Cacciatore, F., Toglia, C.

Optimization of orbital trajectories using genetic algorithms

OPTIMIZATION OF ORBITAL TRAJECTORIES USING GENETIC ALGORITHMS Francesco Cacciatore DEIMOS Space S.L., Tres Cantos (Madrid), Spain This work was carried out at University of Rome ’La Sapienza’ [email protected]

Chiara Toglia PhD Student, Department of Mechanics and Aeronautics University of Rome ’La Sapienza’, Rome, Italy [email protected]

Abstract: This paper deals with a numerical method for obtaining minimum fuel orbit transfers. Impulsive missions are considered, with a constraint on the time of flight. The impulsive transfer can be viewed as a succession of coast arcs separated by thrust points. The initial and final points of the trajectories are supposed to be known. In this work the optimization process is carried out using genetic algorithms (GAs). The mathematical model is described. In particular the Lambert’s problem is studied and a new and alternative formulation for orbit determination is introduced. Characteristics and performance of different GAs operators are analyzed. A comparison with the classical methods is made in order to point out the differences. Numerical results on the optimal mission and on the performance of the optimization process complete the work. 1

Introduction

This paper provides a mathematical model for the optimization of an orbital rendez-vous with circular and arbitrarily oriented departure and arrival orbits. A fixed-time impulsive-thrust manoeuvre is considered. The objective of the optimization process is to minimize the consumption of propellant, i.e. the velocity impulse ∆V for achieving the mission. While indirect optimization methods guarantee excellent numerical accuracy (Colasurdo and Pastrone, 1994; Casalino et al., 1999; Prussing, 1993), recently also GAs (Rosa Sentinella and Casalino, 2006) have been investigated to be applied to this field. In this paper GAs have been used and the performance of different selection operators has been analyzed, in order to define a good search method of the global optimal solution. Considerations about the importance of selective pressure and population scattering are carried out. The gene distribution has been used to monitor and evaluate the performances of the optimization code, and to tune the operator parameters in order to achieve a correct selective pressure. Furthermore, from the point of view of the physical and mathematical model, a new, alternative approach to the problem for fixed time missions is presented. The Lambert’s problem is replaced by a direct algorithm, characterized by a lower computational cost. 2

Genetic Algorithms

Genetic Algorithms (GAs) belong to that branch of numerical computation called Evolutionary Computation, inspired by the principles of the biological evolution. GAs work on a population of individuals, which represent a possible solution to the problem to be solved. The characteristics of each element are described by a given number of genes, that are the problem parameters. The way the problem variables are represented by the genes is called encoding. The binary encoding is widely used (Coley, 1999; Lankhorst, 1996) and it was the first one introduced. In this work the real encoding will be used, due to the need for a good accuracy and to its better characteristic of convergence. The members of the population are distributed in a limited mathematical search space. The population evolves according to the principles of natural evolution. The fitness of an individual is given by a fitness function, obtained from the problem mathematical model. The evolution is performed by genetic operators, the digital analogous of the biological evolution mechanisms. The underlying idea is that the systematical and repetitive application of the operators to subsequent generations of individuals results in a drift of the population towards the optimal areas of the search space, thus reaching the optimal solution. The initial population, used to start the algorithm, is created randomly and not arbitrarily as with the classical methods.

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2.1

Optimization of orbital trajectories using genetic algorithms

Genetic Operators

Three operators compose the basic set used in the standard GA: a Selection operator, whose purpose is to emphasize better solutions and discard the worse ones, a Crossover operator, that in analogy with sexual reproduction creates new individuals crossing the genes of already existing ones, and a Mutation operator, which aims to create new solutions and to increase variety in the population with random changes in the genes of existing individuals. An additional operator will be considered: Elitism, that preserves a restricted population elite for each generation. These operators determine behavior and performance of the algorithm. 2.1.1

