CubeSat Mission- Washinton University

UNIVERSITY OF WASHINGTON DEPARTMENT OF AERONAUTICS & ASTRONAUTICS

RTICC Rapid Terrestrial Imaging

CubeSat Constellation

Preliminary Design Report

AA420/421 Space Design In conjunction with Andrews Space Inc. (SATS)

June 12, 2009

Authors: Michael Bernhardt, Aaron Borth, Rachel Brennan, Enrique Galgana, Peter Gangar, Austin Kemis, Nikolas

Lutzenhiser, Katie Moravec, Skander Mzali, Zahra Nazari, Josh Ross and Eun-Ju Shin-White

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Table of Contents

Table of Contents ..........................................................................................................................................2

Abstract .........................................................................................................................................................3

Mission Design .............................................................................................................................................4

1. Orbital Mechanics .............................................................................................................................4

2. Propulsion .........................................................................................................................................4

3. Image Acquisition............................................................................................................................35

4. Navigation/Control Systems ...........................................................................................................49

5. Communications .............................................................................................................................81

6. Support Systems ............................................................................ Error! Bookmark not defined.93

References.................................................................................................................................................119

Appendices................................................................................................................................................127

A. Orbital Mechanics .........................................................................................................................127

B. Propulsion .....................................................................................................................................144

C. Image Acquisition..........................................................................................................................162

D. Navigation/Control .......................................................................................................................164

E. Communications ...........................................................................................................................182

F. Support Systems ...........................................................................................................................182

G. Biographical Sketches ...................................................................................................................196

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Abstract

The goal of this project was to develop a detailed mission design for an Earth imaging

application using a low-cost constellation of CubeSats. The requirements for this mission include

coverage between 55 degrees North and South, image resolution of 3 meters, image acquisition

within 5 minutes of command, and download to client within 60 minutes. The mission design

concept calls for a Walker constellation of 33 planes with 10 CubeSats per plane, at an altitude of

520 km. A deployment system is proposed to deliver the CubeSats to each orbital plane via a

system of carriers, with each carrier holding 10 CubeSats. Once deployed, the CubeSats will use

a 90mm Maksutov telescope in combination with a 10MP CCD to capture images covering a

ground area of 5km x 5km with a resolution of 3m. In order to acquire images with a nadir

pointing accuracy of 200m, the attitude determination and control system will use GPS and a

custom designed star tracker to provide high accuracy attitude determination. Supporting the star

tracker will be a arrangement of an IMU, sun sensors, and magnetometers. CMG wheels and

magnetorquers will be used to control the attitude of the CubeSats. Once acquired, the images

will be transmitted via a UHF communications link broadcasted using monopole antennae that

will traverse other satellites in the constellation in order to be delivered to the designated ground

station. Providing power to the systems of the CubeSats will be a combination of solar arrays and

batteries. The computing and data handling system will employ commercial-off-the-shelf

hardware including integrated microcontrollers and custom computer solutions. Accuracy,

weight, cost, and efficiency are the primary concerns of this mission and the devised solutions

will be addressed in the following report.

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Mission Design

1. Orbital Mechanics The tasks for orbital mechanics analysis were to design a constellation, develop a deployment

method, and select a suitable launch vehicle. The mission profile of rapid-response imaging with

maximum earth coverage capability drove the selection of the satellite constellation design. The

deployment method was designed to provide the full deployment of all cubesats within a

reasonable time frame, and with achievable delta-V maneuvers. The launch vehicles were chosen

based on their availability and capability for launching to desired inclinations and altitudes.

1.1. Constellation Design The imaging application of the mission calls for many small satellites in low earth orbit

(LEO) to minimize focal length and telescope diameter to achieve the required image resolution.

Two constellation designs were proposed and evaluated based on their ability to provide

maximum coverage and minimize the total number of satellites. The initial design concept was a

constellation of polar orbits, to create “polar streets of coverage” (see Figure 1.1.1 below).

Figure 1.1.1: Polar Streets of coverage

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The primary advantage of this configuration is that it is capable of providing full earth

coverage. There are two disadvantages to this configuration, however. One is that it provides

greater coverage of the poles, where there are less areas of interest for imaging, while it provides

less coverage near the equator. The second difficulty is the large delta-V required to launch to

polar orbit. Due to this high delta-V requirement and the resulting low availability of launches to

polar orbit, the polar streets constellation was ruled out as a feasible option. Choosing an altitude

of 520 km and assuming a launch to 70° would require 20 planes with 20 spacecraft each,

resulting in a delta-V of 2.6 km/s for plane change to 90° plus 45 km/s for 21 plane changes (see

Appendix A.1 for method of calculation). Launching directly into a 90° orbit is also undesirable

as the frequency of launches to polar inclination is around one per year, leading to an

unreasonably long deployment time for 20 planes (see Table 1.3.2 and 1.3.3 below).

The second design proposed was the Walker constellation. In this configuration, the

orbital planes all have the same inclination, but the right ascension of the ascending node for

each plane is equally spaced over 360°, creating a crisscrossing pattern of coverage, shown in

Figure 1.1.2 below. Although the cubesats are still spread out near the equator and condensed

near the higher latitudes, they are distributed more evenly than in the polar streets constellation.

Figure 1.1.2: Walker Constellation

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The Walker constellation involves a loss of coverage near the poles, but allows a

reduction in the total number of spacecraft because of the better distribution at lower latitudes.

The Walker configuration redistributes coverage by halving the number of spacecraft per plane

and almost doubling the number of planes. The number of spacecraft per plane is reduced if the

spacecraft are phased correctly to alternate their crossing at each ascending and descending node.

The number of planes is increased since the ascending nodes must be spread over 360° of the

entire circumference of the Earth, whereas the orbital planes for polar streets are spread out over

180° of the circumference. The reduced number of spacecraft and better coverage provided by

the Walker constellation, and the high delta-V and low launch frequency to 90° associated with

the polar streets constellation, led the team to choose the Walker constellation for this mission.

Coverage analysis for the Walker constellation was initially conducted over an altitude

range of 400 – 550 km. This range was set based on the requirements from propulsion and

imaging. The minimum allowable altitude was set at 400 km, below which the orbits would

degrade too fast, threatening loss of the spacecraft and necessitating large delta-V for orbit

boosting. Above 550 km, the total orbit degradation over the system’s 1-year lifespan would be

within acceptable range, such that the propulsion for orbit maintenance could theoretically be

disregarded. However, altitudes much higher than 550 would require telescopes with a bigger

diameter to achieve the required resolution. Since the diameter is already pushing the dimension

limits for the cubesat structures, increasing the diameter should be avoided. Thus a range from

400-550 km was within the acceptable propulsion and imaging constraints.

This altitude range and an assumed field of view angle of 45° were used to calculate the

satellites’ orbital speed, ground track speed, period, and coverage swath. The swath area and the

speeds determined the minimum number of orbital planes and minimum number of spacecraft

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per plane to achieve full earth coverage, with a maximum revisit time of 5 minutes. In order to

halve the number of spacecraft per plane, they must be phased such that one crosses from each

orbit at 5 minute intervals, as illustrated in Figure 1.1.3. All calculations and derivations are

explained in Appendix A.1.

Figure 1.1.3: Crossing node at equator, with 5-minute revisit time

The number of planes and satellites per plane, calculated over an altitude range of 400 –

500 km, allowed choosing a constellation with a reasonable altitude and minimized number of

spacecraft. The total number of cubesats and the number of planes are given in Table 1.1.1 for

nine selected values of altitude and inclination.

Table 1.1.1: Total number of cubesats and planes for selected alititudes and inclinations.

Inclination (degrees)

Altitude (km) 45° 55° 65°

400 km 360 sats/36 planes 420 sats/42 plane 460 sats/46 planes

450 km 320 sats/32 planes 370 sats/37 planes 410 sats/41 planes

500 km 280 sats/28 planes 330 sats/33 planes 360 sats/36 planes

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The Walker configuration initially chosen was 500 km altitude with 60° inclination, with

36 planes of 10 spacecraft each. The altitude was chosen to achieve 3 m imaging resolution

while minimizing orbit decay. The inclination was chosen to allow reasonable delta-V (~3 km/s

from 35° to 60°) while achieving maximum latitude coverage, since the full earth coverage

requirement must be relaxed to some finite inclination. The full-earth coverage requirement was

relaxed to neglect areas above 60° north latitude and below 60° south latitude since these areas

are of less interest for imaging. The overall configuration was designed to minimize the number

of spacecraft needed and allow some coverage overlap.

Although analysis of the Walker constellation was initially conducted assuming worse-

case coverage at the equator, further research showed that the phasing method described above

(with cubesats making alternating passes over nodes along the equator) did not guarantee 5

minute revisit at every point within the coverage band. Various configurations were analyzed for

the fraction of points that are covered with 5 minutes, as well as the maximum time that any

point has to wait for coverage (See Appendix A.1 for method of analysis). The best coverage

occurs at high altitude and low inclination, since high altitude allows a bigger coverage swath

and low inclination requires less area to be covered. A representative set of options for 520 km

altitude and 55° inclination are shown in Table 1.1.2 below.

Table 1.1.2: Total number of cubesats and planes for selected alititudes and inclinations.

Total # Sats # Planes # Sats per

plane Phasing

Angle (deg) Coverage Fraction

Maximum RevisitGap (min)

330 33 10 19.6 0.979 6.12

360 36 10 22.9 0.997 5.379

363 33 11 16.4 0.993 5.626

390 39 10 26.2 1.000 4.856

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To make the best compromise between minimum planes, minimum satellites, and

maximum coverage fraction, the configuration chosen was a constellation of 33 planes with 10

spacecraft each and 19.6° cubesat phasing between planes, flying at an altitude of 520 km and an

inclination 55°. The fraction of points covered within 5 minutes is 0.979 and the maximum

coverage gap for any point is 6.12 minutes. Since the effects of the increase in altitude and

decrease in latitude were considered acceptable by all other subsystems, this configuration was

the final one chosen for this mission.

1.2. Deployment Method In order to deliver the cubesats to their proper locations in the constellation, a system of

carrier vehicles must be designed. The proposed method is to deploy one carrier into each of the

33 planes by launching groups of 6 to differentially precessing orbits, and then have each carrier

deploy its 10 cubesats on its respective orbit using an elliptical phasing orbit.

Several options were considered for deploying the carriers. The first option was to launch

a master carrier to the desired inclination. This master carrier would hold all the individual

carriers and distribute them to their respective planes by executing plane changes at the

northernmost point of the orbit (at the circle of nodes where the Walker orbits cross, shown in

Figure 1.2.1). The master carrier would drop off the first carrier and then execute a plane change

to transport all the remaining carriers to the next orbit. This option would require 32 plane

changes and would involve carrying unnecessary mass to each plane since only one carrier ends

up on the plane. Assuming a master launch at an inclination of 55° to deploy 33 planes, the total

delta-V would be 38 km/s and the total propellant mass for the plane changes (using the specific

impulse of hydrazine) would be 218 million kg. (Please see Appendix A.2 for method of

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calculation.) The prohibitively large delta-V and extremely high propellant mass required for this

option ruled it out as a feasible method for carrier deployment.

Figure 1.2.1: Close up of polar ring; plane changes for master carrier occur where orbits cross

The second option is to launch in groups of three carriers. For each launch, one carrier

remains in the launch orbit (which has the desired inclination) and the other two carriers execute

plane changes to split off to the right and left so that the three carriers are deployed onto their

respective planes with the correct right ascensions. Each plane change occurs at the node where

the initial launch orbit crosses the desired orbit, as illustrated in Figure 1.2.2. Assuming 33

planes, the delta-V needed for each plane change would be 1.2 km/s, requiring a total delta-V of

26 km/s to deploy all planes. This option would require 11 launches and take up to 1.5 years if

the carriers were launched as secondary payload.

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Figure 1.2.2: Clustered launches

The third option considered was to launch each carrier individually to the desired

inclination. Each launch would be timed so that the right ascension of each orbit would be

correct relative to the others, using the rotation of the Earth to position the launch point correctly.

This option would not require any plane changes, but would take 33 launches and up to 4 years if

each carrier was launched as a secondary payload.

The use of orbit precession was also considered as a possible deployment method, and

was finally chosen as the best option for this mission. All the carriers would be launched to a

particular inclination, and the first carrier would execute a plane change to the desired

inclination. The rest of the carriers would remain on the launch orbit until the orbit had precessed

to the correct right ascension for the next Walker orbit (illustrated in Figure 1.2.3).

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Figure 1.2.3: Using precession for deployment

Assuming a launch to a 50° inclination with a desired inclination of 55°, the delta-V for

one 5° plane change is 660 m/s and the total delta-V for 33 planes is 22 km/s. The precession

time between successive planes is 21 days, resulting in a total deployment time of less than 2

years if all the carriers are launched into a single precession orbit. This deployment time could be

significantly reduced by launching the carriers in groups, allowing several launch orbits to

precess around the earth simultaneously. This would eliminate the need for the last carrier to wait

for all the previous carriers to deploy before it reaches its own orbital plane. (Please see

Appendix A.2 for method of calculation.)

Assuming up to 6 carriers are launched together using a Falcon 1e (further discussed in

the Launch Vehicles section below), 5 launches of 6 carriers and 1 launch of 3 carriers will be

required. With a precession time of 21 days between planes, a group of 6 carriers will take 126

days to deploy and the group of 3 will take 62 days. If one group can be launched per month, the

group of 3 carriers will actually deploy before the fifth group of 6 carriers. Thus the time for

carrier deployment is 5 months until the fifth launch plus 126 days until the last carrier is

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deployed on its orbital plane, or about 9 months total. Based on the lower total delta-V and

potentially much lower deployment time, the precession method with multiple launches was the

chosen carrier deployment method.

Once the carriers are place on their respective orbits, the cubesats must be evenly spaced

out along the orbit. To accomplish this, the carrier will enter an elliptical phasing orbit with a

period slight shorter or longer than that of the cubesat orbit, such that the carrier intersects the

circular cubesat orbit with the appropriate time spacing. The carrier will then provide a delta-V

to enter the circular cubesat orbit, release the cubesat, and the return to its elliptical orbit.

Assuming 10 spacecraft per plane, the time spacing will be 1/10 of the cubesat’s period. If the

elliptical orbit is designed to provide this much time spacing in a single orbit, the delta-V

required to transfer from the elliptical carrier orbit to the circular cubesat orbit is unreasonably

high (~233 m/s). To reduce this delta-V to a reasonable value (5-10 m/s), the carrier must orbit

an integer number of times to provide the correct time spacing, requiring a lower eccentricity for

the elliptical orbit. This lower eccentricity results in a lower delta-V to transfer from the elliptical

carrier orbit to the circular cubesat orbit, and a longer deployment time to deploy all cubesats.

The elliptical phasing orbit used can either be inside or outside the circular cubesat orbit.

An outer elliptical orbit requires a slightly lower delta-V for transfer and a slightly longer

deployment time than an inner elliptical orbit. For the case of 10 cubesats per plane at 520 km

altitude, choosing an outer carrier orbit with 50 orbital periods between cubesat deployments

results in an orbit with a 6898 km perigee radius and a 6916 km apogee radius. The circular and

elliptical orbits are illustrated in Figure 1.2.4, where the outer red line represents the elliptical

carrier orbit and the inner blue line represents the circular cubesat orbit.

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Figure 1.2.4: Cubesat deployment with elliptical carrier orbit

The delta-V to transfer between the elliptical carrier orbit and the circular cubesat orbit is

5 m/s. The time between deployments is 3.3 days. Assuming the carrier starts from the cubesat’s

520 km altitude orbit and carries 10 cubesats, it will need to separate the first cubesat and then

leave and re-enter the circular cubesat orbit 9 times. Each of these 9 times requires two

maneuvers, one to exit the circular orbit (go into an elliptical orbit) and one to enter the circular

orbit (from the elliptical orbit). So the total delta-V is 18 times the delta-V for a single orbit-

changing burn (91 m/s), and the total deployment time is 9 times the time between deployments

(30 days). Thus the total deployment time, from the beginning of the first launch to the time

when the last cubesat is deployed on its orbit, is about 10 months.

Since the coverage provided by the constellation depends on its symmetry, every orbit in

the constellation must decay an equal amount. To ensure equal decay on all orbits, all carriers

will wait in their respective orbits until the last carrier is deployed, and then all carriers will

deploy their cubesat simultaneously. The empty carriers will be for communication and data

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handling, but the specific details of the placement of the carriers after deployment needs further

development. Also, the decommission concept for both the carriers and cubesat has not been

developed, but would probably involve a de-orbit burn to put each spacecraft into an elliptical

orbit where the atmospheric drag will cause the orbit to decay further until the spacecraft burns

up in re-entry.

1.3. Launch Vehicles This mission required either one or several launch vehicles to boost the carriers into orbit,

depending on the deployment method used. The requirements for the launch vehicles were low

cost, high launch frequency, sufficient payload capability to LEO, and capability to achieve

desired inclination.

Initially several U.S. and Russian launch vehicles were considered since the deployment

method was not yet fully developed and the destination orbit that the launch vehicle was to

achieve was not yet decided. The most common U.S. launch vehicles available for commercial

use for LEO transport were the Delta II and Delta IV launch vehicles. Their performance

capabilities are summarized in Table 1.3.1 below.

Table 1.3.1: U.S. Launch Vehicles

Vehicle % P/L mass Cost (see text) Launch Site Inclination Frequency

Delta II 12.36 $13.6 M Cape, Vandenburg

27.8-50 63.8-110

11/yr

Delta IV medium

7.39 $9.82 M (est) Cape, Vandenburg

27.8-50 63.8-110

3/yr

Delta IV medium +


27.8-50 63.8-110

1/yr

Delta IV Heavy


27.8-50 63.8-110

1/yr

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The Russian launch vehicles meeting our mission requirements included the Kosmos 3M,

Proton K, Proton M, Rokot, Dniepr 1, Soyuz, and Zenit 2. Their performance capabilities are

summarized in Table 1.3.2 below.

Table 1.3.2: Russian Launch Vehicles

The launch vehicle availability was based on regularly scheduled launches, using the 2009

launch schedule. Using the Delta II and Delta IV launch vehicles would require a minimum of

seven months to launch all 36 carriers in groups of 3 for the clustered deployment method,

whereas using all of the mentioned Russian launch vehicles would require a minimum of three

months. The launch vehicles’ costs were calculated using launch price per pound and adding on

the percentage of total payload used multiplied by the launch cost. The launch costs for the

Delta IV vehicles are estimated costs and are lower than the actual costs because the launch price

per pound was not available.

Vehicle % P/L mass Cost (see text) Launch Site Inclination Frequency

Kosmos 3M 45.4 $14.00 M Plesetsk 50.6 1/yr

Proton K 3.2 $5.45 M Baikonur 51.6, 63.4, 72.7 5/yr

Proton M 3.0 $5.28 M Baikonur, Kazakhstan

51.6, 63.4, 72.7 7/yr

Rokot KM 33.4 $9.14 M Plesetsk, Baikonur

50.6 63, 73, 82, 86.4

6/yr

Dniepr 1 14.1 $4.28 M Baikonur 63, 73, 82, 86.4 2/yr

Soyuz 10.2 $7.23 M Plesetsk, Baikonur

52,65,70 16/yr

Zenit 2 4.7 $3.96 M Baikonur 46.2, 51.4, 63.9, 89.6, 98.9

1/yr

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Based on the current precession-based deployment method, the launch options were

revisited. The three possible launch methods considered were: a single dedicated launch using a

Delta II, multiple dedicated launches using a Falcon 1, or piggy-back rides on Delta II’s using an

ESPA ring. Assuming a cubesat mass of 6 kg, an empty/dry carrier mass of 40 kg, and 50 kg

propellant mass, the total mass of a single carrier would be 150 kg, and the mass of 33 carriers

would be 4950 kg. The cost of the various methods was calculated for comparison, based on

these payload mass estimates. The payload capabilities and costs for each option are summarized

in Table 1.3.3 below. These payload capabilities listed are rough estimates based on LEO around

200 km and a range of launch inclinations available at Cape Canaveral and Vandenburg Air

Force Base.

Table 1.3.3: Launch Vehicles for Precession Deployment

Vehicle Payload Capability

(kg)

# Carriers per Launch

# Launches Required

Cost per Launch

Total Cost

Delta II 2700-6100 33 1 $55 M $55 M

Delta II (ESPA)

1020 6 6 $10,692/kg $53 M

Falcon 1 240-420 2 17 $7.9 $134 M

Falcon 1e 700-1010 6 6 $9.1 $54.6 M

Multiple dedicated launches using Falcon 1e vehicles is currently considered the best option

because of its low cost. Although piggyback rides are lower cost based on the estimated cost per

kg, they would result in longer deployment due to the wait time for available rides.

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2. Propulsion

The propulsion systems required for successful completion of primary mission requirements

are divided into three major sections, based on the sequence of cubesat deployment and

operation. The first stage of deployment consists of the time interval between launch and

insertion of the cubesat carrier into an elliptical deployment orbit discussed in the

constellation section. Propulsion systems reviewed in this section include launch vehicle

selection as well as integrated carrier propulsion. Stage two of deployment consists of

separating the cubesats from the carrier. Stage three consists of station-keeping and orbital

maintenance performed by the cubesats in order to maintain operation for the required

lifetime. A summary of the delta-V requirements and propellant masses required for each

stage of the mission is given in Table 2.1. Delta-V for disposal of the vehicles is not

included due to the fact that orbital decay should lead to automatic disposal, or alternatively a

non-propulsive method such as a tether may be used.

Table 2.1: Mission Delta-V and Propellant Mass Summary

Stage Carrier Cubesat

1: LEO to 520km 171 m/s 0 m/s

2: Plane Change 660 m/s 0 m/s

3: Cubesat

Deployment 82 m/s 0 m/s

4: Cubesat Launch 10 m/s 5 m/s (imparted)

Total 913 m/s 28 m/s

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2.1. Carrier Propulsion The propulsion system required for the cubesat carriers is sized based on the requirement for

location of the cubesats in the constellation as well as launch vehicle availability. The

original cubesat carrier deployment concept involved the use of two different types of carrier

vehicles, the alpha carrier and the beta carrier. Based on the delta-v trade study conducted, it

was determined that the best solution for deployment was a series of 12 launch vehicles, with

3 carriers per vehicle. Of these, one carrier (alpha type) services the launch plane, while the

remaining pair (beta type) executes ten degree right ascension of the ascending node

(RAAN) plane changes at the intersection nodes of the adjacent planes. The propulsion

requirement for the alpha carriers is far less than that for the beta carriers due to the

extremely high delta-v values required for plane change burns. For the alpha carriers, the

chosen propulsion system consisted of a Northrop MRE-15 Hydrazine Monopropellant

thruster capable of delivering a maximum of 86N of force for approximately 400 seconds,

which was calculated to be sufficient for non-plane change requirements. The beta-type

carrier uses an Aerojet R-42 MMH/NTO bi-propellant thruster rated at 890N. These thrusters

are responsible for raising the carriers from LEO to a 500km altitude circular orbit and

relocating into an elliptical orbit for cubesat deployment. In addition to these requirements,

the beta carriers must complete the aforementioned plane change maneuver. See Table 2.2

for a summary of the total delta-v, thrust required, and engine information for each carrier

vehicle.

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Table 2.2: Alpha Beta Carrier System Requirements

α-Carrier Propulsion System:

β-Carrier Propulsion System:

Total ∆V .65 km/s 1.97 km/s

Thrust Required

Alt 72 N Plane Change 185 N

Alt 109 N Plane Change 282 N Right Ascension 804 N

Propellant Hydrazine MMH/NTO

Thrust 86 N at 400 psia 890 N at 425 psia

Engine Mass 1.1 kg 4.53 kg

Propellant Mass

34 kg 127 kg

Total Mass 169 kg 267 kg

Size 31.8 cm (l) x 11.9 cm (w) 73.66 cm (l) x 38.96 cm (w)

Due to the large amount of delta-v and propellant mass required for cubesat deployment for

the Beta type carrier vehicles, the deployment method was revised to use orbital precession

for cubesat deployment. For this revised method, there would only be one type of carrier

vehicle which would position itself in the correct phasing orbit and then the carrier would be

used to accelerate to the required cubesat orbit for deployment of the satellite. The

maneuvers required to place the carrier vehicle in the correct orbit include: boosting the

carrier up to an altitude of 520 km from a starting drop-off altitude of 200 km using a circular

orbit, performing a 5 degree plane change from the launch plane, and finally placing the

carrier vehicle in its desired elliptical phasing orbit. Once the carrier vehicle is in the correct

orbit for cubesat deployment, the carrier vehicle will launch the first cubesat The actual

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launch method will be a compliant low-acceleration spring similar to that used on the P-Pod

deployment system. Thereafter, the carrier vehicle will be used to impart each cubesat with

the additional required ∆V of 5.076 m/s. Then the carrier will be decelerated by the same

amount for each cubesat deployment to place it back in its original orbit, for a total of 9

cubesat deployments. See Table 2.3 for the details of these maneuvers.

Table 2.3: Carrier Orbital Maneuvers for Hydrazine monopropellant engine, Isp = 230s

Maneuver ∆V Propellant Mass(kg)

Time (sec)

Plane change (5 deg) 663 26.3 236

200 km-->520 km 171 9.4 85

To perform the cubesat orbit positioning maneuvers, both bipropellant and monopropellant

main engine choices were considered. The bipropellant engine was favored at first for the

high amount of specific impulse available with a bipropellant engine (280 sec) as compared

to a specific impulse of 230 sec for a monopropellant engine. With the higher specific

impulse of a bipropellant engine, the total required fuel and oxidizer mass was calculated to

be 35 kg. A large disadvantage of bipropellant engines is the necessity to use two separate

tanks, one for oxidizer, and one for fuel, along with the added complexity of the propulsion

system as a result. For comparison, the amount of propellant required including an 5%

margin of safety/de-commissioning allocation for a monopropellant engine was calculated to

be 47.3 kg for the cubesat orbit positioning maneuvers (including carrier orbital positioning

and cubesat deployment). The propellant mass was calculated using the rocket equation with

an average mass for each given segment, Equation B.2.1 in the Appendix B.2.

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Comparing the two values of propellant mass, the mass of the monopropellant engine was

determined to not be significantly more than the bipropellant system mass. The biggest

advantage of using a monopropellant engine over a bipropellant engine is a much lower level

of complexity with only one propellant tank. Since the amount of fuel mass required for a

monopropellant engine was deemed to be reasonable when compared to the extra dry mass

and failure potential of a bipropellant system, a monopropellant engine was chosen. The

proposed monopropellant engine to be used is the Aerojet MR-107S. See Figure B.4.3 in the

appendix for technical information on this engine.

2.1.1. Carrier Cubesat Deployment Maneuvers

To perform the required acceleration and deceleration maneuvers for cubesat deployment, the

Reaction Control System (RCS) thrusters will be used. The RCS thrusters will also be used

to correct for thrust misalignment of the main engine with the center of mass of the carrier.

The RCS thrusters were sized based on the thrust from the 250 N main engine assuming

offset angle of 1 degree from the center of mass of the carrier vehicle, resulting in a required

thrust of 2.5 N. In order to perform reaction and attitude control, two 4 N monopropellant

engines will be used. The monopropellant engines will be placed as far from the center of

mass as possible to provide the greatest available torque (See Figure 2.1).

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Figure 2.1. Reaction control thruster arrangement.

A cluster of four RCS thrusters will be used in each group. To perform the cubesat

acceleration/deceleration deployment maneuvers, the required thrust will be divided between

the four thrusters. The time to perform each maneuver was calculated based on an estimated

thrust available, average system mass at that time, and delta v requirement. Table 2.4 shows

the delta v, and estimated time required to complete each maneuver.

Table 2.4. Cubesat Deployment Maneuvers.

Cubesat Deployments

∆V (m/s) Propellant Mass (kg) Elapsed Time (sec)

1st 0 0 0 2nd 10.1 0.5 50.5

3rd 10.1 0.4 47.8

4th 10.1 0.4 45.2

5th 10.1 0.4 42.5

6th 10.1 0.4 39.9

7th 10.1 0.3 37.2

8th 10.1 0.3 34.6

9th 10.1 0.3 32.0

10th 10.1 0.3 32.0

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2.1.2. Tankage

In order to cause the least disturbance possible in the center of mass during propellant firing,

the total amount of propellant required for the whole mission will be divided evenly into 4

tanks. The propellant tanks were placed at the back of the carrier vehicle in order to have the

propellant tanks close to the main engine and to avoid interference with the placement of

other components on the carrier. To simplify the system tankage, blowdown tanks were

considered. A spherical tank shape was chosen as this is the most efficient structural shape,

which results in the lowest tank mass. A blowdown tank system was chosen for the

simplicity of a blowdown system design. A blowdown system only requires one propellant

tank, with the pressurant enclosed in a small volume above the fuel, which is separated by a

diaphragm. As the propellant is used, the pressure in the fuel tank decreases, resulting in

decreased thrust. To determine the blowdown ratio for the tanks, the operating pressure

range for the main engine (475-150 psi) was considered. The blowndown ratio was

calculated using Equation B.2.2 (see Appendix B.2).

