Position Estimation in Sensor Networks

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Position Estimation in Sensor Networks

• Brian D O Anderson
• Research School of Information Sciences and
  Engineering, Australian National University
• and
• National ICT Australia
• (Work with A S Morse, D Goldenberg, T Eren)

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OUTLINE

• Aim of Presentation
• Control and Sensor Networks
• The sensor network localization problem
• Rigidity and Global rigidity
• Computational Complexity of Localization
• Conclusions and open problems

Aim of presentation
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AIM OF PRESENTATION

• To introduce problems involving control and
  sensor networks
• To explain the problem of position estimation of
  sensors (sensor network localization)
• To introduce tools of rigidity
• To use tools of rigidity theory to understand the
  essence of the sensor localization problem
• Motivations printers in a building, underwater
  acoustic sensors, sensors in dense foliage, etc

4
OUTLINE

• Aim of Presentation
• Control and Sensor Networks
• The sensor network localization problem
• Rigidity and Global Rigidity
• Computational Complexity of Localization
• Conclusions and open problems

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Sensor Networks

• A collection of sensors is given, in two or three
  dimensions. Warning the earth is not flat!
• Typically, the absolute position of some of the
  sensors (beacons) is known, eg via GPS
• Sensors acquire some other position information,
  eg reciprocally measure distance to neighbours,
  ie those within a radius r.
• Sensors also measure something else--biotoxins,
  water pressure, fire temperature, etc

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Control problems and Sensor Networks

• Covering a region with sensors
• each may see 3 or 4 others
• sensors may fail
• exact positioning may not be possible
• region may have irregular boundaries and/or
  interior obstacles
• Scanning with moving sensors
• There may be an evader
• Evader may destroy sensors
• Sensors with different capabilities
• Dynamic network
• A priori or adaptive strategies?

• Management of energy usage
• Sensing radius depends on power level
• Control architecture for swarm
• What needs to be sensed to control a moving swarm
  (eg birds, fish, UAVs)?
• Allow for robustness
• In warfare, may constrain architecture to avoid
  disclosure of position when transmitting

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OUTLINE

• Aim of Presentation
• Control and Sensor Networks
• The sensor network localization problem
• Rigidity and Global Rigidity
• Computational Complexity of Localization
• Conclusions and open problems

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Sensor Networks
Sensor
r
Depicts sensors with sensing radius r
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Sensor Networks
Sensor graph, with connection between two sensors
if closer than r
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Sensor Networks

• Beacon sensor positions known absolutely
• Inter-neighbour distances known (edge distance
  for each edge of graph) plus inter-beacon
  distances

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Sensor Networks
Beacon sensor
Normal sensor

• Beacon sensor positions known absolutely
• Inter-neighbour distances known (edge distance
  for each edge of graph) plus inter-beacon
  distances

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Sensor Networks-Questions

• What are the conditions for network
  localizability, ie ability to determine the
  absolute position of all sensors--in first
  instance from NOISELESS data?
• What is the computational complexity of network
  localization?
• The first question is an old one (Cayley, Menger,
  chemists)

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Sensor Networks-Questions

• Need to work with a notion of generic
  solvability--need solvability for all values of
  distance round nominal
• Could formulate other problems with different
  inter-sensor information (eg interval of distance
  values, or direction)
• Interest exists in two and three dimensions
• Not yet studying dynamic networks

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Sensor Networks and Formations

• A point formation is a set of points together
  with a set of links and values for the lengths of
  the links.
• A formation determines a graph G (V,L) of
  vertices and edges, and lengths of the edges.
• A formation is like a sensor network with the
  absolute beacon positions thrown away
• A formation that is exactly determined by its
  graph and distance function is globally rigid.
  Any other formation with the same data is
  congruent, ie is determinable by translation
  and/or rotation and/or reflection.

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Congruent Formations
Absolute beacon positions eliminate this residual
uncertainty
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Two dimensional rigidity examples
Not rigid--distances do not determine precise
shape.
Globally Rigid--distances determine shape to
within reflection, rotation or translation
Absolute beacon positions eliminate the
reflection etc uncertainty
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Sensor Networks and Formations

• Suppose m beacons, n-m ordinary nodes, for 2
  dimensions there are at least 3 beacons in
  general position, and in 3 dimensions at least 4
  beacons in general position.
• Then the sensor localization problem is solvable
  if and only if the associated formation is
  globally rigid

Henceforth, we will focus on formations and their
global rigidity
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OUTLINE

• Aim of Presentation
• Control and Sensor Networks
• The sensor localization problem
• Rigidity and Global Rigidity
• Computational Complexity of Localization
• Conclusions and open problems

