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Localization of Sensor Networks Localization of Sensor Networks Jie Gao Computer Science Department Stony Brook University 9/13/05 Jie Gao CSE590-fall05 1 Papers Papers D. Moore, J. Leonard, D. Rus, S. Teller, Robust distributed

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Localization of Sensor Networks Localization of Sensor Networks

Jie Gao

Computer Science Department Stony Brook University

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Papers Papers

  • D. Moore, J. Leonard, D. Rus, S. Teller, Robust

distributed network localization with noisy range measurements, Proc. ACM SenSys 2004.

  • Anchor-free method.
  • Rigidity-aware.
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Trilateration Trilateration without noise without noise

  • If three anchors are not on the same line,

trilateration with accurate distance measurements gives a unique location.

Exercise: verify this.

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Trilateration Trilateration with noise with noise… …

  • With noisy measurements, trilateration can

have flip ambiguity.

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Use quadrilaterals Use quadrilaterals

  • Four nodes with fixed pair-wise distances
  • It is the smallest 3-connected redundantly

rigid graph globally rigid.

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Robust quadrilaterals Robust quadrilaterals

  • If measurement noise is bounded, the

quadrilateral has no incorrect flip.

Incorrect flip: can’t verify whether C or C’ is correct.

  • '

2 2 2

2 sin 4sin ( )sin sin 2sin(2 )

C D CD err AB

d d d d φ θ φ θ φ θ φ − = + + − = +

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Robust quadrilaterals Robust quadrilaterals

  • If measurement noise is bounded, the

quadrilateral has no incorrect flip.

Minimize the error by choosing φ φ φ φ=π/2-2θ.

  • 2

sin

AB err

d d θ =

Robust quadrilateral satisfies

2

sin

err

d b θ <

Where b is the shortest side and θ is the smallest angle.

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Clusters Clusters

  • Robust quads that share three nodes can

be merged into clusters.

  • The cluster is still a 3-connected

redundantly rigid graph globally rigid.

  • Actually it has 3n-6 edges.
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Three phases Three phases

  • Cluster localization

– Each node x find its local robust quadrilaterals. – Merge them to a robust cluster around x.

  • (optional) cluster optimization

– Refine the nodes’ location inside a cluster.

  • Cluster transformation

– Glue the local quadrilaterals together – Transformation to a global coordinate system.

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Algorithm phase I: cluster localization Algorithm phase I: cluster localization

1. Each node x gets the distance measurements between each pair of 1-hop neighbors. 2. Identify the set of robust quadrilaterals. 3. Merge the quads if they share 3 nodes. 4. Estimate the positions of as many nodes as possible by iterative trilateration.

  • Note: Local coordinate system rooted at x.
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Algorithm phase II: cluster optimization Algorithm phase II: cluster optimization

  • For each cluster around node x, refine the

position estimates, for example, by mass- spring relaxation.

  • Optional.
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Algorithm phase III: cluster Algorithm phase III: cluster transformation transformation

  • Align neighboring local coordinates systems.
  • Find the set of nodes in common between two

clusters.

  • Compute the translation, rotation that best align

them.

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Simulations Simulations

  • 183 nodes uniformly inside a building.
  • Connectivity is only between nodes not obstructed

by walls.

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Simulations Simulations

  • Cluster success rate v.s. node degree.
  • Each plot represents a simulation run.
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Algorithm properties Algorithm properties

  • Nodes not included in the robust quadrilaterals are

not localized.

– A wrong location is worse than no location.

  • Even as noise goes to 0, avg degree ≥ 10 to

achieve 100% localization.

  • Not good for sparse networks.
  • The avg degree ≅ 6 for best throughput of the

network.

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Demo Demo

  • Localize mobile nodes
  • Show a video clip
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Observations Observations

  • Localization algorithm performs poorly when

the graph is sparse.

  • Next, we’ll study in the theoretical worst

case what information is sufficient for the localization problem.

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Computational hardness of localization Computational hardness of localization

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Papers Papers

  • [Breu98] H. Breu and D. G. Kirkpatrick. Unit disk graph

recognition is NP-hard. Computational Geometry, Theory and Applications, 9(1-2):3-24, 1998.

  • [Bruck05] J. Bruck, J. Gao and A. Jiang, Localization and

Routing in Sensor Networks by Local Angle Information,

  • Proc. of the Sixth ACM International Symposium on Mobile

Ad Hoc Networking and Computing (MobiHoc'05), 181-192, May, 2005.

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Unit disk graph abstraction Unit disk graph abstraction

  • Unit disk graph: Given a set of points in the plane,

each pair is connected by an edge if their distance is no more than 1.

