Sensor Network Localization: Recent Developments Brian D O Anderson Australian National University...

Sensor Network Localization:Recent Developments

Brian D O Anderson

Australian National University and National ICT Australia

ANU 14 May 2010 2

• Aim of Presentation• Sensor Networks and Operational Problems• The Sensor Network Localization Problem• Rigidity and Global rigidity• Computational Complexity of Localization• Other Problems• Conclusions and open problems

OUTLINE

ANU 14 May 2010 3

Aim of Presentation

• To introduce problems involving sensor networks

• To explain the sensor network localization problem To introduce tools of rigidity to understand the essence of the sensor

network localization problem

• To indicate recent developments in sensor network localization

Collaborations

• I am reporting work by many people• I am reporting work by myself, often with

collaborators, including:

Soura Dasgupta, Tolga Eren, Jia Fang, Baris Fidan, David Goldenberg, Hatem Hmam, Baoqi Huang, Guoqiang Mao, Steve Morse, Sehchun Ng, Alireza Motevallian, Iman Shames, Tyler Summers, Jason Ta, Richard Yang, Brad Yu, Jeffrey Zhang

ANU 14 May 2010 4

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

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OUTLINE

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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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Typical operational problems

• Communications protocols• Conserving power• Loss of sensors• Self-configuration• Scalability• Decentralized operation versus centralized control

and• Knowing where the sensors are: Localization

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 obstaclesScanning with moving sensors• There may be an evader• Evader may destroy sensors• Sensors with different capabilities• Dynamic network• A priori or adaptive strategies?

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Management of energy usage

• Sensing/communications 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

Control Problems with Sensor Networks

Think of a soldier entering a building and emptying a canister of flying sensors the size of bees!

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

OUTLINE

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Sensor Networks

Depicts sensors with sensing radius r

r

Sensor

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Sensor Networks

Depicts sensors with sensing radius r

-highlighting ‘connected’ sensors

r

Sensor

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Sensor Networks

Sensor graph, with connection between two sensors if closer than r

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Beacon sensor

Normal sensor

Sensor Networks

• Beacon (Anchor) sensor positions known absolutely

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

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Beacon sensor

Normal sensor

Sensor Networks

• Beacon (Anchor) sensor positions known absolutely

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

Localization=Figuring out positions of all sensors

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Localization Questions: When and How?

• What are the conditions for network localizability, ie ability to determine the absolute position of all sensors?

• What is the computational complexity of network localization?

• The first question is an old one (Cayley, Menger, chemists)

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Localization Questions--footnotes

• 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, E) of vertices and edges and a length set of the edges.

• A formation is like a sensor network with the absolute beacon positions thrown away

• A graph is a formation with the length values thrown away• A formation with shape exactly determined by its graph and its

length set 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

Translation

ReflectionRotation

Original position

Absolute beacon positions eliminate this residual uncertainty in a globally rigid formation

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Sensor Networks and Formations

• Suppose: m beacons, n-m ordinary nodes; for 2 dimensions there are at least 3 beacons, and in 3 dimensions at least 4 beacons.

• Suppose all sensors and beacons are generically located• Theorem: Under these conditions, the network localization

problem is solvable if and only if the associated formation is globally rigid.

Henceforth, we will focus on formations and their global rigidity

Global rigidity: shape to within congruence is determined by length set and graph

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• Aim of Presentation

• Sensor Networks and operational problems• The sensor localization problem• Rigidity and Global Rigidity• Computational Complexity of Localization• Other Problems• Conclusions and open problems

OUTLINE

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Two dimensional rigidity examples

Not rigid. It can flex.

Fixing distances does not fix shape

Not enough fixed distances

Rigid. It cannot flex. It has more fixed

distances

So if enough distances are known to ensure the formation is rigid, is the shape thereby fully determined?

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Rigidity versus global rigidity

a

dcc

b

d

a

b

Formations are rigid with same distance set but are not congruent. NOT GLOBALLY RIGID!

Rigid, nonglobally rigid formations, can have a finite number of shape ‘ambiguities’.

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• We can repair the previous problem if we additionally fix the distance between b and a.

• This makes the graph redundantly rigid (and 3-connected).

a

c d

b

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• This makes the graph redundantly rigid (and 3-connected).

