OPTIMIZATION OF VERY-LOW-THRUST TRAJECTORIES USING ... · methods that are based on the calculus of...

55th International Astronautical Congress 2004 - Vancouver, Canada

OPTIMIZATION OF VERY-LOW-THRUST TRAJECTORIES

USING EVOLUTIONARY NEUROCONTROL

Bernd Dachwald

German Aerospace Center (DLR), Institute of Space Simulation, Linder Hoehe, 51147 Cologne

[email protected]

ABSTRACT

Searching optimal interplanetary trajectories for low-thrust spacecraft is usually a dif-

ficult and time-consuming task that involves much experience and expert knowledge in

astrodynamics and optimal control theory. This is because the convergence behavior

of traditional local optimizers, which are based on numerical optimal control methods,

depends on an adequate initial guess, which is often hard to find, especially for very-

low-thrust trajectories that necessitate many revolutions around the sun. The obtained

solutions are typically close to the initial guess that is rarely close to the (unknown) global

optimum. Within this paper, trajectory optimization problems are attacked from the per-

spective of artificial intelligence and machine learning. Inspired by natural archetypes, a

smart global method for low-thrust trajectory optimization is proposed that fuses artificial

neural networks and evolutionary algorithms into so-called evolutionary neurocontrollers.

This novel method runs without an initial guess and does not require the attendance of

an expert in astrodynamics and optimal control theory. This paper details how evolu-

tionary neurocontrol works and how it could be implemented. The performance of the

method is assessed for three different interplanetary missions with a thrust to mass ratio

< 0.15mN/kg (solar sail and nuclear electric).

INTRODUCTION

This paper deals with the problem of searching op-timal interplanetary trajectories for very-low-thrustspacecraft. Two propulsion systems are considered:solar sails (large ultra-lightweight reflecting surfacesthat utilize solely the freely available solar radia-tion pressure for propulsion), and a nuclear elec-tric propulsion (NEP) system with constant thrustand specific impulse. The optimality of a trajectorycan be defined according to several objectives, liketransfer time or propellant consumption. Becausesolar sails do not consume any propellant, theirtrajectories are typically optimized with respect totransfer time alone. Trajectory optimization forspacecraft with a SEP system is less straightfor-ward because transfer time minimization and pro-pellant minimization are mostly competing objec-tives, so that one objective can only be optimizedat the cost of the other objective. Spacecraft tra-jectories can also be classified with respect to theterminal constraint. If, at arrival, the position rSC

and the velocity rSC of the spacecraft must matchthat of the target (rT and rT, respectively), one hasa rendezvous problem. If only the position mustmatch, one has a flyby problem. A spacecraft tra-jectory is obtained from the (numerical) integra-tion of the spacecraft’s equations of motion. Be-sides the inalterable external forces, the trajectoryxSC[t] = (rSC[t], rSC[t]) is determined entirely by thevariation of the thrust vector (’[t]’ denotes the timehistory of the preceding variable). The thrust vec-tor F(t) of low-thrust propulsion systems is a con-tinuous function of time. It is manipulated throughthe nu-dimensional spacecraft control function u(t)that is also a continuous function of time. The tra-jectory optimization problem is to find the optimalspacecraft control function u?(t) that yields the op-timal trajectory x?

SC[t]. This problem can not besolved except for very simple cases. What can besolved, at least numerically, however, is a discreteapproximation of the problem. Dividing the allowedtransfer time interval [t0, tf,max] into τ finite ele-ments, the discrete trajectory optimization problemis then to find the optimal spacecraft control his-

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IAC-04-A.6.03


tory u?[t] ∈ Rnuτ that yields the optimal trajectoryx?

SC[t] (the symbol t denotes a discrete time step;note that only the spacecraft control function isdiscrete, whereas the trajectory is still continuous).Through discretization, the problem of finding theoptimal function u?(t) in infinite-dimensional func-tion space is reduced to the problem of finding theoptimal control history u?[t] in a nuτ -dimensionalparameter space (a space which is usually still veryhigh-dimensional). For optimality, some cost func-tion J must be minimized. If the propellant massmP is to be minimized, J = mP(t0)−mP(tf ) = ∆mP

is an appropriate cost function, if the transfer timeis to be minimized, J = tf − t0 = T is an appropri-ate cost function.

