A ROBUST SCHEME TO TRACK MOVING TARGETS IN SENSOR NETS USING AMORPHOUS CLUSTERING AND KALMAN FILTERING

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1 A ROBUST SCHEME TO TRACK MOVING TARGETS IN SENSOR NETS USING AMORPHOUS CLUSTERING AND KALMAN FILTERING Gaurang Mokashi, Hong Huang, Bharath Kuppireddy, and Subin Varghese Klipsch School of Electrical and Computer Engineering New Mexico State University, USA ABSTRACT This paper presents a Kalman-filter-based, distributed processing scheme to track a moving target in a sensor net. There are two major new features about the proposed scheme. First, distributed processing clusters are formed amorphously. That is: no explicit control message exchanges are needed to form a cluster and elect a leader; rather, data packets are routed to a place where leaders are spontaneously generated if they pass certain threshold condition. This leads to great savings in cost associated with cluster and leader management. Second, the proposed scheme loosens the strict requirement in traditional tracking schemes that there exists exactly one cluster leader at a particular step; rather, multiple leaders can exist at the same step. This leads to more robust operation where leaders can fail due to either device or communications links reasons. Further, the creation of multiple leaders is leveraged on the one and same packet flow and no extra cost is incurred. The proposed scheme is simulated in an example tracking application and the effects on error performance of various parameters are studied. I. INTRODUCTION Sensor nets have emerged as a promising technology that has important applications in environment study, homeland security, digital battlefields, etc [][2]. To fully realize the potential of sensor nets, however, significant challenges have to be addressed to deal with the harsh environment sensor nets typically present. Specifically, sensors are tiny, unreliable, resourceconstrained, particularly energy-constrained devices, and communications links between sensors are volatile and unreliable. This work was partially supported through the Sandia National Laboratories SURP Program, grant number Any opinions, findings, and conclusions expressed in this material are those of the author(s) and do not necessarily reflect the views of the Sandia National Laboratories. It is commonly believed that in-network processing is preferable to centralized processing which requires sending all the data to the base station and consumes a large amount of energy. However, to perform robust, low overhead in-network processing is a challenging endeavor in a sensor net which is resource-constrained and where device and communications failures are frequent and common. This paper is intended to make some progress in this direction and presents a low overhead, robust, in-network processing scheme that tracks a moving target in a sensor net. Target tracking is an important application for sensor nets [3]. A classical method for the task is Kalman filtering [4]. One possible implementation is as follows. The system state consists of the position and velocity of the target. At each step, sensors close to the target form clusters and elect a leader to perform the Kalman filtering, and the updated state is forwarded to the cluster leader of the next step. While implementing Kalman filtering in a centralized setting is straight forward, doing so in a distributed manner in a sensor net presents some new difficulties. To form a cluster, determine the membership of a particular sensor, elect a cluster leader, and for the current leader to determine the next leader and to hand off the state, all incur significant communications cost and the correctness of operation is not entirely assured in the harsh environment of a sensor net. This paper proposes a robust scheme for tracking moving targets in sensor nets. The proposed scheme is based on the Kalman filter, but is implemented in such a way that requires little organization, minimal overhead, and is robust against node failure and message loss. There are two major new features about the proposed scheme. First, clusters are formed amorphously. That is: no explicit control message exchanges are needed to form a cluster and elect a leader; rather, data packets are routed to a place where leaders are spontaneously generated if they pass certain threshold condition. This leads to great savings in cost associated with cluster and leader management. Second, the strict requirement that there exists exactly one leader at a particular step is loosened; rather, multiple leaders can exist at the same step, and even no leader at a particular step is allowed of 5

