gtoc5 solution file team 1

8/2/2019 GTOC5 Solution File Team 1

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5th

Global Trajectory Optimization Competition

Team 1 Solution File

TEAM MEMBERS

Dr. Juan S. Senent,Odyssey Space Research (Houston, Texas, USA)

Jacob Williams,ERC, Incorporated. Engineering and Science Contract Group (Houston, Texas,

USA)

Shaun M. Stewart,ERC, Incorporated. Engineering and Science Contract Group (Houston,Texas, USA)

Gregory P. Johnson,The University of Texas at Austin (Austin, Texas, USA)Dr. Cesar Ocampo,The University of Texas at Austin (Austin, Texas, USA)


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DESCRIPTION OF THE METHOD USED

Step 1: The Search Algorithm

A tree-based search algorithm based on a simplified version of the A* search algorithm is used to

find impulsive solutions to the problem. Solutions generated contain an Earth departure, andsegments representing asteroid rendezvous and intercepts. Given an initial epoch, the search

algorithm will attempt to rendezvous or intercept all candidate asteroids at all allowed times of

flight. From the obtained solutions, only those that closely follow a prescribed performance

profile for mass and time will be selected. This performance profile depends on the number of

asteroids targeted in the search. Additionally, solutions are only selected if the maneuvers are

separated in time by a padding based on the maneuver magnitude and rocket equation. Each

selected solution is then refined at different levels kof the sequence. This refinement process

consists of finding solutions that are identical at the k-1 level but vary on the klevel by either

visiting a different asteroid, or doing so with a different time of flight. If in the process of

refining a parent (k-1) solution, a better child (k) solution is found, then that child solution isincorporated into the search pool. Once a solution is processed, it is removed from the search

pool. The search is complete when the pool is empty.

The search algorithm was run for many epochs on computing clusters at NASA Johnson Space

Center and The University of Texas, Texas Advanced Computing Center (TACC). The best

impulsive solutions generated were then used as starting points for finite burn conversion and

optimization.

Step 2: The Finite Burn Conversion and Optimization

The Copernicus trajectory design and optimization tool was used to convert the impulsive

solution from the search process to a low-thrust finite burn solution. As shown in Figure 1, each

phase of the mission, from the ith

asteroid (or Earth) to the jth

asteroid, was divided into 6

segments, following a coast-thrust-coast-thrust-coast-thrust pattern. Propagation of the coast

segments was done using the Gooding Kepler propagation method, and finite burn segments

were integrated using the DLSODE numerical integration package. If the ith

asteroid is a

rendezvous, the first coast represents a waiting period/coast with the asteroid. The durations of

each segment are optimized. The first four segments are propagated forward from the ith

asteroid, and the last two segments are propagated backwards from the jth

asteroid. State, time,

and mass continuity constraints are imposed at the interface of the forward and backward

propagated segments. The positions at both asteroids are fixed, and if it is an intercept, the

velocity vector is free and constrained to be equal to the final velocity vector of the previous

phase. If the jth

asteroid is an intercept, an additional relative velocity inequality constraint is

imposed. The mass at both the ith

and jth

asteroids are free, and constrained so that the initial

mass of the ith phase is equal to the final mass of the (i-1)th phase.


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The finite burn control law used was the single-axis rotation (SAR) control law from Copernicus.

In this model, a thrust vector is rotated about a rotation axis. For this problem, the rotation axis

was fixed to be along the angular momentum vector of the trajectory (note that this axis moves

with the spacecraft and changes as the burn progresses). The initial thrust vector direction was

optimized (using the v direction from the impulsive search as an initial guess). The rate of

change of the thrust vector about the rotation axis was also an optimization variable. Thus, for

each finite burn segment, three optimized parameters determine the thrust pointing history for

that segment. The thrust magnitude was fixed at the maximum of 0.3 Newtons for all finite burn

arcs.

