NON-CENTRALIZED TARGET TRACKING IN NETWORKS OF DIRECTIONAL SENSORS:FURTHER ADVANCES

Li Geng and Petar M. Djuric

Department of Electrical and Computer EngineeringStony Brook University

Stony Brook, NY 11794, USA

ABSTRACT

Directional sensors detect targets within a range and a predefineddirection. In this paper, we assume that directional sensors aredeployed in a sensor field, where at each node there are fourcollocated directional sensors providing a coverage of 360◦. Weextend results of some of our previous work on this subject and studythe effects of different network parameters on the performance of theproposed methods. We also extend the methods so that they can trackmore than one target simultaneously moving in the sensor field.

Index Terms— directional sensors, particle filtering, targettracking

1. INTRODUCTION

In this paper we present our advances in the investigation of non-centralized target tracking in networks of directional sensors. Ourfirst results on this subject were reported in [1]. The interest indistributed processing in wireless sensor networks has been veryhigh lately [2], [3]. The motivation for distributed processing stemsfrom the requirement of avoiding intensive communication overlarge distances so that battery-operated sensor nodes can last longer.Another inducement is that in centralized processing, the tracking iscarried out by the central unit, and when it fails, so does the task oftarget tracking. Therefore, such networks are not robust.

In general, non-centralized processing can be implemented byconsensus-based methods where sensors locally process their dataand exchange their estimates with their neighbors until convergenceis reached [4], [5]. The other possibility is to have the nodesexchange and relay the measurements of all the sensors so that allthe processing nodes in the network (all or a few of them) have thenecessary data to produce the required estimates [6]. In this paper,we deal with the second implementation.

In [1], we proposed two methods for target tracking in networkswith directional sensors and based on binary information (meaningthat the sensors only report if a target was detected). The sensor net-work was composed of nodes, where each node had four directionalsensors providing a coverage of 360◦. We refer to our methods asall-node and one-node methods (ANM and ONM, respectively). Asthe names suggest, with ANM all the nodes in the network performtracking and with ONM, only one node that is in the proximity ofthe target does it. The two methods were implemented with particlefiltering (PF) [7]. The nodes of the network cooperate in that theybroadcast to their neighbors the outcomes of their detection algo-rithms. In this paper, we extend the methods so that they can be ap-

This work was supported by ONR under Award N00014-09-1-1154 andNSF under Award CCF-1018323.

plied with sensors that provide more precise information (than justbinary information) about the location of the detected targets, andthat the sensor nodes can also track more than one target at a time.

The paper is organized as follows. In the next section wedescribe a more general formulation of the problem than in [1]. Thenin Section 3, we present new results. In Section 4, we demonstratethe performance of the proposed method with various computersimulations, and in Section 5, we provide conclusions and some finalremarks.

2. THE PROBLEM

There are J targets that move in a two-dimensional plane accordingto

xt = Axt−1 +But, (1)

where t ∈ N0, xt = [x>1,t x>2,t · · · x>J,t]> ∈ R4J×1 is the

unknown state vector at time instant t, and xj,t ∈ R4×1 is definedby

xj,t = [xj,1,t xj,2,t xj,1,t xj,2,t]>,

where xj,1,t and xj,2,t represent the coordinates of the jth targetin the two-dimensional Cartesian coordinate system, and xj,1,t andxj,2,t are the respective components of the velocity of that target.The matrices A ∈ R4J×4J and B ∈ R4J×2J are known, and theydefine the dynamics of the targets, whereas ut ∈ R2J×1 is a statenoise process whose distribution is known.

The targets move in a field of 4N directional sensors deployed atN different locations given by rn = [r1,n r2n]>, n = 1, 2, · · · , N .At each location, there are four collocated sensors forming a nodeand where each sensor senses targets in a field of view of 90◦, asshown in Fig. 1. We refer to the four sensors as the northeast(NE), northwest (NW), southwest (SW), and southeast (SE), or first,second, third and fourth sensor, respectively.

The sensors make scalar measurements about the targets in thefield, and they are given by

yk,n,t = gk,n(xt) + vk,n,t,

where k = 1, 2, 3, 4 is an index of the sensor at the nth node, andyk,n,t is the measurement of the sensor (identified by k and n), andvk,n,t is noise with distribution that is assumed Gaussian with meanµv and variance σ2

v . We define the functions gk,n(·) by

gk,n(xt) =J∑j=1

Ψdα0||rn − lj,t| |α

I(k, n, lj,t), (2)

2011 4th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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S

SE

NW

E

N

R

W

SW

NE

sensor

3sen

sor 2

sensor 4

sensor 1

Fig. 1. Four co-located sensors at a node.