TRADITIONAL LOCAL LOW-THRUSTTRAJECTORY OPTIMIZATION METHODS

Traditionally, low-thrust trajectories are optimizedby the application of numerical optimal controlmethods that are based on the calculus of varia-tions. These methods can be divided into directmethods, such as nonlinear programming (NLP)methods, and indirect methods, such as neighbor-ing extremal methods and gradient methods. Allthese methods can generally be classified as localtrajectory optimization methods (LTOMs), wherethe term optimization does not mean finding thebest solution but rather finding a solution [1]. Priorto optimization, the NLP methods and the gradientmethods require an initial guess for the control his-tory u[t], whereas the neighboring extremal meth-ods require an initial guess for the starting adjointvector of Lagrange multipliers λλλ(t0) (costate vec-tor) [2]. Unfortunately, the convergence behaviorof LTOMs (especially of indirect methods) is verysensitive to the initial guess, so that an adequateinitial guess is often hard to find, even for an ex-pert in astrodynamics and optimal control theory.Similar initial guesses often produce very dissim-ilar optimization results, so that the initial guesscan not be improved iteratively and trajectory op-timization becomes more of an art than science [3].Even if the optimizer finally converges to an op-timal trajectory, this trajectory is typically closeto the initial guess that is rarely close to the (un-known) global optimum. Because the optimizationprocess requires nearly permanent attendance of theexpert, the search for a good trajectory can becomevery time-consuming and expensive. Another draw-back of LTOMs is the fact that the initial conditions(launch date, initial propellant mass, hyperbolic ex-cess velocity vector, etc.) – although they are cru-cial for mission performance – are generally chosen

according to the expert’s judgment and are there-fore not part of the actual optimization method.

EVOLUTIONARY NEUROCONTROL: ASMART GLOBAL LOW-THRUST

TRAJECTORY OPTIMIZATION METHOD

To evade the drawbacks of LTOMs, a smart globaltrajectory optimization method (GTOM) was de-veloped by the author [4]. This method was termedInTrance, which stands for Intelligent Trajectoryoptimization using neurocontroller evolution. Tofind a near-globally1 optimal trajectory, InTrancerequires only the target body and intervals for theinitial conditions as input. Implementing evolution-ary neurocontrol (ENC), InTrance runs without aninitial guess and does not require the attendanceof a trajectory optimization expert. The remainderof this section will sketch the motivation for ENCand explain the underlying concepts, as well as theapplication of ENC to solve low-thrust trajectoryoptimization problems.

1. Motivation for Evolutionary Neurocontrol

ENC fuses artificial neural networks (ANNs) andevolutionary algorithms (EAs) to so-called evolu-tionary neurocontrollers (ENCs). Like the underly-ing concepts, it is inspired by the natural processesof information processing and optimization. Ani-mal nervous systems incorporate natural evolution-ary neurocontrollers to control their actions, givingthem marvelous capabilities. The smart flight con-trol system of the housefly might provide an exam-ple. The nervous system of the housefly comprisesabout 105 neurons. This small natural neural net-work manages the flight control of the fly, as wellas many even more difficult tasks. Nature has op-timized the performance of the fly’s neurocontrolleron this tasks through the recombination and muta-tion of the fly’s genetic material and through nat-ural selection: smarter flies produce more offspringand there is a high probability that some of themare even smarter than their parents. So, if a naturalevolutionary neurocontroller is able to steer a house-fly optimally from A to B, why should an artificialevolutionary neurocontroller not be able to steer aspacecraft optimally from A to B, which seems to bea much simpler problem? The remainder of this sec-tion will describe how such an artificial evolutionaryneurocontroller could be implemented.

1Near -globally optimal because global optimality canrarely be proved for real-world problems.

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2. Machine Learning

Within the field of artificial intelligence, one im-portant and difficult class of learning problems arereinforcement learning problems, where the opti-mal behavior of the learning system (called agent)has to be learned solely through interaction withthe environment, which gives an immediate or de-layed evaluation2 J (also called reward or reinforce-ment) [5, 6]. The agent’s behavior – it’s strategy –is defined by an associative mapping from situationsto actions S : X 7→ A3. The optimal strategy S? ofthe agent is defined as the one that maximizes thesum of positive reinforcements and minimizes thesum of negative reinforcements over time. If, givena situation X ∈ X , the agent tries an action A ∈ Aand the environment immediately returns a scalarevaluation J(X, A) of the (X, A) pair, one has animmediate reinforcement learning problem. A moredifficult class of learning problems are delayed re-inforcement learning problems, where the environ-ment gives only a single evaluation J(X, A)[t], col-lectively for the sequence of (X, A) pairs occurringin time during the agent’s operation.

