los angeles september 27, 2006 mobicom 2006. localization in sparse networks using sweeps d. k....

Los AngelesSeptember 27, 2006

MOBICOM 2006

Localization in Sparse Networks using Sweeps

D. K. Goldenberg P. Bihler M. Cao J. Fang

B. D. O. Anderson A. S. Morse Y. R. Yang

Los AngelesSeptember 27, 2006

Yale University

MOBICOM 2006

RoadmapMotivation

Problem Formulation

Theoretical Foundation

Related Work

Our Contribution

Experimental Evaluations

Future Work

Location necessary in order for sensed data to be meaningful: e.g., Forest fire detection.

Location information is taken for granted in many network designs: e.g., Geographic routing.

Equipping each node with GPS is not always feasible due to power constraints and other limitations inherent to sensor networks.

Motivation

Localize using inter-node distances!

Nodes can often measure their distances to nearby nodes: Acoustic ranging (e.g. L. Girod et al.), ultra-wideband ranging (e.g. Ubisense), radio interferometry (e.g. Vanderbilt).

RoadmapMotivation

Problem Formulation


Related Work

Our Contribution


Future Work

The network localization problem is to determine the positions of all the nodes.

Anchors are nodes whose positions are known.

The distances between some nodes are known.

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The network is localizable if there exists exactly one position in the plane corresponding to each non-anchor node so that all known inter-node distances are satisfied.

A network in the plane.

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A node is localizable if its position is uniquely determined by the known inter-node distances and anchor positions.

Anchor positions from GPS or manual configuration.

The network localization problem is NP-Hard. (Aspnes et al.)

The localization problem is solvable if and only if the network is localizable.

Even assuming exact distance measurements, there is currently no algorithm that can localize a large class of localizable networks without requiring high connectivity while giving correctness guarantees.

Our contribution – An algorithm that provably and tractably localizes a class of localizable networks with average degree as low as three under the assumption of exact distance measurements.

Techniques to adapt our algorithm to noisy measurements - No proven results.

RoadmapMotivation

Problem Formulation


Related Work

Our Contribution


Future Work

A network in the plane whose node position coordinates are algebraically independent over the rationals is localizable if and only if it has at least three non-collinear anchors and its graph is generically globally rigid in the plane. (Eren et al.)

Consider the network nodes as vertices in a graph.

There is an edge between two vertices if the distance between the corresponding nodes are known.

This is the graph of the network.

There are polynomial time algorithms to determine if a graph is generically globally rigid in the plane.

Can almost always efficiently check if a network in the plane is localizable by analyzing its graph!

Assume the node position coordinates of the networks we consider are algebraically independent over the rationals.

RoadmapMotivation

Problem Formulation


Related Work

Our Contribution


Future Work

Global Approach

Global optimization susceptible to local minimums.

Typically assume uniform deployment of nodes.

May not be effective for networks where average degree is low.

Nodes are localized by processing all nodes at once.

Nodes are localized by sweeping through the network in some order and processing the nodes one by one.

Sequential Approach

Multilateration Trilateration based(Savvides et al.) (e.g. Eren et al., Moore et al.)

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Our work extends the trilateration based methods in order to localize sparse networks where average degree is as low as three.

Experimental evaluations suggest trilateration based method is not effective for sparse networks where average degree is low even assuming exact distances.

A graph has a trilateration ordering if its vertices can be relabeled as v1,...,vn so that

(ii) each vi, i>3, is adjacent to at least three distinct vertices vj, j< i.

(i) the subgraph induced by {v1,v2,v3} is complete.

v1, v2, v3 are the seeds of the ordering.





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Suppose this is the graph of a network.

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A network with three anchors can be localized using just trilaterations followed by an Euclidean transformation if and only if its graph has a trilateration ordering.

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RoadmapMotivation

Problem Formulation


Related Work

Our Contribution


Future Work

Sweeps also identifies all localizable nodes.

Sweeps is a sequential localization algorithm that provably and tractably localizes a class of sparse localizable networks with average degree as low as three assuming exact distance measurements.

We propose techniques to deal with noisy measurements, which experimental evaluations suggest is promising, but no proven results.

We can efficiently check if Sweeps will successfully localize a network by just analyzing the network’s graph.

Experimental evaluations suggest Sweeps is feasible and consistently localizes 90% or more of the nodes in sparse networks of 1000 nodes with average degree five.

