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Transcript
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
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 2
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.
3
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
4
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
5
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
6
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
7
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 worsecase 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
330
33
# Sats per
plane
10
Phasing
Angle (deg)
19.6
Coverage
Fraction
0.979
Maximum Revisit
Gap (min)
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
# Planes
8
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
9
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.
10
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).
11
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
12
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.
13
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 orbitchanging 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
14
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 +
4.67
$6.44 M (est) Cape, Vandenburg 27.8-50 63.8-110 1/yr Delta IV Heavy 2.46$6.25 M (est)
Cape,
Vandenburg
27.8-50
63.8-110
1/yr
15
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
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 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. 16 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. 17 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 82 m/s 0 m/s 4: Cubesat Launch 10 m/s 5 m/s (imparted) Total 913 m/s 28 m/s 3: Cubesat Deployment 18 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. 19 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 20 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. 21 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). 22 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 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th ∆V (m/s) Propellant Mass (kg) 0 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 10.1 23 Elapsed Time (sec) 0 0 0.5 50.5 0.4 47.8 0.4 45.2 0.4 42.5 0.4 39.9 0.3 37.2 0.3 34.6 0.3 32.0 0.3 32.0 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 24 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 (Ti6Al-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 25 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 26 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 27 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 Yaxis. 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 28 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. 29 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. 30 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 8W 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. 31 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 MicroPPT 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 approximately10,000/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.
32
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
33
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
34
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
distance of
. For this mission, the resolution is 3 m at a
, 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.
35
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:
that the spot covers 1.5 pixels (
(for ray optics). With the assumption
, 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
36
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.
37
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.
38
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.
39
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 MaksutovCassegrain 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 40 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 approximately300 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.
41
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,
N pix =
picture length 5000m
=
= 1818 pixels
resolution
3.0m
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.
42
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
43
N pic = N pix − 1
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 KAI10100 (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 44 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. Figure 3.2.1: Picture of Kodak KAI‐10100 CCD 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
45
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:
I vis ~ .4 * I sun = 400
w
m2
Eq. 3.3.1
because only 40% of solar emission is visible light. The intensity on the CCD is:
I CCD =
I vis * ρ *cos(θ ) cos 2 (α )
* Alens
π*f2
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,
46
I CCD =
I vis * ρ * Alens
*b
π*f2
Eq. 3.3.3
where
α
Figure 3.3.1: Light Intensity on Space Craft b = cos(θ ) cos 2 (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,
tsat =
N sat e
.
Ne
=
25, 000
.004
sec
=
6
6.5*10 * b
b
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,
tmin =
N e min
.
Ne
=
500
.0005
sec
=
6
6.5*10 * b
b
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
47
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
48
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
49
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. 50 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.
51
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
52
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
53
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
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
Accuracy = 0.005 deg to 3 deg
Attitude Accuracy = 1 arc sec to 1 arc min
0.0003 deg to 0.01 deg
Attitude Accuracy
0.1 deg to 1 deg (LEO)
<0.1 deg to 0.25 deg
Sun Sensors
Star Sensors
(Scanners and Mappers)
Horizon Sensors
• Scanner/Pipper
(Static)
Magnetometer
Attitude Accuracy = 0.5 deg to 3 deg
Mass
Range (kg)
1 to 15
Power
(W)
10 to 200
0.1 to 2
<1 to 5
0 to 3
<2 to 20
1 to 4
0.5 to 3.5
5 to 10
0.3 to 5
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
54
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.
55
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 threeaxis 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.
