# Passive Sonar Multiple-Target Tracking with Nonlinear Doppler and Bearing Measurements Using Multiple Sensors

Authors: Xiaohua Li; Bo Lu; Wasiq Ali; Jun Su; Haiyan Jin
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

The major advantage of the passive multiple-target tracking is that the sonars do not emit signals and thus they can remain covert, which will reduce the risk of being attacked. However, the nonlinearity of the passive Doppler and bearing measurements, the range unobservability problem, and the measurement to target data association uncertainty make the passive multiple-target tracking problem challenging. To deal with the target to measurement data association uncertainty problem from multiple sensors, this paper proposed a batch recursive extended Rauch-Tung-Striebel smoother- (RTSS-) based probabilistic multiple hypothesis tracker (PMHT) algorithm, which can effectively handle a large number of passive measurements including clutters. The recursive extended RTSS which consists of a forward filter and a backward smoothing is used to deal with the nonlinear Doppler and bearing measurements. The target range unobservability problem is avoided due to using multiple passive sensors. The simulation results show that the proposed algorithm works well in a passive multiple-target tracking system under dense clutter environment, and its computing cost is low.


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## Body

## 1. Introduction

Passive multiple-target tracking has gained more and more attention in the fields of military and civilian, such as navigation, monitoring and early warning, and salvage [1–3]. How to discover targets timely and to track targets accurately becomes one of the hot topics. The aim of the multiple-target tracking is to estimate the expected targets’ states, such as position, velocity, and acceleration, from the linear or nonlinear measurements [4–6]. The advantage of the passive target tracking is that the sonar can remain covert, which will reduce the risk of being attacked. The challenges for underwater passive multiple-target tracking include that the measurements are usually nonlinear, the target range may be unobservable, and the measurement to target data association is complex [5, 6].In general, the passive measurements include bearing, Doppler, and bearing rate. So the passive multiple-target tracking is a typically nonlinear tracking problem [7–15]. One method to handle the nonlinear measurements is the pseudo linearization estimation (PLE) algorithm [16–18]. The other approach is the recursive nonlinear Bayesian filter and smoother, such as the extended Kalman filter (EKF) [19–22], the unscented Kalman filter (UKF) [23, 24], the cubature Kalman filter (CKF) [11, 12], and the particle filter (PF) [16]. EKF locally linearizes the targets’ state and measurement equations using the first-order Taylor series expansion of the nonlinear transformations around the predicted target state. The merit of the EKF is that it is very easy to understand and apply to target tracking problem duo to its linear approximations to the nonlinear system or measurement function, and its computational cost is lower than that of other nonlinear filter, such as UKF and PF. The disadvantage of the EKF is that the tracking performance may be not good if the system model is seriously nonlinear or non-Gaussian.The UKF uses a deterministic sampling method to capture targets’ posterior distribution of mean and covariance based on the unscented transform. So the UKF has better tracking capability than the EKF to some extent, but its computational cost is larger than that of the EKF. The PF algorithm forms the Monte Carlo approximation to the solution of the Bayesian filter. It uses a set of particle samples to approximate the target distribution. The PF is used in the cases that the tracking system is highly nonlinear or non-Gaussian. The Bayesian filters only estimate the state of the targets from the history of measurements. On the contrary, the Bayesian smoother can use the forthcoming measurements to estimate the current state of targets. Corresponding to the Bayesian filter, the Bayesian smoother includes Rauch-Tung-Striebel smoother (RTSS), extended RTSS, unscented RTSS, and particle smoother [25].The other challenge for passive sonar multiple-target tracking is the complexity and particularity of the underwater environment, which is characterized by many of false alarms caused by reverberation and multipath effect [3, 4]. Furthermore, the low target detection will cause tracking uncertainty in target location. All of those will cause a data association uncertainty problem. To handle the data association uncertainty problem, some algorithms are proposed, such as the multiple hypothesis trackers (MHT) [26], the joint PDA filter (JPDA) [27], the probabilistic MHT (PMHT) [28, 29], the random finite set framework-based probability hypothesis density (PHD) [30], the cardinalized PHD (CPHD) [31–33], and multi-Bernoulli filter (MBF). The MHT makes all the data association hypotheses probability optimal. The JPDA makes multiple hypotheses into a single hypothesis and performs the Kalman update with composite measurements. The PMHT is based on the expectation maximization (EM), which optimizes the multiple-target states’ maximum a posteriori (MAP) estimation [34–36]. Different from the data association algorithms, the PHD, CPHD, and MBF are based on the random finite set theory, which makes all the measurements a measurement set and all the targets a target set.The information entropy theories are also used to estimate the target states. The fuzzyc-means clustering method based on maximum information entropy and the probabilistic data association filter (PDA) is proposed in [37], which uses a value optimized by the maximum information entropy to represent the measurement to target association probability. The multiple-target tracking problem is also solved by the maximum entropy intuitionistic fuzzy data association [38], cross entropy [39], maximum fuzzy entropy-based Gaussian clustering algorithm [40], entropy distribution and game theory based on the probability hypothesis density (PHD) method [41], maximum entropy fuzzy based on the fire-fly and PF [42], and the distributed cross entropy-based δ-generalized labelled multi-Bernoulli filter [43].As for the target range unobservability problem, the target range is observable from bearing and Doppler measurements if and only if the bearings are not constant [44, 45]. So in order to avoid the range unobservability problem, this paper uses multiple passive sensors to track targets. In addition, the target tracking accuracy of using multiple sensors is better than that of the single sensor generally.The most commonly used passive measurement is bearing. In this paper, in order to improve the multiple targets’ range observability and tracking performance, we introduce the nonlinear Doppler measurement and use multiple sensors. The extended RTSS method is used to deal with the nonlinear Doppler and bearing measurements. The batch recursive multiple-sensor PMHT algorithm is used to handle the measurement to target data association complexity problem.The remainder of this paper is as follows. The passive multiple-target tracking system model and measurement model are given in Section2. Section 3 develops the multiple-sensor PMHT algorithm which is suitable for multiple-target tracking under dense clutter environment. The simulation result is given in Section 4. At last, a summary is given in Section 5.