The total volume of the propellant tanks was calculated using the density of hydrazine (1010

kg/m3) and the total propellant mass was calculated for all maneuvers, resulting in a total

propellant mass of 47.3 kg, which includes 5% for contingency propellant for additional

attitude control. The total volume required for fuel in the propellant tanks was calculated to

be 0.047 m3. Using Equation B.2.3 (see Appendix B.2), the pressurant volume was

calculated to be 0.014 m3. The chosen tank pressurant to use was helium. Helium was

chosen for its low molecular weight, to add the least amount of weight possible to the

propellant tank. To reduce the tank size, a pressure regulated tank system was chosen. This

system will be made up of four propellant tanks, for a total volume of 0.044 m3 and four

25

pressurant tanks with a total volume of 0.006 m3. Using a pressure regulated system results

in a constant thrust level for the duration of the engine burn.

Using this system, the carrier engines will be operated at a constant inlet pressure of 350 psi.

The material most commonly used for hydrazine propellant tanks is a Titanium alloy (Ti-

6Al-4V) because it is strong, lightweight, and nonreactive.

2.1.3. Carrier Propulsion System Architecture

The following is the proposed propulsion system architecture, shown in Figure 2.2.

Figure 2.2. Carrier Propulsion System Architecture.

This diagram includes propellant service valves for filling and draining the propellant, latch

valves to control the propellant flow to the different engines, propellant feed lines, and the

engines. Each engine also includes a flow regulating solenoid valve. Possible valve sources

include the Moog latch and service valves. To feed propellant to the main engine, the two

26

propellant latch valves would be used. At any one time, up to 4 thrusters may be firing, or 2

RCS thrusters will be fired at the same time as the main engine to compensate for off-axis

thrust vectoring. See Figures B.4.5 and B.4.6 for possible propellant feed control valves.

2.2. Cubesat Deployment Method Trade Study

The separation of the cubesats from the carrier vehicle falls under the mission requirement

for positioning within the constellation. Secondary requirements derived from the assumption of

a desired one month satellite propagation time interval provided a required delta-v for the system

of 5 m/s per cubesat, or 100N-S of impulse per launch. Design candidates for cubesat

deployment systems included a spring-based system, a magnetic “rail-gun” or linear actuator

based system, a pneumatic system utilizing a compressed fluid or cold gas to accelerate the

satellite, and a small hobby solid rocket thruster. Of these, the first was discarded from

consideration due to the extremely high spring constant (k = 4000N/m) required to achieve the

required delta-v, as well as the highly nonlinear nature of the applied force and acceleration load

(20g peak) on the cubesat. The spring constant and acceleration values given above result from

the application of the basic particle kinematic equations and Hooke's spring law, assuming a

10kg cubesat to which is imparted a 5m/s Delta-V relative to the carrier over a 10cm distance

(spring length). The solid-booster option was also ruled impractical due to possible damage to

cubesat and carrier resulting from ignition of the booster with the cubesat still inside the carrier.

Additionally, the solid booster issue is complicated by the low duration of the thrust and the

corresponding difficulty in correctly pointing the cubesat in the desired direction of travel using

27

only the onboard CMGs, which may not be able to adjust attitude quickly enough to accurately

position the satellite. Magnetic actuation was favored during the initial phases of the design;

however, the high power requirements (in the kW range) and the prohibitive tube length to get

appropriate delta-V values meant that it too had to be discarded. Currently, the planned

deployment method is to accelerate the carrier into a circular cubesat orbit and to separate the

satellite using a very low delta-V provided by a small spring. Once separation has occurred, the

carrier would then return to its previous elliptical orbit. This method requires an additional 10m/s

of delta-V budget be allocated to the carrier for each cubesat that must be launched. However, it

has none of the logistical and practical concerns of the alternative methods outlined above, and

allows for the mass of the cubesat to be minimized.

2.3. Cubesat Onboard Propulsion

Cubesat propulsion system selection for this mission is based on the top level design

constraints explained in the mission profile. More specifically, the need for a propulsion system

is due to the satellite lifetime requirement of one year, as well as maintaining position in the

constellation and tumble recovery. Due to performance constraints on the imaging system of the

satellites stemming from the top-level image fidelity requirement, an altitude range from 400 to

500 kilometers is was initially considered, and it is over this range that the following cubesat

propulsion trade studies were conducted. For satellites orbiting in this altitude band, the major

detriment to orbital lifetime is atmospheric drag. The drag on a satellite is proportional to both its

drag coefficient, which is determined by the shape of the satellite, and its frontal area (defined

for this study as the area projected onto a plane normal to the orbital velocity vector). Integration

of the equations governing atmospheric drag result in a projection of orbital lifetime of satellites

based on insertion altitude, drag coefficient, and frontal area. The functional requirements

28

associated with the predicted "worst case" solar panel frontal area of 0.15m2 and Drag

Coefficient of 2.2 are 7.52μN of Drag Force at 500km altitude and 29.68μN at 400km altitude,

based on assumed densities of 7.55*10 -12 kg/m 3 at 400 km and 1.80*10 -12 kg/m 3 at 500km .

These density values were taken from Space Mission Analysis and Design, and correspond to

conditions during a solar maximum, which will be appropriate for a launch window around 2012.

These forces acting over the projected satellite lifetime of one year result in a drag delta-v of

23.7 m/s at 500km and 92.1 m/s at 400km. Please see Appendix B for supporting calculations.

Assuming a desired lifetime of 365-400 days, possible insertion altitudes and frontal areas cover

a range of values, which are shown in Figure 2.3.1 below.

Figure 2.3.1: Optimal 1-Year Lifetime Range (Red)

Due to power generation requirements, it is likely that the satellite’s average frontal area

will be 0.15 m2 or less. This value corresponds to the maximum point of the figure along the Y-

axis. With this frontal area range, an assumed mass of 10kg, and an assumed attitude averaged

drag coefficient of 2.2, the ballistic coefficient of the satellite ranges from 60.5 to approximately

200. The graph above represents the worst-case scenario of the lowest possible ballistic

coefficient, corresponding to a frontal area of .15 m2 and using the density information taken

from SMAD for the case of a solar maxima. The chosen design altitude is 520km, which is

outside of the range considered in the initial study, indicating that propulsion would not be

29

required to maintain altitude for a one-year period. The original proposed solution to counteract

the orbit degradation was to use a nano-satellite scale low Delta-V propulsion method to

maintain altitude for at least one year. Secondary advantages of such a system include orbit

maintenance and station keeping relative to the other satellites in the constellation. The proposed

propulsion system will be unidirectional and balanced with respect to the center of gravity of the

cubesat. Directional control of the thrust will be accomplished by slewing the satellite using the

included inertial attitude control system. However, following calculations performed to

determine the rate of altitude loss, the propulsion system will be now used exclusively for

station-keeping purposes, as the total altitude loss at the chosen 520km altitude over the course

of a yearlong mission is minimal. The loss is approximately 600m, and is constant for the entire

constellation, so that the arrangement of the cubesats in the constellation will not change with

time from this effect. This altitude drop will not compromise the mission, and so will be

neglected for the propulsion system. Please see Figure 2.3.2 for exact altitude change over the

selected range. Additionally, contributions from solar radiation pressure and the non-spherical

shape of the earth were calculated to total less than 2 m/s of drag based delta-V per year , and

were therefore ignored in the preceding analysis.

30

Figure 2.3.2: Altitude Change per Year vs. Initial Altitude

In order to determine the best type of propulsion system for this mission, several

candidate systems were evaluated based on the following aspects: cost, thruster mass, propellant

type, tanks, supporting hardware, thrust required, power required, and commercial off the shelf

(COTS) availability. After narrowing the possibilities due to mass and power constraints, the

most attractive options from a preliminary perspective are the micro-PPT, monopropellant

hydrazine, and cold-gas thrusters. A summary of the pertinent characteristics of each thruster

candidate is given below in Table 2.3.1.

31

Table 2.3.1: Thruster Candidate Comparison

Cold Gas Bi-propellant Mono-propellant Micro-PPT

Specific Impulse (s)

65-70 280-465 220-235 300

Propellant N2 N2H4, H2, O2, Iridium Catalyzed N2H4

Teflon

Power Required 8 W 17 W 15W 2.25 W

Thrust Range 0.02-1 N 4-500N 0.5-400N 20μN

Thruster Mass 0.12-0.2kg 0.35kg 0.5kg 0.1kg

Examples DASA CGT-1 EADS Astrium S10-13

NorthropMRE-01 AFRL Prototype

Cold-gas is on the edge of what is possible given the packaging constraints of a 2 cubesat

unit by 3 cubesat unit chassis. However, a cold-gas-resisto-jet hybrid using liquid propulsion

suffers from extremely low specific impulse. This leads to an increase in propellant mass that

compromises the mass budget of the spacecraft.

Alternatively, a monopropellant hydrazine arrangement was the second design candidate

considered for this application. This candidate requires far less propellant mass than for the cold

gas option; however the size of the thruster itself becomes an issue. The Northrop Grumman

MRE -01 monopropellant thruster may be taken as a representative case. This thruster features a

nominal thrust rating of some 0.8 N, or 800,000μN with a specific impulse of 216s.

Disadvantages of a monopropellant system include the weight of valves and other supporting

hardware, and the relatively high power requirements. The MRE-01 requires 15W of power to

operate which is quite high given the power constraints dictated by the size of the satellite.

32

The third likely option for propulsion application to the cubesat problem is a Micro

Pulsed Plasma Thruster, or Micro-PPT. Currently under development at the Air Force Research

Laboratory, these devices are tiny, self-contained versions of the conventional PPTs that have

been in service since the 1960s. The mass of the AFRL prototype Micro-PPT is 250g, which is

considerably less than the 0.5kg mass of the MRE-01. Supporting system mass for the Micro-

PPT consists only of the capacitor and control systems, of which the masses are included in the

table above. The valves/plumbing/control systems needed for any liquid propulsion option are

therefore added mass to the values given in the Table 2.3.1. Thrust levels are quite low for

micro-PPTs, in the range of 20μN. The Micro-PPT system in use at AFRL uses a nominal power

of 2.25W during cycling, during which the capacitor is fed voltage from the spacecraft bus and

discharges at a rate of 2Hz. Concerns with this design include pricing given the fact that a

proprietary system must be developed and manufactured specifically for this mission. Initial

development costs have been estimated at approximately $10,000/thruster. This is based on an

estimate of the cost of developing the system divided by the proposed number of units to be

manufactured. However, this does not include costs for lifecycle testing, optimization, etc. Costs

may be cut due to the research done by the AFRL from 1999-present on the application of this

technology. This technology is currently at a Technology Readiness Level (TRL) of 6 based on

its current status of prototype systems being tested in a space environment. Additionally,

commercial off the shelf options comparable to the AFRL system are currently available from

Busek, Inc.

Due to the design advantages and disadvantages outlined above, a proprietary system

consisting of two or four Micro-PPTs spaced symmetrically about the satellite's center of gravity

was initially proposed herein as a solution to the station keeping/orbital degradation problem.

33

The reason for using multiple thrusters was that redundancy may be attained at a low mass cost,

since the thruster bodies themselves are so light. Additionally, the required duty cycle of any one

PPT decreases linearly with additional thrusters, so that each thruster would need to be fired less

frequently and would consequently have a longer lifetime. This type of system would be easy to

package due to the nature of these devices, robust (assuming multiple thrusters), and capable of

maintaining the proper orbit for the proscribed lifetime. The packaging of four 7 cm long

thrusters is considerably easier than a single 27cm thruster. The volume dimensions given above

were calculated from the known propellant mass based on an approximate specific impulse of

some 500 seconds and the Tsiolkovsky Rocket Equation, as well as the known density of the

Teflon fuel, 2.2 g/cm 3 . Additionally, if the thrusters are of the self-triggering variety, only a

single Power Processing Unit and Capacitor must be used, further reducing the volume occupied

by the system. This recommendation is of a provisional nature due to the lack of testing and

unknowns regarding cost of development of such a system. If, after a detailed cost analysis is

performed, such a development cycle is deemed to be prohibitively expensive, the fall-back

candidate is a small hydrazine monopropellant thruster such as the MRE-01.

Finally, station-keeping is necessary in order to keep the cubesats in their desired

positions in the constellation. The amount of propellant required for station keeping is

determined by the degree of accuracy of the GPS positioning system. This system has a position

accuracy of 5 meters and a velocity accuracy of 0.1 m/s. Calculation of position correction

behavior is based on a worst-case estimate of drift of the satellite's position using orbit

propagation code written in Matlab. Supporting code may be found in the Appendix C-3. From

this code, the amount of time between correction burns and the impulse needed for each burn

may be determined. The propellant needed for this function is relatively low due to the lack of

34

active forces on the satellite. Assuming a correction burn occurring when a particular satellite

deviates more than 100 meters from its designated position, the amount of propellant required for

one year of operation is 23.71 grams.

Figure 2.3.3: Deviation in Orbital Position vs. Time for GPS Uncertainty 5m, 0.1 m/s

35

3. Image Acquisition

3.1 Lens and Optics System Choosing a lens and optics system for this mission is based on many controlling factors.

The requirements for this mission state that the images acquired should have at least a 3 meter

resolution at the nadir pointing, and be at least 5 km on each side. Furthermore, the satellite

altitude, mass, and volume must be taken into account. Because of the specific resolution

constraints, satellite altitude was the primary driver in this design.

The required diameter of the telescope will play a major role in determining the structural

design of the satellites. A point object imaged by an optical system with aperture diameter D,

produces an image of finite angular spread , given ideally by:

where λ is the wavelength of light. For this application, λ is chosen to be in the middle of the

optical spectrum (λ = 530 nm). To be able to resolve an object that has a finite view angle

from the optical system, we must have . For this mission, the resolution is 3 m at a

distance of , so depending on altitude.

To get this resolution, the optics diameter must be:

This relation is plotted on Figure 3.1.1. For the chosen altitude of 520 km, a 9 cm diameter

telescope was chosen because of the ready availability of this size off-the-shelf.

36

Figure 3.1.1: Diameter of Telescope

The focal length of the optical system must be chosen to give the required scale on the

image recording device, or CCD in our case. As explained in the Image Acquisition section, the

image spot produced for a 3 m object should cover at least one pixel. The spot diameter

produced by an object of dimension s at range h is: (for ray optics). With the assumption

that the spot covers 1.5 pixels ( , where a = pixel size), we obtain the requirement on f:

The initial 50 Megapixel CCD that was chosen had a pixel size of . A plot of the

required focal length vs. altitude h is shown in Figure 3.1.2. For the design altitude

, we must have . Since this exceeds the 1.2-1.3 m focal length of

37

readily available Maksutov telescopes, a magnifying element would have to be added behind the

primary mirror in order to extend the effective focal length.

Figure 3.1.2: Effective Focal Length (50 Megapixel Camera)

Upon further investigation, it was found that a 10 Megapixel CCD with a 4.75 pixel size

would be sufficient to meet these requirements. Figure 3 shows the effective focal length in

relation to satellite altitude considering the 10 Megapixel CCD. Note in figure 3.1.3, the spot size

is assumed to be 7.125 µm, or 1.5 times the pixel size.

38

Figure 3.1.3: Effective Focal Length (10 Megapixel Camera)

From this graph, it can be seen that the effective focal length at an altitude of 520 km is

1.23 m. This effective focal length is much closer to the effective focal lengths provided by off

the shelf telescopes which meet our volume constraints.

The CubeSat can be anywhere from 10 - 30 cm long, but the effective focal length is

much longer. In order to reduce this length, a two- or three-mirror telescope can be used. The

three-mirror designs (Three-Mirror Anastigmatic (TMA)), are more compact lengthwise and

account for coma and spherical aberration, as well as stigmatic and chromatic aberration.

However, two-mirror systems require less mass, both in terms of mirror mass and structural

support mass.1 They also have fewer parts, cost less, and are easier to calibrate mirror alignment.

For simplicity, the two-mirror system is preferred.

1 T. H. Zurbuchen, “A Low-Cost Earth Imaging System,” IEEE Proceedings, 2007.

39

The following two-mirror telescope designs were under consideration: Ritchey-Chretien,

Maksutov-Cassegrain, and Schmidt-Cassegrain. The Ritchey-Chretien design allows for the

correction of both coma and spherical aberrations; however, its two hyperbolic mirrors make it

an expensive commodity. Contrarily, the Maksutov-Cassegrain telescope incorporates spherical

mirrors which are much less expensive, and has a corrector plate which can correct spherical

aberration. A Maksutov-Cassegrain telescope would make an excellent choice for this optics

system because it is an inexpensive, off-the-shelf product. Figure 3.1.4 below is an example of a

two-mirror Maksutov-Cassegrain telescope.

Figure 3.1.4. Two-Mirror Maksutov Telescope

Mission requirements state that the optical system shall have less than or equal to a 3m

resolution (nadir pointing). Figure 3.1.5 illustrates the relationship between the required

resolution of the telescope (3m nadir pointing) and the altitude of the satellite.

40

Figure 3.1.5. Accuracy of the Telescope

It can be seen that the required resolution of the telescope at an altitude of 520 km is

approximately 1.19 arc-sec. Questar Corporation makes a 3.5” Lightweight Maksutov-

Cassegrain Catadioptric Telescope which fits all of these criteria. It is a two mirror optics

system, with a corrector lens diameter of 8.89 cm (< 10 cm). Furthermore, the optical resolution

of this telescope is 1 arc second (claimed by Questar Corp.). According to figure 3.1.5, this

optical resolution will fit mission requirements up to an altitude of 619 km. The estimated costs

for 330 units are approximately $3.16 Million.

Orion produces a Maksutov-Cassegrain telescope which has similar specifications. The

Orion scope has a 9 cm optical diameter and a focal length of 1.25 m. However, its computed

resolving optical power is only 1.29 arc-sec (which results in a ground resolution of

41

approximately 3.25 m @ 520km altitude) and a mass of 1.68 kg. The advantage of Orion’s

telescope is its cost. The optical tubes can be ordered separately from unnecessary hardware

such as eyepiece lenses, tripods, and star mapping for a price of $230. The estimated costs for

330 units are approximately $75,900. Due to the extreme cost reduction compared to the

Questar telescope, it is recommended that the resolution requirement be relaxed.

One aspect which should be considered in preserving image quality is the thermal

expansion of materials. Depending on the working range of the telescopes, thermal expansion

could greatly hinder the image quality due to mirror misalignment and defocusing. Choosing

specific materials for the lenses of the telescopes would be unreasonable because the

manufacturing of lenses is the most difficult and expensive part of constructing a telescope.

However, choosing specific materials for the optical tube is not out of the question. Most optical

tubes for telescopes are aluminum. The density of aluminum is approximately 2700 kg/m³ while

the coefficient of thermal expansion is 23 µm/m-K. Assuming a working range of approximately

70 K, this would result in an expansion of approximately .48 mm in the axial direction, and .46

mm in the hoop direction. This increases the telescope diameter by .14 mm, which can interfere

with both image quality and structural design. On the other hand, for approximately $300 per

unit, the optical tubes can be upgraded to carbon fiber tubes. Carbon fiber has a density range of

approximately 1700-2100 kg/m³ and a coefficient of thermal expansion range of approximately

5-15 µm/m-K. This is beneficial because it results in less expansion in the telescope throughout

the working range, and a lighter construction.

42

3.2 Image Capture The imaging recorder is a critical part for the success of the mission of the spacecraft.

The image recorder will need to be precise in capturing the photo so that the resolution meets the

necessary requirements along with making sure that there is no blur from the speed of the

spacecraft.

For the optics, one of the main factors is the resolution of the photograph. The Request

For Proposal (RFP) states that the photograph must have at least a 3 meter (m) resolution. Thus,

with a digital image recorder such as a Charge Coupled Device (CCD), one pixel will need to be

able to capture an object that is 3m in length and width. For a clear picture, we want to be able to

distinguish a 3m object from that of an object that is smaller. Thus, a 3m object should be able to

be imaged by multiple pixels.

In order to determine the size of the CCD that is needed, the size of the photograph area

will also need to be taken into account. It is stated in the RFP that the photograph must have an

area of 5km x 5km. Using the size of the photograph area and the resolution size, the following

equation is used to determine the amount of pixels needed for the CCD,

pix

picture lengthNresolution

= = 5000 18183.0

m pixelsm

= Eq. 3.2.1

This equation gives the number of pixels along one side of the CCD for a CCD that takes

pictures in black and white. In order to capture a color image (which is requested in the RFP), it

was believed that this number would need to be tripled (red, green, and blue) to allow for the

colored pixels. Thus, it was thought that at least a 30 megapixel CCD was needed for the mission

and our initial choice in a CCD was a 50 megapixel produced by Kodak.

43

In actuality, the calculated number of pixels for the black and white picture was closer to

what would be needed to capture the color picture. After investigating algorithms used to convert

the picture from a mosaic of red, green, and blue, it was found that smaller CCD would be

adequate for our mission.

By looking at a Bayer Demosaicing algorithm

(http://www.cambridgeincolour.com/tutorials.htm), it could be seen for a simple 4 by 4 pixel

arrangement of red, green, and blue pixels, a 3 by 3 picture could be made. This is demonstrated

in the figure below.

The algorithm takes the average of 2 by 2 mosaic pixels to generate one real color pixel. The

picture above shows how a 4 by 4 mosaic of pixels can be split up into 9 total 2 by 2 sections to

yield a 3 by 3 real picture from the 4 by 4 mosaic pixels. This type of algorithm is just one of

many that are in use today but gives a good basis as to the number of pixels that will be needed

for our mission. From this it can be said that at least 1820 pixels per side is required for our CCD

based on

44

1pic pixN N= − Eq. 3.2.2

This is only one of the parameters that affect the choice of the CCD.

Another parameter for choosing the size of the CCD is the contraint that is placed on the

size of the pixel from the telescope. Because of the dimensionsal constraints based on our

telescope, we need a small focal length in order to be able to see the 5 km by 5 km area as stated

in the requirements. The focal length for the Maksutov telescopes is 1.2 to 1.3 meters and it

would be best if the CCD was able to match these numbers so that no extra magnification

devices would be needed. Using the following calculation

*d f ϕ= Eq. 3.2.3

where d is the size of the image, f is the focal length, and φ is the angle subtended by the object

(3m), it was found that for a focal length of 1.3 and a phi of 5.8*10-6 (3m/520km), the spot size

on the CCD would be 7.54 micrometers. In order to obtain a clear picture of the 3 meter object,

we would like the spot size to be seen by more than one pixel. Thus if the spot size were to cover

1.5 pixels, the actual size of the pixel would be about 5 micrometers. With the constraint of the

pixel size being less than 5 micrometers and the number of pixels being at least 1820 pixels, we

were able to better choose our CCD.

Based on the constraints listed above, a reasonable choice for the CCD is the Kodak KAI-

10100 (Figure 3.2.1 below). This CCD is a 10 megapixel CCD that 2840 by 3760 in pixel count.

This CCD also has a pixel size of 4.75 micrometers. Although the cost of this CCD has not been

obtained yet, it should be less than $3,000 per unit (the price of the 50 megapixel CCD). The

CCD has a readout rate of 30 MHz. The noise for the CCD is 10 electrons, which is less than that

45

of the 50 megapixel CCD allowing the shutter times calculated to also be valid for this CCD.

Overall, this CCD meets the requirements of the mission as stated before.

Along with the CCD, a series of components will work to transfer the output of the CCD

to a digital picture. The first of these components is an analog to digital converter. A 12-bit A/D

converter will be needed based on the dynamic range of the CCD. Analog Devices part AD9949

(website http://www.analog.com/en/index.html) will fit our requirements. The converter has a

low power dissipation, 320 mW, and should fit easily into our power system. The price of one

A/D converter is $7.20.

After the picture has been digitalized, the picture must be transferred from a mosaic to a

picture of real colors. This is done in the manner as explained above. Along with the mosaic

algorithm, the image processor can also implement noise reduction and image compression.

Thus, a separate processor will be needed to do the demosaicing and image reduction. The Texas

Instruments TMS320C6457 is satisfactory for this purpose. This processor is rated to 8,000

MIPS, which was a value given to me by Micheal Bernhardt (computer systems) as an adequate

Figure 3.2.1: Picture of Kodak KAI‐10100 CCD

46

number of iterations needed for processing the image. The processor also uses a low voltage, 3V,

and has a wide temperature operating range (-40°C to 100°C).

The data size created by the 10 megapixel camera can reach 100 megabits. To be on the

safe side, we would like to be able to save about 8 pictures at a time on the satellites and thus a 4

gigabyte storage capacity will be used on the satellites. Solid state storage is preferred, so that

there is no reaction moment when a hard drive spins up. A four gigabyte flash drive is

satisfactory.

3.3 Image Optimization

In order to determine the shutter time needed to capture the picture, the amount of visible light to

reach the CCD needed to be found. The following equations were used to calculate the intensity

of light on the CCD. The irradiation of earth by the sun on the visible spectrum is:

2~ .4* 400vis sunwI I

m= Eq. 3.3.1

because only 40% of solar emission is visible light. The intensity on the CCD is:

2

2

* *cos( )cos ( ) **

visCCD lens

II Af

ρ θ απ

= Eq. 3.3.2

where θ is the angle of the, α is the angle that the spacecraft is taking the picture at (seen in

Figure 3.3.1), ρ is the Earth’s albedo and f is the focal length of the telescope. This equation can

be simplified to,

47

2

* * **

vis lensCCD

I AI bf

ρπ

= Eq. 3.3.3

where

2cos( ) cos ( )b θ α=

The time for saturation of the pixels of the CCD can be calculated as the number of electrons at

saturation over the rate of generation of electrons,

. 6

25,000 .004 sec6.5*10 *

sate

sat

e

Ntb bN

= = = Eq. 3.3.4

The minimum time that the shutter needs to be open was found by taking the minimum number

of electrons needed to exceed the noise over the electron generation rate,

minmin . 6

500 .0005 sec6.5*10 *

e

e

Ntb bN

= = = Eq. 3.3.5

Using the equations above and setting the angle that the spacecraft is taking the picture at

to be 45° and the angle of the spacecraft off the equator to be 60°, the minimum time for the

shutter to be open was found to be .002 seconds. Using the same values, the saturation shutter

time was found to be .015 seconds. The CCD uses its own electronic shutter and the controls for

Figure 3.3.1: Light Intensity on Space Craft

α

48

this will need to be calibrated to fit within this time window. Work will need to be done to

calculate the rotation rate for the satellite so that the ground track does not hamper the resolution

of the photograph.

A focusing mechanism will need to be made for the camera system. A possibility for

moving the CCD very small distances is piezoelectric crystals.

Below (Figure 3.3.2) is a picture that shows a possible lay-up for the telescope and CCD

integration.

Figure 3.3.2: Solidworks Lay‐Up of Telescope and CCD Intergration

49

4. Navigation/Control Systems

4.1. Navigation System

The purpose of the navigation system is to determine the position and velocity of each cubesat in

space and propagate its orbit. Position information relative to the Earth is necessary to determine

which cubesat is closest to the target location commanded for imaging. Orbit propagation is

necessary to predict the future location of the cubesat in its orbit for station-keeping purposes.

Navigation System The suggested navigation system to fulfill these requirements is the Global Positioning System

(GPS). Another option for the navigation system that was explored, but eventually dismissed,

was the use of Ground Dish Antennas. It was decided that this system should not be used for this

mission because it is more expensive, it requires a lot more antennas to match GPS coverage, is

more prone to orbit error, and is more labor intensive for scheduling, collecting, and transferring

data. GPS was chosen for this particular mission because it is the most accurate navigation

device that functions within LEO, which is where the cubesat will be operating. A GPS receiver

is a high-accuracy navigation device that obtains amplified signals from a GPS antenna (which

obtains signals from GPS satellites) and outputs data in coordinate format. The solution of the

GPS receiver includes the cubesat’s predicted position above the earth’s surface, velocity vector,

time, and date. The receiver obtains the GPS almanac (GPS satellites’ positions, velocity

vectors, time, and date) and GPS satellites’ ephemerides (highly accurate orbital parameters)

from the antenna. From the almanac, the receiver can produce a rough estimation of the

cubesat’s position. From the almanac and ephemeredes it can produce a very accurate

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estimation. The receiver then calculates the cubesat’s present orbit and predicts its future

position in that orbit.

GPS Hardware The hardware chosen for the GPS receiver was the Cornell Cougar GPS receiver, shown in

Figure 4.1.1.

Figure 4.1.1: Cornell Cougar GPS Receiver

Other receivers that were considered for this mission were the GPS Navigator Receiver and the

SpaceNav GPS Receiver, but were found to be too large to fit within the cubesat. The Cornell

Cougar GPS Receiver operates with a 5 volt DC power supply at 300 mA and uses between 1.5

to 2 Watts of power. It weighs 39 grams and is 9.525cm by 5cm by 1.7cm. Its operational

temperature range is -30ºC to 70ºC. Because it was built by a university, it does not have a listed

price. However, another source indicated that a GPS receiver for high accuracy space missions

would cost around $10k. This GPS receiver has an accuracy of 5 meters, which is adequate for

this mission. Though this GPS receiver has not been space tested, simulation tests have been

performed. The simulation tests were run at altitudes from 300 km to 600 km at 7 km/s.

