GTOC 9: Results from the Astrodynamics ResearchGroup of Penn State

Davide Conte*, Andrew Goodyear, Jason Reiter, Ghanghoon Paik,Guanwei He, Mollik Nayyar, Matthew Shaw, Jeffrey Small, and Jason Everett

Astrodynamics Research Group of Penn State, The Pennsylvania State University,

229 Hammond Building, University Park, PA 16802

May 8, 2017

Abstract

Presented in this paper are methods and resultsof the Astrodynamics Research Group of PennState (ARGoPS) for the 9th Global TrajectoryOptimization Competition [2]. The optimiza-tion strategy utilized a beam search methodto determine the optimal sequence of missionsand transfers between debris, including the or-der in which to visit each group, the timingof each visit throughout the mission windowin order to minimize the number of missions,and the total cost per transfer. Particle SwarmOptimization (PSO) was applied to the opti-mized ordering of debris visits in order to de-termine the departure date and time-of-flightcombination which best minimize the fuel re-quired for each transfer. This method led to asolution that collected all 123 pieces of debrisfrom their Sun-sychronous orbits in 20 totalmissions.

*Corresponding author. E-mail: da-vide.conte90@gmail.com

1 Introduction

The 9th edition of the Global Trajectory Op-timization Competition (GTOC 9) was orga-nized by The European Space Agencys Ad-vanced Concepts Team in the spring of 2017.In this paper, the optimal strategy developedby team ARGoPs (Astrodynamics ResearchGroup of Penn State University) is described.The problem presented for GTOC 9, namedthe ”Kessler Run”, suggests a spacecraft mis-sion design in which a set of 123 pieces of de-bris are removed from orbit in order to preventthe ”Kessler effect”, a scenario in which theexplosion of a satellite leads to cascading col-lisions of resident space objects. The detailedproblem description and equations can be readin the problem description of the 9th GTOC,written by Dr. Dario Izzo [2]. Presented hereis the approach and all formal details on thespace debris removal mission design that wascreated during the month of April, 2017 byteam ARGoPs.

The solution was developed by writing and

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running an optimization procedure in bothMATLAB and the C++ programming lan-guages. The procedure consists of three steps:Phase I (Raiden), Phase II (which is com-prised of both The Butcher (of Groznyj Grad)and Sniper Wolf ), and Phase III. Phase I ofthe program determines the order in whichthe individual debris pieces are visited andtakes the first steps to narrow the search spacefor each transfer by using what is referred toas a beam search clustering method, as ex-plained in Section 2. Phase II of the pro-gram takes the transfer sequence output fromPhase I and further refines the search spacefor individual transfers via a function referredto as The Butcher (of Groznyj Grad), or sim-ply The Butcher, which uses a Lambert solverwithin a Particle Swarm Optimization schemeto search for useful solutions within the timebounds determined by Raiden. Once TheButcher refines each search space even fur-ther, Sniper Wolf takes over and implementsanother search for transfer trajectories usingParticle Swarm Optimization to find and cal-culate the transfers between debris objects us-ing the full J2-affected dynamics to ensure thatthe transfer will converge to a valid and sub-optimal solution (see Section 3 for further de-tails on The Butcher and Sniper Wolf ). Theresults from Sniper Wolf are checked for massfeasibility before any final solution is deemedappropriate. Phase III is then initiated, wherethe verified results are saved in the correct out-put format necessary for upload to the onlinesubmission system. The program structure issummarized in Figure 1.

Each function used is shown in Figure 1.as an individual block. The red colored func-tion blocks indicate lower-fidelity solutions tothe problem, whereas the blue colored func-tion block is the higher-fidelity solution. The

Figure 1: Program Flowchart.

lavender colored function block represents afinal mass check and then the green coloredfunction block indicates the final submissionfile creator function. The search space is pro-gressively narrowed as more fidelity is deter-mined for the solution. A visual representationof this search space is captured in Figure 2.

2 The Beam Search ClusteringMethod

Beam search is a heuristic algorithm thatbuilds search trees with a restricted number ofstates at each level, referred to as beam width.By limiting beam width, beam search can min-imize complexity and memory usage, whichis beneficial for problems with a large searchspace. A basic understanding of this algorithmis shown in Figure 3.

