Fundamentals of Kalman Filtering and Applications to GNSS

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GNSS Clock Modeling and Corrections • On board clock errors – Space vehicle (SV) time • Timing of the signal transmission from each satellite • Directly controlled by its own atomic clock with NO corrections applied – GPS time • Highly accurate • However, errors can be large enough to require corrections • Difficult to directly synchronize clocks in all the satellites • Instead, clocks allowed some degree of relative drift estimated by ground station observations and used to generate clock correction data in GPS navigation message • SV time is corrected using this data, and result is called ―GPS time‖ Kalman Filtering Consultant Associates © 2011 28
Time Calculations (GPS book, p. 64) :: t t SV t SV L1 tSV tSV GD L1 (3.8) , GD Differential bias provided on subframe 1 (Time from pseudorange time tagged) GD (From Subframe 1) _ t SV + t SV (computed) _ L 1 t SV + t a f 0 a f 1 t t a t t OC f 2L 1 OC (3.9) (3.10) (3.11) Kalman Filtering Consultant Associates © 2011 29
Page 1 and 2: Fundamentals of Kalman Filtering an
Page 3 and 4: What Is A Kalman Filter? Kalman Fil
Page 5 and 6: Problem (cont.) • Kalman filter b
Page 7 and 8: What is a Kalman Filter? • Algori
Page 9 and 10: Kalman Filtering Problem • Top le
Page 11 and 12: Example: Curve Fitting • Given -
Page 13 and 14: Example: Curve Fitting (cont.) •
Page 15 and 16: Example: Curve Fitting Summary •
Page 17 and 18: Example—The Kalman Filter Curve-f
Page 19 and 20: Exponentially Correlated Noise •
Page 21 and 22: White Noise • Question: If white
Page 23 and 24: Example • Generating a random wal
Page 25 and 26: Example • Generating an exponenti
Page 27: Error Modeling PSD AUTOCORRELATION
Page 31 and 32: Receiver Crystal Clock Modeling •
Page 33 and 34: Process Noise Covariance for Receiv
Page 35 and 36: GNSS Receiver Clock Modeling • No
Page 37 and 38: RMS Uncertainty After 1 h of Filter
Page 39 and 40: General Measurement Models • Let
Page 41 and 42: GNSS Measurement Models - Measured
Page 43 and 44: Kalman Filtering Consultant Associa
Page 47 and 48: Modeling (cont.) 4 STATES 5 STATES
Page 51 and 52: Discrete Linear Kalman Estimator Sy
Page 53 and 54: Representative Sequence of Values o
Page 55 and 56: Continuous Kalman Filter • Can be
Page 57 and 58: Kalman Filter Examples • Let the
Page 59 and 60: Example (conc.) • Let • Followi
Page 61 and 62: Convergence (cont.) • Typical beh
Page 63 and 64: Causes, Cures of Non-convergence
Page 65 and 66: Chi-Squared Statistic • Detecting
Page 67 and 68: Kalman Filter Engineering • Squar
Page 71 and 72: Alternative Implementations Matrix
Page 73 and 74: Square Root Riccati Equations • O
Page 75 and 76: Linearized Kalman Filter • Partia
Page 77 and 78: Tables of Kalman Filter Equations K
Page 79 and 80:
Examples of Nonlinear KF • Exampl
Page 81 and 82:
Sigma Point Kalman Filters (SPKF)
Page 83 and 84:
Unscented Kalman Filter (UKF) 1) UK
Page 85 and 86:
Unscented Kalman Filter (cont.) c)
Page 87:
References • Kalman Filtering The
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Fundamentals of Kalman Filtering and Applications to GNSS

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