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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 SJTU
April 21-27 Shanghai Jiao Tong University, China.
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Contents
Back ground and problem
Probability model of range difference
Fisher information
Optimal ring formation of sensors
Numerical result
Conclusion
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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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1 (state eq.). (measurement eq.)
whereis 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 sy b G s
s my k
ηε
η ε
+ = + + = + +
−
2ts − 1ts − ts3tη − 2tη − 1tη −
2ty − ty1ty −
2tε − tε1tε −
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Range difference measurements by TDOA method TDOA=Time
Difference Of Arrival
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Range difference ( ) geometry
Range difference between
sensors 0 and i:
An approximation to :
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,0 ,i t t ir d d= −
,
2,0 ,02
2,0 2
,0
1 1 2 cos( )
cos( ) sin ( )
i t t i
i it i t
t t
ii i t i t
t
r d d
d dd
d d
dd
d
θ θ
θ θ θ θ
= −
= − + − −
≈ − − −
ir
ir
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.
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Probability model for range difference
Let , 0,1, 2, , , where is the time of observation
and the light speed.jz c j n
cκ κ = × =
2j,t( + d ,j jz N α σ )
0 , 1, 2, , .j jy z z j n= − =
1 2( , , , ) '.ny y yy =
ir
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Probability model for range difference ir
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Probability model for range difference ir
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Fisher information
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[ ]
1
1
1212
1
Let , , be random variables from ( ; ).Let ( , , ) satisfy
that
E ( ) .Then, under some conditions on ( ; ),
E ( ) E log ( ; ) ( ) .
(
n
n
X X f xW W X X
Wf x
W n f X I
I
θ
θ θ
θ
θθ
θ θ θθ
−
−
=
=
∂ − ≥ = ∂
1) is called the Fisher information for in , , .nX Xθ θ
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Fisher information for target location on a plane (1/2)
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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 tt d φ θ=
( , , )i i ii 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
, 21 1 2 sin sin cos( ) 2 cos cosi i i
i t t i t t i i t t it t t
d d dr d d dd d d
φ φ θ θ φ φ
= − = − − − − +
Target location problem in 3-D
i 0
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Target location problem in 3-D
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Target location problem in 3-D
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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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Optimal ring formation
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Optimal ring formation
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Optimal ring formation
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Optimal ring formation
ˆ ˆ ˆ( ) ( )( ) '.c t c c c cVar t E t t t t= − −
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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 td φ θ
tθ / 4π
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Thanks (謝謝)
Optimal Formation of Sensors for Statistical Target
TrackingContentsMultiple Missile TrackingLinear State-Space
ModelsRange difference measurements by TDOA methodRange difference
( ) geometry Slide Number 9Fisher 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-DTarget location problem
in 3-DSlide Number 16Target location problem in 3-DTarget location
problem in 3-DSlide Number 19Optimal ring formationOptimal ring
formationOptimal ring formationOptimal ring formationOptimal ring
formationOptimal ring formationOptimal ring
formationConclusionsSlide Number 28
