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• W2997301076 abstract "Open AccessEngineering NotesSpacecraft Collision Avoidance Using Aerodynamic DragSanny R. Omar and Riccardo BevilacquaSanny R. OmarUniversity of Florida, Gainesville, Florida 32611*Ph.D. Candidate, Department of Mechanical and Aerospace Engineering, MAE-A Room 211. Student Member AIAA.Search for more papers by this author and Riccardo BevilacquaUniversity of Florida, Gainesville, Florida 32611†Associate Professor, Department of Mechanical and Aerospace Engineering, MAE-A Room 211. Member AIAA.Search for more papers by this authorPublished Online:30 Dec 2019https://doi.org/10.2514/1.G004518SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutNomenclatureasemimajor axisadacceleration due to dragCbballistic coefficientMmean anomalynmean motionttimetctime until collision if no action is takentsballistic coefficient swap pointΔxdesired miss distanceΔϕttotal change in ϕ that occurs by time tcδsemimajor axis difference between perturbed and original trajectoriesμeEarth’s gravitational parameterρatmospheric densityϕmean anomaly difference between perturbed and original trajectoryI. IntroductionThe increasing number of actors in space has led to an increased risk of on-orbit collisions [1]. Predictions have shown that, if action is not taken to mitigate the risks of collisions in space, the additional debris resulting from these collisions may complicate space operations for generations to come [2,3]. Even a satellite in a low Earth orbit that will decay naturally in a few years due to aerodynamic drag can cause long-lasting debris if it collides with an object in a low-perigee elliptical orbit. For example, a small CubeSat in a 500 km circular orbit colliding with a discarded rocket body from a previous launch in a 400 by 35,786 km geostationary transfer orbit can generate a significant debris cloud that remains in orbit for hundreds of years. This reality necessitates that future spacecraft take an active role in debris mitigation and collision avoidance instead of contributing to the problem [4].Large satellites containing propulsion systems have used thrusters to actively avoid debris objects as well as to perform postmission disposal burns. However, many small satellites such as CubeSats do not contain propulsion systems [5] and generally do not perform active collision avoidance or postmission disposal. Drag devices have been developed for small satellites to expedite postmission deorbit [6,7], but the significantly increased surface area from a drag device greatly increases the risk of an onorbit collision per unit time [8]. For satellites containing retractable drag devices such as the drag deorbit device (D3) shown in Fig. 1 and developed at the University of Florida [9] or the Exo-Brake developed by NASA Ames Research Center [10], aerodynamic drag modulations can be used for active collision avoidance, mitigating the increased collision risk caused by the larger surface area.Although much of the existing literature has investigated the use of aerodynamic drag for orbital maneuvering [11–13], there is limited work on aerodynamic drag for collision avoidance. Reference [14] considered the configuration of satellite swarms such that long-term collision risks were minimized, but it did not address the issue of a suddenly apparent, impending impact with an uncooperative and previously unknown space object. Reference [15] considered using aerodynamic drag and solar pressure for avoiding an impending collision, but the focus of that work was on maximizing the miss distance from the debris object. If drag is the only control variable, the maximum collision miss distance can be achieved by configuring the spacecraft with the greatest difference in drag acceleration from its nominal configuration. That is, if a satellite is in a minimum drag configuration and is expected to experience a collision, it should be set to maximum drag; whereas if it is nominally in maximum drag, it should be set to minimum drag. Is it not always practical or desirable, however, to achieve the maximum miss distance. A satellite may interrupt nominal mission operations while maneuvering; so, in many cases, an operator may desire a miss distance large enough to safely avoid the collision with the minimum time spent maneuvering. The existing literature does not provide a means of