### 3.1 The measurement models

Let \(F_{s,k}\) be the observation area for the sensor *s*. That is, only the targets in the field \(F_{s,k}\) may be observed. Assume that the state of sensor *s* is \(y_{s,k}\in R^{n_y}\), where \(R_{n_y}\) is the \(n_y\)-dimensional Euclidean space. The measurement model can be modeled by:

$$\begin{aligned} Z_{s,k} = \cup _{x_k\in X_k}\Theta _{s,k}(x_k, y_{s,k})\cup K_{s,k}(y_{s,k}) \end{aligned}$$

(15)

The measurement model for sensor *s* is given by:

$$\begin{aligned} \Theta ^s_k(y_{s,k},x_k) = \left\{ \begin{array}{l} \quad \{z_{s,k}\}~~ with~~ P_d(y_{s,k},x_k), if~~\mathscr {P}(x_k)\in F_{s,k} \\ \quad \emptyset ~~with~~ 1-P_d(y_{s,k}, x_k)~~\text{ otherwise } \end{array} \right. \end{aligned}$$

where \(\mathscr {P}(\cdot )\) is a projection from certain state space \(R^{n_x}\) to position space \(R^2\), i.e., \(\mathscr {P}: R^{n_x}\rightarrow R^2\) for target state.

### 3.2 The models of multi-sensor probability of detection

In general, radar is a feasible sensor for tracking targets over long distances. Its field of view can be regarded as a fan-shaped area in polar coordinates, as shown in Fig. 1. It is defined by:

$$\begin{aligned} A_{s,k}: [\theta _{s,\min }(y_{s,k}),\theta _{s,\max }(y_{s,k})]\times [0,r_s(y_{s,k})]\subseteq [-\pi ,\pi ] \times \mathbb {R}^+ \end{aligned}$$

(16)

where \(A_{s,k}\) is the FoV of the sensor *s*, \(\mathbb {R}^+\) is the space of positive real numbers. Accordingly, the corresponding Euclidean space position can be gotten by a projection \(q: A_{s,k}\rightarrow R^2\). Thus, the FoV area in Euclidean space is defined by \(F_{s,k}=\{q(a_{s,k})| \text{ for } \text{ all }~ a_{s,k}\in A_{s,k}\}\). The model of probability of detection for sensor *s* is given by

$$\begin{aligned} P_{s,d}(x_k) = \left\{ \begin{array}{l} \quad P_D\text{ exp }\{{-\frac{||{\mathscr {P}}(x_k) - {\mathscr {P}}(y_{s,k})||_2}{2}}\},~~ \text{ if }~~ \mathscr {P}(x_k)\in F_{s,k} \\ \quad 0, ~~\text{ otherwise } \end{array} \right. \end{aligned}$$

where \(P_D\) is a constant, \(||\cdot ||_2\) is 2-norm. This model implies that a target may be detected with a certain probability if it locates in area \(F_{s,k}\) of sensor *s*, and with probability of detection zero otherwise.

### 3.3 Target birth models of multi-sensor measurement driven

In most existing algorithms, the FoV of a sensor is assumed to cover all the track region and there is no any dead zone. Hence, the target birth model is assumed to be prior and known. But for sensors with partly overlapping FoV, a sensor scans only part of the surveillance region. Hence, it is impossible to detect a target when the target is born outside the FoV. In order to build the target birth model, we refer the measurement-driven birth model given in [38,39,40] and extend it to the multi-sensor case.

