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Localization in wireless sensor ad-hoc networks. Xiaobo Long ECSE 6962 course presentation. Introduction. What is localization Determine node locations in ad-hoc sensor networks Distributed Without relying on external infrastructure Without base stations, satellites, etc.

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Localization in wireless sensor ad-hoc networks

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Localization in wireless sensor ad hoc networks l.jpg

Localization in wireless sensor ad-hoc networks

Xiaobo Long

ECSE 6962 course presentation


Introduction l.jpg

Introduction

  • What is localization

    • Determine node locations in ad-hoc sensor networks

      • Distributed

      • Without relying on external infrastructure

        • Without base stations, satellites, etc.

        • GPS: too expensive

          • not suitable for low-cost, ad-hoc sensor networks

  • Why need localization

    • Routing techniques require knowledge of location

    • Sensing tasks require knowledge of location


Algorithms requirements l.jpg

Algorithms requirements

  • Truly distributed

    • employed on large-scale ad-hoc sensor networks

  • Self-organizing

    • do not depend on global infrastructure

  • Robust

    • be tolerant to node failures and range errors

  • Energy efficient

    • require little computation and communication


  • Assumptions l.jpg

    Assumptions

    • Nodes are randomly distributed

    • 2-D environment

    • Static network

      • Nodes don’t move

    • Anchor nodes

      • Have a priori knowledge of their own position

        • with respect to some global coordinate system


    Important parameters l.jpg

    Important parameters

    • Range errors

      • describe accuracy of the distance measurements

      • effect accuracy of localization algorithms

  • Connectivity of the nodes

    • i.e., the average number of neighbors

  • Anchor fraction

    • some anchor nodes have a priori knowledge of their own position

  • Three context parameters are dependent


  • General algorithms lr03 three phases l.jpg

    General algorithms [LR03]-Three phases

    • Distance to anchors

      • Determine the distances between unknowns and anchor nodes

        • starting at the anchor nodes, measure distance to neighbors

        • distance information is flooded into the network

        • flooding limit

        • three algorithms

          • Sum-dist

          • DV-hop

          • Euclidian

  • Node position

    • Derive for each node a position from its anchor distances

      • Lateration

      • Min–max

  • Refinement

    • Refine the node positions

      • using information about the range (distance) to, and positions of, neighboring nodes


  • Phase1 distance to anchors l.jpg

    Phase1: Distance to anchors

    • Sum-dist

      • adding the ranges at each hop during flooding

        • anchors nodes:

          • send a message

            • identity, position, and a path length set to 0

        • receiving node:

          • adds the measured range to the path length

          • forwards (broadcasts) the message

            • if the flood limit allows

            • if the current path length is less than the previous one

        • result

          • each node have stored the position

          • minimum path length

        • drawbacks

          • range errors accumulate when distance information is propagated over multiple hops

          • error is significant for large networks with few anchors and/or poor ranging hardware


    Distance to anchors cont l.jpg

    Distance to anchors (cont.)

    • DV-hop

      - use topological information instead of summing the (erroneous) ranges.

      • counting the number of hops

      • calibration: convert hop counts into distances

        • multiplying the hop count with an average hop distance

        • average hop distance obtained by anchors

      • drawback

        • fails for highly irregular network topologies

        • where the variance in actual hop distances is very large


    Distance to anchors cont9 l.jpg

    Distance to anchors (cont.)

    • Euclidean

      • based on the local geometry of the nodes around an anchor

        • anchors: initiate a flood

        • receiver:

          • receive messages from two neighbors that:

            • know their distance to the anchor

            • know their distance to each other

          • calculate the distance to the anchor

        • result

          • two possible distance to anchor

          • solution

            • neighbor vote: a third neighbor n3 connected to either n1 or n2. replace n1 or n2 with n3


    Distance to anchors cont10 l.jpg

    Distance to anchors (cont.)


    Phase 2 node position l.jpg

    Phase 2: Node position

    • Nodes determine their position

      • based on the distance estimates to a number of anchors

      • provided by one of the three Phase 1 alternatives

        • Sum-dist, DV-hop, or Euclidean

    • Using:

      • the estimated distances (di)

      • known positions (xi; yi)

    • Methods

      • Lateration

      • Min–max


    Lateration algorithm l.jpg

    Lateration algorithm

    (1) unknown position is denoted by (x; y).

    (2) Linear the system by subtracting the last equation from the first n-1 equations.

    (3) Reordering the terms gives a proper system of linear equations in the form Ax = b

    (4) The system is solved using a standard least-squares approach:

    (5) additional sanity check by computing

    the residue between the given distances di and the distances to the location estimate of x

    (6) exceptional cases: the matrix inverse can not be computed and Lateration fails.

