research article asteroid rendezvous mission...

Research ArticleAsteroid Rendezvous Mission Design UsingMultiobjective Particle Swarm Optimization

Ya-zhong Luo1 and Li-ni Zhou2

1 College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China2 Center for National Security and Strategy Studies National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Ya-zhong Luo luoyznudteducn

Received 18 October 2013 Revised 26 January 2014 Accepted 13 February 2014 Published 31 March 2014

Academic Editor Kui Fu Chen

Copyright copy 2014 Y-z Luo and L-n Zhou This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new preliminary trajectory design method for asteroid rendezvous mission using multiobjective optimization techniques isproposedThis method can overcome the disadvantages of the widely employed Pork-ChopmethodThemultiobjective integratedlaunch window and multi-impulse transfer trajectory design model is formulated which employes minimum-fuel cost andminimum-time transfer as two objective functions The multiobjective particle swarm optimization (MOPSO) is employed tolocate the Pareto solution The optimization results of two different asteroid mission designs show that the proposed approachcan effectively and efficiently demonstrate the relations among the mission characteristic parameters such as launch time transfertime propellant cost and number of maneuvers which will provide very useful reference for practical asteroid mission designCompared with the PCP method the proposed approach is demonstrated to be able to provide much more easily used resultsobtain better propellant-optimal solutions and have much better efficiency The MOPSO shows a very competitive performancewith respect to the NSGA-II and the SPEA-II besides a proposed boundary constraint optimization strategy is testified to be ableto improve its performance

1 Introduction

The optimization of interplanetary trajectories to an asteroidcontinues to arouse a great deal of interest [1ndash4] Althoughthe low-thrust propulsion is employed in asteroid rendezvousmissions an impulsive trajectory is always assumed forpreliminary mission design and optimization in whichthe access feasibility is evaluated the launch window isdetermined the gravity-assist maneuvers scheme is plannedand so forth In a preliminary asteroid rendezvous trajec-tory design the Pork-Chop method is widely used Thismethod employs the two-impulse algorithms including theoptimal two-impulse noncoplanar transfer [1 2] the clas-sical Hohmann transfer [3] and the two-impulse Lambertalgorithm [4] to design the transfer trajectory The total ΔVcorresponding to different departure time and arrival timeis then obtained and the contours of minimum total ΔV areplotted to assist the designer to find the best launch windowand transfer trajectory

This type of method is very intuitionistic and easilyexecuted However only the two-impulse trajectory is inves-tigated in this method and as demonstrated by Lawdenrsquostheory [5 6] the two-impulse trajectory is not the propellant-optimal solution undermost conditions Besides thismethodis in essence of an exhaustive searching method its com-putation cost increases exponentially as the search spaceincreases and much human intervention is required as noorderliness exists in most of the contours

In this paper a new preliminary trajectory designmethodfor asteroid rendezvous mission using multiobjective opti-mization techniques is proposed This method can overcomethe disadvantages of the Pork-Chop method The multiob-jective integrated launch window and multi-impulse trans-fer trajectory design model is formulated which employsminimum-fuel cost and minimum-time transfer as twoobjective functions In this model the Earth departure datehyperbolic velocity and the interplanetary transfer impulses

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 823659 13 pageshttpdxdoiorg1011552014823659

2 Mathematical Problems in Engineering

are all chosen optimization design variables The multiobjec-tive particle swarm algorithm is employed to locate the Paretosolution bywhich the relationships characteristics among theoverall mission parameters can be effectively revealed

The particle swarm optimization (PSO) algorithm firstlyintroduced by Kennedy and Eberhart [7] in 1995 has beensuccessfully applied in many fields of research However itis only recently applied to aerospace trajectories optimiza-tion [8ndash11] The single-objective PSO algorithm has beentestified as one successful spacecraft trajectory optimizerin designing impulsive interplanetary trajectories [8] low-thrust trajectories for asteroid exploration [9] Lyapunov andHalo (periodic) orbits [10] and multiple-burn rendezvoustrajectories [11]

The implementation of the PSO algorithm adopts a pop-ulation of particles whose behavior is affected by either thebest local (ie within a certain neighborhood) or the bestglobal individual The relative simplicity of PSO and the factthat it is a population-based technique have made it a naturalcandidate to be extended for multiobjective optimizationIn a survey paper in 2006 on multiobjective particle swarmoptimization (MOPSO) [12] it was reported that there werecurrently over twenty-five different proposals of MOPSOreported in the specialized literatureThe studies onMOPSOremain a very active area of research and the MOPSO hasbeen successfully applied to many practical multiobjectiveoptimization problems recently applied to robotics [13]industrial management [14] and chemical engineering [15]It also has been applied in the domain of aerospace includingairfoil shape optimization complex physicsshape optimiza-tion and multidisplinary design optimization [16] Howeverit was seldom applied in solving multiobjective spacecrafttrajectory optimization problems The recent studies showthat the single-objective PSO algorithm is one effectivereliable and accurate spacecraft trajectory optimizer [8ndash11]In this study we will show that the MOPSO could be onesuccessful optimizer for multiobjective spacecraft trajectoryoptimization problems

In summary the main contribution of this paper is two-fold (1) A novel asteroid rendezvous mission design methodusing the multiobjective techniques is proposed Comparedwith the current widely employed Pork-Chop method theproposed approach is demonstrated to be able to providemuch more easily used results obtain better propellant-optimal solutions and have much better efficiency (2) Asfar as we know it is the first time to apply the MOPSO tospacecraft trajectory optimization The MOPSO proves tobe quite effective in finding the Pareto-optimal solutions toasteroid rendezvous multiobjective optimization problemsThe MOPSO shows a very competitive performance withrespect to two highly competitivemultiobjective evolutionaryalgorithms the nondominated sorting genetic algorithm-II (NSGA-II) [17] and the strength Pareto evolutionaryalgorithm-II (SPEA-II) [18]

2 Optimization Problem

21 Asteroid Rendezvous Design Problem The interplanetarytransfer trajectory is always divided into three different

segments that is planet departure segment heliocentrictransfer segment and capture segment by using the conceptof influencing sphere As the influencing sphere of the planetis much smaller than that of the sun the flight path and flighttime of the departure and capture segments are much smallcompared with that of the heliocentric transfer Thereforefor a preliminary trajectory design of an asteroid explorationmission the design emphasis is firstly focused on the helio-centric transfer while the planet departure is assumed asinstantaneous process with an impulsive maneuver and thecapture segment is omitted

This paper studies the asteroid rendezvous problemdeparture from the Earth 119905

0represents the spacecraft launch

date which is to be designed and 119881infin 119906 and V define the

departure hyperbolic velocity according to the formulas

120579 = 2120587119906

120593 = arccos (2V minus 1) minus120587

2

kinfin

119881infin

= cos (120593) cos (120579) i + cos (120593) sin (120579) j + sin (120593) k

(1)

