Location Discovery – Part II
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Location Discovery – Part II Lecture 5 September 16, 2004 EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems & Sensor Networks. Andreas Savvides andreas.savvides@yale.edu Office: AKW 212 Tel 432-1275 Course Website http://www.eng.yale.edu/enalab/courses/eeng460a. Today.

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Location Discovery – Part IILecture 5 September 16, 2004EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems &Sensor Networks

Andreas Savvides

andreas.savvides@yale.edu

Office: AKW 212

Tel 432-1275

Course Website

http://www.eng.yale.edu/enalab/courses/eeng460a


Today

  • Presentation topics scheduling

  • Stop by Ed Jackson’s office so that he can swipe your ID for the lab

  • Internal website access

  • Project and presentation discussions

  • Any issues with graduate student registrations?

  • Today’s discussion topics

    • Quick recap from last time

      • GDOP – Angles matter

      • Conditions for position uniqueness (another presentation on this later)

    • Improved MDS Localization

      Material for this lecture from:

      [Shang04] Y. Shang, W. Ruml, Improved MDS Localization, Proceedings of Infocom 2004

      [Savvides04b] A. Savvides, W, Garber, R. L. Moses and M. B. Srivastava, An Analysis of Error Inducing Parameters in Multihop Sensor Node Localization, to appear in the IEEE Transcations on Mobile Computing


Taxonomy of Localization Mechanisms

  • Active Localization

    • System sends signals to localize target

  • Cooperative Localization

    • The target cooperates with the system

  • Passive Localization

    • System deduces location from observation of signals that are “already present”

  • Blind Localization

    • System deduces location of target without a priori knowledge of its characteristics


Active Mechanisms

Target

Synchronization channel

Ranging channel

  • Non-cooperative

    • System emits signal, deduces target location from distortions in signal returns

    • e.g. radar and reflective sonar systems

  • Cooperative Target

    • Target emits a signal with known characteristics; system deduces location by detecting signal

    • e.g. ORL Active Bat, GALORE Panel, AHLoS, MIT Cricket

  • Cooperative Infrastructure

    • Elements of infrastructure emit signals; target deduces location from detection of signals

    • e.g. GPS, MIT Cricket


Passive Mechanisms

Target

Synchronization channel

Ranging channel

?

  • Passive Target Localization

    • Signals normally emitted by the target are detected (e.g. birdcall)

    • Several nodes detect candidate events and cooperate to localize it by cross-correlation

  • Passive Self-Localization

    • A single node estimates distance to a set of beacons (e.g. 802.11 bases in RADAR [Bahl et al.], Ricochet in Bulusu et al.)

  • Blind Localization

    • Passive localization without a priori knowledge of target characteristics

    • Acoustic “blind beamforming” (Yao et al.)


Active vs. Passive

  • Active techniques tend to work best

    • Signal is well characterized, can be engineered for noise and interference rejection

    • Cooperative systems can synchronize with the target to enable accurate time-of-flight estimation

  • Passive techniques

    • Detection quality depends on characterization of signal

    • Time difference of arrivals only; must surround target with sensors or sensor clusters

      • TDOA requires precise knowledge of sensor positions

  • Blind techniques

    • Cross-correlation only; may increase communication cost

    • Tends to detect “loudest” event.. May not be noise immune


Measurement Technologies

  • Ultrasonic time-of-flight

    • Common frequencies 25 – 40KHz, range few meters (or tens of meters), avg. case accuracy ~ 2-5 cm, lobe-shaped beam angle in most of the cases

    • Wide-band ultrasonic transducers also available, mostly in prototype phases

  • Acoustic ToF

    • Range – tens of meters, accuracy =10cm

  • RF Time-of-flight

    • Ubinet UWB claims = ~ 6 inches

  • Acoustic angle of arrival

    • Average accuracy = ~ 5 degrees (e.g acoustic beamformer, MIT Cricket)

  • Received Signal Strength Indicator

    • Motes: Accuracy 2-3 m, Range = ~ 10m

    • 802.11: Accuracy = ~30m

  • Laser Time-of-Flight Range Measurement

    • Range =~ 200, accuracy =~ 2cm very directional

  • RFIDs and Infrared Sensors – many different technologies

    • Mostly used as a proximity metric


Possible Implementations/ Computation Models

Computing Nodes

  • Centralized

  • Only one node computes

2. Locally Centralized

Some of unknown nodes compute

3. (Fully) Distributed

Every unknown node computes

  • Each approach may be appropriate for a different application

  • Centralized approaches require routing and leader election

  • Fully distributed approach does not have this requirement


Different Problem Setups & Algorithms

  • Absolute vs. relative frame of reference

    • Beacons or no beacons

    • Infrastructure vs. ad-hoc

    • Single hop vs. multihop

  • Many candidate approaches and solution methods (depending on problem setup, measurement technology and computation resources)

    • Least-squares optimization

    • Approaches based on radio connectivity

    • Learning based approaches

    • Semi definite programming approaches

      • Both measurement based and connectivity based

    • Vision based algorithms


Obtaining a Coordinate System from Distance Measurements: Introduction to MDS

  • MDS maps objects from a high-dimensional space to a

  • low-dimensional space,

  • while preserving distances between objects.

  • similarity between objectscoordinates of points

  • Classical metric MDS:

  • The simplest MDS: the proximities are treated as distances in an Euclidean space

  • Optimality: LSE sense. Exact reconstruction if the proximity data are from an Euclidean space

  • Efficiency: singular value decomposition, O(n3)


Applying Classical MDS

  • Create a proximity matrix of distances D

  • Convert into a double-centered matrix B

  • Take the Singular Value Decomposition of B

  • Compute the coordinate matrix X (2D coordinates will be in the first 2 columns)

NxN matrix of 1s

NxN identity matrix

NxN matrix of 1s


Example: Localization Using Multidimensional Scaling (MDS) (Yi Shang et. al)

  • The basic MDS-MAP algorithm:

  • Compute shortest paths between all pairs of nodes.

