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

GNC/Aug. ‘09 2 of 21 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 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 Strategies

GNC/Aug. ‘09 3 of 21 Outline of Talk I.Related research II.Problem definition III.Mixed real/integer Kalman filter  Optimal, retains all past integers  Suboptimal, retains finite window of past integers IV.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 V.Truth-model simulation & results VI.Conclusions

GNC/Aug. ‘09 4 of 21 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 of 21 Dynamics Model Real-state dynamics: Partitioning of integer states by affected measurement sample times (past, past & present, past, present & future): Growth of integer state with sample number Or dynamic re-partitioning

GNC/Aug. ‘09 6 of 21 Measurement Model … using integer vector partitions … using full integer vector

GNC/Aug. ‘09 7 of 21 Example Sensitivities of Different Measurement Types to Different Integers

GNC/Aug. ‘09 8 of 21 Kalman Filtering/Smoothing Problem find: x 0, …, x k+1, w 0, …, w k, & n k+1 = [  n 0 ; …;  n k ] to minimize: subject to: x j+1 =  j x j +  j w j +  j for j = 0, 1, 2,..., k n k+1 is an integer-valued vector

GNC/Aug. ‘09 9 of 21 Stage-k a posterior info: Combined information eqs. w/dynamics substitution for x k : New stage-(k+1) a posterior info after QR factorization: Optimal SRIF Kalman Filter

GNC/Aug. ‘09 10 of 21 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 of 21 Suboptimal KF Retention of Exact Integers within a Window of Samples

GNC/Aug. ‘09 12 of 21 Stage-k a posterior info: Combined information eqs. w/dynamics substitution for x k & m k New stage-(k+1) a posterior info after QR factorization: Suboptimal SRIF Kalman Filter

GNC/Aug. ‘09 13 of 21 Terminal sample K initialization: 1-sample backwards recursion starts w/filtered w k & smoothed x k+1 info. eqs. & uses dynamics to get QR factorize to isolate smoothed x k info. eq. Optimal RTS Smoother in SRIF Form

GNC/Aug. ‘09 14 of 21 Suboptimal RTS Smoother Retention of Exact Integers within a Window of Samples

GNC/Aug. ‘09 15 of 21 Terminal sample K initialization: 1-sample backwards recursion starts w/filtered w k &  n k-i & smoothed x k+1 & l k+1 info. eqs. & uses dynamics & integer permutation/partitions to get Suboptimal RTS Smoother (1 of 2)

GNC/Aug. ‘09 16 of 21 Suboptimal RTS Smoother (2 of 2) New stage-k smoothed x k & l k 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 of 21 Example 1-Dimensional CDGPS-Type Problem with 3 rd -Order Dynamics Dynamics: Measurements:

GNC/Aug. ‘09 18 of 21 x 1 Errors for Three Kalman Filters

GNC/Aug. ‘09 19 of 21 x 1 Errors for Three Smoothers

GNC/Aug. ‘09 20 of 21 Integer-Part Computational Cost of Optimal & Suboptimal Algorithms

GNC/Aug. ‘09 21 of 21 Summary & Conclusions 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