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Low-Thrust Mission Trade Studieswith Parallel, Evolutionary Computing1,2

Seungwon Lee, Ryan P. Russell, Wolfgang Fink,Richard J. Terrile, Anastassios E. Petropoulos, and Paul von Allmen

Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109818-393-7720

Seungwon.Lee@jpl.nasa.gov

1 1 0-7803-9546-8/06/$20.00© 2006 IEEE

2 IEEEAC paper #1042, Version 5, Updated Oct, 25 2005

Abs t rac t—New mission concepts are increasinglyconsidering the use of ion propulsion for fuel-efficientnavigation in deep space. The development of new low-thrust mission concepts requires efficient methods to rapidlydetermine feasibility and thoroughly explore trade spaces.This paper presents parallel, evolutionary computingmethods to assess a trade-off between delivered payloadmass and required flight time. The developed methodsutilize a distributed computing environment in order tospeed up computation, and use evolutionary algorithms toapproximate optimal solutions. The methods are coupledwith the Primer Vector theory, where a thrust controlproblem is transformed into a co-state control problem andthe initial values of the co-state vector are optimized. Thedeveloped methods are applied to two mission scenarios: i)an orbit transfer around Earth and ii) a transfer between twodistant retrograde orbits around Europa. The solutions foundwith the present methods are comparable to those obtainedby other state-of-the-art trajectory optimizers. The requiredcomputational time can be up to several orders of magnitudeshorter than that of other optimizers thanks to the utilizationof the distributed computing environment, the significantreduction of the search space dimension with the PrimerVector theory, and the efficient and synergistic explorationof the remaining search space with evolutionary computing.

TABLE OF CONTENTS

1. INTRODUCTION ............................................ 12. METHODOLOGIES.......................................... 13. MISSION TRADE STUDY RESULTS..................... 44. PARALLEL COMPUTING PERFORMANCE........... 75. COMPARISON WITH CURRENT PRACTICES ........ 75. CONCLUSIONS .............................................. 8ACKNOWLEDGEMENTS ...................................... 8REFERENCES.................................................... 8BIOGRAPHY..................................................... 9

1. INTRODUCTION

The high fuel efficiency of low-thrust propulsiontechnologies enables new kinds of missions, since thedelivered payload mass can be increased and/or trip time

reduced compared to chemical propulsion systems. Whilebeing highly fuel efficient, the thrust provided by the low-thrust propulsion system is relatively small, typically onthe order of one Newton. As a result, any significantmaneuver of a spacecraft with the electric propulsion systemrequires continuous thrust over long periods of time. Thismakes the low-thrust trajectory optimization morechallenging than the chemical-propulsion spacecrafttrajectory optimization, where only a few impulsivemaneuvers need to be optimized.

The development of new low-thrust mission conceptsrequires methodologies to rapidly determine feasibility andthoroughly explore trade spaces. In particular, a broaderassessment of the feasible trade space during early missiondesign reduces the risk of proceeding with a point-designsolution that may be sub-optimal or difficult to implement.The broad assessments also mitigate the risk of overlookingimportant design trades solutions that could reduce cost andschedule risk in later mission design phases.

In a prior study, Lee et al demonstrated that parallel,evolutionary computing methods reliably and rapidly assessa trade-off between flight-time and propellant mass for low-thrust missions [1]. In the current study, the developedmethod is further applied to a different mission scenario,and the performance of the developed methodology isanalyzed in comparison with state-of-the-art tools andcurrent practices.

2. METHODOLOGIES

In the problem of finding optimal trajectories for low-thrustmissions, a common goal is to find the minimum-time,minimum-fuel, or Pareto-optimal trajectories, where Pareto-optimality means that no other solutions are superior tothem in terms of both flight time and fuel consumption.When maximizing the deliverable payload mass is anequally attractive mission objective as minimizing time offlight, the Pareto-optimal solutions that demonstrate thetrade-off between flight time and deliverable payload massare desired. In general, the Pareto optimization problem forlow-thrust missions is difficult to solve not only due to

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continuous thrust over long periods of time but also due tothe search for Pareto optimality.