56
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 57
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
AeroAstro
Miniature Star
Tracker
A-STR
Galileo
Avionica
Ball
Aerospace
Sodern
SunSpace
Size
[mm]
5.4 x 5.4 x 7.6
Weight
[kg]
0.425
Power
[W]
<2
Accuracy
[arcsec]
± 70
Rate
[Hz]
~1
195 x 175 x 288
3
8.9 ~ 13.5
< 10
10
< 5.5
8~9
< 10
10
SED 16
Dia. 203.2
H = 198.12
170 x 160 x 290
< 3.0
7.5
< 15
10
Sun-Star
136 x 136 x 280
1.63
< 3.5
22 (1σ)
1
CT-602
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 58 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 59 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. 60 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 61 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 62 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 5 H. Lee and H. Bang, “Star Pattern Identification technique by Modified Grid Algorithm”, IEEE, VOL. 43, NO. 3, July 2007 R. Zenick and T. J. McGuire, “Lightweight, Low‐power Coarse Star Tracker”, 17th Annual AIAA/USU Conference on small Satellites 63 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 64 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 65 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. 66 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 Acquisition of Sun, Earth, Star 3 to 10% of total Propellant mass Low total impulse, typically <5000 N.s, 1 K to 10 K pulses, 0.01 to 5.0 sec pulse width 100 K to 200 K pulses, minimum impulse bit of 0.01 N.s, 0.01 to 0.25 sec pulse width Low total impulse, typically <7000 N.s, 1 K to 10 K pulses, 0.02 to 0.20 sec pulse width 5 to 10 pulse trains every few days, 0.02 to 0.10 sec pulse width On/off pulsing, 10 K to 100 K pulses, 0.05 to 0.20 sec pulse width On-orbit normal mode control with 3-axis stabilization, limit cycle Precession control (spinners only) Momentum Management (wheel unloading) 3-axis control during ΔV (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) 67 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 Thrusters Hot Gas (Hydrazine) Cold Gas Reaction and Momentum Wheels 0.5 to 9000 N** <5 N** 0.4 to 400 N.m.s for momentum wheels at 1200 to 5000 rpm; max torques from 0.01 to 1 N.m 68 Mass (kg) Power (W) ~2 to 20 10 to 110 Control Moment Gyros 25 to 500 N.m of torque >10 Magnetic Torquers 1 to 4000 A.m2‡ 0.4 to 50 90 to 150 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. 69 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. 70 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. 71 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 72 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 73 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] Star Tracker ±0.0085Degrees 3-axes (3σ) 0.110** IMU Complete 6 DOF Max update rate 64Hz 120 μaguss ±1Degrees 2-axes (2σ) Torque of about 0.003Nm 3 wheel Wheel motor Gimbal motor Torque of 0.024mN-m 318 turns of coil 0.030 Camera 4.4 x 4.4 x 2.54 Lens Dia. 3.35 Length 3.60 3.5 x 2.2 x 1.2 Magnetometer Sun Sensor CMG Magnetorquer 25.6 x 10-6 0.036 0.5 2V None *** 0.05 x 3 0.001 0.036 ~d=3, t=0.25 0.5 D=~3.5 1 0.08 ** Mass and power required based on AeroAstro’s Miniature Star Tracker ***Design specifications will be determined later on. 0.3 x 0.3 x 0.14 Dia. 2.43 Height 3.49 Power Required [Watt] <2** 74 5.5 x 5.5 x 0.5 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 Star Tracker • Camera • Lens IMU Magnetometer Sun Sensor GPS • Receiver • Antenna CMG Magnetorquers Hardware will be designed EO-1312M 1/2" CMOS Monochrome USB Lite Edition Camera 25mm Megapixel Fixed Focal Length Lens MICRO-ISU BP3010 3-Axis magnetic sensor HMC 1043 Medium sun sensor Cornell Cougar GPS receiver Toko DAX Dielectric Patch Antenna Wheels: designed (3) Wheel motor Gimbal Motor Designed (3) Cost [each]725
Company
UW
Edmundoptics
$195$700
$45$5640
Edmundoptics
Honeywell
AeroAstro
<$20000 Cornell Cougar undetermined UW Micromo Electronics Applimotion, Inc. UW undetermined 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. 75 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. 76 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 Pitch with folded solar panel for unfolded solar panel for folded solar panel for unfolded solar panel for folded solar panel Gravity-gradient Solar Radiation Aerodynamic 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. 77 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 Tc = −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 Tc = Tp sin θ . Another option is Proportional-plus-derivative (PD) Control such as Tc = K1θ + K 2θ , 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 Tc = K1θ + K 2θ + K3 θ dt . The PID controller is a sub-class of a full state feedback controller. 78 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. 79 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. 80 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 81 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. C3 C1 C2 CN Ground Station 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, 82 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. (a) (b) (c) 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. 83 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 84 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. 85 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 highgain 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. 86 U/L: Single target, directional signal X/L: Multiple neighbors, ideally isotropic signal D/L: Single target, maximize fly‐by access time 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. 87 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. 88 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 crossplane 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. 89 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. 90 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 91 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. 