GUESS VALUE FOR INTERPLANETARY TRANSFER DESIGN THROUGH GENETIC ALGORITHMS

Transcription

1 AAS GUESS VALUE FOR INTERPLANETARY TRANSFER DESIGN THROUGH GENETIC ALGORITHMS Paola Rogata, Emanuele Di Sotto, Mariella Graziano, Filippo Graziani INTRODUCTION A global search technique, as genetic algorithms (GAs), can be very useful in order to provide guess values for interplanetary trajectories computation. The interplanetary trajectory design is generally accomplished using sophisticated dynamic models and optimisation algorithms often involving gradient computations. Their convergence to local minima depends upon a good initial guess, which, generally, is difficult to identify. GAs can be very efficient in this context if the cost function computation, usually the required delta-v, is performed without excessive computational efforts. The adoption of simplified models, such as patched conics approximation and analytical ephemeredes, allow us to obtain this guess value in reasonable computational time. The cost function is characterised by handling the different arcs defining the particular mission scenario and computing all the delta-v needed to match two successive arcs including departure manoeuvre, and insertion manoeuvre at arrival. The software has been tested through well-known missions, to both internal and external planets, and a new interesting trajectory to Pluto has been computed as well. The design of an interplanetary transfer is carried out with very sophisticated tools that implement complex optimisation algorithms. Generally, these involve a series of gradient computations in the optimisation process so that only those problems with the necessary smoothness properties are easy to solve. Furthermore, these techniques can converge to local minima or not converge at all depending upon the initial guess. The availability of guess values, sufficiently close to the searched solution, represents a key point in this process. The problem to be solved, that could require multiple-arc trajectories, fly-bys and deep-space manoeuvres, is to obtain an interplanetary transfer with the lowest delta-v. Frequently the complexity of the problem does not allow the designer to get guess values through easy procedures. This task can be efficiently faced by means of genetic algorithms. To use the genetic algorithms a proper cost function, providing the total mission delta-v, has been formulated. Taking into account the huge number of function evaluations, required to get an optimum solution, simplified models and assumptions have been implemented to perform a global search within reasonable time. This cost function is completely general and is characterised by handling the different arcs defining the particular mission scenario. In such a way, the tool is very flexible allowing the user to define any mission as a sequence of distinct arcs whose terminations can include different kinds of manoeuvres, Aerospace Engineers, GMV S.A., Isaac Newton 11, P.T.M. Tres Cantos, Madrid Full Professor of Astrodynamics of Aerospace Systems at School of Aerospace Engineering, University of Rome La Sapienza. 1

2 such as: departure manoeuvre, deep space, fly-by with massive bodies and insertion manoeuvre at arrival. The genetic algorithm will look for possible solutions included in a grid (the range of the parameters) established by the user to properly bound the problem under analysis. In this way, the optimum found is only the closest to the true optimum. The optimal trajectory is the first guess for subsequent local optimisation with more conventional algorithms. Missions of increasing complexity can been analysed starting from the simple case of a launch window computation up to more complex mission including multiple revolution arcs, fly by with several massive bodies and deep space manoeuvres. This paper reports the obtained results concerning with several trajectories to both the internal and external planets. One interesting trajectory to Pluto is particularly highlighted in the paper; this mission includes two intermediate fly-bys to Jupiter and Saturn with a total transfer time of almost 9 years. A local optimiser, using more detailed dynamic models, has refined the guess values provided by the GA, with final results very similar to the guess provided by the GA. GENETIC SEARCH OVERVIEW GAs are a class of non-deterministic methods that incorporate the ideas of natural evolution through biological-like mechanisms, such as crossover and mutation, expressed in a mathematical formulation. One generic set of the parameters describing the problem has to be encoded to form a numerical string (chromosome). Several tens of chromosomes generate a population that evolves in response to the chromosome recombination and random mutations affecting a subset of them. Figure 1 shows an illustrative example of how a GA works. Consider the task of maximising a function f(x,y) of two variables; in this case an individual is a point (x,y) in a 2-D parameter space. A multimodal surface represents the cost function and the GA is in charge of finding the maximum. The search starts from a random set of initial points. The picture at the right shows the initial population (points on the surface), whereas the last population, obtained at the end of the generational cycle, is shown at the left. As one can notice, the GA search has been effective on a particular surface characterised by several local maxima (ring plateaux). A more conventional algorithm would be probably stuck at whatever of these local extremes if the initial guess was not sufficiently close to the true global maximum. Moreover, it is important to remark, that even if a great number of individuals stay close to the surface maximum, there are several individuals that still look for a solution in other far points of the searching field. This behaviour is due to the mutation operator that is fundamental to avoid a premature convergence to a local function extreme. When a lot of individuals come up to a specific point of the searching space, the mutation probability automatically increases, and more individuals begin to explore regions far away from this possible solution. 2