## 2. System Model and Measurement Model

We consider the passive multiple-target tracking problem in a two-dimensional space.


### 2.1. Tracking System Model

Assume there areM targets in the tracking space. The mthm=1,2,⋯,M target’s state is xmt=xmt,ẋmt,ymt,ẏmtT, where xmt and ymt are the location of target m in the x and y coordinate, respectively, and ẋmt and ẏmt are the velocity of target m. Assume the tracking time is from t=1 to t=N. The overview of the multiple-target tracking using Doppler and bearing measurements is shown in Figure 1.Figure 1
Multiple-target tracking scenario using Doppler and bearing measurement.Assume that all the targets move according to the following constant velocity (CV) model or constant acceleration (CA) model [46].
(1)xmt=Fxmt−1+umt,where umt is the system process noise which is assumed as Gaussian white noise. F is the system state transition matrix. For the CV and CA models, F is denoted as FCV and FCA, respectively, and
(2)FCV=1Δt000100001Δt0001,(3)FCA=1Δt0.5Δt200001Δt0000010000001Δt0.5Δt200001Δt000001,where Δt is the sampling interval.The process noise covariance matrices for the CV and CA models are denoted asQCV and QCA, respectively, which are given by
(4)QCV=δp2Δt44Δt2200Δt22Δt20000Δt44Δt2200Δt22Δt2,(5)QCA=δp2Δt520Δt48Δt36000Δt48Δt33Δt22000Δt36Δt22Δt000000Δt520Δt48Δt36000Δt48Δt33Δt22000Δt36Δt22Δt,where δp2 is the process noise intensity.


### 2.2. Measurement Model for Multiple Sensors

Assume there areS passive sensors in the target tracking space, and the sth sensor’s states is xst=xst,ẋst,yst,ẏstT, in which xst and yst are the location of the sensor s and ẋst and ẏst are the velocity of passive sensor s.The Doppler and bearing measurements are nonlinear with respect to targets’ states and sensors’ states, which are given by(6)zm,st=hm,st+wm,st,in which wm,st is the measurement noise with a covariance matrix Rm,st.The Doppler and bearing measurement functionhm,st is given by
(7)hm,st=1−ẋmt−ẋstsinθm,st+ẏmt−ẏstcosθm,stcf0arctanxmt−xstymt−yst,where c is the sound speed in water, f0 is targets’ radiation frequency, and θm,st is bearing from target m and sensor s.


## 2.1. Tracking System Model

Assume there areM targets in the tracking space. The mthm=1,2,⋯,M target’s state is xmt=xmt,ẋmt,ymt,ẏmtT, where xmt and ymt are the location of target m in the x and y coordinate, respectively, and ẋmt and ẏmt are the velocity of target m. Assume the tracking time is from t=1 to t=N. The overview of the multiple-target tracking using Doppler and bearing measurements is shown in Figure 1.Figure 1
Multiple-target tracking scenario using Doppler and bearing measurement.Assume that all the targets move according to the following constant velocity (CV) model or constant acceleration (CA) model [46].
(1)xmt=Fxmt−1+umt,where umt is the system process noise which is assumed as Gaussian white noise. F is the system state transition matrix. For the CV and CA models, F is denoted as FCV and FCA, respectively, and
(2)FCV=1Δt000100001Δt0001,(3)FCA=1Δt0.5Δt200001Δt0000010000001Δt0.5Δt200001Δt000001,where Δt is the sampling interval.The process noise covariance matrices for the CV and CA models are denoted asQCV and QCA, respectively, which are given by
(4)QCV=δp2Δt44Δt2200Δt22Δt20000Δt44Δt2200Δt22Δt2,(5)QCA=δp2Δt520Δt48Δt36000Δt48Δt33Δt22000Δt36Δt22Δt000000Δt520Δt48Δt36000Δt48Δt33Δt22000Δt36Δt22Δt,where δp2 is the process noise intensity.