calculating the drag profile necessary to achieve a precise desired miss distance with minimal maneuvering time. The existing literature also does not provide a way of analytically characterizing the orbital maneuvering capabilities of a satellite based on the achievable drag modulation, the ambient density, and the time until collision.This Note discusses a time-optimal solution for collision avoidance using aerodynamic drag based on an analytical solution iteratively refined using a high-fidelity orbit propagator. Although the aerodynamic collision avoidance methods discussed in this Note may be the only feasible means of active collision avoidance for small satellites without thrusters, the methods are applicable to any satellite, regardless of size, that is capable of modulating its ballistic coefficient. The case considered is one where a satellite’s operator receives a conjunction notice (generally one to seven days before the expected collision) and must decide what desired drag profile to uplink to the satellite in order to avoid the collision by a given distance. Section II first presents a method of analytically computing the effect that a perturbation to the nominal drag profile will have on an orbit. Section III then shows how this solution is used with numerical orbit propagation techniques to compute the drag profile needed to achieve a desired miss distance. The optimality of this solution is also discussed. Section IV discusses the control capabilities of aerodynamic collision avoidance, Sec. V presents the results of a case study and Monte Carlo simulations used to validate the method, and Sec. VI presents the conclusions of this study.II. Analytical Estimation of Drag-Induced Orbital PerturbationsThe first step of the aerodynamic collision avoidance algorithm involves analytically estimating the perturbation to an orbit resulting from a change in drag. This is done by computing the difference in mean anomaly that would occur between the original orbit and a new orbit starting from the same initial conditions but with a different drag profile. This analytical solution is then used to calculate the change in the drag profile necessary to achieve a desired miss distance from a debris object. In the analytical solution, a circular orbit around a spherical Earth with a nonrotating atmosphere is assumed. The density is treated as constant throughout the maneuver. The change in semimajor axis during the course of the maneuver is also assumed to be small as compared to the original semimajor axis. Although these assumptions are not accurate, they allow for an initial estimate of the drag-induced orbital perturbations. This estimate can later be refined through numerical orbit propagation techniques. Define ϕ=Mr−Mn(1)where Mn is the mean anomaly of the new trajectory (altered drag profile), and Mr is the mean anomaly of the original or reference trajectory. Note that, through the rest of this analysis, the subscript “r” will refer to the original unperturbed trajectory and the subscript “n” will refer to the new trajectory: ϕ˙=dϕdt=nr−nn(2)where n is mean motion and is calculated based on Earth’s gravitational parameter and the current semimajor axis as n=μea3(3)The first step is to derive a relationship between ϕ˙ and the difference between ar and an (the semimajor axes in the original and new trajectories, respectively): ϕ˙=μear3−μean3=μe(1ar3/2−1an3/2)(4)δ=an−ar(5)ϕ˙=μe(ar−3/2−(ar+δ)−3/2)(6)Equation (6) can be linearized in terms of δ using the binomial expansion theorem. This theorem states that, in general, for some real numbers z and j [16], (1+z)j=∑k=0jj!k!(j−k)!zk(7)For z≪1, only the first two terms in Eq. (7) can be kept with reasonable accuracy, yielding (1+z)j≈1+jz(8)Using Eq. (8) and the assumption that δ≪a0 allows for the simplification (ar+δ)−3/2=ar−3/2(1+δar)−3/2≈ar−5/2(ar−3δ2)(9)Substituting Eq. (9) into Eq. (6) and rearranging yields the following first-order linear differential equation that describes ϕ˙ in terms of δ, μe, and ar: ϕ˙=μear5/2(ar−(ar−3δ2))=3δμe2ar5/2(10)Differentiating Eq. (10) with respect to time and assuming that ϕ and δ are the only variables changing appreciably over time yields ϕ¨=3μe2a05/2δ˙(11)where a0 is the initial semimajor axis. Because the drag-induced semimajor axis change is small, it is reasonable to use a0 instead of ar when evaluating and computing the derivative of