INVESTIGATION OF THE DEISGN AND PERFORMANCE OF REPEATING SPACE TRACK CONSTELLATIONS

Transcription

1 INVESTIGATION OF THE DEISGN AND PEFOMANCE OF EPEATING SPACE TACK CONSTELLATIONS Michelle K. Perez Avisor: Dr. Christoher Hall Virginia Polytechnic Institute an State University, Blacsburg, VA, Abstract A constellation is a grou of satellites that are coorinate to carry out a common mission. There are many ways to esign satellite constellations, an this aer investigates an comares a oular metho of esign nown as Flower Constellations, with a relatively newer aroach nown as Parametric Constellations. This aer also introuces tools that were eveloe in MATLAB an Satellite Tool Kit to ai in the esign an islay of Parametric Constellations. Following this tool resentation, the aer also iscusses how the tools, were use to show that a Parametric Constellation can be more effective than a Flower Constellation for a secific Geoscience emote Sensing mission nown as FLOAD (FLOwer constellation eloying ADiometers). From this investigation an tool eveloment rocess, we conclue that the Parametric Constellations can be more effective because the esign arameters of this system allow for the use of real numbers. a B e i F F h F n h M N N r γ θ ω Nomenclature = semimajor axis = base satellite (Parametric) = eccentricity = inclination = inex number of satellite = hase enominator arameter (Flower) = hasing ste arameter (Flower) = hase numerator arameter (Flower) = height of eriasis = Mean Anomaly = ays to comlete one orbit (Flower) = number of etals (Flower) = relative to the base satellite = relative orbit frequency (Parametric) = ight Ascension of Ascening Noe = ascening noe sacing (Parametric) = argument of eriasis II. Introuction constellation is an imortant observation metho A in which a grou of satellites is launche, synchronize an coorinate to carry out a common mission. This common mission coul inclue erforming tass that enable technologies lie global telecommunications, satellite raio, global ositioning, an isaster monitoring. In aition to enabling the above technologies, constellations can also be use for scientific missions such as Earth observation. Constellation esign is riven by the mission an the goal of the constellation s erformance. Common constellation erformance arameters an esign stanars inclue the erio of the satellite, the amount of time between revisits of an area, the amount of coverage, the allowable ga of coverage over an area, an the access time for secific groun stations. Deening on the mission objectives, a constellation can be esigne to otimize any one of these erformance arameters. A. Flower Constellations There are many ifferent strategies to esign satellite constellations an the first of these strategies is a metho nown as Flower Constellations (FCs). Flower Constellations were recently eveloe an are escribe in eferences (1) an (2). Constellations that emloy this esign metho are calle FC because the orbit ath of the satellites aears as an outline of flower etals as the satellites orbit a boy. This set of constellations is esigne so that a system of satellites can orbit aroun a Planet Centere Planet Fixe Frame. This form of esign is base on the selection of eleven arameters that efine the orbits of the satellite an their hasing. Within the eleven arameters, five are Kelerian orbital element arameters an the other six are integer arameters. In an FC system, all of the satellites have common values of the semimajor axis a, eccentricity e, inclination i, an argument of eriasis ω. The constellation esigner selects values for these common arameters, but instea of choosing a value for the eccentricity, a height of erigee, h is selecte. The esigner also selects values for the mean anomaly, M 0, an the ight Ascension of Ascening Noe (AAN) 0 for the first satellite. These five selecte orbital element arameters efine the shae, an orientation of the satellites with resect to the Earth or lanet being orbite. The integer arameters become involve with the constellation to control the hasing of the satellites. The hasing rules of the FC efine the AAN an the mean anomaly for the th satellite as Perez 1

2 M F n + 1 = + 2π (1) F Fn N + F Fh 1 = M + 2 (2) F N + π The first three integer arameters seen in Equations (1) an (2), which are the hase numerator F n, the hase enominator F, an the hase ste F h, control the satellite hasing an istribution of the satellites along the sace trac in the constellation. The next two integer arameters, N the number of etals, an N the number of ays neee to comlete a full orbit, hel to efine the orbital erio of each satellite an the semimajor axis of the satellites. A major limitation of the FC system is that all of the hasing arameters are limite to integers. Currently, alications for FCs are being researche in GPS, Dee sace observation an Earth observation systems. One such Earth observation alication is resente in Section III of this aer. B. Parametric Constellations A secon strategy that is use to esign constellations is the Parametric Constellation (PC) esign metho, which is relatively newer an not as well nown as the FC metho. This system of constellations is escribe in eference (3). This metho is base on arametric relative equations which result from the relative equations of motion for Kelerian orbits. Similar to the FC system, this constellation esign roceure leas to a system of target satellites which has a reeating sace trac with resect to a base boy or a base satellite. This system of esign has eight initial arameters, five of which are again base on the Kelerian orbital element set an three aitional arameters that are use to hel efine the orbit size an hasing of the satellites. In a PC, all target satellites also have an ientical semimajor axis an eccentricity lie the FC system. The other three orbital element arameters in the PC system vary base on the hasing of the satellites an the choice of the aitional three arameters in the system. Each target satellite in a PC is efine by its inclination, argument of eriasis, an mean anomaly in terms of a real number system which controls the AAN sacing for the satellite. The satellite hasing rules that control the PC system can be seen in Equations (3), (4), (5), an (6) where the subscrit is the integer inex for the satellite, the subscrit r means relative to the base satellite an the subscrit B is for the base satellite. ( ) = 1 + θ 1 (3) cosi cos + 1 B i i = cos sin ib sin i cosδ 1 cosi i = cos + (4) sin ib sin Δ 1 tan (tanib cosδ ) ω = ω1 φ1 sin Δ sin i φ = tan 1 cos ib cos i ω = φ + ω φ1( M ) 0 = tan = γ 1 B sin i cosi sin Δ sin ib sin i cos i + cos ib cosi ( M ) B0 φ1( ) ω (5) (6) In the equations above, the only arameters in aition to the orbital elements of the first satellite, which are selecte by the esigner to control the system are N, the number of satellites, γ, the relative orbit frequency an θ, the sacing between each ascening noe. The last two arameters are base on the real number system. In terms of selecting arameters an esigning constellations, the PC system is more user frienly because it involves less extraneous arameters. The PC system also only requires eight arameters comare to the FC s eleven. The PC system is also a stronger constellation esign metho because it is not limite to creating relative sace tracs about a Planet Center Planet Fixe Frame. The PC system can easily be use to esign a constellation of satellites aroun another satellite which is harer an more ifficult to o in an FC system. Finally, the PC metho is also more owerful because its aitional arameters, asie from the orbital elements, are base on real numbers rather than integers. The PC system of esigning constellations has romising alications in intersatellite constellation esign an formation flying, but is not currently in use in any alication. II. Develoment of Parametric Constellation Tools Given that Parametric Constellations are a relatively new esign metho, not an extensive amount of Perez 2

3 information is nown about their otential alications. In orer to learn more about the otential hel in this esign theory, we evelo an use a series of tools to ai in the esign, islay an further unerstaning of this metho. A. MATLAB Grahical User Interface an Satellite Tool Kit Parametric Constellation Dislay We first evelo a series of tools in MATLAB an Satellite Tool Kit (STK) that allow the user to easily esign an islay a PC system. Satellite Tool Kit is owerful software that lets users visualize an calculate erformance arameters for satellites in 2D an 3D saces. The urose of this first set of tools is to allow a user to enter initial PC esign arameters an visualize the constellation that results from those arameters. The best way to achieve this goal is by writing a Grahical User Interface (GUI) rogram in MATLAB that accets PC arameters from the user an connects to STK, so that STK can islay the user s constellation. We create the MATLAB GUI that can be seen in Figure 1 to serve this urose an comlete this rocess. the ata an runs through the calculations that were illustrate 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 efine, MATLAB then interfaces with STK through a lug in. Using this interface, MATLAB creates an oulates a scenario with the comute satellite orbital elements in STK in orer to visualize the constellation. MATLAB controls the STK scenario through a series of Connect commans that are given to STK through the interface. These Connect commans inclue the ability to create a scenario, initialize a time erio, establish a reference frame, create a satellite, an initialize the orbital elements of the satellite. Once MATLAB finishes creating an oulating the scenario, STK islays the PC system on a 2D ma an on a 3D globe. An examle of one such constellation that was create from the above GUI on a 2D ma can be seen in Figure 2. In aition to the still image in this Figure, STK also shows an animation of the constellation for the time erio that is set by the user for the scenario using the Connect commans. The GUI in Figure 1 an the image in Figure 2, are Figure 1. eeating Groun Trac Grahical User Interface. This is the GUI that we evelo an use for a eeating Groun Trac PC system. The arameters that the user is romte for in the GUI are the eight governing arameters for the PC system. Once the user enters the arameters, an clics on the ush button Calculate, MATLAB can retrieve both reresentative of a eeating Groun Trac PC system. In aition to forming a constellation with a eeating Groun Trac aroun a Planet Centere Planet Fixe Frame, it is also ossible to create a PC system aroun a base satellite in a circular orbit, as was Perez 3

