TARGET TRACKING BY USING PARTICLE FILTER IN SENSOR NETWORKS

TARGET TRACKING BY USING PARTICLE FILTER IN SENSOR NETWORKS D. Gu and H. Hu Department of Computing and Electronic Systems University of Essex, UK Abstract This paper presents a distributed particle filter over sensor networs. We propose two major steps to mae a particle filter to wor in a distributed way. The first step is the estimation of global mean and covariance of weighted particles by using an average consensus filter. Through this consensus filter, each sensor node can gradually diffuse its local mean and covariance of weighted particles over the entire networ and asymptotically obtain the estimated global mean and covariance. The second step is the propagation of the estimated global mean and covariance through state transition distribution and lielihood distribution by using an unscented transformation. Through this transformation, partial high order information of the estimated global mean and covariance can be incorporated into the estimates for non-linear models. Simulations of tracing tass in a sensor networ with 100 sensor nodes are given. Keywords: Distributed particle filter, unscented transformation, distributed estimation, sensor networs. 1

1 Introduction Sensor networs consist of massively distributed, small devices that have some limited sensing, processing and communication capabilities. They have a broad range of environmental sensing applications, including environment monitoring, vehicle tracing, collaborative processing of information, gathering data from spatially distributed sources, etc. [1] [2]. One of the major goals in sensor networs is to detect and trac changes in the monitored environment [3]. Object tracing in sensor networs was implemented by an information driven approach in [4]. Recently it has been conducted by using a distributed Kalman filter [5]. Particle filter is one of the widely used tracing algorithms due to its applicability to non-linear and non- Gaussian dynamic systems [6] [7]. It is difficult to transmit particles across the networs due to its large scale. However, the parameter based representation of particle filters can be used in sensor networs. By transmitting the model parameters, a distributed particle filter can be implemented. The aim is to reduce the energy cost in computation and communication, which can significantly increase the node lifespan [8]. In a particle filter, moving objects are modeled as a simple Marov process, specified by their state transition probabilities. Observations about states of moving objects are modeled by their lielihood probabilities. The aim of the algorithm is to estimate the posterior probability density function (pdf). Particle filter is applied as a Monte Carlo approximation to posterior pdf, i.e. the posterior pdf is represented by a set of weighted samples (or particles) [6] [7]. Due to the use of weighted particles to approximate pdf, particle filter is computational expensive. And the cost in a sensor networ to exchange weighted particles with neighbors is very significant for large scale networs. Further, the algorithm robustness in sensor networs is also very important. Failure of a sensor node should not cause failure of the entire networ. Currently there are several distributed particle filters (DP F s) that have been developed [9] [10] [11] [12]. In these algorithms, the distributed nature is achieved by either transmitting local statistics of particles to a centralized unit or using the message passing method. Transmitting local statistics of particles to a centralized unit is not an efficient approach. It is also not robust. 2

Failure of the centralized unit is vital to the entire networ. In the message passing method, the algorithms construct a path through the networs, which passes through all nodes. Global statistics of particles are accumulated by adding local statistics in each node through a forward pass. Then there needs a bacward pass, which run the important sampling and selection steps in each sensor node by using the accumulated global statistics. In [9], the factorized lielihood function is used and each partial lielihood function is updated at individual sensors using only local observations and partial lielihood function estimated in the preceding sensors. The partial lielihood function is represented by a parameterized model and the parameters are transmitted through the path. The same strategy to communicate the highly compact distribution is also used in robotics for map building in [13]. In [10], a set of uncorrelated sensor cliques is used and they are automatically constructed according to moving target trajectories. The algorithm uses a low dimensional Gaussian mixture model (GM M) to describe the posterior pdf. Model parameters rather than weighted particles are transmitted over the networ. Using a GM M to approximate the posteriori pdf is also adopted in [11] where the estimated parameters of GM M are transmitted to a fusion center. In [12], the particles are distributed in a sensor networ, i.e. each sensor node holds part of particles. Local statistics of particles are calculated and transmitted to a centralized unit. In this paper, we propose to distribute the whole particle set evenly across the entire networ and exchange local statistics of particles between neighbor nodes to estimate global statistics of all particles. Because the calculations of global statistics are in the average form, global statistics can be estimated by using an average consensus filter. The consensus filter can diffuse local statistics over the entire networ through communication with neighbor nodes [14] [15] [5] and estimate global statistics using local statistics and neighbor s local statistics. By using the estimated global statistics, each node can predict the next step particles, update weights and resample particles. The consensus filter only requires local communication, i.e. each node only needs to communicate with its neighbors and gradually gains global statistics. Failures of any nodes do not affect the algorithm performance given the networ is still connected. The estimated results can be accessed from any nodes in the networ. In this paper, the mean and covariance are 3