3 Figure 1: Left: First population, randomly generated. Right: last population, at the end of the generational cycle. The algorithm provides the best solution of each population that could be improved by a further generation. GAs do not need guess values because the initial set of solutions is randomly generated at the beginning of the optimisation process. The search for the interplanetary trajectories has been carried out with the GA Pikaia 1 ; Pikaia is a Fortran optimisation subroutine and its source code can be freely downloaded from NCAR s ( Pikaia encodes the problem parameters using a decimal alphabet, namely the simple 1-digit base 10 integers. The number of digits, used for parameter encoding is not fixed and can be changed depending upon the specific problem. The number of digits affects the accuracy related to parameter encoding (phenotype), the higher the number of digits in the encoding, the denser the grid where the coded parameters (genotypes) are located. Nevertheless, the number of digits notably affects the mutation process performance: increasing the number of digits reduces the probability of each gene being changed by the mutation process. In particular, this reduces the probability that a significant gene of the chromosomes is affected by mutation and, thus, the benefits of this process are also reduced. So, the choice inherent to the number of digits is a key point in this GAs setting. In general, a number of digits equal to 4 or 5 has been used in the studied cases. Pikaia incorporates a single crossover operator known as one-point crossover. This operator acts on a pair of parent-chromosomes to produce a pair of offspring-chromosomes. Each parent is chosen through a selection technique; the selection procedure is such that the probability of an individual being selected for breeding is proportional to that individual s fitness (the value of the cost function computed for each individual). Pikaia incorporates a mutation operator to avoid a premature convergence to a local minimum. This is a single mutation operator, known as uniform one-point mutation, but allows the mutation rate to vary dynamically in the course of the evolutionary run. Reproduction planes included in PIKAIA permit either a full generation replacement or a steady-state generation replacement. In the latter case an offspring is inserted in the population only if: its fitness is superior to that of the last member of the population; its genotype differs in at least one gene from any other genotype already present in the population. 3