    * quite expensive in the number of floating point operations that is required.


    Min max algorithm l.jpg

    Min–max algorithm

    • For each anchor:

      • construct a bounding box

      • using its position & distance to estimate

      • [xi-di, yi-di] x [xi+di, yi+di]

    • Determine the intersection of these boxes

      • [max(xi-di), max(yi-di)] x

        [min(xi+di), min(yi+di)]

    • Position of the node

      = center of the intersection box


    Phase 3 refinement l.jpg

    Phase 3: Refinement

    • Refine the (initial) node positions computed during phase 2

      • not all available information used in the first two phases

      • positions are not very accurate, even under good conditions

        • (high connectivity, small range errors)

  • Iterative refinement procedure

    • take into account all inter-node ranges

    • nodes update their positions

      • a node broadcasts its position estimate

      • receives the positions and range estimates from its neighbors

      • performs Lateration procedure of Phase 2 to determine its new position

      • refinement stops when position update becomes small -> reports the final position

  • Problem

    • errors propagate quickly through the network

      • a single error from 1 node needs only d iterations to affect all nodes (d: network diameter)


  • Examples of localization algorithms l.jpg

    Examples of localization algorithms

    • Ad-hoc positioning by Niculescu and Nath [NN01]

    • Robust positioning by Savvides, Langendoen and Rabaey [SLR02]

    • N-hop multilateration by Savarese, Park and Srivastava [SPS02]

    • compare various alternatives for each phase

      • simulation on the same platform

    • conclusion

      • no single algorithm performs best

      • which algorithm be preferred depends on the conditions

        • range errors, connectivity, anchor fraction, etc.

      • still significant room for improving accuracy & increasing coverage


    General problems for localization l.jpg

    General problems for localization

    • insufficient data

      • lack of absolute reference points or anchors

    • distance measurements are noisy

      • creating additional uncertainty

    • difficult for scalability

      • algorithms that scale linearly with the size of the network are hard to devise

        • data must be broadcast through wireless channel

          • limited communications capacity.


    Localization with noisy range measurements mlrt04 l.jpg

    Localization with Noisy Range Measurements [MLRT04]

    • Challenges of network localization with noise

      • only numerical optimization of distance constraints ---- fails

        • knowing the length of each graph edge

          ---- does NOT guarantee a unique realization

      • need to handle nodes with ambiguous positions

      • non-rigid graph

        • can be continuously deformed to produce an infinite number of different realizations

      • rigid graph

        • two kinds of ambiguity

          • flip ambiguities

          • discontinuous flex ambiguities

      • Can NOT be solved by graph rigidity theory or tests

        when distance measurements are noisy


    Two kinds of ambiguity l.jpg

    Two kinds of ambiguity

    For (b):

    If edge AD is removed, then reinserted, the graph can flex in the direction of the arrow, taking on a different configuration but exactly preserving all distance constraints.

    For (a):

    Vertex A can be reflected across the line connecting B and C with no change in the distance constraints.


    Solution for ambiguity problem l.jpg

    Solution for ambiguity problem

    • only localize those vertices that:

      • have a small probability of being flip or flex ambiguity

    • robust quadrilaterals

      • construct robust quadrilaterals regions to locate node

        • prevent incorrect realizations of flip ambiguities

          • would otherwise corrupt localization computations

        • cope with measurement noise in the system

        • drawback

          • bad performance under low node connectivity


    Robust quadrilaterals algorithm l.jpg

    Robust quadrilaterals algorithm

    • Define: cluster

      • a node and its set of neighbors

    • Three phases

      • Cluster localization

        • Quadrilaterals

          • the smallest possible sub-graph that can be unambiguously localized in isolation

        • identify all robust quadrilaterals

        • find the largest sub-graph

          • composed solely of overlapping robust quads

        • minimizes the probability of realizing a flip ambiguity

      • Cluster optimization (optional)

        • refine the position estimates for each cluster

          • using numerical optimization

      • Cluster transformation

        • compute transformations between neighboring clusters

          • finding the set of nodes in common between two clusters

          • solving for the rotation, translation, and possible reflection that best aligns the clusters


    Slide21 l.jpg

    • Quadrilaterals:

      • knowing the locations of any three vertices

        • sufficient to compute the location of the fourth using trilateration

      • problem

        • but still NOT sufficient to guarantee a unique graph realization

          • when distance measurements are noisy

        • If the smallest angle θi is near zero, there is a risk that measurement error

      • solution

        • restrict our quadrilateral to be robust

          ---> only those triangles with a sufficiently large minimum angle as robust

          • b is the length of the shortest side and θ is the smallest angle

        • use the robust quadrilateral as a starting point

        • localize additional nodes by chaining together connected robust quads

          • whenever two quads have three nodes in common & the first quad is fully localized

          • can localize the second quad by trilaterating from the three known positions


    Slide22 l.jpg

    (a) robust four-vertex quadrilateral

    (b) decomposition of the robust quadrilateral into four triangles.

