# Information-Based Sensor Control for Multitask Planning via Multi-Bernoulli Filter

Authors: Li Wang; Dongming Deng; Zhongliang He
Journal: Security and Communication Networks
 (2022)

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## Abstract

In multitarget tracking, the issue of sensor control is a challenging problem in theoretical analysis and calculation. In this paper, we study the sensor control strategies for multitask planning based on information criteria and propose two sensor control strategies according to the two different task objectives. Initially, we propose a sensor control strategy to improve the overall multitarget tracking performance within the partially observed Markov decision process (POMDP) framework, where the reward function is calculated by the Bhattacharyya distance between the prior and the posterior multitarget densities. In this strategy, we present a target-oriented multi-Bernoulli (TOMB) particle sampling method to approximate the multitarget density and then derive the solution of Bhattacharyya distance in detail. Subsequently, as another important contribution of this paper, we propose a threat-based sensor control strategy, which is still solved under the information theory where the goal is to prioritize multiple threat targets and then to track preferentially the maximum threat target. These strategies are finally used to optimize the sensor trajectory for range-bearing multitarget tracking.


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## Body

## 1. Introduction

In the field of target tracking, the scheme of sensor control is of great significance to the quality of information obtained in the overall target tracking system. Usually, the control of the sensor includes changing the position, direction, or motion state of the sensor platform. The core idea of sensor control is to select the working mode and operating parameters of the sensor according to certain optimal criteria to ensure that the performance of target tracking can be optimized globally. For example, a single sensor, controlling its position, velocity, and course, may affect the estimation accuracy of multitarget tracking in which not only the number of targets changes over time but also the measurements are confronted with missed detections and false alarms. In essence, sensor control is an optimal nonlinear control problem [1]. The solution to such problems is usually in the framework of partially observable Markov decision process (POMDP) [2–4]. However, due to the uncertainty of the target and measurement, it is a very challenging problem to solve the optimal sensor control in multitarget tracking.In recent years, the multitarget tracking algorithm based on random finite set (RFS) [5, 6] has been paid more attention. This method uses the RFS to model the multitarget state and the measurement. By introducing the finite set statistics (FISST) [6–8] and the generalized FISST [7], the multi-target state estimation under the clutter environment is described as Bayesian filtering problem, thus avoiding the intractable data association process of the traditional multitarget tracking. RFS-based multitarget tracking method can be used to update the posterior multitarget density according to the measurement set at each moment in the framework of Bayesian filtering [9]. In order to simplify the complexity of solving multitarget Bayesian filter directly in multi-target state space, Mahler and Vo et al. proposed a series of approximate multi-target filters, including multitarget moment recursive filters [6–12] and a variety of (labeled and unlabeled) multi-Bernoulli (MBer) filters [13–15]. It is noteworthy that the cardinality balanced multitarget multi-Bernoulli (CBMeMBer) filter, which is conceptually different from the probability hypothesis density (PHD) [7] filter and the cardinalized probability hypothesis density (CPHD) [10] filter, directly recurses the multitarget density. Therefore, the CBMeMBer filter is inherently advantageous for solving many information divergences which are essentially characterized by the multitarget densities.At present, some scholars have proposed some sensor control strategies [16–23] in the framework of FISST. Most of these strategies are devoted to improving the multitarget estimation accuracy, which is achieved by maximizing information gain or information divergence between the two multitarget densities. Often, these strategies are categorized as information-based criteria, which may jointly consider multiple task objectives to optimize multitarget tracking performance. Moreover, information-based sensor control is generally solved by a reward function which may be closely related to the multi-target filter. As previously noted, the CBMeMBer filter is inherently advantageous when it is in conjunction with sensor control, and some scholars have proposed some sensor control strategies [16, 23, 24] based on this filter. As a typical example, literature [16] used Rényi divergence as the reward function to solve the information-based sensor control. However, literature [16] does not show definitely how to sample to approximate the multi-target density, which usually plays a crucial role in solving reward function and sensor control strategy. For this reason, this paper further refines the previous work [24] and presents a target-oriented multi-Bernoulli (TOMB) particle sampling method where the goal is to approximate the multi-Bernoulli density and the multi-target density. This method provides a great convenience for solving the many reward functions based on the MeMBer filter, such as Kullback–Leibler divergence [25, 26] and more general Rényi divergence [18, 19]. Here, we choose the Bhattacharyya distance [27] as the reward function to decide the optimal sensor control strategy.In addition, information-based sensor control can be used not only to jointly optimize multiple tasks but also to perform some specific tasks to achieve the operator requirements. Target threat assessment is an important part of the threat assessment [28–30] in decision-level fusion. It is widely used in the military and civil fields. For example, in the military environment, it will play an important role in weapon allocation against enemy targets if we can accurately recognize and track preferentially the maximum threat target in real time. Under the circumstances, threat-based sensor control is a two-level decision-making process [31], which includes deciding the maximum threat target and deciding a sensor control strategy for the maximum threat target. Here, we will solve the threat-based sensor control strategy under information theory. Therefore, threat-based sensor control can be decomposed into two steps: assessing the target threat level and extracting the probability distribution of the maximum threat target from multi-target density. The assessment of the target threat level needs to consider various threat factors. From this perspective, threat-based sensor control is a multi-attribute decision problem [32]. Moreover, extracting the probability distribution of the maximum threat target is closely related to the multi-target filtering algorithm. As previously noted, the CBMeMBer filter directly recurses the posterior multi-target probability distribution by a set of fixed and independent Bernoulli parameters, which leads to extract the probability distribution of the maximum threat target from the multi-target density more inexpensive and reliable in contrast to the PHD and CPHD filter. Hence, as another important contribution, we propose a threat-based sensor control via SMC-CBMeMBer filter in this paper.The main content of this paper is to study the sensor control information based on the multitask planning via the multi-Bernoulli filter. For improving the multi-target tracking performance, we propose a sensor control strategy where the reward function is calculated by the Bhattacharyya distance between the prior and the posterior multi-target densities. In this strategy, we present a target-oriented multi-Bernoulli (TOMB) particle sampling method to approximate the multi-target density and then to derive the solution of Bhattacharyya distance in detail. In addition, for special task planning, we propose a threat-based sensor control strategy. The proposed strategy, which focuses on the maximum threat target, is finally carried out by Bhattacharyya distance.This paper is organised as follows. In Section2, we retrospect RFS modeling of multitarget systems and the general sensor control method based on information theory. The CBMeMBer filter is shown in Section 3. In Section 4, we present a target-oriented multi-Bernoulli (TOMB) particle sampling method to approximate the multi-target density and then to derive the solution of Bhattacharyya distance in detail. Subsequently, two sensor control strategies are formulated in Section 5, including the multi-target information gain based and threat based. Simulation analysis is described in Section 6. Finally, we discuss some conclusions in Section 7.


## 2. Problem Formulation

### 2.1. Multi-Target Model Based on Random Finite Set

Assume that there arenxk targets at time k and their states are x1,k,…,xnxk,k, respectively. Each state takes the value in the target state space X⊆ℝnx. Moreover, the sensor receive nzk measurements z1,k,…,znzk,k, each taking values in the measurement space Z⊆ℝnz. Then, the multi-target state and the multi-target measurement can be modeled at time k as(1)Xk=x1,k,…,xnxk,ka∈ℱX,Zk=z1,k,…,znzk,k∈ℱZ,where ℱX and ℱZ represent the sets of all finite subsets of X and Z, respectively.Suppose that at timek−1, a target with state xk−1∈Xk−1 continues to survive at time k with probability pS,kxk−1. If the spawn target is not taken into account, the set of target state at the time k is modeled as(2)Xk=∪xk−1∈Xk−1Sk|k−1xk−1∪Γk,where Sk|k−1 denotes the RFSs of survival targets at time k and Γk represents the RFSs of birth targets at time k.Provided that the state transition equation for each target is(3)xk+1=Fk⋅xk+wk,where Fk denotes the state transition matrix and wk∼N0,Qk denotes the process noise vector.In addition, at timek, for target state xk∈Xk, the sensor detects its corresponding measurement zk∈Zk with probability pD,kxk, and if the measurement noise is vk, it can be assumed that the measurement equation has the following form:(4)zk=hxk,xs,kν+vk,where the sensor position xs,kν=xs,kν,ys,kνT is determined by the control action ν. The sensor measurement of target is a RFS, which can be expressed as Θkxk. If the effect of clutter is taken into account, the multi-target measurements Zk received at time k are modeled as(5)Zk=∪xk∈XkΘkxk∪κk,where κk represents the clutter distribution at time k.


