Interacting multiple sensor filter

Signal Processing 92 (2012) 2180–2186

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Interacting multiple sensor filter$ Zhigang Liu a,n, Jinkuan Wang a, Yanbo Xue b a b

Insitute of Engineering Optimization and Smart Antenna, Northeastern University, Qinhuangdao 066004, China Cognitive Systems Laboratory, McMaster University, Hamilton, Ontario, Canada L8S 4K1

a r t i c l e in f o

abstract

Article history: Received 15 November 2010 Received in revised form 31 January 2012 Accepted 8 February 2012 Available online 17 February 2012

Due to the limited sensing range of sensors, moving target tracking has to be realized by relaying from one sensor to the other in wireless sensor networks. Therefore, the tracking procedure can be modelled as a Markov chain system. By reconstructing the innovation equation, the relaying Kalman filter (RKF) algorithm is designed in light of Bayesian theory. To deal with nonlinear cases, the interacting multiple sensor filter (IMSF) is proposed in this paper by using the unscented Kalman filter (UKF), the extended Kalman filter (EKF) or the particle filter (PF). Then, the posterior Crame´r–Rao lower bound (PCRLB) is derived for multisensor collaborative tracking. Finally, simulation results show the effectiveness of the proposed IMSF algorithm. & 2012 Elsevier B.V. All rights reserved.

Keywords: Wireless sensor networks Collaborative tracking Kalman filter Beyesian theory

1. Introduction Collaborative tracking is one of the most fundamental collaborative information processing problems in sensor networks [1–9,9]. The limited energy, computational power and communication resources of a sensor node require the use of a large number of sensor nodes in a wide region. Collaboration among these nodes also allows the sensor network to estimate the state of the dynamic process continuously with improved accuracy. Olfati-Saber [3] proposed distributed Kalman filter (DKF) with embedded consensus filters that a central Kalman filter for homogeneous sensor networks can be decomposed into n microfilters with inputs that are provided by two types of consensus filters. By introducing the novel microfilter architecture with identical high-pass consensus filters, DKF was extended for heterogeneous sensor networks in [4]. Due to

$ This work was supported by the National Natural Science Foundation of China under Grant 60874108, the Natural Science Foundation of Hebei Province under Grant F2011501021, and the Fundamental Research Funds for the Central Universities under Grant N110423005. n Corresponding author. Tel.: þ 86 335 8066033; fax: þ86 335 8051795. E-mail address: [email protected] (Z. Liu).

0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2012.02.004

system modelling uncertainties, Ahmad et al. designed the decentralized robust Kalman filtering scheme for uncertain stochastic systems over heterogeneous sensor networks [5]. Using the sign of innovations, Ribeiro et al. proposed the distributed Kalman filter with low-cost communications and its performance is very close to the clairvoyant Kalman filer [6]. Adhering to the limited power and bandwidth resources, Msechu et al. designed two quantizer-estimator structures and further developed the decentralized quantized Kalman filers with scalable communication cost [7], which can trade off between bandwidth requirements and overall tracking performance. Due to the limited sensing range of sensor nodes, target tracking in sensor networks has to be realized by relaying from one node to the other. Therefore, the tracking procedure can be modelled as a Markov jump linear system. Motivated by the Interacting Multiple Model algorithm [10–12], we reconstruct the innovation equation from a Bayesian perspective, and design the relaying Kalman filter (RKF) algorithm for distributed tracking in sensor networks [13]. In order to generalize the above results, we propose the interacting multiple sensor filter (IMSF) in nonlinear cases. Finally, we derive the posterior Crame´r–Rao lower bound (PCRLB) for the above algorithms.