4 PIKAIA algorithm permits the use of elitism, too. Elitism prevents the best element of the population from being deleted to make room for a new offspring. GENETIC ALGORITHMS FOR INTERPLANETRAY TRANSFERS DESIGN The objective of any interplanetary transfer is to reach a target celestial body after leaving the Earth. These missions may require sizeable energy and long flight time. Aiming to obtain reasonable values for the necessary energy, fly-bys with planets and/or deep-space manoeuvres may be inserted in the mission profile. These fly-bys and manoeuvres, while representing useful means to reduce the delta-v budget, make the mission design more difficult. The computation of an interplanetary transfer involves determining not only departure and arrival dates, but also fly-by and deep-space dates and conditions; these have to be identified to satisfy mission requirements while using the minimum delta-v budget. Furthermore, the motion of a probe has to be considered as a multi-body problem. Even with only the gravitational attractions of the Earth, the Sun and the target planet, the trajectory analysis is quite complex and sophisticated tools are required to perform the trajectory design. Traditional optimisation methods rely on some kind of gradient search technique at some point in the computational process. In order to apply gradient techniques, the system dynamics and the cost function must at least have continuous first derivatives with respect to state and control (e.g. when low thrust propulsion is used). Nevertheless, even when the required smoothness conditions are met, gradient techniques have two significant disadvantages: Gradient and recursive quadratic methods may have small domains of convergence, resulting in convergence sensitivity to initial guess. For complicated dynamics optimisation problems, gradient and quadratic methods may prematurely terminate at local minima. These two drawbacks are significant when not enough is known of the solution structure to form a reasonable initial guess. In the case of a complex interplanetary trajectory, with several manoeuvres and fly-bys, the formulation of the initial guess is not an easy task. It could be useful to adopt a global optimisation technique, like Genetic Algorithms are, which does not rely on gradient information and does not require an a priori knowledge of the solution structure. GAs perform a heuristic search through a huge number of the cost function evaluations. This search requires considerable computational effort when a complete dynamic model is adopted and long numerical orbit propagations are required. Simplified models have been implemented to avoid unaffordable computational times that would provide, in any case, an approximate solution. A central gravity field has been considered neglecting any kind of orbit perturbations and assuming fixed conic approximation. Under these simplified assumptions, the problem to attain a prescribed position starting from a fixed point (two-point boundary value problem) can be regarded as a Lambert s problem whose solution is straightforward. (The particular algorithm implemented to solve the Lambert s problem is the one proposed by Simò that not only avoids any ambiguity related to the Lagrange s form of the transfer time equation but also uses only one formulation for all the conics, either elliptic, hyperbolic or parabolic arcs). Furthermore, it allows the computation of multi-revolution arcs. The whole trajectory, including planet fly-bys and deep-space manoeuvres, has been split into different arcs, and each interplanetary arc has been solved with the Lambert s method. Planets positions have been analytically computed using polynomial expansions of the planetary orbital elements. Simplified models for fly-by and deep-space manoeuvres have been implemented too, paying a particular attention to avoiding any kind of terminal constraints. It is important to remark that GAs are 4

5 well suited to solving unconstrained optimisation problems, whereas they are not so efficient when some equality and inequality constraints are imposed by means of penalty functions. The constraints, limiting the search field where the same are satisfied, affect one of the most important features of these algorithms, namely, the capability to perform a global search. PROGRAM DESCRIPTION Genetic Algorithms have been implemented in an in-house trajectory optimization program that adopts a central gravitational field and fixed conics model. This software is highly flexible allowing the user to define multiple phase trajectories by handling the single phase as an elementary brick to construct the whole trajectory. Each phase can be defined by assigning departure and arrival conditions, whereas extent launch windows can be provided for the trajectory terminal conditions. Different terminal conditions can be defined for each arc (arrival, fly-by, rendezvous, deep space manoeuvre) so that a wide range of possible mission scenarios could be analysed: starting from the simple direct transfer to a single planet up to multiple flybys resonant transfers. In order to match each arc with the following one an impulsive manoeuvre is introduced, this manoeuvre is accounted in the cost function along with the departure and arrival V. The GAs, trying to reduce the total mission V, determines the proper condition for fly-bys and deep space manoeuvres. Fly-by conditions are established by properly selecting fly by date that univocally determines the planet state vector (see dedicated section for more details). When these conditions are not met, the code will insert a manoeuvre to take into account the trajectory mismatching (concerning with the velocity vector). A similar approach is followed to establish a deep-space manoeuvre, the main difference being in the number of involved parameters. The code has to select not only the proper space position where to insert the deep-space manoeuvre but also a convenient date, in this way the optimisation parameters introduced by a deep-space manoeuvre are four. The software capability to select a convenient deep space manoeuvre has been tested with a 180 degrees Earth-Venus transfer. In this case a broken plane manoeuvre is necessary to allow the probe to pass from the ecliptic to the Venus orbital plane. GAs find an optimum solution by inserting the manoeuvre just at the node of the Venus orbit in the ecliptic. Assigned the mission profile, namely the sequence of arcs that make up the whole trajectory, mission parameters, like departure and arrival dates, positions and dates of deep-space manoeuvres and fly-bys dates, are searched. The search is carried out into a range of dates (for departure and arrival) and a limited space volume for the location of the deep-space manoeuvre. Once assigned the launch window at departure and arrival and the mission profile, the program searches for the trajectory parameters providing the minimum delta-v. The following two sub sections describe the simple models adopted for the deep-space manoeuvre as well as for the fly-by conditions. Deep-space manoeuvre The mission profile may include impulsive manoeuvres, namely, manoeuvres in which an increment of velocity ( V) is added instantaneously. When a probe has to go from point P 1 (for example a planet) to P 2 (another planet); it is possible that this transfer is not achievable with one arc of conic, so a manoeuvre could be useful in some unknown point P (Figure 2). GAs choose the manoeuvre timing, within the defined mission timeframe, and the position inside a spherical shell volume determined by assigning an internal and external radius from the central body. 5

