1. Lesson 2 – kalman Filters

2. Agenda Applications What is a Kalman Filter Conceptual Overview
Design Kalman Filter
Example
Demo MATLAB

3. Applications Tracking Objects Economics Navigation (Google car)
Missiles
Lip reading
Hands
Economics
Navigation (Google car)

4. What is a Kalman Filter Recursive data processing algorithm Recursive?
KF does not require all previous data to be kept in storage
Kalman filter finds the most optimum averaging factor for each consequent state. Also somehow remembers a little bit about the past states

5. Previous lesson Monte Carlo localization Now we look at KF
World divided info discrete grids
Approximates the posterior distribution as histogram
Now we look at KF
The posterior distribution is given by a Gaussian
Gaussian is a cont. Function
Area underneath sums to one

6. Conceptual overview Lost on the 1D line
A measurement at t1: Mean = z1 and Variance = 2z1
Optimal estimate of position is: 𝑥 (t1) = z1
Variance of error in estimate: 2x (t1) = 2z1
Boat in same position at time t2

7. Conceptual overview So we have the prediction 𝑥 -(t2)
Better Measurement at t2: Mean = z2 and Variance = z2
Need to correct the prediction due to measurement to get 𝑥 (t2)
Closer to more trusted measurement
prediction 𝑥 -(t2)
measurement z(t2)

8. Conceptual overview
Corrected mean is the new optimal estimate of position
New variance is smaller than either of the previous two variances
corrected optimal estimate 𝑥 (t2)
prediction 𝑥 -(t2)
measurement z(t2)

9. Conceptual overview Optimal estimate 𝑥 (t2) = µ
𝑥 − : prediction (a priori estimate)
𝑥 : update (a posteriori estimate)
K: Kalman gain

10. Conceptual overview At time t3, boat moves with velocity dx/dt=u
Naïve Prediction 𝑥 -(t3)
At time t3, boat moves with velocity dx/dt=u
Naïve approach: Shift probability to the right to predict
This would work if we knew the velocity exactly (perfect model)

11. Conceptual overview
Naïve Prediction 𝑥 -(t3)
𝑥 (t2)
Better to assume imperfect model by adding Gaussian noise
dx/dt = u + w
u= nominal velocity
w= noise term or uncertainty (variance)
Distribution for prediction moves and spreads out
Prediction 𝑥 -(t3)

12. Conceptual overview Now we take a measurement at t3
Need to once again correct the prediction
Same as before
Corrected optimal estimate 𝑥 (t3)
Measurement z(t3)
Prediction 𝑥 -(t3)

13. Design Kalman Filters

14. Design Kalman Filters Process to be estimated: (state space)
xk = Axk-1 + Buk + wk-1
Process Noise (w) with covariance Q
zk = Hxk + vk
Measurement Noise (v) with covariance R
Prediction: 𝑥 - is estimate based on measurements at previous time-steps
𝑥 -k = Axk-1 + Buk
P-k = APk-1AT + Q
P-k: Prior error covariance
Correction: 𝑥 k has additional information – the measurement at time k
𝑥 k = 𝑥 -k + K(zk - H 𝑥 -k )
K = P-kHT(HP-kHT + R)-1
Pk = (I - KH)P-k

15. Design Kalman Filters

16. Design Kalman Filters

17. Kalman Filters - Example
Consider an object falling under a constant gravitational field. Let y(t) denote the height of the object, then:

18. Kalman Filters - Example
Construct state space model from equations, when we are able to perform measurements, zk, of the height.
that is, find A, B, uk and H in:
xk = Axk-1 + Buk
zk = Hxk

19. Kalman Filters - Example
Construct state space model from equations, when we are able to perform measurements, zk, of the height.
Solution: A B H u
Xk-1 Xk zk

20. Example
MATLAB demo