Z. Liu et al. / Signal Processing 92 (2012) 2180–2186

The remainder of the paper is organized as follows. In Section 2, collaborative target tracking problem is formulated. Section 3 presents the RKF algorithm in light of Bayesian theory, followed by the IMSF algorithm in nonlinear cases. In Section 5, we derive the PCRLB for multisensor collaborative tracking. Simulation results that demonstrate the performance of the proposed method are given in Section 6. Section 7 concludes the paper. 2. Problem formulation A typical sensor network is composed of hundreds of or even thousands of sensor nodes which are classified as generic nodes and sink nodes. A generic node’s task is to obtain measurements from the monitored environment. Sink nodes gather data from generic nodes and relay them to the base station. Consider an ad hoc sensor network of n sensor nodes at positions, s1 ,s2 , . . . ,sn . There is only one target of interest whose state equation is corrupted by process noise. At discrete time k, the state of the target evolves according to the following equation: xðkÞ ¼ Fxðk1Þ þ GwðkÞ,

ð1Þ

where F is the transition matrix that takes the state x(k) from time k1 to time k, G is the process noise matrix, w(k) is zero-mean white Gaussian noise (WGN) vector of variance matrix Q, which is due to disturbances and modelling errors. The sensing model of the sensor si is given by zi ðkÞ ¼ Hi xðkÞ þvi ðkÞ,

ð2Þ

where Hi is the measurement matrix for sensor i, zi ðkÞ is the corresponding observation, which is range-only in this paper, vi ðkÞ is the zero-mean additive WGN of variance R. In general, vi ðkÞ and w(k) are assumed to be mutually independent. When zi ðkÞ is target-originated measurement, Eq. (2) is validated. If the target moves beyond the sensing range of the ith sensor, Eq. (2) may be modified into zi ðkÞ ¼ vi ðkÞ:

ð4Þ

where n denotes the number of sensors, and the cumulative set of measurements is Z

k

¼ fZðjÞgkj ¼ 1 :

are mutually independent for n Z2. Using the total probability theorem, the conditional mean of the state at time k can be written as [14] ^ xðkÞ ¼ E½xðkÞ9Zk  ¼

n X

E½xðkÞ9Eki ,Zk PfEki 9Zk g ¼

i¼1

n X

x^ i ðkÞbi ðkÞ,

i¼1

ð6Þ where x^ i ðkÞ is the updated state conditioned on the event that the target is located in the sensing range of the ith sensor, and

bi ðkÞ ¼ PfEki 9Zk g

ð7Þ

is the corresponding sensor probability of event Eki . The Chapman–Kolmogorov equation for the Markov chain Eki is written as PfEki 9Zk1 g ¼

n X

PfEki 9Ek1 gPfEk1 9Zk1 g j j

j¼1

¼

n X

Mij PfEk1 9Zk1 g, j

ð8Þ

j¼1

g is an element from the Markov where the term PfEki 9Ek1 j transition matrix M. Using Bayes’ formula, Eq. (7) is rewritten as

bi ðkÞ ¼ PfEki 9zi ðkÞ,Zk1 g ¼

1 pfzi ðkÞ9Eki ,Zk1 gPfEki 9Zk1 g, c

ð9Þ

where c ¼ Pfzi ðkÞ9Zk1 g, pfg is the probability density function, and i ¼ 1, . . . ,n. The updated state conditioned on the current measurements Z(k) is expressed as ^ ^ xðkÞ ¼ xðk9k1Þ þ

n X

bi ðkÞKi ðkÞz~ i ðkÞ,

ð10Þ

i¼1

where the filter gain Ki ðkÞ of the ith sensor is Ki ðkÞ ¼ Pðk9k1ÞHi ½Si ðkÞ1 ,

ð11Þ

ð3Þ

The measurements from all the sensors at time k may be collected as ZðkÞ ¼ fzi ðkÞgni¼ 1 ,

2181

ð5Þ

and the corresponding innovation and innovation covariance are z~ i ðkÞ ¼ zi ðkÞz^ i ðk9k1Þ

ð12Þ

and Si ðkÞ ¼ Hi Pðk9k1ÞHTi þ Ri :

ð13Þ

The covariance associated with the updated state is PðkÞ ¼ Pðk9k1Þ

n X

bi ðkÞKi ðkÞHi Pðk9k1Þ:

ð14Þ

i¼1

3. Relaying Kalman filter revisited 3.1. State and covariance update

3.2. Prediction equations

Assume an area of interest which is covered by all the sensors in senor networks is divided into n parts according to the sensor sensing range. When the target appears in this area, the associated events of the target, i.e.,

The predicted state and measurement are computed as follows: ^ ^ xðk9k1Þ ¼ Fxðk1Þ,

ð15Þ

Eki ¼ fEk ¼ i,i ¼ 1, . . . ,ng

^ z^ i ðk9k1Þ ¼ Hi xðk9k1Þ,

ð16Þ

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Z. Liu et al. / Signal Processing 92 (2012) 2180–2186

and the covariance of the predicted state is T

T

Pðk9k1Þ ¼ FPðk1ÞF þ GQG :

5. Estimation bound ð17Þ

3.3. Sensor probability update Combining (8) and (9) into (7) yields the updated equation of the sensor probability

b ðk9k1ÞLi ðkÞ , bi ðkÞ ¼ Pn i j ¼ 1 bj ðk9k1ÞLj ðkÞ

ð18Þ

where the corresponding sensor likelihood is Li ðkÞ ¼ pfzi ðkÞ9Eki ,Zk1 g ¼ N ½z~ i ðkÞ; 0,Si ðkÞ,

T

ð19Þ

and bi ðk9k1Þ is the predicted probability of the ith sensor

bi ðk9k1Þ ¼

n X

Mij bj ðk1Þ:

The performance of the IMSF tracker can be quantified by the PCRLB, which is defined as the inverse of the Fisher information matrix (FIM) [15]. The PCRLB provides a mean square error lower bound on the performance of any unbiased estimator of the unknown target state. Let r and n be operators of the first and second-order partial derivatives, respectively   @ @ T rxðkÞ ¼ , ..., , ð25Þ @x1 @xr ¼ rxðkÞ rxðkÞ , nxðkÞ xðkÞ

ð26Þ

where r denotes the dimension of target state. If we assume that all the sensors have independent measurement processes, the overall FIM at time kþ1 can be expressed as

ð20Þ

j¼1

21 11 1 12 Jk þ 1 ¼ D22 Dk þ k Dk ½Jk þ Dk 

n X

rik þ 1 Jzkiþ 1 ,

ð27Þ

i¼1

where

4. Interacting multiple sensor filter The relaying Kalman filter has addressed the state estimation of moving target with linear state-space models in sensor networks. However, most dynamical systems are nonlinear in practice, which inspires us to extend the relaying Kalman filter to nonlinear cases. The resulting filter is referred to as the interacting multiple sensor filter (IMSF). Consider a nonlinear dynamical system described by the state-space model xðkÞ ¼ f ½k1,xðk1Þ þ wðkÞ

ð21Þ

and zi ðkÞ ¼ hi ½k,xðkÞ þvi ðkÞ,

ð22Þ

where f ½k1,xðk1Þ denotes a nonlinear transition function that is often time-variant, and hi ½k,xðkÞ denotes a nonlinear measurement function of the ith sensor that may be time-variant, too. Similarly, when the target moves beyond the sensing range of the ith sensor, the measurement equation can be simplified as Eq. (3). After receiving the messages from all the nodes, the sink node firstly computes the innovation z~ i ðkÞ, the innovation variance Si(k), the updated state x^ i ðkÞ, and the updated state covariance Pi(k) by using the unscented Kalman filter (UKF), the extended Kalman filter (EKF) or the particle filter (PF), where the subscript i denotes the node i in the network. Secondly, the sensor probability bi ðkÞ can be obtained by making use of Eqs. (18)–(20). ^ Finally, the overall estimate xðkÞ and the overall covariance P(k) are computed for output purpose only, according to the fusion equations: ^ xðkÞ ¼

