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Geomagnetic Field Tracker for Deorbiting a CubeSat Using Electric Thrusters

2019; American Institute of Aeronautics and Astronautics; Volume: 42; Issue: 5 Linguagem: Inglês

10.2514/1.g003908

1533-3884

Jose E. Morales, Jongrae Kim, Robert R. Richardson,

Spacecraft Design and Technology

Open AccessEngineering NotesGeomagnetic Field Tracker for Deorbiting a CubeSat Using Electric ThrustersJose E. Morales, Jongrae Kim and Robert R. RichardsonJose E. MoralesUniversity of Leeds, Leeds, England LS2 9JT, United Kingdom*Ph.D. Researcher, Institute of Design, Robotics and Optimisation, School of Mechanical Engineering.Search for more papers by this author, Jongrae KimUniversity of Leeds, Leeds, England LS2 9JT, United Kingdom†Associate Professor, Institute of Design, Robotics and Optimisation, School of Mechanical Engineering.Search for more papers by this author and Robert R. RichardsonUniversity of Leeds, Leeds, England LS2 9JT, United Kingdom‡Professor, Institute of Design, Robotics and Optimisation, School of Mechanical Engineering.Search for more papers by this authorPublished Online:4 Mar 2019https://doi.org/10.2514/1.G003908SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionCubesats have gained popularity due to their small size, low cost, and relative simplicity. CubeSats are deployed as 10 cm cubic blocks of up to 1.33 kg each, which can be combined in order to assemble systems of multiple units, abbreviated as U (1U, 2U, 3U, and beyond). They provide easier access to space than traditional spacecraft because they are considerable cheaper and their development time is also much shorter. Current and proposed applications include Earth observation [1], telecommunications [2], astronomy [3], and technology demonstrations in general. One of the main concerns, however, is their potential negative impact on the space debris problem.It is estimated that a total of 24,000 objects that are 10 cm wide or larger were in orbit around the Earth as of 2011 [4]. Major collisions in space have already occurred; for instance, in 2009, American communications satellite Iridium 33 and the retired Russian Kosmos-2251 crashed, resulting in the destruction of both spacecraft and the generation of a large amount of orbital debris [5]. The Kessler syndrome [6] predicts that a critical density of objects in low Earth orbit could be reached and that collisions between them could cause a cascade of impacts that would eventually render some orbital ranges unusable within this century. Taking this into account, as well as the expected increase in CubeSat launch rates, it is critical that technologies are developed in order to deorbit these devices at the end of their operational life, thus preventing their accumulation in space. In a response to this, international guidelines require spacecraft to be deorbited or moved to a disposal orbit within 25 years of the end of their mission.Several CubeSat deorbiting systems have been proposed over the past years, with some of the most popular being the following: solar sails [7], inflatables [8], and electric tethers [9]. Sails rely on aerodynamic drag augmentation. NASA has been working in deorbiting sails technology with their NanoSail-D and NanoSail-D2 [10] experiments. NanoSail-D was lost shortly after launch because of an anomaly in the launch vehicle, whereas its ground spare, NanoSail-D2, was successfully launched to space in November 2010. After being deployed into a circular orbit of 650 km, the spacecraft deployed its sail, finally reentering Earth’s atmosphere on 29 November 2011 after spending 240 days in space and becoming the first successful demonstration of a deorbiting sail. The University of Surrey’s DeorbitSail satellite was successfully launched into orbit in 2015; however, the deployment of the sail was unsuccessful [11]. Finally, the University of Glasgow together with the company AAC Clyde Space have been working toward the development of the Aerodynamic End of Life Deorbit System for CubeSats (AEOLDOS) [7]. The system is currently commercially available; however, no references to deployment in space have been found to date.Inflatable structures also rely on aerodynamic drag augmentation. Maessen et al. explored different shapes and determined