Selection

The selection operator pushes ahead in the evolution the best solutions and stops the poor ones. The effect of the selection is a pressure on the population (called selective pressure): the higher this pressure, the faster the tendency of the members of the population to converge to the first areas of higher fitness encountered during the computation. On the other hand, a lower pressure will permit a better spreading of the population and better exploration of the search space. A trade off is needed, since, if the pressure is too high, the algorithm can converge to a local optimum, and if too low, a sufficient evolution is not reached and the optimal solution can be missed. Fitness Proportionate Selection (FPS). It has been the first selection operator proposed: the expected number of times that an individual will be selected (expected value) is equal to the ratio between the fitness of the individual and the average fitness of the population. The main drawback of the FPS is the high selection pressure, and its inconvenient variation during the evolution: at the beginning of the computation a small amount of individuals will have a fitness above the average, and thus will be selected a big number of times, reducing the exploration of the search space; at the end of the computation, when all the individuals will be quite similar, the pressure will become very small, impeding a further evolution. Sigma Scaling. An improvement of the FPS can be obtained using in the selection process information relative to the variations inside the population. The Sigma Scaling selection scales the fitness function using the standard deviation σ to calculate the expected value. The effect of the scaling is a modulation of the selective pressure: at the beginning of the evolution, when the population is wide-ranging (σ is high), the above-average individuals will still be preferred, but the selection pressure will not be as high as it was for FPS; at the end of the computation the variance will be low, and the differences among individuals will be enhanced, balancing the decrease of the selective pressure. Ranking Selection. The absolute values of fitness are used to order the individuals in a ranking and then ignored: the expected values of the solutions depend only on the position inside the ranking. The aim of this strategy is to impede to particularly fit individuals to takeover the population and steer the algorithm in a restricted area of the search space in too early phases of the computations. A mapping function is used to transform an individual’s position in its expected value. Tournament Selection. A small number of individuals is randomly extracted from the population, and a tournament is played between them: the fitness of the individuals and a victory probability r are used to establish the winner, that will be selected. A number of tournaments equal to the population dimension is played, so the dimension of the population remains the same. Typically, tournament selection with dimension equal to two is used; the victory probability indicates the probability that the fittest individual will win the tournament. A value of r lower than one will reduce the selective pressure. This selection is simple, with fast computation and easily controllable pressure. Furthermore also poorer individuals may be selected, reducing the selection pressure and improving the explorative capabilities of the algorithm. For these reasons it is usually the preferred operator. 2.1.2

Crossover

The crossover operator algorithm depends strongly on the gene encoding used. In the present real-coded optimization good results have been obtained using the Simulated Binary Recombination (SBR) operator proposed by Deb (Deb, 1998; Deb, 1999; Deb, 2005): a probability distribution is used to establish the distance of the offsprings from the parents. Define the ratio of the distance between the parents’ genes and the distance between offsprings’ genes as β; for the j-th gene, the effective value of β used (indicated as β¯j ) is obtained setting equal to u (a randomly extracted number between 0 and 1) the integral of an

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opportune probability distribution, as follows: Z

β¯j

β¯j

P(βj )dβj = u

is such that

(1)

0

In the present work a unique value of β¯ has been considered for all the genes; the formula used is:  1  if u ≤ 0.5 (2u) ηc +1  ! η 1+1 ¯ c β= 1  otherwise   2(1 − u)

(2)

After calculating β¯ the parent individuals are crossed as follows: xj (i1 )t+1 =

 1 ¯ j (i1 )t + (1 − β)x ¯ j (i2 )t (1 + β)x 2

(3a)

xj (i2 )t+1 =

 1 ¯ j (i1 )t + (1 + β)x ¯ j (i2 )t (1 − β)x 2

(3b)

In the above written formulae ηc is a non-negative real number that indicates the probability for the offsprings to be similar to the parents, x(i1 ) e x(i2 ) are respectively the first and the second individuals chosen for crossover, the subscript j refers to the considered gene of the solution, and t is the time, namely the generation. The operator parameters are tuned in such a way that at the beginning of the computation the individuals will be quite different between each other, and therefore the offsprings will be well distributed in the search space, allowing a good explorative ability of the algorithm. With the increase of the generations the individuals will get closer, and the offspring will be more likely located in the same area of the parents, providing the optimizer with a good ability to exploit the higher fitness areas. 2.1.3

Mutation

The structure of the operator depends strongly on the solution encoding chosen for the GA: binary coded mutation randomly changes some of the bits of the genome of an individual, while for the real coded operator different mutation functions can be used, and two main architectures are possible: total random mutation (Chambers, 2000), that assigns to the mutated gene a totally new value, or the creep mutation, that introduces a small casual change in the mutated gene value; this generates a behavior analogous to the hill climbing used in more classical optimization methods. For the optimizer discussed here a creep mutation operator has been chosen, and the preferred mutation function has been the polynomial probability distribution proposed by Deb (1998). After extracting a random number u(im ) for every individual to be mutated, the parameter δq is computed: (  1 2u(im ) + (1 − 2u(im ))(1 − δ)ηm +1 ηm +1 − 1 δq =   1 1 − 2(1 − u(im )) + (2u(im ) − 1)(1 − δ)ηm +1 ηm +1

if u(im ) ≤ 0.5

(4)

otherwise

where: δ=

min



  x(im )j − gjmin , gjmax − x(im )j  gjmax − gjmin

(5)

The parameter ηm is, similarly to ηc for crossover, a distribution index for the mutated individual. Only a part of the population is mutated: the number of individuals extracted for mutation is a tuning parameter of the operator, and influences its performances. 2.1.4

Elitism

The fittest individuals of the population are saved before the application of the other operators. The elite is reinserted in the population after selection, crossover and mutation, substituting the worse individuals. Elitism strongly increases the performances of the GA, avoiding good solutions to be destroyed by the casual factors of the algorithm. The dimension of the elite is extremely reduced, and the dimension of the population is kept constant.