Eq. (10). Recall that these analytical relations are not meant as substitutes for high-fidelity numerical orbit propagation but rather serve as first-order estimates of the effects of drag changes on an orbit. The next step is to relate δ˙ to the spacecraft Cb. This can be accomplished via the work energy theorem that states that the change in orbital energy is equal to the total work done by aerodynamic drag because drag is the only significant nonconservative force. From equation 2–80 (p. 92) in Ref. [17], the orbital energy per unit mass and its derivative with respect to time are defined based on the semimajor axis of an orbit as E=−μe2a(12)dEdt=μe2a2dadt(13)From the work energy theorem and the circular orbit assumption, we also have dEdt=adv≈−Cbρv3≈−Cbρ(μea)3/2(14)where ad is the acceleration due to drag, and the circular orbit velocity is given by v=(μe/a). In a nonrotating atmosphere, ad is computed by ad=−CdA2mρv2=−Cbρv2(15)where Cd is the drag coefficient, m is mass, ρ is the atmospheric density, and A is a reference area. Substituting Eq. (14) into Eq. (13) and solving for (da/dt) yields dadt=−2Cbρμea(16)Taking the time derivative of Eq. (5), substituting Eq. (16) in for (da/dt), and assuming ρ is constant and an≈ar≈a0 yields δ˙=dandt−dardt=−2ρμea0(Cbn−Cbr)(17)Equation (17) can be substituted directly into Eq. (11) to get ϕ¨=−3ρμea02(Cbn−Cbr)=−3ρμea02ΔCb(18)To analyze the effects of drag changes on real orbits, the initial mean semimajor axis of the original orbit can be used for a0 in Eq. (18). Also, ρavg can be substituted for ρ, where ρavg is the time-averaged value of density for a given analysis period as determined by numerical orbit propagation and the use of a high-fidelity density model like NRLMSISE-00 (Naval Research Lab Mass Spectrometer and Incoherent Scatter Radar) [18].III. Computing the Drag Profile Necessary to Achieve a Desired Miss DistanceA. Analytical SolutionGiven a desired miss distance from the original collision point Δx that will occur at time tc, the desired difference in mean anomaly from the original orbit can be computed as Δϕd=Δxa(19)Note that only the along-track component of the miss distance is considered because, in general, it will be much larger than the radial miss distance. For example, if the drag is perturbed such that the original and new trajectories have a 200 km along-track separation, the semimajor axis difference between the trajectories will generally be less than 1 km. Also, ignoring the radial separation gives a “worst-case” collision miss distance. For example, if the along-track separation is 200 km, the absolute separation will generally be greater than or equal to 200 km due to the radial component. Additionally, although Eq. (19) assumes that Δx is measured along the orbit track, this arc length and the absolute position difference between two orbital positions will be almost identical as long as Δx is relatively small as compared to the orbit circumference (less than 1000 km). It is unlikely that an operator would require a collision miss distance greater than this and, if they do, they likely would not be concerned with achieving it precisely.If ΔCb is maintained for time Δt, then the total Δϕ will be Δϕ=12ϕ¨(Δt)2+ϕ˙0Δt(20)where ϕ˙0 is ϕ˙ at the start of the analysis interval and equals zero at the beginning of the trajectory. Achieving Δϕd with the minimum amount of time spent in an off-nominal drag configuration will involve maintaining Cbmd until time ts and then maintaining the nominal ballistic coefficient Cbnom until tc, where Cbmd is the Cb that maximizes the magnitude of ΔCb=Cb−Cbnom. Minimizing time spent in the off-nominal drag configuration is often desirable because applying Cbmd may impose operational constraints such as requiring an attitude change that temporarily renders the satellite unable to fulfill mission objects. With this maneuver, a constant ϕ¨=−[(3ρμe)/a2]ΔCb will be maintained until ts and ϕ¨=0 will occur between ts and tc. Using Eq. (20) to describe the behavior before and after ts leads to Δϕt=12ϕ¨ts2+ϕ¨ts(tc−ts)(21)where Δϕt represents the total change in ϕ at time tc. This can be rewritten as (−12ϕ¨)ts2+(ϕ¨tc)ts−Δϕt=0(22)Given a desired Δϕtd, Eq. (22) can be solved using the quadratic formula to get the ts required to achieve Δϕtd. From the two ts values returned by the quadratic formula, the smallest positive value should be chosen. Recall