A multi-Bernoulli RFS *X* is a union of a fixed number of independent Bernoulli RFSs \(X^{(i)}\) with existence probability \(r^{(i)}_B \in (0,1)\) and probability density \(p^{(i)}_B(i = 1,\ldots ,M)\)[41]. So, the probability density of a multi-Bernoulli RFS can be abbreviated as:

$$\begin{aligned} \pi = \left\{ ({r_B}^{\left( i \right) },{p_B}^{\left( i \right) }) \right\} _{i = 1}^M \end{aligned}$$

(17)

Assume that the form of the birth model with generalized labeled multi-Bernoulli (GLMB) is as follows:

$$\begin{aligned} {\pi _B}({\textbf{X}_+}) = \Delta ({\textbf{X}_+}){w_B}(\mathcal{L}({\textbf{X}_+ })){[{p_B}]^{{\textbf{X}_+}}} \end{aligned}$$

(18)

where

$$\begin{aligned} {w_B}(I) = \prod \limits _{i \in \mathbb {B}} {(1 - r_B^{(i)})} \prod \limits _{l \in I} {\frac{{{1_{\mathbb {B}}}(l)r_B^{(l)}}}{{1 - r_B^{(l)}}}} \end{aligned}$$

(19)

The existence probability can be initialized by the following equation:

$$\begin{aligned} {r_B}(z^1_k,\cdots ,z^{S}_k) = \min \left( {r_{B\max }},\prod _s\frac{\lambda ^s _B{r_U}(z^1_k,\cdots ,z^{S}_k)}{\sum \nolimits _{(\varsigma ^1_k,\cdots ,\varsigma ^S_k)\in Z^1_k\times \cdots \times Z^S_k} {{r_U}(\varsigma ^1_k,\cdots ,\varsigma ^S_k)} }\right) \end{aligned}$$

(20)

where

$$\begin{aligned} {r_U}(z^1_k,\cdots ,z^{S}_k) = 1 - \sum \limits _{(I,\xi ) \in \mathcal{F}\left( {\mathbb {L}} \right) \times \Xi } {\sum \limits _{\vartheta \in \Theta (I,S)} {{1_{{z_\theta }}}} } (z){w^{(I,\xi ,\vartheta )}} \end{aligned}$$

(21)

where \(\Theta (I,S)\) is the space of multi-sensor association map. \(\vartheta\) is a multi-sensor association map (defined next subsection).

### 3.4 Multi-sensor association map

In the GLMB filter, the association map plays a crucial role in the RFS multi-objective likelihood function.

###
**Definition 1**

[10]: The association map is a mapping: \(\theta\): \(\mathbb {L}\rightarrow \left\{ {0,1, \cdots ,|Z|} \right\}\). If \(\theta \left( i \right) = \theta \left( {i'} \right) > 0\), it means \(i = i'\), the set \(\Theta\) represents the associated map space, and its subset *I* can be represented by \(\Theta (I)\). The association diagram describes the correspondence between the trajectory and the measurement, the trajectory *l* produces a measure of \({z_{\theta \left( l \right) }} \in Z\), and the undetected trajectory is denoted by 0.

###
**Definition 2**

[42]: The multi-sensor association map is a mapping: \(\vartheta\): \(\mathbb {L}\rightarrow {S_1} \times {S_2} \times \cdots \times {S_m}\), \({S_s} = \left\{ {0,1 \cdots ,\left| {{Z_s}} \right| } \right\}\), where \(\vartheta \left( i \right) = \vartheta \left( {i'} \right) ,\theta \left( {i,l} \right) \ne 0\) means \(i = i'\), \(\theta \left( {i,l} \right)\) is the rth element in vector \(\theta \left( i \right)\). The set \(\Theta (I,S)\) represents the space of multi-sensor associated map, and its subset *I* can be represented by \(\Theta (I,S)\).

