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INVESTIGATION OF THE DEISGN AND PERFORMANCE OF REPEATING SPACE TRACK CONSTELLATIONS

Michelle K. Perez Advisor: Dr. Christopher Hall

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24060

Abstract

A constellation is a group of satellites that are coordinated to carry out a common mission. There are many ways to design satellite constellations, and this paper investigates and compares a popular method of design known as Flower Constellations, with a relatively newer approach known as Parametric Constellations. This paper also introduces tools that were developed in MATLAB and Satellite Tool Kit to aid in the design and display of Parametric Constellations. Following this tool presentation, the paper also discusses how the tools, were used to show that a Parametric Constellation can be more effective than a Flower Constellation for a specific Geoscience Remote Sensing mission known as FLORAD (FLOwer constellation deploying RADiometers). From this investigation and tool development process, we conclude that the Parametric Constellations can be more effective because the design parameters of this system allow for the use of real numbers.

Nomenclature a = semimajor axis B = base satellite (Parametric) e = eccentricity i = inclination k = index number of satellite Fd = phase denominator parameter (Flower) Fh = phasing step parameter (Flower) Fn = phase numerator parameter (Flower) hp = height of periapsis M = Mean Anomaly Nd = days to complete one orbit (Flower) Np = number of petals (Flower) r = relative to the base satellite γ = relative orbit frequency (Parametric) Ω = Right Ascension of Ascending Node

Ωθ = ascending node spacing (Parametric) ω = argument of periapsis

II. Introduction constellation is an important observation method in which a group of satellites is launched,

synchronized and coordinated to carry out a common mission. This common mission could include performing tasks that enable technologies like global

telecommunications, satellite radio, global positioning, and disaster monitoring. In addition to enabling the above technologies, constellations can also be used for scientific missions such as Earth observation. Constellation design is driven by the mission and the goal of the constellation’s performance. Common constellation performance parameters and design standards include the period of the satellite, the amount of time between revisits of an area, the amount of coverage, the allowable gap of coverage over an area, and the access time for specific ground stations. Depending on the mission objectives, a constellation can be designed to optimize any one of these performance parameters.

A. Flower Constellations There are many different strategies to design

satellite constellations and the first of these strategies is a method known as Flower Constellations (FCs). Flower Constellations were recently developed and are described in References (1) and (2). Constellations that employ this design method are called FC because the orbit path of the satellites appears as an outline of flower petals as the satellites orbit a body. This set of constellations is designed so that a system of satellites can orbit around a Planet Centered Planet Fixed Frame.

This form of design is based on the selection of eleven parameters that define the orbits of the satellite and their phasing. Within the eleven parameters, five are Keplerian orbital element parameters and the other six are integer parameters. In an FC system, all of the satellites have common values of the semimajor axis a, eccentricity e, inclination i, and argument of periapsis ω . The constellation designer selects values for these common parameters, but instead of choosing a value for the eccentricity, a height of perigee, hp is selected. The designer also selects values for the mean anomaly, M0, and the Right Ascension of Ascending Node (RAAN)

0Ω for the first satellite. These five selected orbital element parameters define the shape, and orientation of the satellites with respect to the Earth or planet being orbited.

The integer parameters become involved with the constellation to control the phasing of the satellites. The phasing rules of the FC define the RAAN and the mean anomaly for the kth satellite as

A

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d

nkk F

Fπ21 +Ω=Ω + (1)

dd

hdpnkk NF

FFNFMM

++=+ π21 (2)

The first three integer parameters seen in Equations (1) and (2), which are the phase numerator Fn, the phase denominator Fd, and the phase step Fh, control the satellite phasing and distribution of the satellites along the space track in the constellation. The next two integer parameters, Np the number of petals, and Nd the number of days needed to complete a full orbit, help to define the orbital period of each satellite and the semimajor axis of the satellites. A major limitation of the FC system is that all of the phasing parameters are limited to integers. Currently, applications for FCs are being researched in GPS, Deep space observation and Earth observation systems. One such Earth observation application is presented in Section III of this paper.

