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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

Localization in wireless sensor ad-hoc networks

Xiaobo Long

ECSE 6962 course presentation

introduction
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
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
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
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
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
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
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
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
phase 2 node position
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
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
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
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
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
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
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
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
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
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
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
(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
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
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
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
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
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
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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