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Rigidity

• Let F be a formation with vertex and edge sets V
  and L. Imagine it is moving. Let qi denote the
  position at time t of the i-th vertex. For each
  edge (i,j) in L, let ?(i,j) denote the fixed
  distance. Then
• (qi - qj) (Dqi - Dqj ) 0
• Can write this equation for every edge
• R(F)(Dq) 0
• Here R(F) is the rigidity matrix. For a rigid
  formation
• One rotation and two translations give nullspace
  of dimension 3 in two dimensions
• Three rotations and three translations give
  nullspace of dimension 6 in two dimensions

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Two dimensional rigidity examples
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Three dimensional rigidity examples
Not rigid. R(F) has 7 dimensional nullspace
Rigid. R(F) has 6 dimensional nullspace
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Rigid Formations
Sample two dimensional Rigidity Matrix--a Matrix
Net ? xi Mi yi Ni in coordinates xi and yi of
points.
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More on rigidity

• Rank R(F) for a fixed graph will have the same
  value for almost all lengths
• One has to focus on genericity issues and work
  with generic rigidity
• In two dimensions, there is a combinatorial
  characterization of generically rigid
  graphs-Lamans theorem, with fast algorithm for
  testing
• No such result is available in three dimensions.
  (Partial results exist)

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Rigidity versus global rigidity
a
b
c
d
Both formations are rigid. Neither can be changed
into the other by translation, rotation or
reflection.They have the same edge lengths. So
they are not globally rigid! It is possible to
have a strictly finite number greater than one of
solutions to the formation realization problem
--this connotes rigidity but not global rigidity.
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Rigidity versus global rigidity

• We can fix the previous problem if we fix the
  distance between b and a.
• This makes the graph redundantly rigid (and
  3-connected, see next slide)

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Rigidity versus global rigidity

• Formally, a graph is redundantly rigid if the
  removal of any single edge gives a graph that is
  also generically rigid.
• A graph is k-connected if the removal of any
  set of less than k vertices means that it is
  still connected.
• Equivalently, it is k-connected if for any pair
  of vertices, one can find k paths joining them,
  with no common vertices except the end vertices.
• Theorem In two dimensions a graph with at least
  4 vertices is generically globally rigid if and
  only if it is 3-connected and redundantly rigid.
  

This connects a global property needed to solve
estimation problem to a local property holding
almost everywhere, for 2D graphs.
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2D Global rigidity--examples

• Theorem In two dimensions a graph with at least
  4 vertices is generically globally rigid if and
  only if it is 3-connected and redundantly rigid.
  
• Nontrivial consequence 6-connectivity is
  sufficient for global rigidity in two dimensions.

Wheel graphs with at least four vertices are
globally rigid
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2D Global rigidity --examples

• Theorem Let G(V,E) be a 2-connected graph. Let
  G2 (V,E? E2) be the graph formed from G by
  adding an edge between any two vertices with a
  common neighbor vertex in G. Then G2 is globally
  rigid.
• One gets G2 by doubling sensor radius!

Example where G is a cycle
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3D global rigidity

• In three dimensions If a graph is generically
  globally rigid, then it is redundantly rigid and
  at least 4-connected. There is a counterexample
  to the converse bipartite graph K5,5
• Necessary and sufficient conditions for 3D global
  rigidity are not known!
• In three dimensions, if a particular formation
  (graph plus distances) is globally rigid, it is
  not known whether almost all formations with the
  same graph are globally rigid.
• 12-connected 3D graphs might be always globally
  rigid

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Two dimensional trilateration
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Trilateration

• One way to construct globally rigid formations
  add a new node to a globally rigid formation,
  connecting it to d 1 nodes of the existing
  formation in general position (d spatial
  dimension). Then the new formation is generically
  globally rigid.
• A trilateration graph G in dimension d is one
  with an ordering of the vertices 1,d1,d2,.n
  such that the complete graph on the initial d1
  vertices is in G and from every vertex j gt d1,
  there are at least d1 edges to vertices earlier
  in the sequence.
• Trilateration graphs are generically globally
  rigid.

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Two dimensional trilateration
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Nongeneric behaviour
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Trilateration

• Theorem Let G(V,E) be a connected graph. Let
  G3 (V,E? E2 ? E3) be the graph formed from G by
  adding an edge between any two vertices at the
  ends of a path of 1,2 or 3 edges. Then G3 is a
  trilateration graph in 2 dimensions.
• Also G4 is a trilateration graph in 3 dimensions.

