y^2}$ is the speed of the target, and $\alpha$ is the angle between the target velocity and the RLOS. Then, the Doppler shift is expressed as

$$z=-C{v}_r=-Cvcos\alpha ,$$

(36)

where $C\triangleq 2f_c/c$. The radial velocity can also be expressed as

$$v_r=\frac{{\dot{p}}_x{x}_{\mathrm{r}}+{\dot{p}}_y{y}_{\mathrm{r}}}{\rho }.$$

(37)

Next, we define physics-based constraints in the unobservable subspace of the state space to reduce the space of admissible solutions of $x_z^{\left(i\right)}$ in (31).

For a tracking scenario, the minimum and maximum values of the range $({\rho }_{min},{\rho }_{max})$ and maximum possible target speed $\left(v_{max}\right)$ are usually available from the prior information [59,60]. If we assume that the random variable (RV) $\rho$ is uniformly distributed in the interval $[{\rho }_{min},{\rho }_{max}]$, then the mean and SD are given by

$$\overline{\rho }=({\rho }_{min}+{\rho }_{max})/2$$

(38)

$${\sigma }_{\rho }=({\rho }_{max}-{\rho }_{min})/\sqrt{12}$$

(39)

The uniform distribution has a finite support. The mean and SD are commonly used in a prior Gaussian prior distribution, which has an infinite support. If we use these parameters in a Gaussian distribution, in some cases, the samples drawn from the Gaussian distribution may lie outside the interval and may be negative. To avoid such problems, we use the mean $\overline{\rho }$ and assume that the interval contains the RV with a high probability, e.g., 0.999. Then, ${\sigma }_{\rho ,0.999}$ is given by

$${\sigma }_{\rho ,0.999}=({\rho }_{max}-{\rho }_{min})/6.581.$$

(40)

Therefore, the distribution of the range $\rho$ is denoted as $\mathcal{N}(\rho ;\overline{\rho },{\sigma }_{\rho ,0.999}^2)$.

The maximum value of $\cos \alpha$ in (31) is 1. Therefore, from (36), for a given measurement z, the minimum value of speed is

$$v_{min}=|z/C|.$$

(41)

Given the maximum possible target speed $\left(v_{max}\right)$ and the minimum value of speed, a similar argument applies to the RV speed, and the distribution of the target speed v is denoted as $\mathcal{N}(v;\overline{v},{\sigma }_{v,0.999}^2)$.

Let $\theta$ (see Figure 2) be the target heading (azimuth angle of target velocity measured from th X-axis in the counter-clockwise direction). Since $\theta$ has a circular distribution, we assume that $\theta$ is uniformly distributed in $[0,2\pi )$. We draw a sample for speed from $\mathcal{N}(v;\overline{v},{\sigma }_{v,0.999}^2)$ and a sample for $\theta$. Then, the X and Y components of velocity are given by

$${\dot{p}}_x=vcos\theta ,{\dot{p}}_y=vsin\theta .$$

(42)

Once we sample the distance $\rho$ form $\mathcal{N}(\rho ;\overline{\rho },{\sigma }_{\rho ,0.999}^2)$ and sample the target velocity $\dot{p}={[{\dot{p}}_x,{\dot{p}}_y]}^{\mathrm{T}}$ using (42), a finite number of feasible solutions for the target position $p={[p_x,p_y]}^{\mathrm{T}}$ can be obtained using the following proposition.

Proposition 1.

