This paper presents a novel distributed Bayesian filtering (DBF) method using measurement dissemination (MD) for multiple unmanned ground vehicles (UGVs) with dynamically changing interaction topologies. Different from statistics dissemination (SD)-based algorithms that transmit posterior distributions or likelihood functions, this method relies on a full-in and full-out (FIFO) transmission protocol, which significantly reduces the transmission burden between each pair of UGVs. Each UGV only sends a communication buffer (CB) and a track list (TL) to its neighbors, in which the former contains a history of sensor measurements from all UGVs, and the latter is used to trim the redundant measurements in the CB to reduce communication overhead. It is proved that by using FIFO, each UGV can disseminate its measurements over the whole network within a finite time, and the FIFO-based DBF is able to achieve consistent estimation of the environment state. The effectiveness of this method is validated by comparing with the consensus-based distributed filter (CbDF) and the centralized filter (CF) in a multitarget tracking problem.

Introduction

Estimation using a group of networked unmanned ground vehicles (UGVs) has been widely utilized to collectively measure environment status [1], such as intruder detection [2], signal source seeking [3], and pollution field estimation [4], due to its merits on low cost, high efficiency, and good reliability. The commonly adopted estimation approaches include the Kalman filter, extended Kalman filter, and particle filter [5], and the most generic scheme might be the Bayesian filter because of its applicability for nonlinear systems with arbitrary noise distributions [6,7]. In fact, a Bayesian filter can be reduced to different methods in certain conditions. For example, under the assumption of linearity and Gaussian noise, a Bayesian filter can be reduced to the Kalman filter [8]; for general nonlinear systems, a Bayesian filter can be numerically implemented as a particle filter [8]. Because of this generality, this study focuses on its networked variant, and uses it for tracking targets via local communication between neighboring UGVs.

The interaction topology [9,10] plays a central role on the design of networked Bayesian filter, of which two types are widely investigated in literature: centralized filters (CF) and distributed filters. In the former, local statistics estimated by each agent is transmitted to a single fusion center, where a global posterior distribution is calculated at each filtering cycle [11,12]. In the latter, each agent individually executes distributed estimation, and the agreement of local estimates is achieved by certain consensus strategies [1315]. In general, the distributed filters are more suitable in practice since they do not require a fusion center with powerful computation capability and are more robust to changes in network topology and link failures. So far, the distributed filters have two mainstream schemes in terms of the transmitted data among agents, i.e., statistics dissemination (SD) and measurement dissemination (MD). In the SD scheme, each agent exchanges statistics, such as posterior distributions and likelihood functions, within neighboring agents [16]. In the MD scheme, instead of exchanging statistics, each agent sends sensor measurements to neighboring agents.

The statistics dissemination scheme has been widely investigated during the last decade, especially in the field of signal processing, network control, and robotics. Madhavan et al. presented a distributed extended Kalman filter for nonlinear systems [17]. This filter was used to generate local terrain maps by using pose estimates to combine elevation gradient and vision-based depth with environmental features. Olfati-Saber proposed a distributed linear Kalman filter for estimating states of linear systems with Gaussian process and measurement noise [18]. Each distributed linear Kalman filter used additional low-pass and band-pass consensus filters to compute the average of weighted measurements and inverse covariance matrices. Gu proposed a distributed particle filter for Markovian target tracking over an undirected sensor network [19]. Gaussian mixture models were adopted to approximate the posterior distribution from weighted particles and the parameters of Gaussian mixture models were exchanged via average consensus filter. Hlinka et al. proposed a distributed method for computing an approximation of the joint (all-sensors) likelihood function by means of weighted-linear-average consensus algorithm when local likelihood functions belong to the exponential family of distributions [20]. Bandyopadhyay and Chung presented a Bayesian consensus filter that uses logarithmic opinion pool for fusing posterior distributions of the tracked target [6]. Other examples can be found in Refs. [7,21].

Despite the popularity of statistics dissemination, exchanging statistics can cause high communication burden if the environment to be detected is relatively large in space and complicated in structure. One remedy is to approximate statistics with parametric models, e.g., Gaussian mixture model [22], which can reduce communication burden to a certain extent. However, such manipulation increases the computation burden of each agent and sacrifices filtering accuracy due to approximation. The measurement dissemination scheme is an alternative solution to address the issue of exchanging large-scale statistics. An early work on measurement dissemination was done by Coates who used adaptive encoding of observations to minimize communication overhead [23]. Ribeiro and Giannakis exchanged quantized observations along with error-variance limits considering more pragmatic signal models [24]. A recent work was conducted by Djurić et al., who proposed to broadcast raw measurements to other agents, and therefore each agent has a complete set of observations of other agents for executing particle filtering [25]. A shortcoming of aforementioned works is that their communication topologies are assumed to be a fixed and complete graph that every pair of distinct agents is constantly connected by a unique edge. In many real applications, the interaction topology can be time varying due to unreliable links, external disturbances, or range limits [26,27]. In such cases, dynamically changing topologies can cause random packet loss, variable transmission delay, and out-of-sequence measurement (OOSM) issues [28], thus decreasing the performance of distributed estimation. Leung et al. has explored a decentralized Bayesian filter for dynamic robot networks [29] in order to achieve centralized-equivalent filtering performance. However, it requires the communication of both measurements and statistics, which can still incur large communication overhead.