### 2.2. Sensor Control Based on Information Theory

The research of sensor control strategy based on information theory is generally carried out in the framework of POMDP which is a generalized form of Markov decision process. In principle, a POMDP usually includes several elements, i.e., a set of available sensor control actions, the current state information, and the reward function with respect to each sensor actionν. Let Uk denote the set of available sensor control vectors at time k and each sensor control ν∈Uk determine a sensor position at the next moment. The current state information can be depicted by the posterior multi-target density pkXk|Z1:k at time k. Often, the optimal control action uk is selected through the following criteria:(6)uk=argmaxν∈UkEℛν,pk|k−1Xk|Z1:k−1,Zkν,where ℛν,p,Z is the reward function with relation to the control ν, and it depends on the future measurement set Zk+1ν. In general, the future measurement set can be obtained by (4) and (5). However, using the method is computationally expensive. To reduce the computational cost, we refer to the traditional processing method [19, 33], and only consider generating a predicted ideal measurement set (PIMS) [33–36] for each control ν. Often, The PIMS may be obtained in the case of no clutter, noise, and the detection probability pD=1.In addition, it is noteworthy that (6) maximizes the reward function based on a single step ahead (the “Myopic” policy [33]). Ideally, the sensor control strategy may look an infinite number of steps ahead. However, as the number of steps ahead increases, the computational cost may increase exponentially. And the future motion model and measurement of the target are unknown so that the multistep prediction brings a lot of uncertainty. Hence, many practical solutions are based on the one-step approximation.In this paper, Bhattacharyya distance is chosen as the reward function, which is an entropy measure of information gain between the two multi-target densities. It can be expressed as(7)ℛν=−ln∫pk|kX;νpk|k−1XδX,where pk+1|k+1X;ν is the posterior multi-target density and pk+1|kX is the prior multi-target density. They can be calculated by Bayesian recursion formulas.(8)pk|k−1Xk|Z1:k−1=∫fk|k−1Xk|Xk−1pk−1|k−1Xk−1|Z1:k−1δXk−1,(9)pk|kXk|Z1:k=gkZk|Xkpk|k−1Xk|Z1:k−1∫gkZk|Xkpk|k−1Xk|Z1:k−1δXk.In formula (9) fk|k−1Xk|Xk−1 and gkZk|Xk are the multi-target transition density and the multi-target likelihood function at time k, respectively.


## 2.1. Multi-Target Model Based on Random Finite Set

Assume that there arenxk targets at time k and their states are x1,k,…,xnxk,k, respectively. Each state takes the value in the target state space X⊆ℝnx. Moreover, the sensor receive nzk measurements z1,k,…,znzk,k, each taking values in the measurement space Z⊆ℝnz. Then, the multi-target state and the multi-target measurement can be modeled at time k as(1)Xk=x1,k,…,xnxk,ka∈ℱX,Zk=z1,k,…,znzk,k∈ℱZ,where ℱX and ℱZ represent the sets of all finite subsets of X and Z, respectively.Suppose that at timek−1, a target with state xk−1∈Xk−1 continues to survive at time k with probability pS,kxk−1. If the spawn target is not taken into account, the set of target state at the time k is modeled as(2)Xk=∪xk−1∈Xk−1Sk|k−1xk−1∪Γk,where Sk|k−1 denotes the RFSs of survival targets at time k and Γk represents the RFSs of birth targets at time k.Provided that the state transition equation for each target is(3)xk+1=Fk⋅xk+wk,where Fk denotes the state transition matrix and wk∼N0,Qk denotes the process noise vector.In addition, at timek, for target state xk∈Xk, the sensor detects its corresponding measurement zk∈Zk with probability pD,kxk, and if the measurement noise is vk, it can be assumed that the measurement equation has the following form:(4)zk=hxk,xs,kν+vk,where the sensor position xs,kν=xs,kν,ys,kνT is determined by the control action ν. The sensor measurement of target is a RFS, which can be expressed as Θkxk. If the effect of clutter is taken into account, the multi-target measurements Zk received at time k are modeled as(5)Zk=∪xk∈XkΘkxk∪κk,where κk represents the clutter distribution at time k.


## 2.2. Sensor Control Based on Information Theory

The research of sensor control strategy based on information theory is generally carried out in the framework of POMDP which is a generalized form of Markov decision process. In principle, a POMDP usually includes several elements, i.e., a set of available sensor control actions, the current state information, and the reward function with respect to each sensor actionν. Let Uk denote the set of available sensor control vectors at time k and each sensor control ν∈Uk determine a sensor position at the next moment. The current state information can be depicted by the posterior multi-target density pkXk|Z1:k at time k. Often, the optimal control action uk is selected through the following criteria:(6)uk=argmaxν∈UkEℛν,pk|k−1Xk|Z1:k−1,Zkν,where ℛν,p,Z is the reward function with relation to the control ν, and it depends on the future measurement set Zk+1ν. In general, the future measurement set can be obtained by (4) and (5). However, using the method is computationally expensive. To reduce the computational cost, we refer to the traditional processing method [19, 33], and only consider generating a predicted ideal measurement set (PIMS) [33–36] for each control ν. Often, The PIMS may be obtained in the case of no clutter, noise, and the detection probability pD=1.In addition, it is noteworthy that (6) maximizes the reward function based on a single step ahead (the “Myopic” policy [33]). Ideally, the sensor control strategy may look an infinite number of steps ahead. However, as the number of steps ahead increases, the computational cost may increase exponentially. And the future motion model and measurement of the target are unknown so that the multistep prediction brings a lot of uncertainty. Hence, many practical solutions are based on the one-step approximation.In this paper, Bhattacharyya distance is chosen as the reward function, which is an entropy measure of information gain between the two multi-target densities. It can be expressed as(7)ℛν=−ln∫pk|kX;νpk|k−1XδX,where pk+1|k+1X;ν is the posterior multi-target density and pk+1|kX is the prior multi-target density. They can be calculated by Bayesian recursion formulas.(8)pk|k−1Xk|Z1:k−1=∫fk|k−1Xk|Xk−1pk−1|k−1Xk−1|Z1:k−1δXk−1,(9)pk|kXk|Z1:k=gkZk|Xkpk|k−1Xk|Z1:k−1∫gkZk|Xkpk|k−1Xk|Z1:k−1δXk.In formula (9) fk|k−1Xk|Xk−1 and gkZk|Xk are the multi-target transition density and the multi-target likelihood function at time k, respectively.


## 3. CBMeMBer Filter

### 3.1. Multi-Bernoulli RFS

A single Bernoulli RFSX, which is in the state space X, is either a singleton or an empty with probabilities r and 1−r, respectively. If X is a singleton, its element is distributed in accordance with a probability density function π. In addition, the cardinality distribution of X is a Bernoulli distribution whose parameter is r. And the probability density of X can be expressed as(10)πX=1−r,X=∅,r⋅px,X=x.A multi-Bernoulli RFSX, which is in the state space X, is a union of independent Bernoulli RFSs with a fixed number, i.e., X=∪i=1MXi, where Xi denotes the i−th Bernoulli RFS, and its existence probability and the probability density function are ri and pi, respectively. Obviously, the average cardinality of X is ∑i=1Mri. And the probability density π of X can be written by(11)πx1,…,xn=π∅∑1≤i1≠…≠in≤M∏j=1nrijpijxj1−rij,where π∅=∏j=1M1−rj. For the sake of simplicity, (11) is abbreviated as π=ri,pii=1M.


### 3.2. SMC-CBMeMBer Recursion

The multi-target multi-Bernoulli (MeMBer) filter, which is different from PHD and CPHD filter, directly recurses the posterior multi-target probability distribution so that solving the multi-target tracking problem is more facilitated. Moreover, the traditional SMC-PHD/CPHD filter is necessary to classify the particle set and extract the target state through a less stable and computationally costly clustering method. Thus, the accuracy of multi-target state estimation heavily depends on the stability of the clustering algorithm. Vo et al. have proved that the MeMBer filter overestimates the cardinality (number of targets), and the CBMeMBer filter [15] is proposed based on the cardinality correction strategy. The following will give its sequential Monte Carlo (SMC) implementation [15].


#### 3.2.1. Prediction

Assume that the posterior multi-Bernoulli density at timek−1 is expressed as(12)πk−1=rk−1i,pk−1ii=1Mk−1,where pk−1i denotes the state distribution of the i−th Bernoulli RFS and is given by(13)pk−1ix=∑j=1Lk−1iwk−1i,jδxk−1i,jx.The predicted multi-Bernoulli densityπk|k−1 at time k is given by(14)πk|k−1=rP,k|k−1i,pP,k|k−1ii=1Mk−1∪rΓ,ki,pΓ,kii=1MΓ,k,where rΓ,ki,pΓ,kii=1MΓ,k denotes the multi-Bernoulli density of birth targets at time k. If given proposal density qkixk|xk−1, then the predicted particle of survival target is(15)xP,k|k−1i,j∼qkixk|xk−1i,j.The predicted multi-Bernoulli density of survival targets can be expressed as follows:(16)rP,k|k−1i=rk−1i∑j=1Lk−1iwk−1i,jps,kxk−1i,j,pP,k|k−1ix=∑j=1Lk−1iw˜P,k|k−1i,jδxP,k|k−1i,jx,where(17)w˜P,k|k−1i,j=wP,k|k−1i,j∑j=1Lk−1iwP,k|k−1i,j,wP,k|k−1i,j=wk−1i,jfk|k−1xP,k|k−1i,j|xk−1i,jps,kxk−1i,jqkixP,k|k−1i,j|xk−1i,j.For the multi-Bernoulli density of birth targets at timek, the probability density of the i−th Bernoulli RFS is characterized by(18)pΓ,ki=∑j=1LΓ,kiw˜Γ,ki,jδxΓ,ki,jx.If(19)xΓ,ki,j∼qΓ,ki⋅|Zk,j=1,…,LΓ,ki,then the particle weight of the birth target is(20)w˜Γ,ki,j=wΓ,ki,j∑j=1LΓ,kiwΓ,ki,j,wΓ,ki,j=pΓ,kixΓ,ki,jqΓ,kixΓ,ki,j|Zk.


