Alberta Ingenuity & CMASE
Lesson 1: Robot Movement
Purpose: We want to look at the movement of a group of robots (robot swarm). Robots
are generally programmed to travel in straight lines. When several robots are moving in
the same area, they need to stay in contact with each other to perform their duties
effectively and to avoid colliding. We will look at two situations: 1) when the robots
are travelling in a formation and 2) when the robots are travelling in different directions.
Both of these types of movements can be seen in the following videos at
http://videolectures.net/aaai07/, specifically:
1) Two-on-two Robot Soccer (Li, Keyong et al)
2) Robot Swarm Localization Using Trilateration (Maxim, P. and Spears B.)
One robot of the group is generating (triggering) pulses and the other robots use 3 pulses
to compute (trilaterate) the coordinates of the triggering robot. Using this method, the
robots can regenerate their formation when it weakens, and they can also avoid collisions
when travelling in different directions. This is a feature of Machine Learning and
Artificial Intelligence called Reinforcement Learning in which feedback causes the robot
to change (or not change) its behavior.
The programming allows each robot its turn to trigger a pulse to allow the other robots to
compute its position relative to their own coordinates. Each robot has its own ID, which
is sent out when the robot triggers. The receiving robots increment this ID by one and
compare it to their own ID. The robot that matches the two ID’s is now able to start
triggering. The other robots continue to trilaterate as they move.
Problem: Our objective is to interpret the movement of the robots with respect to
coordinate geometry, vector algebra, and linear equations and to solve related problems.
Hypothesis: With our mathematical techniques we can solve problems related to robots
moving in formation and also in different directions.
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Let’s look at a situation in which three robots travel in formation:
6
5
4
3
A(-1,3)
2
1
-6
-5
-4
-3
-2
y
x
-1 -10 B(0,0)1 2
-2
-3
-4
-5
C(1,-5)
-6
3
4
5
6
Created with
an unregistered
of C
Advanced
Grapher -athttp:/
www.serpik.com/
Three
robots A,version
B, and
are travelling
the/ same
speed in agrapher/
the same direction, trying to
  1
maintain the original formation. Robot A starts with position vector   , robot B with
3 
0
1 
position vector   and robot C with position vector   .
  5
0
1) If robot A reaches the point (3,6) in two seconds:
a) find the points that robots B and C will reach in 2 seconds.
b) find the speed of the robots in units/s.
c) write the velocity vector for the three robots.
d) Find the location of robot A after 5 seconds
e) How long will it take robot C to reach the point (51,32.5)?
f) At some time robot B is at point (30,21.5). Is robot B still in formation? Justify.
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2) The same robots are relocated with A at (-2,3), B at (5,2) and C at (-1,-2). This time
they are travelling in different directions, at different speeds. Robot A has a velocity
vector 3i - j, robot B has velocity vector -2i - j, and robot C has velocity vector i + 2j.
6
5
4
3
A(-2,3) 2
1
-6
-5
-4
-3
-2
y
-1 -10
-2
C(-1,-2)
-3
-4
-5
-6
B(5,2) x
1
2
3
4
5
6
robot is version
travelling
the fastest?
Createda.withWhich
an unregistered
of Advanced
Grapher - http:/ / www.serpik.com/ agrapher/
b. At what point will the paths of robots A and C cross?
c. Are they on a collision course? Justify.
The robots are programmed so that if they come too close to each other, they
will stop and change direction.
d. How far apart are robots A and B after 1 second of movement?
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e. If robot B changes direction at D(2, 0.5) and then travels to E(6,-4), find:

i. the vector DE


ii. the angle between vector BD and vector DE , to the nearest
degree.
Sources:
1) IB Diploma Program, Mathematics SL Standards Document, IBO, revised 2005
2) http://videolectures.net/aaai07/, Robot Soccer, Keyong Li et al, 2007
3) http://videolectures.net/aaai07/, Robot Swarm Localization using Trilateration,
Paul Maxim and Bill Spears, 2007
4) http://www.cs.ualberta.ca/research/areas/robotics/profiles/index.php
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