Evolved Spacecraft Trajectories for Low Earth Orbit

David W. Hinckley Jr.Mechanical Engineering

University of VermontBurlington, VT 05405 USA

dhinckle@uvm.edu

Karol ZiebaComputer Science

University of VermontBurlington, VT 05405 USA

kzieba@uvm.edu

Darren L. HittMechanical Engineering

University of VermontBurlington, VT 05405 USA

dhitt@uvm.edu

Margaret J. EppsteinComputer Science

University of VermontBurlington, VT 05405 USA

maggie.eppstein@uvm.edu

ABSTRACTIn this paper we use Differential Evolution (DE), with bestevolved results refined using a Nelder-Mead optimization, tosolve complex problems in orbital mechanics relevant to lowEarth orbits (LEO). A class of so-called ‘Lambert Problems’is examined. We evolve impulsive initial velocity vectors giv-ing rise to intercept trajectories that take a spacecraft fromgiven initial positions to specified target positions. We seekto minimize final positional error subject to time-of-flightand/or energy (fuel) constraints. We first validate that themethod can recover known analytical solutions obtainablewith the assumption of Keplerian motion. We then apply themethod to more complex and realistic non-Keplerian prob-lems incorporating trajectory perturbations arising in LEOdue to the Earth’s oblateness and rarefied atmospheric drag.The viable trajectories obtained for these difficult problemssuggest the robustness of our computational approach forreal-world orbital trajectory design in LEO situations whereno analytical solution exists.

Categories and Subject DescriptorsI.2.m [Artificial Intelligence]: Differential Evolution; J.2 [PhysicalSciences and Engineering]: Engineering; Aerospace

KeywordsOrbital Mechanics; Lambert’s Problem; Differential Evolution;Orbital Trajectory Planning; Low Earth Orbit

1. INTRODUCTIONThe planning of orbital maneuvers and/or trajectories for space-

craft represents a design optimization problem that is associatedwith multiple engineering constraints (e.g., time of flight, fuel con-sumption, and positional accuracy). Due to the inherently non-linear equations of classical orbital motion, analytical approaches

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to most trajectory optimization problems are generally not avail-able and numerical optimization is required. This is especiallytrue in the case of modern problems of practical interest for de-signing low Earth orbits (LEOs), which are further complicatedby various sources of non-Keplarian perturbations that becomesignificant near to the Earth such as planetary oblateness, atmo-spheric drag, and solar radiation pressure among others.

Owing to the multiple objectives and system complexity, vari-ous evolutionary approaches for trajectory optimization have beenexplored over the past decade. Cacciatore & Toglia [2] consideredminimum fuel orbital trajectories resulting from a finite seriesof impulsive thrusts using a genetic algorithm (GA). Lee et al.[13] also used GAs to evolve orbital elements (semi-major axis,eccentricity, inclination) instead of an initial trajectory velocity.As such, their approach was necessarily limited to the idealizedtwo-body problem of Keplerian theory and lacks the ability toincorporate perturbations that arise in LEO scenarios. Bessette& Spencer [1] used both DE [18] and Covariance Matrix Adap-tation Evolution Strategy (CMA-ES) [8] approaches to optimizemulti-objective Keplerian orbital transfers in LEO; in their study,the optimal trajectory was based on the simultaneous consider-ations of fuel consumption and time of flight. Sentinella andCasalino [17] used both GA and DE approaches to examine in-terplanetary orbit trajectory optimization (e.g., as opposed toLEO maneuvers) involving intermediate planetary fly-by gravi-tational assists. Englander [5] posed the problem of choosing asequence of fly-bys as an integer programming problem that wassolved using a GA approach, and nested DE and particle swarmoptimization inside a GA to reproduce solutions to the Galileoand Cassini missions [6]. Izzo et al. [11] utilized DE to design ahighly complex trajectory among the Galilean moons of Jupiterthat optimized observational conditions at time of spacecraft atthe time of flyby. However, all of these prior efforts assumed ide-alized physics for the governing equations of motion and, as such,are not sufficient for trajectory planning for LEO.

