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KR ISHNA KALYANAM( INFOSC ITEX CORP. )
I N C O L L A B O RAT I O N W I T H
S . D A R B H A ( TA M U )P. P. K H A R G O N E K A R ( U F , E - A R PA )
M . PA C H T E R ( A F I T / E N G )P. C H A N D L E R A N D D . C A S B E E R ( A F R L / R Q C A )
A F R L / R Q C A U AV T E A M M E E T I N GO C T 3 1 , 2 0 1 2
Optimal Min-max Pursuit Evasion on a Manhattan Grid
RQCA Conf. Rm. 2
UGS Sensor Range
UGS Communication Range
Valid Intruder PathScenario
UAV Communication Range
BASE
10/31/12
RQCA Conf. Rm. 3
Pursuit-Evasion Framework
• Pursuer engaged in search and capture of intruder on a Manhattan road network
• Intersections in road instrumented with Unattended Ground Sensors (UGSs)
• Pursuer has a 2x speed advantage over the evader• Pursuer has no on-board sensing capability• Evader triggers UGS and the event is time-stamped
and stored in the UGS• Pursuer interrogates UGSs to get evader location
information• Capture occurs when pursuer and evader are co-
located at an UGS location
10/31/12
RQCA Conf. Rm. 4
Manhattan Grid (3 row corridor)
All edges of the grid are of same length Purser arrives at node (t/c/b,0) with delay D>0 (time steps) behind the evader Evader dynamics - move North, East or South but cannot re-visit a node Pursuer actions - move North, East or South or Loiter/ Wait at current location Pursuer has a 2x speed advantage over the evader
c
0 1 2 n
b
t
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D
RQCA Conf. Rm. 5
Governing Equations
10/31/12
RQCA Conf. Rm. 6
Problem FrameworkPose the problem as a Partially Observable Markov
Decision Process (POMDP) unconventional POMDP since observations give
delayed intruder location information with random time delays!
Use observations to compute the set of possible intruder locations
Dual control problem Pursuer’s action in addition to aiding capture
also affects the future uncertainty associated with evader’s location (exploration vs. exploitation)
10/31/12
RQCA Conf. Rm. 7
Partial and delayed state information
10/31/12
RQCA Conf. Rm. 8
Optimization Problem
10/31/12
t
c
b
D
0 1 2
RQCA Conf. Rm. 9
Bellman recursion
10/31/12
RQCA Conf. Rm. 10 10/31/12
Induction - Motivation
cD
0 1 2 D-1 D
D-1 D-2 1 0
single row: capture in exactly D steps T(D)=1+T(D-1);T(1)=1 => T(D) = D
two rows: capture in exactly D+2 steps T(D)=1+T(D-1);T(1)=3 => T(D) = D+2
pursuer
evader
t
bD D-1 D-2 1
0
RQCA Conf. Rm. 11
A Feasible Policy (upper bound)
t
c
b
D
0 1 2
10/31/12
RQCA Conf. Rm. 12
Bottom/Top row - delay 1
1
0
pursuer
evader0 1
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RQCA Conf. Rm. 13
Bottom/Top row - delay 2
1
00 1 2
2
10/31/12
RQCA Conf. Rm. 14
Center row - delay 1
1
1
00 1 2 3
2
10/31/12
RQCA Conf. Rm. 15
Center row - delay 2
01 2 3 40
2
2 1
1
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RQCA Conf. Rm. 16
Bottom row - delay 3
10/31/12
Center row - delay 3
t
c
b
D
0 1 2
RQCA Conf. Rm. 17 10/31/12
Specification of the policyμ
Delay (D) Sequence Max Steps
1 ENLNL 5
2 EN2L 6
3 EN2 13
≥4 EN2? D+10
Delay (D) Sequence Max Steps
1 ENLS2 11
2 ENS2 12
3 ENSES 13
≥4 ?? D+10
bottom row:
center row:
RQCA Conf. Rm. 18
Induction argument for D>=4
Basic step: Tμ(r,3)=13
Induction hypothesis:
10/31/12
RQCA Conf. Rm. 19 10/31/12
Specification of the policyμ
Delay (D) Sequence Min-Max Steps
1 ENLNL 5
2 EN2L 6
≥3 EN2 D+10
Delay (D) Sequence Min-Max Steps
1 ENLS2 11
2 ENS2 12
3 ENSES 13
≥4 ED-3NSE2S D+10
bottom row:
center row:
RQCA Conf. Rm. 20
Center row, delay D>=4
10/31/12
D
k=D k=D+1 k=2D-4
k=2D+2
k=2D
k=2D-20 1 D-4 D-3 D-2 D-1
(D-3) moves E
RQCA Conf. Rm. 21
Center row, delay D>=4 (contd.)