51

A GPS antenna is needed to receive signals from the GPS satellites, amplify those signals, and

send them to the GPS receiver. An option for the GPS antenna is the Synergy Systems SMK-4

GPS antenna, shown in Figure 4.1.2.

Figure 4.1.2: SMK-4 GPS Antenna

Another antenna that was looked into was the Toko DAX Dielectric Patch Antenna, but was

replaced because it did not provide enough signal amplification for the receiver. The Synergy

Systems antenna is 34mm by 25.3mm by 10.9mm and weighs 30 grams. The bandwidth is 2

MHz and the gain is 24dB. It operates at 11mA with a 5 volt power supply. Its operational

temperature range is -30ºC to 85ºC and costs only $25. The combined GPS receiver, antenna,

wiring, and screws, weigh about 80 grams.

Data Output The navigation solution of this GPS system includes the cubesat’s position in earth-centered

earth-fixed (ECEF) coordinates, velocity vector, GPS time, GPS week (date), and dilution of

precision. A diagram of ECEF coordinates is shown in Figure 4.1.3.

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Figure 4.1.3: ECEF coordinates

After the receiver is given the GPS satellites’ positions, velocity vectors, time, week, it sends this

information to the propagator, which is built into the receiver. The propagator calculates a

Keplerian orbit to predict where the cubesat will be at a given future time. It then sends the

cubesat’s predicted position, velocity vector, time, and week back to the receiver to be sent to

other subsystems. The accuracy of the velocity vector is 0.1 m/s. A block diagram of the data

output is shown in Figure 4.1.4.

Figure 4.1.4: Receiver Block Diagram

53

The first run is called a “cold start,” where the receiver takes about 10 minutes to download the

GPS almanac (GPS satellite coordinates, velocity vector, time, week) and roughly estimates the

cubesat’s position. The GPS almanac data is good for several months before it must be

discarded and re-downloaded by the receiver. A “warm start” is performed every 4-6 hours

where the receiver takes about 3 minutes do download the GPS satellites’ ephemeredes and

produce a highly accurate estimation of the cubesat’s position The receiver uses the almanac and

ephemeredes to predict future cubesat positions using Doppler shifts. It listens for Doppler

shifted signals (shifts occurring in the electromagnetic spectrum) about the L1 (1575.42 MHz)

frequency. It outputs a navigation solution at a rate of 0.1 Hz. The output of the receiver is an

ASCII string where each character requires 8 bits. The receiver output rate is 19200 bps. As

functionality measure, the receiver outputs data every 10 seconds, even when it is not being used

or the navigation solution is invalid.

4.2 Attitude Determination and Control

The attitude determination and control system (ADCS) is responsible for determining and

controlling the rotational parameters of the spacecraft. It is essential for cubesat imaging,

communications, and other purposes. The ADCS must employ sensors capable of ensuring ±200

meters pointing accuracy for the ground target, corresponding to an accuracy of 0.022 degrees,

and actuators that can slew the spacecraft to the desired angle at a rate of 540 deg/min.

Since high accuracy is a requirement of this mission, three-axis control mode is

necessary. Active control with system feedback will make corrections via the control system as

frequently as required to maintain attitude and position accuracy. However, passive control can

provide coarse control. There should be a backup detection system for redundancy. This backup

54

will be independent of the primary system, hence providing a full coverage of attitude detection

regardless of the position of the satellite. The backup system will also rely on analog signals and

preferably introduce minimal additional mass and volume.

Sensors A summary of typical devices, as well as their performance and physical characteristics, are

given in Table 4.2.1.

Table 4.2.1: Typical ADCS Sensors.

Sensor Typical Performance Range Mass Range (kg)

Power (W)

Inertial Measurement Unit (Gyros & Accelerometers)

Gyro drift rate = 0.003 to 1 deg/hr, accel. Linearity = 1 to 5x10-6 g/g2 over range of

20 to 60 g

1 to 15 10 to 200

Sun Sensors Accuracy = 0.005 deg to 3 deg 0.1 to 2 0 to 3 Star Sensors

(Scanners and Mappers) Attitude Accuracy = 1 arc sec to 1 arc min

0.0003 deg to 0.01 deg <1 to 5 <2 to 20

Horizon Sensors • Scanner/Pipper • Fixed Head

(Static)

Attitude Accuracy 0.1 deg to 1 deg (LEO)

<0.1 deg to 0.25 deg

1 to 4

0.5 to 3.5

5 to 10 0.3 to 5

Magnetometer Attitude Accuracy = 0.5 deg to 3 deg 0.3 to 1.2 <1 (Wertz and Larson, Space Mission Analysis and design, third edition)

1. Inertial Measurement Unit

An inertial measurement unit (IMU) consists of accelerometers and gyros. Individual gyros

provide one or two axes of rotational acceleration information and are grouped together along

with the accelerometers, which sense the translational acceleration. The sensed acceleration is

then sent to the central processing unit (CPU). The data is integrated once to acquire velocity and

a second time to determine the position. Gyroscope accuracy is limited by instrument drift, so the

IMU must be used in conjunction with an absolute reference such as star sensors.

2. Sun Sensors

55

Sun sensors are visible-light detectors which measure one or two angles between their

mounting base and incident sunlight. They are popular, accurate and reliable, but require clear

fields of view. They can be used as part of the normal attitude determination system, part of the

initial acquisition and recovery system, or part of an independent solar array orientation system.

3. Star Sensors

Star sensors provide high-precision measurements. Star sensors can be scanners or trackers.

Scanners are used on spinning spacecraft, whereas trackers are used on 3-axis attitude stabilized

spacecraft to track one or more stars to derive 2- or 3-axis attitude information. Even though star

trackers are very accurate, care is required in their specification and use. The Cubesat must be

stabilized to some extent before the trackers can determine where they point. This stabilization

requires alternate sensors. Also, the star tracker is sensitive to the sun, moon, and planets, so that

the sensor is blinded while it is exposed to them. Therefore, even though the star trackers have

very high accuracy, they must be used with an additional sensor type (i.e. IMU or/and sun

sensor). Because the star tracker delivers an accuracy that meets mission requirements, this

option was chosen for the subsystem design for fine pointing.

4. Magnetometers

Magnetometers are simple, reliable, lightweight sensors that measure both the direction and

size of the Earth's magnetic field. When compared to the Earth's known fields, their output helps

establish the spacecraft's attitude. This option was chosen for the subsystem design for coarse

pointing.

56

Proposed Hardware for Sensors In order to meet the accuracy of 0.022 degrees, star trackers must be used. Between the

star tracker data updates, IMU will provide the fine attitude. However, in-house star tracker

needs an initial reference and in case of blocking by the Sun; therefore, magnetometers and sun

sensors are used for coarse pointing.

For the magnetometer, a Honeywell 3-axis magnetic sensor could be used. This sensor

has an ultra-compact, size of 3.0 x 3.0 x 1.4mm3 and a mass of 25.6 milli-grams. It has a three-

axis surface mount sensor which is designed to be very sensitive so that it can measure low

magnetic field. This sensor provides wide magnetic field range of ± 6 gauss, and has linearity

error of 0.1% in conditions of ± 1gauss. This choice of sensor is shown in Figure 4.2.1.

Figure 4.2.1: Honeywell Magnetometer

Three orthogonally aligned sensors for three-axis measurements will be used. These will

measure the direction and magnitude of the magnetic field. Generally, only direction is required

for attitude determination.

57

For the sun sensor, AeroAstro Medium Sun Sensor could be used. This sensor provides

accuracy of ± 1 degree and 2-aixs attitude determination, and it has a full angle circular field of

view of 60 degrees. This sensor is shown in Figure 4.2.2.

Figure 4.2.2: AeroAstro Medium Sun Sensor

For fine pointing with rapid update, an IMU will be used. The MICRO-ISU BP3010 from

Bec Navigation Ltd. is currently the best option available. This IMU has a reliable cost with an

accuracy of 0.5° and weights 0.03 kg. It also has a very small size, 35 mm x 22 mm x 12 mm. It

has a worse case drift rate of 0.1 degrees/seconds. This drift rate can be compensated by position

update from GPS and angle update from star tracker. This choice of IMU is shown in Figure

4.2.3.

Figure 4.2.3: IMU2 for Fine Pointing

2 http://www.becnav.co.uk/imu.html?gclid=CN69nNXXlJkCFRwpawodP2Q8Zw

58

Star trackers will be used for fine pointing because they can provide the attitude accuracy

required: at least 0.022 degrees during image acquisition. The star tracker uses the following

process: the camera takes the image of a field of bright stars and the computer goes through an

algorithm to identify the star pattern with a star catalog in memory to determine the attitude of

the satellite. Most star trackers in the market have volume, weight, cost, or power consumption

that is too large for this Cubesat mission. A comparison of star trackers is shown in Table 4.2.2.

Table 4.2.2: Comparison of startrackers.

Company Model Size [mm]

Weight [kg]

Power [W]

Accuracy [arcsec]

Rate [Hz]

AeroAstro Miniature Star Tracker

5.4 x 5.4 x 7.6 0.425 < 2 ± 70 ~ 1

Galileo Avionica

A-STR 195 x 175 x 288 3 8.9 ~ 13.5 < 10 10

Ball Aerospace

CT-602 Dia. 203.2 H = 198.12

< 5.5 8 ~ 9 < 10 10

EADS Sodern

SED 16 170 x 160 x 290 < 3.0 7.5 < 15 10

SunSpace Sun-Star 136 x 136 x 280 1.63 < 3.5 22 (1σ) 1

As shown in Table 4.2.2, most of the star trackers are not suitable for the cubesat. Only

AeroAstro’s Miniature Star Tracker meets requirements but has a large cost (~$200,000).

Therefore, independent design of a star tracker is currently under consideration.

Designing Star Tracker

Sizing camera lens and CCD chip The following is a summary of analysis for sizing a camera lens and CCD chip.

Calculations and results are provided in Appendix D.2.3.

With lens equation, the focal length of lens can be determined. For example, if the acquisition

pointing accuracy is 20 arcsecond and the pixel size of 10 µm, the focal length of lens would be

59

about 100 mm. Also, the number of pixels on the CCD can be determined with an equation for a

reasonable size of field of view (FOV: the angle of exposure field for the CCD) chosen with

assumption of one pixel corresponding to the accuracy of pointing.

The results indicate that any range of pointing accuracy and field of view would possibly be

satisfied by the pixel size of the existing small digital camera. Therefore, a choice must be made

between picking a large field of view or higher resolution of CCD. These results are summarized

in Figure 4.2.4.

Figure 4.2.4: Pointing accuracy vs. total pixels on CCD.

To collect enough starlight, the diameter of the lens must be sufficiently large. The

diameter of the lens can be obtained from the equation of the light power, the light power that

collected by the camera lens from a star with assumption of all photons hit a single pixel and

60

CCD quantum efficiency of 0.25 and the minimum number of electrons of 400 electrons.

Detailed calculations are presented in Appendix D.2.3 Diameter of Lens section. Figure 4.2.5

shows the exposure time versus the diameter of the lens.

Figure 4.2.5: Exposure time vs. lens diameter.

As seen in Figure 4.2.5, as exposure time decreases, the diameter of the lens increases.

The lens focal ratio can be obtained with the focal length of the lens and the diameter of

the lens. Figure 4.2.6 shows the pointing accuracy versus the lens focal ratio.

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Figure 4.2.6: Pointing accuracy vs. lens focal ratio.

As shown in Figure 4.2.6, for higher pointing accuracy shorter exposure time corresponds

to smaller lens focal ratio. The practical focal ratio would be ≥ 1.4 because we can obtain a

shorter exposure time at 1/60 sec with a pointing accuracy of 0.013 degrees, a focal length of

45.5mm and a lens diameter of 3 cm, and the total pixels on CCD is 1.42 Megapixels with field

of view of 15°.

Star Pattern Recognition Algorithms Some researches have been done for the possible choices of algorithms. There are two

similar but different kinds of algorithms to recognize a star pattern: oriented triangles algorithm

and grid algorithm. Both algorithms would provide high accuracy.

The oriented triangles algorithm, first selects a star as a pivot star and two closer stars as

neighbor stars from the image as shown in Figure 4.2.8. The computer initializes the stars in the

image by giving local numbers, and then generates a list of the potential triangles. The stars

given local numbers 1, 2, and 3 become the local triangle, and the surrounding stars become the

reference triangle. Using the distance between stars, the computer algorithm rebuilds the

62

constellation then measures the distance between stars to verify the local number then compare

with star identification number.

(a) (b)

(Rousseau, Bostel, and Mazari, Star Recognition Algorithm for APS Star Tracker: Oriented Triangles) (c) (d)

Figure 4.2.7: The triangle algorithm3.

The grid algorithm is similar to the triangle algorithm but it uses polar coordinate to

construct the pattern. From the image, a reference star is selected as a pivot star and is identified

from the given database. The position of the pivot star becomes the center of the circle with

pattern radius, rp . The surrounding sky is then partitioned over this circle. Then the pivot star is

translated to the center of the FOV, and the related reference stars are translated in the same

distance as the pivot star. Then an alignment star is oriented to the reference frame so that the

related reference stars are rotated at the same angle; then a grid pattern is constructed. The

constructed pattern is compared with patterns in the database for identification. Figure 4.2.9

3 G. L. Rousseau, J. Bostel, and B. Mazari, “Star Recognition Algorithm for APS Star Tracker: Oriented Triangles”, IEEE, February 2005

63

shows a basic definition of a coordinate system on the left and the principal of a grid pattern in

polar coordinates on the right.

(Lee and Bang, Star Pattern Identification technique by Modified Grid Algorithm)

(a) Definition of inertial and CCD body frames (b) Grid pattern is polar coordinate

Figure 4.2.8: The grid algorithm4.

Research has been done by AeroAstro and the MIT space systems laboratory, for

developing a coarse star tracker, showed that this algorithm is more efficient because it has the

potential to reduce the power drawn by processor to factor by 10. This algorithm was claimed to

be more accurate.5

Through research, a technique was found that would increase the accuracy- hyperacuity

technique, called subpixel precision. In a focused image, a star can appear as a point (e.g. one

pixel). If we defocus the image slightly, the star will spread into several pixels, so the

measurement of the center of the star will be more accurate for distance calculation. In this way,

4 H. Lee and H. Bang, “Star Pattern Identification technique by Modified Grid Algorithm”, IEEE, VOL. 43, NO. 3, July 2007 5 R. Zenick and T. J. McGuire, “Lightweight, Low‐power Coarse Star Tracker”, 17th Annual AIAA/USU Conference on small Satellites

64

the determination of the position of a star is more accurate than using one pixel. Figure 4.2.7

shows the hyperacuity technique. However, this technique would take a lot of iterations.

(Liebe, Star Trackers for Attitude Determination)

Figure 4.2.9: Hyperacuity technique6

Inhouse Star Tracker A prototype in-house star tracker has been designed. The camera chosen is the

EdmundOptics' EO-1312m 1/2" mono CMOS USB Lite. This camera provides resolution of

1280 x 1024 pixels and uses USB 2.0 board. The included software provides the images in JPEG

and Bitmap file format. The best lens for this camera has been chosen to be the EdmundOptics'

25mm Megapixel fixed focal length. The size of camera is 4.4 cm x 4.4 cm x 2.54 cm, and the

lens has diameter of 3.35 cm and height of 3.6 cm. The lens has a field of view of 14.6 degrees,

focal length of 25mm, and focal ratio of 1.4. The focal ratio of 1.4 was chosen for capturing

enough light into CMOS, so it would be wide enough for least 3 magnitude of 5.7 stars. Camera

and lens are shown in Figure 4.2.11.

5 C. C. Liebe, “Star Trackers for Attitude Determination”, IEEE, June 1995

65

Figure 4.2.11: EdmundOptics Camera and Lens

The algorithm for the star tracker was designed by Professor A.T. Mattick of the University of

Washington. The 3500 brightest stars are used as a star catalog for the oriented triangles

algorithm. Figure 4.2.12 shows the schematic of in-house star tracker.

Figure 4.2.12 Schematic of in-house star tacker

As seen in Figure 4.2.12, the camera takes a snapshot of the star field and saves this image as a

bitmap file. Then a program reads the image file and finds the locations and brightnesses of

pixels illuminated by starlight and saves them for the main algorithm program. With initial

reference (right ascension and declination) of pointing axis of camera, the algorithm program

looks through the star catalog to find all the possible stars within the angular uncertainty of 10

degrees. Note that uncertainty of the inputs initial reference should be with in 10 degrees. Then

the program tries to match 3 of these stars to 3 stars from camera image with the brightness, ratio

66

of angles and distance. Then it does a least-squares fit using CCD pointing, minimizing the

difference between the positions of predicted and actual illuminated pixcel positions to determine

the pointing axis of the camera and rotation angle of camera about that axis. This rotation angle

will be used by the control system for pointing acquisition. This algorithm provides an accuracy

of 10 arcsecond in camera pointing axis and about 80 arcsecond in rotation of camera axis.

Figure 4.2.13 shows the inertial and CCD body frames of an in‐house star tracker.

Figure 4.2.13 Inertial and CCD body frames of inhouse star tracker

As seen in Figure 4.2.13, the inertial body frame is using the geocentric frame. ζ is declination

(Dec) in degrees, φ is right ascension (RA) in hours, and η is rotational angle of CCD plane. The

u, v, and w are unit vectors. w is the direction of initial reference. CCD plane is parallel to the

X-Y plane and perpendicular to w. The angles (θ1, θ2, θ3) and the distances (d1, d2, d3) will be

measured and used for the star identification process in the algorithm.

Further tasks would be determining whether it will be sharing a CPU with the main computer or

have a separate one integrated into the control system.

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Actuators 1. Thrusters

If the orbital maintenance thrusters outlined in the propulsion section are also used for

control, it must be considered that 3 to 10%7 of the total propellant mass would be consumed for

attitude control purposes. In order to control the attitude of the cubesats, a system of thrusters

must provide the attitude maintenance requirements outlined in Table 4.2.3.

Table 4.2.3: Attitude Control Requirements for Space Propulsion

Attitude Control 3 to 10% of total Propellant mass Acquisition of Sun, Earth, Star Low total impulse, typically <5000 N.s, 1 K to

10 K pulses, 0.01 to 5.0 sec pulse width On-orbit normal mode control with 3-axis

stabilization, limit cycle 100 K to 200 K pulses, minimum impulse bit

of 0.01 N.s, 0.01 to 0.25 sec pulse width Precession control (spinners only) Low total impulse, typically <7000 N.s, 1 K to

10 K pulses, 0.02 to 0.20 sec pulse width Momentum Management (wheel unloading) 5 to 10 pulse trains every few days, 0.02 to

0.10 sec pulse width 3-axis control during ΔV On/off pulsing, 10 K to 100 K pulses, 0.05 to

0.20 sec pulse width (Wertz and Larson, Space Mission Analysis and design, third edition)

2. Reaction and Momentum Wheels

Reaction wheels are torque motors with high-inertia rotors. They can spin in either direction,

and provide one axis of control for each wheel. Momentum wheels are reaction wheels with a

nominal spin rate to provide a nearly constant angular momentum. The torque capability of the

wheels usually is determined by the slew requirements or the need for control authority above

the peak disturbance torque in order for the wheels to maintain pointing accuracy. Reaction

wheels were initially considered for this mission; however, the problem with the

6 Wertz, James R., “Space Mission Analysis and design”, Space Technology Library, third edition)

68

reaction/momentum wheels is that they saturate rapidly and they need to be desaturated

periodically, which can add complexity and difficulty to the process. Therefore, this option was

not considered as the final solution.

3. Control Moment Gyros

Control moment gyros are single- or double-gimbaled wheels spinning at constant speed.

Control systems with control moment gyros can produce large torques about all three of the

spacecraft's orthogonal axes. However, they require a complex control law and momentum

exchange for desaturation. Almost all the CMGs found so far exceed the weight tolerance of this

mission. A suitable CMG is currently under development at Andrews Space Inc.

4. Magnetorquers

Magnetorquers use magnetic coils to generate magnetic dipole moments. Magnetorquers

can compensate for the spacecraft's residual magnetic fields or attitude drift from minor

disturbance torques. They produce torques proportional and perpendicular to the Earth's varying

magnetic field. Because they use the Earth's natural magnetic fields, they are less effective at

higher orbits.

A summary of possible actuators, as well as their performance and physical

characteristics are given in Table 4.2.4.

Table 4.2.4: Typical ADCS Actuators.

Actuator Typical Performance Range Mass (kg)

Power (W)

Thrusters Hot Gas (Hydrazine)

Cold Gas

0.5 to 9000 N**

<5 N**

Reaction and Momentum Wheels

0.4 to 400 N.m.s for momentum wheels at 1200 to 5000 rpm; max torques from 0.01 to 1 N.m

~2 to 20 10 to 110

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Control Moment Gyros 25 to 500 N.m of torque >10 90 to 150

Magnetic Torquers 1 to 4000 A.m2‡ 0.4 to 50 0.6 to 16 (Wertz and Larson, Space Mission Analysis and design, third edition) **Multiply by moment arm to get torque. ‡For 500-km orbit and maximum Earth field of 0.4 gauss, the maximum torques would be 4.9*10-15N.m.

For the actuators, CMGs seem to be the best option for the main control. Magnetorquers are

chosen to serve for the fine control. As there are no off-the-shelf CMGs available that can meet

the cost, weight, and size requirements of this mission, the team worked in conjunction with

Andrews Space to utilize the nano-CMGs currently under development. Also, as trade study

revealed, the only options for the of-the-shelf magnetorquers are long rods of at least 30 cm.

Since these options are not feasible for this mission, the team decided to design magnetorquers,

which better fit the requirements.

The CMG wheel is sized as follows . Calculations are given in Appendix D.2.4.

Designing the CMG Wheel: The moments of inertia for the Cubesat were determined first in order to start the calculations

for sizing the CMGs. For these calculations, the center of mass is assumed to be at the center of

the body with uniform density. In order to find the required moment of inertia for the CMG, the

required slew rate is determined. Since the cubesat will accelerate for half of this angle and

decelerate for half, only half of the angle and time are used in the calculations. The angular

acceleration of the Cubesat can be determined. The calculations were completed for a range of

slew rates from 1° - 15° per second. Then using the moments of inertia of the Cubesat and the

angular acceleration, the required wheel torque is determined, and then the required momentum

of the wheel.

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The CMG cluster for an x-axis maneuver is shown in Figure 4.2.14. The same convention is

used for the other two directions.

(Lappas, Steyn, and Underwood, Attitude Control for Small Satellites using Control Moment Gyros)

Figure 4.2.14: CMG Cluster for an x-axis maneuver

Once the angular momentum is determined, assuming an angular velocity of 60,000 rpm

for the DC motor (the specifications of the DC motor are presented in the Appendix), the CMG's

required moment of inertia is determined. Note that this DC motor is for the wheel only. A ULT

Applimotion frameless motor (shown in the Appendix) is going to be used for the gimbals.

The required moments of inertia for four cases are of slew rates, 9 deg/sec, 8 deg/sec, 7

deg/ sec, and 5 deg/sec, with three different cases of 6 rad/sec, 10 rad/sec, and 25 rad/sec

maximum gimbal angle rates are calculated and summarized in Appendix D.2.4.

71

As an example, the required moments of inertia for the maximum gimbal angle rate of 6

rad/sec are plotted in Figure 4.2.15.

Figure 4.2.15: Required Moment of Inertia Vs Slew Rates for a Maximum Slew Rate of 6 rad/sec

An important factor in deciding the slew rate is the time it takes for the CMG to reach

that slew rate with assuming a torque of 0.003 N-m, from Andrews, the time in each direction is

determined. The summary results are plotted in Figure 4.2.16.

72

Figure 4.2.16: Slew Rate vs Time it Takes to Reach to that Slew Rate

It is observed that the higher the slew rate, the longer it takes to get to that slew rate.

Then in order to size the wheel and find the diameter that can produce the required slew rate, the

moments of inertia for a brass wheel are calculated for four different combinations, with density

of 8400 kg/m^3 and 8700 kg/m^3 and two thicknesses of 0.005 m and 0.0025m. The diameter is

varied from 0.01m to 0.07 m. As an example, the results obtained from a density of 8400

kg/m^3 is plotted in Figure 4.2.17.

Figure 4.2.17:MOI Obtained from the wheel vs. diameter

73

Now, the possible moments of inertia obtained from the wheel are compared to the

required moments of inertia, and it can be decided which diameter for the wheel is necessary.

To sum up, from these calculations it is observed that for 9 degrees per second, the

longest time to reach the desired slew rate is about 6 seconds along the 0.1m side. It looks like a

wheel with a diameter of 0.03m and thickness of 0.0025 m can produce the worst case MOI

required.

A simple simulink block diagram can be designed to model the dynamics of the DC

motor, which was chosen to run the CMG wheel, as shown in Figure D.2.5.1. Running Simulink

model with the motor parameters, given in Appendix B.1, and a voltage input of 1 volts, an

output shown in Figure 4.2.18 is obtained. As shown, with 1V input an angular velocity of about

850 rad/sec or 8116.9 rpm is obtained.

Figure 4.2.18: Response to an Input of 1 volts

74

Designing the Magnetorquers: As mentioned before, the of-the-shelf magnetorquers found in the trade study did not fit

the requirements for this mission. Therefore, the team designed square plates with wires wound

up around it to be used as magnetorquers. For these calculations, the earth magnetic strength is

taken to be about 5*10^-5 Tesla. After studying a range of different areas for the plate, it is

decided to use a plate with an area of 0.003025 m^2. This will require 318 turns of wires to

produce 0.024 mN-m torque, using AWG 26 copper with a diameter of 4.06*10^-4 m and a

resistivity of 1.7*10^-9 ohm-m. For these calculations, current of 1 Amp and power of 1 Watt

are assumed.

A summary of the proposed hardware that has been located so far and their physical

properties is provided in Table 4.2.5.

Table 4.2.5: Proposed Hardware for Attitude Determination and Navigation Systems.

Part Performance Mass [kg]

Size [cm]

Power Required

[Watt] Star Tracker ±0.0085Degrees

3-axes (3σ) 0.110** Camera 4.4 x 4.4 x 2.54

Lens Dia. 3.35 Length 3.60

<2**

IMU Complete 6 DOF Max update rate 64Hz

0.030 3.5 x 2.2 x 1.2 0.5

Magnetometer 120 μaguss 25.6 x 10-6 0.3 x 0.3 x 0.14 2V Sun Sensor ±1Degrees

2-axes (2σ) 0.036 Dia. 2.43

Height 3.49 None

CMG Torque of about 0.003Nm 3 wheel

Wheel motor Gimbal motor

0.05 x 3 0.001 0.036

~d=3, t=0.25

D=~3.5

***

0.5

Magnetorquer Torque of 0.024mN-m 318 turns of coil

0.08

5.5 x 5.5 x 0.5

1

** Mass and power required based on AeroAstro’s Miniature Star Tracker ***Design specifications will be determined later on.

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A summary of the cost of the chosen sensors and actuators for the attitude determination

and navigation systems is given in Table 4.2.6.

Table 4.2.6: Costs Estimates and Companies for Parts used in ADCS and Navigation System.

Type Hardware Cost [each] Company Star Tracker

• Camera

• Lens

will be designed EO-1312M 1/2" CMOS Monochrome USB Lite Edition Camera 25mm Megapixel Fixed Focal Length Lens

$725 $195

UW Edmundoptics Edmundoptics

IMU MICRO-ISU BP3010 $700 Bec Navigation Ltd. Magnetometer 3-Axis magnetic sensor HMC 1043 $45 Honeywell Sun Sensor Medium sun sensor $5640 AeroAstro GPS

• Receiver • Antenna

Cornell Cougar GPS receiver Toko DAX Dielectric Patch Antenna

<$20000

Cornell Cougar

CMG Wheels: designed (3) Wheel motor Gimbal Motor

undetermined UW Micromo Electronics Applimotion, Inc.

Magnetorquers Designed (3) undetermined UW

ADCS Details The formation of spacecraft attitude dynamics and control problems involves

consideration of kinematics. In kinematics, we are primarily interested in describing the

orientation of a body that is in rotational motion. One scheme for orienting a rigid body to a

desired attitude is called a body-axis rotation; it involves successively rotating three times about

the axes of the rotated, body-fixed reference frame. The first rotation is about any axis. The

second rotation is about either of the two axes not used for the first rotation. The third rotation is

then about either of the two axes of the two axes not used for the second rotation. There are 12

sets of Euler angles for such successive rotations about the axes fixed in the body. A picture of

the frame of reference is shown in Figure 4.2.19.

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Figure 4.2.19: Frame of Reference (Wertz and Larson, Space Mission Analysis and design, third edition)

The functional schematic for the control is shown in Figure 4.2.20.

Figure 4.2.20: Functional Schematics

As shown in the figure above, the ADCS sensors send their inputs to the convertor as

analog signals. The convertor then converts these signals to digital signals. These are sent to the

processor, which contains the control algorithm and process the data. Then the output is sent to

the pulse width modulator and then to the actuators to execute the command.