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Figure 2: A visual representation of the solu-tion search space for transfers between debris.

A fast-paced beam search method capableof dividing the given timeline into a sequenceof missions was used. This method, referredto as Raiden, was implemented to definethe sequence by linking together debrisbased on the right ascension of ascendingnode (RAAN) and inclination of each orbit.Each mission was constructed such that noindividual transfer exceeded a maximumplane change magnitude while attempting todistribute the required propellant mass evenlybetween missions. Raiden is structured tocontine to link pieces of debris together untileither a maximum amount of objects permission is achieved, or the estimated propel-lant usage exceeds the allowable estimatedpropellant usage per mission. The departureand arrival epochs in between each piece ofdebris were chosen such that the transfer isperformed when the two debris pieces arenear their closest approach and the transfercould be completed with plenty of time tospare given the required plane change.

Figure 3: Standard beam search algorithm. [1]

The beam search algorithm was appliedon both the debris-to-debris scope, andthe mission-to-mission scope. The startingepoch of each mission was probed randomlythroughout the allowable mission timeline fora set amount of iterations and, at each probedstarting epoch, Raiden was used to find thelocally next best missions and store them in acustom beam search map structure designedto branch from in future iterations. Afterthe stopping criteria is met that terminatesthe debris-level beam search algorithm, themission-level algorithm branches from alllocally optimal missions to find the next bestset of missions. Each branch of the mission-level beam search algorithm is partnered witha custom time cell structure that containsinformation about available time intervalsfor future missions based on the previousmissions in that specific branch. The onlystopping criteria of the mission-level beamsearch algorithm is the remaining number ofdebris available to be linked in a mission,based on the locations of the debris through-

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out the allowable time intervals. The variousmission sequences that were developed usingRaiden were named as follows (in order ofincreasing optimality, and amount of debriscaptured): The Pain, The Fear, The End, andThe Fury. The Fury is the mission sequencethat was ultimately chosen.

3 Trajectory OptimizationIn this section, the trajectory optimizationmethods used to compute the individualorbital transfers between debris pieces, re-ferred to as The Butcher and Sniper Wolf,are presented. A Keplerian approximation(solution to Lambert’s problem) was usedwith a particle swarm algorithm to narrowthe search space and give Sniper Wolf betterinitial guesses (The Butcher). The particleswarm algorithm is then given the deriveddepartures and time-of-flights from the Kep-lerian approximation as intial guesses to findthe optimal time of flight and correspondingtransfer velocity in the J2 dynamic model(Sniper Wolf ).

Given the computationally expensiveprocess involved in optimizing trajectorieswhere J2 perturbations are considered, itwas necessary to find a way to decrease thesearch space by utilizing a lower-fidelityapproximation to each orbital transfer. Infact, this lowers computational time whilegiving the high-fidelity optimizer a betterinitial guess used to eventually minimizepropellant consumption. It was determinedthat the solution to the Keplerian Lambert’sproblem was a ”good enough” estimate to bepassed on to the optimizer that would take

into account J2 effects. Oblateness effectsthe spacecraft position drift over time fromthe two-body problem model (see Figure 4).In order to account for this perturbation, adeviation in the initial spacecraft velocity isneeded. Thus, an accurate search space forthe perturbed Lambert’s problem solution wasneeded in order to ensure that the spacecraftwould rendezvous with the debris and thatthe propellant needed to accomplish such ma-neuvers would be minimized. After multipletests, the average transfer time between debrispieces corresponding to the optimal solutionswas found to be less than half of a day, thusvalidating the fact that the Keplerian solutioncan successfully narrow down the searchspace for the majority of transfers (except forthose transfers requiring much longer time offlight). In order to determine such a searchspace, a Particle Swarm Optimization (PSO)technique was applied. [4] Given the desireddebris ID numbers and the window availablefor the transfer as determined by the beamsearch algorithm, the available departureepoch in the transfer window and the transfertime (based on the remaining time available tocomplete the transfer) were used as particlesin the optimization. The subsequent cost foreach particle at each iteration was found byusing the Keplerian Lambert’s solution todetermine the ∆v cost of the transfer. Thisoptimization method resulted in a singledeparture epoch and transfer time that wouldminimize the ∆v cost for the transfer underpurely Keplerian dynamics. Bounds werethen applied to these numbers (based on theexpected convergence under J2 oblatenessdynamics) in order to provide a search regionfor Sniper Wolf. This algorithm was namedThe Butcher due to the fact that it is supposedto return rough estimates that needed to be

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refined. Additionally, running this optimizerfor all possible ranges of departure and arrivaldates would result in porkchop plots, whichare contour plots of ∆v on departure vs.arrival date axes.

