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Problem A: Drone Clusters as Light Displays

Summary

In an unspecified city, the mayor wonders whether an outdoor aerial three-

display light show should be incorporated into this year’s annual festival. The

model presented to the mayor must include: three different displays (a Ferris

wheel, a dragon, and an image of the modelers’ choosing), the number and

placement of drones needed to make each display as a static image, and the

mathematical descriptions of the flight paths for the animations in each display.

The memo to the mayor must discuss: the number of drones needed for the entire

show, required launch area, required air space, safety considerations, and duration

of the aerial light show. The memo must also recommend to the mayor whether to

proceed and include the aerial light show in the festival.

The models described in this paper were designed to fulfill the requirements

of including an aerial drone light show during the festival. The displays are

animated by a variety of moving and blinking lights. The drone displays are all

projected onto an imaginary Cartesian Coordinate Plane angled 45.0 degrees

towards the audience of the show. The model also considers safety regulations for

professional and recreational drone use. Each of the drone displays fit into a 100.0

m by 125.0 m by 15.0 m area, and at no time does a drone fly 120.0 m above

ground level.

After careful and detailed analysis of the model, a conclusion was reached

that incorporating an aerial light show into the city’s annual festival would not be a

feasible alternative than a traditional firework show. While drones may be safer

than fireworks, the cost and overall preparation time needed to create a drone show

outweighs the potential safety benefits.

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Table of Contents

Givens and Assumptions ............................................................ 3

The Model ......................................................................................... 4

Part I ............................................................................................. 4

Part II .......................................................................................... 10

Part III ......................................................................................... 12

Strengths and Weaknesses .................................................. 14

Extensions ...................................................................................... 15

Appendices ..................................................................................... 16

Bibliography .................................................................................. 30

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1. Givens and Assumptions

Givens:

1. The light show must contain three separate displays: a Ferris wheel, a

dragon, and an image of the teams choosing.

2. For each display, the number of drones required for the static image of the

display as well as the movements of each drone to implement the animation

of the display must be determined.

3. The number of drones needed for the light show, required launch area for the

drones, required air space, safety considerations, and the duration of the light

show must be explained.

4. A memo to the mayor of the city must be written where the team will

explain whether the light show should take place.

Assumptions:

1. The Intel® Shooting Star™ Drones, as shown in the video provided in the

problem statement, will be used in this model. The drones have a maximum

speed of 3.00 m/s and a fly life of 20.0 minutes.

2. All the drones are fully charged at the beginning of each show.

3. Even though the same drones will be used, the size, battery life, GPS

reception, and LED brightness are the same for each device.

4. These drones have LED lights that can be programed to blink at specific

times and coordinate locations. The LED lights have over 4 billon color

combinations that are based on the RGBW (red, green, blue, and white)

LED.

5. Wind resistance is negligible in the calculations used.

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6. The number of drones that can be used at once cannot exceed 500 because

500 drones in the air at once is the world record. This assumption is made

considering that this city lacks the financial and other means to break the

world record.

7. A display is “animated” if any of the drones in it move, blink, or perform

both actions simultaneously. Animation is defined as the illusion of motion,

so this assumption is justified.

8. All three displays makeup one larger show, but at no time is one display

shown simultaneously in the air with another display. Individual displays

will follow one after another, and the drones will return to their launch area

after each display.

2. The Models

Part I

The Coordinate Plane:

The Cartesian Coordinate System is used in this mathematical model to

account for all three dimensions of the displays through the x, y, and z indices.

When projecting the displays during the aerial light show, the plane must be angled

at 45.0 degrees towards the audience to optimize the visuals. The x-index

represents the width of the display, the y-index represents the height of the display,

and the z-index represents the depth of the display.

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Ferris Wheel:

Figure 1. Ferris Wheel from 2D to 3D.

The model of the Ferris wheel display was developed as follows: plotted on

graph paper, 2D plotted in Mathematica, and then 3D plotted in Mathematica. The

display is made up of 178 drones and consists of two parts: the fulcrum and the

wheel. The fulcrum is illuminated with red light and the wheel is illuminated with

blue light. The fulcrum extends both in front of and behind the Ferris wheel from

the center point of the wheel and has a height of 75.0 m. The wheel, behind the

fulcrum, consists of an outer circle and eight spokes. The outer circle has a radius

of 50.0 m. 64 drones illuminate the circumference of the outer circle. Each spoke is

made of 20 drones. The spokes rotate clockwise at a speed of 0.100 degrees per

1.00 second. The speeds of each drone in the spokes (in order from closest to the

center of the outer circle to farthest) are as follows: 0.550 m/s, 0.820 m/s, 1.10 m/s,

1.37 m/s, 1.64 m/s, 1.92 m/s, 2.19 m/s, 2.47 m/s, 2.74 m/s. (For calculations of the

speed, see the appendix.) The drones closest to the circumference of the outer

circle must travel a further distance per each degree, but in the same time as the

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drones in the inner circles. Therefore, the outer drones must move faster than the

inner drones.