## 2.2. Measurement Model for Multiple Sensors

Assume there areS passive sensors in the target tracking space, and the sth sensor’s states is xst=xst,ẋst,yst,ẏstT, in which xst and yst are the location of the sensor s and ẋst and ẏst are the velocity of passive sensor s.The Doppler and bearing measurements are nonlinear with respect to targets’ states and sensors’ states, which are given by(6)zm,st=hm,st+wm,st,in which wm,st is the measurement noise with a covariance matrix Rm,st.The Doppler and bearing measurement functionhm,st is given by
(7)hm,st=1−ẋmt−ẋstsinθm,st+ẏmt−ẏstcosθm,stcf0arctanxmt−xstymt−yst,where c is the sound speed in water, f0 is targets’ radiation frequency, and θm,st is bearing from target m and sensor s.


## 3. Multiple-Sensor PMHT

### 3.1. PMHT for Multiple Sensors

The following notations are used in this section.S is the total number of passive observers.M is the total number of targets.nt is the number of measurements.T is the total number of tracking time.Xt=xm,st is the set of target states at time t.Zt=zm,st is the set of measurements at time t.Kt=kr,st is the set of measurement-target assignments at time t.X=Xt is the set of the target state for time 1,2,⋯,T.Z=Zt is the set of measurements for time 1,2,⋯,T.K=Kt is the set of measurement-target assignments for time 1,2,⋯,T.The PMHT is a Bayesian framework-based batch recursive algorithm, which obtains the maximum a posteriori estimation of the target states based on the expectation maximization (EM) method [47].Letkr,st=m indicate that measurement r from sensor s at time t is associated with target m. Assume the measurement to target assignment is independent from each other.Define the prior probability of therth measurement from the sth sensor and mth target as
(8)pkr,st=m=πmt.The MAP estimate ofX is
(9)X^MAP=argmaxXElogpX∣Z.In order to calculate the MAP, define the following auxiliary function as(10)QXn+1;Xn=∫ΚlogpXn+1,Κ∣ZpK∣Xn,ZdK,where n is the number of EM iteration.The goal is to maximize the auxiliary function overXn+1 using an initialized target state X0. In each EM iteration step, the goal of the PMHT is to solve the following equation.
(11)Xn+1=argmaxXn+1QXn+1;Xn.We have(12)pX,Z=∏m=1Mpxm1∏t=2T∏m=1Mpxmt∣xmt−1∏s=1S∏t=1T∏r=1nt∑p=1PπpNzr,st;z^m,st,Rm,st,(13)pK,Z,X=∏m=1Mpxm1∏t=2T∏m=1Mpxmt∣xmt−1·∏s=1S∏t=1T∏r=1ntπmNzr,st;z^m,st,Rm,st.According to the conditional probabilistic theory,(14)pK∣Z,X=pK,Z,XpX,Z=∏s=1S∏t=1N∏r=1ntwm,rnt,s.From (12), (13), and (14), the posterior association probability is
(15)wm,rnt,s≜pkmt∣xmt,zr,st=πmNzr,st;z^m,st,Rm,st∑m=1MπmNzr,st;z^m,st,Rm,st,in which
(16)z^m,st=hxmt,xst.According to the conditional probabilistic theory,(17)pΚ,X∣Z=pΚ,X,ZpZ.Substituting equation (17) into equation (10), the defined auxiliary function is calculated as
(18)QXn+1;Xn=∫ΚlogpΚ,Xn+1,ZpK∣Xn,ZdK−logpZ.ThelogpZ has no effect with regard to maximizing the auxiliary function QXn+1;Xn in (18). So it can be removed from (18).So the auxiliary function is given by(19)QXn+1;Xn=∑s=1S∑t=1T∑r=1nt∑m=1Mlogπmwm,rnt,s⏞QΠ+log∏m=1Mpxmn+11∏t=2Tpxmn+1t∣xmn+1t−1+∑s=1S∑t=1T∑r=1nt∑m=1Mlogpzrt∣xmntwm,rnt,s⏟QX.In order to maximize the auxiliary function, we compute its derivative and set the derivative to zero.It turns out that theXn+1 is given by applying the extended RTSS method, in which the synthetic measurement z~m,st and R~m,st are calculated by
(20)z~m,st=∑r=1ntwm,rnt,szr,st∑r=1ntwm,rnt,s,(21)R~m,stRm,st∑r=1ntwm,rnt,s.The implementation of the batch recursive multiple-sensor extended RTSS-based PMHT is depicted in Table1.Table 1
Batch recursive multiple-sensor PMHT.
1. InitializationInitialize the target states x^m0∣0 and Pm0∣0.2. Set the EM iteration n=1. Calculate the target state prediction xmnt∣t−1 and state covariance Pmnt∣t−1.3. Calculate the posterior association probabilities wm,rnt,s in (15).4. Evaluate the synthetic Doppler and bearing measurements z~m,st and covariance R~m,st in (20) and (21).5. Evaluate the innovation covariance and filter gain for each target and passive sensor.6. Update the target state xmnt∣t and state prediction covariance Pmnt∣t for each target m according to the extended RTSS.7. Forward n=n+1. Repeat the EM algorithm of steps 3 to 6 until the iteration convergent.