n X

x^ i ðkÞÞbi ðkÞ

ð23Þ

i¼1

xðkÞ D11 k ¼ EfnxðkÞ log p½xðk þ 1Þ9xðkÞg,

ð28Þ

xðk þ 1Þ D12 log p½xðk þ1Þ9xðkÞg, k ¼ EfnxðkÞ

ð29Þ

12 T D21 k ¼ ½Dk  ,

ð30Þ

xðk þ 1Þ D22 k ¼ Efnxðk þ 1Þ log p½xðk þ1Þ9xðkÞg,

ð31Þ

and xðk þ 1Þ Jzkiþ 1 ¼ Efnxðk log p½zi ðk þ1Þ9xðk þ 1Þg: þ 1Þ

The factor rik þ 1 2 f0; 1g implies the ith sensor’s contribution to the PCRLB. Due to the uncertainty of target’s motion, it is not possible to know in advance which sensors will attend the incoming tracking event. Thus, the factor rik þ 1 can be computed according to the following equation: ( 0, bi ðk þ1Þ o d; i rk þ 1 ¼ ð33Þ 1, bi ðk þ1Þ Z d, where d 2 ½0; 1 is a threshold value. We usually choose d ¼ 0:1 for a value that is too high would bring down the contribution to target tracking. When rik þ 1 ¼ 1, it means that the ith sensor attends the tracking event at time kþ1. Otherwise, rik þ 1 ¼ 0. Using Eqs. (1) and 2, the conditional densities can be written as  1 1 exp  ½xðk þ 1Þ p½xðk þ 1Þ9xðkÞ ¼ 1=2 2 ð2pÞr=2 9Q 9  1 FxðkÞT Q ½xðk þ1ÞFxðkÞ ð34Þ and  1 exp  ½zi ðk þ 1Þ 2 ð2p9R9Þ  Hi xðk þ 1ÞT R1 ½zi ðk þ 1ÞHi xðkþ 1Þ

p½zi ðk þ1Þ9xðk þ 1Þ ¼ and PðkÞ ¼

n X i¼1

^ ^ ½P i ðkÞ þðxðkÞ x^ i ðkÞÞðxðkÞ x^ i ðkÞÞT bi ðkÞ:

ð24Þ

ð32Þ

where Q ¼ GQGT .

1

1=2

ð35Þ

Z. Liu et al. / Signal Processing 92 (2012) 2180–2186

Consequently, T D11 k ¼F Q

1

T D12 k ¼ F Q

D22 k ¼Q

1

F,

1

ð36Þ ,

ð37Þ

,

ð38Þ

2183

follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 zi ðkÞ ¼ ðx1 ðkÞai Þ2 þ bi þvi ðkÞ,

ð43Þ

where the measurement noise vi ðkÞ  N ð0,RÞ with R ¼ s2v . In the following experiments, the process noise standard deviation is sw ¼ 0:0001 ms2 . The initial state is xð0Þ ¼ ½1 m,0:1 ms1 ,0:025 ms2 T and the associated covariance is Pð0Þ ¼ diag½1 m2 ,0:01 m2 s2 ,0:0001 m2 s4 :

and Jzkiþ 1

¼ HTi R1 Hi :

ð39Þ

If we apply the matrix inversion lemma to (27), then T 1 Jk þ 1 ¼ ðQ þFJ1 þ k F Þ

n X

rik þ 1 HTi R1 Hi :

ð40Þ

i¼1

However, the measurement models of most of dynamical systems are nonlinear in practice. For the rangebased measurement model, Eq. (2) is rewritten as zi ðkÞ ¼ Di ðxðkÞÞ þ vi ðkÞ,

ð41Þ

The number of iterations matrix is defined as 2 0:93 0:01 0:01 6 0:01 0:93 0:01 6 6 6 0:01 0:01 0:93 6 6 0:01 0:01 0:01 6 M¼6 6 0:01 0:01 0:01 6 6 0:01 0:01 0:01 6 6 4 0:01 0:01 0:01 0:01

where the distance between the ith sensor and the target is given by Di ðxðkÞÞ ¼ JxðkÞsi J1=2 ¼ ½ðak ai Þ2 þ ðbk bi Þ2 1=2 ,

