Some Results on Source Localization Soura Dasgupta, U of Iowa With: Baris Fidan, Brian Anderson, Shree Divya Chitte and Zhi Ding

Some Results on Some Results on Source LocalizationSource Localization

Soura Dasgupta, U of IowaSoura Dasgupta, U of IowaWith: Baris Fidan, Brian Anderson, With: Baris Fidan, Brian Anderson,

Shree Divya Chitte and Zhi DingShree Divya Chitte and Zhi Ding

NICTA/ANU August 8, 2008

Outline

• Localization– What– Why– How

• Issues• Linear algorithm

– Conceptually simple– Poor performance

• Goal• New nonlinear algorithm

– Characterize conditions for convergence

• Estimating Distance from RSS


What is Localization?Source Localization:• Sensors with known position• Source at unknown location• Sensors must estimate source

location Sensor Localization:• Anchors with known position• Sensor at unknown location• Sensor must estimate its location Need some relative position

information


Why localize?

• To process a signal in a sensor network sensors must locate the signal source– Bioterrorism

• Pervasive Computing– Locating printers/computers

• A sensor in a sensor network must know its own position for– Routing

– Rescue

– Target tracking

– Enhanced network coverage


Wireless Localization

• Emerging multibillion dollar market• E911• Mobile advertising• Asset tracking for advanced public safety• Fleet management: taxi, emergency vehicles• Location based access authority for network

security• Location specific billing


Some Existing Technology

• Manual configuration– Infeasible in large scale sensor networks

– Nodes move frequently

• GPS– LOS problems

– Expensive hardware and power

– Ineffective for indoor problems


What Information?

Bearing

Power level

TDOA

Distance


How to measure distance?Many methods

One example• Emit a signal• Wait for reflection to returnSecond example• Source emits a signal• Signal strength=A/dc

– A=signal strength at d=1– d=distance– c a constant– Received Signal Strength (RSS)


How to localize from distances?

• One distance– Circle

• Two distances– Flip ambiguity

• Three distances– Specified

– Unless collinear

• In 3-d– Need 4

– Noncoplanar


Outline








Issues

• Sensor/Anchor Placement• Fast efficient localization• Achieve large geographical coverage• Past work

– Distance from an anchor available if within range

– Place anchors in a way that sufficient number of distances available to each sensor

• Enough to have the right number of distances?• With Linear algorithms yes

– Linear algorithms have problems


Outline


• Issues

• Linear algorithm– Conceptually simple– Poor performance





Linear Algorithm

• Three anchors in 2-d (xi,yi)

• Sensor at (x,y)

(x-x1)2+ (y-y1)2= d12

(x-x2)2+ (y-y2)2= d22

(x-x3)2+ (y-y3)2= d32

2 (x1 - x2)x+2 (y1 - y2)y= d22- d1

2 +x12-x2

2

2 (x1 – x3)x+2 (y1 – y3)y= d32- d1

2 +x12-x3

2

Gives (x,y)


Linear Algorithm

• Three anchors in 2-d (xi,yi)

• Sensor at (x,y)

(x-x1)2+ (y-y1)2= d12+n

(x-x2)2+ (y-y2)2= d22+n

(x-x3)2+ (y-y3)2= d32+n

2 (x1 - x2)x+2 (y1 - y2)y= d22- d1

2 +x12-x2

2

2 (x1 – x3)x+2 (y1 – y3)y= d32- d1

2 +x12-x3

2

Can have noise problems


Example of bust

• Three anchors (0,0), (43,7) and (47,0)

• Sensor at (17.9719,-29.3227)

• True distances: 34.392, 44.1106, 41.2608

• Measured distances: 35, 42, 43

• Linear estimate: (16.8617,-6.5076)


Outline








Goal

• Need nonlinear algorithms– Hero et. al., Nowak et. al., Rydstrom et. al.

• Minimum of nonconvex cost functions– Cannot guarantee fast convergence

• Propose new algorithm• Characterize geographical regions around small

numbers of sensors/anchors– If source/sensor lies in these regions

– Guaranteed exponential convergence

– Gradient descent

• Practical Convergence


Outline








New algorithm

Notation:

• xi are vectors containing anchor coordinates

• y* vectors containing sensor coordinates

• di distance between xi and y*: ||xi-y*||

Find y to minimize weighted cost:

.0,)(222

1

iii

n

ii xydyJ


Good News

• Three anchors (0,0), (43,7) and (47,0)

• Sensor at (17.9719,-29.3227)

• True distances: 34.392, 44.1106, 41.2608

• Measured distances: 35, 42, 43

• Minimizing estimate: (18.2190,-29.2123)


Bad News

May have local minima

.0,)(222

1

iii

n

ii xydyJ


Example

y*=0

False minimum at y=[3,3]T.

TTT xxx 3,1,1,3,1,1 321


Level Surface


Goal

• Anchor/ Sensor placement• How to distribute anchors to achieve large

geographical coverage

• Problem 1: Given xi, i find S1 so that if y* is in S1, J can be minimized easily.

• Problem 2: Given xi, find S2 so that for every y* in S2, one can find i for which J can be minimized easily.


When is convergence easy

Gradient descent minimization globally convergent

.0,)(222

1

iii

n

ii xydyJ

n

iiiii xyxyd

y

yJ

1

222)(

.

n

iiiii xkyxkydkyky

1

22 ][][][]1[


Necessary and sufficient condition

•The gradient below is zero iff y=y*.