• Theorem: Globally rigid = redundantly rigid + 3-connected (in two dimensions)

• There is a counting condition (combinatorial test) for redundant rigidity—and so global rigidity.

a

c d

b

Three dimensional rigidity examples

• Again, there is a global rigidity notion, which is more than rigidity. 　　　　

• There is NO combinatorial test known for 3D global rigidity

• There is a test involving linear algebra for 2D and 3D global rigidity.

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Not rigid Rigid. But it has an ambiguity.

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2D Global rigidity--examples

“Wheel” graphs with at least four vertices are globally rigid

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2D Global rigidity --examples

• Consider a graph G where there are two non-intersecting paths between every pair of nodes (i.e. 2-connected G).

• Connect each node to a neighbor of its neighbor. This give G2.

• Theorem: All such G2 graphs are globally rigid.• One gets G2 by doubling sensor radius, i.e. from G(2r)!

Example where G is a cycle

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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 globally rigid.

Globally Rigid

Globally Rigid

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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 globally rigid.

Globally Rigid

Globally Rigid

Whole is globally rigid (2D case)

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Two dimensional trilateration

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Making Trilateration Graphs

• 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 quadrilateration graph in 3 dimensions.

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

quadrilateration in three dimensions

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OUTLINE

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• Brute force—with unclear complexity: let qi denote position of sensor i, d(i,j) distance between sensors i and j:

Minimize {d(i,j) - || qi – qj ||}2 or {d(i,j)2 - || qi – qj ||2}2

(i,j) E

Computational Complexity of Localization

• 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.

(i,j) E

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Trilateration graphs

• Once a seed for a trilateration graph is known, the localization of the nodes proceeds sequentially, and on a distributed basis.

• This means it is linear in the number of nodes• However, if there are errors in the distance

measurements then These may propagate (effects not well understood), but

countered by having more anchors than three Localization of any one node ought to somehow take

account of presence of noise! (more comment later)

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Bilateration graphs

• Localization can involve growing possibilities and resolving which possibility at the end.

• A wheel graph is an example.

• Start with 1,2,3. • 4 has two possibilities• Then 5 has four

possibilities

2

3

4

5

6

1

• Using 16 and 56, see that 6 has eight possibilities• Resolve by using 62.• Then resolve 5’s ambiguity • Finally resolve 4.

With N rim nodes, there are 2N-2 possibilities before resolution!

Although if there is a common sensing radius, with more than 12 rim nodes, there are more connections

Recursive Localization

• Assume that There are at least three anchors and all sensors are in

the convex hull of the anchors Every sensor is in the convex hull of three other

neighbor sensors (which means it is OK if it is in the convex hull of three or more neighbor sensors)

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Coordinatization Trick

• Coordinates of 1 can be expressed in terms of those of 2,3,4:

• Weighting coefficients are nonnegative, sum to 1

• Distances give weights!• Now use a recursion:

37

2

3

4

1

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Overall Equations

• Stack together equations like this for every node• Decentralized structure is maintained• Each row of update matrix sums to 1. Update matrix has all

nonnegative entries.• So it is a stochastic matrix.

This fact assures convergence—exponentially fast Anchors are fed in to the process.

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In 2D, a sensor adjacent to 3 non-collinear beacons can be uniquely localized if the distance measurements are accurate.

Localization with precise distances

Anchor 1

Anchor 2

Anchor 3

Sensor 0

d02

d03

d01

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Imprecise distances lead to inconsistency with respect to geometric relations, and sometimes cause localization algorithms to collapse.

Imprecise distances can be made more accurate and consistent by exploiting the geometric and algebraic relations between nodes.

Anchor 1

Anchor 2

Anchor 3

d03+ε3

d01+ε1

d02+ε2

Localization with imprecise distances

What point should we pick???

Noisy Localization

• Approach 1: Let x,y be the coordinates of the unknown agent. Let xi,yi be coordinates of i-th anchor or pseudo anchor at measured distance di. Choose x,y to minimize something like

• Approach 2:Use Cayley-Menger determinant: this gives a constrained optimization problem. For 3 anchors,

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Noisy Localization

• Is localizability a property that is robust to the presence of noise?