Let r119864(1199050) k119864(1199050) be the heliocentric position and velocity

of the Earth at 1199050 then the initial position and velocity vector

of the spacecraft r(1199050) and k(119905

0) entering the heliocentric

transfer trajectory are defined as

r (1199050) = r119864(1199050) k (119905

0) = k119864(1199050) + kinfin (2)

The heliocentric transfer trajectory is modeled by a two-body dynamicmodel with the following governing equations

= V119909

119910 = V119910

= V119911

V119909= minus

120583119909

(1199092 + 1199102 + 1199112)32

+ 119886119909

V119910= minus

120583119910

(1199092 + 1199102 + 1199112)32

+ 119886119910

V119911= minus

120583119911

(1199092 + 1199102 + 1199112)32

+ 119886119911

(3)

where 119909 119910 and 119911 are position components of spacecraft V119909

V119910 and V

119911are velocity components of spacecraft 119886

119909 119886119910 and 119886

119911

are acceleration component of spacecraft and 120583 is the gravityparameter of the sun

The thrust acceleration Γ(119905) = (119886119909 119886119910 119886119911)119879 can be approx-

imated as 119899 impulses

Γ (119905) =

119899

sum

119894=1

Δk119894120575 (119905 minus 119905

119894) (4)

where 119905119894is the time where an impulse is applied Δk

119894=

(ΔV119909119894 ΔV119910119894 ΔV119911119894)119879(119894 = 1 2 119899) and they are all the

variables to be designedThe final state conditions are defined

r (119905119891) = r119860(119905119891) k (119905

119891) = k119860(119905119891) (5)

Mathematical Problems in Engineering 3

where 119905119891is the rendezvous time of spacecraft with the target

asteroid and r119860(119905119891) and k

119860(119905119891) are the position and velocity

vector of the target asteroid at the rendezvous time

22 Feasible-Solution Iteration Model In order to avoid deal-ingwith the equality constraints described as (5) the Lambertalgorithm is employed to establish the infeasible iterationoptimization model The chosen independent variables thatis the optimization variables are impulse times and the first119899 minus 2 impulses

119905119894(119894 = 1 2 119899) Δk

119894(119894 = 1 2 119899 minus 2) (6)

The last two impulses are determined by solving theLambert problem constrained by (5) In this feasible iterationmodel each evaluation of the objective function producesa feasible solution that satisfies implicitly the rendezvousconditions Detail on this multi-impulse rendezvous opti-mization model using the Lambert algorithm can be foundin [19 20]

23 Multiobjective Optimization Model The total velocitycharacteristic is chosen as the first objective function

min1198911(x) = ΔV = 119881

infin+

119899

sum

119894=1

1003816100381610038161003816Δk1198941003816100381610038161003816 (7)

The heliocentric transfer time is chosen as the secondobjective function

min1198912(x) = 119905

119891minus 1199050 (8)

The constraint on the time of impulse is considered Thegeneral constraint on 119905

119894(119894 = 1 2 119899) is

1199050le 1199051lt 1199052sdot sdot sdot lt 119905

119899le 119905119891 (9)

As the transfer time is one of the objective functions 119905119891

is chosen as an optimization variable The total optimizationvariables include four parts 119905

0 119881infin 119906 V 119905

119894(119894 = 1 2 119899)

Δk119894(119894 = 1 2 119899 minus 2) and 119905

119891 In order to improve

optimization performance the variable-scaling method isimposed on 119905

119894 Let 120572

119894= (119905119894minus 1199050)(119905119891minus 1199050)120572119894le 1 Therefore the

final optimization variables x are

x = (1199050 119881infin 119906 V 120572

1 120572

119899 Δk1 Δk

119899minus2 119905119891)119879

(10)

3 Optimization Algorithms

31 Introduction of MOEA A general multiobjective opti-mization problem is to find the design variables that optimizea vector objective function over the feasible design spaceTheobjective functions are the quantities that the designer wishestominimizemaximize or attain a certain valueThe problemformulation in standard form for a minimization is givenhere which is similar for the other cases

Minimize f (x) = (1198911(x) 119891

2(x) 119891

119898(x))119879 (11)

subject to 119892119894 (x) le 0 119894 = 1 2 119901

ℎ119895(x) = 0 119895 = 1 2 119902

(12)

where

x = (1199091 1199092 119909V)

119879isin X sub RV

(13)

For the multiobjective asteroid rendezvous design prob-lem the two objective functions are described by (7) and (8)the constraints are described by (9) and the optimizationvariables are described as (10)

The classical optimization method for a multiobjectiveoptimization problem is the weighting method In recentyears the multiobjective evolutionary algorithms (MOEA)have been greatly investigated in the domain of multiob-jective optimization There are many variants of MOEAreported in the literature a recent survey onMOEA and theirapplication in aeronautical and aerospace engineering hasbeen made in [16]

In the study except for theMOPSO we also test two othermostly popular MOEA The first is the NSGA-II algorithmwhich is proposed by Deb et al [17] This algorithm uses theidea of transforming the119898 objectives to a single fitness mea-sure by the creation of a number of fronts sorted accordingto nondomination During the fitness assignment the firstfront is created as the set of solutions that is not dominatedby any solutions in the population These solutions aregiven the highest fitness and temporarily removed from thepopulation then a second nondominated front consistingof the solutions that are now nondominated is built andassigned the second-highest fitness and so forth This isrepeated until each of the solutions has been assigned afitness After each front has been created its members areassigned crowding distances (normalized distance to closestneighbors in the front in the objective space) later to be usedfor niching The NSGA-II has been successfully applied inspacecraft trajectory optimization for example in designinga three-objective impulse rendezvous problem [20 21] and atwo-objective robust rendezvous problem with consideringuncertainty [22]

The second is the SPEA-II proposed by Zitzler et al[18] It uses an archive containing nondominated solutionspreviously found (the so-called external nondominated set)At each generation nondominated individuals are copiedto the external nondominated set removing the dominatedsolutions For each individual in this external set a strengthvalue is computed Pareto dominance is adopted to ensurethat the solutions are properly distributed along the Paretofront It also uses a nearest neighbor density estimationtechnique and a fine-grained fitness assignment strategywhich guide the search more efficiently

32 Brief Description of the MOPSO The MOPSO appliedin this study is the algorithm proposed by Pulido andCoello [23] which was competitive against the most pop-ular MOEA such as the NSGA-II the PAES and otherMOPSO on typical benchmark problems under com-mon performance metrics [23] The source code of theMOPSO is available from the EMOO repository located athttpdeltacscinvestavmxsimccoelloEMOO

The MOPSO is based on the use of Pareto ranking and asubdivision of decision variable space into several subswarms


BeginFor each swarm

(1) Initialize its particles(2) Initialize the set of global leaders 119892leader

End ForDO

For each swarmDo

For each particle(3) Select a leader(4) Perform the flight(5) Update the valueIf it is a leader then add to the 119892leader