that Cb should be set to Cbmd for t≤ts and should be set to Cbnom for t>ts. If both ts values are imaginary, this means that it is not possible to achieve Δϕtd. In this case, Cb should be set to Cbmd until time tc to achieve the greatest Δϕt possible.B. Optimality of the ApproachThis maneuvering approach is optimal in the sense that it results in the minimum time spent in an off-nominal drag configuration for a desired collision miss distance. Put another way, this approach results in the largest possible miss distance for a given amount of time a spacecraft can spend maneuvering. Equation (21) shows that, for a given ts, Δϕt increases directly with ϕ¨. From Eq. (18), ϕ¨ is maximized when ΔCb is maximized. Thus, to maximize the Δϕt for a given ts and to minimize the ts needed to achieve a desired Δϕt, Cb should be commanded such that |ΔCb| (difference between nominal and perturbed Cb) is maximized during the maneuvering time. Equation (21) also shows that, the greater the tc, the smaller the ts that will be required for a given Δϕt. This shows that maneuvering should be initiated immediately to minimize ts. If maneuvering is not initiated immediately, this is equivalent to having a smaller tc, which will result in a larger ts required to achieve the desired Δϕt. This demonstrates that achieving a desired Δϕt with the minimum time spent in an off-nominal drag configuration involves completing all maneuvering as early as possible in the trajectory and commanding a maximum magnitude of ΔCb during the maneuver (time ts). After ts, the satellite is returned to the nominal drag configuration.C. Numerical RefinementEquation (22) is dependent on ϕ¨ as given by Eq. (18), where a constant density and a circular orbit around a spherical Earth are assumed. Because these assumptions are not valid in a realistic environment, the total Δϕ that will occur if a prescribed ballistic coefficient profile is applied will be different than that which is predicted by the analytical theory. To address this concern, the analytically estimated Cb profile can be iteratively refined using a high-fidelity numerical orbit propagator. Because knowledge of the satellite and debris object orbits is required to use this algorithm, calculations would likely be completed on the ground and the final Cb profile would be uplinked to the satellite; so, the computation time associated with orbit propagation is not an issue.First, the required ts can be analytically estimated by solving Eq. (22) based on the density at the initial conditions, Δϕtd, and Cbmd. Two trajectories of the satellite can then be numerically propagated in the high-fidelity simulation environment. In one, the satellite applies Cbmd until ts and Cbnom until tc; in the other, the satellite maintains Cbnom during the entire trajectory until tc. The actual miss distance Δϕta is the difference in mean anomaly between the two satellite trajectories at tc. The aiming miss distance Δϕtaim is originally set equal to the desired miss distance Δϕtd and can be adjusted iteratively based on difference between Δϕtd and Δϕta as Δϕtaim=Δϕtaim+(Δϕtd−Δϕta)(23)The average density ρavg from the simulation start until tc can also be computed based on the density at each point during the numerical integration process. Then, ρavg can be used to compute ϕ¨ in Eq. (18). This ϕ¨ can then be used to solve Eq. (22) for the ts required to achieve the updated Δϕtaim. The perturbed satellite trajectory is simulated as before with the new ts, and the new Δϕta is recorded. The process of updating ρavg and Δϕtaim [Eq. (23)], computing ts, and propagating a new trajectory continues until a ts is obtained that yields |Δϕtd−Δϕta| less than some specified tolerance. Recall that, to achieve the desired Δϕtd, the satellite applies Cbmd until this ts, and then it returns to Cbnom afterward.Sometimes, a limit-cycling behavior occurs where Δϕta is initially greater than Δϕtd; but, when the correction prescribed by Eq. (23) is applied, the new Δϕta is less than Δϕtd. A correction to Δϕtaim is applied again, but this results in Δϕta greater than Δϕtd. In some cases, this overshoot–undershoot cycle continues indefinitely. To remedy this, if more than five iterations of numerical propagation occur, the Δϕtaim update law becomes Δϕtaim=Δϕtaim+0.5(Δϕtd−Δϕta)(24)The added factor of 0.5 lessens the magnitude of the correction to Δϕtaim and