6 P 2 V V 2 V 1 P P 1 Figure 2: Deep-space manoeuvre. When the position of P is obtained, two Lambert s problems are solved: the first one defines the conic arc to go from P 1 to P, the second one defines the conic arc to go from P to P 2. Therefore, increment V necessary to change the probe orbit is given by: V = V 2 V 1 GA searches for a position and a time that minimise this increment V. The number of parameters to optimise for each deep-space manoeuvre is four: three for the position in the space and one for the time at which the manoeuvre is performed. Gravity assists manoeuvre To reduce the launch energy a mission can include one or several encounters (or gravity assist) with planets. In this program, the gravity assist model is based on the approximation that the manoeuvre is instantaneous and sphere of influence of planets has zero radius. Fly by conditions impose that the incoming relative velocity to a planet has to be equal to the outgoing relative velocity while a minimum altitude (h) from the planet is assured: VrIN = VrOUT = V r > R + h p p These two constraints are generally fulfilled in the code by introducing a fly-by manoeuvre. In other terms, a V is introduced to force the previous conditions. The ephemerides provide the planet state vector for a specific date so that relative velocities with respect to the planet are known and the deviation angle δ can be computed. These two parameters are directly linked to the semi-major axis and the eccentricity of the hyperbolic path around the planet that, in turn, provide the perigee altitude. 6

7 M V V 2 V 1 V E Figure 3: Gravity assists conditions Two cases can take place: Perigee altitude is higher than the minimum altitude established from the planet. In this case the tool will add (if necessary) a manoeuvre only to ensure that the modulus of the incoming relative velocity coincides with the outgoing one (Figure 4 a). Perigee altitude is lower than the minimum altitude established from the planet. In this case the matching manoeuvre not only will ensure the equality of the relative velocity modulus but also the minimum altitude (Figure 4 b). V rin V rin δ<δ MAX δ MAX δ>δ MAX V V rout V (a) V rout (b) Figure 4: Relative velocity in the hyperbolic swingby. APPLICATIONS The results provided by the implemented software are hereafter presented. Firstly, well-known missions have been reproduced in order to test the program, finally two new trajectories are reported: the first one consisting in a planetary tour involving Jupiter Saturn and Neptune, the second one presents a mission to Pluto with intermediate fly-bys of Jupiter and Saturn.In the studied cases the number of generation cycles that was necessary to achieve good results depended on the complexity of each missions; i.e., when many parameters had to be optimised the necessary number of cycles increased. 7

