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The Cat and The Mouse -The Case of Mobile Sensors and Targets
David K. Y. Yau
Lab for Advanced Network Systems
Dept of Computer Science
Purdue University
(Joint work with J. C. Chin, Y. Dong, and W. K. Hon)
Project Background
 Sensor-cyber project in national defense
 Near real-time detection, tracking, and analysis of plumes
(nuclear, chemical, biological, …)
 Multi-university partnership funded by Oak Ridge
National Lab
 Sensor testbed design and implementation
 Research team: Purdue, UIUC, LSU, U of Florida, Syracuse
 Personel
 Purdue: Jren-Chit Chin, Yu Dong, David Yau, Wing-Kai Hon
 Oak Ridge National Lab: Nageswara Rao
 Partnership with SensorNet initiative
SensorNet Initiative
 Building comprehensive
incident management
system
 Coordinate knowledge and
response effectively
 Provide data highway for
processing sensor data
 Deliver near-real-time
information for effective
counter-measure
Analysis, modeling
and prediction
Biological
Radiation
Chemical
Why Mobile?
 The mouse
 Evasion of detection
 Nature of “mission”
 The cat
 Improved coverage with fewer sensors
 Robustness against contingencies
 Planned or random movement (randomness
useful)
Scenario—UAV Surveillance
 UAV detect
radioactive plume
 Estimate position of
plume source
 Control center
predicts movement
 Emergency
response
Mobility Model
 Four-tuple <N, M, T, R>
 N: network area
 M: accessibility constraints
-- the “map”
 T: trip selection
 R: route selection
 Random waypoint model is a special case
 Null accessibility constraints
 Uniform random trip selection
 Cartesian straight line route selection
Problem Formulation
 Two player game
 Payoff is time until detection (zero sum)
 Cat plays detection strategy
 Stochastic, characterized by per-cell presence
probabilities
 Mouse plays evasion strategy
 Knows statistical process of cat’s movement, but not
necessarily exact routes (exact positions at given
times)
Best Mouse Play
 Cat’s presence matrix given
 Network region divided into 2D cells
 Pi,j gives probability for mouse to find cat in cell (i, j)
 Expected detection time “long” compared with
trip from point A to point B
 Dynamic programming solution to maximize
detection time
 Local greedy strategy does not always work
Optimal Escape Path Formulation
 For each cell j, mouse decides whether to stay or to move to a
neighbor cell (and which one)
 If stay, expected max time until detection is Ej[Tstay]
 If move to neighbor cell k, expected max time until detection is
Ej[Tmove(k)]
 For cell j, expected max time until detection, Ej[T], is largest
of Ej[Tstay] and Ej[Tmove(k)] for each neighbor cell k of j
 Ej[Tstay] determined by cat’s presence matrix and expected cat’s sojourn
time in each cell
 Optimal escape path is sequence of safest neighbors to move
to, until mouse decides to stay
 How to compute Ej[T] for each cell j?
Computing Ej[T]
 Initialize Ej[T] as Ej[Tstay]
 Insert all the cells into heap sorted by decreasing
Ej[T]
 Delete root cell 0 from heap
 For each neighbor cell k of 0, update Ek[T] as
Ek[T] := max(Ek[T], Ek[Tmove(0)])
 Reorder heap in decreasing Ej[T] order
 Repeat until heap becomes empty
Example Optimal Paths
0.007
0.009
0.01
0.009
0.007
0.009
0.05
0.1
0.05
0.009
0.01
0.1
0.08
0.1
0.01
0.009
0.05
0.1
0.05
0.009
0.0075
0.009
0.01
0.009
0.0075
Path when mouse moves
slowly
Path when mouse moves
quickly
Comparison with Local Greedy Strategy
0.0075
0.0075
0.0075
0.0075
0.0075
0.0075
0.1
0.1
0.1
0.0075
0.0075
0.1
0.08
0.1
0.0075
0.0075
0.1
0.1
0.1
0.0075
0.0075
0.0075
0.0075
0.0075
0.0075
Current mouse position
• Local greedy strategy:
mouse will stay
• Dynamic programming
strategy: mouse moves to
cell with small probability of
cat’s presence (0.0075)
If Cat Plays Random Waypoint Strategy
 Highest presence probability at the center of
the network area
 Lowest presence probabilities at the corners
and perimeters
 Good “safe havens” for mouse to hide
 Sum of presence probabilities is one
 n cats  sum of probabilities  n
 Equality for disjoint cats’ surveillance areas
Distribution of Movement Direction
in 150 m by 150 m Network Area
Cat’s Presence Matrix in 500500 m Network
for Random Waypoint Movement
Distribution of Movement Direction
(a) Calculated
probabilities of sensor
moving towards the
center cell from
different current cells
(b) Measured probabilities
of sensor moving towards
the center cell from different
current cells
Analytical Cell Coverage Statistics
(a) Expected number of
trips before covering a
cell (average = 11.431,
maximum = 18.667)
(b) Expected time
before covering a cell
(average = 59.604 s,
maximum = 97.353 s)
Measured Cell Coverage Statistics
(b) Expected number of
trips before covering a
cell (average = 10.301,
maximum = 20.482)
(b) Expected time
before covering a cell
(average = 52.721 s,
maximum = 105.169 s)
Optimal Cat Strategy
 Maximize minimum presence probability
among all the cells
 Eliminate safe haven
 Achieved by equal presence probabilities in
each cell
 Will lead to Nash Equilibrium
 Zero sum game  Pareto optimality
Presence Matrices
Random Waypoint Model
Bouncing
Seeing Mouse Strategy
Blind Mouse Strategies
Compared
Average
Detection Time
DP
Mouse
RWP
Strategies
Stay
Cat Strategies
Scan
1083.26
Bouncing
RWP
628.66 2823.26
415.31
442.23
271.73
511.50
305.03
226.13
Vc = 10 m/s, Vm = 10 m/s, Rc = 25 m, Rm = 0 m
Seeing Mouse Strategies
Compared
Average
Detection Time
Bouncing
Centric
Mouse
Strategies Static
Stay
Cat Strategies
Bouncing
RWP
149.53
1455.28
340.85
1092.29
92.39
899.07
10.23
21.99
Vc = 10 m/s, Vm = 10 m/s, Rc = 5 m, Rm = 10 m
Effect of Mouse Sensing Range
Effect of Mouse Speed
Effect of Number of Cats
Minimum Sensing Range for Expected
Random Waypoint Coverage
 Stationary mouse; cat in random waypoint
movement
 Expected coverage desired by given
deadline
 What is minimum sensing distance
required?
 Stochastic analysis of shortest distance between
cat and mouse within deadline
Lower Bound Cat-mouse Distance
• Network divided into m by n cells; each has fixed size
s by s
• D(i, j): Euclidean distance between cell i and cell j
• N sets of cells sorted by set’s distance to mouse
• Each set of cells denoted as Sj, 0 ≤ j ≤ N - 1
• Each cell in Sj is equidistant from the mouse; distance is DSj
• Distances sorted in increasing order; i.e., DSj < DSj+1
Example Equidistant Sets of Cells
Mouse located at center of network area
Correlation between Cells Visited
• Pi: probability that cat may visit cell i
• PSj: probability that cat may visit any cell in set Sj
PS j 
P
lS j
l
Shortest Distance Probability Matrix
from Cell i to Cell j
3-D probability matrix B
 b0,0