was observed to break the limit-cycling behavior in all tested scenarios.D. Returning to the Original TrajectoryIn many cases, such as that of a single satellite trying to perform some nominal Earth observation mission while avoiding collisions, it is not necessary to return to some predetermined reference trajectory after the end of the collision avoidance maneuver. In other scenarios, such as that of a satellite in a constellation faced with avoiding a collision, it may be necessary to return to the nominal orbit after the danger is over. In such a case, prior algorithms developed by Omar and Wersinger [13] can be used to maneuver the satellite back to the original orbit using aerodynamic drag. To return to the original orbit, however, the satellite must be capable of achieving positive and negative ϕ¨ values [see Eq. (18)], which require ballistic coefficients less than and greater than that of the nominal trajectory. To see this, consider the case where a satellite applies a below nominal ballistic coefficient to avoid the collision. The satellite does not lose altitude as fast as in the nominal trajectory, and thus lags “behind” the nominal trajectory. To return to the original orbit, the satellite must apply a ballistic coefficient greater than the nominal Cb to drop to a lower semimajor axis than, and ultimately catch up to, the nominal trajectory. Similarly, If a satellite initially applies an above nominal Cb to avoid a collision, it will drop faster and get “ahead” of the original orbit. A below nominal Cb will then be needed to rendezvous with the original orbit. The time required to return to the nominal trajectory, if such a return is possible, will depend on the minimum and maximum achievable Cb values as well as the nominal Cb.IV. Controllability AnalysisAs can be verified from Eq. (21), the maximum Δϕt is achieved when ts=tc. In this situation, Cbmd is maintained all the way until tc. The maximum Δϕt is dependent on tc and the atmospheric density, which is a function of altitude. Figure 2 shows the ts required to achieve a desired miss distance for tc values of one day and three days based on Eq. (22). Figure 2 uses the 1976 standard atmospheric densities [19] at 400, 500, and 600 km altitudes. If the desired miss distance cannot be achieved, ts is set equal to tc. Note that Fig. 2 is for a 4 kg D3-equipped CubeSat [9] with Cbnom=0.1375 m2/kg and Cbmd=0.002750 m2/kg. A photograph of the D3 attached to a CubeSat is shown in Fig. 1, but other drag devices or attitude changes could be used to produce the drag modulations needed for maneuvering. The controllability analysis will be identical for any satellite with the same ballistic coefficient values but will differ for different ballistic coefficients. Figure 3 shows the maximum achievable miss distance for a D3-equipped 4 kg CubeSat at various altitudes in standard atmospheric conditions (left chart) and for various satellites in 500 km circular orbits as a function of tc (right chart). The Cbnom and Cbmd values used to create the right-side chart in Fig. 3 correspond to a 3-unit (3U) CubeSat with no deployables, a 3U with single-deployed long-edge solar panels (3U sdp), a 3U with double-deployed long-edge solar panels (3U ddp), and a 3U CubeSat equipped with a D3 (3U D3): all nominally in a maximum drag configuration.Fig. 1 D3 attached to a CubeSat.Fig. 2 Swap time required for desired miss distances with tc=1 day (left) and tc=3 days (right) for a 3U D3-equipped CubeSat.Fig. 3 Maximum achievable miss distance vs time to collision for 3U D3-equipped CubeSat at various altitudes (left) and for different satellites in 500 km circular orbits (right).V. Numerical SimulationsA. Orbit Propagation MethodsSimulations were conducted with a high-fidelity orbit propagator including a rotating atmosphere, historic NRLMSISE-00 [18] density values, and gravitational perturbations from the EGM2008 (2008 Earth Gravity Model) model [20] through the degree and order of four. To simulate an impending collision, a satellite and a debris object were created with the same initial positions but different initial velocities to simulate the conditions at a collision. The satellite and debris object trajectories were then backward propagated for time tc to simulate the conditions at which a satellite operator would receive a conjunction notice [21]. Backward propagation was completed by