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
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
Data Fusion: Motivation
•
Ad-hoc/sensor networks may
estimate processes within the
network
–
node locations, route feasibility
or in the surrounding environment
–
•
Comms./
ranging
Data
collection
In centralized estimation, data is
relayed to a central sink
–
•
object motion, quantity gradients
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: Approach
•
Each node gathers data
–
•
•
Comms./
ranging
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
–
–
Data
collection
•
•
•
•
e.g., RF, acoustic, EO/IR, temp.
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 resourceefficient and scalable.
Data Fusion/Distributed Estimation:
Survey of Past Work
•
Research in data/estimate/track fusion dates back at least to the 1970s [BarShalom & 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.
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
Comms./
ranging
–
•
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: Approach
•
Each node measures ranges to
other nearby nodes using round-trip
travel time (RTT) measurements
–
•
Dynamic node states (position and
velocity coordinates) are modeled
in state space
–
Comms./
RTT ranging
–
–
•
•
•
relatively simple and affordable
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: Survey of Past Work
•
A variety of measurements can be used for localization:
–
Received signal strength indicator (RSSI): inexpensive, but requires environmentspecific 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.
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 < 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]
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).
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.
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
Efficient Bayesian Information Fusion
Data
collection
Communication
and Ranging
Data
Extract New
Information
New
Info.
Update
Local View
•
Information
Update
Local View
Communicate
Compress/
Aggregate
Information
Node j receiving the SSP fuses it with its most recently-computed sufficient
statistics
for
, where
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:
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.
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.
Data
collection
Communication
and Ranging
Data
Extract New
Information
New
Info.
Update
Local View
Information
Update
Local View
Communicate
Compress/
Aggregate
Information
SSP block formation
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 EKFbased 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.
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.
Random Node Mobility Model
(Discretized Continuous White Noise Acceleration Model)
667 m
North
200 m
• 20 nodes
• Initial velocity
s. dev: 10 m/s
• Acceleration
s. dev.: 1 m/s
200 m
0m
0m
East
667 m
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.
Position Estimation Error (m)
150
Mean |Error|, Centralized
Mean |Error|, Distributed
RMS Error, Centralized
RMS Error, Distributed
100
74
73
50
0
0
47
35
2
4
6
8
10
12
Mean Absolute Estimation Error (m)
Simulation Results
70
60
50
40
30
20
1 hop
2 hops
3 hops
10
0
0
2
4
6
8
10
Time (sec.)
Number of Measurements per SSP
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)
Future Research Directions
•
Development of more efficient optimal and approximate algorithms for
fusing delayed information.
•
Development of algorithms for adaptive subnetwork formation
(for localization)
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