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The control model must meet some functional requirements. First it shall fulfill

positioning requirements of solar cells, communications, and payload. Second it shall be able to

counteract the maximum external torque of the environment at the selected orbit at 500 km.

From the calculations, shown in Appendix D.2.1, the worst-case disturbance torques are

determined and summarized in Table 4.2.7.

Table 4.2.7: Worst-Case Disturbance Torques.

Disturbance Torque [ ] Note

Roll with folded solar panel Gravity-gradient

Pitch with folded solar panel

for unfolded solar panel Solar Radiation

for folded solar panel

for unfolded solar panel Aerodynamic

for folded solar panel Magnetic Field

As it is observed in Table 4.2.7, these disturbance torques are negligible compared to

torques required to point the camera for image acquisition.

Lastly, it shall have differing modes of operation, namely, Detumbling mode, Acquisition

mode, Normal mode, Slew mode, and Contingency mode. The detumbling mode shall occur

immediately after the satellite is ejected from the carrier module, or after strong environmental

deflections. For this mode, no attitude knowledge shall be required prior to operation. Also, this

mode shall operate until the satellite's rotation rate reduces to an acceptable level. The

acquisition mode shall operate immediately after the first detumbling mode. This mode will

initially determine the attitude of the satellite. The mode shall subsequently nadir point the

antenna until the first transmission is complete. Normal mode shall operate when normal

environment deflections are occurring. Requirements for this mode shall drive the system design.

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Slew mode shall operate as a transition mode from sun pointing to earth pointing and back again.

Contingency mode shall operate in emergencies if normal mode fails. This mode shall use less

power to meet power or thermal constraints.

The control design is illustrated in Figure 4.2.21.

Figure 4.2.21: Control Design

There are several options for the control law, which relates control torque to the error

signal. One option is proportional control, which is the simplest kind and has the form

cT Kθ= − where Tc is the control torque and K is the system gain and theta is the error signal.

Another option is the Bang-Bang Control, which has the thruster pulse as output and the

direction is determined by the sine of the error signal sinc pT T θ= . Another option is

Proportional-plus-derivative (PD) Control such as 1 2c K KT θ θ= + , where coefficient of the time

derivative of the error signal provides damping and reduces the angular excursions. The final and

most desirable option is the Proportional-plus-Integral-plus-Derivative (PID) Control

as 1 2 3c K K K dtT θ θ θ= + + ∫ . The PID controller is a sub-class of a full state feedback

controller.

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In order to design the simulation, the desired attitude is defined via attitude

parameterization methods. The most common attitude parameterization methods are Euler angles

and Quaternion. For space applications, Quaternion is the preferred option in order to avoid

singularities and minimize the use of Sine and Cosine functions. In order to compensate for the

errors, a Full State Feedback controller is used. The error then is presented as shown below.

where matrix A captures the dynamics of the Cubesat, matrix B captures the dynamics of the

actuators, vector u is the input to the actuators, y is the observations made by the sensors, and

matrix C captures the dynamics of the sensors.

Then to make sure that the system is immune to the saturation in the actuators, we need to

check for stability as shown below

where Kp is the gains needed for the Full State Feedback controller to make the system stable

and it also needs to capture the efficiency factor of the controller, control saturation, and slew

rate constraints. For simulation purposes, the linearized controller can be used to simulate the

non-linear system. On the actual Cubesat, there shall be different controllers for different modes

of operations.

In order to fully determine the position and attitude of the Cubesat, the navigation system

must be integrated into the ADCS system. A possible case is shown in Figure 4.2.22.

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Figure 4.2.22: ADCS Architecture Diagram

As it is observed in the ADCS architecture block diagram, the attitude sensors provide

vector measurements that are passed through an optimal estimator such as the QUEST algorithm

to determine a solution for an attitude estimate. This estimate is then passed to a fine estimator

such as an extended Kalman Filter, along with angular velocity measurements to obtain the

attitude solution. The attitude controller compares the determined attitude with the desired

attitude and calculates appropriate control torque to minimize the error. These torques are sent to

the appropriate actuators to exert moments on the Cubesat. The GPS sensor provides a state

vector to the orbit controller. This position information is compared with the position indicated

from the master controller, which is driven with the mission requirements.

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5. Communications

The constellation of satellites will work as a system with several individual agents. These

agents include ground stations, CubeSat carriers and the CubeSats within the constellation. The

transmission of data is required for this mission, and the data is transmitted via communication

systems aboard each of the system agents. A properly designed system allows the time from

image request to ground reception to be minimized while preserving image quality and security.

5.1. Communications Architecture

The communication network consists of a ground station and the constellation of CubeSats.

To optimize the power usage for communication, only predetermined CubeSats will be used in a

relay chain from request to delivery. The ground station will have updated knowledge of the

constellation, including the position and velocity vectors of the CubeSats. With this knowledge,

the ground station will compute the CubeSat node order to be used in the relay chain from the

ground station, to the target CubeSat for data acquisition, and back to the ground station. Figure

5.1.1 shows this communication relay architecture. The node order is then stored into a queue

along with a data request (imaging, position, health status, etc.) and sent to the nearest CubeSat.

This CubeSat, C1, pings a confirmation signal to the ground station, dequeues the first node in

the node order queue, and relays the modified queue. The relayed signal propagates radially to

neighboring CubeSats. Several CubeSats may receive the signal, and they will each execute a

Boolean check to see if they are the intended next node. Only the intended CubeSat, C2, will

satisfy this check, and the rest will ignore the signal. C2 then pings a confirmation signal to C1,

dequeues the first node, and relays the newly modified queue. This process repeats as necessary

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until the target CubeSat receives the signal. It then collects the requested data, appends the

collected data to the queue, and relays the newly modified queue along the predetermined return

path. The ground station then receives the queue with the requested data but stripped of the

communication node path.

Figure 5.1.1. Communication architecture outline.

Approximately 40% of the orbit is spent in the shadow of Earth where images cannot be

collected. Because the image collection is one of the most power and computationally intensive

operations that occur onboard the CubeSats, the communication node queue will prioritize the

dark-side CubeSats, where no imaging occurs, over the light-side ones. This will help to

dedicate more power to communication.

5.2. Communications Overview

The primary factors of the communications system are the unit costs, range, and power

requirements. The system hardware consists of antennas, receivers, and transmitters. The

receivers and transmitters are often combined into a transceiver. Before analyzing the hardware,

Ground Station

C1C2 C3 CN

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the system constraints must be defined. These include constraints from the geometry of the

constellation design as well as constraints from data requirements.

There are two communication links analyzed in detail: crosslink and downlink. For crosslink

communication, the orbit geometry is derived from the current constellation design; a Walker

constellation with an inclination of 55° and an altitude of 520 km. There are four

communication paths that need to be analyzed within the crosslink communication, shown in

Figure 5.2.1.

Figure 5.2.1. Four crosslink communication paths: (a) In-plane (r1 = 4000 km), (b) Cross-plane

parallel (r2 = 1375 km, r3 = 3200 km), (c) Cross-plane ascending-descending (r4 = 5400 km)

With 10 CubeSats per plane the in-plane spacing is approximately 4000 km. There are two

scenarios for cross-plane communications: ascending/descending and parallel communication.

For both scenarios the greatest distance is at the equator. To communicate between an ascending

and descending CubeSat as they cross the equator requires initializing communication at a

distance of approximately 5400 km. There are two possible communication links for parallel

cross-plane communications, as shown above in Figure 5.2.1. As one CubeSat C1, crosses the

equator the adjacent plane will have one CubeSat, C2, phased 19.6° above the equator and one

CubeSat, C3, phased 10.4° below it. The distance from C1 to C2 is approximately 3200 km and

that between C1 and C3 is approximately 1400 km.

(a) (b) (c)

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For downlink it is assumed that the ground station is a on-meter parabolic dish with tracking

capabilities to 10° above the horizon, as shown in Figure 5.2.2.

Figure 5.2.2. Downlink access range with r = 1820 km.

Using an altitude of 520 km, the maximum distance is 1820 km as the CubeSat first comes into

view. Along with the relative distances imposed by the constellation design, the system’s

capabilities are also constrained by the top-level data requirements.

Each CubeSat is required to acquire and transmit up to 60 images per day with full color and

at least 8-bit resolution. Because approximately 30% of the orbit is in the Earth’s shadow, the

available time to acquire the 60 images is 14.4 hours rather than 24. The Imaging group suggests

the use of a 10-Megapixel CCD with 10-bit resolution, resulting in approximately 100 Mb per

image. In addition to images, the CubeSats will be required to send data including onboard

system health (i.e. power levels, propellant levels, etc.), ADCS and navigation data, and

command data from the ground station. These additional data packages are significantly smaller

than the images themselves, each no more than a few kilobits per day. For 60 images per day,

the required data transfer rate for images alone is about 116 kbps of uncompressed data. Adding

the other data sources and including a small safety margin, this transfer rate simply becomes 120

kbps. Compression formats such as JPEG have compression ratios from about 60% through

98% or higher. Compression ratios of 90% are commonly used without significantly damaging

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the image quality. Utilizing this JPEG format would decrease the required data transfer rate to

12 kbps, significantly decreasing the load requirements of the communications system.

Downlink communication operates with constraints independent of crosslink. For a circular

orbit with an altitude of 520 km, the ground velocity is approximately 7.6 km/s. For a tracking

range of 160° (10° above the horizon), this results in a maximum in-view time of 450 seconds.

Allowing for two minutes for the ground station to initiate communication with an overhead

CubeSat and two minutes for the CubeSat to transfer a single image, there are over three and a

half minutes left of access time to allow for additional communication (impulsive system health

check, navigation corrections, etc.) or other interruptions such as tracking errors. For one image

to be transmitted in two minutes requires a transfer rate of 84 kbps, seven times the data rate of

crosslink communication.

The well-known Link equation provides a relationship between the required power of the

transmitter and the constraints discussed above as shown in Appendix E. Typical values for

several terms in the equation are assumed at this stage of analysis, and are also summarized in

Appendix E. Using these estimated values, the Link equation is simplified to relate the design

parameters, i.e. transmitter power, frequency, and antenna gains. The transmitter power is

estimated to be half of the input DC power, which is targeted to be no more than 5W for

crosslink and no more than 1W for downlink. The antenna types are chosen using analysis via

the Link equation and by inspection of the necessary gain patterns for each of the links in the

communication architecture.

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5.3. Communications Hardware

The driving factors for the communication hardware on the CubeSat are mass, volume and

cost. For antennas, these are met using simple designs. The uplink communication is not

constrained by these same factors and was not analyzed. Figure 5.3.1 shows the general gain

patterns that provide the desired communication performances. The crosslink communication

requires the ability to communicate with any of the neighboring CubeSats, according to the

queue of nodes. This suggests the use of monopole antennas, which have toroidal gain patterns

about the monopole axis. With ambiguity in the relative attitudes of neighboring CubeSats, a

more isotropic gain pattern is desirable. This is easily accomplished using multiple monopole

antennas arranged orthogonal to one another. The downlink communication is towards the

ground station, which only requires a unidirectional gain pattern. Though antennas such as the

patch antenna have such unidirectional gain patterns, the added hardware and system complexity

does not make them the most viable option. Instead, by arranging three monopole antennas to be

mutually orthogonal to each other, the attitude of the CubeSats becomes a nonissue for any of the

communication links, up, cross or down. Additionally, the idea to use the CubeSat carrier

vehicles as dedicated downlink nodes has been proposed. With this approach, a dedicated high-

gain antenna will be used along with accurate ADCS pointing to allow greater access time and a

higher data transfer rate for downlink communications. Further research and analysis will be

required for this approach.

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Figure 5.3.1. Desired communication gain patterns.

Monopole antennas are commonly in lengths of λ/2 or λ/4, where λ is the signal’s

wavelength. After a quick trade study comparing the half and quarter wavelength monopoles, it

was concluded that there is only a small power benefit to using a half wavelength monopole due

to its slightly higher gain. The smaller size of the half wavelength antenna is more attractive for

the mission than the slight decrease in power usage of the full wavelength antennas. For this

reason, quarter wavelength monopole antennas are used for analysis using the Link equation. To

minimize flexing and oscillation in the antennas, the length of the monopoles should be

constrained to a reasonably small number. For analysis, a maximum antenna length of 1 m was

used. Figure 5.3.2 shows the relationship between inter-CubeSat distance, wavelength and

transmission power for crosslink communication using the quarter wave monopole antenna.

U/L: Single target, directional signalX/L: Multiple neighbors, ideally isotropic signal D/L: Single target, maximize fly‐by access time

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Figure 5.3.2. Power requirements for crosslink comm using half wave monopole antennas.

From Figure 5.3.2 it can be seen for a maximum transmit power of ~2.5W (input power

~5W) the wavelength must be greater than about 0.8 m (375 MHz) for the ascending/descending

cross-plane communication, but only about 0.6 m (500 MHz) for in-plane communication, both

of which correspond to the UHF frequency band. Similarly, Figure 5.3.3 shows the relationship

between inter-CubeSat distance, wavelength and transmission power for downlink

communication using the half wave monopole antenna. From this plot it can be seen that the

downlink antenna requirements can be met using the same antenna size as the crosslink.

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Figure 5.3.3. Power requirements for downlink comm using half wave monopole antennas.

A particular transceiver that may meet the mission requirements is the RF DataTech

LRT470 radio module. It operates at frequencies of 406-475 MHz, which equates to a

wavelength range of 63-72 cm. Using this, however, prevents the ascending/descending cross-

plane communication at the equator. Figure 5.3.4 shows the communication power requirements

as a function of communication range for crosslink using the extremities of the transceiver’s

range. Higher frequencies help transfer data quicker but require higher power, as shown in

Figure 5.3.4. To conserve power, the low frequency, 406 MHz, will be used for communication.

As this will be the primary communication frequency, the antennas will be appropriately sized at

15.75 cm. These antennas will be tape-spring antennas, deployed from a coiled position. The

downlink communication is assumed to be the transfer of a large data packet to ground station

and will, by default, use the high frequency of 475 MHz. The power requirements as a function

of communication range for downlink communication is shown in Figure 5.3.5.

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Figure 5.3.4. Crosslink communication power requirements using the LRT470 transceiver’s

maximum and minimum frequencies.

Figure 5.3.5. Downlink communication power requirements using the LRT470 transceiver’s

maximum frequency.

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5.4. Central Processing

A central processor provides the satellite with the needed processing for navigation,

control, communication, and general system tasks. The processing throughput required for the

central processor was estimated using Table 16-13 “Size and Throughput Estimates for Common

Onboard Applications” in Space Mission Analysis and Design by James R. Wertz and Wiley J.

Larson. The estimated throughput required for navigation, control, communication, and general

system tasks is less than 2 MIPS. However, this estimate is subject to change as individual

algorithms for tasks are identified. An estimate of 5 MIPS to account for intensive processes

such as the star tracking algorithm provides a conservative value for analysis.

The throughput required for image processing was many orders of magnitude greater than

the throughput required for navigation, control, communication, and general system tasks.

Instead of having the central processor perform all tasks in the satellite, a separate processor will

perform the image processing. This configuration would allow the central processor to have

relatively low capabilities and, consequently, consume less energy. The image processor would

be turned off or in a sleep state when not needed. This configuration would decrease the energy

consumed at the expense of increased complexity.

The hardware chosen to satisfy a throughput of 5 MIPS is manufactured by Pumpkin, Inc.

Pumpkin, Inc. hardware was chosen because of the ease of incorporation into the CubeSat

structure. The central processor is the MSP430 series microcontroller by Texas Instruments.

The MSP430 series microcontroller has a throughput up to 25 MIPS and consumes less than 60

mW. Also, it can operate at temperatures between -40°C and 85°C. Pumpkin, Inc. incorporates

the microcontroller into a pluggable module. The price of the Pumpkin, Inc. pluggable processor

92

module based on the MSP430 microcontroller is $500. The motherboard manufactured by

Pumpkin, Inc. accepts their pluggable processor modules and provides convenient connectors to

peripherals. The motherboard price is $1,200. Pumpkin, Inc. produces an operating system,

called Salvo™, designed to run on low performance embedded systems. The Salvo™ operating

system costs between $750 and $1,500 depending on the features.

The integration of the communication hardware and signal flow can be seen in Figure

5.4.1.

Figure 5.4.1. Signal Flow Diagram for CubeSat Communications System.

As seen in Figure 5.3.6 there are two outputs from the primary processor. If the CubeSat is not

intended to acquire any data but merely relay a command signal, the processor will simply direct

the signal back to the transmitter. Conversely, if the CubeSat is intended to acquire data, the

primary processor will send the signal to the appropriate hardware, collect the data, send it to the

image processor for compression, then to temporary storage and finally sent to the transmitter.

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6. Support Systems

6.1. Power

Power systems are required to support the other components of the CubeSat. Displayed below in Table

6.1.0 is the supported hardware for the mission, their operating voltage, and their duty cycles. A dash

indicates unknown or inapplicable values. The sum of power requirements total 20.82 Watts, but the

power required for the entire system is estimated at 10 Watts on average as a conservative estimate;

the CubeSats will not be acquiring images for the entirety of the mission, nor will they be constantly

readjusting their orientation or their orbits.

Solar Power The preliminary analysis involves the determination of the type of power system required for the

mission. The most practical option among radio‐isotope, electrochemical, and solar power will be a

combination of solar power using photovoltaic cells for primary power during light‐side operation and

electrochemical (battery cells) providing power on the dark side of the CubeSat orbits.

Several solar cell semi conductors are available for this mission, including Ge, Si, GaAs, or triple‐junction

cells using GaInP2, listed below in Table 6.1.1.

The determining factor in material selection in this case is efficiency, so the triple‐junction cell has been

chosen for this application. Preliminary research shows the Spectrolab NeXt Triple Junction (XTJ) Solar

cells to give the best performance at AM0 (standard space) conditions.

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Table 6.1.0 CubeSat Systems Power Requirements

System Subsystem Component Power required Operating

V l

Duty cycle

GNC Navigation GPS 2 Watts 5 Volts 1

ADCS DC motor (CMG) x3 0.5 Watts (ea.) 7.5 Volts .01

Magnetorquer x2 1 Watt (ea.) ‐ ‐

Star Tracker 1 Watt 5 Volts ‐

Magnetometer 20 mW 2.0 Volts ‐

IMU 0.5 Watts ‐ ‐

Sun Sensor ‐ ‐ ‐

Propulsion Micro PPT 2.5 Watts 5.5 Volts ‐

Carrier Propulsion Valves, valve heaters, 72 Watts (max) 28 Volts ‐

Imaging, CDH CDH FM430 Flight Module 50 mW 5 Volts ‐

Communications Transceiver 5 Watts 10‐15 Volts ‐

Imaging CCD 3 Watts 15 Volts .01

A/D Converter 153 mW 3 Volts .1

Support Systems Power Battery (charging) 3.1 Watts 8.4 Volts ‐

Thermal ‐ ‐ ‐ ‐

Structures ‐ ‐ ‐ ‐

Total 20.82 Watts

Estimated Max Power Used 10 Watts

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Table 6.1.1 Semiconductor Material Properties

Semiconductor Band gap Wavelength, λ Responsivity, Efficiency,

Ge .66 1.9 .57 6%

Si 1.1 1.1 .41 15%

GaAs 1.43 .85 .32 19%

Ga InP2 1.9 .65 ‐ ‐

Triple junction (Ge‐GaAs‐GaInP2)

‐ ‐ ‐ 29%

Table 6.1.2 Triple Junction Solar Cell Properties

Triple‐Junction Cell Vmp (V)

Jmp (mA/cm2)

Power per 20 cm2 cell (W)

Bare mass per 20 cm2 cell (g)

Efficiency, η

Spectrolab NeXt Triple Junction1 (GaInP2/GaAs/Ge )

2.333 17.32 .81 1.68 29.9%

Spectrolab Ultra Triple Junction2 (GaInP2/GaAs/Ge )

2.35 16.3 .77 1.68 28.3%

Spectrolab Improved Triple Junction3 (GaInP2/GaAs/Ge )

2.27 16 .73 1.68 26.8%

Iterations of analysis have shown that a 9V bus will be satisfactory, as the solar array bus voltage must

be slightly higher than battery charge voltage and the solar cells provide discreet increases in voltage. It

is planned for the solar cells to cover all sides except for the top and bottom‐most side, shown in Figure

6.1.3. The top surface is assumed to have 100 cm2 of solar panel area, leaving room for a star tracker.

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Table 6.1.3 Simplified diagram of CubeSat, with one solar wing

The system will require a simple power conditioning scheme using charge controllers, and DC‐DC

converters for hardware that requires these additions. Current technologies of low‐voltage, low‐power

DC‐DC converters show a range of 90‐98% efficiency. Figure 6.1.4, shown below, illustrates the

preliminary design of the power architecture. Solar arrays provide power to the system and charge the

battery, adjusted by a controller.

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Figure 6.1.4 Preliminary power architecture

Although the total power from hardware sums to 20.82 Watts (Table 6.1.0), it is estimated that 10 Watts

maximum will be used at any given time. Adding a safety margin of 20% results in 12 Watts of solar

power generation required. To achieve solar power generation of 12W, 320 cm2 of solar panels directly

facing the sun is required. Costs for the solar array system, with an estimate of $250/W4 for current

technologies, total around $5350 (2700 cm2 total cell area at 1366 W/m2 and 29% efficiency).

The mass of the bare solar cells is 84 mg/cm2. For 2700 cm2 of solar cells, this amounts to 227 g of

mass. However, this figure represents only the bare cell so a safety margin of 20% is added to account

for other structural support that may be required, producing 272g of solar power system mass. Each 20

cm2 cell (assumed 4 x 5 cm) can provide 2.33 V and 346 mA (17.32 mA/cm2) at maximum power

generation.

To attain a 9V bus, 4 of these must be placed in series, and to achieve the required current, there must

be 6 strings of 4 cells, totaling 24 cells.

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To determine the solar wing area required, if at all, for the CubeSat, analysis was done based on

averaging the power generated for all possible orientations. Additionally, this power generation is

weighted by the fraction of time in light to the total orbit period (worst case of 56 minutes out of a 94

minute orbit). It has been determined (see Appendix F1.3) that orientation for maximum power yields

16.6 W with 400 cm2 of extra solar panel area perpendicular to the largest CubeSat faces (as shown in

Figure 6.1.3) with the previous factors accounted for. During light‐side operation, the maximum power

attainable with active pointing is 27.9W with a 400 cm2 solar wing. Figure 6.1.4, below, shows the

relationship between the solar wing area and the average power generated during an orbit using active

maximum power pointing, the average power required, and the power generated based on the average

of all orientations.

Table 6.1.4 Average power generated during an orbit

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Carrier power has not been fully explored, as the supported hardware has not been yet identified.

However, similar analysis was done (see Appendix F1.3) to show that by covering 4 sides of the carrier

(60 cm on a side, 14400 cm2 total, 12kg), it is possible to generate an average 70 W of power (based on

the average of all orientations), or 120 W of power using maximized pointing. Current power

requirement estimates require 80 Watts of power for propulsion during the infrequent orbit changes.

It must be emphasized that many of these analyses have taken the worst‐case requirements and

expanded on them by extra margins of safety. The orbits at their specific inclination and altitude (55º

inclination and 520 km altitude) experience a worst‐case light‐side phase of 56 minutes in a 94 minute

orbit.

Electrochemical Power The design parameters of the mission require a power source when solar power is not available, namely

on the dark side of the CubeSat orbits. The operational requirements for dark‐side power is

minimization of mass and volume and the ability to withstand the approximate 5600 charging cycles of

light‐to‐dark (with 94 minute orbit periods at 520 km altitude) during the one year lifetime of the

mission. Li‐ion batteries were chosen since they provide high specific energy (on the order of 180 W‐

h/kg), have little to no self‐discharge, are low volume, and have long cycle lives. With an estimate of 5 W

of required power, a 7.4 V bus (due to the combined battery voltage), and a worst‐case estimate of 36

minutes of discharge time during the dark, 2 Li‐ion cells will be required, having at least 405 mAh of

capacity each. As an example, the Sanyo Li‐ion battery UF634042F holds 1200 mAh of capacity per

battery cell and can charge at high currents (1230 mA). Due to unknown cycle life characteristics of

these batteries, a higher energy capacity is used to decrease the required depth of discharge. Using this

battery as an example, the depth of discharge is 34%. Additionally, the recharge time estimate of 58

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minutes requires 3.1 W of additional solar power (see Appendix F1.1) at 731 mA and 4.2 V. Each cell

weighs 25g.

For the carrier vehicle, an estimate of 50 Watts of power required throughout the orbit and a

28V bus, 64 batteries will be required (8 in series, 8 strings in parallel). This provides 30 W‐h of energy

and only a 10.6% (126.7 mA‐h) depth of discharge per battery. Each battery discharges at an average

rate of 211 mA during the dark period of orbit and charges at an average rate of 914 mA during the light

period of orbit. The average power required during the light period of orbit to charge the batteries is

30.7 W, and the total mass of all batteries together is 1.6 kg.

6.2. Thermal

Thermal control is necessary to provide an operational temperature for the various

components on the satellite. Methods for thermal control are categorized in passive and

active systems. Passive thermal control systems include surface finishes, insulation, and

radiators. Heaters are considered an active thermal control system. The satellites will attempt

to mainly utilize a passive system to maintain an operation temperature.

Requirements The thermal requirements for various components of the satellite were compiled from

“Table 11‐43. Examples of Typical Thermal Requirements for Spacecraft Components” in Space

Mission Analysis and Design and various component manufacturers. Table 6.2.1 summarizes

the thermal requirements for the satellite’s components.

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Table 6.2.1 Thermal Requirements for Spacecraft Components Component Operational Temperature Range (°C)

Batteries 0 to 60

Optics ‐30 to 60

Star Tracker ‐30 to 60

CMG ‐30 to 70

CCD ‐50 to 70

Computer Hardware ‐40 to 80

Sun Sensor ‐40 to 90

Antennas ‐100 to 100

Solar Panels ‐150 to 110

Based on the operational temperature ranges of each component, an average thermal

equilibrium from 0°C to 20°C of the components inside the satellite’s structure would be

sufficient. The batteries are the most sensitive to temperature because the charge and

discharge voltage varies with temperature.

Orbit Environment The environment enveloping the satellite during its orbits is influenced by the

inclination, longitude of the ascending node, and altitude of the satellite. Radiation

encountered by the satellite originates from the sun, earth, and reflected sunlight off the earth.

The thermal parameters for an altitude of 520 km and 55° inclination are calculated in Appendix

F.2.1 and summarized in Table 6.2.2.

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Table 6.2.2 Thermal flux encountered by the satellite

Solar Radiation Earth Radiation Reflected Solar Radiation

1,370 W/m² 200 W/m² 175 W/m²

Thermal flux from solar radiation is in the direction away from the sun and thermal flux from

earth radiation and reflected solar radiation is in the direction away from the earth.

The period of darkness during the satellite’s orbit at an altitude of 520 km and 55°

inclination is shown in Figure 6.2.1. The coordinate system used to determine the period of

darkness is with the x‐axis and y‐axis in the ecliptic plane. The y‐axis is in the direction of the

sun and the x‐axis is parallel to the velocity of the earth. The longitude of the ascending node is

measured from the x‐axis.

Figure 6.2.1 Dark period versus longitude of the ascending node

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The satellite will experience between approximately 22 and 36 minutes of darkness during its

life time. The average dark period is about 30 minutes.

Satellite Geometry and Material The geometry of the satellite affects the amount of heat absorbed and radiated from

the satellite. The geometry of the imaging and carrier satellites is shown in Figure 6.2.2. The

imaging and carrier satellites were assumed to maintain an attitude of nadir pointing during an

orbit. The projected area was averaged over an orbit and was assumed constant. The

materials covering both satellites are mainly photovoltaic cells and aluminum.

Figure 6.2.2 Imaging and carrier satellite

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The geometry and material parameters for the imaging and carrier satellites are summarized in

Table 6.2.3.

Table 6.2.3. Geometric and Material Parameters

Geometric Parameters Imaging Satellite Carrier Satellite

Mass 6 kg 60 kg

Surface Area 0.38 m² 0.80 m²

Sun Projected Area 0.07 m² 0.18 m²

Earth Projected Area 0.10 m² 0.15 m²

Photovoltaic Cell Coverage 80% 40%

Material Parameters Absorptivity Emissivity

Photovoltaic Cell 0.90 0.85

Aluminum 0.13 0.065

The photovoltaic cells were assumed to gather energy for 50% of the time during the light side

of the orbit.

Temperature Variations Using the finite difference method as described in Appendix F.2.2, the temperature of

the satellites can be approximated. The temperature of each satellite is assumed to be the

same throughout the entire satellite. Also, each satellites is assumed to have the parameters

described before and a specific heat capacity of 500 J/kg∙K. The simulation was started at 20°C

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and used the average orbit dark period according to Figure 6.2.1. Figure 6.2.3 and Figure 6.2.4

shows the temperature variations of the imaging and carrier satellites.