4 reviously mentione. In orer to create this tye of constellation, which is nown as a eeating elative Orbit constellation, we evelo an use a ifferent GUI tool similar to the one shown in Figure 1. To calculate this eeating elative Orbit constellation, the user must follow the same rocess as before by entering their PC esign arameters into a GUI. Once the user enters the arameters, MATLAB can again retrieve the arameters, calculate the iniviual satellite orbital elements accoring to the PC hasing rules, an finally use Connect commans to oulate a scenario in STK. Even though the rocess is the same, we evelo an use a ifferent GUI rogram because the user must enter the roerties of the base satellite, unlie before in the eeating Groun Trac constellation. Figure 2. Two Dimensional eeating Groun Trac PC. This is an examle of the 2D image that STK rouces once the user enters all the PC arameters into the GUI interface in Figure 1. B. Transformation Functions Flower an Parametric Constellations have the ability to mimic each other an can rouce the same exact constellation with their iniviual arameter efinitions. Using this fact, we evelo a series of functions that allows the user to convert FCs to PCs an vice a versa. Parametric Constellations however have a bigger scoe than the FC system, because their aitional hasing arameters are base on a real number system, while the FC system is base on integer arameters. Keeing this limitation in min, we evelo two transformation functions in MATLAB. These two functions o not lot the constellation or interface with STK, they are simly builing bloc functions that allow the user to enter the efining initial arameters of one system an return the initial arameters of the secon system for the same constellation. In the eveloment of these functions, we wor with both sets of hasing equations an arameters for the two systems simultaneously. To mae the functions wor roerly, each hasing arameter in the two systems must be efine in terms of the other, which is extremely comlicate. As a result of the comlexity, we only evelo these transformation functions for the eeating Groun Trac scenario aroun the Earth. In the eeating Groun Trac scenario, the equations are more manageable to eal with because the arguments of eriasis an the inclinations of the satellite are also ientical for all satellites in the constellation, in aition to the re-existing common values of the semimajor axis an eccentricity. We also create these functions for the case where the hasing ste, F h for the FC system is equal to zero. This simlification is mae, because the original functions an efinitions observe in eference (1) i not inclue the F h arameter. The easier of the two functions to create is the FC to PC transformation because we convert an conense eleven (ten without F h ) arameters into eight. In this conversion, some arameters such as the argument of eriasis, the inclination, the initial mean anomaly, an the initial AAN are the same in both systems so they o not vary in the transformation from one system to another. Also the number of satellites between the systems is the same where N s in the FC system becomes the N arameter in the PC system. Not all arameters are the same though, an it is necessary to calculate three quantities from FC arameters to comlete the PC arameter efinition. The first arameter we efine for the PC system is the eccentricity. We can calculate this arameter from the FC arameters of N, N, an h using equation set (7). In this equation set, T reresents the erio of the iniviual satellite, ω Earth is the angular see of the Earth, μ is the stanar gravitational arameter of the Earth, an Earth is the raius of the Earth. 2π N T = ωearth N T a = μ 2π Earth + h e = 1 a (7) Following the eccentricity, we calculate values for the two real number hasing arameters of the PC system, starting with the relative orbit frequency as can be seen in Equation (8). This is the simler arameter to efine an involves only the semimajor axis of the satellites. Perez 4

