Robust Extended Kalman Filtering in Hybrid Positioning Applications
1
Robust Extended Kalman Filtering in Hybrid Positioning Applications WPNC 2007 Tommi Perälä Tampere University of Technology, Finland
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
2
Outline
• Motivation • Robust WLS-filter • Robust bayesian filter • Simulations • Tests • Conclusions
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
3
Motivation
• Kalman filter (KF, EKF, EKF2, …) well suited to small positioning devices – low complexity • OK for small nonlinearities and nearly gaussian noise • Outliers can cause KF to fail badly – need robust KF
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
4
Deterministic filtering
• State evolution
• Measurements
• Minimize
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
5
The weighted least squares filter (WLS-filter)
• Instead of squares, minimize some other score function • Kalman-type recursions, variances scaled by measurement’s ”likeliness” (score function)
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
6
WLS-filter: The algorithm
1.
Choose the score function
2.
Calculate the innovations
3.
Modify the Kalman covariances
4.
Calculate estimate
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
7
Bayesian filtering
• State evolution
• Measurements
• Solve
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
8
The approximate bayesian filter (B-filter)
• Approximate the innovation pdfs with heavy-tailed nongaussian densities
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
9
B-filter: The algorithm
1. Choose the innovation pdf 2. Calculate the likelihood score of the innovation 3. Calculate the posterior mean and the covariance 4. Approximate the posterior with a normal density
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
10
Simulations
• MATLAB simulation test bench (different filters, geometries, measurements) • 100 tracks, 100 timesteps, 1 Hz • 9 cases: 0 – 5 GPS, 0 – 2.5 base stations • Outlier probability 0.00 or 0.05 • Filters’ ability to detect outliers also tested
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
11
Simulation results
• Criteria: 2D MSE, 95th percentile, inconsistency • B-filter outperforms EKF and EKF2 in contaminated cases • In contaminated GPS cases the WLS-filter fails completely • In contaminated basestation-only cases the WLS-filter sometimes outperform all filters. • Fewer than half of outlier identifications were correct
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
12
Tests
• Walking & bus, GPS receiver EKF
Robust
50 m Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
13
Conclusions
• Robust filters outperform EKF in contaminated cases and do almost as well in uncontaminated cases • The approximate bayesian filter outperforms WLS-filter in GPS cases • Score function choice • Outlier identification based on innovation not reliable (yet)
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
14
Conclusions
• Robust filters outperform EKF in contaminated cases and do almost as well in uncontaminated cases • The approximate bayesian filter outperforms WLS-filter in GPS cases • Score function choice • Outlier identification based on innovation not reliable (yet)
Thank you for your attention! Questions?
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
Robust Extended Kalman Filtering in Hybrid Positioning Applications WPNC 2007 Tommi Perälä Tampere University of Technology, Finland
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
2
Outline
• Motivation • Robust WLS-filter • Robust bayesian filter • Simulations • Tests • Conclusions
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
3
Motivation
• Kalman filter (KF, EKF, EKF2, …) well suited to small positioning devices – low complexity • OK for small nonlinearities and nearly gaussian noise • Outliers can cause KF to fail badly – need robust KF
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
4
Deterministic filtering
• State evolution
• Measurements
• Minimize
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
5
The weighted least squares filter (WLS-filter)
• Instead of squares, minimize some other score function • Kalman-type recursions, variances scaled by measurement’s ”likeliness” (score function)
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
6
WLS-filter: The algorithm
1.
Choose the score function
2.
Calculate the innovations
3.
Modify the Kalman covariances
4.
Calculate estimate
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
7
Bayesian filtering
• State evolution
• Measurements
• Solve
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
8
The approximate bayesian filter (B-filter)
• Approximate the innovation pdfs with heavy-tailed nongaussian densities
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
9
B-filter: The algorithm
1. Choose the innovation pdf 2. Calculate the likelihood score of the innovation 3. Calculate the posterior mean and the covariance 4. Approximate the posterior with a normal density
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
10
Simulations
• MATLAB simulation test bench (different filters, geometries, measurements) • 100 tracks, 100 timesteps, 1 Hz • 9 cases: 0 – 5 GPS, 0 – 2.5 base stations • Outlier probability 0.00 or 0.05 • Filters’ ability to detect outliers also tested
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
11
Simulation results
• Criteria: 2D MSE, 95th percentile, inconsistency • B-filter outperforms EKF and EKF2 in contaminated cases • In contaminated GPS cases the WLS-filter fails completely • In contaminated basestation-only cases the WLS-filter sometimes outperform all filters. • Fewer than half of outlier identifications were correct
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
12
Tests
• Walking & bus, GPS receiver EKF
Robust
50 m Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
13
Conclusions
• Robust filters outperform EKF in contaminated cases and do almost as well in uncontaminated cases • The approximate bayesian filter outperforms WLS-filter in GPS cases • Score function choice • Outlier identification based on innovation not reliable (yet)
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
14
Conclusions
• Robust filters outperform EKF in contaminated cases and do almost as well in uncontaminated cases • The approximate bayesian filter outperforms WLS-filter in GPS cases • Score function choice • Outlier identification based on innovation not reliable (yet)
Thank you for your attention! Questions?
Department of Mathematics
Robust Extended Kalman Filtering in Hybrid Positioning Applications
5/4/07