Figure 6.2.3 Temperature versus duration of the imaging satellite

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Figure 6.2.4 Temperature versus duration of the carrier satellite

The imaging satellite reached an equilibrium temperature of 14±15°C after about 7 hours. The

carrier satellite reached an equilibrium temperature of ‐8±2°C after about 35 hours. The

sensitivity of the equilibrium temperature due to variation of dark period and solar constant are

summarized in Table 6.2.4.

Table 6.2.3. Equilibrium Temperature Sensitivity

Imaging Satellite Carrier Satellite

Variation of dark period ±7.7°C ±7.0°C

Variation of solar constant ±1.5°C ±1.4°C

Most of the components aboard the satellites would operate under these temperature

conditions. However, the batteries would not perform adequately under these temperatures.

Hardware The thermal control for the majority of components is passive and no special treatment

is needed for surfaces to maintain operational temperature. The batteries will require patch

heaters to raise their temperature to perform within their specifications. The batteries should

be thermally insulated to reduce heat transfers between the batteries and other surfaces. This

will reduce the amount of power consumed by the patch heaters.

Temperature sensors provide information on the temperature of components.

Temperature sensors placed throughout the satellite will provide a temperature map of the

satellite and will determine if a component is approaching a nonoperational temperature.

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Common commercial temperature sensors are inexpensive and are accurate to within a degree

Celsius.

Conclusion The equilibrium temperatures of the imaging and carrier satellites will provide

operational temperatures for the majority of components. The batteries in each satellite will

require patch heaters to maintain an operational temperature. The major assumptions made

were perfect thermal conduction between components and constant projected areas during an

orbit. Further analysis needs to be conducted to size the patch heaters for the batteries and

investigate internal heat transfers in the satellites.

6.3 Structures The structural design and internal layout of the CubeSats and carrier vehicles is important to establishing

a cost effective constellation. The structural aspect of this mission is comprised of the CubeSat design

and the carrier design. These components have different sets of governing requirements and are built to

both similar and different structural objectives. The design processes for both the CubeSat and carrier

are detailed below.

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CubeSat Design

The design of the CubeSats was based on the following primary parameters:

• Accommodating the large size of the telescope and attached CCD

• Balancing the layout of components to allow alignment between the propulsion and the

center of mass

• Withstanding the forces experienced during launch

• Providing protection from launch and space environments

In order to accomplish this while minimizing the volume of each CubeSat, a starting design was selected

that used the large dimensions of the telescope and CCD as the overall length of the satellite. The

component dimensions shown in Table 6.3.1 were then used to determine the additional CubeSat units

needed to accommodate the necessary hardware.

Table 6.3.1: CubeSat internal hardware dimensions

System Subsystem Component Volume Dimensions (cm)

GNC Nav GPS board 9.525 x 5 x 1.7 ADCS DC Motor for CMG

CMG Wheels (3 wheels) 3 Magnetorquers Magnetometer Star Tracker Camera Lens IMU Sun Sensor

2 x 2 x 3 r=2 2.54 x 2.54 x 1.9 0.3 x 0.3 x 0.14 4.4 x 4.4 x 2.54 3.35 dia, 3,6 length 3.5 x 2.2 x 1.2 2.43 dia, 3.49 length

Propulsion PPT Unit 2 x 1 x 8

Imaging, CDH CDH Motherboard 9 x 9 x 2 Communications Transceiver 7.8 x 5.2 x 2 Imaging CCD 2x1.5x0.75 Telescope 26.3 long, 10.6 diameter Support Systems Power 2 Batteries 3.9 x 4.2 x 0.06

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Based on the volume and layout estimations of the CubeSat hardware, a configuration comprising a 3U

structure attached in parallel to the telescope mounted in a 1U cube was chosen. Figure 6.3.1 shows a

dimensioned CAD rendition of this design with attached solar wings.

Figure 6.3.1: CubeSat design

An exploded view of the CubeSat hardware is shown in Figure 6.3.2.

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Figure 6.3.2: CubeSat exploded view

The total mass of each CubeSat is 4.6kg including all of the hardware shown in Figure 6.3.2. Allowing for

a margin of error, the CubeSat mass used in all calculations was 6kg. Based on the arrangement of

components, the center of mass is shown in Figure 6.3.3 with the principle axis of inertia shown.

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Figure 6.3.3: CubeSat center of mass

The moments of inertia about the principal axis shown in Figure 6.3.3 are as follows:

Ix=193000 g*cm2

Iy=523000 g*cm2

Iz=605000 g*cm2

All exterior surfaces of the CubeSat except the base are covered with solar cells. The wing configuration

enables the CubeSat to employ greater rotational freedom with maximum power collection capability

and not be constricted to an attitude at nadir pointing. It is still to be determined what the most

efficient default attitude will be or whether it would be possible to generate sufficient power at all times

without returning the CubeSat to nadir pointing after each picture is taken.

Carrier Design

The layout of the carriers was designed around the following parameters:

• Accommodating the 10 CubeSats required to populate each plane

• Containing the propulsion and instrumentation needed to ensure proper positioning of the

carriers.

• Balancing the layout of components to allow alignment between the propulsion and the center

of mass

• Withstanding the forces experienced during launch

• Providing protection from launch and space environments

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In order to accomplish this while minimizing the volume of each carrier, a starting design was selected

that held the 10 CubeSats in a grid fashion, with one side as the ejection side for all CubeSats. Figure

6.3.4 shows a dimensioned CAD rendition of this design with a CubeSat protruding from its slot.

Figure 6.3.4: Front view of carrier design

An illustration of the CubeSat containment and release mechanisms is shown in Figure 6.3.5. The

CubeSat is held in its pigeon hole by a spring loaded catch. When the CubeSat is released, the

compressed spring shown is released to deliver the required delta V to the CubeSat to transfer it into

orbit.

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Figure 6.3.5: CubeSat held in carrier slot

Additional volume was added to the rear of the CubeSat holding grid to include the propulsion and

instrumentation required to position the carriers to launch the CubeSats into their correct orbits. An

artist rendition of the propulsion system is shown in Figure 6.3.6.

Figure 6.3.6: Rear view of carrier design

In order to deliver the carriers to their respective orbits, Falcon 1e launch vehicles will be used to carry 6

carriers at a time in an arrangement shown in Figure 6.3.7.

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Figure 6.3.7: Falcon 1e nosecone packing arrangement

An additional option considered for deploying the carriers was to launch them as a secondary payload

on a Delta model launch vehicle. The Evolved Expendable Launch Vehicle (EELV) Secondary Payload

Adapter (ESPA) ring shown in Figure 6.3.8 would be used to attach the carriers to the launch vehicle.

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Figure 6.3.8: ESPA ring

ESPA is a joint program developed by the DoD Space Test Program and the Space Vehicles Directorate of

the Air Force Research Laboratory. It is designed to carry six small satellites as secondary payloads, each

weighing up to 170kg. The ESPA ring is fitted between the launch vehicle and the primary payload as

shown in Figure 6.3.9.

Figure 6.3.9: ESPA payload configuration

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Advantages of using this launch system include ejecting secondary payloads into orbits independent of

the primary payload and ease of accessibility of the secondary spacecraft after encapsulation through

fairing access doors. This provides opportunity for checking spacecraft batteries, health, and removing

inhibit plugs within only several days of launch.

The significant disadvantage to using the ESPA ring is that it mounts the carriers in a cantilever type

fashion. Unlike a typical launch system where the thrust axis is parallel to the separation system axis and

produces a compressive force on the spacecraft, a cantilever-mounted structure generates significant

bending moment. This imposes more stringent structural requirements on the carrier. The dimensions and

mass of the carrier are also limited when using the ESPA ring. The ESPA payload requirements are

shown in Table 6.3.2

Table 6.3.2: ESPA payload requirements

Dimension Requirement Width 60cm Height 60cm Length (from flange mount) 96cm

C.G. (from flange mount) 48cm

Flange Mount Diameter 38cm Mass 170kg

The carriers have been designed to fit within the requirements of the ESPA ring to be able to consider

this as a viable option.

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Structural Support

In order for the carriers and CubeSats to survive the launch environment, a force profile must be

generated based on the information detailed in the Payload Planner’s Guide. Using a Delta IV Heavy as

an example launch vehicle, Figure 6.3.6 shows the “stop sign” graph for the design load factors for

dynamic envelope requirements.

Figure 6.3.6: Delta IV Heavy Design Load Factors for Dynamic Envelope Requirements

Carrier and CubeSat load models must be generated and evaluated based on the acceleration dynamic

envelope. These models will be used to determine the structural requirements of the satellites.

The most viable material to use for the structural elements of both CubeSats and carriers is the

aluminum alloy 6061‐T6. This is attributed to its commercial availability, ease of manufacture, and

relatively low cost. The material properties for 6061‐T6 aluminum are detailed in Table 6.3.3 below.

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Table 6.3.3: Properties of aluminum alloy 6061‐T6

Material Property Magnitude

Modulus of Elasticity 68.9 GPa

Tensile Yield Strength 276 MPa

Poisson’s Ratio 0.33

Density 2700 kg/m^3

Coefficient of Thermal Expansion (20‐100 C) 23.6 x 10^‐6

Composite materials may be a viable alternative to aluminum in select locations. Composites have the

valuable benefits of having a much lower density than metals and yield strength around 10 times the

yield strength of aluminum. Despite their advantages, composites are prohibitively expensive, requiring

that the benefits outweigh the added cost. Due to these qualities, the primary components under

consideration for composite construction are fuel tanks and structural members requiring high strength

characteristics. Additional studies must be conducted to determine the plausibility of using composite

materials in the construction of the CubeSats and/or carriers.

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7. Systems Overview The design of this mission as outlined in this report has satisfied the requirements outlined. In addition,

the constellation has remained within the mass and cost limits imposed on the mission. Following are

the details of these mass and cost allocations.

Satellite mass The mass requirement per CubeSat is 10kg, and based on the values shown in Table 7.1, the actual

mass of each CubeSat will be significantly less than the limit. This will be advantageous for evaluating

launch cost and structural design.

Table 7.1: CubeSat mass analysis

Subsystem Component Mass Imaging CCD 50g Telescope 1.6kg Navigation GPS 80g Propulsion PPT 320g ADCS 2 Star Trackers 220g CMG (Wheels and

Motor) 100g

3 Magnetorquers 240g

IMU 30g 2 Sun Sensors 80g

CDH Computing 100g Communications 200g Power Batteries 40g Solar Arrays 300g Structures 4 CubeSat Units 600g Total 3.9kg

Allowing for Uncertainty 6.0kg

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Using a large percentage of the same components as in the CubeSats as well as the additional

hardware required for each carrier, the total mass of each carrier is detailed in Table 7.2 below.

Table 7.2: Carrier mass analysis

Subsystem Component Mass Navigation GPS 80g Propulsion Main Engine 1.01kg Valves 4.56kg

16 Thrusters 5.28kg

Fuel/Pressurant 44.2kg

ADCS Star Tracker 220g IMU 30g 2 Sun Sensors 80g

CDH Computing 100g Communications 200g Power Batteries 2.5g Solar Arrays 10kg Structures Structural Support 600g Cargo 10 CubeSat

Satellites 30kg

Total 160kg Allowing for Uncertainty 170kg

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Mission Cost With a maximum mission cost of $200 million, the cost analysis was broken into individual CubeSat

cost, individual carrier cost, and total CubeSat, carrier, and launch cost.

The individual CubeSat cost detailed in Table 7.3 is based on the hardware currently under

consideration. Additional system development is still required, so the CubeSat component costs may

change as the design is perfected.

Table 7.3: CubeSat cost analysis

Subsystem Component Cost Imaging CCD + Processor $10K Telescope $1K Navigation GPS $10K Propulsion PPT $10K ADCS 2 Star Trackers $2K 3 Sun Sensor $17K

CMG $2K IMU $1K CDH Computing $4K Communications $3K Power Batteries $1K Solar Arrays $7K Thermal Heaters, MLI $1K

Structures 4 CubeSat Units $16K Total $85K

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The individual carrier cost detailed in Table 7.4 is based using the same hardware architecture as for

the CubeSats. Due to the size difference, components such as the propulsion system and attitude

control system will need to be scaled up.

Table 7.4: Carrier cost analysis

Subsystem Component Cost Propulsion Engine/Propellant $200K Navigation GPS $20K ADCS Star Tracker 2pcs $2K Sun Sensor 3pcs $15K

IMU $2K CDH Computing $4K Communications $3K Power Batteries/Solar Cells $50K Structures $80K Total $426K

The total mission cost estimate shown in Table 7.5 calculates the total mission cost based on the

individual CubeSat and individual carrier costs, and the average estimated launch cost per 3 carriers

from Tables A.3.1 and A.3.2. As shown, our estimated total mission cost is well under our $200 million

limit, allowing reasonable room for additional cost requirements. A point to be taken into account is

that the costs of engineering and labor are not included in the total mission cost estimate. This may

considerably affect the total cost.

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Table 7.5: Total Mission Cost

Stage # Units Cost

CubeSats 330 $30M

Carriers 33 $15M

Launch $100K Total $145K

From our analysis over the last two quarters, we have met our revised requirements; a constellation

design has been established that will provide 5 minute coverage intervals, an imaging system that is

capable of 3m nadir resolution, and a communications architecture has been designed that will allow

images to be delivered within 1 hour of a command. The problems encountered while designing a

system to meet these criteria have brought about many inventive solutions. Additional perfecting must

be done in order to bring this mission into reality, but significant progress has been made toward that

goal.

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Appendices

A. Orbital Mechanics

A.1. Constellation Design The following constants and equations were used to analyze the coverage of each constellation configuration considered. The radius Rearth of the Earth is (Eqn.A.1.1) The gravitational constant µearth of the Earth is (Eqn.A.1.2) The radius of the satellite’s orbit Rsat is defined as the sum of Earth’s radius Rearth and altitude h. (Eqn.A.1.3) The velocity Vsat of the satellite in orbit is calculated as follows.

(Eqn.A.1.4)

The period of the resulting orbit Torbit is calculated from the circumference and velocity of the orbit. (Eqn.A.1.5)

The velocity of the ground track Vground can be determined from circumference of the Earth and the period of the orbit. (Eqn.A.1.6)

As the cubesat travels above the earth, it sweep out a certain area that is within its field of view. This swept out area is called its coverage swath, illustrated in Figure A.2.1 below.

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Figure A.2.1: Side and top view of coverage swath for each cubesat

The swath width wswath is calculated based on the altitude h and the maximum slew angle θ from nadir. (Eqn.A.1.7) The necessary swath length lswath is determined from the ground track velocity Vground and the required revisit time trevisit . (Eqn.A.1.8) The above parameters are used to determine the number of planes and satellites per plane necessary for a polar constellation with full earth coverage. The number of planes Nplanes is determined from the circumference of the Earth and the swath width wswath. Half the circumference is used since each orbital plane provides coverage over two opposite points on the Earth in ascension and descension. (Eqn.A.1.9)

The number of satellites per plane Nsats/plane is determined from the total circumference of the Earth and the swath length lswath. (Eqn.A.1.10)

For a Walker constellation, the number of planes is determined differently due to the inclination. The spacing snodes of the right ascension nodes depends on wswath and the sine of the inclination i. This is illustrated in Figure A.2.2 below for i = 60°and wswath = 1000 km.

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(Eqn.A.1.11)

Figure A.2.2: Nodes spacing for inclined orbits

The entire circumference of the Earth is used to calculate the number of planes since the ascending nodes must be spread over 360° for a Walker constellation. (Eqn.A.1.12)

For a Walker constellation, each location where two orbits cross will have two spacecraft passing over within the revisit time (5 min), allowing the revisit time on an individual orbit to be twice as long (10 min). At the equator, each ascension node is paired with a descension node. Assuming worst-case coverage occurs at the equator, these ascension/descension nodes pairs must be spaced closely enough to eliminate any coverage gap between orbits. Therefore, the number of satellites per plane can be reduced to half the number needed for a polar constellation. (Eqn.A.1.13)

To ensure that one spacecraft always crosses the ascension/descension node within the revisit time, they must be phased such that they cross alternately at that point, as illustrated in Figure A.2.3. This figure shows one crossing node, with a 10-minute revisit time on a single orbit and a 5-minute revisit time at the crossing node.

Figure A.2.3: Crossing node at equator, with 5-minute revisit time

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This phasing is best explained by example. Assume 36 planes and 10 spacecraft per plane. The 10 spacecraft with have true anomalies spacing of 36° around the single orbit. The two orbits that cross at given point along the equator will have a right ascension Ω that differs by 180°. The spacecraft on the second of these orbits must cross halfway in between the spacecraft of the first orbit, so their true anomalies ν will be spaced out by 36°, but must be offset by 18°. Orbit 0: Ω = 0° ν = 0°, 36°, 72°, etc. Orbit 18: Ω = 180° ν = 18°, 54°, 90°, etc. Thus the phasing varies 18° over 18 planes, so each plane must have its spacecraft phasing by 1° from the previous plane. This phasing angle is based on the assumption of worst-case coverage at the equator, but since this assumption is incorrect, full coverage at the equator actually does not guarantee full coverage everywhere else. Thus other phasing options must be considered. Polar Streets Analysis The MATLAB code presented below was used to conduct analysis for the polar streets constellation, to find the number of planes and satellites per plane needed for a given altitude, as well as the associated delta-V required to deploy the constellation. % AA 421 Polar Streets Analysis % Peter Gangar % 04/13/09 clear all close all clc % Earth constants Re=6378; % km, radius of the earth mu=398600; % km^3/s^2, gravitational constant of the earth Ce=2*pi*Re; % km, circumference of the earth g0=9.81; % m/s^2, gravitational acceleration of the earth Ihyd=325; % sec, Isp for hydrazine % Orbital parameters h=520; % km, altitude for 20 planes with 20 sats each fov=45; % deg, field of view tr=5; % min, revisit time % Derived parameters rs=Re+h; % km, radius of sat orbit vs=sqrt(mu/rs); % km/s, velocity of sat orbit cs=2*pi*rs; % km, circumference of sat orbit

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T=cs/vs/60; % min, period of sat orbit vg=Ce/(T*60); % km/s, ground speed of sat ws=2*h*tan(fov*pi/180); % km, swath width ls=vg*tr*60; % km, swath length np=ceil(Ce/2/ws); % integer # of planes ns=ceil(Ce/ls); % integer # of sats per plane npc=np-1; % integer # of plane changes % Cubesat and Carrier masses ms=6; % kg, mass of cubesat mc=40; % kg, mass of carrier me=10*np; % kg, extra mass for supporting structure mt=me+np*(mc+ns*ms); % kg, total mass to orbit i1=70; % degrees, inclination of launch orbit i2=90; % degrees, inclination of final orbit di=(i2-i1)*pi/180; % radians, inclination change dvpc=2*vs*sin(di/2); % km/s, delta-V for plane change to polar ra=2*pi/np; % deg, RAAN spacing dvra1 = 2*vs*sin(ra/2); % km/s, delta-V for a RAAN change MR= exp(-(dvra1*1000)/(g0*Ihyd)); % mass ratio (initial over final) % Loop backward for each plane change to find initial mass for p = 1:npc m(1) =(mc+ns*ms); % kg, final mass after plane change m(p+1) = m(p)/MR; % kg, initial mass before plane change after carrier jettison m(p+1) =m(p+1) + ns*ms + np*mc; % kg, initial mass before carrier jettison end dvratot=npc*dvra1; mptot = m(npc)-(ns*ms)-(np*mc)-me; dv1=vs; % km/s, delta-V to orbital velocity dv2=dvpc; % km/s, delta-V for plane change to polar dv3=dvratot; % km/s, delta-V for RAAN changes dvtot=dv1+dv2+dv3; % km/s, total delta-V

Walker Analysis The FORTRAN code presented below was used to conduct analysis for the Walker constellation, to find the fraction of points covered with the specified revisit time, and the maximum revisit time gap for given configurations of the constellation.

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C WALKERA.FOR C ANALYSIS OF WALKER CONSTELLATIONS. C USER SPECIFIES ALTITUDE, INCLINATION, AND FIELD-OF-VIEW OF SPACECRAFT C [FOV IS MAX ANGLE OF SPACECRAFT FROM NADIR POINTING] C USER SPECIFIES PARAMETER RANGE FOR WALKER CONSTELLATIONS TO ANALYZE: C MIN AND MAX VALUES OF # PLANES, NP, MIN AND MAX # SATS/PLANE, NS, C AND MAX # OF SATS IN CONSTELLATION, NSATS. C USER SPECIFIES A "TARGET" REVISIT TIME (FOR COMPUTING FRACTION OF C COVERAGE AREA MEETING THAT TIME). C THIS ROUTINE RUNS THROUGH NP AND NS VALUES. C AT EACH NP,NS, IT RUNS THROUGH ALL ALLOWABLE VALUES OF PHASE INTEGER NF. C AT EACH NP/NS/NF (A SPECIFIC WALKER CONSTELLATION), IT RUNS THROUGH ALL C LATITUDES FROM 0 TO ORBIT INCLINATION, AND A RANGE OF LONGITUDES. C AT EACH LAT/LONG IT FINDS MAX TIME THAT THIS LOCATION IS NOT IN VIEW OF C A SPACECRAFT. C FOR A GIVEN NP/NS IT REPORTS THE "BEST" NF - THE ONE WITH MINIMUM C VALUE OF MAX OUT-OF-VIEW TIME WITHIN COVERAGE AREA. C C OUTPUT FILES: C WTIME.TXT: MAX-TIME-OUT-OF-VIEW(MINUTES) NSATS NP NS NF(BEST) C WFRAC.TXT: FRAC NSATS NP NS NF(BEST) C WHERE FRAC=FRACTION OF COVERAGE AREA HAVING OUT-OF-VIEW TIME C LESS THAN TARGET VALUE SPECIFIED BY USER. C INFO: WALKER CONSTELLATION HAS NP PLANES, EACH AT SAME INCLINATION. C PLANES ARE EQUALLY SPACED IN RA. THERE ARE NS SATS IN EACH PLANE, C SPACED EQUALLY AROUND PLANE. PHASE DIFFERENCE BETWEEN PLANES SPECIFIED C BY INTEGER NF. PHASE DIFFERENCE (FROM ASCENDING NODE) BETWEEN ADJACENT C PLANES GIVEN BY D_PHASE(DEG)=NF*360./NSATS, NSATS=NP*NS. C NF CAN TAKE ON VALUES 0 TO NP-1. PROGRAM WALKERA IMPLICIT REAL (A-H,Q-Z) DIMENSION TLV(400),TGV(400),TLO(400),TGO(400) DIMENSION IUSE(400) DIMENSION AL(0:100),P(0:100,0:100) DIMENSION NPV(10000),NSV(10000),NFV(10000) DIMENSION FRAC(10000) DIMENSION NPL(1000),NSL(1000),NFL(1000) DIMENSION TM4(1000) 1 FORMAT(A,\) RE=6380E3 WRITE(*,1) ' ENTER ALTITUDE IN KM: ' READ(*,*) HKM H=HKM*1000 R=RE+H XMU=4E14 Z=1 PI=4*ATAN(Z) PI2=2*PI TP_SEC=PI2*SQRT(R**3/XMU) TP=TP_SEC/60 W=PI2/TP

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WRITE(*,1) ' ENTER INCLINATION IN DEG: ' READ(*,*) XINC_DEG LINC=XINC_DEG+.001 XINC=XINC_DEG*PI/180 SI=SIN(XINC) CI=COS(XINC) WRITE(*,1) ' ENTER FOV IN DEG: ' READ(*,*) FOV_DEG FOV=FOV_DEG*PI/180 SF=SIN(FOV) CF=COS(FOV) Y=1+H/RE SA=SF*(Y*CF-SQRT(1-(Y*SF)**2)) A=ASIN(SA) CA=COS(A) CA2=CA*CA WRITE(*,1) ' ENTER TARGET REVISIT TIME IN MINUTES: ' READ(*,*) TRV WRITE(*,1) ' ENTER MIN AND MAX NUMBER OF PLANES: ' READ(*,*) NPMIN,NPMAX WRITE(*,1) ' ENTER MIN AND MAX # SATS/PLANE: ' READ(*,*) NSMIN,NSMAX WRITE(*,1) ' ENTER MAXIMUM # OF SATS IN CONSTELLATION: ' READ(*,*) NTMAX TMAX=20 NLONG=100 ! # LONGITUDE POINTS TO SAMPLE AT A GIVEN LAT KTOT=0 DO 600 NP=NPMIN,NPMAX NSLAST=NSMAX NS2=NTMAX/NP IF(NS2 .LT. NSMAX) NSLAST=NS2 DO 500 NS=NSMIN,NSLAST NT=NS*NP DTHETA=PI2/NS ! SAT-SAT ANGLE IN A PLANE DRA=PI2/NP ! PLANE-PLANE ANGLE IN RA DA_LONG=DRA/NLONG ! LONGITUDE SAMPLE INCREMENT PU=PI2/NT TMIN4=1000 ! TMIN4=MIN OFF TIME OVER ALL SAMPLED NF VALUES DO 400 NF=0,NP-1 ATRY=0 ARV=0 DPHI=PU*NF

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DO 20 N=0,NP-1 AL(N)=DRA*N DO 10 M=0,NS-1 P(N,M)=(N*DPHI+M*DTHETA) 10 CONTINUE 20 CONTINUE TOFF3=0 ! TOFF3=MAX OFF TIME AT THIS NF DO 300 ILAT=0,LINC+5 ALAT_DEG=ILAT ALAT=ALAT_DEG*PI/180 SLAT=SIN(ALAT) CLAT=COS(ALAT) IF(ILAT .EQ. 0) THEN AREA=.5 ELSE AREA=CLAT ENDIF TOFF2=0 ! TOFF2= MAX OFF TIME AT THIS LAT DO 200 ILONG=0,NLONG ALONG=ILONG*DA_LONG TOFF1=0 ! MAX OFF TIME AT THIS LAT AND LONG L=0 DO 95 N=0,NP-1 DO 94 M=0,NS-1 QC=CLAT*COS(ALONG-AL(N)) QS=CLAT*CI*SIN(ALONG-AL(N))+SLAT*SI Q2=QC*QC+QS*QS IF(Q2 .LE. CA2) GOTO 94 Q=SQRT(Q2) AV=ACOS(CA/Q) QSA=ABS(QS) QCA=ABS(QC) IF(QSA .LT. QCA) THEN E=ATAN(QSA/QCA) ELSE E=PI/2-ATAN(QCA/QSA) ENDIF IF(QC .GT. 0) THEN IF(QS .GT. 0) THEN B=-E ELSE B=E ENDIF ELSE IF(QS .GT. 0) THEN B=E+PI ELSE B=PI-E ENDIF ENDIF P0=-P(N,M)-B

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TL=(P0-AV)/W TG=(P0+AV)/W IF(TL .GT. TP) THEN 92 CONTINUE TL=TL-TP TG=TG-TP IF(TL .GT. TP) GOTO 92 ENDIF IF(TG .LT. 0) THEN 93 CONTINUE TL=TL+TP TG=TG+TP IF(TG .LT. 0) GOTO 93 ENDIF IF(TL .GT. TMAX) THEN TL=0 TG=TG-TP IF(TG .GT. 0) THEN L=L+1 TLV(L)=TL TGV(L)=TG ENDIF ELSE IF(TL .LT. 0) TL=0 IF(TG .GT. 0) THEN IF(TG .GT. TMAX) TG=TMAX L=L+1 TLV(L)=TL TGV(L)=TG ENDIF ENDIF 94 CONTINUE 95 CONTINUE IF(L .EQ. 0) THEN TOFF1=TMAX GOTO 180 ENDIF C ORDE R THE TIME PERIODS BY TL: DO 110 I=1,L IUSE(I)=0 110 CONTINUE DO 120 I=1,L TMIN=10000 DO 115 J=1,L IF(IUSE(J) .NE. 0) GOTO 115 IF(TLV(J) .LT. TMIN) THEN TMIN=TLV(J) JMIN=J ENDIF 115 CONTINUE TLO(I)=TMIN TGO(I)=TGV(JMIN) IUSE(JMIN)=1

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120 CONTINUE IF(TLO(1) .GT. .0001) THEN DT=TLO(1) IF(DT .GT. TOFF1) TOFF1=DT TON=TLO(1) ELSE TON=0 ENDIF TOFF=TGO(1) DO 140 I=2,L IF(TLO(I) .LE. TOFF) THEN IF(TGO(I) .GT. TOFF) TOFF=TGO(I) ELSE TON=TLO(I) DT=TON-TOFF IF(DT .GT. TOFF1) TOFF1=DT TOFF=TGO(I) ENDIF 140 CONTINUE 180 CONTINUE IF(TOFF1 .GT. TOFF2) TOFF2=TOFF1 IF(ILAT .LE. LINC) THEN ATRY=ATRY+AREA IF(TOFF1 .LE. TRV) ARV=ARV+AREA ENDIF 200 CONTINUE IF(ILAT .LE. LINC) THEN IF(TOFF2 .GT. TOFF3) THEN TOFF3=TOFF2 C ILAT_TOPS=ILAT ENDIF ENDIF 300 CONTINUE IF(TOFF3 .LT. TMIN4) THEN TMIN4=TOFF3 NFMIN=NF ENDIF KTOT=KTOT+1 NPV(KTOT)=NP NSV(KTOT)=NS NFV(KTOT)=NF FRAC(KTOT)=ARV/ATRY 400 CONTINUE C WRITE(1,2) NP,NS,NT,NFMIN,TMIN4 WRITE(*,2) NP,NS,NT,NFMIN,TMIN4 LTOT=LTOT+1 NPL(LTOT)=NP NSL(LTOT)=NS NFL(LTOT)=NFMIN