5 1 2 μ 1 γ = 3 (8) a ω Earth Next we efine the secon real number hasing arameter, the ascening noe sacing in terms of the hasing FC arameters using Equation (9). n = γω μ a = 2 n h = a Earth 1 3 ( ) 1 e Earth (12) Fn θ = 2π (9) F The final arameter that we efine an calculate, which has not yet been iscusse is M 10. This is an exclusive fit arameter that allows a FC system to be ulicate by a PC system. This fit arameter serves as a fuge factor for the transformation between the two constellations an is calculate using Equation (10). To use this arameter in PC hasing, it is simly ae onto the quantities of the left han sie of Equation (6) as seen rewritten in Equation (11). These final equations comlete the relationshis that we use to transform an FC system to a PC system in MATLAB for the first transformation function. M M =M + γ + ) (10) 0 ( 0 ω 0 = M 10 + γ ( M B 0 φ1( ) ) ω (11) Next we efine the relations that are necessary as we create the secon PC to FC transformation function. This function cannot be comlete if the relative orbit frequency of the PC system is a real number an not an integer. This is a result of the bigger scoe of the PCs, as was reviously iscusse. This function is more ifficult to comlete because there are so many FC arameters that it is not ossible to efine all of them ineenently. Many values for these arameters are chosen out of convenience instea of necessity by the user in the original FC efinition, which maes it ifficult when writing an automate comuter rogram. As with the revious transformation function in changing from FC to PC, the argument of eriasis, the inclination, the initial mean anomaly, an the initial AAN are the same, so they o 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 N s in the FC system. To comlete the FC efinition, we calculate an efine five aitional FC arameters in terms of PC arameters. The first arameter that we transfer an efine in terms of the PC arameters γ, an e, is the height of erigee using Equation (12). The next FC arameters that we calculate are the N an N integer arameters. In the FC system, these two arameters are efine together an not ineenently. They are efine in the first fraction as can be seen in Equation (13), where the enominator references an aroximate Siereal Perio. N T = N (24 *3600) N = a T = 2π μ (24 *3600) N = T (13) As a result of this simultaneous efinition, the values of N an N are set equal to the reuce fraction in the transformation function. Thus in the function, the arameter N is arbitrarily set equal to 1, an we assign N to the corresoning enominator to mae the last equation in the Equation (13) set true. These arameters coul be set to ifferent values if the user selects a value for N other than 1. The final arameters that we efine are the hasing arameters: hasing numerator an hasing enominator. No hasing ste arameter is secifie in this function because as was note reviously, this transfer function is efine for FC systems where F h is equal to zero. The remaining hasing arameters, F n an F, are again efine simultaneously an are eenent on each other in the FC system. Usually for convenience the F arameter is selecte first to reresent the number of orbit lanes an then F n is set to one by the user. As we create the transformation function, we use the same efinition an general iea, as can be seen in Equation (14). Perez 5

6 θ F F n F n = 2π F 2π = θ θ F = = 1 2π (14) This final equation efinition comletes the PC to FC transformation relations that we use in MATLAB to create the secon transformation function. Clearly the PC to FC transformation is not as rigorous as the revious FC to PC function, which is a result mainly of the number of arameters require an the intereenence of those arameters. egarless of this fact, the eveloment of these two functions is imortant because it shows that it is ossible to ulicate any FC system (for a F h = 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 an real number arameter ifferences between the two systems. This further emonstrates the avantage of a PC esign metho. C. Automate Access Calculator The last tool that we evelo is an automate access calculator GUI rogram. We evelo this tool using the first tool, the eeating Groun Trac GUI tool, as a basic framewor. This secific calculator allows a user to enter PC arameters, comute the constellation in MATLAB, visualize the constellation in STK, an comute the access time for the constellation for a chosen groun station an time erio. Access calculations can be one for each iniviual satellite with a secific groun station in STK searately, but this rocess is time consuming an reetitive to comlete. This tool automates all access calculations for a constellation so that a user can simly run the GUI, which will return the access time for the constellation without further maniulation in either MATLAB or STK. In aition to erforming access calculations for a secific constellation, this GUI also has the caability to hel the user fin an otimum constellation base on access time. In this calculator the user enters the arameters for the initial satellite or the orbital elements, an then the user can enter a range of values for the two PC hasing arameters, the relative orbit frequency an the ascening noe sacing between the satellites. The GUI is then able to comute for the same set of orbital elements, all ossible combinations of the hasing arameters for the range of values entere in the GUI. The GUI then connects to STK, as with the revious tools, an comutes the access time for each ossible hasing