where Ψ(·) denotes the emitted power of the target measured atsome predefined distance d0, lj,t = [xj,1,t xj,2,t]

> is the locationof the jth target at time instant t, and α is a path-loss parameter. Thesymbol I(k, n, lj,t) is an indicator function given by

I(k, n, lj,t) =

{1, lj,t ∈ Rk,n0, otherwise

,

whereRk,n is the area of sensitivity of the sensor.The nodes transmit signals to their neighbors, where the signals

are defined by

sk,n,t = i, γi ≤ yk,n,t < γi+1, i = 0, 1, 2, · · · , L− 2,

where L ≥ 3, and γi are thresholds used for constructing sk,n,t,with γ0 = −∞, γL−1 = ∞, and γ0 < γi < γi+1 < γL−1,i = 1, 2, · · · , L − 3. In the simplest case, when L = 3, the signalsk,n,t takes only one of two values, that is, then the sensors provideonly binary information. While broadcasting sk,n,t, the nodes alsobroadcast the identification (ID) number of the sensors that detecteda target. The nodes also have the capability to relay such informationfrom their neighbors. The communication between nodes is assumedto be error free. The objective is that the nodes, given the availableinformation, track the targets in the field with particle filters that runat all the nodes or only at some predefined nodes.

3. NON-CENTRALIZED TARGET TRACKING

In [1] we addressed the same problem as described in the previoussection except that we allowed for only one target in the field at atime and where L = 3. We considered two scenarios. In one ofthem, the nodes kept relaying information about the sensed targetuntil all the nodes in the network had complete information aboutwhat was sensed. It was a setup where each and every node waslike a central unit, possessing all the ID numbers of the sensors thatdetected the target at a time instant t. In the second scenario, thebroadcasting continued only a limited number of times, so that onlythe nodes in the proximity of the target had all the sensed informationabout the target.

The first setup was equivalent to centralized target tracking inbinary sensor networks (in the sense that every node was performingthe tasks of a centralized unit). We followed the line of reasoningfrom [8], and obtained the necessary expressions for computing theweights of the generated particles.

In the second setup, only one node at a time performed particlefiltering. This was always a node that was in the proximity of the

target. The information used for processing was collected from thenodes in the neighborhood of the processing node. We proposed anon-centralized scheme for deciding which node would take over theprocessing at the next time instant. We also provided details aboutwhat information had to be transferred to the next processing node.

We compared the two setups both in terms of average meansquare error of the estimated target location and average total numberof transferred bits during tracking. We found that the loss inperformance of the second setup was almost negligible although itsnumber of transmitted bits was somewhat less than 50% of that ofthe first setup.

As already pointed out, previously we only focused on binarysensors. Here we extend the analysis to sensors that use multiplethresholds. Since we apply PF, we have to obtain the equation forupdating the weights of the generated particles. We can show that

w(m)t ∝ w

(m)t−1

∏k,n∈Dt

p(sk,n,t|l(m)1:J,t),

where l(m)1:J,t ≡

{l(m)1,t , l

(m)2,t , · · · , l

(m)J,t

}, is the mth particle and

w(m)t is the associated weight at time instant t, and

p(sk,n,t = i|l(m)

1:J,t

)= Q

(θ(m)k,n,i

)−Q

(θ(m)k,n,i+1

),

where Q(·) is the complementary distribution function of the stan-dard Gaussian distribution, θ(m)

k,n,i = [l(m)1:J,t µv σv γi k n]>, and

where

Q(θ(m)k,n,i) = Q

γi − gk,n(l(m)1:J,t

)− µv

σv

with g(·) being given by (2).

An interesting question is the choice of range for the directionalsensors. We address a sensor network with nodes that are equidis-tantly located and with separation equal to d. Let ρ be the distancethat defines the range of the sensor. For a given ρ ∈ (

√2d2, d), we can

partition the square formed by four adjacent nodes with sets createdby the ranges of the sensors monitoring parts of that square. Let thesets that form the partition be denoted by Ai, i = 1, 2, · · · , I . Theset Ai simply denotes the uncertainty area if some of the neighbor-ing four sensors (that form the square) detect the target. Clearly, ifwe assume that a target may be anywhere in the square with uniformdistribution, then the probability that the target is in area Ai is givenby pi = Ai/d

2. We define the expected uncertainty E(U) when atarget is in the square by

E(U) =

I∑i=1

piAi.

This sum is a nonlinear function of ρ. The objective is to find theoptimal ρ, the one that minimizes the expected uncertainty. Whend ∈

(√2d/2, d

), we obtained that the optimal ρ is about 0.8d.