B. Parametric Constellations A second strategy that is used to design

constellations is the Parametric Constellation (PC) design method, which is relatively newer and not as well known as the FC method. This system of constellations is described in Reference (3). This method is based on parametric relative equations which result from the relative equations of motion for Keplerian orbits. Similar to the FC system, this constellation design procedure leads to a system of target satellites which has a repeating space track with respect to a base body or a base satellite. This system of design has eight initial parameters, five of which are again based on the Keplerian orbital element set and three additional parameters that are used to help define the orbit size and phasing of the satellites. In a PC, all target satellites also have an identical semimajor axis and eccentricity like the FC system. The other three orbital element parameters in the PC system vary based on the phasing of the satellites and the choice of the additional three parameters in the system.

Each target satellite in a PC is defined by its inclination, argument of periapsis, and mean anomaly in terms of a real number system which controls the RAAN spacing for the satellite. The satellite phasing rules that control the PC system can be seen in Equations (3), (4), (5), and (6) where the subscript k is the integer index for the satellite, the subscript r means relative to the base satellite and the subscript B is for the base satellite.

( )11 −+Ω=Ω Ω kk θ (3)

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cos

1

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kkB

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iii

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iiiii

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11

(5)

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ΔΩ= −

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)(100

1)(1 coscoscos

sinsinsintan (6)

In the equations above, the only parameters in addition to the orbital elements of the first satellite, which are selected by the designer to control the system are N, the number of satellites, γ , the relative orbit

frequency and Ωθ , the spacing between each ascending node. The last two parameters are based on the real number system.

In terms of selecting parameters and designing constellations, the PC system is more user friendly because it involves less extraneous parameters. The PC system also only requires eight parameters compared to the FC’s eleven. The PC system is also a stronger constellation design method because it is not limited to creating relative space tracks about a Planet Center Planet Fixed Frame. The PC system can easily be used to design a constellation of satellites around another satellite which is harder and more difficult to do in an FC system. Finally, the PC method is also more powerful because its additional parameters, aside from the orbital elements, are based on real numbers rather than integers. The PC system of designing constellations has promising applications in inter-satellite constellation design and formation flying, but is not currently in use in any application.

II. Development of Parametric Constellation Tools

Given that Parametric Constellations are a relatively new design method, not an extensive amount of

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information is known about their potential applications. In order to learn more about the potential held in this design theory, we develop and use a series of tools to aid in the design, display and further understanding of this method.

A. MATLAB Graphical User Interface and Satellite Tool Kit Parametric Constellation Display

We first develop a series of tools in MATLAB and Satellite Tool Kit (STK) that allow the user to easily design and display a PC system. Satellite Tool Kit is powerful software that lets users visualize and calculate performance parameters for satellites in 2D and 3D spaces.

The purpose of this first set of tools is to allow a user to enter initial PC design parameters and visualize the constellation that results from those parameters. The best way to achieve this goal is by writing a Graphical User Interface (GUI) program in MATLAB that accepts PC parameters from the user and connects to STK, so that STK can display the user’s constellation. We create the MATLAB GUI that can be seen in Figure 1 to serve this purpose and complete this process.

The parameters that the user is prompted for in the GUI are the eight governing parameters for the PC system. Once the user enters the parameters, and clicks on the push button “Calculate,” MATLAB can retrieve

the data and runs through the calculations that were illustrated above in Equations (3) through (6) to calculate the orbital elements for each successive satellite in the constellation. Once all the orbital elements for each satellite are defined, MATLAB then interfaces with STK through a plug in. Using this interface, MATLAB creates and populates a scenario with the computed satellite orbital elements in STK in order to visualize the constellation. MATLAB controls the STK scenario through a series of Connect commands that are given to STK through the interface. These Connect commands include the ability to create a scenario, initialize a time period, establish a reference frame, create a satellite, and initialize the orbital elements of the satellite.

Once MATLAB finishes creating and populating the scenario, STK displays the PC system on a 2D map and on a 3D globe. An example of one such constellation that was created from the above GUI on a 2D map can be seen in Figure 2. In addition to the still image in this Figure, STK also shows an animation of the constellation for the time period that is set by the user for the scenario using the Connect commands.

