Local Node Selection for Localization in a Distributed ... - IEEE Xplore

I. INTRODUCTION

Local Node Selection for Localization in a Distributed Sensor Network LANCE M. KAPLAN, Senior Member, IEEE U.S. Army Research Laboratory

This paper discusses a new localized resource manager for a wireless sensor network of bearings-only sensors. Specifically, each node uses knowledge of the target under surveillance to determine whether it should actively collect measurements and how far to disseminate the data in order for the sensor network to maintain track of the target. At each node, the resource manager requires only knowledge of the relative location to the target for itself and the active nodes from the previous snapshot. The decentralized strategy represents a modification to the global node selection (GNS) method that exploits knowledge of the location of all nodes in the network. Simulations show that despite the lack of global network knowledge, the new localized management method is almost as effective as GNS in terms of balancing the tradeoff between energy usage and localization accuracy.

Manuscript received June 17, 2004; revised March 2, 2005; released for publication June 20, 2005. IEEE Log No. T-AES/42/1/870596. Refereeing of this contribution was handled by W. D. Blair. This research was prepared through collaborative participation in the Advanced Sensors and the Communications and Networks Collaborative Technology Alliances sponsored by the U.S. Army Research Laboratory under Cooperative Agreements DAAD19-01-2-008 and DAAD19-01-2-0011, respectively. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as presenting the official policies either express or implied of the Army Research Laboratory or the U.S. Government. Author’s address: ATTN: AMSRD-ARL-SE-SE, U.S. Army Research Laboratory, 2800 Powder Mill Rd., Adelphi, MD 20783, E-mail: ([email protected]).

c 2006 IEEE 0018-9251/06/$17.00 ° 136

Wireless sensor networks formed using unattended ground sensors promise to provide an effective low cost solution for battle space surveillance [1]. For such a network to be effective over an extended time, power conservation is an absolute necessity [2]. This work presents a new sensor management strategy for a sensor network consisting of passive sensors collecting bearing measurements, i.e., direction of arrival (DOA) estimates. The processing to achieve sensor management is done at every node within communication reach of all the active nodes. Specifically, each node determines whether or not to be an active participant based only on local information about its additive utility relative to the known active set of nodes. By utility, we mean the reciprocal of the expected mean squared (MS) error performance of the extended Kalman filter (EKF). In addition, these additive utilities determine the communication reach of the active nodes. The management algorithm limits the reach of the node while maintaining the same localization performance as if the node was able to disseminate its data to all other nodes in the network. It will be shown that the communication reach of the node is inversely proportional to how well the active nodes can localize the target. We refer to the distributed management strategy that uses only local node information as autonomous node selection (ANS). The new management approach represents a modification of the global node selection (GNS) method [3]. In GNS, each node within reach of all the active nodes uses knowledge about the location of every node in the network to make an activation decision for the next snapshot. Despite only using localized information, ANS is shown through simulations to be almost as effective as GNS in terms of managing the tradeoff between geolocation accuracy and energy usage. In a realistic environment, the ANS is more desirable than GNS because the whole network does not need to recalibrate as nodes burn out or as more nodes are inserted into the network. Furthermore, when the number of nodes in the network is large, the memory requirements of GNS can become burdensome. The overall node selection methodology of ANS and GNS was inspired by the geometrical dilution of precision (GDOP) metric presented in [4] and implemented in [5]. The earlier work in [4] and [5] represent a special case of GNS where 1) prior information from the tracker is not considered, 2) the network is always fully connected in the sense that all nodes are potential candidates for activation between each snapshot, and 3) the number of active nodes per snapshot is set to three. The GDOP metric is simply the expected MS error of a maximum likelihood estimator that only incorporates measurements from a

