Worksheet

U6.3 The Kalman gain

State estimation · University · about 35 min

What this lesson is about

Where the weighting comes from: two variances, one line of arithmetic, no taste involved.

• The scalar filter, in full
• Reading the gain
• The steady state gain

Questions 6 marks in all

1. [1 mark]A scalar Kalman filter has p = 100 and R = 25. What is the Kalman gain k?

2. [1 mark]One full predict and correct of the scalar filter. What does it print?

   x, p = 0.0, 100.0
   v, dt, Q, R = 20.0, 0.1, 0.5, 25.0
   measured = 5.0
   x = x + v * dt
   p = p + Q
   k = p / (p + R)
   x = x + k * (measured - x)
   p = (1 - k) * p
   print(round(k, 3), round(x, 3), round(p, 3))

3. [1 mark]The steady state gain from the lesson's formula, checked by running the filter. What does it print?

   Q, R = 0.5, 25.0
   p = (Q + (Q ** 2 + 4 * Q * R) ** 0.5) / 2
   print(round(p / (p + R), 3))
   
   p = 100.0
   for i in range(200):
       p += Q
       k = p / (p + R)
       p = (1 - k) * p
   print(round(k, 3))

4. [1 mark]Q stays at 0.5 and R is doubled from 25 to 50. What happens to the steady state gain?

   1. AIt falls, because the measurement is now trusted less
   2. BIt rises, because there is more noise to correct
   3. CIt halves exactly
   4. DIt is unchanged

5. [1 mark]The filter is started with p = 0.01 while the estimate is in fact badly wrong. What happens for the first second or so?

   1. AThe gain is near 0, so it largely ignores the measurements
   2. BThe gain is near 1, so it jumps to each measurement
   3. CIt diverges immediately
   4. DNothing different from starting at p = 100

6. [1 mark]Once the gain has settled, which statements are true?

   Tick every answer that is true.

   1. AThe filter behaves like an exponential filter with alpha equal to the settled gain
   2. BThe gain can be computed once and hard-coded, skipping the variance arithmetic
   3. CThe filter has stopped using the measurements
   4. DThe gain is always 0.5 at steady state

The task: a Kalman filter in one dimension

Run the scalar filter while driving at the wall. Plot measured, estimate and gain, with the gain as a percentage so that it shows. Print my y: and final gain:, the gain itself as a number between 0 and 1. No position().

from bugbot import *
connect()

DT, START_GAP = 0.1, 110.0
R = 25.0
Q = 0.5
est, p = 0.0, 100.0

Plan your program here, then type it in and press Run.

Do it on the robot
www.bugbotlab.com/learn/u6-3-the-kalman-gain/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

1. Start est at 30, which is wrong, and p at 0.01, which says you are sure of it. How long does the filter take to believe the measurements, and how does that compare with p at 100?
2. Work out the steady state gain from the formula and check it against the one you measured.
3. Double R and halve it. Which way does the gain go, and does that match the sentence "R is how noisy the measurement is"?