#### 3.2.2. Update

Let the predicted multi-Bernoulli densityπk|k−1 at time k is(21)πk|k−1=rk|k−1i,pk|k−1ii=1Mk|k−1,where(22)pk|k−1i=∑j=1Lk|k−1iwk|k−1i,jδxk|k−1i,jxk|k−1.Then, the posterior multi-Bernoulli densityπk can be expressed as(23)πk≈rL,ki,pL,kii=1Mk|k−1∪rU,kz,pU,kx;zz∈Zk,where the parameters of the inheritance track (missed) part $rL,ki,pL,kii=1Mk|k−1 are(24)rL,ki=rk|k−1i1−ρL,ki1−rk|k−1iρL,ki,pL,kix=∑j=1Lk|k−1iw˜L,ki,jδxk|k−1i,jx,where(25)ρL,kix=∑j=1Lk|k−1iwk|k−1i,jpD,kxk|k−1i,j,w˜L,ki,j=wL,ki,j∑j=1Lk|k−1iwL,ki,j,wL,ki,j=wk|k−1i,j1−pD,kxk|k−1i,j.rU,kz,pU,kx;zz∈Zk are the parameters of the updated multi-Bernoulli density, then(26)rU,kz=∑i=1Mk|k−1rk|k−1i1−rk|k−1iU,kiz/1−rk|k−1iρL,kix2κkz+∑i=1Mk|k−1rk|k−1iρU,kiz/1−rk|k−1iρL,kix,(27)pU,kx;z=∑i=1Mk|k−1∑j=1Lk|k−1iw˜U,k∗i,jzδxk|k−1i,jx,where(28)ρU,kiz=∑j=1Lk|k−1iwk|k−1i,jψk,zxk|k−1i,j,w˜U,k∗i,jz=wU,k∗i,jz∑i=1Mk|k−1∑j=1Lk|k−1iwU,k∗i,jz,wU,k∗i,jz=wk|k−1i,jrk|k−1i1−rk|k−1iψk,zxk|k−1i,j,ψk,zxki,j=gkz|xk|k−1i,jpD,kxk|k−1i,j.


#### 3.2.3. Resampling

Similar to the particle filter, the SMC-CBMeMBer filter cannot still avoid the particle degeneracy. In order to reduce the effect of degeneracy, the usual practice is to resample the particles, which can be referred to [15].


## 3.1. Multi-Bernoulli RFS

A single Bernoulli RFSX, which is in the state space X, is either a singleton or an empty with probabilities r and 1−r, respectively. If X is a singleton, its element is distributed in accordance with a probability density function π. In addition, the cardinality distribution of X is a Bernoulli distribution whose parameter is r. And the probability density of X can be expressed as(10)πX=1−r,X=∅,r⋅px,X=x.A multi-Bernoulli RFSX, which is in the state space X, is a union of independent Bernoulli RFSs with a fixed number, i.e., X=∪i=1MXi, where Xi denotes the i−th Bernoulli RFS, and its existence probability and the probability density function are ri and pi, respectively. Obviously, the average cardinality of X is ∑i=1Mri. And the probability density π of X can be written by(11)πx1,…,xn=π∅∑1≤i1≠…≠in≤M∏j=1nrijpijxj1−rij,where π∅=∏j=1M1−rj. For the sake of simplicity, (11) is abbreviated as π=ri,pii=1M.


## 3.2. SMC-CBMeMBer Recursion

The multi-target multi-Bernoulli (MeMBer) filter, which is different from PHD and CPHD filter, directly recurses the posterior multi-target probability distribution so that solving the multi-target tracking problem is more facilitated. Moreover, the traditional SMC-PHD/CPHD filter is necessary to classify the particle set and extract the target state through a less stable and computationally costly clustering method. Thus, the accuracy of multi-target state estimation heavily depends on the stability of the clustering algorithm. Vo et al. have proved that the MeMBer filter overestimates the cardinality (number of targets), and the CBMeMBer filter [15] is proposed based on the cardinality correction strategy. The following will give its sequential Monte Carlo (SMC) implementation [15].


### 3.2.1. Prediction

Assume that the posterior multi-Bernoulli density at timek−1 is expressed as(12)πk−1=rk−1i,pk−1ii=1Mk−1,where pk−1i denotes the state distribution of the i−th Bernoulli RFS and is given by(13)pk−1ix=∑j=1Lk−1iwk−1i,jδxk−1i,jx.The predicted multi-Bernoulli densityπk|k−1 at time k is given by(14)πk|k−1=rP,k|k−1i,pP,k|k−1ii=1Mk−1∪rΓ,ki,pΓ,kii=1MΓ,k,where rΓ,ki,pΓ,kii=1MΓ,k denotes the multi-Bernoulli density of birth targets at time k. If given proposal density qkixk|xk−1, then the predicted particle of survival target is(15)xP,k|k−1i,j∼qkixk|xk−1i,j.The predicted multi-Bernoulli density of survival targets can be expressed as follows:(16)rP,k|k−1i=rk−1i∑j=1Lk−1iwk−1i,jps,kxk−1i,j,pP,k|k−1ix=∑j=1Lk−1iw˜P,k|k−1i,jδxP,k|k−1i,jx,where(17)w˜P,k|k−1i,j=wP,k|k−1i,j∑j=1Lk−1iwP,k|k−1i,j,wP,k|k−1i,j=wk−1i,jfk|k−1xP,k|k−1i,j|xk−1i,jps,kxk−1i,jqkixP,k|k−1i,j|xk−1i,j.For the multi-Bernoulli density of birth targets at timek, the probability density of the i−th Bernoulli RFS is characterized by(18)pΓ,ki=∑j=1LΓ,kiw˜Γ,ki,jδxΓ,ki,jx.If(19)xΓ,ki,j∼qΓ,ki⋅|Zk,j=1,…,LΓ,ki,then the particle weight of the birth target is(20)w˜Γ,ki,j=wΓ,ki,j∑j=1LΓ,kiwΓ,ki,j,wΓ,ki,j=pΓ,kixΓ,ki,jqΓ,kixΓ,ki,j|Zk.


### 3.2.2. Update

Let the predicted multi-Bernoulli densityπk|k−1 at time k is(21)πk|k−1=rk|k−1i,pk|k−1ii=1Mk|k−1,where(22)pk|k−1i=∑j=1Lk|k−1iwk|k−1i,jδxk|k−1i,jxk|k−1.Then, the posterior multi-Bernoulli densityπk can be expressed as(23)πk≈rL,ki,pL,kii=1Mk|k−1∪rU,kz,pU,kx;zz∈Zk,where the parameters of the inheritance track (missed) part $rL,ki,pL,kii=1Mk|k−1 are(24)rL,ki=rk|k−1i1−ρL,ki1−rk|k−1iρL,ki,pL,kix=∑j=1Lk|k−1iw˜L,ki,jδxk|k−1i,jx,where(25)ρL,kix=∑j=1Lk|k−1iwk|k−1i,jpD,kxk|k−1i,j,w˜L,ki,j=wL,ki,j∑j=1Lk|k−1iwL,ki,j,wL,ki,j=wk|k−1i,j1−pD,kxk|k−1i,j.rU,kz,pU,kx;zz∈Zk are the parameters of the updated multi-Bernoulli density, then(26)rU,kz=∑i=1Mk|k−1rk|k−1i1−rk|k−1iU,kiz/1−rk|k−1iρL,kix2κkz+∑i=1Mk|k−1rk|k−1iρU,kiz/1−rk|k−1iρL,kix,(27)pU,kx;z=∑i=1Mk|k−1∑j=1Lk|k−1iw˜U,k∗i,jzδxk|k−1i,jx,where(28)ρU,kiz=∑j=1Lk|k−1iwk|k−1i,jψk,zxk|k−1i,j,w˜U,k∗i,jz=wU,k∗i,jz∑i=1Mk|k−1∑j=1Lk|k−1iwU,k∗i,jz,wU,k∗i,jz=wk|k−1i,jrk|k−1i1−rk|k−1iψk,zxk|k−1i,j,ψk,zxki,j=gkz|xk|k−1i,jpD,kxk|k−1i,j.


### 3.2.3. Resampling

Similar to the particle filter, the SMC-CBMeMBer filter cannot still avoid the particle degeneracy. In order to reduce the effect of degeneracy, the usual practice is to resample the particles, which can be referred to [15].


## 3.2.1. Prediction

Assume that the posterior multi-Bernoulli density at timek−1 is expressed as(12)πk−1=rk−1i,pk−1ii=1Mk−1,where pk−1i denotes the state distribution of the i−th Bernoulli RFS and is given by(13)pk−1ix=∑j=1Lk−1iwk−1i,jδxk−1i,jx.The predicted multi-Bernoulli densityπk|k−1 at time k is given by(14)πk|k−1=rP,k|k−1i,pP,k|k−1ii=1Mk−1∪rΓ,ki,pΓ,kii=1MΓ,k,where rΓ,ki,pΓ,kii=1MΓ,k denotes the multi-Bernoulli density of birth targets at time k. If given proposal density qkixk|xk−1, then the predicted particle of survival target is(15)xP,k|k−1i,j∼qkixk|xk−1i,j.The predicted multi-Bernoulli density of survival targets can be expressed as follows:(16)rP,k|k−1i=rk−1i∑j=1Lk−1iwk−1i,jps,kxk−1i,j,pP,k|k−1ix=∑j=1Lk−1iw˜P,k|k−1i,jδxP,k|k−1i,jx,where(17)w˜P,k|k−1i,j=wP,k|k−1i,j∑j=1Lk−1iwP,k|k−1i,j,wP,k|k−1i,j=wk−1i,jfk|k−1xP,k|k−1i,j|xk−1i,jps,kxk−1i,jqkixP,k|k−1i,j|xk−1i,j.For the multi-Bernoulli density of birth targets at timek, the probability density of the i−th Bernoulli RFS is characterized by(18)pΓ,ki=∑j=1LΓ,kiw˜Γ,ki,jδxΓ,ki,jx.If(19)xΓ,ki,j∼qΓ,ki⋅|Zk,j=1,…,LΓ,ki,then the particle weight of the birth target is(20)w˜Γ,ki,j=wΓ,ki,j∑j=1LΓ,kiwΓ,ki,j,wΓ,ki,j=pΓ,kixΓ,ki,jqΓ,kixΓ,ki,j|Zk.


