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Volume 2006, Article ID 83042, Pages 1 16DOI 10.1155/ASP/2006/83042 Particle Filtering Algorithms for Tracking a Maneuvering Target Using a Network of Wireless Dynamic Sensors Joaqu´ın M

Volume 2006, Article ID 83042, Pages 1 16

DOI 10.1155/ASP/2006/83042

Particle Filtering Algorithms for Tracking a Maneuvering

Target Using a Network of Wireless Dynamic Sensors

Joaqu´ın M´ıguez and Antonio Art ´es-Rodr´ıguez

Departamento de Teor´ıa de la Se˜nal y Comunicaciones, Universidad Carlos III de Madrid, Avenida de la Universidad 30,

Legan´es, 28911 Madrid, Spain

Received 16 June 2005; Revised 24 January 2006; Accepted 30 April 2006

We investigate the problem of tracking a maneuvering target using a wireless sensor network We assume that the sensors are binary (they transmit ’1’ for target detection and ’0’ for target absence) and capable of motion, in order to enable the tracking

of targets that move over large regions The sensor velocity is governed by the tracker, but subject to random perturbations that make the actual sensor locations uncertain The binary local decisions are transmitted over the network to a fusion center that recursively integrates them in order to sequentially produce estimates of the target position, its velocity, and the sensor locations

We investigate the application of particle filtering techniques (namely, sequential importance sampling, auxiliary particle filtering and cost-reference particle filtering) in order to efficiently perform data fusion, and propose new sampling schemes tailored to the problem under study The validity of the resulting algorithms is illustrated by means of computer simulations

Copyright © 2006 J M´ıguez and A Art´es-Rodr´ıguez This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 INTRODUCTION

Recently, there has been a surge of interest in the

applica-tion of networks of wireless microsensors in diverse areas,

including manufacturing, health and medicine,

transporta-tion, environmental monitoring, scientific instrumentation

is that they involve the detection, classification, and

track-ing of signals, with the outstandtrack-ing peculiarity that the large

amounts of (possibly multimodal) data acquired by the

sen-sors must be handled and integrated in order to perform

are usually depicted as a collection of data-acquiring devices

(sensors) and one or more fusion centers which are in charge

of integrating the data to extract the information of

inter-est

This paper deals with the problem of tracking a

As well as in most WSN applications, the main constraints

are related to the network cost of deployment and

opera-tion It is desirable that the sensors be inexpensive and, as a

consequence, devices with limited processing capabilities are

consump-tion restricconsump-tions must be met for the continued and

reli-able operation of networks consisting of battery-supported

sensors In this respect, radio communication is a major

However, this implies that only a small fraction of the data collected by the sensors can be transmitted to the fusion cen-ter, which results in a decrease of the tracking capability One solution is to deploy dense networks (i.e., networks with a large number of sensors per unit area/volume) in order to boost performance but, with this approach, it may turn out prohibitive to provide adequate coverage for large regions Another strong requirement which is peculiar to target track-ing and target localization [6,7] applications is the need to accurately estimate the position of the sensors in the network [8], a task which is often included in network calibration [9] When the number of sensors is large, the accurate estimation

of their locations becomes hard

Bearing in mind the above considerations, we investi-gate the tracking of a maneuvering target on a 2-dimensional

mea-sure some distance-related physical magnitude (e.g., re-ceived signal strength) and use it to make a binary de-cision regarding the presence of the target within a cer-tain range The resulting bit (“1” if the target is detected within the sensor range, “0” otherwise) is transmitted to the fusion center, where the local decisions are integrated to

recursively estimate the current position and velocity of the

target

Since the target can move over a large region and it

may not be possible to deploy such a widespread

net-work with the required sensor density for adequate

per-formance, we propose to use a relatively small number

of dynamic sensors Specifically, we assume that the

sen-sors are deployed randomly (there is uncertainty in the

knowledge of their initial positions) and the network is

endowed with a control system that allows the tracker to

command the sensors to move with a certain speed

(in-cluding magnitude and phase) The speed of motion

as-signed to each sensor is determined by the tracker using

the target trajectory estimates provided by the fusion

cen-ter, but the actual sensor movement is also subject to

ran-dom perturbations We will discuss several strategies for

sen-sor speed assignment that differ in complexity and

perfor-mance

The novel target tracking algorithms that we introduce in

this paper are largely based on the sequential Monte Carlo

(SMC) methodology, also known as particle filtering (PF)