8 A Simple Launch Window Initially the GA capability to solve the problem of a launch window has been tested with an Earth- Venus transfer. Possible dates for this transfer has been searched in a range of departure dates from up to and in a range of arrival dates from up to Figure 5 reports the contour lines plot representing the level lines of the surface describing the total V as a function of the departure and arrival dates (pork-chop plots). Those individuals (possible solutions) provided by the GAs are overlapped to the contour plots (little points in the domain) at the beginning of the evolutional cycle (on the left) and after 500 cycles (on the right). At the beginning, the individuals are scattered due to the pseudo-random initialisation of the population, whereas, after 500 generation the majority of them stay close to the minimum even if some individuals do not thrust in the majority and keep on searching for a possible global minimum. Figure 5: First population randomly generated (left) and last population at the end of the generational cycle (right). The tool, after 500 generations and with only 10 individuals, has individuated the departure date on the and the arrival date on the with a total V of 7.32 Km/s, including V at departure and to insert the probe in a circular orbit around Venus. Two regions are evident in Figure 5: the lower one corresponds to trajectories characterised by a transfer angle smaller than 180º, while the upper region corresponds to trajectories characterised by a transfer angle greater than 180º. Between these two regions there is an energy wall corresponding transfer angle equal to 180º. The visible bridge, connecting the two regions, represents the 180º transfers whose apse line overlaps with the Venus orbit node on the ecliptic. By restricting the departure and arrival dates to values within the energy wall (the crest in Figure 6), only high inclination direct transfers are possible. Such a transfer requires remarkable V and makes unfeasible the found trajectories. 8

9 Figure 6: Direct transfer Earth-Venus with a transfer angle of 180º. In order to reduce the necessary V, it could be useful to introduce an intermediate manoeuvre inside the transfer. The program is able to the get Venus node on the ecliptic as the best position where performs the deep-space manoeuvre, minimising, in this way, the required energy for the transfer. Two arcs constitute this transfer: the first one in the ecliptic, while the second one is on the same plane of the Venus orbit (see Figure 7). In this mission there are 6 parameters to be optimised: one allocated for the departure date, 4 for the deep space manoeuvre (one for the date and 3 for the position in the space) and the last one for the arrival date. Transfer Earth-Venus (transfer angle 180º) with one manoeuvre x Z Venus Earth x Y X 1 2 x 10 8 Figure 7: Transfer Earth-Venus with a transfer angle of 180º and an intermediate manoeuvre. The GAs capability to solve trajectory optimisation problem has been tested in more complex missions, including multiple fly-bys, like the Voyager missions: Voyager 1 Two arcs of trajectory constitute this mission; the probe leaves the Earth and before to arrive to Saturn makes a fly-by with Jupiter. Departure date was fixed on September 1977, and the probe arrived to Saturn on November 1980; the date for the intermediate fly-by was foreseen on March

10 At the beginning, the possible dates have been searched in large departure and arrival window, namely four years. In this mission the optimisation variables are three: departure date, fly-by date with Jupiter and date of encounter with Saturn. In Table 1 the dates obtained through the GA after 1000 generations are compared with the official data: Table 1 VOYAGER1 E.Departure J. Fly-by S.Arrival GA Official dates This trajectory requires a C3 of Km 2 /s 2 corresponding to a V = 7.3Km/s for the insertion manoeuvre from a parking orbit of 200Km. The trajectory found by the program is presented in Figure 8 in the ecliptic 2000 reference frame: Figure 8: Voyager1 trajectory found by the GA. More complex was the mission of the Voyager 2 probe; this was constituted by four arcs and included a greater number of fly-bys. Voyager 2 The probe leaves the Earth and before encounter with Neptune makes fly-bys with Jupiter, Saturn and Uranus. Departure date was fixed on August 1977, and the probe encountered Neptune on November The parameters to be optimised in this case are five dates: one for Earth departure, one for each of the three intermediate swingbys and one more for the last encounter. Table 2 shows the results obtained after 5000 generation cycles of the GAs. GAs performs the search of the dates determining the trajectory within a timeframe as long as the entire mission, namely 12 years. It should be noticed that the arrival date, provided by the program, is quite different from the official one, but the considered planet (Neptune) has a very long period, so that an interval of several days means a little amount in terms of true anomaly errors along the planet orbit. 10