2 (in that case, the state update is forwarded to the step after that). This leads to more robust operation where leaders can fail due to either device or communications links reasons. It is important to note here that the creation of multiple leaders is leveraged on the one and same packet flow and no extra cost is incurred. In the following, we provide a general description of the proposed scheme in Section II, discuss parameter settings in Section III, present simulation results in Section IV, and conclude in Section V. II. DESCRIPTION OF THE PROPOSED SCHEME The proposed scheme is based on Kalman filter, and the state of the dynamical system is the position and velocity of the target. At each step, sensors form amorphous clusters to aggregate sensor measurements to the cluster leaders, which update state estimations and hand off to the next potential cluster leaders. At any particular time, there may be multiple clusters working in parallel to estimate the state, so that the scheme is robust against message loss and device failure. The tracking results are periodically transmitted to the base station. We assume a two-dimensional space, though it not difficult to extend our results to higher dimension. The target is assumed to emit acoustic signal, which decays according to an inversely squared law, and the sensors measure the strength of this signal in noise. The sensors are assumed to know their positions using any of localization methods currently available. Sensors exchange their position information periodically among one-hop neighbors. Below, we provide more details about two crucial components in the proposed scheme: amorphous clustering and Kalman filtering. A. Amorphous Clustering Amorphous clusters are formed by using signallevel-based routing (SLR). A sensor transmits a message, which consists of the signal level and the sensor location, only if the signal level is above a threshold s th. A message is sent to the one-hop neighbor that has the highest signal level. A sensor relays a message only if there is a neighbor which has a higher signal level, otherwise it only receives message and consider itself a potential candidate for a cluster leader. A potential candidate decides itself to be a cluster leader if it received at least m th distinct messages by the end of the time slot. In cases of multiple local maximum, we will have multiple cluster leaders. Message transmissions occur randomly during the first α fraction of the slot T to avoid synchronization and to allow all messages to be delivered before the end of the slot. In essence, SLR marshals sensory data to cluster leaders, which have local maxima in signal strength, but does that without explicitly forming clusters and incurring the associated cost. To be selected as a cluster leader, a sensor needs to satisfy two conditions: a) it is around a local signal maximum; b) it has received sensory data from at least m th other sensors. The combination of these two conditions acts as a deterrent against false selection of a cluster leader without enough information to make an estimate. The above method allows cluster leaders to be created spontaneously simply by marshaling the packet flow towards local signal maxima, and requires virtually zero cluster management and maintenance overhead. This greatly reduces communications cost. The method also allows multiple cluster leaders to be created at one particular step, but that is leveraged on the same packet flow and incurs no extra cost. Allowing multiple cluster leaders is more robust as compared with the traditional approach that insists on exactly one cluster leader at a time, because a cluster leader failure, due to malfunction of device or break of communications links, etc., is no longer that much detrimental. To deal with the case that no cluster leader is formed in a particular step due to low signal strength, low density of sensors or device failures, we make the packet carrying the estimation state persistent for certain number of hops. That is: if the packet could not find a leader at a particular step, it continues to the next predicated target position using geographical routing, and so continues until a leader is found or the hop limit is reached. B. Kalman filtering Kalman filtering occurs in cluster leaders, which update the prior state estimates and error covariance, compute Kalman gains, and compute posterior state estimates and error covariance. A cluster leader located at x k then sends a message, which include the posterior state estimate and error covariance, addressed to a location x k+ +vt, using geographical routing [5] with a little variation. The variation is that the message is delivered to a sensor in the neighborhood of x k+, and after which it continues to use SLR to reach the next cluster leader. More specifically, the state of the systems is defined as the two dimensional coordinates and velocity of the target: x k = [x, y, vx, vy]. And, we assume a constant velocity motion. The Kalman filter consists of two groups of equations, the time update equations for calculating the priori state and the measurement equations for incorporating the measurements into the 2 of 5

3 priori estimate to get the posteriori estimate. The two groups of equations are as follow: Time update equations, Measurement update equations, xˆk - = Axˆ k + Bu k () P k - = AP k A T + Q (2) K k = P k - H T ( HP k - H T + R ) (3) xˆk = xˆk- + K k ( z k -Hxˆk- ) (4) P k = (I- K k H )P k - (5) In the above, xˆk is the priori state estimate at step k, A is the state update matrix: A = T T There is no forcing, so B and u are zero. P k - is the priori estimate of error covariance at step k. Q is the process noise and is assumed to be zero. K k is Kalman filter gain at step k. H is the observation matrix: H = Lastly, R is the measurement noise covariance. Initialization of states and noise covariance is around somewhere not far away from the true values to expedite convergence. Least mean square error estimation is performed on the received sensory data at the cluster leader to produce the measurement (target coordinates z k ) at current step, which, together with the prior estimate coming from cluster leads of the previous step, is feed into Kalman Filter to produce a posterior estimate and to predict the position of the target at the next step. This new position estimate and estimated error covariance are then sent to the predicted new position using geographical routing. The cycle is repeated in next step, and so forth. III. PARAMETERS DETERMINATION IN THE PROPOSED SCHEME The proposed method uses the notion of a step. At the start of each step, all the sensors wake up and measure the acoustic signal strength in their neighborhood. The time period of a step is T, which is roughly the time taken by the target to travel from one cluster leader to another. Hence T is dependent on two parameters, the target speed v and the distance between two cluster leaders. The distance between two cluster leaders is assumed to be some multiple of the communication radius or in other words multiple hop distance. To deal with the situation that the target exhibits litter motion and to feed information to the base station at a minimal frequency, T is set to be no larger that a certain value. The value of T generally adapts to the target speed, but is constant if the target exhibits a uniform speed motion, which is the case, considered in the simulation that follows. The measured signal strength is compared with the signal threshold s th to decide whether the current sensor should transmit its sensory data and participate in further processing or go back to sleep until the start of the next step. This keeps only the sensors that received strong enough signals awake and put sensors that received ambiguous signals to sleep, thus saving energy and extending the network lifetime. A lower value of s th activates more number of sensors and vice versa. Hence the value of the s th is an important parameter as it decides the tradeoff between conserving energy and enhancing sensitivity and robustness of detection. The effect of s th on the estimation performance is studied in the simulation. A sensor can consider it self as a cluster leader if all the signal strength values accumulated in its buffer are less its own signal strength value and the number of these data values in the buffer is greater the value m th. The parameter m th acts as deterrent against formation of a false cluster leader due to noise, etc. The value of m th should be large enough to prevent false triggering but it should also be small enough to ensure that a cluster leader is formed in each step. Even by taking this precaution and choosing the m th properly, there is a possibility that the cluster leader could not be formed as the number of data values in the buffer did not exceed the m th. This problem is dealt with by making the packet carrying the estimation state persistent for certain number of hops. That is: if the packet could not find a leader at a particular step, it continues to the next predicated target position using geographical routing, and so continues until a leader is found or the hop limit is reached. There is also the possibility of forming false cluster leaders whose estimation is not accurate enough. This is deal with by requiring a cluster leader assimilate an estimate in Kalman filtering from a previous step only if it is not far from its own measurement. If it is, the previous estimate is forwarded to the next step until either it is close to the measurement of some cluster 3 of 5

