Optimal Formation of Sensors for Statistical Target Tracking April - PowerPoint PPT Presentation

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 ( 金聲浩 ) Korea Advanced Institute of Science and Technology (KAIST) South Korea 1/27

Contents  Back ground and problem  Probability model of range difference  Fisher information  Optimal ring formation of sensors  Numerical result  Conclusion 2/27

Multiple Missile Tracking cooperative sensing Loitering Missiles and (with LADAR seeker) precise target tracking cooperative Attack Missiles target attack (with semi-active seeker) Moving Target Launcher 3/27

Linear State-Space Models + = + + η s a F s (state eq.) t 1 t t t t = + + ε y b G s . (measurement eq.) t t t t t where s is an -dimensional state vector, m t − y k dimensional oberservation vector, t η ε and Gaussian white noises. t t η − η − η − t 1 t 3 t 2 s − s − s t 2 t 1 t y − y − y t 2 t 1 t ε − ε − ε t 2 t 1 t 4/27

Range difference measurements by TDOA method  TDOA=Time Difference Of Arrival 5/27

r Range difference ( ) geometry i  Range difference between sensors 0 and i : = − r d d i t ,0 t i ,  An approximation to : r i = − r d d i t t i ,   2 d d   = − + − θ − θ i ,0 i ,0 d 1 1 2 cos( )   t i t 2 d d   t t 2 d ≈ θ − θ − θ − θ i ,0 2 d cos( ) sin ( ) i ,0 i t i t d t 6/27

Proba bability mo model del f for range d differ eren ence r i = × = κ κ  Let z c , j 0,1,2, , , where n is the time of observation  j and the light speed. c N α σ )  2 z ( + d ,  . j j,t j = − =   y z z , j 1,2, , . n j 0 j y =   ( y , y , , y )'. 1 2 n 7/27

Proba bability mo model del for r range d e differ erence e i r   8/27

r Proba bability mo model del f for range d differ eren ence i    9/27

Fisher information   θ  Let X , , X be random variables from f x ( ; ). 1 n =  Let W W X ( , , X ) satisfy that 1 n = θ E ( W ) . θ θ Then, under some conditions on f x ( ; ), − 1   ∂ 2   [ ] − − θ ≥ θ = θ 1   2   E ( W ) n E log f X ( ; ) I ( ) . θ θ ∂ θ 1       θ θ  I ( ) is called the Fisher information for in X , , X . 1 n 10/27

Fisher information for target location on a plane (1/2) 11/27

Fisher information for target location on a plane (2/2) 12/27

Fisher information for target location on a plane (2/2) 13/27

Target location problem in 3-D • Target location = d φ θ t ( , , ) t t t • location of sensor i = d φ θ i ( , , ) i i i • location of sensor 0 = origin 14/27

Target location problem in 3-D  Range difference between sensors 0 and i :   2 d d d = − = − − φ φ θ − θ − φ φ +   i i i r d d d 1 1 2 sin sin cos( ) 2 cos cos   i t t i , t t i i t t i 2 d d d   t t t i 0 15/27

Target location problem in 3-D 16/27

Target location problem in 3-D 17/27

Target location problem in 3-D 18/27

Geometric Interpretation of I φφ 19/27

Optimal ring formation 20/27

Optimal ring formation 21/27

Optimal ring formation 22/27

Optimal ring formation 23/27

Optimal ring formation  = − − ˆ ˆ ˆ Var t ( ) E t ( t )( t t )'. c t c c c c 24/27

Optimal ring formation 25/27

Optimal ring formation MSE from (24, K) ring formations 26/27

Conclusions d φ θ  The ring formation renders the estimators , , t t t 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 π θ approaches from either direction. / 4 t 27/27

Thanks ( 謝謝 )