Coalitions for Distributed Sensor Fusion 1

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1 Coalitions for Distributed Sensor Fusion 1 Abstract - We address the problem of efficient use of communication bandwidth in a network of distributed sensors. Each sensor node has enough computational power to fuse its own estimates with those of other nodes using an optimal filter, but message passing is expensive. The goal is to give each node an identical tactical picture of the area of interest without compromising target track accuracy. In the current leading approach, each sensing node sends an Associated Measurement Report (AMR: a sensor reading matched to a track ID) to every other node. This paper considers a method of decimation of reporting nodes by choosing a subset (coalition) of those nodes that have the best information on the target. Only coalition nodes share AMRs with each other; then the coalition sends a report to non-coalition nodes. Simulations and analytical studies are used to support the promise of the Coalition approach. Keywords: Target tracking, sensor data fusion, sensor management, distributed systems. 1 Introduction Coordinated distributed sensors have many advantages over isolated sensors, including greater area coverage, robustness, and cooperative sensing. This paper considers sensor nodes that have enough computational power to fuse their own estimates with those of the other nodes in the network using an optimal filter. Systems of concern include JTIDS, the Cooperative Engagement Capability (CEC), and the proposed Tactical Component Network (TCN). The goal is to give each node an identical tactical picture of the area of interest without compromising target track accuracy. The best estimate of target position is achieved if each sensing node sends an Associated Measurement Report (AMR: a sensor reading matched to a track ID) to every other node at each sensing period. The AMR approach gives an «optimal» tracking estimate, and is thus appropriate in cases where communications bandwidth is not a limited resource. However, in cases where inter-node communication bandwidth is a limited resource, the challenge is to reduce the required bandwidth without degrading tracking accuracy. Two key strategies, decimation and tracklets, Michael Howard, David Payton, Regina Estkowski HRL Laboratories, LLC Malibu, CA 9265 {howard, payton, regina}@hrl.com have recently been compared in [1]. In decimation, a subset of sensing nodes sends their AMRs to every other node. Each node makes a decision whether or not to transmit each AMR. A tracklet is an equivalent measurement report, an accumulation of several sensor measurements into a single message. One or more AMRs can be combined into a single tracklet that contains all the kinematic and state information of the AMRs. In the tracklet approach, every sensing node sends a tracklet to every other node, but at a multiple of the sensing rate. This paper considers a strategy that is a blend of the tracklet and decimation approaches described above. As in decimation, only a subset of nodes that sense the target actually send reports. We call this subset a coalition. The coalition periodically sends a tracklet to every noncoalition node. In the next section, we describe the coalition approach. Section 3 discusses the analysis methodology, and in Section 4 we present analytic and empirical results. 2 Distributed Tracking and the Coalition Approach It is beyond the scope of this paper to detail the mathematics behind each of the alternative approaches. [2][3] cover tracking networks and various types of filters. [1][2] have detailed descriptions of AMRs, tracklets, and decimation. The use of tracklets was defined in [2][4][5][6]. In this section we will provide a brief description of the Kalman filter and measurement updates, then discuss some aspects of the coalition approach. 2.1 Kalman Filter Formulation A Kalman filter can be represented in its information form, which has some nice properties for combining evidence. The following briefly shows the form of the information filter and the information each node has to work with. The target state vector is 3D position and velocity: x = [p x, p y, p z, v x, v y, v z ] T. F(k) is the state transition matrix, and there is a zero mean Gaussian noise w k so 1 This work was supported by funding from Raytheon Corporation and sponsored by Raytheon Joint Sensor Networking, nd Street North, St. Petersburg, FL,