## 3.2.2. Update

Let the predicted multi-Bernoulli densityπk|k−1 at time k is(21)πk|k−1=rk|k−1i,pk|k−1ii=1Mk|k−1,where(22)pk|k−1i=∑j=1Lk|k−1iwk|k−1i,jδxk|k−1i,jxk|k−1.Then, the posterior multi-Bernoulli densityπk can be expressed as(23)πk≈rL,ki,pL,kii=1Mk|k−1∪rU,kz,pU,kx;zz∈Zk,where the parameters of the inheritance track (missed) part $rL,ki,pL,kii=1Mk|k−1 are(24)rL,ki=rk|k−1i1−ρL,ki1−rk|k−1iρL,ki,pL,kix=∑j=1Lk|k−1iw˜L,ki,jδxk|k−1i,jx,where(25)ρL,kix=∑j=1Lk|k−1iwk|k−1i,jpD,kxk|k−1i,j,w˜L,ki,j=wL,ki,j∑j=1Lk|k−1iwL,ki,j,wL,ki,j=wk|k−1i,j1−pD,kxk|k−1i,j.rU,kz,pU,kx;zz∈Zk are the parameters of the updated multi-Bernoulli density, then(26)rU,kz=∑i=1Mk|k−1rk|k−1i1−rk|k−1iU,kiz/1−rk|k−1iρL,kix2κkz+∑i=1Mk|k−1rk|k−1iρU,kiz/1−rk|k−1iρL,kix,(27)pU,kx;z=∑i=1Mk|k−1∑j=1Lk|k−1iw˜U,k∗i,jzδxk|k−1i,jx,where(28)ρU,kiz=∑j=1Lk|k−1iwk|k−1i,jψk,zxk|k−1i,j,w˜U,k∗i,jz=wU,k∗i,jz∑i=1Mk|k−1∑j=1Lk|k−1iwU,k∗i,jz,wU,k∗i,jz=wk|k−1i,jrk|k−1i1−rk|k−1iψk,zxk|k−1i,j,ψk,zxki,j=gkz|xk|k−1i,jpD,kxk|k−1i,j.


## 3.2.3. Resampling

Similar to the particle filter, the SMC-CBMeMBer filter cannot still avoid the particle degeneracy. In order to reduce the effect of degeneracy, the usual practice is to resample the particles, which can be referred to [15].


## 4. Reward Function

The reward function plays a vital role in the sensor control problem.In this paper, Bhattacharyya distance, as a reward function, is used in information-based sensor control, which in essence is a relative entropy measure (information gain) between the two multi-target densities. An increase in the value of Bhattacharyya distance implies that more information may be obtained from the next measurements. Further, the maximum value can determine the optimal sensor position at the next moment. Especially, (7) shows that Bhattacharyya distance depends on the prior and the posterior multi-target densities. Although (11) has given the multi-target density approximated by a set of fixed and independent Bernoulli parameters, it is computationally intractable for solving the Bhattacharyya distance. For this reason, this section presents a TOMB particle sampling method that approximates the multi-target density by sampling the multi-Bernoulli parameters and then to derive the solution of Bhattacharyya distance in detail.


### 4.1. TOMB Particle Sampling

In essence, the CBMeMBer filter propagates the multi-Bernoulli parameters consisting of the existence probabilitiesri and the probability distribution pi, so the TOMB particle sampling method can be divided into two steps: sampling the hypothesized track according to the ri and its corresponding probability distribution pi. For example, for the l−th sample, 1≤l≤Lk×∑i=1Mri, the sampling method can be expressed as(29)rl,xl∼qr×qx|r,where qr is the sampling distribution of Bernoulli RFS (track) and qx|r is the sampling distribution of the corresponding state. The specific sampling process is as follows.


#### 4.1.1. Bernoulli RFS (Track) Sampling

ri represents the existence probability of the i−th Bernoulli RFS, and the sampling distribution of the track is(30)qr=ri∑i=1Mri.


#### 4.1.2. State Sampling

Assume that the probability densitypi of the i−th Bernoulli RFS can be approximated by a set of weighted particles, then the sampling distribution of state is given by(31)qx|ri=wi,j,where wi,j is the weight of the corresponding state particle.


### 4.2. Bhattacharyya Distance

As previously noted, we use the TOMB sampling method to approximate the predicted multi-target density, which can be expressed as(32)pk|k−1X≈∑j=1Lk|k−1wk|k−1jδxk|k−1jx,where Lk|k−1=Lk×∑i=1Mk|k−1rk|k−1i and wk|k−1j=1/Lk|k−1.In addition, as another important component of Bhattacharyya distance, the posterior multi-target density also needs to be approximated by using the TOMB to sample the updated Bernoulli RFSs that is usually updated by the PIMS. According to (4), the PIMS is comprised of the predicted multi-target state set X⌢k|k−1=x⌢k|k−1ii=1Nk|k−1 and sensor position xs,kν for each ν∈Uk. For the former, the predicted multi-target states can be extracted from the predicted multi-Bernoulli density.Since the PIMS does not consider clutter, noise, and the detection probabilitypD=1, there is no the Bernoulli RFS of inherited track (missed) part, and (26) can be simplified to(33)rU,kz=∑i=1Mk|k−1rk|k−1i1−rk|k−1ipk|k−1i,ψk,z/1−rk|k−1i2∑i=1Mk|k−1rk|k−1ipk|k−1i,ψk,z/1−rk|k−1i=1.It can be seen that all the existence probabilities of the updated Bernoulli RFSs are equal. According to the TOMB sampling method, theqr of all the updated tracks are also equal. Comparing the updated Bernoulli RFSs and the predicted Bernoulli RFSs, we found that they have the same particles, and only the particle weights are different. Therefore, according to (27), on the basis of the TOMB sampling method, the posterior multi-target density can be deduced as follows:(34)pk|kX≈∑j=1Lk|k−1w⌣k|kjδxk|k−1jx,(35)w⌣k|kj=w⌣k|k∗j∑i=1Lk|k−1w⌣k|k∗j,(36)w⌣k|k∗j=∑z∈Zkνw⌣U,kjz,(37)w⌣U,kjz=w⌣U,k∗jz∑j=1Lk|k−1w⌣U,k∗jz,(38)w⌣U,k∗jz=wk|k−1jMjrgkz|xk|k−1j,(39)Mjr=rk|k−1j∗1−rk|k−1j∗,where j∗ represents the index of the predicted Bernoulli RFSs (track).In addition, Bhattacharyya distance is given as follows:(40)ℛν=−ln∫pk|kX;νpk|k−1XδX.According to the TOMB sampling method, we obtain two approximate multi-target densities (32) and (34).Subsequently, substitute (32) and (34) into (40) and we obtain(41)ℛν=−ln∫∑j=1Lk|k−1wk|kjδxk|k−1jx∑j=1Lk|k−1wk|k−1jδxk|k−1jxδx.Note that when a sum of delta functions reaches to an exponent, the cross-terms will disappear due to the characteristic of the delta function. Hence, (41) is simplified as(42)ℛν=−ln∫∑j=1Lk|k−1w⌣k|kj1/2δxk|k−1jx1/2∑j=1Lk|k−1wk|k−1j1/2δxk|k−1jx1/2δx.When the two sums of delta functions are multiplied, the cross-terms also vanish and the approximation simplifies to(43)ℛν=−ln∫∑j=1Lk|k−1w⌣k|kj1/2wk|k−1j1/2δxk|k−1jxδx.According to the integration property of the delta function, (43) is simplified as(44)ℛν−ln∑j=1Lk|k−1w⌣k|kjwk|k−1j.


## 4.1. TOMB Particle Sampling

In essence, the CBMeMBer filter propagates the multi-Bernoulli parameters consisting of the existence probabilitiesri and the probability distribution pi, so the TOMB particle sampling method can be divided into two steps: sampling the hypothesized track according to the ri and its corresponding probability distribution pi. For example, for the l−th sample, 1≤l≤Lk×∑i=1Mri, the sampling method can be expressed as(29)rl,xl∼qr×qx|r,where qr is the sampling distribution of Bernoulli RFS (track) and qx|r is the sampling distribution of the corresponding state. The specific sampling process is as follows.


### 4.1.1. Bernoulli RFS (Track) Sampling

ri represents the existence probability of the i−th Bernoulli RFS, and the sampling distribution of the track is(30)qr=ri∑i=1Mri.


### 4.1.2. State Sampling

Assume that the probability densitypi of the i−th Bernoulli RFS can be approximated by a set of weighted particles, then the sampling distribution of state is given by(31)qx|ri=wi,j,where wi,j is the weight of the corresponding state particle.


## 4.1.1. Bernoulli RFS (Track) Sampling

ri represents the existence probability of the i−th Bernoulli RFS, and the sampling distribution of the track is(30)qr=ri∑i=1Mri.