End ForWhile (number of iterations le sgmax)(6) Store leaders in 119892leader in 119899swarms groups

End For(7) Assign each leader group to a swarm

While (number of iterations le GMax)End

Algorithm 1

which is done using clustering techniques The completeexecution process of this algorithm can be divided into threestages initialization flight and generation of results [23]

At the first stage every swarm is initialized Each swarmcreates and initializes its own particles and generates theleaders set among the particle swarm set by using Paretoranking In the second stage it firstly performs the executionof the flight of every swarm next it applies a clusteringalgorithm to group the guide particles This is performeduntil reaching a total of GMax iterations The execution ofthe flight of each swarm can be seen as an entire PSO process(with the difference that it will only optimize a specific regionof the search space) First each particle will select a leaderto which it will follow At the same time each particle willtry to outperform its leader and to update its position Ifthe updated particle is not dominated by any member of theleaders set then it will become a new leader The executionof the swarm will start again until a total of sgmax iterationsare reached Once all the swarms have finished theirs flights aclustering algorithm takes the control by grouping the closestparticle guides into 119899swarms swarms Each resulting group willbe assigned to a different swarmThe third and final stage willpresent all the nondominated solutions found

Details of this algorithm can be found in [23] and itspseudocode code is shown in Algorithm 1

The MOPSO algorithm requires the following param-eters (1) GMax the total number of generations that thealgorithmwill be executed (2) sgmax the number of internalgenerations that the particles of each swarm will run beforesharing their leaders (3) 119899particles the total number of parti-cles and (4) 119899swarms the number of particle groups

33 Constraints Optimization Method The multiobjectiveasteroidmission design problem is a highly constrained prob-lem whose constraints are described in (9) The simulationexperiments show that these constraints strongly influencethe convergence In the MOPSO the constraints are alwayshandled in checking Pareto dominance [23 24] When wecompare two individuals we first check their feasibility Ifthey are both feasible then the comparison is done usingPareto dominance If one is feasible and the other is infeasiblethe feasible individual wins If both are infeasible then theindividual with the lower amount of total constraint violationwins

The total constraint is calculated by making use of a non-differentiable penalty function For the general constrainedproblem in (10)ndash(13) the penalized total constraint function119862(x119872) is

119862 (x119872) = 119872[

[

119901

sum

119894=1

max (0 119892119894(x)) +

119902

sum

119895=1

10038161003816100381610038161003816ℎ119895(x)10038161003816100381610038161003816]

]

(14)

In the present work 119872 = 1000000 is used as a penaltycoefficient

Formost optimization problems each design variable hasits own upper and low valuesThus a strategy tomaintain theparticles within the search space in case they go beyond theirboundaries is necessary for the MOPSO algorithm

In [23 24] the following strategy was employed Whena decision variable goes beyond its boundaries the decisionvariable takes the value of its corresponding boundary (eitherthe lower or the upper boundary) The strategy will belikely effective when the Pareto solutions are located in theboundary of variables

However our simulation experiments show that thisstrategy is not very effective in solving our problems There-fore another simple strategy is proposed When a decisionvariable goes beyond its boundaries the decision variabletakes a random value from its feasible design space Theprobability-based disposal to boundary constraint couldenrich the diversity of swarm flight

The MOPSO with these two different boundary con-straint optimization strategies is respectively called asMOPSO-I and MOPSO-II Their performance is comparedin Section 43

34 Algorithm Assessment Metrics In order to allow a quan-titative assessment of the performance of the MSOPO weadopted the following two metrics

The first one is the epsilon indicator [25] Given a refer-ence Pareto front (ideally the true Pareto front if available)the epsilon indicator measures the minimum amount 120576

necessary to translate all the points of the found Pareto frontto weakly dominate the reference set The epsilon indicatorhas two types that is the additive type and the multiplicativetype and the multiplicative type is used here

The second one is the hypervolume indicator [26] Thehypervolume of a set of solutions measures the size of theportion of objective space that is dominated by those solu-tions collectively Generally hypervolume is favored because


Table 1 Orbit elements of asteroid

Index Asteroid 1 Asteroid 2Name 1999YR14 2340a (AU) 165365126892224 084421076388332195e 040069261757759106 044975834146342486i (deg) 37221930161441943 58547882390182853Ω (deg) 31338963493744654 21150460158030430120596 (deg) 94143875285008676 39994195753797953119872 (deg) 11473402134869427 24044827444641544Epoch (MJD2000) 32550000 3255000000

it captures in a single scalar both the closeness of the solutionsto the optimal set and to some extent the spread of thesolutions across objective space

The statistical test chosen for result evaluation is theMann-Whitney test [27] This is a nonparametric rank-basedtest that can be used to compare two independent sets ofsampled data It outputs119875 values that estimate the probabilityof a failure to reject the null hypothesis of the study questionHere the (1 minus 119875) values can be interpreted as the probabilitythat the performance of one algorithm is different (superioror junior) with statistical significance to that of the other

4 Simulation Results

41 Problem Configuration In order to testify the effective-ness of the proposed method two different asteroid missiondesigns are illustrated Table 1 lists the orbit elements of thetwo asteroids The upper and lower space of the optimizationvariables are provided in Table 2 Three different cases withthe number of impulses of 2 3 and 4 are respectively testedfor each asteroid mission

42 Pareto Fronts Analysis From our experiments thismultiobjective asteroid rendezvous design problem is verydifficult to be solved and the true Pareto fronts of thismultiobjective problem are not known In order to obtain thePareto fronts as possible close to the true ones theMOPSO isexecuted with a much larger number of function evaluationsThe parameters of the MOPSO are 119899particles = 400 119866119872119886119909 =

400 119904119892119898119886119909 = 5 and 119899swarms = 20Considering the stochastic characteristic of the MOPSO

10 independent runs for each test case are completed All thePareto solutions of the 10 independent runs are comparedand the repeated and non-Pareto solutions are deleted andthe revised Pareto solutions are selected as the final solutionsIn the following examples the figured Pareto fronts are allobtained in the same method The MOPSO with differ-ent boundary constraint optimization methods that is theMOPSO-I and MOPSO-II is both tested The performancecomparisons between the MOPSO-I and MOPSO-II will beanalyzed in the next section

Figure 1 compares the Pareto solution front for two-impulse three-impulse and four-impulse of the Asteroid1999YR14 mission Figure 2 compares this for the Aster-oid 2340 mission The tradeoffs between the total ΔV and

the transfer time are clearly demonstrated by Figures 1 and 2which will be useful for a mission designer Several inherentpeculiarities regarding the optimal multiobjective asteroidrendezvous trajectories have been observed from the opti-mization results For the second asteroid the correspondingtotal ΔV reduces evidently as the transfer time increaseswhen the transfer time is in the range of [100 140] daybut small change in other ranges therefore the Pareto frontconcentrates on a narrow range of 40 days For the firstasteroid the corresponding total ΔV reduces evidently as thetransfer time increases in a large range of [100 900] daytherefore the Pareto solutions distribute in a much largerspace but the Pareto front is discontinuous in small range asdemonstrated in Figure 1