a pyramidal structure to be optimal in their approach [8]. Similarly, Nakasuka et al. investigated the effectiveness of a spherical structure, proposing a system that would be effective for satellites at an initial orbit of up to 800 km [12]. Lokcu and Ash explored the design of inflatable structures with spherical, pyramidal, and pillow shapes [13]. They claimed that, with this system, orbital decay times of 30 years could be attained for CubeSats in an initial 900 km orbit. Andrews et al. explored the concept of a cone-shaped inflatable structure, which would not only allow the deorbiting of a CubeSat but also its survival during atmospheric reentry phase, and it would make possible the retrieval of payloads from CubeSats [14]. And, Viquerat et al. proposed a combination between inflatable structures and sails for its InflateSail CubeSat concept [15].Finally, tethers interact with either the geomagnetic field or the plasma environment to generate drag forces. Voronka et al. proposed the deployment of a 1-km-long tether in order to study the feasibility of their application to the deorbit of CubeSats [16]. Their simulations suggested that CubeSats at an initial orbit of 1000 km could be deorbited within 25 years. Zhu and Zhong [17] and Zhong and Zhu [18–21] have devoted much work to the application of space tethers to the deorbit of small spacecraft problem. With their analysis of the dynamics of an electrodynamic tether, they first explored the practicality of the technology [17–19]. Later, Zhong and Zhu proposed an optimal on–off control approach in which the simulations showed that the satellite system could lose 100 km of altitude within 60 days [20]. They developed a control system using a finite receding horizon control in order to achieve the optimal trajectory, which potentially would bring a tethered system from 800 to 700 km within 25 days [21]. Janhunen [22] and Khurshid et al. [23] focused on the tether interaction with space plasma rather than the Earth’s magnetic field. Their approach was applied in the Aalto-1 CubeSat as a technology demonstration mission [23].A major disadvantage of these three approaches is that all of them require the deployment of actuators in space, which is technically challenging and makes the system prone to failures. Further complications can come from the folding of sails and inflatables, which in turn require a considerable amount of volume. Recently, a new approach has emerged that uses electric engines. Descriptions and working principles of five different types of electric engines can be found in Refs. [24–28]. These electric engines have low mass and volume, high specific impulse, do not require the deployment of actuators, and contain no movable parts: except for microcathode arc thrusters and micropulsed plasma thrusters, which contain springs, with these being major advantages over the approaches already discussed. The idea of electric thrusters has been explored since the 1960s. The first in-space demonstration of electric propulsion was executed by the Space Electric Rocket Test (known as SERT-1) mission in 1964 [29]. Initially too bulky for its application in nanosatellites, this technology has evolved; and now, electric engines are available in sizes and within budgets that make their integration in this class of satellites possible.The application of electric thrusters to the CubeSat deorbit problem requires us, however, to keep the thruster in the right direction to dissipate the kinetic energy of the CubeSat. The geomagnetic field tracker algorithm to be presented in this Note provides an effective and simple solution to control the orientation of a CubeSat for deorbiting purposes. The algorithm only requires magnetometers, a Global Positioning System (GPS) receiver, and magnetorquers, which are standard components in most CubeSats. Because of the minimal requirements, together with its simplicity to implement the algorithm, it is an ideal algorithm for use within the stringent constraints present in nanosatellites.This Note is organized as follows: Section II gives a summary of magnetorquers and electric propulsion. Section III describes the proposed deorbiting algorithm, as well as the respective stability and robustness analysis. Deorbiting phase simulations are presented in