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TM4(LTOT)=TMIN4 500 CONTINUE 600 CONTINUE 2 FORMAT(1X,4I4,F8.4) OPEN(1,FILE='WTIME.TXT') DO 900 I=1,LTOT TMN=1000 JMN=0 DO 850 J=1,LTOT IF(TM4(J) .LT. 0) GOTO 850 IF(TM4(J) .LT. TMN) THEN TMN=TM4(J) JMN=J ENDIF 850 CONTINUE IF(JMN .EQ. 0) GOTO 901 NP=NPL(JMN) NS=NSL(JMN) NF=NFL(JMN) NT=NP*NS WRITE(1,6) TM4(JMN),NT,NP,NS,NF TM4(JMN)=-1 900 CONTINUE 901 CONTINUE CLOSE(1) OPEN(1,FILE='WFRAC.TXT') DO 700 I=1,KTOT F=0 JMX=0 DO 650 J=1,KTOT IF(FRAC(J) .GT. 2) GOTO 650 IF(FRAC(J) .GT. F) THEN F=FRAC(J) JMX=J ENDIF 650 CONTINUE IF(JMX .EQ. 0) GOTO 710 NP=NPV(JMX) NS=NSV(JMX) NF=NFV(JMX) NT=NP*NS WRITE(1,6) FRAC(JMX),NT,NP,NS,NF FRAC(JMX)=5 DO 680 J=1,KTOT IF(NPV(J) .EQ. NP .AND. NSV(J) .EQ. NS) FRAC(J)=5 680 CONTINUE 700 CONTINUE 710 CONTINUE CLOSE(1) 6 FORMAT(1X,F7.3,1X,I5,2X,3I3) STOP END

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A.2. Deployment Method Carrier Deployment The MATLAB code presented below was used to conduct analysis of the various deployment methods once a specific Walker constellation configuration had been chosen. It is used to find delta-V and propellant mass for the master carrier method, delta-V for the clustered launch method, and delta-V and deployment vs. launch inclination for the precession method. %% AA 421 Walker Analysis % Peter Gangar % 05/13/09 clear all close all clc % Earth constants Re=6378; % km, radius of the earth mu=398600; % km^3/s^2, gravitational constant of the earth Ce=2*pi*Re; % km, circumference of the earth g0=9.81; % m/s^2, gravitational acceleration of the earth Ihyd=325; % sec, Isp for hydrazine % Orbital parameters h=520; % km, altitude for 20 planes with 20 sats each fov=45; % deg, field of view tr=5; % min, revisit time i=55; % deg, inclination of orbital planes % Derived parameters rs=Re+h; % km, radius of sat orbit vs=sqrt(mu/rs); % km/s, velocity of sat orbit cs=2*pi*rs; % km, circumference of sat orbit T=cs/vs/60; % min, period of sat orbit vg=Ce/(T*60); % km/s, ground speed of sat ws=2*h*tan(fov*pi/180); % km, swath width ls=vg*tr*60; % km, swath length np=33; % integer # of planes ns=10; % integer # of sats per plane % Cubesat and Carrier masses ms=10; % kg, mass of cubesat mc=25; % kg, mass of carrier

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me=5*np; % kg, extra mass for structure supporting carriers mt=me+np*(mc+ns*ms); % kg, total mass to orbit % Spacing distances and angles ra=360/np; % deg, spacing of right ascension sa=Ce/np; % km, spacing of right ascension so=sa*sin(i*pi/180); % km, spacing of planes (orthogonal to flight path) dt=so/Ce*360; % deg, angle for plane change at polar node % Deployment Methods %% Master Carrier npc=np-1; % integer # of plane changes dvp1 = 2*vs*sin(dt*pi/180/2); % km/s, delta-V for plane change at polar node MR= exp(-(dvp1*1000)/(g0*Ihyd)); % mass ratio (mf/m0) % Loop backward for each plane change to find initial mass for p = 1:npc m(1) =(mc+ns*ms); % kg, final mass after plane change m(p+1) = m(p)/MR; % kg, initial mass before plane change after carrier jettison m(p+1) =m(p+1) + ns*ms + np*mc; % kg, initial mass before carrier jettison end dvptot=npc*dvp1; % km/s, total delta-V for deployment mptot = m(npc)-(ns*ms)-(np*mc)-me; % kg, mass of propellant %% Clustered Launches nps=np*2/3; % integer # of plane change split-offs dvc1=dvp1; % km/s, delta-V for plane change at polar node dvctot=nps*dvc1; % km/s, total delta-V for deployment %% Precession i0=[45:0.1:i-1 , i+1:0.1:65]; % degrees, inclination of insertion orbit for n=1:length(i0) thtpc = abs((i-i0(n))*pi/180); % radians, theta for plane change dv1(n) = 2*vs*sin(thtpc/2); % km/s, delta-V for plane change dvtot(n) = np*dv1(n); % km/s, delta-V for all plane changes j2 = -1.083*10^-3; di(n) = ((3*j2*(2*pi/(T*60))*6378^2)/(2*rs^2))*cos(i*pi/180)*(3600*24)*(180/pi); di0(n) = ((3*j2*(2*pi/(T*60))*6378^2)/(2*rs^2))*cos(i0(n)*pi/180)*(3600*24)*(180/pi);

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ddomega(n) = di(n) - di0(n); % degrees/day, relative precession rate t1(n)=ra/abs(ddomega(n)); % days, precession time for one right ascension ttot(n) = npc*ra/abs(ddomega(n)); % days, precession time for all planes to deploy end figure(1) plot(i0,dv1) grid xlabel('Launch Inclination (degrees)') ylabel('Delta-V for single plane change (km/s)') figure(2) plot(i0,dvtot) grid xlabel('Launch Inclination (degrees)') ylabel('Total Delta-V for all plane changes (km/s)') figure(3) plot(i0,t1) grid xlabel('Launch Inclination (degrees)') ylabel('Precession Time Between Planes (days)') figure(4) plot(i0,ttot) grid xlabel('Launch Inclination (degrees)') ylabel('Total Precession Time (days)') figure(5) plot3(i0,dv1,t1) grid xlabel('Launch Inclination (degrees)') ylabel('Delta-V for single plane change (km/s)') zlabel('Precession Time Between Planes (days)') Cubesat Deployment The cubesats must be spaced out evenly along their orbit. The time spacing Tspacing within a single orbital is the number of cubesats Nsats divided by the period of the cubesat orbit Tsat. (Eqn.A.2.1)

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To deploy each cubesat with the correct phasing, the elliptical orbit of the carrier must be designed to return to the circular cubesat orbit either ahead or behind the cubesat orbit by the amount Tspacing. (For this analysis, the case of a shorter period is assumed since the required delta-V does not vary significantly between the two cases.) The carrier must orbit several times between each deployment so that the elliptical orbit is close to circular and the delta-V at the deployment is achievable. Assuming an integer number Norbs of carrier orbits, the period of the elliptical carrier orbit is as follows. (Eqn.A.2.2)

The time T1 to deploy one cubesat, and the time Tall to deploy all cubesats can now be found. (Eqn.A.2.3) (Eqn.A.2.4) The semi-major axis a, the perigee and apogee radius rp and ra respectively, and the eccentricity e are then calculated as follows, where rsat is the radius of the circular cubesat orbit. (Eqn.A.2.5)

(Eqn.A.2.6) (Eqn.A.2.7) (Eqn.A.2.8)

The velocity at the deployment point can then be calculated by finding the angular momentum h, using it to find the velocity at apogee va of the carrier orbit. (Eqn.A.2.9) (Eqn.A.2.10)

The resulting delta-V is the difference between the velocity vc of the circular cubesat orbit and the velocity at apogee va of the elliptical carrier orbit.

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(Eqn.A.2.11) The MATLAB code presented below was used to conduct analysis of the possible options for elliptical carrier orbits to be used to deploy all cubesats on a given orbital plane with the correct spacing. It is used to calculate the perigee or apogee radius, single-maneuver delta-V, total delta-V, single-cubesat deployment time, and total deployment time corresponding to the integer number of orbits allowed between deployments. % AA 420 Cubesat Deployment % Peter Gangar % 03/09/09 clear all close all clc % Analysis for eliptical carrier orbit for cubesat deployment % Physical constants mu=398600; % km^3/s^2, GM of the earth re=6378; % km, radius of the earth % Set parameters h=520; % km, altitude of cubesat orbit Ns=10; % number of cubesats to deploy % Cubesat orbit rc=re+h; % km, radius of cubesat orbit vc=sqrt(mu/rc); % km/s, velocity of cubesat orbit Tc=2*pi*rc^(3/2)/sqrt(mu); % s, period of cubesat orbit % Find transfer orbit for deployment at apogee of smaller orbit for Ne=1:50 % integer number of carrier orbits between deployment Te=Tc*(1-1/(Ne*Ns)); % s, period of carrier orbit less than circular cubesat orbit a=(Te*sqrt(mu)/2/pi)^(2/3); % semi-major axis of carrier orbit smaller than cubesat orbit rp=2*a-rc; % km, radius less than cubesat orbit, rp for carrier orbit ra=rc; % km, radius equal to cubesat orbit, ra for carrier orbit e=(ra-rp)/(ra+rp); % eccentricity of carrier orbit h=sqrt(rp*mu*(1+e)); % km^2/s, angular momentum va=h/ra; % km/s, velocity at apogee dv=(vc-va)*1000; % m/s, delta-V for one orbit change maneuver DV=2*(Ns-1)*dv; % m/s, delta-V to deploy all cubesats T1=Ne*Te/3600/24; % s, time to deploy one cubesat Td=(Ns-1)*T1; % days, time to deploy all cubesats output1(Ne,:)=[Ne rp dv DV T1 Td]; % writes results to matrix end

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% Find transfer orbit for deployment at perigee of larger orbit for Ne=1:50 % integer number of carrier orbits between deployment Te=Tc*(1+1/(Ne*Ns)); % s, period of carrier orbit more than circular cubesat orbit a=(Te*sqrt(mu)/2/pi)^(2/3); % semi-major axis of carrier orbit larger than cubesat orbit ra=2*a-rc; % km, radius more than cubesat orbit, ra for carrier orbit rp=rc; % km, radius equal to cubesat orbit, rp for carrier orbit e=(ra-rp)/(ra+rp); % eccentricity of carrier orbit h=sqrt(rp*mu*(1+e)); % km^2/s, angular momentum vp=h/rp; % km/s, velocity at apogee dv=(vp-vc)*1000; % m/s, delta-V for one orbit change maneuever DV=2*(Ns-1)*dv; % m/s, delta-V to deploy all cubesats T1=Ne*Te/3600/24; % s, time to deploy one cubesat Td=(Ns-1)*T1; % days, time to deploy all cubesats output2(Ne,:)=[Ne ra dv DV T1 Td]; % writes results to matrix end figure(1) plot(output1(:,1),output1(:,3),output2(:,1),output2(:,3)) grid xlabel('Number of Carrier Orbits') ylabel('Delta-V to Change Orbits (m/s)') legend('Smaller Orbit', 'Larger Orbit') figure(2) plot(output1(:,1),output1(:,4),output2(:,1),output2(:,4)) grid xlabel('Number of Carrier Orbits') ylabel('Total Delta-V for Deployment (m/s)') legend('Smaller Orbit', 'Larger Orbit') figure(3) plot(output1(:,1),output1(:,5),output2(:,1),output2(:,5)) grid xlabel('Number of Carrier Orbits') ylabel('Time Between Deployments (days)') legend('Smaller Orbit', 'Larger Orbit') figure(4) plot(output1(:,1),output1(:,6),output2(:,1),output2(:,6)) grid xlabel('Number of Carrier Orbits') ylabel('Total Deployment Time (days)') legend('Smaller Orbit', 'Larger Orbit') output2(50,:)

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B. Propulsion

B.1 References Capacitor Sizing: https://www.avx.com/docs/catalogs/bestcap.pdf

Capacitor Equations: q =5J = cV, V = 4V (http://www.mae.cornell.edu/campbell/pubs/IEEE002.pdf), required capacitance = 5J/4V = 1.2F

(AVX BestCap BZ-12 1000 microfarad at 5.5V input)

Dimensions: 48x30x2.9 (mm)

B.2 Equations (Note: Specific ranges of variables studied are given in the Matlab code)

Carrier Propulsion

The following set of equations were used to size the propellant and pressurant tankage system for carrier propulsion.

The propellant mass required for each maneuver, based upon the initial mass and delta v required can be calculated using Equation 1:

(Eqn.B.2.1)

For a blowdown tankage system, the blowdown ratio can be cacluated based upon the operating pressure range of the engine using Equation 2:

(Eqn.B.2.2)

The pressurant tank volume can be calculated, based upon the propellant tank volume and the blowdown ratio, as given in Equation 3:

(Eqn.B.2.3)

Satellite Lifetime:

To determine the satellite lifetimes without propulsion based on a range of insertion altitudes (Figure 8), the following expressions were used. First, utilizing Newton’s Second Law, we state that the change in kinetic energy of an orbital body is equal to the negative product of its velocity and the drag force acting upon it.

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(Eqn.B.2.4)

Next, from orbital mechanics we state that the energy of a body in a circular orbit is equal to minus the quotient of the gravitational parameter mu divided by twice the radius of the orbit. Differentiating, we arrive at an alternate expression for the drag power.

(Eqn.B.2.5)

(Eqn.B.2.6)

Once we have this expression, we can take as an assumption an exponential scaling of density with altitude, and also use the definition of the drag coefficient and the circular velocity equation from orbital mechanics to define all of the component terms in the equation immediately above.

(Eqn.B.2.7)

(Eqn.B.2.8)

(Eqn.B.2.9)

Equating the two sides of the preceding expression and cancelling terms, we attain an expression for the change in radius with time. This expression may then be rearranged and integrated to give the lifetime of the satellite, based on the starting height, density, frontal area, and drag coefficient.

(Eqn.B.2.10)

(Eqn.B.2.11)

(Eqn.B.2.12)

Propulsion Requirements

Determining the thrust required to hold orbit at the injection altitude begins with equating the thrust force needed to the drag force on the satellite. We can then simply use the result from the definition of the drag coefficient above to determine this required force.

(Eqn.B.2.13)

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To arrive at the total delta-v of the mission per year, we simply assume a constant deceleration from the drag force in the absence of any thrust, and multiply this acceleration by the time interval of interest.

(Eqn.B.2.14)

Once the delta-v for one year has been calculated, we can use this result to determine the amount of propellant required for our design cases, as follows: initially, the rocket equation is used to calculate the required mass fraction, which is then multiplied by the wet mass of the cube-sat to give the total propellant mass.

(Eqn.B.2.15)

(Eqn.B.2.16)

The mass of the propellant, as well as its volume, are used as metrics for evaluating the applicability of a design candidate. The volume of the propellant is calculated using the ideal gas law, as follows.

(Eqn.B.2.17)

Where R is the specific gas constant, and tank temperature and pressure were assumed to be 300K, and 10 MPa. For the case of solid propellants such as the Teflon used in the PPTs, the propellant volume was calculated by multiplying the known propellant mass by the density of the material. The length of the stick required was then calculated by assuming a one-centimeter diameter cylindrical propellant charge.

Matlab Implementation of Cube-sat Orbital Decay and Propulsion Requirements

clear all, close all; %AA420 Orbital Decay/Lifetime Trade Study Calculator %Rewrite 1/29/09 %Inputs m = 10; % Mass in kg Cdmin = 2.2; % Minimum Estimated Drag Coefficient Cdmax = 2.75; % Maximum Estimated Drag Coefficient Amin = 0.02; % Minimum Frontal Area (Incl. Solar Panels) in m^2 Amax = 0.15; % Maximum Frontal Area in m^2

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Hmin = 350; % Minimum Injection Altitude in km Hmax = 500; % Maximum Injection Altitude in km rho0 = 6.98*10^-12; % Density at Minimum Injection Altitude in kg/m^3 hstarmin = 54.8; % Scale Height at Minimum Altitude in km hstarmax = 68.7; %Scale Height at Maximum Altitude in km FOS = 1.2; %Margin for Orbital Lifetime rppt = 10; %PPT Radius in mm i = 65; %Inclination in degrees % Physical Constants rE = 6370; %Radius of Earth in km mu = 398000; %Earth Gravitational Constant g0 = 9.81; %Earth Gravitational Acceleration in m/s^2 % I: Propulsionless Decay Analysis %--------------------------------- %Calculate Density Variation over the specified range h0 = [Hmin:1:Hmax]; for j =1:length(h0) r(j) = h0(j)+rE; v(j) = sqrt(mu/r(j)); hstar(j) = hstarmin + ((hstarmax-hstarmin)/length(h0))*j; rho(j) = rho0*exp(-(h0(j)-Hmin)/hstar(j)); end %Perform Time Integration over range of Injection Altitudes and Area %Low Cd Case for u = 1:length(h0) Area(u) = Amin +((Amax-Amin)/length(h0))*u; for b = 1:length(h0) tlow(u,b) = (m/(r(b)*1000*v(b)*1000*Area(u)*Cdmin*rho(b))*hstar(b)*1000);%*(1-exp((h0(b)-Hmin)/hstar(b))); dayslow(u,b) = tlow(u,b)/(24*3600); if dayslow(u,b)>365*FOS && dayslow(u,b) < 400*FOS optlow(u,b) = dayslow(u,b); else optlow(u,b) = 0; end thigh(u,b) = (m/(r(b)*1000*v(b)*1000*Area(u)*Cdmax*rho(b))*hstar(b)*1000);%*(1-exp((h0(b)-Hmin)/hstar(b))); dayshigh(u,b) = tlow(u,b)/(24*3600); if dayshigh(u,b)>365*FOS && dayslow(u,b) < 400*FOS opthigh(u,b) = dayslow(u,b); else opthigh(u,b) = 0; end end end %Plot Data - No Propulsion

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figure(1), surf(h0,Area,dayslow), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Lifetime(Days)','FontSize',12) title('\itSatellite Lifetime vs. Injection Altitude','FontSize',16) figure(2), surf(h0,Area,optlow), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Lifetime(Days)','FontSize',12) title('\itOptimum Range for 1 year Satellite Lifetime-Low Cd','FontSize',16) % II: Propulsion Requirements %---------------------------- % Propellant Data Icg = 65; %Cold Gas Specific Impulse Ibi = 450; %Bipropellant Specific Impulse Ippt = 500; %Pulsed Plasma Thruster Specific Impulse Iion = 1000; %Ion Thruster Specific Impulse Ihyd = 240; %Hydrazine Monoprop Specific Impulse %Delta-V Requirement for y = 1:length(h0) for k = 1:length(h0) Drag(k) = 0.5*rho(k)*((v(k)*1000)^2)*Area(y)*Cdmin; Adrag(k) = Drag(k)/m; T(k) = 2*pi*sqrt((r(k)^3)/mu); N(k) = (3600*24*365)/T(k); vf(k) = v(k)-(Adrag(k)*T(k)); dv(k) = v(k)-vf(k); deltav(y,k) = dv(k)*N(k); end end %Propellant Mass, Volume and Tankage Info. appt = pi*(rppt/1000)^2; for z = 1:length(h0) for x = 1:length(h0) mpcg(z,x) = m-m/(exp(deltav(z,x)/(g0*Icg))); mpppt(z,x) = m-m/(exp(deltav(z,x)/(g0*Ippt))); mpion(z,x) = m-m/(exp(deltav(z,x)/(g0*Iion))); mphyd(z,x) = m-m/(exp(deltav(z,x)/(g0*Ihyd))); rtcg(z,x) = 100*((3/(4*pi))*((mpcg(z,x)*(8314/28)*300)/(2*10^7)))^(1/3); rthyd(z,x) = 100*((3/(4*pi))*(mphyd(z,x)*(8314/32)*300)/(2*10^7))^(1/3);

149

vppt(z,x) = mpppt(z,x)/2200; %Density of Teflon = 2.2 g/cm^3 =2200kg/m^3 lppt(z,x) = (vppt(z,x)/appt)*100; %Length of PPT in cm end end %Plot Data - Propulsion figure(3), surf(h0,Area,deltav), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Required Delta-V(m/s)','FontSize',12) title('\itDelta-V Requirement for Range of Altitudes and Areas','FontSize',16) figure(4), surf(h0,Area,mpcg), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Propellant Mass (kg)','FontSize',12) title('\itCold-Gas Required Propellant Mass','FontSize',16) figure(5), surf(h0,Area,rtcg), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Tank Radius (cm)','FontSize',12) title('\itCold-Gas Required Tank Radius (Spherical)','FontSize',16) figure(6), surf(h0,Area,mphyd), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Propellant Mass (kg)','FontSize',12) title('\itHydrazine Required Propellant Mass','FontSize',16) figure(7), surf(h0,Area,rthyd), shading interp, xlabel('Injection Altitude(km)','FontSize',12) ylabel('Frontal Area (m^2)','FontSize',12) zlabel('Tank Radius (cm)','FontSize',12) title('\itHydrazine Monopropellant Tank Radius (Spherical)','FontSize',16) %III: X-Y Station Keeping Monte Carlo Calculation - commented so that I can %understand it later clear all; close all; %AA420 Stationkeeping Worst Case Code %Modified 4/20/09 %Inputs m = 10; %Cubesat mass (kg) h0 = 500; %Assumed Initial Starting Altitude i0 = 60 * (pi/180); %Assumed Initial Inclination in Degrees

150

sam = 1000; %Number of Random Samples dpos = 5; %GPS position uncertainty (meters) dvel = 0.5; %GPS velocity uncertainty (m/s) tstep = 1; %Solution step time (s) numsteps = 10000; %Number of Iterations Rbox = 1; %Position discrepany boundary (km) Vbox = 100; %Velocity discrepancy boundary (m/s) corrdur = 500; %Correction Burn Duration (s) ibit = 50*10^-6; %Impulse Bit (N-s) Isp = 500 %Specific Impulse (s) % Physical Constants rE = 6370; %Radius of Earth in km mu = 398000; %Earth Gravitational Constant g0 = 9.81; %Earth Gravitational Acceleration in m/s^2 xearth = 0; yearth = 0; zearth = 0; xhat = [1 0 0]; %Unit Vectors along Vernal Equinox, RHR, Pole. yhat = [0 1 0]; zhat = [0 0 1]; r0 = (h0 + rE); %Initial Scalar Radius (km) v0 = sqrt(mu/r0); %Initial Scalar Velocity (km/s) R0 = [0; r0; 0]; %Postion Vector V0 = [-v0*sin(i0);0;v0*cos(i0)]; %Velocity Vector R1 = R0+(dpos/1000*ones(3,1)); %Worst-Case Position Vector V1 = V0+(dvel/1000*ones(3,1)); %Worst-Case Velocity Vector H0 = cross(R0,V0); %Specific Angular Momentum Vector N0 = cross(zhat, H0); %Nodal Vector H1 = cross(R1,V1); %Specific Angular Momentum Vector N1 = cross(zhat, H1); %Nodal Vector r0mag =sqrt(R0(1)^2+R0(2)^2+R0(3)^2); r1=sqrt(R1(1)^2+R1(2)^2+R1(3)^2); v0mag =sqrt(V0(1)^2+V0(2)^2+V0(3)^2); v1 =sqrt(V1(1)^2+V1(2)^2+V1(3)^2); %Convert to Keplerian Elements eps0 = ((v0)^2)/2 - mu/r0; eps1 = ((v1)^2)/2 - mu/r1; a0 = -mu/(2*eps0); a1 = -mu/(2*eps1); e0 = (1/mu)*((v0^2-(mu/r0))*R0-(dot(R0,V0))*V0); e1 = (1/mu)*((v1^2-(mu/r1))*R1-(dot(R1,V1))*V1); nu0test = dot(R0,V0); nu1test = dot(R1,V1); e0mag = sqrt(e0(1)^2+e0(2)^2+e0(3)^2); e1mag = sqrt(e1(1)^2+e1(2)^2+e1(3)^2); i0 = (pi/2)-acos(H0(3)/sqrt(H0(1)^2+H0(2)^2+H0(3)^2)); i1 = (pi/2)-acos(H1(3)/sqrt(H1(1)^2+H1(2)^2+H1(3)^2));

151

om0 = acos(N0(1)/sqrt(N0(1)^2+N0(2)^2+N0(3)^2)); om1 = acos(N1(1)/sqrt(N1(1)^2+N1(2)^2+N1(3)^2)); ap0 = acos(dot(N0,e0)/(sqrt(N0(1)^2+N0(2)^2+N0(3)^2)*e0mag)); ap1 = (2*pi)-acos(dot(N1,e1)/(sqrt(N1(1)^2+N1(2)^2+N1(3)^2)*e1mag)); nu00 = acos(dot(e0,R0)/(e0mag*r0)); nu10 = (2*pi)-acos(dot(e1,R1)/(e1mag*r1)); p0 = 2*pi*sqrt(a0^3/mu); p1 = 2*pi*sqrt(a1^3/mu); E0 = acos((e0mag+cos(nu00))/(1+e0mag*cos(nu00))); E1 = acos((e1mag+cos(nu10))/(1+e1mag*cos(nu10))); M00 = E0-e0mag*sin(E0); M10 = E1-e1mag*sin(E1); %Propagate Orbits pl0 = a0*(1-e0mag^2); pl1 = a1*(1-e1mag^2); for i = 1:numsteps n0 = (mu/a0^3)^0.5; n1 = (mu/a1^3)^0.5; M0(i) = M00+ n0*(tstep*i); M1(i) = M10+ n1*(tstep*i); nu0(i) = M0(i)+2*e0mag*sin(M0(i))+1.25*e0mag^2*sin(2*M0(i)); nu1(i) = M1(i)+2*e1mag*sin(M1(i))+1.25*e1mag^2*sin(2*M1(i)); if nu0(i) > (2*pi) nu0(i) = nu0(i)-(2*pi); end if nu1(i) > (2*pi) nu1(i) = nu1(i)-(2*pi); end %Re-Build Orbital State Vectors Rx0(i) = pl0*(cos(om0)*cos(ap0 + nu0(i)) - sin(om0)*cos(i0)* sin(ap0 + nu0(i))); Rx1(i) = pl1*(cos(om1)*cos(ap1 + nu1(i)) - sin(om1)*cos(i1)* sin(ap1 + nu1(i))); Ry0(i) = pl0*(sin(om0)*cos(ap0 + nu0(i)) + cos(om0)*cos(i0)* sin(ap0 + nu0(i))); Ry1(i) = pl1*(sin(om1)*cos(ap1 + nu1(i)) + cos(om1)*cos(i1)* sin(ap1 + nu1(i))); Rz0(i) = pl0*sin(i0)* sin(ap0 + nu0(i)); Rz1(i) = pl1*sin(i1)* sin(ap1 + nu1(i));

152

Vx0(i) = sqrt(mu/pl0)*(cos(om0)*(sin(ap0 + nu0(i)) + e0mag* sin(ap0)) + sin(om0)*cos(i0)*(cos(ap0 + nu0(i)) + e0mag* cos(ap0))); Vx1(i) = sqrt(mu/pl1)*(cos(om1)*(sin(ap1 + nu1(i)) + e1mag* sin(ap1)) + sin(om1)*cos(i1)*(cos(ap1 + nu1(i)) + e1mag* cos(ap1))); Vy0(i) = sqrt(mu/pl0)*(sin(om0)*(sin(ap0 + nu0(i)) + e0mag* sin(ap0)) + cos(om0)*cos(i0)*(cos(ap0 + nu0(i)) + e0mag* cos(ap0))); Vy1(i) = sqrt(mu/pl1)*(sin(om1)*(sin(ap1 + nu1(i)) + e1mag* sin(ap1)) + cos(om1)*cos(i1)*(cos(ap1 + nu1(i)) + e1mag* cos(ap1))); Vz0(i) = sqrt(mu/pl0)*(sin(i0)*(cos(ap0 + nu0(i)) + e0mag* cos(ap0))); Vz1(i) = sqrt(mu/pl1)*(sin(i1)*(cos(ap1 + nu1(i)) + e1mag* cos(ap1))); dRx(i) = Rx1(i)-Rx0(i); dRy(i) = Ry1(i)-Ry0(i); dRz(i) = Rz1(i)-Rz0(i); dR(i) = sqrt(dRx(i)^2+dRy(i)^2+dRz(i)^2); dVx(i) = Vx1(i)-Vx0(i); dVy(i) = Vy1(i)-Vy0(i); dVz(i) = Vz1(i)-Vz0(i); dV(i) = sqrt(dVx(i)^2+dVy(i)^2+dVz(i)^2)*1000; end numvect = [1:numsteps]; figure (1), plot(numvect,dRx,'g',numvect,dRy,'b',numvect,dRz,'k') xlabel('Elapsed Time of Flight (sec)','FontSize',12) ylabel('Position Deviation (km)','FontSize',12) title('\itDeviation in Position vs. Elapsed Time','FontSize',16) figure (2), plot3(Rx0,Ry0,Rz0,'b',Rx1,Ry1,Rz1,'g',xearth,yearth,zearth,'ok') xlabel('X Position(km)','FontSize',12) ylabel('Y Position (km)','FontSize',12) zlabel('Z Position (km)','FontSize',12) title('\itPosition Trace of Optimum(Blue) and Deviated(Green) Orbits','FontSize',16) figure (3), plot(numvect,nu0,'g',numvect,nu1,'b') xlabel('Elapsed Time of Flight (min)','FontSize',12) ylabel('True Anomaly (radians)','FontSize',12) title('\itTrue Anomaly','FontSize',16) figure (4), plot(numvect,M0,'g',numvect,M1,'b') xlabel('Elapsed Time of Flight (min)','FontSize',12) ylabel('Mean Anomaly (radians)','FontSize',12) title('\itMean Anomaly','FontSize',16)