The GUI in Figure 1 and the image in Figure 2, are

both representative of a Repeating Ground Track PC system. In addition to forming a constellation with a Repeating Ground Track around a Planet Centered Planet Fixed Frame, it is also possible to create a PC system around a base satellite in a circular orbit, as was

Figure 1. Repeating Ground Track Graphical User Interface. This is the GUI that we develop and use for a Repeating Ground Track PC system.

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previously mentioned. In order to create this type of constellation, which is known as a Repeating Relative Orbit constellation, we develop and use a different GUI tool similar to the one shown in Figure 1.

To calculate this Repeating Relative Orbit constellation, the user must follow the same process as before by entering their PC design parameters into a GUI. Once the user enters the parameters, MATLAB can again retrieve the parameters, calculate the individual satellite orbital elements according to the PC phasing rules, and finally use Connect commands to populate a scenario in STK. Even though the process is the same, we develop and use a different GUI program because the user must enter the properties of the base satellite, unlike before in the Repeating Ground Track constellation.

B. Transformation Functions Flower and Parametric Constellations have the

ability to mimic each other and can produce the same exact constellation with their individual parameter definitions. Using this fact, we develop a series of functions that allows the user to convert FCs to PCs and vice a versa. Parametric Constellations however have a bigger scope than the FC system, because their additional phasing parameters are based on a real number system, while the FC system is based on integer parameters. Keeping this limitation in mind, we develop two transformation functions in MATLAB. These two functions do not plot the constellation or interface with STK, they are simply building block functions that allow the user to enter the defining initial parameters of one system and return the initial parameters of the second system for the same constellation.

In the development of these functions, we work with both sets of phasing equations and parameters for the two systems simultaneously. To make the functions

work properly, each phasing parameter in the two systems must be defined in terms of the other, which is extremely complicated. As a result of the complexity, we only develop these transformation functions for the Repeating Ground Track scenario around the Earth. In the Repeating Ground Track scenario, the equations are more manageable to deal with because the arguments of periapsis and the inclinations of the satellite are also identical for all satellites in the constellation, in addition to the pre-existing common values of the semimajor axis and eccentricity. We also create these functions for the case where the phasing step, Fh for the FC system is equal to zero. This simplification is made, because the original functions and definitions observed in Reference (1) did not include the Fh parameter.

The easier of the two functions to create is the FC to PC transformation because we convert and condense eleven (ten without Fh) parameters into eight. In this conversion, some parameters such as the argument of periapsis, the inclination, the initial mean anomaly, and the initial RAAN are the same in both systems so they do not vary in the transformation from one system to another. Also the number of satellites between the systems is the same where Ns in the FC system becomes the N parameter in the PC system. Not all parameters are the same though, and it is necessary to calculate three quantities from FC parameters to complete the PC parameter definition.

The first parameter we define for the PC system is the eccentricity. We can calculate this parameter from the FC parameters of Nd, Np, and hp using equation set (7). In this equation set, T represents the period of the individual satellite, Earthω is the angular speed of the Earth, μ is the standard gravitational parameter of the Earth, and REarth is the radius of the Earth.

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1

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ωπ

(7)

Following the eccentricity, we calculate values for the two real number phasing parameters of the PC system, starting with the relative orbit frequency as can be seen in Equation (8). This is the simpler parameter to define and involves only the semimajor axis of the satellites.

Figure 2. Two Dimensional Repeating Ground Track PC. This is an example of the 2D image that STK produces once the user enters all the PC parameters into the GUI interface in Figure 1.

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Eartha ω

μγ 121

3 ⎟⎠⎞

⎜⎝⎛= (8)

Next we define the second real number phasing parameter, the ascending node spacing in terms of the phasing FC parameters using Equation (9).

d

n

FFπθ 2−=Ω (9)

The final parameter that we define and calculate, which has not yet been discussed is 10M . This is an exclusive fit parameter that allows a FC system to be duplicated by a PC system. This fit parameter serves as a fudge factor for the transformation between the two constellations and is calculated using Equation (10). To use this parameter in PC phasing, it is simply added onto the quantities of the left hand side of Equation (6) as seen rewritten in Equation (11). These final equations complete the relationships that we use to transform an FC system to a PC system in MATLAB for the first transformation function.