## 4.1.2. State Sampling

Assume that the probability densitypi of the i−th Bernoulli RFS can be approximated by a set of weighted particles, then the sampling distribution of state is given by(31)qx|ri=wi,j,where wi,j is the weight of the corresponding state particle.


## 4.2. Bhattacharyya Distance

As previously noted, we use the TOMB sampling method to approximate the predicted multi-target density, which can be expressed as(32)pk|k−1X≈∑j=1Lk|k−1wk|k−1jδxk|k−1jx,where Lk|k−1=Lk×∑i=1Mk|k−1rk|k−1i and wk|k−1j=1/Lk|k−1.In addition, as another important component of Bhattacharyya distance, the posterior multi-target density also needs to be approximated by using the TOMB to sample the updated Bernoulli RFSs that is usually updated by the PIMS. According to (4), the PIMS is comprised of the predicted multi-target state set X⌢k|k−1=x⌢k|k−1ii=1Nk|k−1 and sensor position xs,kν for each ν∈Uk. For the former, the predicted multi-target states can be extracted from the predicted multi-Bernoulli density.Since the PIMS does not consider clutter, noise, and the detection probabilitypD=1, there is no the Bernoulli RFS of inherited track (missed) part, and (26) can be simplified to(33)rU,kz=∑i=1Mk|k−1rk|k−1i1−rk|k−1ipk|k−1i,ψk,z/1−rk|k−1i2∑i=1Mk|k−1rk|k−1ipk|k−1i,ψk,z/1−rk|k−1i=1.It can be seen that all the existence probabilities of the updated Bernoulli RFSs are equal. According to the TOMB sampling method, theqr of all the updated tracks are also equal. Comparing the updated Bernoulli RFSs and the predicted Bernoulli RFSs, we found that they have the same particles, and only the particle weights are different. Therefore, according to (27), on the basis of the TOMB sampling method, the posterior multi-target density can be deduced as follows:(34)pk|kX≈∑j=1Lk|k−1w⌣k|kjδxk|k−1jx,(35)w⌣k|kj=w⌣k|k∗j∑i=1Lk|k−1w⌣k|k∗j,(36)w⌣k|k∗j=∑z∈Zkνw⌣U,kjz,(37)w⌣U,kjz=w⌣U,k∗jz∑j=1Lk|k−1w⌣U,k∗jz,(38)w⌣U,k∗jz=wk|k−1jMjrgkz|xk|k−1j,(39)Mjr=rk|k−1j∗1−rk|k−1j∗,where j∗ represents the index of the predicted Bernoulli RFSs (track).In addition, Bhattacharyya distance is given as follows:(40)ℛν=−ln∫pk|kX;νpk|k−1XδX.According to the TOMB sampling method, we obtain two approximate multi-target densities (32) and (34).Subsequently, substitute (32) and (34) into (40) and we obtain(41)ℛν=−ln∫∑j=1Lk|k−1wk|kjδxk|k−1jx∑j=1Lk|k−1wk|k−1jδxk|k−1jxδx.Note that when a sum of delta functions reaches to an exponent, the cross-terms will disappear due to the characteristic of the delta function. Hence, (41) is simplified as(42)ℛν=−ln∫∑j=1Lk|k−1w⌣k|kj1/2δxk|k−1jx1/2∑j=1Lk|k−1wk|k−1j1/2δxk|k−1jx1/2δx.When the two sums of delta functions are multiplied, the cross-terms also vanish and the approximation simplifies to(43)ℛν=−ln∫∑j=1Lk|k−1w⌣k|kj1/2wk|k−1j1/2δxk|k−1jxδx.According to the integration property of the delta function, (43) is simplified as(44)ℛν−ln∑j=1Lk|k−1w⌣k|kjwk|k−1j.


## 5. Multi-Task Planning Based Sensor Control

### 5.1. Sensor Control Based on the Multi-Target Information Gain

In this subsection, we employ Bhattacharyya distance as the measure of information gain between the two multi-target densities obtained by the TOMB particle sampling method. The purpose of this strategy is to maximize the Bhattacharyya distance between the two multi-target densities and then to decide the optimal control strategy. The specific sensor control pseudocode is described in Algorithm 1.Algorithm 1: Sensor control subroutine.
Input:rk|k−1i,pk|k−1ii=1Mk|k−1,pk|k−1ix=∑j=1Lk|k−1iwk−1i,jδxk|k−1i.jxOutput: The optimal sensor control uk.(1)
Extract the multi-target stateX⌢k|k−1=x⌢k|k−1ii=1N⌢k|k−1.(2)
LetUk=νll=1N.(3)
forl=1:Ndo(4)
Get a PIMSZkν based on the X⌢k|k−1 and ν.(5)
Use the TOMB sampling method to approximate the (32) and (34)–(39).(6)
Calculate theℛν based on the (44).(7)
end for(8)
Returnuk←argmaxEℛν for all the ν.


### 5.2. Sensor Control Based on Target Threat Assessment

Target threat assessment plays an important role in the military fields. For example, it is a very significant factor for hostile surveillance and analysis of enemy tactics. In general, the purpose of target threat assessment is to prioritize multiple targets according to the target threat level. Moreover, obtaining an accurate target threat assessment needs to analyze various threat factors. For this reason, this subsection will use the TSM function [37, 38], which is a typical threat assessment function, to recognize the maximum threat target. Our ultimate purpose is to propose a sensor control strategy based on the maximum threat target.The following will elaborate on the threat-based sensor control in two steps.


#### 5.2.1. TSM Function

As an important component of the target threat assessment, assessing the target threat level is closely related to the target movement situation. Often, there are two ways to assess the target threat level according to the target movement parameters. The first method is to weight the threat level of various factors such as distance and speed [30]. However, this method requires to assign value to weight under different tracking scenarios and assessment conditions, and it is poor adaptability to the environment. The other is the classic TSM (Tactical Significance Map) function [31], which takes into account the contribution of each factor to the target threat level, and it is tractable and effective. Therefore, this paper selects the TSM function to assess the target threat level at the current time.Assume that the state vector of the current targeti can be expressed as xi=px,i,py,i,px,i',py,i', where pi and pi′ represent the position and velocity, respectively. And the sensor position is xs=px,s,py,sT. The distance from the target to the monitoring center (sensor) is expressed as(45)di=px,i−px,s2+py,i−py,s2.Given that the tactical importance of the target is related to the distancedi from the target to the monitoring center, then the TSM function expression is(46)fTSMxi∼exp−di22σTSM2.It can be seen from (46) that the smaller the distance between the target and the monitoring center, the higher the threat degree of the target. σTSM is related to the speed and course of the target, and it can be described as follows:(47)σTSMpi,pi′=1−θpi,pi′πk0pi′+m0,where k0 and m0 are constants. θpi,pi′ is the course angle of the target i. It can be expressed as(48)θpi,pi′=cos−1pi,pi′pipi′.In summary, the final expression of the TSM function is(49)fTSMxi=exp−d2221−θpi,pi′/π2k0pi′+m02.Obviously, according to the TSM function, we can determine the maximum threat targetx⌢th,k|k−1 among the multi-target state set X⌢k|k−1.


#### 5.2.2. Threat-Based Sensor Control

As an important work, we study to use the Bhattacharyya distance to solve the threat-based sensor control. Obviously, the prior and the posterior densities of the maximum threat target are two essential components. Hence, we need to extract the probability density of the maximum threat target from the multi-Bernoulli density. Also encouraging, the CBMeMBer filter provides us with great convenience. In essence, the multi-Bernoulli RFS is characterized by a set of fixed and independent Bernoulli parameters. If the maximum threat targetx⌢th,k|k−1 comes from the i−th Bernoulli parameter rk|k−1i,pk|k−1i, it is clear that pk|k−1i is the predicted probability density of the maximum threat target, and we can assume that the pk|k−1i is(50)pth,k|k−1ix=∑j=1Lk|k−1iwth,k|k−1i,jδxth,k|k−1i,jx,where wk|k−1i,j is the weight of the maximum threat target particle.As another component, the posterior probability density of the maximum threat target is usually obtained by updating (50) with the PIMS. According to (27), the posterior probability density can be expressed as(51)pth,k|kix=∑j=1Lk|k−1iw˜th,U,k∗i,jzδxth,k|k−1i,jx,where(52)w˜th,U,k∗i,jz=wth,U,k∗i,jz∑j=1Lk|k−1iwth,U,k∗i,jz,wth,U,k∗i,jz=wth,k|k−1i,jgkz|xth,k|k−1i,j.For each controlν∈Uk, we can directly use (50) and (51) to calculate the Bhattacharyya distance and then decide the optimal sensor control strategy. Figure 1 shows the schematic block diagram of threat-based sensor control, and subroutine is described in Algorithm 2.Figure 1
The schematic block diagram of threat-based sensor control.Algorithm 2: Threat-based sensor control subroutine.
Input:rk|k−1i,pkk−1ii=1Mk|k−1,pk|k−1ix=∑j=1Lk|k−1iwk−1i,jδxk|k−1i.jx.Output: The optimal sensor control uk.(1)
Extract the multi-target stateX⌢k|k−1=x⌢k|k−1ii=1N⌢k|k−1.(2)
Use the TSM function to recognize the maximum threat targetx⌢th,k|k−1.(3)
Extract thepth,k|k−1ix from the predicted multi-Bernoulli density.(4)
LetUk=νll=1N.(5)
forl=1:Ndo(6)
Get a PIMSZkν based on the X⌢th,k|k−1 and ν.(7)
Use the PIMS to update thepth,k|k−1i.(8)
Calculate theℛν based on the (44).(9)
end for(10)
Returnuk←argmaxEℛν for all the ν.


