Transcript Using Cramer-Rao-Lower-Bound to Reduce Complexity of

Using Cramer-Rao-Lower-Bound
to Reduce Complexity of
Localization in
Wireless Sensor Networks
Dominik Lieckfeldt, Dirk Timmermann
Department of Computer Science and Electrical Engineering
Institute of Applied Microelectronics and Computer Engineering
University of Rostock
[email protected]
Outline
1. Introduction
2. Goal
3. Localization in wireless sensor networks

Overview

Cramer-Rao-Lower-Bound

Complexity and energy consumption
4. Characterizing Potential Benefits
5. Conclusions / Outlook
6. Literature
Using CRLB to Reduce Complexity of Localization in WSNs
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Introduction
•
•
Wireless Sensor Network (WSN):
 Random deployment of large
number of tiny devices
 Communication via radio
frequencies
 Sense parameters of environment
Applications



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•
Forest fire
Volcanic activity
Precision farming
Flood protection
Location of sensed information  important parameter in WSNs
Using CRLB to Reduce Complexity of Localization in WSNs
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Introduction – Localization
Example
Parameters:
 m … Number of beacons
 n … Number of unknowns
 N=m+n … Total number
of nodes
Beacon
Unknown
Error ellipse
Using CRLB to Reduce Complexity of Localization in WSNs
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Goal of this Work
• Investigate potential impact and applicability of
adapting and scaling localization accuracy to:




Activity
Importance
Energy level
Other parameters (context)
• Obey fundamental trade-off between:
• Benefits:
accuracy <-> complexity
 Decreased communication
 Prolonged lifetime of WSN
Using CRLB to Reduce Complexity of Localization in WSNs
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Localization in WSN
•
Possible approaches
 Lateration (typically used)
 Angulation
 Proximity
•
Beacon
Unknown
2
3
d1, 2
1
Lateration
d1, 3
d1, 4
 Use received signal strength (RSS) to
estimate distances di,j :
RSS ~ 1/d²
 Idea:
– Estimate distances to beacons
– Solve non-linear system of equations
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( x1  x2 ) 2  ( y1  y2 ) 2  d12, 2
( x1  x3 ) 2  ( y1  y3 ) 2  d12, 3

( x1  x4 ) 2  ( y1  y4 ) 2  d12, 4
Using CRLB to Reduce Complexity of Localization in WSNs
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Localization in WSN
•
Measurements of RSS are disturbed:
 Interference
 Noise
•
How accurate can estimates of
position be?
 Cramer-Rao-Lower-Bound (CRLB) poses lower bound on
variance of any unbiased estimator
Distance
CRLBrss 
Geometry
1
b
N
d
2
1, j
i2
 d1 i , j d i , j 
 2 2 




i  2 j  i 1  d1, i d1, j 
N 1 N
2

 E ( x  x~ ) 2  ( y  ~y ) 2

 10np 

b  

ln
10
 rss

2
np … Path loss coefficient
 rss … standard deviation of
x …
x~ …
Using CRLB to Reduce Complexity of Localization in WSNs
RSS measurements
true parameter
estimated parameter
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Cramer-Rao-Lower-Bound
Probability 
Error model of
RSS
measurements
Number of
beacons
Geometry
CRLB
Lower bound on
variance of

RSS [dBm]
position
error
Using CRLB to Reduce Complexity of Localization in WSNs
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•
Example
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Probability 
Cramer-Rao-Lower-Bound
0.4
0.2
2
1 dimension
0
True position at x=0
-5
-2.5
0
2.5
x
Disturbed position estimates
Probability density of position estimates
Standard deviation or root mean square error more
intuitive than variance
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  CRLBrss  E( xˆ  x ) 2  ( yˆ  y) 2 
Using CRLB to Reduce Complexity of Localization in WSNs
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Cramer-Rao-Lower-Bound – An
Example
• 2 beacons, 1 unknown
Beacon
Unknown
300
 /4
Increase of  rss [%] 
250
200
150
100
50