Figure 4: Drift vector effects due to J2 dynam-ics. [3]

In order to find the exact necessary ∆v andtime-of-flight between two given debris for arange of departure and arrival dates (account-ing for J2 effects), a multi-objective PSO wasimplemented. This algorithm, referred to asSniper Wolf, i.e. the precise, patient and (al-most) infallible J2-perturbed Lambert solver,is tasked with finding valid transfer solutionsthat satisfy the relative position and velocityconstraints for rendezvous (100 meters and 1m/s, respectively) while minimizing the total∆v needed to accomplish such maneuver. Thebasic pseudocode is shown in Figure 5. Notethat only two-burn maneuvers were consid-ered to keep the optimization tradespace di-mensions (and thus computational time) to aminimum. In order to increase the robust-ness of Sniper Wolf, the optimizer would rununtil a valid (δr < δrTOL) and/or optimal

(∆v < ∆vestimated) solution was found. Theinputs for Sniper Wolf are directly taken fromthe outputs of The Butcher as described aboveand are:

1. IDs of the departure and arrival debrispieces.

2. Departure and arrival time ranges.

3. Estimated ∆v and orbit type (long vs.short way solution, etc.).

The outputs of Sniper Wolf are:

1. Optimal ∆v1 and ∆v2 needed to initiateand complete the rendezvous maneuver.

2. Departure time and TOF correspondingto the optimal transfer found.

3. A flag corresponding to whether the op-timizer was able to find a valid solutionwithin the prescribed number of itera-tions.

4 Results

After running Raiden, the top-level optimizer,multiple times using the beam search methoddescribed in Section 2, the most optimal solu-tion that was found, The Fury, consisted of 20missions with a total of 103 transfers betweendebris pieces. The timeline of the mission isshown in Figure 6, where each segment corre-sponds to each individual mission (as definedby the number above it) launched as part ofThe Fury. The first mission in chronologicalorder, Mission 20, starts on MDJ 23476.38.A summary of all of the missions and trans-fers ∆v and propellant mass is given in Table1. The average required ∆v and transfer time

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Sniper Wolf - Pseudocode(P )00 while δr > δrTOL && ∆v > ∆vestimated

01 PSO particles are randomly initialized using departure time, TOF,and 3-component velocity vector deviations as particle elements

02 for 1 : 1 : iterations03 for 1 : 1 : particles04 r1, r2 ← Propagate starting and arrival debris pieces05 ∆v1, ∆v2← Solve Keplerian Labert’s problem using r1, r206 if Periapse constraint is not met07 J = 107 // Particle receives a large penalty

08 else09 Integrate s/c trajectory using the solution to Lambert’s

problem with the particle’s velocity vector deviations10 δr← Compute drift vector magnitude11 J = 0.001∗ [10∗ (δr−δrTOL)∗u(δr−δrTOL)+ ...

∆v]← Compute cost12 end13 end // end of particle computations

14 Determine the best particle based on J15 Update ”velocity” and ”position” vectors for each particle16 if J stagnates17 Reset half of the particles which have the highest J18 end19 end // end of PSO iterations

20 PBest, Jbest← Find best solution among the particles20 if new Jbest > old Jbest21 Jbest = new Jbest and Solution = Pbest

22 end23 end // end of the while loop

24 Return the best Solution

Figure 5: Sniper Wolf basic pseudocode.

values per transfer for each mission are shownin Figure 7.

