Recursive Triangulation Using BearingsOnly Sensors G Hendeby Li

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

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

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

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 § G. Hendeby Recursive Triangulation Using Bearings-Only Sensors Generalized Gaussian noise 50 measurements 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 Filter 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. 98 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. 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 G. Hendeby Recursive Triangulation Using Bearings-Only Sensors TARGET ‘ 06 Austin Court, Birmingham

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