Under the independence of all sensors, the multi-sensor association map \(\vartheta\) can be expressed as

$$\begin{aligned} \vartheta \triangleq (\vartheta ^T_1,\cdots , \vartheta ^T_S) \end{aligned}$$

(22)

where \(\vartheta ^T_s\) can be seen as components of \(\vartheta\). Assume that multi-sensor measurements can be represented by \(Z_k \triangleq Z_{1,k}\cup \dots \cup Z_{S,k}\), multi-sensor posterior probability:

$$\begin{aligned} f(X_k|Z_k) = \frac{{g(Z_k|X_k)f(X_k)}}{{\int {g(Z_k|X_k)f(X_k)\delta X_k}}} \end{aligned}$$

(23)

where the integral is defined as:

$$\begin{aligned} \int {f(X_k)\delta X_k} = \sum \limits _{i = 0}^\infty {\frac{1}{{i!}}} \sum \limits _{\left( {{l_1}, \cdots {l_i}} \right) \in {{\mathbb {L}}^i}} {\int \limits _{{{\mathbb {X}}^i}} {f(\{ ({x_1},{l_1}), \cdots ({x_i},{l_i})\} )} } d({x_1}, \cdots {x_i}) \end{aligned}$$

(24)

Assume that the multi-sensor are independent, then multi-sensor likelihood function is given by:

$$\begin{aligned} g\left( {Z_k|X_k} \right) = \prod ^S_{s=1}{e^{-\left\langle {K_s,1} \right\rangle }}{K_s^{Z_{s,k}}}\sum \limits _{\vartheta _s \in \Theta (\mathcal L\left( X \right) ,S)} {{{\left[ {{\psi _{Z_{s,k}}}\left( {\cdot ;\vartheta _s } \right) } \right] }^X}} \end{aligned}$$

(25)

where

$$\begin{aligned} {\psi _{Z_{s,k}}}\left( {x,l;\vartheta _s } \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{P_{s,d}}\left( {x,l} \right) g\left( {{z_{\vartheta _s \left( l \right) }}|x,l} \right) }}{{k\left( {{z_{\vartheta _s \left( l \right) }}} \right) }},\;\;\;\;if~~\vartheta _s \left( l \right) > 0}\\ {1 - {P_{s,d}}\left( {x,l} \right) ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;if~~\vartheta _s \left( l \right) = 0} \end{array}} \right. \end{aligned}$$

(26)

### 3.5 GLMB estimation algorithm for multi-sensor with partly overlapping FoV

The basic principle of the multi-sensor tracking algorithm is to distribute the multi-sensor reasonably. At different times, the detection results of the individual sensors may be different. At each moment, we need to judge the target, to determine the target in which the sensor within the scope of observation. The specific multi-sensor GLMB estimation algorithm is as follows:

#### 3.5.1 Update step

If the multi-sensor multi-target a priori density function is the form of generalized labeled multi-Bernoulli, then the multi-sensor multi-target posterior probability density is also the form of generalized labeled multi-Bernoulli:

$$\begin{aligned} \begin{aligned} \pi \left( {\textbf{X}_k|Z_k} \right) =\Delta \left( {\textbf{X}_k} \right) \sum \limits _{\left( {I,\xi } \right) \in \mathcal{F}\left( {\mathbb {L}} \right) \times \Xi } {\sum \limits _{\vartheta \in \Theta (I,S) } {{w^{\left( {I,\xi ,\vartheta } \right) }}\left( Z_k \right) } } \times {\delta _I}\left( {\mathcal{L}\left( {\textbf{X}_k} \right) }\right) {\left[ {{p^{\left( {\xi ,\vartheta } \right) }}\left( {\cdot |Z_k} \right) } \right] ^{\textbf{X}_k}} \end{aligned} \end{aligned}$$

(27)

where \(\vartheta\) represents the current multi-sensor association map. The association parameters are defined as follows:

$$\begin{aligned} \begin{array}{lll} {w^{\left( {I,\xi ,\vartheta } \right) }}\left( Z_k \right) &{}=&{} \frac{{\delta _{{\vartheta ^{-1}}\left( {\left\{ \prod ^S_{s=1} (0:\left| Z_{s,k} \right| ) \right\} } \right) }\left( I \right) {w^{\left( {I,\xi } \right) }}{{\left[ {\eta _{Z_k}^{\left( {\xi ,\vartheta } \right) }} \right] }^I}}}{{\sum \limits _{\left( {I,\xi } \right) \in \mathcal{F}\left( {\mathbb {L}} \right) \times \Xi } {\sum \limits _{\vartheta \in \Theta (I,S) } {{\delta _{{\vartheta ^{ - 1}}\left( {\left\{ {0:\left| Z_k \right| } \right\} } \right) }}\left( I\right) {w^{\left( {I,\xi } \right) }}{{\left[ {\eta _{Z_k}^{\left( {\xi ,\vartheta } \right) }} \right] }^I}} } }}\\ {p^{\left( {\xi ,\vartheta } \right) }}\left( {x,l|Z_k} \right) &{}=&{} \frac{{{p^{\left( \xi \right) }}\left( {x,l} \right) {\psi _{Z_k}}\left( {x,l;\vartheta } \right) }}{{\eta _Z^{\left( {\xi ,\vartheta } \right) }\left( l \right) }}\\ \eta _{Z_k}^{\left( {\zeta ,\vartheta } \right) }\left( l \right) &{}=&{} \left\langle {{p^{\left( \xi \right) }}\left( { \cdot ,l} \right) ,{\psi _{Z_k}}\left( {\cdot ,l;\vartheta } \right) } \right\rangle \\ {\psi _{Z_k}}\left( {x,l;\vartheta } \right) &{}=&{} {\delta _0}\left( {\vartheta \left( l \right) } \right) {q_D}\left( {x,l} \right) \mathrm{{ }}+ \left( {1 - {\delta _0}\left( {\vartheta _s \left( l \right) } \right) } \right) \frac{\prod ^S_{s=1}{{P_{s,d}}\left( {x,l} \right) g\left( {{z_{\vartheta _s \left( l \right) }}|x,l} \right) }}{{\prod ^S_{s=1}K\left( {{z_{\vartheta _s \left( l \right) }}} \right) }} \end{array} \end{aligned}$$

(28)

where \(\mathop \Psi \nolimits _{Z_k} \left( {x,l;\vartheta }\right)\) is the likelihood function in the case of multi-sensor; assume that the sensors are independent of each other, it can be simplified as:\(\prod {\psi _{Z_{1,k}}\cdots {\psi _{{Z_{S,k}}}}\left( {\;:\;;\vartheta _s } \right) }\), suppose that the likelihood function is a Gaussian distribution,\({P_{s,d}}(x_k,l) = {P_{s,d}}\), \(g(z_{s,k}|x_k,l) = \mathscr {N}\left( {z_{s,k};\mathop H_{s,k} ,\mathop R_{s,k} }\right)\). \({H_{s,d}}\) and \({R_{s,d}}\) are the observation matrix and measurement noise covariance for sensor *s*. Assume that probability density \({p^{(\xi )}}( \cdot ,\ell )\) of target \(\ell\) follows a Gaussian mixture distribution:

$$\begin{aligned} {p^{(\xi )}}(\cdot ,\ell ) = \sum \limits _{i = 1}^{{J^{(\xi )}}(\ell )} {w_i^{(\xi )}}(\ell ,0)N(x;\mu _i^{(\xi )}(\ell ,0),P_i^{(\xi )}(\ell ,0)) \end{aligned}$$

(29)

Then we can get the following multi-sensor cost matrix:

$$\begin{aligned} {C_{i,S}} = - Ln\left[ \frac{{\prod ^S_{s=1}{P_{s,d}}\sum \limits _{j=1}^{{J^{(\xi )}}({l_i})} {{q_k}^{(\xi )}({z^s_{k,j}};{l_i})} }}{\prod ^S_{s=1}{(1 - {P_{s,d}})K({z^s_k})}}\right] \end{aligned}$$

