1. 1 Localization Technologies for Sensor Networks Craig Gotsman, Technion/Harvard Collaboration with: Yehuda Koren, AT&T Labs

2. 2 Sensor networks Set of nodes with ability to: –Measure parameters related to environment –Process information –Communicate / route –Estimate location –Communicate information to central processor Create a smart environment

3. 3 Hardware architecture sensors CPU/ memory radio battery Acoustic, seismic, image, magnetic, etc. interface Electro-magnetic interface Event detection Wireless communication with neighboring nodes In-node processing Limited battery supply Simple, small, cheap

4. 4 Potential applications Warning TornadoFire

5. 5 Potential applications Transportation –Monitor traffic conditions –Plan routes –Parking allocation

6. 6 Limitations Power is the bottleneck –Long distance communication impossible No pre-configuration or global knowledge –Achieve global goals through local interaction and self organization Limited computational power Price Use a very large number of sensors in a wide region

7. 7 Location-aware sensors Data should be location-stamped Geographic routing Region-targeted querying (123,456) (134,778) (234,466) (294,666) (372,862) (362,423) (432,553) (519,450) (589,703)

8. 8 Find a fully distributed algorithm for sensor localization Why not simply use GPS ??? The problem we address:

9. 9 Limitations of GPS Power Price Line of sight conditions Accuracy ? What else can we use ?

10. 10 Distance to neighboring sensors Received Signal Strength Indicator (RSSI) Time of Arrival (ToA) Technologies: 3 6 7 5 5 3 5 6 8 3 7 4 4 5 6 4 May be noisy Local distances  coordinates (??)

11. 11 Previous Solutions Anchor-based: –Some beacon nodes know their exact location –Other sensors estimate their location from nearby beacons

12. 12 Previous Solutions Incremental approaches: –Assign coordinates to a small core of sensors –Repeatedly assign coordinates to more sensors based on local calculations –Prone to error accumulation

13. 13 Previous Solutions Anchor Free Localization (AFL) [Priyantha et al., 2004] A two-stage, distributed approach: Based on connectivity, elect central, north, south, east and west sensors Estimate coordinates for rest of sensors 1. Initial coordinate assignment Optimization using gradient-descent to approximate measured distances 2. Accurate distributed layout

14. 14 Graph layout interpretation Given a graph with edge lengths A layout that realizes all edge lengths exist Only close nodes are connected ( “ disk graph ” ) Goal: Find this layout ! Two issues: 1.Layout existence - measured lengths are noisy ! 2.Layout uniqueness – graph ’ s rigidity

15. 15 Graph rigidity In 2D: –Global rigidity  3-connectivity –6-connectivity  Global rigidity Computing layout of rigid graph is NP-hard [Eren et al., 2004] 12 43 A graph is globally rigid if it has a unique embedding (up to distance preserving transformations) 2,3 1,4 1 42,3 Non rigid Globally rigid

16. 16 Beyond classical rigidity In disk-graphs - close nodes must be connected (up to noise) Non-adjacent nodes should be placed further apart Optimal layout Prunes redundant embeddings

17. 17 Graph drawing algorithms Energy-minimization algorithm using localized stress energy: Known problem: foldovers Why ???

18. 18 Localized stress and foldovers Graph not rigid – the energy does not address nonadjacent nodes Local minima – global optimization infeasible We must treat nonadjacent nodes explicitly

19. 19 Algorithm overview 1.“ Spectral ” initialization –Convex optimization – insensitive to initialization –Tends to generate fold-free layouts –Uses given distances inaccurately 2.Local stress optimization –Sensitive to initialization –Accurate use of distances – produces optimal layout when initialized properly –Global optimization – overcomes local noise A two-phase approach:

20. 20 Spectral layout - Laplacian  Based on [Hall, 1970]  Given a weighted graph with n nodes, w ij being the edge weights ( w ij =0 for non-adjacent nodes)  The Laplacian of the graph is the matrix L, where:

21. 21 Spectral layout - Goal Solve: - coordinates of node i Locate related nodes closely, while spreading nodes well Weighted squared distances between nodes Squared distances between nodes Edge weights express similarity/proximity Solution is Laplacian eigenvector(s) - scale invariant Does not use distances directly…

22. 22 Local stress minimization Relocate the nodes to minimize: Accurate optimization process, addressing measured distances directly Effective only when initialized smartly Generally, the spectral initialization is good

23. 23 Example Original layout 1000 sensors on 10x10 square, R=0.8 Stress with spectral initialization

24. 24 Example Original placement 715 sensors on 10-3 ring, R=0.8 Stress with spectral initialization

25. 25 Conclusions A fully distributed algorithm for sensor network layout Based on graph drawing methods Main challenge: layout computation with only local communication Still need to represent distances better in spectral layout

26. 26 Future Work Higher dimensions (3D ?) Improve spectral embedding using LLE Implement on real systems Incorporate more geometric info (e.g. angles) Multi-camera calibration Dynamic systems