## 5.1. Sensor Control Based on the Multi-Target Information Gain

In this subsection, we employ Bhattacharyya distance as the measure of information gain between the two multi-target densities obtained by the TOMB particle sampling method. The purpose of this strategy is to maximize the Bhattacharyya distance between the two multi-target densities and then to decide the optimal control strategy. The specific sensor control pseudocode is described in Algorithm 1.Algorithm 1: Sensor control subroutine.
Input:rk|k−1i,pk|k−1ii=1Mk|k−1,pk|k−1ix=∑j=1Lk|k−1iwk−1i,jδxk|k−1i.jxOutput: The optimal sensor control uk.(1)
Extract the multi-target stateX⌢k|k−1=x⌢k|k−1ii=1N⌢k|k−1.(2)
LetUk=νll=1N.(3)
forl=1:Ndo(4)
Get a PIMSZkν based on the X⌢k|k−1 and ν.(5)
Use the TOMB sampling method to approximate the (32) and (34)–(39).(6)
Calculate theℛν based on the (44).(7)
end for(8)
Returnuk←argmaxEℛν for all the ν.


## 5.2. Sensor Control Based on Target Threat Assessment

Target threat assessment plays an important role in the military fields. For example, it is a very significant factor for hostile surveillance and analysis of enemy tactics. In general, the purpose of target threat assessment is to prioritize multiple targets according to the target threat level. Moreover, obtaining an accurate target threat assessment needs to analyze various threat factors. For this reason, this subsection will use the TSM function [37, 38], which is a typical threat assessment function, to recognize the maximum threat target. Our ultimate purpose is to propose a sensor control strategy based on the maximum threat target.The following will elaborate on the threat-based sensor control in two steps.


### 5.2.1. TSM Function

As an important component of the target threat assessment, assessing the target threat level is closely related to the target movement situation. Often, there are two ways to assess the target threat level according to the target movement parameters. The first method is to weight the threat level of various factors such as distance and speed [30]. However, this method requires to assign value to weight under different tracking scenarios and assessment conditions, and it is poor adaptability to the environment. The other is the classic TSM (Tactical Significance Map) function [31], which takes into account the contribution of each factor to the target threat level, and it is tractable and effective. Therefore, this paper selects the TSM function to assess the target threat level at the current time.Assume that the state vector of the current targeti can be expressed as xi=px,i,py,i,px,i',py,i', where pi and pi′ represent the position and velocity, respectively. And the sensor position is xs=px,s,py,sT. The distance from the target to the monitoring center (sensor) is expressed as(45)di=px,i−px,s2+py,i−py,s2.Given that the tactical importance of the target is related to the distancedi from the target to the monitoring center, then the TSM function expression is(46)fTSMxi∼exp−di22σTSM2.It can be seen from (46) that the smaller the distance between the target and the monitoring center, the higher the threat degree of the target. σTSM is related to the speed and course of the target, and it can be described as follows:(47)σTSMpi,pi′=1−θpi,pi′πk0pi′+m0,where k0 and m0 are constants. θpi,pi′ is the course angle of the target i. It can be expressed as(48)θpi,pi′=cos−1pi,pi′pipi′.In summary, the final expression of the TSM function is(49)fTSMxi=exp−d2221−θpi,pi′/π2k0pi′+m02.Obviously, according to the TSM function, we can determine the maximum threat targetx⌢th,k|k−1 among the multi-target state set X⌢k|k−1.


### 5.2.2. Threat-Based Sensor Control

As an important work, we study to use the Bhattacharyya distance to solve the threat-based sensor control. Obviously, the prior and the posterior densities of the maximum threat target are two essential components. Hence, we need to extract the probability density of the maximum threat target from the multi-Bernoulli density. Also encouraging, the CBMeMBer filter provides us with great convenience. In essence, the multi-Bernoulli RFS is characterized by a set of fixed and independent Bernoulli parameters. If the maximum threat targetx⌢th,k|k−1 comes from the i−th Bernoulli parameter rk|k−1i,pk|k−1i, it is clear that pk|k−1i is the predicted probability density of the maximum threat target, and we can assume that the pk|k−1i is(50)pth,k|k−1ix=∑j=1Lk|k−1iwth,k|k−1i,jδxth,k|k−1i,jx,where wk|k−1i,j is the weight of the maximum threat target particle.As another component, the posterior probability density of the maximum threat target is usually obtained by updating (50) with the PIMS. According to (27), the posterior probability density can be expressed as(51)pth,k|kix=∑j=1Lk|k−1iw˜th,U,k∗i,jzδxth,k|k−1i,jx,where(52)w˜th,U,k∗i,jz=wth,U,k∗i,jz∑j=1Lk|k−1iwth,U,k∗i,jz,wth,U,k∗i,jz=wth,k|k−1i,jgkz|xth,k|k−1i,j.For each controlν∈Uk, we can directly use (50) and (51) to calculate the Bhattacharyya distance and then decide the optimal sensor control strategy. Figure 1 shows the schematic block diagram of threat-based sensor control, and subroutine is described in Algorithm 2.Figure 1
The schematic block diagram of threat-based sensor control.Algorithm 2: Threat-based sensor control subroutine.
Input:rk|k−1i,pkk−1ii=1Mk|k−1,pk|k−1ix=∑j=1Lk|k−1iwk−1i,jδxk|k−1i.jx.Output: The optimal sensor control uk.(1)
Extract the multi-target stateX⌢k|k−1=x⌢k|k−1ii=1N⌢k|k−1.(2)
Use the TSM function to recognize the maximum threat targetx⌢th,k|k−1.(3)
Extract thepth,k|k−1ix from the predicted multi-Bernoulli density.(4)
LetUk=νll=1N.(5)
forl=1:Ndo(6)
Get a PIMSZkν based on the X⌢th,k|k−1 and ν.(7)
Use the PIMS to update thepth,k|k−1i.(8)
Calculate theℛν based on the (44).(9)
end for(10)
Returnuk←argmaxEℛν for all the ν.


## 5.2.1. TSM Function

As an important component of the target threat assessment, assessing the target threat level is closely related to the target movement situation. Often, there are two ways to assess the target threat level according to the target movement parameters. The first method is to weight the threat level of various factors such as distance and speed [30]. However, this method requires to assign value to weight under different tracking scenarios and assessment conditions, and it is poor adaptability to the environment. The other is the classic TSM (Tactical Significance Map) function [31], which takes into account the contribution of each factor to the target threat level, and it is tractable and effective. Therefore, this paper selects the TSM function to assess the target threat level at the current time.Assume that the state vector of the current targeti can be expressed as xi=px,i,py,i,px,i',py,i', where pi and pi′ represent the position and velocity, respectively. And the sensor position is xs=px,s,py,sT. The distance from the target to the monitoring center (sensor) is expressed as(45)di=px,i−px,s2+py,i−py,s2.Given that the tactical importance of the target is related to the distancedi from the target to the monitoring center, then the TSM function expression is(46)fTSMxi∼exp−di22σTSM2.It can be seen from (46) that the smaller the distance between the target and the monitoring center, the higher the threat degree of the target. σTSM is related to the speed and course of the target, and it can be described as follows:(47)σTSMpi,pi′=1−θpi,pi′πk0pi′+m0,where k0 and m0 are constants. θpi,pi′ is the course angle of the target i. It can be expressed as(48)θpi,pi′=cos−1pi,pi′pipi′.In summary, the final expression of the TSM function is(49)fTSMxi=exp−d2221−θpi,pi′/π2k0pi′+m02.Obviously, according to the TSM function, we can determine the maximum threat targetx⌢th,k|k−1 among the multi-target state set X⌢k|k−1.


## 5.2.2. Threat-Based Sensor Control

As an important work, we study to use the Bhattacharyya distance to solve the threat-based sensor control. Obviously, the prior and the posterior densities of the maximum threat target are two essential components. Hence, we need to extract the probability density of the maximum threat target from the multi-Bernoulli density. Also encouraging, the CBMeMBer filter provides us with great convenience. In essence, the multi-Bernoulli RFS is characterized by a set of fixed and independent Bernoulli parameters. If the maximum threat targetx⌢th,k|k−1 comes from the i−th Bernoulli parameter rk|k−1i,pk|k−1i, it is clear that pk|k−1i is the predicted probability density of the maximum threat target, and we can assume that the pk|k−1i is(50)pth,k|k−1ix=∑j=1Lk|k−1iwth,k|k−1i,jδxth,k|k−1i,jx,where wk|k−1i,j is the weight of the maximum threat target particle.As another component, the posterior probability density of the maximum threat target is usually obtained by updating (50) with the PIMS. According to (27), the posterior probability density can be expressed as(51)pth,k|kix=∑j=1Lk|k−1iw˜th,U,k∗i,jzδxth,k|k−1i,jx,where(52)w˜th,U,k∗i,jz=wth,U,k∗i,jz∑j=1Lk|k−1iwth,U,k∗i,jz,wth,U,k∗i,jz=wth,k|k−1i,jgkz|xth,k|k−1i,j.For each controlν∈Uk, we can directly use (50) and (51) to calculate the Bhattacharyya distance and then decide the optimal sensor control strategy. Figure 1 shows the schematic block diagram of threat-based sensor control, and subroutine is described in Algorithm 2.Figure 1
The schematic block diagram of threat-based sensor control.Algorithm 2: Threat-based sensor control subroutine.
Input:rk|k−1i,pkk−1ii=1Mk|k−1,pk|k−1ix=∑j=1Lk|k−1iwk−1i,jδxk|k−1i.jx.Output: The optimal sensor control uk.(1)
Extract the multi-target stateX⌢k|k−1=x⌢k|k−1ii=1N⌢k|k−1.(2)
Use the TSM function to recognize the maximum threat targetx⌢th,k|k−1.(3)
Extract thepth,k|k−1ix from the predicted multi-Bernoulli density.(4)
LetUk=νll=1N.(5)
forl=1:Ndo(6)
Get a PIMSZkν based on the X⌢th,k|k−1 and ν.(7)
Use the PIMS to update thepth,k|k−1i.(8)
Calculate theℛν based on the (44).(9)
end for(10)
Returnuk←argmaxEℛν for all the ν.