An important extension of our previous work is the applicationof our algorithms to multiple target tracking. The ANM practicallyperforms in the same way as in the case of only one target. Animportant difference is that the dimension of the state space is now4J (J > 1), instead of just four. This obviously leads to deterioratedperformance of the PF when the same number of particles are used.

More challenging is the problem of extending the equivalentof ONM when there is more than one target in the sensor field.

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Obviously, the number of nodes tasked with tracking will be afunction of time, and theoretically, it can vary between one and J .When all the targets are in close proximity to each other in time andspace, only one node may be employed for tracking, and by contrast,when they are all separated, J different nodes will be tracking thetargets. The process of node assignment for tracking builds on themethod proposed in [1].

We now briefly describe this method. Let the number of nodestracking the targets be Jt ≤ J . After estimating the predictedlocations of their targets at time instant t − 1, all of these nodesbroadcast the obtained estimates and ID numbers of the targets. Theneighboring nodes of the transmitting nodes receive this information,and based on their own ID numbers and the predicted locations of thetargets in their proximity decide without ambiguity which nodes willdo processing at time instant t. Once a decision is made, the PF atthese nodes proceeds as described in [1].

4. SIMULATION RESULTS

In this section, we present simulations of our tracking algorithmsand study the effect on their performance on both, the number ofthresholds used for detection and the range of the sensors. We alsopresent results of tracking two targets.

In the simulations we used a network with N = 100 sensorsdeployed on a square grid, where the neighboring nodes wereseparated by d = 40 m. We set some of the parameters as follows:α = 2.5, Ψ = 5, 000, and d0 = 1 m. The threshold was chosenso that the sensor range was ρ = 32 m (that is, ρ = 0.8d). Thestate noise process had a covariance matrixCu = diag{0.05, 0.01},whereas the measurement noises had a mean µv = 1 and a varianceσ2v = 0.01. The sampling interval was Ts = 1 s. The matrices A

andB from (1) were defined by

A = I ⊗

1 0 Ts 00 1 0 Ts0 0 1 00 0 0 1

, B = I ⊗

T2s2

0

0T2s2

Ts 00 Ts

(3)

where⊗ denotes the Kronecker product, and I is the identity matrixwith size J × J .

The PF algorithm used M = 500 particles. The initialdistribution of the particles was Gaussian with a mean x0 =[220 220 0.01 0.01]> and a covariance matrix C0 =diag{10, 10, 0.1, 0.1}. Each experiment consisted of K = 50 trials(realizations).

We found that the tracking performances in this network werebasically the same as those shown in [1], where the total numberof sensors was 25 only. We note that the density of nodes of thetwo networks was the same. The obtained results are not surprisingbecause the performance should not depend on the size of thenetwork when we assume error free communication and no latency.

We also compared the communication costs of ANM and ONMwhen they are implemented on the two networks. The comparison isgiven in Table 1, where we listed the average of the total numberof transferred bits for the two methods. It can be seen that theANM does not scale well. Its average of total number of transferredbits increased by about 4.3 times when the number of nodes is 100instead of 25. The ONM had a factor of increase of about only 1.3when we moved from a network of 25 to a network of 100 nodes.The increase of communication cost of the ANM when N = 100 isdue to the larger network size and the need for longer ID numbers of

Average total of transferred bitsMethod BSN1 BSN2 DBSN1 DBSN2ANM 27586 120589 38621 168834ONM 15351 21302 18995 25975

Table 1. Average total number of transferred bits for the ANMand ONM in binary sensor networks (BSNs) and directional binarysensor networks (DBSNs). Here BSN1/DBSN1 indicates a networkwith 25 nodes, and BSN2/DBSN2 a network with 100 nodes.

the nodes, whereas ONM increases its communication cost becauseof the longer ID numbers.

In all previous experiments, we have simulated the performancefor networks with binary directional sensors. Here, we conductedexperiments where the sensors used multiple thresholds (one, two,three, and four thresholds). The results are shown in Fig. 2 (rootmean square error (RMSE) of the position of the target as a functionof time, i.e., of

√(x1,t − x1,t)2 + (x2,t − x2,t)2) and in Fig. 3

(RMSEs of the four different states of the target as functions oftime). As expected, the best results were obtained when the sensorsused four thresholds and the worst, for one threshold. In practicalimplementations, one would use three thresholds rather than four.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10

time (s)

RM

SE in

pos

ition

(m)

Average Position RMSE

one thresholdtwo thresholdsthree thresholdsfour thresholds

Fig. 2. RMSE of position of one target tracking with multiplethresholds as functions of time.

