Kalman Filtering Tutorial

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What is a Covariance Matrix? The covariance of two random variables x1 and x2 is cov( x , x ) ≡ E[( x − x )( x − x )] 1 2 1 1 2 2 ∞ ∞ = ( x − x )( x − x ) p( x , x ) dx dx ≡ ∫ −∞ ∫ −∞ 1 1 2 2 1 1 1 2 where p is the joint probability density function of x1 and x2. 6 σ 2 x x The correlation coefficient is the normalised quantity ρ 12 2 σ x1x2 ≡ , −1 ≤ ρ12 ≤ + 1 σ σ x x 1 2 1 2
The covariance of a column vector x=[x1 .. xn]’ is defined as cov( x) ≡ E[( x − x)( x − x)'] ∞ ∞ = ∫ ... ∫ ( x − x)( x − x)' p( x) dx1.. dxn −∞ −∞ ≡ 7 P xx and is a symmetric n by n matrix and is positive definite unless there is a linear dependence among the components of x. The (i,j) th element of Pxx is σ 2 xi x j Interpreting a covariance matrix: diagonal elements are the variances, off-diagonal encode correlations.
Page 1 and 2: Understanding and Applying Kalman F
Page 3 and 4: What is a Kalman Filter and What Ca
Page 5: Word examples: • Determination of
Page 9 and 10: Error Ellipses corresponding to 50
Page 11 and 12: Formulating a Kalman Filter Problem
Page 13 and 14: FALLING BODY EXAMPLE Consider an ob
Page 15 and 16: ⎡1 1⎤ ⎡0.5⎤ x(k +1) = ⎢ x
Page 17 and 18: State Estimation 0. Known are x$( k
Page 19 and 20: Page 219 Bar-Shalom ANATOMY OF KALM
Page 21 and 22: {F, H} is completely observable if
Page 23 and 24: 101 99 97 95 93 91 89 87 True posit
Page 25 and 26: Kalman Filter Extensions • Valida
Page 27 and 28: Sequential Measurement Processing I
Page 29 and 30: Extended Kalman Filter (EKF) Many p
Page 31 and 32: Iterated Extended Kalman Filter (IE
Page 33 and 34: Kalman filtering classes, for defin
Page 35 and 36: Odometry Error Covariance Estimatio
Page 37: Scanned Monocular Sonar Sensing Sma

Kalman Filtering Tutorial

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