SSC20-WKI-09

A Non-Hohmann Method for Orbital Element Database Pre-Processing

William O. S. Hudnut, Cameron P. Mehlman, Kurt S. Anderson Department of Mechanical, Aerospace, and Nuclear Engineering Rensselaer Polytechnic Institute JEC 2049 Jonsson Engineering Center 110 8th Street Troy, NY 12180 USA 802-356-0329 [email protected]

May 2020

ABSTRACT

One major obstacle to successful orbital debris remediation is the determination of which pieces of debris are the most viable targets for capture and de-orbit. The viability of a target is deter- mined by some combination of the debris’ risk factor (a combination of its size, composition, and the orbit it occupies), the anticipated resource cost to ﬁnd and capture the debris, and the un- derlying probability of successful intercept and capture of that target. The problem of selecting debris for capture by a multi-capture capable spacecraft is fundamentally a traveling salesman problem in which the traveler only has the resources to reach a very limited subset of the available destinations. Therefore, rapidly identifying the sets of destinations (i.e. pieces of debris) which are either too expensive to reach or insuﬃciently valuable to justify targeting will reduce the tar- get destination set; this would signiﬁcantly enhance the eﬃciency of the solution. This problem of intelligently reducing the space of possible solutions can be partially solved by performing a preliminary ﬁltering and sorting of orbital debris database entries using known spacecraft or- bital parameters and maneuvering ∆V-budget to reduce the number of possible destinations for an optimizer to those which are in fact accessible from the spacecraft’s initial orbit. The chosen algorithm for analyzing and ﬁltering the data is a two-burn node-to-node non-Hohmann transfer, which was used to estimate the ∆V-cost for transfer from the capture spacecraft’s initial orbit to an orbit near the target piece of debris. Once the ∆V-cost was calculated for each transfer orbit, entries with excessive fuel costs were removed from consideration, and the fuel cost to access each remaining orbit was appended to its entry. This method was capable of reducing a 10,400- item list of debris to less than 100 accessible targets in under 3 seconds on an ordinary laptop computer. This reduction in database size brought the number of targets down to a practical size for processing by a more computationally expensive optimization algorithm suitable for selecting ﬁnal targets for a multi-capture spacecraft.

Hudnut1 34th Annual Small Satellite Conference INTRODUCTION (each OSCaR is equipped with only 3 to 4 nets depending on the CubeSat conﬁguration). The In 1958, the United States’ Vanguard I satel- ephemeris data of cataloged pieces of debris in lite became the ﬁrst man-made object to en- orbit around the earth is recorded by organiza- ter orbit around earth without reentering the tions such as NORAD, and can be obtained in atmosphere. Since then, due to the increase the form of two-line elements [10]. To facilitate of man made objects in orbit as well as satel- the greatest probability of success of the mis- lite collisions, the amount of detritus in orbit sion, this list must be reduced to those pieces around earth has increased exponentially. The of debris that require acceptable quantities of ESA estimates over 100 million pieces of debris propellant to ﬁnd, capture, and deorbit given orbit Earth at every moment, most are smaller an OSCaR’s initial orbit. The ﬁrst step in nar- than 10 cm [7]. Debris of this size creates huge rowing this list to the ﬁnal target set is to use risks for any space endeavor within Earth’s an algorithm that can remove pieces of debris orbit because they are diﬃcult to avoid and that cannot be obtained without exceeding a travel at hypersonic speeds (up to 7.8 km/s). given ∆V budget. A collision at these speeds can cause severe damage to a spacecraft; even ﬂecks of paint The purpose of this algorithm is to conduct a have been known to crater and crack windows. preliminary search, such that the reduced list The most imminent threat from space debris is is of a manageable size for a more accurate and that with increasing numbers of debris in orbit, computationally expensive optimizer that will there is an increasing chance of debris perpet- determine the ﬁnal targets. The sorter esti- ually colliding and breaking down into smaller mates the amount of fuel required to complete pieces. As a result, vast regions of low-earth or- a two-burn node-to-node non-Hohmann trans- bit (LEO) space could be deemed unusable due fer from the satellite’s orbit to the debris’ orbit. to ﬁelds of small yet highly dangerous parti- Debris that required more ∆V than the prede- cles. This hypothesized cascading of collisions termined limit are removed from the list. The yielding an exponential growth in the number algorithm described herein does not currently of pieces of fragmented debris is also known as consider additional propellant required for any Kessler’s Syndrome [20]. phasing burns required to realize the subse- quent OSCaR-debris rendezvous once OSCaR In the past, there has been little to no eﬀort has placed itself in the debris piece orbit. The to actively reduce the amount of debris in or- sorter returns a ﬁnal sorted list of all debris bit around Earth, and most proposals only tar- within an optimal ∆V range of the satellite. get larger intact spacecraft. OSCaR (Obsolete Spacecraft Capture and Removal) is a 3-unit METHOD CubeSat satellite system designed to target, capture, and de-orbit smaller debris in LEO The overall goal of the sorter program was (of approximately 1 to 20 cm in size). OSCaR to produce a reduced list of accessible ren- is being developed to pursue individual pieces dezvous targets from a database ﬁle of debris of debris and then, capture its target using targets, an initial set of orbital parameters for a weighted net. The debris and net is slowly the spacecraft, and an onboard fuel budget for pulled into the atmosphere, through the use the spacecraft. In order to achieve this goal, of electromagnetic tethers [19], where they will four steps needed to be undertaken. First, burn up upon re-entry. The ﬁnal piece of de- a suitable target set had to be obtained from bris captured by OSCaR is deorbited along with existing databases. Second, the data thus the spacecraft as OSCaR uses its remaining obtained needed to be manipulated in order fuel reserve to place itself into an eliptical orbit to produce classical orbital parameters in a which dips into the atmosphere and decays. convenient format for large-scale processing. Third, an algorithmic method of calculating the OSCaR’s target set is received from a grounded delta-V required to reach a target orbit needed station where thousands of known pieces of to be implemented. Finally, the data set needed debris are taken into consideration, evaluated to be cleared of impractical targets and pro- and the ﬁnal debris target set is selected. Be- vided in a convenient format for later process- cause of the size of CubeSat satellites, OSCaR ing. has both substantial limits on fuel and nets