)( 00 ωγ +Ω+=MM (10)

RkBk MMM ωφγ −−+= )( )(10100 (11)

Next we define the relations that are necessary as we create the second PC to FC transformation function. This function cannot be completed if the relative orbit frequency of the PC system is a real number and not an integer. This is a result of the bigger scope of the PCs, as was previously discussed. This function is more difficult to complete because there are so many FC parameters that it is not possible to define all of them independently. Many values for these parameters are chosen out of convenience instead of necessity by the user in the original FC definition, which makes it difficult when writing an automated computer program. As with the previous transformation function in changing from FC to PC, the argument of periapsis, the inclination, the initial mean anomaly, and the initial RAAN are the same, so they do not change in the transfer from one system to another. Also the number of satellites between the systems is the same, where N in the PC system becomes Ns in the FC system.

To complete the FC definition, we calculate and define five additional FC parameters in terms of PC parameters. The first parameter that we transfer and define in terms of the PC parameters γ , and e, is the height of perigee using Equation (12).

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Earth

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(12)

The next FC parameters that we calculate are the Np and Nd integer parameters. In the FC system, these two parameters are defined together and not independently. They are defined in the first fraction as can be seen in Equation (13), where the denominator references an approximate Sidereal Period.

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)3600*24(

2

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21

3

μπ

(13)

As a result of this simultaneous definition, the values of Nd and Np are set equal to the reduced fraction in the transformation function. Thus in the function, the parameter Nd is arbitrarily set equal to 1, and we assign Np to the corresponding denominator to make the last equation in the Equation (13) set true. These parameters could be set to different values if the user selects a value for Nd other than 1.

The final parameters that we define are the phasing parameters: phasing numerator and phasing denominator. No phasing step parameter is specified in this function because as was noted previously, this transfer function is defined for FC systems where Fh is equal to zero. The remaining phasing parameters, Fn and Fd, are again defined simultaneously and are dependent on each other in the FC system. Usually for convenience the Fd parameter is selected first to represent the number of orbit planes and then Fn is set to one by the user. As we create the transformation function, we use the same definition and general idea, as can be seen in Equation (14).

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==

=

−=

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Ω

Ω

12

2

2

πθ

θπ

πθ

dn

d

d

n

FF

F

FF

(14)

This final equation definition completes the PC to FC transformation relations that we use in MATLAB to create the second transformation function. Clearly the PC to FC transformation is not as rigorous as the previous FC to PC function, which is a result mainly of the number of parameters required and the interdependence of those parameters. Regardless of this fact, the development of these two functions is important because it shows that it is possible to duplicate any FC system (for a Fh = 0) with a PC system, but not necessarily any PC system with an FC system. This limitation of the FC results because of the integer and real number parameter differences between the two systems. This further demonstrates the advantage of a PC design method.

C. Automated Access Calculator The last tool that we develop is an automated access

calculator GUI program. We develop this tool using the first tool, the Repeating Ground Track GUI tool, as a basic framework. This specific calculator allows a user to enter PC parameters, compute the constellation in MATLAB, visualize the constellation in STK, and compute the access time for the constellation for a chosen ground station and time period. Access calculations can be done for each individual satellite with a specific ground station in STK separately, but this process is time consuming and repetitive to complete. This tool automates all access calculations for a constellation so that a user can simply run the GUI, which will return the access time for the constellation without further manipulation in either MATLAB or STK.

In addition to performing access calculations for a specific constellation, this GUI also has the capability to help the user find an optimum constellation based on access time. In this calculator the user enters the parameters for the initial satellite or the orbital elements, and then the user can enter a range of values for the two PC phasing parameters, the relative orbit frequency and the ascending node spacing between the satellites. The GUI is then able to compute for the same set of orbital elements, all possible combinations of the phasing parameters for the range of values entered in the GUI. The GUI then connects to STK, as with the

previous tools, and computes the access time for each possible phasing combination with the common orbital elements. Once STK has computed all of the access calculations for each constellation combination, MATLAB creates a surface plot of the access time for each constellation as a function of the two phasing parameters. From this surface plot, the user can pick the optimum relative orbit frequency and ascending node spacing for a specific set of orbital element parameters based on the access time performance of the satellites.