using an 8th order Runge Kutta numerical integration scheme (RK78) with an adaptive step size (Ref. [22] pp. 117–123) and multiplying the increment function by a negative Δt instead of a positive Δt for each time step.B. Simulation Results1. Case StudyAs an initial test, a satellite and a debris object were created with an epoch of 3 January 2014 and initial state vectors of xs=[67780004.73666.0347]Tand xd=[67780007.74860]Trespectively, where the first three state components are Cartesian position in kilometers and the last three are velocity in kilometers per second expressed in an Earth-centered inertial (ECI) frame initially aligned with the Earth-centered Earth-fixed frame at the simulation epoch. This corresponds to an approximately 400-km-altitude satellite orbit. Both trajectories were backward propagated for two days using the high-fidelity orbit model discussed in Sec. V.A. The debris object had a ballistic coefficient of 0.008271 m2/kg, which corresponds to a 1.33 kg, 1-unit (1U) CubeSat; and the satellite had a ballistic coefficient of 0.1375 m2/kg, which corresponds to the maximum drag ballistic coefficient for a 4 kg, D3-equipped CubeSat [9]. This simulates the case where a conjunction notice is received two days before an impending collision. The method from Sec. III was used to determine the ts necessary to achieve a miss distance of 200 km, assuming that Cbnom=0.1375 m2/kg and Cbmd=0.002750 m2/kg (D3 fully retracted for minimum drag). The algorithm determined that ts=11134.7 s was necessary to achieve the 200 km along-track miss distance within a tolerance of 0.1 km in the high-fidelity simulation environment. Figure 4 shows the orbits of the satellite and the debris object, and Fig. 5 shows the absolute separation in kilometers between the satellite and the debris object position vectors over time, with and without aerodynamic collision avoidance. The graph on the right in Fig. 5 shows a zoomed-in section of the trajectory near the critical two-day mark. Note that the trajectory with ts=0 represents the trajectory that maintains the nominal drag profile and experiences a collision at tc=2 days. Performing the aerodynamic maneuvering (ts=3.093 h) results in a separation of 200 km after two days.Fig. 4 Satellite and debris object positions (pos) in ECI frame during simulation interval.Fig. 5 Satellite–debris separation over time with (ts=3.093 h) and without (ts=0 h) maneuvering.Figure 6 displays the semimajor axis over time of the nominal and perturbed trajectories. This illustrates that the total semimajor axis change over the maneuver is small (less than 2%) and the difference in the semimajor axis between the trajectories is also small (less than 2 km), justifying the assumptions made in the analytical relations in Sec. II.Fig. 6 Satellite semimajor axis over time in perturbed (maneuvering) and nominal trajectories.2. Monte Carlo SimulationsTo further validate the performance of the algorithm, 1000 Monte Carlo simulations were conducted for the scenario of a 4 kg, 3U CubeSat equipped with the drag deorbit device discussed in Ref. [9] (Cbmin=0.002750 m2/kg and Cbmax=0.1375 m2/kg). In each scenario, the satellite begins in the maximum drag configuration and switches to minimum drag for a computed time ts in order to avoid the collision. Each simulation was initialized with mean orbital elements, an epoch, and a time to collision randomly selected from the distributions given in Table 1. The average density for the selected satellite was computed and was used with Eq. (21) to compute the maximum achievable miss distance Δxmax. The desired miss distance was then selected from a uniform distribution between 0.25Δxmax and 0.75Δxmax: not to exceed 300 km. In each simulation, the nominal satellite trajectory (maximum drag configuration) was first numerically propagated up to time tc. The procedure in Sec. III was then used to compute ts such that a propagation of the satellite trajectory with Cbmin until ts and Cbmax between ts and tc yielded a desired miss distance (with a tolerance of 0.1 km) from the nominal trajectory at tc. Because the nominal trajectory was assumed to collide with a debris object at tc, there was no need to simulate the debris object and the miss distance could be computed relative to the nominal trajectory at tc. In all Monte Carlo simulations, the satellite was able to achieve the desired