Efficient Distributed Algorithms for Data Fusion and Node Localization in Mobile Ad-hoc Networks

1
Efficient Distributed Algorithms for Data Fusion
and Node Localization in Mobile Ad-hoc Networks

• Andrew P. Brown, Ronald A. Iltis, and Ryan
  Kastner

University of California, Santa
Barbara http//stnlabs.ece.ucsb.edu

This work was supported in part by NSF grant No.
CNS-0411321
2
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

3
Data Fusion Motivation

• 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

Comms./ ranging
Data collection
4
Data Fusion Approach

• Each node gathers data
• e.g., RF, acoustic, EO/IR, temp.
• Extracted information is used to update local
  estimates
• Information is compressedwithout lossinto
  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.

Comms./ ranging
Data collection
5
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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Node Localization Motivation

• 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

Comms./ ranging
7
Node Localization Approach

• 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

Comms./ RTT ranging
8
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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Linear Gaussian State Space Model

• 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 tn
  tn 1 is arbitrary)
• or, for m lt 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

10
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 n3.

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Efficient Bayesian Information Fusion

• There is an important case in which the fusion of
  Gaussians formula can be usedwhen 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 lt 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
  Nb 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.

18
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.

19
Random Node Mobility Model(Discretized
Continuous White Noise Acceleration Model)
667 m
200 m
North

• 20 nodes
• Initial velocity
• s. dev 10 m/s
• Acceleration
• s. dev. 1 m/s

0 m
200 m
East
667 m
0 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 Nh 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
Communications efficiency improved, with little
degradation in accuracy, for block sizes of up
to at least 5 (depending on meas. frequency)

• Note some divergence observed due
• to decreasing connectivity
• subnetwork membership
• adaptation is required

22
Future Research Directions

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

23
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