Maurizio A. Spirito and Francesco Sottile Lausanne, November 4, 2005

1. Performance Analysis and Enhancement of Certain Range-based Localization Algorithms for Wireless Ad-Hoc Sensor Networks Maurizio A. Spirito and Francesco Sottile Lausanne, November 4, 2005

2. Outline • Introduction on the Range-based Localization • Classical MDS vs Distributed Weighted MDS • Graph Realization Analogy • Robust Quadrilateral Localization Algorithm • New Test of Quadrilateral Robustness • Performance and Simulation Results

3. Range-based Localization Statement of the Problem Given: • A set of N points in the plane (nodes coordinates) • Coordinates of 0  K<N points (anchors) • M  N x (N-1) distances between pairs of points Find: • Positions of all N- K points of unknown coordinates Relative Localization (anchor-less): If K = 0, the estimated topology is subject to translation, rotation, reflection

4. unknown coordinates of nodes i and j range measurement between nodes iandj Classical Multidimensional Scaling • Objective: the Classical MDS algorithm minimizes the so-called “stress” 2 • Least Squares Optimality Criterion in Euclidean space: minimization of differences between ALL estimated distances and measured distances from ALL edges • Efficiency: Classical MDS uses Singular Value Decomposition (SVD), O(N3) • Drawbacks • Full connectivity and symmetric links needed (M = N x (N-1) / 2) • Centralized processing • Weakness toward measurement errors • Anchor nodes not taken into account

5. 3. Minimize global CF by iterative distributed minimization of local CFs at each node 2. Write CF as sum of local CFs, associated to each node to be located 1. Build Cost Function (CF) weight0 given to observed range between nodes i and j (weight=0 ~ no measurement available) . . . K anchor nodes (K0) andN-K sensors with unknown coordinates The Distributed Weighted MDS (dwMDS) algorithm addresses all challenges posed by the application of Classical MDS to wireless sensor networks: • dwMDS allows for Anchor Nodes and for Missing Measurements • dwMDS enables Distributed Processing • dwMDS accounts for Measurement Errors • dwMDSlower complexity:O(NxL), L = number of iterations Distributed Weighted MDS vs. Classical MDS J. A. Costa, N. Patwari, A. O. Hero III, ” Adaptive Distributed Multidimensional Scaling for Localization in Sensor Networks”, 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 05), March 18-23, 2005, Philadelphia, PA, USA

6. 3a: • Analogy: Sensor nodes localization analogous to graphrealization (~ finding out coordinates of graphvertices based on the constraints of edge lengths) WSNlocalization is unique (up to rotation, translation, reflection) iff its underlying graph is globally rigid (i.e., enough well distributed constraints) 1: 3: Globallyrigid Flexible Rigid a b 2a: d c b 2b: a d c Localization and Graph Realization Analogy • Abstraction: Wireless SensorNetwork (WSN) abstracted as a graph where WSN nodes~ Graph vertices Inter-node ranges ~ Graph edgelengths

7. cluster 1 cluster 2 cluster 3 Robust Quad (RQ): A globally rigid quadrilateral (4 nodes, 6 edges) that, in absence of measurement errors, can be unambiguously localized in isolation D C • Cluster Localization: Each node searches for all RQs in its cluster, finds the largest sub-graph made only of overlapping RQs and estimates the coordinates of its neighbors that can be unambiguously localized • Cluster Optimization (optional): Refine position estimates in each cluster with numerical optimization  dwMDS in our study A B Robust Quadrilateral Localization Extend Graph Realization Analogy Introducing Noisy Ranges:localize only nodes with high likelihood of unambiguous realization • Cluster Transformation: Shift, rotate, reflect local coordinate systems of pairs of adjacent clusters using shared nodes D. Moore, J. Leonard, D. Rus, S. Teller, “Robust Distributed Network Localization with Noisy Range Measurements,” SenSys’04, November 3–5, 2004, Baltimore, Maryland, USA