Sec. IV, and the conclusions are given in Sec. V.II. Environment and HardwareCubeSats are subject to volume, mass, and budget constraints; and it is not always possible to equip them with the desired sensors and actuators. For example, traditional attitude determination algorithms, such as TRIAD and QUEST (quaternion estimation), require a minimum of two independent vector measurements [30]. These vectors are commonly obtained by a combination of magnetometers, Earth sensors, sun sensors, or star sensors. This requirement for multiple attitude sensors might be prohibitive for CubeSats. CubeSats usually also lack the means for propulsion, i.e., orbit housekeeping and orbital maneuvers in general are not possible. These restrictions have made the development of an effective deorbiting system for this class of satellites a rather challenging task.A. Coordinates Systems and Geomagnetic FieldThree different coordinate systems are used in this work, which are the Earth-centered inertial (ECI) frame, the local-vertical/local-horizon (LVLH) frame, and the body B frame. The ECI frame has its origin at the center of the Earth, the positive x axis points toward the point of the vernal equinox, the positive z axis is aligned with the Earth rotation axis, and the positive y axis completes the right-handed frame. The LVLH frame has its origin at the center of mass of the CubeSat, the positive z axis points along the negative position vector, the positive y axis points along the negative orbit normal, and the positive x axis is defined by the cross product of y and z vectors. And, the B frame has its origin at the center of mass of the satellite, and its axes in this Note are assumed to be aligned with the principal axes of the satellite.Earth has a stable and steady magnetic field for which the variations are negligible as compared to the usual time length of a satellite mission. The International Geomagnetic Reference Field (IGRF-12) model provided by the International Association of Geomagnetism and Aeronomy is used in the numerical simulations to be presented later [31].B. Magnetorquers and MagnetometersMagnetorquers are the most common attitude actuators for CubeSats. Magnetorquers are light and small, do not have movable parts, and do not need fuel. These features make them ideal for integration in nanosatellites. Magnetorquers generate a magnetic dipole, which interacts with Earth’s magnetic field. The torque generated by this interaction is given by τm=m×b(1)where τm is the torque in newton meters, m is the 3×1 magnetic dipole vector produced by the magnetic torquers in amperes per square meter, and b represents the 3×1 Earth’s magnetic field vector in tesla. Magnetorquers work in conjunction with magnetometers, which are the common attitude sensors in CubeSats because of their small size and light weight, and which are in charge of measuring the geomagnetic field b in the B frame.C. Electric PropulsionIn terms of propulsion, several types of electrical engines are being developed for their integration in nanosatellites. A list of these engines includes electrospray [24], pulsed plasma thrusters [25], the Hall effect thruster [26], the CubeSat ambipolar thruster [27], and the microcathode arc thruster [28]. Table 1 summarizes some of the main features for each particular type of electric engine, with the aim to provide an insight of the potential that this technology has for applications in the field of CubeSats [27,32].Note that the mass of these engines is still somewhat prohibitive for their use in 1U CubeSats, except in some unusual missions because little or no room would be left for the rest of the systems and payload. However, their integration in the popular 3U or larger configurations seems feasible.These electric propulsion technologies open the possibility for nanosatellite orbital control, including the crucial phase of deorbiting. In the following section, an algorithm that enables the use of electrical engines for CubeSats with this objective is presented.D. Attitude Kinematics and Attitude DynamicsSatellite attitude kinematics are given as q˙=12[−[ω×]ω−ωT0]q(2)where q is the 4×1 quaternion representing the relative attitude of the B frame with respect to the ECI frame; q˙ is the time derivative of the quaternion; ω is the 3×1 angular