time-varying state of a dynamic system that cannot be observed

directly, but through some related measurements The a

pos-teriori probability distribution of the state given the

using a probability mass function (pmf) with random

sup-port This pmf is composed of samples in the state-space

and associated weights, which can be intuitively seen as

ap-proximations of the a posteriori probability of the

sam-ples The PF approach lends itself naturally to the

prob-lem at hand, where the unobserved dynamic state consists

of the target position and speed together with the sensor

locations, and the measurements are the local sensor

deci-sions and the power of the communication signals

trans-mitted by the sensors and received at the fusion center

From this point of view, a PF algorithm performs a data

fusion process that consists of approximating the a

poste-riori distribution of the target and sensor trajectories, as

well as any estimators that can be derived from it, given

families of PF techniques (in particular, sequential

impor-tance sampling, auxiliary particle filtering and cost-reference

particle filtering) and propose new sampling schemes

tai-lored to the problem of joint sensor and target

track-ing

The remaining of this paper is organized as follows

Section 2introduces some background material on PF The

problem of tracking a maneuvering target using dynamic

model suitable for application of the PF methods is

de-rived Four target-tracking PF algorithms are introduced in

Section 4and, since these procedures are based on the

net-work capability to govern the speed of motion of the

sen-sors, different velocity assignment strategies are discussed in

Section 5 Illustrative computer simulation results are

pre-sented inSection 6and, finally,Section 7is devoted to a brief

discussion and concluding remarks

2 BACKGROUND: BAYESIAN TRACKING AND PARTICLE FILTERING

Let us consider the dynamic system in state-space form

x t = f x



x t−1,u t



y t = f y



x t,φ, n t



, t =1, 2, (observation equation),

(2)

wherex tis the system state,y tis the corresponding

(possibly non-Gaussian) system noise and observation noise processes, respectively For the sake of deriving estimation

usually made on (1)–(2) These include the pair-wise statisti-cal independence ofx t,u t, andn t, the availability of a known prior probability density function (pdf) for the initial state,

p(x0), and the fixed parameters,p(φ), and the ability to

sam-ple the transition pdf p(x t | x t−1) and to evaluate the likeli-hood functionp(y t | x0:t,y1:t−1) (which reduces top(y t | x t)

Many problems can be stated as the estimation of the se-quence of statesx0:t = { x0, , x t }given the series of obser-vationsy1:t= { y1, , y t } From the Bayesian point of view, all relevant information is contained in the smoothing pdf,

p(x0:t | y1:t), but it is in general impossible to find it (or its moments) in a closed form The sequential importance sam-pling (SIS) algorithm is a recursive Monte Carlo technique

recursive decomposition

p

x0:t| y1:t



∝ p

y t | x0:t,y1:t−1

p

x t | x t−1

p

x0:t−1| y1:t−1

(3)

particu-lar, we are interested in approximatingp(x0:t | y1:t) from its samples, but it is obviously impossible to draw from the lat-ter pdf directly Instead, we choose an importance pdf (also

do-main that includes that ofp(x0:t| y1:t), and drawM samples

x0:t(i) ∼ q(x0:t | y1:t),i = 1, , M Each sample is weighted

asw(i)t = p(x0:t(i)| y1:t)/q(x(i)0:t | y1:t) and the pair (x(i)0:t,w(i)t ) is called a particle

However, the computational cost of drawing particles in this way grows with time, which is unacceptable for online

importance function to be factored asq(x0:t | y1:t)∝ q(x t |

x0:t−1,y1:t)q(x0:t−1 | y1:t−1) and, using (3), it turns out that both sampling and weighting of the particles can be

{ x0:t−1(i) ,w t−1(i)} M

i=1, at timet we proceed to draw x(i)t and update

w t−1(i) as

x t(i)∼ q

x t | x0:t−1(i) ,y1:t





w t(i)= p



y t | x(i)0:t,y1:t−1



p

x t(i)| x(i)t−1

q

x(i)t | x(i)0:t−1,y1:t

 w(i)t−1, (5)

w(i)t = w

(i)

t

M

k=1 w(k)t

en-sure that the collection of weights yields a proper pmf for

finiteM The new setΩt = { x(i)0:t,w t(i)} M

i=1readily yields a dis-crete approximation of the smoothing pdf,

p

x0:t| y1:t



≈  p

x0:t| y1:t



=

M

i=1

w(i)t δ

x0:t− x0:t(i)





p(x0:t | y1:t)→ p(x0:t | y1:t), asM → ∞[17] Any desired

for example, the minimum mean square error (MMSE)

es-timate of x0:t is xmmse

0:t = M

i=1 w(i)t x(i)0:t Equations (4)–(6) constitute the sequential importance sampling (SIS)

algo-rithm

One major practical limitation of the SIS method is

that the variance of the weights increases stochastically over

time until one single weight accumulates the unit

proba-bility mass, rendering the approximation p(x 0:t | y1:t)