In order to explain this phenomenon the propellant-optimal solutions are obtained by using the approachemployed in [28] In this optimization the transfer timeis fixed and ΔV is calculated every 5 days in the transfertime of range [100 1100] days The relations between thetransfer time and ΔV of the propellant-optimal solutions areillustrated in Figure 3 Some transfer time ranges with much-higher propellant cost are located for example the ΔV isabout 95 kms for the transfer time of 300 days while beingonly 62 kms for the transfer time of 280 days The formersolution is larger than the latter solution by about 50 in ΔVbesides its transfer time is also larger than the latterThis issuecan explain why the Pareto fronts of two-impulse solutionsdiscontinue near the transfer time of 300 days

The influence of different number of impulses on thetotal ΔV has been also demonstrated by Figures 1 and 2It is obvious that a three-impulse trajectory and a four-impulse trajectory cost less propellant in compared with atwo-impulse trajectory It is necessary to increase the numberof impulse to find a much better trajectory in the preliminaryasteroid rendezvous mission design

43 Performance Analysis of the MOPSO

431 Boundary Constraint Optimization Comparisons Asseen from Figures 1 and 2 the boundary constraint opti-mization method of the MOPSO has a great influence on thePareto fronts Graphically the MOPSO-II has much betterperformance in diversity and the spread of its solutionsfound is much larger than that of the MOPSO-I This caneasily explain that probability-based disposal to boundaryconstraint can enrich the diversity of swarm flight

It is not easily to determine which one is much closerto the true Pareto fronts from Figure 1 and the two-impulsecomparison case is shown in Figure 4(a) while it is easy to seefromFigure 2 that the Pareto fronts obtained byMOPSO-I aremuch closer to the true Pareto fronts and the three-impulsecomparison case is redrawn in Figure 4(b) to demonstratethis clearly

Therefore the quantitative metrics are calculated and thestatistical results are provided inTable 3 As seen fromTable 3the MOPSO-II has better performance with respect to thehybervolume indicator (larger valuemeans better) for the twocases While for the epsilon indicator (smaller value means


Table 2 Design space of variables

Variables Space Units1199050

[4000 10000] MJD2000119881infin

[0 5] kms119906 [0 1] naV [0 1] na

120572119894(119894 = 1 2 119899)

119899 = 2 [0 06] [03 1]na119899 = 3 [0 04] [01 08] [05 1]

119899 = 4 [0 04] [01 08] [05 09] [07 1]ΔV119909119894 ΔV119910119894 ΔV119911119894(119894 = 1 2 119899 minus 2) [minus4 4] kms

119905119891minus 1199050

[100 1500] days

100 200 300 400 500 600 700 8006000

6200

6400

6600

6800

7000

7200

7400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Two-impulseThree-impulse

Four-impulse

(a) Produced by MOPSO-I

200 400 600 800 10005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

(b) Produced by MOPSO-II

Figure 1 Pareto fronts of Asteroid 1

better) the MOPSO-II is much better in case of Asteroid 1two-impulse problem and the MOPSO-I are much better incase of Asteroid 2 three-impulse

Through the comparisons provided in Table 2 andFigure 4 our proposed boundary constraint demonstratesmuch better performance in total Besides these comparisonsshow that the same algorithmwill have different performancein solving the same type of problem with different configura-tions

432 Algorithm Parameters Analysis Our simulation exper-iments show that 119899particles and119866119872119886119909 are the twomain param-eters affecting the performance of theMOPSO in solving thiscomplex multiobjective problem In order to quantitativelyevaluate their influence 119900 119899particles is respectively set as 400200 and 100 119866119872119886119909 is also respectively set as 400 200 and100 and a total of nine groups of parameters (in Table 4) arefurthered tested Other parameters are the same as used inSection 42

Ten independent runs for the MOPSO-II with eachgroup of parameters in solving the three-impulse Asteroid1 problem are executed

Figure 5(a) compares the Pareto fronts of Cases 1 5and 9 Figure 5(b) compares that of Cases 2 5 and 8

and Figure 5(c) compares that of Cases 7 8 and 9 Thesecomparisons are to show the influence of the total number offunction evaluations It is clearly seen from Figure 5 that thelarger number of function evaluations evidently increases theperformance of theMOPSOThus a larger number of swarmsize and iteration are necessary for obtaining the optimalPareto fronts for this practical multiobjective optimizationproblem

Figure 6(a) compares the Pareto fronts of Cases 2 and 4Figure 6(b) compares that of Cases 6 and 8 and Figure 6(c)compares that of Cases 3 5 and 7 These comparisons areto show the influence of 119899particles and 119866119872119886119909 with the samenumber of function evaluations Evidently we cannot deter-mine which case is better by the graphical results providedin Figure 6Therefore the quantitative metrics are calculatedand the statistical results are provided in Tables 5 6 7 and 8

The comparisons between Case 2 and Case 4 show thatlarger size of swarm can obtain better average epsilon indica-tor with a 119875 value of 6776 but worse average hybervolumeindicator with a 119875 value of 1405The comparisons betweenCase 6 and Case 8 show that larger size of swarm bothincrease average performance in epsilon and hybervolumewith a 119875 value of 4727 and 757 By comparing Cases 35 and 7 Case 5 demonstrates the best performance in total


200 400 600 800 1000 1200 14008900

9000

9100

9200

9300

9400

9500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


100 150 200 250 300 350 400 450 5009000

9200

9400

9600

9800

10000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse



100 200 300 400 500 600 700 800 900 1000 1100Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

times104

04

06

08

1

12

14

16

Figure 3 Relations between transfer time and total characteristic velocity (propellant-optimal solutions Asteroid 1)

Table 3 MOPSO performance with different boundary constraint optimization

Test problem Algorithm Epsilon HybervolumeMean std P value Mean std P value

Asteroid 1 two-impulse MOPSO-I 0011987 0008378 01620 0972618 0006829 00257MOPSO-II 0006833 0004066 0977958 0001301

Asteroid 2 three-impulse MOPSO-I 0025630 00278051 01620 087509 0308357 06232MOPSO-II 00399274 0026254 0968537 0016548

Table 4 MOPSO parameters configurations

Index Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9119899particles 400 400 400 200 200 200 100 100 100GMax 400 200 100 400 200 100 400 200 100


100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II

(a) Asteroid 1 two-impulse

0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II

(b) Asteroid 2 three-impulse

Figure 4 Pareto fronts produced by MOPSO with different boundary constraint optimization

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9

Figure 5 Pareto fronts produced by MOPSO-II with different number of function evaluations


0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7

Figure 6 Pareto fronts produced by MOPSO-II with same number of function evaluations

Table 5 MOPSO-II performance with different parameters (Cases 2 and 4)


