Introduction to Location Discovery Lecture 6

September 20, 2005

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/2005f/eeng460a

Lecture Outline

Recap from last lecture, Kalman Filters & time update phase

The SCAAT Kalman Filter Programming project overview Computing locations using

multidimensional scaling (MDS)

Kalman Filter Equations

From Greg Welch

http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf

KF, EKF & SCAAT

“A Kalman Filter(KF) is a recursive mathematical procedure for least-squares estimation of a linear system”

Follows a predictor-corrector model An Extended Kalman Filter is a KF variant

of non-linear systems A SCAAT Kalman Filter operates on a

single measurement at a time.

SCAAT Kalman Filter

Proposed by Greg Welsh and Gary Bishop at UNC in 1997

Used in Hi-Ball Optoelectronic Tracker System

[Slide created from Welsh & Bishop, “SCAAT: Incremental Tracking with Incomplete Information”, Sigraph 97]

SCAAT Kalman Filter

Single-constraint-at-a-time(SCAAT)

• Process one measurement at a time

C- # of independent constraints for a unique solution

N – # of constraints used to derive an estimate

Nind - # of independent constraints

[Slide from Welsh & Bishop, “SCAAT: Incremental Tracking with Incomplete Information”, Sigraph 97]

SCAAT Aims at Temporal Improvements

Computation is faster KF matrices H, x and z are smaller The state prediction model described by

matrix A needs to change

Example system that uses all measurements at once

Programming Project

Implement a SCAAT filter in SOS Need to worry about

1. Correctness of implementation

2. Layering & modules

(See project handout)

Back to Distance Based Localization…

What if the distances between nodes are know?

Can we compute a coordinate system without beacons (or anchors)?

Yes, one possible way is to use MDS…

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)

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)

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 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

Where is the challenge?

The results are numerically correct based on the used measurement model

Two main assumptions• Hops are proportional to distance• Error represented as a function of distance

Is that the case in practice?

Back to the Physical Layer: RSSI In Node Localization

Ultrasound ToA RSSI in football field

Max range 3m, accuracy 2cm Max range 20m, accuracy 7mPower levels: 7,10

01

23

45

6

0

1

2

3

4

50

0.5

1

1.5

2

2.5

7

8

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6

27

10

5

28

12

X coordinate

42

33

4

30

35

29

11

13

14

3

34

32

37

36

17

31

Connectivity at Power Level 7

41

25

15

1

2

38

39

18

26

24

16

Y coordinate

40

19

22

23

20

21

Z c

oo

rdin

ate

RSSI Characterization Study(D.Lymberopoulos et. al. based on Chipcon 2420 radio)

01

23

45

6

0

1

2

3

4

50

0.5

1

1.5

2

2.5

7

8

9

6

27

10

5

28

12

X coordinate

42

33

4

30

35

29

11

13

14

3

34

32

37

36

17

31

Connectivity at Power Level 6

41

25

15

1

2

3839

18

26

24

16

Y coordinate

40

19

22

23

20

21

Z c

oo

rdin

ate

01

23

45

6

0

1

2

3

4

50

0.5

1

1.5

2

2.5

7

8

9

6

27

10

5

28

12

X coordinate

42

33

4

30

35

29

11

13

14

3

34

32

37

36

17

31

Connectivity at Power Level 4

41

25

15

1

2

3839

18

26

24

16

Y coordinate

40

19

22

23

20

21

Z c

oo

rdin

ate

RSSI: The Log Normal Shadowing Model

Problem with indoor environments:

Path loss exponent varies in different areas

Significantly affected by reflections at higher power levels

Antenna factor dominates

),0(X

exponent losspath -

d distance reference aat losspath )PL(d

power, transmit P

Xd

d log 10)PL(dPRSS(d)

2

00

T

0100T

N

Transmit Power Levels in Chipcon CC2420

= 1mW = 43.5mW

Radio supply voltage= 2.5VAnd Power = I*V

Going from Watts to dBm

1mW

mW)P(in 10logdBm)P(in

+20dBm=100mW

+10dBm=10mW

+7dBm=5mW

+6dBm = 4mW

+4dBm=2.5mW

+3dBm=2mW

0dBm=1mW

-3dBm=.5mW

-10dBm=.1mW

Sources of RSSI Variability

Intrinsic• Radio receiver and transmitter calibration

Extrinsic • Relative antenna orientation for

Receiver/Transmitter • Multipath and shadowing effects

Indoor Path Loss Measurements

Floor measurements in a 24 x 20 lounge – no obstacles

-100

-90

-80

-70

-60

-50

-40

0 5 10 15 20

Distance (feet)

RS

SI(

dB

m) Same power level using suboptimal antenna

η=3

RSSI Measurement Configuration: RSSI vs Distance

0 1 2 3 4 5 6 7 8-50

-45

-40

-35

-30

-25Distance Vs Average RSSI at a low power level

DISTANCE (m)

Avera

ge R

SS

I (d

bm

)

Data cannot be fitted to the widely used log-normal shadowing signal propagation model Reflections Antenna Orientation

Antenna Radiation Pattern

Side View Top View

Communication rangeSymmetric Region Antenna orientation

independent regions

Communication range

Interesting Properties of RSSI

Moving away from the antenna active region

Link Asymmetry in 3D-scenarios

1 2 3 4 5 6 7 820

22

24

26

28

30

32

34

36

Power Level ( 1 - Maximum power level )

One way Links

% o

f on

e-w

ay lin

ks

RSSI Study Results & Conclusions

The RX/TX radio calibration issues have minimal effect on localization • If you were using the log normal shadowing model, that would

result in approximately 0.5m of error Antenna orientation is a major source of variability There is a small region around the antenna, that is less

susceptible to orientation effects• This allows to say when 2 radios are in close proximity to each

other Interesting radio property

• There are some signal strength measurements that can only be obtained if two antennas(& radios) are very close to each other!

Understanding Fundamental Behaviors

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