Intuitively, resampling amounts to stochastically discarding

particles with small weights while replicating those with

larger weights Although several schemes have been proposed

[13, 14,19], we will restrict ourselves to the conceptually

y1:t)

One appealing version of the SIS approach is obtained when

we integrate the importance sampling and resampling steps

using the auxiliary particle filtering (APF) technique [20] In

particular, we define

q

x t,k | x(k)0:t−1,y1:t



∝ ρ(k)t p

x t | x t−1(k)

w(k)t−1, (8)

where k is an “auxiliary index,” ρ t(k) = p(y t |  x(k)t ,x(k)0:t−1),

andxkis an estimate ofx k givenx(k)t−1(typically the mean of

p(x t | x t−1(k))) The sampling step is then carried out in two

stages,

k(i)∼ q t(k), whereq t(k = j) ∝ w(j)t−1 ρ(j)t ,

x t(i)∼ p

x t | x t−1(k(i))

, i =1, , M,

(9)

that is, we randomly select the indices of the particles with a higher predicted likelihood (according to the estimatesx(i)t ) and then propagate the selected particles using the transition pdf The new samples are joined with the former ones ac-cording to the auxiliary indices,x(i)0:t = { x(k0:t−1(i)),x(i)t }, and the weights are updated as

w t(i)∝ p



y t | x(i)0:t,y1:t−1



ρ(kt−1(i)) . (10)

carried out at each time step, but this scheme has the

through the predictive likelihoodρ(j)t

When a reliable and tractable probabilistic model of system (1)–(2) is not available, the PF approach can still be exploited using the cost-reference particle filtering (CRPF)

let us aim at computing the sequence of states that attains the minimization of an arbitrary cost function with additive structure,

C

x0:t,y1:t



x0:t−1,y1:t−1

x t,y t



where 0 < λ < 1 is a forgetting factor and  C is an

incre-mental cost function If a risk function is also defined that consists of the cost of a predicted state, that is,R(x t−1,y t)=

λC(x0:t−1) + C( f x(x t−1, 0)) (zero-mean system noise is as-sumed), then we can describe the basic steps of a CRPF algo-rithm

(1) Initialization: draw { x(i)0 ,C(i)0 =0} M

i=1from an arbitrary pdf The only constraint is that the particles should not

be infinitely far away from the initial state Notation

C0(i)represents the cost of theith particle in the absence

of observations, that is,C(i)0 = C(x0,y1:0)

(2) Selection: givenΩt−1 = { x(i)t−1,C t−1(i) } M

ob-servationy t, compute the risksR(i)t = R(x t−1(i) ,y t) and resample the particles according to the probabilities

p

x t−1(i)

R(i)t



nonin-creasing function, termed the generating function The resampled particles preserve their original costs and yieldΩt−1 = { x(i)t−1,C(i)

t−1 } M i=1

(3) Propagation: generate new particles by drawing x(i)t ∼

p t(x t | x t−1,y t) from an arbitrary propagation pdf p t

Assign new weights, C(i)t = λ C(i)

t−1 + C(x(i)t ,y t), to buildΩt = { x(i)t ,C t(i)} M

i=1 Estimation of the state can be performed at any time,

ei-ther by choosing the minimum cost particle or by

assign-ing probabilities to the particles of the formπ t(i) ∝ μ(C(i)t )

M

i=1 π t(i)x t(i)

3 SYSTEM MODEL

We address the problem of tracking a maneuvering

tar-get that moves along a 2-dimensional space using a WSN

It is assumed that each sensor can measure some

physi-cal magnitude related to the distance between the target

and the sensor itself, and then uses the measurement to

determine whether the target is within a predefined range

or not (we will henceforth refer to the sensors as

“bi-nary”) The distinctive features of the system under study

are the following: (a) the sensor locations are not exactly

known, so they must be estimated together with the

tar-get trajectory, and (b) the sensors are dynamic, that is,

they are able of motion with adjustable speed (magnitude

and phase) With this setup, it is possible to use a

rela-tively small network to follow a target over a large area

by making the sensors move according to the estimated

track

We can describe the tracking problem using a dynamic

state-space model formulation similar to (1)–(2) Letr(τ) and v(τ)