Given the range ρ, the target velocity ${[{\dot{p}}_x,{\dot{p}}_y]}^{\mathrm{T}}$, and the target-originated Doppler measurement z at the sensor located at $s={[s_x,s_y]}^{\mathrm{T}}$, there are two solutions for the target position

$$\left\{\begin{array}{c}{p}_x={\displaystyle \frac{z\rho {\dot{p}}_x\pm I_{{\dot{p}}_x}{\dot{p}}_y\rho \sqrt{C^2{v}^2-z^2}}{C{v}^2}}+s_x\hfill \\ p_y={\displaystyle \frac{z\rho {\dot{p}}_y\pm \left|{\dot{p}}_x\right|\rho \sqrt{C^2{v}^2-z^2}}{C{v}^2}}+s_y\hfill \end{array}\right.$$

(43)

where $I_{{\dot{p}}_x}$ is an indicator of the sign of ${\dot{p}}_x$ (positive and negative). If ${\dot{p}}_x\ge 0$, $I_{{\dot{p}}_x}=1$ and if ${\dot{p}}_x<0$, $I_{{\dot{p}}_x}=-1$. The first and second pair of solutions for $p_x$ and $p_y$ use the positive and negative signs, respectively, in (43). The proof of (43) is given in the Appendix A.

For each measurement $z\in {\tilde{Z}}_{s,k}$, the above procedure will be repeated until $M_b$ samples representing new-born states have been obtained. Example 1 below illustrates the generation of the new-born target states.

Example 1.

We assume that the transmitting frequency of a sensor s located at ${[0,0]}^{\mathrm{T}}$ is $f_c=900\mathrm{MHz}$, the maximum detection range is ${\rho }_{max}=\mathrm{25,000}\mathrm{m}$, the minimum detection range is ${\rho }_{min}=500\mathrm{m}$, and the maximum speed of the target is $v_{max}=35\mathrm{m}/\mathrm{s}$. The SD of measurement noise is equal to ${\sigma }_{\epsilon }=1\mathrm{Hz}$. At the current time, a 4-dimensional target with state $x={[3000\mathrm{m},15\mathrm{m}/\mathrm{s},-3500\mathrm{m},10\mathrm{m}/\mathrm{s}]}^{\mathrm{T}}$ generates a Doppler measurement $z=13.0158\mathrm{Hz}$ at sensor s. To compare the performance of the CAR and A-R methods, both methods are used to generate $M_b=5000$ samples representing new-born states $x_z^{\left(i\right)}={[p_x,{\dot{p}}_x,p_y,{\dot{p}}_y]}^{\mathrm{T}}(i=1,2,\dots ,5000)$ based on the measurement z.

For the CAR method, the distribution of range $\rho$ is $\mathcal{N}(\rho ;\mathrm{14,000}\mathrm{m},{\left(3342.95\mathrm{m}\right)}^2)$ (obtained using (38) and (40)) and the distribution of speed v is $\mathcal{N}(v;18.58\mathrm{m}/\mathrm{s},{(4.99\mathrm{m}/\mathrm{s})}^2)$. For the A-R method, we draw samples from a multivariate Gaussian $\mathcal{N}(\overline{x};{\mu }_x,C_x),$ where ${\mu }_x={[0\mathrm{m},0\mathrm{m}/\mathrm{s},0\mathrm{m},0\mathrm{m}/\mathrm{s}]}^{\mathrm{T}}$ and $C_x=\mathrm{diag}[{\left(\mathrm{15,000}\mathrm{m}\right)}^2,{(50\mathrm{m}/\mathrm{s})}^2,{\left(\mathrm{15,000}\mathrm{m}\right)}^2,{(50\mathrm{m}/\mathrm{s})}^2{]}^{\mathrm{T}}$. Only those samples whose likelihood larger than a threshold $\eta =1\times {10}^{-10}$ are accepted in the A-R method. The samples generated by the A-R and CAR methods are illustrated in Figure 3 and Figure 4, respectively. Comparing Figure 3 with Figure 4, it is clear that the CAR method reduces the velocity space and position space significantly.

The Doppler measurements of the samples are shown in Figure 5. It can be observed from Figure 5b that the measurement range of samples is about $[6.51,19.52]\mathrm{Hz}$ for the A-R method. Meanwhile, for the CAR method, the measurements of all samples are equal to $z=13.02\mathrm{Hz}$. The computing time for the A-R and CAR methods are 1.31 s and 0.05 s, respectively. The computing time of the CAR method is about 26.20 times faster than the A-R method. For the A-R method, a total number of 291,204 samples are generated but only 1.72% samples are accepted. Of course, decreasing the threshold $\eta$ can improve the acceptance rate of the A-R method and reduce the computing time, but will further increase the uncertainty of the samples.