velocity vector of the body frame with respect to the inertial frame expressed in the body frame; and [ω×] is the 3×3 matrix defined such that [a×]b=a×b, where a and b are 3×1 vectors. The attitude dynamics of the satellite are expressed through the Euler’s rotation equation are as follows: ω˙=J−1(−ω×J+τtot)=g(ω,τtot)(3)where J is the 3×3 inertia tensor of the satellite; and τtot is the 3×1 vector representing the sum of all external torques, which includes control torques as well as perturbation torques caused by the environment, such as solar radiation pressure, gravity gradient, and atmospheric drag.III. Geomagnetic Field Tracker and Stability AnalysisIn this section, the geomagnetic field tracker is presented for deorbiting a CubeSat. The closed-loop system with the geomagnetic field tracker is derived, which is given by a linear periodically time-varying system. The stability analysis is performed for the time-varying system using the Floquet theorem.A. Control LawTo deorbit satellites using a thruster, it first needs to be oriented in a desired direction that ensures that the thrust vector opposes the direction of the orbital velocity. The control algorithm aligns the satellite attitude to the geomagnetic field vector using magnetorquers so that the thruster can dissipate the kinetic energy of the nanosatellite during half of the orbit. Only one attitude sensor (the magnetometer) is required, which is relatively small, compact, and cheap; and it is integrated in many CubeSats missions. This will provide the magnetic field vector reading in the B frame. A GPS receiver is needed to determine when the thruster must be fired to deorbit the satellite. The orbital position and velocity in the ECI frame are provided by the GPS receiver.The objective of the attitude control algorithm is to align the +x body axis with the local magnetic vector. To accomplish this, the desired torque is given by τdes=k1e(4)where k1 is a control gain to be designed: e=i×b^(5)i is the unit vector toward the +x direction of the B frame; and b^ is the normalized magnetic field in the B frame, which is equal to b/‖b‖, where ‖b‖ is the magnitude of b. The commanded magnetic dipole for the magnetorquers to generate the desired torque can be obtained using Eq. (1). Including additional damping effect, the geomagnetic field tracker dipole becomes mctrl=k1[b^×]e−k2dBdtb^(6)where k2 is another control gain to be designed; and dB/dt is the time derivative in the B frame, which could be implemented using the finite difference approximation the same way that the B-dot controller is implemented [33]. Note that a similar control algorithm was proposed by Jan and Tsai for initial attitude acquisition [34]. Equation (6) can be directly implemented with a numerical differentiator to approximate the time derivative of geomagnetic field vector measurement. If a rate gyroscope sensor is available, then dB(b^)/dt can be replaced by [b^×]ω. Hence, mctrl=k1[b^×]e−k2[b^×]ω(7)Figure 1 shows the CubeSat alignment with the geomagnetic field vector, as well as the relative attitude to the spacecraft velocity vector, which determines when the thrusters (mounted in the +x face of the satellite) can be fired and when they should be turned off. An interesting instance is observed in Fig. 1 when the satellite is above the North Pole. Intuitively, it might seem that having the thruster activated over the north pole would push the satellite upward, i.e., gaining the altitude. However, the objective of the deorbiting algorithm is to dissipate the kinetic energy of the satellite, and it leads to lowering the altitude. This is achieved whenever the thrust vector has a component that opposes the velocity vector.Fig. 1 CubeSat +x axis aligned with magnetic field and the relative attitude with respect to the velocity vector.Once the CubeSat’s +x face is tracking the magnetic field vector, it can be determined when to activate the thrusters in order to cause the loss of orbital energy. This knowledge comes from the fact that the magnetic field vector is available in the LVLH frame of blvlh=[bxlvlhbylvlhbzlvlh]T, where (⋅)T is the transpose. Because of the relationship between the orbital velocity vector v and the +x axis in the LVLH frame, the thrust vector opposes the direction of travel at some degree