In the present study, differential evolution is used to solve aclass of ‘Lambert-type’ orbital trajectory problems (Figure 1).We incorporate orbital perturbations due to planetary oblatenessand atmospheric drag that are necessary for accurate planning ofLEO trajectories. In the classical Lambert problem (i.e., with-out perturbations) a trajectory is sought that takes a spacecraftfrom an initial orbit location P1 to a new position P2 duringsome time interval Δt [4]. The end condition of the arrival at P2

may involve a simple positional requirement (intercept trajectory)or a positional and velocity requirement (rendezvous trajectory).Multiple objectives may include minimizing positional accuracyof the trajectory endpoint (i.e., relative to P2), fuel consumption,and time-of-flight accuracy. In particular, the fuel constraint hashistorically represented a key limitation in orbit planning. In thiswork we combine these objectives into a single weighted fitnessfunction, with various weights depending on the particular prob-lem being solved. The resulting fitness function thus represents a

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Figure 1: Schematic diagram of the classical Lam-bert problem involving the orbital transfer from aninitial position P1 to a new location P2 with a pre-scribed time-of-flight Δt.

trajectory optimization problem. The solution sought in this classof problems is ultimately the initial velocity vector at P1 that ini-tiates the transfer trajectory; from this initial velocity vector, allorbital elements of the of the trajectory may be determined (e.g.,[4]).

This investigation consists of two essential parts. In the firstpart, we show that it is possible to recover classic solutions toLambert’s problem for either (1) a specified time-of-flight or (2) aminimum energy orbital transfer, thus validating the evolved tra-jectories against known analytical results. In the second part, weevolve trajectories in the presence of orbital perturbations arisingfrom the oblateness of the Earth and from rarefied atmosphericdrag, for which there are no known analytical solutions. Both ofthese effects are most pronounced in LEO (∼100-400 km), whichis the altitude range occupied by the new generation of smallsatellites (‘cubesats’, ‘nanosats’) being developed by NASA, theEuropean Space Agency, and universities [9]. The ability to au-tomate the optimization of these nonlinear problems offers real-world benefits for designing LEO trajectories.

2. OVERVIEW OF GOVERNING PHYSICS

2.1 Newtonian Gravitational PotentialA Newtonian gravitational potential is the canonical model

used in orbital mechanics. In this model, gravity is a assumedto be a spherically-symmetric, attractive force that is inverselyproportional to the square of the distance between the centers ofmass between two bodies. For problems involving a spacecraftand the Earth, the mass of the spacecraft is inconsequential andthe gravitational constant is given by μ = G Me where G is theuniversal gravitational constant and Me is the mass of the Earth.The governing equation for the spacecraft motion relative to aframe centered at the Earth is given by

d2r

dt2= − μ

|r|3 r (1)

where r is the position vector of the spacecraft from the Earth’scenter and μ is the gravitational parameter defined above.

2.2 Classical Lambert ProblemAs the perturbation-free, classical Lambert Problem will be

used for validation of the numerical simulations, a brief overviewof the analytical solution is warranted. The Lambert problem isa boundary value problem in time for the governing differentialequation for the spacecraft motion. The solution represents atrajectory that takes a spacecraft from an initial position P1 toa new position P2 in a specified time interval Δt (see Figure 1).According to Lambert’s Theorem [15], the time-of-flight dependsonly on the geometry of the space triangle formed between P1, P2

and the central body center of gravity, and the semi-major axisa of the elliptical path connecting the initial and final positions.This analytical result is expressed as

rμ

a3Δt = α − β − (sin α − sin β) (2)

where μ is a gravitational constant and a is the semi-major axisof the transfer ellipse. The angles α, β are defined by

sinα

2=

rs

2a, sin

β

2=

rs − c

2a(3)