Asteroid 1 three-impulse Case 2 0010692 0006797 06776 0984063 0002738 01405Case 4 0012918 0012848 0985643 0002736




Table 7 MOPSO-II performance with different parameters (Cases 3 5 and 7)

Test problem Algorithm Epsilon HybervolumeMean std Mean std

Asteroid 1 three-impulseCase 3 000583413 00036399 0994062 000135846Case 5 000438474 000129004 0994755 000100777Case 7 000564449 000274183 0994917 000119178


0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II


Figure 7 Pareto fronts produced by MOPSO NSGA-II and SPEA-II

Table 8 119875 value of MOPSO (Cases 3 5 and 7)

Case 3 Case 5 Case 7Case 3 04727 09097Case 5 03447 04727Case 7 02123 07337Bold indicates epsilon the other is hybervolume

However the improvements are not evident as the 119875 valuesare all a large value

Although the result obtained from this experiment seemsto be inconclusive on which parameter is the best we canargue that the MOPSOrsquos performance is not sensitive to themain PSO parameters under the condition that the totalnumber of function evaluation retains a large value We haveobtained competitive results in most cases without payingspecial attentions on PSO parameters

44 Comparison of the MOPSO with Other MOEA To dem-onstrate the performance of the employedMOPSO two otherpopular algorithms are tested and compared the NSGA-IIand the SPEA-II

In the following examples the total number of functionevaluations was set to 20000 for all the algorithms comparedThe NSGA-II and the SPEA-II were run using a populationsize of 200 a maximum number of generations of 100 TheMOPSO-I and MOPSO-II used 200 particles a maximumnumber of generations of 20 a maximum number of gener-ations per swarm of 5 and a total of 5 swarms The sourcecode of the NSGA-II and SPEA-II provided in the EMOOrepository is also employed in this study

The two test problems are the same to those employedin Section 43 to compare boundary constraint optimizationTen independent runs for each test case are executed Thefinal Pareto fronts are compared in Figure 7 and the statisticalresults with respect to epsilon and hybervolume indicatorsare provided in Tables 9 10 and 11

Evidently seen from Figure 7(a) the MOPSO-I producesbetter Pareto fronts than the NSGA-II and SPEA-II for thefirst test problem The statistical results of the two indicatorsalso support this The 119875 value with respect to epsilon iscalculated as 058 and 113 respectively and 073 and017 for hybervolume This shows that the MOPSO-I isabsolutely super to the NSGA-II and SPEA-II

Also evidently seen from Figure 7(b) the MOPSO-Iproduces much closer Pareto fronts than the NSGA-II andSPEA-II for the second test problem However its Paretofronts are a little narrower whichmay result in that its epsilonand hybervolume indicator are junior to the NSGA-II andSPEA-II Furthermore the fact that the Pareto fronts arehighly discontinuous would make these metrics irrelevantIn the view of practical spacecraft mission design obtainingmuch closer Pareto fronts from which the designer canchoose one single satisfying solution for engineering designwould be much desirable performance for a multiobjectiveoptimizerThus we can say that theMOPSO-I is a little betterthan the NSGA-II and SPEA-II even the statistical results onquantitative metrics do not support this

In conclusion the MOPSO showed a better performancewith respect to the NSGA-II and the SPEA-II for the two testproblems For other test problems the MOPSO is not alwaysbetter but its performance is very competitive

45 Comparison with PCP Method The widely used Pork-Chop (PCP) method is also tested here for comparison TheΔV is calculated every 1 day in both Earth departure time andasteroid arrival time and the two-impulse Lambert algorithmis employed The contours of time of flight corresponding tothe optimal transfers for Asteroid 1 are illustrated in Figure 8For convenience only the solutions with a ΔV less than18 kms and Earth departure and asteroid arrive times in[5000 7000] (MJD2000) are presented

We analyze the comparisons between the proposed mul-tiobjective optimization approach and the PCPmethod in thefollowing three aspects


Table 9 Performance comparisons of MOPSO NSGA-II and SPEA-II


Asteroid 1 two-impulseMOPSO-I 0035041 0038848 0876849 0026781NSGA-II 0113438 0094049 0821205 0080834SPEA-II 0100881 0032094 0811347 0035008

Asteroid 2 three-impulseMOPSO-II 0159179 0081796 0875023 0078886NSGA-II 00594614 0058546 0838198 0301149SPEA-II 012172 0055917 0902724 0072393

Table 10 119875 value of MOPSO-I NSGA-II and SPEA-II

MOPSO-I NSGA-II SPEA-IIMOPSO-I 00058 00113NSGA-II 00073 05205SPEA-II 00017 01620Bold indicates epsilon the other is hybervolume

Table 11 119875 value of MOPSO-II NSGA-II and SPEA-II

MOPSO-II NSGA-II SPEA-IIMOPSO-II 00113 03847NSGA-II 03447 00211SPEA-II 04274 05205Bold indicates epsilon the other is hybervolume

451 Uses of the Results In the PCP method the contours oftheΔV are plotted to assist the designer to find the best launchwindow and transfer trajectory However the contours havemany local peaks and it is not very convenient to determinewhich solution is the best one Besides the size of the regioncontaining the global minimum is a very important factor infinding real trajectories in any given year This is not easilyobserved from the Pork-Chop figures

In contrast to the PCP method the proposed multi-objective optimization approach can provide friendly thedesigner of this information The relations between totalcharacteristic velocity transfer time and earth departuretime of those obtained Pareto-optimal solutions for two testcases are provided in Figure 8 From Figure 8 it can beclearly observed that the earth departure time for all theoptimal rendezvous trajectories concentrates on one arrowdomain for these two cases The optimized Earth launchwindow and its size can be easily determined through themultiobjective optimization design which will provide veryuseful information for engineering design

452 Global Optimality The minimum-propellant solutionsearched by the PCP method for the first asteroid missionis calculated with a ΔV of 60153ms and it is located inFigure 9 (lowast denotes) As seen in Figure 1 the MOPSO haslocated a set of Pareto solutions with a ΔV about this valueThis issue can also demonstrate the global convergence abilityof the MOPSO Besides the PCP method obtains only the

two-impulse trajectory As is well known the two-impulsetrajectory is not the propellant-optimal solution under mostconditions and increasing the number of impulses willreduce the propellant cost Our proposed approach candesign the multi-impulse trajectory thus it locates bettersolution than the PCP method

453 Efficiency The PCP is in essence of an exhaustivesearching method In our test the search space for departuretime and arrive time is [4000 10000] day With the searchstep of 1 day the PCP method calculates a total number of6000 lowast 6000 trajectories while only 400 lowast 400 lowast 5 trajectoriesare calculated in the proposed method for the most highlycost case The proposed approach improves the calculationefficiency by about 40 times Besides the PCP method needsmuch human intention to determine the final solutions whilethe proposed approach is an automated search method

