Performance metrics for anchorless localization metrics for anchorless localization Henrik Holm, Honeywell ACS Labs PPL Workshop, Aug 6 2007

Performance metrics for anchorless localization

Henrik Holm, Honeywell ACS Labs

PPL Workshop, Aug 6 2007

Honeywell Confidential and Proprietary ACS Laboratories

Outline

• What is our goal?

• Anchorless localization.

• Localization metrics.

• Especially for anchorless.

• Summary.

2


What is our goal?• Firefighter (or other first responder)

localization system.

• No pre-existing infrastructure.

• Each responder carries device.

• Range data.

• Dead reckoning data.

• GPS if available.

• Devices cooperate to obtain (relative) topology.

3


Range Data

• Several means to obtain range data with distinct shortcomings.

• RSSI (multipath, shadowing, fading.)

• Ultrasound TOA (penetration, sound characteristic vs. temperature & draft.)

• Base algorithm: Trilateration.

• Optimization to deal with overdeterminism.

• Optimization/overdeterminism increases accuracy.

4


Anchorless Localization

5

A

U

A

A

A

A

3

5

4

1

6

2

No anchors or “reference nodes” with known location initially.


ALL: Previous work

• Savarese et al.: “Assumption Based Coordinates”

• Priyantha et al.: “Anchor-Free Localization”

• Moses et al.: “Self-Localization”

• Shang et al.: Connectivity-based localization

6


ALL - Optimal Localization

Simultaneously localize all nodes

• Patwari, Hero, et al.

• Moses et al.

7

1 Tests

Vector: xNorm: ||xn ! xi||hello {these} are curly brackets

2 Equations

2.1 Global optimization

arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)

In this equation, xi is the coordinates of node i, N(i) theneighborhood of node i, and dij is the (measured) distancebetween nodes j and i.

2.2 Euclidean Distance1N

#!

i

||x̂i ! xi||2 (2)

Here, x̂i is the estimated position of node i and N is thetotal number of nodes.

2.3 Priyantha’s Global Energy Ratio

GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)

Where dij is the true and d̂ij is the measured distance.

2.4 Minimal A!ne Distance

minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)

Here, Tz,!,",#(·) denotes an a!ne transformation consist-ing of translation by z, rotation according to ! and ", andpossible reflection according to #.

1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)



#!

i

||x̂i ! xi||2 (2)



GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)



Performance Metric (PM)

• Need for accurate and realistic performance assessment.

• Comparing different algorithms and approaches.

• Improving/tuning during development.

8


PM in anchored system

9

Euclidean Distance

1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)

In this equation, xi is the coordinates of node i, N(i) the neighborhood of node i,and dij is the (measured) distance between nodes j and i.


#!

i

||x̂i ! xi||2 (2)

Here, x̂i is the estimated position of node i and N is the total number of nodes.


GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)

Here, Tz,!,",#(·) denotes an a!ne transformation consisting of translation by z,rotation according to ! and ", and possible reflection according to !.

1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)



#!

i

||x̂i ! xi||2 (2)



GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)



Euclidean Norm in ALL

For ALL: Calculated topology has arbitrary rotation, translation, reflection wrt. target topology.

10


“Distance Difference”?

11

Priyantha et al.: “Global Energy Ratio”

1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)



#!

i

||x̂i ! xi||2 (2)



GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)


1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)



#!

i

||x̂i ! xi||2 (2)



GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)



“Distance Difference”?

• In our experience: not working well.

• Increased number of nodes not decreasing average error.

• Goes against intuition & theory (Patwari).

• Not measuring what we are really interested in.

• Would have preferred Euclidean Norm.

12


Euclidean Norm in ALL

NO -- unfair emphasis on a few nodes. What if the nodes you choose to anchor have worse location than anyone else?

13

Fix at origin,1st coordinate,2nd coordinate?(“Virtual anchors”.)

U

U

U

U

U

U

3

5

4

1

6

2


U

U

U

U

U

U

3

5

4

1

6

2

Affine Euclidean Distance

Instead: translate/rotate/reflect local topology until euclidean distance is minimized.

14

1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)



#!

i

||x̂i ! xi||2 (2)



GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)


1 Tests


2 Equations


arg min{xi}

!

i

!

j!N(i)j<i

!||xj ! xi||

2 ! d2ij

"2 (1)



#!

i

||x̂i ! xi||2 (2)



GER =1

N(N ! 1)/2

$%%%&!

i

!

j!N(i)j<i

'd̂ij ! dij

dij

(2

(3)



minz,!,"#

'1N

#!

i

||Tz,!,",#(x̂i) ! xi||2

((4)



Summary

• In anchored localization: use Euclidean Distance (ED) to true positions.

• No straightforward way to assess performance of anchorless localization.

• Metric based on distance differences works, but does not represent what we really aim for.

• Use ED, however minimize distance by rigidly transforming the result of localization.

15

Documents

Performance metrics for anchorless localization metrics for anchorless localization Henrik Holm, Honeywell ACS Labs PPL Workshop, Aug 6 2007