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2.2

Optimization of orbital trajectories using genetic algorithms

Selective pressure and GA behavior

In the previous sections the concepts of selective pressure and search space exploration have been introduced. These factors are of primary importance for the performance and characteristics of a genetic optimizer. A GA carries out the evolution of a population of solutions starting from a first generation created on random bases. It reasonable to think the first population as uniformly spread in all the search space; the evolution will cause a drift of the individuals towards the areas of higher fitness, thus changing the distribution of the elements of the population, until the optimal solution is found. In a well designed GA there is a good balance between the ability of the algorithm of concentrating the population in the better zones of the search space (that is, its exploitation ability), and the ability of scattering the individuals around (that is, its exploration ability). The selection (and also the elitism) operator is mainly responsible for the exploitation, concentrating the population around its best members. If this push is big, the potential solutions that compose the population will tend to become all very similar to each other, and will be located in a small zone of the search space. The pressure exerted by the selection is the main contribution to the overall selective pressure of the GA. The task of creating new individuals belongs exclusively to crossover and mutation. A remarkable difference between the two last mentioned operators is that the former creates new individuals from the old ones presents in the population, while the latter can create new members independent of the already existing ones. The creation of new individuals is fundamental, and is what gives to the GAs the ability of exploring the search space. The explorative ability is what makes the GA especially good in finding the global optimum area even in a complicated search space. The scattering of the solutions allows the collection of information about the shape of the fitness function, from which derives the global ability of the GA. It has been noticed how the lack of one of crossover or mutation operators determines a considerable decrease of the performances of the algorithm. The variety in the population is the driving force of GAs: the differences among individuals are used to steer the research. If all the population components became identical, there would be no fitter individuals to select, and no poorer individuals to discard: the algorithm would necessarily stop. This is exactly what happens when the area close to the optimal solution is reached: the potential solutions represented by the individuals tend to become all identical, and no relevant improvement in the results are obtained. This is why the solution provided by a GA is poorly refined: the real optimum is extremely hard to obtain (to the limit, impossible) because the evolution will stop very close to it. The GA behavior and the concepts presented in this section will be analyzed and described more in depth with the analysis of the gene-distribution evolution during the computation. 2.3

GAs vs classical optimization methods

Optimization methods can be classified into two main, big classes: classical deterministic methods and stochastic methods. The choice of the most suitable method to be used concerns the kind of problem in exam. A brief comparison of the two methods will be done in the following, pointing out their principal drawbacks. Concerning the classical methods, the main problems are due to the fact that the optimal solution strongly depends on the choice of the initial guess. For this reason it is possible that the algorithm converges toward a suboptimal solution, i.e. a local and not global optimum. Furthermore this kind of optimization algorithm are characterized by a poor efficiency and flexibility to work with problems of different nature. In many ’practical’ problems, the evaluation function is irregular or complicated, so it is not approachable with classical methods. Using GAs the optimization process can result slow, since there is not a preferential direction of research. Moreover the found solution may be unsatisfying, because of a not high accuracy. In this case an improvement of the solution is necessary, using classical methods. To conclude it is suitable to use classical methods for regular objective function and genetic methods for irregular or complicated function. A good compromise can be using both the methods, i.e., looking for the initial guess with GAs and refining the optimal solution with classical methods. 3

Lambert’s problem

The mathematical model of the orbital trajectory in a fixed flight time needs the implementation of a method for determining the characteristics of the transfer orbit knowing two position vectors and the time