Various methods have been used to solve this optimizationproblem. A majority of the work has utilized either direct orindirect methods [2]. The direct method approaches theproblem by adjusting the control variables iteratively toreduce/optimize a performance index such as flight time andpropellant mass [3,4]. The continuous control and statevariables are often discretized, which results in a nonlinearprogramming. The indirect method, on the other hand,makes use of the control law that arises when the low-thrustspacecraft problem is formulated using calculus of variations[5–8]. A co-state vector is introduced and the thrust historyis completely determined by the initial values of the co-states and the governing differential equations. The problemis transformed to a boundary value problem and is typicallysolved via a root-solving algorithm.

For the nonlinear programming problems in the directmethod and for the indirect method, local gradient-basedalgorithms such as Newton’s method and sequentialquadratic programming are popular because they are widelyavailable and proven very effective for many applications[9,10]. However, the traditional algorithms find locallyoptimal solutions typically in the vicinity of the initialguesses. Additionally, the algorithms are unstable when theobjective function landscape is rugged and the functiongradient is discontinuous. The trajectory optimizationproblem tends to have many locally optimal solutions,which make it difficult to find the globally optimalsolution. Furthermore, the traditional optimizationalgorithms do not directly handle the multi-objectiveproblem but convert it into a single scalar objective, the so-called weighting method [9,10]. The resulting solution ishighly sensitive to the weighting factors and is a singlesolution rather than a set of Pareto-optimal solutions.Therefore, the low-thrust orbit transfer problem calls for amore robust, global, and Pareto-optimal optimizationalgorithm.

This paper presents an innovative coupling between theindirect problem formulation of Primer Vector theory andevolutionary algorithms to solve the Pareto optimizationproblem arising in low-thrust mission trade studiesinvolving orbit transfers. The present method includesseveral innovative features. First, the present methodsconsist of two global-search algorithms: a genetic algorithmand simulated annealing [11–15] in contrast to a traditionalapproach where local-search algorithms are used. Neitheralgorithm requires the objective function gradient and bothare likely to find a globally optimal solution in a ruggedsearch domain, as opposed to gradient-based algorithms.Second, the genetic algorithm takes advantage of apopulation-based search to directly solve the multi-objectiveoptimization problem in a single run. Third, the simulatedannealing algorithm exploits a highly non-local ensemblesearch namely “shotgun” mode (described in a later section)to efficiently solve the problem. Finally, the present methodexploits parallel computing to speed up the optimizationprocess. Each component of the complementary methods isdescribed in detail below.

Primer Vector Theory

The Primer Vector theory introduces a co-state vector and“indirectly” optimizes the thrust control variables byadjusting the initial values of the co-state vector [6]. Thismethod uses the optimality conditions that arise fromcalculus of variations to transform the control vector into afunction of the co-states and states. Traditionally, thistransformation combined with the Euler-Lagrange boundaryconditions, or the so-called transversality conditions, leadsto a two-point boundary-value problem, whose solutioninherently satisfies the conditions for local optimality. Inthe current approach, the transversality conditions areignored, and the initial values of the co-states are iterated todirectly optimize the desired objective. This hybrid methodis termed indirect however, because the co-states are thecontrol parameters in lieu of a direct parameterization of thethrust vector history itself. The approach as stated is wellsuited for integration into any general constrainedoptimization framework, such as the current multi-objectiveevolutionary computing approaches. This approach isparticularly attractive because it avoids one of the problemstypically associated with indirect methods: i.e. gradientsand transversality conditions must be tediously re-derivedeach time an objective or constraint is changed. Thetransversality conditions are indirectly satisfied byoptimizing the initial co-states, and the gradients are notrequired for the proposed evolutionary optimizationmethods. Therefore, the problem can be tailored for customapplications with relative ease compared to traditionalmethods.

Global Optimization Methods

In order to solve the global optimization problems, whichappear in the nonlinear programming problem of the PrimerVector theory, the present method uses a genetic algorithmand simulated annealing. The genetic algorithm is inspiredby the natural selection and sexual reproduction process ofliving organisms [11–13], and the simulated annealingmimics the thermodynamic process of cooling moltenmetals [14,15]. Both methods have mechanisms to escapefrom local minima in order to find a globally optimalsolution. The global search mechanism is a reproductionoperator with a stochastic selection mechanism in the caseof the genetic algorithm and a mutation operator with theMetropolis algorithm in the case of the simulated annealing.

