Dynamic Sensor Coverage

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Dynamic Sensor Coverage

• Abhishek Tiwari
• California Institute of Technology

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Collaborators

• Dr. Myungsoo Jun, GERC, UFL
• Dr. David E. Jeffcoat, AFRL-MN Eglin Air Force
  Base.
• Dr. Richard M. Murray, Caltech
• Dr. Babak Hassibi, Caltech

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Outline

• Introduction, motivation and applications of the
  problem.
• Related previous work.
• Problem description.
• Kalman filtering review.
• Markov chains review.

• Some results for the single sensor case.
• New results under coupled environment.
• Future work and discussion.

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Introduction

• Given An uncertain region in space.
• Task Maintain an appreciable estimate of
  uncertain parameters at each point in the region.
  
• Apparatus/tools A FEW limited range
  mobile sensors.

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Motivation/Applications
Weather monitoring Parameters
Temperature, Pressure, Humidity, Wind
speed/direction.
Surveillance Parameters Intruders
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Motivation/Applications
Situational Awareness Parameters friends, enemies
Roboflag Parameters Location of opponent robots
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Past Work

• Static coverage techniques based on Voronoi
  partitions. (Cortes et. al.) .
• ITAC 2004
• Dynamic sensor coverage experiments. (Batalin,
  Sukhatme). SPIE 02
• Kalman filtering with intermittent observations.
  (Sinupoli et. al.) CDC03
• Markov chains for search and surveillance.
  (Jeffcoat, Stone).

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Problem Description
N discrete time systems.

• Assumptions --------------
• The systems are decoupled.
• The noise processes are independent at different
  locations.

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Problem Description
Limited range A sensor has access to the
measurement of system i, iff it is currently
at system i. Algorithm Sensors send their
measurements to the base station which employs
a kalman filter to estimate the states of all N
systems.
Kalman Filter
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Kalman filter review

• Sensor at system i ? Execute both time and
  measurement updates.
• No sensor at system i ? Execute only time update
  .

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Problem Description
Sensor motion modeled by a Discrete Time
Discrete State Markov chain.
We need some additional properties
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Markov chain review
Defn. A DTDS Markov chain is called irreducible
if, starting from any one of the states, it is
possible to get to any other state (not
necessarily in one jump) with a non-zero
probability.
Defn. A state i of a Markov chain has period
d if, given that , we can only have
when n is a multiple of d. We call
i periodic if it has some period gt 1. If the
markov chain is irreducible, then either all
states are periodic or none are.
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Markov chain review

• Defn Suppose is a DTDS Markov chain with
  finite number of states,
• is ergodic if
• is irreducible
• is not perodic.

Ergodicity Theorem
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Problem Description
The idea is to write down the kalman filter
equations and then let .
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Modified Riccati Recursion
To maintain an appreciable estimate of the states
of all N systems we want that
remains bounded for all i.
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Modified Riccati Recursion
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Problem Statement
Defn. We say that the dynamic sensor coverage
problem has been successfully solved if for any
initial probability distribution of the sensors
the N limits
are finite for any set if initial conditions
If there exists an
such that is unbounded
for some ,then the sensors have
failed to solve the dynamic coverage problem.
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Single sensor case
N discrete time systems.

• Assumptions --------------
• The systems are decoupled.
• The noise processes are independent at different
  locations.

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Failure result
Failure Theorem
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Results for a single sensor
Success result Quasi-convex feasibility
condition. No closed form condition.
http//www.cds.caltech.edu/atiwari/
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Coupled Environment
Assumption does not hold anymore
1
2
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Coupled Environment
Unstable
Unstable
Stable
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Coupled Environment
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Coupled Environment
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Multiple Sensors
Sensor 1
Measure of cooperation
Sensor 2
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Conclusions

• Whats new? Theoretical framework for the
  Dynamic Coverage problem.
• Insightful Results are intuitive and provide
  deeper insight.
• Backward compatibility A lot of already existing
  tools in Kalman filter and Markov chain
  literature can be used.
• Other areas Coupled Environment case can be
  used to get meaningful results for packet based
  networks.
• Scalability The approach can be extended to
  multiple sensors and systems with ease.

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Future Work

• Stability region in the coupled environment case.
• Synthesis of the transition probability matrix.
• Multiple sensor case.
• Convergence to static coverage results as the
  number of sensors increases.
• Experiments. (Multi Vehicle Wireless Testbed).
• Networked control systems.

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Thank you ?

• Questions ?