used as statistics of particles. This will limit the proposed DP F to Gaussian dynamic systems. However, it can handle non-linear models by using the unscented transformation (UT ) as UT incorporates partial high order information of mean and covariance into the estimates for nonlinear models [16] [17]. In the rest of this paper, a centralized particle filter in sensor networ environment is described in Section 2. Section 3 presents the estimates of global statistics by using an average consensus filter. The unscented transformation and the distributed particle filter are given in Section 4. Section 5 provides simulation results. Finally, conclusions are summarized in Section 6. 2 Particle Filter Tracing in Sensor Networs We consider a networ of M sensors, which is used to trac a moving object. The moving object is modeled by a discrete state equation: x = f(x 1 ) + v 1 (1) where x is the n x dimensional state vector and v is the Gaussian noise with mean zero and covariance Q. The state is also modeled as a Marov process with initial distribution p(x 0 ) N (x 0, Q 0 ) and state transition probability p(x x 1 ). When we consider Gaussian dynamic systems, the state transition probability is a Gaussian function N (f(x 1 ), P x) with mean f(x 1) and covariance P x. All sensors should now the state function f and the noise vector v. Each sensor m can mae a n z dimensional observation z m, (m = 1,..., M) at time. The observation state equation is assumed as follows: z m, = h m (x ) + w (2) where w is the Gaussian noise with mean zero and covariance R. The observation equation (2) is also modeled as a lielihood function p(z m, x ). When we consider Gaussian dynamic systems, the lielihood function is a Gaussian function N (h m (x ), P z ) with mean h m(x ) 4

and covariance P z. It is assumed that the state noise and observation noise are independent, E[v w T ] = 0. We build up a centralized particle filter (CP F ) for the tracing purpose first in this section. Then, we describe the distributed version in the next section. Let x 0: denote {x t, t = 0,..., } and z m,1: denote {z m,t, t = 0,..., }. Then, the tracing purpose of a particle filter is to estimate the posteriori pdf p(x 0: z m,1: ), or p(x z m,1: ). In CP F, observations from all sensors are transmitted to a centralized unit. It is assumed that the centralized unit receives only one observation from one sensor at each, The centralized unit maintains a set of weighted particles (x (n), ω(n) observation z m, at, it uses the CP F to predict particles x (n) 1,..., N), where N is the number of particles. ). When the centralized unit receives an and update weights ω (n), (n = Let {x (n) 0:, ω(n) }N n=1 denote a random measure that characterizes the posterior pdf p(x 0: z m,1: ), where the weights are normalized, N n=1 ω(n) = 1. The posterior pdf at can be approximated by p(x 0: z m,1: ) N n=1 ω (n) δ(x 0: x (n) 0: ) (3) where δ denotes the Dirac delta function. Expectation of a function g(x ) can be approximated by E[g(x 0: )] N g(x 0: )p(x 0: z m,1: ) = n=1 N n=1 ω (n) g(x(n) 0: ) (4) Since it is often impossible to sample directly from the posterior pdf, it is a normal practice to sample from a nown proposal distribution q(x 0: z m,1: ), x (n) 0: q(x(n) 0: z m,1:). Therefore, the weights in (3) are defined to be ω (n) p(x(n) 0: z m,1:) q(x (n) 0: z m,1:) (5) 5

