G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Recursive Triangulation UsingBearings-Only Sensors

G. Hendeby, LiU, Sweden

R. Karlsson, LiU, Sweden

F. Gustafsson, LiU, Sweden

N. Gordon, DSTO, Australia

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Motivating Problem

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

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

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

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)

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

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.

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Filters: UKF

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

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Filters: PF

Simulate several possible states and compare to the measurements obtained.

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

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

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Test Setup

Measurements from: Initial estimate: Initial estimate covariance: Different target positions along the -axis have been

evaluated. Poor initial information

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Test Setup: Measurement Noise

Gaussian noise:

Gaussian mixture noise:

Generalized Gaussian noise:

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

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:

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Comparison: RMSE

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

Gaussian mixture noise

Generalized Gaussian noise

50 measurements

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

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. measurements0 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

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

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.

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham

Conclusions, cont’d

(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

G. HendebyRecursive Triangulation Using Bearings-Only Sensors

TARGET ‘06Austin Court, Birmingham