Using the Parrot Drone in Mathematics Classesictcm.pearsontc.net/files/ictcm-drone.pdf · Figure1shows the Parrot AR.Drone 2.01 in ight. This drone can be operated from an iOS (iPad,

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Figure 1: Flying a Drone

Using the Parrot Drone in Mathematics Classes

Figure 1 shows the Parrot AR.Drone 2.01 in flight. This drone can be operated froman iOS (iPad, iPhone, iTouch) or Android tablet. It is readily available at many malls andthrough the Internet,2 is easy to fly, and is equipped with cameras that can record andstream live photographs and video during a flight. Figure 2 shows a photograph taken fromthe drone. More sophisticated (and expensive) drones are often used for reconnaissance –for example, soldiers might use a drone to look for IEDs along an intended route or rescueworkers might search for people in danger. Figures 3, 4, and 5 show the drone on theground and in the air.

We are using the drone in several different mathematics courses to teach standardtopics. It is sturdy enough so that we are able to give all of our students the opportunityto fly it. In addition to the drones, we use free mission simulation software developedby NSF-supported DIYModeling project. As one example, we used the drone in a onehour introduction to the West Point Mathematics Program for high school students in the

1http://ardrone.parrot.com/parrot-ar-drone/usa/ Accessed 15 June 2012.2http://www.amazon.com/Parrot-AR-Drone-Quadricopter-Controlled-Android/dp/B003ZVSHB0 Ac-

cessed 15 June 2012.

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Figure 2: An Aerial Photograph from a Quadricopter Drone

Figure 3: A Quadricopter Drone on the Ground

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Figure 4: A Quadricopter Drone Flying

summer between their junior and senior years. The questions below are sample questionsthat students considered in that introduction. In this minicourse we will consider many,many different questions based on the drone for courses ranging from the middle and highschool level through the advanced undergraduate.

Drone – Sample Exercises I: Range Questions

This particular drone has an air speed of about 18 kilometers per hour (or 300 metersper minute). It operates on a single rechargeable battery and has a flight time of aboutten minutes starting with a fully charged battery. This gives it an effective one-way flyingrange of 3,000 meters if it is flying in a straight line in calm air for the full ten minutesand there is no wind. In practice the flight time on a charge varies and we need to plan forless than ten minutes of flight time. For the following questions suppose you plan a flighttime of at most eight minutes.

Question 1

(a) How far can you fly in calm air?

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Figure 5: A Quadricopter Drone Flying

(b) Suppose you have a 10 kph head wind – that is, you are flying against a 10 kph wind.How far can you fly?

(c) Suppose you have a 10 kph tail wind – that is, you are flying with a 10 kph wind.How far can you fly?

(d) Suppose your mission involves flying from point A to point B and returning. Supposea ten kph wind is blowing from point B directly toward point A. What is the maximumdistance from point A to point B for this kind of mission?

Question 2 Suppose you plan to fly the drone for eight minutes directly north and thereis a crosswind blowing from east to west of 10 kph. Because the wind will blow the drone“off-course” it will wind up northwest of its starting point. Where will it wind up?

Question 3 Suppose that you plan to fly the drone from a base at point A to a destinationat point B directly north of your base. Suppose as in the previous question that there is across wind blowing from east to west of 10 kph. If you aim the drone straight north youwill be blown off course.

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-2 km 2 km

2 km

-2 km

Figure 6: Area of operations for a round trip mission

(a) At what angle should you aim (measured clockwise from straight north) should youaim the drone to reach the destination?

(b) What is the maximum possible distance from point A to point B for this mission?

(c) Suppose another mission involves flying from point A to point B and returning. Whatis the maximum distance from point A to point B for this kind of mission?

Suppose you have a base at point A and are interested in three kinds of missions –delivery missions, recovery missions, and round-trip missions. For a delivery mission thedrone flies one-way from the base to a destination at point B. For a recovery mission thedrone flies one-way from an origin at point B to the base. For a round-trip mission thedrone flies from the base at point A to a destination at point B and then immediatelyreturns to point A. In calm air delivery and recovery missions can be accomplished as longas point B is inside a circle of radius 2400 meters. This is called the area of operations fora delivery mission or the area of operations for a recovery mission. In calm air the areaof operations for a round-trip mission is 1200 meters. Figure 6 is a diagram of the area ofoperations for a round trip mission in calm air.

Question 4 Draw a diagram showing the area of operations for a delivery mission if thereis a 10 kph wind blowing from north to south.

Question 5 Draw a diagram showing the area of operations for a recovery mission if thereis a 10 kph wind blowing from north to south.

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N

B

A

θ

Figure 7: Bearing

For the next few questions it is helpful to think of the direction from point A to pointB in terms of the angle measured clockwise from due north. See Figure 7. This angle issometimes called the bearing.

Question 6 Find the bearing and distance from the point (0, 0) to the point (3, 4).

In still air you aim the drone toward the destination but if there is a wind blowing –for example, from the north toward the south – then you have to adjust the direction inwhich the drone is flying to compensate for the wind. See Figure 8.

Question 7 Suppose you are flying a delivery mission from the point (0, 0) to the point(3, 4) and there is a 10 kph wind blowing from the north toward the south. Find the directionin which you should aim the drone and the flight time required for the mission.

Question 8 Draw a diagram showing the area of operations for a round-trip mission ifthere is a 10 kph wind blowing from north to south.

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N

B

A

head toward this point

φ

Figure 8: Heading