Here, s is the semi-perimeter of the space triangle defined byP1, P2 and the Earth’s center, c is the chord length of the seg-ment P1P2, and a is the semi-major axis of the elliptical pathfrom P1 to P2. It is known that a is inversely proportional tothe total mechanical energy of the trajectory and thus is linkedwith the fuel requirements for the maneuver. An important corol-lary to Lambert’s Theorem is the existence of a minimum energytrajectory for which:

amin =s

2(4)

For a given semi-major axis a of the ellipse, the correspondinginitial velocity vector v1 can be determined according to [15]:

v1 =

rμ

4a

»„cot

α

2+ cot

β

2

«P2 − P1

c+

„cot

β

2− cot

α

2

«P1

|P1|–

(5)The inherent utility of Eq. (5) for this study is that the vectorcomponents of the evolved initial velocities for Classical Lambertproblems can be validated against these exact analytical results.For example, if a solution is sought that emphasizes positional ac-curacy and time-of-flight constraints (i.e., no consideration givento energy costs) one expects to recover the classical Lambert so-lution given by subsituting Eq. (3) into Eq. (2), solving non-linearly for a, and substituting a into Eq. (5) to solve for theinitial velocity vector v1 (note that in this case Δt, P1, and P2

are specified, so by the geometry of the problem s and c are alsoknown). Alternatively, if one seeks to emphasize positional ac-curacy and minimize energy consumption (i.e., no considerationgiven to time-of-flight), the minimum energy transfer orbit withamin given by Eq. (4) results. One can then substitute Eq. (4)into Eq. (5) to solve for the initial velocity vector v1. In this case,Δt can be solved for using Eqs. (2) and (3), given amin.

In reality, LEO trajectories are subject to non-negligible per-turbative effects – including gravitational variations due to plan-etary oblateness and rarefied atmospheric drag – that result innon-Keplerian motion. The details and forms of these perturba-tions are described below. A consequence of the non-Keplerianmotion is that the analytical results for the Lambert Problem areno longer valid.

2.3 Perturbation from Earth’s OblatenessA correction to the Newtonian gravitational potential is re-

quired to account for the fact that the Earth is actually an oblatespheroid, as a consequence of its axial rotation. This oblatenessgives rise to an axisymmetric gravitational potential that can beformally expressed in terms of a series expansion of zonal spheri-cally harmonics [16]. For the Earth, the oblateness perturbation

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can be well modeled by the addition of a single correction term,called the second zonal harmonic J2, to the spherically symmetricNewtonian potential. The corresponding perturbative accelera-tion aoblate is given by [16]:

aoblate =3

2

J2μR2E

|r|5»x

„5

z2

|r|2 − 1

«i + y

„5

z2

|r|2 − 1

«j

+ z

„5

z2

|r|2 − 3

«k

–(6)

where RE is the equatorial radius of the Earth and i, j,k are unitvectors aligned in the directions of the coordinate axes. The mag-nitude of this perturbation is zero at the equator and increaseswith latitude; further, the magnitude is seen to diminish withthe fourth-power of the altitude. Therefore, its effect is mostpronounced for LEO scenarios and especially those with high-latitude inclinations. A dynamical consequence of this perturba-tion is that a slow precession of the orbit results – that is, theorbit is no longer closed as in Keplerian motion. Although theprecession rate is typically no more that a few degrees per orbit,over the span of many orbits the impact of this perturbation effectbecomes significant.

2.4 LEO Atmospheric DragThe so-called ‘Karman line’, defined to be at an altitude of 100

km above the Earth, is commonly regarded as the boundary be-tween the Earth’s atmosphere and space. In reality, the density ofthe Earth’s atmosphere undergoes a significant decay beginningat an altitude of 20 km and a highly rarefied vestige of the atmo-sphere extends well beyond the Karman line and into the regionof LEO [19], as illustrated in Figure 2. As a consequence, space-craft in very low orbits are subject to a ballistic drag force fromthe residual atmosphere. While small in magnitude, the effect ofatmospheric drag is to reduce the velocity of an orbiting objectand, given sufficient time, leads to an orbital decay and eventualatmospheric reentry. A model of ballistic drag adrag is typicallyemployed to describe this effect and is given by [19]:

adrag = −1

2

ρ

B|v|2v (7)

where ρ is the altitude-dependent atmospheric density, v is theinstantaneous velocity vector and B is the ballistic coefficient forthe spacecraft geometry. As this effect is a retarding force, theacceleration is always negative and in the direction opposite ofthe current velocity.