This paper proposes a distributed Bayesian filtering (DBF) method that only uses measurement dissemination for a group of networked UGVs with dynamically changing interaction topologies. In our previous works [3032], we have proposed a latest-in-and-full-out protocol for measurement exchange and developed a corresponding DBF algorithm. However, it is only applicable to either tracking moving targets when the interaction topology is time-invariant or to localizing static targets. In this work, we substantially extend the previous works and make the following contributions: (1) We introduce a new protocol called the full-in-and-full-out (FIFO) that allows each UGV to broadcast a history of measurements to its neighbors via single hopping, enabling the localization and tracking of targets using general nonlinear sensor models under time-varying topologies. (2) We propose the frequently jointly strongly connectedness condition of the interaction topology and show that, under this condition, FIFO can disseminate UGVs' measurements over the network within a finite time. (3) We develop a FIFO-based distributed Bayesian filter (FIFO-DBF) for each UGV to implement locally. A track list is designed to reduce the computational complexity of FIFO-DBF and the communication burden. The FIFO-DBF can avoid the OOSM issue. (4) We prove the consistency of FIFO-DBF: each UGV's estimate of target position converges in probability to the true target position asymptotically if the interaction topologies are frequently jointly strongly connected.

The rest of this paper is organized as follows: Section 2 formulates the target tracking problem using multiple UGVs; Section 3 proposes the FIFO protocol for measurement dissemination in dynamically changing interaction topologies; Section 4 introduces the FIFO-DBF algorithm and the track list; Section 5 proves the consistency of FIFO-DBF; Section 6 presents simulation results; and Sec. 7 concludes the paper.

Problem Formulation

Consider a network of N UGVs in a bounded two-dimensional space S, as shown in Fig. 1. The interaction topology can be dynamically changing due to limited communication range, team reconfiguration, or intermittently link failure. Each UGV is equipped with a sensor for target detection. Due to the limit of communication range, each UGV can only exchange sensor measurements with its local neighbors. Every UGV locally runs a Bayesian filter to estimate the target position in S utilizing its own measurements and the received measurements from other UGVs.

Target and Sensor Model.

The target motion uses a stochastic discrete-time model 
xk+1g=f(xkg,vk)
(1)

where the superscript g represents the target, xkgS is the target position at time k, and vk is the white process noise.

The sensor measurement is described by a stochastic model 
zki=hi(xkg,xki)+wki
(2)

where the superscript i{1,,N} represents the index of the UGV, xkiS is the sensor position, and wki is the white measurement noise. The measurement function hi depends on the type of the sensor.

The design of the Bayesian filter relies on the conditional probability of obtaining a certain measurement zki given the current target and sensor states, which is denoted by P(zki|xkg;xki) [5]. The conditional probability P(zki|xkg;xki) depends on both hi and wki in Eq. (2). For example, if wki is a zero-mean Gaussian white noise with covariance Γki, then P(zki|xkg;xki) can be described as P(zki|xkg;xki)=N(hi(xkg,xki),Γki). For non-Gaussian noise, such as Poisson noise or Cauchy noise [33], P(zki|xkg;xki) can also be similarly defined (for the purpose of simplicity, P(zki|xkg;xki) is shorted as P(zki|xkg) for the rest of the paper). It should be noted that this work is not confined to any specific distribution of the noise. The measurement function hi for several typical sensors is listed as follows [34]:

Range-only sensors:hi is a function of the relative Euclidean distance between the sensor and the target 
hi(xkg,xki)=||xkgxki||2

where ||·||2 is the Euclidean distance in S.

Bearing-only sensors:hi is a function of the relative bearing between the sensor and the target 
hi(xkg,xki)=(xkgxki)

where denotes the angle from the sensor to the target.

Range-bearing sensors:hi includes both the relative distance and bearing 
hi(xkg,xki)=xkgxki

Graphical Model of Interaction Topology.

We consider a simple2 graph G=(V,E) to represent the interaction topology of N networked UGVs, where the vertex set V={1,,N} represents the index set of UGVs and E=V×V denotes the edge set. For the purpose of narrative simplicity, we use directed graphs to describe our approach in this work. The undirected graphs can actually be treated as bidirectional directed graphs.

The adjacency matrixA=[aij] of the graph G describes the interaction topology 
aij={1if(i,j)E0if(i,j)E

where aij is the entry on the ith row and jth column of the adjacency matrix. The notation aij = 1 indicates that the ith UGV can directly communicate to the jth UGV and aij = 0 indicates no direct communication from i to j. A directed graph is strongly connected if there is a directed path connecting any two arbitrary vertices in V.

Define the union of a set of simple directed graphs as the graph with the vertices in V and the edge set given by the union of each member's edge sets. Such collection is jointly strongly connected if the union of its members forms a strongly connected graph.3 We use Gk to represent the interaction topology of time k and define the inbound neighbors and outbound neighbors of the ith UGV under Gk as the set Niin(Gk)={j|ajik=1,jV} and Niout(Gk)={j|aijk=1,jV}. All UGVs in Niout(Gk) can directly receive information from the ith UGV via the single hopping.