5 Conclusions

The paper formulates the asteroid rendezvous preliminarytrajectory design as a multiobjective optimization problemand employs the multiobjective particle swarm optimization(MOPSO) algorithm to locate the Pareto-optimal solutionset Comparedwith thewidely employed Pork-Chopmethodthe proposed approach is demonstrated to be able to providemuch more easily used results obtain better propellant-optimal solutions and have much better efficiency Theresults show that the proposed approach can effectivelyand efficiently demonstrate the relations among the missioncharacteristic parameters such as launch time transfer timepropellant cost and number of maneuvers which will pro-vide useful reference for practical asteroid mission designTheMOPSOproves to be quite effective in finding the Pareto-optimal solutions and its performance can be improved bya proposed boundary constraint optimization strategy TheMOPSO is found to be very competitive with respect to twohighly competitive multiobjective evolutionary algorithmsthe NSGA-II and the SPEA-II

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200

Transfer time (day)Total characteristic velocity (ms)

Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)

(b) Asteroid 2 two-impulse

Figure 8 Relations between total characteristic velocity transfer time and earth departure time

Earth departure time (MJD2000)

Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800

Figure 9 Contours of time of flight corresponding to the optimaltransfers (Asteroid 1)

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (no 11222215) the 973 Project(no 2013CB733100) and the Hunan Provincial Natural Sci-ence Foundation of China (no 13JJ1001)

References

[1] N D Hulkower C O Lau and D F Bender ldquoOptimumtwo-impulse transfer for preliminary interplanetary trajectorydesignrdquo Journal of Guidance Control and Dynamics vol 7 no4 pp 458ndash461 1984

[2] C O Lau and N D Hulkower ldquoAccessibility of near-earthasteroidsrdquo Journal of Guidance Control and Dynamics vol 10no 3 pp 225ndash232 1987

[3] E Perozzi A Rossi and G B Valsecchi ldquoBasic targetingstrategies for rendezvous and flyby missions to the near-earthasteroidsrdquo Planetary and Space Science vol 49 no 1 pp 3ndash222001

[4] D Qiao H T Cui and P Y Cui ldquoEvaluating accessibilityof near-earth asteroids via earth gravity assistsrdquo Journal of

Guidance Control and Dynamics vol 29 no 2 pp 502ndash5052006

[5] D F LawdenOptimal Trajectories for Space Navigation Butter-worths London UK 1963

[6] J E Pussing ldquoPrimer vector theory and applicationsrdquo inSpacecraft Trajectory Optimization B A Conway Ed pp 16ndash36 Cambridge University Press New York NY USA 2010

[7] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 IEEE Piscataway NJ USADecember 1995

[8] C R Bessette andD B Spencer ldquoIdentifying optimal interplan-etary trajectories through a genetic approachrdquo in Proceedings ofthe AIAAAAS Astrodynamics Specialist Conference and ExhibitAIAA Paper 2006-6306 pp 809ndash826 Keystone Colo USAAugust 2006

[9] K Zhu F Jiang J Li and H Baoyin ldquoTrajectory optimizationof multi-asteroids exploration with low thrustrdquo Transactions ofthe Japan Society for Aeronautical and Space Sciences vol 52 no175 pp 47ndash54 2009

[10] M Pontani and B A Conway ldquoParticle swarm optimizationapplied to space trajectoriesrdquo Journal of Guidance Control andDynamics vol 33 no 5 pp 1429ndash1441 2010

[11] M Pontani P Ghosh and B A Conway ldquoParticle swarmoptimization of multiple-burn rendezvous trajectoriesrdquo Journalof Guidance Control andDynamics vol 35 no 4 pp 1192ndash12072012

[12] M Reyes-Sierra and C A C Coello ldquoMulti-objective particleswarm optimizers a survey of the state-of-the-artrdquo Interna-tional Journal of Computational Intelligence Research vol 2 no3 pp 287ndash308 2006

[13] K B Lee and J H Kim ldquoMultiobjective particle swarmoptimization with preference-based sort and its application topath following footstep optimization for humanoid robotsrdquoIEEE Transactions on Evolutionary Computation vol 17 no 6pp 755ndash766 2013

[14] X Y Shao W Q Liu Q Liu and C Y Zhang ldquoHybrid discreteparticle swarm optimization for multi-objective flexible job-shop scheduling problemrdquo International Journal of AdvancedManufacturing Technology vol 67 no 9ndash12 pp 2885ndash29012013

[15] M Shokrian and K A High ldquoApplication of a multi objectivemulti-leader particle swarm optimization algorithm on NLP


andMINLP problemsrdquoComputers ampChemical Engineering vol60 pp 57ndash75 2014

[16] A Arias-Montano C A C Coello and E Mezura-MontesldquoMultiobjective evolutionary algorithms in aeronautical andaerospace engineeringrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 5 pp 662ndash694 2012

[17] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[18] E Zitzler M Laumanns and L Thiele ldquoSPEA2 improvingthe strength Pareto evolutionary algorithmrdquo in EvolutionaryMethods for Design Optimization and Control with Applicationsto Industrial Problems K Giannakoglou D T Tsahalis JPeriaux K D Papailiou and T Fogarty Eds pp 95ndash100Springer Berlin Germany 2002

[19] S P Hughes L M Mailhe and J J Guzman ldquoA comparisonof trajectory optimizationmethods for the impulsive minimumfuel rendezvous problemrdquo in Guidance and Control 2003 IJ Gravseth and R D Culp Eds vol 113 of Advances in theAstronautical Sciences pp 85ndash104 2003

[20] Y Z Luo G J Tang and Y J Lei ldquoOptimal multi-objectivenonlinear impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 4 pp 994ndash1002 2007

[21] Y Z Luo G J Tang and Y J Lei ldquoOptimal multi-objectivelinearized impulsive rendezvousrdquo Journal of Guidance Controland Dynamics vol 30 no 2 pp 383ndash389 2007

[22] Y Z Luo Z Yang and H N Li ldquoRobust optimizationof nonlinear impulsive rendezvous with uncertaintyrdquo ScienceChina Physics Mechanics amp Astronomy vol 57 no 3 pp 1ndash102014

[23] G T Pulido and C A C Coello ldquoUsing clustering techniquesto improve the performance of a particle swarm optimizerrdquoin Genetic and Evolutionary ComputationmdashGECCO 2004 vol3102 of LectureNotes in Computer Science pp 225ndash237 SpringerBerlin Germany 2004

[24] C A C Coello G T Pulido and M S Lechuga ldquoHandlingmultiple objectives with particle swarm optimizationrdquo IEEETransactions on Evolutionary Computation vol 8 no 3 pp256ndash279 2004

[25] E Zitzler L Thiele M Laumanns C M Fonseca and V G daFonseca ldquoPerformance assessment of multiobjective optimiz-ers an analysis and reviewrdquo IEEE Transactions on EvolutionaryComputation vol 7 no 2 pp 117ndash132 2003