50 100 150 200

10−6

10−4

10−2

100

Air

Den

sity

(lo

g−sc

ale

kg/m

3 )

Altitude (km)

Figure 2: A power law fit of atmospheric densitydata showing the density decay with altitude.

3. METHODS

3.1 Hybrid Evolutionary ApproachWe initially considered both CMA-ES [8] and DE [18], since

both are well-suited for evolving real-valued genomes. In pre-liminary experimentation, we found that both methods were of-ten successful. When CMA-ES was able to find a correct solu-tion, it was able to do so more quickly than DE. However, DE

more consistently found correct solutions whereas CMA-ES had agreater tendency to become trapped in local minima. Moreover,DE has been used successfully for other types of space missionplanning, as reviewed in Section 1. Thus, for the remainder ofthis study we opted to use DE (Matlab implementation availableat http://www1.icsi.berkeley.edu/~storn/code.html).

For all experiments reported here we used DE/rand/1/bin [18].For each solution vector in the current population (xi) a newvector (vi) is created as vi = xr0 + F (xr1 − xr2 ) where xr0 ,xr1 , and xr2 are distinct random vectors from the population.Next, a new trial vector ui is formed by performing crossover,randomly selecting each allele from either vi or xi based on acrossover probability Cr (a separate binomial trial is performedfor each allele). The target vector xi is replaced by ui in the nextgeneration, if the fitness of the latter is better than or equal tothe fitness of the former.

In this work, DE was used to find solutions in the correct basinof attraction of the global optimum. Once DE had terminated,we then refined the single best solution from the DE run us-ing Matlab’s fminsearch (a Nelder-Mead unconstrained nonlinearoptimization method). The use of the gradient-based solver per-mitted more rapid convergence to an accurate solution once apotential solution was in the correct basin of attraction.

3.2 Fitness FunctionThe final position of the spacecraft P2 is estimated by inte-

grating the governing equations using Matlab’s ode45 based onthe evolved intial velocity vector v1 and the prescribed or evolvedtime-of-flight Δt. Because the fitness function is necessarily basedon this simulation of flight trajectories, the time required to eval-uate fitness is proportional to the time of flight of the trajectorybeing simulated, which varies for different individuals in the pop-ulation. We created a single objective fitness function f to beminimized by weighting multiple objectives as follows:

f = w1‖(P2 − P2)‖ + w2‖(v1 − vp)‖2 + w3 min(C, 0) (8)

In the above, wi are the weights for each term and ‖...‖ denotesthe 2-norm of the vector (Euclidean distances between estimates

and targets). ‖(P2 − P2)‖ represents the positional error relativeto the target location P2. The square of the necessary change invelocity relative to the initial velocity ‖(v1 − vp)‖2 is propor-tional to the amount of energy required to change the velocity ofthe spacecraft for the new trajectory. The final term is a crash

penalty, where C is the maximum depth of P2 below the sur-face of the Earth (this term is only non-zero when the aircrafthas crashed, and helps provide a gradient back to the feasible re-gion). The values of the three weights wi vary with the particulartest problem (and are sometimes 0), as described in Section 3.4.

3.3 Test ProblemsA set of four test problems were identified for the application

of the evolutionary algorithm. Described below, these problemsconsist of both Keplerian motion, for which analytical solutionsexist and non-Keplerian motion:

Problem 1: Classical Lambert Problem for a Given Timeof Flight. In this problem only Newtonian physics areconsidered and the goal is to minimize positional error atP2 for some prespecified time of flight. Only the initialvelocity vector is evolved. The exact analytical solution forthis problem is known.