4.4. Detailed Implementation Steps

The samples generated in Section 4.3 are not only from the birth targets but also include “ghost samples”. Those “ghost samples” may occupy a large part of total samples because of the high nonlinearity of the measurement model. To reduce the false samples in one iteration, we use the iterated-corrector strategy for fusion information from multi-sensor. The iterated-corrector update may also keep “ghost samples” in one iteration, while these samples will be eliminated at the following time steps. The flow chart of the proposed CAR-GLMB approach is shown in Figure 6.

Algorithm 1 shows the pseudocode of a single run $k=2,3,\dots$ for the proposed approach. The following parameters are always assumed available to the MTT system:

• sensor model parameters: position $s^{\left(j\right)}={[s_x,s_y]}^T$, carrier frequency $f_{c,j}$, detection probability $p_D^{\left(j\right)}(·)$, and clutter intensity $\kappa (·)$ for sensor $j(j=1,\dots ,N_s)$;
• birth model parameters: the maximum speed $v_{\max }$, the minimum ${\rho }_{min}$ and maximum ${\rho }_{max}$ detection ranges of the sensor, and the number $M_b$ of birth samples;
• likelihood function $g\left(z\right|x,\ell )$ and transition density $f\left(x\right|·,\ell )$;
• survival probability function $p_S(x,\ell )$.

Algorithm 1 Step-by-step pseudocode for a single run of the CAR-GLMB method.

INPUT: → The posterior $\mathit{\pi }\left(\mathit{X}\right|Z)$ and birth distribution ${\mathit{\pi }}_B$ from previous time step

OUTPUT: → The posterior $\mathit{\pi }\left(\mathit{X}\right|Z)$ and birth distribution ${\mathit{\pi }}_B$ at the current time

1:  Predict persistent and birth distributions to obtain the prior distribution ${\mathit{\pi }}_{+}={\mathit{\pi }}_S\cup {\mathit{\pi }}_B$

2:  Select P sensors: $A_k=\mathrm{selec}-\mathrm{sensor}(P,{\mathit{\pi }}_{+})$

3:  Collect measurement set $Z_k^{\left(A_k\right)}$ from sensors $A_k$

4:  for sensor $i\in A_k$ do

5:        if $i=1$ then

6:             Predict ${\mathit{\pi }}_{+}^i={\mathit{\pi }}^{i-1}\left(\mathbf{X}\right|Z)$

7:       else

8:            Pseudo-predict ${\mathit{\pi }}_{+}^i={\mathit{\pi }}^{i-1}\left(\mathit{X}\right|Z)$

9:        end if

10:       Calculate parameters of the posterior to obtain ${\mathit{\pi }}^i\left(X\right|Z)$

11:  end for

12:  The fused posterior $\mathit{\pi }\left(X\right|Z)={\mathit{\pi }}^i\left(X\right|Z)$

13:  Collect the complementary measurement set ${\tilde{Z}}_{s,k}^{\left(A_{k,}\right)}$ from sensors $A_k$

14:  for $z\in {\tilde{Z}}_{s,k}^{\left(A_{k,}\right)}$do

15:        $n\leftarrow 0$

16:        while $n<M_b$ do

17:             Generate a birth sample using (38)–(43).

18:             $n\leftarrow n+1$.

