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2015 Workshop on Combinatorics and Applications at S JTU April 21-27 S hanghai Jiao Tong University, China. Optimal Formation of Sensors for Statistical Target Tracking April 24, 2015 Sung-Ho Kim ( ) Korea Advanced Institute of

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Optimal Formation of Sensors for Statistical Target Tracking

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April 24, 2015

Sung-Ho Kim (金聲浩) Korea Advanced Institute of Science and Technology (KAIST) South Korea

2015 Workshop on Combinatorics and Applications at S JTU

April 21-27 S hanghai Jiao Tong University, China.

SLIDE 2

 Back ground and problem  Probability model of range difference  Fisher information  Optimal ring formation of sensors  Numerical result  Conclusion

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SLIDE 3

Multiple Missile Tracking

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Launcher Moving Target Loitering Missiles (with LADAR seeker) cooperative sensing and precise target tracking Attack Missiles (with semi-active seeker) cooperative target attack

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Linear State-Space Models

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(state eq.) . (measurement eq.) where is an -dimensional state vector, dimensional oberservation vector, and Gaussian white noises.

t t t t t t t t t t t t t t

s a F s y b G s s m y k η ε η ε

+ =

+ + = + + −

2 t

s −

1 t

s −

s

3 t

η −

2 t

η −

1 t

η −

2 t

y −

y

1 t

y −

2 t

ε −

ε

1 t

ε −

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Range difference measurements by TDOA method

 TDOA=Time Difference Of Arrival

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SLIDE 6

Range difference ( ) geometry

 Range difference between

sensors 0 and i:

 An approximation to :

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,0 , i t t i

r d d = −

, 2 ,0 ,0 2 2 ,0 2 ,0

1 1 2 cos( ) cos( ) sin ( )

i t t i i i t i t t t i i i t i t t

r d d d d d d d d d d θ θ θ θ θ θ = −     = − + − −     ≈ − − −

r

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  .

 

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Proba bability mo model del f for range d differ eren ence

Let , 0,1,2, , , where is the time of observation and the light speed.

z c j n c κ κ = × = 

2 j,t

( + d ,

j j

z N α σ )  , 1,2, , .

j j

y z z j n = − = 

1 2

( , , , )'.

y y y

y =



r

SLIDE 8

 

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Proba bability mo model del for r range d e differ erence e i

r

SLIDE 9

  

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Proba bability mo model del f for range d differ eren ence i

r

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Fisher information

 

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[ ]

1 1 1 2 1 2 1

Let , , be random variables from ( ; ). Let ( , , ) satisfy that E ( ) . Then, under some conditions on ( ; ), E ( ) E log ( ; ) ( ) . (

n n

X X f x W W X X W f x W n f X I I

θ θ θ

θ θ θ θ θ θ θ

− −

= =   ∂   − ≥ =     ∂        

) is called the Fisher information for in , , .

X X θ θ 

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Fisher information for target location on a plane (1/2)

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SLIDE 12

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Fisher information for target location on a plane (2/2)

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Fisher information for target location on a plane (2/2)

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Target location problem in 3-D

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( , , )

t t t

t d φ θ =

( , , )

i i i

i d φ θ =

Target location
location of sensor i
location of sensor 0 = origin

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 Range difference between sensors 0 and i:

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2 , 2

1 1 2 sin sin cos( ) 2 cos cos

i i i i t t i t t i i t t i t t t

d d d r d d d d d d φ φ θ θ φ φ   = − = − − − − +      

Target location problem in 3-D

SLIDE 16

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Target location problem in 3-D

SLIDE 17

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Target location problem in 3-D

SLIDE 18

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Target location problem in 3-D

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Geometric Interpretation of Iφφ

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Optimal ring formation

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SLIDE 21

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Optimal ring formation

SLIDE 22

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Optimal ring formation

SLIDE 23

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Optimal ring formation

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

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Optimal ring formation

ˆ ˆ ˆ ( ) ( )( )'.

c t c c c c

Var t E t t t t = − −

SLIDE 25

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Optimal ring formation

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MSE from (24, K) ring formations

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Optimal ring formation

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Conclusions

 The ring formation renders the estimators

stochastically independent.

 Optimal sensor formations are half-and-half arrangement

between the center and the outer-most ring.

 (n,4)-ring performs better to worse than (n,3)-ring as

approaches from either direction.

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, ,

t t t

d φ θ

θ / 4 π

SLIDE 28

Optimal Formation of Sensors for Statistical Target Tracking

Contents

 Back ground and problem  Probability model of range difference  Fisher information  Optimal ring formation of sensors  Numerical result  Conclusion

Multiple Missile Tracking

Linear State-Space Models

s −

s −

s

η −

η −

η −

y −

y

y −

ε −

ε

ε −

Range difference measurements by TDOA method

Range difference ( ) geometry

 Range difference between

sensors 0 and i:

 An approximation to :

r d d = −

1 1 2 cos( ) cos( ) sin ( )

r d d d d d d d d d d θ θ θ θ θ θ = −     = − + − −     ≈ − − −

r

r

 

Proba bability mo model del f for range d differ eren ence

( + d ,

z N α σ )  , 1,2, , .

y z z j n = − = 

( , , , )'.

y y y

y =



r

 

Proba bability mo model del for r range d e differ erence e i

r

  

Proba bability mo model del f for range d differ eren ence i

r

Fisher information

 

[ ]

Fisher information for target location on a plane (1/2)

Fisher information for target location on a plane (2/2)

Fisher information for target location on a plane (2/2)

Target location problem in 3-D

( , , )

t d φ θ =

( , , )

i d φ θ =

 Range difference between sensors 0 and i:

Target location problem in 3-D

Target location problem in 3-D

Target location problem in 3-D

Target location problem in 3-D

Geometric Interpretation of Iφφ

Optimal ring formation

Optimal ring formation

Optimal ring formation

Optimal ring formation



Optimal ring formation

ˆ ˆ ˆ ( ) ( )( )'.

Var t E t t t t = − −

Optimal ring formation

MSE from (24, K) ring formations

Optimal ring formation

Conclusions

 The ring formation renders the estimators

stochastically independent.

 Optimal sensor formations are half-and-half arrangement

between the center and the outer-most ring.

 (n,4)-ring performs better to worse than (n,3)-ring as

approaches from either direction.

, ,

d φ θ