2 x k = F k x k-1 + w k. Since most radars are not good at velocity measurement, the observation vector is ust position: z = [p x, p y, p z ] T, and z k = H k x k + v k, where z k is the observation at time k, H k the observation matrix, and v k is the observation noise. R k is the observation noise covariance. These equations apply equally for linear and non-linear forms, by use of different terms in the H k and F k matrices. The estimate of the target state is the expectation of x given all the past observations: x ˆ( k k) = Ε{ x( k) Z k }, where the covariance is called P(k k). Intuitively P -1, the inverse of the covariance of the state estimate xˆ, tells us how good the estimate is. The information state vector [7] at time step ( l) is given by 1 yˆ( l) P ( l) xˆ( l). P -1 ( l) is called the information matrix. We ll use this form of the filter because it clarifies the presentation, but in practice either form may be used. Each node i maintains a global filter by combining the contributions of each node : T 1 y ˆ k k = yˆ k k 1 + H k R ( k) z ( k). ( ) i( ) ( ) i The information matrix is given by 1 1 T 1 P k k = P k k 1 + H ( k) R ( k) H ( k) i ( ) i ( ). The decimation and coalition strategies are called reporting responsibility approaches [2]. In these, a metric is used to decide whether to send each sensor measurement out to other nodes. There may be several components to the metric; for example, one decimation strategy reported in [1] is to compare the trace of the covariance matrix P before and after adding the sensor measurement. If the decrease is more than some prespecified threshold, the measurement is sent ; otherwise it is dropped. We discuss some other ideas for a metric in the next section. Tracklet strategies (including the coalition strategy as described in the next section) require that each node maintain a global filter and a private local filter. Each node adds the measurement into the tracklet equivalent measurement report, which may be based on all or some portion of the local track. It then makes a decision on whether to send the tracklet based on whether the covariance of the global filter is greater by some threshold than the new local state s covariance. 2.2 The Coalition Approach We very briefly defined three approaches in the introduction : Associated Measurement Report, or AMR (our baseline, where every sensor broadcasts every measurement), tracklets (an aggregation of multiple sensor measurements, broadcast periodically), and decimation (where only a subset of the sensors broadcast AMRs). Tracklets and decimation are reasonable ways of reducing bandwidth requirements by reporting more intelligently. We now propose a new strategy. A Sensor Coalition is a group of sensor nodes that cooperate to track a single target. It is a form of decimation in that only a subset of sensing nodes actually sends a report. But rather than broadcasting their AMRs to all nodes as in decimation, coalition nodes only send AMRs to other coalition nodes and then the coalition sends an update tracklet, an aggregation of the coalition AMRs, to every non-coalition node. A different coalition would be formed to track each unique target. Since non-coalition nodes are updated only after coalition nodes update each other, the coalition approach does not achieve the goal of giving every node an equivalent tactical picture. In comparison with a decimation approach, coalition nodes have exactly the same error as decimation nodes. However, non-coalition nodes get information that is delayed by the time it takes to collect the coalition AMRs into a tracklet, and send the tracklet. In this approach, nodes that need the most accurate data (e.g. weapons systems) must be able to be at least silent members of the coalition. Nodes that want to monitor the tactical picture could receive the slightly older updates. The sensor coalition we propose can be a very transitory organization, whose lifetime is limited to the time a et or a missile moves across an area of interest. The membership is likely to be dynamic, because as a target moves, obscuration and reflectivity levels of sensors of different types in different locations will change. Multiagent systems research has considered protocols for generating and optimizing organizational structure, often called coalition formation [8]. However, protocols that require negotiation and consensus are not suitable for the multi-sensor tracking domain, due to the critical time constraints. In the next section we discuss some components of a coalition membership function. 2.3 Considerations for Coalition Membership At one extreme, any node that can sense the target is a member of the coalition tracking that target; this is the current AMR method. At the other extreme, only one node tracks the target and periodically updates the rest of the nodes. The goal is to strike a balance by finding a subset of nodes based primarily on the ability of a node to provide unique information on the target. The metric reported in [1] compares the trace of the position covariance matrix before and after adding the sensor measurement to the global filter. 637