Problem 2: Classical Lambert Problem with MinimumEnergy. As above, only Newtonian physics are considered,however in this case the goal is to minimize the amount ofenergy necessary to reach P2. In this case, the time of flightalong the desired trajectory is unknown in advance, so boththe initial velocity vector and the time of flight are evolved.The exact analytical solution for this problem is known.

Problem 3: Intercept Trajectory Accounting for Oblate-ness and Drag. In this non-Keplerian problem, perturba-tions are introduced through the inclusion of the J2 correc-tion for non-spherical oblateness of the Earth (Eq. (6)) as

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well as the correction for atmospheric drag (Eq. (7)). Thetime of flight is specified, so only the initial velocity vectoris evolved. The goal is to minimize positional error at P2

for some prespecified time of flight. No analytical solutionexists for this problem.

Problem 4: Multi-Orbit Intercept Trajectory Account-ing for Oblateness. In this problem, the Keplerian orbitis perturbed through the inclusion of a J2 gravitational cor-rection for non-spherical oblateness of the Earth (Eq. (6)).A pre-specified time of flight is imposed that will requirethe spacecraft to complete multiple orbits of the Earth be-fore reaching the target position. While perturbations dueto atmospheric drag could be included as incorporated asin Test Problem 3, when drag is present it is impossible toguarantee that a trajectory exists that will reach the tar-get without entering the Earth’s atmosphere. Thus, in thisproof-of-concept study where we wish to quantify how oftenwe can find a correct solution, we neglected perturbationsdue to atmospheric drag for this problem to ensure that asolution existed. The time of flight is specified, so only theinitial velocity vector is evolved. The goal is to minimizepositional error at P2 for some prespecified time of flight.No analytical solution exists for this problem.

3.4 ExperimentsFor all experiments reported here, we used the same initial

and final locations. Centering a 3-dimensional Cartesian coordi-nate system at the center of the Earth, the initial position wasspecified to be P1 = 〈6500, 0, 0〉 km. This places the space-craft approximately 122 km above the surface of the Earth, arealistic value for a LEO. The final position was specified to beP2 = 〈−3591.7, 4024.3, 4024.3〉 km, which has a simple ClassicalLambert solution of v1 = 〈0, 5.6, 5.6〉 km/s, assuming a time offlight Δt = 30 min (as specified for test problems 1 and 3). Fortest problem 4, Δt = 479.895 min, which forces the trajectory toorbit the Earth 5 times before arriving at the target location P2.For test problem 2, vp is specified as 〈0, 1, 1〉 (vp is not neededfor the other 3 test problems) and the evolving Δt is preventedfrom going negative. Parameter values used for each of the fourtest problems are summarized in Table 1. Appropriate values forthe weights wi were determined empirically, since accurate nor-malization of the ranges of the terms is not possible. Note thatwe have set some weights to zero so that not all terms are usedin all test problems.

Table 1: Parameter Values for the 4 Test Problems(NA = not applicable).

Problem M Δt (min) w1 w2 w3

1 3 30 10 0 02 4 NA 1 25 03 3 30 100 0 1E+084 3 479.895 100 0 1E+08

The four test problems were each run for 12 repetitions fromdifferent initial random populations. DE runs were terminatedafter a maximum of 20 min of CPU time (on an HP Pavilion g7with an intel core i3 processor), but were allowed to terminateearly if either (i) the maximum absolute error of any dimensionin the evolving estimate of the intial velocity vector v1 was lessthan 0.01 km/s (for those cases where the true vi was known)or (ii) fitness f fell below 1e-9. We then applied fminsearch tothe best evolved solution for 200 iterations (with all tolerancesand other options left at the default settings). A trial was con-sidered successful if the final best solution was within 1 m of thetarget location (this is a very conservative criterion for real LEOmissions).

For the DE, we used a scaling parameter F = 0.85, a crossoverrate of Cr = 0.8, and population size of N = 5 × M , where Mis the number of objective variables. These parameter settingswere chosen based on limited preliminary experimentation using

recommended ranges in [18]. For example, we experimented withcrossover rates between 0.1 and 1.0 in increments of 0.1 and foundthat Cr = 0.8 performed the best for this problem (gave lowerfitnesses in a shorter amount of time). We also experimented withpopulation sizes N based on using either 5 or 10 individuals pergene, but did not see appreciable differences in resulting fitnessesand so went with the smaller population sizes.