The optimization problem was solved (which included all mission phases) in Copernicus, using

SNOPT. The problem contained 579 optimization variables and 404 constraints. First, the

objective function was to maximize the final spacecraft mass, with the total time of flight

constrained. After this converged, the objective function was changed to minimize the total

mission time, with the final minimum mass constraint imposed. Finally, to drive the constraint

violations as low as possible, the problem was resolved with no objective function and very tight

tolerances on the remaining constraints.

Figure 1: Mission Phase from Asteroid I to Asteroid J.


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SUMMARY OF THE BEST SOLUTION FOUND

Included in this section is an overview of the solution, which departs from Earth on July 5, 2020,

continuing on to rendezvous and intercept 15 asteroids in 5394 days after the initial departure.

The trajectory is broken into phases which begin at the Earth or previous asteroid and terminate

after flyby or rendezvous with the subsequent asteroid. Each phase can include multiple finiteburn and coast arcs (note that some of the arcs can have zero duration due to the

optimization process, as explained in the previous section). The durations for each individual

finite burn arc are provided. The trajectory results in a final spacecraft mass of 500 kg.

A separately attached file contains the spacecraft position, velocity, mass, and thrust history of

the solution provided in one-day increments (except at the interface between phases, since the

phase durations are not integer numbers of days) . Although the solution ASCII file shows

all times moving forward, due to the fact that some segments are integrated backward, the initial

step also may not be a full day increment. The first state in the file is at the Earth just after

the departure V-infinity.

The thrust model contains a smoothly-varying thrust vector direction for each finite burn

segment. Thus, the thrust direction is not constant between the one-day data points in the

solution file.

Below are tables describing the details of the solution, as requested in the problem statement.


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GTOC5 Names of the visited asteroids and Visit Type:

1 (2007 UN12) [Rendezvous]

2 (2001 GP2) [Rendezvous]3 (2001 GP2) [Intercept]

4 (2007 UN12) [Intercept]

5 (1991 VG) [Rendezvous]6 (1991 VG) [Intercept]

7 (2008 EA9) [Rendezvous]

8 (2008 EA9) [Intercept]9 (2008 EL68) [Rendezvous]10 (2008 EL68) [Intercept]

11 (2006 UQ216) [Rendezvous]

12 (2006 UQ216) [Intercept]

13 (2009 BD) [Rendezvous]14 (2009 BD) [Intercept]

15 (2006 RH120) [Rendezvous]

16 (2006 RH120) [Intercept]17 (2008 UA202) [Rendezvous]

18 (2008 UA202) [Intercept]

19 (2008 UD95) [Rendezvous]

20 (2008 UD95) [Intercept]21 (2007 CS5) [Rendezvous]

22 (2007 CS5) [Intercept]

23 (2003 LN6) [Rendezvous]24 (2003 LN6) [Intercept]

25 (2006 FH36) [Rendezvous]

26 (2006 FH36) [Intercept]

27 (2010 LG61) [Rendezvous]28 (2010 LG61) [Intercept]

29 (2002 JR100) [Rendezvous]

30 (2002 JR100) [Intercept]

Launch Date (MJD):

MJD = 59035.15866036806

Launch V-infinity (km/s):

V-infinity vector at Earth on the launch date:

[0.2280480279606559, 1.812918737500153, 0.108445890003707] km/s


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Launch V-infinity Magnitude (km/s):

V-infinity magnitude at Earth departure: 1.8304209250592 km/s

Date and Spacecraft Mass at Each Asteroid Rendezvous or Flyby: (spacecraft mass reported

is the mass BEFORE(m-) and AFTER(m+) the drop off of either the science mass or the

penetrator mass):

1. MJD = 59284.968927. SC MASS (m-) = 3979.005924, SC MASS (m+) = 3939.0059242. MJD = 59572.452325. SC MASS (m-) = 3769.184201, SC MASS (m+) = 3729.1842013. MJD = 59708.908486. SC MASS (m-) = 3655.167703, SC MASS (m+) = 3654.1677034. MJD = 60017.748365. SC MASS (m-) = 3529.673747, SC MASS (m+) = 3528.6737475. MJD = 60267.759457. SC MASS (m-) = 3337.110684, SC MASS (m+) = 3297.1106846. MJD = 60373.573985. SC MASS (m-) = 3207.712165, SC MASS (m+) = 3206.7121657.

