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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

- 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

- Quick recap from last time

- 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

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

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 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

- 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

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

- 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)

- 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

- 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

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

Channel

Effects

Setup

Error

Computation

Error

Induced by intrinsic

measurement error

Position

Error

- 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”

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

6 neighbors

12 neighbors

RMS Error(m)

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)

- 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

Wishful thinking… some of it running on XYZ Node…

- 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!!!