0:01

0:01

0:01

0:01

0:01

0:01

0:01

0:01

0:01

0:01

3

0:01

0:01

0:01

0:01

0:93 0:01

0:01 0:93

0:01 0:01

0:01 0:01

0:01

0:01

0:93

0:01

0:01

0:01

0:01

0:93

0:01 7 7 7 0:01 7 7 0:01 7 7 7: 0:01 7 7 0:01 7 7 7 0:01 5

0:01

0:01

0:01

0:01

0:93

101

PCRLB

where fak , bk g denotes the target’s position at time k, and fai , bi g is the position of sensor si, which is generally fixed. When the measurement noise covariance is R ¼ s2v and the target’s state is given by xk ¼ ½ak þ 1 , a_ k þ 1 , bk þ 1 , b_ k þ 1 T , Eq. (39) should be modified into 0 1 2 ˇ ˇ ˇ a 0 a b 0 k þ 1 k þ 1 k þ 1 B C B 0 0 0 0C 1 B C Jzkiþ 1 ¼ ð42Þ B C, 2 ˇ ˇ C ˇ s2v D2i ðxðkÞÞ B bk þ 1 0A @ ak þ 1 bk þ 1 0 0 0 0 0

0:01

is 300, and the Markov transition

δ=0.4 δ=0.2 δ=0.1 δ=0

100

ˇ

where aˇ k þ 1 ¼ ak þ 1 ai and bk þ 1 ¼ bk þ 1 bi . 10−1

6. Simulation

0

0

1

and 0

T 2 =2

B G¼@ T

50

100

150

200

250

300

Number of iterations Fig. 1. PCRLB versus d, the threshold value of the sensor probability.

101

σv=10 σ =1 v

σ =0.1 v

100 PCRLB

Assume that eight identical sensors in a sensor network are located at (  1 m, 10 m), (150 m, 10 m), (300 m, 10 m), (450 m, 10 m), (600 m, 10 m), (750 m, 10 m), (900 m, 10 m) and (1000 m, 10 m), respectively. The sensing range of each sensor is 400 m, and a target accelerates along the x-axis. In Eq. (1), the state vector is defined as xðkÞ ¼ ½x1 ðkÞ,x2 ðkÞ,x3 ðkÞT , where x1 ðkÞ, x2 ðkÞ and x3 ðkÞ are the location, speed and acceleration, respectively. F and G are known matrices given by 0 1 1 T T 2 =2 B C F¼@0 1 T A

0

10−1

1 C A,

1 where T¼1 s is the sampling interval, and the process noise wðkÞ  N ð0,Q Þ with a nonsingular covariance Q ¼ s2w . The measurement equation can be displayed as

10−2

0

50

100 150 200 Number of iterations

250

300

Fig. 2. PCRLB versus measurement noise standard deviation sv .

2184

Z. Liu et al. / Signal Processing 92 (2012) 2180–2186

6.1. PCRLB

100

DKF IMSF PCRLB

10−1 RMSEac(ms−2)

Fig. 1 depicts the relationship between the PCRLB and d, the threshold value of the sensor probability. When the threshold value d varies from 0.4 to 0, more sensors can attend the tracking event. As such the PCRLB is minimal at d ¼ 0. But in practice, it is impossible that all the sensors take part in the tracking event at the same time, due to the limited sensing ranges. Therefore we choose the threshold value to be 0.1 in the following simulation experiments. Similarly, Fig. 2 shows the effect of the measurement noise standard deviation sv on the PCRLB. As expected, the PCRLB continues to increase with sv rising. From above, this PCRLB would be limited by the measurement noise covariance s2v . As such, a senor that has smaller measurement noise covariance would outperform another one that has larger measurement noise.

