probabilistic methods: kalman filtering (intelligent
TRANSCRIPT
![Page 1: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/1.jpg)
Probabilistic Methods:
Kalman Filtering
(Intelligent Autonomous Robotics)
Subramanian Ramamoorthy
School of Informatics
![Page 2: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/2.jpg)
Story so far…
Estimation: Extract clean signal from noisy measurement
Assume a sensor model (y = Hx + e) where the state is
linearly transformed and corrupted by noise
Use one of many estimation procedures
Least squares
Weighted least squares
Recursive formulation of WLS
Maximum likelihood and Bayesian estimation
November 17, 2008 Kalman Filtering 2
![Page 3: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/3.jpg)
Recap: Estimate Coefficients of a Cubic
Polynomial
Use k measurements at different t
to estimate coefficients x :
February 22, 2008 Estimation and Sensor Fusion 3
![Page 4: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/4.jpg)
Main Theme for this lecture
Introduction of dynamics
How to estimate a state that evolves over time?
Incrementally - as measurements arrive one at a time
The Kalman filter algorithm: extends the least-squares
estimation concept to the case of dynamic systems
Discrete time, linear dynamics
Continuous time, linear dynamics
Brief discussion - Extension for nonlinear dynamics
November 17, 2008 Kalman Filtering 4
![Page 5: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/5.jpg)
Historical Remarks
November 17, 2008 Kalman Filtering 5
Authoritative Resource on the subject:
http://www.cs.unc.edu/~welch/kalman/index.html
![Page 6: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/6.jpg)
Let’s get started: Simple Estimation Problem
Consider a ship in choppy waters
Needs to navigate using a distant landmark (star)
To keep things simple, assume the model x = z + e
Can make multiple measurements of the same ground
truth (let us say – successively in time)
We know (a priori) one crucial fact – the variance of each
measurement
Then, what is the true value of the
distance to the landmark?
November 17, 2008 Kalman Filtering 6
![Page 7: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/7.jpg)
Core Problem: Updating Estimated State
How to combine multiple sequential measurements?
November 17, 2008 Kalman Filtering 7
![Page 8: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/8.jpg)
Answer: Variance-weighted Sum
November 17, 2008 Kalman Filtering 8
Remember this.We’ll see something similar again.
![Page 9: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/9.jpg)
In pictures…
November 17, 2008 Kalman Filtering 9
![Page 10: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/10.jpg)
What happens if we Add Dynamics?
November 17, 2008 Kalman Filtering 10
![Page 11: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/11.jpg)
Effect of Adding System Dynamics
The uncertainty is amplified –
density function gets “squashed”
November 17, 2008 Kalman Filtering 11
![Page 12: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/12.jpg)
Idea of the Kalman Filter
November 17, 2008 Kalman Filtering 12
![Page 13: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/13.jpg)
Key Variables: First Two Statistical Moments
November 17, 2008 Kalman Filtering 13
![Page 14: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/14.jpg)
Discrete Kalman Filter: Key Ingredients
Discrete process model
State change over time
Linear difference equation
Discrete measurement model
Relationship between state and measurement
Linear function
Model parameters
Process noise characteristics
Measurement noise characteristics
November 17, 2008 Kalman Filtering 14
![Page 15: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/15.jpg)
Discrete Kalman Filter: Two Models
November 17, 2008 Kalman Filtering 15
![Page 16: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/16.jpg)
Discrete Kalman Filter: Model Equations
November 17, 2008 Kalman Filtering 16
![Page 17: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/17.jpg)
Complete Model Specification
November 17, 2008 Kalman Filtering 17
![Page 18: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/18.jpg)
How the Filter Works
Time Update or
Prediction (a priori
estimates) - Project state
and covariance forward in
time
Measurement Update or
Correction (a posteriori
estimates) - Update
variables based on a
noise measurement
November 17, 2008 Kalman Filtering 18
![Page 19: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/19.jpg)
Discrete Kalman Filter – In Equations
November 17, 2008 Kalman Filtering 19
Same idea as inslide #8
![Page 20: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/20.jpg)
Discrete Kalman Filter: The Complete Loop
November 17, 2008 Kalman Filtering 20
![Page 21: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/21.jpg)
Example: Estimating a (Noisy) Constant
November 17, 2008 Kalman Filtering 21
![Page 22: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/22.jpg)
Recap: What is the Kalman Filter?
An optimal, recursive data processing algorithm
Optimal in what sense?
If the system dynamics are linear
And system/sensor noise is Gaussian and white
Then, there is no alternate algorithm with lower MSE
Optimal Best Linear Unbiased Estimator
Bayesian View of the Kalman Filter:
Propagate conditional probability density of desired
quantities, conditioned on knowledge of actual data
November 17, 2008 Kalman Filtering 22
![Page 23: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/23.jpg)
Dealing with Nonlinearities
November 17, 2008 Kalman Filtering 23
![Page 24: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/24.jpg)
Extended Kalman Filter - Setup
November 17, 2008 Kalman Filtering 24
![Page 25: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/25.jpg)
Extended Kalman Filter - Computation
Notice that the above equations look a lot like the linear
system dynamics and measurement equations
So, we can define a Kalman filter over this error
And then, use that linear error estimate to drive the main
estimation loop
November 17, 2008 Kalman Filtering 25
![Page 26: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/26.jpg)
Extended Kalman Filter: The Loop
November 17, 2008 Kalman Filtering 26
![Page 27: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/27.jpg)
Reference:
S. Thrun et al., Probabilistic Robotics, MIT Press, 2005.
Several pictures and equations in this presentation are
taken from the tutorial by Welch and Bishop
(http://www.cs.unc.edu/~welch/kalman/)
November 17, 2008 Kalman Filtering 27
![Page 28: Probabilistic Methods: Kalman Filtering (Intelligent](https://reader030.vdocument.in/reader030/viewer/2022032511/6235b19f2d847b1352599c66/html5/thumbnails/28.jpg)
Kalman Filter: Continuous Time-variant case
November 17, 2008 Kalman Filtering 28