For the genetic algorithm, the following parameters aretypically used: population size of 1000, maximum numberof generations of 100, crossover probability of 0.8,mutation rate per gene of 1/N, where N is the number ofgenes, and elitist fraction of 30%.

For the simulated annealing, a “shotgun” approach isapplied to improve the computational efficiency. In thismode, we start with a random set of values for the co-stateparameters and expose them to a fixed (user-defined) numberof iterations while the “temperature” of the annealingprocess is oscillating. If, within the specified number ofiterations, the target orbit is reached with a user-defined

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accuracy based on the current set of parameters that prescribean optimal trajectory, then a "solution" is found and stored.Otherwise, the algorithm starts over with another set ofrandom initial values for the parameters to be optimized. Inthe results presented here, the number of simulatedannealing iterations used for the indirect methodoptimization was 100.

Pareto-Optimization Methods

For the Pareto-optimization, the genetic algorithm handlesdirectly multiple objectives with non-dominated sorting in asingle run [16–18]. The non-dominated sorting uses theconcepts of non-dominance and dominance to rank thepopulation composed of candidate solutions. Whencomparing two solutions, a solution is termed dominated ifthe solution is inferior to the other solution in allobjectives. Otherwise, the solution is termed non-dominated. Assuming that the optimization direction is tomaximize every objective, the conditions for non-dominanceand dominance between two solution (objective) vectors

(

€

a,b ) are mathematically given by

€

if ∀i ai < bi, then a is dominated by b.

else, a is not dominated by b.

The non-dominated sorting finds solutions that are non-dominated in comparison with the rest of the candidatesolutions in the population. The non-dominated solutionsconstitute a first Pareto-front and are assigned the bestfitness value. The sorting continues with the dominatedsolutions (i.e., the complement of the non-dominatedsolutions) by finding the next Pareto-front and assigning aslightly worse fitness value. Since the non-dominatedsorting does not involve a weighting process of aggregatingthe multi-objectives into a single scalar objective function, acareful, educated initial guess of the weighting factors is notneeded. The genetic algorithm accompanied by the non-dominated sorting can generate Pareto-optimal solutions ina single synergistic optimization run.

One of the desired outcomes of the Pareto optimization isthat the generated Pareto front in the solution space iswidely spread and uniformly spaced. In order to achieve thefeature, the optimization process must have a mechanism toexplore a new solution space rather than to exploit adiscovered solution space for an incremental improvement.Although the genetic algorithm has an explorative searchmechanism through reproduction operators such as crossoverand mutation, they are designed to increase the diversity inthe design space (parameter space) rather than the diversityin the solution space. The diversity in the design space andthat in the solution space are loosely related, but notnecessarily be maintainable by a one common operator.

To obtain a diverse Pareto front, we introduce diversity-encouraging mechanisms that are crowding-distancecomparison and fitness sharing. The crowding-distancecomparison is to rank a group in the same Pareto frontbased on crowding distance, which measures the distance

between the neighboring solutions. The ranking processassigns a higher fitness value to a solution with a highercrowding distance, because the trade space near the solutionis less occupied than that with a lower crowding distance.This mechanism is efficient when the population of thePareto front is large and their crowding distances are not tooinhomogeneous. However, t he crowding-distancecomparison fails to produce a wider and uniform Paretofront when the Pareto front is scanty and narrow due to thedifficulty of finding solutions and the lack of solution inearly generations. The difficulty is particularly severe in thelow-thrust mission optimization problem. The distressingproblem calls for a stronger strategy to encourage anexplorative search and diversity in population.

In order to overcome the shortcoming of the crowding-distance comparison, fitness-sharing mechanism is added tothe genetic algorithm [11]. The candidate solutions aredivided into several groups according to their flight times,and the fitness values of the solutions in the same group areshared in a way such that the sum of the fitness values inevery group is the same. Since this process is irrelevant tothe population size of the Pareto solutions, it performs welleven when the Pareto front is scanty and narrow. Also, thesharing is applied to the entire population regardless of theirPareto front. Therefore, the diversity of the population isencouraged even when a few solutions in a narrow range offlight time are substantially better than other solutions.

The concepts of non-dominated sorting and fitness sharingdo not apply to the simulated annealing approach used here.Instead, a traditional weighting method and the highly non-local ensemble search named shotgun mode is used to guidethe multi-objective optimization process.