 
B   bi ,0

 
b
 mn1,0


b0, j


bi , j


 bmn1, j
b0,mn1 



 bi ,mn1 




 bmn1,mn1 


Each element bi,j
• gives cat’s shortest distance distribution from mouse after trip from cell i to j
• is a size N vector: bi,j[k] is the probability that the shortest distance during
the trip is DS
k
Shortest Distance Probability
Matrix after l Trips
•
•
Bl is the shortest distance probability matrix after l trips
Computed by * operator
Bl  Bl 1  B
• Each element of Bl is calculated as:
1 mn1 l 1
b 
bi , x * bx , j

mn x 0
l
i, j
• Let
l
l
b(i , j )' x denote bil,x1  bx, j, then b(i , j )' x is calculated as:
l 1
b(li , j )' x [0]  1  (1  bi , x [0])(1  bx , j [0]);
n
l
( i , j )' x
b
n
n 1
k 0
k 0
[n]  1  (1   b [k ])(1   bx , j [k ])   b(li , j )' x [k ],1  n  N  1
k 0
l
( i , j )' x
l 1
i,x
[0] is the probability that DS is shortest distance for trip l,
where b
0
l
and b(i , j )' [n] is probability that DSn is shortest distance for the trip, 1 ≤ n ≤ N - 1
x
Expected shortest distance
 The expected shortest distance between cat
and mouse after l trips:
mn1
mn1 N

1
1
l
l


E[d min ] 
D

b
[
k
]


S
i
,
j
k

mn i 0  mn j 0 k 0

Approximate Expected Shortest Distance
 Approximate expected shortest distance from mouse
after cat has visited k cells:
N 1
E[d min (k )]   PD j (k ) DS j
j 0
 PDj(k) is probability that after visiting k cells, a cell in Sj
is visited, but no cell in Si, i< j, is visited
Lower Bound Cat-mouse Distance
for Random Waypoint Model
(a) Expected speed = 5 m/s
(b) Expected speed = 10 m/s
(c) Expected speed = 25 m/s
Lower Bound Cat-mouse Distance
for Indiana Map-based Model
Conclusions
 Considered cat and mouse game between
mobile sensors and mobile target
 For random waypoint model, other
coverage properties can be obtained
analytically
 Expected cell sojourn time, expected time to
cover general AOI, number of sensors to
achieve coverage by given deadline, …
Conclusions (cont’d)
 Many extensions possible
 Explicit account for plume explosion /
dispersion models
 Model for sensor (un)reliability, interference,
etc
 Explicit quantification of sensing uncertainty
and its reduction