be the continuous-time complex random processes of the

target position and velocity, respectively If measurements

straightforward to derive the discrete-time linear kinematic

model [22]

xt =Axt−1+ Qut, (13)

where xt = [r t,v t] ∈ C2is the target state at discrete time

t, r t = r (τ = tT s) andv t = v (τ = tT s) are samples of the

position and speed random processes, respectively,

A= 1 T s

0 1

(14)

uI2

N(u t |0,σ2

uI2), and

Q=

⎡

⎢12T2

s 0

⎤

The target probabilistic description is completed with a known a priori pdf for the initial target location and speed,

before any observations are collected

The system state includes the target position and velocity,

as well as the sensors positions, and we also assume a linear motion model for the latter Specifically, there areN ssensors

s i,t = s i,t−1+T s v s

i,t+m i,t, i =1, 2, , N s, (16)

wheres i,t andv i,t s are complex values that represent theith

N(m i,t | 0,σ2

velocity is assigned by the tracking algorithm Several

i,t given an estimate of xt are possible

i =1, , N s, that account for the randomness in the initial deployment of the network

By defining the vector of sensor locations, st =[s1,t, ,

s N s,] , and taking together (13) and (16), we can write the complete system state equation as

xt =Axt−1+ Qut,

st =st−1+T svs

where vs t = [v s

1,t, , v s

N s,] and mt ∼ N(m t | 0,σ2

mIN s) When needed, we will denote the complete system state as

σ t =[x t, s t] ∈ C N s+2

decisions produced by the sensors, whereas the sensor trajec-tory itself is estimated from the received power of the signals transmitted by the sensors To be specific, lety i,t ∈ {0, 1}be

probabil-ities

p

y i,t =1| r t,s i,t



= p d



d i,t,α

,

p

y i,t =0| r t,s i,t





d i,t,α

,

(18)

whered i,t = | r t − s i,t |is the distance between the target and theith sensor location at time t Therefore, the probability of

param-eterα > 0 which represents the probability of a false alarm

(i.e., the probability of a positive output when there is no

de-pends on the physical magnitude that is measured, the sensor range of coverage as well as other practical considerations Notwithstanding, we assume the following general proper-ties ofp d[10]:

(i) p d(d i,t,α) ≥0 (since it is a probability), (ii) limd i,t →∞ p d(d i,t,α) = α (this is the definition of the

false alarm probability), and (iii) p dis a monotonically decreasing function ofd i,t

The vector of local decisions is subsequently denoted as

yt =[y1,t, , y N s,]

For the measurements of the received signal powers, we

adopt the log-normal noise model commonly used in mobile

π i,t s =10 log10

⎛

s i,t − r oβ

⎞

⎠+l i,t

=10 log10

⎛

⎝s i,t −1r oβ

⎞

⎠+ i+l i,t,

(19)

transmitted power, the carrier frequency, and other

phe-nomena causing power attenuation, such as multipath

fad-ing, except the distance),π s

l)

pa-rameter but, in order to account for unknown power

(de-pending on the practical setup of the communication

sys-tem we may wish to define the intermediate powers as

π s

t =[π1,ts , , π N s s,]

We use notationθ t =[yt ,π

t ] for the complete 2N s ×

1 observation vector Notice that the observation function

(equivalent to f y in (2)) for this system is highly nonlinear

and it is hard even to write it in some compact form

How-ever, assuming statistical independence among the various

noise processes in the state and observation equations, it is

straightforward to derive the likelihood function, namely,

p

θ t | σ0:t,θ1:t−1

= p

yt |xt, st



p

π s

t |s0:t,πs

1:t−1



in

p

yt |xt, st



=

N s



k=1

p

y k,t | r t,s k,t



If we assume that the random variables k,k =1, , N s,

have Gaussian a priori densitiesN( k | k,0,σ2

k,0), the second factor in (21),

p

π s

t |s0:t,π s

1:t−1



=

N s



k=1

p

π s k,t | s k,0:t,π s

k,1:t−1



can be calculated recursively using the formulae



π s k,t = π s k,t −10 log10

s k,t − r o−β

k,t = σ

2

l k,t−1+ k,t−1 πs

k,t

σ2

l +σ2

k,t−1

σ2

k,t = σ

2

l σ2

k,t−1

σ2

l +σ2

k,t−1

p

π k,t s | s k,0:t,π k,1:t−1 s 

∝



 σ2

k,t

σ2

l σ2

k,t−1

exp



2



π s2

k,t

σ2

l

+2k,t−1

σ2

k,t−1 − 

2

k,t

σ2

k,t

, (26)

