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.