Full-In-and-Full-Out Protocol

This study proposes a FIFO protocol for measurement exchange in dynamically changing interaction topologies. The use of FIFO allows us to apply measurement dissemination-based distributed filters to time-varying networks. Let YKi={[xki,zki]|kK} be the set of state-measurement pairs of the ith UGV, where K is an index set of time steps. Each UGV contains a communication buffer (CB) and a track list (TL). A CB stores state-measurement pairs of all UGVs 
Bki=[YKki11,,YKkiNN]

where Bki is the CB of ith UGV at time k and Kkij(jV) is the time index set. YKkijj represents the set of jth UGV's state-measurement pairs of time steps in Kkij that are stored in ith UGV's CB at time k. A UGV's TL stores the information of this UGV's reception of all UGVs' measurements, and is used for trimming old state-measurement pairs in the CB to reduce the communication burden. The details of TL will be introduced in Sec. 4.2. Each UGV sends its CB and TL to its outbound neighbors at every time step.

Algorithm 1

FIFO Protocol

 (1) Initialization. 
   CB : The CB of ith UGV is initialized as an empty set at k = 0: 
   B0i=[YK0i11,,YK0iNN],whereYK0ijj={[ø,ø]}. 
   TL : The TL of ith UGV is initialized at k = 0: 
   q1ij=[0,,0,1],i.e.q1jl=0,j,l{1,N}. 
 (2) At time k(k1) for ith UGV: 
(2.1) Receiving Step. 
   CB : The ith UGV receives all CBs of its inbound neighbors Niin(Gk1). The received CB from the lth UGV is Bk1l(lNiin(Gk1))
   TL : The ith UGV receives all TLs of its inbound neighbors Niin(Gk1). The received TL from the lth UGV is Qk1l(lNiin(Gk1))
(2.2) Observation Step. 
   CB : The ith UGV updates YKkiii by its own state-measurement pair: 
   YKkiii=YKk1iii{[xki,zki]}. 
(2.3) Updating Step. 
   CB : The ith UGV updates other entries of its own CB, YKkijj(ji), by merging with all received CBs: 
   YKkijj=YKk1ijjYKk1ljj,ji,lNiin(Gk1). 
   TL : The ith UGV updates its own TL, Qki, using the received TLs (see Algorithm 3). The CB is trimmed based on the updated track list (see Algorithm 4). 
(2.4) Sending Step. 
   CB : The ith UGV sends its updated CB, Bki, to all of its outbound neighbors defined in Niout(Gk)
   TL : The ith UGV sends its updated track list, Qki, to its outbound neighbors Niout(Gk)
 (3)kk+1 until stop 
 (1) Initialization. 
   CB : The CB of ith UGV is initialized as an empty set at k = 0: 
   B0i=[YK0i11,,YK0iNN],whereYK0ijj={[ø,ø]}. 
   TL : The TL of ith UGV is initialized at k = 0: 
   q1ij=[0,,0,1],i.e.q1jl=0,j,l{1,N}. 
 (2) At time k(k1) for ith UGV: 
(2.1) Receiving Step. 
   CB : The ith UGV receives all CBs of its inbound neighbors Niin(Gk1). The received CB from the lth UGV is Bk1l(lNiin(Gk1))
   TL : The ith UGV receives all TLs of its inbound neighbors Niin(Gk1). The received TL from the lth UGV is Qk1l(lNiin(Gk1))
(2.2) Observation Step. 
   CB : The ith UGV updates YKkiii by its own state-measurement pair: 
   YKkiii=YKk1iii{[xki,zki]}. 
(2.3) Updating Step. 
   CB : The ith UGV updates other entries of its own CB, YKkijj(ji), by merging with all received CBs: 
   YKkijj=YKk1ijjYKk1ljj,ji,lNiin(Gk1). 
   TL : The ith UGV updates its own TL, Qki, using the received TLs (see Algorithm 3). The CB is trimmed based on the updated track list (see Algorithm 4). 
(2.4) Sending Step. 
   CB : The ith UGV sends its updated CB, Bki, to all of its outbound neighbors defined in Niout(Gk)
   TL : The ith UGV sends its updated track list, Qki, to its outbound neighbors Niout(Gk)
 (3)kk+1 until stop 

The FIFO protocol is stated in Algorithm 1, which consists of the CB and TL parts. For the purpose of clarity, we focus on the CB parts in this section and leave the description of the TL parts to Sec. 4.2. Figure 2 illustrates the FIFO cycles in a network of three UGVs with dynamically changing topologies. The following facts can be observed from Fig. 2: (1) the topologies are jointly strongly connected in the time interval [0,6); (2) each UGV can receive the state-measurement pairs of other UGVs within finite steps. Extending these facts to a network of N UGVs, we have the following properties of FIFO:

Theorem 1.Consider a network of N UGVs with dynamically changing interaction topologiesG={G1,G2,G3,}. IfGis frequently jointly strongly connected, i.e.,

  1. (1)

    there exists an infinite sequence of time intervals[km,km+1),m=1,2,, starting atk1=0and are contiguous, nonempty, and uniformly bounded;

  2. (2)

    the union of graphs across each such interval is jointly strongly connected,

then each pair of UGVs can exchange measurements under FIFO. In addition, it takes no more thanNTusteps for a UGV to communicate to another one, whereTu=supm=1,2,(km+1km)Tis the upper bound of interval lengths.

Proof. Without loss of generality, we consider the transmission of Bt1i from the ith UGV to an arbitrary jth UGV (jV{i}), where t1[k1,k2). Since each UGV will receive inbound neighbors' CBs and send the merged CB to its outbound neighbors at the next time step, the ith UGV can transmit Bt1i to jth UGV if and only if a path [l1,,ln] exists, with l1=i, ln = j, l2,,ln1V{i,j}, and the edge (ls,ls+1) appears no later than (ls+1,ls+2),s=1,,n2.