92 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. 93 Table 6.1.0 CubeSat Systems Power Requirements System Subsystem Component Power required Operating Duty cycle GNC Navigation GPS 2 Watts V l 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 10 Watts Estimated Max Power Used 94 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 Power per 20 (mA/cm2) cm2 cell (W) Efficiency, η .81 Bare mass per 20 cm2 cell (g) 1.68 Spectrolab NeXt Triple Junction1 (GaInP2/GaAs/Ge ) Spectrolab Ultra Triple Junction2 (GaInP2/GaAs/Ge ) Spectrolab Improved Triple Junction3 (GaInP2/GaAs/Ge ) 2.333 17.32 2.35 16.3 .77 1.68 28.3% 2.27 16 .73 1.68 26.8% 29.9% 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. 95 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. 96 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. 97 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 98 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 99 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. 100 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. 101 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 102 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 103 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 104 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 105 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. 106 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. 107 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 GNC Nav ADCS Propulsion GPS board DC Motor for CMG CMG Wheels (3 wheels) 3 Magnetorquers Magnetometer Star Tracker Camera Lens IMU Sun Sensor PPT Unit Imaging, CDH Support Systems CDH Communications Imaging Power Motherboard Transceiver CCD Telescope 2 Batteries 108 Volume Dimensions (cm) 9.525 x 5 x 1.7 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 2 x 1 x 8 9 x 9 x 2 7.8 x 5.2 x 2 2x1.5x0.75 26.3 long, 10.6 diameter 3.9 x 4.2 x 0.06 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. 109 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. 110 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 111 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. 112 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. 113 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. 114 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 115 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 Width Height Length (from flange mount) Requirement 60cm 60cm 96cm C.G. (from flange mount) 48cm Flange Mount Diameter Mass 38cm 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. 116 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. 117 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. 118 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 100g CMG (Wheels and Motor) 3 Magnetorquers 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 119 240g 6.0kg 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 Total 30kg Allowing for Uncertainty 120 160kg 170kg Mission Cost With a maximum mission cost of200 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 121 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. 122
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. 123
References
http://www.mae.cornell.edu/cubesat/
AFIT/GAE/ENY/07-D01,"VACUUM CHAMBER CONSTRUCTION AND
CONTAMINATION STUDY OF A MICRO PULSED PLASMA THRUSTER", Jacob H.
Debevec, BS 2d Lt, USAF, December 2006 pp.25,26
http://cubesat.atl.calpoly.edu/media/Documents/Papers/stensat_hist.pdf
http://informationvision.net/spacecraft.aspx
http://www.aoainc.com/capabilities/SiteFiles/docs/MRE-01_MonoProp_Thruster.pdf
http://www.futron.com/pdf/resource_center/white_papers/FutronLaunchCostWP.pdf
http://www.mae.cornell.edu/cubesat/past/Science-GPS.doc.
http://www.saftbatteries.com/doc/Documents/liion/Cube572/VL34480_1107.5a3fabf4-c54d489b-a4f2-2062e5a60c34.pdf
http://www.spacequest.com/
http://www.spacequest.com/product.php
http://www.spectrolab.com/prd/space/cell-main.asp www.matweb.com http://ieeexplore.ieee.org/ielx5/4344970/4344971/04345026.pdf?isnumber=4344971 http://www.aeroastro.com/publications/SSC03‐II‐7.pdf http://spacecraft.ssl.umd.edu/design_lib/Delta4.pl.guide.pdf Hopkins, Joshua B, et. al. International Reference Guide to Space Launch Systems. 3 ed.
Reston, VA: AIAA, 1999.
“DeltaIV.” Encyclopedia Astronautica. 1 Mar 09.
<http://www.astronautix.com/lvs/deltaiv.htm>
Space Transportation Costs: Trends in Price Per Pound to Orbit 1990-2000”
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September 6, 2002
<http://www.futron.com/pdf/resource_center/white_papers/FutronLaunchCostWP.pdf>
Caffrey, Robert, Gary Mitchell, Zeno Wahl, and Ray Zenick. Product Platform Concepts Applied to Small Satellites: A New Multipurpose Radio Concept by AeroAstro Inc. Aug. 2002. AeroAstro Inc. 31 Jan. 2009 <http://www.aeroastro.com/publications/SSC02‐X‐8.pdf>. "Dipole antenna." Wikipedia, the free encyclopedia. 29 Jan. 2009 <http://en.wikipedia.org/wiki/Dipole_antenna>. Klofas, Bryan, Jason Anderson, and Kyle Leveque. A Survey of Cubesat Communication Systems. 18 Apr. 2008. California Institute of Technology. 02 Feb. 2009 <http://atl.calpoly.edu/~bklofas/Presentations/DevelopersWorkshop2008/CommSurvey
‐Bryan_Klofas.pdf>. "Patch antenna." Wikipedia, the free encyclopedia. 29 Jan. 2009 <http://en.wikipedia.org/wiki/Patch_antenna>. Pedtke, Dan, Mark Lofquist, and Kimberly Kohlhepp. The Modular S‐Band Radio Suite. Aug. 2004. AeroAstro Inc. 31 Jan. 2009 <http://www.aeroastro.com/publications/SSC04‐V‐
4.pdf>. "Transceivers." Satellite Communications Equipment. Ed. Mark Termondt. Satcom Services. 03 Feb. 2009 <http://www.satcom‐services.com/transceivers.htm#ac>. Wetrz, James R., and Wiley J. Larson, eds. Space Mission Analysis and Design. New York: Springer London, Limited, 1992. 125
Mattick, Arthur T., Associate Professor, University of Washington.