Assume the proposal distribution meets the following condition: q(x x 0: 1, z m,1: ) = q(x x 1, z m,1: ) (6) Then, we have q(x 0: z m,1: ) = q(x x 0: 1, z m,1: )q(x 0: 1 z m,1: 1 ) = q(x x 1, z m,1: )q(x 0: 1 z m,1: 1 ) The weight updating equation (5) can be rewritten in a recursive form: ω (n) ω (n) 1 p(z m, x (n) )p(x(n) x(n) 1 ) q(x (n) x(n) 1, z m,) (7) Finally, the filtering posterior pdf p(x z m,1: ) can be approximated as p(x z m,1: ) ω (n) δ(x x (n) ) (8) And expectation of a function g(x ) can be approximated as follows: E[g(x )] N n=1 ω (n) g(x(n) ) (9) The choice of the proposal distribution is one of the important steps in particle filters. The most popular practice is to choose the state transition probability as the proposal distribution. q(x x 1, z m,1: ) = p(x x 1 ) (10) This choice minimizes the variance of the importance weights [7]. By this choice, the weights updating equation (7) can be easily implemented: ω (n) ω (n) 1 p(z m, x (n) ) (11) 6

3 Estimation of Global Statistics via Consensus Filter To implement a distributed particle filter (DP F ), each sensor should maintain N particles x (n) m, and weights ω (n) m,. posterior probability to be estimated. Ideally, particles and weights from all sensor nodes should represent the One way to avoid transmitting all particles and weights across the entire networ is to transmit the highly compact statistics of weighted particles. The mean, covariance or any other high order moments of the particles can be used as highly compact statistics These statistics are calculated through the average operation. It has been shown that the average operation can be implemented in a distributed way by using an average consensus filter [18]. The average consensus filter does not rely on the message passing approach. It estimates the global average by using local and neighbors information. Eventually, the local information is diffused into the entire networ through consensus protocol. Failures of any sensor nodes do not crash the entire networ as long as the networ is still connected. Therefore it is more robust than the message passing approach. And the power consumed on the communication is less than the centralized approach as it only requires local communication. In the following, the global mean and covariance of particles are estimated through an average consensus filter. The estimated global mean and covariance are used in the next section to implement DP F. In a Gaussian dynamic model. The mean and covariance are used to characterize the pdf. Each sensor m can calculate the local mean x m, and covariance P x m, : x m, = P x m, = N n=1 N n=1 ω (n) m, x(n) m, (12) ω (n) m, (x(n) m, x m,)(x (n) m, x m,) T The global mean and covariance of the entire networ can be calculated by using the local means 7

and covariances: x = 1 M P x = 1 M M N m=1 n=1 M N m=1 n=1 ω (n) m, x(n) m, = 1 M M x m, (13) m=1 ω (n) m, (x(n) m, x m,)(x (n) m, x m,) T = 1 M M Pm, x (14) m=1 From above equations, it can be seen that the global mean and covariance are the averages of the local means and covariances. These averages can be estimated in each sensor node by using an average consensus filter. In a sensor node m, let ˆ x m, denote the estimate of global mean x and ˆP x m, denote the estimate of global covariance P x. Let y m, and u m, denote the output vector and input vector of the consensus filter discussed below, respectively. y m, represents either estimated global mean ˆ x m, or estimated global covariance ˆP m, x. Correspondingly, u m, represents either local mean x m, or local covariance Pm, x. A sensor networ can be modeled by using algebraic graph theory. A graph can be used to represent interconnections between sensor nodes. A vertex of the graph corresponds to a node and edges of the graph capture the dependence of interconnections. Formally, a graph G = (V, E) consists of a set of vertices V = {v 1,..., v M }, indexed by nodes in the networ, and a set of edges E = {(v i, v j ) V V}, containing unordered pairs of distinct vertices. Assuming the graph has no loops, i.e. (v i, v j ) E implies v i v j. Let R denote the distance that a node can communicate via wireless radio lins. Edge (v i, v j ) is connected if the Euclidean distance d ij between nodes i and j is less than or equal to R. A graph is connected if for any vertices (v i, v j ) V, there exists a path of edges in E from v i to v j. The set of neighbors of vertex i is defined as N i = {j V : (i, j) E}. The degree of vertex i is defined as d i = N i and maximum degree is d max = max i d i. Let be the degree matrix, = diag(d i ). The adjacency matrix A is the integer matrix with rows and columns indexed by the vertices, such as the ij-entry of A is equal to the number of edges from i to j. Following [19], Laplacian matrix of a graph G is defined as, L = A For a connected graph, Laplacian matrix L is symmetric and positive semi-definite. Its mini- 8

