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ESTIMATING TARGET STATE DISTRIBUTIONS IN A DISTRIBUTED SENSOR NETWORK USING A MONTE-CARLO APPROACH

Milind Borkar, Volkan Cevher and James H. McClellan

Georgia Institute of Technology Atlanta, GA 30332-0250

ABSTRACT

Distributed processing algorithms are attractive alternatives to cen- tralized algorithms for target tracking applications in sensor net- works. In this paper, we address the issue of determining a ini- tial probability distribution of multiple target states in a distributed manner to initialize distributed trackers. Our approach is based on Monte-Carlo methods, where the state distributions are repre- sented as a discrete set of weighted particles. The target state vec- tor is the target positions and velocities in the 2D plane. Our ap- proach can determine the state vector distribution even if the indi- vidual elements are not capable of observing it. The only condition is that the network as a whole can observe the state vector. A ro- bust weighting strategy is formulated to account for mis-detections and clutter. To demonstate the effectiveness of the algorithm, we use direction-of-arrival nodes and range-doppler nodes.

1. INTRODUCTION

In sensor networks, distributed processing is becoming more pop- ular than the centralized approaches [1]. This is because in cen- tralized networks, since there is only one processing node in the network, if that particular node is incapacitated, the entire sys- tem fails. The communication overhead is also significant. More- over, if all the sensing nodes are trying to transmit raw data to the central processing node, the required bandwidth increases signifi- cantly with the number of nodes. To overcome these drawbacks, a distributed processing approach is attractive.

Distributed processing stipulates processing capabilities at in- dividual sensors. We denote a sensor that has the ability to process data in addition to sensing as a smart sensor. Distributed process- ing eliminates the need for a central processing node. Thus the system is not fully dependent on a single node for processing thus eliminating the computational bottleneck. Since a smart sensor can process its own data, it only transmits sufficient statistics in the communication channel, minimizing the communication among sensors. Communication consumes more battery power than com- putation, hence smart sensor networks with distributed processing have additional advantages.

In this paper, a novel method for determining initial multiple target state distributions in a smart sensor network is proposed in a distributed framework. A Monte-Carlo method is used to generate a discretized approximation to the target state distribution. This distribution is represented using hypothesized target states called

Prepared through collaborative participation in the Advanced Sen- sors Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-02-0008.

particles and their associated weights. The resulting distribution can be used to initialize various distributed joint tracking (DJT) algorithms such as the ones in [2–6]. The algorithm satisfies the typical constraints of a distributed system. The communication be- tween individual sensors has fixed bandwidth. Since the informa- tion propagated between sensors is the cumulative state informa- tion, the amount of information passed between individual sensors does not increase. The sensor types focused on are Direction of Ar- rival (DOA) nodes (e.g., acoustic arrays with known microphone positions) and range-doppler nodes (e.g., a radar sensor). How- ever, the results are general and can be extended to networks with different sensor modalities. Each sensor runs a tracking algorithm that operates in a different state space determined by the sensor modality. We shall refer to the tracking algorithms running at the different sensors as the organic trackers. The DJT operates in a state space which may be different from the state spaces of the or- ganic trackers at the individual nodes. We assume that each tracker is capable of detecting a new target. When an organic tracker de- tects a new target in its limited subspace, it transmits information throughout the network to generate the target state distribution. We also have a robust weighting strategy that can accommodate clutter as well as missing data.

Communication takes place between neighboring sensors only and there is a predefined path for the information flow through the network from the first sensor to the last sensor.

The organization of the paper is as follows. Section 2 briefly introduces the acoustic and radar trackers. Section 3 proposes a Monte-Carlo approach for the distributed estimation of the target state distribution. Section 4 demonstrates the effectiveness of the proposed algorithms on synthetic data. Conclusions and future work follow in Section 5.

2. ACOUSTIC AND RADAR TRACKERS

The two types of sensor nodes used to demonstrate the initializa- tion algorithm are DOA sensors and Range-Doppler sensors. The DOA tracker operates in the [θ q φ]’ space whereθ is the direc- tion towards the target, q is the ratio of the targets velocity to the targets range andφ is the heading direction of the target. The range-doppler tracker operates in the [r vr]’ space where r is the range to the target and vr is the targets radial velocity. Detailed descriptions about these trackers can be found in [7–15].