as shown inAppendix 7, where k,tandσ2

k,tdenote the a

measure-mentsπ k,1:t s

Our aim is to estimate the trajectory of the target and the

sen-sors, x0:tand s0:t, respectively, from the series of observations,

posteriori smoothing density p( σ0:t | θ1:t) and, in

0:t = E p( σ0:t | θ1:t)[σ0:t],

the pdf in the subscript

Because of the strong nonlinearities in the system model, neither the smoothing density nor the MMSE estimator have

a closed form and, as a consequence, numerical methods are necessary In the sequel, we explore the application of the SIS

desired track estimates

If the probabilistic assumptions on the model (1)–(2) are not reliable enough (e.g., we may suspect that the noise pro-cesses that appear both in the state and observation equations are non-Gaussian), it may be desirable to resort to the more robust CRPF methodology For this reason, we also extend

incor-porate the sensor motion and location uncertainty

4 PF ALGORITHMS FOR MANEUVERING TARGET TRACKING

Ωt−1 = { σ(i)

0:t−1,w(i)t−1 } M

notation σ(i)

t = x(t i)

s(t i)

importance function consists of two steps:

t ∼ q

σ t | σ(i) 0:t−1,θ1:t



,

w(i)t ∝ w(i)t−1 p



θ t | σ(i) 0:t,θ1:t−1



p

σ(i)

t | σ(i)

t−1



q

σ(i)

t | σ(i) 0:t−1,θ1:t

 = w t−1(i) p



yt |x(i)t , s(i)t 

p

π s

t |s(i)0:t,π s

1:t−1



p

xt(i)|x(i)t−1

p

s(i)t |s(i)t−1

q

σ(i)

t | σ(i) 0:t−1,θ1:t

(27)

st−1) = p(x t | xt−1)p(s t | st−1) The simplest form of the

σ t−1)= p(x t |xt−1)p(s t |st−1) as a trial density [14] In such

case, the weight update equation is significantly simplified,

w(i)t ∝ w(i)t−1 p

yt |x(i)t , s(i)t 

p

π s

t |s(i)0:t,π s

1:t−1



that is, the weights are sequentially computed according to

the likelihoods alone Note at this point that the

compu-tation of p( π s

t | s(i)0:t,π s

1:t−1), by means of (26), requires

up-date the posterior mean and variance of the log-powers

k,k = 1, , N s, according to (23)–(25) Unfortunately,

a proposal pdf leads to simple but inefficient algorithms,

that usually need a large number of particles to achieve

an adequate performance Our computer simulations (see

Section 6) show that this is the case, indeed, and the

rea-son is that new particles are generated “blindly,” without

regard of the information contained in the new

observa-tions,θ t[14] We hereafter use the term standard SIS (SSIS)

algorithm for the procedure based on the prior proposal

pdf

In general, the performance of an SIS algorithm is highly

dependent on the design of an efficient importance

func-tion, that is, one that yields particles in the regions of the

state-space with large a posteriori probability density [13,14]

This is normally accomplished by exploiting the latest

obser-vations when sampling the particles In our case, and

un-less the variance σ2

we can expect that sampling the sensor positions from the

prior, s(i)t ∼ p(s t |s(i)t−1), yield acceptable results This is

the PF algorithm remains locked to the sequence of positions

continu-ous raw data and yield a highly informative likelihood

How-ever, tracking the target state is considerably more involved

avail-able observations are binary and they do not directly

pro-vide any information on the target velocity (only on its

posi-tion)

Intuitively, we propose to overcome these difficulties by

building an importance function that generates trial

a region of the state-space with a high likelihood for the

im-provement of the SIS algorithm both in terms of efficiency (a lower number of particles is required for a certain de-gree of estimation accuracy) and robustness (the percent-age of track losses becomes negligible), as will be shown by our computer simulations The details of the procedure are

as follows Let{ κ n } N1

the sensors that produce a positive decision at timet, that is,

y κ1 , = · · · = y κ N1, = 1 and all other sensors transmit a 0

value Then, for each sample x(i)t , we deterministically con-struct a set ofN1predicted target states of the form



x(i)t,κ n =

⎡

⎣r

(i)

t−1+T s v(i)κ n,

v κ(i)n,

⎤

where

v(i)κ n, = s(i)κ n, − r t−1(i)

is the speed value that partially forces the target position to-wards theκ nth sensor, with 0 <  < 1 For each one of the

predicted target states, a likelihood is computed,

(i)