MJD = 60831.771859. SC MASS (m-) = 2945.439175, SC MASS (m+) = 2905.439175

8. MJD = 60928.047708. SC MASS (m-) = 2829.169364, SC MASS (m+) = 2828.1693649. MJD = 61306.381289. SC MASS (m-) = 2588.275495, SC MASS (m+) = 2548.27549510.MJD = 61395.546319. SC MASS (m-) = 2484.321266, SC MASS (m+) = 2483.32126611.MJD = 61744.626326. SC MASS (m-) = 2374.320668, SC MASS (m+) = 2334.32066812.MJD = 61856.927112. SC MASS (m-) = 2291.671436, SC MASS (m+) = 2290.67143613.MJD = 62237.997062. SC MASS (m-) = 2109.438232, SC MASS (m+) = 2069.43823214.MJD = 62316.298187. SC MASS (m-) = 2018.329010, SC MASS (m+) = 2017.32901015.MJD = 62476.785517. SC MASS (m-) = 1943.487995, SC MASS (m+) = 1903.48799516.MJD = 62567.401346. SC MASS (m-) = 1868.321625, SC MASS (m+) = 1867.32162517.MJD = 62730.063256. SC MASS (m-) = 1807.705784, SC MASS (m+) = 1767.70578418.MJD = 62799.489007. SC MASS (m-) = 1729.087966, SC MASS (m+) = 1728.08796619.MJD = 63113.393527. SC MASS (m-) = 1571.560829, SC MASS (m+) = 1531.56082920.MJD = 63181.306236. SC MASS (m-) = 1500.602597, SC MASS (m+) = 1499.60259721.MJD = 63486.943773. SC MASS (m-) = 1249.277961, SC MASS (m+) = 1209.27796122.MJD = 63531.237285. SC MASS (m-) = 1175.868862, SC MASS (m+) = 1174.86886223.MJD = 63708.648109. SC MASS (m-) = 1078.807702, SC MASS (m+) = 1038.80770224.MJD = 63759.286340. SC MASS (m-) = 1018.529103, SC MASS (m+) = 1017.52912025.MJD = 63981.536635. SC MASS (m-) = 958.052941, SC MASS (m+) = 918.05294126.MJD = 64026.560179. SC MASS (m-) = 899.233667, SC MASS (m+) = 898.23366727.MJD = 64227.152359. SC MASS (m-) = 744.938194, SC MASS (m+) = 704.93819428.

MJD = 64253.567910. SC MASS (m-) = 684.008486, SC MASS (m+) = 683.008486

29.MJD = 64410.486986. SC MASS (m-) = 556.802641, SC MASS (m+) = 516.80264130.MJD = 64429.321514. SC MASS (m-) = 501.000000, SC MASS (m+) = 500.000000


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Date and Spacecraft Mass at the Final Rendezvous as asked for in the Problem Description

(note that this is not the end of the mission) :

29. MJD = 64410.486986. SC MASS (m-) = 556.802641, SC MASS (m+) = 516.802641

Date and Spacecraft Mass of Final Asteroid Visit (FLYBY of 2002 JR100):

30. MJD = 64429.321514. SC MASS (m-) = 501.000000, SC MASS (m+) = 500.000000

Thrust Durations Per Phase (days):

Note that each phase has three thrust arcs (DT1, DT2, and DT3), but some thrust arcs may have

zero duration due to the optimization process, as explained above.

