Teacher Answer Key: Measured Turns
Introduction to Mobile Robotics > Measured Turns Investigation
Phase 1: Swing Turn Path – Evaluate the Hypothesis (1)
1. When you ran your robot, which wheel spun?
The left wheel, the wheel driven by the motor on port C, should be the one to spin.
2. Observe the shape traced by the tracer on the paper. Remember that the tracer
approximates the position of the robot’s left wheel, but is not exactly on top of it.
i. What shape did the wheel trace on the paper?
The wheel should have traced out a circle on the paper, which is represented by the mark that the pen
left, but is not exactly the same circle. If the tracer did not leave a perfect circle, encourage students to try
running the robot again. The robot or the paper may have shifted somewhere during the turn, causing the
circle to not line up correctly. It is also possible that the surface is slanted, which will lead to a skewed
tracing.
ii. What part of the robot is at the center of the circle?
At this point, the robot should have pivoted around its right wheel, which stays in place as it turns.
iii. What part of the robot ran along the circumference of the circle?
The left wheel, which is attached to Motor C, ran along the circumference of the circle.
iv. Are these observations consistent with the first part of the hypothesis.
Yes, the wheel traced out a circle on the paper, which is exactly what was predicted in the first part of Dr.
Turner’s hypothesis.
Hypothesis (part 1)
As the robot swing-turns, the moving wheel traces out a portion of a circle.
3. Measure the path traced by the robot’s wheel.
i. What is the diameter of the traced circle?
The radius of the traced circle should be just about the distance between the two wheels on the robot. It is
measured to be approximately 14.5cm (15cm if you’re rounding) on Taskbot. Therefore, the diameter
should be 29cm (30cm if you’re rounding).
ii. What is the circumference of the traced circle?
The circumference is equal to the diameter times pi. In our example, it is about 91.1cm.
iii. Rewrite Dr. Turner’s hypothesis equation, filling in the values you have now
calculated. The hypothesis equation is as follows:
Hypothesis (equation form)
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(Angle Turned / 360 degrees) = (Distance traveled by wheel / 91.1cm)
We know that the circumference of the traced circle is 91.1cm, so we can fill that in. We also know that a
full circle is equal to 360 degrees.
Phase 2: Hypothesis – Part 2
Answer the following:
4. There is an important distinction between the circumference of a wheel on your robot
and the circumference of the circle your robot traces.
i. Explain the difference.
The circumference of the wheel is the distance around the perimeter of the wheel. The
circumference of the circle your robot traces is the distance along the perimeter of the circle
around which the robot turns. The circumference of the wheel should be much smaller than the
circumference of the circle.
ii. Which one is used, along with motor axle rotations, to calculate distance traveled?
The circumference of the robot’s wheels is used to calculate distance traveled. Recall that distance
traveled is equal to the number of motor axle rotations multiplied by the circumference of the robot’s
wheels.
iii. Which one can be measured by placing a marker near the outside wheel of your
robot and programming it to turn?
The circumference of the circle your robot traces can be measured by placing a marker or pen near
the outside wheel of your robot and having it turn. The robot will drag the marker along with it, thereby
tracing the path of its outside wheel, and nicely illustrating the circle around which the wheel travels as
the robot turns.
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Phase 2: Calculate for a 90-degree Turn
Calculate the following:
5. Following along with the calculations in the video (using your own numbers),
determine how many motor degrees you need to set in your program to make the
robot turn 90 degrees.
Students will calculate this following the example in the video. Below is a sample calculation.
The sample numbers used are that we want the robot to turn 90° (AngleTurned = 90°), the circumference
of the traced circle is 72cm, and the circumference of the wheel is 25cm. Replace these numbers with
ones specific to the problem when checking student answers.
From the hypothesis equation:
AngleTurned
DistanceTraveled
90° DistanceTraveled
=
=
=
Full Circle
Circumference of Traced Circle 360°
72cm
Now we can use that equation to solve for Distance Traveled
DistanceTraveled =
90°
× 72cm = 18cm
360°
Now that we have the distance that the wheel needs to travel, we can use the equation from the Wheels
and Distance Investigation to determine the number of rotations necessary to make the robot go this
distance.
DistanceTraveled = Wheel Circumference × Wheel Rotations
Substituting known values into this equation gives us
DistanceTraveled = Wheel Circumference × Wheel Rotations = 18cm = 25cm × Wheel Rotations
We can solve that equation for Wheel Rotations by dividing both sides by the Wheel Circumference.
Wheel Rotations =
18cm
= 0.72
25cm
And now, to convert the number of rotations into the necessary number of degrees, we need to multiply
the rotations by 360, which is the number of degrees in a full circle.