As the union of graphs across the time interval [k2,k3) is jointly connected, ith UGV can directly send Bt1i to at least one another UGV at a time instance, i.e., l2V{i},t2[k2,k3) s.t. l2Niout(Gt2). If l2=j, then Bt1i has been sent to j. If l2j,Bt1i has been merged into Bt2+1l2 and will be sent out in the next time step.

Using the similar reasoning for time intervals [km,km+1),,m=3,4,, it can be shown that all UGVs can receive the state-measurement pairs in Bt1i no later by kN+1. Therefore, the transmission time from an arbitrary UGV to any other UGVs is no greater than NTu. ◻

Corollary 1. For a frequently jointly strongly connected network, each UGV receives the CBs of all other UGVs under FIFO within finite time.

Proof. According to Theorem 1, each UGV is guaranteed to receive Btj(t0,jV) when kt+NTu. ◻

Remark 1. Theorem 1 and Corollary 1 are the consequences of FIFO and the use of the CB. In fact, if we use the traditional methods that each UGV only sends the current sensor measurement to neighboring UGVs without the use of CB, it can happen that two UGVs may never exchange their sensor measurements, even if there exists a path connecting them. The condition of frequently jointly strongly connectedness is also crucial for guaranteeing the consistency of the FIFO-based distributed Bayesian filter, as shown in Sec. 5.

Distributed Bayesian Filter Via Full-In-and-Full-Out Protocol

We first introduce the generic DBF. Let XkS be the random variable representing the position of the target at time k. Define Zki as the set of measurements at time k that are in the ith UGV's CB, i.e., Zki={zkj|[xkj,zkj]Bki,jV} and let Z1:ki=t=1kZti. It is easy to notice that Z1:ki is the set of all measurements in Bki. We also define z1:ki=[z1i,,zki] as the set of the ith UGV's measurements at times 1 through k. The probability density function (PDF) of Xk, called individual PDF, of the ith UGV is represented by Ppdfi(Xk|Z1:ki). It is the estimation of the target position given all the measurements that the ith UGV has received. The initial individual PDF, Ppdfi(X0), is constructed given prior information including past experience and environment knowledge. It is necessary to initialize Ppdfi(X0) such that the probability density of the true target position is nonzero, i.e., Ppdfi(X0=x0g)>0.

Under the framework of DBF, the individual PDF is recursively estimated by two steps: the prediction step and the updating step.

Prediction. At time k, the prior individual PDF Ppdfi(Xk1|Z1:k1i) is first predicted forward by using the Chapman–Kolmogorov equation 
Ppdfi(Xk|Z1:k1i)=Xk1SP(Xk|Xk1)Ppdfi(Xk1|Z1:k1i)dXk1
(3)

where P(Xk|Xk1) represents the state transition probability of the target, based on the Markovian motion model (Eq. (1)).

Updating. The ith individual PDF is then updated by the Bayes' rule using the set of newly received measurements at time k, i.e., Zki 
Ppdfi(Xk|Z1:ki)=KiPpdfi(Xk|Z1:k1i)P(Zki|Xk)
(4a)
 
=KiPpdfi(Xk|Z1:k1i)zkjZkiP(zkj|Xk)
(4b)
where P(zkj|Xk) is the sensor model and Ki is a normalization factor, given by 
Ki=[XkSPpdfi(Xk|Z1:k1i)P(Zki|Xk)dXk]1

Here, we have utilized the commonly adopted assumption [19,22,35] in the distributed filtering literature that the sensor measurement of each UGV at current time is conditionally independent from its own previous measurements and the measurements of other UGVs given the target and the UGV's current position. This assumption allows us to simplify P(Zki|Xk,Z1:k1i) as P(Zki|Xk) in Eq. (4a) and factorize P(Zki|Xk) as zkjZkiP(zkj|Xk) in Eq. (4b).

Algorithm 2

FIFO-DBF Algorithm

For ith UGV at kth step (iV): 
 (1) Initialize a temporary PDF by assigning the stored individual PDF to it: 
   Ptmpi(Xt)=Pstoi(Xt), 
where 
   Pstoi(Xt)=Ppdfi(Xt|z1:t1,,z1:tN). 
 (2) For ξ=t+1 to k, iteratively repeat two steps of Bayesian filtering: 
(2.1) Prediction 
   Ptmppre(Xξ)=SP(Xξ|Xξ1)Ptmpi(Xξ1)dXξ1. 
(2.2) Updating 
   Ptmpi(Xξ)=KξPtmppre(Xξ)P(Zξi|Xξ), 
   Kξ=[SPtmppre(Xξ)P(Zξi|Xξ)dXξ]1. 
(2.3) If zξjø for jV, update the stored PDF: 
   Pstoi(Xξ)=Ptmpi(Xξ). 
 (3) The individual PDF of ith UGV at time k is Ppdfi(Xk|Z1:ki)=Ptmpi(Xk)
For ith UGV at kth step (iV): 
 (1) Initialize a temporary PDF by assigning the stored individual PDF to it: 
   Ptmpi(Xt)=Pstoi(Xt), 
where 
   Pstoi(Xt)=Ppdfi(Xt|z1:t1,,z1:tN). 
 (2) For ξ=t+1 to k, iteratively repeat two steps of Bayesian filtering: 
(2.1) Prediction 
   Ptmppre(Xξ)=SP(Xξ|Xξ1)Ptmpi(Xξ1)dXξ1. 
(2.2) Updating 
   Ptmpi(Xξ)=KξPtmppre(Xξ)P(Zξi|Xξ), 
   Kξ=[SPtmppre(Xξ)P(Zξi|Xξ)dXξ]1. 
(2.3) If zξjø for jV, update the stored PDF: 
   Pstoi(Xξ)=Ptmpi(Xξ). 
 (3) The individual PDF of ith UGV at time k is Ppdfi(Xk|Z1:ki)=Ptmpi(Xk)

The Full-In and Full-Out-Distributed Bayesian Filtering Algorithm.