### 3.2. Recursive Extended RTSS

The recursive extended RTSS is a linearized RTS smoother, which is based on analogous approximation to the EKF. It is a Gaussian approximation to the Bayesian smoother for the nonlinear target state and measurement model. The extended RTSS firstly performs the extended Kalman filter; then, a Kalman smoother is applied. That is, the extended RTSS consists of a forward filter and a backward smoother, which is summarized as follows:Forward filter:Fori=1,⋯,T,Time update(22)x^mt∣t−1=Fx^mt−1∣t−1,(23)Pmt∣t−1=FPmt−1∣t−1FT+Qmt.Measurement update(24)x^mt∣t=x^mt∣t−1+Gm,stzm,st−hm,st∣t−1,(25)Pmt∣t=I−Gm,stHm,st⋅Pmt∣t−1I−Gm,stHm,stT+Gm,stRm,stGm,sTt,where the filter gain is
(26)Gm,st=Pmt∣t−1Hm,sTt⋅Hm,stPmt∣t−1Hm,sTt+Rm,st−1,(27)Hm,st=∂hm,st∂xmtxmt=x^mt∣t−1.Backward smoothing:Fori=T‐1,⋯,1,
(28)x^mt∣T=x^mt∣t+Cmtx^mt+1∣T−x^mt+1∣t,(29)Pmt∣T=Pmt∣t+CmtPmt+1∣T−Pmt+1∣tCmTt,where
(30)Cmt=Pmt∣tFTPm−1t+1∣t.For the Doppler and bearing measurement, the measurement Jacobian matrix is given by(31)Hm,st=∂hm,st∂xmt∂hm,st∂ẋmt∂hm,st∂ymt∂hm,st∂ẏmt=ymt−ystr20−xmt−xstr200−f0xmt−xstcr0−f0ymt−ystcr,where r=xmt−xst2+ymt−yst2.


### 3.3. Data Fusion for Multiple Sensors

For the tracking problem using multiple sensors, we need to fuse the target and measurement information from multiple sensors. The most used data fusion methods are parallel fusion and sequential fusion [48].In this paper, we use the sequential data fusion for multiple sensors which updates the target state and covariance by each sensor sequentially. The multiple sensors deal with the measurements one after another with an intermediate target state and covariance at one time.Letx^m,st∣t and Pm,st∣t denote the target state estimation and corresponding covariance processed by the sensor s. The Doppler and bearing measurements from the first passive sensor are used to calculate the first intermediate target state x^m,1nt∣t and covariance Pm,1nt∣t for each of the targets using a single sensor filter. Then, the measurements from the next passive sensor are used to update the intermediate target state and covariance as follows:
(32)x^m,st∣t=x^m,s−1t∣t+Gm,stzm,st−hm,st∣t,(33)Pm,st∣t=I−Gm,stHm,st⋅Pm,s−1t∣tI−Gm,stHm,stT+Gm,stRm,stGm,sTt,where Gm,st is the filter gain, and
(34)x^m,0t∣t=x^mt∣t−1,Pm,0t∣t=Pmt∣t−1,(35)x^mt∣t=x^m,St∣t,Pmt∣t=Pm,St∣t.The sequential fusion implementation of multiple sensors is shown in Figure2.Figure 2
Sequential fusion implementation of multiple sensors.