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 42, NO. 1

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given snapshot. Just as in [5], we linearize the bearing measurement equation to implement the EKF. The EKF is computationally efficient. Although it is a suboptimal solution, it does generate reasonable state estimates for our sensor network scenario. The information-driven sensor querying and data routing (IDSQ) technique of Zhao, et al. considers node selection using the optimal framework of the sequential Bayesian filter [6—8]. Specifically, IDSQ selects a single node that minimizes the resulting entropy of the posterior state distribution. Clearly, both the filtering and measure calculations are computationally demanding. In [8], a more computationally efficient version of the IDSQ is postulated where the tracker becomes an EKF and the entropy measure simplifies to the determinant of the filtered covariance matrix. Such a strategy is similar to that adopted by ANS and GNS. However, that paper does not implement or analyze the more efficient version of the IDSQ. Furthermore, efficient IDSQ is designed to operate with one active node per snapshot, and the active node must possess global knowledge in order to determine which node to activate for the subsequent snapshot. This paper details and rationalizes the development of ANS. Preliminary work describing former manifestations of the ANS appear in [9] and [10]. Because ANS is a modification of GNS, many of the details regarding GNS and the simulations used to evaluate performance can be found in the companion paper [3]. This paper is organized as follows. Section II defines the problem and summarizes the models used by the EKF, node selection, and the simulations. The mathematical foundation to implement node selection is laid out in Section III. Section IV details the implementation of ANS and briefly describes GNS. Simulations are used to evaluate the functionality of ANS in Section V. Finally, Section VI provides concluding remarks. II. PROBLEM DEFINITION A wireless sensor network composed of bearings-only sensors is tracking a target. The node may include an array of microphones that can estimate the DOA of a source signal. The node also contains a computer to process the raw data, a two-way radio to share information, and a battery to provide power. To conserve power, only a small subset of nodes is actively estimating the DOA. During periodic intervals, which we refer to as snapshots, the active nodes share their measurements. Then, a decentralized EKF [11, 12] maintains the target track. After a given snapshot, all nodes within the communication range of the active nodes must decide whether or not to be active for the next snapshot. An effective resource manager must allow for each node to make an activation decision using only knowledge obtained

by the active nodes from the previous snapshot. The active nodes for the current snapshot should be well positioned relative to the target in order to collect useful measurements that optimize geolocation performance. Otherwise, if the active nodes are nearly collinear with respect to the target and/or the nodes are too far from the target, the localization accuracy can be poor. Furthermore, an effective resource manager must limit the communication reach of a node without sacrificing overall geolocation performance. This paper makes the following assumptions about the sensor network. First, the location and orientation of each node is known within a universal coordinate system. In practice, the sensor network will need to perform a calibration process such as described in [13], [14], [15], and [16]. Once a node enters into the network, its geographical position remains constant. Next, all nodes are time synchronized using an algorithm such as described in [17], [18], and [19]. The probability of detection is one, the probability of false alarm is zero, the probability of node failure is zero and the probability of communication error is zero. The bearing errors follow an additive white Gaussian noise (AWGN) model. In general, the standard deviation ¾ of the measurement error is a function of the position of the node relative to the target. Finally, the decentralized EKF is tracking a single target. Some remarks about the relaxation of these assumptions are provided in the concluding section. We use a four state EKF where the first two and last two states correspond to the 2-D position and velocity of the target, respectively. The EKF employs a constant velocity dynamical model for the target where acceleration is treated as process noise. The measurement model is the true DOA between the target and the node at the snapshot time embedded in AWGN where ¾ is a constant value of 5 deg, which is approximately the error observed in real data collected by the U.S. Army Research Laboratory [20]. In [3], results for more sophisticated bearing error models using GNS were very similar to the constant ¾ model. We do not consider the more sophisticated models in this work. The simulations generate the measurements and test the geolocation performance and energy usage of the node selection technique integrated into the decentralized EKF. The measurements are the retarded DOA embedded in AWGN with ¾ = 5± . By retarded DOA, we mean the DOA that accounts for the propagation delay between the target and the node. We assume the nodes measure acoustic energy so that the propagation speed is c = 347 m/s. The simulation also tabulates energy usage at each node by accumulating the energy it uses to transmit and receive measurements as predicted by a model introduced in [21]. It has been reported that the