(30)

where \(z^s_{k,j}\) represents the *j*th measurement of the *s*th sensor, the update history \((\xi ,\vartheta )\) and \(\eta _{Z,k}^{(\xi ,\vartheta )}\) of the multi-sensor association map are as follows:

$$\begin{aligned}{} & {} \eta _{Z_k}^{(\xi ,\vartheta )} = \prod ^S_{s=1}\sum \limits _{i = 1}^{{J^{(\xi ,s)}}(\ell )} {w_{z^s_{k,i}}^{(\xi ,\vartheta _s)}} (\ell ) \end{aligned}$$

(31)

$$\begin{aligned}{} & {} {p^{(\xi ,\vartheta )}}(x_k,\ell |Z_k) = \prod ^S_{s=1}\sum \limits _{i = 1}^{{J^{(\xi ,s)}}(\ell )} {\frac{{w_{z^s_{k,j}}^{(\xi ,\vartheta _s)}(\ell )}}{{\eta _{Z^s_k}^{(\xi ,\vartheta _s)}}}} \mathcal{N}(x;\mu _{z^s_{k,j}}^{(\xi ,\vartheta _s)}(\ell ),P_i^{(\xi ,\vartheta _s)}(\ell )) \end{aligned}$$

(32)

Although the multi-sensor labeled multi-Bernoulli filter has the same representation as the single-sensor, there are many differences between the multi-sensor and the single-sensor for the specific parameters:

$$\begin{aligned} w_{Z^s_k,i}^{(\xi ,\vartheta _s)}(\ell )= & {} w_{Z^s_k,i}^{(\xi ,\vartheta )}(\ell ,j - 1) \times \left\{ {\begin{array}{*{20}{c}} {\frac{{{P_{D,s}}q_i^\xi ({z_{\vartheta _s(\ell ,j)}};\ell )}}{{\kappa ({z_{\vartheta _s (\ell ,j)}})}}, if \vartheta _s (\ell ,j) > 0}\\ {1 - {P_{D,s}}, \quad if \vartheta _s (\ell ,j) = 0} \end{array}} \right. \end{aligned}$$

(33)

$$\begin{aligned} q_i^\xi ({z_{\vartheta _s (\ell ,j)}};\ell )= & {} \mathscr {N}(z^s_{k,j};{H_{k,s}}\mu _{Z^s_k,i}^{(\xi ,\vartheta _s)}(\ell , m-1),{H_{k,s}}P_i^{(\xi ,\vartheta _s )}(\ell ,j - 1)H_{k,s}^T + {R_{k,s}})\nonumber \\ \mu _{Z^s_k,i}^{(\xi ,\vartheta _s )}(\ell )= & {} \nonumber \\{} & {} \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} \mu _{Z,i}^{(\xi ,\vartheta )}(\ell ,j- 1) + K_i^{(\xi ,\vartheta _s )}\\ \times ({z_{\vartheta (\ell ,j)}} - {H_{k,s}}\mu _{z^s_{k,j}}^{(\xi ,\vartheta _s)}(\ell ,j - 1))H_{k,s}^T,\;\;if \vartheta _s((\ell ,j)> 0 \end{array}\\ {\mu _{Z^s_{k},i}^{(\xi ,\vartheta _s )}(\ell ,j - 1), \quad \quad if \vartheta (\ell ,j) = 0} \end{array}} \right. \nonumber \\ K_i^{(\xi ,\vartheta _s)}(\ell )= & {} \nonumber \\{} & {} \left\{ {\begin{array}{*{20}{c}} {P_i^{\xi ,\vartheta _s}(\ell , j-1)H_{k,s}^T{{({H_{k,s}}P_i^{(\xi ,\vartheta _s )}(\ell ,j - 1)H_{k,s}^T)}^{ - 1}} ,\quad if \vartheta _s(\ell ,j) > 0}\\ {0, \quad \quad if \vartheta _s (\ell ,j) = 0} \end{array}} \right. \end{aligned}$$

(34)

The above formula gives the recursive process of the measurement \({Z_1}\) to \({Z_m}\) of the multi-sensor association map.