• Unique stationary point.

n

iiiii xyxyd

y

yJ

1

222)(

.


Refined Goal

• Problem 1: Given xi, i find S1 so that if y* is in S1, J has a unique stationary point.

• Problem 2: Given xi, find S2 so that for every y* in S2, one can find i for which J has a unique stationary point.

• In fact UTC gradient descent minimization is exponentially convergent.


A Preliminary Setup/Problem 1

Assume:

• xi not collinear in 2-d

• xi not coplanar in 3-d

Then there exist i such that

n

ii

n

iii yx

1

1

1

*


Observation

If i nonnegative then y* in convex hull of xi.

n

ii

n

iii yx

1

1

1

*


A Sufficient condition

Unique stationary point if P+PT>0.

P=diag{...., n}(I-[1,…1]T[...., n])

n

ii

n

iii yx

1

1

1

*


Only sufficient condition

•Actually with E(y) dependent on y

•Gradient is–E(y)PET(y)(y-y*)

• Substantial slop

• But provides nontrivial regions for guaranteed exponential convergence

–Robustness to noise

–Convergence in distribution with noise

.


A more precise condition

No false stationary points exist if:

n

ii

n

iii yx

1

1

1

*

911

2

n

ii

n

i i

i


A special case

No false stationary points exist if n<9 , y* is in convex hull of the anchors and i=1

n

ii

n

iii yx

1

1

1

*

911

2

n

ii

n

i i

i


Implications

• For small n, much larger than convex hull• Recall in 2-d need n>2 and in 3-d, n>3• Can achieve wide coverage with small number of

anchors

• Condition in terms of i.

• Need direct characterization in terms of y*

• This set is an ellipse, determined from the xi.– Using a simple matrix inversion

– 3x3 or 4x4


Problem 2

•Changing i always moves/removes false stationary points unless one anchor has the same distance as the sensor from all other agents

n

iiiii xyxyd

y

yJ

1

222)( .


Problem 2

• Given xi, find S2 so that for every y* in S2, one can find i for which J has a unique stationary point.

• Can find such a i if the following holds.

• And i=| i| guarantees unique stationary point

3||1

n

ii


Implication

If y* is in the convex hull of anchors then i

nonnegative

Condition always holds

n

ii

n

ii

1

1

1

3||


Further Implication

• And i=| i| guarantees unique stationary point– Not the only choice

• If y* is close to xi, i greater Larger i. – Accords with intuition

n

ii

n

iii yx

1

1

1

*


Direct characterization

• Given xi, find S2 so that for every y* in S2, one can find i for which J has a unique stationary point.

• S2 is a polygon containing the convex hull– Obtained by solving a linear program

• Interior has many i and substantial regions where the same i satisfy requirement

1

2

3

6

5

98

7

4


Simulation Particulars

• Five anchors:– [1, 0, 0] T , [0, 2, 0] T , [-2, -1, 0] T, [0, 0, 2] T[0, 0, -1] T

– Example 1: y*=0. In convex hull

– Example 2: y*=[-1,1,1]T. Not in convex hull


Simulation 1


Simulation 2


Outline







Log Normal Shadowing• RSS: s=A/dc

– A, c known

– s measured

• In far field:– ln s = ln A –c ln d + w

– w~N(0,)

– Estimate for some m, dm

• Reformulation– a=c/m, z=(A/s)a p=dm

– ln z= ln p – a w

– z= e–aw p


Efficient Estimation• Cramer Rao Lower Bound (CRLB)

– Best achievable error variance among unbiased estimators

– Unbiased: Mean of estimate=Parameter

• Efficient Estimator is unbiased and meets CRLB• CRLB=p2 a2

• Does an efficient estimator exist? NO• ln z= ln p – a w

• Affine in Gaussian noise, nonaffine in p


Maximum Likelihood Estimator• ln z= ln p – a w

• w~N(0,)

• f(z|p)=exp(-(lnz –lnp)2 /2a2)/((2) 1/2a)

• pML = z

• Bias:– E[z]- p = ( exp(a2) -1)p

• Error Variance:– (exp(2a2)- exp(a2) +1)p2

• Both grow exponentially with variance


Unbiased Estimator• z= e–aw p, w~N(0,)

• Given z, a and find g(z, a, ) such that for all p: E[g(z, a, )]=p

• Unique unbiased estimator– pU = exp(-a2) z

– Linear in z

• Error Variance:(exp(a2) -1)p2 c.f.

(exp(2a2)- exp(a2) +1)p2

• Better but grows exponentially with variance


Another estimate• Unique unbiased estimator linear in z• Find linear estimator that has the smallest error

variance• pV = exp(-3a2/2) z

• Bias:– ( exp(-a2) -1)p

• Error Variance:– (1- exp(-a2))p2

• Both bounded in the variance• cf CRLB=p2 a2

• MMSE?NICTA/ANU August 8, 2008


Conclusion

Localization from distancesShowed Linear algorithms are bad

Proposed new cost function

Characterized conditions for exponential convergence

Implications to anchor/sensor deployment

Practical convergence

Estimating distance form RSS under lognormal shadowingUnbiased and ML estimation Large error variance

New estimate Error variance and bias bounded in variance

Documents

Some Results on Source Localization Soura Dasgupta, U of Iowa With: Baris Fidan, Brian Anderson, Shree Divya Chitte and Zhi Ding