Recursive Triangulation Using Bearings-Only Sensors

1
Recursive Triangulation UsingBearings-Only
Sensors

• G. Hendeby, LiU, Sweden
• R. Karlsson, LiU, Sweden
• F. Gustafsson, LiU, Sweden
• N. Gordon, DSTO, Australia

2
Motivating Problem
Track a target during close fly-by using bearings
only sensors

• Known to be difficult to estimate
• Highly nonlinear, especially at short range
• Previously used to demonstrate usefulness of new
  methods
• Methods and performance measures will be discussed

3
Filters

• The following filters have been evaluated and
  compared
• Local approximation
• Extended Kalman Filter (EKF)
• Iterated Extended Kalman Filter (IEKF)
• Unscented Kalman Filter (UKF)
• Global approximation
• Particle Filter (PF)

4
Filters (I)EKF

• EKF Linearize the model around the best
  estimate and apply the Kalman filter (KF) to
  the resulting system.
• IEKF Relinearize the model after a measurement
  update with a (hopefully) improved estimate, and
  restart the update with this linear model.

5
Filters UKF

• Simulate carefully chosen sigma points to
  transform involved covariance matrices and use in
  the KF.

6
Filters PF

• Simulate several possible states and compare to
  the measurements obtained.

7
Filter Evaluation

• Root mean square error (RMSE)
• Standard performance measure
• Bounded by the Cramér-Rao Lower Bound (CRLB)
• Ignores higher order moments
• Kullback divergence
• Compares the distance between two distributions
• Captures effects not seen in the RMSE

8
Test Setup

• Measurements from
• Initial estimate
• Initial estimate covariance
• Different target positions along the -axis
  have been evaluated.
• Poor initial information

9
Test Setup Measurement Noise

• Gaussian noise
• Gaussian mixture noise
• Generalized Gaussian noise

10
Test Setup True Inferred Distribution

• True inferred state distribution for one noise
  realization,
• Some non-Gaussian features
• Computed using gridding, not feasible for use in
  practice
• CRLB for this situation

11
Comparison RMSE
Gaussian mixture noise

• The PF is overall best, however CRLB is not
  reached
• (I)EKF sometimes diverges, iterating then could
  be catastrophic
• Difficult to extract information from
  non-Gaussian measurements
• Higher moments are ignored in this comparison

Generalized Gaussian noise
50 measurements
12
Comparison Kullback divergence

• The Kullback divergence has been used to capture
  other differences between estimated and true
  distribution. Note, the results represents only
  one realization.
• Here Gaussian mixture noise and

Filter No. measurements No. measurements No. measurements No. measurements No. measurements No. measurements
0 1 2 3 4 5
EKF 3.16 10.15 10.64 11.53 10.81 11.23
IEKF 3.16 10.12 10.40 11.55 11.14 11.61
UKF 3.16 10.15 10.62 11.53 11.14 11.63
PF 3.32 9.17 8.99 8.87 9.87 9.98
13
Conclusions

• A bearings-only estimation problem, with large
  initial uncertainty, has been studied using
  different filters.
• As a complement to comparing RMSE, the Kullback
  divergence has been used to capture more than the
  variance aspects of the obtained estimates.

14
Conclusions, contd

• (Iterated) Extended Kalman Filter ((I)EKF)
• Works acceptable with good initial information,
  but has difficulties with bad initial information
• Iterating often slightly improve performance, but
  sometimes backfires badly
• Unscented Kalman Filter (UKF)
• Results are not bad, but not as impressive as
  suggested in recent literature
• Particle Filter (PF)
• Works well at the price of higher computational
  effort

15
(No Transcript)