mum eigenvalue is 0 and the corresponding eigenvector is 1 = [1,..., 1] T or L1 = 0 [19]. An average consensus filter in a sensor node m is designed as follows in the discrete form: [ ] y m,+1 = y m, + ɛ (y j, y m, ) + (u m, y m, ) j N m (15) where ɛ is the updating rate and should be ɛ 1/d max. This requirement guarantees the stability of the discrete consensus filter according to Gersgorin theorem. y m, can asymptotically converge to the average of local inputs u m, : y m, 1 M M u m, (16) m=1 Since y m, represents either ˆ x m, or ˆP m, x and u m, represents either x m, or Pm, x, we have: ˆ x m, x = 1 M ˆP x m, P x = 1 M M m=1 x m, M Pm, x (17) The continuous form of the average consensus filter (15) is expressed as follows: ẏ m (t) = (y j (t) y m (t)) + (u m (t) y m (t)) (18) j N m where y m (t) is the estimated global statistics and u m (t) is the local statistics at time t. We can stac all y m (t) and inputs u m (t) into vectors y(t) and u(t), respectively and get a matrix form m=1 ẏ(t) = Ly(t) + u(t) y(t) (19) Its transfer function is given by H(s) = (si + I + L) 1 (20) 9

Applying Gersgorin theorem to the square matrix I + L for the connected graph, we have 1 λ(i +L) 1+2d max and λ min (I +L) = 1, where λ(i +L) is one of eigenvalues of matrix I +L. It means that all poles of H(s) are strictly negative and fall within the interval [ 1, (1+2d max )]. Thus H(s) is stable. As lim s H(s) = 0, it is a low-pass filter. This consensus filter (18) is a stable low-pass filter given the graph is connected. 4 Distributed Particle Filter Once the estimated global mean and covariance are obtained, they should be propagated through the state transition probability to generate global mean and covariance of the predicted particles. Accordingly the predicted particles can be drawn from the Gaussian distribution with the propagated global mean and covariance. With the predicted particles, the weight updating step and resampling step are the same as the steps in the CP F. The scaled unscented transformation (UT ) can propagate a n a dimensional random variable a through a non-linear function d : R na R n b to generate b [16], b = d(a) In the UT, a set of 2n a +1 points whose sample mean ā and covariance P a are generated first. These points are called sigma points S (i) = {W µ (i), W σ (i), A (i), i = 0,..., 2n a } and calculated as 10

follows: A (0) = ā A (i) = ā + ( (na + κ)p a ) (i), i = 1,..., na (21) A (i) = ā ( (na + κ)p a ) (i), i = na + 1,..., 2n a W (0) µ = λ/(n a + λ) W (0) σ = λ/(n a + λ) + (1 α 2 + β) W (i) µ = W (i) σ = 1/[2(n a + λ)], i = 1,..., 2n a λ = α 2 (n a + κ) n a ( (na ) (i) where κ, α, and β are the scaling parameters and + κ)p a is the ith row or column of the matrix square root of (n a + κ)p a. W (i) is the weight associated with the ith point. More details on the parameter selection can be found in [17]. It has been proved in [16] that the sigma points chosen by (21) have the same sample mean, covariance, and all higher odd-ordered central moments as the distribution of a. The non-linear function d is then applied to each of these points to yield the transformed points B (i) : B (i) = d(a (i) ) (22) The transformed mean and covariance are calculated from the transformed points. b = P b = 2n a i=0 2n a i=0 W (i) µ B (i) (23) W (i) σ (B (i) b)(b (i) b) T (24) It has been proved in [16] that the errors in transformed mean and covariance are at the forth and higher order in the Taylor series. This approximation is better than the linearized model used by extended Kalman filter. In DP F, the sigma points of particles at 1 are defined as S (i) (i) m, 1 = {W µ, W σ (i), X (i) m, 1 }, 11

