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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
      weight0 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 (K0) 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

								
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