GTOC9: Methods and Results from the CaliforniaPolytechnic State University Team

Max Rosenberg∗, Paul Gilles, Adam Frye, Liam Smith

California Polytechnic State University, 1 Grand Avenue, San Luis Obispo, California 93407 (United States)

May 12, 2017

Abstract

The solution methodology created by studentsof California Polytechnic State University forthe 9th Global Trajectory Optimization Com-petition is presented. Pertinent details ofthe optimization program are explained, andthe program’s system architecture is presentedboth in its current form and in an improvedstate to guide future work.

1 Introduction

The 9th Global Trajectory OptimizationCompetition (GTOC 9) tasked entrants todesign a mission to place de-orbit packageson 123 simulated pieces of space debris.[2]A list of debris ephemerides and a missioncost calculator were provided to entrants,along with conditions for submitting a validsolution. We will discuss these conditions asthey apply to our submissions. A completeenumeration of the conditions, as well as

∗Corresponding author. E-mail:mcrosenb@calpoly.edu

an explanation of the nomenclature andabbreviations used throughout, can be foundin the GTOC 9 Prompt.The mission consists of simulated ’launches’that contribute some initial sunk cost to everysub-mission required to generate a solution,as well as sub-missions to place de-orbitpackages on each object. Each sub-mission’scontribution to the total cost is evaluated as afunction of the fuel required to guide a 2000kgdry-weight probe along a chosen trajectory,plus a number of 30kg de-orbit packages tobe placed on each object visited during anygiven sub-mission. Due to the large numberof debris objects, missions consist generallyof more than one sub-mission. The disparateorbits of the objects are illustrated visually inFigure 1.

As can be inferred from the figure, theorbits of the debris objects differ from oneanother significantly in two specific orbitalelements: Right-Ascension of the AscendingNode, Ω, and inclination, i. Transferringbetween orbits differing greatly in thesetwo elements requires maneuvers known

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Figure 1: One orbit of each debris object isshown beginning at MJD=20376.6438.

as Node-Line Shifts and Plane Changes.These maneuvers are known to be extremelyexpensive in terms of fuel, and thus i of anEarth-orbiting satellite is normally determinedduring launch. For the purposes of this com-petition, a crucial factor would be groupingdebris pieces based on the similarity of theseparameters associated with each piece.

2 Approach

The first conceptual step taken was to splitthe problem into two parts. The first part,trajectory optimization, would involve gener-ating optimal transfers between debris piecesduring a sub-mission. The degree of optimal-ity, or cost, of the trajectory optimization partwould be evaluated by the ∆V required to per-form each transfer. The second part, MissionOptimization, would ideally break the list of123 debris objects into smaller subsets of ob-jects, which could be de-orbited during 1 sub-mission.

The initial work focused on generating sub-missions through the program designed tosolve the first part of the problem. This pro-gram consisted of three parts:

• A master script, which sets the debrisobjects of interest to a particular sub-mission, as well as the starting Epoch

• An optimization function and script,which generates the optimal sub-missiontrajectory and tabulates the spacecraftstate vectors at every impulsive maneuver

• A cost function, which computes the ∆Vfor each transfer and state vectors at eachimpulsive maneuver, as output from analgorithm designed to solve Lambert’sproblem

The actual cost being optimized in this systemis the sub-mission cost, not the missioncost. Writing the mission cost optimizer wassomething that eluded our team due to timeconstraints. The system architecture of ourprogram is presented in Figure 2.

Figure 2: The flow of information through thesolution code.

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At the core of the optimization was DavidEagle’s algorithm to solve the J2-PerturbedLambert Problem.[1]. The use of a solutionto Lambert’s problem that included the J2perturbation was imperative to meet therelative distance tolerance required by thecontest rules. The use of a fast solver thatdid not include the perturbation resulted inrelative distances in excess of 5 km, while thetolerance was 100 m. This distance frustratedour efforts to validate our own solutions untilthe root of the discrepancy was discovered.The J2-Perturbed Lambert Solver was op-erated on by one of MATLAB’s built-inoptimization functions, FMINCON. Thisfunction searches for a local minimum of thecost function when subjected to constraints.We chose to constrain only the states ofthe spacecraft at arrival and departure to bethe same as the debris piece. Constrainingadditional parameters to reduce the numberof invalid solutions generated would havemade implementing a higher level optimizermuch easier, but would have slowed the runtimes of the code significantly, somethingwe could not afford in the final days of thecompetition. Consequently the final mass wasnot constrained, and sub-missions requiredinspection to ensure that they did not violatemass requirements. Likewise the pericenter ofthe transfer trajectories was not constrained,leading to a large number of solutions markedas invalid because of a close encounter withthe Earth. The resulting tool was lean, if notentirely robust. We attempted to generatea tool to implement some sort of geneticalgorithm to provide us with an optimalmission profile, as opposed to generating asmany optimal trajectories as we could. Wewere not successful in implementing this stepbefore the competition deadline. The task fell

to us to become the mission optimizers our-selves. As reflected in the system architectureshown in Figure 2, the Epoch and Debris IDsfor each sub-mission are user inputs to theprocess. What this meant for us was a franticscramble to generate valid solutions in thehours leading up to the competition deadline,essentially by hand-picking debris objects foreach sub-mission.

3 Next StepsAs mentioned previously, the system wedesigned to compute solutions to the problemwas not fully enclosed. Manual entry of thedebris ID and solution start Epoch were stillnecessary, precluding a truly global solution,or even anything close to that. As we workto improve our methods, a suitable next stepwould be to refine the system architecture asshown in Figure 3.

Figure 3: The desired flow of informationthrough the solution code. Note the elimina-tion of the manual step of choosing the Epochand Initial Object.

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As can be seen there are several differencesbetween the current iteration of our softwareand what we believe to be an effective meansof finding a global solution. The most salientof these differences can be seen to the upperleft of Figure 3. The block where the Epochand initial object are selected is changed froma ’manual input’ block to a ’process’ block.Under this new system, a choice is madebased on the validity and quality of a solutiongenerated from the main script and used torefine the initial conditions, rather than relyingon the heuristic understanding of the systemoperators. The sub-missions are eventuallycurated into a single stack of documents thatthen form the ultimate submission.This stage could take the form of a simpleloop, or become the cost function inside ofa new optimizer. Genetic algorithms areespecially suited to problems of this nature.In this case, the list of debris objects to bevisited on each sub-mission would constitute a’chromosome.’ We investigated implementingan algorithm of this nature for this purpose,however we were unsuccessful in reachinga valid solution via this method prior to thecompetition deadline.

4 ConclusionCal Poly is honored to have taken part inthis competition. The submission of severalsub-missions constituting valid solutions tothe GTOC 9 problem represents a significantmilestone for our team. While the originalcontribution made to the literature may besmall, we feel that we have achieved a strongfoothold of understanding how to workproblems like the one posed by GTOC 9.

This experience will allow us to refine ourapproach for future competitions, and willserve to grow the body of knowledge at ourinstitution. We see our participation in thiscompetition as a true manifestation of ourUniversity’s motto, Discere Faciendo.

AcknowledgementThe authors thank Dr. Kira Abercromby forher guidance and inspiration. Her uncompro-mising demand that each student reach theirmaximum potential and her tireless efforts tothat end motivated the authors to perseverewhen the problem seemed insurmountable,and she is ultimately responsible for usreaching our goal of finding a solution.

References[1] David Eagle. Orbital Mechanics with

MATLAB: Lambert’s Problem, July2014.

[2] Dario Izzo. Problem description for the9th Global Trajectory Optimisation Com-petition, May 2017.

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