Hudnut2 34th Annual Small Satellite Conference Orbital ephemeris data can be retrieved from transfer orbit is usually optimal for co-planar several sources, including the European Space initial and ﬁnal orbits, in the case of orbital de- Agency [7] and Space-Track.org [16]. These bris rendezvous it is likely that the debris will databases usually store the ephemeris data for be out-of-plane with respect to the initial orbit observed satellites and debris in the two-line of the spacecraft, and will have randomly ori- element (TLE) format. In order to achieve the ented apse lines. ﬁrst step of the process, the acquisition of or- bital data for debris, the team turned to a M.S. In most cases for a non-Hohmann orbital thesis by Philip Hoddinott associated with the transfer, time-optimal transfers are preferred. OSCaR project. This thesis was on the devel- These transfer maneuvers are usually calcu- opment of a method for retrieving open-source lated by a method which iteratively solves Lam- orbital data in a program usable by anyone fa- bert’s problem [18]. However, while these so- miliar with basic orbital mechanics and MAT- lutions are good for exact intercepts between LAB coding [10]. This program scanned the spacecraft, they are computationally intensive Space-Track.org TLE database for orbiting bod- due to their iterative nature [5]. Therefore, an ies categorized as "debris" appearing after a analytical method of trajectory and ∆V calcula- given date, and then generated a MATLAB data tion was derived in order to save computational ﬁle and populated it with the TLE data thus re- time for the purposes of the initial ﬁltering al- trieved. gorithm.

Wrapper The following derivation is a method to cal- culate the ∆V required for a two-burn non- The second and fourth steps of the process are Hohmann transfer between two inclined ellip- handled in the wrapper script, which intakes tical orbits, called orbits 1 and 2, where these the TLE data ﬁle, processes it into a useful orbits have arbitrarily oriented apse lines. The format, passes it to the calculation algorithm, departure burn occurs at a point on the line and then cleans the resultant data of target or- of nodes on the initial orbit, and the insertion bits which the spacecraft cannot reach. It also burn occurs at a point on the line of nodes on contains the user-input orbital parameters and the other orbit, nominally the ﬁnal orbit, op- ∆V budget of the OSCaR spacecraft. The pro- posing the initial point. cessing of the incoming data is relatively sim- ple, as the TLE format contains all the neces- Beginning with the orbit elements for orbits 1 sary information to determine the orbits of the a1, e1, Ω1, i1, ω1, θ1 and 2 a2, e2, Ω2, i2, ω2, body it concerns. The necessary processing is θ2 ; where aj is the orbital semimajor axis of simply the extraction of the classical orbit ele- the trajectory, ej is the orbital eccentricity, Ωj ments from the appropriate indices of the TLE- is the right of ascension of the ascending node, formatted data set to pass to the ∆V calcula- ij is the orbital inclination with respect to the tion. Once transfer ∆V has been calculated for equatorial plane, ωj is the argument of periap- each orbit, the orbits with excessive transfer sis, and θj is the actual anomaly; the orbital el- costs are deleted from the ﬁnal output ﬁle. ements of the transfer orbit T at, et, Ωt, it, ωt, θt ∆V Calculation Algorithm must be calculated in order to determine the ∆V cost to intercept a given piece of debris. The main objective of the ﬁlter-sorter algorithm was to provide a timely, accurate estimation of From the orbital elements of orbits 1 and 2, the the ∆V cost to access any given piece of debris direction cosine matrices N CPF1 and N CPF2 from a given initial spacecraft orbit. In order which relate the orientations of the perifocal to make this estimate, some trajectory method reference frames of orbits 1 and 2 respectively must be chosen which suits the most likely to the inertial reference frame N, with its basis maneuvers which will be needed to access the vectors nˆ1, nˆ2, nˆ3 can be calculated. debris. While a Hohmann or Hohmann-like

