location discovery – part ii lecture 5 september 16, 2004 eeng 460a / cpsc 436 / enas 960...
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Location Discovery – Part IILecture 5
September 16, 2004
EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems &
Sensor Networks
Andreas [email protected]
Office: AKW 212Tel 432-1275
Course Websitehttp://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 timeo GDOP – Angles mattero 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
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
TargetSynchronization channelRanging channel
Passive Mechanisms
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.)
?
TargetSynchronization channelRanging channel
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 clusterso 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
1. CentralizedOnly one node computes
2. Locally Centralized Some of unknown nodes compute
3. (Fully) DistributedEvery unknown node computes
Computing Nodes
• 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
o 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 objects coordinates 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
1. Create a proximity matrix of distances D2. Convert into a double-centered matrix B
3. Take the Singular Value Decomposition of B
4. Compute the coordinate matrix X (2D coordinates will be in the first 2 columns)
U
NIDU
NI
11
2
1-B 2
NxN matrix of 1s
NxN matrix of 1s
NxN identity matrix
TVAVB
2
1
VAX
The basic MDS-MAP algorithm:1. Compute shortest paths between all pairs of nodes.2. Apply classical MDS and use its result to construct a relative map.
3. Given sufficient anchor nodes, transform the relative map to an absolute map.
Example: Localization Using Multidimensional Scaling (MDS) (Yi Shang et. al)
MDS-MAP ALGORITHM
1. 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
2. Apply classical MDS and use its result to construct a 2-D (or 3-D) relative map. O(n3)
3. 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
1. Set-up the range for local maps Rlm (# of hops to consider in a map)
2. Compute maps of individual nodes1. Compute shortest paths between all pairs of nodes2. Apply MDS3. Least-squares refinement
3. Patch the maps together• Randomly pick a node and build a local map, then merge the
neighbors and continue until the whole network is completed4. 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
The basic MDS-MAP algorithm:
1. Given connectivity or local distance measurement, compute shortest paths between all pairs of nodes.
2. Apply multidimentional scaling (MDS) to construct a relative map containing the positions of nodes in a local coordinate system.
3. 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.
LOCALIZATION USING MDS-MAP (Shang, et al., Mobihoc’03)
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)
1. For each node, compute a local relative map using MDS
2. Merge/align local maps to form a big relative map
3. Refine the relative map based on the relative positions (optional). (When used, referred to as MDS-MAP(P,R) )
4. Given sufficient anchors, compute absolute positions
5. Refine the positions of individual nodes based on the absolution positions (optional)
MDS-MAP(P) (Shang and Ruml, Infocom’04)
1. For each node, compute a local relative map using MDS• Size of local maps: fixed or adaptive
2. Merge/align local maps to form a big relative map• Sequential or distributed; scaling or not
3. Refine the relative map based on the relative positions• Least squares minimization: what information to use
4. Given sufficient anchors, compute absolute positions • Anchor selection; centralized or distributed
5. Refine the positions of individual nodes based on the absolution positions• Minimizing squared errors or absolute errors
SOME IMPLEMENTATION DETAILS OF MDS-MAP(P)
AN EXAMPLE OF C-SHAPE GRID NETWORKS
MDS-MAP(P) without both optional refinement steps.
Known 1-hop distances with 5% range error
Connectivity information only
2 4 6 8 10
2
4
6
8
10
2 4 6 8 10
2
4
6
8
10
RANDOM UNIFORM PLACEMENT
200 nodes; 4 random anchors
Connectivity information only Known 1-hop distances with 5% range error
5 10 15 20 25 30 350
50
100
150
200
Connectivity
Me
an
err
or
(%R
)
MDS-MAP(P)MDS-MAP(P,R)DV-hop
RANDOM C-SHAPE PLACEMENT
160 nodes; 4 random anchors
Connectivity information only Known 1-hop distances with 5% range error
5 10 15 20 25 300
50
100
150
200
Connectivity
Me
an
err
or
(%R
)
MDS-MAP(P)MDS-MAP(P,R)DV-hop
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 uncertaintyIntrinsic vs. Extrinsic Error Component
Estimated Location Error Decomposition
PositionError
ChannelEffects
ComputationError
SetupError
Induced by intrinsic measurement 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
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410
-1
100
101
102
angle measurementsdistance measurements
Density (node/m2)
RM
S L
ocat
ion
Err
or
20mm distance measurement certainty == 0.27 angular certainty
100
101
102
103
10-1
100
101
102
103
104
distancesangles
Range Error Scaling Factor
RM
S L
ocat
i on
Err
or/s
igm
a
Range Tangential Error
Results from Cramer-Rao Bound Simulations based on White Gaussian Error
m/rad
m/m
Density Effects with Different Ranging Technologies
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Ranging Error Variance(m)
D=0.030D=0.035D=0.040D=0.045D=0.050D=0.055
RMS Error(m)
6 neighbors
12 neighbors
Network Scalability
x-coordinate(m) y-coordinate(m)
RM
S L
o cat
ion
Err
or x
10
Error propagation on a hexagon scenario (angle measurement)Rate of error propagation faster with distance measurements but Much smaller magnitude than angles
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!!!