From the perspective of machine learning, aspacecraft steering strategy may be defined as anassociative mapping S that gives – at any time alongthe trajectory – the current spacecraft control u(t)from some input X(t) ∈ X that comprises the vari-ables that are relevant for the optimal steering ofthe spacecraft (the current state of the relevant en-vironment). Because the trajectory is the result ofthe spacecraft steering strategy, the trajectory op-timization problem is actually a problem of findingthe optimal spacecraft steering strategy S?. This isa delayed reinforcement problem because a space-craft steering strategy can not be evaluated beforeits trajectory is known. Only then a reward can begiven according to the fulfillment of the optimiza-tion objective(s) and constraints. One obvious wayto implement spacecraft steering strategies is to useartificial neural networks because they have beensuccessfully applied to learn associative mappingsfor a wide range of problems.

3. Artificial Neural Networks and Neurocon-trol

Being inspired by the processing of information inanimal nervous systems, ANNs are a computabil-ity paradigm that is alternative to conventional se-rial digital computers. ANNs are massively paral-

2This evaluation is analogous to the cost function in opti-mal control theory. To emphasize this fact, it will be denotedby the same symbol, J .

3X is called state space and A is called action space.

lel, analog, fault tolerant, and adaptive [7]. Theyare composed of processing elements (called neu-rons) that model the most elementary functions ofbiological neurons. Linked together, those elementsshow some characteristics of the brain, for example,learning from experience, generalizing from previ-ous examples to new ones and extracting essentialcharacteristics from inputs containing noisy and/orirrelevant data, so that they are relatively insen-sitive to minor variations in its input to produceconsistent output [8].

Because the neurons can be connected in manyways, ANNs exist in a wide variety. Here, however,only feedforward ANNs are considered. Feedfor-ward ANNs have typically a layered topology, wherethe neurons are organized in a number of neuronlayers. The first neuron layer is called the inputlayer and has ni input neurons that receive the net-work’s input. The last neuron layer is called theoutput layer and has no output neurons that pro-vide the network’s output. All intermediate lay-ers/neurons are called hidden layers/neurons. Afeedforward ANN, as it is used here, can be regardedas a continuous parameterized function (called net-work function)

Nπππ : X ⊆ Rni → Y ⊆ (0, 1)no

that maps from a set of inputs X onto a set of out-puts Y. The parameter set πππ = {π1, . . . , πm} ofthe network function comprises the m internal pa-rameters of the ANN (the weights of the neuronconnections and the biases of the neurons).

ANNs have been successfully applied as neuro-controllers (NCs) to reinforcement learning prob-lems [8]. An ANN controls a dynamical systemby providing a control Y(t) ∈ Y from some inputX(t) ∈ X that contains the relevant informationfor the control task. Note that the NC’s behavior iscompletely characterized by its network function Nπππ

(that is again completely characterized by its para-meter set πππ). If the correct output is known for aset of given inputs (the training set), the differencebetween the given output and the correct outputcan be utilized to learn the optimal network func-tion N? := Nπππ? by adapting πππ in a way that mini-mizes this difference for all input/output pairs in thetraining set. A variety of learning algorithms hasbeen developed for this kind of learning, the back-propagation algorithm – a gradient-based method –being the most widely known. Unfortunately, learn-ing algorithms that rely on a training set fail whenthe correct output for a given input is not known,as it is the case for delayed reinforcement learningproblems. The next section will address a learning

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method that may be used for determining N? in thiscase.

4. Evolutionary Algorithms and EvolutionaryNeurocontrol

EAs are robust methods for finding global optimain very high dimensional search spaces. They havebeen successfully applied as a learning method forANNs, as well as for a wide range of other opti-mization problems [9, 10, 11]. EAs use a vocab-ulary borrowed from biology. The key element ofan EA is a population that comprises numerous in-dividuals ξj∈{1,...,q}, which are potential solutionsfor the given optimization problem. All individu-als of the (initially randomly created) populationare evaluated according to a fitness function4 J fortheir suitability to solve the problem. The fitnessof each individual J(ξj) is crucial for its probabilityto reproduce and to create offspring into a newlycreated population because fitter individuals are se-lected with a greater probability for reproductionthan less fit ones. The selected parents undergoa series of genetic transformations (mutation, re-combination) to produce offspring that consists ofa mixture of the parents genetic material. Underthis selection pressure, the individuals – also calledchromosomes or strings – strive for survival. Aftersome reproduction cycles, the population convergesagainst a single solution ξ?, which is in the best casethe globally optimal solution for the given problem.EAs can be employed for searching the NC’s opti-mal network function because a NC parameter setcan be mapped onto a real-valued string that pro-vides an equivalent description of the network func-tion. By searching for the fittest individual, theEA searches for the optimal spacecraft trajectory.Fig. 1 sketches the transformation of a chromosomeinto a trajectory.