when the component bxlvlh is positive, which implies the thruster points in the direction of travel as shown in Fig. 2. In Fig. 2, it is important to clarify that the +x axis of the LVLH frame does not necessarily coincide with the velocity vector; however, in a quasi-circular orbit, this is a reasonable simplification.Fig. 2 CubeSat attitude cases to deorbit, showing attitudes at which the thruster is a) on or b) off.Nonlinear simulations were performed to evaluate the tracking error of the algorithm, which is the angle difference between the magnetic field vector and the +x body axis of the satellite. In all the following simulations within this Note, we consider a 3U CubeSat equipped with three orthogonal magnetorquers; a three-axis magnetometer; a three-axe rate gyroscope; a GPS receiver, which is one of the standard CubeSat configurations; and an electric engine for propulsion, for which the thruster vector direction is assumed to be aligned with the +x face of the satellite. The CubeSat has a mass of 3.5 kg, and the diagonal terms of the inertia tensor are [0.01;0.0506;0.0506] kg⋅m2. It has an initial orbital altitude of 900 km, in a near-circular orbit. The inclination is set to 65 deg. The control gains are tuned such that k1=0.5 and k2=25 through multiple trial-and-error procedures. Initially, the gains are varied to identify a coarse range of the feasible gains, and a fine tuning is performed to determine the final gains in aligning the body axis to the magnetic field vector. It is also important to mention that, in practice, magnetometers and magnetorquers cannot be activated at the same time because the magnetometers measurements would be affected by the magnetorquers. Therefore, a duty cycle is implemented in which, during a period of 5 s, magnetorquers are active for the first 4 s and magnetometers are active for the remaining 1 s. All simulations were performed using MATLAB/Simulink with the default numerical integration algorithm and the relative tolerance of 0.001.Figure 3 shows the evolution of the tracking angle error through the whole deorbiting process with the electrospray thruster. It can be seen that, after an initial error of about 100 deg, the CubeSat is able to track the magnetic vector within an accuracy of roughly 5 deg. High pointing accuracy is not required for the deorbit phase of the mission; therefore, this performance is acceptable.Fig. 3 Magnetic vector tracking error.B. Stability AnalysisTo provide a stability analysis of the geomagnetic field tracking mode of a CubeSat, a linearization is performed. First, take the time derivative of Eq. (5) as follows: e˙=(ω×i)×b^=−[b^×][ω×]i=f(b^,ω)(8)Note that the desired equilibrium states are e=[000]T and ω=[ωx00]T, where ωx is a rotation rate allowed in the +x axis of the satellite, which is not required to be zero. The equilibrium point also implies that the nominal normalized magnetic field vector in the B frame is given by b^=[100]. The linearized system is expressed as follows: x˙=[∂f(b^,ω)∂e∂f(b^,ω)∂ω∂g(ω,τtot)∂e∂g(ω,τtot)∂ω]x+[∂f(b^,ω)∂mctrl∂g(ω,τtot)∂mctrl]mctrl(9)Once the Jacobian equations are solved, a 6×6 state matrix is obtained, and it is noticed that the first and the fourth rows and columns are all zero. This is due to the fact that the control torque [Eq. (4)] is always perpendicular to the +x axis of the satellite at the equilibrium point. Therefore, it has no components in the +x axis of the B frame, and the rotational motion about this axis is allowed in the geomagnetic field tracking mode. Hence, a reduced dimension linearized system is given by x˙r=Arxr+Br(t)Krxr(10)where Krxr is a compact form of Eq. (7): Ar=[0−ωx−10ωx00−1000σ2ωx00σ3ωx0](11)Br(t)=[000−bxJzz00bxJyy0]T(12)Kr=[0−k10k2k10−k20](13)xr=[eyezωyωz]T(14)ωx is assumed to be a constant, σ2=(−Jxx+Jzz)/Jyy, and σ3=(Jxx−Jyy)/Jzz.The resultant closed-loop system can then be expressed as follows: x˙r=ACL(t)xr(15)where ACL(t)≔Ar+Br(t)Kr(16)As the geomagnetic field is periodic along the orbit, the time-varying matrix ACL(t) is also periodic as follows: ACL(t)=ACL(t+T)(17)where T is the period of the system equal to the orbit period. Because the system is now expressed in linear periodic time-varying form, the Floquet theory can be used for the stability analysis. A brief summary of the Floquet