11 Table 2 VOYAGER 2 E.Departure J. Fly-by S. Fly-by U. Fly-by Arrival to N. GA Official dates This trajectory requires C3 is 90.5 Km 2 /s 2 corresponding to a V = 6.76Km/s for the departing manoeuvre from a parking orbit of 200Km. The trajectory found by the program is drawn in Figure 9. x 10 9 Voyager Saturn Jupiter Z 0 Uranus Neptune Earth 0 x X -2 Y x 10 9 Figure 9: Voyager2 trajectory found by the GA. Mariner 10 Finally another well-known mission has been studied to check the tool on a more complex problem. Mariner 10 is a mission that included fly-bys (by Venus and Mercury) and deep-space manoeuvres. Two deep-space manoeuvres were inserted in the mission profile to take into account the change to the nominal mission profile, made during the actual flight, which allowed the probe to attain Mercury planet more than once. The optimisation parameters in this mission are 13: one for departure date, one date for each fly-by (3), 4 for each deep-space manoeuvre (2x4) and the last one for the date of the last encounter with Mercury. Table 3 shows a comparison between the actual timeline and the results obtained through the GA after generations. 11

12 Table 3 MARINER 10 Departure V. Fly-by M. Fly-by M. Fly-by Arrival GA Official dates This trajectory requires a C3 of 18.83Km 2 /s 2 corresponding to a V = 4.05Km/s for the escape manoeuvre from a parking orbit of 200Km. In order to get with the actual trajectory two manoeuvres have been inserted in the mission profile between the fly-bys with Mercury; for the first one the V required was of 25m/s, whereas for the second one was necessary a V = 88 m/s. The trajectory found by the program is presented in Figure 10. x 10 8 Mariner 10 Earth Y Mercury X x 10 8 Figure 10: Mariner 10 trajectory found by the GA. In the two next sections two original missions will be presented: a tour to reach Neptune and a mission to Pluto with intermediate fly-by with Jupiter and Saturn. A tour from Earth to Neptune through fly-by with Jupiter and Saturn GAs have found a new tour, a probe leaving the Earth in January 2018 could reach Neptune in June 2030, making fly-by with Jupiter (in October 2019) and with Saturn (in February 2022). The GAs have found these dates within a 10 years timeframe for departure, from 2015 up to 2025, and 5 years for arrival, from 2025 up to In Table 4 the results obtained by the GAs after generations are shown: Table 4 E-J-S-N TOUR Departure J. Fly-by S. Fly-by Arrival

13 This trajectory requires a C3 of 86.15Km 2 /s 2 corresponding to a V = 6.61Km/s for the insertion manoeuvre from a parking orbit of 200Km. The trajectory found by the program is presented in Figure 11. Figure 11: New tour from The Earth to Neptune. Pluto mission Pluto is the only planet in the solar system that has not been explored by spacecraft. Pluto orbits the Sun in an elliptical, inclined, 248-years orbit. Perihelion was reached in 1989; the planet is now receding from the Sun. To permit the scientific investigation, Pluto has to be reached before that his tenuous atmosphere can refreeze onto surface due to the planet receding from the Sun. To find a new possible mission to Pluto, large departure and arrival windows were considered; in particular the departure and arrival dates have been chosen by the GA within 10 years from 2015 to The mission profile examined foresees two intermediate fly-bys, the first one with Jupiter planet and the second one with Saturn. The software has identified a possible trajectory with departure on the 19 th of December 2016 and a flight time of 9 years. Table 5 FLY-BY INFORMATIONS Distance Date Jupiter 1.98R J 2018/04/21 Saturn 13.8R S 2020/01/10 where 1R J = 71370Km and 1R S = 60268Km. These results have been obtained after generation cycles. The number of variables to optimise in this mission was four: one for the departure date, one for each fly-by and one for arrival date. This trajectory requires a C3 of Km 2 /s 2 corresponding to a V = 7.82Km/s for the insertion manoeuvre from a parking orbit of 200Km. 13