Constraint Handling Methods

The low-thrust trajectory optimization problems involve notonly multiple objectives but also multiple constraints suchas the boundary condition for a spacecraft final state to meeta given target state. Typically, the constraints are treatedwith a penalty function as part of the fitness/energy function[9,10]. The penalty function approach requires a weightingprocess when combining the penalty function and theobjective function into a single scalar fitness function. Thisapproach is used for the simulated annealing application. The resulting fitness function is a linear combination of twoobjectives (flight time and propellant mass) and severalpenalty functions given by the constraint violation degree ofmeeting the target state in the trajectory problem.

A different approach, stochastic ranking, is used in thegenetic algorithm application [19]. The stochastic rankingmethod strikes a balance between the objectives and theconstraints in their contributions to the population rankingprocess by randomly choosing the ranking criterion betweenthe two. A user-defined parameter determines how probableit is to choose one criterion versus the other. The stochasticranking process is inserted inside a bubble-sort loop. In theloop, the individuals are ranked by comparing adjacentindividuals. The comparison criterion is randomly chosen

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between the objective-based one or the constraint-basedone.

Figure 1 shows the stochastic ranking procedure used torank individuals in a population with size N. U(0,1) is auniform random number generator, N is the population size,and P is the user-defined parameter that determines thelikelihood of the ranking criterion to be an objective basedone. The function f(Ij) is the objective-based fitness value ofthe individual Ij and the value c(Ij) is the constraint-basedfitness value of the individual Ij. The zero value of c(Ij)means that the individual solution does not violate anyconstraints. The result of this ranking process is to assign ahigher index (j) to the individual with a higher value of f(I)and a lower value of c(I) . When the value of P is zero, thisranking procedure is entirely based on the constraintfunction c(I), and for P=1, the ranking is entirely based onthe objective function f(I). A typical value of P used in thiswork is 0.45, which means that 45% of the comparison isbased on the objective function and 55% of the comparisonis based on the constraint function.

Figure 1. Stochastic ranking inside a bubble-sort-loop.

Parallel Computing Methods

The genetic algorithm uses a population-based search andthus is amenable to parallel computing [20]. When thefitness evaluation is one of the computationally moreexpensive parts of the overall optimization, parallelcomputing becomes an ideal choice to reduce thecomputation time. The fitness evaluation of the candidatesolutions in the population is distributed among severalprocessors in the distributed memory system. Theevaluation result is sent to the master processor on whichthe rest of the algorithmic process such as parent selection,offspring creation, and population replacement is executed.This type of parallel-computing implementation is calledmaster-slave parallel genetic algorithms [20]. The fitness-value passing is the only message passing between the

processors in the parallel genetic algorithm run. As a result,the computational overhead due to parallel computing ismarginal.

For the simulated annealing, an “embarrassingly” parallelapproach is taken for parallel computing. Since eachprocessor performs an independent optimization run withoutcross-communication (message passing) between processors,an almost perfect linear speed up is demonstrated. We havedemonstrated a 1002 CPU run on JPL’s institutional clustercomputer (Cosmos, 37th largest super computer in the worldto date). The analyses of the parallel computing performanceare presented in Section 4.

3. MISSION TRADE STUDY RESULTS

The present method is applied to two types of trajectoryproblems: A) two-body orbit transfer problem and B)restricted three-body orbit transfer problem. The optimiza-tion results are presented and compared with solutionsfound with other state-of-the-art optimizers in terms ofsolution optimality and computational efficiency.

Orbit Transfer around the Earth

As an example of two-body orbit transfer problems, a low-thrust orbit transfer around the Earth, is considered. TheEarth is the only gravitational body in this problem and isapproximated as a point mass. The optimization problem isto find Pareto-optimal solutions for the transfer from a low-eccentricity, small orbit to a high-eccentricity, coplanar,larger orbit. The initial orbit has a semimajor axis of9,222.7 km and an eccentricity of 0.2, and the final orbithas a semimajor axis of 30,000 km and an eccentricity of0.7. A relatively high thrust magnitude of 9.3 N is used forthis transfer problem. The specific impulse of the thrustengine is set to 3,100 s. The initial mass of the spacecraft is300 kg.

