Fundamentals of Kalman Filtering and Applications to GNSS

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Chi-Squared Statistics (cont.) • Detecting Anomalous Sensor Data (conc.) – The equivalent statistic 2 X is nonnegative with a minimum value of zero. – If the Kalman filter were perfectly modeled and all white-noise sources were Gaussian, this would be a chi-squared statistic with distribution. – An upper limit threshold value on X can be used to detect anomalous sensor data, but a practical value of that threshold 2 should be determined by the operational values of X , not the theoretical values. – That is, first its range of values should be determined by monitoring the system in operation, then a threshold value chosen such that the fraction of good data rejected when will be acceptable. T vk Y vk v 2 k X 2 X max 2 2 Xmax Kalman Filtering Consultant Associates © 2011 66
Kalman Filter Engineering • Square root filtering – Robust against round off • KF implementation requirements – Memory & OPS • Nuisance Variable Examples – Some can be ignored (at some cost) • Correlated noise states (e.g., S/A) • Anything not appearing elsewhere in model – Some cannot be ignored • Sensor biases • Sensor scale factors Kalman Filtering Consultant Associates © 2011 67
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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 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: Chi-Squared Statistic • Detecting
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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