Hudnut3 34th Annual Small Satellite Conference Figure 1: Inclined orbit, showing relevant unit vectors and angles

The unit vector of the line of nodes γˆ is given Since orbital velocities, and therefore required by ∆V s for turning maneuvers, are higher at lower orbits, the method assumes that the ∆V at wˆ1 × wˆ2 γˆ = , (1) the lower terminal radius is performed entirely kwˆ × wˆ k 1 2 along the ﬂight path of the spacecraft at that as shown in ﬁgure 1 where wˆ1, wˆ2 are the unit point, and that all turning required for the vectors ﬁxed in the inertial reference frame par- transfer is performed at the higher terminal ra- allel to the orbital angular momentum vectors dius. This will adequately constrain the prob- of orbits 1 and 2 respectively. The matrix rep- lem of solving for three parameters at, et, θγt of resentations of these vectors in the N refer- the transfer orbit. It should be noted that in all ence frame correspond to the third columns of situations during the implementation of this N PF N PF C 1 and C 2 respectively. algorithm in code, the inverse tangent opera- tion was carried out using the atan2 function Deﬁning θγ1 and θγ2 as the angles between the to avoid ambiguities associated with the nor- pˆ1, pˆ2 unit vectors the vector representations mal inverse tangent operation. of which in the N basis form ﬁrst columns of N PF N PF C 1 and C 2 respectively and γˆ, it can To solve for these three unknowns, the follow- be shown that the two terminal radii are r1(θγ1) ing three equations are used assuming r1 < and r2(θγ2), using the formula r2, although the equations can be modiﬁed to 2 function using the reverse assumption : aj(1 − ej ) rj(θ) = , (2) 1 + ej cos θγj 2 2 at(1 − et ) a1(1 − e1) K1 = = = r1(θγ1), (3) using appropriate values of a, e, θ. 1 + et cos θγt 1 + e1 cos θγ1

Hudnut4 34th Annual Small Satellite Conference 2 2 at(1 − et ) a2(1 − e2) value calculated for θγt can be used in equation K2 = = = r2(θγ2), (4) 1 − et cos θγt 1 + e2 cos θγ2 (9), and these values for θγt and et are used in equation (5) to get at.

et sin θγt e1 sin θγ1 α = tan−1 = tan−1 = α . Using the equations for orbital angular mo- t 1 + e cos θ 1 + e cos θ 1 t γt 1 γ1 mentum, radial, and tangential velocity, specif- (5) ically

h = a(1 − e2), (11) K1 and K2 are placeholder variables for the known values of r (θ ) and r (θ ), while α is µ 1 γ1 2 γ2 t v = e sin θ, & (12) the ﬂight path angle of the transfer orbit at the r h point of departure, A, which is deﬁned as equal µ vθ = (1 + e cos θ). (13) to α1, the ﬂight path angle of the original orbit h at the point of departure. v1rA, v1θA, the radial and tangential velocities The following is a rearrangement of equation of orbit 1 at departure point A; vtrA, vtθA, the (5 which will be useful) in the solution, specif- radial and tangential velocities of the transfer ically orbit at departure point A; vtθB, vtrB, the radial and tangential velocities of the transfer orbit at αt + αtet cos θγt = et sin θγt. (6) arrival point B; and v2rA, v2θB, the radial and tangential velocities of orbit 2 at arrival point Equation (3) can be rearranged to acquire an B can all be calculated. expression for at as #»A #»A Since v 1 · v t = v1AvtA is known because the K1(1 + et cos θγt) initial burn is constrained to be in the direc- at = 2 . (7) 1 − et tion of the original velocity vector at point A, this can be used to ﬁnd Substituting this into equation (4) yields #»A v 1 #»A K2(1 − et cos θγt) = K1(1 + et cos θγt), (8) vtA = v t , (14) v1A #» #» #» N r a = N r A N v A which can be rearranged into an expression for and since 1 t , t can then be found using the following manipulation et