Figure 2 shows the Pareto-optimal solutions found with thepresent methods. The Pareto-optimal solutions are obtainedwith various error tolerances between the computed finalorbit and the given target orbit, ranging from 1 to 10percent. As the error tolerance decreases, the obtainedsolutions converge to “accurate” solutions, which requiremore flight time and propellant mass as shown in Figure 2.

The present solutions are compared with the solutions foundwith GA-Q-Law, which is an optimized heuristic controllaw based on a Lyapunov feedback control law named Q-lawand a genetic algorithm [21–24]. It has been demonstratedthat the GA-Q-Law finds nearly Pareto-optimal solutions ina reasonable computation time. The present methodefficiently yields solutions comparable to GA-Q-lawsolutions for a wide range of flight times. Note that GA-Q-law solutions (error<0.03%) have a much lower errortolerance than the present solutions. Figure 3 shows fourPareto-optimal trajectories selected among the Pareto-optimal solutions found with the present indirect method.As the flight time increases, several coast (no-thrust) arcs(green dashed lines) are inserted around the apoapsis.

For i=1 to N doFor j=1 to N-1 do

Sample u∈U(0,1)If(c(Ij)=c(Ij+1)=0) or u<P then

If(f(Ij) > f(Ij+1) thenSwap (Ij, Ij+1)

End ifElse

If(c(Ij)>c(Ij+1) then Swap(Ij, Ij+1) Endif Endif

End forIf no swap done break the i loop

End for

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Figure 2. Pareto-optimal solutions found with the present method incomparison with GA-Q-law solutions for the orbit transfer around theEarth. The error tolerances between the computed final orbit and thegiven target orbit range from 1 to 10 percent for the present method.

.

Figure 3. Pareto-optimal trajectories found with thepresent indirect method for the orbit transfer aroundthe Earth. The red solid and the green dashed linesrepresent thrusting and coasting arcs, respectively.

.

Figure 4. Pareto-optimal solutions found with the presentmethod in comparison with Mystic solutions for the distantretrograde orbit transfer around Europa. The error tolerances

between the computed final state and the given target state rangefrom 1 to 5 percent for the present method.

Figure 5. Pareto-optimal trajectories found with thepresent indirect method for the distant retrograde

orbit transfer around Europa. The red solid and thegreen dashed lines represent thrusting and coasting

arcs, respectively.

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The computation time used to obtain the Pareto-optimalsolutions is of the same order of magnitude for GA-Q-Lawand the present methods. Both methods utilize parallelcomputing. With Intel 3.2 GHz Xeon processors, GA-Q-Law used about 5 minutes of wall-clock time on 8processors, while the present methods used about 5 minuteson 16 processors for each of the 1%, 5%, and 10%-errorsolution sets.

Distant Retrograde Orbit (DRO) Transfer around Europa

As a restricted three-body orbit transfer problem, a DROtransfer around Europa, the icy Galilean moon closest toJupiter, is considered. The gravitational fields of Jupiter andEuropa are included (both being approximated as pointmasses), while the gravitational fields of the other moonsare excluded. The dynamics of the spacecraft are described inthe rotating frame where Europa is at the center, the x-axispoints along the Jupiter-Europa line, and the z-axis pointsalong Europa’s angular momentum vector with respect toJupiter. The initial DRO is given by the position vector(0.07518, 0) and the velocity vector (0, -0.14992) in the x-y plane with the unit length of 67,0988 km and the unittime of 48831.6 seconds. Similarly, the final DRO has theposition vector (0.03067, 0) and the velocity vector (0, -0.07274). The thruster is modeled with a specific impulseof 7,365 s and thrust magnitude of 4.984 N, and the initialspacecraft mass is taken as 25,000 kg, figures, which aretypical for a nuclear electric propulsion mission.

Figure 4 shows the Pareto-optimal solutions found with thedirect and indirect methods. The Pareto-optimal solutionsare obtained with error tolerances between the computedfinal state and the given target state, ranging from 1 to 5percent. As the error tolerance decreases, the obtainedsolutions converge to “accurate” solutions. The presentoptimization results are compared with the solutions foundby Mystic, a high-fidelity trajectory optimization softwarepackage based on the static/dynamic control algorithm[25,26]. The present method yields results that arecomparable to optimal Mystic solutions. Figure 5 showsthe variations of the trajectory and control profile for a fewselected Pareto-optimal solutions. As the flight timeincreases, several coast arcs (green dashed lines) are insertedaround the y-axis.

