The Cat and The Mouse --The Case of Mobile Sensors and Targets

The Cat and The Mouse --The Case of Mobile Sensors and Targets

David K. Y. YauLab for Advanced Network Systems

Dept of Computer Science

Purdue University

(Joint work with J. C. Chin, Y. Dong, and W. K. Hon)

David K. Y. YauLab for Advanced Network Systems

Dept of Computer Science

Purdue University

(Joint work with J. C. Chin, Y. Dong, and W. K. Hon)

Why Mobile?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)

The mouse Evasion of detection Nature of “mission”

The cat Improved coverage with fewer sensors Robustness against contingencies Planned or random movement (randomness

useful)

Mobility ModelMobility 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

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 FormulationProblem 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)

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 PlayBest 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

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 FormulationOptimal 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?

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]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] asEk[T] := max(Ek[T], Ek[Tmove(0)])

Reorder heap in decreasing Ej[T] order Repeat until heap becomes empty

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] asEk[T] := max(Ek[T], Ek[Tmove(0)])

Reorder heap in decreasing Ej[T] order Repeat until heap becomes empty

Example Optimal PathsExample 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 StrategyComparison 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

• Local greedy strategy: mouse will stay

• Dynamic programming strategy: mouse moves to cell with small probability of cat’s presence (0.0075)

Current mouse position

If Cat Plays Random Waypoint StrategyIf 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

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

Distribution of Movement Direction in 150 m by 150 m Network Area

Cat’s Presence Matrix in 500500 m Network for Random Waypoint Movement

Cat’s Presence Matrix in 500500 m Network for Random Waypoint Movement

Distribution of Movement DirectionDistribution 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 StatisticsAnalytical 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 StrategyOptimal 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

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

Minimum Sensing Range for Expected Random Waypoint Coverage

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

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 DistanceLower 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

• 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 CellsExample Equidistant Sets of Cells

Mouse located at center of network area

Correlation between Cells VisitedCorrelation between Cells Visited

• Pi: probability that cat may visit cell i

• PSj: probability that cat may visit any cell in set Sj

• Pi: probability that cat may visit cell i

• PSj: probability that cat may visit any cell in set Sj

∑∈

≈j

jSl

lS PP

Shortest Distance Probability Matrix from Cell i to Cell j

Shortest Distance Probability Matrix from Cell i to Cell j

3-D probability matrix B 3-D probability matrix B

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

−−−−

−

−

1,1,10,1

1,,0,

1,0,00,0

mnmnjmnmn

mnijii

mnj

bbb

bbb

bbb

B

LL

MOOOM

OO

MOOOM

LL

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

where is the probability that DS0 is shortest distance for trip l,

and is probability that DSn is shortest distance for the trip, 1 ≤ n ≤ N

- 1

• Let denote , then is calculated as:

Shortest Distance Probability Matrix after l Trips

Shortest Distance Probability Matrix after l Trips

• Bl is the shortest distance probability matrix after l trips• Computed by * operator

• Bl is the shortest distance probability matrix after l trips• Computed by * operator

BBB ll ∗= −1

• Each element of Bl is calculated as:

∑−

=

−=1

0,

1,, *

1 mn

xjx

lxi

lji bb

mnb

lji x

b )',( jxl

xi bb ,1

, ∗− lji x

b )',(

∑ ∑ ∑= =

−

=

−

−

−≤≤−−−−=

−−−=n

k

n

k

n

k

ljijx

lxi

lji

jx

l

xil

ji

Nnkbkbkbnb

bbb

xx

x

0 0

1

0)',(,

1,)',(

,

1

,)',(

11],[])[1])([1(1][

]);0[1])(0[1(1]0[

]0[)',(l

ji xb

][)',( nblji x

Expected shortest distanceExpected shortest distance

The expected shortest distance between cat and mouse after l trips:

The expected shortest distance between cat and mouse after l trips:

∑ ∑∑−

=

−

= =⎟⎟⎠

⎞⎜⎜⎝

⎛×=

1

0

1

0 0,min ][

11][

mn

i

mn

j

N

k

ljiS

l kbDmnmn

dEk

Approximate Expected Shortest DistanceApproximate Expected Shortest Distance

Approximate expected shortest distance from mouse after cat has visited k cells:

PDj(k) is probability that after visiting k cells, a cell in Sj is visited, but no cell in Si, i< j, is visited

Approximate expected shortest distance from mouse after cat has visited k cells:

PDj(k) is probability that after visiting k cells, a cell in Sj is visited, but no cell in Si, i< j, is visited

∑−

=

=1

0min )()]([

N

jSD jj

DkPkdE

Lower Bound Cat-mouse Distance for Random Waypoint Model

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

Lower Bound Cat-mouse Distance for Indiana Map-based Model

ConclusionsConclusions

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, …

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)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 Validation with empirical data

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 Validation with empirical data