Degrees toTurn = Wheel Rotations × 360° = 0.72 × 360° = 259°
For this example robot, the wheel needs to be set to go 259° to make the robot turn 90 degrees.
Analysis and Conclusions: Conclusions
6. Now that you’ve run your program, answer the following questions:
i. Did your robot turn about 90 degrees? If not, what factors may have caused the
difference?
This will depend on the student.
It is unlikely that they will have calculated what is necessary to make the robot turn 90 degrees on the
very first try. Most students will want to use trial and error to revise their calculated number, until they
have the robot turn what they see as ‘exactly’ 90 degrees. In fact, it is almost impossible to measure the
exact amount the robot turned at all without highly sensitive scientific equipment, so it is mostly a matter
of judgment.
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Factors such as surface friction, kickback in the motors, and squishiness of the tires all play a roll in the
precision of the robot’s turn. Also, note that we used measurements from the traced circle to determine
the radius of the turn, and not from the robot itself. So even if the calculations were spot-on, a robot still
may not have made a precise 90-degree turn.
ii. Does this behavior support the hypothesis? Why or why not?
Yes, the hypothesis is supported by this behavior.
In most cases, the students will have been ‘close’ the first time. It will be the factors of the physical world
that will have impeded the robot’s performance. Because many students will see their robot as having
done a turn ‘close enough’ to a 90-degree turn, this would support the hypothesis.
The hypothesis states that as the robot turns, each of its wheels traces out a portion of a circle. This is
evident by the path left on the paper by the tracer. The left wheel traces out a circle, and the right wheel
traces out a point (or a circle with radius equal to zero).
The hypothesis also states that the fraction of a full circle that the wheel travels is equal to the fraction of
a full turn that the robot will do. Students worked backwards to get to this point. They started by being
asked how to get the robot to do one quarter of a full turn, and worked out how many degrees the wheel
needed to travel in order to turn one quarter of the circumference of the circle. And, when they ran their
programs, the robot did go approximately one quarter of a turn, so this all supports the hypothesis.
iii. Do you now have enough information to say whether or not the hypothesis is valid, or
do you need more information?
The information from this experiment clearly supports the first part of the hypothesis: that as the
robot swing-turns, the moving wheel traces out a portion of a circle. This can be seen simply by looking at
the shape left on the paper by the tracer.
It also supports the second part of the hypothesis: the amount the robot turns is proportional to the
portion of the circle that the wheel travels. Students have made the wheel travel a specific amount along
the circumference of the circle (one fourth of the circumference), and they can see that to robot makes on
fourth of a full turn, or a 90-degree turn.
However, this does not give us enough information to say definitively whether or not the hypothesis is
valid. This is only one example. For real scientific validity, many more examples of this behavior must be
found and recorded that fit the model proposed by the hypothesis. More information is most certainly
needed.
7. Recall the number of motor degrees you used to make the robot turn 90 degrees
using a swing turn. Estimate how many degrees the motor would need to spin to turn
the robot:
All of these ‘estimates’ are multiples of the original value they have to make the robot turn 90 degrees that
they wrote in Question 5. We will assume that it takes about 450 degrees of motor rotation to make
the robot turn 90-degrees.
i. 180 degrees
To turn the robot 180 degrees, students would need to double the number for 90-degrees, which would
give them 900 degrees that the motor would need to turn.
ii. 270 degrees
For a 270-degree turn, the original number would be tripled, or 1350.
iii. 360 degrees
A full turn of 360 degrees is four times the original number, or 1800 degrees.
iv. 2 full turns
Two full turns is equal to the robot turning 720 degrees, which is 8 times as much as the original 90
degree turn, so students would multiply the original number by 8 to get 3600 degrees.
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8. Modify your program and run the robot for the amounts you predicted for each of the
angles in the Question 7.
i. 180 degrees
ii. 270 degrees
iii. 360 degrees
iv. 2 full turns
v. Do these new results support or refute the hypothesis?
The students’ answers will vary, but if they obtained the correct results in Question 7, the new results
ought to support the hypothesis.
9. The tracer was close to, but not exactly under the wheel when you ran the robot. As
such, the circle it traced was not exactly the path the wheel traveled. Your robot
attempted to turn for 90 degrees based on calculations made using the tracer’s path
instead of the actual wheel.
i. What was the actual distance between the tracer tip and the point where the wheel
touched the ground?
The students must measure this distance. The value should be no more than 5cm.
ii. How successful was your robot in turning 90 degrees using the estimate rather than
the exact value?