The computation time used for the Pareto-optimal solutionscompares as follows: With Intel 3.2 GHz Xeon processors,Mystic used about 360 minutes on one processor for each ofthe 0.1%- and 5%-error solution sets, while the presentindirect method used about 5 minutes of wall-clock time on16 processors for the genetic algorithm and 300 minutes on256 processors for the simulated annealing for each of the1%- and 5%-error solution sets.

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Figure 6. Load distribution among 8 processors in aparallel, genetic-algorithm optimization run.

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Figure 7. Computational time distribution betweenmessage-passing work (MPI functions) and the rest (otherfunctions) in a parallel genetic-algorithm optimization run.

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Figure 8. Computational speed up obtained with aparallel genetic algorithm run and a parallel simulated

annealing run.

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4. PARALLEL COMPUTING PERFORMANCE

The parallel-computing performance of the present methodsis analyzed in terms of load balance among processors,message-passing overhead, and computational speed up. Forthe parallel-computing implementation, the geneticalgorithm uses the master-slave architecture, while thesimulated annealing uses an “embarrassingly” parallelarchitecture. Therefore, load balance and massage-passingoverhead should be carefully supervised in the geneticalgorithm. On the other hand, they are irrelevant in thesimulated annealing, since there is no master processorplaying a different role among processors and there is nomassage-passing overhead.

Figure 6 shows the load distribution among 8 processors ina genetic algorithm optimization run. The master node(processor ID =1) has about 10% more load, and the slavenodes (processor ID >1) have uniform load. Since themaster node plays a different role in the master-slavearchitecture, the 10% overload in the master node isreasonable. The computational time distribution between themessage-passing-related work (MPI functions) and the restof the work is analyzed for 8 processors in Figure 7. Themaster node has a different time distribution than othernodes do. The master node spends about 95% of its time inMPI functions, while the slave nodes spend only about 8%

in MPI function and the rest of their time in fitness functionevaluation. These distributions are anticipated since themain role of the master node is to distribute the fitness-function evaluation work to the slave nodes, to collect theresults, and to proceed with ranking and reproduction for thenext generation/iteration.

Finally, the computational speed-up factor obtained with theparallel computing is plotted in Figure 8 for both a geneticalgorithm and simulated annealing optimization run. Thesimulated annealing run demonstrates an almost ideal speedup thanks to no message-passing between processors. Thegenetic algorithm run shows an 85x speed up with 128processors. The speed up is reasonable as there is a marginalmessage-passing between master and slave nodes. The speedup of the genetic algorithm run is directly determined by theratio of the fitness-evaluation time to the message-passingtime. As the ratio increases, the speed up becomes moreideal.

5. COMPARISON WITH CURRENT PRACTICES

The present methods are compared with the current practicesin terms of design capability, computational efficiency, anddomain expertise requirement. Table 1 highlights severalaspects of the direct comparison between the present methodand four current state-of-the-art low-thrust tools used at the

Table 1. Comparison between the present method and other state-of-the-art tools for low-thrust mission design.

Tool Present Method SEPTOP Mystic MALTO GA-Q-Law

Design Capability

Low-fidelitymission design

for planetocentricand restricted

three-bodytrajectories

Low-fidelitymission

design forheliocentrictrajectories

High-fidelitymission design

for gravity-assistheliocentric,

multibody, andplanetocentric

trajectories

Low-fidelitymission design

for gravity-assist

heliocentrictrajectories

Low-fidelitymission designfor planetocen-tric trajectories

OptimizationMethod

Evolutionaryalgorithms withPrimer Vector

theory

Local-searchalgorithms

with calculusof variations

Static/dynamiccontrol algorithm

Quadraticprogrammingwith direct

method

Heuristiccontrol law and

geneticalgorithm

Trade StudySingle

synergistic runSingle

parametric runMultiple

independent runsSingle

parametric runSingle

synergistic run

ParallelComputing

Yes No No No Yes

Domain ExpertiseRequirement

Reasonablebound of each

variable

Educatedinitial guessfor co-states

Educated initialguess for thrustprofile is useful

Educatedinitial guess

for thrustprofile

Reasonablebound of each

variable

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Jet Propulsion Laboratory: SEPTOP, Mystic, MALTO, andGA-Q-Law. These tools have proven successful in allstages of low-thrust mission design, from preliminaryconcept studies and proposals to flight projects such as theDeep Space 1 mission.