3 The trace metric would allow a non-member node to compare its ability to reduce position uncertainty beyond a threshold. If it passes the metric, it oins the coalition of its own accord by starting to send the coalition its AMRs. The details of oining are dependent on the network; for example, if something like multicast is available, it can simply oin the multicast address for the coalition, recorded in the last tracklet message from the coalition. The trace metric is based entirely on current data as represented by the current state of the global filter and the latest local measurement. The rest of this section discusses three other terms that might be added to such a metric to possibly improve the selection of coalition members. The value of these terms, however, remains to be investigated in future work. The metric would become a weighted sum of each term, and the weights would be decided after some experimentation on a particular network. First, if a node s contribution is increasing at every time step, it should be preferred for membership over another node with equal information but a decreasing contribution. A node could calculate the derivative of the trace metric mentioned abover, over several reporting periods. Second, since target reflectivity is often different at different angles, geometric location of sensors with respect to the present and predicted position of the target could be added. Intuitively, we would like to choose a small set of sensors with significantly different viewing angles to the target. One way to do this is with a Delauny triangulation, where the vertices of the triangulation include the target and all sensors within a given distance of the target. The sensors that are adacent to the target in the triangulation would then be such a set of sensors. To calculate this, a node must know the locations of every other node, which is a reasonable expectation for allied tracking systems. Delauny triangulations can be quickly calculated and locally repaired. Figure 1a shows how the Delauny neighborhood of a target T changes as T moves in the direction of node S. Within the next instant, the topology of the triangulation will change and S will lie in the Delauny neighborhood of T. If T continues in the predicted direction, the Delauny neighborhood will consist of the sensors on the dotted line. Even if T changes course, in terms of geometry, S is still a reasonable sensor to include in the coalition. The frequency of topology changes due to target motion can be reduced by selecting sensors that are far from the target. Figure 1b indicates how the Delauny neighborhood can be extended by eliminating immediate neighbors until desired distances are obtained. A third consideration for a coalition metric requires knowledge of the capabilities of current coalition a S members. For example, phased array sensor and rotators have different ranges and reporting frequencies. Even among the same type of sensor, they typically run banks of filters optimized for different hypotheses of target movement [2]; one may detect a maneuver before another or lose a track for short periods of time. Therefore, we would like to select a set of coalition members with a variety of capabilities. This decision requires a knowledge of the current coalition members and their capabilities. Non-coalition members could be informed of the coalition membership in a short coded message sent only when membership changed, so this might not add appreciably to bandwidth. Of course, coalition members know the membership because they receive AMRs from each other coalition member directly. 3 Analysis Methodology T Figure 1. The Delauny neighborhood of T includes nodes on the heavy solid line. Direct neighbors of T can be removed leaving the extended neighborhood, on the dashed line. 3.1 Bandwidth Analysis The bandwidth requirements of each approach (AMR, decimation, tracklets, and coalition) were calculated analytically. To simplify the analysis and clarify the issues, we make the worst-case assumption that all nodes can see all tracks at all times. In reality, only a subset of nodes would be able to see a target at any one time due to the earth s curvature alone, if not occluded by local terrain features. Also, this analysis assumes a perfect network with bandwidth on demand. It does not account for the real world constraints of deployed military systems, which may incur overhead for transmission security. So AMR and Tracklet bandwidths in this analysis are an upper bound, and bandwidth savings of decimation and coalition over AMR are also an upper bound. We use message size estimates in Table 1, ustified in [1]. A CCS target is Constant Course and Speed which is simulated using a stochastic differential equation with noise of 18 m/sec 3/2, so it s not perfectly constant. Using these estimates, if n nodes send AMRs to each other at a b T 638

