localization of objects using stochastic tunneling
DESCRIPTION
A novel wireless localization scheme in the three dimensional domain that employs stochastic optimization with tunneling transformation to recursively estimate the location of wireless tags in a network from pair wise signal strength measurements. Spatially co-located wireless tags, receiving signals from a common transmitter, exhibit correlation in their Received Signal Strength Indicator (RSSI) values. Hence in a network of wireless tags, with pair wise correlation coefficients available, posterior distribution of the unknown tag separation is used to relatively localize them using maximum a posteriori (MAP) Estimator. However, due to the non-convex/non-tractable nature of this posterior distribution, deterministic optimization methods will end in one of the many local maxima unless the initial guess is close to the region of attraction of the global maximum. In this paper, a novel stochastic localization method called LOCalization Using Stochastic Tunneling (LOCUST) is proposed which utilizes constrained simulated annealing with tunneling transformation to solve this non-tractable posterior distribution. The tunneling transformation allows the optimization search operation to circumvent or “tunnel” through ill-shaped regions in the posterior distribution resulting in faster convergence to global maximum. Finally, simulation results of our localization method are presentedTRANSCRIPT
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Localization of Objects using Stochastic
Tunneling
Mohammed Rana Basheer and S. Jagannathan
Department of Electrical and Computer Engineering,
Missouri University of Science & Technology
300 W 16th Street, Rolla, MO 65401
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Real Time Location System (RTLS)
• Used for locating or tracking assets in places where GPS signals are not readily available
• Uses Time of Arrival (ToA), Time Difference of Arrival (TDoA), Angle of Arrival (AoA) or Received Signal Strength Indicator (RSSI)
• Cost, accuracy and calibration issues have limited their adoption in factory environment.
Figure 1. Boeing factory floor*
*http://www.ce.washington.edu/sm03/boeingtour.htm
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RTLS using RSSI• Uses signal strength of received
radio signals to locate objects
• In free space, Friis transmission equation gives the relation between signal strength and distance between a transmitter and receiver at far field
Figure 3. Ideal Variation of RSSI with DistanceAsset
Shop Floor
area
RF
Receivers
Figure 2. Asset Tracking using RTLS
RSSIdB = A – nlog(r)
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Limitations of RTLS using RSSI• Multipath fading noise results in
fast variation of received signal
strength
• Bounded localization error only
under Line of Sight (LoS)
condition
• Shadow fading under LoS
• Costly periodic radio profiling
required for Non-LoS condition
Figure 4. Localizing wireless tags in
a room/container
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Problem Statement
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Previous Work
1. Yi Shang, Wheeler Ruml, Ying Zhang, and Markus P. J. Fromherz, “Localization from mere connectivity,” in Mobihoc „03, June
2003, pp. 201–212.
2. N. Patwari and A. O. Hero, “Manifold learning algorithms forlocalization in wireless sensor networks,” in Proceedings of theIEEE
International Conference on Acoustics, Speech and SignalProcessing (ICASSP), May 2004, volume 3, pp. 857–860.
3. Wang C, Chen J, Sun Y, Shen X. Wireless sensor networks localization with Isomap. IEEE International Conference on
Communications, 2009.
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Limitations
Figure 5. Multipath noise correlation with distance
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Limitations (Contd.)• Highly non-linear surface at high frequency violates
linearity assumptions used by MDS, LLE and Isomap
Figure 6. Multi-tag correlation variance at Frequency =27MHz
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Proposed Idea• Localization problem is expressed as estimating the radial distance
parameter between wireless tags using maximum-a-posteriori estimator
• Non-linear relationship between RSSI fading correlation coefficients and
radial distance is used instead of linear approximations used in MDS, LLE
and Isomap
• Parameter estimation space is reduced by imposing triangular inequality
constraints
• Finally, the high convergence time arising due to uneven non-convex
terrain of the localization cost function is improved by stochastic tunneling
operator
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Steps• The joint distribution of RSSI between a pair of co-located tags that are
under NLoS condition with a common transmitter is derived to find the
estimator for correlation coefficient
• Large sample PDF of correlation coefficient estimator under NLoS
conditions is derived to generate the localization cost function
• For a network with greater than 3 wireless tags, triangle inequality
constraints are added to reduce the estimation space
• Monte Carlo Markov Chain optimization method called Constrained
Simulated Annealing is used to solved this highly uneven cost function
• Finally, stochastic tunneling operator is applied to improve convergence
speed
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Joint PDF of Received Signal Strength
Under NLoS
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Estimating DBED Parameters
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PDF of Correlation Coefficient Estimate
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Triangle Inequality Constraint
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Radial separation Estimation
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Stochastic Optimization• Non-linear greedy optimization techniques can get stuck in
local maxima
• Monte Carlo Markov Chain optimization using Constrained
Simulated Annealing (CSA) is used
• Simulated annealing is guaranteed to convergence at the
expense of computation time
Figure 7. Local localization likelihood function at various frequencies
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Constrained Simulated Annealing*• CSA is a variant of the popular Simulated Annealing (SA) optimization
• Generation function generates sample parameter values where the
optimization objective function is evaluated
• Acceptance function that depends on a parameter Tacc called the
“Annealing Temperature” determines the probability of moving away from
a local maxima
• CSA looks for saddle points (local maxima) that occur at the local maxima
in radial distance space and local minima in Lagrange multiplier space
• Separate acceptance functions for radial separation and Lagrange
multiplier space to account for their different optimization objectives
*B. W. Wah, Y. Chen, and T. Wang, "Simulated annealing with asymptotic convergence for nonlinear constrained
optimization," J. of Global Optimization, Vol. 39, No. 1, pp. 1-37, Sep. 2007
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Stochastic Tunneling*
Figure 8. Tunneling Effect
*W. Wenzel, and K. Hamacher, "Stochastic Tunneling Approach for
Global Minimization of Complex Potential Energy Landscapes,"
American Physical Society, vol. 82, No. 5, pp. 3003-3007, Apr. 1999.
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Simulation Results• m=20 wireless tags and n=4 anchor nodes in a 20m x 20m x 20m
cubical workspace
• Anchor nodes positioned at corners of this cubical workspace
• Wireless tags were distributed randomly in the cubical workspace
• correlation coefficient matrix was generated from double truncated
normal random variables with mean and variance computed from
N=100 pair wise RSSI samples and true radial separation between
tags
• 50 Monte Carlo simulation trials were performed to determine the
mean, median, standard deviation and 90th percentile of localization
errors.
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CDF of Localization Errors
Figure 9. CDF of localization error at 27MHz
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Table 1. Summary of Localization Error
MethodF
(MHz)
Localization Error (m)
Mean Median90th
percentileStandard deviation
LOCUST
100
2.562 2.399 4.739 1.487
LLE 24.378 23.967 30.628 6.121
MDS 23.232 22.841 29.597 5.096
LOCUST
70
1.865 1.942 2.568 0.665
LLE 17.42 17.832 23.223 4.324
MDS 13.44 13.243 17.806 3.426
LOCUST
27
0.545 0.503 0.864 0.256
LLE 3.245 3.286 5.229 1.485
MDS 2.888 2.929 3.984 0.893
LOCUST
10
0.231 0.246 0.483 0.124
LLE 0.173 0.154 0.304 0.112
MDS 0.181 0.165 0.298 0.092
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Summary
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Questions