A N PF1 A K − K {r1 }N = C {r1 }PF1 , (15) e = 2 1 . (9) t (K + K ) cos θ 2 1 γt A N PF1 A {v1 }N = C {v1 }PF1 , (16) #» #» N v A Substitution of the expression for et into equa- N A 1 v t = vtA. (17) tion (6) gives v1A #» #» (K2 − K1) cos θγt (K2 − K1) sin θγt N A N A N PFt αt+αt = −→ With r t , v t , [ C ] can be found to be (K2 + K1) cos θγt (K2 + K1) cos θγt #» N N #»A N #»A h t = r t × v t , (18) K2 − K1 K2 − K1 αt 1 + = tan θγt −→ #» #» #» K2 + K1 K2 + K1 N A N N A N #» v t × h t r t e t = − . (19) K2 + K1 µ rtA αt + 1 = tan θγt −→ K2 − K1

αt(2K2) Using these values, the perifocal frame unit tan θγt = −→ K2 − K1 vectors and the direction cosine matrix of the transfer orbit can be quickly assembled: −1 αt(2K2) θγt = tan . (10) #» K − K N #» N 2 1 e t h t pˆt = , wˆt = , qˆt =w ˆt × pˆt, et ht Now that an expression has been found to cal- N PFt culate θγt from known quantities K1, K2, αt, the C = pˆt, qˆt, wˆt ,

Hudnut5 34th Annual Small Satellite Conference which can be used to get RESULTS

B N PFt B When the algorithm was given a low- {vt }N = C {vt }PFt . (20) eccentricity initial orbit that intersected an area of space densely populated with debris, The following relation is also found often 20 or more suitable pieces of debris were found (pieces that could be reached within B N PF2 B the propellant budget). When an initial orbit {v2 }N = C {v2 }PF2 . (21) with the orbital elements a = 7.287 ∗ 106m, e = 0.00014, Ω = 90◦, ω = 230◦, i = 98◦ and max- N #»A N #»A ∆V1 = v t − v 1 , and ∆V2 = imum ∆V of 450m/s was put into the algo- N #»B N #»B v 2 − v t . The sum of these two values, rithm a total of 23 possible pieces of debris ∆V1 and ∆V2, is the total cost of the transfer were returned. The initial list of two-line ele- maneuver, ∆Vtot. ments used by the code is a list of all of NO- RAD’s non-protected cataloged objects in orbit The above method was implemented as a func- around earth between 1960 to 2020. The dis- tion in MATLAB R2018b, and used to estimate tribution of how many of these pieces and what the individual ∆V to transfer from a speciﬁed range of ∆V is required to maneuver to the de- initial orbit to each orbit in a target data set. bris orbit can be seen in ﬁgure 2.

Figure 2: Suitable Pieces of Debris vs. ∆V range

The orbital elements of the ﬁnal list of debris elements and the initial orbit is expected. have a minimal diﬀerential from the initial or- bital elements except for argument of periapsis. In order to evaluate the results of the sorting However, because the orbits will nominally be algorithm, some expectations are required for nearly circular, changing the argument of pe- where to expect the highest amount of suitable riapsis should require a minor amount of ∆V, debris for de-orbit. Using data from the ESA’s even if the change in angle is greater than one MASTER software, the distribution of debris in radian. Because the maximum ∆V is small in Earth’s orbit can be analyzed. A diagram plot- relation to the ∆V required for large orbital ma- ting the spatial debris density within Earth’s neuvers, the similarity between the ﬁnal list of orbit against altitude and angle of declination can be seen below. Hudnut6 34th Annual Small Satellite Conference Figure 3: Spatial Debris Density vs. Altitude and Declination [2]

