A 25 Asteroid Solution to the GTOC7 Problem

Team #28

NASA Goddard Space Flight Center, The University of Illinois, a.i. solutions Inc.

Ryne Beeson∗, Abel Brown †, Bruce Conway∗, Victoria Coverstone∗, John Downing‡, Donald Ellision∗,

Jacob Englander‡, Alexander Ghosh∗, Tiffany Heyd †, Brandon Holladay †, Bindu Jagannatha∗,

Jeremy Petersen †, John Reagoso †, Daniel Skeberdis †, Matthew Vavrina†, Benjamin Villac †,

Geoffrey Wawrzyniak †

June 17, 2014

The solution method employed by the NASA Goddard — University of Illinois — a.i. solutions

team was centered around the Evolutionary Mission Trajectory Generator (EMTG) preliminary

mission design tool that is in use at NASA Goddard and is being jointly developed with the

University of Illinois at Urbana-Champaign. The first incarnation of EMTG was developed by Dr.

Jacob Englander as a doctoral student at the University of Illinois at Urbana-Champaign. The

EMTG software handles several mission types including high (impulsive) and low thrust varieties

as well as a wide range of boundary conditions and spacecraft/launch vehicle hardware models. The

medium-fidelity impulsive mission design feature employs the multiple gravity assist with one deep

space maneuver (MGADSM) model [1, 2]. Its counterpart low-thrust design feature uses the Sims-

Flanagan transcription based multiple gravity assist low-thrust (MGALT) model [3, 4, 5, 6, 7, 8, 9].

The first step in our solution method was a search space pruning of the GTOC7 asteroid list. A plot

of inclination (INC) vs. semimajor axis (SMA) was constructed that highlighted several groupings

of asteroids. Specifically, asteroids located in the 2.83 — 2.94 AU SMA band with an inclination

of less than 4 degrees were focused on. Many of the 834 asteroids in this set are members of the

Koronis family of main belt asteroids.

Next a minimum ∆v optimization was performed, sending the mothership to each of these

asteroids. The mothership was allowed to depart Earth at any point from MJD 59215 to 62867.5 and

rendezvous with its target asteroid, Figure 1. All three probes left from the starting asteroid, with

only one counting that asteroid towards its sequence score. For each of the arrival epoch/asteroid

ID pairs, a greedy tree search algorithm was performed using that epoch and asteroid as the first

node in the tree. This algorithm performed a minimum propellant usage MGALT optimization

from the starting epoch/asteroid to each of the asteroids in the 834 asteroid “Koronis” list. The

optimizer was allowed an initial wait period of up to two years after the mothership rendezvous with

the first asteroid before starting the search algorithm. The probe’s time of flight was constrained

∗The University of Illinois at Urbana-Champaign†a.i. solutions, Inc.‡NASA Goddard Space Flight Center

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to be 120 days. If feasible solutions were found, then the best-cost solution was chosen as the next

destination for probe 1. If a feasible solution was not identified, then the maximum allowed time

of flight was increased in 30 day increments up to a maximum of 480 days, until such a solution

was found. Probe 1 was then forced to remain at its destination asteroid for a minimum of 30 days

before the algorithm performed a subsequent tree search this time allowing for a wait time of up to

60 days. This process was repeated until probe 1 exhausted all of its propellant or a probe flight

time of six years was exceeded. The identical procedure was performed for probe 2, except that

all asteroids visited by probe 1 were removed from contention. Probe 3 was arbitrarily selected to

remain at the first asteroid for 30 days to count it towards its sequence score (probes 1 and 2 do

not count the mothership rendezvous asteroid in their sequences). This greedy tree search method

formed the basis for our final solution.

The top candidate sequence from the greedy tree search resulted in the three probes at different

asteroids, yet not far from each other or the mothership. To create a feasible final solution, the

final asteroids of each probe’s sequence were removed so that each probe could rendezvous with

the mothership, which was still at the starting asteroid. Plots of the thrust history of each probe

provided insight that additional asteroids could be visited during long coast arcs. To identify these

potential asteroids, a spherical filter on position was applied at the match-point constraint for the

Sims-Flanagan transcription. Insertion of an additional asteroid in probe 1’s sequence resulted in a

slightly infeasible solution at first, but was quickly corrected by performing another spherical filter

on position at an opportune asteroid.

To improve the final solution and exhaust the mothership capability, a final spherical filter

on position was performed on the final rendezvous location of the first probe returning to the

mothership. The final filtering identified a nearby asteroid that the mothership and probes could

rendezvous with, Figure 2. Importantly, probe 3 was able to rendezvous with the asteroid for 30

days and count it towards its total. The final probe trajectories and control histories are appended

in Figures 3, 4, 5, 6, 7 and 8.