## 3.1. PMHT for Multiple Sensors

The following notations are used in this section.S is the total number of passive observers.M is the total number of targets.nt is the number of measurements.T is the total number of tracking time.Xt=xm,st is the set of target states at time t.Zt=zm,st is the set of measurements at time t.Kt=kr,st is the set of measurement-target assignments at time t.X=Xt is the set of the target state for time 1,2,⋯,T.Z=Zt is the set of measurements for time 1,2,⋯,T.K=Kt is the set of measurement-target assignments for time 1,2,⋯,T.The PMHT is a Bayesian framework-based batch recursive algorithm, which obtains the maximum a posteriori estimation of the target states based on the expectation maximization (EM) method [47].Letkr,st=m indicate that measurement r from sensor s at time t is associated with target m. Assume the measurement to target assignment is independent from each other.Define the prior probability of therth measurement from the sth sensor and mth target as
(8)pkr,st=m=πmt.The MAP estimate ofX is
(9)X^MAP=argmaxXElogpX∣Z.In order to calculate the MAP, define the following auxiliary function as(10)QXn+1;Xn=∫ΚlogpXn+1,Κ∣ZpK∣Xn,ZdK,where n is the number of EM iteration.The goal is to maximize the auxiliary function overXn+1 using an initialized target state X0. In each EM iteration step, the goal of the PMHT is to solve the following equation.
(11)Xn+1=argmaxXn+1QXn+1;Xn.We have(12)pX,Z=∏m=1Mpxm1∏t=2T∏m=1Mpxmt∣xmt−1∏s=1S∏t=1T∏r=1nt∑p=1PπpNzr,st;z^m,st,Rm,st,(13)pK,Z,X=∏m=1Mpxm1∏t=2T∏m=1Mpxmt∣xmt−1·∏s=1S∏t=1T∏r=1ntπmNzr,st;z^m,st,Rm,st.According to the conditional probabilistic theory,(14)pK∣Z,X=pK,Z,XpX,Z=∏s=1S∏t=1N∏r=1ntwm,rnt,s.From (12), (13), and (14), the posterior association probability is
(15)wm,rnt,s≜pkmt∣xmt,zr,st=πmNzr,st;z^m,st,Rm,st∑m=1MπmNzr,st;z^m,st,Rm,st,in which
(16)z^m,st=hxmt,xst.According to the conditional probabilistic theory,(17)pΚ,X∣Z=pΚ,X,ZpZ.Substituting equation (17) into equation (10), the defined auxiliary function is calculated as
(18)QXn+1;Xn=∫ΚlogpΚ,Xn+1,ZpK∣Xn,ZdK−logpZ.ThelogpZ has no effect with regard to maximizing the auxiliary function QXn+1;Xn in (18). So it can be removed from (18).So the auxiliary function is given by(19)QXn+1;Xn=∑s=1S∑t=1T∑r=1nt∑m=1Mlogπmwm,rnt,s⏞QΠ+log∏m=1Mpxmn+11∏t=2Tpxmn+1t∣xmn+1t−1+∑s=1S∑t=1T∑r=1nt∑m=1Mlogpzrt∣xmntwm,rnt,s⏟QX.In order to maximize the auxiliary function, we compute its derivative and set the derivative to zero.It turns out that theXn+1 is given by applying the extended RTSS method, in which the synthetic measurement z~m,st and R~m,st are calculated by
(20)z~m,st=∑r=1ntwm,rnt,szr,st∑r=1ntwm,rnt,s,(21)R~m,stRm,st∑r=1ntwm,rnt,s.The implementation of the batch recursive multiple-sensor extended RTSS-based PMHT is depicted in Table1.Table 1
Batch recursive multiple-sensor PMHT.
1. InitializationInitialize the target states x^m0∣0 and Pm0∣0.2. Set the EM iteration n=1. Calculate the target state prediction xmnt∣t−1 and state covariance Pmnt∣t−1.3. Calculate the posterior association probabilities wm,rnt,s in (15).4. Evaluate the synthetic Doppler and bearing measurements z~m,st and covariance R~m,st in (20) and (21).5. Evaluate the innovation covariance and filter gain for each target and passive sensor.6. Update the target state xmnt∣t and state prediction covariance Pmnt∣t for each target m according to the extended RTSS.7. Forward n=n+1. Repeat the EM algorithm of steps 3 to 6 until the iteration convergent.


