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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 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 Lausanne, November 4, 2005 2 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 B1 Anchor node Unknown node B2 A3 B4 B3 A2 B6 A1 B5 Relative Localization (anchor-less): If K = 0, the estimated topology is subject to translation, rotation, reflection Lausanne, November 4, 2005 3 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 unknown coordinates of nodes i and j d ij N N 2 ij 2 ij z i z j i 1 j i 1 range measurement between nodes i and j • Efficiency: Classical MDS uses Singular Value Decomposition (SVD), O(N3) 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 Lausanne, November 4, 2005 4 Distributed Weighted MDS vs. Classical MDS 1. Build Cost Function (CF) 2. Write CF as sum 3. Minimize global of local CFs, CF by iterative weight0 given to observed range distributed associated to each between nodes i and j (weight=0 ~ no minimization of local node to be located measurement available) CFs at each node N K N K w d ij N 2 ij ij 2 . . . 2 i2 c i 1 i 1 j i 1 K anchor nodes (K0) and N-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: 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 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 Lausanne, November 4, 2005 5 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) Globally rigid Rigid Flexible a b 1: 2a: 3a: d c b 2b: a 3: d c Lausanne, November 4, 2005 6 Robust Quadrilateral Localization Extend Graph Realization Analogy Introducing Noisy Ranges: localize only nodes with high likelihood of unambiguous realization D C Robust Quad (RQ): A globally rigid quadrilateral (4 nodes, 6 edges) that, in absence of measurement errors, can be unambiguously localized in isolation A B 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 3. Cluster Transformation: Shift, rotate, cluster 3 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 Lausanne, November 4, 2005 7 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 C C 1° D 2° D D 3° D = + + B B B F F E E E A Cluster Head A A A Cluster Nodes Localization 1° C (Cx, Cy) 2° D = Tril(A,B,C) D 3° D x B B (dAB, 0) F y Tril(A,D,E) A(0, 0) A A E E = Tril(A,B,D) Lausanne, November 4, 2005 8 Quad Robustness and Trilateration: Theory Assumptions: 1. A, B, D exact coordinates known dDC (True distance) 2. exact A-C and B-C distances known ^ dDC 3. noisy D-C distance available (Noisy Measured distance) Problem: estimate coords. of 4th unknown node C ˆ d DC d DC eDC eDC ~ N (0, dist ) true d1 D C ˆ ˆ ˆ C if d DC d1 d DC d 2 C correct vertex dBC C otherwise flipped vertex A` B ` 1 P(flip C ) P eDC (d 2 d1 ) d 2 d1 d2 2 1 dAC Notation: d err (d 2 d1 ) 2 C’ flipped Solution: Minimize P(flip C ) Maximize d err d1 dist (C , D) Condition of robustness: d err d min d 2 dist (C , D) Lausanne, November 4, 2005 9 Quad Robustness: Original Test Problem: assess robustness of quad Assumption: All coordinates unknown 4 possible choices for “fourth unknown node” D C D C D C 4 triangles BCD ACD ABD ABC A B A B A B robTriangle(ABC) & Quad(ABCD) is Robust if for any choice of 4th node to be located, estimation is robust robTriangle(ABD) & robTriangle(ACD) & (~ flip probability bounded) robTriangle(BCD) Recall Condition of robustness: d err 1 d 2 d1 d min 2 Observe that