8. D C D D C B 1° 2° D D 3° B F = + + B F F E A E A E A Cluster Head E A A Cluster Nodes Localization C (Cx, Cy) 2° 3° D = Tril(A,B,C) D 1° x B B Þ Þ (dAB, 0) y Tril(A,D,E) A A(0, 0) E = Tril(A,B,D) Example of Cluster Localization Node A estimates incrementally relative location of neighbors that can be unambiguously localized, following the chain of RQs and trilaterating along the way Node A searches for robust quads

9. dDC ^ (True distance) dDC (Noisy Measured distance) ` d1 ` dBC correct vertex flipped vertex d2 dAC C’ Notation: flipped Solution: Condition of robustness: Quad Robustness and Trilateration: Theory • Assumptions: • A, B, D exact coordinates known • exact A-C and B-C distances known • noisy D-C distance available • Problem: estimate coords. of 4th unknown node C true D C A B

10. D C D C 4 triangles A A B B robTriangle(ABC) & robTriangle(ABD) & robTriangle(ACD) & robTriangle(BCD) if for any choice of 4th node to be located, estimation is robust (~ flip probability bounded) Quad(ABCD) is Robust Recall Condition of robustness: The shortest side robTriangle The smallest angle Observe that Quad Robustness: Original Test Problem: assess robustness of quad Assumption: All coordinates unknown  4 possible choices for “fourth unknown node” D C A B D. Moore, J. Leonard, D. Rus, S. Teller, “Robust Distributed Network Localization with Noisy Range Measurements,” SenSys’04, November 3–5, 2004, Baltimore, Maryland, USA

11. ` true d1 D C ` dBC correct vertex A B flipped vertex d2 dAC correct vertex C’ New Rob Test flipped vertex flipped Must be outside of the ambiguous interval correct vertex C flipped vertex C’ Ambiguous interval centered aroun d0 12 Rob Test in total 6 Rob Test (the other 6 are coincident) Quad Robustness: New Test Due to all 6 measured distances, we need 3 Rob Test for a sigle node

12. Settings: • Terrain Dimension: X=30 m , Y = 30 m; N=16(number of nodes);Fixed Random Topology • dist = 1 m(std. dev. range error);NT=200(number of simulation trials) • Full Connectivity 1 cluster only RobQuad New Test RobQuad Original Test Without MDS alg. Refinement RMSE(RQ originalTest) = 3.34 m RMSE(RQ newTest) = 1.43 m RMSE(CRB) = 0.52 m New vs. Original Robustness Test - 1

13. 2 iterations 5 iterations RMSE(RQ originalTest) = 1.21 m RMSE(RQ newTest) = 0.56 m RMSE(RQ originalTest) = 1.78 m RMSE(RQ newTest) = 0.62 m RMSE(CRB) = 0.52 m New vs. Original Robustness Test - 2 With MDS alg. Refinement

14. RMSE vs Number of Iterations 1.21 m 0.56 m 0.52 New vs. Original Robustness Test - 3 • New Test Advantages: • Better location accuracy • Allows faster convergence of MDS alg. (refinement)  Lower energy consumption (in terms of wireless transmissions)

15. Comments: • Accuracy strongly affected by network connectivity • With lower connectivity not all nodes’ located • New test outperforms Original test even with lower connectivity Accuracy vs. Connectivity • Settings: • Terrain Dimension: X=30 m , Y = 30 m; N=16(number of nodes);NT=200 Random Topologies • dist = [0.3, 1, 2, 3] m(std. dev. range error) • Niter = 15 (both for cluster and global MDS refinement) • Connectivity: • Full Connectivity  15 neighbors/node • Maximum Ranging Distance 24 m  avg(Nbors) = 13 • Maximum Ranging Distance 19 m  avg(Nbors) = 10

16. Summary

17. Thank you!

18. Performance evaluation