A Passive Approach to Sensor Network Localization

Rahul Biswas and Sebastian Thrun

International Conference on Intelligent Robots and Systems 2004

Presented by Ayan Banerjee

Localization

• Target Localization : Sensor nodes detect the location of the target. (assumes sensor nodes know their own location)

• Self Localization : Sensor nodes gains knowledge about their own location

Active and Passive Self Localization

• Active Self Localization : System sends signals (beacon signals) to localize sensor nodes

• Passive Localization : System uses already existing signals (environment noise) to determine location of sensor nodes

Active vs. Passive• Active :

– Advantages :• Accurate• Robust

– Disadvantages : • More Power requirement • Increased Overhead• Requires Infrastructure• Sensor nodes Detectable

• Passive : – Advantages :

• Requires Less power• Less overhead• No infrastructure required• Sensor nodes less detectable

– Disadvantages :• Less Accurate• Lack of unique solution• Solution depends on characterization of the signals

State of the art• Localization by GPS systems : lack of desired

accuracy• Many beacon based localization schemes :

Accurate but incurs high cost and has high power requirements

• Passive localization schemes :– Radio signal strength based localization– Optical retro-reflector equipped sensors

Contribution of the paper

• Provides a Novel Passive Localization scheme • Implements the Localization scheme using

Crossbow Mica 2 motes • Defines Error metrics and evaluates the

algorithm with respect to changes in Sensor noise, and scalability

Background

• Probability Theory• Bayesian Networks

Bayesian Networks

• It is a Probabilistic Graphical Model.

Conditional Independence

• Two sets of nodes A and B are conditionally independent in a Bayesian Network given a third set of nodes C if all the paths between nodes in A and B are separated by a node in C.[1]

[1] http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html

Example of Conditional Independence

Example of Conditional Independence

How does Conditional Independence help?

• Joint probability of all nodes:

• Storage space reduces from O(2n ) to O(n2k ) where k is the maximum number of parents of a node

( , , , ) ( ) ( | ) ( | , ) ( | , , )

( , , , ) ( ) ( | ) ( | ) ( | , )

P A B C D P A P B A P C A B P D A B C

P A B C D P A P B A P C A P D B C

Likelihood in Bayesian Networks

• Given an observed event A and a probability model find the events B,C,D…. that are most likely to produce the event A.

( |{ , , ...}) ( |{ , , ...})L A B C D P A B C D

In Bayesian Networks

• Given an Event set R and its parent set P then

• This takes advantage of the fact that all Ri s are conditionally independent

( | ) ( | ) ( | )i ii

L P R P R P P R P

Why Bayesian Network?

• We can compute the answer to any question such as in linear time to the number of nodes

• Also we can derive the cause to any event which is a node in the Bayesian network

• We can store all joint probability information in O(n2k ) space

( | , ), ( | , , ),.............P A B C P B E F G etc

Problem Statement

• N Sensor Nodes at Location (S1 , S2 , …… Sn ) (unknown)

• M sound generating events at locations (E1 , E2 , …… Em ) (unknown)

• M time values of the sound generating events (T1 , T2 , …… Tm ) (unknown)

• M.N number of variables of the form which signifies the time recorded by the n th sensor for the m th event. (known)

mnR

Problem is to find the location of the sensors from the measured matrix R

Sensing ModelmnR is considered to be normally distributed

2 2

( , )( , )

( , ) ( ) ( )

m m nn n

x x y ym n m n m n

d S ER N T

s

d S E S E S E

Approach

• Formulate the problem as a Bayesian Network

Approach• Measure the different times recorded by all the

sensors for the different sound generating events in the surroundings

• Calculate :

• Calculate the likelihood of the data R given the model.

• S that maximizes the likelihood is the best prediction of the location.

2

2

( , )( )1( | , , ) exp( )

22

mn mm nm

n n m m

d S ET RsP R S E T

Maximum Likelihood Estimation• Given the measured data R the likelihood that the set S, E and T are its

parents in the Bayesian Network is given by:

• Maximizing likelihood is the same as maximizing log likelihood as log likelihood is monotonic in likelihood. Thus it maximizes

using gradient descent algorithm

,

2

2,

( | , , ) ( | , , )

( , )( )exp( )

2

mn m n n

m n

mn mm n

m n

L R S E T P R S E T

d S ET Rs

2

,

( , )( )mn mm n

m n

d S El T R

s

Condition for unique solution• The unknowns are

– The vector S which is 2 dimensional so it poses 2M unknowns– The vector E which also is 2 dimensional and so it poses 2N unknowns – The vector T which is 1 dimensional and poses N unknowns

• Now in order to have a global reference the any one of the sensor say S1 is considered as origin

• Another sensor S2 is considered to lie on the x axis• So S1 has a coordinate (0,0) and S2 has a coordinate (x,0). This

eliminates 3 of the unknowns.• The number of equations obtained is equal to the number of

measured.• If number of measured is P• We must have

mnR

mnR

2. 3. 3P M N

Analysis

• P can be at most M.N• But sensors may not pick up each and every

sound that occurs in the surroundings• So P < M.N• Even if there exists degenerate

cases (Bayesian Network Unconnected)2. 3. 3P M N

Experimental Setup

• 7 Crossbow Mica 2 motes placed randomly • 7 sound sources• The Mica2 motes are time synchronized using the Reference

Broadcast Protocol (RBS)– This is necessary in order to assume that the time recorded at one

sensor node is directly comparable to time recorded at another

• Paper does not give any information regarding the environment

Results• Error metrics :

– Metric 1: Standard deviation of the expected distance between the sound sources and sensor nodes.

– Metric 2: Actual Error, the error in the predicted location of the sensor as compared to the actual location of the sensors

'.l s

Simulation• Simulator emulates the observed characteristics of the sensor nodes• There were 20 sensor nodes in simulation • 20 sound sources• Standard deviation in the recording error by the sensors were set to a

value that corresponds to 10 cm

Variation of network parameters

• Changing Sensor noise

Variation of Network Parameters• Changing the number of sensor nodes with sensor noise set to a value

corresponding to an error of 10cm.

Variation of Network Parameter

• Effect of Changing the number of sound sources

Conclusion

• Localization of sensor nodes using uncontrolled environmental sound

• A probabilistic generative model for the problem• Localization => Maximum Likelihood Estimation• Algorithm works well when there a large fraction

of sensor nodes here all the sound• The scaled likelihood which is also the predicted

error is not a good error metric• Gradient Descent may get stuck at local minima

Critique

• Strong points:– Strong mathematical background– A novel endeavor

• Weak Points:– Experimental and Simulation environment not

presented well– Implementation not well done– Too naïve optimization algorithm

Impact on Class

• Localization is an important problem in Mobile Computing

• This paper introduces the class to a novel passive localization scheme

Impact on Project

• Project : “Finding Interaction Through Sound”• Finding interacting groups is similar to

localization problem• Classroom environment, it is likely that the

algorithm works well• But in order to use this algorithm we need to

improve it.

Thank You

Questions?