The intention in the mission planning wasto attempt to gather as many debris piecesas possible while keeping fuel requirementswithin the mass restrictions defined by theproblem. Figure 7 shows that this objectivewas achieved with success for larger missions,such as Missions 1 through 8. These largermissions maintained higher efficacy in usingthe fuel that was allocated for their duration.However, as large missions are created by sort-ing debris pieces into similar orbital planes,the likelihood of being able to group the restof the debris similarly decreases. There is a

Figure 6: The Fury - Mission timeline.

trend that appears, where an increase in the ra-tio of fuel cost per debris captured manifests as

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Figure 7: The Fury averaged results.

the number of debris available decreases to asmall number (such as a single transfer). Somemissions consisted of lower quantities of de-bris removed with larger fuel masses, such asMissions 17 through 20. Although this is anundesirable effect, it is inherently a part of theproblem.

In order to remove larger quantities of de-bris in less missions, the conditions that de-termine how similar orbital planes are classi-fied for groups of debris needed to be relaxed.The relaxation of these conditions can result incostly transfers that are high in fuel consump-tion. The higher fuel consumption at timeswould fall outside of the bounds of the masscapability set forth by the problem statement.The alternative approach was to take a largernumber of missions in order to be capable oftransferring between all debris pieces withoutexceeding the mass restrictions that govern theproblem. The required mass ended up beinglower than the maximum carrying capabilityavailable at times, but this was useful in low-ering the cost function for specific missions di-rectly.

Generally, the missions consisting of longer

Table 1: Summary of all of the mission of thesequence The Fury.

The Fury - Mission SummaryMissionNumber

Numberof Trans-fers

∆v[m/s]

PropellantMass[kg]

1 11 3978.62 4951.582 10 3929.44 4881.753 9 3316.76 3599.014 10 3923.04 4764.825 8 3408.66 3776.526 7 3984.92 4841.947 6 2258.99 2027.758 6 2765.71 2685.579 6 2313.49 2102.7610 5 2785.08 2715.9811 5 2508.40 2339.2412 4 2506.96 2308.1613 4 2787.16 2701.6014 3 1467.72 1140.7215 3 1251.12 941.4216 2 1384.10 1049.3117 1 2373.11 2106.1718 1 1479.99 1134.2319 1 2861.90 2759.2220 1 2479.13 2239.80

average times of flight between objects alsorequired less fuel, as seen in Figure 7. How-ever, outliers like Mission 3 exist that do notfit that pattern. This likely occured becausenot only are variation in semimajor axis andeccentricity not accounted for when determin-ing the mission sequence, but the optimizationmethods used are not guaranteed to find globalminima.

Future improvement to this method of so-lution could consist of isolating the high fuelsingle transfer missions in order to find a way

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to group them with nearby missions that fallwithin similar time-frames. Another aspectof the solution that could be analyzed fur-ther is the number of burns per transfer. Theability to increase the number of burns be-yond two could yield more favorable fuel con-sumption requirements for most transfers, es-pecially those with larger plane change mag-nitudes.

5 Discussion and Conclusions

In this paper the methods and results of the As-trodynamics Research Group of Penn State inthe 9th Global Trajectory Optimization Com-petition were described [2]. An optimiza-tion strategy using a beam search clusteringmethod (Raiden) was used to group orbitaldebris with similar orbital planes and RAANto determine the order and timing of eachvisit. A Keplerian Lambert’s problem solution(The Butcher) was applied to narrow downthe search space for a high-fidelity optimizerusing Particle Swarm Optimization (SniperWolf ), which determined the best departuredate and time of flight for each visit. Whilethe method did result in four one-transfer mis-sions with high fuel costs, all 123 objects werecaptured in 20 missions, giving ARGoPS a fi-nal score of 1512.60176793564.

Further improvements require more com-putational power to find a better mission se-quence and trajectories.

References[1] H. Akeb, M. Hifi, and R. MHallah. A

beam search algorithm for the circularpacking problem. Computers & Opera-tions Research, 2009.

[2] Dario Izzo. Problem description for the9th Global Trajectory Optimisation Com-petition.

[3] H. Ma, S. Xu, and Y. Liang. Global op-timization of fuel consumption in j2 ren-dezvous using interval analysis. Advancesin Space Research, 59:1577–1598, 2017.

[4] M. Pontani and B. Conway. Particleswarm optimization applied to space tra-jectories. Journal of Guidance, Control,and Dynamics, 33(5):1429–1441, 2010.

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