5. Neurocontroller Input and Output

Two fundamental questions concerning the utiliza-tion of a NC for spacecraft steering are: (1) ”Whatinput should the NC get?” (or ”What should the NCknow to steer the spacecraft?”) and (2) ”What out-put should the NC give?” (or ”What should the NCdo to steer the spacecraft?”). To be robust, a space-craft steering strategy should be time-independent:to determine the currently optimal spacecraft con-trol u(ti), the spacecraft steering strategy should

4This fitness function is also analogous to the cost func-tion in optimal control theory. To emphasize this fact, it willbe denoted by the same symbol, J .

��

��

chromosome/individual/string ξ=

NC parameter set πππ

��

��

NC network function N=

spacecraft steering strategy S

�� spacecraft control function u[t]

�� spacecraft trajectory xSC[t]

?

?

?

Fig. 1: Transformation of a chromosome into a trajec-tory

have to know – at any time step ti – only the cur-rent spacecraft state xSC(ti) and the current targetstate xT(ti), hence S : X = {(xSC,xT)} 7→ {u}. Ifa propulsion system other than a solar sail is em-ployed, the current propellant mass mP(ti) mightbe considered as an additional input, S : X ={(xSC,xT,mP)} 7→ {u}. The number of potentialinput sets, however, is still large because xSC andxT may be given in coordinates of any referenceframe and in combinations of them. The differencexT−xSC may be used as well, also in coordinates ofany reference frame and in combinations of them.Two potential input sets are depicted in Figs. 2 and3.

Fig. 2: Example for a NC that implements a solar sailsteering strategy

Each output neuron gives a value Yi ∈ (0, 1). Thenumber of potential output sets is also large becausethere are many alternatives to define u, and to cal-culate u from Y. The following approach gave goodresults for the majority of problems: the NC pro-vides a three-dimensional output vector d′′ ∈ (0, 1)3

from which a unit vector d in the desired thrust di-

4


Fig. 3: Example for a NC that implements an SEPspacecraft steering strategy

rection (called direction unit vector) is calculatedvia

d′ = 2d′′ −

111

∈ (−1, 1)3 (1)

and

d = d′/|d′| (2)

For solar sails, u = d, hence S : {(xSC,xT)} 7→ {d}(see Fig. 2). For SEP spacecraft, the output mustinclude the engine throttle χ, so that u = (d, χ),hence S : {(xSC,xT,mP)} 7→ {d, χ} (see Fig. 3).

6. Evolutionary Neurocontroller Design

Fig. 4 shows how an ENC may be applied for low-thrust trajectory optimization. To find the optimalspacecraft trajectory, the ENC method runs in twoloops.

Fig. 4: Low-thrust trajectory optimization using ENC

Within the (inner) trajectory integration loop, aNC steers the spacecraft according to its network

function Nπππj that is completely defined by the NC’sparameter set πππj . The EA in the (outer) NC opti-mization loop holds a population of NC parametersets, Ξ = {πππ1, . . . ,πππq}, and examines all of them fortheir suitability to generate an optimal trajectory.Within the trajectory optimization loop, the NCtakes the current spacecraft state xSC(ti∈{0,...,τ−1})and that of the target xT(ti) as input, and mapsthem onto some output. For SEP spacecraft, theinput includes the current propellant mass mP(ti)and the output includes the current throttle χ(ti).The first three output values are interpreted as thecomponents of d′′(ti), from which the direction unitvector d(ti) is calculated via Eq. (1) and (2). Now,the spacecraft control u(ti) is calculated from theNC output. Then, xSC(ti) and u(ti) are insertedinto the equations of motion and (numerically) in-tegrated over one time step ∆t = ti+1 − ti to yieldxSC(ti+1). The new state is fed back into the NC.The trajectory integration loop stops when the ac-curacy of the trajectory is sufficient or when a giventime limit is reached. Then, back in the NC opti-mization loop, the NC’s trajectory is rated by thefitness function J(πππj). The fitness of πππj is crucial forits probability to reproduce and to create offspring.Under this selection pressure, the EA breeds moreand more suitable steering strategies that generatebetter and better trajectories. Finally, the EA con-verges against a single steering strategy, which givesin the best case a near-globally optimal trajectoryx?