Hence if G(r) is connected, G(3r) is a
trilateration in two dimensions, and G(4r) is a
trilateration in three dimensions
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OUTLINE

• Aim of Presentation
• Control and Sensor Networks
• The sensor localization problem
• Rigidity and Global Rigidity
• Computational Complexity of Localization
• Conclusions and open problems

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Computational Complexity of Localization

• Brute force
• Minimize ? ?(i,j) - qi - qj 2

(i,j)? E

• Theorem Trilateration graph is realizable in
  polynomial time. (Proof relies on finding a seed
  in polynomial time--choose 3 out of n--and
  then realizing starting with seed, which is
  linear time)
• Theorem Realization for globally rigid weighted
  graphs (formations) that are realizable is
  NP-hard. (Proof relies on wheel graph and
  NP-hardness of set-partition -search problem.
  Heuristic argument on next slide)

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Computational Complexity

• Reflection possibilities are linked with
  computational complexity

Suppose all edge distances known for small
triangles. Localization goes working out from any
beacon. Triangle reflection possibilities grow
exponentially.
and reflection possibilities are only sorted out
when one gets to another beacon
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Trilateration localization protocol

• Sensors have 2 modes, localized and unlocalized
• Sensors determine distance from heard transmitter
• All sensors are pre-placed and listening

Localized mode Broadcast position Unlocalized
mode listen for broadcast IF broadcast from
(x,y) heard, determine distance to (x,y) IF 3
broadcasts heard, determine position and switch
to unlocalized mode
Decentralized algorithm!
But how fast?
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OUTLINE

• Aim of Presentation
• Control and Sensor Networks
• The sensor localization problem
• Rigidity and Global Rigidity
• Computational Complexity of Localization
• Conclusions and open problems

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Conclusions

• Rigidity is not enough you need global rigidity
  to localize ( beacons)
• Even then, computational complexity may be
  terrifying
• Polynomial or linear time localization is
  possible, given trilateration
• Change of sensing radius converts connectedness
  to global rigidity/trilateration
• For a class of random sensor graphs, there is not
  much difference between rigid, globally rigid and
  trilateration.
• Results for 3D are less developed.

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Some Open Problems

• Three dimensional graphs
• Partial localizability
• Islands of localizability in random graphs
• Asymmetric sensing radii
• Angular sensing
• Measures of health graphical, and geometric
• Motion of sensors
• Random graphs.

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Random sensor networks

• Sensors may be deployed randomly. We are
  interested in localization.
• The tool is random graph theory (which has been
  heavily studied)
• The random geometric graphs Gn(r) are the graphs
  associated with two dimensional formations with n
  vertices with all links of length less than r,
  where the vertices are points in 0,12
  generated by a two dimensional Poisson point
  process of intensity n

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Random geometric graphs

• There is a phase transition at which the graph
  becomes connected with high probability
• r O(sqrt(log n)/n)
• Connected means if Gn(r) has a minimum vertex
  degree of k then with high probability it is
  k-connected.
• Since 6-connectivity guarantees global rigidity,
  r O(sqrt(log n)/n) implies global rigidity
  with high connectivity.

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2D Random geometric graphs

• If Gn(r) is 2-connected, then Gn(2r) is
  globally rigid
• If Gn(r) is connected, then Gn(3r) is a
  trilateration
• Let r1 ,r2, r3, rg and rt denote the radius at
  which Gn(r) is connected, 2-connected,
  3-connected, globally rigid and a trilateration
  with probability 1-?. Then for large n,
• r6 ? rg ? r3 ? r2
• and 3r1 ? rt ? 2r2 ? rg

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Illustration of phase transition
Probability that Gn(r) is k-connected or globally
rigid
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Random geometric graphs

• All the above have an underlying condition of
  type
• r O(sqrt(log n)/n)
• If nr2/(log n) gt 8, then with high probability
  G is a trilateration graph, and is localizable in
  linear time given the positions of 3 connected
  nodes.
• Key observation for proof the density of nodes
  guarantees one can pick an initial triangle of 3,
  and then one at a time a new node connected to 3
  of those already chosen
• It is also localizable in a sort of decentralized
  fashion.

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Beacons and localization time

• Suppose sensors placed with Poisson intensity n
  and sensing radius r O(sqrt(log n)/n).
• If 3 beacons are placed closer than r, can
  localize in O(sqrtn/(log n) steps
• If beacons are placed on the unit square by
  Poisson process with intensity O(n/log n), can
  localize in O(sqrt(log n)) steps (Key idea
  probability that square of side O(r) has 3
  beacons is constant p so some such square has 3
  with very high probability)
• If beacons are placed by Poisson process of
  intensity O(n), localization can be effected in
  O(1) time with very high probability

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Illustration of phase transition
This and next graphs for 3D!
Phase transition is sharper for bigger n (Beacons
all sense one another)
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Theory vs simulation
Sensing radius required to get 95 localization
via trilateration (Beacons all sense one another)
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Speed of localization
Steps required to complete localization vs
sensing radius (Beacons all sense one another)