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University of California, Santa Barbara http://stnlabs.ece.ucsb.edu Efficient Distributed Algorithms for Data Fusion and Node Localization in Mobile Ad-hoc Networks Andrew P. Brown, Ronald A. Iltis, and Ryan Kastner This work was supported in part by NSF grant No. CNS-0411321

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Overview Data fusion Node localization Linear Gaussian state space model and Bayesian estimation Resource-efficient distributed estimation/data fusion Extension to non-linear models Localization in mobile ad-hoc networks Directions for future research and conclusions

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Ad-hoc/sensor networks may estimate processes within the network –node locations, route feasibility or in the surrounding environment –object motion, quantity gradients In centralized estimation, data is relayed to a central sink –relay node energy depletion, data congestion Distributed data processing enables power conservation and network scalability –Packet transmission is the most power-expensive operation –First and foremost, the algorithms maximize communication efficiency –Computation and storage resource efficiency are maintained –The algorithms are scalable to large and huge networks Data Fusion: Motivation Data collection Comms./ ranging

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Each node gathers data –e.g., RF, acoustic, EO/IR, temp. Extracted information is used to update local estimates Information is compressed— without loss—into sufficient statistics packets (SSPs), which are forwarded, multi-hop to other nodes –frequently, to nearby nodes –Infrequently, to more distant nodes (or to a sink) Nodes receiving SSPs fuse the information (update local estimates) Data fusion with communication delays is addressed Estimation of time-varying processes is handled naturally The algorithms are resource- efficient and scalable. Data Fusion: Approach Data collection Comms./ ranging

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Data Fusion/Distributed Estimation: Survey of Past Work Research in data/estimate/track fusion dates back at least to the 1970s [Bar- Shalom & Tse, 1975] Many early approaches assumed errors were uncorrelated across quantities to be fused can lead to inaccurate estimation and even instability [Widnall & Gobbini, 1983] C. Y. Chong, E. Tse, and S. Mori [1983 and many later papers] have shown how to optimally account for correlations due to common information. Application for time-varying states is very challenging Multiple existing approaches for optimal fusion with time-invariant states have been unified [e.g., X. R. Li, 2003] For time-varying states, the decentralized information filter has provided a useful framework for many applications [e.g., Mutambara & Durrant-Whyte, 2000] In this paper, we analyze and provide a solution to the problem of optimal estimate fusion for time-varying states. We also address the problem of fusion of delayed information (due to finite communication and processing delays), which poses the current greatest research challenge for high-accuracy, real-time distributed estimation.

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We present node localization as an example of distributed data fusion Node position information is valuable for internal network use –efficient routing, position dependent services, network security, E911 and for providing data context in sensor network applications –environment monitoring, object tracking, etc. GPS is not always an option due to node design constraints –cost, power, form factor and reliabillity –jamming, shadowing, multipath Node mobility poses a challenging problem, which we effectively address Our distributed approach provides real-time location awareness Node Localization: Motivation Comms./ ranging

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Each node measures ranges to other nearby nodes using round-trip travel time (RTT) measurements –relatively simple and affordable Dynamic node states (position and velocity coordinates) are modeled in state space –a priori knowledge of environment/ terrain not required –uncertainties modeled statistically –kinematics used to predict node movements The EKF is used to process the nonlinear range measurements and track the node positions Cross-correlations between node estimate errors are accounted for Information is shared, as needed –frequently with nearby-nodes, less frequently with more distant nodes Node Localization: Approach Comms./ RTT ranging

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Node Localization: Survey of Past Work A variety of measurements can be used for localization: –Received signal strength indicator (RSSI): inexpensive, but requires environment- specific calibration –Connectivity: inexpensive, but high node density is required for high accuracy –Angle of arrival (AOA)/bearing: fewer measurements required for localization, but more costly and vulnerable to scattering near antennas –Range/time-of-flight measurements: can be based on round-trip travel time (RTT) or time difference of arrival (TDOA), so no sensor or RF front end modifications are required additional signal processing may be required for multipath mitigation: actually a problem for all measurement types, but most easily mitigated for range/time-of-flight measurements A wide variety of position estimation algorithms have been proposed. For tracking mobile nodes, Kalman filter-based methods seem most advantageous. Savvides, Srivastava, et. al., 2001/2 have proposed geometric combined with Kalman filter-based algorithms. See further:Kim, Brown, Pals, Iltis, Lee, JSAC, May 2005. J. J. Caffery, Jr., Wireless location… Kluwer, 2000.

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The variation of the process (e.g., node or tracked object position, quantity gradient) is modeled as linear kinematic, subject to white Gaussian random perturbations: (the interval t n – t n – 1 is arbitrary) or, for m < n, Likewise, the measurement error is modeled as additive white Gaussian: The extension to non-linear models, as required for localization, will be discussed. Note that for time-varying states, network-wide clock synchronization is required can be estimated, along with the states [e.g., Widnall & Gobbini, 1983] Linear Gaussian State Space Model

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Bayesian Estimation denotes the cumulative measurement set, i.e., the set of all measurements recorded at node i, along with the set of all measurements for which sufficient statistics are received via communication with other nodes, up to and including time m. denotes the a posteriori probability distribution on x(n), given the cumulative information available at node i at time m. In the linear Gaussian case, with mean and covariance The a posteriori distribution depends on the data only through the mean and covariance; thus, the mean and covariance constitute sufficient statistics for the distribution. The mean and covariance can be efficiently computed used the well-known Kalman filter. The complexity in the mobile node localization application is (due to estimate prediction).