Mesbahi, Mehran., Associate Professor, University of Washington.
“Space Transportation Costs: Trends in Price Per Pound to Orbit 1990-2000”
September 6, 2002.
Wertz, James R. and Wiley J. Larson, ed. Space Mission Analysis and Design. Hawthorne, CA:
Microcosm Press, 2007
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Springer and California: Microcosm Press, 1999.
126
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.
127
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
(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.
128
(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
129
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:
Orbit 18:
Ω = 0°
Ω = 180°
ν = 0°, 36°, 72°, etc.
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;
mu=398600;
Ce=2*pi*Re;
g0=9.81;
Ihyd=325;
%
%
%
%
%
km, radius of the earth
km^3/s^2, gravitational constant of the earth
km, circumference of the earth
m/s^2, gravitational acceleration of the earth
sec, Isp for hydrazine
% Orbital parameters
h=520;
fov=45;
tr=5;
% km, altitude for 20 planes with 20 sats each
% deg, field of view
% min, revisit time
% Derived parameters
rs=Re+h;
vs=sqrt(mu/rs);
cs=2*pi*rs;
% km, radius of sat orbit
% km/s, velocity of sat orbit
% km, circumference of sat orbit
130
T=cs/vs/60;
vg=Ce/(T*60);
ws=2*h*tan(fov*pi/180);
ls=vg*tr*60;
%
%
%
%
min, period of sat orbit
km/s, ground speed of sat
km, swath width
km, swath length
np=ceil(Ce/2/ws);
ns=ceil(Ce/ls);
npc=np-1;
% integer # of planes
% integer # of sats per plane
% integer # of plane changes
% Cubesat and Carrier masses
ms=6;
mc=40;
me=10*np;
mt=me+np*(mc+ns*ms);
%
%
%
%
kg,
kg,
kg,
kg,
mass of cubesat
mass of carrier
extra mass for supporting structure
total mass to orbit
i1=70;
i2=90;
di=(i2-i1)*pi/180;
% degrees, inclination of launch orbit
% degrees, inclination of final orbit
% radians, inclination change
dvpc=2*vs*sin(di/2);
% km/s, delta-V for plane change to polar
ra=2*pi/np;
dvra1 = 2*vs*sin(ra/2);
MR= exp(-(dvra1*1000)/(g0*Ihyd));
% deg, RAAN spacing
% km/s, delta-V for a RAAN change
% 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;
dv2=dvpc;
dv3=dvratot;
% km/s, delta-V to orbital velocity
% km/s, delta-V for plane change to polar
% 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.
131
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: '
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
132
WRITE(*,1) ' ENTER INCLINATION IN DEG: '
LINC=XINC_DEG+.001
XINC=XINC_DEG*PI/180
SI=SIN(XINC)
CI=COS(XINC)
WRITE(*,1) ' ENTER FOV IN 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: '
WRITE(*,1) ' ENTER MIN AND MAX NUMBER OF PLANES: '
WRITE(*,1) ' ENTER MIN AND MAX # SATS/PLANE: '
WRITE(*,1) ' ENTER MAXIMUM # OF SATS IN CONSTELLATION: '
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
133
10
20
DO 20 N=0,NP-1
AL(N)=DRA*N
DO 10 M=0,NS-1
P(N,M)=(N*DPHI+M*DTHETA)
CONTINUE
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
134
TL=(P0-AV)/W
TG=(P0+AV)/W
92
93
IF(TL .GT. TP) THEN
CONTINUE
TL=TL-TP
TG=TG-TP
IF(TL .GT. TP) GOTO 92
ENDIF
IF(TG .LT. 0) THEN
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
95
CONTINUE
CONTINUE
IF(L .EQ. 0) THEN
TOFF1=TMAX
GOTO 180
ENDIF
C ORDE
110
115
R THE TIME PERIODS BY TL:
DO 110 I=1,L
IUSE(I)=0
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
CONTINUE
TLO(I)=TMIN
TGO(I)=TGV(JMIN)
IUSE(JMIN)=1
135
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)
140
180
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
CONTINUE
CONTINUE
IF(TOFF1 .GT. TOFF2) TOFF2=TOFF1
IF(ILAT .LE. LINC) THEN
ATRY=ATRY+AREA
IF(TOFF1 .LE. TRV) ARV=ARV+AREA
ENDIF
200
C
300
CONTINUE
IF(ILAT .LE. LINC) THEN
IF(TOFF2 .GT. TOFF3) THEN
TOFF3=TOFF2
ILAT_TOPS=ILAT
ENDIF
ENDIF
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
C
CONTINUE
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
136
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
137
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;
mu=398600;
Ce=2*pi*Re;
g0=9.81;
Ihyd=325;
%
%
%
%
%
km, radius of the earth
km^3/s^2, gravitational constant of the earth
km, circumference of the earth
m/s^2, gravitational acceleration of the earth
sec, Isp for hydrazine
% Orbital parameters
h=520;
fov=45;
tr=5;
i=55;
%
%
%
%
km, altitude for 20 planes with 20 sats each
deg, field of view
min, revisit time
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;
ns=10;
% integer # of planes
% integer # of sats per plane
% Cubesat and Carrier masses
ms=10;
mc=25;
% kg, mass of cubesat
% kg, mass of carrier
138