#### 3.5.2 Predict step

The predict step is the same as the case of single sensor. That is, if the multi-sensor multi-target a priori density function is the form of the generalized labeled multi-Bernoulli, then the multi-objective prediction is:

$$\begin{aligned} {\pi _ + }\left( {{\mathrm{{X}}_ + }} \right) \mathrm{{ = }}\Delta \left( {{\mathrm{{X}}_\mathrm{{ + }}}} \right) \sum \limits _{\left( {{I_\mathrm{{ + }}},\xi } \right) \in \mathcal{F}\left( {\mathbb {L}} \right) \times \Xi } {{w_\mathrm{{ + }}}^{\left( {{I_\mathrm{{ + }}},\xi } \right) }} {\delta _{{I_\mathrm{{ + }}}}}\left( {\mathcal{L}\left( {{\mathrm{{X}}_\mathrm{{ + }}}} \right) } \right) {\left[ {{p_\mathrm{{ + }}}^{\left( \xi \right) }} \right] ^{{\mathrm{{X}}^\mathrm{{ + }}}}} \end{aligned}$$

(35)

where

$$\begin{aligned} \begin{array}{*{20}{l}} {{w_ + }^{\left( {{I_ + },\xi } \right) } = {w_B}\left( {{I_ + } \cap \mathbb {B}} \right) {w_s}^{\left( \xi \right) }\left( {{I_ + } \cap \mathbb {L}} \right) }\\ {{p_ + }^{\left( \xi \right) }\left( {x,l} \right) = {1_{\mathbb {L}}}\left( l \right) p_s^{\left( \xi \right) }\left( {x,l} \right) + \left( {1 - {1_{\mathbb {L}}}\left( l \right) } \right) {p_B}\left( {x,l} \right) }\\ {p_s^{\left( \xi \right) }\left( {x,l} \right) = \frac{{\left\langle {{p_s}\left( { \cdot ,l} \right) f\left( {x| \cdot ,l} \right) ,{p^{\left( \xi \right) }}\left( { \cdot ,l} \right) } \right\rangle }}{{\eta _s^{\left( \xi \right) }\left( l \right) }}}\\ {\eta _s^{\left( \xi \right) }\left( l \right) = \int {\left\langle {{p_s}\left( { \cdot ,l} \right) f\left( {x| \cdot ,l} \right) ,{p^{\left( \xi \right) }}\left( { \cdot ,l} \right) } \right\rangle dx} }\\ {{w_s}^{\left( \xi \right) }\left( L \right) = {{\left[ {\eta _s^{\left( \xi \right) }} \right] }^L}\sum \limits _{I \in L} {{1_I}\left( L \right) } {{\left[ {{q_s}^{\left( \xi \right) }} \right] }^{I - L}}{w^{\left( {I,\xi } \right) }}}\\ {{q_s}^{\left( \xi \right) }\left( l \right) = \left\langle {{q_s}\left( { \cdot ,l} \right) ,p_s^{\left( \xi \right) }\left( { \cdot ,l} \right) } \right\rangle } \end{array} \end{aligned}$$

(36)

In this step, the multi-sensor has the same form as the single-sensor.Let \({w_B}({I_ + } \cap \mathbb {B})\) be the weight of new label \({I_+} \cap \mathbb {B}\),\({w_s}^{\left( \xi \right) }({I_ + } \cap \mathbb {L})\) is the weight of the survival label \(({I_+} \cap \mathbb {L})\), \({p_B}\left( {x,l} \right)\) is the probability density of the new target. \(p_s^{\left( \xi \right) }\left( {x,l}\right)\) is the density of the survival target obtained from the prior density \({p^{(\xi )}}\left( { \cdot ,l} \right)\).\(f\left( {x| \cdot ,l} \right)\) represents the probability density of the survival target.

Algorithm 1 provides pseudocode for a multi-sensor multi-target tracking algorithm with partly overlapping fields of view.