PHASE 1:

o DT1 = 0.000000o DT2 = 5.520256o DT3 = 18.308627

PHASE 2:

o DT1 = 16.414142o DT2 = 59.842510o DT3 = 116.495918

PHASE 3:

o DT1 = 21.962502o DT2 = 0.000000o DT3 = 62.048364

PHASE 4:o DT1 = 65.085443o DT2 = 76.174628o DT3 = 0.000000

PHASE 5:

o DT1 = 114.560651o DT2 = 74.169986o DT3 = 28.698983

PHASE 6:

o DT1 = 22.888293o DT2 = 8.646581o DT3 = 69.935032

PHASE 7:

o DT1 = 127.317973o DT2 = 47.072883o DT3 = 122.161547

PHASE 8:

o DT1 = 22.462619


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o DT2 = 3.196519o DT3 = 60.909305

PHASE 9:

o DT1 = 105.704703o DT2 = 41.905736o

DT3 = 124.676044

PHASE 10:

o DT1 = 20.326601o DT2 = 0.000000o DT3 = 51.868525

PHASE 11:

o DT1 = 79.484387o DT2 = 10.297473o DT3 = 33.936974

PHASE 12:

o DT1 = 9.544052o DT2 = 0.000000o DT3 = 38.864060

PHASE 13:

o DT1 = 25.992863o DT2 = 12.650456o DT3 = 167.061612

PHASE 14:

o DT1 = 15.449726o DT2 = 0.000000o DT3 = 42.560721

PHASE 15:

o DT1 = 65.407176o DT2 = 0.000000o DT3 = 18.404513

PHASE 16:

o DT1 = 7.502091o DT2 = 0.000000o DT3 = 32.412755

PHASE 17:

o DT1 = 14.741652o DT2 = 25.511044o DT3 = 27.413008

PHASE 18:o DT1 = 10.299928o DT2 = 0.000000o DT3 = 33.532414

PHASE 19:

o DT1 = 33.903103o DT2 = 47.977819o DT3 = 95.781907


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PHASE 20:

o DT1 = 6.852131o DT2 = 0.000000o DT3 = 28.286358

PHASE 21:

o DT1 = 36.103759o DT2 = 135.076640o DT3 = 112.945305

PHASE 22:

o DT1 = 10.211653o DT2 = 0.000000o DT3 = 27.708642

PHASE 23:

o DT1 = 88.115135o DT2 = 16.011315o DT3 = 4.905746

PHASE 24:o DT1 = 4.136477o DT2 = 0.000000o DT3 = 18.880320

PHASE 25:

o DT1 = 18.066068o DT2 = 24.318437o DT3 = 25.122681

PHASE 26:

o DT1 = 0.000000o DT2 = 2.865503o DT3 = 17.813220

PHASE 27:

o DT1 = 58.181467o DT2 = 54.156768o DT3 = 61.656561

PHASE 28:

o DT1 = 5.157659o DT2 = 1.280235o DT3 = 17.317931

PHASE 29:

o DT1 = 18.605514o DT2 = 28.563584o DT3 = 96.078188

PHASE 30:

o DT1 = 3.470171o DT2 = 1.665394o DT3 = 12.800890


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Total Flight Time Tau (Days):

FLIGHT TIME = 5394.16285338689 DAYS.

Value of the Performance Index J:

J = 15.0

Value of the Final Spacecraft Mass (kg):

SC FINAL MASS = 500.000 KG.


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VISUAL REPRESENTATION OF THE TRAJECTORY

Plots of the solution are shown on the following pages. In these plots, the cyan arcs are coast

arcs, and the red arcs are thrust arcs. The white arcs are the Earth and asteroid orbits. Segment

markers (S1, S2, etc.) indicate the start and end of thrust and coast arcs (there are a total of 180

segments in the mission, associated with 30 phases).


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Solution Plot 1: Overview


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Solution Plot 2: Oblique View


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Solution Plot 3: Initial and Final Points

Earth Departure

Final Intercept


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CONCLUSION

We would like to thank our colleagues at the NASA Johnson Space Center, Odyssey Space

Research, ERC Inc., and UT-TACC, including Jerry Condon, J.D. Frick, Ernie Wu and KennyFrame for all of their support and encouragement during this competition. We also thank the

GTOC5 authors: Ilia Grigoriev, Julia Plotnikova, Maxim Zapletin, Elena Zapletin of Lomonosov

Moscow State University, for proposing such a complex and fascinating problem. It has

consumed most of our free time over the past month.

gtoc5 solution file team 1

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