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;
dvp1 = 2*vs*sin(dt*pi/180/2);
node
MR= exp(-(dvp1*1000)/(g0*Ihyd));
% integer # of plane changes
% km/s, delta-V for plane change at polar
% 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;
dvc1=dvp1;
% integer # of plane change split-offs
% 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);
dvtot(n) = np*dv1(n);
% km/s, delta-V for plane change
% 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);
139
ddomega(n) = di(n) - di0(n);
rate
t1(n)=ra/abs(ddomega(n));
right ascension
ttot(n) = npc*ra/abs(ddomega(n));
planes to deploy
% degrees/day, relative precession
% days, precession time for one
% days, precession time for all
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)
140
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.
141
(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 deltaV, 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;
re=6378;
% km^3/s^2, GM of the earth
% km, radius of the earth
% Set parameters
h=520;
Ns=10;
% km, altitude of cubesat orbit
% number of cubesats to deploy
% Cubesat orbit
rc=re+h;
vc=sqrt(mu/rc);
Tc=2*pi*rc^(3/2)/sqrt(mu);
% km, radius of cubesat orbit
% km/s, velocity of cubesat orbit
% 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
142
% 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,:)
143
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)
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.
144
(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)
145
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;
%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
146
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
147
figure(1), surf(h0,Area,dayslow), shading interp,
xlabel('Injection Altitude(km)','FontSize',12)
ylabel('Frontal Area (m^2)','FontSize',12)
title('\it{Satellite 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)
title('\it{Optimum 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;
T(k) = 2*pi*sqrt((r(k)^3)/mu);
N(k) = (3600*24*365)/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);
148
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('\it{Delta-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('\it{Cold-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)
title('\it{Cold-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('\it{Hydrazine 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)
title('\it{Hydrazine 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
149
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));
150
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));
151
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('\it{Deviation 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('\it{Position 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)
title('\it{True Anomaly}','FontSize',16)
figure (4), plot(numvect,M0,'g',numvect,M1,'b')
xlabel('Elapsed Time of Flight (min)','FontSize',12)
title('\it{Mean Anomaly}','FontSize',16)
152
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('\it{Deviation in Position and Velocity Vector Sum vs. Elapsed
Time}','FontSize',16)
legend('Position','Velocity')
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
153
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);
%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
154
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('\it{Delta-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('\it{Time 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('\it{Carrier 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
155
end
Delta_V_45_400
Delta_V_45_450
Delta_V_45_500
Delta_V_55_400
Delta_V_55_450
Delta_V_55_500
Delta_V_65_400
Delta_V_65_450
Delta_V_65_500
=
=
=
=
=
=
=
=
=
ddvt(1,1)
ddvt(1,2)
ddvt(1,3)
ddvt(2,1)
ddvt(2,2)
ddvt(2,3)
ddvt(3,1)
ddvt(3,2)
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('\it{Required 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('\it{Carrier Wet Mass vs. Inclination at h = 500km}','FontSize',16)
legend('LH2/LOX Bipropellant','N2O4/MMH Bipropellant','UDMH Solid
Propellant')
156
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.
157
Figure B.4.2.Spec sheet for Aerojet R-42.
158
Figure B.4.3 Carrier Main Engine.
159
Figure B.4.4 Carrier RCS Thruster.
160
Figure B.4.5 Moog Latch Valve.
161
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.
162
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,
I vis
.4 * I sun = 400
w
m2
Eq. C.2.3
because only 40% of total light is visible light, which is what we are going for.
