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Bayes Filter Rem inder
- Prediction
- Correction
I ntroduction to Mobile Robotics Bayes Filter Extended Kalm an - - PowerPoint PPT Presentation
I ntroduction to Mobile Robotics Bayes Filter Extended Kalm an - - PowerPoint PPT Presentation
I ntroduction to Mobile Robotics Bayes Filter Extended Kalm an Filter Wolfram Burgard 1 Bayes Filter Rem inder Prediction Correction Discrete Kalm an Filter Estimates the state x of a discrete-time controlled process with a
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Discrete Kalm an Filter
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Estimates the state x of a discrete-time controlled process with a measurement
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Com ponents of a Kalm an Filter
Matrix (nxn) that describes how the state evolves from t-1 to t without controls or noise. Matrix (nxl) that describes how the control ut changes the state from t-1 to t. Matrix (kxn) 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 Update Exam ple
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prediction measurement correction It's a weighted mean!
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Kalm an Filter Update Exam ple
prediction correction measurement
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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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Nonlinear Dynam ic System s
- Most realistic robotic problems involve
nonlinear functions
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Linearity Assum ption Revisited
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Non-Linear Function
Non-Gaussian!
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Non-Gaussian Distributions
- The non-linear functions lead to non-
Gaussian distributions
- Kalman filter is not applicable anymore!
W hat can be done to resolve this?
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Non-Gaussian Distributions
- The non-linear functions lead to non-
Gaussian distributions
- Kalman filter is not applicable anymore!
W hat can be done to resolve this? Local linearization!
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EKF Linearization: First Order Taylor Expansion
- Prediction:
- Correction:
Jacobian matrices
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Rem inder: Jacobian Matrix
- It is a non-square m atrix in general
- Given a vector-valued function
- The Jacobian m atrix is defined as
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Rem inder: Jacobian Matrix
- It is the orientation of the tangent plane to
the vector-valued function at a given point
- Generalizes the gradient of a scalar valued
function
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EKF Linearization: First Order Taylor Expansion
- Prediction:
- Correction:
Linear function!
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Linearity Assum ption Revisited
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Non-Linear Function
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EKF Linearization ( 1 )
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EKF Linearization ( 2 )
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EKF Linearization ( 3 )
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EKF Algorithm
1 . Extended_ 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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Exam ple: EKF Localization
- EKF localization with landmarks (point features)
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1 . EKF_ localization ( µt-1, Σt-1, ut, zt, m):
Prediction: 3. 5. 1. 2. 3. Motion noise Jacobian of g w.r.t location Predicted mean Predicted covariance (V maps Q into state space) Jacobian of g w.r.t control
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1 . EKF_ localization ( µt-1, Σt-1, ut, zt, m):
Correction:
3. 5. 6. 7. 8. 9. 10.
Predicted measurement mean (depends on observation type) Innovation covariance Kalman gain Updated mean Updated covariance Jacobian of h w.r.t location
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EKF Prediction Step Exam ples
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EKF Observation Prediction Step
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EKF Correction Step
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Estim ation Sequence ( 1 )
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Estim ation Sequence ( 2 )
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Extended Kalm an Filter Sum m ary
- Ad-hoc solution to deal with non-linearities
- Performs local linearization in each step
- Works well in practice for moderate non-
linearities
- Example: landmark localization
- There exist better ways for dealing with
non-linearities such as the unscented Kalman filter called UKF
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