SC[t].

7. Additionally Encoded Problem Parameters

If an EA is already employed for the optimiza-tion of the NC, it is manifest to use it also forthe co-optimization of additional problem parame-ters. This can be done without major additionaleffort. InTrance encodes the following parametersadditionally on the chromosome, making them anexplicit part of the optimization problem: (1) thelaunch date, (2) the launch velocity vector (hyper-bolic excess velocity vector), and (3) the initial pro-pellant mass (except for solar sails).

RESULTS

Within this section, ENC (InTrance) is applied tothree interplanetary very-low-thrust trajectory op-timization problems. The first problem is a near-Earth asteroid rendezvous using a solar sail (witha characteristic acceleration5 of ac = 0.14 mm/s2),the second problem is a MESSENGER-like mis-sion to Mercury but using a solar sail (with ac =

5maximum acceleration at Earth distance from the sun

5


0.1 mm/s2), and the third problem is a Jupiter flybyusing a NEP spacecraft (with an acceleration – orthrust to mass ratio – of 0.0357 mm/s2 < a(t) =F/m(t) ≤ 0.05 mm/s2).

1. Near-Earth Asteroid Rendezvous MissionUsing a Solar Sail

Within this section, the convergence behavior ofENC and the quality of the obtained solutions isassessed for an exemplary rendezvous mission to anear-Earth asteroid (1996FG3). For solar sailcraftwith ac = 0.14 mm/s2 (ideal6 50 m×50 m solar sail,launch mass 148 kg, useful mass7 75 kg) a trajectorywas calculated in [12, 13] using a LTOM. This ref-erence trajectory launches from Earth on 13Aug 06and takes 1640 days to rendezvous 1996FG3, if thesolar sailcraft is inserted directly into an interplane-tary trajectory with an hyperbolic excess energy ofC3 = 4km2/s2.

In the first experiment, InTrance was run fivetimes for the reference launch date but with zerohyperbolic excess energy. The maximum transfertime was set to Tmax = 1 800 days. For discretiza-tion, this time interval was cut into τ = 360 finiteelements of equal length, so that the solar sail isallowed to change its attitude once every 5 days.The accuracy limit for the distance and the rela-tive velocity at the target was set to ∆rf,max =0.3 · 106 km and ∆vf,max = 0.1 km/s, respectively,which is compatible with the reference trajectory.The best found InTrance-trajectory (Fig. 5) is 135days faster than the reference trajectory, while re-ducing at the same time the C3-requirement from4 km2/s2 to 0 km2/s2, thus permitting a reduc-tion of the launcher requirements and eventuallyof launch costs. The final distance to 1996FG3 is∆rf = 0.200 · 106 km and the final relative velocityis ∆vf = 0.065 km/s, both being better than therequired values.

In the second experiment, InTrance was used tofind the optimal launch date for the 1996FG3 ren-dezvous problem (with C3 = 0 km2/s2). Fig. 6(a)shows the solutions of five InTrance runs with dif-ferent initial NC populations. The small vari-ance of the five solutions gives evidence for a goodconvergence behavior of ENC. Taking 1435 daysto rendezvous 1996FG3, the best found trajec-tory (Fig. 6(b)) is 205 days faster than the refer-ence trajectory (∆rf = 0.267 · 106 km and ∆vf =0.089 km/s). The optimal launch date was foundto be 22Oct 05, 295 days earlier than the referencelaunch date.

6where the incident radiation is reflected specularly7spacecraft bus plus scientific payload

Fig. 5: Best InTrance-trajectory for the 1996FG3 ren-dezvous (reference launch date)

(a) Trajectories for five different initial populations

(b) Best InTrance-trajectory

Fig. 6: 1996FG3 rendezvous (optimized launch date)

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The third experiment was to find out whethera given C3 of 4 km2/s2 could be spent more effi-ciently than done by the reference trajectory. Theoptimal launch date for this problem was found tobe 12 Feb 06, a half year earlier than the referencelaunch date. Fig. 7 shows the best found trajec-tory. It takes only 944 days to rendezvous 1996FG3,being 696 days faster than the reference trajectory(∆rf = 0.272 · 106 km and ∆vf = 0.091 km/s). Forthis calculation, the solar sail was allowed to changeits attitude once every 4 days (Tmax = 1000 days,τ = 250).

