Assignment 2 information

Assignment information

Announcements
Support code
Sample inputs

whuang@cs.rpi.edu; email
Last updates: March 1; February 28; February 23; February 15, 2001

Announcements

Here are the loose ends from class today (3/1) for those of you who left before we figured them all out.
- The problem with the formula for the angle to the major axis is that the quantity 2b/(a-c) is the tangent of 2*theta which is zero for both theta = 0 and for theta = 90.
  Instead, when the error ellipse is degenerate, use one of these formulas:
```
tan theta = b / a      if a != 0
cot theta = b / c      if c != 0
```
  These formulas only work when the error ellipse is degenerate!
- If you are using degrees to represent angles, then you have probably calculated the derivatives incorrectly:
```
if theta is in radians, then d/dtheta(cos_rad theta) = - sin_rad theta

if theta is in degrees, then first let phi = theta * pi / 180

d/dtheta(cos_deg theta) = d/dtheta(cos_rad phi)
                      = - (sin_rad phi) * dphi/dtheta
                      = - (sin_deg theta) * pi / 180
```
  An alternative, of course, is to do all your calculations in radians.
- Another testing suggestion for part B: Create a sequence of inputs which allows the error ellips to get large before sensing so that you can see a dramatic decrease in the size of the ellipse. You should specify different sensing angles to see that it reduces uncertainty in the proper direction.
Someone just showed me some pretty wild uncertainty ellipses, and I made a few observations and suggestions:
- Develop an intuitive sense of what the uncertainty ellipses should look like and how they should grow. Remember that if DSigmaFrac is, say 0.005, then a 95% uncertainty (in distance only) is just 1% on each side of the mean. Your angular uncertainty should grow throughout Part A. The area of your ellipse should also grow.
- To make it easier to see what's going on, I suggest you create a path for the robot that traverses a square with some number of small steps on each side. (That is, say, take 10 steps in a straight line, then turn 90 degrees, take another 10 steps, etc.)
  Another useful path to try is to trace out a circle in a number of discrete steps. Your first step should be a turn, and then you should have a number of identical steps followed by a turn. You might take 20 or 50 steps to go around the circle. The position uncertainty should be circular after you travel around the circle once.
To draw the ellipse for the uncertainty bound in equation 17 in the "Localization & Estimation (Draft, continued)" handout, substitute x = cos(theta) and y = sin(theta) into the equation. You will then be able to factor out an r^2 from all terms on the left side of the equation and solve for r as a function of theta. You can then approximate the ellipse by drawing lines between points that you compute for some increment of theta.
On the subject of this ellipse, when the covariance matrix does not have full rank (i.e. a*c - b*b == 0) the underlying distribution is really one dimensional, so the equations 19 and 20 in the "Localization & Estimation (Draft, continued)" handout do not work.
There are two ways I can explain this. One is that these formulas are trying to draw an ellipse that captures a certain volume under the probability density function. As the minor axis radius approaches 0, the major axis radius approaches infinity to make up the difference. Of course, the reason that the minor axis radius would be 0 is when you know the position of the robot exactly in some direction (as you do at the start of a run!) The other explanation is that with a one dimensional distribution, you have to use a different set of formulas.
In order to draw the degenerate 95% error ellipse, you will simply draw a line of length 4*sigma (i.e. 2*sigma on each side of the mean) along the major axis. You can get the angle of the major axis by using the same formula as for the nondegenerate case (i.e. equation 18 in the aforementioned handout). The variance of the distribution will be equal to a + c.
There are new versions of the support code out as of late Friday February 23. There are also a number of new files.

Support Code for reading input files

Here are the support code files for the basic robot simulator and the random number stuff (updated 2/23/01 to include the simulated sensor)

rand.cc (this was updated to correct an error)
rand.h
robot.cc (define IM_USING_STUPID_WINDOWS if you are)
robot.h
cgal.h

Here's a little test program which I used to test the simulated laser rangefinder:

robottest.cc
Makefile (for the CS environment)

Sample inputs

Here's the problem description file from the assignment:

testprob.txt (this is just something I was using for testing)
prob1.txt (use this one)
world1.txt (from assignment 1)
world2.txt (test world)
world3.txt (this is a "real" world)