4 leader and gets assimilated in Kalman filtering or a certain hop limit has been reached. In this way, outliners will die quietly. The effect of m th on estimation performance is investigated in the simulation. IV. SIMULATION EXPERIMENTS We simulated a tracking application using our method. The setup involved sensors spread around an area of square meter field. The sensors were perturbed from the grid positions with a variance of. The target is assumed to move within the sensor area with constant velocity. Various parameters such as s th, m th, measurement noise variance and the number of cluster leaders are evaluated to study their effect on the estimation error. Figure shows the effect of signal threshold s th and the measurement noise variance on the average mean square error (MSE) in the position estimation of the target. The signal power is set to 2 units, signal threshold 2 or 4 units, m th value (the minimal number of messages received to qualify as a cluster leader) is equal to 6. It is no surprise that increase in measure noise variance increases MSE. However, s th also has some modest effect on MSE. Generally, lower s th leads to smaller MSE, since more sensors are involved in the estimation processes. But there is anomaly at the noise variance value of 4, whose explanation is not entirely clear to us and the guess is it originates from statistical fluctuation. the average MSE. The smaller the value of m th ; the more likely multiple cluster leaders are formed, and the less likely that no cluster leader is formed at particular step. Thus, smaller values of m th generally lead to smaller MSE as seen in Figure 2. Figure 3 shows the relationship between m th and the number of cluster leaders formed. Obviously, more cluster leaders are formed with a lower m th threshold. But this has to be cautioned against the risk of false triggering. In the setting of our simulation, a m th value of 4 seems to be an optimal one, which almost always guarantees at least one cluster leader at each step without much false triggering. Average MSE Average MSE vs Measurement Noise Variance Measuremnt Noise Variance mth = 4 mth = 6 mth = 8 Figure 2: The effect of m th and the measurement noise variance on the average mean square error (MSE) in the position estimation of the target. Average MSE Average MSE vs Measurement Noise Variance Measurement Noise Variance sth = 2 sth = 4 sth = 6 Figure : the effect of signal threshold s th and the measurement noise variance on the average mean square error (MSE) in the position estimation of the target Average number of cluster heads per cycle Average No. of Cluster heads vs Mth mth Avg. # cluster heads Figure 3: The effect of m th on the average number of cluster leaders formed per step. V. CONCLUSION The m th is a very important parameter as it controls the size of a cluster. Figure 2 shows the effect of m th on We have presented a Kalman-filter-based, distributed processing scheme to track a moving target in a sensor net. The proposed scheme incorporates two major new features: amorphous clustering for reducing overhead 4 of 5

5 and allowing multiple leaders for robustness. The proposed scheme is simulated in an example tracking application and the effects on error performance of various parameters are studied. REFERENCE [] D. Estrin, R. Govindan, J. Heidemann and S. Kumar, Next Century Challenges: Scalable Coordination in Sensor Networks, in Proc. of Mobicom, 999. [2] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci., A Survey on Sensor Networks., IEEE Communication Magazine, August 22 [3] Dan Li, Kerry Wong, Yu Hen Hu, and Akbar Sayeed. Detection, Classification and Tracking of Targets in Distributed Sensor Networks, IEEE Signal Processing Magazine, March 22. [4] Peter S. Maybeck, Stochastic Models, Estimation, and Control, Volume, 979 Academic Press, Inc. [5] B. Karp and H. T. Kung, GPSR: Greedy Perimeter Stateless Routing for Wireless Networks, in Proc. of Mobicom, 2. 5 of 5