1. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin Court, Birmingham Recursive Triangulation Using Bearings-Only Sensors G. Hendeby, LiU, Sweden R. Karlsson, LiU, Sweden F. Gustafsson, LiU, Sweden N. Gordon, DSTO, Australia

2. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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

3. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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 )

4. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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.

5. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin Court, Birmingham Filters: UKF Simulate carefully chosen “sigma points” to transform involved covariance matrices and use in the KF.

6. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin Court, Birmingham Filters: PF Simulate several possible states and compare to the measurements obtained.

7. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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 

8. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin Court, Birmingham Test Setup  Measurements from:  Initial estimate:  Initial estimate covariance:  Different target positions along the -axis have been evaluated.  Poor initial information

9. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin Court, Birmingham Test Setup: Measurement Noise  Gaussian noise:  Gaussian mixture noise:  Generalized Gaussian noise:

10. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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:

11. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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

12. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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 FilterNo. measurements 012345 EKF3.1610.1510.6411.5310.8111.23 IEKF3.1610.1210.4011.5511.1411.61 UKF3.1610.1510.6211.5311.1411.63 PF3.329.178.998.879.879.98

13. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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.

14. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin 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

15. G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘06 Austin Court, Birmingham