## 6. Simulation

### 6.1. Performance Evaluation of Multi-Target Tracking

In this paper, we use the OSPA metric [39] to evaluate the multi-target tracking performance because it jointly captures differences in cardinality and individual elements between two finite sets. It is defined as follows. Let the real multi-target state set and the estimated multi-target state set are x1,…,xn and Y=y1,…,ym, respectively. If n≤m, c>0, and p≥1, then the OSPA distance is given by(53)dp¯cX,Y=1mminπ∈Πm∑i=1ndcxi,yπip+cpm−n1/p,where d¯cx,y=minc,x−y and Πk denotes the set of permutations on 1,…,k for any positive integer k. If n>m, then dp¯cX,Y=dp¯cY,X.


### 6.2. Sensor Control Set

If the actual sensor position at timek is xs,k=xs,k,ys,kT and sensor control set is Uk+1 at time k+1, then all possible locations for the sensor are(54)Uk+1=xs,k+jvs,cTNRcosl2π/Nθ;ys,k+jvs,cTNRsinl2π/Nθ,j=1,…,NR;l=1,…,Nθ.In this paper, selectNR=2 and Nθ=8, then Uk+1 contains a total of 17 kinds of control strategies (including the sensor in a fixed position). vs,c represents the sensor speed. In fact, the NR and Nθ correlate with the quality of the tracking optimization. If they increase further, there is limited room for estimation improvement, but the computational burden (running time) has increased significantly.


### 6.3. Scene Parameters

This paper considers the range-bearing tracking (RBT). Set the monitoring area as−π,π×0,10002m. The target trajectories in the area are the near-normal motion model (Nearly constant velocity model, NCVM) [40], and there are a total of five targets. The state transition density of the motion model is(55)fxk|xk−1=Nxk;Fkxk−1,Qk,where(56)Fk=I2TI202I2,Qk=σv2T44I2T32I2T32I2T2I2,where In represents n×n unit matrix, σv=5m/s2 is the standard deviation of the process noise, and T=1s is the sampling period with a total of 50 samples. The birth process is a multi-Bernoulli RFS with density πΓ=rΓi,pΓii=15, where rΓi=0.01, pΓix=Nx;mγi,Pγ, mγ1=−450;−450;0;0T, mγ2=300,−300,0,0T, mγ3=400,300,0,0T, mγ4=−300,220,0,0T, mγ5=300,−200,0,0T, and Pγ=diag100,100,100,100T. The probability of target survival and detection are pS,k=0.99 and pD,k=0.98, respectively. Clutter is Poisson with λ=5 (clutter number) and is uniformly distributed over the region −π,π×0,10002m. Pruning of hypothesized tracks is performed with a weight threshold of rT=10−3. The number of sampled particles per track is Lk=500, and the minimum number of particles per track is Lmin=100.For range-bearing tracking (RBT), (4) has the following form:(57)zk=xk−xs,k2+yk−ys,k2tan−1yk−ys,kxk−xs,k+vk,where vk∼N⋅;02×1,Rk, Rk=diagσr2,σθ2T, σr=5m, and σθ=π/180rad/s.In this simulation, the software and hardware conditions are MATLAB 2016, Windows 7, Inter Core i5-4590 CPU 3.30 GHz, and RAM 8 GB.


### 6.4. Simulation Analysis

In this simulation, Figure2 shows the actual target trajectories with initial sensor position xs,0=0,0T. In order to illustrate the control performance, we compare the estimation accuracy for five sensor control strategies in the same scenario. The first control strategy is “Stationary,” which denotes the sensor in a fixed position. “Max MT divergence,” which represents the sensor control based on maximizing the information divergence between two multi-target densities, is described in Subsection 5.1. The third is “Random control,” that is to say we choose ν randomly from Uk. “Threat-based” is presented in Section 5.2. The fifth control strategy is “PENT.”Figure 2
Actual target trajectories and maximum threat target.Figure3 shows the sensor control trajectory of “Max MT divergence” in a single experiment. In the whole control process, the sensor automatically adjusts its position according to the current filtering result with the aim of acquiring the best observation for multi-target. Furthermore, with the birth and death of the target, the sensor position will produce obvious maneuvering control to meet the change of multi-target density. In the following process, the sensor will always be controlled through the criterion of information gain maximization.Figure 3
Sensor movement trajectory for Max MT divergence policy.In order to verify the control performance of “Max MT divergence” in RBT, 200 Monte Carlo (MC) simulations are performed. The OSPA statistics of multi-target position estimation for all control strategies are shown in Figure4. Obviously, “Max MT divergence” is clearly superior to the other four strategies. It is hardly surprising because this control strategy solves the Bhattacharyya distance between the prior and the posterior multi-target densities based on the SMC-CBMeMBer filter. Then, under the criterion of maximizing the information gain, the optimal sensor control strategy is obtained. In addition, “PENT” performs poorly. The main reason is that “PENT” is based on the maximization of the posterior expected number of targets as the evaluation criterion and not concerned with localization uncertainty. “Threat-based” performs slightly worse than “Max MT divergence,” where the reason is that this strategy is decided only by the Bhattacharyya distance between two probability densities of the maximum threat target. Surprisingly, “Stationary” also performs better. One possible explanation for this result is that, in our scenario, the sensor is surrounded by multi-target, i.e., the multi-target is distributed in the four quadrants with the sensor as origin. Simultaneously, all targets have better movement trend to this fixed sensor. Thus, the sensor obtains better measurement information of multi-target. It is worth noting that “random control” also has a good effect. Maybe, the “random irregular” motion gives the tracking system more multi-target information gain, and the multi-target tracking performance is correspondingly improved.Figure 4
Position OSPA statistics of multi-target in RBT.In addition, for “Threat-based,” Figure2 also shows the maximum threat target each moment (as indicated by the red “+” in Figure 2), which is recognized by the TSM function. Obviously, the maximum threat target changes over time. In the first twenty seconds, target 1 is recognized as the maximum threat target. Afterwards, with the birth of the target (target 4), the maximum threat target shifts the target 4. Subsequently, target 4 dies and target 1 shifts the maximum threat target again. Especially, with the changing of target movement situation such as speed, course, and distance, the target threat level is continuously assessed based on the value of the TSM function, which determines the maximum threat target.Figure5 shows the sensor control trajectory of “Threat-based” in a single simulation. As we can see, in the whole control process, the sensor automatically adjusts its position, where the goal is to improve tracking performance for the maximum threat target. In the first twenty seconds, the sensor moves toward target 1 which is the maximum threat target during this period. With target 4 turning into the maximum threat target, the sensor clearly maneuvers itself toward target 4. Afterwards, the sensor repeats the previous behavior. The main reason is that “Threat-based” always tries to adjust sensor position to maximize the information gain between the probability distributions of the maximum threat target. Therefore, when the maximum threat target changes, the sensor always adapts quickly to this change.Figure 5
Sensor movement trajectory for threat-based policy.Figure6 shows the OSPA statistics of the maximum threat target position estimation for the five control strategies. Especially, “Threat-based” is clearly superior to the other four strategies. The reason is that “Threat-based” is decided by the Bhattacharyya distance between the probability distributions of the maximum threat target while the other four control strategies are not.Figure 6
Position OSPA statistics of the maximum threat target in RBT.In addition, Figure7 shows the targets cardinality estimation and its standard deviation. It can be seen that cardinality estimation statistics of all control strategies are close to the actual values. However, by comparing the standard deviation of all control strategies, “PENT” is obviously superior to the other four strategies. The main reason is that PENT has to maximize the posterior expectation of the target cardinality.Figure 7
Cardinality estimation statistics.To illustrate the computational burden for each control strategy, we calculate the average running time for all control strategies in 200 MC simulations as shown in Figure8. It can be seen that the average running time of “Stationary” and “Random control” is very short; the reason is that they have no computational cost of solving sensor control strategy. “Threat-based” takes more time, which mainly comes from recognizing the maximum threat target and calculating the Bhattacharyya distance. Since this divergence is only calculated for the threat target, the running time is not as long as “Max MT divergence,” which solves the Bhattacharyya distance between the two multi-target densities. “PENT” performs slowest; this is because it needs to update all predicted particles and then calculates the corresponding cost function. In addition, it is noteworthy that the average running time of all control strategies increases drastically as the target number increases.Figure 8
Average running time for five control strategies.Figure9 shows all the possible sensor position of “Threat-based” in 200 MC simulations. Although the sensor trajectory may differ for each MC simulation, the sensor trajectory cloud can still reflect the overall sensor movement trend of “Threat-based.”Figure 9
Sensor trajectory cloud for threat-based policy.