The generic DBF is not directly applicable to time-varying interaction topologies. This is because changing topologies can cause intermittent and out-of-sequence arrival of measurements from different UGVs, giving rise to the OOSM problem. One possible solution is to ignore all measurements that are out of the temporal order. This is undesirable since this will cause significant information loss. Another possible remedy is to fuse all measurements by running the filtering algorithm from the beginning at each time step. However, this solution causes excessive computational burden. To avoid both OOSM problem and unnecessary computational complexity, we add a new PDF, namely, the stored PDF, Pstoi(Xt), that is updated from the ith UGV's initial PDF by fusing the state-measurement pairs of all UGVs up to a certain time tk. The choice of t is described in Sec. 4.2. The individual PDF, Ppdfi(Xk|Z1:ki), is then computed by fusing the measurements from time t + 1 to k in the CB into Pstoi(Xt), running the Bayesian filter (Eqs. (3) and (4)). Note that initially, Pstoi(X0)  =  Ppdfi(X0).

The FIFO-DBF algorithm is stated in Algorithm 2. Each UGV runs FIFO-DBF after its CB is updated in the Updating Step in Algorithm 1. At the beginning, we assign the stored PDF to a temporary PDF, which will then be updated by sequentially fusing measurements in the CB to obtain the individual PDF. It should be noted that, when the UGV's CB contains all UGVs' state-measurement pairs from t to ξ, the temporary PDF corresponding to time ξ is assigned as the new stored PDF. Figure 3 illustrates the FIFO-DBF procedure for the first UGV as an example. It can be noticed that the purpose of using the stored PDF is to avoid running the Bayesian filtering from the initial PDF at every time step. Since the stored PDF has incorporated all UGVs' measurements up to time step t, the information loss is prevented. We point out that the time t of each UGV's stored PDF can be different from others. The stored PDF is saved locally by each UGV and not transmitted to others. FIFO-DBF is able to avoid the OOSM issue since all measurements are fused in the correct temporal order.

Track Lists for Trimming Communication Buffers.

The size of CBs can keep increasing as measurements cumulate over time. The use of the stored PDF has made it feasible to trim excessive state-measurement pairs from the CBs. To avoid information loss, a state-measurement pair can only be trimmed from a UGV's CB when all UGVs have received it. We design the track list (TL) for each UGV to keep track of all UGVs' reception of other UGVs' measurements. We first define a binary term qkijjl(i,j,lV) : qkijjl=1 if the ith robot knows that the jth UGV has received the state-measurement pair of the lth UGV of time kij, [xkijl,zkijl]; qkijjl=0 if the ith robot cannot determine whether [xkijl,zkijl] has been received by the jth robot. Therefore, it can happen that [xkijl,zkijl] has been received by the jth UGV but the ith UGV does not know this and thus qkijjl=0. Now, we define the ith UGV's track list as Qki=[qki1i1,,qkiNiN]T(iV), which is a N×(N+1) binary matrix with qkijij=[qkijj1,,qkijjN,kij]T (jV). The last column of Qki,[ki1,,kiN], corresponds to measurement times.

The exchange and updating of TLs are described in Algorithm 1, with the updating details presented in Algorithm 3. For the ith UGV, it updates the ith row of its TL matrix using the entries of its CB, and updates other rows of the TL using the received TLs from inbound neighbors. The updating rule guarantees that, if the last term of the jth row is kij, the ith UGV is ensured that every UGV has received all UGVs' state-measurement pairs of times earlier than kij. Figure 4 shows the updating of each UGV's TL using Algorithm 3. We can use TLs to trim CBs, which is described in Algorithm 4. In the example of Fig. 4, the first and third UGV's CB will be trimmed at k = 6 and the trimmed state-measurement pairs correspond to times 1, 2, and 3.

The use of TLs can avoid the excessive size of CBs and guarantee that trimming the CBs will not lose any information; the trimmed measurements have been encoded into the stored PDF. The following theorem formalizes this property.

Theorem 2.Each UGV's estimation result using the trimmed CB is the same as that using the nontrimmed CB.

Proof. Consider the ith UGV. Let kmi=minjkij. Trimming Bki happens when all entries in Qki corresponding to time kmi equal 1. This indicates that each UGV has received all UGVs' state-measurement pairs of time kmi. A UGV has either stored the pairs in its CB or already fused them to obtain the stored PDF. In both cases, such pairs are no longer needed to be transmitted. Therefore, it causes no loss to trim these measurements. ◻

The following theorem describes when CBs get trimmed, and it provides an upper bound of the communication burden that FIFO-DBF will incur. A detailed complexity analysis of FIFO-DBF is presented in Sec. 4.3. Consider trimming all the state-measurement pairs of time t in the ith UGV's CB. Let ktlj(>t) be the first time that the lth UGV communicates to the jth UGV in the time interval (t,). Define k̃tj=maxlktlj, which is the time that the jth UGV receives all other UGVs' measurements of t. Similarly, let ktji(>k̃tj) be the first time that the jth UGV communicates to the ith UGV in the time interval (k̃tj,) and define k̃ti=maxjktji. The following theorem gives the time when the ith UGV (iV) trims all state-measurement pairs of time t in its own CB.