κ n, = p

yt | x(i)t,κ n,s(i)t 

wheres(i)t =s(i)t−1+T svs t−1are the predicted sensor positions From the likelihoods, a mean velocity component is calcu-lated for theith particle,

v(i)t−1 =

N1

n=1

v κ(i)n,

(i)

κ n,

N1

r=1 κ(i)r,

(32)

and, finally, the mean velocity values are used to generate new particles using a Gaussian proposal pdf

x(i)t ∼ N

xt |Ax(i)t−1,σ u2QQH

where x(i)t−1 =r(t i) −1

v(t i) −1 Sensor locations are still sampled from the priorp(s t |st−1)

Initialization Fori =1, , M,

x(0i) ∼ p(x0), s(0i) ∼ p(s0), choose(n,0 i) = n,0(n =1, , N s),w(0i) =1/M.

(1) for each one of theN1sensors with positive decisions and indices

{ κ n } N1

n=1, compute predicted target locations

v(κ i) n,t = s(κ i) n,t − r(t−1 i)

 + (1− )v(t−1 i), 0<  < 1,



(i) t−1+T s v(κ i) n,t

v(κ i) n,t

;

(2) compute predicted likelihoods and use them for averaging the speed



−1, (i)

κ n,t = p

 ,

v(t−1 i) =

N1

n=1

v(κ i) n,t

(i)

κ n,t

N1

r=1 (κ i) r,t

; (3) sample a new target state using the mean speed

⎡

⎣r(t−1 i)

v(t−1 i)

⎤

⎦,

uQQH

; (4) draw new sensor locations from the prior pdf,



.

Weight update.

(1) Update the mean and variance of the log-powers Forn =1, , N s,



π n,t(i) = π s n,t −10 log10s(i)

n,t − r o−β

,

(n,t i) = σ

2

l (n,t−1 i) +(n,t−1 i) πs(i)

n,t

σ2

l +σ2 (i) n,t−1

, σ2 (i) n,t = σ l2σ2 (i)

n,t−1

σ2

l +σ2 (i) n,t−1

.

(2) Update the weights

w t(i) ∝ w(t−1 i) p



p

π s |s(0:i) t,π s

1:t−1

N

uQQH

N

uQQH.

MMSE estimation.

σmmse

M

i=1

w(t i) σ(i)

t

Resampling.

LetM!eff=(M

i=1 w t(i)2)−1, ifM!eff< ηM, 0 < η < 1, then

perform multinomial resampling and setw(t i) =1/M, for all i.

Algorithm 1: MSIS tracking algorithm

Since we use a different importance function for drawing

xt(i),i =1, , M, the weight update equation is not as simple

as in the SSIS technique In particular,

w(i)t ∝ w(i)t−1 p

yt |x(i)t , s(i)t



p

π s

t |s(i)0:t,π s

1:t−1





x(i)t |Ax(i)t−1,σ2

uQQH

N

x(i)t |Ax(i)t−1,σ2QQH,

(34)

Gaussian density

Algorithm 1shows a summary of the SIS method with the proposed importance function, that we will refer to in the sequel as mean-velocity SIS (MSIS) algorithm Note that resampling steps are carried out whenever the estimated ef-fective sample size [14],M!e f f = (M

i=1 w(i)t 2)−1, falls below

a certain threshold,ηM (with 0 < η < 1) Intuitively, M!eff

Initialization Fori =1, , M,

x0∼ p(x0), s0∼ p(s0), choose(n,0 i) = n,0(n =1, , N s),w0=1/M.



x(t l) =Ax(t−1 l), s(t l) =s(t−1 l) +T svs, l =1, , M,



π n,t(i) = π s n,t −10 log10s(n,t i) − r o−β

,

(n,t i) = σ

2

l (n,t−1 i) +(n,t−1 i) πs(i)

n,t

σ2

l +σ2 (i) n,t−1

, σ2

n,t = σ l2σ2

n,t−1

σ2

l +σ2

n,t−1, forn =1, , N s,

ρ(t i) = p

p

π s |s(i) t−1,s(t i),π s

1:t−1

 ,

p (k = l) ∝ w(t−1 l) ρ(t l),

k(i) ∼ p(k).

 ,



.