## 3.2. Recursive Extended RTSS

The recursive extended RTSS is a linearized RTS smoother, which is based on analogous approximation to the EKF. It is a Gaussian approximation to the Bayesian smoother for the nonlinear target state and measurement model. The extended RTSS firstly performs the extended Kalman filter; then, a Kalman smoother is applied. That is, the extended RTSS consists of a forward filter and a backward smoother, which is summarized as follows:Forward filter:Fori=1,⋯,T,Time update(22)x^mt∣t−1=Fx^mt−1∣t−1,(23)Pmt∣t−1=FPmt−1∣t−1FT+Qmt.Measurement update(24)x^mt∣t=x^mt∣t−1+Gm,stzm,st−hm,st∣t−1,(25)Pmt∣t=I−Gm,stHm,st⋅Pmt∣t−1I−Gm,stHm,stT+Gm,stRm,stGm,sTt,where the filter gain is
(26)Gm,st=Pmt∣t−1Hm,sTt⋅Hm,stPmt∣t−1Hm,sTt+Rm,st−1,(27)Hm,st=∂hm,st∂xmtxmt=x^mt∣t−1.Backward smoothing:Fori=T‐1,⋯,1,
(28)x^mt∣T=x^mt∣t+Cmtx^mt+1∣T−x^mt+1∣t,(29)Pmt∣T=Pmt∣t+CmtPmt+1∣T−Pmt+1∣tCmTt,where
(30)Cmt=Pmt∣tFTPm−1t+1∣t.For the Doppler and bearing measurement, the measurement Jacobian matrix is given by(31)Hm,st=∂hm,st∂xmt∂hm,st∂ẋmt∂hm,st∂ymt∂hm,st∂ẏmt=ymt−ystr20−xmt−xstr200−f0xmt−xstcr0−f0ymt−ystcr,where r=xmt−xst2+ymt−yst2.


## 3.3. Data Fusion for Multiple Sensors

For the tracking problem using multiple sensors, we need to fuse the target and measurement information from multiple sensors. The most used data fusion methods are parallel fusion and sequential fusion [48].In this paper, we use the sequential data fusion for multiple sensors which updates the target state and covariance by each sensor sequentially. The multiple sensors deal with the measurements one after another with an intermediate target state and covariance at one time.Letx^m,st∣t and Pm,st∣t denote the target state estimation and corresponding covariance processed by the sensor s. The Doppler and bearing measurements from the first passive sensor are used to calculate the first intermediate target state x^m,1nt∣t and covariance Pm,1nt∣t for each of the targets using a single sensor filter. Then, the measurements from the next passive sensor are used to update the intermediate target state and covariance as follows:
(32)x^m,st∣t=x^m,s−1t∣t+Gm,stzm,st−hm,st∣t,(33)Pm,st∣t=I−Gm,stHm,st⋅Pm,s−1t∣tI−Gm,stHm,stT+Gm,stRm,stGm,sTt,where Gm,st is the filter gain, and
(34)x^m,0t∣t=x^mt∣t−1,Pm,0t∣t=Pmt∣t−1,(35)x^mt∣t=x^m,St∣t,Pmt∣t=Pm,St∣t.The sequential fusion implementation of multiple sensors is shown in Figure2.Figure 2
Sequential fusion implementation of multiple sensors.


## 4.Simulation

Assume the targets move in the two-dimensional space, and there are two static passive sensors. The two passive sensors deal with Doppler and measurements from targets and clutter. The two passive sensors located at (0, -3500) m and (0, -5500) m, respectively.Assume that the clutter number is a Poisson distribution, and they are uniformly distributed in the Doppler and bearing measurement space. The average number of clutter in each sampling scan is 20 in the Doppler and bearing measurements domain. The target detection probability is the same for all targets; here, we set it to 0.8. The tracking time is 160 s with a sampling interval 1 s. The Monte Carlo run is 200. The process noise intensity is 1 m. The Doppler measurement noise variance is 1 Hz, and the bearing noise variance is0.8∘.


