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

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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)

- 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

Analysis, modeling and prediction

Biological

Radiation

Chemical

- 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

- The mouse
- Evasion of detection
- Nature of “mission”

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

- UAV detect radioactive plume
- Estimate position of plume source
- Control center predicts movement
- Emergency response

- 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

- 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)

- 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

- 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?

- 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

- For each neighbor cell k of 0, update Ek[T] as
- Repeat until heap becomes empty

Path when mouse moves slowly

Path when mouse moves quickly

- 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

- 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

(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

(b) Expected time before covering a cell (average = 59.604 s, maximum = 97.353 s)

(a) Expected number of trips before covering a cell (average = 11.431, maximum = 18.667)

(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)

- 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

Random Waypoint Model

Bouncing

Vc = 10 m/s, Vm = 10 m/s, Rc = 25 m, Rm = 0 m

Vc = 10 m/s, Vm = 10 m/s, Rc = 5 m, Rm = 10 m

- 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

- 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
- Nsets 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

Mouse located at center of network area

- Pi: probability that cat may visit cell i
- PSj: probability that cat may visit any cell in set Sj

3-D probability matrix B

- 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

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

- Each element of Bl is calculated as:

- Let denote , then is calculated as:

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

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

- 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

(b) Expected speed = 10 m/s

(c) Expected speed = 25 m/s

(a) Expected speed = 5 m/s

- 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, …

- 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