[26] E Zitzler and L Thiele ldquoMultiobjective optimization usingevolutionary algorithmsmdasha comparative case studyrdquo in ParallelProblem Solving from NaturemdashPPSN V A E Eiben Edvol 1498 of Lecture Notes in Computer Science pp 292ndash301Springer Amsterdam The Netherlands 1998

[27] M Hollander and D A Wolfe Nonparametric Statistical Meth-ods Wiley Series in Probability and Statistics John Wiley ampSons Hoboken NJ USA 2nd edition 1999

[28] Y Z Luo J Zhang H Y Li and G J Tang ldquoInteractiveoptimization approach for optimal impulsive rendezvous usingprimer vector and evolutionary algorithmsrdquo Acta Astronauticavol 67 no 3-4 pp 396ndash405 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


are all chosen optimization design variables The multiobjec-tive particle swarm algorithm is employed to locate the Paretosolution bywhich the relationships characteristics among theoverall mission parameters can be effectively revealed

The particle swarm optimization (PSO) algorithm firstlyintroduced by Kennedy and Eberhart [7] in 1995 has beensuccessfully applied in many fields of research However itis only recently applied to aerospace trajectories optimiza-tion [8ndash11] The single-objective PSO algorithm has beentestified as one successful spacecraft trajectory optimizerin designing impulsive interplanetary trajectories [8] low-thrust trajectories for asteroid exploration [9] Lyapunov andHalo (periodic) orbits [10] and multiple-burn rendezvoustrajectories [11]

The implementation of the PSO algorithm adopts a pop-ulation of particles whose behavior is affected by either thebest local (ie within a certain neighborhood) or the bestglobal individual The relative simplicity of PSO and the factthat it is a population-based technique have made it a naturalcandidate to be extended for multiobjective optimizationIn a survey paper in 2006 on multiobjective particle swarmoptimization (MOPSO) [12] it was reported that there werecurrently over twenty-five different proposals of MOPSOreported in the specialized literatureThe studies onMOPSOremain a very active area of research and the MOPSO hasbeen successfully applied to many practical multiobjectiveoptimization problems recently applied to robotics [13]industrial management [14] and chemical engineering [15]It also has been applied in the domain of aerospace includingairfoil shape optimization complex physicsshape optimiza-tion and multidisplinary design optimization [16] Howeverit was seldom applied in solving multiobjective spacecrafttrajectory optimization problems The recent studies showthat the single-objective PSO algorithm is one effectivereliable and accurate spacecraft trajectory optimizer [8ndash11]In this study we will show that the MOPSO could be onesuccessful optimizer for multiobjective spacecraft trajectoryoptimization problems

In summary the main contribution of this paper is two-fold (1) A novel asteroid rendezvous mission design methodusing the multiobjective techniques is proposed Comparedwith the current widely employed Pork-Chop method theproposed approach is demonstrated to be able to providemuch more easily used results obtain better propellant-optimal solutions and have much better efficiency (2) Asfar as we know it is the first time to apply the MOPSO tospacecraft trajectory optimization The MOPSO proves tobe quite effective in finding the Pareto-optimal solutions toasteroid rendezvous multiobjective optimization problemsThe MOPSO shows a very competitive performance withrespect to two highly competitivemultiobjective evolutionaryalgorithms the nondominated sorting genetic algorithm-II (NSGA-II) [17] and the strength Pareto evolutionaryalgorithm-II (SPEA-II) [18]

2 Optimization Problem

21 Asteroid Rendezvous Design Problem The interplanetarytransfer trajectory is always divided into three different

segments that is planet departure segment heliocentrictransfer segment and capture segment by using the conceptof influencing sphere As the influencing sphere of the planetis much smaller than that of the sun the flight path and flighttime of the departure and capture segments are much smallcompared with that of the heliocentric transfer Thereforefor a preliminary trajectory design of an asteroid explorationmission the design emphasis is firstly focused on the helio-centric transfer while the planet departure is assumed asinstantaneous process with an impulsive maneuver and thecapture segment is omitted

This paper studies the asteroid rendezvous problemdeparture from the Earth 119905

0represents the spacecraft launch

date which is to be designed and 119881infin 119906 and V define the

departure hyperbolic velocity according to the formulas

120579 = 2120587119906

120593 = arccos (2V minus 1) minus120587

2

kinfin

119881infin

= cos (120593) cos (120579) i + cos (120593) sin (120579) j + sin (120593) k

(1)

Let r119864(1199050) k119864(1199050) be the heliocentric position and velocity

of the Earth at 1199050 then the initial position and velocity vector

of the spacecraft r(1199050) and k(119905

0) entering the heliocentric

transfer trajectory are defined as

r (1199050) = r119864(1199050) k (119905

0) = k119864(1199050) + kinfin (2)

The heliocentric transfer trajectory is modeled by a two-body dynamicmodel with the following governing equations

= V119909

119910 = V119910

= V119911

V119909= minus

120583119909

(1199092 + 1199102 + 1199112)32

+ 119886119909

V119910= minus

120583119910

(1199092 + 1199102 + 1199112)32

+ 119886119910

V119911= minus

120583119911

(1199092 + 1199102 + 1199112)32

+ 119886119911

(3)

where 119909 119910 and 119911 are position components of spacecraft V119909

V119910 and V

119911are velocity components of spacecraft 119886

119909 119886119910 and 119886

119911

are acceleration component of spacecraft and 120583 is the gravityparameter of the sun

The thrust acceleration Γ(119905) = (119886119909 119886119910 119886119911)119879 can be approx-

imated as 119899 impulses

Γ (119905) =

119899

sum

119894=1

Δk119894120575 (119905 minus 119905

119894) (4)

where 119905119894is the time where an impulse is applied Δk

119894=

(ΔV119909119894 ΔV119910119894 ΔV119911119894)119879(119894 = 1 2 119899) and they are all the

variables to be designedThe final state conditions are defined

r (119905119891) = r119860(119905119891) k (119905

119891) = k119860(119905119891) (5)







119905119894(119894 = 1 2 119899) Δk

119894(119894 = 1 2 119899 minus 2) (6)



min1198911(x) = ΔV = 119881

infin+

119899

sum

119894=1

1003816100381610038161003816Δk1198941003816100381610038161003816 (7)


min1198912(x) = 119905

119891minus 1199050 (8)


119894(119894 = 1 2 119899) is


119899le 119905119891 (9)



0 119881infin 119906 V 119905

119894(119894 = 1 2 119899)

Δk119894(119894 = 1 2 119899 minus 2) and 119905



119894 Let 120572



x = (1199050 119881infin 119906 V 120572

1 120572

119899 Δk1 Δk

119899minus2 119905119891)119879

(10)



Minimize f (x) = (1198911(x) 119891

2(x) 119891

119898(x))119879 (11)

subject to 119892119894 (x) le 0 119894 = 1 2 119901

ℎ119895(x) = 0 119895 = 1 2 119902

(12)

where

x = (1199091 1199092 119909V)

119879isin X sub RV

(13)