Due to experimental errors, this answer may vary from student to student. The actual distance that
matters, however, is the distance from the stationary wheel to the pen tip, as opposed to the distance
from the stationary wheel to the moving wheel. If you were to measure these two distances, you would
find that they are approximately a half centimeter different. Adding a half centimeter to the radius of the
circle (which is what you do when you measure the traced circle) adds about 3cm to the circumference.
So the circumference that the students found is about 3cm larger than the circumference that the actual
wheel travels (which is obtained by measuring the wheel-to-wheel distance). Over one-quarter of the
circle, the change in distance is approximately 0.75cm, which leads to an extra 50 degrees of motor
rotation. So, if anything, the robots should have gone farther than they needed to (though not by much!)
because of the simplification of measuring the traced circle and not the robot’s wheel-to-wheel distance.
However, due to random factors like friction, it is possible that the robot did run very close to 90 degrees.
So answers will depend on the student.
iii. Would you say this was a good estimate, based on the results you got from using it?
Since the estimate will in general be close to a 90degree turn, it should be considered a good
estimate.
iv. Since the right wheel stays planted and the left wheel moves, what is the actual
physical distance of the radius or diameter of the turn?
The actual physical distance of the radius of the turn equals the distance between the two wheels on the
robot. This should be approximately 13.8cm.
10. Vehicle designers in the real world deal with turning issues all the time. It is
important for both safety and performance that they know exactly what their vehicles
can and cannot handle. Think about real world vehicles while you answer the
following questions.
i. What are the limitations of swing turns?
One of the limitations of swing turns is that they are very wide turns, and take up a lot of space for the
robot to get all the way around and back to its starting point. This can be a problem in tight spaces.
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ii. Name another mobile vehicle that is capable of a swing turn, and estimate its turning
radius (or diameter).
Almost any mobile vehicle will do swing turns. Cars have turning radii of between 25 and 50 feet. Small
mowing tractors have radii of between 2 and 5 feet. Larger farm equipment have turning radii of
between 10 and 30 feet. The zamboni at the ice rink has a turn radius as small as 10 feet.
iii. Do normal cars use this type of turn?
Yes, normal cars can swing turn. They cannot point turn.
Analysis and Conclusions: Exercises
11. Shante’s robot has wheels that are 4.5 cm in diameter and traces a turning circle
with a 12.4 cm radius. Her wheels are driven directly by the motors on each side.
She needs her robot to make a swing turn of 210 degrees. How many degrees do
her motors need to run in order to make the robot make this turn? Show your work.
1188 degrees, or approximately 1200 degrees.
To do this problem, let’s first divide up what we know, and what we want to find.
We want to find:
Degrees the motor needs to spin
We have:
Wheel diameter
Wheel-to-wheel distance
Amount robot must turn
We still need:
Circle circumference
Wheel circumference
The circumference of the circle that the robot turns can be found from the wheel-to-wheel distance and
the fact that it is a swing turn. For the swing turn, the radius of the circle was equal to the wheel-to-wheel
distance on the robot. So the circumference of the circle is equal to the wheel-to-wheel distance times 2
times pi.
Circle Circumference = Wheel-to-wheel distance x 2 x pi = 12.4cm x 2 x 3.14 = 77.9cm
The wheel circumference is found from the diameter of the wheel.
Wheel Circumference = wheel diameter x pi = 4.5cm x 3.14 = 14.1cm
The robot must turn 215 degrees, which is (215/360), or 0.6 revolutions.
If 77.9cm represents the circumference of the whole circle, then (77.9cm x 0.6) represents 0.6 of a turn
around that circle, which is the amount covered by 215 degrees.
Distance wheel must go = Circle Circumference x amount robot must turn = 77.9cm x 0.6 = 46.7cm
Now we divide the distance the wheel must go by the distance the wheel goes per revolution (the wheels
circumference) to get the number of revolutions that the wheel must do.
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Revolutions = Distance wheel must go / wheel circumference = 46.7cm / 14.1cm = 3.3 revolutions
Degrees the motor needs to spin = Revolutions x 360 = 1188 degrees
12. Larry has forgotten to save his program, and now he can’t remember what he had
his Wait For block set to. He knows that he wanted his robot to do a swing turn, and
he measured his wheels to find out that they are 2.5 cm in diameter and they are 9.0
cm apart. He also knows that the robot turned exactly 180 degrees. How many
degrees should Larry set his Wait For block to when he rebuilds his program?
Both motors will need to be set to spin 1290 degrees.
Again, let’s first divide up what we know, and what we want to find.