(i = 0,..., 2n x ). ( (i) X (i) m, 1 = {ˆ x m, 1, ˆ x m, 1 ± (n x + κ) m, 1) ˆP x } (25) Based on the state equation (1), the transformed sigma points are X (i) m, : X (i) (i) m, = f(x m, 1 ) (26) The sigma points of the noise v are {0, ± ( (nx + κ)q ) (i)}. As it is the additive noise, the predicted mean and covariance of particles at are as follows: ˆ x m, = 2n x i=0 2n x ˆP m, x = Q + W (i) µ X (i) m, (27) i=0 W σ (i) (X (i) m, ˆ x m, )(X (i) m, ˆ x m, ) T Based on the predicted mean and covariance, the predicted particles x m, can be drawn from the Gaussian distribution N (ˆ x m,, ˆP m, x ). From the observation equation (2), the predicted observations are z m, : z m, = h m (ˆ x m, ) (28) The energy consumed for communicating a bit in a node can be many orders of magnitude greater than the energy required for a single local computation. So the analysis of energy consumed in communication is important in sensor networs. Assume the nodes are distributed over a squared area with nodes on a uniform grid. The scenario where the nodes are distributed randomly over a squared area is equivalent to that with uniform grid. By denoting N b as the number of bytes communicated between two nodes per time step, it can be found that the communication in bytes for the centralized method in which all nodes send their data to the center of the networ is: M(1 + 2 +... + M/2)N b = O(M 3/2 ). The worst case in this method is the centralized unit is not in the center of the networ, but is at the end of the area. The communication in bytes for such a case is: (M 1 + M 2 +... + 1)N b = O(M 2 ). Once the centralized unit receives 12

all data, it can run the standard particle filter. For the message passing method and our proposed method, the communication and computation are executed iteratively. The communication cost is related to the number of loops, i.e. the accuracy of the estimated results. Generally, both the standard particle and the distributed particle filter using the message passing method converge linearly. As our proposed method is an approximation to the standard particle filter, we can use the same number of loops to represent the same accuracy. By denoting T as the number of loops, the communication in bytes for the message passing method is 2MN b T = O(M). The communication in bytes for our proposed method is N a MN b T = O(M), where N a is the average of number of neighbors, for example, N a can be four in the networs with uniform grid. In summary, the centralized method is not scalable, while both of the message passing method and our proposed method are scalable. Furthermore the message passing method depends on a path which should be planned in advance. Any failures of nodes require to re-plan the path and so does the addition of nodes. Our proposed method is more robust than the message passing method to this end. 5 Simulations A sensor networ with 100 nodes (M = 100) is used for simulation. The sensors are randomly placed in a square 5 5 shown in figure 1. We tae the communication distance in figure 1 as R = 0.9. This results in a connected graph and its maximum degree is d max = 14. The connectivity can be verified by finding that the ran of Laplacian matrix is 99 or (M 1). We assume that a moving object has a state equation: x = Ax 1 + Bv 1 13

3 2 1 0 1 2 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 Figure 1: networ connection 1 0 T 0 T 2 /2 0 0 1 0 T 0 T 2 /2 where A =, B =. T is the discrete sampling time and 0 0 1 0 T 0 0 0 0 1 0 T selected as 0.05. The observation equation is a sonar-lie model, which can observe the distance and angle between a sensor positioned in q m and the moving object. z m, = x m, q m (x m, q m ) + w where x m, q m denotes the Euclidean distance between x m, and q m. (x m, q m ) denotes the angle of vector (x m, q m ). The covariances of Gaussian noises v and w are Q = 0.25 0 0 0.25 and R = 0.01 0 0 0.0004 1, β = 0, κ = 2. The moving object in all tests starts from (0, 0).. The parameters of UT is also selected as α = The CP F is tested first. Limited number of particles (10 particles) are used. 500 step tracing result is shown in figure 2. It can be seen that the estimated trajectory (dotted line) is not very close to the true trajectory (solid line), specially at the rapid changing places where the tracing loses its target. The estimated particles at three different time steps are also shown in figure 2 (see 14