The sensor range ρ affects the network coverage and the trackingperformance. In the previous section, we defined the optimal rangeof the sensors and found its value numerically. In Fig. 4 we show theperformance of the ANM for the following sensor ranges: ρ = 0.5d,ρ = 0.8d, ρ = d, and ρ = 1.5d. The results confirm that the bestperformance is for ρ = 0.8d.

Finally, we did experiments with two targets. The parameters ofthe model were selected in a way to allow for having the targets intime-space be in proximity to each other. The matrices A and B ofthe targets were defined by (3). We studied two scenarios, where inone the targets had low initial velocities, and in the other, they hadhigh initial velocities. More specifically, the initial states for the lowvelocity targets were set to

x1,0 = [180 260 0.01 0.01]>, x2,0 = [180 220 0.01 0.01]>

and for high velocity to

x1,0 = [180 260 2 − 2]>, x2,0 = [180 220 2 2]>.

87

0 10 20 30 40 500

2

4

6

8

10

time (s)

RM

SE

ofx

1,t

(m)

0 10 20 30 40 500

2

4

6

8

10

time (s)

RM

SE

ofx

2,t

(m)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time (s)

RM

SE

ofx

1,t

(m/s

)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time (s)

RM

SE

ofx

2,t

(m/s

)

Fig. 3. RMSEs of the individual states of one target tracking withmultiple thresholds as functions of time. The legends of the curvesare identical to that of Fig. 2

0 5 10 15 20 25 30 35 40 45 500

5

10

15

time (s)

RM

SE in

pos

iton

(m)

r = 0.5dr = 0.8dr = dr = 1.5d

Fig. 4. RMSE of position of one target tracking with different sensorranges as functions of time obtained with ANM.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

10

time (s)

RM

SE in

pos

iton

(m)

T1 Low veloT2 Low veloT1 High veloT2 High velo

Fig. 5. RMSE of two targets as functions of time.

The RMSEs of the targets’ positions obtained by ANM areshown in Fig. 5. A better performance was obtained for the highvelocity targets. This can further be confirmed from Fig. 6, wherewe plotted the number of lost tracks in the two cases. The definitionof lost tracks was based on et =

√(x1,t − x1,t)2 + (x2,t − x2,t)2

exceeding a threshold Λ within at least 10 consecutive samplingperiods [9]. In our experiments, we set Λ to 20 m, 50 m, 80 m, 100

20 50 80 100 1200

5

10

15

(m)

num

ber o

f tra

ck lo

ss

Low velocityHigh velocity

Fig. 6. Number of lost tracks for different criteria.

m, and 120 m. From the histogram, we can see that the algorithmwas more successful in tracking the targets with high velocity.

5. CONCLUSIONS

In this paper, we addressed non-centralized target tracking in net-works of directional sensors. We extended some of our previouswork on this subject by allowing the sensors to use more than onethreshold in making decisions. We also investigated the effects ofthe sensor ranges on the performance of the algorithms. Finally, wemodified the methods so that they can handle the more challengingtask of tracking two or more targets, which are simultaneously mov-ing in the sensor field.

6. REFERENCES

[1] P. M. Djuric and L. Geng, “Non-centralized target tracking innetworks of directional sensors,” in the Proceedings of the IEEEAerospace Conference, Big Sky, Montana, 2011.

[2] G. Ferrari, Sensor Networks: Where Theory Meets Practice,Springer, Heidelberg, Germany, 2010.

[3] S. Haykin and K. J. R. Liu, Eds., Handbook on Array Processingand Sensor Networks, John Wiley & Sons, 2010.

[4] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus andcooperation in networked multi-agent systems,” Proceedings ofthe IEEE, vol. 95, no. 10, pp. 319–339, 2007.

[5] A. G. Dimakis, S. Kar, J. M. F. Moura, M. G. Rabbat, andA. Scaglione, “Gossip algorithms for distributed signal process-ing,” Proceedings of the IEEE, vol. 98, no. 11, pp. 1847–1864,2010.

[6] T. Zhao and A. Nehorai, “Distributed sequential Bayesianestimation of a diffusive source in wireless sensor networks,”IEEE Transactions on Signal Processing, vol. 55, no. 4, pp.1511–1524, 2007.

[7] A. Doucet, N. de Freitas, and N. Gordon, Eds., SequentialMonte Carlo Methods in Practice, Springer, New York, 2001.

[8] P. M. Djuric, M. Vemula, and M. F. Bugallo, “Target tracking byparticle filtering in binary sensor networks,” IEEE Transactionson Signal Processing, vol. 56, no. 6, pp. 2229–2238, 2008.

[9] M. F. Bugallo, S. Xu, and P. M. Djuric, “Performance com-parison of EKF and particle filtering methods for maneuveringtargets,” Digital Signal Processing, vol. 17, pp. 774–786, 2004.

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