D
(D-3) moves E
2
k=0,k=D
k=D+1 k=2D-4
k=2 k=4 k=2D-4 k=2D-2
k=2D+2k=2D
k=2D
k=2D-20 1 D-4 D-3 D-2 D-1
10/31/12
RQCA Conf. Rm. 22
Center row, delay D>=4 (contd.)
D
k=0,k=D
k=D+1 k=2D-4
k=2D+2
k=2D
k=2D-20 1 D-4 D-3 D-2 D-1
10/31/12
RQCA Conf. Rm. 23
Center row, delay D>=4
Bottom row, delay D>=4
D
0 1
k=D+1
D-2k=4,k=D+2
k=0,k=D
10/31/12
RQCA Conf. Rm. 24
Lower Bound on Steps to capture
10/31/12
t
c
b
D
0 1 2
RQCA Conf. Rm. 25
Lower bound on optimal time to capture
10/31/12
RQCA Conf. Rm. 26
Optimal (min-max) Steps to Capture
10/31/12
RQCA Conf. Rm. 27
East is optimal at red UGS
sketch of proof:
10/31/12
28
Optimal trajectory
There is an optimal trajectory, referred to as a ``turnpike”, which both the pursuer and the evader strive to reach and stay in, for most of the encounter.
Here, the turnpike is the center row of the symmetric 3 row grid. The pursuer, after initially going east, if not already on the turnpike,
immediately heads towards it. The evader initially heads to the turnpike, unless it is already on it,
until the ``end game", whence it swerves and gets off the turnpike to avoid immediate capture.
The pursuer stays on the turnpike, monitoring the delays, until he observers delay 1. At this point, he also executes the ``end game" maneuver, and captures the evader in exactly 11 more steps.
RQCA Conf. Rm. 10/31/12
29
Summary
Advantages Policy is dependent only on the delay at, and time elapsed since, the last
red UGS (sufficient statistic?) Policy is optimal despite not relying on the entire information history of
pursuer
Disadvantages Policy is not in analytical form i.e., function from information state to
action space (and so not extendable to other graphs) what is the intuition (exploration vs. exploitation, does separation exist?)
Extension(s) Can policy be approximated by a feedback policy that minimizes suitable
norm of the error (distance to evader + size of uncertainty) Capture can no longer be guaranteed (by a single pursuer) if number of
rows exceeds 3 With 2 pursuers, capture can be guaranteed in D+4 steps on any number
of rows (including infinity)!
RQCA Conf. Rm. 10/31/12
RQCA Conf. Rm. 30
Extras
10/31/12
RQCA Conf. Rm. 31
Center row, delay D>=4 (contd.)
D
k=0,k=D
k=D+1 k=2D-4
k=2D+2
k=2D
k=2D-20 1 D-4 D-3 D-2 D-1
10/31/12
conservative bound: D-1+11=D+10 (see extra slide)
RQCA Conf. Rm. 32 10/31/12
D
0
k=0,k=D
k=D+1 k=2D-4
k=2 k=4 k=2D-4
k=2D-2
k=2D
k=2D-2k=2D
0 1 D-4 D-3 D-2 D-1
1
steps to capture: D-1+3=D+2conservative bound (per policy) = D-1+11=D+10