### 4.1. Case of the CV Model

The five targets’ initial position and velocity for the CV model are given in Table2.Table 2
The five targets’ initial position and velocity.
TargetPositionVelocity1(800, 2000) m(-22, -20) m/s2(1500, 1500) m(20, -30) m/s3(500, -500) m(23, 17) m/s4(0, 0) m(-13, -27) m/s5(500, -1000) m(13, -20) m/sFor the CV model, the simulated initial positionp and velocity v are generated from the position p0 and velocity v0 of ground truth with an estimation error such that p=p0+e1 and v=v0+e2, where e1=30 m and e2=2 m/s.The tracking scenario of true target trajectories and PMHT-estimated trajectories for the CV model is shown in Figure3. The true Doppler and measurements for two static passive sensors without clutter and the PMHT synthetic Doppler and bearing measurements under dense clutter are given in Figures 4 and 5, respectively.Figure 3
The tracking scenario of multiple targets, given two passive static sensors, true target trajectories, and estimated tracks.Figure 4
Bearing measurements for two static passive observers: (a) the true bearing measurements without clutter; (b) synthetic bearing measurements with clutter.
(a)(b)Figure 5
Doppler measurements for two static passive sensors: (a) the true Doppler measurements without clutter; (b) synthetic Doppler measurements with clutter.
(a)(b)In Figure3, the extended RTSS-based PMHT algorithm can track all the five targets effectively; even if the target 1 and target 4 tracks are away from the true trajectory, the proposed algorithm can get the true target track after several time scans in the middle tracking time scan.As seen in Figures4 and 5, the density of Doppler and bearing measurement from clutter is high, and the synthetic Doppler and bearing measurement of the proposed algorithm broadly consists of the true Doppler and bearing measurements without clutter, which means that the proposed algorithm has good declutter ability.The position and velocity RMSE of the five targets are given in Figures6 and 7. Accordingly, the average position and velocity RMSE are shown in Tables 3 and 4. As shown in Figures 6 and 7, the position RMSE and velocity RMSE for target 1 are increasing in the middle sampling scans, and as time goes on, the RMSE is decreased to a low lever, which is similar to other targets. This also can be seen from Tables 3 and 4. The average position and velocity RMSE is small which can meet the tracking accuracy requirement.Figure 6
The position RMSE versus time scans of five targets.Figure 7
The velocity RMSE versus time scans of five targets.Table 3
The average position RMSE of five targets.
Targets12345RMSE (m)34.949.9323.7120.2313.25Table 4
The average velocity RMSE of five targets.
Targets12345RMSE (m/s)2.1160.9911.2161.0020.590This paper uses the average normalized estimation error squared (ANEES) to evaluate the consistency of the proposed algorithm. For one target, the ANEES is defined as follows:(36)ANEESt=1Nm∑i=1Nmxit−x^itTPit−1xit−x^it,where Nm is Monte Carlo runs, xit is the true target state, x^it is the estimated target state, and Pit is the target state covariance.The ANEES for fives targets is shown in Figure8. As can be seen in Figure 8, the consistency of the proposed algorithm is good.Figure 8
The average NEES versus time scans for five targets.


### 4.2. Case of the CA Model

The four targets’ initial position, velocity, and acceleration for the CA model are given in Table5.Table 5
The five targets’ initial position and velocity.
TargetPositionVelocityAcceleration1(800, 2000) m(-22, -20) m/s(0.3, 0) m/s22(1500, 1500) m(20, -30) m/s(0, -0.3) m/s23(500, -500) m(23, 17) m/s(0.4, 0.4) m/s24(0, 0) m(-13, -27) m/s(0.1, 0.1) m/s2For the CA model, the simulated initial positionp, velocity v, and acceleration a are generated from the position p0, velocity v0, and acceleration a0 of ground truth with an estimation error such that p=p0+e1, v=v0+e2, and a=a0+e3, where e1=30 m, e2=2 m/s, and e3=0.5 m/s2.The tracking scenario of the CA model for target true trajectories and the estimated tracks is shown in Figure9. Similar to the CV model, the estimated tracks are consistent with the true targets’ trajectories.Figure 9
The tracking scenario of multiple targets, given two passive static sensors, CA model, true target trajectories, and estimated tracks.The position RMSE and velocity RMSE for the CA model are shown in Figures10 and 11. As shown in Figures 10 and 11, the position RMSE and velocity RMSE for target 3 are increasing in the middle sampling scans, and as time goes on, the RMSE is decreased to a low lever. The average position and velocity RMSE for the CA model is small which can meet the tracking accuracy requirement.Figure 10
The position RMSE for four targets (CA model).Figure 11
The velocity RMSE for four targets (CA model).