KAPLAN: LOCAL NODE SELECTION FOR LOCALIZATION IN A DISTRIBUTED SENSOR NETWORK

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energy to transmit a bit of data is orders of magnitude larger than the energy to implement a floating point operation [2]. Therefore, the simulation ignores the energy a node uses to collect measurements. Technically, the communication model provides the energy required for a node to reach another node in one hop. The actual communication protocol, e.g., single hop or multiple hops with routing, is beyond the scope of this paper. We simply use the energy model to provide a rough order of magnitude (ROM) estimate of the energy burned in the implementation of node selection. Whether or not a node reaches another node in one or more hops, the energy is monotonically increasing with respect to the communication reach of the node. Given l bits of data, the model states that the energy to transmit the data a distance of d meters is Et = l"elec + l"amp d4

(1)

where

(2)

The energy per bit to run the electronics such as the filters is represented by "elec , and the energy to run the power amplifier is encapsulated by "amp . As suggested in [21], the values for these parameters are "elec = 50 nJ/bit and "amp = 0:0013 pJ/bit/m4 . The model assumes 1=d4 propagation loss in the channel due to the complicated transmission medium. It is anticipated that the node antenna will be low to ground, possibly hidden under the grass. Finally, we set the packet size to be l = 384 bits to encompass the information required to perform decentralized tracking along with the overhead for error correcting and security. III. FOUNDATION FOR NODE SELECTION The overall goal of the node selection is to use the predicted node/target geometry at a given snapshot to determine the best set of active nodes Na of desired cardinality Nd from the entire set of Ns nodes available in the network. By best, we mean the set of nodes that leads to the least expected MS position error for the filtered estimate of the target state. In practice, only the subset of nodes within the communication reach of the active nodes from the previous snapshot are available to be picked for the next active set. In the sequel, we refer to such an available set of nodes as the candidate set Nc of cardinality Nc . The MS error can be determined by the filtered Fisher information matrix (FIM). As shown in [3] and [12], the covariance update equation of the EKF can be expressed in its information form using FIMs that represent the inverse of the covariance error. The update equation is ¸ · Jm,Na 0 Jf,Na = Jp + (3) 0 0 138

(4) represents the inverse state covariance error if no prior information is used, i.e., single snapshot localization. Note that (ri , Ái ) represents the location of the ith node relative to the predicted target position in polar coordinates, and ¾i is the rms bearing error of the ith node. The first two elements of the target state corresponds to the x ¡ y position. We define the utility of the active node set Na as the reciprocal of the MS error, detfJ˜ f,Na g ¹(Na ) = (5) trfJ˜ g f,Na

and the energy to receive the data is Er = l"elec :

where Jp and Jf,Na represent the inverse of the predicted and filtered state covariance, respectively. The measurement FIM matrix, ¸ X 1 · sin2 Ái ¡ sin Ái cos Ái Jm,Na = ¾ 2 r2 ¡ sin Ái cos Ái cos2 Ái i2Na i i

J˜ f,Na = [Jp ]1 : 2, 1 : 2 ¡ [Jp ]1 : 2, 3 : 4 ([Jp ]3 : 4, 3 : 4 )¡1 [Jp ]3 : 4, 1 : 2 + Jm,Na

(6) and [A]i : j,k : l represents the (j ¡ i + 1) £ (l ¡ k + 1) subblock of A that consists of the intersection of rows i through j and columns k through l. Because J˜ f,Na is a 2 £ 2 positive definite matrix, its two eigenvalues can be expressed as T T (1 ¡ °) and ¸max = (1 + °) 2 2 where T = trfJ˜ f,Na g and 0 · ° · 1. We refer to ° as the angular diversity parameter. Now, the utility can be rewritten as T (7) ¹(Na ) = (1 ¡ ° 2 ): 4 ¸min =

When ° = 0, the angular diversity is maximized in the sense that the area of the error ellipse is minimized for a given value of T. For this case, the error ellipse is a circle because the eigenvalues are equal. This scenario can only be achieved if the actives nodes surround the target. At the other extreme, when ° = 1, the sensors only provide a single view of the target where the area of the error ellipse is infinite because the major axis has infinite length. This scenario occurs when the nodes are colinear with the target. Node selection that incorporates the prior information collected by the tracker attempts to maximize (5). We also consider the case where node selection ignores the prior by setting Jp = 0 so that (5) simplifies to detfJm,Na g ¹(Na ) = (8) trfJm,Na g which is simply the reciprocal of the GDOP metric proposed in [4]. Ignoring the prior information realizes a modest computational savings by avoiding


