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

2. ANU 14 May 20102 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

3. ANU 14 May 20103 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

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 20104

5. 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 20105 OUTLINE

6. ANU 14 May 20106 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

7. ANU 14 May 20107 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

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? ANU 14 May 20108 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!

9. ANU 14 May 20109 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

10. ANU 14 May 201010 Sensor Networks Depicts sensors with sensing radius r r Sensor

11. ANU 14 May 201011 Sensor Networks Depicts sensors with sensing radius r -highlighting ‘connected’ sensors r Sensor

12. ANU 14 May 201012 Sensor Networks Sensor graph, with connection between two sensors if closer than r

13. ANU 14 May 201013 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

14. ANU 14 May 201014 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

15. ANU 14 May 201015 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)

16. ANU 14 May 201016 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

17. ANU 14 May 201017 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.

18. ANU 14 May 201018 Congruent Formations Translation Reflection Rotation Original position Absolute beacon positions eliminate this residual uncertainty in a globally rigid formation

19. ANU 14 May 201019 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

20. ANU 14 May 201020 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

21. ANU 14 May 201021 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?

22. ANU 14 May 201022 Rigidity versus global rigidity a d c c 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’.

23. ANU 14 May 201023 Rigidity versus global rigidity 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

24. ANU 14 May 201024 Rigidity versus global rigidity 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

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. ANU 14 May 201025 Not rigid Rigid. But it has an ambiguity.

26. ANU 14 May 201026 2D Global rigidity--examples “Wheel” graphs with at least four vertices are globally rigid

27. ANU 14 May 201027 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 G 2. Theorem: All such G 2 graphs are globally rigid. One gets G 2 by doubling sensor radius, i.e. from G(2r)! Example where G is a cycle

28. ANU 14 May 201028 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

29. ANU 14 May 201029 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)

30. ANU 14 May 201030 Two dimensional trilateration

31. ANU 14 May 201031 Making Trilateration Graphs Theorem: Let G=(V,E) be a connected graph. Let G 3 = (V,E  E 2  E 3 ) 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 G 3 is a trilateration graph in 2 dimensions. Also G 4 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

32. ANU 14 May 201032 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

33. ANU 14 May 201033 Brute force—with unclear complexity: let q i denote position of sensor i, d(i,j) distance between sensors i and j: Minimize  { d(i,j) - || q i – q j || } 2 or  { d(i,j) 2 - || q i – q j || 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

34. ANU 14 May 201034 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)

35. ANU 14 May 2010 35 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 2 N-2 possibilities before resolution! Although if there is a common sensing radius, with more than 12 rim nodes, there are more connections

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 201036

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: 37 2 3 4 1 ANU 14 May 2010

38. 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. 39 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 d 02 d 03 d 01

40. ANU 14 May 201040 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 d 03 +ε 3 d 01 +ε 1 d 02 +ε 2 Localization with imprecise distances What point should we pick???

41. Noisy Localization Approach 1: Let x,y be the coordinates of the unknown agent. Let x i,y i be coordinates of i-th anchor or pseudo anchor at measured distance d i. 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 201041

42. 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. ANU 14 May 201042

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. ANU 14 May 201043 It’s early days for error propagation research.

44. 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 201044 OUTLINE

45. ANU 14 May 201045 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 G n (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

46. ANU 14 May 201046 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 G n (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]

47. 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)  S 1 and S 2 are common and non-common sensing areas  M, P, Q are independent Poisson r.v.’s with means λS 1, λS 2, λS 2 Define a parameter M P Q d ρ is also a function of d, which enables us to estimate d through the numbers M, P, Q 47ANU 14 May 2010

48. 48 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. 48ANU 14 May 2010

49. 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 49ANU 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) 50ANU 14 May 2010

51. 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. ANU 14 May 201051

52. 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 ANU 14 May 201052

53. ANU 14 May 201053 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

54. ANU 14 May 201054 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.

55. ANU 14 May 201055 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

56. ANU 14 May 201056 Questions