After optimizing each probe trajectory using EMTG’s MGALT model, which is based on the

Sims-Flanagan low thrust transcription, the trajectories were re-optimized using EMTG’s medium-

high fidelity finite burn low-thrust (FBLT) mode [7]. The FBLT transcription is identical to Sims-

Flanagan except that each low-thrust step is continuously integrated using an rk8713M adaptive-

step integrator [10].

The team would like to acknowledge several organizations and individuals for support during this

project. Special thanks is given to the Navigation and Mission Design Branch at NASA Goddard,

Space Science Mission Operations at NASA Goddard, and the Illinois Space Grant Consortium.

The Illinois contingent would like to thank the exceptional guidance and generous allocation from

the Blue Waters sustained-petascale computing staff as well as the Illinois Campus Cluster Program

and the Computational Science and Engineering Program. This research is part of the Blue Waters

sustained-petascale computing project, which is supported by the National Science Foundation

(awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort

of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing

Applications. Lastly, the authors would like to extend a special thanks to their families for providing

support during the competition.

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Table 1: GTOC7 25 asteroid trajectory summary

Mother ship launch date MJD 59494.46Mother ship final mass at mission end 6485.22 kgLaunch v∞ 6.0 km/sProbe 1 release MJD 60617.50Probe 1 sequence ID’s 4990-2345-1401-15941-1883-3445-3918-10973Probe 1 re-acquisition MJD 62808.50Probe 2 release MJD 60415.92Probe 2 sequence ID’s 2660-4989-4396-7405-8456-4453-1048-2978Probe 2 re-acquisition MJD 62606.92Probe 3 release MJD 60077.0Probe 3 sequence ID’s 9664-9624-12542-6557-3411-3920-16038-12481-2096Probe 3 re-acquisition MJD 62268.0Primary Performance Index 25Secondary Performance Index 2586.036877213732 kg

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Figure 1: Mother-ship trajectory from Earth to Asteroid 9664

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Figure 2: Mother-ship trajectory from Asteroid 9664 to Asteroid 2096

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Figure 3: Probe 1 trajectory

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Figure 5: Probe 2 trajectory

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Figure 7: Probe 3 trajectory

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References

[1] T. Vinko and D. Izzo, “Global Optimisation Heuristics and Test Problems for PreliminarySpacecraft Trajectory Design,” Tech. Rep. GOHTPPSTD, European Space Agency, the Ad-vanced Concepts Team, 2008. Available on line at www.esa.int/act.

[2] J. Englander, B. Conway, and T. Williams, “Automated Mission Planning via EvolutionaryAlgorithms,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 6, 2012, pp. 1878–1887.

[3] J. A. Sims and S. N. Flanagan, “Preliminary Design of Low-Thrust Interplanetary Missions,”AAS/AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, August 1999.

[4] C. Yam, D. d. Lorenzo, and D. Izzo, “Low-Thrust Trajectory Design as a Constrained GlobalOptimization Problem,” Proceedings of the Institution of Mechanical Engineers, Part G: Jour-nal of Aerospace Engineering, Vol. 225, 2011, pp. 1243–1251.

[5] J. A. Englander, B. A. Conway, and T. Williams, “Automated Interplanetary Mission Plan-ning,” AAS/AIAA Astrodynamics Specialist Conference, Minneapolis, MN, August 2012.

[6] D. H. Ellison, J. A. Englander, and B. A. Conway, “Robust Global Optimzation of Low-Thrust,Multiple-Flyby Trajectories,” AAS/AIAA Astrodynamics Specialist Conference, Hilton Head,SC, August 2013.

[7] J. A. Englander, D. H. Ellison, and B. A. Conway, “Global Optimization of Low-Thrust,Multiple-Flyby Trajectories and Medium and Medium-High Fidelity,” AAS/AIAA Space FlightMechanics Meeting, Santa Fe, NM, January 26 - 30 2014.

[8] D. H. Ellison, J. A. Englander, M. T. Ozimek, and B. A. Conway, “Analytical Partial DerivativeCalculation of the Sims-Flanagan Transcription Match Point Constraints,” AAS/AIAA Space-Flight Mechanics Meeting, Santa Fe, NM, January 2014.

[9] J. A. Englander and A. C. Englander, “Tuning Monotonic Basin Hopping: Improving theEfficiency of Stochastic Search as Applied to Low-Thrust Trajectory Optimization,” 24th In-ternational Symposium on Space Flight Dynamics, Laurel, MD, May 2014.

[10] J. R. Dormand and P. J. Prince, “Higher Order Embedded Runge-Kutta Formulae,” Journalof Computational and Applied Mathematics, Vol. 7, March 1981, pp. 67–75.

Figure 9: Team mascot

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