Fig. 7: 1996FG3 rendezvous (optimized launch date):Best InTrance-trajectory for C3 = 4km2/s2

2. Mercury Rendezvous Mission Using a SolarSail

Within this section, a rendezvous mission to Mer-cury is assessed, comparable to the MESSENGERmission (same spacecraft dry mass to Mercury, ap-proximately same transfer time) but using a so-lar sail instead of the chemical propulsion sys-tem. MESSENGER makes an Earth-Venus-Venus-Mercury-Mercury-Mercury gravity assist and fivedeep space maneuvers to reach the planet withinabout 6.6 years. Table 1 gives the most importantmission parameters [14].

InTrance was used to derive the performance re-quirements for solar sails that would be able totransport a spacecraft with the dry mass of MES-SENGER to Mercury within approximately thesame transfer time. The maximum transfer timewas set to Tmax = 2600 days. For discretization,this time interval was cut into τ = 520 finite ele-ments of equal length, so that the solar sail is al-lowed to change its attitude once every 5 days. The

Spacecraft dry mass 499.4 kg

Launch 02 Aug 04

Launch mass 1099.5 kg

Launcher Delta II, Model 7925-H

C3 16.8 km2/s2

Gravity assists Earth-(2×)Venus-

(3×)Mercury

Transfer Time 6.623 years

Mercury orbit insertion 18 Mar 11

Table 1: MESSENGER mission parameters [14]

accuracy limit for the distance and the relative ve-locity at the target was set to ∆rf,max = 106 kmand ∆vf,max = 0.25 km/s, respectively. For a faircomparison, MESSENGER’s C3 was also used forthe solar sail option. Fig. 8 shows a possible solarsail trajectory for such a mission.

Fig. 8: Trajectory for MESSENGER-like mission toMercury using a solar sail

Fig. 8 demonstrates that a solar sail with a mod-erate characteristic acceleration of 0.1 mm/s2 is ableto reach Mercury within approximately the MES-SENGER transfer time, if it is injected with thesame hyperbolic excess energy of 16.8 km2/s2. Incontrast to the chemical mission scenario, no grav-ity assist maneuver is necessary to achieve the re-quired ∆V , which results in a more flexible missionprofile, with a practically permanent launch win-dow [4]. Table 2 summarizes the significant missionparameters.

The launch date for the solar sail mission wasnot optimized but adopted from the actual MES-SENGER mission. An optimization of the launchdate might yield a slightly shorter flight time. TheMESSENGER-trajectory, however, is also not glob-

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Solar sail payload 499.4 kg

Launch 02Aug 04

Characteristic acceleration 0.1mm/s2

C3 16.8 km2/s2

Gravity assists –

Transfer Time 5.914 years

Mercury orbit insertion 02 Sep 10

Table 2: Significant parameters for a MESSENGER-like mission using a solar sail

ally optimal because the originally envisaged launchwindow (10 Mar 04) was missed. A launch on10 Mar 04 with C3 = 15.2 km2/s2 would have re-sulted in a shorter flight time of 5.5 years, leadingto arrival at Mercury on 06 Apr 09 [15].

Assuming a quadratic solar sail, Fig. 9 gives thesail side length s that is required to achieve ac =0.1 mm/s2 for different sail assembly8 loadings σSA

and payload masses mPL, where the bold line relatesto MESSENGER’s dry mass of 500 kg.

Fig. 9: Solar sail technology requirements forMESSENGER-like mission to Mercury using a solar sail

It can be seen that the required sail size increasesdrastically for σSA ' 45 g/m2, and that it ap-proaches infinity for σSA ≈ 83 g/m2 because the so-lar sail would be to heavy to achieve ac = 0.1 mm/s2

even without any payload. Table 3 gives values forfive points on the 500 kg-curve.

It can be seen that the sail assembly loading mustonly be / 45 g/m2 to yield a benefit in launch masswith respect to the MESSENGER mission. For anadvanced solar sail with a low sail assembly loadingof about 10 g/m2 the launch mass is only about onehalf of the actual MESSENGER launch mass.

