MAS115: SEMESTER 2 INDIVIDUAL PROJECT
PAUL BLACKWELL
Random Robots
A robot moves on a rectangular array of squares, in accordance with a
‘program’ made up of simple instructions. The instructions are interpreted
according to the way that the robot is facing—in the positive x direction
(‘east’), the negative x direction (‘west’), the positive y direction (‘north’),
or the negative y direction (‘south’). There are three basic instructions,
assigned at random, according to some given set of probabilities.
Rotate Left: The robot makes a quarter turn left i.e. anticlockwise.
Rotate Right: The robot makes a quarter turn right i.e. clockwise.
Move 1: The robot goes forward one square, in the direction in which
it is currently facing.
Background. This project is inspired by the excellent board game ‘RoboRally’ (see e.g. https://boardgamegeek.com/boardgame/18/roborally).
In the game, instructions are represented by cards, and a player programs
their robot by selecting cards from a hand that they have been dealt. The
robots race around a hazardous factory environment. No knowledge of the
game is needed to do the project.
The basic task
Your task is to write R or Python code that will simulate the movement
of a robot as it obeys a sequence of k instructions, with each one being
Move 1 with probability p (0 < p < 1), and otherwise equally likely to
be Rotate Left or Rotate Right. Assume that the robot moves on a
(2n + 1) × (2m + 1) board, starting on the centre square and facing east,
with n and m large enough to avoid it reaching the edge of the board.
Your program should specify values of k and p, and then simulate a large
number of runs, each consisting of selecting a random set of instructions and
carrying them out. It should record the distribution of the robot’s location
after carrying out the k instructions; that is, for each square, the proportion
of runs in which the robot finishes in that square. A suitable default value
is k = 5, matching the original game.
You should make sure your code is producing sensible results by testing it
with several different cases that you can check by hand.
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PAUL BLACKWELL
Extending the project
In addition to the basic task, you should investigate a way of extending the
project further. Here are a few ideas: some are quite open-ended, and some
may be hard to complete, but the idea is to investigate!
Features of the original game.
• Additional instructions might include moving forward by more than
one square, or moving backwards one square.
• If the robot does go off the edge of the board, it falls to destruction!
How does the probability depend on the size of the board?
• The board might contain other hazards—squares where the robot
will be destroyed—or target squares. What is the probability of
surviving k instructions? What is the probability of hitting the
target in k instructions? Or ever? Without being destroyed first?
More mathematical ideas.
• Assuming n and m to be large, can you count the number of different
squares in which the robot might end up, after k moves? Can you
gives a general expression as a function of k, or some upper or lower
bounds?
• If k is large enough, the robot will eventually hit an edge of the
board. Can you say anything mathematically about the probabilities
of hitting different edges?
• Can you exploit some of the symmetry in the original problem to
improve the accuracy of your estimates based on simulation? Conversely, what happens if you drop the symmetry between left and
right turns?
Applications in biology. The process described above could be seen as
a very simplified model of the movement a wild animal, for example as
recorded by a GPS tag, or of a micro-organism in a laboratory experiment.
Some of the extensions above may lead to slightly more realistic models.
Alternatively, consider these ideas.
• An animal is searching for randomly located patches that contain
food, with each patch of size h × h squares. Do the probabilities of
the different types of ‘instruction’ make a difference to how quickly
it is likely to find a patch?
• What if rotations of less than a quarter turn are allowed? You will
need to think about how to ensure that makes sense. You could even
dispense with the grid of squares completely.
Feel free to extend in any other way you can think of. What you choose is
up to you. But don’t try to do all of the above; something that is narrower
but deeper is likely to be more interesting.
MAS115: SEMESTER 2 INDIVIDUAL PROJECT
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The project write-up
Write up your project, including a description of how your code works, in a
LATEX or markdown report of around 4 pages, and strictly no more than 6
pages. The title should be “MAS115: Semester 2 individual project”, and
the ‘author’ should be your registration number. Do not include your name.
Your report will be peer-assessed ; that is, it will be marked by your fellow students, with some oversight from staff. Your involvement in carrying
out the peer assessment of other students’ reports is a required part of the
project.
You will submit the report online via the course website. The deadline for
uploading is midnight at the end of Wednesday 13th May.
Late work and plagiarism
Late work. Due to the peer assessment of this work, it is important that
work is submitted on time. Any work submitted after the deadline may be
given a mark of zero. Anybody with circumstances affecting their ability
to hand in the work must contact Prof Blackwell in advance of the hand-in
date.
Plagiarism. The project is an individual assignment and must be your
own work. All the University rules on non-invigilated examinations apply.
You must not copy work from other students. You can ask for help on the
discussion board, but must not ask for help on any other internet discussion
forum or email list or anything of a similar nature.
Yellow stickers. If you have been assessed by the Disability and DyslexiaSupport Service (DDSS) and have been given yellow stickers to put on your
work, please let Prof Blackwell know so that we can attach a virtual sticker
to your mini-project.