BeginFor each swarm


End ForDO

For each swarmDo





Algorithm 1







119862 (x119872) = 119872[

[

119901

sum

119894=1

max (0 119892119894(x)) +

119902

sum

119895=1

10038161003816100381610038161003816ℎ119895(x)10038161003816100381610038161003816]

]

(14)































[4000 10000] MJD2000119881infin

[0 5] kms119906 [0 1] naV [0 1] na

120572119894(119894 = 1 2 119899)

119899 = 2 [0 06] [03 1]na119899 = 3 [0 04] [01 08] [05 1]


119905119891minus 1199050

[100 1500] days

100 200 300 400 500 600 700 8006000

6200

6400

6600

6800

7000

7200

7400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


200 400 600 800 10005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse












200 400 600 800 1000 1200 14008900

9000

9100

9200

9300

9400

9500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


100 150 200 250 300 350 400 450 5009000

9200

9400

9600

9800

10000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse




Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

times104

04

06

08

1

12

14

16









100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II


0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II



0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9



0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119905119894(119894 = 1 2 119899) Δk

119894(119894 = 1 2 119899 minus 2) (6)



min1198911(x) = ΔV = 119881

infin+

119899

sum

119894=1

1003816100381610038161003816Δk1198941003816100381610038161003816 (7)


min1198912(x) = 119905

119891minus 1199050 (8)


119894(119894 = 1 2 119899) is


119899le 119905119891 (9)



0 119881infin 119906 V 119905

119894(119894 = 1 2 119899)

Δk119894(119894 = 1 2 119899 minus 2) and 119905



119894 Let 120572



x = (1199050 119881infin 119906 V 120572

1 120572

119899 Δk1 Δk

119899minus2 119905119891)119879

(10)



Minimize f (x) = (1198911(x) 119891

2(x) 119891

119898(x))119879 (11)

subject to 119892119894 (x) le 0 119894 = 1 2 119901

ℎ119895(x) = 0 119895 = 1 2 119902

(12)

where

x = (1199091 1199092 119909V)

119879isin X sub RV

(13)








BeginFor each swarm


End ForDO

For each swarmDo





Algorithm 1







119862 (x119872) = 119872[

[

119901

sum

119894=1

max (0 119892119894(x)) +

119902

sum

119895=1

10038161003816100381610038161003816ℎ119895(x)10038161003816100381610038161003816]

]

(14)































[4000 10000] MJD2000119881infin

[0 5] kms119906 [0 1] naV [0 1] na

120572119894(119894 = 1 2 119899)

119899 = 2 [0 06] [03 1]na119899 = 3 [0 04] [01 08] [05 1]


119905119891minus 1199050

[100 1500] days

100 200 300 400 500 600 700 8006000

6200

6400

6600

6800

7000

7200

7400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


200 400 600 800 10005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse












200 400 600 800 1000 1200 14008900

9000

9100

9200

9300

9400

9500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


100 150 200 250 300 350 400 450 5009000

9200

9400

9600

9800

10000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse




Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

times104

04

06

08

1

12

14

16









100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II


0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II



0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9



0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










BeginFor each swarm


End ForDO

For each swarmDo





Algorithm 1







119862 (x119872) = 119872[

[

119901

sum

119894=1

max (0 119892119894(x)) +

119902

sum

119895=1

10038161003816100381610038161003816ℎ119895(x)10038161003816100381610038161003816]

]

(14)































[4000 10000] MJD2000119881infin

[0 5] kms119906 [0 1] naV [0 1] na

120572119894(119894 = 1 2 119899)

119899 = 2 [0 06] [03 1]na119899 = 3 [0 04] [01 08] [05 1]


119905119891minus 1199050

[100 1500] days

100 200 300 400 500 600 700 8006000

6200

6400

6600

6800

7000

7200

7400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


200 400 600 800 10005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse












200 400 600 800 1000 1200 14008900

9000

9100

9200

9300

9400

9500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


100 150 200 250 300 350 400 450 5009000

9200

9400

9600

9800

10000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse




Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

times104

04

06

08

1

12

14

16









100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II


0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II



0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9



0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























[4000 10000] MJD2000119881infin

[0 5] kms119906 [0 1] naV [0 1] na

120572119894(119894 = 1 2 119899)

119899 = 2 [0 06] [03 1]na119899 = 3 [0 04] [01 08] [05 1]


119905119891minus 1199050

[100 1500] days

100 200 300 400 500 600 700 8006000

6200

6400

6600

6800

7000

7200

7400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


200 400 600 800 10005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse












200 400 600 800 1000 1200 14008900

9000

9100

9200

9300

9400

9500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


100 150 200 250 300 350 400 450 5009000

9200

9400

9600

9800

10000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse




Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

times104

04

06

08

1

12

14

16









100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II


0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II



0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9



0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200 400 600 800 1000 1200 14008900

9000

9100

9200

9300

9400

9500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse


100 150 200 250 300 350 400 450 5009000

9200

9400

9600

9800

10000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse




Tota

l cha

ract

erist

ic v

eloci

ty (m

s)


Four-impulse

times104

04

06

08

1

12

14

16









100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II


0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II



0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9



0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100 200 300 400 500 600 700 800 9005500

6000

6500

7000

7500

8000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II


0 200 400 600 800 1000 1200 14008500

9000

9500

10000

10500

11000

Transfer time (day)

Tota

l ch

arac

teris

tic v

eloci

ty (m

s)

MOPSO-IMOPSO-II



0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 1Case 5

Case 9

(a) Cases 1 5 and 9

9000

Case 2Case 5

Case 8

0 200 400 600 800 1000 1200 14006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

(b) Cases 2 5 and 8

0 200 400 600 800 10006000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 7Case 8

Case 9

(c) Cases 7 8 and 9



0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000 12005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 2Case 4

(a) Cases 2 4

0 200 400 600 800 10005500

6000

6500

7000

7500

8000

8500

9000

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 6Case 8

(b) Cases 6 8

0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

8000

8500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

Case 3Case 5

Case 7

(c) Cases 3 5 and 7












0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000 1200 14005500

6000

6500

7000

7500

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-INSGA-II

SPEA-II


100 120 140 160 180 200 220 2409000

9200

9400

9600

9800

10000

10200

10400

Transfer time (day)

Tota

l cha

ract

erist

ic v

eloci

ty (m

s)

MOPSO-IINSGA-II

SPEA-II





























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























5 Conclusions





0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 1000

50006000

700080005800

5900

6000

6100

6200


Eart

h de

part

ure t

ime

(MJD

2000

)


100110

120130

90009200

94009600

532053255330533553405345


Eart

h de

part

ure t

ime

(MJD

2000

)




Aste

roid

arriv

e tim

e (M

JD20

00)

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

5200

5400

5600

5800

6000

6200

6400

6600

6800

7000

6060 6080 6100 6120 6140 6160 61806200

6300

6400

6500

6600

6700

6800


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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