10−2 10−3 10−4 10−5

0

50

100 150 200 Number of iterations

250

300

Fig. 5. The RMSE in acceleration versus number of iterations when sv ¼ 1 m.

6.2. Performance of the IMSF To validate the performance of the IMSF algorithm, we compare our approach to the DKF algorithm and the

103

103

RMSEpos(m)

102

RMSEpos(m)

102

10

DKF IMSF PCRLB

1

DKF IMSF PCRLB

100 10−1

100

10

101

10−2

0

50

−1

0

50

100 150 200 Number of iterations

250

300

100 150 200 Number of iterations

250

300

Fig. 6. The RMSE in position versus number of iterations when sv ¼ 0:1 m.

Fig. 3. The RMSE in position versus number of iterations when sv ¼ 1 m.

102

101

RMSEvel(ms−1)

100

DKF IMSF PCRLB

10−1

100

DKF IMSF PCRLB

10−1 10−2

10−2

10−3

RMSEvel(ms−1)

101

10−3 0

50

100 150 200 Number of iterations

250

300

Fig. 4. The RMSE in velocity versus number of iterations when sv ¼ 1 m.

0

50

100 150 200 Number of iterations

250

300

Fig. 7. The RMSE in velocity versus number of iterations when sv ¼ 0:1 m.

Z. Liu et al. / Signal Processing 92 (2012) 2180–2186

PCRLB. The simulations are based on 20 independent Monte Carlo runs for a fair comparison. Figs. 3–5 show the root-mean square errors (RMSEs) of position, velocity and acceleration when the measurement noise standard deviation sv ¼ 1 m, and Figs. 6–8 show the RMSEs in position, velocity and acceleration when sv ¼ 0:1 m. As can be seen from Figs. 3–8, the IMSF algorithm has lower RMSEs in position, velocity and acceleration than the DKF algorithm over an interval of 70–300. Due to the fact the DKF algorithm is sensitive to limited sensing range, it diverges abruptly. When the number of iterations is greater than 70, the position error curves for the IMSF algorithm are 101

DKF IMSF PCRLB

RMSEac(ms−2)

100 10−1

2185

above the theoretical PCRLBs, and the curves in velocity and acceleration approximate the relative PCRLBs. In addition, the proposed algorithm has a consistent performance under different levels of the measurement noise. 6.3. Sensor probability From a sensor probability’s perspective, the updated sensor probabilities of the IMSF algorithm for sv ¼ 1 m and sv ¼ 0:1 m evolve with the number of iterations, as depicted by Figs. 9 and 10 respectively, where bi denotes the sensor probability that the ith sensor contributes to target tracking. During this procedure, the sensor probability of each sensor varies in a consequent order, and this implies that the proposed algorithm works by collaborative tracking within the networks. Most importantly, Figs. 9 and 10 show that the sensor probabilities of our approach are not sensitive to the changes of measurement noise.

10−2

7. Conclusion 10−3 10

−4

10−5

0

50

100 150 200 Number of iterations

250

300

1 0.5 0 1 0.5 0

β5 β6

0 1 0.5 0 1 0.5 0 1 0.5 0

β8

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150 200 Number of iterations

250

300

0.5

β7

β4

β2

1 0.5 0

β3

β1

Fig. 8. The RMSE in acceleration versus number of iterations when sv ¼ 0:1 m.

For wireless sensor networks composed of sensors with limited sensing ranges, the target tracking procedure can be modelled as a Markov jump system. Inspired by the Bayesian theory, we design the relaying Kalman filter (RKF) algorithm by reconstructing the innovation equation for collaborative tracking recursively. However, real-world systems are nonlinear in general. We therefore extend the RKF algorithm to the interacting multiple sensor filter (IMSF) algorithm in dealing with nonlinearity, and also derive the

1 0.5 0

Fig. 9. The updated sensor probability versus number of iterations when sv ¼ 1 m.

