Answers

U6.5 Covariance and the ellipse

State estimation · University · about 30 min

What this lesson is about

Uncertainty in two dimensions has a shape, and the shape is the useful part.

• Why it is not a circle
• The covariance matrix
• Why this earns its keep
• In practice

Questions 6 marks in all

1. [1 mark]A robot drives 200 cm straight with a 1 degree heading error. How large, in cm to one decimal place, is the across-track error?

   Answer: 3.5 (accept within 0.06). Across-track error is heading error in radians times distance: 200 x 0.01745 = 3.49 cm.

2. [1 mark]In two dimensions, about what percentage of the probability lies inside the one sigma ellipse?

   Answer: 39 (accept within 1). About 39 percent, not the 68 percent of one dimension. The 95 percent ellipse is about 2.45 sigma.

3. [1 mark]The ellipse axes of a covariance matrix [[5, 2], [2, 2]], from the 2 x 2 eigenvalue formula. What does it print?

   a, b, d = 5.0, 2.0, 2.0
   tr, det = a + d, a * d - b * b
   root = (tr * tr / 4 - det) ** 0.5
   l1, l2 = tr / 2 + root, tr / 2 - root
   print(l1, l2)
   print(round(l1 ** 0.5, 2), round(l2 ** 0.5, 2))

   Answer:

   6.0 1.0
   2.45 1.0

   The trace is 7 and the determinant 6, so the eigenvalues are 6 and 1. The one sigma axes are their square roots, 2.45 and 1: a tilted ellipse, because the off-diagonal is not zero.

4. [1 mark]A tag straight ahead of the robot gives a good range and a poor bearing. Which part of the uncertainty ellipse does a fix mainly shrink?

   1. AAlong the line of sight
   2. BAcross the line of sight
   3. CBoth equally
   4. DNeither, bearings are needed to shrink anything
   Answer: A. A range measures distance along the line to the tag. A filter carrying the full matrix takes exactly that direction from the fix and leaves the sideways uncertainty.

5. [1 mark]What does the off-diagonal term cov(x, y) of P tell you?

   1. AWhether the x and y errors move together, which tilts the ellipse
   2. BHow large the x uncertainty is
   3. CThe total uncertainty
   4. DThe heading error
   Answer: A. The diagonal gives each axis's variance. The off-diagonal says whether the errors are correlated, which rotates the ellipse away from the axes.

6. [1 mark]The uncertainty ellipse has grown long across the direction of travel. What kind of landmark would help most?

   1. AOne to the side of the track
   2. BOne straight ahead
   3. COne directly behind
   4. DAny landmark helps equally
   Answer: A. A range to a landmark to the side measures across-track position. Choosing routes to keep the ellipse small is active perception.

The task: the shape of the uncertainty

from bugbot import *
import math
connect()

DT = 0.1
along = across = 0.0
head_sigma = 0.5

from bugbot import *
import math
connect()

DT = 0.1
SCALE_ERR = 0.04            # the flow sensor's scale, a few percent
HEAD_ERR = 0.03             # radians of heading uncertainty, growing slowly

along = across = 0.0
head_sigma = 0.5
forward(70)
for i in range(60):
    stepped = abs(flow()[1]) * DT
    head_sigma += HEAD_ERR * DT * 10
    along += SCALE_ERR * stepped
    across += stepped * math.radians(head_sigma)
    plot("along", along)
    plot("across", across)
    wait(DT)
stop()
print("along:", round(along, 2))
print("across:", round(across, 2))