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(6). However, the storage requirement is not reduced because the prior information is already incorporated in the EKF. When implementing the ANS method, the active nodes have no knowledge of the other nodes. The inactive nodes within earshot of the active nodes can determine their added value to the utility. These nodes should only decide to become active if they can increase the overall utility by replacing one of the active nodes. To this end, the inactive node i calculates its differential utility as ¢ ¤ £ ¡ d¹(i j Na ) = max ¹ (Na nfjg) [ fig ¡ ¹(Na nfjg) : j2Na

(9) This differential utility can be compared with the differential utility of an active node j, d¹(j j Na ) = ¹(Na ) ¡ ¹(Na nfjg):

(10)

When the differential utility of an inactive node i is greater than that of active node j, the overall utility of the active set can be increased by replacing node j with node i. It is shown that the possible differential utility of a node is bounded above by a quantity that is a function of the target/node distance and an angular diversity parameter. This bound helps to determine how far an active node must communicate in order to reach a node that might also decide to activate. The following theorem introduces the upper bound on the differential utility. THEOREM 1 Given an inactive node i whose distance to the target is ri and whose measurement standard deviation is ¾i , its differential utility for any active set Na of cardinality Na is bounded above by d¹(i j Na )
Nd . In any event, the resulting Na active nodes collect their measurements for the current snapshot and transmit this information using the transmission range as calculated by (13) for i 2 Na+ . Finally, the process repeats itself for the subsequent snapshot where Na+ becomes Na¡ .

Fig. 2 illustrates stages of the ANS method where Nd = 2. In this example, three active nodes communicate their bearing measurements and update their tracking filters at snapshot k ¡ 1. Then, the active nodes implement the search stage and one node becomes inactive. The two remaining nodes begin the discovery stage by computing the threshold. Then, they communicate the threshold, their location and the target track information, i.e., state and covariance. Nodes within communication range have enough information to compute their differential utilites and make an activation decision. The one node that decides to become active also has enough information to start its tracking filter. Finally, all active nodes collect and communicate their measurements for snapshot k with each other. The ANS is parameterized by Nd and ·. The parameter Nd represents the minimum number of nodes that will be active per a snapshot. The · parameter correlates to the enthusiasm for a nearby dormant node to become active. When · is small, a node will need to provide higher added value to join the active set than it would if · is larger. For a fixed Nd , the average number of active nodes per snapshot that result from ANS increases as · increases. The transmission control via (13) is able to maintain the same localization performance as a fully connected implementation of the ANS where data from the active nodes reaches every node in the network. This is due to the fact that Theorem 1 ensures that the nodes not reached cannot exceed the threshold ¿ . In addition, the transmission control strategy provides a natural mechanism to alert inactive nodes only when they can possibly contribute to the geolocation of the target. The nodes that are currently useless do not waste energy by trying to decode unnecessary data packets. C.

Global Node Selection

GNS effectively incorporates the search stage of ANS with the exception that the candidate set consists of all nodes within communication reach of the active nodes from the previous snapshot. In other words, all nodes within earshot of the Na active nodes implement the search technique described in Section IVA. The resulting Nd active nodes can include inactive nodes from the previous set because

Fig. 2. Illustration of active sets during one snapshot of ANS for Nd = 2. (a) Na¡ , the active set determined from the previous snapshot. (b) Na0 , the intermediate set determined by the search stage. (c) Na+ , the finalized active set that communicates and shares measurements for the current snapshot. 140


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of the expanded candidate set. Therefore, there is no need to implement a discovery stage. As a result, Na = Nd . The Na active nodes then collect and share their measurements for the upcoming snapshot. They also use (13) to determine their broadcast range. The process repeats for the next snapshot.