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Quasi unit disk graph model Quasi unit disk graph model

  • A better model of the radio coverage range.
  • Quasi-UDG model.

∃ uv if |uv|<α<1; no uv if |uv|>1; unclear otherwise. α 1

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Localization and embedding Localization and embedding

  • Embedding: Given a graph in the plane, assign

coordinates to the vertices such that neighboring nodes are within distance 1 and non-neighboring nodes are of distance α apart.

  • α=1: UDG embedding.
  • α<1: 1/α-approximate UDG embedding or α-Quasi

UDG embedding.

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Input of the embedding Input of the embedding

  • Connectivity information: only

the combinatorial graph.

  • Local distance information:

(accurate) edge lengths.

  • Local angle information:

(accurate) angle between any two incoming links.

  • We study whether UDG

embedding is NP-hard.

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Local measurements do not suffice Local measurements do not suffice

  • With connectivity information:

– UDG embedding is NP-hard. – α-Quasi UDG embedding is NP-hard, for α>

  • With local distance information:

– UDG embedding is NP-hard. – If the graph has Ω(n2) edges, then there is a unique solution which can be found by semi-definite programming.

  • With local angle information:

– UDG embedding is NP-hard. – α-Quasi UDG embedding is NP-hard, for α>

  • With both local distance & angle information:

– UDG embedding is in P.

2/3 1/ 2

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Hardness of UDG embedding Hardness of UDG embedding

  • Reduction from 3-SAT problem.
  • A 3-SAT instance:
  • A variable appears at most 3 times; A clause has at

most 3 literals.

  • A 3-SAT instance is satisfied if there is an

assignment to the variables such that all clauses are 1. 0/1 variable clause literal

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Graph representation of a 3 Graph representation of a 3-

  • SAT instance

SAT instance

clauses variables An edge connects a literal to a clause if the literal appears in that clause.

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Find orientations of the edges Find orientations of the edges

  • Define an edge directed from a clause c to a literal

u to be that c chooses u to satisfy it. Or, u is assigned 1.

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A feasible orientation A feasible orientation

1. If a literal has incoming edges, then its negated version has outgoing edges. 2. Each clause has at least 1 outgoing edge. “every clause is satisfied.”

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Find orientations of the edges Find orientations of the edges

  • The graph is orientable if and only if the 3SAT

instance is satisfiable.

  • Now we realize the graph by a unit disk graph s.t. a

valid embedding in 2D gives a valid orientation of the edges.

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A widely used lemma about UDG A widely used lemma about UDG

  • Crossing lemma: if two edges cross in a UDG, then
  • ne node has edges to the three other nodes in

UDG. |uw| ≤ |wp|+|up| |vx| ≤ |vp|+|xp| |wu|+|vx| ≤ |wx|+|ux| ≤ 2

Also, |wv|+|ux| ≤ |wx|+|ux| ≤ 2 There must be 2 edges on the quad adjacent to the same node.

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How to represent orientation? How to represent orientation?

  • “Cage” with “beads” inside.

By the crossing lemma: 1. The cage can’t self-intersect in the embedding. 2. The whole chain is completely inside or completely

  • utside the cage.

Think of the small chain as a little arrow indicating the

  • rientation.
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How to represent orientation? How to represent orientation?

  • “Cage” with “beads”.
  • Any two beads are independent vertices they

are at least distance 1 away.

  • Each bead is at least 1 away from nodes on the

cage.

  • By a packing argument, you can not put too many

beads inside a fixed-sized cage.

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Implementation of cages Implementation of cages

  • An n-cage can contain at most n beads.

Radius ½ disk Optimal packing A 6-vertex cage is a 0-cage. An 8-vertex cage is a 1-cage. A 9-vertex cage is a 2-cage. A10-vertex cage is a 3-cage.

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An oriented edge An oriented edge

  • A chain of cages.
  • All the cages have the same “orientation”, since no

two chains can be inside one cage.

  • This represents a directed edge in the 3SAT graph.
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An oriented path that turns An oriented path that turns

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Cross Cross-

  • overs
  • vers

Fix the two ends

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Literals and clauses Literals and clauses

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Variables Variables

3 outgoing edges for each literal If one edge is an incoming edge, then the edges of the other literal must all be outgoing.

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Clauses Clauses

At least one edge is an

  • utgoing one.

A clause with three literals. A clause with two literals.

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UDG embedding is NP UDG embedding is NP-

  • hard

hard

  • The 3SAT instance is satisfiable if and only if there is

a valid orientation, if and only if the unit disk graph has a valid embedding.

  • 3SAT is NP-hard. Thus UDG embedding is NP-hard