We want to find:
Degrees the motor needs to spin
We have:
Wheel diameter
Wheel-to-wheel distance
Amount robot must turn
We still need:
Circle circumference
Wheel circumference
The circumference of the circle that the robot turns can be found from the wheel-to-wheel distance and
the fact that it is a swing turn. For the swing turn, the radius of the circle is equal to the wheel-to-wheel
distance on the robot. So the circumference of the circle is equal to the wheel-to-wheel distance times 2
times pi.
Circle Circumference = Wheel-to-wheel distance x 2 x pi = 9.0cm x 2 x 3.14 = 56.5cm
The wheel circumference is found from the diameter of the wheel.
Wheel Circumference = wheel diameter x pi = 2.5cm x 3.14 = 7.9cm
The robot must turn 180 degrees, which is (180/360), or 0.5 revolutions.
If 56.5cm represents the circumference of the whole circle, then (56.5cm x 0.5) represents 0.5 of a turn
around that circle, which is the amount covered by 180 degrees.
Distance wheel must go = Circle Circumference x amount robot must turn = 56.5cm x 0.5 = 28.3cm
Now we divide the distance the wheel must go by the distance the wheel goes per revolution (the wheels
circumference) to get the number of revolutions that the wheel must do.
Revolutions = Distance wheel must go / wheel circumference = 28.3cm / 7.9cm = 3.6 revolutions
Degrees the motor needs to spin = Revolutions x 360 = 1.8 x 360 = 1290 degrees
13. Below is Peter’s program. Assume that the comment he wrote accurately describes
the amount the Wait For block will wait. He wants his robot to make a full 180 degree
swing turn, and his robot traces a circle with a radius of 10.8 cm. He has a choice
between two wheel sizes: 4.6 cm in diameter and 7.2 cm in diameter.
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i. Which wheels should he choose to come closest to making the 180 degree turn with
his current program?
The 4.6cm diameter wheels are the best ones to use. To find this out, students will need to do the math
for both wheels. Some things are the same for both wheels, like the diameter of the circle that the rolling
wheel will travel.
Because this is a swing turn, the large circle’s radius is the distance between the two wheels.
Circle Diameter = wheel-to-wheel distance x 2 = 10.8cm x 2 = 21.6cm
Circle Circumference = Circle Diameter x pi = 21.6cm x 3.14 = 67.8cm
And for both wheels, we know we want them to travel around 180 degrees of that circle, which is
(180/360) or 0.5 of the circle.
Distance the robot needs to travel = Circle circumference x 0.5 = 67.8cm x 0.5 = 33.9cm
OK, so now we know that whatever wheel we use has to travel 33.9cm in order to go around half the
circle, which would turn the robot 180 degrees in a swing turn. Now we have to figure out which wheel
comes closest to this as it goes the number of degrees that Peter set in his program.
In his program, Peter has the Wait For block waiting for 760 degrees.
Number of motor revolutions = 760/360 = 2.11 turns of the motor
We can do the last part one of two ways.
First method:
Divide the distance the robot needs to go by the number of turns the wheel will make to get the necessary
wheel circumference.
Wheel Circumference = Distance robot needs to go / Turns of the motor = 33.9cm / 2.11 revs = 16.07 cm
And now find which wheel has a circumference closest to 16.07cm.
Wheel 1 Circumference = Wheel 1 Diameter x pi = 4.6cm x 3.14 = 14.4cm
Wheel 2 Circumference = Wheel 2 Diameter x pi = 7.2cm x 3.14 = 22.6cm
Second Method:
Find the circumference of each wheel, and multiply them each by the number of revolutions to find out
how far they will travel under that number of revolutions.
Wheel 1 Circumference = Wheel 1 Diameter x pi = 4.6cm x 3.14 = 14.4cm
Wheel 1 Distance traveled = Wheel 1 Circumference x revolutions = 14.4cm x 2.11 rev = 30.4cm
Wheel 2 Circumference = Wheel 2 Diameter x pi = 7.2cm x 3.14 = 22.6cm
Wheel 2 Distance traveled = Wheel 2 Circumference x revolutions = 22.6cm x 2.11 rev = 47.7cm
The distance the wheel needs to travel was found above to be 33.9cm, and 30.4cm is clearly closer to
33.9cm than 47.7cm is. He should choose the wheels that are 4.6cm in diameter.
ii. What else can he change on the robot to make the turn?
Peter can also move the wheels farther apart by adding bushings to the axle. This will make the turn
wider, increasing the circumference of the turned circle. Or he can change the gear ratio, so that even if
the motor spins one full revolution, the wheel may turn less than one full revolution (or more than one full
revolution), which will change the amount the robot goes.
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