4 rate r (reports/minute) over a total time period T, the bandwidth required for the baseline AMR approach is: bw amr = 224 n (n-1) r T (1) In an AMR decimation approach, only a subset c of the n nodes will transmit an AMR to each of the other n-1 nodes, based on a decision made by each node at each reporting time. The metric for sending should select nodes that have the highest quality sensor reading to report. It will provide less information than the full AMR approach, and tracking error will be somewhat higher, but if the decimation is done carefully the loss of precision will be minimized. The bandwidth used by the decimation approach would be: bw decimation = 224 c (n-1) r T (2) Note that this bandwidth calculation, and also the coalition bandwidth below, are idealized. If the decision to report or not (in effect, oining the c group) is based on metrics whose value may change from moment-to-moment, then c could vary. But for the purposes of comparison, we are holding c constant. In a tracklet approach, each of the n nodes periodically (every t time steps) sends a tracklet to each of the other nodes. Fewer words are required for a tracklet on a maneuvering target because the velocity terms for a maneuvering target are of little value in estimating the state. Assume for simplicity that tracklets for maneuvering targets are being sent at a higher rate than for the CCS targets, yielding the same bandwidth. Then the bandwidth used by the tracklet approach would be: bw tracklet = 56 n (n-1) r T/ t (3) The coalition approach is a combination of the decimation and tracklet approach. Like decimation, only a subset c (the coalition) of nodes decides to send an AMR to each of the other (c-1) coalition nodes. One of the coalition nodes periodically (every t timesteps) sends a tracklet to each of the n c non-coalition nodes. Assume for simplicity that tracklets for maneuvering targets are being sent at twice the rate of the CCS targets, yielding the same bandwidth. Then over a total time period T, if each sensor is making measurements at this rate r (reports/minute), the bandwidth used by the coalition approach is: bw coalition = 224 (# AMRs within coalition) + 56 (# tracklets to each non-coalition node) Required Bandwidth 2.5 x Bandwidth Comparison O(n 2 ) comm AMR Decimation Tracklet Coalition AMR bandwidth Tracklet Bandwidth Reporting Nodes (out of 1 ) Figure 2. Bandwidth comparison for sample situation in which all 1 nodes can hold the target for 2 iterations. Note that AMR and Tracklet are only defined for all reporting nodes, bw coalition = 224 c (c-1) r T + 56 (n-c) r T/ t (4) This represents one AMR at each timestep sent from every coalition node to every other, plus a tracklet from the coalition to every non-coalition node at each tracklet time t. 3.2 Tracking Error Simulation We analyze error trends empirically by means of a Cartesian Kalman filter simulation. A target moves at constant course and speed across an area of interest. For simplicity, all nodes can track the target throughout the 2 time steps. Both measurement and process noise were modeled as mean white Gaussian noise. Process noise was unit variance for all sensors. A number of factors can reduce a sensor s accuracy for a period of time, such as weather and terrain obscuration, aspect of target to sensor, calibration changes due to heating, etc. We modeled that by adding extra noise to a subset of the nodes. Other simplifications were made in this analysis that may affect the details of the results. Reporting nodes all have the same reporting rate. The simulation does not account for processing and communication time lags. Tracklets were sent at fixed time intervals; in practice they would only be sent when the error of the coalition s local filter falls some threshold below that of the global filter. Table 1: Bandwidth of Sensor Update Messages Type of Update Type of target 32 bit words total bits / message AMR All types 7ea x 32 bits 224 bits Tracklet CCS target 17.5ea x 32 bits 56 bits Maneuvering target 1.5 x 32 bits 336 bits 639

5 4 Results 4.1 Bandwidth Savings and Error Characterization of Coalition Approach The total required bandwidth of each of the four approaches (per equations 1-4 above) are plotted in Figure 2 as a function of the number of reporting nodes. Required coalition bandwidth is much less than the baseline AMR approach, but it is also less than the decimation approach because it is a combination of the decimation and tracklet approach. The tracking errors versus bandwidth for optimal tracklet policies and optimal AMR policies (i.e., decimation) were compared in [1]. We will start by comparing the coalition approach to the baseline AMR approach, and then briefly discuss the comparison for decimation and tracklets. A different way to look at the bandwidth savings of the coalition approach over the AMR approach is to plot the relative savings of the coalition vs. AMR approach (bw amr bw coalition ) as a percent of bw amr versus coalition size. Figure 3 shows that plot. The bandwidth savings over any time period T that is a multiple of the tracklet interval t, is related to the size of the coalition c 2. Figure 3 shows the set of curves varying coalition size c from 1 to the maximum number of sensing nodes 1, where tracklets are sent from the coalition to each of the n-c non-coalition nodes, every 1-7 time periods. The graph is parabolic, with