β1 β2

1 0.5 0 1 0.5 0 1 0.5 0 0.5

β6

1 0.5 0

β7

0

1 0.5 0

β8

β5

1 0.5 0

β3

Z. Liu et al. / Signal Processing 92 (2012) 2180–2186

β4

2186

1 0.5 0

0

50

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300

0

50

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300

0

50

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300

0

50

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300

0

50

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300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150 200 Number of iterations

250

300

Fig. 10. The updated sensor probability versus number of iterations when sv ¼ 0:1 m.

estimation bound which is consistent with the proposed algorithm. We have conducted computer experiments to show that the PCRLB is influenced by the measurement noise covariance and the threshold value of the sensor probability. Furthermore, the IMSF algorithm has better performance than the DKF algorithm in aspects of position, velocity and acceleration. Most importantly, the proposed algorithm makes the collaborative tracking within sensor networks more efficient. Simulation results are consistent with the derivation of the proposed IMSF algorithm. However, only a linear trajectory is considered, and the results for more realistic situations will be exploited in future works. References [1] T. Vercauteren, X. Wang, Decentralized sigma-point information filters for target tracking in collaborative sensor networks, IEEE Transactions on Signal Processing 53 (2008) 2997–3009. [2] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control 49 (2004) 1520–1533. [3] R. Olfati-Saber, Distributed Kalman filter with embedded consensus filters, in: 44th IEEE Conference on Decision and Control 2005, December 2005, pp. 8179–8184. [4] R. Olfati-Saber, Distributed Kalman filtering for sensor networks, in: Proceedings of the 46th IEEE Conference on Decision and Control, December 2007, pp. 5492–5498.

[5] A. Ahmad, M. Gani, F. Yang, Decentralized robust Kalman filtering for uncertain stochastic system over heterogeneous sensor networks, Signal Processing 88 (2008) 1919–1928. [6] A. Ribeiro, G.B. Giannakis, S.I. Roumeliotis, SOI-KF: distributed Kalman filtering with low-cost communications using the sign of innovations, IEEE Transactions on Signal Processing 54 (2006) 4782–4795. [7] E.J. Msechu, S.I. Roumeliotis, A. Ribeiro, G.B. Giannakis, Decentralized quantized Kalman filtering with scalable communication cost, IEEE Transactions on Signal Processing 56 (2008) 3727–3741. [8] L. Zhang, Q. Cheng, Y. Wang, S. Zeadally, Novel distributed sensor positioning system using the dual of target tracking, IEEE Transactions on Computers 57 (2008) 246–260. [9] N. Patwari, J. Ash, S. Kyperountas, A. Hero III, R. Moses, N. Correal, Locating the nodes: cooperative localization in wireless sensor networks, IEEE Signal Processing Magazine 22 (2005) 54–69. [10] H.A.P. Blom, Y. Bar-Shalom, The interacting multiple model algorithm for systems with Markovian switching coefficients, IEEE Transactions on Automatic Control 33 (1988) 780–783. [11] E. Mazor, A. Averbuch, Y. Bar-Shalom, J. Dayan, Interacting multiple model methods in target tracking: a survey, IEEE Transactions on Aerospace and Electronic Systems 34 (1998) 103–123. [12] X.R. Li, V.P. Jilkov, Survey of maneuvering target tracking, part V: multiple-model methods, IEEE Transactions on Aerospace and Electronic Systems 41 (2005) 1255–1321. [13] Z.G. Liu, J.K. Wang, W. Qu, Relaying Kalman filters for sensor networks, in: Proceedings of ICCDA 2010, Qinhuangdao, China, June 2010, pp. 618–621. [14] T. Kirubarajan, Y. Bar-Shalom, Probabilistic data association techniques for target tracking in clutter, Proceedings of the IEEE 92 (2004) 536–557. [15] P. Tichavsky´, C.H. Muravchik, A. Nehorai, Posterior Crame´r–Rao bounds for discrete-time nonlinear filtering, IEEE Transactions on Signal Processing 46 (1998) 1386–1396.