Dragon:

Figure 2. Dragon from image to coordinate points. ("Toothless Smile Cliparts #2928639")

The dragon is made up of 330 drones. (The coordinates for each individual

drone are listed by quadrant in the appendix.) The drones that make up the eyes of

the dragon are programmed to give the illusion of a blinking motion by timing the

lights on the drones to turn on and off. To obtain the coordinates, an image of a

dragon was taken off the internet. The website Dot Pattern Effect was then used to

edit the image into individual dots. Dots were then selectively erased until an

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image consisting of 330 dots (one dot per drone) remained. The image was then

pasted over a coordinate plane using the website Desmos.

Figure 3. Dragon converted to 3D perspective.

The coordinates were found in Desmos using the sliding tool. The dots'

coordinates were then copied into Mathematica and plotted using the 3D plot

function to account for the z-index. The dragon, like the Ferris wheel, while rotate

around the origin on the x and y-axes at a rate of 1.00 degree for every 0.100

seconds. The speeds of the drones making up the dragon are determined with the

same method used to determine the speeds of the drones making up the spokes of

the Ferris wheel. (See appendix for the code determining the speed of the spokes of

the Ferris wheel.)

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Team’s Display Choice:

Figure 4. Process to convert rocket from 2D to 3D. ("Rocket clip art free clip art Microsoft")

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The rocket is made up of 160 drones. (For coordinates of individual drones,

see appendix.) The first model of the rocket was made using a website called Dot

Pattern Effect to create an image out of individual dots. Each dot represents one

drone. The second model of the rocket was created using Desmos to plot the dots

on a coordinate plane and get their (x, y) values. The third model of the rocket was

created with Mathematica. The (x, y) values were added to a list plot, and then a z

value was determined for each point to add depth to the display. The rocket

consists of three layers of drones to give the illusion of a three-dimensional image.

Figure 5. Rocket's path of motion.

Rocket’s Path of Motion:

1) The rocket will start centered on point (0,0) tilted 45.0° clockwise. The point

(0,0) in real life will be 70.0 m above ground level that the drone as launched from.

2) The rocket will travel 10.0 m x 10.5 m to the right at a 45.0° angle above the

horizontal.

3) The rocket will rotate 90.0° counter clockwise around the point (10.0,10.0).

4) The rocket will finally travel 10.0 x 10.5 m to the left at a 45.0° angle above

the horizontal.

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Part II

Required Air Space:

All three of the displays in the light show are designed to fit in a 100.0 m by 100.0

m by 15.0 m area in the air. The area fits inside of the parameters set by the safety

regulations.

Safety Considerations:

The area under the drones will be blocked off so that the viewers will not be able to

walk under the moving drones.

Drones will stay under the 120.0 m (400.0 feet) height limit.

Drones cannot fly directly over a crowd of spectators.

The display should not take place within 8047 m (5.00 mi) of an airport.

The drones cannot fly over 120.0 m above ground level at any time.

The drones cannot fly over an open stadium, if there is an event taking place in that

stadium.

Duration of Show:

The show cannot last more than 20.0 minutes (including launch time, transition

times, and landing time) because the battery life of Intel® Shooting Star™ drones

are 20.0 minutes.

Launch Area:

Each drone is 382 mm long and wide, and will have 127 mm between each drone.

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The most drones used are used in the dragon animation. This animation uses a total

of 330 drones, so the launch area will consist of 18 drones by 18 drones with the

remaining 6 in a row under the last.

((382*18) + (127*17))2 = 81,631,225 mm2

(127+382) ((382*6) + (127*5)) = 420,943 mm2

81,631,225+420,943= 82,052,168 mm2 --> 82,053 m2

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Part III

To whom it may concern:

We understand that this city is considering the incorporation of an aerial

drone light show into this year’s annual festival. Drones may appear to be a safe

and modern alternative to a traditional firework display; however, our team

strongly recommends that this city uses a firework display instead of an aerial

drone display because of the extensive time, cost, and preparation a drone display

entails.