The development of this tool saves the user a tremendous amount of time. For example, by entering in a range of ten values for each phasing parameter, the GUI will link to Matlab and complete 100 automated access calculations with only one user action. This volume of calculations if not automated could take hours. This tool serves an easy and quick way to obtain an initial estimate of how the access time varies with respect to the phasing parameters for a constellation with a constant set of orbital elements.

III. Parametric Constellation Application to FLORAD Mission

Finally, we attempt to show that PCs are a more advantageous constellation design strategy than FCs. In order to do this, we redesign an FC application with a PC system to show that the PC provides better performance than the FC for the same mission requirements. This process is still in its early stages, but results completed thus far will now follow.

A. Overview of FLORAD Mission The FC application that we select to compare and

explore is an Italian Space Agency mission that is in Phase A, known as FLORAD or FLOwer constellation deploying RADiometers. This constellation is described

in Reference (2). The goal of this mission is to measure thermal and hydrological properties of the troposphere over the Mediterranean region by deploying millimeter-

Figure 3. FLORAD Target Area Window. The MWM and larger RSE window for the FLORAD Mission.2

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wave (MMW) scanning radiometers. This mission has a goal of frequent revisit time at regional scale over the Mesoscale Western Mediterranean (MWM) window and quasi-global coverage over the Regional Scale European (RSE) window. The combined target area for this mission is shown in Figure 3.

In order to adequately complete the science goals of this mission a number of constellation requirements have been developed. The driving factor for the constellation design is to maximize the spatial coverage at a regional scale and to maximize the time coverage over the target area for atmospheric monitoring purposes. It is also preferable to narrow the revisit time to the area to be less than two hours over Southern Europe. Some of the constraints for the mission are that the constellation must last for two years, be deployed in one launch, and have less than or equal to four satellites. One additional constraint from the sensors is that the orbit height of the satellites must be between 450 and 1250 km, so that the sensors have an average linear Field of View less than 25 km.

B. Comparison of Selected Flower Constellation with Parametric Constellation Design

At this point in the planning stage of the mission no constellation has been selected yet, but according to

Reference (2) there are three major options of Flower Constellations that are going through an optimization process. Here, we take the first FC option that has been proposed, redesign the FC option with a PC, and then compare the two solutions.

The first FC option is made up of four satellites with slightly elliptical orbits, a critical inclination of 63.4 degrees and a perigee to apogee ratio of about 450 km to 850 km. This critical inclination is used to avoid the need for perigee control and to satisfy the need to focus mission coverage on the MWM region. The parameters of the FC for this first FLORAD solution can be seen in Table 1.

To demonstrate that a PC system is better than this current FC option, the PC must have better performance and it must not be able to be duplicated by an FC constellation. The easiest way to design a PC that cannot be duplicated by an FC is to use phasing parameters that are real numbers. In order to determine whether or not the PC has better performance, we use a metric of total access time for the constellation. We use the access time because one of the major goals of the mission is to maximize the time the satellites are in the target region. The access time for this example is calculated using an ESA tracking station in STK known as Weilheim II, which is in the middle of the MWM target region. We select this ground station because of

Table 1. FC Parameters for FLORAD Option 1. Np Nd Ns Fn Fd Fh hp (km) i (deg) ω (deg) RAAN (deg) M0 (deg) 44 3 4 1 4 1 476 63.4 251.7 282.6 287.6

Table 2. PC Parameters for FLORAD Option 1.

γ Ωθ (deg) N i (deg) ω (deg) e RAAN (deg) M0 (deg) 14.6 90 4 63.4 251.7 0.0259 282.6 287.6

Figure 4. FC FLORAD Option 1 Coverage Area. The white shaded area represents the coverage over the tracking area.

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its location and because it is a likely choice for a ground station for this constellation’s mission. The coverage results over the area with the access times shaded on the orbit path from STK can be seen in Figure 4 for the FC solution. The total access time for this FC constellation with Weilheim II is 7.2273 hours.

In designing the improved PC system, we use the automated access calculator tool. We are able to create and find a better PC system by running iterations with different relative orbit frequencies with real numbers in this tool. The better PC solution results in an increased performance in access time over the original FC system. The parameters for the resulting PC can be seen in Table 2.