I CCD =
I vis * ρ *cos(θ ) cos 2 (α )
* Alens
π*f2
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,
I CCD =
I vis * ρ * Alens
*b
π*f2
Eq. C.2.5
where
b = cos(θ ) cos 2 (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,
tsat =
N pixel
Ne
=
40, 000
.006
sec
=
6
6.5*10 * b
b
Eq. C.2.7
163
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,
tmin =
N min
500
.0005
sec
= 6
=
10 * b
Ne
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.
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.
164
Eq. (D.2.1.3)
Eq. (D.2.1.4)
, 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
is the frontal surface area,
, c is the speed of light,
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,
center of mass,
, and the difference of center -of solar pressure to
.
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.
165
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;
aerodynamic pressure; and
the center of gravity.
Assumption:
and
for unfolded solar panel,
Eq. (D.2.1.11)
for unfolded solar panel,
Eq. (D.2.1.12)
D.2.2. Calculations for Worst Disturbance Torque Estimation
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
Fs=1358;c=3*108;Amax=0.15;Amin=0.02;q=0.5;i=0;
cps-cg=0.05;
the center of
166
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;
Cd Amax V2);Tamax=Fmax (0.025);Print["Ta_max = ",Tamax]
Fmax=0.5 (
Ta_max = 3.91215×10-7
Cd Amin V2);Tamin=Fmin (0.025);Print["Ta_min = ",Tamin]
Fmin=0.5 (
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.
167
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
,
.
168
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.
Accuracy of pointing [δθ in degrees] 0.0057 0.0069 0.0080 0.0092 0.010 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.020 0.022 0.023 * assumption:
Total pixel on CCD [Megapixels] 10° 3.05 2.12 1.55 1.19 0.94 0.76 0.63 0.53 0.45 0.39 0.34 0.30 0.26 0.24 0.21 0.19 11° 3.69 2.56 1.88 1.44 1.14 0.92 0.76 0.64 0.55 0.47 0.41 0.36 0.32 0.28 0.26 0.23 12° 4.39 3.05 2.24 1.71 1.35 1.10 0.91 0.76 0.65 0.56 0.49 0.43 0.38 0.34 0.30 0.27 13° 5.15 3.58 2.63 2.01 1.59 1.29 1.06 0.89 0.76 0.66 0.57 0.50 0.45 0.40 0.36 0.32 14° 5.97 4.15 3.05 2.33 1.84 1.49 1.23 1.04 0.88 0.76 0.66 0.58 0.52 0.46 0.41 0.37 15° 6.85 4.76 3.50 2.68 2.12 1.71 1.42 1.19 1.01 0.87 0.76 0.67 0.60 0.53 0.48 0.43 16° 7.80 5.42 3.98 3.05 2.41 1.95 1.61 1.35 1.15 1.00 0.87 0.76 0.67 0.60 0.54 0.49 17° 8.80 6.11 4.49 3.44 2.72 2.20 1.82 1.53 1.30 1.12 0.98 0.86 0.76 0.68 0.61 0.55 18° 9.87 6.85 5.04 3.86 3.05 2.47 2.04 1.71 1.46 1.26 1.10 0.96 0.85 0.76 0.68 0.62 19° 11.0 7.64 5.61 4.30 3.39 2.75 2.27 1.91 1.63 1.40 1.22 1.07 0.95 0.85 0.76 0.69 20° 12.1 8.46 6.22 4.76 3.76 3.05 2.52 2.12 1.80 1.55 1.35 1.19 1.05 0.94 0.84 0.76 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.
169
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.
170
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] 1/2 1/4 1/8 1/15 1/30 1/60 1/125 1/250 1/500 * assumption: the magnitude of brightness of star, M = 5.5
Diameter [cm] 0.56296 0.79614 1.12591 1.54172 2.18032 3.08344 4.45056 6.29404 8.90111 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
171
Accuracy of pointing [δθ in degrees] 0.0057 0.0069 0.0080 0.0092 0.010 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.020 0.022 0.023 1/2 17.76 14.80 12.69 11.10 9.87 8.88 8.07 7.40 6.83 6.34 5.92 5.55 5.22 4.93 4.67 4.44 1/4 12.56 10.47 8.97 7.85 6.98 6.28 5.71 5.23 4.83 4.49 4.19 3.93 3.69 3.49 3.31 3.14 1/8 8.88 7.40 6.34 5.55 4.93 4.44 4.04 3.70 3.42 3.17 2.96 2.78 2.61 2.47 2.34 2.22 Exposure Time [sec] 1/15 1/30 1/60 6.49 4.59 3.24 5.41 3.82 2.70 4.63 3.28 2.32 4.05 2.87 2.03 3.60 2.55 1.80 3.24 2.29 1.62 2.95 2.08 1.47 2.70 1.91 1.35 2.49 1.76 1.25 2.32 1.64 1.16 2.16 1.53 1.08 2.03 1.43 1.01 1.91 1.35 0.95 1.80 1.27 0.90 1.71 1.21 0.85 1.62 1.15 0.81 1/125 2.25 1.87 1.60 1.40 1.25 1.12 1.02 0.94 0.86 0.80 0.75 0.70 0.66 0.62 0.59 0.56 1/250 1/500 1.59 1.12 1.32 0.94 1.13 0.80 0.99 0.70 0.88 0.62 0.79 0.56 0.72 0.51 0.66 0.47 0.61 0.43 0.57 0.40 0.53 0.37 0.50 0.35 0.47 0.33 0.44 0.31 0.42 0.30 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.
