Kalman Filtering & Smoothing to Estimate Real-Valued States & Integer Constants

1. Kalman Filtering & Smoothing to Estimate Real-Valued States & Integer Constants Mark L. Psiaki, Sibley School of Mechanical & Aerospace EngineeringCornell University AIAA GNC, 11 Aug. 2009

2. Goal • Improve estimation algorithms for systems that have integer measurement ambiguities • CDGPS with double-differenced integer ambiguities • Systems using carrier-phase measurements of TDMA signals Strategies • Use SRIF/LAMBDA-type formulation to deal with mixed real/integer problem • Develop optimal & suboptimal Kalman filter & smoother algorithms • Optimal: keep all ambiguities & treat as integers • Suboptimal: retain integers in a finite time window GNC/Aug. ‘09

3. Outline of Talk • Related research • Problem definition • Mixed real/integer Kalman filter • Optimal, retains all past integers • Suboptimal, retains finite window of past integers • Mixed real/integer fixed-interval smoother • Optimal, retains all integers of fixed interval • Suboptimal, retains finite window of past & future integers relative to each time point • Truth-model simulation & results • Conclusions GNC/Aug. ‘09

4. Related Research: • Batch estimation w/integer ambiguities • The LAMBDA method, Teunissen (1995) & follow-ons • Other methods, e.g., Chen & Lachapelle (1995) • SRIF LAMBDA-like method, Psiaki & Mohiuddin (2007) • Kalman filtering w/integer ambiguities • Standard Covariance EKF, Kroes et al. (2005) • SRIF-based EKF, Mohiuddin & Psiaki (2008) • Sub-optimal dropping of each integer ambiguity immediately after its last occurrence in a measurement • Smoothing w/integer ambiguities • Nothing GNC/Aug. ‘09

5. Dynamics Model Real-state dynamics: Growth of integer state with sample number Partitioning of integer states by affected measurement sample times (past, past & present, past, present & future): Or dynamic re-partitioning GNC/Aug. ‘09

6. Measurement Model … using integer vector partitions … using full integer vector GNC/Aug. ‘09

7. -3 x 10 11 10 9 8 7 Ambiguity Sensitivities, htilde (m) 6 5 4 3 2 1 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (sec) Example Sensitivities of Different Measurement Types to Different Integers GNC/Aug. ‘09

8. find: x0, …, xk+1, w0, …, wk, & nk+1 = [dn0; …; dnk] to minimize: subject to:xj+1 = Fjxj + Gjwj + hj for j = 0, 1, 2, ..., knk+1 is an integer-valued vector Kalman Filtering/Smoothing Problem GNC/Aug. ‘09

9. Stage-k a posterior info: Combined information eqs. w/dynamics substitution for xk: New stage-(k+1) a posterior info after QR factorization: Optimal SRIF Kalman Filter GNC/Aug. ‘09

10. Measurement Update via Integer Linear Least-Squares Solution • Solve integer linear least-squares problem to determine integer a posteriori estimate • Back-substitute to compute real-valued states: GNC/Aug. ‘09

11. -3 x 10 11 Measurements used in t = 3000 sec k 10 sub-optimal filter 9 8 7 Ambiguity Sensitivities, htilde (m) 6 5 4 3 t = 3000 sec +0/- i*deltat window k for considering exact integers 2 1 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Time (sec) Suboptimal KF Retention of Exact Integers within a Window of Samples GNC/Aug. ‘09

12. Stage-k a posterior info: Combined information eqs. w/dynamics substitution for xk & mk New stage-(k+1) a posterior info after QR factorization: Suboptimal SRIF Kalman Filter GNC/Aug. ‘09

13. Optimal RTS Smoother in SRIF Form • Terminal sample K initialization: • 1-sample backwards recursion starts w/filtered wk & smoothed xk+1 info. eqs. & uses dynamics to get • QR factorize to isolate smoothed xk info. eq. GNC/Aug. ‘09

14. Suboptimal RTS Smoother Retention of Exact Integers within a Window of Samples GNC/Aug. ‘09

15. Suboptimal RTS Smoother (1 of 2) • Terminal sample K initialization: • 1-sample backwards recursion starts w/filtered wk & Dnk-i & smoothed xk+1 & lk+1 info. eqs. & uses dynamics & integer permutation/partitions to get GNC/Aug. ‘09

16. Suboptimal RTS Smoother (2 of 2) • New stage-k smoothed xk & lk square-root information equations after QR factorization • is the integer vector that minimizes • The real part of the state is determined by back substitution: GNC/Aug. ‘09

17. Example 1-Dimensional CDGPS-Type Problem with 3rd-Order Dynamics • Dynamics: • Measurements: GNC/Aug. ‘09

18. x1 Errors for Three Kalman Filters GNC/Aug. ‘09

19. x1 Errors for Three Smoothers GNC/Aug. ‘09

20. Integer-Part Computational Cost of Optimal & Suboptimal Algorithms GNC/Aug. ‘09

21. Developed optimal & suboptimal Kalman filters & fixed-interval smoothers for mixed real/integer estimation problems Constant integer ambiguities enter only measurements Optimal algorithms consider all integers in data batch Suboptimal algorithms drop integers that affect measurements only in remote past or future Tested using data from truth-model simulation Optimal & suboptimal filter achieve modest accuracy gains vs. all-reals approximate filter Filter accuracy gains may be greater for different problem Optimal & suboptimal smoother significantly more accurate than all-reals smoother Suboptimal smoother nearly as accurate as optimal smoother Suboptimal algorithms reduce required processing by at least 65% through reductions in dimensions of measurement update integer linear least-squares problems Summary & Conclusions GNC/Aug. ‘09