Case Study: An Information Theoretic Approach to Observer Path Design for BearingsOnly Tracking A' L

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Case StudyAn Information Theoretic Approach to
Observer Path Design for Bearings-Only
Tracking(A. Logothetis, A. Isaksson, R. Evans)

• Sensor Reading Group
• 30 October 2003
• Timothy Chung

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Main contributions

• Given an array of bearings-only sensors, the
  authors optimally solve the 2D target tracking
  problem, using cost as mutual information between
  measurements sequence and
• (a) final observer state, xT and
• (b) observer trajectory sequence, XT.

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Previous work

• Some of the previous work done
• Didnt include process noise in the target model
• Assumed constant measurement noise
• i e. sensor measurements are range INdependent
• Metric is a function of the Fisher Information
  Matrix (FIM)
• CRLB FIM-1, which is the lowest estimation
  error
• e.g. maximize det, trace, smallest ?

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Modeling

• Target
• Statistical model of the kinematics
• Constant velocity with zero-mean random
  accelerations
• where

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Modeling, cont.

• Measurement
• The sensor array has a preference in orientation
  relative to the target!
• Measurement noise is given by

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Information Theory Tools

• Recall the definitions of entropy and mutual
  information for continuous r.v.s
• Entropy
• Conditional Entropy
• Mutual Information

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Preliminaries

• Given a linear Gauss-Markov system
• Notation k 2 0,T
• Xk ( x(1), x(2), , x(k) )
• Yk ( y(1), y(2), , y(k) )
• Statistical properties

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Lemma 3.1 I(x(T)YT)

• The mutual information between the final state
  x(T) and the measurement sequence YT is given by
• where

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Proof of Lemma 3.1

• First, we note that the state x(T) is a Gaussian
  f(x(T)) with mean
• and covariance is given by the recursive
  covariance extrapolation equation

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Proof of Lemma 3.1, cont.

• Similarly, f(x(T)YT) is a Gaussian with mean
  (defined below) and covariance P(TT)
• The a posteriori mean and uncertainty are given
  by Kalman filter update equations

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Proof of Lemma 3.1, cont.

• Now, we plug these distributions into the
  definition of the mutual information
• Recall f(x(T)) N(?1,?1) f(x(T)YT) N(?2,?2).
• The entropy of a multi-variate Gaussian
  distribution is given by

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Proof of Lemma 3.1, cont.

• The result of Lemma 3.1 follows immediately

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Maximization of Mutual Information

• Formulate this as a constrained optimization
  problem
• Given dynamics equations (target, observer,
  measurement models)
• Initial conditions
• Constraints (e.g. dynamics of observer vehicle)
• Find
• where UT-1 is the sequence of control inputs
  from k1,..T-1.

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Application to Mobile Robotics

• Bearings-only problem

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References

• Logothetis, EM Algorithms for State and
  Parameter Estimation of Stochastic Dynamical
  Systems, Ph.D thesis, U. Melbourne,
  Australia,1997.
• Anderson and Moore, Optimal Filtering, Prentice
  Hall, New Jersey, 1979.
• Cover and Thomas, Elements of Information
  Theory, John Wiley Sons, 1991.