153

figure (5), plot(numvect,dR,'g',numvect,dV,'b') xlabel('Elapsed Time of Flight (sec)','FontSize',12) ylabel('Deviation (km and m/s)','FontSize',12) title('\itDeviation in Position and Velocity Vector Sum vs. Elapsed Time','FontSize',16) legend('Position','Velocity') %Correction Burn Trade j = 1; while dR(j) < Rbox intcorr = tstep*j; %time interval between corrections vavgcorr = dR(j)/corrdur; %average velocity of correction burn impulsecorr = 2*vavgcorr*m; %required correction impulse numpulses = impulsecorr/ibit; %number of pulses per correction freqpulses = corrdur/numpulses; %required pulse frequency numcorr = ((365*24*3600)/(corrdur+intcorr)); %number of corrections required totimpulse = impulsecorr*numcorr; %total impulse required for 1 year (N-s) totdvcorr = totimpulse/m; %total delta-V for one year (m/s) j = j+1; end intcorr vavgcorr impulsecorr numpulses freqpulses numcorr totimpulse totdvcorr mp = (m-m/(exp(totdvcorr/(g0*Isp))))*1000%propellant mass from rocket equation (g) vp = mp*(2.2) %propellant volume from density of teflon (cm^3) Carrier Code: This code calculates the required delta-v for each carrier type and determines the amount of propellant mass required for orbital maneuvering as a function of the specific impulse. Discrete values are then calculated for several key points. %AA420 Cube-sat Carrier Propulsion %LEO from SMAD p.730 - 185km %Inputs mc = 10; %Cube-sat Mass in kg ns = 10; %# of cube-sats in carrier nc = 1; %# of carriers per launch ms = 50; %Carrier Structural Mass in kg leo = 185; % Low Earth Orbit in km

154

h0 = [350:500]; % Desired Final Orbital Altitude in km theta = 55; %Launch Site Latitude in degrees cp = 2; %Number of Planes per Carrier ratheta = 10; %Physical Constants m0 = (mc*ns+ms)*nc; g0 = 9.81; rleo = leo + 6370; mu = 398000; vleo = sqrt(mu/rleo); Ih2ox = 450; Ihyd =325; Ihtpb = 300; thtr = theta * (pi/180); rathtr = ratheta*(pi/180); %Continuous Trade Analysis %Create Final Radius Vector and delta-V for m = 1:length(h0) r(m) = h0(m)+6370; v(m) = sqrt(mu/r(m)); at(m) = (r(m)+rleo)/2; dvt(m) = 631.3481*(abs(sqrt(2/rleo-1/at(m))-sqrt(1/rleo))+abs(sqrt((2/r(m))-(1/at(m)))-sqrt(1/r(m)))); %Plane Change Delta V dvpc(m) = 2*v(m)*sin(((pi/2)-thtr)/2); dvm(m) = dvt(m)+dvpc(m); dvsm(m) = dvm(m)*1000; end %Plane Change Delta-V Requirement from launch inclination as a Function of Apogee eha = [500:100:42000]; for b = 1:length(eha) eRa(b) = eha(b)+6370; ea(b) = (eRa(b)+rleo)/2; eE(b) = -mu/(2*ea(b)); eVa(b) = sqrt(mu*((2/eRa(b))-(1/ea(b)))); eVp(b) = sqrt(mu*((2/rleo)-(1/ea(b)))); edvpc(b) = 2*eVa(b)*sin(((pi/2)-thtr)/2); edvup(b) = eVp(b)-vleo; edvt(b) = edvpc(b)+2*edvup(b); edvm(b) = edvt(b)*1000; eT(b) =2*pi*sqrt((ea(b)^3)/mu); %Propellant Calculation lamdah2ox(b) = exp(edvm(b)/(g0*Ih2ox)); lamdahyd(b) = exp(edvm(b)/(g0*Ihyd)); lamdahtpb(b) = exp(edvm(b)/(g0*Ihtpb)); mh2ox(b) = m0*lamdah2ox(b); mhyd(b) = m0*lamdahyd(b); mhtpb(b) = m0*lamdahtpb(b); end

155

eTA = (eT*0.5)/3600; % Ellipse Time to Apogee in Hours figure (1), plot (eha,edvt,'g') xlabel('Elliptical Orbit Apogee Height(km)','FontSize',12) ylabel('Delta-V Required for Plane Change(km/s)','FontSize',12) title('\itDelta-V for Polar Orbit Plane Change vs. Apogee Alititude','FontSize',16) figure (2), plot (eha,eTA,'g') xlabel('Elliptical Orbit Apogee Height(km)','FontSize',12) ylabel('Time to Apogee(h)','FontSize',12) title('\itTime to Apogee from Burn vs. Apogee Alititude','FontSize',16) figure (3), plot (eha,mh2ox,'g',eha,mhyd,'b',eha,mhtpb,'k'); xlabel('Elliptical Orbit Apogee Height(km)','FontSize',12) ylabel('Carrier Mass in kg','FontSize',12) title('\itCarrier Wet Mass vs. Apogee Alititude','FontSize',16) legend('LH2/LOX Bipropellant','N2O4/MMH Bipropellant','UDMH Solid Propellant') %Discrete Analysis:Define Nine Data Points for Trade Analysis of Walker %Const. (2-18-09) tf = [45 55 65]; %Final Inclination in Deg hf = [400 450 500]; %Final Altitude in Km for f = 1:length(tf) for g = 1:length(hf) rd(g) = hf(g)+6370; vd(g) = sqrt(mu/rd(g)); ddthtr(f) = (pi/180)*(tf(f)-theta); ddvpc(f,g) = sqrt(vd(g)^2+vd(g)^2-(2*vd(g)*vd(g)*cos(ddthtr(f)))); ddvpcs(f,g) = 2*vd(g)*sin(ddthtr(f)/2); dea(f,g) = (rd(g)+rleo)/2; deVa(f,g) = sqrt(mu*((2/rd(g))-(1/dea(f,g)))); deVp(f,g) = sqrt(mu*((2/rleo)-(1/dea(f,g)))); ddvp(f,g) = abs(deVp(f,g)-vleo); ddva(f,g) = abs(vd(g)-deVa(f,g)); ddvtr(f,g) = ddvp(f,g)+ddva(f,g); dvra(f,g) =2*vd(g)*sin(rathtr/2)*(cp-1); ddvt(f,g) = ddvtr(f,g)+ddvpc(f,g)+dvra(f,g); %Propellant Calculation lamh2ox(f,g) = exp((ddvt(f,g)*1000)/(g0*Ih2ox)); lamhyd(f,g) = exp((ddvt(f,g)*1000)/(g0*Ihyd)); lamhtpb(f,g) = exp((ddvt(f,g)*1000)/(g0*Ihtpb)); mdh2ox(f,g) = m0*lamh2ox(f,g); mdhyd(f,g) = m0*lamhyd(f,g); mdhtpb(f,g) = m0*lamhtpb(f,g); %Number of Carriers per Plane/Planes per Carrier Analysis (2-19-09) end

156

end Delta_V_45_400 = ddvt(1,1) Delta_V_45_450 = ddvt(1,2) Delta_V_45_500 = ddvt(1,3) Delta_V_55_400 = ddvt(2,1) Delta_V_55_450 = ddvt(2,2) Delta_V_55_500 = ddvt(2,3) Delta_V_65_400 = ddvt(3,1) Delta_V_65_450 = ddvt(3,2) Delta_V_65_500 = ddvt(3,3) %Thrust Required for Carrier tT =7*60; %Thrust Time in (s) aT = (Delta_V_55_500*1000)/tT; fT = aT*mdhyd(2,2) figure (4), plot (tf,ddvpcs(:,1),'g',tf,ddvpcs(:,2),'b',tf,ddvpcs(:,3),'m'); xlabel('Final Orbit Inclination(deg)','FontSize',12) ylabel('Required Delta-V (km/s)','FontSize',12) title('\itRequired Delta-V for Inclination Change, Discrete Case','FontSize',16) legend('400km Altitude','450km Altitude','500km Altitude') figure (5), plot (tf,mdh2ox(:,3),'g',tf,mdhyd(:,3),'b',tf,mdhtpb(:,3),'k'); xlabel('Orbital Inclination(deg)','FontSize',12) ylabel('Carrier Mass in kg','FontSize',12) title('\itCarrier Wet Mass vs. Inclination at h = 500km','FontSize',16) legend('LH2/LOX Bipropellant','N2O4/MMH Bipropellant','UDMH Solid Propellant')

157

Thruster Specification Sheets

Northrop Grumman MRE‐01 Monopropellant Thruster

For satellite attitude and velocity control. Technical Data Propellant: Hydrazine Thrust at maximum operating Pressure: 1.0 N at 350 psia Thrust at 275 psia inlet pressure: 0.8 N Steady state specific impulse at 275 psia inlet pressure: 216 seconds Operating pressure range: 5-600 psia Life (demonstrated) Maximum throughput: 34 kg Maximum cycles: 370,000 Thrust valve power at 28 Vdc: 15 W Weight (STM/DTM): 0.5 kg/0.9 kg Envelope (width x length): 114 mm x 175 mm Spacecraft Programs Chandra X-ray Observatory, DSP, STEP4

Figure B.4.1. MRE-01.

158

Figure B.4.2.Spec sheet for Aerojet R-42.

159

Figure B.4.3 Carrier Main Engine.

160

Figure B.4.4 Carrier RCS Thruster.

161

Figure B.4.5 Moog Latch Valve.

162

Figure B.4.5. Moog Service Valve.

C. Image Acquisition

C.1. Optics The RFP states that the image resolution shall be less than 3m, with an image dimension of at least 5000m per side. Thus, equation (1) relates the object distance, image distance, and focal length.

(Eq. C.1.1)

Since the object distance is much greater than the image distance, the image distance is essentially the effective focal length.

163

The relationship between the angle of the telescope to the image, the wavelength of light, and the diameter of the lens is:

(Eq. C.1.2)

The wavelengths for visible light range from 400-750 nm. Since green is in the middle of the light spectrum, a wavelength of 530 nm was used to estimate the diameter of the telescope.

C.2. Image Capture

In order to figure out the shutter time needed to capture the picture, the amount of visible light to reach the spacecraft needed to be found. The following equations were used to calculate the intensity of light on the CCD,

2.4* 400vis sunwI I

m= Eq. C.2.3

because only 40% of total light is visible light, which is what we are going for.

2

2

* *cos( )cos ( ) **

visCCD lens

II Af

ρ θ απ

= Eq. C.2.4

where θ is the angle of the spacecraft off the equator, α is the angle that the spacecraft is taking the picture at, ρ is the rate of reflection and f is the focal length of the telescope. This equation can be simplified to,

2

* * **

vis lensCCD

I AI bf

ρπ

= Eq. C.2.5

where

2cos( ) cos ( )b θ α= Eq. C.2.6

The time for saturation of the pixels of the CCD can be calculated as the number of pixels over the fraction of electrons that hit the CCD over time,

6

40,000 .006 sec6.5*10 *

pixelsat

e

Nt

N b b= = = Eq. C.2.7

164

The minimum time that the shutter needed to be open was found by taking the minimum number of electrons needed to hit the CCD to overcome noise over the fraction of electrons that hit the CCD over time,

minmin 6

500 .0005 sec10 *e

NtN b b

= = = Eq. C.2.8

D. Navigation/Control

D.1. Navigation System No material posted

D.2. Attitude Determination and Control

D.2.1. Worst-Case Disturbance Torques Estimation

In this section, the worst-case disturbance torques are estimated using simplified equations.

Gravity-gradient Eq. (D.2.1.1)

Eq. (D.2.1.2)

where is the max gravity torque; µ is the Earth’s gravity constant ; R is

orbit radius (m), θ is the maximum deviation of the Z-axis from local vertical in radians, and and are moments of inertia about z and y (or x, if smaller) axes in .

For Dimensions

Assumption for the moment of inertia: the solar panel is folded.

165

Eq. (D.2.1.3)

Eq. (D.2.1.4)

Solar Radiation

Solar radiation pressure, , is highly dependent on the type of surface being illuminated. A

surface is either transparent, absorbent, or a reflector, but most surfaces are a combination of the three. Reflectors are classed as diffuse or specular. In general, solar arrays are absorbers and the spacecraft body us a reflector. The worst case solar radiation torque is

Eq. (D.2.1.5)

where and is the solar constant, 1,358 , c is the speed of light,

, is the frontal surface area, is the location of the center of solar pressure,

is the center of gravity, q is the reflectance factor (ranging from 0 to 1), and is the angle of incidence of the Sun.

Assumption: the reflectance factor, , and the difference of center -of solar pressure to center of mass, .

for unfolded solar panel, Eq. (D.2.1.6)

for folded solar panel, Eq. (D.2.1.7)

Magnetic Field

Eq. (D.2.1.8)

where is the magnetic torque on the spacecraft; D is the residual dipole of the vehicle in , and B is the Earth’s magnetic field in tesla. B can be approximated as

for a polar orbit to half that at the equator. M is the magnetic moment of the Earth, , and R is the radius from dipole (Earth) center to spacecraft in .

Assumption: the residual dipole of the vehicle for small sized spacecraft.

Eq. (D.2.1.9)

Aerodynamic

Atmospheric density for low orbits varies significantly with solar activity.

166

Eq. (D.2.1.10)

where ; F being the force; the drag coefficient (usually between 2 and 2.5); ρ the atmospheric density; A, the surface area; V, the spacecraft velocity; the center of

aerodynamic pressure; and the center of gravity.

Assumption: and



D.2.2. Calculations for Worst Disturbance Torque Estimation

Gravity-gradient

Moment of Inertia

h=0.3;d=0.2;w=0.1;m=10;

h = z, w = y; d = x;

Ix=1/12 m (h2+w2);Print["Ix = ",Ix]

Iy=1/12 m (h2+d2);Print["Iy = ",Iy]

Iz=1/12 m (w2+d2);Print["Iz = ",Iz]

Ix = 0.0833333

Iy = 0.108333

Iz = 0.0416667

=3986*1014;R=(6378+500) 1000;=0.002865*/180;

TgX=(3 )/(2 R3) Abs[Iz-Iy] Sin[2 ];Print["TgX = ",TgX]

TgX = 1.22513×10-8

TgY=(3 )/(2 R3) Abs[Iz-Ix] Sin[2 ];Print["TgY = ",TgY]

TgY = 7.65707×10-9

Solar Radiation

Fs=1358;c=3*108;Amax=0.15;Amin=0.02;q=0.5;i=0;

cps-cg=0.05;

167

F=Fs/c Amax (1+q) Cos[i];

Tspmax=F (0.05);Print["Tsp_max = ",Tspmax]

Tsp_max = 5.0925×10-8

F=Fs/c Amin (1+q) Cos[i];

Tspmin=F (0.05);Print["Tsp_min = ",Tspmin]

Tsp_min = 6.79×10-9

Magnetic Field

Dipole=1;M=7.96*1015;

B=(2 M)/R3;

Tm=Dipole B;Print["Tm = ",Tm]

Tm = 0.0000489279

Aerodynamic

=1.80*10-12; Cd=2;Amax=0.15;Amin=0.02;V=7613;

cpa-cg=0.025;

Fmax=0.5 ( Cd Amax V2);Tamax=Fmax (0.025);Print["Ta_max = ",Tamax]

Ta_max = 3.91215×10-7

Fmin=0.5 ( Cd Amin V2);Tamin=Fmin (0.025);Print["Ta_min = ",Tamin]

Ta_min = 5.2162×10-8

D.2.3. Sizing a camera lens and CCD chip for Star Tracker

Accuracy

Use lens equation to calculate focal length, the length between lens and CCD.

Eq. (D.2.3.1)

where is pixel size, is the focal length, and is the accuracy of pointing.

168

The assume that the accuracy of pointing is the field angle corresponding to 1 pixel and the pixel

size is 10µm. Table D.2.3.1 shows the range of the pointing accuracy with corresponding focal

length of lens.

Table D.2.3.1: Range of pointing accuracy and corresponding lens focal length..

Pointing accuracy [δθ in degrees]

Pointing accuracy [δθ in radians]

Pointing accuracy [δθ in arcsec] focal length of lens[mm]

0.0057 1E‐04 20.6 100.0

0.0069 0.00012 24.8 83.3

0.0080 0.00014 28.9 71.4

0.0092 0.00016 33.0 62.5

0.010 0.00018 37.1 55. 6

0.012 0.0002 41.3 50.0

0.013 0.00022 45.4 45.5

0.014 0.00024 49.5 41. 7

0.015 0.00026 53.6 38.5

0.016 0.00028 57.8 35.7

0.017 0.0003 61.9 33.3

0.018 0.00032 66.0 31.3

0.019 0.00034 70.1 29.4

0.020 0.00036 74.3 27. 8

0.022 0.00038 78.4 26.3

0.023 0.0004 82.5 25.0

The pointing accuracy is calculated for range of 50 m to 200 m of the ground distance and the

altitude of 500 km.

Field of view

A reasonable size of field angle should be chosen, and it can be used to determine the number of

pixels on the CCD using the following equation for one direction, with the assumption of one

pixel corresponding to , .

169

Eq. (D.2.3.2)

where Nx is the number of pixels in x-direction, θfield is the field of view, and is the accuracy

of pointing. If both directions have the same number of pixels, the total pixels on CCD will be

. Table D.2.3.2 shows various pixel sizes for the CCD, with corresponding pointing

accuracy and field of view.

Table D.2.3.2: Total pixels on CCD vs. pointing accuracy for selects fields of views.

Total pixel on CCD [Megapixels] Accuracy of pointing

[δθ in degrees] 10° 11° 12° 13° 14° 15° 16° 17° 18° 19° 20° 0.0057 3.05 3.69 4.39 5.15 5.97 6.85 7.80 8.80 9.87 11.0 12.1 0.0069 2.12 2.56 3.05 3.58 4.15 4.76 5.42 6.11 6.85 7.64 8.46 0.0080 1.55 1.88 2.24 2.63 3.05 3.50 3.98 4.49 5.04 5.61 6.22 0.0092 1.19 1.44 1.71 2.01 2.33 2.68 3.05 3.44 3.86 4.30 4.76 0.010 0.94 1.14 1.35 1.59 1.84 2.12 2.41 2.72 3.05 3.39 3.76 0.012 0.76 0.92 1.10 1.29 1.49 1.71 1.95 2.20 2.47 2.75 3.05 0.013 0.63 0.76 0.91 1.06 1.23 1.42 1.61 1.82 2.04 2.27 2.52 0.014 0.53 0.64 0.76 0.89 1.04 1.19 1.35 1.53 1.71 1.91 2.12 0.015 0.45 0.55 0.65 0.76 0.88 1.01 1.15 1.30 1.46 1.63 1.80 0.016 0.39 0.47 0.56 0.66 0.76 0.87 1.00 1.12 1.26 1.40 1.55 0.017 0.34 0.41 0.49 0.57 0.66 0.76 0.87 0.98 1.10 1.22 1.35 0.018 0.30 0.36 0.43 0.50 0.58 0.67 0.76 0.86 0.96 1.07 1.19 0.019 0.26 0.32 0.38 0.45 0.52 0.60 0.67 0.76 0.85 0.95 1.05 0.020 0.24 0.28 0.34 0.40 0.46 0.53 0.60 0.68 0.76 0.85 0.94 0.022 0.21 0.26 0.30 0.36 0.41 0.48 0.54 0.61 0.68 0.76 0.84 0.023 0.19 0.23 0.27 0.32 0.37 0.43 0.49 0.55 0.62 0.69 0.76

* assumption:

Diameter of Lens

The diameter of lens can be obtained from the equation of the light power from a star, where D is

lens diameter in meters.

170

Eq. (D.2.3.3)

This power is the light power collected by the camera lens. The equation for the intensity of

visible light from a star is as follows, where M is magnitude of brightness of star.

Eq. (D.2.3.4)

Substituting the intensity equation into light power equation and change unit of diameter to

centimeter gives the following.

Eq. (D.2.3.5)

The light power from a star can be related to the number of photons per second by the following

equation, where .

Eq. (D.2.3.6)

Assuming that all photons hit a single pixel, the number of electrons generated per second, is

given below, where is CCD quantum efficiency.

Eq. (D.2.3.7)

A typical value of for CCD is 0.25. Using this value into equation then becomes

Eq. (D.2.3.8)

Therefore, the number of electrons generated during exposure can be calculated by the following

equation, where t is exposure time.

171

Eq. (D.2.3.9)

The equation for the number of electrons generated during exposure becomes

Eq. (D.2.3.10)

The minimum number of electrons to have a good star image is about 400 electrons.

Therefore,

Solving for D,

Table D.2.3.3 shows the different diameter along the exposure time.

Table D.2.3.3: Exposure time vs. lens diameter.

Exposure Time [sec] Diameter [cm] 1/2 0.56296 1/4 0.79614 1/8 1.12591 1/15 1.54172 1/30 2.18032 1/60 3.08344 1/125 4.45056 1/250 6.29404 1/500 8.90111

* assumption: the magnitude of brightness of star, M = 5.5

The lens focal ratio can be obtained with the following equation, where is focal length of lens

and D is the diameter of lens.

Eq. (D.2.3.11)

Table D.2.3.4 shows the lens focal ratio of various accuracy of pointing and exposure time.

Table D.2.3.4: Lens focal ratio for various pointing accuracies and exposure times

172

Exposure Time [sec] Accuracy of pointing [δθ in degrees] 1/2 1/4 1/8 1/15 1/30 1/60 1/125 1/250 1/500

0.0057 17.76 12.56 8.88 6.49 4.59 3.24 2.25 1.59 1.12 0.0069 14.80 10.47 7.40 5.41 3.82 2.70 1.87 1.32 0.94 0.0080 12.69 8.97 6.34 4.63 3.28 2.32 1.60 1.13 0.80 0.0092 11.10 7.85 5.55 4.05 2.87 2.03 1.40 0.99 0.70 0.010 9.87 6.98 4.93 3.60 2.55 1.80 1.25 0.88 0.62 0.012 8.88 6.28 4.44 3.24 2.29 1.62 1.12 0.79 0.56 0.013 8.07 5.71 4.04 2.95 2.08 1.47 1.02 0.72 0.51 0.014 7.40 5.23 3.70 2.70 1.91 1.35 0.94 0.66 0.47 0.015 6.83 4.83 3.42 2.49 1.76 1.25 0.86 0.61 0.43 0.016 6.34 4.49 3.17 2.32 1.64 1.16 0.80 0.57 0.40 0.017 5.92 4.19 2.96 2.16 1.53 1.08 0.75 0.53 0.37 0.018 5.55 3.93 2.78 2.03 1.43 1.01 0.70 0.50 0.35 0.019 5.22 3.69 2.61 1.91 1.35 0.95 0.66 0.47 0.33 0.020 4.93 3.49 2.47 1.80 1.27 0.90 0.62 0.44 0.31 0.022 4.67 3.31 2.34 1.71 1.21 0.85 0.59 0.42 0.30 0.023 4.44 3.14 2.22 1.62 1.15 0.81 0.56 0.40 0.28

D.2.4. Designing CMG Wheel

To determine the moments of inertias of the cubesat, the conventions used is shown in Figure

D.2.4.1, with d=0.2 m, w=0.1m, and h=0.3m.

Figure D.2.4.1: Geometry of the cubesat

Then the moments of inertia of the body with assuming uniform density and center of mass is to

be at the center of the body as calculated as shown below.

173

2 2 2

2 2 2

2 2 2

1 ( ) 0.0417[ . ]121 ( ) 0.108[ . ]

121 ( ) 0.083[ . ]

12

h

w

d

I m w d kg m

I m h d kg m

I m h w kg m

= + ≈

= + ≈

= + ≈

Eq. (D.2.4.1)

In order to find the required moment of inertia for the CMG, first the required slew rate is

determined. Since the cubesat will accelerate for half of this angle and decelerate for the half,

only half of the angle and time are used in the calculations. Using the equation below, the

angular acceleration of the cubesat can be determined. Please note that calculations were

completed for a range of slew rates from 1° - 15° per second.

2

2

1 22

ttθθ θ θ= ⇒ =

ii ii

Eq. (D.2.4.2)

Then using the moments of inertia of the cubesat calculated earlier and this angular acceleration,

the required wheel torque is determined with the following equation.

w req sN I θ− =ii

Eq. (D.2.4.3)

where Nw-req is the required torque needed to achieve the specified maneuver, Is is the spacecraft

moment of inertia, and θii

is the spacecraft’s angular acceleration.

A summary of these results is shown in Table D.2.4.1.

Table D.2.4.1: Slew Rates and Required Wheel Torque

174

Then using the equation below to determine the momentum of the wheel.

0 02 cos cos 2

cos cosx

xNN h hδ β δ

δ β δ= ⇒ =

i

i

Eq. (D.2.4.4)

where Nx is the wheel torque on x-axis maneuver, h is angular momentum of the CMG,

δi

is the

gimbal angles rate, and angles of β and δ are corresponding to Figure D.2.4.2. The CMG cluster

for an x-axis maneuver is shown in Figure D.2.4.2. The same convention is used for the other

two directions. Note that for these calculations, β is assumed to be 60 degrees and δ, which is the

initial gimbal angle position, is assumed to be zero for all directions. The calculations are carried

out for three cases with maximum gimbal angle rates of 6 rad/sec, 10 rad/sec, and 25 rad/sec.

175

Figure D.2.4.2: CMG Cluster for an x-axis maneuver

Once the angular momentum is determined, assuming an angular velocity of 60,000 rpm for the

DC motor, the following equation can be used to determine the CMG's required moment of

inertia. Note that this DC motor is for the wheel only. We will use a ULT Applimotion frameless

motor for the gimbals.

0

0 CMG CMGhh I Iωω

= ⇒ =

Eq. (D.2.4.5)

The required moments of inertias for four cases are of slew rates, 9 deg/sec, 8 deg/sec, 7 deg/

sec, and 5 deg/sec, with all three different cases of maximum gimbal angle rates are calculated

and summarized in Table D.2.4.2.

Table D.2.4.2: Required MOI for Four Slew Rates using Three Different Maximum Gimbal Angle Rates

176

An important factor in deciding the slew rate is the time it takes for the CMG to reach that slew

rate, determined from the equation below.

( _ )( )Slew Rate MOItTorque

=

Eq. (D.2.4.6)

Assuming a torque of 0.003 N-m, from Andrews Space, the time in each direction is determined.

The results are summarized in Table D.2.4.3.

Table D.2.4.3. time to reach desired slew rate

177

Then in order to size the wheel and find the diameter that can produce the required slew rate, the

moments of inertia for a brass wheel are calculated for four different combinations, with density

of 8400 kg/m^3 and 8700 kg/m^3 and two thicknesses of 0.005 m and 0.0025m. The diameter is

varied from 0.01m to 0.02 m. The formulas used are as follows

2

2 2

21 (3 )

12

z

x y

mrI

I I m r h

=

= = +

Eq. (D.2.4.7)

corresponding to Figure D.2.4.3.

178

Figure D.2.4.3. Coordinates of the Cylinder for MOI Calculations

The results are summarized in Table D.2.4.4.