From this table it is obvious that the successful PC utilizes the same Keplerian orbital element parameters as the FC. This is done because the positioning and orientation of the orbit needs to be the same in order to achieve adequate coverage over the MWM target area. The difference between the two types of constellations comes with the choice of the phasing parameters and how the satellites move through the orbit. Since PCs allow the use of real number parameters to control the phasing, the PC is able to achieve additional coverage by using a real number to represent the phasing parameter of γ , the relative orbit frequency. Using these tools and methods leads to the demonstrated PC, which has an access time of 7.4529 hours, and results in a 3.12% increase in access over the target area. The associated coverage area for this access time for the PC can be seen in Figure 5. This small increase shows that the PC system can be made to out perform an FC system.

Looking at Figures 4 and 5, it can be seen that the FC and PC both completely cover the MWM region and RSE scale window adequately. The only difference in the type of coverage is that the PC satellites stay on the same relative path and cover the same area, whereas the FC satellites tend to canvas slightly different parts of the target region as the satellites move through their orbits. This should not make much of a difference and does not put the PC system at a disadvantage because the sensors on board the satellites are scanners and will scan +/- 50 degrees around the boresight. This scanning should make up for difference in the area coverage. Thus, for this FLORAD FC option, the PC system developed is a usable solution that out performs this FC in terms of total access time with this ground station.

C. FLORAD Comparison Conclusion There are still more performance comparisons to

make to ensure that the PC system developed for this first FLORAD FC option truly is a more advantageous system in all mission performance criteria. It is clear from the work that has been done though that the developed PC does spend 3% more time in the region and does adequately cover the MWM and RSE window. This PC also as a result of its real number parameters is not duplicable in the form of a FC. The advantage that is provided in this application by using a PC over an FC is small, but may turn out to be larger for the other two planned FC options for this mission when we complete further comparisons.

Figure 5. PC Option Coverage Area for FLORAD. The white shaded area represents the coverage over the tracking area.

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III. Conclusion Overall, constellations are an integral part of space

and Earth observation. We have shown that there are many different ways to design constellations, two of which are Flower Constellations and Parametric Constellations. Flower Constellations are more widely known and are starting to be used in Earth observing applications. Parametric Constellations have just recently been developed, and are important to consider because their basis on the real number system gives the constellation set a wider range of possibilities.

We also have demonstrated in this paper that a series of tools have been developed for Parametric Constellations. The first set of tools that has been developed can be used to aid in the design, and display of Parametric Constellations in MATLAB and STK. The second set of tools that has been developed are functions that allow users to convert back and forth from Flower Constellation to Parametric Constellation parameters. The final tool that has been developed is a GUI that automates access calculations for a constellation and creates a surface plot of the total access time for a range phasing parameters.

Finally with the tools developed, Parametric Constellations were applied to a planned Flower Constellation mission known as FLORAD. It was shown in this application, that the Parametric Constellation method has been able to meet and exceed the performance of one of the potential Flower Constellation options by 3% in terms of access time for a specific ground station. This demonstrates that the Parametric Constellations can be superior to Flower Constellation systems depending on the application and desired target area.

Overall Parametric Constellations are a new and fascinating constellation design strategy that appears to have great potential and advantages in comparison to other current constellation design methods.

Acknowledgments I would like to acknowledge my advisor, Dr.

Christopher Hall for his help and continued support throughout my research.

References 1Wilkins, M., “The Flower Constellations – Theory,

Design Process, and Applications,” Ph.D. Dissertation, Aerospace Engineering, Texas A&M University, College Station, TX, 2004.

2Marzano, F., Cimini, D., Memmo, A., Montopoli, M., Rossi, T., et al. “Flower Constellation of Millimeter-Wave Radiometers for Tropospheric Monitoring at Pseudogeostationary Scale,” IEEE

Xplore - Geoscience and Remote Sensing, Vol. 47, No. 9, 2009, pp. 3107 – 3122.

3Lee, S., “Dynamics and Control of Satellite Relative Motion: Designs and Applications,” Ph.D. Dissertation, Aerospace Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, 2009.