4. RESULTSUpon examination of the results, three types of results were

apparent: (i) infeasible trajectories that ended up at P2 but thatintersected the Earth (this occurred only once, for test problem 1),(ii) trajectories that ended up hundreds or thousands of km fromP2 (this occurred in 7 of the 12 trials of the difficult mulit-orbittest problem 4), and (iii) feasible trajectories that ended close tothe target location P2 (in all but one case these were much lessthan 1 m from the target and were considered successful trials).

Histograms of the positional error ‖(P2 − P2)‖ are shown inFigure 3 for all solutions after DE (before fminsearch) within 50km of the target, and in Figure 4 for all solutions within 1 m of thetarget after refinement using fminsearch. Quantitative results forall successful runs of DE + fminsearch are summarized in Table2. Positional errors represent the average Euclidean distance (in

m) between P2 and P2 for all successful runs.For test problems 1,2, and 3, we were able to find successful

solutions in all but two trials (Table 2). In the one unsuccessfultrial for problem 1, the solution was in a local optimum that sentthe spacecraft on a trajectory through the Earth (note that thiscould have been precluded if we had explicitly formulated thefitness function to guard against such trajectories, but we optedto check for this in post-processing rather than further slow downthe fitness function). In the one trial considered unsuccesful forproblem 2, the final positional error was still only 4.6 m fromthe target; this level of error may actually be within tolerance,depending on the mission, and probably could have been further

0 5 10 15 20 25 30 35 400

5

a) Test Problem 1

0 5 10 15 20 25 30 35 400

5

10

b) Test Problem 2

0 5 10 15 20 25 30 35 400

2

4

c) Test Problem 3

0 5 10 15 20 25 30 35 400

1

2

positional error after DE alone (km)

succ

essf

ul r

uns

out o

f 12

d) Test Problem 4

Figure 3: Histograms of the positional error‖(P2 − P2)‖ (in km) after DE but before refinementwith fminsearch, for all trials that were within 50km of the target after DE (out of 12 trials for eachof the four test problems.

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0 0.01 0.02 0.03 0.04 0.05 0.06

123

a) Test Problem 1

0 0.05 0.1 0.15 0.2 0.25 0.31357

b) Test Problem 2

0 0.001 0.003 0.005 0.007

1

3

5

succ

essf

ul r

uns

out o

f 12

c) Test Problem 3

0 0.0005 0.0010 0.0015 0.0020 0.0025 0.00300

1

2

3

final positional error after DE + fminsearch (m)

d) Test Problem 4

Figure 4: Histograms of the positional error‖(P2 − P2)‖ (in m) after DE followed by refinementwith fminsearch, for all trials that were within 1 mof the target after DE + fminsearch (out of 12 trialsfor each of the four test problems).

reduced with additional iterations of fminsearch, although we didnot try the latter. For the more difficult test problem 4, we werestill able to find successful solutions in 5 out of 12 trials (Table2).

If one views the 12 trials as restarts, the best solutions foundare quite impressive, as summarized in the rightmost column ofTable 2. Here, it can be seen that final positional errors of the bestsolutions found were negligible. For test problem 1 the analyticalsolution for the initial velocity was recovered to within 9 decimalplaces of accuracy (in km/s), and for test problem 2 the analyticalsolution for the optimal time-of-flight was recovered to within4.8s.

Table 2: Number of successful trials (out of 12), av-erage positional of successful trials, positional errorof best trial out of 12.