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Bayesian Information Fusion From C. Y. Chong, E. Tse, and S. Mori [1983], holds if but this is not the case, in general, for time-varying states. The independence assumption does hold for but it is computationally intractable to jointly estimate the states at all measurement times, since the complexity grows with n 3.

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Efficient Bayesian Information Fusion There is an important case in which the fusion of Gaussian’s formula can be used—when one measurement set is the current measurement vector: which can be computed as or, if the information form of the Kalman filter is used, using only add/subtract operations... in either case, the overall algorithm complexity is Node i obtaining measurement at time n computes the sufficient statistics and transmits them to other nearby nodes, in the form of a sufficient statistics packet (SSP), stamped with the asynchronous measurement time

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Efficient Bayesian Information Fusion Node j receiving the SSP fuses it with its most recently-computed sufficient statistics for, where

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Optimal Delayed Information Fusion Due to finite communication and processing delays, the case n < m is common in practice; however, optimal information fusion is much more difficult:

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Sub-Optimal Delayed Information Fusion A computation and storage-efficient fusion algorithm is obtained using the approximation which holds exactly if the states are time-invariant or if the delay is 0. The development of more efficient optimal and sub-optimal algorithms for delayed information fusion is an open research problem. Many useful results have been obtained in the closely-related field of out-of-sequence-measurement (OOSM) fusion.

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Improved Communication Efficiency Locally aggregating information over a block of N b measurements, before transmitting a compact representation to other nodes, provides a parameterizable tradeoff of improved communication efficiency for increased latency in information propagation. SSP block formation

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Extension to Nonlinear State Estimation To meet the low-power, low-complexity requirements of ad-hoc sensor networks, current practical approaches to non-linear estimation typically rely on EKF-based or, possibly, “unscented”/sigma-point Kalman filter-based algorithms which adaptively approximate the non-linear state and/or measurement equations as linear, using the most recent state estimates. The distributed data fusion and localization algorithms are directly applicable. In fact, the algorithms were designed for robustness, with the non-linear case in mind: –In the linear case, can be obtained directly from the a priori information and the measurement using the Kalman filter, but in the non-linear case, the linearization (about ) would be too inaccurate. –In the non-linear case, is obtained from the predicted and updated EKF estimates, and, and thus is accurate, assuming the EKF is tracking the states.

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Range Measurement Model Measurement model: where the noise is assumed additive Gaussian (an important practical concern is non-line-of-sight error mitigation [e.g., Kim, Brown, Pals, Iltis, Lee, JSAC, May 2005]), and The EKF linearization is specified in the above reference. Because the range between nodes i and j depends on the positions of both nodes, the estimation errors for node i and j positions are correlated. As nodes range to each other, the estimation errors for all node positions become correlated! If unaccounted for, this can lead to inaccuracy and even instability [Widnall & Gobbini, 1983]. The positions of all nodes should be estimated jointly, which is costly. Sub-optimal algorithms for adaptive subnetwork formation are required. State coupling... Linearization...

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Random Node Mobility Model (Discretized Continuous White Noise Acceleration Model) 0 m 200 m 0 m 20 nodes Initial velocity s. dev: 10 m/s Acceleration s. dev.: 1 m/s 667 m East North 200 m 667 m

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Simulation Parameters One-hop communication range: 275 m (required for this low-density network) Each node ranged to its nearest 5 neighbors, if within range, at 1 Hz (average) Range measurements were obtained with 10-m standard deviation Nodes communicated SSPs to neighbors located a maximum of N h = 1, 2, or 3 hops away (delivery to all nodes not guaranteed) The processing + communication delay was modeled as 0.3 sec., or more, for the first hop, and 0.2 sec., or more, for subsequent hops. For 70% of the nodes, the initial position and velocity estimates had error standard deviations of 150 m and 5 m/s, respectively. The remaining 30% of the nodes obtained independent estimates of their own position and velocity once per second with s. devs. of 10 m and 0.333 m/s.

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Simulation Results Note: some divergence observed due to decreasing connectivity subnetwork membership adaptation is required Communications efficiency improved, with little degradation in accuracy, for block sizes of up to at least 5 (depending on meas. frequency)

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Future Research Directions Development of more efficient optimal and approximate algorithms for fusing delayed information. Development of algorithms for adaptive subnetwork formation (for localization)

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Conclusions Resource-efficient Bayesian data fusion can be achieved by communicating sufficient statistics packets (SSP)s representing information extracted from the most recent local measurements A tool has been provided for trading off improved communications efficiency for information propagation latency The problem of accurately fusing delayed information has been presented, along with exact and approximate solutions The feasibility of localizing and tracking highly-mobile nodes with distributed algorithms has been demonstrated http://stnlabs.ece.ucsb.edu

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