TABLE I Statistics for Tracking Results Illustrated in Fig. 3

Implementation No Prior With Prior

Average RMS Nodes Average Power Position Error Per Snapshot Per Node (meters) mean max (Joules) 4.50 3.66

2.66 2.10

49 3

0.4587 0.0046

V. SIMULATION RESULTS We have integrated ANS and GNS into the EKF tracker. For both node selection techniques, we considered both the implementation using the prior (or predicted) information from the tracker where (5) is used as the utility or the implementation that ignores prior information so that (8) is the utility. In addition, we considered either a fully connected implementation where the transmission range of a node is large enough to reach the entire network, or where transmission control is implemented via (13). These implementations were evaluated over simulated data representing a target that is moving at a constant velocity of 10 m/s through a field of Ns = 50 nodes. The nodes are randomly placed via a uniform distribution over a 1 km £ 2 km region. Fig. 3 shows the active nodes during different snapshots of target tracking when implementing the ANS using Nd = 2 with and without prior information for a value of · = 1. Overall, the number of snapshots is 100 occurring in intervals of 1 s. The rms process noise was set to ¾º = 0 m/s2 . The “ * ” symbol marks the active nodes, the “+” symbol marks inactive nodes within communication range of all active nodes, and the “ ¢ ” symbol marks nodes within communication range of some of the active nodes. When the active nodes achieve a good measurement geometry, the communication reach is smaller. The communication reach of the active nodes is higher when ignoring the prior information. Interestingly, the differential utility of the active set of the ANS without the prior can become so small that the communication range reaches the entire network and most of the nodes decide to become active. After the subsequent snapshot, the number of active nodes will reduce to two since most nodes are in the candidate set of the search stage. Experiments indicate that this refreshing phenomena occurs only when Nd = 2 and no prior information is exploited. The refreshing phenomena occurs in the priorless ANS when a good active set of two nodes for one snapshot becomes colinear with the target over the subsequent snapshot. Then, the differential utility for the active set and the resulting threshold goes to zero. The active nodes reach the entire network and all nodes decide to activate because of the low threshold. When incorporating prior information, the ANS avoids a low threshold because the angular diversity of the prior information tempers the poor measurement geometry. The GNS method also avoids

the problem because the candidate set of the search process is expanded so that it is not forced to pick an active set with a poor measurement geometry. Table I provides statistics averaged over all 100 snapshots for the two examples illustrated in Fig. 3. Specifically, it provides the rms position error, average node usage, maximum node usage and energy burned. On average, the priorless ANS employs more nodes and burns more energy. In this case, the higher energy of the priorless ANS does not translate to better localization accuracy. Next, we explore the average performance of ANS and GNS over multiple tracking trials. Specifically, for a given parameter setting of the node selection technique, we generated 100 different Monte Carlo trials over 40 different random node configurations for a total of 4000 trials. Just as in Fig. 3, a single trial represents the target traveling 1 km through the field where the snapshot interval is 1 s. The process noise of the EKF is set to zero, i.e., ¾º = 0 m/s2 . The geolocation performance of ANS and GNS with and without prior information is shown in Fig. 4. This figure plots the average rms position error versus the average number of active nodes per snapshot. The curves for GNS represent the fully connected case. However, the curves are not much different when implementing transmission control. For the ANS case, the curves are representative of either transmission implementation. The numbers in the markers for these curves represent the (Nd , ·) parameter settings for the ANS method. The value of Na for the GNS method corresponds directly with the average number of snapshots because Na = Nd . The figures indicates that the node selection achieves slightly better localization performance when prior information is used. Usually, as Nd increases, the rms position error decreases. However, this trend is not always true. Perhaps this is an artifact of the linearization of the measurement equation in the EKF. As expected with the ANS methods, as · increases for a fixed Nd , the number of active nodes per snapshot increases. The increase in average Na as · increases leads to better geolocation accuracy for cases that Nd < 5 and prior information is used. However, the trend is reversed when prior information is ignored. This may be due to the fact that optimization of the utility does not explicitly match to the minimization of the rms position error when ignoring prior information. Overall, the tradeoff between localization accuracy and node usage is


141

Fig. 3. ANS results during decentralized EKF tracking with Nd = 2 for different snapshots where the left and right columns represent results without and with the use of prior information, respectively.