## 6.1. Performance Evaluation of Multi-Target Tracking

In this paper, we use the OSPA metric [39] to evaluate the multi-target tracking performance because it jointly captures differences in cardinality and individual elements between two finite sets. It is defined as follows. Let the real multi-target state set and the estimated multi-target state set are x1,…,xn and Y=y1,…,ym, respectively. If n≤m, c>0, and p≥1, then the OSPA distance is given by(53)dp¯cX,Y=1mminπ∈Πm∑i=1ndcxi,yπip+cpm−n1/p,where d¯cx,y=minc,x−y and Πk denotes the set of permutations on 1,…,k for any positive integer k. If n>m, then dp¯cX,Y=dp¯cY,X.


## 6.2. Sensor Control Set

If the actual sensor position at timek is xs,k=xs,k,ys,kT and sensor control set is Uk+1 at time k+1, then all possible locations for the sensor are(54)Uk+1=xs,k+jvs,cTNRcosl2π/Nθ;ys,k+jvs,cTNRsinl2π/Nθ,j=1,…,NR;l=1,…,Nθ.In this paper, selectNR=2 and Nθ=8, then Uk+1 contains a total of 17 kinds of control strategies (including the sensor in a fixed position). vs,c represents the sensor speed. In fact, the NR and Nθ correlate with the quality of the tracking optimization. If they increase further, there is limited room for estimation improvement, but the computational burden (running time) has increased significantly.


## 6.3. Scene Parameters

This paper considers the range-bearing tracking (RBT). Set the monitoring area as−π,π×0,10002m. The target trajectories in the area are the near-normal motion model (Nearly constant velocity model, NCVM) [40], and there are a total of five targets. The state transition density of the motion model is(55)fxk|xk−1=Nxk;Fkxk−1,Qk,where(56)Fk=I2TI202I2,Qk=σv2T44I2T32I2T32I2T2I2,where In represents n×n unit matrix, σv=5m/s2 is the standard deviation of the process noise, and T=1s is the sampling period with a total of 50 samples. The birth process is a multi-Bernoulli RFS with density πΓ=rΓi,pΓii=15, where rΓi=0.01, pΓix=Nx;mγi,Pγ, mγ1=−450;−450;0;0T, mγ2=300,−300,0,0T, mγ3=400,300,0,0T, mγ4=−300,220,0,0T, mγ5=300,−200,0,0T, and Pγ=diag100,100,100,100T. The probability of target survival and detection are pS,k=0.99 and pD,k=0.98, respectively. Clutter is Poisson with λ=5 (clutter number) and is uniformly distributed over the region −π,π×0,10002m. Pruning of hypothesized tracks is performed with a weight threshold of rT=10−3. The number of sampled particles per track is Lk=500, and the minimum number of particles per track is Lmin=100.For range-bearing tracking (RBT), (4) has the following form:(57)zk=xk−xs,k2+yk−ys,k2tan−1yk−ys,kxk−xs,k+vk,where vk∼N⋅;02×1,Rk, Rk=diagσr2,σθ2T, σr=5m, and σθ=π/180rad/s.In this simulation, the software and hardware conditions are MATLAB 2016, Windows 7, Inter Core i5-4590 CPU 3.30 GHz, and RAM 8 GB.


## 6.4. Simulation Analysis

In this simulation, Figure2 shows the actual target trajectories with initial sensor position xs,0=0,0T. In order to illustrate the control performance, we compare the estimation accuracy for five sensor control strategies in the same scenario. The first control strategy is “Stationary,” which denotes the sensor in a fixed position. “Max MT divergence,” which represents the sensor control based on maximizing the information divergence between two multi-target densities, is described in Subsection 5.1. The third is “Random control,” that is to say we choose ν randomly from Uk. “Threat-based” is presented in Section 5.2. The fifth control strategy is “PENT.”Figure 2
Actual target trajectories and maximum threat target.Figure3 shows the sensor control trajectory of “Max MT divergence” in a single experiment. In the whole control process, the sensor automatically adjusts its position according to the current filtering result with the aim of acquiring the best observation for multi-target. Furthermore, with the birth and death of the target, the sensor position will produce obvious maneuvering control to meet the change of multi-target density. In the following process, the sensor will always be controlled through the criterion of information gain maximization.Figure 3
Sensor movement trajectory for Max MT divergence policy.In order to verify the control performance of “Max MT divergence” in RBT, 200 Monte Carlo (MC) simulations are performed. The OSPA statistics of multi-target position estimation for all control strategies are shown in Figure4. Obviously, “Max MT divergence” is clearly superior to the other four strategies. It is hardly surprising because this control strategy solves the Bhattacharyya distance between the prior and the posterior multi-target densities based on the SMC-CBMeMBer filter. Then, under the criterion of maximizing the information gain, the optimal sensor control strategy is obtained. In addition, “PENT” performs poorly. The main reason is that “PENT” is based on the maximization of the posterior expected number of targets as the evaluation criterion and not concerned with localization uncertainty. “Threat-based” performs slightly worse than “Max MT divergence,” where the reason is that this strategy is decided only by the Bhattacharyya distance between two probability densities of the maximum threat target. Surprisingly, “Stationary” also performs better. One possible explanation for this result is that, in our scenario, the sensor is surrounded by multi-target, i.e., the multi-target is distributed in the four quadrants with the sensor as origin. Simultaneously, all targets have better movement trend to this fixed sensor. Thus, the sensor obtains better measurement information of multi-target. It is worth noting that “random control” also has a good effect. Maybe, the “random irregular” motion gives the tracking system more multi-target information gain, and the multi-target tracking performance is correspondingly improved.Figure 4
Position OSPA statistics of multi-target in RBT.In addition, for “Threat-based,” Figure2 also shows the maximum threat target each moment (as indicated by the red “+” in Figure 2), which is recognized by the TSM function. Obviously, the maximum threat target changes over time. In the first twenty seconds, target 1 is recognized as the maximum threat target. Afterwards, with the birth of the target (target 4), the maximum threat target shifts the target 4. Subsequently, target 4 dies and target 1 shifts the maximum threat target again. Especially, with the changing of target movement situation such as speed, course, and distance, the target threat level is continuously assessed based on the value of the TSM function, which determines the maximum threat target.Figure5 shows the sensor control trajectory of “Threat-based” in a single simulation. As we can see, in the whole control process, the sensor automatically adjusts its position, where the goal is to improve tracking performance for the maximum threat target. In the first twenty seconds, the sensor moves toward target 1 which is the maximum threat target during this period. With target 4 turning into the maximum threat target, the sensor clearly maneuvers itself toward target 4. Afterwards, the sensor repeats the previous behavior. The main reason is that “Threat-based” always tries to adjust sensor position to maximize the information gain between the probability distributions of the maximum threat target. Therefore, when the maximum threat target changes, the sensor always adapts quickly to this change.Figure 5
Sensor movement trajectory for threat-based policy.Figure6 shows the OSPA statistics of the maximum threat target position estimation for the five control strategies. Especially, “Threat-based” is clearly superior to the other four strategies. The reason is that “Threat-based” is decided by the Bhattacharyya distance between the probability distributions of the maximum threat target while the other four control strategies are not.Figure 6
Position OSPA statistics of the maximum threat target in RBT.In addition, Figure7 shows the targets cardinality estimation and its standard deviation. It can be seen that cardinality estimation statistics of all control strategies are close to the actual values. However, by comparing the standard deviation of all control strategies, “PENT” is obviously superior to the other four strategies. The main reason is that PENT has to maximize the posterior expectation of the target cardinality.Figure 7
Cardinality estimation statistics.To illustrate the computational burden for each control strategy, we calculate the average running time for all control strategies in 200 MC simulations as shown in Figure8. It can be seen that the average running time of “Stationary” and “Random control” is very short; the reason is that they have no computational cost of solving sensor control strategy. “Threat-based” takes more time, which mainly comes from recognizing the maximum threat target and calculating the Bhattacharyya distance. Since this divergence is only calculated for the threat target, the running time is not as long as “Max MT divergence,” which solves the Bhattacharyya distance between the two multi-target densities. “PENT” performs slowest; this is because it needs to update all predicted particles and then calculates the corresponding cost function. In addition, it is noteworthy that the average running time of all control strategies increases drastically as the target number increases.Figure 8
Average running time for five control strategies.Figure9 shows all the possible sensor position of “Threat-based” in 200 MC simulations. Although the sensor trajectory may differ for each MC simulation, the sensor trajectory cloud can still reflect the overall sensor movement trend of “Threat-based.”Figure 9
Sensor trajectory cloud for threat-based policy.


## 7. Conclusion

In this paper, we propose two sensor control strategies based on information criteria in the framework of POMDP. The first proposed strategy is solved by the Bhattacharyya distance between the prior and the posterior multi-target densities where the goal is to optimize the overall multi-target tracking performance. In this strategy, we present a TOMB particle sampling method to approximate the multi-target density and use the method to derive the solution of Bhattacharyya distance in detail. As another important contribution of this paper, the second strategy is proposed based on target threat assessment, where the goal is to prioritize multiple threat targets and then to track preferentially the maximum threat target. The key of this proposed strategy is that we innovatively extract the probability distribution of the maximum threat target from the multi-Bernoulli density based on the TSM function. Simulation results verify the effectiveness of the proposed strategies. Obviously, these control strategies can be extended to other approximate multi-target filters based on RFS, such as the PHD filter, CPHD filter, and generalized labeled multi-Bernoulli (GLMB) filter. In future work, we will further research the multi-sensor management for multi-target tracking, including the rational resources allocation and the relative control for multi-sensor.



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*Source: 3536618-2022-07-08.xml*