19:        end while

20:        Collect the birth distribution for z as $\{r_B^{(\ell )}\left(z\right),p_B^{(\ell )}(·,·;z):\ell ={\ell }_B\left(z\right)\}$

21:  end for

22:  The birth distribution is obtained as ${\mathit{\pi }}_B=▵\left({\mathit{X}}_{+}\right)$${\sum }_{L\in \mathcal{F}\left(\mathbb{B}\right)}{\omega }_B\left(L\right)\times {\delta }_L\left(\mathcal{L}\left({\mathit{X}}_{+}\right)\right){\left[p_B\right]}^{{\mathit{X}}_{+}}$

Lines 1 of Algorithm 1 implement (13)–(19). Since the number of Doppler sensors is generally large, a subset of sensors can be selected in line 2, to ensure that the communication and real-time constraints are met. The sensor management problem is beyond the scope of this work; please refer to [61,62,63] for more information. The selected sensors in line 3 supply their measurement set $Z_k^{\left(A_k\right)}$ and lines 4–12 implement the iterated-corrector scheme. Finally, the generation of birth distribution is implemented in lines 13–22.

5. Numerical Simulations and Results

Numerical results are presented for 2D tracking scenarios where targets with the time-varying and unknown number are observed by Doppler-only sensors, as shown in Figure 7. The carrier frequency $f_c$ and measurement time interval T of each sensor are 900 MHz and 10 s. The probability of detection $P_D$ and measurement noise SD of each sensor are 0.95 and 1 Hz. The measurement space is $\mathcal{Z}=[-200,200]\mathrm{Hz}$. The false detections are uniformly distributed over the measurement space with the Poisson rate $\lambda =2$ per scan. The minimum and maximum detection ranges of a sensor are 500 m and 25,000 m, and the maximum speed of a target is 35 m/s. For the adaptive birth model, we choose $r_{Bmax}=0.05$ and ${\lambda }_B=0.3$ [47]. The survival probability is $p_S=0.99$ [31,32]. The A-R method (regarded as the state-of-the art method) is widely used to generate the birth distribution using Doppler-only measurements and is used as a comparative algorithm. Parameters used in the CAR and A-R methods are the same as the ones used in Example 1.

We use the OSPA [64] and OSPA${}^{\left(2\right)}$ [37] distances to measure the tracking error which are widely used in the field of RFS based MTT. The OSPA estimates’ errors in the cardinality and localization by measuring the distance between two sets of states. Let $d_p^{\left(c\right)}(\varphi ,\psi )$ be the OSPA between two sets $\varphi ,\psi \in \mathcal{F}\left(\mathbb{X}\right)$ with order p and cutoff c. For $\varphi =\{{\varphi }^{\left(1\right)},{\varphi }^{\left(2\right)},\dots ,{\varphi }^{\left(m\right)}\}$ and $\psi =\{{\psi }^{\left(1\right)},{\psi }^{\left(2\right)},\dots ,{\psi }^{\left(n\right)}\}$ with $m\le n$, the OSPA $d_p^{\left(c\right)}(\varphi ,\psi )$ is measured as follows [64]:

$$d_p^{\left(c\right)}(\varphi ,\psi )\triangleq {\left(\frac{1}{n}\left(\underset{\pi \in {\mathsf{\Pi }}_n}{min}\sum _{i=1}^m{\overline{d}}^{\left(c\right)}{\left({\varphi }^{\left(i\right)},{\psi }^{\left(\pi \right(i\left)\right)}\right)}^p+c^p\left(n-m\right)\right)\right)}^{\frac{1}{p}},$$

(44)

where ${\overline{d}}^{\left(c\right)}\left({\varphi }^{\left(i\right)},{\psi }^{\left(\pi \right(i\left)\right)}\right)=min\left(c,∥{\varphi }^{\left(i\right)}-{\psi }^{\left(\pi \right(i\left)\right)}∥\right)$ and $\pi$ is a permutation function in the set ${\mathsf{\Pi }}_n$ of all the permutations of $\psi$. If $m>n$, then $d_p^{\left(c\right)}(\varphi ,\psi )=d_p^{\left(c\right)}(\psi ,\varphi )$. However, the OSPA error does not consider the track labeling issue [37]. The OSPA${}^{\left(2\right)}$ [37] is an adaptation of the OSPA which contains the interpretation of a per-track per-time error and can accommodate sets of tracks. Let $X=\{x^{\left(1\right)},x^{\left(2\right)},\dots ,x^{\left(m\right)}\}$ and $Y=\{y^{\left(1\right)},y^{\left(2\right)},\dots ,y^{\left(n\right)}\}$ denote two sets of tracks, where $m\le n$. The OSPA${}^{\left(2\right)}$ ${\tilde{d}}_p^{\left(c\right)}(X,Y)$ between X and Y is defined as follows [37]:

$${\tilde{d}}_p^{\left(c\right)}(X,Y)\triangleq {\left(\frac{1}{n}\left(\underset{\pi \in {\mathsf{\Pi }}_n}{min}\sum _{i=1}^m{\tilde{d}}^{\left(c\right)}{\left(x^{\left(i\right)},y^{\left(\pi \right(i\left)\right)}\right)}^p+c^p\left(n-m\right)\right)\right)}^{\frac{1}{p}},$$

(45)

where ${\tilde{d}}^{\left(c\right)}\left(·,·\right)$ is the base-distance, which is the time average OSPA given by

$${\tilde{d}}_p^{\left(c\right)}(x,y)\triangleq \left\{\begin{array}{cc}\sum _{t\in {\mathcal{D}}_x\cup {\mathcal{D}}_y}\frac{d^{\left(c\right)}(\left\{x\left(t\right)\right\},\left\{y\left(t\right)\right\})}{|{\mathcal{D}}_x\cup {\mathcal{D}}_y|},\hfill & {\mathcal{D}}_x\cup {\mathcal{D}}_y\ne \oslash \hfill \\ 0,\hfill & {\mathcal{D}}_x\cup {\mathcal{D}}_y=\oslash \hfill \end{array}\right.$$

(46)

where ${\mathcal{D}}_x$ and ${\mathcal{D}}_y$ denote the time domain. If $m>n$, then ${\tilde{d}}_p^{\left(c\right)}(X,Y)={\tilde{d}}_p^{\left(c\right)}(Y,X)$. All experiments are tested in Matlab R2010a and implemented on a computer with a 3.40 GHz processor.

5.1. Scenario 1: Single Target Simulation

We first consider a scenario for tracking a single target, which exists for time indices $k=1,2,\dots ,30$ with the initial state vector $x={[3000\mathrm{m},15\mathrm{m}/\mathrm{s},-3500\mathrm{m},10\mathrm{m}/\mathrm{s}]}^{\mathrm{T}}$. The SD of the process noise is ${\sigma }_w=0.1\mathrm{m}/{\mathrm{s}}^2$. The purpose of this scenario is to clearly observe the change of the particles representing the posterior density over time, since particles are the focus of this paper. To verify the performance of the proposed CAR based tracking method, we compare it with the A-R based tracking method.

We use a simple sensor selection approach, in which sensors are randomly selected at each measurement time. For the GLMB recursion parameters, the number of components calculated and stored in each forward propagation is 100. When the number of sensors selected at each measurement time k is two, the positions of particles representing the posterior density at different time steps are illustrated in Figure 8. It can be observed from Figure 8 that the distribution of particles is relatively scattered at the initial measurement times. Hence, the initial estimated target position is less accurate. As time progresses, the distribution of particles becomes concentrated, and the estimated target position becomes closer to the true position.

Figure 9 plots the true and CAR based estimated x and y coordinates versus time of the true trajectory and estimates of the tacking method. These results are consistent with results in Figure 8. We observe that the tracker takes some time to detect the target. However, the target’s position estimates become more accurate over time.

