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Introduction to Location Discovery Lecture 4
September 14, 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
Announcement of student presentations Make sure you get the additions from last set of lecture slides Make sure your name is on signup sheet for the lab Stop by Ed Jackson’s office so that he can swipe your ID for the lab Internal website access
• Some work-in-progress and copyrighted material will only appear on this site. Note the password.
Reference material for this lecture:[Savvides 04a]A. Savvides and M. B. Srivastava, Location Discovery, pp 231 – 238
[Savvides 03] A. Savvides, H. Park and M. B. Srivastava, The n-Hop Multilateration Primitive for Node Localization Problems
[Goldenberg04] D. Goldenberg, A. Krishnamurthy, W. Maness, Y. R. Yang, A. Young and A.
Savvides, Network Localization in Partially Localizable Networks (slide 22 only)
Why is Location Discovery(LD) Important?
Very fundamental component for many other services• GPS does not work everywhere• Smart Systems – devices need to know where they are• Geographic routing & coverage problems• People and asset tracking• Need spatial reference when monitoring spatial
phenomena We will use the node localization problem as a
platform for illustrating basic concepts from the course
Why spend so much time on LD?
LD captures multiple aspects of sensor networks:• Physical layer imposes measurement challenges
o Multipath, shadowing, sensor imperfections, changes in propagation properties and more• Extensive computation aspects
o Many formulations of localization problems, how do you solve the optimization problem?
o How do you solve the problem in a distributed manner?– You may have to solve the problem on a memory constrained processor…
• Networking and coordination issueso Nodes have to collaborate and communicate to solve the problemo If you are using it for routing, it means you don’t have routing support to solve the
problem! How do you do it?• System Integration issues
o How do you build a whole system for localization?o How do you integrate location services with other applications?o Different implementation for each setup, sensor, integration issue
Base Case: Atomic Multilateration
Base stations advertise their coordinates & transmit a reference signal
PDA uses the reference signal to estimate distances to each of the base stations
Note: Distance measurements are noisy!
Base Station 1
Base Station 3
Base Station 2
u
Problem Formulation
Need to minimize the sum of squares of the residuals
The objective function is
This a non-linear optimization problem• Many ways to solve (e.g a forces formulation, gradient
descent methods etc
22,, )ˆ()ˆ( uiuiiuiu yyxxrf
2,min),( iuuu fyxF
A Solution Suitable for an Embedded Processor
Linearize the measurement equations using Taylor expansion
where
Now this is in linear form
)( 22,, Oyxfr yixiiuiu
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Solve using the Least Square Equation
The linearized equations in matrix form become
Now we can use the least squares equation to compute a correction to our initial estimate
Update the current position estimate
Repeat the same process until δ comes very close to 0
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How do you solve this problem?
Check conditions• Beacon nodes must not lie on the same line• Assuming measurement error follows a white
gaussian distribution
Create a system of equations• Solve to get the solution – how would you
solve this in an embedded system?• How do you solve for the speed of sound?
Acoustic case: Also solve for the speed of sound
Minimize over all
This can be linearized to the form
where2
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The Node Localization Problem
Beacon
Unkown Location
Randomly Deployed Sensor Network
Beacon nodes
• Localize nodes in an ad-hoc multihop network• Based on a set of inter-node distance measurements
Solving over multiple hops
Iterative Multilateration
Beacon node(known position)
Unknown node(known position)
Iterative Multilateration
Iterative Multilateration
ProblemsError accumulationMay get stuck!!!
% of initial beacons
Localized nodes
total nodes
Collaborative Mutlilateration
• All available measurements are used as constraints
• Solve for the positions of multiple unknowns simultaneously
• Catch: This is a non-linear optimization problem!
• How do we solve this?
Known position
Uknown position
Problem Formulation
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The objective function is
Can be solved using iterative least squares utilizing the initial Estimates from phase 2 - we use a Kalman Filter
1
2
3 4
5
6
How do we solve this problem?
In an embedded system? Backboard material here… One possible solution would use a Kalman
Filter• This was found to work well in practice, can
“easily” implemented on an embedded processor
[more details see Savvides03]
Initial Estimates (Phase 2)
Use the accurate distance measurements to impose constraints in the x and y coordinates – bounding box
Use the distance to a beacon as bounds on the x and y coordinates
a
a ax
U
Initial Estimates (Phase 2)
Use the accurate distance measurements to impose constraints in the x and y coordinates – bounding box
Use the distance to a beacon as bounds on the x and y coordinates
Do the same for beacons that are multiple hops away
Select the most constraining bounds
a
b
c
b+c b+c
X
Y
U
U is between [Y-(b+c)] and [X+a]
Initial Estimates (Phase 2)
Use the accurate distance measurements to impose constraints in the x and y coordinates – bounding box
Use the distance to a beacon as bounds on the x and y coordinates
Do the same for beacons that are multiple hops away
Select the most constraining bounds
Set the center of the bounding box as the initial estimate
a
a a
b
c
b+c b+c
X
Y
U
Initial Estimates (Phase 2)
Example:• 4 beacons• 16 unknowns
To get good initial estimates, beacons should be placed on the perimeter of the network
Observation: If the unknown nodes are outside the beacon perimeter then initial estimates are on or very close to the convex hull of the beacons
Overview: Collaborative Multilateration
Collaborative Multilateration
ChallengesComputation constraintsCommunication cost
1
2
3
45
2
1
3
45
1
2
3
45
Overview: Collaborative Multilateration
Collaborative Multilateration
ChallengesComputation constraintsCommunication cost
0
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
10,000
10 20 30 40 50 60 70 80 90 100
No. of Unknown Nodes
MF
lop
s
Distributed Centralized
Distributed reduces computation cost Even sharing of communication cost
Satisfy Global Constraints with Local Computation
From SensorSim simulation 40 nodes, 4 beacons IEEE 802.11 MAC 10Kbps radio Average 6 neighbors per node
Kalman Filter
From Greg Welch
• We only use measurement update since the nodes are static• We know R (ranging noise distribution)• Not really using the KF for now, no notion of time
Global Kalman Filter
Matrices grow with density and number of nodes => so does computation cost
Computation is not feasible on small processors with limited computation and memory
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# of unknown nodes x 2
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Beware of Geometry Effects!
Known as Geometric Dilution of Precision(GDOP)
Position accuracy depends on measurement accuracy and geometric conditioning
i ijj ij
NNGDOPGDOP
,
2|sin|),(
ijjk i
kj
bQAAQAx bT
bT 111 )(ˆ From pseudoinverse equation
Geometry: It’s the angles not the distance!
0
510
15
0
5
10
150
0.2
0.4
0.6
0.8
x coordinatey coordinate
RMS Error(m)
CR-Bound Evaluation on a 10 x 10 grid
(0,0)
10
10
beacon unknown
(x,y)
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sin
2
Beware of Solution Uniqueness Requirements
In a 2D scenario a network is uniquely localizable if:1. It belongs to a subgraph that is redundantly rigid
2. The subgraph is 3-connected
3. It contains at least 3 beacons
More details in future lectures
Nodes can be exchanged without violating the measurement constraints!!!
[conditions from Goldenberg04]
Does this solve the problem?
No! Several other challenges Solution depends on
• Problem setupo Infrastructure assisted (beacons), fully ad-hoc & beaconless, hybrid
• Measurement technologyo Distances vs. angles, acoustic vs. rf, connectivity based, proximity basedo The underlying measurement error distribution changes with each technology
The algorithm will also change• Fully distributed computation or centralized• How big is the network and what networking support do you have to solve
the problem?• Mobile vs. static scenarios• Many other possibilities and many different approaches
More next time…