0
0
0.1
0.3
0.2
Distance/ [rad] 
Using CRLB to Reduce Complexity of Localization in WSNs
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Complexity of Localization
• Complexity depends on:
 Dimensionality (2D/3D)
 Number of Beacons
 Number of nodes with unknown position
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Energy Consumption and
Localization
•
Communication
 Two-way communication beacon <-> unknown
 Main contribution to total energy
consumption
Calculation
 Simplest case: Energy spend ~ number of
beacons
Energy 
•
Number of beacons 
Using CRLB to Reduce Complexity of Localization in WSNs
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Reducing Complexity of
Localization in WSNs
• How to reduce Complexity?
 Constrain number of beacons used
 Idea:
Select those beacons first that contribute
most to localization accuracy!
Using CRLB to Reduce Complexity of Localization in WSNs
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Related Work
Beacon
Placement
Simulate
localization error
Weighting range
measurements
Choose nearest k
beacons
[CTL05]
Variance/Distance
Detect outliers
[LZZ06, CPI06, BRT06]
[OLT04, PCB00]
• Impact of geometry not considered
• No local rule which prevents insignificant
beacons from broadcasting their position
Using CRLB to Reduce Complexity of Localization in WSNs
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Characterizing Potential Benefits
• Simulations using Matlab
• Aim:
 Proof of Concept
 Determine how likely it is that
constraining the number of beacons is
possible without increasing CRLB
significantly
Using CRLB to Reduce Complexity of Localization in WSNs
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Characterizing Potential Benefits
•
Simulation setup:
 Random deployment of m beacons and 1 unknown
 For every deployment calculate:
 m
k
–  ( i ) (k ) i  1,,  ; k  1,, m
– k=m: consider all beacons
 m
– k<m: consider all  k  combinations
of subsets of beacons
 determine ratio
 ( i ) (k )   ( i ) (k) /  ( i ) (m)  1
Using CRLB to Reduce Complexity of Localization in WSNs
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Characterizing Potential Benefits
• Potential of approach
 m=13 beacons
Event: “CRLBok“  
(i)
( k )  0,05
(equals 5% increase)
100
optimal
random
90
80
P(CRLBok )) [%]

[%] 
ok

70
Potentially highest savings
in terms of energy and
50
communication effort
60
40
30
20
10
3
5
7
k
9
Using CRLB to Reduce Complexity of Localization in WSNs
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Conclusion / Outlook
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•
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Preliminary study based on CRLB
 Considers strong impact of geometry on
localization accuracy
Selection of subsets of beacons for
localization is feasible in terms of:
 Prolonging lifetime of sensor network
 Decreasing communication
Outlook
 Determine/investigate local rules for selecting
subset of beacons
Using CRLB to Reduce Complexity of Localization in WSNs
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Literature
[BHE01] Nirupama Bulusu, John Heidemann, and Deborah Estrin. Adaptive beacon placement. In ICDCS '01:
Proceedings of the The 21st International Conference on Distributed Computing Systems, pages 489–503,
Washington, DC, USA, 2001. IEEE Computer Society.
[BRT06] Jan Blumenthal, Frank Reichenbach, and Dirk Timmermann. Minimal transmission power vs. signal strength
as distance estimation for localization in wireless sensor networks. In 3rd IEEE International Workshop on
Wireless Ad-hoc and Sensor Networks, pages 761–766, Juni 2006. New York, USA.
[CPI06] Jose A. Costa, Neal Patwari, and Alfred O. Hero III. Distributed weighted-multidimensional scaling for node
localization in sensor networks. ACM Transactions on Sensor Networks, 2(1):39–64, February 2006.
[CTL05] King-Yip Cheng, Vincent Tam, and King-Shan Lui. Improving aps with anchor selection in anisotropic sensor
networks. Joint International Conference on Autonomic and Autonomous Systems and International
Conference on Networking and Services, page 49, 2005.
[LZZ06] Juan Liu, Ying Zhang, and Feng Zhao. Robust distributed node localization with error management. In
MobiHoc '06: Proceedings of the seventh ACM international symposium on Mobile ad hoc networking and
computing, pages 250–261, New York, NY, USA, 2006. ACM Press.
[OLT04] E. Olson, J. J. Leonard, and S. Teller. Robust range-only beacon localization. In Proceedings of Autonomous
Underwater Vehicles, 2004.
[PCB00] Nissanka B. Priyantha, Anit Chakraborty, and Hari Balakrishnan. The cricket location-support system. In 6th
ACM International Conference on Mobile Computing and Networking (ACM MOBICOM), 2000.
[PIP+03] N. Patwari, A. III, M. Perkins, N. Correal, and R. O'Dea. Relative location estimation in wireless sensor
networks. In IEEE TRANSACTIONS ON SIGNAL PROCESSING, volume 51, pages 2137–2148, August 2003.
[SHS01] Andreas Savvides, Chih-Chieh Han, and Mani B. Strivastava. Dynamic fine-grained localization in ad-hoc
networks of sensors. Pages 166–179, 2001.
Using CRLB to Reduce Complexity of Localization in WSNs
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Thank you for your Attention!
Questions?
Introduction – Localization
Example
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Start-up phase:
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•
N=10000 nodes with 10% beacons
Area: (1000x1000)m
Transmission range is chosen to provide
connection to at least 3 beacons
Minimum transmission power
Initial localization of nodes in range of at
least 3 beacons
Every node has connections to 50 other
nodes
-> need to select subset of beacons for
localization
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
Transmission radius [m]
80
100
1
X: 50
Y: 0.954
0.9
0.8
In refinement phase:

X: 45
Y: 0.9526
0.9
P(more than 2 beacons in range)
•
Example Scenario:
0.7
0.6
P(n)
•
1
0.5
0.4
0.3
0.2
0.1
0
20
30
40
50
60
n (Number of connections to any node)
Using CRLB to Reduce Complexity of Localization in WSNs
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