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

Distributed Selection of
References for Localization in
Wireless Sensor Networks
Dominik Lieckfeldt, Jiaxi You, Dirk Timmermann
Institute of Applied Microelectronics and Computer Engineering
University of Rostock, 18119 Rostock, Germany
Email: {dominik.lieckfeldt, jiaxi.you}@uni-rostock.de
Introduction > Selecting References > Simulations > Conclusion
Outline
1. Introduction


Localization in Sensor Networks
Sources of errors regarding localization
2. Selecting references for localization
 Finding a criteria for selection
 Description of the algorithm
3. Simulation results
4. Summary and conclusions
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Introduction > Selecting References > Simulations > Conclusion
Localization in Wireless Sensor
Networks
• Why?
 Mapping of location ↔ sensor data
•
Problem:
 Nodes randomly deployed
 GPS not on every node possible
•
Solution:
 Few nodes with GPS → Beacons
 Remaining nodes → Unknowns
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Introduction > Selecting References > Simulations > Conclusion
Baseline Algorithm for Localization
Unknown
Beacon
TX range
Reference/Beacon
1. Phase
Refinement
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Introduction > Selecting References > Simulations > Conclusion
Sources of Error
Error
Systematic
Random
RF
Shadowing, orientation of
antenna
Noise , Fading (interference)
Hardware
Tolerances
Noise
Environment
•
Temperature, Humidity,
Location of References
Selection
of
beacons
that
contribute
most
to accurate
(Geometry )
localization

Distributed Beacon Selection1
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Introduction > Selecting References > Simulations > Conclusion
Finding a Selection Criteria
Theory of Estimation
 Comparison of estimators based on
variance of estimates
 Fundamental lower bound on Variance
→ Cramer-Rao-Lower-Bound (CRLB)
•
CRLB
•
subset
Selection using CRLB
Here: Use CRLB as selection criteria
Need 3 reference points
for localization!
?
CRLB
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Introduction > Selecting References > Simulations > Conclusion
Inequality of Cramér and Rao
• Poses lower bound on variance of any
2
estimator
3
d1, 2
1
• CRLB for localization based on:
•
d1, 3
d1, 4
4
 Time-of-Arrival (ToA) or received signal
strength (RSS) derived by Patwari et al.2
RSS:
Distances
…
…
…
…
path loss coefficient
deviation of RSS
true parameter
estimated parameter
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Introduction > Selecting References > Simulations > Conclusion
Impact of Geometry on CRLB
Example: 2 references, 1 unknown
100
Linear vector
CRLB normalized
•
Reference
10
1
0
Circular vector
linear
circular
0.1
0.2
0.3
Distance/ [rad]
0.4
Unknown
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Introduction > Selecting References > Simulations > Conclusion
Distributed Selection Procedure
Need 5 reference
points for
localization.
• Phase I:


Inquiry send by unknown
All beacons compute
response probability
(


… maximal tx range)
TDMA: Beacon i responds
with probability
and
broadcasts its position
and estimated distance
End condition:
–
One beacon has
responded
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Introduction > Selecting References > Simulations > Conclusion
Distributed Selection Procedure
• Phase II:

After first response:
–
–

Use estimated distances
and position of first
responder to avoid
collinear beacons
How? Utilize CRLB
End condition:
–
2 beacons have
responded
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Introduction > Selecting References > Simulations > Conclusion
Distributed Selection Procedure
• Phase III:

Recalculation of
based on previous
responses and on CRLB

Reference i responds with
probability
End condition:

–
Sufficient number of
references has
responded
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Introduction > Selecting References > Simulations > Conclusion
Error of location estimates:
•
Power-Error-Product (PEP):
•
Simple Energy Model (TDMA):
More
efficient
•
PEP
Performance Metrics
PEP schematic
= 0.3 mJ
= 0.81 mJ
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Introduction > Selecting References > Simulations > Conclusion
Simulation Results (RSS)
Reference
Unknown
10
3
18
16
2
14
location error [m]
Distance-based
PEP [mJ]
10
10
10
Local-CRLB
Local-Distance
Local-CRLB with circle rule
Local-CRLB
Local-Distance
Local-CRLB with circle rule
Global-Distance
Global-CRLB
12
1
10
8
0
6
10
CRG-based
-1
4
2 2
10
12
4 4
66
88
10
12
number
referencesininsubset
subset
number
ofof
references
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14
13
Introduction > Selecting References > Simulations > Conclusion
Simulation Results (TOA)
Reference
Unknown
1
10 2
1.8
Distance-based
PEP [mJ]
location error [m]
1.6
0
10
1.4
Local-CRLBLocal-Distance
Local-CRLBGlobal-Distance
with circle rule
Local-Distance
Local-CRLB
Local-CRLB with circle rule
Global-CRLB
1.2
10
-11
0.8
0.6
-2
CRG-based
100.4
22
44
66
88
10
12
number
number of
of references
references in subset
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14
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Introduction > Selecting References > Simulations > Conclusion
•
•
Contribution:
 Analysis of distributed algorithms for selecting
references for localization
 Investigation of error of localization
 Comparison regarding Power-Energy-Product
Conclusions:
 Use of CRLB can improve selection regarding
accuracy
 Convergence of CRLB-based algorithms
should be improved to increase energy
efficiency
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- Thank you for your attention -
Questions?
Literature:
1
2
Lieckfeldt, D; You, Jiaxi; Timmermann, D.: “An algorithm for distributed for
distributed beacon selection”, IEEE PerSeNS, 2008
Patwari, N.; O. Hero III, A.; Perkins, M.; Correal, N. & O'Dea, R.: “Relative location
estimation in wireless sensor networks“, IEEE TSP, 2003
Introduction > Selecting References > Simulations > Conclusion
Summary
Wireless Sensor
Networks
Localization
Accuracy
Limited
resources
Auswahl von Referenzen
Distance
CRLB
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Formeln

T X 
 
E T  X 




Var T  X   
I  
N
1
CRLBrss 
b
i
resp
P
d
i 2
2
1, j
 d1i , j d i , j 
 2 2 




i  2 j i 1  d1,i d1, j 
N 1 N
 d1,i 

 1  
 d tx 
2
2
 E( x  ~
x )2  ( y  ~
y )2 
i
resp
P

ΔE (t )   δEi (t, n )
i 1
(i)
tx
d tx
CRLBrss Rresp  
i   CRLBrss ( Rresp )
CRLBrss Rresp  
i
1 T
PEP  ΔE  e
e   (x  ~
x )2  ( y  ~
y )2
T i 1
δEi (t, ntx(i) )  (t  ntx(i) ) Erx  ntx(i) Etx
M
d1,i

 10np 

b  

ln
10
 rss

i
Presp
2
2
np
 rss
x
x~
Etx
Erx
Motivation > SotA > Beacon Selection > Conclusion
Beacon Selection: CRLB explained
Probability 
Error model of
RSS
measurements
Number of
beacons
Geometry
CRLB
Lower bound on
variance of

RSS [dBm]
position
error
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•
Beispiel





Probability 
Cramer-Rao-Lower-Bound
0.4
0.2
2
1 Dimension
0
Wahre Position: x=0
-5
-2.5
0
2.5
x
Fehlerhafte Positionsschätzungen
PDF der Positionsschätzungen
Standardabweichung -> intuitives Maß um Fehler zu
charakterisieren
5
  CRLBrss  E( xˆ  x ) 2  ( yˆ  y) 2 
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Localization in WSN > Distributed Beacon Selection > Conclusion
Baseline Algorithm for Localization
Unknown
Reference
Beacon/Reference
1. Phase
Refinement
( x1  x2 )  ( y1  y2 )  d
2
2
3
d1, 2
1
d1, 3
Tx range
2
2
1, 2
( x1  x3 )2  ( y1  y3 )2  d12,3
y
( x1 , y1 )
( x1  x4 )2  ( y1  y4 )2  d12, 4
d1, 4
4
WPNC 2008 - "Distributed Selection of References for Localization in Wireless Sensor Networks"
x
21
Introduction > Selecting References > Simulations > Conclusion
Distributed Selection Procedure
Need 5 reference
points for
localization.
• Phase I:




Inquiry sent by unknown
References calculate
i
response probability Pant
2
d


i
Pant
 1   1,i 
 d tx 
TDMA: Reference i
i
response with probability Pant
After first response:
–
Utilize CRLB to avoid
collinear references
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Introduction > Selecting References > Simulations > Conclusion
Distributed Selection Procedure
• Phase II:

i
Pant



i
Recalculation of Pant
based on the decrease of
CRLB
CRGrss Rant  
i   CRGrss ( Rant )
CRGrss Rant  
i
Reference i response with
i
probability Pant
End condition:
–
Sufficient number of
references has
responded
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Einleitung > Positionsbestimmung > Auswahlverfahren > Zusammenfassung
Drahtlose Sensornetzwerke
•
Definition:




•
Netz aus kleinsten Knoten
Zufällige Positionierung
Drahtlose Kommunikation
Erfassung von Umweltparametern
Eigenschaften:
 Ressourcenarm
 Fehleranfällig
• Anwendungsbereiche:
Analyse, Beobachtung, Überwachung
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