172
1
m( w2 + d 2 ) ≈ 0.0417[kg.m 2 ]
12
1
I w = m(h 2 + d 2 ) ≈ 0.108[kg.m2 ]
12
1
I d = m(h 2 + w2 ) ≈ 0.083[kg.m2 ]
12
Ih =
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.
1
2
ii
ii
θ = θ t2 ⇒ θ =
2θ
t2
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.
ii
N w−req = I s θ
Eq. (D.2.4.3)
where Nw-req is the required torque needed to achieve the specified maneuver, Is is the spacecraft
ii
moment of inertia, and
θ
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
173
Then using the equation below to determine the momentum of the wheel.
Nx
i
N x = 2h0 δ cos β cos δ ⇒ h0 = 2
i
δ cos β cos δ
Eq. (D.2.4.4)
i
δ
where Nx is the wheel torque on x-axis maneuver, h is angular momentum of the CMG,
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.
174
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.
h0 = I CMGω ⇒ I CMG =
h0
ω
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
175
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.
t=
( Slew _ Rate)( MOI )
Torque
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
176
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
Iz =
mr 2
2
1
I x = I y = m(3r 2 + h 2 )
12
corresponding to Figure D.2.4.3.
177
Eq. (D.2.4.7)
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] Mass[kg] I_z 3
(thickness of 0.005m) (8400kg/m ) [kg‐m2] I_x=I_y Mass[kg] I_z 2
[kg‐m ] (8700kg/m^3) [kg‐m2] I_x=I_y [kg‐m2] 0.01 3.93E‐07 3.30E‐03 0.012 5.65E‐07 4.75E‐03 0.014 7.70E‐07 6.47E‐03 0.016 1.01E‐06 8.44E‐03 0.018 1.27E‐06 1.07E‐02 0.02 1.57E‐06 1.32E‐02 3
Dia. Volume[m ] Mass[kg] [m] (thickness of 0.0025 m) (8400kg/m3) 4.12E‐08
8.55E‐08
1.58E‐07
2.70E‐07
4.33E‐07
6.60E‐07
I_z [kg‐m2] 2.75E‐08
3.42E‐03 4.27E‐08 5.26E‐08
4.92E‐03 8.86E‐08 9.27E‐08
6.70E‐03 1.64E‐07 1.53E‐07
8.75E‐03 2.80E‐07 2.39E‐07
1.11E‐02 4.48E‐07 3.57E‐07
1.37E‐02 6.83E‐07 I_x=I_y Mass[kg] I_z 2
[kg‐m ] (8700kg/m^3) [kg‐m2] 2.85E‐08 5.45E‐08 9.60E‐08 1.58E‐07 2.47E‐07 3.70E‐07 I_x=I_y [kg‐m2] 0.01 0.012 0.014 0.016 0.018 0.02 8.25E‐08
1.71E‐07
3.17E‐07
5.40E‐07
8.66E‐07
1.32E‐06
4.47E‐08
9.04E‐08
1.65E‐07
2.79E‐07
4.44E‐07
6.73E‐07
4.63E‐08 9.37E‐08 1.71E‐07 2.89E‐07 4.60E‐07 6.98E‐07 7.85E‐07 1.13E‐06 1.54E‐06 2.01E‐06 2.54E‐06 3.14E‐06 6.60E‐03 9.50E‐03 1.29E‐02 1.69E‐02 2.14E‐02 2.64E‐02 178
6.83E‐03 9.84E‐03 1.34E‐02 1.75E‐02 2.21E‐02 2.73E‐02 8.54E‐08 1.77E‐07 3.28E‐07 5.60E‐07 8.97E‐07 1.37E‐06 Figure D.2.4.4.Spec sheet for DC motor
179
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.
i
θ =ω
Eq. (D.2.5.1)
ω = (−bω + KI )
Eq. (D.2.5.2)
i
1
J
i
1
I = (v − Kω − RI )
L
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.