A drone show is not practical for this event because the drones currently

available on the market do not have a long enough battery life. The current battery

life of the Intel® Shooting Star™ drone is about twenty minutes. There are limited

options for other drones to be used in the display because the Shooting Star has one

the longest battery lives of all modern drones-- not only drones meant for light

displays. Unfortunately, the Intel® Shooting Star™ drone is not available on the

market today, and it is currently the only drone designed exclusively for light

shows. The unavailability of the drones and shortness of flight time severely limits

the potential of the aerial light show that this city would be looking to put on at the

festival.

Drone displays are a new technology, which means there is no library of pre-

existing programs for the drones to execute. Not only would the event planners

have to design the entirety of the show themselves, but they would also have to

individually code each drone from scratch. The extra time and effort to plan and

write the code for the show produces a result which, in our professional opinion, is

hardly more engaging than a firework display. Although a drone show may seem

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more impressive than traditional fireworks, the work required simply is not worth a

20-minute drone show.

We understand that a drone display appears to be safer than a firework

display, but if all of the correct procedures regulations are followed, a fireworks

show will be a completely safe and fun activity for everyone to enjoy. If the city is

still convinced that drones are the superior alternative to fireworks because of their

safety appeal, it should know that drones are not entirely safe either. Drones have

their own list of regulations which we would have to follow, and even then, we

cannot completely guarantee that the show will run as planned. Drone technology

is simply too new to make an aerial light show for this city feasible this year. An

aerial drone display may easily be substituted in for a firework display 10 or 20

years in the future, but not at the present. Instead of devoting great amounts of

resources to a light display using drones, the city should spend more time ensuring

the proper safety guidelines are taken with a firework display.

Sincerely,

Team 7417

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4. Strengths and Weaknesses

Strengths:

Simple yet detailed design

-The model is straightforward and uncomplicated, so it will take a very short time

for each display program to be coded and applied to the drones. The animations are

described in detail using diagrams and the process using rotations and translations

is also described.

Location on Cartesian Coordinate Plane

-The models accurately show where the drones will be located at the start of the

animation using a cartesian coordinate plane.

Weaknesses:

Efficiency of drone usage

-The number of drones used for each light display is very different. Future

extensions of this model could implement a uniform number of drones to power

each image to save time and be cost efficient.

Transition Between Displays

-Between each display, the drones must return to the launch pad. If a uniform

number of drones was used in each of the displays, then the drones could be

programmed to seamlessly transition into the next display. Depending on the

number of drones needed for each display, drones could be programmed to either

come from or return to their designated locations on the launch pad to improve

efficiency between transitions of the displays.

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Pricing

-Because the drones that were assumed to be used are currently not on the market,

the cost of the light show was unable to be modelled or determined. Future

extensions would provide projections of prices based off the prices of the drones

that are currently on the market.

5. Extensions

3D:

Future models could more effectively utilize the z-index. Although one of the

Ferris wheel models specified the placement of the fulcrum to be in front and

behind of the wheel, only two axes were used in the animation for the display as

well as for the other two displays described in this paper.

Color:

The Intel® Shooting Star™ drone is capable of displaying 4 billion different

colors. Only a few of the available colors were used in this model. Future models

could incorporate a larger variety of colors in the displays and animations of the

light show.

Programming:

This model did not include the programming of the individual drones needed to

execute the animations. Future models could include a computer simulation of the

programmed paths of the drones.

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6. Appendices

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Figure 6. Finding the Point of Origin.

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Fingas, J. (2017, July 22). UK drone rules will require you to take safety tests. Retrieved

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safety-tests/

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checklist_1200x627_ae05.pdf

Intel News Fact Sheet. (n.d.). Retrieved November 9, 2017, from

https://newsroom.intel.com/wp-content/uploads/sites/11/2017/07/Intel-Shooting-Star-

Tech-Fact-Sheet-073117-1.pdf

Rocket clip art free clip art Microsoft clip art Christmas clip 2. (n.d.). Retrieved November 09,

2017, from http://clipartix.com/rocket-clipart-image-11792/

Share This Post. (n.d.). Retrieved November 09, 2017, from

http://www.techcrackers.com/featured/intel-has-unveiled-its-shooting-star-drone/

Toothless Smile Cliparts #2928639. (n.d.). Retrieved November 09, 2017, from http://clipart-

library.com/clipart/1707199.htm