1. Rennes 081004 1 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)

2. Rennes 081004 2 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

3. Rennes 081004 3 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. Rennes 081004 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

5. Rennes 081004 5 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

6. Rennes 081004 6 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

7. Rennes 081004 7 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

8. Rennes 081004 8 Sensor Networks Depicts sensors with sensing radius r r Sensor

9. Rennes 081004 9 Sensor Networks Sensor graph, with connection between two sensors if closer than r

10. Rennes 081004 10 Beacon sensor Normal sensor Sensor Networks Beacon sensor positions known absolutely Inter-neighbour distances known (edge distance for each edge of graph) plus inter-beacon distances

11. Rennes 081004 11 Sensor Networks Beacon sensor positions known absolutely Inter-neighbour distances known (edge distance for each edge of graph) plus inter-beacon distances Beacon sensor Normal sensor

12. Rennes 081004 12 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)

13. Rennes 081004 13 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

14. Rennes 081004 14 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.

15. Rennes 081004 15 Congruent Formations Translation Reflection Original position Absolute beacon positions eliminate this residual uncertainty Rotation

16. Rennes 081004 16 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

17. Rennes 081004 17 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

18. Rennes 081004 18 Aim of Presentation Control and Sensor Networks The sensor localization problem Rigidity and Global Rigidity Computational Complexity of Localization Conclusions and open problems OUTLINE

19. Rennes 081004 19 Let F be a formation with vertex and edge sets V and L. Imagine it is moving. Let q i 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: (q i - q j ) (Dq i - Dq j )= 0 Can write this equation for every edge: R(F)(Dq) = 0 Here R(F) is the rigidity matrix. For a rigid formation: 1. One rotation and two translations give nullspace of dimension 3 in two dimensions 2. Three rotations and three translations give nullspace of dimension 6 in two dimensions Rigidity

20. Rennes 081004 20 Two dimensional rigidity examples Not rigid. One degree of freedom “floppiness”. R(F) has 4 dimensional nullspace Rigid. R(F) has 3 dimensional nullspace

21. Rennes 081004 21 Three dimensional rigidity examples Not rigid. R(F) has 7 dimensional nullspace Rigid. R(F) has 6 dimensional nullspace

22. Rennes 081004 22 Rigid Formations v1v1 v2v2 v3v3 v4v4 (1,2)x 1 - x 2 y 1 - y 2 00 (1,3)x 1 - x 2 y 1 - y 2 0 0 (1,4)000x 1 - x 2 y 1 - y 2 (2,3)0x 1 - x 2 y 1 - y 2 0 (2,4)0x 1 - x 2 y 1 - y 2 0 (3,4)00x 1 - x 2 y 1 - y 2 Sample two dimensional Rigidity Matrix--a Matrix Net ∑ x i M i +y i N i in coordinates x i and y i of points.

23. Rennes 081004 23 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- Laman’s theorem, with fast algorithm for testing No such result is available in three dimensions. (Partial results exist)

24. Rennes 081004 24 Rigidity versus global rigidity a d c a c d b b 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.

25. Rennes 081004 25 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) a c d b

26. Rennes 081004 26 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.

27. Rennes 081004 27 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

28. Rennes 081004 28 2D Global rigidity --examples Theorem: Let G=(V,E) be a 2-connected graph. Let G 2 = (V,E  E 2 ) be the graph formed from G by adding an edge between any two vertices with a common neighbor vertex in G. Then G 2 is globally rigid. One gets G 2 by doubling sensor radius! Example where G is a cycle G G2G2

29. Rennes 081004 29 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 K 5,5 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

30. Rennes 081004 30 Two dimensional trilateration

31. Rennes 081004 31 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,…d+1,d+2,….n such that the complete graph on the initial d+1 vertices is in G and from every vertex j > d+1, there are at least d+1 edges to vertices earlier in the sequence. Trilateration graphs are generically globally rigid.

32. Rennes 081004 32 Two dimensional trilateration

33. Rennes 081004 33 Nongeneric behaviour Globally rigid a b c da b c d’ But not globally rigid when a,b,c are collinear! Globally rigid

34. Rennes 081004 34 Trilateration 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 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

35. Rennes 081004 35 Aim of Presentation Control and Sensor Networks The sensor localization problem Rigidity and Global Rigidity Computational Complexity of Localization Conclusions and open problems OUTLINE

36. Rennes 081004 36 Brute force: Minimize  {  (i,j) - || q i - q j ||} 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. (Proof relies on wheel graph and NP-hardness of set-partition -search problem. Heuristic argument on next slide)

37. Rennes 081004 37 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

38. Rennes 081004 38 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?

39. Rennes 081004 39 Aim of Presentation Control and Sensor Networks The sensor localization problem Rigidity and Global Rigidity Computational Complexity of Localization Conclusions and open problems OUTLINE

40. Rennes 081004 40 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.

41. Rennes 081004 41 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.

42. Rennes 081004 42 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 n vertices 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

43. Rennes 081004 43 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 G n (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.

44. Rennes 081004 44 2D Random geometric graphs If G n (r) is 2-connected, then G n (2r) is globally rigid If G n (r) is connected, then G n (3r) is a trilateration Let r 1,r 2, r 3, r g and r t denote the radius at which G n (r) is connected, 2-connected, 3- connected, globally rigid and a trilateration with probability 1- . Then for large n, r 6  r g  r 3  r 2 and3r 1  r t  2r 2  r g

45. Rennes 081004 45 Illustration of phase transition Probability that G n (r) is k-connected or globally rigid

46. Rennes 081004 46 Random geometric graphs All the above have an underlying condition of type r = O(sqrt[(log n)/n]) If nr 2 /(log n) > 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.

47. Rennes 081004 47 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(sqrt[n/(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

48. Rennes 081004 48 Illustration of phase transition Phase transition is sharper for bigger n (Beacons all sense one another) This and next graphs for 3D!

49. Rennes 081004 49 Theory vs simulation Sensing radius required to get 95% localization via trilateration (Beacons all sense one another)

50. Rennes 081004 50 Speed of localization Steps required to complete localization vs sensing radius (Beacons all sense one another)