Test Successful Avg Pos Err Best Pos ErrProblem Trials (m) (m)

1 11 0.023 9.3e-32 11 0.071 6.7e-63 12 0.003 9.6e-44 5 0.002 1.0e-3

The trajectories for all successful runs for Test Problems 1, 2,and 3 are shown in Figures 5, 6, and 7, respectively. In somecases trajectories lie on top of each other and are not visuallydistinguishable. For Test Problem 4 we show all 5 orbits of onlythe single best trajectory in Figure 8; note the precession of theorbits toward the target point P2. The right-hand panels of Fig-ures 5-8 show close-ups of the successful trajectories after DE butbefore refinement with fminsearch (note the zoomed in scales ofthe axes). After refinement with fminsearch, the successful tra-jectories are all < 0.3 m from the target (Figure 4), so they wouldappear indistinguishable on these figures.

5. DISCUSSIONThe focus of this study has been to investigate the utility of a

DE-based approach for spacecraft trajectory planning under therealistic orbital conditions that would be present in the LEO en-vironment. With this in mind, it is appropriate to examine boththe performance and limitations of the present model within thecontext of actual mission planning performed by space agenciessuch as NASA.

5.1 Traditional Mission Planning ApproachesHistorically, the computational design of mission trajectories

has been based on problem-specific algorithms that employedclassical numerical approaches for orbital mechanics (e.g. boundary-value problems [7]). Within the past decade or so, however, therehas been an effort to develop more generalized and robust tra-jectory design approaches including optimization. Johnson et al.[12] developed the Copernicus program for NASA Johnson SpaceCenter that incorporated many existing case-specific NASA tra-jectory codes into a single mission design and optimization tool.Beginning in the 1990s, modern dynamical systems approachesusing invariant manifolds were introduced for mission planningin perturbed systems [10]; this approach was realized in missionplanning for the NASA Genesis mission launched in 2001 [14].Most recently, NASA Goddard Space Flight Center has intro-duced the Evolutionary Mission Trajectory Generator softwarewhich designs and optimizes complex interplanetary trajectoriesinvolving multiple planetary flybys using a GA [5]. However, todate these methods have largely focused on planning orbital tra-jectories for problems where near-Earth perturbations are negli-gible.

With the recent increase in small satellites in LEO, such as‘cubesats’ and ‘nanosats’, it is important to extend automateddesign approaches to include realistic LEO perturbations. Thecurrent study serves as proof-of-concept that DE can be effectivein automating accurate LEO trajectory planning, which repre-sents a new contribution to the current astrodynamics literature.

5.2 Trajectory AccuracyIn real LEO maneuvers, the required positional accuracy for

modern trajectory plannign is on the order of meters (Jacob Eng-lander, personal communication). Of course, the specific errortolerance is necessarily dependent on the particular mission. Inthe results presented here, all succesful runs had positional errorsof less than 0.3 m (with best runs out of 12 trials within 0.01m of the target position) for all four problems (Table 2); thesepositional errors are well within the required tolerances for LEOmissions.

5.3 Opportunities for Model ImprovementAlthough the computational model presented in this work has

incorporated a number of realistic and important perturbationsfor LEO maneuvers, there remain opportunities for further im-provement, including: (a) incorporating additional perturbations;(b) solving more difficult LEO problems, such as multi-impulse,trajectories, continuous thrust trajectories, and multi-spacecraftswarm trajectories; and (c) evolving Pareto sets of non-dominatedsolutions with respect to the multiple objectives. We discuss thesethree areas below.

There are two particular sources of perturbation that are notaccounted for in the present model. The first is the effect of solarradiation (photon) pressure, which can either accelerate or decel-erate the spacecraft depending on its orientation. For very loworbits, this effect is small compared to atmospheric drag; how-ever, for higher orbits the reverse is true. The second source ofperturbation is caused by gravitational effects of the Moon. Thiseffect is negligible for LEO scenarios but is relevant for higher or-bits (e.g., geosynchronous orbits, GEO). Therefore, to extend thecapabilities of the current framework beyond LEO, the inclusionof lunar gravity will be required.