maximum bandwidth savings for small coalition sizes. A coalition size of zero is meaningless for the coalition approach, but a coalition of 1 means only one node tracks the target and periodically sends a tracklet to the rest of the n-1 nodes (56 (n-1) r T / t). This requires about 95% less bandwidth than the other extreme, where every node is in the coalition equivalent to the AMR approach. Perhaps the most notable feature is that the curve approaches zero slope for small size coalitions, and only gradually decreases. Thus even if 3% of the nodes are in the coalition, the bandwidth savings can still be 9% of the savings for a coalition of 1. Furthermore, this curve changes very little as the number of measurements in a tracklet changes. With a larger numbers of nodes, the curves in Figure 3 are very close together. With less than 1 nodes, the tracklet curves start to separate, but even with only 1 node in the coalition, sending tracklets of size 1, it still has 75% of the potential savings over the AMR scenario. In other words, we can increase the frequency at which we send tracklets from the coalition to non-coalition nodes, without losing much of our bandwidth savings over the AMR approach. This is because we only send O(n-c) tracklets, whereas AMRs are broadcast from every node to every other, O(n 2 ). Thus bandwidth savings is dominated by the size of the % of maximum BW savings Position Error (meters) Non-Coalition Error With Updates Without Update Iteration Figure 3. Tracking error of non-coalition nodes, with and without coalition tracklet updates every 7 periods. Bandwidth Savings of Coalition over AMR O(n 2 ) comm 1 C(*,1) 9 C(*,3) C(*,5) 8 C(*,7) Coalition Size (out of 1 reporting nodes) Figure 4. (AMR bandwidth minus Coalition bandwidth) divided by AMR bw. C(*,m) means a plot for a coalition of varying sizes, using a tracklet of m measurements. Bandwidth Savings of Coalition over Decimation O(n log n) co % of maximum BW savings Coalition Size (out of 1 reporting nodes) C(*,1) C(*,3) C(*,5) C(*,7) Figure 5. C(*,m) means a plot for a coalition of varying sizes, using a tracklet of m measurements to update noncoaliton nodes. 64

6 coalition, and the frequency of updates to the n-c noncoalition nodes is less important. Therefore, we can even send a coalition tracklet every time step and keep the noncoalition error down, without significantly increasing bandwidth requirements. In the coalition approach, coalition nodes send AMRs to each other, and one coalition node sends a tracklet to each non-coalition node every 7 th time step (immediately after the 7 th AMR update with no time lag). Seven time steps is a long time to go without an update, but it was chosen to make the error trend clear. Figure 4 shows the XY position error of the target as seen by a non-coalition node that is updated by periodic coalition tracklets, and one that is not updated, averaged over five runs. In both cases the non-coalition nodes make their own noisy measurements every timestep, and add them into their Kalman filters. The error of the non-coalition nodes exhibits the characteristic sawtooth error pattern: immediately after updating with the more accurate coalition tracklet, the error of the non-coalition node will be close to that of the coalition node. In between updates, the error increases. In Figure 5, coalition error is plotted against non-coalition error, showing that when a tracklet arrives, every seventh iteration, non-coalition error is reset to match that of the coalition nodes. The coalition system should be carefully compared with the closest existing system : decimation. In a decimation system, the best nodes are chosen to send their AMRs to all the other nodes. The bandwidth follows equation 2 above. The metric for choosing a node to be in the decimation group is likely to be very similar to the metric used to oin a coalition. The difference is that in the coalition system, coalition members only send AMRs to other coalition members, and then the coalition sends a tracklet to the non-coalition nodes. Figure 6 shows the coalition savings with respect to the decimation approach. Error (meters) Coalition vs. Non-Coalition Error Coalition Non-Coalition Iteration Figure 6. This plot shows non-coalition nodes getting updated every 7 nodes. In general, their error is greater than coalition nodes. The appearance of maxima gives a clear goal for coalition size. In this comparison the number of measurements in the tracklet makes a bigger difference, particularly in the smaller coalition sizes. We have not compared the error of the two systems for this paper, due to its dependence on the time each update is received, which is not in our current simulation. If there is a node that needs highly accurate tracking data, but has no sensor of its own (perhaps a weapons system), it can be added to the coalition as a silent partner. It receives AMRs from each member but contributes no sensor measurements of its own. This