Next, we analyze the average tracking performance of the CAR and A-R methods. To fully study the influence of the threshold in the A-R method, three specific different thresholds $\eta =0$, $\eta =1\times {10}^{-10}$, and $\eta =1\times {10}^{-3}$ are used. Note that using the threshold $\eta =0$ means that the samples are generated over the entire surveillance space and all the generated samples are accepted. To have the average performance, we have performed 60 Monte Carlo runs. When two sensors are selected at each measurement time, the averaged OSPA distance ($p=1$ and $c=\mathrm{10,000}$ m) and OSPA${}^{\left(2\right)}$ distance (with the same c, p, and window length $w=10$) are shown in Figure 10a,b, respectively. These figures show that the CAR method is more accurate than the A-R method with different thresholds in terms of the OSPA and OSPA${}^{\left(2\right)}$ distances. In addition, the OSPA and OSPA${}^{\left(2\right)}$ errors of the A-R method decrease as the threshold $\eta$ increases from 0 to $1\times {10}^{-3}$. A larger threshold indicates that the measurement of the sample is closer to the received measurement, i.e., the sample is more accurate. Therefore, the A-R method with a large threshold provides better tracking accuracy. However, when the threshold is large, the acceptance rate of samples is low, and the A-R method requires a long computation time to obtain a specific number of samples.

To compare the computation time and tracking accuracy of CAR and A-R methods with different number of selected sensors, the averaged computation time for executing a complete MC simulation, and the averaged OSPA and OSPA${}^{\left(2\right)}$ errors averaging over all measurement times, are shown in

Table 1. Average performance comparison in Scenario 1.

Method	Number of Sensors	Computing Time (s)	Relative Computing Time	OSPA Error (m)	OSPA${}^{\left(2\right)}$ Error (m)
A-R (threshold = 0)	1	5.34	1.02	5818.91	6883.98
	2	20.23	1.07	3832.67	5375.01
	3	36.54	1.07	3488.99	4758.76
A-R (threshold = $1\times {10}^{-10}$)	1	20.05	3.84	4991.90	6257.66
	2	70.84	3.74	3496.93	4934.66
	3	167.75	4.92	2360.85	3668.70
A-R (threshold = 1$1\times {10}^{-3}$)	1	38.27	7.33	4500.24	5893.79
	2	130.31	6.87	3072.37	4542.32
	3	325.87	9.56	2141.50	3417.82
CAR	1	5.22	1.00	4135.93	5581.78
	2	18.96	1.00	2145.42	3613.66
	3	34.08	1.00	1438.66	2688.03

. The relative computation time is determined relative to the corresponding CAR based computation time. The results demonstrate that, selecting more sensors at each measurement time, the values of the OSPA and OSPA${}^{\left(2\right)}$ distances decrease. As the number of sensors increases, the computation time increases accordingly. When the same number of sensors is selected at each measurement time, the CAR method always performs better than the A-R method in the OSPA and OSPA${}^{\left(2\right)}$ errors. When the number of sensors is 1, 2, and 3, our OSPA error drops by 8.10–28.92%, 30.17–44.02%, and 32.82–58.77% respectively, compared to the A-R method. Our OSPA${}^{\left(2\right)}$ error drops by 5.29–18.92%, 20.44–32.77%, and 21.35–43.51%, respectively. For each sensor, the birth distribution generated by our approach is more accurate than that of the A-R method. This advantage becomes more significant as more sensors are used, as seen by the reduction percentage in error. When the threshold $\eta =0$, the computation time of the CAR method is comparable with that of the A-R method. This means that the CAR method takes little time to generate samples. When the threshold $\eta =1\times {10}^{-10}$, the CAR method runs about 3.74–4.92 times faster than the A-R method. When the threshold $\eta =1\times {10}^{-3}$, the CAR method runs about 6.87–9.56 times faster than the A-R method. These results are consistent with the theoretical analysis, showing the superiority of the CAR method with regard to the state-of-the-art A-R method.

5.2. Scenario 2: Multi-Target Simulation

In this scenario, all targets in the surveillance area travel with the NCV motion but with different velocities. A challenging tracking scenario is considered where targets move close to each other and their initial states are ${[3000\mathrm{m},15\mathrm{m}/\mathrm{s},-3500\mathrm{m},10\mathrm{m}/\mathrm{s}]}^{\mathrm{T}}$, ${[3000\mathrm{m},25\mathrm{m}/\mathrm{s},-2000\mathrm{m},-5\mathrm{m}/\mathrm{s}]}^{\mathrm{T}}$, and [${4500\mathrm{m},30\mathrm{m}/\mathrm{s},-5500\mathrm{m},0\mathrm{m}/\mathrm{s}]}^{\mathrm{T}}$. The SD of the process noise is ${\sigma }_w=0.1\mathrm{m}/{\mathrm{s}}^2$.

