Fundamentals of Kalman Filtering and Applications to GNSS

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Example: Curve Fitting Summary • Curve-fitting least squares fit Z HX T T T H H H Z H H 0 ˆ 1 X 1. Results predicated upon assumed model Z = HX Z 1 Model validity is always a problem 2. All residuals Z i Z ˆ i are weighted equally. No provisions for ―de-weighting‖ some points. 3. No information regarding ―a priori‖ knowledge of parameters used. 4. Batch processing is implied by 2 t MODEL 5. Criteria is one of fitting data; not minimizing estimation error, X Xˆ 1 2 Z t t Z t Xˆ 2 T 1 T H H H Z Kalman Filtering Consultant Associates © 2011 16
Example—The Kalman Filter Curve-fitting least squares fit • The Kalman filter – Kalman filtering brings into consideration 2, 3, 4, 5. – Modeling remains a problem. – The least squares curve fit and the Kalman filter yield the same estimates when • Initial uncertainty in X is large. • All observations are of equal quality. • System is overdetermined or exactly determined. – The Kalman filter is recursive. – The Kalman filter accommodates a dynamical model for X. Kalman Filtering Consultant Associates © 2011 17
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: Example: Curve Fitting Summary •
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 and 28: Error Modeling PSD AUTOCORRELATION
Page 29 and 30: Time Calculations (GPS book, p. 64)
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 69 and 70:
Kalman Filtering Consultant Associa
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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