## 4.1. Case of the CV Model

The five targets’ initial position and velocity for the CV model are given in Table2.Table 2
The five targets’ initial position and velocity.
TargetPositionVelocity1(800, 2000) m(-22, -20) m/s2(1500, 1500) m(20, -30) m/s3(500, -500) m(23, 17) m/s4(0, 0) m(-13, -27) m/s5(500, -1000) m(13, -20) m/sFor the CV model, the simulated initial positionp and velocity v are generated from the position p0 and velocity v0 of ground truth with an estimation error such that p=p0+e1 and v=v0+e2, where e1=30 m and e2=2 m/s.The tracking scenario of true target trajectories and PMHT-estimated trajectories for the CV model is shown in Figure3. The true Doppler and measurements for two static passive sensors without clutter and the PMHT synthetic Doppler and bearing measurements under dense clutter are given in Figures 4 and 5, respectively.Figure 3
The tracking scenario of multiple targets, given two passive static sensors, true target trajectories, and estimated tracks.Figure 4
Bearing measurements for two static passive observers: (a) the true bearing measurements without clutter; (b) synthetic bearing measurements with clutter.
(a)(b)Figure 5
Doppler measurements for two static passive sensors: (a) the true Doppler measurements without clutter; (b) synthetic Doppler measurements with clutter.
(a)(b)In Figure3, the extended RTSS-based PMHT algorithm can track all the five targets effectively; even if the target 1 and target 4 tracks are away from the true trajectory, the proposed algorithm can get the true target track after several time scans in the middle tracking time scan.As seen in Figures4 and 5, the density of Doppler and bearing measurement from clutter is high, and the synthetic Doppler and bearing measurement of the proposed algorithm broadly consists of the true Doppler and bearing measurements without clutter, which means that the proposed algorithm has good declutter ability.The position and velocity RMSE of the five targets are given in Figures6 and 7. Accordingly, the average position and velocity RMSE are shown in Tables 3 and 4. As shown in Figures 6 and 7, the position RMSE and velocity RMSE for target 1 are increasing in the middle sampling scans, and as time goes on, the RMSE is decreased to a low lever, which is similar to other targets. This also can be seen from Tables 3 and 4. The average position and velocity RMSE is small which can meet the tracking accuracy requirement.Figure 6
The position RMSE versus time scans of five targets.Figure 7
The velocity RMSE versus time scans of five targets.Table 3
The average position RMSE of five targets.
Targets12345RMSE (m)34.949.9323.7120.2313.25Table 4
The average velocity RMSE of five targets.
Targets12345RMSE (m/s)2.1160.9911.2161.0020.590This paper uses the average normalized estimation error squared (ANEES) to evaluate the consistency of the proposed algorithm. For one target, the ANEES is defined as follows:(36)ANEESt=1Nm∑i=1Nmxit−x^itTPit−1xit−x^it,where Nm is Monte Carlo runs, xit is the true target state, x^it is the estimated target state, and Pit is the target state covariance.The ANEES for fives targets is shown in Figure8. As can be seen in Figure 8, the consistency of the proposed algorithm is good.Figure 8
The average NEES versus time scans for five targets.


## 4.2. Case of the CA Model

The four targets’ initial position, velocity, and acceleration for the CA model are given in Table5.Table 5
The five targets’ initial position and velocity.
TargetPositionVelocityAcceleration1(800, 2000) m(-22, -20) m/s(0.3, 0) m/s22(1500, 1500) m(20, -30) m/s(0, -0.3) m/s23(500, -500) m(23, 17) m/s(0.4, 0.4) m/s24(0, 0) m(-13, -27) m/s(0.1, 0.1) m/s2For the CA model, the simulated initial positionp, velocity v, and acceleration a are generated from the position p0, velocity v0, and acceleration a0 of ground truth with an estimation error such that p=p0+e1, v=v0+e2, and a=a0+e3, where e1=30 m, e2=2 m/s, and e3=0.5 m/s2.The tracking scenario of the CA model for target true trajectories and the estimated tracks is shown in Figure9. Similar to the CV model, the estimated tracks are consistent with the true targets’ trajectories.Figure 9
The tracking scenario of multiple targets, given two passive static sensors, CA model, true target trajectories, and estimated tracks.The position RMSE and velocity RMSE for the CA model are shown in Figures10 and 11. As shown in Figures 10 and 11, the position RMSE and velocity RMSE for target 3 are increasing in the middle sampling scans, and as time goes on, the RMSE is decreased to a low lever. The average position and velocity RMSE for the CA model is small which can meet the tracking accuracy requirement.Figure 10
The position RMSE for four targets (CA model).Figure 11
The velocity RMSE for four targets (CA model).


## 5. Conclusion

The major advantage of the passive sonars multiple-target tracking is that the sonars do not emit signals, and thus they can remain covert, which will reduce the risk of being attacked. But there are also challenges. Firstly, the Doppler and bearing measurements are nonlinear which makes the multiple-target tracking difficult. Secondly, the target states may be unobservable. Thirdly, the underwater environment is with dense clutter which will cause the measurement to target data association uncertainty problem. To deal with those problems, this paper proposed the extended RTSS-based batch PMHT method for multiple sensors and applied it to the passive multiple-sensor tracking system under dense clutter environment. This paper uses the extended RTSS algorithm to handle the nonlinear Doppler and bearing measurements. Multiple passive sonars are used to avoid the target state range unobservable problem. The multiple-sensor batch PMHT is used to deal with the data association uncertainty problem under dense clutter. The experiment results demonstrated that the proposed extended RTSS-based multiple-sensor PMHT algorithm can track multiple targets efficiently in the dense clutter environment, and the computing time is low.



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*Source: 4163766-2021-10-11.xml*