d err b sin ( ) 2 robTriangle b sin 2 () dmin b b 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 Lausanne, November 4, 2005 10 Quad Robustness: New Test d1 true ˆ C ˆ C if d DC d1 d DC d 2 ˆ correct vertex D C C otherwise flipped vertex dBC ˆ A` B` ˆ C if d DC d 0 correct vertex d d1 d 2 C C otherwise flipped vertex 0 2 d2 derr 2 (d2 d1) dmin dAC 1 C’ flipped New Rob Test dˆ Must be outside of the DC ambiguous interval correct vertex C flipped vertex C’ Due to all 6 measured distances, Ambiguous interval we need 3 Rob Test for a sigle node ˆ d DC centered aroun d0 R% (d 2 d1 ) 12 Rob Test in total 0 d1 d2 R% [0,1) d1 d 2 d0 2 6 Rob Test (the other 6 are coincident) Lausanne, November 4, 2005 11 New vs. Original Robustness Test - 1 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 Without MDS alg. Refinement RobQuad New Test RobQuad Original Test 35 RMSE(RQ newTest) = 1.43 m RMSE(CRB) = 0.52 m 35 RMSE(RQ originalTest) = 3.34 m 30 9 30 9 3 3 7 7 25 5 25 5 4 4 8 8 20 20 y, meters y, meters 12 12 15 15 2 2 15 0 15 0 13 13 11 10 11 10 1 14 1 14 10 10 5 5 6 0 6 0 -5 -5 -5 0 5 10 15 20 25 30 35 -5 0 5 10 15 20 25 30 35 x, meters x, meters Lausanne, November 4, 2005 12 New vs. Original Robustness Test - 2 With MDS alg. Refinement 2 iterations 5 iterations 35 35 30 9 30 9 3 3 7 7 25 5 25 5 4 4 8 8 20 20 y, meters y, meters 12 12 15 2 15 2 15 0 15 0 13 13 11 11 10 10 1 14 1 14 10 10 5 5 6 6 0 0 -5 -5 -5 0 5 10 15 20 25 30 35 -5 0 5 10 15 20 25 30 35 x, meters x, meters RMSE(RQ originalTest) = 1.78 m RMSE(RQ originalTest) = 1.21 m RMSE(RQ newTest) = 0.62 m RMSE(CRB) = 0.52 m RMSE(RQ newTest) = 0.56 m Lausanne, November 4, 2005 13 New vs. Original Robustness Test - 3 RMSE vs Number of Iterations 3.5 RQ oldTest RQ newTest 3 CRB 2.5 1.21 m RMSE [m] 2 loc 1.5 0.56 m 1 0.52 0.5 0 0 5 10 15 20 Number of iterations New Test Advantages: 1. Better location accuracy 2. Allows faster convergence of MDS alg. (refinement) Lower energy consumption (in terms of wireless transmissions) Lausanne, November 4, 2005 14 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) 6 Connectivity: Full connect CMDS 1. Full Connectivity 15 Full connect RQnewTest 5 Full connect RQoldTest neighbors/node 13 nbrs/node RQnewTest 2. Maximum Ranging Distance 13 nbrs/node RQoldTest 24 m avg(Nbors) = 13 RMSEloc [m] 4 10 nbrs/node RQnewTest 3. Maximum Ranging Distance 10 nbrs/node RQoldTest 19 m avg(Nbors) = 10 3 Comments: 2 1. Accuracy strongly affected by network connectivity 2. With lower connectivity not 1 all nodes’ located 3. New test outperforms 0 Original test even with 0 0.5 1 1.5 2 2.5 3 lower connectivity [m] dist Lausanne, November 4, 2005 15 Summary Lausanne, November 4, 2005 16 Thank you! Lausanne, November 4, 2005 17 Performance evaluation (a) (b) Exact Topology Estimated Topology 20 13 14 12 12 13 15 5 8 15 9 4 8 9 11 0 14 5 0 eiP ( xi xi ) 2 ( yi yi ) 2 ˆ ˆ i 1, N y [m] y [m] 10 10 5 4 15 e ( xi xi ) 2 ( yi yi ) 2 -5 i 1, N 6 7 10 6 P ˆ ˆ 1 1 N e i P 5 11 P 0 1 2 3 i 2 -10 7 N i 1 0 3 0 5 10 15 20 -10 -5 0 5 x [m] x [m] ei P 1 N P P 2 (c) (d) N 1 i 1 Adjusted Estimated Topology Exact and Location Error 20 13 14 20 12 13 12 14 RMSE P P 15 2 2 15 15 15 9 9 11 11 8 10 y [m] y [m] 10 8 i 1, N 5 10 4 eiP (7xi xi ) 2 ( yi yi ) 2 ˆ ˆ 4 5 6 10 7 6 5 1 5 0 1 0 3 2 3 2 0 0 0 5 10 15 20 0 5 10 15 20 x [m] x [m] Lausanne, November 4, 2005 18