  • Apply classical MDS and use its result to construct a relative map.

  • Given sufficient anchor nodes, transform the relative map to an absolute map.


MDS-MAP ALGORITHM

  • Compute all-pair shortest paths. O(n3)

    • Assigning values to the edges in the connectivity graph:

    • Known connectivity only: all edges have value 1 (or R/2)

    • Known neighbor distances: the edges have the distance values

  • Apply classical MDS and use its result to construct a 2-D (or 3-D) relative map. O(n3)

  • Given sufficient anchor nodes, convert the relative map to an absolute map via a linear transformation. O(n+m3)

  • Compute the LSE transformation based on the positions of anchors.

  • O(m3),m is the number of anchors

  • Apply the transformation to the other unknown nodes. O(n)


MDS-MAP (P) – The Distributed Version

  • Set-up the range for local maps Rlm (# of hops to consider in a map)

  • Compute maps of individual nodes

    • Compute shortest paths between all pairs of nodes

    • Apply MDS

    • Least-squares refinement

  • Patch the maps together

    • Randomly pick a node and build a local map, then merge the neighbors and continue until the whole network is completed

  • If sufficient anchor nodes are present, transform the relative map to an absolute map

  • MDS-MAP(P,R) – Same as MDS-MAP(P) followed by a refinement phase


LOCALIZATION USING MDS-MAP

(Shang, et al., Mobihoc’03)

  • The basic MDS-MAP algorithm:

  • Given connectivity or local distance measurement, compute shortest paths between all pairs of nodes.

  • Apply multidimentional scaling (MDS) to construct a relative map containing the positions of nodes in a local coordinate system.

  • Given sufficient anchors (nodes with known positions), e.g, 3 for 2-D or 4 for 3-D networks, transform the relative map and determine the absolute the positions of the nodes.

  • It works for any n-dimensional networks, e.g., 2-D or 3-D.


MDS-MAP(P) (Shang and Ruml, Infocom’04)

  • The basic MDS-MAP works well on regularly shaped networks, but not on irregularly shaped networks.

  • MDS-MAP(P) (or MDS-MAP based on patches of local maps)

  • For each node, compute a local relative map using MDS

  • Merge/align local maps to form a big relative map

  • Refine the relative map based on the relative positions (optional). (When used, referred to as MDS-MAP(P,R) )

  • Given sufficient anchors, compute absolute positions

  • Refinethe positions of individual nodes based on the absolution positions (optional)


SOME IMPLEMENTATION DETAILS OF MDS-MAP(P)

  • For each node, compute a local relative map using MDS

    • Size of local maps: fixed or adaptive

  • Merge/align local maps to form a big relative map

    • Sequential or distributed; scaling or not

  • Refine the relative map based on the relative positions

    • Least squares minimization: what information to use

  • Given sufficient anchors, compute absolute positions

    • Anchor selection; centralized or distributed

  • Refinethe positions of individual nodes based on the absolution positions

    • Minimizing squared errors or absolute errors


AN EXAMPLE OF C-SHAPE GRID NETWORKS

Known 1-hop distances with 5% range error

Connectivity information only

MDS-MAP(P) without both optional refinement steps.


RANDOM UNIFORMPLACEMENT

Connectivity information only

Known 1-hop distances with 5% range error

200 nodes; 4 random anchors


RANDOM C-SHAPEPLACEMENT

Connectivity information only

Known 1-hop distances with 5% range error

160 nodes; 4 random anchors


Understanding Fundamental Behaviors(Savvides04b)

What is the fundamental error behavior?

Measurement technology perspective

  • Acoustic vs. RF ToF (2cm – 1.5m measurement accuracy)

  • Distances vs. Angules

    Deployment - what density?

    Scalability How does error propagate?

    Beacon density & beacon position uncertainty

    Intrinsic vs. Extrinsic Error Component


Estimated Location Error Decomposition

Channel

Effects

Setup

Error

Computation

Error

Induced by intrinsic

measurement error

Position

Error


Cramer Rao Bound Analysis

  • Cramer-Rao Bound Analysis on carefully controlled scenarios

    • Classical result from statistics that gives a lower bound on the error covariance matrix of an unbiased estimate

  • Assuming White Gaussian Measurement Error

  • Related work

    • N. Patwari et. al, “Relative Location Estimation in Wireless Sensor Networks”


Density Effects

Results from Cramer-Rao Bound Simulations based on White Gaussian Error

Range Tangential Error

m/rad

RMS Location Error

RMS Location Error/sigma

m/m

Range Error Scaling Factor

Density (node/m2)

20mm distance measurement certainty == 0.27 angular certainty


Density Effects with Different Ranging Technologies

6 neighbors

12 neighbors

RMS Error(m)


Network Scalability

Error propagation on a hexagon scenario (angle measurement)

Rate of error propagation faster with distance measurements but

Much smaller magnitude than angles

RMS Location Error x 10

y-coordinate(m)

x-coordinate(m)


More Observations on Network Scalability…

  • Performance degrades gracefully as the number of unknown nodes increases.

  • Increasing the number of beacon nodes does not make a significant improvement

  • Error in beacons results in an overall translation of the network

  • Error due to geometry is the major component in propagated error


Localization Service Middleware

Wishful thinking… some of it running on XYZ Node…


Are we done with localization?

  • Well there is more…

    • Computation using angles

    • Mobility and tracking

    • Probabilistic approaches

  • More about localization in future lectures

  • Next time – embedded programming tutorial

    • Read programming assignment 1 before coming to class!!!


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