⎛
⎛ i⎞ ⎜
θ
⎜ ⎟ ⎜0
⎜ i⎟ ⎜
⎜ω ⎟ = ⎜ 0
⎜ i⎟ ⎜
⎜⎜ I ⎟⎟ ⎜
⎝ ⎠ ⎜0
⎝
1
−b
J
−K
L
180
⎞
⎛ ⎞
0 ⎟
⎜
0⎟
⎟⎛θ ⎞
⎜
⎟
K ⎟⎜ ⎟
0
ω
=
v
⎜
⎟
⎜
⎟
J ⎟⎜ ⎟ ⎜ ⎟
I
− R ⎟⎟ ⎝ ⎠ ⎜⎜ 1 ⎟⎟
⎝ L⎠
L ⎟⎠
Eq. (D.2.5.4)
⎛θ ⎞
⎜ ⎟
y = ( 0 1 0) ⎜ ω ⎟
⎜I⎟
⎝ ⎠
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.
⎛
⎜s
−1
⎜
b
T (s) = ( 0 1 0 ) ⎜⎜ 0 s +
J
⎜
K
⎜⎜ 0
L
⎝
⇒ T ( s) =
⎞
0 ⎟
⎟
−K ⎟
J ⎟
R ⎟⎟
s+ ⎟
L⎠
−1
⎛ ⎞
⎜0⎟
⎜ ⎟
⎜0⎟
⎜1⎟
⎜⎜ ⎟⎟
⎝ L⎠
Ks
JLs + ( JR + Lb)s 2 + ( K 2 + bR)s
3
Eq. (D.2.5.6)
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
181
E. Communications The Link Equation is given as Eb Pt Ll Gt Ls La Gr
=
N0
kTN R
(Eq. E.1) which can be rearranged to isolate Pt and substitute efficiencies for the loss terms to give Pt ≥
Eb kTN R (1 − C ) 4π r 2
N0
Gtηt
ηr Aeff
(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‐
182
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) 183
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] 184
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. 185
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 186
(Eq. F.1.5) 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 187
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: 188
Figure F.1.8 Power properties of carrier vehicle 189
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
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
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;
%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
Vmp = 2.33;
Jmp = 17.32/1000; %A/cm^2
Imp = Jmp*A_cell/.0001; %A
P_cell = Imp*Vmp;
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');
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);
190
q_level = 1-dod;
E_bat_Sanyo = 187*.0246; %W-h/kg * kg
title('Number of cells required by power
required')
N_bat = floor(V_bus_solar/V_bat);
V_bat_total = N_bat*V_bat;
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
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
costrange = 240*[20];
massrange = 1.68*39*2*[1 2.5]/1000; %kg
I_disc = P_req_dark/V_bat_total; %discharge
current
% Batteries
T = 94/60; %Orbital period, hours
cycles = 365*(24/T); %cycles of light-todark
mass_bat_req = E_bat/175; %Batteries over
specific energy
mass_bat_total = N_bat*mass_bat_req; %total
mass of batteries
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;
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 ylabel('Power (W)');
legend('Average power with pointing',
'Average power', 'Required power',
'Location', 'SouthEast');
title('Power generation as a function of
solar wing area')
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;
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')
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);
% 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
figure(1)
plot(extra, .9*maxpower, extra, .9*avgpower,
extra, 12*ones(length(extra)), 'r')
xlabel('Area of solar panel wings (cm^2)');
191
axis([0 3600 0 200])
avgpower2 =
(56/94)*.29*1366*(1/(100^2))*(4*(7200 +
extra1))/(pi^2);
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])
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')
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
V_bus_guess = 28; %A guess
P_req_guess = 48;
e_cell = .2861;
V_cell = Vmp;
N_str = ceil(V_bus_guess/V_cell);
I_sun = 1370; %W/m^2
A_cell = .0001*20; %m^2
P_sun = A_cell*I_sun;
%P_cell = e_cell*P_sun;
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
Vmp = 2.33;
Jmp = 17.32/1000; %A/cm^2
Imp = Jmp*A_cell/.0001; %A
P_cell = Imp*Vmp;
%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
N_cell_guess = ceil(P_req_guess/P_cell);
A_array_guess = N_cell_guess*A_cell/.0001;
%cm^2
192
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;
193
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) 194 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.
195 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
196 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.
197 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 MultiElement 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
198 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
199 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.
200 
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