14 The parameters computed by the genetic optimisation have been used as input for a tool (INTNAV, from GMV) implementing complete solar system dynamic models and using numerical ephemerides. This program solves a TBVP, formulated as a constrained parameter optimisation, using a software package OPRQP from the Numerical Optimization Centre (Hatfield). The trajectory is sectioned in several phases, typically heliocentric arcs joining hyperbolic trajectories inside the sphere of influence of the visited celestial bodies. At the departure phase, the Earth s gravity field is considered constituted by 30 tesseral and zonal harmonics. In the other phases non-sphericity is not considered. Furthermore in this phase the air drag perturbation has been considered. During the whole mission the third body perturbations from Sun, Jupiter, Saturn and Pluto have been considered. Note that this tool does not consider either the fly-by as an instantaneous manoeuvre or sphere of influence of planets with zero radius. The fly-bys dates provided by the GAs match, approximately, dates of exit from the planets sphere of influence. The probe leaves the Jupiter s sphere of influence on the 22 nd of April 2018 and Saturn s sphere of influence on the 2 nd of December In Figure 12 the trajectory found by the GA for this mission and the trajectory computed with the more sophisticated tool are presented. Figure 12: EJSP trajectory found by the GA (on the left) and by a local optimiser (on the right) CONCLUSIONS The developed software has given good results in term of guess values; it has been able to determine reasonable trajectory parameters, like departure, arrival and fly-by dates, without need for any initial solution. The computational time necessary to perform each analysis depends on mission complexity; namely, increasing the number of optimisation parameters, the software needs much more time to achieve good results. The approximated results, provided by the GAs, have been used as starting points to a classical optimisation method implementing a complete dynamic model This has showed a rapid convergence towards a more realistic solution that does not differ too much from the GAs approximated solution. 14

15 REFERENCES 1. P. Charbonneau, B Knapp User s guide to Pikaia, NCAR Technical Note, C. Simò, Solución del problema de Lambert mediante regularización, Collectanea Mathematica, VOL. XXIV, Pagès J., Vergès M., Villa R., Study of the Generation of Trajectories for Multiple Target Fly-by Mission, Draft of Final Report, Facultat d Informatica U.P.C. ESOC No5646,83 4. Battin R.H, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Inc, New York Meeus J., Astronautical Algorithms, Second Edition, Willmann-Bell, Inc Sturms F. M., Polynomial Expressions for Planetary Equators and Orbit Elements With Respect to the Mean Coordinate System, JPL, California Institute of Technology, Pasadena, California January 15, Crain T., Bishop R.H., Fowler W., Interplanetary FlyBy Mission Optimization Using a Hybrid Global-Local Search Method, Journal of Spacecraft and Rockets, Vol. 37, No. 4, July-August Janin G., Gòmez-Tierno M.A., The Genetic Algorithms for Trajectory optimization, XXXVI Congress International Astronautical Federation, Stockholm, Swe-den, October 7-12, Petropoulos A.E., Longuski J.M., Bonfiglio E.P., Trajectories to Jupiter via Gravity Assist from Venus, Earth, and Mars, Journal of Spacecraft and Rockets, Vol. 37, No. 6, November-December W.Hartmann, V.Coverstone-Carrol, S.N. Williams, Optimal Interplanetary Spacecraft Trajectories via a Pareto Genetic Algorithm, The Journal of the Astro-nautical Sciences, Vol 46, No3, July-September G.Rauwolf, V.Coverstone-Carrol, Near-Optimal Low-Thrust Orbit Transfers Generated by a Genetic Algorithm, Journal of Spacecraft and Rockets, Vol33, No.6, November-December V.Coverstone-Carrol, Near-Optimal Low-Thrust Trajectories via a Micro Genetic Algorithms, Journal of Guidance, Control and Dynamics, Vol. 20, No. 1,