I ntroduction to Mobile Robotics Bayes Filter Kalm an Filter - - PowerPoint PPT Presentation

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i ntroduction to mobile robotics bayes filter kalm an
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I ntroduction to Mobile Robotics Bayes Filter Kalm an Filter - - PowerPoint PPT Presentation

Oct 18, 2023 47 likes •302 views

I ntroduction to Mobile Robotics Bayes Filter Kalm an Filter Wolfram Burgard 1 Bayes Filter Rem inder Prediction Correction Bayes Filter Rem inder Prediction Correction Kalm an Filter Bayes filter with Gaussians

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Bayes Filter – Kalm an Filter I ntroduction to Mobile Robotics

Wolfram Burgard

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  • Prediction
  • Correction

Bayes Filter Rem inder

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  • Prediction
  • Correction

Bayes Filter Rem inder

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Kalm an Filter

  • Bayes filter with Gaussians
  • Developed in the late 1950's
  • Most relevant Bayes filter variant in practice
  • Applications range from economics, weather

forecasting, satellite navigation to robotics and many more.

  • The Kalman filter “algorithm” is

a couple of m atrix m ultiplications!

5

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Gaussians

  • σ

σ µ

Univariate

µ

Multivariate

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Gaussians

1 D 2 D 3 D

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Properties of Gaussians

  • Univariate case
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  • Multivariate case

(where division "–" denotes matrix inversion)

  • We stay Gaussian as long as we start with

Gaussians and perform only linear transform ations

Properties of Gaussians

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Discrete Kalm an Filter

Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation with a measurement

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Com ponents of a Kalm an Filter

Matrix (n×n) that describes how the state evolves from t-1 to t without controls or noise. Matrix (n×l) that describes how the control ut changes the state from t-1 to t. Matrix (k×n) that describes how to map the state xt to an observation zt. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Qt and Rt respectively.

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Kalm an Filter Updates in 1 D

prediction measurement correction It's a weighted mean!

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Kalm an Filter Updates in 1 D

How to get the blue one? Kalm an correction step

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Kalm an Filter Updates in 1 D

How to get the magenta one? State prediction step

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Kalm an Filter Updates

prediction correction measurement

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Linear Gaussian System s: I nitialization

Initial belief is normally distributed:

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Dynamics are linear functions of the state and the control plus additive noise:

Linear Gaussian System s: Dynam ics

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Linear Gaussian System s: Dynam ics

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Observations are a linear function of the state plus additive noise:

Linear Gaussian System s: Observations

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Linear Gaussian System s: Observations

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Kalm an Filter Algorithm

1. Algorithm Kalm an_ filter( µt-1, Σt-1, ut, zt):

2. Prediction: 3. 4. 5. Correction: 6. 7. 8. 9. Return µt, Σt

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Kalm an Filter Algorithm

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The Prediction-Correction-Cycle

Prediction

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The Prediction-Correction-Cycle

Correction

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The Prediction-Correction-Cycle

Correction Prediction

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Kalm an Filter Sum m ary

  • Only two parameters describe belief about

the state of the system

  • Highly efficient: Polynomial in the

measurement dimensionality k and state dimensionality n: O(k2.376 + n2)

  • Optim al for linear Gaussian system s!
  • However: Most robotics systems are

nonlinear!

  • Can only model unimodal beliefs