For the GLMB recursion parameters, the number of components calculated and stored in each forward propagation is 500. Two sensors are selected at each measurement time. Figure 11 plots the x and y coordinates versus time for the true and CAR estimated trajectories. These plots show that the CAR method is able to detect and track targets successfully.

The averaged OSPA distance ($p=1$ and $c=\mathrm{10,000}$ m) and ${\mathrm{OSPA}}^{\left(2\right)}$ distance (with the same c, p, and window length $w=10$) are shown in Figure 12a,b, respectively. These figures show that the CAR method is more accurate than the A-R method with different thresholds in terms of the OSPA and ${\mathrm{OSPA}}^{\left(2\right)}$ distance.

To compare the computational efficiencies and tracking accuracies of the CAR and A-R methods as a function of number of selected sensors, the average computation time for executing a complete MC simulation, and the average OSPA and OSPA${}^{\left(2\right)}$ errors are shown in

Table 2. Average performance comparison in Scenario 2.

Method	Number of Sensors	Computing Time (s)	Relative Computing Time	OSPA Error (m)	OSPA${}^{\left(2\right)}$ Error (m)
A-R (threshold = 0)	1	81.85	1.06	5949.45	6697.29
	2	206.51	1.04	4729.47	6533.19
	3	301.63	1.02	4184.66	6116.06
A-R (threshold = $1\times {10}^{-10}$)	1	127.71	1.66	5824.85	6600.18
	2	311.41	1.57	4188.66	6078.71
	3	484.16	1.63	2975.35	4950.81
A-R (threshold = $1\times {10}^{-3}$)	1	185.02	2.41	5558.52	6204.03
	2	475.90	2.40	3456.05	5550.20
	3	773.32	2.61	2198.64	4167.35
CAR	1	76.92	1.00	4341.22	5331.12
	2	198.32	1.00	2634.28	4709.37
	3	296.51	1.00	1556.83	3446.18

. Consistent with the results in Scenario 1, the values of the OSPA and OSPA${}^{\left(2\right)}$ distances are lower when more sensors are selected at each measurement time. When the number of sensors is 1, 2, and 3, our OSPA error drops by 21.90–27.03%, 23.78–44.30%, and 29.19–62.80% respectively, compared to the A-R method. Our OSPA${}^{\left(2\right)}$ error drops by 14.07–20.40%, 15.15–27.92%, and 17.31–43.65%, respectively. There is an overall increase of the computation time with an increase in the number of sensors. The CAR method always outperforms the A-R method in terms of the OSPA and OSPA${}^{\left(2\right)}$ distances, when the same number of sensors is selected. When the threshold is $\eta =1\times {10}^{-10}$, the CAR method runs about 1.57–1.66 times faster than the A-R method. When the threshold is $\eta =1\times {10}^{-3}$, the CAR method runs about 2.40–2.61 times faster than the A-R method. Compared with Scenario 1, Scenario 2 uses more components in the GLMB recursion to accurately track multiple targets.

6. Conclusions

MTT using Doppler-only measurements is a challenging problem in the multi-target multi-sensor tracking system due to the poor observability of the target state. In this paper, we have developed a novel CAR based birth model that adaptively generates the birth distribution using the received Doppler-only measurements. The CAR based model uses physics-based constraints in the unobservable subspace of the state space to reduce ambiguity in the initial state. We have implemented the CAR based birth model in the multi-sensor GLMB filter to demonstrate its effectiveness. Our simulation results show that the novel birth model works significantly faster than the approach used by the state-of-the-art methods, with superior tracking accuracy. Future work will investigate other types of measurements, such as Bearing-only measurements, for tracking the distinguishable and non-cooperative targets.