8the sail film and the required structure for storing, de-ploying and tensioning the sail

sail sail sail launch

assembly size assembly mass

loading mass

σSA

�g/m2

�s [m] mSA [kg] m [kg]

10 83 67 568

20 89 159 658

30 97 284 783

40 108 466 966

45 115 594 1093

Table 3: Trade-off between sail size s and sail assemblyloading σSA for ac = 0.1mm/s2

3. Jupiter Flyby Mission Using Nuclear Elec-tric Propulsion

InTrance was originally developed for the optimiza-tion of solar sail trajectories and later extendedto optimize also trajectories for electric spacecraft.Within this section, it is applied for an exemplaryJupiter flyby mission with a NEP spacecraft, to as-sess the suitability of ENC also for the optimiza-tion of electric very-low-thrust trajectories. Un-like for the previous examples, the optimization ob-jective was to minimize the used propellant massfor a transfer within a maximum flight time ofTmax = 10 000 days (27.4 years, τ = 2000). Tofind the absolute minimum – independent of the ac-tual constellation of Earth and Jupiter – no flyby atJupiter itself but only a crossing of Jupiter’s orbitwithin a distance of less than 106 km was required,and InTrance was allowed to vary the launch datewithin a one year interval.

Without consideration of technical feasibility, alaunch mass of m = 14 t (for C3 = 0km2/s2)was chosen, including a minimum dry mass ofmdry = 10 t. The propulsion system was assumedto provide a thrust of F = 0.5 N, so that thethrust to mass ratio is 0.0357 mN/kg < F/m(t) ≤0.05 mN/kg. The specific impulse was assumed tobe that of a state-of-the-art ion thruster, Isp =4500 s [16]. Fig. 10 shows a possible trajectoryfor this mission. The velocity increment of thistrajectory is ∆V = 10.5 km/s, whereas a mini-mum ∆V Hohmann-type trajectory to Jupiter9

would require ∆V = 8.6 km/s. Therefore, thegravitational loss of the NEP system is only about21%. Due to the high specific impulse of the elec-tric thruster, only about 3 t of propellant are re-quired. In contrast, a propellant mass of about 94 twould be required for a Hohmann-type transfer ofa 11 t-spacecraft with a chemical propulsion system

9at perihelion

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Fig. 10: Jupiter flyby using NEP

(H2/LOX, Isp = 390 s [17]). Note also that the rela-tive velocity at Jupiter is larger for the Hohmann-type transfer (∆vf = 5.9 km/s) than for the very-low-thrust transfer (∆vf = 5.1 km/s). This resultdemonstrates that a NEP system allows very goodpayload ratios for high-∆V missions, if the transfertime plays a subordinate role, so that the gravita-tional losses can be kept small.

SUMMARY AND CONCLUSIONS

Within this paper, low-thrust trajectory optimiza-tion problems have been attacked from the per-spective of artificial intelligence and machine learn-ing. Inspired by natural archetypes, a novelmethod for spacecraft trajectory optimization wasproposed. It fuses artificial neural networks andevolutionary algorithms into evolutionary neuro-controllers. Evolutionary neurocontrol was imple-mented within a program termed InTrance, whichstands for Intelligent Trajectory optimization usingneurocontroller evolution. InTrance was applied tofind near-globally optimal trajectories for three ex-emplary interplanetary very-low-thrust problems: anear-Earth asteroid rendezvous using a solar sail, aMESSENGER-like mission to Mercury but using asolar sail, and a Jupiter flyby using a spacecraftwith a nuclear electric propulsion system. For thenear-Earth asteroid rendezvous problem, InTrancefound a trajectory that is considerably better thana reference trajectory found by a human trajectoryoptimization expert using a local trajectory opti-mization method. It was already shown in previouspapers that evolutionary neurocontrollers are ableto find spacecraft steering strategies that generatebetter trajectories, which are closer to the global

optimum because they explore the trajectory searchspace more exhaustively than a human expert cando by using traditional optimal control methods.Here, it was shown that evolutionary neurocontrolcan also be applied successfully for very-low-thrustproblems, where it is very difficult to generate anadequate initial guess for local trajectory optimiza-tion methods. Evolutionary neurocontrol runs with-out an initial guess and does not require the atten-dance of an expert in astrodynamics and optimalcontrol theory. It may be also used to find theoptimal initial conditions for the mission. Beingproblem-independent, the application field of evolu-tionary neurocontrol may be extended to a varietyof other optimal control problems.

References

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