This data from Figure 3 shows that the dens- meters, maximum ∆V of 450 m/s and varying est areas of debris are between 800 and 1000 right ascension of ascending node between 0◦ kilometers of altitude, and peaks at an angle of and 180◦ with step size of 1◦ the following re- declination of approximately 98◦. sults were obtained. The angle of periapsis is not relevant because the orbit is near circular. When the ﬁlter-sorter algorithm was run with The same initial list of all cataloged items be- an initial orbit with near 0 eccentricity, an in- tween 1960 and 2020 was also used. clination of 98.6◦, semi-major axis of 7.287 × 106

Figure 4: Number of suitable pieces of debris vs. right ascension of ascending node (i = 98.6◦)

Hudnut7 34th Annual Small Satellite Conference The data in Figure 4 shows an average of ap- tained. In Figure 5 it is seen that the average proximately 20 pieces of debris at most values amount of debris within a suitable range of the of Ω, with a sharp increase as Ω approaches initial orbit is about 1 to 2 pieces of debris, and 180◦. The considerable amount of obtainable multiple orbits have no debris within range of debris at all values of right ascension of as- the initial orbit. Again, this this strongly cor- cending node coincides with the ESA data from relates with the data obtained from the ESA Figure 3. and indicates that the sorter algorithm is ac- curately identifying possible targets given OS- When a similar simulation was run using an CaR’s initial orbit. inclination of 0◦, a very diﬀerent result was ob-

Figure 5: Number of suitable pieces of debris vs. right ascension of ascending node (i = 0◦)

In addition to accurately identifying possible curate analysis method designed to select the targets, the algorithm is quite fast. In testing targets and generate maneuvers for a multi- on a laptop computer running Windows 10 on capture spacecraft, based on the spacecraft’s an Intel i7-6700HQ at 2.60 GHz with 16 GB of initial orbit. This objective has clearly been RAM, the wrapper program was able to parse achieved: with the 450 m/s fuel budget al- and sort a 10,400 item database down to the located for the OSCaR multi-capture CubeSat same list of 82 accessible orbits from an ini- [2], fewer than 100 debris items were found for tial near-circular orbit at 800 km altitude, 180◦ any initial orbit out of the 10,400 debris TLE right ascension, and 98.6◦ inclination, with 450 ﬁles downloaded from Space-Track.org [16]. m/s ∆V, in an average of 2.0966 seconds over While this is still a large number of nodes, it is a 10 trials. greater than hundredfold reduction in problem size and a huge boon to any attempt to produce CONCLUSIONS a viable target set for a multi-capture, or even single-capture, spacecraft. The goal of this sorter program was to reduce orbital debris ephemeris data ﬁles down to a A valuable feature of the wrapper program is size suitable for use by a more complex and ac- that it was made to be adaptable to database