The current version of the model is restricted to trajectoriesproduced by a single initial impulse. While this is acceptablefor preliminary orbit design, it excludes the possibility of mid-

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course correction(s) to account for perturbative effects. A moreflexible and realistic approach would be to allow for a finite num-ber of impulses during the trajectory. One could also allow forthe possibility of a continuous thrust trajectory, where the space-craft produces a thrust during the entire maneuver. Continuousthrust maneuvers are consistent with spacecraft utilizing electricpropulsion systems (e.g., ion engines) as well as solar sails.

Another area of interest is the design of trajectories of smallswarms of LEO spacecraft flying together; these spacecraft mustnot only reach a particular target area, but must also remainwithin certain distances of each other and, in some cases, maintainspecific formations (as in the Magnetosspheric Multiscale (MMS)mission http://mms.gsfc.nasa.gov/). We have recently begunwork on this challenging problem.

In the current approach we chose to combine the multiple ob-jectives into a single weighted fitness function, and have empha-sized trajectory planning where the primary objective is eithertime-of-flight or fuel considerations. Realistic mission planningcould benefit from a multi-objective version of DE (e.g., [3]) thatreturns a relatively uniformly spaced set of non-dominated solu-tions. Mission planners could then consider the trade-offs betweenthe various objectives and constraints in selecting an appropriatetrajectory to implement.

In the proof-of-concept study described here, we implementedthe model in Matlab and allowed each trial of DE to evolve so-lutions for 20 minutes each on a laptop computer. In the con-text of real mission planning, this is an insignificant amount ofcomputation, especially given the computational capabilities atplaces such as NASA Goddard. However, there is no doubt thatthe efficiency of the code could also be improved dramatically innumerous ways, such as converting from Matlab to C or C++.Moreover, for real mission planning one should use a higher fi-delity forward trajectory simulator, such as the publically avail-able GMAT code (http://gmat.gsfc.nasa.gov/), and a higherorder ordinary differential equation solver.

6. CONCLUSIONSIn this study, Differential Evolution (DE) was investigated as

a tool for the design of intercept and rendezvous problems in lowEarth orbit (LEO). The accuracy of the technique was demon-strated in two test cases involving Keplerian orbits that could bebenchmarked against known solutions. The method was then ap-plied to two cases involving non-Keplerian orbits due to two typesof realistic perturbations encountered in LEO. The DE was foundto be very robust in that it was rarely trapped in local optima andwas often able to get very close enough to the global optimum thatsubsequent refinement with Nelder-Mead optimization was ableto reduce final positional errors to within less than 0.3 m, whichis well within the acceptable tolerance for real LEO mission plan-ning. More challenging problems, such as the multi-orbit problemstudied here, may require multiple restarts. However, even in thisdifficult problem we were successful in 5 out of 12 trials. Becausewe only evolve the initial velocity (and possibly time of flight), ad-ditional complex physics effects can be readily incorporated intofitness evaluation without affecting the remainder of the evolu-tionary code. We conclude that a hybrid method using DE tofind the global basin of attraction, followed by refinement with alocal optimization method such as Nelder-Mead to hone in on theglobal optimum, is a promising approach to mission design andoptimization of LEO trajectories.

7. ACKNOWLEDGEMENTSThis work was partially supported by the Vermont Space Grant

under NASA Cooperative Agreement # NNX10AK674. We thankJacob Englander (NASA Goddard) for helpful discussions thatimproved this manuscript.

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Figure 5: Trajectories of the 11 successful runs for Test Problem 1. The analytical trajectory is shown inblack. Left: Earth scale view. Right: Close up of the best solutions from DE (before fminsearch) near targetP2.

Figure 6: Trajectories of the 11 successful runs for Test Problem 2. The analytical trajectory is shown inblack. Left: Earth scale view. Right: Close up of the best solutions from DE (before fminsearch) near targetP2.

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Figure 7: Trajectories of the 12 successful runs for Test Problem 3. Left: Earth scale view. Right: Close upof the best solutions from DE (before fminsearch) near target P2.

Figure 8: Trajectory of most successful run for Test Problem 4. To avoid confusion, only the 5 orbits of themost successful trial are shown. Left: Earth scale view. Right: Close up of the best solution from DE (beforefminsearch) near target P2.

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