(1) update the mean and variance of the log-powers Forn =1, , N s,



π n,t(i) = π s n,t −10 log10s(i)

n,t − r o−β

,

(n,t i) = σ l2(n,t−1 i) +(n,t−1 i) πs(i)

n,t

σ2

l +σ2 (i) n,t−1

;

(2) update the weights

w t(i) ∝ p





p

π s |s(i) t,π s

1:t−1



ρ(t k(i))

.

MMSE estimation.

σmmse

M

i=1

w(t i) σ(i)

t

Algorithm 2: APF tracking algorithm

is an estimate of how many independent samples from the

true smoothing distribution would be necessary to obtain a

Monte Carlo approximation with the same accuracy as that

given byΩt = { σ(i)

t ,w(i)t } M

i=1

The extension of the APF given by (9)–(10) to the joint

track-ing of the target and the sensor trajectories yields

k(i)∼ p t(k),

σ(i)

t ∼ p

σ t | σ(k (i))

t−1



,

σ(i) 0:t="σ(k (i)) 0:t−1,σ(i)

t

#

,

w(i)t ∝ p



yt | σ(i)

t



p

π s

t |s(i)0:t,π s

1:t−1



ρ(k(i)) ,

(35)

where p t (k = l) ∝ w(l)t−1 ρ(l)t ,ρ(l)t = p(y t |  σ(l)

t )p( π s

t | s(l)t ,

s(l)0:t−1,π s

1:t−1), andσ(l)

state vector at timet computed from σ(l)

sam-pling) As indicated inSection 2.2, the straightforward way of computing such a prediction is to take the mean of the prior pdf, in particular,



σ(i)

t =

⎡

⎣x

(i)

t

s(i)t

⎤

⎦ = E p( σt | σ(i)

t −1 )

xt

st

=

⎡

(i)

t−1

s(i)t−1+T svs t

⎤

⎦. (36)

The likelihood computations are carried out as indicated

in (20)–(26) A summary of the APF is given inAlgorithm 2

In order to apply the CRPF methodology, we extend the algo-rithm proposed in [11], originally devised for tracking a sin-gle target using a network of fixed binary sensors Specifically,

we choose an incremental cost function of the form

 C

σ t,θ t





yt,y

σ t



+ζ

N s

n=1



π s n,t −10 log10

s n,t − r o−β

−  n,t,

(37) where 0< ζ < 1, D H(a, b) denotes the Hamming distance1

y( σ t) is a vector of “artificial observations” generated

deter-ministically as

y n



σ t



=

⎧

⎨

⎩

1, ifr t − s n,t< γ

0, ifr t − s n,t  ≥ γ, n =1, , N s, (38)

whereγ > 0 is the range of coverage of a sensor The estimate



n,tof the log-power nis recursively computed as



n,t = ξ n,t−1+ (1− ξ) πs

wheren,0 =  n,0,πn,t s is defined in (23), and 0< ξ < 1 This

choice of cost function enables the algorithm user to adjust

the relative weight of the local binary decisions, yt, and the

t, on the overall cost of particles Correspondingly, the risk function is constructed

as

R

σ t−1,θ t



σ0:t−1,θ1:t−1

t−1+T svs t

,θ t

.

(40) For the generating function we have chosen

μ

C(i)t



C(i)t −mink

"

C t(k)

#M k=1+ 1/M

(3, (41)

which was shown to work well for a related vehicle navigation

com-putations The selection step is carried out by the standard

multinomial resampling using the pmf

p

σ(i)

0:t





R(i)t



M

k=1 μ

R(k)t

, i =1, , M, (42)

although alternatives exist that enable the parallelization

scheme forσ tis given by the pair of equations

xt =Axt−1+ν x T szx,t,

st =st−1+T svs

t+ν s T szs,t, (43) whereν x,ν s > 0 are adjustable parameters that control the

variance of the propagation process, and zx,t, zs,tare complex

be-tween simplicity (the sampling scheme is similar to the basic

SIS algorithm with prior proposal pdf) and performance

1 The number of bits that di ffer between the two arguments.

The resulting instance of the CRPF method for the

5 SENSOR MOTION

For the sake of the derivation of the PF algorithms in the pre-vious section, we have assumed that the sensor speed values

contained in vs t ∈ C N sare given It has been mentioned, how-ever, that it is also a task of the tracker to compute these val-ues and transmit them to the sensors, so that they move to locations which are (in some sense) suitable for detecting the target position and provide informative local decisions There are several possible strategies for the assignment of velocities to the sensors Let us just mention some of them, all assuming that the sensors can move approximately at the same space as the target itself