combination with the common orbital elements. Once STK has comute all of the access calculations for each constellation combination, MATLAB creates a surface lot of the access time for each constellation as a function of the two hasing arameters. From this surface lot, the user can ic the otimum relative orbit frequency an ascening noe sacing for a secific set of orbital element arameters base on the access time erformance of the satellites. The eveloment of this tool saves the user a tremenous amount of time. For examle, by entering in a range of ten values for each hasing arameter, the GUI will lin to Matlab an comlete 100 automate access calculations with only one user action. This volume of calculations if not automate coul tae hours. This tool serves an easy an quic way to obtain an initial estimate of how the access time varies with resect to the hasing arameters for a constellation with a constant set of orbital elements. III. Parametric Constellation Alication to FLOAD Mission Finally, we attemt to show that PCs are a more avantageous constellation esign strategy than FCs. In orer to o this, we reesign an FC alication with a PC system to show that the PC rovies better erformance than the FC for the same mission requirements. This rocess is still in its early stages, but results comlete thus far will now follow. A. Overview of FLOAD Mission The FC alication that we select to comare an exlore is an Italian Sace Agency mission that is in Phase A, nown as FLOAD or FLOwer constellation eloying ADiometers. This constellation is escribe Figure 3. FLOAD Target Area Winow. The MWM an larger SE winow for the FLOAD Mission. 2 in eference (2). The goal of this mission is to measure thermal an hyrological roerties of the trooshere over the Meiterranean region by eloying millimeter- Perez 6

7 wave (MMW) scanning raiometers. This mission has a goal of frequent revisit time at regional scale over the Mesoscale Western Meiterranean (MWM) winow an quasi-global coverage over the egional Scale Euroean (SE) winow. The combine target area for this mission is shown in Figure 3. In orer to aequately comlete the science goals of this mission a number of constellation requirements have been eveloe. The riving factor for the constellation esign is to maximize the satial coverage at a regional scale an to maximize the time coverage over the target area for atmosheric monitoring uroses. It is also referable to narrow the revisit time to the area to be less than two hours over Southern Euroe. Some of the constraints for the mission are that the constellation must last for two years, be eloye in one launch, an have less than or equal to four satellites. One aitional constraint from the sensors is that the orbit height of the satellites must be between 450 an 1250 m, so that the sensors have an average linear Fiel of View less than 25 m. B. Comarison of Selecte Flower Constellation with Parametric Constellation Design At this oint in the lanning stage of the mission no constellation has been selecte yet, but accoring to eference (2) there are three major otions of Flower Constellations that are going through an otimization rocess. Here, we tae the first FC otion that has been roose, reesign the FC otion with a PC, an then comare the two solutions. The first FC otion is mae u of four satellites with slightly ellitical orbits, a critical inclination of 63.4 egrees an a erigee to aogee ratio of about 450 m to 850 m. This critical inclination is use to avoi the nee for erigee control an to satisfy the nee to focus mission coverage on the MWM region. The arameters of the FC for this first FLOAD solution can be seen in Table 1. To emonstrate that a PC system is better than this current FC otion, the PC must have better erformance an it must not be able to be ulicate by an FC constellation. The easiest way to esign a PC that cannot be ulicate by an FC is to use hasing arameters that are real numbers. In orer to etermine whether or not the PC has better erformance, 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 examle is calculate using an ESA tracing station in STK nown as Weilheim II, which is in the mile of the MWM target region. We select this groun station because of Table 1. FC Parameters for FLOAD Otion 1. N N N s F n F F h h (m) i (eg) ω (eg) AAN (eg) M 0 (eg) γ Table 2. PC Parameters for FLOAD Otion 1. θ (eg) N i (eg) ω (eg) e AAN (eg) M 0 (eg) Figure 4. FC FLOAD Otion 1 Coverage Area. The white shae area reresents the coverage over the tracing area. Perez 7