    If θ3 (smallest)is near zero:

    say in edge AD, will cause vertex D to be reflected over this sliver of a triangle


    Localization with mere connectivity srz03 l.jpg

    Localization with mere connectivity [SRZ03]

    • Goal

      • using fewer anchor nodes to derive the locations of the nodes

        • even yields relative coordinates when no anchor nodes are available

  • Method

    • MDS (multi-dimensional scaling)

      • starts with one or more distance matrices

        • derived from points in a multidimensional space

      • find a placement of the points in a low-dimensional space

        • usually two or three-dimensional

      • closely related to PCA (principal component analysis)

      • types of MDS techniques

        • classical metric MDS, replicated MDS, weighted MDS, etc.

  • Classical metric MDS

    • tolerates error gracefully

      • due to the over-determined nature of the solution

    • it can be performed efficiently on large matrices

      • a closed-form solution


  • Mds map algorithm based on mds l.jpg

    MDS-MAP algorithm- Based on MDS

    • First step

      • estimate distance between each possible pair of nodes

        • use shortest-paths algorithm

        • shortest path distances are used to construct the distance matrix for MDS

    • Second step

      • apply classical MDS to the distance matrix

        • core of classical MDS

          • SVD (singular value decomposition)

        • result of MDS

          • a relative map that gives a location for each node

  • Third step

    • if given sufficient anchor nodes

      • transform the relative map to an absolute map

      • based on the absolute positions of anchors

  • Drawback

    • requires centralized computation


  • Localization for mobile sensor network he04 l.jpg

    Localization for mobile sensor network [HE04]

    • Usually

      • mobility make localization more difficult

        • none of above mechanism consider mobile nodes and anchors

    • Sequential Monte Carlo localization

      • take advantage of mobility

        • to improve the accuracy of localization

        • reduce the number of anchors required

      • based on MCL (Monte Carlo Localization)

        • used for robots localization


    Sequential monte carlo smc l.jpg

    Sequential Monte Carlo (SMC)

    • Key idea

      • estimate the posterior distribution of discrete time dynamic models

    • Algorithm

      • t: discrete time

      • l(t): position distribution of the node at time t

      • o(t): observations from anchor nodes received between time t-1 and time t

      • p(l(t) | l(t-1)): transition equation

        • prediction of node’s current position based on previous position

      • p(l(t) | o(t)): observation equation

        • describes the likelihood of the node being at the location l(t) given the observations

        • filter impossible positions

      • estimate recursively in time the filtering distribution p(l(t) | o(0), o(1), …, o(t))

      • A set of N samples L(t) is used to represent the distribution l(t)

      • recursively computes the set of samples at each time step

      • since L(t-1) reflects all previous observations, can compute l(t) using only L(t-1) and o(t).


    Conclusion l.jpg

    Conclusion

    • Goal

      • determine node locations in ad-hoc sensor networks

      • can use a small number of anchors

    • Three phases

      • various alternatives for each phase

    • Challenges

      • noisy distance measurements

      • mere connectivity

      • mobility


    Reference l.jpg

    Reference

    • Ian F. Akyildiz, Weilian Su, Yogesh Sankarasubramaniam, and Erdal Cayirci, A Survey on Sensor Networks.

    • [LR02] Koen Langendoen, Niels Reijers, Distributed localization in wireless sensor networks: a quantitative comparison, Computer Networks, 2003, pp. 499-518.

    • [NN01] D. Niculescu, B. Nath, Ad-hoc positioning system, IEEE GlobeCom, 2001.

    • [SLR02] C. Savarese, K. Langendoen, J. Rabaey, Robust positioning algorithms for distributed ad-hoc wireless sensor networks, USENIX Technical Annual Conference, 2002, pp. 317–328.

    • [SPS02] A. Savvides, H. Park, M. Srivastava, The bits and flops of the N-hop multilateration primitive for node localization problems, in: First ACM International Workshop on Wireless Sensor Networks and Application (WSNA), 2002, pp. 112–121.

    • [MLRT04] David Moore, John Leonard, Daniela Rus and Seth Teller, Robust Distributed Network Localization with Noisy Range Measurements, ACM, 2004.

    • [SRZ03] Yi Shang, Wheeler Ruml, Ying Zhang, Markus P. J. Fromherz, Localization from Mere Connectivity, MobiHoc, 2003.

    • [HE04] Lingxuan Hu, David Evans, Localization for Mobile Sensor Networks, MobiCom, 2004.


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