Theorem 3.Theith UGV trims{[xtl,ztl](lV)}from its CB at the timek̃ti.

Proof. The ith UGV can trim {[xtl,ztl](lV)} only when it is sure that all other UGVs have also received these state-measurement pairs. This happens at k̃ti and thus k̃ti is the time when the trim occurs.◻

Corollary 2.Under the frequently jointly strongly connectedness condition, the size of any UGV's CB is no greater than2N(N1)Tu.

Proof. We consider an arbitrary ith(iV) UGV. According to Theorem 1, a UGV can communicate to any other UGV within NTu steps. Therefore, k̃ti2NTu, since it first requires each UGV communicate to all other UGVs and then each UGV communicate to the ith UGV. This implies that the state-measurement pairs of a certain time of all UGVs will be trimmed from each UGV's CB within 2NTu steps.

The maximum size of the CB occurs when the state-measurement pairs of a certain time from all but one UGV are saved in the ith UGV's CB. Therefore, the size of any UGV's CB is no greater than 2N(N1)Tu. ◻

Algorithm 4

Trimming CBs using TLs

For the ith UGV, find the smallest time in Qki : kmi=min{ki1,,kiN}
 1. Remove state-measurement pairs in Bki that corresponds to measurement times earlier than kmi, i.e., Bki=Bki{[xtl,ztl]},t<kmi,lV. 
 2. If entries associated with time kmi in Qki are 1s, then 
  1. set these entries to be 0. 
  2. update the ith row of Qki using the current CB, i.e., qkiiil=1 if [xkiil,zkiil]Bki,lV
  3. remove all corresponding state-measurement pairs in Bki, i.e., Bki=Bki{[xkmil,zkmil]},lV. 
  4. kmikmi+1. 
For the ith UGV, find the smallest time in Qki : kmi=min{ki1,,kiN}
 1. Remove state-measurement pairs in Bki that corresponds to measurement times earlier than kmi, i.e., Bki=Bki{[xtl,ztl]},t<kmi,lV. 
 2. If entries associated with time kmi in Qki are 1s, then 
  1. set these entries to be 0. 
  2. update the ith row of Qki using the current CB, i.e., qkiiil=1 if [xkiil,zkiil]Bki,lV
  3. remove all corresponding state-measurement pairs in Bki, i.e., Bki=Bki{[xkmil,zkmil]},lV. 
  4. kmikmi+1. 

Complexity Analysis of Full-In and Full-Out-Distributed Bayesian Filtering.

Compared to statistics dissemination, FIFO is usually more communication-efficient for distributed filtering. To be specific, consider a grid representation of the environment with the size D × D. The transmitted data between each pair of UGVs are the CB and TL of each UGV. The size of the CB is upper bounded by O(N2Tu), according to Corollary 2. On the contrary, the communicated datum of a statistics dissemination approach that transmits unparameterized posterior distributions or likelihood functions is O(D2). In applications such as the target localization, D is generally much larger than N. Besides, the consensus filter usually requires multiple rounds to arrive at consensual results. Therefore, when Tu is not comparable to D2, the FIFO protocol requires much less communication burden.

It is worth noting that each UGV needs to store an individual PDF and a stored PDF, each of which has size O(D2). In addition, each UGV needs to keep the CB and TL. This is generally larger than that of statistics dissemination-based methods, which only stores the individual PDF. Therefore, the FIFO-DBF sacrifices the local memory for saving the communication resource. This is actually desirable for real applications as local memory of vehicles is usually abundant compared to the limited bandwidth for communication.

Remark 2. Under certain interaction topologies, CBs can grow to undesirable sizes and cause excessive communication burden if the trim cannot happen frequently. In this case, we can use a time window to constrain the measurements that are saved in CBs. This will cause information loss to the measurements. However, with a decently long time window, FIFO-DBF can still effectively estimate the target position.

Proof of Consistency

This section proves the consistency of the maximum a posteriori (MAP) estimator of latest-in-and-full-out-DBF under unbiased sensors (sensors without offset). A state estimator is consistent if it converges in probability to the true value of the state [36]. Consistency is an important metric for stochastic filtering approaches [8], which not only implies achieving consensus asymptotically but also requires the estimated value converge to the true value. We first prove the consistency for static UGVs and then for moving UGVs. Here, we assume that S is a finite set (e.g., a finely discretized field) and the target is relatively slow compared to the filtering dynamics. In addition, the target position can be uniquely determined by the multi-UGV network with proper placement (i.e., excluding the special case of ghost targets [37]).

Static Unmanned Ground Vehicles.