miss distance within 0.1 km. The average computation time on a laptop computer running MATLAB R2016a was 52 s, and the maximum computation time was 246 s. This Monte Carlo analysis proves that the collision avoidance algorithm is robust enough to be used in an operational scenario and that the assumptions made in deriving the analytical solutions do not lead to a convergence failure in the numerical refinement portion of the algorithm.VI. ConclusionsThe steadily increasing number of small satellites in orbit demands that future spacecraft actively avoid on-orbit collisions instead of being a part of the debris problem. This Note shows that modulations in aerodynamic drag using a drag device or attitude changes can be used to maneuver a satellite away from an impending collision. This method is particularly valuable for small satellites such as CubeSats that do not contain thrusters and generally have no other way to maneuver. This Note first presents a means of analytically estimating the effect that a ballistic coefficient change will have on an orbit. The analytical solution is then used along with numerical orbit propagation techniques to compute the ballistic coefficient profile that a satellite must follow to achieve a desired miss distance from an impending collision with the minimum amount of time spent maneuvering. The robustness of this solution was verified via 1000 Monte Carlo simulation runs where the desired collision miss distance was achieved within 0.1 km in all simulations. Ultimately, the use of retractable drag devices and the algorithms discussed in this Note enable small satellites to avoid collisions without using thrusters and play an active role in orbital debris mitigation.AcknowledgmentsThe authors wish to thank a.i. solutions for initially sponsoring this investigation under a NASA Kennedy Space Center subcontract (project LSP 15-025: “A Drag Device for Controlled De-Orbiting of LEO Spacecraft”). This work was also supported by a NASA Space Technology Research Fellowship (grant number 80NSSC17 K0232). References [1] Rossi A. and Valsecchi G. 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Google ScholarTablesTable 1 Monte Carlo simulation parametersVariableRangeDistributionMean semimajor axis[6778, 6878] kmUniformMean true anomaly[0, 360] degUniformMean eccentricity[0, 0.004]UniformMean right ascension[0, 360] degUniformMean argument of the perigee[0, 360] degUniformMean inclination[1, 97] degUniformEpoch[11/1/2003, 11/1/2014]UniformTime to collision[2, 5] daysUniform Previous article Next article FiguresReferencesRelatedDetailsCited byComparison between convex and non-convex optimization methods for collision avoidance maneuvers by a spacecraftActa Astronautica, Vol. 202Convex optimization of collision avoidance maneuvers in the presence of uncertaintyActa Astronautica, Vol. 197Geostationary Orbital Debris Collision Hazard after a Collision10 May 2022 | Aerospace, Vol. 9, No. 5Simulation of the orbital decay of a spacecraft in low Earth orbit due to aerodynamic drag7 October 2021 | The Aeronautical Journal, Vol. 22CubeSat Adaptive Attitude Control with Uncertain Drag Coefficient and Atmospheric DensityRunhan Sun, Camilo Riano-Rios, Riccardo Bevilacqua, Norman G. Fitz-Coy and Warren E. Dixon29 December 2020 | Journal of Guidance, Control, and Dynamics, Vol. 44, No. 2 What's Popular Volume 43, Number 3March 2020 CrossmarkInformationCopyright © 2019 by Sanny Omar. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAerodynamicsAeronautical EngineeringAeronauticsAerospace SciencesAstrodynamicsAstronauticsAtmospheric ScienceComputational Fluid DynamicsFluid DynamicsMonte Carlo MethodNumerical AnalysisOrbital ManeuversOrbital PropertySpace DebrisSpace Orbit KeywordsAerodynamic DragSpacecraftsSatellite OperatorBallistic CoefficientEarthMonte Carlo SimulationComputingEarth Centered InertialElliptical OrbitGeostationary Transfer OrbitAcknowledgmentsThe authors wish to thank a.i. solutions for initially sponsoring this investigation under a NASA Kennedy Space Center subcontract (project LSP 15-025: “A Drag Device for Controlled De-Orbiting of LEO Spacecraft”). This work was also supported by a NASA Space Technology Research Fellowship (grant number 80NSSC17 K0232).PDF Received8 April 2019Accepted6 November 2019Published online30 December 2019" @default.
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