Introduction to Location Discovery Lecture 4

September 14, 2004

EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems &

Sensor Networks

Andreas Savvidesandreas.savvides@yale.edu

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

i

uii

i

uii r

yyy

r

xxx

ˆ,

ˆ

22 )ˆ()ˆ( uiuii yyxxr

zA

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

)(3

)(2

)(1

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22

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, ,u

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x

f

f

f

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yuuxuu yyxx ˆˆ and ˆˆ

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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000 )()(),,( yyxxstsxxf iiii

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yXXXb TT 1)( MMSE Solution:

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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254

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2343,43,4

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232

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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

00)5(ˆ)5(ˆ

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H # of edges

# of unknown nodes x 2

254

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214

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ˆ

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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…