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Sensor Network Localization: Recent Developments

Sensor Network Localization: Recent Developments. Brian D O Anderson Australian National University and National ICT Australia. OUTLINE. Aim of Presentation Sensor Networks and Operational Problems The Sensor Network Localization Problem Rigidity and Global rigidity

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Sensor Network Localization: Recent Developments

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  1. Sensor Network Localization:Recent Developments Brian D O Anderson Australian National University and National ICT Australia

  2. OUTLINE • 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 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 ANU 14 May 2010

  4. 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

  5. OUTLINE • 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 ANU 14 May 2010

  6. 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 ANU 14 May 2010

  7. 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 ANU 14 May 2010

  8. 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/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! ANU 14 May 2010

  9. OUTLINE • 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 ANU 14 May 2010

  10. Sensor Networks Sensor r Depicts sensors with sensing radius r ANU 14 May 2010

  11. Sensor Networks Sensor r Depicts sensors with sensing radius r -highlighting ‘connected’ sensors ANU 14 May 2010

  12. Sensor Networks Sensor graph, with connection between two sensors if closer than r ANU 14 May 2010

  13. Sensor Networks Beacon sensor Normal sensor • Beacon (Anchor) sensor positions known absolutely • Inter-neighbour distances known (edge distance for each edge of graph) plus inter-beacon distances ANU 14 May 2010

  14. Sensor Networks Beacon sensor Normal sensor Localization = Figuring out positions of all sensors • Beacon (Anchor) sensor positions known absolutely • Inter-neighbour distances known (edge distance for each edge of graph) plus inter-beacon distances ANU 14 May 2010

  15. 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) ANU 14 May 2010

  16. 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 ANU 14 May 2010

  17. 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 rotationand/or reflection. ANU 14 May 2010

  18. Congruent Formations Original position Translation Rotation Reflection Absolute beacon positions eliminate this residual uncertainty in a globally rigid formation ANU 14 May 2010

  19. 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 ANU 14 May 2010

  20. OUTLINE • 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 ANU 14 May 2010

  21. 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? ANU 14 May 2010

  22. Rigidity versus global rigidity a c d b a b c d 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’. ANU 14 May 2010

  23. Rigidity versus global rigidity a c d b • 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). ANU 14 May 2010

  24. Rigidity versus global rigidity a c d b • 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. ANU 14 May 2010

  25. 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. Not rigid Rigid. But it has an ambiguity. ANU 14 May 2010

  26. 2D Global rigidity--examples “Wheel” graphs with at least four vertices are globally rigid ANU 14 May 2010

  27. 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 ANU 14 May 2010

  28. 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 ANU 14 May 2010

  29. 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) ANU 14 May 2010

  30. Two dimensional trilateration ANU 14 May 2010

  31. 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 ANU 14 May 2010

  32. OUTLINE • 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 ANU 14 May 2010

  33. Computational Complexity of Localization • Brute force—with unclear complexity: let qidenote 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 (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. ANU 14 May 2010

  34. 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) ANU 14 May 2010

  35. Bilateration graphs 5 6 4 1 3 2 • 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 • 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 ANU 14 May 2010

  36. 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) ANU 14 May 2010

  37. 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: 2 1 3 4 ANU 14 May 2010

  38. 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. ANU 14 May 2010

  39. Localization with precise distances In 2D, a sensor adjacent to 3 non-collinear beacons can be uniquely localized if the distance measurements are accurate. Anchor 2 d02 Sensor 0 d03 d01 Anchor 3 Anchor 1 ANU 14 May 2010

  40. Localization with imprecisedistances Imprecise distances lead to inconsistency with respect to geometric relations, and sometimes cause localization algorithms to collapse. Anchor 2 What point should we pick??? d02+ε2 d03+ε3 d01+ε1 Anchor 3 Anchor 1 Imprecise distances can be made more accurate and consistent by exploiting the geometric and algebraic relations between nodes. ANU 14 May 2010

  41. Noisy Localization • Approach 1: Let x,y be the coordinates of the unknown agent. Let xi,yibe 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, ANU 14 May 2010

  42. Noisy Localization • Is localizability a property that is robust to the presence of noise? • TheoremIf • 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. ANU 14 May 2010

  43. 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. It’s early days for error propagation research. ANU 14 May 2010

  44. OUTLINE • Aim of Presentation • 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 ANU 14 May 2010

  45. 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 ANU 14 May 2010

  46. Random geometric graphs • There is a phase transition of sensing radiusat 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] ANU 14 May 2010

  47. Estimating distances via connectivity d Q P M • 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 S2are common and non-common sensing areas • M, P, Q are independentPoisson r.v.’s with means λS1, λS2, λS2 • Define a parameter ρ is also a function of d , which enables us to estimate d through the numbers M, P, Q ANU 14 May 2010

  48. Distance estimator Theorem: If M, P and Q are independentPoisson random variables and their expected values define the Maximum-Likelihood Estimator for ρis The estimator for d is The theorem can be expanded in its application to handle modelling of noise in the sensing radius. 48 ANU 14 May 2010

  49. 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 49 ANU 14 May 2010

  50. 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) ANU 14 May 2010

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