The consistency of FIFO-DBF for static UGVs is stated as follows:

Theorem 4.Assume the UGVs are static and the sensors are unbiased. If the network of N UGVs is frequently jointly strongly connected, then the MAP estimator of target position converges in probability to the true position of the target using FIFO-DBF, i.e., 
limkP(XkMAP=xg)=1,iV
where 
XkMAP=argmaxXPpdfi(X|Z1:ki)
Proof. The DBF can be transformed into the batch form by recursively applying Eq. (4) from k to the initial time 1 (back in time) 
Ppdfi(X|Z1:ki)=KiPpdfi(X|Z1:k1i)zkjZkiP(zkj|X)=KiPpdfi(X|Z1:k2i)zk1jZk1iP(zk1j|X)zkjZkiP(zkj|X)==KiPpdfi(X)z1jZ1iP(z1j|X)zkjZkiP(zkj|X)=KiPpdfi(X)j=1NtKkijP(ztj|X)

The last step is obtained by using the relation Bki=[YKki11,,YKkiNN] and Z1:ki is the set of all measurements in Bki.

Comparing Ppdfi(Xk=x|Z1:ki) with Ppdfi(Xk=xg|Z1:ki)5 yields 
Ppdfi(x|Z1:ki)Ppdfi(xg|Z1:ki)=Ppdfi(x)j=1NtKkijP(ztj|x)Ppdfi(xg)j=1NtKkijP(ztj|xg)
(5)
Take the logarithm of Eq. (5) and average it over k steps: 
1klnPpdfi(x|Z1:ki)Ppdfi(xg|Z1:ki)=1klnPpdfi(x)Ppdfi(xg)+j=1N1ktKkijlnP(ztj|x)P(ztj|xg)
(6)
Since Ppdfi(x) and Ppdfi(xg) are bounded and nonzero by the choice of the initial PDF, limk1kln(Ppdfi(x)/Ppdfi(xg))=0. Note that the sensor measurement is a random variable drawn from the underlying distribution associated with the sensor model, i.e., ztjP(·|xg),jV. Therefore, ln(P(ztj|x)/P(ztj|xg)) is a random variable associated with ztj. Due to the finite delay of measurement arrival (Corollary 1), i.e., kNTu|Kkij|k, where |·| is the set cardinality, we can use the law of large numbers to study the asymptotic behavior of the series in Eq. (6) 
1ktKkijlnP(ztj|x)P(ztj|xg)PEztj[P(ztj|x)P(ztj|xg)]
(7a)
 
=ztjP(ztj|xg)P(ztj|x)P(ztj|xg)dztj
(7b)
 
=DKL(P(ztj|x)||P(ztj|xg))
(7c)
where “P” represents “convergence in probability” and DKL(P1||P2) denotes the Kullback–Leibler (KL) divergence between two probability distribution P1 and P2. KL divergence has the property that P1,P2,DKL(P1||P2)0, and the equality holds if and only if P1=P2. Therefore 
limk1ktKkijlnP(ztj|x)P(ztj|xg)<0,xxglimk1ktKkijlnP(ztj|x)P(ztj|xg)=0,x=xg
Considering the limiting case of Eq. (6), we get 
limk1klnPpdfi(x|Z1:ki)Ppdfi(xg|Z1:ki)<0,xxg
(8)
 
limk1klnPpdfi(x|Z1:ki)Ppdfi(xg|Z1:ki)=0,x=xg
(9)
Equations (8) and (9) imply that 
Ppdfi(x|Z1:ki)Ppdfi(xg|Z1:ki)P{0xxg1x=xg
Therefore 
limkP(XkMAP=xg)=1

Moving Unmanned Ground Vehicles.

The consistency proof for the moving UGVs case is different from the static UGVs case in that each moving UGV makes measurements at multiple different positions. We classify UGV measurement positions into two disjoint sets: infinite-measurement spots SI that contain positions where a UGV keeps revisiting as time tends to infinity, and finite-measurement spots SF that contain positions where the UGV visits finitely many times (i.e., the UGV does not visit again after a finite time period). It is easy to know that each UGV has at least one position where it revisits infinitely many times as k tends to infinity.

Theorem 5.Assume the UGVs move within a finite set of positions and the sensors are unbiased. If the network of N UGVs is frequently jointly strongly connected, then the MAP estimator of target position converges in probability to the true position of the target using FIFO-DBF, i.e., 
limkP(XkMAP=xg)=1,iV
where 
XkMAP=argmaxXPpdfi(X|Z1:ki)
Proof. Similar to Eq. (5), comparing Ppdfi(x|Zki) and Ppdfi(xg|Zki) yields 
Ppdfi(x|Zki)Ppdfi(xg|Zki)=Ppdfi(x)j=1NtKkijP(ztj|x;xtj)Ppdfi(xg)j=1NtKkijP(ztj|xg;xtj)
(10)
The only difference from Eq. (5) is that P(·|x;xlj) varies as the jth UGV moves since xlj changes over time. Similar to Eq. (6), we obtain 
1klnPpdfi(x|Z1:ki)Ppdfi(xg|Z1:ki)=1klnPpdfi(x)Ppdfi(xg)+j=1N1ktKkij,xtjSFlnP(ztj|x;xtj)P(ztj|xg;xtj)+j=1N1ktKkij,xtjSIlnP(ztj|x;xtj)P(ztj|xg;xtj)
where the second summand corresponds to the sensor positions that are in the finite-measurement spots set, and the third summand corresponds to the positions in the infinite-measurement spots set. By referring to Eq. (7), it is straightforward to know 
j=1N1ktKkij,xtjSFlnP(ztj|x;xtj)P(ztj|xg;xtj)P0j=1N1ktKkij,xtjSIlnP(ztj|x;xtj)P(ztj|xg;xtj)PEztj[P(ztj|x)P(ztj|xg)]

since only finitely many observations associated with sensor positions in SF are obtained but infinitely many observations associated with sensor positions in SI are received. The rest of the proof is similar to that of Theorem 4. ◻