loosens the traditional ties binding a specific sensor to a specific weapon system, and makes the estimate more robust. When one silent node (snode) is added, if we still use c to represent the number of reporting coalition nodes, the inter-coalition AMR message is now sent from c nodes to c+1 other nodes, so the coalition part of the bandwidth becomes 224 c ((c+1)-1)rt, or 224c 2 rt. The non-coalition update now has 1 less node to update, so it becomes 56 (n-(c+1))rt/t. Therefore the bandwidth bw coalition (4) will change to : bw coalition +snode = coalition part + non-coalition part = 224 c 2 r T + 56 (n-(c+1)) r T/ t (5) The difference between equation (5) and equation (4) is 224crT + 56rT/t. We conclude that high priority systems can be added to the coalition as silent partners without significantly increasing the bandwidth (only O(c) increase). In the coalition approach, non-coalition nodes should experience about the same error as in a pure tracklet approach. However, coalition nodes would experience about the same error as the AMR approach. Intelligent routing systems will do better than O(n 2 ) Bandwidth Savings of Coalition over AMR O(n log n) comm 1 C(*,1) C(*,3) C(*,5) 9 C(*,7) % of maximum BW savings Coalition Size (out of 1 reporting nodes) Figure 7. The savings are plotted for varying tracklet sizes of 1 to 7 measurements each. This plot is similar to Figure 3, except that communication is O(n log n) here rather than O(n 2 ). 641

7 performance, and can approach O(n log n). In such a network the dominating portion of coalition bandwidth, which is a broadcast from every coalition member to every other, will be less expensive. Figure 7 shows that there is still a bandwidth savings, but it drops off more linearly, so there is nearly an equal penalty to pay for each node that is added to the coalition. However, as mentioned in section 3.1, there is also an overhead to be paid for transmission security, and it would surely offset the gains made by intelligent routing. In each of the bandwidth analyses in this paper, we used tracklet sizes from 1 to 7. Tracklets of size 1 are similar to an AMR. The coalition AMRs must be collected and fused, and the resulting estimate sent to the non-coalition nodes. 5 Conclusions The preceding analysis indicates that the coalition approach is promising. The bandwidth trends have been defined analytically. The study compared coalitions of different sizes, from a coalition of 1 that uses the least bandwidth, to a coalition that contains all nodes, which is the same as the AMR system. We showed that, with O(n 2 ) communication, even if the coalition includes 3% of the sensing nodes, we still save 9% of the maximum savings for a coalition of 1 (8% for O(n log n) communication). Error trends were indicated via a very simple simulation, and more work should be done to try to find a bound on the error analytically, and to improve the simulation to get more convincing experimental results. This study has not yet produced a graph of error versus bandwidth for the coalition strategy, which is required for a thorough treatment. This would be the subect of future work, since there are issues of network design and update timing that complicate such an analysis. Instead, since the coalition scheme is a combination of decimation and tracklet strategies, we suggest that the error of coalition nodes would be nearly that of decimation schemes. The error of non-coalition nodes would be comparable to tracklet systems. Fire Control Symposium, Kauai, Hawaii, pp August 21. [2] Y. Bar-Shalom and W.D. Blair, eds., Multitarget- Multisensor Tracking, Norwood, MA, Artech House, 2. [3] S. S. Blackman and R. Popoli, Design and Analysis of Modern Tracking Systems, Norwood, MA, Artech House, [4] Oliver E. Drummond, Feedback in Track Fusion Without Process Noise, Signal and Data Processing of Small Targets 1995, Proc. SPIE, Vol. 2561, p July [5] Oliver E. Drummond, A Hybrid Sensor Fusion Algorithm Architecture and Tracklets, Signal and Data Processing of Small Targets 1997, Proc. SPIE, Vol. 3163, p. 485, July [6] Oliver E. Drummond, Tracklets and a Hybrid Fusion With Process Noise, Signal and Data Processing of Small Targets 1997, Proc. SPIE, Vol. 3163, p. 512, July [7] J. Manyika and H. Durrant-Whyte, Data Fusion and Sensor Management: A Decentralized Information- Theoretic Approach. Ellis Horwood Series in Electrical and Electronic Engineering, Ellis Horwood, New York, [8] Gerhard Weiss, ed., Multiagent Systems, a Modern Approach to Distributed Artificial Intelligence, MIT Press, Acknowledgements Raytheon Corporation supported this research, and we are indebted in particular to Bill Barker, Tom Nichols, Dennis King and Chris Eck for their deep understanding of this domain and their excellent comments and suggestions. References [1] W. H. Barker, T. S. Nichols, C. R. Eck, S. Kadambe, A Tradeoff Analysis of Bandwidth Utilization Strategies in Distributed Sensor Networks. In proceedings of National 642