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. 2 Outline 1.Introduction on the Range-based Localization 2.Classical MDS vs Distributed Weighted MDS 3.Graph Realization Analogy 4.Robust Quadrilateral Localization Algorithm 5.New Test of Quadrilateral Robustness 6.Performance and Simulation Results

3. Lausanne, November 4, 2005 3 Range-based Localization 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 Statement of the Problem

4. Lausanne, November 4, 2005 4 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 Drawbacks 1.Full connectivity and symmetric links needed ( M = N x (N-1) / 2 ) 2.Centralized processing 3.Weakness toward measurement errors 4.Anchor nodes not taken into account Efficiency: Classical MDS uses Singular Value Decomposition (SVD), O(N 3 ) range measurement between nodes i and j unknown coordinates of nodes i and j

5. Lausanne, November 4, 2005 5 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 K anchor nodes (K  0) and N-K sensors with unknown coordinates weight  0 given to observed range between nodes i and j (weight=0 ~ no measurement available) 1. Build Cost Function (CF) The Distributed Weighted MDS (dwMDS) algorithm addresses all challenges posed by the application of Classical MDS to wireless sensor networks: 1.dwMDS allows for Anchor Nodes and for Missing Measurements 2.dwMDS enables Distributed Processing 3.dwMDS accounts for Measurement Errors 4.dwMDS lower complexity: O(NxL), L = number of iterations 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...

6. Lausanne, November 4, 2005 6 d b c a 2a: 3a: 3:3: 1: Globally rigid Rigid Flexible a d b c 2b: Localization and Graph Realization Analogy Abstraction: Wireless Sensor Network (WSN) abstracted as a graph where WSN nodes ~ Graph vertices Inter-node ranges ~ Graph edge lengths Analogy: Sensor nodes localization analogous to graph realization (~ finding out coordinates of graph vertices based on the constraints of edge lengths) WSN localization is unique (up to rotation, translation, reflection) iff its underlying graph is globally rigid (i.e., enough well distributed constraints)

7. Lausanne, November 4, 2005 7 Robust Quadrilateral Localization 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 Robust Quad (RQ): A globally rigid quadrilateral (4 nodes, 6 edges) that, in absence of measurement errors, can be unambiguously localized in isolation D A B C Extend Graph Realization Analogy Introducing Noisy Ranges: localize only nodes with high likelihood of unambiguous realization 1.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 2.Cluster Optimization (optional): Refine position estimates in each cluster with numerical optimization  dwMDS in our study cluster 1 cluster 2 cluster 3 3.Cluster Transformation: Shift, rotate, reflect local coordinate systems of pairs of adjacent clusters using shared nodes

8. Lausanne, November 4, 2005 8 Example of Cluster Localization Node A searches for robust quads = A B C D A D E F A B D E + + 1° 2° 3° A(0, 0) B C (C x, C y ) D = Tril(A,B,C) x y (d AB, 0) Cluster Nodes Localization 1° E = Tril(A,B,D) A B D  2° Tril(A,D,E)  A D E F 3° Node A estimates incrementally relative location of neighbors that can be unambiguously localized, following the chain of RQs and trilaterating along the way A C D E F B Cluster Head

9. Lausanne, November 4, 2005 9 Quad Robustness and Trilateration: Theory D A B C true ` d BC d AC ` C’ flipped correct vertex flipped vertex d1d1 d2d2 d DC ^ (Noisy Measured distance) d DC (True distance) Assumptions: 1.A, B, D exact coordinates known 2.exact A-C and B-C distances known 3.noisy D-C distance available Problem: estimate coords. of 4th unknown node C Solution: Notation: Condition of robustness:

10. Lausanne, November 4, 2005 10 Quad Robustness: Original Test D A B C robTriangle The shortest side The smallest angle 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 Problem: assess robustness of quad Assumption: All coordinates unknown  4 possible choices for “fourth unknown node” D A B C D A B C 4 triangles Quad(ABCD) is Robust robTriangle(ABC) & robTriangle(ABD) & robTriangle(ACD) & robTriangle(BCD) if for any choice of 4 th node to be located, estimation is robust (~ flip probability bounded) Recall Condition of robustness: Observe that

11. Lausanne, November 4, 2005 11 Ambiguous interval centered aroun d 0 Quad Robustness: New Test C’ D A B C true ` d BC d AC ` flipped d1d1 d2d2 correct vertex flipped vertex flipped vertex C’ correct vertex C Due to all 6 measured distances, we need 3 Rob Test for a sigle node 12 Rob Test in total 6 Rob Test (the other 6 are coincident) Must be outside of the ambiguous interval New Rob Test correct vertex flipped vertex

12. Lausanne, November 4, 2005 12 New vs. Original Robustness Test - 1 RMSE(RQ newTest) = 1.43 m RobQuad New Test RobQuad Original Test RMSE(RQ originalTest) = 3.34 m RMSE(CRB) = 0.52 m Without MDS alg. Refinement Settings: Terrain Dimension: X=30 m, Y = 30 m; N=16 (number of nodes); Fixed Random Topology  dist = 1 m (std. dev. range error); N T =200 (number of simulation trials) Full Connectivity  1 cluster only

13. Lausanne, November 4, 2005 13 New vs. Original Robustness Test - 2 RMSE(CRB) = 0.52 m 2 iterations RMSE(RQ originalTest) = 1.78 m RMSE(RQ newTest) = 0.62 m RMSE(RQ originalTest) = 1.21 m RMSE(RQ newTest) = 0.56 m 5 iterations With MDS alg. Refinement

14. Lausanne, November 4, 2005 14 New vs. Original Robustness Test - 3 New Test Advantages: 1.Better location accuracy 2.Allows faster convergence of MDS alg. (refinement)  Lower energy consumption (in terms of wireless transmissions) 1.21 m 0.56 m 0.52 RMSE vs Number of Iterations

15. Lausanne, November 4, 2005 15 Comments: 1.Accuracy strongly affected by network connectivity 2.With lower connectivity not all nodes’ located 3.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); N T =200 Random Topologies  dist = [0.3, 1, 2, 3] m (std. dev. range error) N iter = 15 (both for cluster and global MDS refinement) Connectivity: 1.Full Connectivity  15 neighbors/node 2.Maximum Ranging Distance 24 m  avg(Nbors) = 13 3.Maximum Ranging Distance 19 m  avg(Nbors) = 10

16. Lausanne, November 4, 2005 16 Summary

17. Lausanne, November 4, 2005 17 Thank you!

18. Lausanne, November 4, 2005 18 Performance evaluation