Fundamentals of Kalman Filtering and Applications to GNSS

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Bad Data Rejection • Data rejection – Assuming that adequate knowledge of the innovations—vector Z HXˆ exists—data rejection filters can be implemented – For example • Excess amplitude If Z HXˆ A i MAX reject data • Excess rate (or change) If Z HXˆ Z HXˆ δA i1 i MAX reject data • Other – Many ingenious techniques have been used, but often depend on the specifics involved – For example, Chi-Squared Distribution Kalman Filtering Consultant Associates © 2011 64
Chi-Squared Statistic • Detecting anomalous sensor data – The Kalman gain matrix K P – includes the factor Yvk HkPk Hk R k the information matrix of innovations. The innovations are the measurement residuals v k zk Hk xˆ k , the differences between the apparent sensor outputs and predicted sensor outputs. – The associated likelihood function for innovations is 1 T L vk exp vk Yvk vk , 2 k – and the log-likelihood is log L vk vk Yvk vk which can be easily calculated. k H T k T 1 , H k P k Hk R k Y vk T , T 1 Kalman Filtering Consultant Associates © 2011 65
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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: Causes, Cures of Non-convergence
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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