the blac dot clusters). They represent the covariance changes from large cluster at the beginning to small cluster at the end. For the same true trajectory, the DP F is tested in 100 sensor nodes allocated as shown in figure 1. Each of sensor nodes contains 10 particles. So the computation time is nearly the same as the previous test. The tracing result in one of the sensor nodes is shown in figure 3. Comparing figure 3 with figure 2, it clearly shows that the DP F outperforms the CP F. It can trac the rapid turnings of the true trajectory. The downside is that the estimated result has larger covariance than the result estimated by the CP F. This is due to the use of the consensus filter, which only provides an estimated global distribution for the DP F. 2 1 0 y(m) 1 2 3 Estimated trajectory True trajectory 4 7 6 5 4 3 2 1 0 1 x(m) Figure 2: tracing result of CP F with 10 particles 2 1 0 y(m) 1 2 Estimated trajectory True trajectory 3 4 7 6 5 4 3 2 1 0 1 x(m) Figure 3: tracing result of DP F with 10 particles We increase the number of particles from 10 to 100 to compare the DP F with the CP F again. The tracing results are shown in figure 4 and 5, respectively. In general, the estimated trajectory by the DP F is comparable to the result by the CDF. The estimated trajectory by the 15

DP F is closer to the true trajectory with larger covariance than the result estimated by the CDF. 9 8 Estimated trajectory True trajectory 7 6 5 y(m) 4 3 2 1 0 1 5 4 3 2 1 0 1 x(m) Figure 4: tracing result of CP F with 100 particles 10 Estimated trajectory True trajectory 8 6 y(m) 4 2 0 2 6 5 4 3 2 1 0 1 x(m) Figure 5: tracing result of DP F with 100 particles Finally, the CP F with 500 particles and the DP F with 5 particles in each node are compared. This means that both algorithms use the same number of particles. The tracing results are shown in figure 6 and 7, respectively. Also for comparison purpose, the estimated trajectory by the CP F with 5 particles is shown in figure 8. Since the DP F is an approximation to the CP F, its performance is an approximation to the performance of the CP F. Its tracing result is close to the CP F with 500 particles while it uses less computation time in each node. Its tracing result is far better than the result estimated by the CP F with 5 particles while it uses nearly the same computation time. 16

1 0 True trajectory Estimated trajectory 1 2 3 y(m) 4 5 6 7 8 12 10 8 6 4 2 0 2 x(m) Figure 6: tracing result of CP F with 500 particles 1 0 Estimated trajectory True trajectory 1 2 3 y(m) 4 5 6 7 8 12 10 8 6 4 2 0 2 x(m) 6 Conclusions Figure 7: tracing result of DP F with 5 particles This paper proposed a distributed particle filter. The main idea is to use an average consensus filter to estimate global mean and covariance of posterior probability and use a UT to propagate the mean and covariance to generate the predicted statistics. Due to the use of the average consensus filter, this DP F is an approximation to the CP F. When the networ connectivity is guaranteed and the updating rate of the consensus filter meets a condition, the estimated global mean and covariance converge to the true values. This DP F only needs information exchanges between neighbor sensor nodes. The global information can be diffused over the entire networ through local information exchanges. It is scalable as the adding of more nodes does not affect the algorithm performance. It is also robust as it can still produce the right results even if failures of some nodes occur. The use of UT in the DP F can incorporate partial high order information of the mean and 17

1 0 Estimated trajectory True trajectory 1 2 3 y(m) 4 5 6 7 8 12 10 8 6 4 2 0 2 x(m) Figure 8: tracing result of CP F with 5 particles covariance into the estimates for non-linear models. However this DP F is currently limited to Gaussian dynamic systems. Further exploration of the use of other parameter models, such as Gaussian mixture model, to characterize the posterior probability could mae the DP F to be applied to non-gaussian dynamic systems. References [1] I. Ayildiz and W. Su and Y. Sanarasubramniam, A surevy on sensor networs, IEEE Communications Magazine, pp.102-114, Aug. 2002. [2] D. Estrin and R. Govindan and J. Heidemann and S. Kumar, Next century challenges: scalable coordination in sensor networs, Proc. of the ACM/IEEE Int. Conf. on Mobile Computing and Networing, pp.263-270, 1999, Seattle, Washington, USA. [3] F. Martinerie, Data fusion and tracing using HMMs in a distributed sensor networ, IEEE Trans. Aerospace and Electronic Systems, vol.33, pp.11-28, 1997. [4] F. Zhao and J. Shin and J. Reich, Information-driven dynamic sensor collaboration for tracing applications, IEEE Signal Processing Magazine, vol.19, pp.61-72, 2002. [5] D. P. Spanos and R. Olfati-Saber and R. M. Murray, Dynamic consensus on mobile networs, Proc. of the 16th IFAC World Congress, Prague, Czech, April 2005. 18