The focus of this paper is to generate a probability distribu- tion for the target in the [x y vx vy]’ space where x and y are the Cartesian coordinates of the targets location and vx and vy are the velocity components along the x-y directions. Notice that the true location and velocity of the target is not observable at any of the individual nodes and that the organic trackers operate in dif-

ferent state spaces that have lower dimensions than the state space in which the targets distribution is desired. This means there is a many to one mapping from the states used by the organic trackers to the state space in which the final target distribution is gener- ated. It is assumed that the organic trackers are available at the different nodes and the outputs of the organic trackers are used to generate the desired probability distribution. The sensor network is assumed to be calibrated so each sensor is aware of its own lo- cation.

3. A MONTE-CARLO APPROACH FOR THE DISTRIBUTED ESTIMATION OF THE TARGETS

PROBABILITY DISTRIBUTION

To have an optimal particle distribution, one must sample from the true posterior distribution [16]. Using Bayes’ rule, the posterior distribution can be expressed as

p(xt|zt) = p(zt|xt)p(xt) p(zt)

(1)

wherext is the target state at timet andzt is the vector of mea- surements from allM sensors at timet. We assume that the mea- surements at the individual nodes are independent conditioned on the state. Hence, the combined data likelihood for all sensors can be factored into the product of the data likelihoods at the individual sensor nodes. Since no prior information is available,p(xt) is uni- form and is dropped from the equation.p(zt) is simply a propor- tionality constant since it does not depend on the state. Therefore, (1) can be simplified as

p(xt|zt) ∝ M∏

m=1

p(zm,t|xt) (2)

wherezm,t is the measurement from themth sensor at timet. We choose not to communicate raw data between nodes to limit communication bandwidth. Thus determining the posterior dis- tribution analytically is impossible. Therefore, we chose as our proposal function

π(xt|zt) = 1 M

M∑ m=1

p(xt|zm,t) (3)

Our choice of the proposal function is an equally weighted mixture of the individual posterior distributions from the different nodes. For the different nodes, particles can be sampled from the individ- ual posterior distributions as follows:

For DOA nodes:

r(i) ∼ U(0, rmax) (4)

θ(i) ∼ N(θt, Σθt) (5) q(i) ∼ N(qt, Σqt) (6) φ(i) ∼ N(φt, Σφt) (7) x

(i) t = r

(i) cos(θ(i)) (8)

y (i) t = r

(i) sin(θ(i)) (9)

v(i)xt = q (i)r(i) cos(φ(i)) (10)

v(i)yt = q (i)r(i) sin(φ(i)) (11)

For Range-Doppler nodes:

r(i) ∼ N(rt, Σrt) (12)

θ(i) ∼ U(0, 2π) (13) v(i)r ∼ N(vrt , Σvrt ) (14)

v (i) t ∼ U(−(v2max − (v(i)r )2)0.5, (v2max − (v(i)r )2)0.5) (15)

x (i) t = r

(i) cos(θ(i)) (16)

y (i) t = r

(i) sin(θ(i)) (17)

v(i)xt = v (i) r cos(θ

(i)) + v (i) t sin(θ

(i)) (18)

v(i)yt = v (i) r sin(θ

(i))− v(i)t cos(θ(i)) (19) Estimates of(θt, Σθt), (qt, Σqt), and(φt, Σφt) are available

from the organic trackers at the DOA nodes. Similarly, estimates of (rt, Σrt) and(vrt , Σvrt ) are available from the organic trackers at the range-doppler nodes. There is a range ambiguity in the DOA node. Similarly there is a DOA ambiguity and a tangential velocity ambiguity in the range-doppler node. Therefore, these values are drawn from appropriate uniform distributions. Here,rmax is the assumed maximum range at which the target is visible to the DOA node, andvmax is the assumed maximum velocity of the target. Radial velocity is considered positive if the target is moving away from the node. Tangential velocity is considered positive if the tangential component points in the counterclockwise direction.

Using (4) through (19) one can sample particles from the in- dividual posteriors. If the total number of nodes isM , then to sampleD particles from the mixture given by (3), one can sample D/M particles from each individual posterior and combine these particles to