Hudnut8 34th Annual Small Satellite Conference ﬁles in any format. It was designed to use mulgated, such as the method put forward by the ﬁles output by Paul Hoddinott’s thesis code Jean-Pierre Marec in his doctoral thesis [12], [10], but so long as MATLAB can parse the ﬁle they are iterative methods which would signif- type, it can accept ﬁles formatted in any man- icantly increase the computational cost of this ner, simply by altering the ﬁle indices which algorithm. are passed to the ∆V algorithm by the wrap- per. So long as the code’s user is familiar with References the inputs required by the ∆V algorithm and [1] Australian Space Academy. the format of their input ﬁles, any input format (n.d.). Retrieved from can be used, making this program applicable http://www.spaceacademy.net.au/watch/ for any project which would need its capabili- track/orbspec.htm ties. [2] Baker, J., Canter T., Gollin B., Kirchner C., It is worth noting, however, that the algorithm Lyons J., Roman J., Shen J., (2016). OSCAR is not without ﬂaws. Speciﬁcally, the ∆V al- II Critical Design Report. Rensselaer Poly- gorithm only used the two-burn node-to-node technic Institute non-Hohmann transfer algorithm for its trans- fer ∆V estimation. While this is a good trans- [3] Battin, R. H. (1999). An Introduction to the fer solution for two elliptical orbits with signif- Mathematics and Methods of Astrodynamics icant diﬀerences in inclination, there are some (2nd ed.). Reston, VA: American Institute of cases where it is not optimal. One of these is Aeronautics and Astronautics. a near-coplanar case. If the spacecraft is in an orbit with very small variance in inclination, [4] Chobotov, V. A., Karrenberg, H. K., Chao, but large variance in altitude, from the orbit C., Miyamoto, J. Y., & Lang, T. J. (1991). Or- of a piece of debris, a Hohmann-like periapsis bital Mechanics (V. A. Chobotov, Ed.). Wash- or apoapsis transfer will likely have a lower ∆V ington, DC: American Inst. of Aeronautics cost than the node-to-node transfer. The other and Astronautics. major case is if either the initial or target orbit is in a highly eccentric orbit. This case will, [5] Der, G. J. (2011). The Superior however, generally not be a concern, as even Lambert Algorithm. Retrieved from an optimal transfer between such orbits will in- AMOS Technical Conference website: variably require more propellant than a small https://www.amostech.com/TechnicalPaper capture satellite such as OSCaR will carry, and s/2011/Poster/DER.pdf, on 5 June 2019 as such they are unlikely to be valid targets for such an operation. [6] Edfors, A. (2010). COE Visualizer Tool (Ver- sion 1.01) [Computer software]. Retrieved However, given that most of the debris with June 19, 2019. which the OSCaR spacecraft is concerned with are in low-eccentricity orbits within a well- [7] ESA Meteoroid and Space Debris Terres- constrained band of altitudes, and that the trial Environment Reference (2020) improvement of eﬃciency of a Hohmann-like transfer in the case of a coplanar orbit is small [8] Space Debris by the Numbers due to the low eccentricities of the expected tar- Retrieved from ESA website: get orbits, this issue is unlikely to cause sig- https://www.esa.int/Safety_Security/Space niﬁcant losses in ability to identify targetable _Debris/Space_debris_by_the_numbers, on debris, and was accepted in order to reduce 12 May 2020 computational time. Solving for a Hohmann- [9] Gooding, R. H. (1990). A Procedure for the like transfer would require multiple additional Solution of Lambert’s Orbital Boundary Value solutions, to check for transfers from the peri- Problem. Celestial Mechanics and Dynamical apsis and apoapsis of the initial orbit, and to Astronomy, 48, 145-165. Retrieved June 6, the periapsis and apoapsis of the target orbit, 2019. as well as any other points that might actually be optimal. While optimal solutions for trans- [10] Hoddinott, P. (2018). Tracking of Space De- fers between close elliptic orbits have been pro- bris from Publically Available Data. Thesis.

Hudnut9 34th Annual Small Satellite Conference Rensselaer Polytechnic Institute, Troy, NY. Lecture. Retrieved May 31, 2019, from http://control.asu.edu/Classes/MAE462/ [11] Leitmann, G., et al. (1962). Optimization 462Lecture09.pdf Techniques With Applications to Aerospace Systems (G. Leitmann, Ed.). London: Aca- [16] SAIC (n.d.). Two-Line Element Bulk demic Press. Database. Retrieved Summer 2019, from http://www.space-track.org. [12] Marec, J. P. (1967). Transferts Optimaux Entre Orbits Elliptiques Proches (Doctoral [17] Stahl, M., & Raghavan, M. (n.d.). SCAT- thesis, Faculty of Sciences of Paris, 1967) TERBAR3 (Version 1.0.0.1) [Computer (NASA, Trans.). Chatillon: Oﬃce National software]. Retrieved June 19, 2019, from D’Études et de Recherches Aérospatiale https://www.mathworks.com/matlabcentral /ﬁleexchange/1420-scatterbar3 [13] Oldenhuis, Rody, orcid.org/0000-0002- 3162-3660. "Lambert" v1.3.0.0, 2019- [18] Sun, F., & Vinh, N. X. (1983). Lambertian 06-05. MATLAB Robust solver for Lam- Invariance and Application to the Problem of bert’s orbital-boundary value problem. Optimal Fixed-Time Impulsive Orbital Trans- https://nl.mathworks.com/matlabcentral/ fer. Acta Astronautica, 10(5-6), 319-330. doi: ﬁleexchange/26348 10.1016/0094-5765(83)90083-8

[14] Peet, M. M. (n.d.). MAE 462 Lecture [19] Tethers Unlimited. http://tethers.com 8: Impulsive Orbital Maneuvers. Lec- ture. Retrieved May 31, 2019, from [20] Corbett, J. Micrometeoroids and Orbital http://control.asu.edu/Classes/MAE462/ Debris (MMOD). August 6, 2017, from 462Lecture08.pdf https://www.nasa.gov/centers/wstf/site _tour/remote_hypervelocity_test_laboratory [15] Peet, M. M. (n.d.). MAE 462 Lecture /micrometeoroid_and_orbital_debris.html 9: Bi-elliptics and Out-of-Plane Maneuvers.

Hudnut 10 34th Annual Small Satellite Conference