comparable between ANS and GNS. In fact, the ANS appears to provide a better tradeoff when incorporating the prior. The advantages for transmission control are illustrated in Fig. 5(a). The figure plots the average energy usage per node versus the rms position error for various parameter settings of the ANS. Again, the parameter settings are indicated inside the markers. For a given parameter setting, fully connected and transmission controlled versions of ANS lead to the same geolocation accuracy. However, the energy usage is greatly reduced when incorporating the transmission control. On average, the energy reduction is in the order 17x and 27x when ignoring the prior or implementing the prior, respectively. Fig. 5(b) zooms into the curves representing the implementation of transmission control, and Fig. 5(c) provides the convex hull for various parameter setting of the ANS. These curves clearly indicate that its better to incorporate the prior in terms of achieving a better tradeoff between energy usage and geolocation accuracy. For most cases, it is best to set · = 1 and · = Nd when ignoring and when using the prior, respectively. In other words, for these settings of ·, 142

Fig. 4. Comparison of the tradeoff between geolocation accuracy and node usage for various implementations of the ANS and GNS approaches.

the resulting point on the energy versus localization curve will be close to the convex hull. Fig. 6 compares the energy versus localization performance of ANS with that of GNS when implementing transmission control. The numbers inside the markers indicate the value of Nd . For the


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Fig. 5. Tradeoff between energy usage and localization accuracy. (a) Fully connected versus transmission control. (b) Transmission control implementations. (c) Convex hull of transmission control implementations.

Fig. 6. Comparison of the tradeoff between energy usage and localization accuracy for GNS and ANS. (a) Without prior. (b) With prior.

ANS method, · = 1 or · = Nd when ignoring or using the prior information, respectively. When using no prior, the ANS and GNS performance curves are almost identical except when Nd = 2. On the other hand, the GNS conserves slightly more energy than ANS when incorporating prior information Clearly, the ANS methods achieves almost as good as a tradeoff of performance, and the ANS only uses localized knowledge of the network. The higher energy usage of the ANS, with prior or when Nd = 2 for no prior, is more than can be explained by transmission of the packets from the active nodes between the search and discovery stages

(see Section IV and Fig. 2(b)). For each trial, we calculated the minimum, average, and maximum value of the critical range (see (12)) over the 100 snapshots and averaged these values over all 4000 trials. Fig. 7 shows the resulting statistics versus average node usage for the GNS and ANS method. On average, the critical range is slightly higher for GNS method for a fixed node usage when implementing the prior information. However, the average maximum critical range is significantly higher for the ANS method. Due to the quadratic relationship in the energy transmission model (see (1)), the difference between maximum critical ranges causes the ANS to suffer from more


143

Fig. 7. Critical range versus node usage for GNS and ANS. (a) Without prior. (b) With prior.

energy usage for a given localization accuracy. We suspect that because the ANS calculates the critical range after the search stage before it can discover the inactive nodes, there are snapshots where the best Nd nodes provide very poor localization. The poor nodes lead to a small differential utility and large critical range. On the other hand, the GNS, by incorporating the inactive node in the search stage, is able to find Nd nodes that lead to a smaller critical range. The figure also demonstrates that by implementing the prior information, the critical range is usually reduced on average by a factor of 15%. Finally, an outlier occurs in the maximum critical range curve for the priorless ANS when Nd = 2. This can be explained by the refreshing phenomena explained earlier. VI. CONCLUSIONS This paper introduces the ANS technique. This technique is distributed on each node that is within communication range of the active set of nodes for a given collection snapshot. At each node, the ANS determines whether or not that node should be active for the subsequent snapshot by only incorporating geometrical knowledge of itself and the active set of nodes from the previous snapshot. The ANS can be viewed as a modification of the GNS technique introduced in [3] that considers the location information of every node in the network to make an activation decision. The ANS and GNS methods can be implemented with or without the consideration of the predicted FIM (or covariance error). These two selection techniques can also limit the broadcast range of the node or set the broadcast range so that the node communicates to every node in the network. Simulations show that it is best to implement node selection by incorporating the prior information and transmission range control. Both of these two options helps to reduce the energy usage. In fact, using prior information also leads to a slight improvement of localization accuracy. In terms of tradeoff between energy usage and localization 144

