1 Localization Technologies for Sensor Networks Craig Gotsman, Technion/Harvard Collaboration with: Yehuda Koren, AT&T Labs.

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Localization Technologies for Sensor Networks

Craig Gotsman, Technion/Harvard

Collaboration with:

Yehuda Koren, AT&T Labs

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Sensor networks

• Set of nodes with ability to:

– Measure parameters related to environment

– Process information– Communicate / route– Estimate location– Communicate information to central

processorCreate a smart environment

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Hardware architecture

sensors CPU/memory

radio

battery

Acoustic, seismic, image, magnetic, etc.

interface

Electro-magnetic interface

Event detectionWireless communication with neighboring nodes

In-node processing

Limited battery supply

Simple, small, cheap

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Potential applications

• Warning

Tornado Fire

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Potential applications

• Transportation

– Monitor traffic conditions

– Plan routes– Parking

allocation

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Limitations

• Power is the bottleneck– Long distance communication

impossible• No pre-configuration or global knowledge

– Achieve global goals through local interaction and self organization

• Limited computational power• Price

Use a very large number of sensors in a wide region

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Location-aware sensors

• Data should be location-stamped• Geographic routing• Region-targeted querying

(123,456)

(134,778)

(234,466)

(294,666)

(372,862)

(362,423)

(432,553)

(519,450)

(589,703)

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Find a fully distributed algorithm for sensor localization

Why not simply use GPS ???

The problem we address:

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Limitations of GPS

• Power• Price• Line of sight

conditions• Accuracy ?

What else can we use ?

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Distance to neighboring sensors

• Received Signal Strength Indicator (RSSI)• Time of Arrival (ToA)

Technologies:

3

6

75

5

3

5

6

8

3

74

45

6

4

May be noisy

Local distances coordinates (??)

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Previous Solutions• Anchor-based:

– Some beacon nodes know their exact location

– Other sensors estimate their location from nearby beacons

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Previous Solutions• Incremental approaches:

– Assign coordinates to a small core of sensors– Repeatedly assign coordinates to more sensors

based on local calculations– Prone to error accumulation

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Previous SolutionsAnchor Free Localization (AFL) [Priyantha et al., 2004]

A two-stage, distributed approach:

• Based on connectivity, elect central, north, south, east and west sensors

• Estimate coordinates for rest of sensors

1. Initial coordinate assignment

• Optimization using gradient-descent to approximate measured distances

2. Accurate distributed layout

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Graph layout interpretation • Given a graph with edge

lengths• A layout that realizes all edge

lengths exist• Only close nodes are

connected (“disk graph”) • Goal:

Find this layout !

2

3

6

1 4

2

4

Two issues:

1. Layout existence - measured lengths are noisy !

2. Layout uniqueness – graph’s rigidity

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Graph rigidity

• In 2D:

– Global rigidity 3-connectivity

– 6-connectivity Global rigidity • Computing layout of rigid graph is NP-hard [Eren et al.,

2004]

1 2

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• A graph is globally rigid if it has a unique embedding (up to distance preserving transformations)

2,31,4

1

42,3Non rigid

Globally rigid

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Beyond classical rigidity• In disk-graphs - close nodes must be connected (up

to noise)• Non-adjacent nodes should be placed further apart

• Optimal layout 1, , , d

n ip p p R

Prunes redundant embeddings

,

,

i j ij

i j

p p l i j Ei j

p p r i j E

,edge length- max,ij ij

i j El r l

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Graph drawing algorithms• Energy-minimization algorithm using localized stress

energy:

2

,i j ij

i j E

p p l

• Known problem: foldovers

Why ???

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Localized stress and foldovers

• Graph not rigid – the energy does not address nonadjacent nodes

• Local minima – global optimization infeasible

2

,i j ij

i j E

p p l

We must treat nonadjacent nodes explicitly

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Algorithm overview

1. “Spectral” initialization– Convex optimization – insensitive to

initialization– Tends to generate fold-free layouts– Uses given distances inaccurately

2. Local stress optimization– Sensitive to initialization– Accurate use of distances – produces

optimal layout when initialized properly

– Global optimization – overcomes local noise

A two-phase approach:

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Spectral layout - Laplacian Based on [Hall, 1970]

Given a weighted graph with n nodes, wij being the edge weights (wij=0 for non-adjacent nodes)

The Laplacian of the graph is the matrix L, where:n n

deg

ij

ij

i

w i jL

i j

5 3 2 0 0

3 10 1 6 0

2 1 9 4 2

0 6 4 14 4

0 0 2 4 6

2

3

6

1 4

2

4

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Spectral layout - Goal

2 2

,

2 2

,

Mini jij i j

i jn

i ji ji jx y

x y

w x x y y

x x y y

R

Solve:

- coordinates of node i,i iyx

Locate related nodes closely, while spreading nodes wellLocate related nodes closely, while spreading nodes well

Weighted squared distances between

nodes

Squared distances between nodes

• Edge weights express similarity/proximity• Solution is Laplacian eigenvector(s) - scale invariant• Does not use distances directly…

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Local stress minimization• Relocate the nodes to minimize:

2

,i j ij

i j E

p p l

• Accurate optimization process, addressing measured distances directly

• Effective only when initialized smartly• Generally, the spectral initialization is good

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Example

Original layout

1000 sensors on 10x10 square, R=0.8

Stress with spectral initialization

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Example

Original placement

715 sensors on 10-3 ring, R=0.8

Stress with spectral initialization

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Conclusions• A fully distributed algorithm for sensor

network layout• Based on graph drawing methods• Main challenge: layout computation with only

local communication• Still need to represent distances better in

spectral layout

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Future Work

• Higher dimensions (3D ?)• Improve spectral embedding using LLE• Implement on real systems

• Incorporate more geometric info (e.g. angles)• Multi-camera calibration

• Dynamic systems