Simulation

We conduct a simulation that uses a team of six UGVs to localize three moving targets. Every UGV maintains three individual PDFs, each corresponding to a target. At each time step, a UGV's sensor can measure the positions of three targets. We assume that the UGVs know the association between the measurement and the corresponding target. The targets have different motion models, including the linear motion (target 1), sinusoidal motion (target 2), and circular motion (target 3). Three of the UGVs have range-only sensors and the other three UGVs have bearing-only sensors. The interaction topology of the UGVs is time varying and consists of four types, as shown in Figs. 5(a)5(d). A randomly generated sequence of topologies is used (Fig. 5(f)). It can be noticed that the interaction topology is frequently jointly strongly connected when all four types appear repeatedly (Fig. 5(e)). Ten test trials are used, with the randomly generated initial positions of UGVs and targets. There exist different methods to implement a Bayesian filter, including the histogram filter and the particle filter [5]. The histogram filter is easy to implement and can keep track of the probability mass over the whole field, but can be computationally heavy for large fields. The particle filter, on the contrary, is advantageous when the field is very large, but can introduce inaccuracy due to particle deprivation [5]. We use both methods to implement the Bayesian filter in the simulation, and their results are very similar. For the purpose of clarity, we only include the results from the histogram filter here.

We compare FIFO-DBF with two commonly adopted approaches in multi-agent filtering: the consensus-based distributed filter (CbDF) [38] and the CF [39]. The CbDF requires UGVs to continually exchange their individual PDFs with neighbors, computing the average of its own and the received PDFs. Multiple rounds of communication and averaging are needed at each step to ensure the convergence of UGVs' individual PDFs. The CF assumes a central unit that can constantly receive and fuse all UGVs' latest measurements into a single PDF.

Figure 6 shows the simulation results of a specific trial. The sum of the first UGV's three individual PDFs is shown in the figures. It shows that the FIFO-DBF can successfully localize and track moving target's positions and effectively reduce the estimation uncertainty, which is similar to the performance of the CF (Fig. 7(b)). On the contrary, CbDF is less effective in reducing the estimation uncertainty (Fig. 7(a)).

We quantitatively compare the three filters in terms of the estimation error and entropy of the uncertainty. The estimation error is defined as the difference between the true target position and the MAP estimate of the individual PDF 
Δk=||XkMAPxkg||2
The entropy of the uncertainty is 
Hk=XkSPpdf(Xk)log(Ppdf(Xk))

The average of the estimation error and entropy of each target across ten trials are shown in Fig. 8. It can be noticed that the CF achieves the most accurate position estimation and fastest entropy reduction. This is an expected result since the CF utilizes all sensor measurements. The FIFO-DBF achieves similar results as the CF asymptotically. This is a very interesting result, since FIFO-DBF only communicates with neighboring UGVs and has a subset of other UGVs' measurements. The CbDF achieves similar position estimation performance as the CF and FIFO-DBF. However, it fails to effectively reduce the estimation entropy. This is because the linear combination of PDFs used in the CbDF does not follow the nonlinear nature of Bayesian filtering; thus information is lost during the combination. The FIFO-DBF, on the other hand, rigorously follows the procedure of Bayesian filtering, and therefore achieves better performance. Besides, CbDF requires multiple rounds of exchanging individual PDFs, which incurs much higher communication burden than FIFO-DBF at each time step. Therefore, FIFO-DBF is more preferable than CbDF.

Conclusion

This paper presents a general measurement dissemination-based DBF for a network of multiple UGVs under dynamically changing interaction topologies. The information exchange among UGVs relies on the FIFO protocol, under which UGVs exchange the communication buffers and track lists with neighbors. Under the condition that the union of the interaction topologies is frequently jointly strongly connected, FIFO can disseminate measurements over the network within finite time. By using the track list, the CBs can be trimmed to save communication resource without causing information loss. The FIFO-DBF algorithm is developed to estimate individual probability density function for target localization. The FIFO-DBF can significantly reduce the transmission burden between each pair of UGVs compared to the statistics dissemination methods, and can achieve consistent state estimation. Simulations comparing FIFO-DBF with CbDF and the CF show that FIFO-DBF achieves similar performance as the CF and more superior performance than the CbDF.

Acknowledgment

This paper is in memory of Prof. J. Karl Hedrick, who dedicated valuable time to contributing to this work while he was fighting with cancer. His intellectual contribution to nonlinear control and transportation engineering and his dedication to students and colleagues have always inspired, and will continue inspiring us to explore the unknown and educate the next generation of scholars.

Funding Data

  • National Natural Science Foundation of China (Grant Nos. 51622504 and 51575293).

  • Office of Naval Research (Grant No. MURI N00014-13-1-03).

  • International Sci&Tech Cooperation Program of China (Grant No. 2016YFE0102200).

2

A (directed/undirected) graph G=(V,E) is simple if it has no self-loops (i.e., (i,j)Eonlyifij) or multiple edges with the same source and target nodes (i.e., E only contains distinct elements).

3

The counterpart definition for undirected graphs is given in Ref. [13].

4

” is the notation of the logical “OR” operator.

5

For the purpose of simplicity, we use Ppdfi(x|Z1:ki) to represent Ppdfi(Xk=x|Z1:ki) in this proof.

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