(i) Greedy sensors: given estimates of the target state and

the sensor locations,xtandst, respectively, the tracker commands each sensor to move towards the target, that is,v s i,t+1 =[(rt+T sv t)− s i,t]ϑ t, whereϑ t > 0 is a real

scale factor used to adjust the absolute value| v s i,t+1 | The main weakness of this approach is the high prob-ability of loosing the target track, either when the es-timatesxt andst are poor or when the target makes

a sharp maneuver that leads it far away from the esti-mated positionrtin a short period of time

(ii) Inclosing the target: a more robust strategy is to define

some distance threshold around the estimated target

differ-ently depending on whether the sensors are below the threshold or above it In particular, we can bring far located sensors closer to the target while those already

tar-get, that is,

v s i,t+1 =

⎧

⎨

⎩

)



r t+T sv t



−  s i,t

*

ϑ t, ifr t −  s i,t> d o,



v t, ifr t −  s i,t< d o . (44) This is more robust to estimation mismatches and sharp target maneuvers than the greedy approach, but still has some obvious limitations Indeed, after a few time steps, the network consists of a large number of

a relatively small number of sensors which are closer

to the target position Ifd ois too large, the outer sen-sors are basically useless (they always produce 0 out-puts, except when the target maneuvers away from the estimated track) and the inner ones are comparatively very few, so the network resources are wasted

(iii) Uniform coverage: the aim is to uniformly cover with

sensors the regionS( rt,d o) := { x ∈ C:| x −  r t | < d o },

thresh-old This strategy overcomes the limitations of the

greedy and inclosing approaches, but it can be

com-putationally complex to implement In particular, if

we wish a statistically uniform coverage of S( rt,d o), then it is necessary to perform a sequence of statistical tests on the population of sensor locations If we seek

Initialization Fori =1, , M,

draw x(0i)and s(0i)from a uniform distribution in the region of interest, and setC(0i) =0,(n,0 i) = n,0,n =1, , N s

(1) setπs(i) n,t = π s n,t −10 log10| s(n,t−1 i) +T s v s

n,t − r o | −β, and



n,t(i) = ξ n,t−1(i) + (1− ξ) πs(i)

n,t; (2) compute the risks

R(t i) = λC t−1(i) + C Ax

(i) t−1

 ,θ t

; (3) resample the particles according to the probabilities

p

σ(i) t−1





R(t i)



M k=1 μ

R(t k)

, i =1, , M;

(4) build the selected particle set



Ωt−1 ="σ(i)

t−1,C(t−1 i)

#M i=1, where



C t−1(i) = C(t−1 k), (n,t−1 i) =  (n,t−1 k) ifσ(i)

t−1 = σ(k) t−1

ν x T s

 2

,

ν s T s

 2

 ,



π s(i) n,t = π s n,t −10 log10s(i)

n,t − r o−β

,



(n,t i) = ξ (n,t−1 i) + (1− ξ) πs(i)

n,t,

C t(i) = λ C (i) t−1+ C

σ(i)

t ,θ t



.

Estimation.

σmean

M

i=1

μ

C t(i)



M k=1 μ

C(t k)σ(i)

t

Algorithm 3: CRPF tracking algorithm

a deterministic coverage, for example, a regular grid

around the target, the required computations may also

(iv) Following the target: one of the simplest methods, and

the one we have adopted for the numerical

experi-ments inSection 6, is to preserve the relative positions

of the sensors, which are due to their initial

deploy-ment, and simply move the complete network with the

latest velocity estimated for the target, that is,

v s i,t+1 =  v t, ∀ i ∈+1, , N s

,

If the initial distribution of the sensors is good enough

(there are no significant areas which are

overpopu-lated, while others remain poorly covered), then this

simple and computationally inexpensive strategy

mit-igates the drawbacks of the greedy and inclosing

tech-niques

6 COMPUTER SIMULATIONS

requires a specific choice of the detection probability func-tion,p d Letγ > 0 be the range of coverage of a single sensor.

The local decision probabilities are

p

y i,t =1| r t,s i,t



= p d



d i,t,α

=(1− α) Prob+

d i,t+g i,t < γ,

+α,

p

y i,t =0| r t,s i,t





d i,t,α

,

(46)

braces, d i,t = | r t − s i,t |,g i,t ∼ N(0, σ2

There-fore, the detection probability can be compactly written as

p d(d i,t,α) =(1− α)F N(γ − d i,t |0,σ2

g)+α, where F N(· | μ, σ2)

is the Gaussian cumulative distribution function with mean

μ and variance σ2