Table D.2.4.4: Summary of the MOI for the Disk Corresponding to Different Diameters

Dia. [m]

Volume[m3] (thickness of 0.005m)

Mass[kg] (8400kg/m3)

I_z [kg‐m2]

I_x=I_y [kg‐m2]

Mass[kg] (8700kg/m^3)

I_z [kg‐m2]

I_x=I_y [kg‐m2]

0.01 3.93E‐07 3.30E‐03 4.12E‐08 2.75E‐08 3.42E‐03 4.27E‐08 2.85E‐08 0.012 5.65E‐07 4.75E‐03 8.55E‐08 5.26E‐08 4.92E‐03 8.86E‐08 5.45E‐08 0.014 7.70E‐07 6.47E‐03 1.58E‐07 9.27E‐08 6.70E‐03 1.64E‐07 9.60E‐08 0.016 1.01E‐06 8.44E‐03 2.70E‐07 1.53E‐07 8.75E‐03 2.80E‐07 1.58E‐07 0.018 1.27E‐06 1.07E‐02 4.33E‐07 2.39E‐07 1.11E‐02 4.48E‐07 2.47E‐07 0.02 1.57E‐06 1.32E‐02 6.60E‐07 3.57E‐07 1.37E‐02 6.83E‐07 3.70E‐07

Dia. [m]

Volume[m3] (thickness of 0.0025 m)

Mass[kg] (8400kg/m3)

I_z [kg‐m2]

I_x=I_y [kg‐m2]

Mass[kg] (8700kg/m^3)

I_z [kg‐m2]

I_x=I_y [kg‐m2]

0.01 7.85E‐07 6.60E‐03 8.25E‐08 4.47E‐08 6.83E‐03 8.54E‐08 4.63E‐08 0.012 1.13E‐06 9.50E‐03 1.71E‐07 9.04E‐08 9.84E‐03 1.77E‐07 9.37E‐08 0.014 1.54E‐06 1.29E‐02 3.17E‐07 1.65E‐07 1.34E‐02 3.28E‐07 1.71E‐07 0.016 2.01E‐06 1.69E‐02 5.40E‐07 2.79E‐07 1.75E‐02 5.60E‐07 2.89E‐07 0.018 2.54E‐06 2.14E‐02 8.66E‐07 4.44E‐07 2.21E‐02 8.97E‐07 4.60E‐07 0.02 3.14E‐06 2.64E‐02 1.32E‐06 6.73E‐07 2.73E‐02 1.37E‐06 6.98E‐07

179

Figure D.2.4.4.Spec sheet for DC motor

180

Figure D.2.4.5. Gimbal Motor

D.2.5. Dynamics

In order to model the DC motor, Newton's equations and Kirchhoff's laws are combined to

obtain the equations below.

θ ω=i

Eq. (D.2.5.1)

1 ( )b KIJ

ω ω= − +i

Eq. (D.2.5.2)

1 ( )I v K RIL

ω= − −i

Eq. (D.2.5.3)

where ω is the angular velocity, J is the moment of inertia, b is the rotational friction, K is the

motor constant, I is the current, L is constant matrix, v is voltage, R is the motor resistance, and

θ is the angular position.

The state space representation of the above equations is shown below.

0 1 0 00 0

10

b K vJ J IK RI

LL L

θ θω ω

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠

−= =

− −

i

i

i

Eq. (D.2.5.4)

181

( )0 1 0yI

θω⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

=

Eq. (D.2.5.5)

where the voltage is input and the angular velocity is the output. Then from these, the transfer

function can be obtained as shown below.

( )

1

1 0 0( ) 0 1 0 0 0

10

sb KT s sJ J

K Rs LL L

−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

−−= +

+

Eq. (D.2.5.6)

3 2 2( )

( ) ( )KsT s

JLs JR Lb s K bR s⇒ =

+ + + + Eq. (D.2.5.7)

This can be modeled in Simulink for further simulations, as shown in Figure D.2.5.1.

Figure D.2.5.1: Simulink Block Diagram with Voltage Input and Angular Velocity Output

182

E. Communications The Link Equation is given as

0

b t l t s a r

N

E PL G L L GN kT R

= (Eq. E.1)

which can be rearranged to isolate Pt and substitute efficiencies for the loss terms to give

( ) 2

0

1 4Nbt

t t r eff

kT R CE rPN G A

πη η−

≥ (Eq. E.2)

The ‘≥’ rather than a ‘=’ is to assure that the power at the receiving end is at least as great as

the minimal power required. In the Link Equation, Aeff is the effective area of the receiving

antenna. This is commonly approximated as

(Eq. E.3)

In this, Gr is the gain of the receiving antenna. For the onboard antennas, quarter‐wave

monopoles, the peak gain is 1.6 and for the ground station’s antenna, a 1‐meter parabolic dish,

the peak gain is 25.

The modulation chosen is the differential phase shift key (DPSK). This modulation can be

either binary or quadriphased, but this has not been analyzed into a trade study as of yet. The

primary benefit of the DPSK modulation is it is not susceptible to phase disturbances in the

transmission. However, this comes at the cost of sensitivity to noise. To achieve a bit‐error‐

183

rate (BER) of no more than 10‐5, the signal‐to‐noise ratio must be approximately 10, as found

in Figure 13‐9 in the SMAD book.

The crosslink raw data rate is found using the imaging hardware, a 10 megapixel CCD with 10‐

bit resolution, a mission requirement of 60 images per day, and an inherent limitation of only

imaging during the light side of the CubeSat orbits. The orbits are in the light 61% of the time

on average, which limits the transmission of the 60 images to 14.4 hours. With this, the raw

data rate in bits/second is found as

(Eq. E.4)

The downlink raw data rate is found by taking the size of a single image and the accessible

range of the fly‐by time. At 520 km the ground speed of a circular orbit is 7.6 km/s. The

ground station is assumed to be able to track to within 10° of the horizon, or a span of 160°.

This leads to a maximum accessible distance of 1820 km and a maximum in‐view time of

approximately seven minutes. It was assumed that communication initiation would take two

minutes and the transmission of a single image was restricted to two minutes, leaving three

minutes of margin. As for the crosslink raw data rate, the image size for the downlink is

dictated by the 10 megapixel CCD with 10 bit resolution. From this, the resulting downlink raw

data rate is

(Eq. E.5)

184

To compress the data, JPEG image compression will be used. Images can be compressed up to

compression ratios of 98% using JPEG compression, which means that the final size is only 2%

of the initial size. However, at such high compression ratios the image quality suffers

drastically. A compression ratio of 90% is considered “lossless” and does not sacrifice quality.

The general values used in the Link Equation analysis are summarized in Table E.1 below. The

crosslink‐specific values are summarized in Table E.2 and the downlink‐specific in E.3.

Table E.1. Link Equation Values Symbol Definition Assumed Value Eb/N0 Signal‐to‐noise ratio 10 Ll Line loss N/A Ls Space loss N/A La Transmission path loss N/A k Boltzmann’s constant 1.38x10‐23 [(m2kg)/(s2k)] TN System noise temperature 300 [k] C Data compression ratio 90% ηt Transmit antenna efficiency 0.8 ηr Receive antenna efficiency 0.8

Table E.2. Crosslink Link Equation Values Symbol Definition Assumed Value

Pt Transmitter power ≤2.5W (5W DC) Gt Transmit antenna gain 1.6 Gr Receive antenna gain 1.6 R Raw Data rate 11.6x104 [bps] r Furthest communication distance 4000 [km] λ Wavelength 63 [cm]

Table E.3. Downlink Link Equation Values Symbol Definition Assumed Value

Pt Transmitter power ≤1W (2W DC) Gt Transmit antenna gain 1.6 Gr Receive antenna gain 25 R Raw Data rate 11.6x104 [bps] r Furthest communication distance 2000 [km] λ Wavelength 74 [cm]

185

F. Support Systems

F.1. Power F1.1 Charge power required

With a power requirement estimate of 5 W during the dark period of each orbit and a

discharge time of 36 minutes, the required total energy is defined as

Edisc = Preqtdark (Eq. F.1.1)

With a charging time of 56 minutes, the power required to charge is defined by

Pcharge = Edisc / tcharge (Eq. F.1.2)

The charge power necessary in this case is 3.1 W average throughout the light‐side period of

orbit. However, to charge the SANYO UF634042F batteries, the current must be controlled

specifically at 1230 mA according to the specifications sheet and reduced gradually after about

50 minutes (if needed). The charge time for 810 mA‐h of discharged capacity at 1230 mA is 40

minutes.

F1.2 Orientation Power Analysis

To determine the average power generated during an orbit and to compare this figure to the

average power required, the following equation us used to find the frontal area shown to the

sun from any arbitrary orientation, with "extra" area added due to solar wings.

186

Aarb = 300cos(θ)cos(ψ) + (100+extra)sin(ψ)sin(φ) + 600sin(θ)cos(φ) (Eq. F.1.3)

Figure F.1.4 below illustrates the assumed axes of rotation. Each number represents the area

of solar cells assumed on each face of the CubeSat, with 100 cm2 allotted to the top portion to

leave room for sensors and external equipment.

Figure F.1.4 Orientation of CubeSat

To find the average area exposed to the sun, the triple integral is found with limits from 0 to

π/2 and then divided by π3/8, the integral without the function. The average area exposed to

the sun is then found to be

Aavg = 4(1000 + extra)/ π2 (Eq. F.1.5)

187

Using this area and the given solar panel properties and averaging the time in light over the

total orbit time, the Figures F.1.6 and F.1.7 are produced. One thing to note is that the powers

calculated for maximum power generation

Figure F.1.6 Power properties of given orientation

188

Figure F.1.7 Orientation required to achieve maximum power

To obtain the maximum power, Matlab’s “fminsearch” function was used on the reciprocal of

Eq. F.1.3 to find the maximum area shown to the sun. Matlab’s “fminsearch” function runs a

minimization code to find the absolute minimum of n‐variable functions based on initial

conditions to determine the value of the variables that produce a minimum. It can be seen in

the second figure that past a certain amount of extra solar wing area, the orientation directly

faces the largest area of the CubeSat. As the wing area grows, the maximum power generation

is achieved from increasing inclination toward facing the wing area.

Similar analysis was done for each carrier vehicle, and the power generated from the

sun and maximum power orientation can be seen in the following figures:

189

Figure F.1.8 Power properties of carrier vehicle

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Figure F.1.9 Orientation required to achieve maximum power

F1.3 Power Analysis MATLAB Code CubeSat power requirements analysis clear all; close all; clc %%Power analysis P_req_guess = 12; %:20;%15:45; e_cell = .2861; I_sun = 1370; %W/m^2 A_cell = .0001*20; %m^2 P_sun = A_cell*I_sun; %P_cell = e_cell*P_sun; Vmp = 2.33; Jmp = 17.32/1000; %A/cm^2 Imp = Jmp*A_cell/.0001; %A P_cell = Imp*Vmp; N_cell_guess = ceil(P_req_guess/P_cell); A_array_guess = N_cell_guess*A_cell/.0001; %cm^2 V_bus_guess = 9; %A guess V_cell = Vmp; N_str = ceil(V_bus_guess/V_cell);

V_bus_solar = N_str * V_cell; I_bus_guess = P_req_guess/V_bus_solar; I_cell = Imp; %.34 %Amp generated by a single cell %Number of strings M_str = ceil(I_bus_guess/I_cell); I_bus = I_cell*M_str; N_cell = N_str.*M_str; P_bus = N_cell*P_cell; A_array = N_cell*A_cell/.0001; %cm^2 if length(P_req_guess) > 1 figure(1) plot(P_req_guess,N_cell) grid on axis equal xlabel('Power required by spacecraft (W)'); ylabel('Total number of solar cells required');

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title('Number of cells required by power required') figure(2) plot(P_req_guess,A_array) grid on xlabel('Power required by spacecraft (W)'); ylabel('Area of solar panels required by spacecraft (cm^2)'); title('Area of solar cells required to generate power required'); end costrange = 240*[20]; massrange = 1.68*39*2*[1 2.5]/1000; %kg % Batteries T = 94/60; %Orbital period, hours cycles = 365*(24/T); %cycles of light-to-dark P_req_dark = 5; %W, a guess for power required in the dark t_dark = 36*60; %42 minutes worst case in seconds V_bat = 3.7; V_charge = 4.2; t_charge = (94.6-36)*60; dod = .5;

q_level = 1-dod; E_bat_Sanyo = 187*.0246; %W-h/kg * kg N_bat = floor(V_bus_solar/V_bat); V_bat_total = N_bat*V_bat; V_bus_bat = N_bat*V_bat; E_disc = P_req_dark*t_dark/3600; %watt-hour Cap_disc = E_disc/V_bat; %Amp-hour E_bat = E_disc/(N_bat*q_level); %watt-hour Cap_bat_req = Cap_disc/(N_bat); Cap_bat = Cap_bat_req/q_level; P_charge = 3600*E_disc/t_charge; %Power needed to charge I_disc = P_req_dark/V_bat_total; %discharge current mass_bat_req = E_bat/175; %Batteries over specific energy mass_bat_total = N_bat*mass_bat_req; %total mass of batteries P_req = P_charge + 11; %total power required for the cubesat solar panels to support I_charge = P_charge/V_charge; Cap_Sanyo = 1.2; %A dod_true = Cap_bat_req/Cap_Sanyo;

Orientation analysis

clear all close all theta = 0:.1:90; phi = 0:.1:90; psi = 0:.1:90; angles = (pi/180)*[theta; phi; psi]; extra = 0:10:1000; for i = 1:length(extra) [maxang(:,i), p1(i)] = fminsearch(@(x) powerval(x,extra(i)), [pi/5, pi/3, pi/3]); maxpower(i) = (56/94)*1/p1(i); %Max power averaged over a worst-case orbit. 56 min out of 94 maxangles(:,i) = (180/pi)*maxang(:,i); end % To change the top base area of solar panels, change the number next to extra % from 1000 to 1100 if going from a top area of 100 cm^2 to 200 cm^2 % (also in powerval function) avgpower = (56/94)*.29*1366*(1/(100^2))*(4*(1000 + extra))/(pi^2); figure(1) plot(extra, .9*maxpower, extra, .9*avgpower, extra, 12*ones(length(extra)), 'r') xlabel('Area of solar panel wings (cm^2)');

ylabel('Power (W)'); legend('Average power with pointing', 'Average power', 'Required power', 'Location', 'SouthEast'); title('Power generation as a function of solar wing area') figure(2) plot(extra, maxangles(1, :), extra, maxangles(2, :), extra, maxangles(3, :)) legend('Angle theta at max power', 'Angle phi at max power', ... 'Angle psi at max power', 'Location', 'SouthEast'); xlabel('Area of solar panel wings (cm^2)') ylabel('Angle in degrees') title('Orientation of spacecraft at maximum power') % Carriers extra1 = 0:60:3600; %extra2 = 0:50:2500; for j = 1:length(extra1) [maxang2(:,j),p2(j)] = fminsearch(@(x) powerval2(x,extra1(j),0), [pi/5, pi/3, pi/3]); maxpower2(j) = (56/94)*1/p2(j); maxangles2(:,j) = (180/pi)*maxang2(:,j); end

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avgpower2 = (56/94)*.29*1366*(1/(100^2))*(4*(7200 + extra1))/(pi^2); P_req = 50; figure(3) plot(extra1, maxpower2, extra1, avgpower2)%, extra1, P_req*ones(length(extra1)), 'r') xlabel('Area of extra solar panel wings (cm^2)'); ylabel('Power (W)'); legend('Average power with pointing', 'Average power', 'Location', 'SouthEast') title('Power of carrier based on extra solar panel area from wings')

axis([0 3600 0 200]) figure(4) plot(extra1, maxangles2(1, :), extra1, maxangles2(2, :), extra1, maxangles2(3, :)) legend('Angle theta at max power', 'Angle phi at max power', ... 'Angle psi at max power', 'Location', 'Best'); xlabel('Area of extra solar panel wings (cm^2)') ylabel('Angle in degrees') title('Orientation of carrier at maximum power') axis([0 3600 -10 100])

Cubesat orientation function

function p = powerval(x0, extra) theta = x0(1); phi = x0(2); psi = x0(3); A = 300*cos(theta)*cos(psi) + (100+extra)*sin(psi)*sin(phi) + 600*sin(theta)*cos(phi); p = 1/(.29*A*1366*(1/(100^2)));

Carrier orientation function

function p = powerval(x0, extra1, extra2) %Carrier numbers theta = x0(1); phi = x0(2); psi = x0(3); A = (3600+extra1)*cos(theta)*cos(psi) + (3600+extra2)*sin(psi)*sin(phi); p = 1/(.29*A*1366*(1/(100^2)));

Carrier power requirements analysis

clear all close all P_req_guess = 48; e_cell = .2861; I_sun = 1370; %W/m^2 A_cell = .0001*20; %m^2 P_sun = A_cell*I_sun; %P_cell = e_cell*P_sun; Vmp = 2.33; Jmp = 17.32/1000; %A/cm^2 Imp = Jmp*A_cell/.0001; %A P_cell = Imp*Vmp; N_cell_guess = ceil(P_req_guess/P_cell); A_array_guess = N_cell_guess*A_cell/.0001; %cm^2

V_bus_guess = 28; %A guess V_cell = Vmp; N_str = ceil(V_bus_guess/V_cell); V_bus_solar = N_str * V_cell; I_bus_guess = P_req_guess/V_bus_solar; I_cell = Imp; %.34 %Amp generated by a single cell %Number of strings M_str = ceil(I_bus_guess/I_cell); I_bus = I_cell*M_str; N_cell = N_str.*M_str; P_bus = N_cell*P_cell; A_array = N_cell*A_cell/.0001; %cm^2

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if length(P_req_guess) > 1 figure(1) plot(P_req_guess,N_cell) grid on axis equal xlabel('Power required by spacecraft (W)'); ylabel('Total number of solar cells required'); title('Number of cells required by power required') figure(2) plot(P_req_guess,A_array) grid on xlabel('Power required by spacecraft (W)'); ylabel('Area of solar panels required by spacecraft (cm^2)'); title('Area of solar cells required to generate power required'); end % Carriers P_req_dark = 50; %W, a guess for power required in the dark t_dark = 36*60; %36 minutes worst case, in seconds V_bat = 3.7; V_charge = 4.2; t_charge = (94.6-36)*60; dod = .5; %depth of discharge q_level = 1-dod; E_bat_Sanyo = 187*.0246; %W-h/kg * kg N_bat_series = floor(V_bus_solar/V_bat); V_bat_total = N_bat_series*V_bat; V_bus_bat = N_bat_series*V_bat; E_disc = P_req_dark*t_dark/3600; %watt-hour M_bat_parallel = 8; N_bat = M_bat_parallel*N_bat_series; Cap_disc = E_disc/V_bat; %Amp-hour E_bat = E_disc/(N_bat*q_level); %watt-hour Cap_bat_req = Cap_disc/(N_bat); Cap_bat = Cap_bat_req/q_level; P_charge = 3600*E_disc/t_charge; %Power needed to charge I_disc = P_req_dark/V_bat_total; I_disc_bat = I_disc/M_bat_parallel; mass_bat_req = E_bat/175; %Batteries over specific energy mass_bat_total = N_bat*mass_bat_req; P_req = P_charge + P_req_guess; I_charge = P_charge/V_charge; I_charge_bat = I_charge/M_bat_parallel; Cap_Sanyo = 1.2; %A dod_true = Cap_bat_req/Cap_Sanyo;

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F.2. Thermal

F.2.1. Orbit Environment

From Equations (11.17) and (11.18) in Space Mission Analysis and Design,

(Eq. F.2.1)

(Eq. F.2.2)

Parameters for obits near 520 km altitude and 55° inclination,

= 231 W/m²

= 1367 W/m²

= 0.86

= 0.43

= 0.30

F.2.2. Time Dependent Heat Transfers

The total thermal power being absorbed by a satellite is as follows.

(Eq. F.2.3)

(Eq. F.2.4)

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Breaking the orbit into the light and dark period results in the following equations.

(Eq. F.2.5)

(Eq. F.2.6)

The relationship between the change of an object’s temperature to a change in absorbed heat

is as follows.

(Eq. F.2.7)

Starting at an initial temperature, the change in absorbed heat during the light or dark phase of

the orbit can be approximated by the following, where Δt is a finite time step.

(Eq. F.2.8)

The resulting temperature after a finite time step is as follows.

(Eq. F.2.9)

Repeating the processes and shifting between the equation for the light and dark period of the

orbits when appropriate results in an approximation for the satellite’s temperature as the

satellite orbits.

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G. Biographical Sketches

Michael (Mike) Bernhardt

Michael Bernhardt is a senior in the Aeronautics and Astronautics department at the University of Washington. He has an amateur interest in rocketry and space. He has participated as a consultant for high school rocket competitions. He possesses rudimentary skills in computer programming and electronics. Michael Bernhardt currently spends his free time making computer renders and animations. Michael Bernhardt investigated and researched thermal control systems and computer hardware solutions. He wrote feasibility and cost analyses for the thermal and computer subsystems.

Aaron Borth

My name is Aaron Borth and I am part of the imaging group for the proposal. Last summer as an intern I had the opportunity to work at a small design company, PCSI Design. I was able to do design work on aerospace products as well as consumer products (handheld firestarter). I spent a lot of time doing design work in Solidworks and also transfering engineering data from CATIA V5 to the Solidworks environment (737 door seals). As for classes, on top of the classes required for the A&A Undergraduate, I have taken Advanced Propulsion, Systems Engineering, FEA Analysis, and am currently taking Heat Transfers. For the proposal, I wrote the section about the image recording device (CCD) and shutter time and helped set up MATLAB codes for calculating the shutter time and focal length of the telescope.

Rachel Brennan

Rachel Brennan is currently a senior in the Aeronautics/Astronautics Engineering department at the University of Washington. She is a transfer student from Everett Community college, where she graduated with an A.S. in Engineering in 2007. Her relevant coursework completed includes Oribital Mechanics, Introduction to Propulsion, Systems Engineering, and Advanced Propulsion. In addition, Rachel plans to take the Rocket Propulsion course in the spring.

Rachel has completed four summer internships in the aerospace industry. She completed internships with GE Aircraft Engines in the areas of Structural and Heat Transfer/Fluid Systems Engineering. She also completed an internship with B/E Aerospace in Mechanical Design and

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an internship with Boeing Commercial Airplanes in Aerodynamic Performance. Rachel is interested in the areas of space systems engineering, structures, and propulsion. Rachel is a member of the propulsion system team, with Nik Lutzenhiser, which is responsible for the design of the cube satellite and carrier propulsion systems.

In addition to her Aeronautics/Astronautics Engineering courses, Rachel is very active with the Society of Women Engineers as Vice President and Region J conference lead. Rachel’s hobbies include: traveling, hiking, camping, and dancing.

Enrique (Gally) Galgana

Enrique Galgana is a senior in the University of Washington Aerospace Engineering Department. His relevant coursework to this mission include: Technical Writing & Presentation, Atmospheric Flight Sciences, Orbital Mechanics, Aerospace Flight Instrumentation, Vibrations, and Controls. Enrique has been working has an undergraduate researcher since March 2007 in the Composites/Structures Lab. A few of the job activities include: Designing and manufacturing test fixtures, communicating with material dealers (metals/composites), conducting experiments, data analysis, C-Scanning, and microscopy. Because of his interest in imaging and cameras, he is responsible for the imaging and optics system part of this proposal.

Peter Gangar

Peter Gangar is a senior in the Aeronautics and Astronautics department at the University of Washington. He graduated from Bellevue Community College with an Associate of Arts and Sciences degree in 2005. Besides the foundational aerospace courses, his relevant coursework includes orbital mechanics, controls, and systems engineering. In addition to aerospace engineering, Peter is pursuing a Bachelor of Arts degree in Classics to be completed in 2010. He joined the University of Washington's 2007-08 Design Build Fly Team as part of the structures team, manufacturing control surfaces and wings. He has also served as treasurer for AIAA student branch since early 2008. During the summer of 2008, Peter worked as systems engineering intern for Phantom Works, assisting on a project to implement RFID technology on aircraft parts.

Peter served under the navigation/controls team for the 2009 senior design project. During the proposal stage, he was responsible for constellation design, coverage analysis, and orbital mechanics. He contributed the constellation design section to the proposal.

In his non-existent free time, Peter enjoys reading, playing the violin, and spending time with his family. He hopes to earn a private pilot license, and dreams of becoming an astronaut. Whether or not he achieves this dream, his ultimate goal is to serve God in whatever he does.

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Douglas (Austin) Kemis

Douglas Kemis is currently a senior in the Department of Aeronautical and Astronautical Engineering at the University of Washington. He received his A.S. in Engineering from Everett Community College concurrently with graduation from high school in 2007, propelling him to graduate with a B.S. in Aerospace engineering at the age of 20. Douglas relishes challenging projects and has engaged in several team and single endeavors requiring extensive planning and execution. These include finishing two 1000ft2 home basements, singlehandedly wiring the first at eleven years of age, constructing a three stall horse barn, operating heavy machinery, and providing general maintenance at home and in industrial settings on all types of systems. Douglas applies his knowledge and experience to developing inventive conceptual designs to topics ranging from professional home design to hydrogen production plants using electrolysis. He enjoys reading, playing the saxophone, and spending time with family.

Relevant coursework to this project includes Orbital Mechanics, Propulsion, and Advanced Propulsion. At this time, Douglas is acting as a Research Assistant for the Composites Lab. Douglas is the Systems Engineer for the Rapid Response Earth Imaging Constellation, University of Washington design team. Along with compiling the proposal, he was responsible for the Deployment Subsystem research and design.

Nikolas Lutzenhiser

Nikolas Lutzenhiser is currently a senior pursuing a degree in Aeronautics/ Astronautics at the University of Washington. Relevant coursework/specialization undertaken in the aforementioned degree has been centered around fluid dynamics, propulsion, and heat transfer, and plasmas. I am interested in performing research in one of these fields either prior to or post graduation. I am the Aerodynamics Team Leader for the 2009 Formula SAE Design Project. Participation in the Formula SAE program has provided me with experience in the fields of Computational Fluid Dynamics analysis as well as composites manufacturing, personnel and time management, and the engineering design cycle. Components developed by the Aerodynamics Team include Multi-Element Wings, an Underbody Diffuser Tray, and a CFD-Optimized Engine Cooling System. Each of these systems works to improve the performance of the vehicle, which will be tested against other universities at the 2009 Formula SAE competition. I also worked as an intern for the United States Military Sealift Command's Engineering Support Division during the summer of 2008, performing testing and inspection of several vessels and redesign and implementation of systems in port. This position allowed me to gain valuable experience in practical application of my academic knowledge. I am planning on working in the space industry after graduation, or attending graduate school. Together with Rachel Brennan, Nikolas is responsible for the

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propulsion system design of both the satellite constellation and the satellite carriers. He enjoys video gaming, motorsports, and soccer.

Katie Moravec

The relevant coursework I have taken includes Orbital and Space Flight Mechanics, Control in Aerospace Systems, Propulsion, Advanced Propulsion, Structural Vibrations, Structural Analysis I & II, and Aerospace Lab I & II. The course I enjoyed the most was controls, which was also my strongest subject area. For the past three years, I have had summer-long internships at Boeing. The first two summers I spent in Flight Test Engineering, and the third summer I spent in Product Development Weights Engineering. During my third internship, I worked in a team that came up with design improvements for the 777 freighter. During our Structural Analysis II course, I was part of a team that designed a wooden wingbox. In this proposal, I was responsible for the GPS system. I spent a number of hours researching GPS systems that would best fit the requirements for this project. After gathering information on cost, accuracy, power requirements, weight, size, and electrical interface, I found a possible GPS system that meets the needs of this project.

Skander Mzali

Skander Mzali is a Senior in Aeronautics and Astronautics and the University of Washington, pursing a Bachelor's of Science in Aeronautic and Astronautic Engineering at 19 years old. He hopes to continue his Aerospace studies in Graduate School to attain a Master's degree. Skander worked on the research and analysis of power systems for the spacecraft.

Zahra (Bita) Nazari

Zahra researched about sensors and actuators, to be used in attitude determination and navigation systems. She studied the performance, physical characteristic, and costs for each of these devices. She also prepared a summary of her and Eun-Ju 's collected information for the proposal. In the write-up, she discusses the cons and pros of the devices and suggests a possible control mode and navigation system. During autumn quarter she worked on the Spring-Mass System project in professor Ly's control lab. The system consisted of four sliders attached to each other via three springs. The objective was to move the furthest slider from the motor for one inch in the shortest time. She successfully modeled the system and designed a PID controller to

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improve the response of the system. Zahra is now taking AA448, which is a control systems class on sensors and actuators.

Josh Ross

Josh is a senior in the Aeronautics and Astronautics department at the University of Washington. He is focusing his studies on spacecraft dynamics, trajectory determination laws, and distributed space system communications and applications. He currently works in the Distributed Space Systems Lab. Coursework relevant to the mission includes orbital mechanics, spacecraft dynamics, plasma physics, and aerodynamics of bodies in rarefied flow.

Josh is in the Communications group and is responsible for designing a communication algorithm that optimizes time, power, and computation. Josh works with Miguel Carrion in researching the hardware to be used on the Cubesat communication system. Outside of communications, Josh has worked with Peter Gangar for constellation design. Josh originally proposed a Walker constellation, and then worked with Professor Mattick and Peter to determine the impracticality of such a complex constellation.

Eun-Ju (Zetta) Shin-White

Eun-Ju researched about sensors and actuators, which can be used for attitude determination and navigation. She studied their performance, physical characteristics, and costs. She calculated the worst-case disturbance torques and reported them on the proposal. During the autumn quarter, she worked on the Magnetic Ball Levitation project in professor Ly's control lab. She succeeded in designing a PID controller in order to improve the model, so that the magnetic ball would ascend while under the influence of the magnetic field. Eun-Ju is now taking AA448, which is a control systems class on sensors and actuators.

CubeSat Mission- Washinton University

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