8 its location an because it is a liely choice for a groun station for this constellation s mission. The coverage results over the area with the access times shae on the orbit ath from STK can be seen in Figure 4 for the FC solution. The total access time for this FC constellation with Weilheim II is hours. In esigning the imrove PC system, we use the automate access calculator tool. We are able to create an fin a better PC system by running iterations with ifferent relative orbit frequencies with real numbers in this tool. The better PC solution results in an increase erformance in access time over the original FC system. The arameters for the resulting PC can be seen in Table 2. From this table it is obvious that the successful PC utilizes the same Kelerian orbital element arameters as the FC. This is one because the ositioning an orientation of the orbit nees to be the same in orer to achieve aequate coverage over the MWM target area. The ifference between the two tyes of constellations comes with the choice of the hasing arameters an how the satellites move through the orbit. Since PCs allow the use of real number arameters to control the hasing, the PC is able to achieve aitional coverage by using a real number to reresent the hasing arameter of γ, the relative orbit frequency. Using these tools an methos leas to the emonstrate PC, which has an access time of hours, an results in a 3.12% increase in access over the target area. The associate 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 mae to out erform an FC system. Looing at Figures 4 an 5, it can be seen that the FC an PC both comletely cover the MWM region an SE scale winow aequately. The only ifference in the tye of coverage is that the PC satellites stay on the same relative ath an cover the same area, whereas the FC satellites ten to canvas slightly ifferent arts of the target region as the satellites move through their orbits. This shoul not mae much of a ifference an oes not ut the PC system at a isavantage because the sensors on boar the satellites are scanners an will scan +/- 50 egrees aroun the boresight. This scanning shoul mae u for ifference in the area coverage. Thus, for this FLOAD FC otion, the PC system eveloe is a usable solution that out erforms this FC in terms of total access time with this groun station. C. FLOAD Comarison Conclusion There are still more erformance comarisons to mae to ensure that the PC system eveloe for this first FLOAD FC otion truly is a more avantageous system in all mission erformance criteria. It is clear from the wor that has been one though that the eveloe PC oes sen 3% more time in the region an oes aequately cover the MWM an SE winow. This PC also as a result of its real number arameters is not ulicable in the form of a FC. The avantage that is rovie in this alication by using a PC over an FC is small, but may turn out to be larger for the other two lanne FC otions for this mission when we comlete further comarisons. Figure 5. PC Otion Coverage Area for FLOAD. The white shae area reresents the coverage over the tracing area. Perez 8

9 III. Conclusion Overall, constellations are an integral art of sace an Earth observation. We have shown that there are many ifferent ways to esign constellations, two of which are Flower Constellations an Parametric Constellations. Flower Constellations are more wiely nown an are starting to be use in Earth observing alications. Parametric Constellations have just recently been eveloe, an are imortant to consier because their basis on the real number system gives the constellation set a wier range of ossibilities. We also have emonstrate in this aer that a series of tools have been eveloe for Parametric Constellations. The first set of tools that has been eveloe can be use to ai in the esign, an islay of Parametric Constellations in MATLAB an STK. The secon set of tools that has been eveloe are functions that allow users to convert bac an forth from Flower Constellation to Parametric Constellation arameters. The final tool that has been eveloe is a GUI that automates access calculations for a constellation an creates a surface lot of the total access time for a range hasing arameters. Finally with the tools eveloe, Parametric Constellations were alie to a lanne Flower Constellation mission nown as FLOAD. It was shown in this alication, that the Parametric Constellation metho has been able to meet an excee the erformance of one of the otential Flower Constellation otions by 3% in terms of access time for a secific groun station. This emonstrates that the Parametric Constellations can be suerior to Flower Constellation systems eening on the alication an esire target area. Overall Parametric Constellations are a new an fascinating constellation esign strategy that aears to have great otential an avantages in comarison to other current constellation esign methos. Xlore - Geoscience an emote Sensing, Vol. 47, No. 9, 2009, Lee, S., Dynamics an Control of Satellite elative Motion: Designs an Alications, Ph.D. Dissertation, Aerosace Engineering, Virginia Polytechnic Institute an State University, Blacsburg, VA, Acnowlegments I woul lie to acnowlege my avisor, Dr. Christoher Hall for his hel an continue suort throughout my research. eferences 1 Wilins, M., The Flower Constellations Theory, Design Process, an Alications, Ph.D. Dissertation, Aerosace Engineering, Texas A&M University, College Station, TX, Marzano, F., Cimini, D., Memmo, A., Montooli, M., ossi, T., et al. Flower Constellation of Millimeter-Wave aiometers for Troosheric Monitoring at Pseuogeostationary Scale, IEEE Perez 9