[6] M.S. Arulampalam and S. Masell and N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracing, IEEE Trans. Signal Processing, vol.50, pp.174-188, 2002. [7] A. Doucet and N. de Freitas and N. Gordon, Sequential Monte Carlo methods, Springer- Verlag, New Yor, 2001. [8] P. Gupta and P. R. Kumar, The capacity of wireless networs, IEEE Trans. Inf. Theory, vol.46, No.3, pp.388-404, 2000. [9] M. J. Coates, Distributed particle filtering for sensor networs, Proc. of Int. Symp. Information Processing in Sensor Networs (IPSN), Bereley, CA, April 2004. [10] Y. Sheng and X. Hu and P. Ramanathan, Distributed particle filter with GMM approximation for multiple targets localization and tracing in wireless sensor networs, Proc. of the 4th Int. Symposium on Information Processing in Sensor Networs,April 2005. [11] L. Zuo and K. Mehrotra and P. Varshney and C. Mohan, Bandwidth-efficient target tracing in distributed sensor networs using particle filters, Proc. of 14th European Signal Processing Conference EURASIP2006, Florence, Italy, 2006. [12] A. S. Bashi and V. P. Jilov and X. R. Li and H. Chen, Distributed implementations of particle filters, Proc. 2003 International Conf. Information Fusion, Cairns, Australia, July 2003. [13] M. Rosencrantz and G. Gordon and S. Thrun, Decentralized sensor fusion with distributed particle filters, Proc. of Conf. Uncertainty in Artificial Intelligence, Acapulco, Mexico, Aug. 2003. [14] N. A. Lynch, Distributed Algorithms, Morgan Kaufmann Publishers, Inc., San Francisco, CA, 1996. [15] R. Olfati-Saber and J. S. Shamma, Consensus filters for sensor networs and distributed sensor fusion, Proc. of the 44th IEEE Conference on Decision and Control, 2005. 19

[16] S. J. Julier and J. K. Uhlmann and H. F. Durrant-Whyte, New approach for the nonlinear transformation of means and covariances in linear filters, IEEE Trans. Automatic Control, vol.5, No.3, pp.477-482, 2000. [17] R. van der Merwe and A. Doucet and N. de Freitas and E. Wan, The unscented particle filter, Technical report,dept. of Engineering, University of Cambridge, 2000. [18] R. Olfati-Saber and R. M. Murray, Consensus problems in networs of agents with switching topology and time-delay, IEEE Trans. Automatic Control, vol.49, No.9, pp.101-115, 2004. [19] C. Godsil and G. Royle, Algebraic graph theory, Springer-Verlag, 2001. Brief Biographies Dongbing Gu received the BSc and MSc degrees in control engineering from Beijing Institute of Technology, China. He received the PhD degrees from University of Essex, U.K. He joined Department of Electronic Engineering, Changchun Institute of Optics and Fine Mechanics, China as a lecturer in 1988, and became an Associate professor in 1993 and a professor in 1999. He was an academic visiting scholar in the Department of Engineering Science, University of Oxford in the U.K. during October 1996 and October 1997. Currently He is a senior lecturer in the Department of Computing and Electronic Systems, University of Essex, U.K. His current research interests include predictive control, distributed formation control, machine learning, autonomous robots and statistical image processing. He has published about 100 papers in international journals and conferences. He is a senior member of IEEE. 20

Huosheng Hu received the MSc degree in industrial automation from the Central South University in China and the PhD degree in robotics from the University of Oxford in the U.K. Currently, He is a Professor in Computer Science at the University of Essex, leading the Human Centred Robotics (HCR) Group. His research interests include mobile robotics, sensors integration, data fusion, distributed computing, intelligent control, behavior and hybrid control, cooperative robotics, tele-robotics and service robots. He has published over 200 papers in journals, boos and conferences within these areas. Prof. Hu is currently Editor-in-chief for International Journal of Automation and Computing. He has been a Guest Professor at 6 universities in China such as Central South University, Shanghai University, Wuhan University of Science and Engineering, Kunming University of Science and Technology, and Northeast Normal University since 2000. He is a Chartered Engineer, a senior member of IEEE, and a member of IEE, AAAI, ACM, IAS, and IASTED. 21