theory is provided next, and more details can be found in Ref. [35].Define a constant matrix F such thateFT=ϕ(T,0)(18)where the following equations must be satisfied: ϕ(t,τ)=L(t)e(t−τ)FL−1(τ)(19)L(t+T)=L(t)(20)ϕ(⋅,⋅) is the state transition matrix, and L(t) is the solution of the following two point boundary value problem [36]: L˙(t)=ACL(t)L(t)−L(t)F(21a)F˙(t)=0(21b)The boundary conditions are given by L(t0)=I4 and L(t0+2T)=I4, and I4 is the 4×4 identity matrix. The linear periodic time-varying system given by Eq. (15) is exponentially stable if, and only if, F is Hurwitz, i.e., all real parts of the eigenvalues are negative.A Floquet analysis was performed for a range of ωx values that went from −10 to 10 deg/s. Figure 4 shows the maximum real part of the eigenvalues of F, which are always negative. Hence, the stability of the geomagnetic field tracking system for the given range of ωx is verified.Fig. 4 Floquet stability analysis results.IV. Deorbiting Performance and Robustness AnalysisA deorbit scenario was simulated in order to test the performance of the proposed algorithm. Aerodynamic drag accelerations are taken into account for this numerical analysis using the U.S. Standard Atmosphere, 1976 model [37]. Gaussian perturbation equations were used in order to compute the evolution of the orbital parameters [38]. The deorbit process is simulated for the five different types of electric engines shown in Table 1. The time span for each simulation scenario is 20 days, unless the deorbiting operation is achieved in less time.A. DeorbitingFigure 5 shows the evolution of the orbital altitudes for each of the five engine scenarios. The first effect that is apparent from these charts is that the eccentricity of the orbit is gradually increased. This comes from the fact that the thrust is applied during only one-half of the orbit. It also can be seen from Fig. 5a that, once the perigee of the orbit reaches denser layers of the atmosphere, the orbit quickly falls into a critical altitude of around 100 km, which is when the CubeSat can be considered to be deorbited.Fig. 5 Deorbiting rates of the a) electrospray thruster; b) micropulsed plasma thruster; c) Hall effect thruster; d) CubeSat ambipolar thruster; and e) microcathode arc thruster.The deorbit times are of course inversely proportional to the thrust that the engines can provide, with the better performance obtained with electrospray engines (with a deorbit time of roughly 16 days), followed by the CubeSat ambipolar thruster that decreases the perigee of the satellite by about 250 km in 20 days. An important aspect to take into account here is the operational lifetime of each engine, which has the potential to determine if the engine can be active during the whole deorbit operation or if it can be used to lower the perigee to an altitude at which aerodynamic drag can take over and deorbit the satellite within the time span required by international guidelines. According to the references in Table 1, the electrospray engine has a lifetime of about 650 h, which would be enough to carry the entire deorbit operation. In the case of the micropulsed plasma thruster, only state-of-the-art technology would allow a decrease in the perigee of a 3U CubeSat by about 40 km; therefore, further advancements in this specific engine technology are necessary for their practical use in deorbit operations. Finally, for Hall effect thrusters, it is claimed that they are expected to be able to deorbit a 3U CubeSat from an initial orbit of 750 km. Although no mention of an algorithm is made, this gives confidence that the application of this type of engine to the deorbit problem is practical. No information regarding the lifetime of the CubeSat ambipolar thruster and microcathode arc thruster was found.Another interesting aspect to look at is the evolution of the semimajor axis. Figure 6 depicts this metric for the electrospray scenario, where a step pattern can be observed. This is also an effect of the thrust being applied during only half of the orbit.Fig. 6 Semimajor axis evolution using electrospray thruster.The system state and control inputs for the electrospray scenario case are shown Figs 7–10. Figure 7 shows the quaternion evolution. As shown in Fig. 8, the angular rates are kept at low values, including