For trade studies, the present method and GA-Q-Law canconduct the study in a single synergistic run while othertools require either multiple independent runs or a single yetparametric run. Although the per-study time varies widelywith the type and extent of studies, it can be as long asseveral months for one broad trade study. By utilizingdistributed computing resources, which are abundantnowadays, the per-study time can be reduced to severalhours or even several minutes. Both the present method andGA-Q-Law capitalize on the parallel computing resources.

The challenge of using current low-thrust mission designtools often arises from the extensive requirement of domainexpertise. Current tools such as SEPTOP, Mystic, andMALTO require an initial guess to start their optimizationprocess. The total computational time of these tools greatlydepends on the quality of the initial guess [27]. Without aneducated guess, the computational time may increase by upto several orders of magnitude or may fail to convergeentirely. In contrast, the present method automaticallyconducts an optimization process using only minimaldomain expertise for a reasonable bound of each variable.Furthermore, the computational efficiency and solutionoptimality are less sensitive to the quality of the inputs inthe present method.

5. CONCLUSIONS

We have developed a robust and efficient method for low-thrust mission design trade studies by applying a geneticalgorithm and simulated annealing to the Primer Vectortheory. The present method uses the co-state vector and“indirectly” optimizes the thrust control variables byadjusting the initial values of the co-states. The investigatedmission scenarios demonstrate that this method finds nearlyPareto-optimal solutions with a high computationalefficiency and minimal guidance from domain expertise.The high computational efficiency is obtained by takingadvantage of the utilization of the distributed computingenvironment, the significant reduction of the search spacedimension by the Primer Vector theory, and the efficientand synergistic exploration of the reduced search space byevolutionary computing. Future applications of the presentmethod include the optimization of more complex andrealistic mission scenarios.

ACKNOWLEDGEMENTS

We thank Christoph Adami for useful discussions aboutperformance analysis for genetic algorithms, and we alsothank Robert Tisdale for his support on the parallelizationof the simulated annealing. This work was carried out at theJet Propulsion Laboratory, California Institute ofTechnology, under a contract with the National Aeronautics

and Space Administration. The research was supported byJPL’s Research and Technology Development program.

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[19] T. P. Runarsson and X. Yao, “Stochastic Ranking forConstrained Evolutionary Optimization,” IEEETransactions on Evolutionary Computation, Vol. 4, No.3, 2000, pp. 284-294.

[20] E. Cantu-Paz, Efficient and Accurate Parallel GeneticAlgorithms, Kluwer Academic Publishers,Boston/Dordrecht/London, 2000.

[21] S. Lee, P. von Allmen, W. Fink, A. E. Petropoulos,and R. J. Terrile, “Design and Optimization of Low-Thrust Orbit Transfers using the Q-law and EvolutionaryAlgorithms,” IEEE Aerospace Conference Proceedings,Big Sky, Montana, Mar. 7-11, 2005.

[22] S. Lee, P. von Allmen, W. Fink, A. E. Petropoulos,and R. J. Terrile, “Comparison of Multi-ObjectiveGenetic Algorithms in Optimizing Q-Law Low-ThrustOrbit Transfers,” Late-Breaking Paper, Genetic andEvolutionary Computation Conference, Washington DC.June 2005.

[23] S. Lee, A. E. Petropoulos, and P. von Allmen, “Low-Thrust Orbit Transfers Optimization with Refined Q-Lawand Multi-Objective Genetic Algorithms,” AAS 05-393Paper, AAS/AIAA Astrodynamics Specialist Conference,Lake Tahoe, CA, August 8-11, 2005.

[24] A. E. Petropoulos, “Low-Thrust Orbit Transfers UsingCandidate Lyapunov Functions with a Mechanism forCoasting,” AAS/AIAA Astrodynamics SpecialistConference, AAS Paper 04-5089, Providence, RhodeIsland, Aug. 16-19, 2004.

[25] G. J. Whiffen and J. A. Sims, “Application of a NovelOptimal Control Algorithm to Low-Thrust TrajectoryOptimization,” AAS/AIAA Space Flight MechanicsMeeting, AAS Paper 01-209, Santa Barbara, California,Feb. 11-15, 2001.