error, the ANS achieves almost as good performance as GNS. While the simulations in this paper considered a constant velocity target, the node selection methods can accommodate maneuvering targets. This can be accomplished by simply allowing for high process noise in the EKF or by using more complicated filters. In fact, we have studied the performance of GNS using real data generated from targets traveling around a track when implementing multiple mode filters that exploit the constant turn rate model [22]. The ANS and GNS methods represent two extremes in terms of the network information available at a single node. A compromise of performance might be achievable by allowing nodes to store information about their inactive neighbors. By this approach, the node selection uses partial network information. When the neighborhood reduces to no neighbors, then the method simplifies to ANS, and when the number of neighbors is equal to the network size, the method becomes GNS. We are currently investigating the performance of such a partial node selection strategy. Future and current work is also investigating the consequence of relaxing the assumptions of Section II. For example, the node selection methods were developed under the assumption that the nodes are stationary. Actually, the constraint of local information for ANS provides it with the flexibility to accommodate a node that has moved before it decides to become active. We believe that the ANS can be easily modified to also accommodate nodes that are moving while they are active. In its current form, ANS can handle multiple targets if the targets are significantly separated. For this case, different clusters of nodes will be tracking the different targets where each cluster is only receiving measurements from a single target. However, when multiple targets are within detection range of the same nodes, the tracking and node selection algorithms must consider the data association and track initiation problems. We plan to modify the ANS algorithm to consider the multi-target tracking


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issues. We also plan to improve the robustness of node selection for cases of miss detections, false alarms and communication errors. Other future and current work includes modification of the utility to include a term that helps to balance the energy usage of the nodes in order the maximize the effective lifetime of the network. In addition, we are also developing a multiple access control (MAC) protocol that schedules the communication of active nodes between snapshots. Finally, we also plan to adapt the value of the Nd parameter based on the absolute residual error between the measured and predicted observations. In the end, we hope to develop a joint communication and sensing protocol for tracking targets in a wireless sensor network.

Then, the difference between the utilities of Kk+ and Kk is Tk2 T 1 1 (1 ¡ °k2 ) + k 2 2 (1 + °k ¡ 2°k bk2 ) 4 2 ¾j rj T dk = ¡ k (1 ¡ °k2 ) 1 1 4 Tk + 2 2 ¾j rj =

1 1 ¾j2 rj2

(1 + °k )2 ¡ 4°k bk2 Tk : 1 1 4 Tk + 2 2 ¾j rj

Because 0 · bk2 · 1 and Tk > 0, then dk < By (9),

APPENDIX A. PROOF OF THEOREM 1

d¹(i j Na ) = max dk

Lets label the Na nodes in the active set Na as k = 1, : : : , Na . The calculation of the differential utility via (9) for inactive node j requires the calculation of the utility of the following sets Kk = Na nfkg

Kk+

for k = 1, : : : , Na :

= Kk [ fjg

Lets also use °k to represent the angular diversity parameter for Kk . The utility for set Kk is ¹(Kk ) = where

The utility of the set Kk+ is derived from the filtered FIM, 1 1 J˜ f,K+ = J˜ f,Kk + 2 2 ~nj ~nTj k ¾j rj 1 1 ~n ~nT ¾j2 rj2 j j

REFERENCES [1]

[3]

[5]

[6]

bk2 q bk 1 ¡ bk2

bk

q 3 1 ¡ bk2 5 2 1 ¡ bk

and the determinant of J˜ f,K+ is k

Tk2 T 1 1 (1 ¡ °k2 ) + k 2 2 (1 + °k ¡ 2°k bk2 ): 4 2 ¾j rj

(1 + ° ¤ )2 1 1 4 ¾j2 rj2

where ° ¤ = maxk °k .

[4]

where U is an orthonormal matrix, ¤ = diagf(Tk =2)(1 + °k ), (Tk =2)(1 ¡ °k )g, and ~nj = [¡ sin Áj , cos Áj ]T . Because ~nj is normal, · ¸ q ~nTj U = bk , § 1 ¡ bk2

detfJ˜ + fg=