ωx, which is an axis where little or no control torques can be applied. Figure 9 depicts the control magnetic dipoles, it can be seen that, after an initial saturation in mx and mz, the control action is very small during the rest of the process: on the order of 0.02 A⋅m2. As expected, mx control action is virtually zero once the satellite is tracking the magnetic vector. Finally, Fig. 10 shows the action of the thruster, in which the thrusters are only active during half the orbital period.Fig. 7 CubeSat attitude in quaternions.Fig. 8 CubeSat angular velocities.Fig. 9 Control magnetic dipoles.Fig. 10 Thruster activation.B. Robustness AnalysisA full nonlinear robustness analysis of the deorbiting algorithm is performed through Monte Carlo simulations. An uncertainty vector Δ is defined by considering uncertainties in the mass and the inertia tensor, as well as the initial conditions in terms of the orbital parameters and the initial attitude. A detailed explanation of the Δ vector components is given in Table 2. The work of Cortiella et al. was taken as a reference for the values of the nondiagonal elements of the inertia tensor [39]. The electrospray scenario was considered, and a total of 1000 runs was executed, with the results shown in Fig. 11.Fig. 11 Robustness analysis with Monte Carlo simulations.It can be seen that, even in the presence of uncertainties, the deorbiting algorithm performs well, and a maximum deorbit time of less than 20 days in the electrospray engine case can be achieved.V. ConclusionsAn efficient and simple attitude tracking algorithm for deorbiting a CubeSat is presented. The stability of the proposed tracking algorithm is proved through a Floquet analysis. With the tracking algorithm, the deorbiting performances for five different types of electric engines are demonstrated. Robustness of the deorbiting algorithm in the presence of uncertainties is shown through the Monte Carlo simulations. Future works include showing the efficiency of the algorithm for various CubeSat configurations and developing optimal control gain procedures to minimize the usage of control energy. The simplicity of the proposed algorithm and the minimal requirements for sensors and actuators are the most important advantages. It allows this algorithm to be suitable for implementation in many CubeSats with the existing specifications.AcknowledgmentsThe authors would like to thank Consejo Nacional de Ciencia y Tecnología for their financial support during this work. The Monte Carlo simulations were undertaken on the Advanced Research Computing 2 (ARC2), which is part of the high-performance computing facilities at the University of Leeds. References [1] Nascetti A., Pittella E., Teofilatto P. and Pisa S., “High-Gain S-Band Patch Antenna System for Earth-Observation CubeSat Satellites,” IEEE Antennas and Wireless Propagation Letters, Vol. 14, Feb. 2015, pp. 434–437. doi:https://doi.org/10.1109/LAWP.2014.2366791 IAWPA7 1536-1225 CrossrefGoogle Scholar[2] Hodges R. E., Hoppe D. J., Radway M. J. and Chahat N. E., “Novel Deployable Reflectarray Antennas for CubeSat Communications,” 2015 IEEE MTT-S International Microwave Symposium IMS 2015, IEEE Publ., Piscataway, NJ, 2015, pp. 4–7. doi:https://doi.org/10.1109/MWSYM.2015.7167153 Google Scholar[3] Park J.-P. and et al., “CubeSat Development for CANYVAL-X Mission,” SpaceOps Conferences, AIAA Paper 2016-2493, 2016. doi:https://doi.org/10.2514/6.2016-2493 Google Scholar[4] Chen S., “The Space Debris Problem,” Asian Perspectives, Vol. 35, No. 4, 2011, pp. 537–558. CrossrefGoogle Scholar[5] Oltrogge D. L. and Leveque K., “An Evaluation of CubeSat Orbital Decay,” 25th Annual AIAA/USU Conference on Small Satellites, Paper SC11-VII-2, Aug. 2011. Google Scholar[6] Kessler D. J. and Cour-Palais B. G., “Collision Frequency of Artificial Satellites: The Creation of a Debris Belt,” Journal of Geophysical Research: Space Physics, Vol. 83, No. A6, June 1978,pp. 2637–2646. doi:https://doi.org/10.1029/JA083iA06p02637 JGREA2 0148-0227 CrossrefGoogle Scholar[7] Harkness P., McRobb M., Lützkendorf P., Milligan R., Feeney A. and Clark C., “Development Status of AEOLDOS—A Deorbit Module for Small Satellites,” Advances in Space Research, Vol. 54, No. 1, July 2014, pp. 82–91. doi:https://doi.org/10.1016/j.asr.2014