[26] G. J. Whiffen, “Optimal Low-Thrust Orbit Transfersaround a Rotating Non-Spherical Body,” AAS/AIAASpace Flight Mechanics Meeting, AAS Paper 04-264,Maui, Hawaii, Feb. 8-12, 2004.

[27] A. E. Petropoulos and S. Lee, “Optimization of Low-Thrust Orbit Transfers Using the Q-Law for the InitialGuess,” AAS/AIAA Astrodynamics SpecialistConference, AAS Paper 05-392, Lake Tahoe, Aug. 7-11,2005

BIOGRAPHY

Seungwon Lee is a member of thetechnical staff at the Jet PropulsionLaboratory of California Institute ofTechnology. Her research interestincludes evolutionary computingmethods, Astrodynamics, materials andnanostructure simulation, and parallelcluster computing. She received herB.S. and M.S in Physics from the SeoulNational University in 1995 and 1997,

and her Ph.D. in Physics from the Ohio State University in2002. Her work is documented in numerous journals andconference proceedings.

Ryan Russell is a member of thetechnical staff at the Jet PropulsionLaboratory. His research interestsinclude Astrodynamics, dynamicalsystems, and trajectory design andoptimization. He received a B.S. inAerospace Engineering from Texas A&MUniversity in 1999 and a M.S and Ph.D.in Aerospace Engineering from the

University of Texas in 2000 and 2004. His work isdocumented in numerous journals and conferenceproceedings.

Wolfgang Fink is a SeniorResearcher at NASA’s Jet PropulsionLaboratory, Pasadena, CA, VisitingResearch Associate Professor of bothOphthalmology and NeurologicalSurgery at the University of SouthernCalifornia, Los Angeles, CA, andVisiting Associate in Physics at theCalifornia Institute of Technology,Pasadena, CA. He is the founderand head of the Visual and

Autonomous Exploration Systems Research Lab at Caltech.His research interests include theoretical and applied

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physics, biomedicine, astrobiology, computational fieldgeology, and autonomous planetary and space exploration.Dr. Fink obtained an M.S. degree in Physics from theUniversity of Göttingen in 1993 and a Ph.D. in TheoreticalPhysics from the University of Tübingen in 1997. His workis documented in numerous publications and patents.

Richard J. Terrile created anddirects the Center for EvolutionaryComputation and Automated Designat NASA’s Jet Propulsion Labora-tory. His group has developedgenetic algorithm based tools toimprove on human design of spacesystems and has demonstrated thatcomputer aided design tools can also

be used for automated innovation and design of complexsystems. He is an astronomer, the Mars Sample ReturnStudy Scientist and the co-discoverer of the Beta Pictoriscircumstellar disk. Dr. Terrile has B.S. degrees in Physicsand Astronomy from the State University of New York atStony Brook and an M.S. and a Ph.D. in Planetary Sciencefrom the California Institute of Technology in 1978.

Anastassios E. Petropoulos is aSenior Member of the EngineeringStaff in the Outer Planet MissionAnalysis Group at the Jet PropulsionLaboratory. He is actively involved indeveloping algorithms for designinglow-thrust orbit transfers, and also asingle-body and multi-body low-thrustmission design. In 1991 he receivedhis B.S. in Aeronautical and

Astronautical Engineering from the Massachusetts Instituteof Technology, and in 1993 and 2001 he received his M.S.and Ph.D. degrees from Purdue University. Hisdissertation focuses on the problem of low-thrust, gravity-assist trajectory design.

Paul von Allmen is the supervisor ofthe High Capability Computing andModeling group and a principalresearcher at the Jet PropulsionLaboratory. Paul is currently leadingresearch in thermoelectric andnonlinear optics material and devicedesign, nano-scale chemical sensors,and semiconductor optical detectors.

Prior to joining JPL in 2002, Paul worked at the IBMZurich Research Lab on semiconductor diode lasers, at theUniversity of Illinois on the silicon device sinteringprocess, and at the Motorola Flat Panel Display Divisionas manager of the theory and simulation group and asprincipal scientist on micro-scale gas discharge UV lasers.Paul received his Ph.D. in physics from the Swiss FederalInstitute of Technology in Lausanne, Switzerland in 1990.His work is documented in numerous publications andpatents.

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