Fundamentals of Kalman Filtering and Applications to GNSS

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Alternative Problems, Issues • What is the required radar accuracy to achieve a prescribed accuracy in the final estimate of P t • What are the implications of having range rate information available in addition to range data? • How about samples every 0.05 sec, 0.01 sec., etc.? • How sensitive is the algorithm to deviations from assumed model? – Suppose A is not constant – Suppose measurement errors not ―independent‖ P( t 0 6) ( 0 6). • Given the initial conditions, how accurately could one estimate without any data? Kalman Filtering Consultant Associates © 2011 6
What is a Kalman Filter? • Algorithm for generating estimates of the state of a system based upon – A mathematical model • System states are governed by linear differential or difference equations driven by white noise • Measurements are linear functions of the states + white measurement noise – An initial estimate • Initial estimates (can be very poor) are required. Also level of uncertainty needs specification – A set of measurements • Data from real hardware (sensors) such as GNSS – Pseudoranges – Delta Pseudoranges Kalman Filtering Consultant Associates © 2011 7
Page 1 and 2: Fundamentals of Kalman Filtering an
Page 3 and 4: What Is A Kalman Filter? Kalman Fil
Page 5: Problem (cont.) • Kalman filter b
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 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
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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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