• Theorem If there are three or more noncollinear anchors graph is globally rigid distance measurement errors are suitably small

then there is a unique solution to least squares minimization problem for the sensor coordinates which is close to their correct positions. Position errors goes to zero as measurement errors go to zero.

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Error Propagation

• When one sensor is being localized using other sensors as pseudo-anchors, their position errors will propagate to the position error of the sensor being localized

• General propagation laws are not well understood• In random networks, it is understood there must be

a relation between anchor density, ordinary node density and localization errors.

• The relation is not known.

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It’s early days for error propagation research.


• Sensor Networks and operational problems• The sensor localization problem• Rigidity and Global Rigidity• Computational Complexity of Localization• Other problems:

Random sensor networks Connectivity based localization Robustness with link or sensor loss

• Conclusions and open problems

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OUTLINE

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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 all links of length less than r, where the vertices are points in [0,1]2 generated by a two-dimensional Poisson point process of intensity n

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Random geometric graphs• There is a phase transition of sensing radius at which the graph

becomes connected with high probability:

r = O([(log n)/n])

• At this order of sensing radius, the graph also becomes 2-connected, 3-connected,…

• For large n, connected implies (nontrivially): if Gn(r) has minimum vertex degree k then with high probability it is k-connected.

• Since 6-connectivity nontrivially guarantees global rigidity, r = O(√[(log n)/n]) implies global rigidity with high probability.

• Similar results apply for trilateration, which means computationally easily localization. One wants:

r > [8(log n)/n]

Estimating distances via connectivity

• Suppose a sensor network produced by a homogeneous Poisson process of density λ and conforming to the unit disk model of radius r

• Define M,P,Q to be the numbers of common neighbors and non-common neighbors of two nodes with distance d (d<r) S1 and S2 are common and non-common sensing areas

M, P, Q are independent Poisson r.v.’s with means λS1, λS2, λS2

• Define a parameter

21

1

)()()(2

)(2

SS

S

QEPEME

ME

MP Q

dρ is also a function of d , which enables us to estimate d through the numbers M, P, Q

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Distance estimator

• Theorem: If M, P and Q are independent Poisson random variables and their expected values define

the Maximum-Likelihood Estimator for ρ is

• The estimator for d is

• The theorem can be expanded in its application to handle modelling of noise in the sensing radius.

)ˆ1(6.1ˆ rd

otherwise,

0,

2

21

ˆ

QPM

QPM

M

)()()(2

)(2

QEPEME

ME

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Simulations of sensor localization using the distance estimation

• Sensor networks are with the log-normal shadowing model (standard two parameter model for noise in sensing radius) α: path-loss exponent σ: standard deviation of

shadowing effect λ: sensor density

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Robustness in Localization

• What is robustness? Tolerance of sensor or link loss.

• Why do we need robustness? Sensors nodes may die, due to power depletion or mechanical

malfunction. Communication Links may be disconnected, obstructed. Distance Measurements may not be accurate enough

• Typical questions: What sensor networks remain localizable after the loss of p

sensors, or q measurement links, or both.

• Early result: if loss of p sensors and q links can be tolerated, so can loss of p-s sensors and q + s links (s>0)

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Robustness to Edge Loss

• A counting criterion is available for graphs which remain rigid after losing k edges.

• A sensor network will remain localizable after the loss of p - 1 edges (links) with p = 2,3 if and only if: It remains connected after removal of any 2 vertices

and It remains rigid after losing p edges

• Differing but close necessary and sufficient conditions apply for p > 3.

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Robustness to Vertex Loss

• For a sensor network with N sensors to be localizable after loss of one vertex, it must have at least 2N links (i.e. edges in the graph).

• If a sensor network has N sensors and exactly 2N links, a necessary and sufficient condition for retention of localizability after loss of one vertex is It is 4-connected Removing any edge (link) results in a graph satisfying

certain counting conditions.

• There is little progress dealing with loss of >1 sensor

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OUTLINE

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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 and sometimes bilateration• Change of sensing radius converts connectedness to global

rigidity/trilateration• Noise in measurements in only beginning to be addressed• 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’• Motion of sensors• Error propagation due to inaccurate distance measurement• Coping with probability of sensor failure• Characterizing robustness in face of link/sensor loss• Malicious agents inserting incorrect measurements• …etc

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