Our work is an extension of the trilateration based localization method for networks whose graphs may not have a trilateration ordering.

(ii) each vi, i>3, is adjacent to at least two distinct vertices vj , j < i.


A graph has a bilateration ordering if its vertices can be relabeled as v1,...,vn so that


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Sweeps is for localizable networks whose graphs have a bilateration ordering, but not necessarily a trilateration ordering.

Experimental evaluations suggest such networks occur with high probability even in networks with average degree as low as three.

No trilateration ordering.

Bilateration - Determining a finite candidate positions set for a node using its distances to two or more nodes with finite candidate positions sets.

Candidate position set of a node is a set of points in the plane which contains the node's position.

Bilateration with consistency checking is where only a subset of a finite candidate positions set is chosen to use in a bilateration operation.

{pa p'a} {pb}

{pc p'c1 p'c2 p'c3}

If nodes A and B are positioned at pa and pb, then that determines at most two positions for node C, one of which must be the position of node C.

If nodes A and B are positioned at p'a and pb, then that determines at most two positions for node C.

A B

C

The algorithm Sweeps consists of performing a sequence of bilaterations and set reductions, combined with consistency checking.

Bilateration at node c

Set reduction - Removing points from a node's finite candidate positions set using its distances to one or more nodes with finite candidate positions sets.

{p2, p'2}

1

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{p1, p'1}

d

║p'2 - p1║ d

║p'2 - p'1║ d

Remove point p'2 from the candidate positions set of node 2 because the true position of node 2 must be distance d to at least one point in the candidate positions set of node 1.

Set reduction with consistency checking reduces a node’s candidate positions set even further using an additional criteria. (See paper for details)

Set reduction at node 2

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pv3

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Sweep through the network according to the bilateration ordering, and perform a bilateration operation at each unlocalized node.

Localizable network whose graph has a bilateration ordering.

Sweeps

This graph does not have a trilateration ordering, so cannot be localized by a trilateration based method!

Assign positions to seed vertices so their inter-node distances are satisfied.

This determines a unique position for each vertex relative to the seed vertices.

pv6

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p1 p2

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Sweep through the network in a different order performing set reduction at each unlocalized node.

QSweep through the network according to the bilateration ordering, and perform a bilateration operation at each unlocalized node.



Assign positions to seed vertices so their inter-sensor distances are satisfied.


Sweeps

Sweep through the network in a different order performing set reduction at each unlocalized node.

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Localizable in two sweeps plus Euclidean transformation.

QSweep through the network according to the bilateration ordering, and perform a bilateration operation at each unlocalized node.



Assign positions to seed vertices so their inter-sensor distances are satisfied.


Sweeps

There are localizable networks whose graphs do not have bilateration orderings, and so cannot be sweepable.

A network’s graph must have a bilateration ordering for the network to be sweepable.

Theorem: A localizable network whose graph has a bilateration ordering can be localized with two sweeps followed by a Euclidean transformation.

We say such networks are sweepable.

However, experimental evaluations suggest sweepable networks occur with high probability even in networks with average degree as low as three.

RoadmapMotivation

Problem Formulation


Related Work

Our Contribution


Future Work

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Average Degree

Percentage of localizable nodes localized by Sweeps.

Percentage of localizable nodes localized by Trilateration.

Uniformly random 250 node network.

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Maximum Size of Candidate Positions Sets

Worst case complexity of Sweeps is exponential, but experimental evaluations suggest Sweeps is practically feasible.

Sweeps localizes more nodes than trilateration at the expense of computational complexity.

At average degrees 3 and 9.5, the maximum size of the candidate positions sets is at most 8 for 95% of the nodes.

At average degree 6, the maximum size of the candidate positions sets is at most 8 for just 75% of the nodes.

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Position Error (% of Sensing Range)

Proportion of nodes with less than given position error

Uniformly random deployment of 100 nodes, 5 anchors, average degree 8.

Zero-mean Gaussian noise with std 5% of sensing range added to distance measurements.

90% of nodes localized by Sweeps have position error less than 50%.

RoadmapMotivation

Problem Formulation


Related Work

Our Contribution


Future Work

Future Work

Obtain theoretical results on the effectiveness of Sweeps in the presence of noisy distance measurements.

Extending sweeps to 3-D.

Obtain theoretical results relating the probability of a network’s graph having bilateration ordering with the average degree of the network.

los angeles september 27, 2006 mobicom 2006. localization in sparse networks using sweeps d. k....

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