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of flight. In astrodynamics this problem is classically known as Lambert’s problem. From a mathematical point of view it is a two-point boundary-value problem, which has to be solved in an iterative way. Several methods for solving it have been studied in literature. The problem was born at the beginning of the 19th century and called Gauss problem, for determining the orbit of the Ceres planetoid from the observation of its motion. Later on it was simplified and used for problem of intercept and rendezvous. Bate et al. (1971) directly calculate the velocity vector in the extreme points of the orbital arc from their position vectors and the time of flight. It is an iterative problem which uses the value of one of the characteristic parameters of the orbit for the convergence criterion. Battin and Vaughan (1984) propose as solution an elegant algorithm, which uses the simplicity of Gauss method but eliminates its two main defects, i.e. the singularity when a π angle separates the points and the low convergence rate for not small angles. Prussing (1979) introduces two angles, whose geometrical meaning can be related to the eccentric anomaly. From the classical astrodynamics notions, due to Lagrange, the time of flight for an elliptical orbit has been calculated involving two angles, α and β in this notation, whose numerical values come from the geometry of the terminal radius vector and the semimajor axis of the transfer orbit. Prussing finds a simple geometrical interpretation, which can be used to geometrically construct these angles in position space. The determination of these angles involves a certain ambiguity, which can be solved for by applying a criterion that allows to choose among one of the four possible solutions, being the two point positions, the flight time and the transfer angle ν fixed. To solve the Lambert’s problem following Prussing approach, one can introduce the Lambert’s theorem, which states that the flight time ∆t between points P1 and P2 along a transfer ellipse depends on the semimajor axes a, the chord distance c and the sum of the radii r1 + r2 of the points with respect to the focus of the orbit: ∆t = ∆t(a, c, r1 + r2 ) In particular this theorem states that ∆t does not depend on the value of the eccentricity e. The demonstration of this theorem can be found in Mengali (2001). The time of flight can be calculated as: s a3 [α − β − (sin α − sin β)] ∆t = µ where the angles α and β are defined by: r r α r1 + r2 + c β r1 + r2 − c sin = ; sin = 2 4a 2 4a Since the values of α and β come from the values of their sines, an ambiguity occurs, that can be solved using Prussing criterion:   α = α0 if ∆t ≤ tm β = β0 if ∆ν ∈ [0, π] α = 2π − α0 if ∆t > tm β = −β0 if ∆ν ∈ [π, 2π]

Lambert’s problem geometry

The term tm represents the flight time of the minimum energy ellipse, i.e. the ellipse connecting the point P1 and P2 characterized by the minimum value of the semimajor axes amin = (r1 + r2 + c)/4. In fact, fixed the extreme points of the orbital arc, two orbits can be found for each value of a > amin , which depend on the flight time (there is a fast and a slow solution). For a = amin the two solutions coincide. For a < amin no solution exists. 4

Problem Description

The problem analyzed consists of the optimization of an orbital rendez-vous with arbitrarily oriented departure and arrival orbits. A fixed time, impulsive thrust manoeuvre is considered. A maximum of four impulses is used, corresponding to three orbital branches, with an additional waiting arc lying on the departure orbit. Excluding the waiting trajectory, every orbital branch is included between two thrust points, i.e., the points were burns occur. The fitness function chosen is the total velocity impulse (∆V ) necessary to perform the orbital transfer. The total impulse is given by the sum of the single impulses provided in each thrust point. The ∆V necessary to pass from an orbital arc to the following is given by the velocity difference in each thrust point. ∆V is directly linked to the mass variation, and thus to the propellant consumption, through the Tsiolkovski equation. The velocity in each point depends on

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Optimization of orbital trajectories using genetic algorithms

the orbital parameters, that are computed using the variables that have been selected to describe the transfer. The variables are, from the point of view of the genetic optimizer, the genes of the members of the population. In this work the following genes are used: 1. r, radius vector module, in thrust point 1 and 2; 2. ∆u, real anomaly variation in the orbital plane for the first and the third trajectory arc; 3. ψ, azimuth of the trajectory branch in thrust point 0 (starting point) and thrust point 3 (final point); 4. η, parameter to determine the semi-major axis of the first and the third trajectory arc; 5. twi , waiting time on the departure orbit; 6. twf , waiting time on the arrival orbit; The parameters r, ∆u and ψ have been used to determine the position of the thrust points in space. The parameters η complete and uniquely determine the geometry of the trajectory. The meaning of the gene η and the reason for its introduction will be described in section 4.1. The manoeuvre is characterized by 10 parameters to be optimized. Therefore 10 genes identify the peculiar features of each individual in the population. 4.1

Alternative resolution to orbit determination for closed-time mission

Since a fixed-time problem is considered, the flight time of each orbital arc could be assumed as a gene or an optimization parameter. If this were the choice, then Lambert’s problem should be solved, in order to determine the characteristics of each arc and the velocities at its extreme points. This is the classical approach, that involves as many iterative processes (i.e. Lambert’s problems) as the number of orbital arcs composing the transfer. This determines a growth of the computational cost, that can be reduced by using a different approach. The choice of another orbital parameter, like the value of the semimajor axis, as optimization parameter makes the determination problem a direct and no longer iterative process. Following this method, the determination of the transfer orbit is obtained directly for all the orbital arc minus one, for which the flight time can be easily derived and the Lambert’s problem, as described in section 3, has to be applied. Doing so, the computational cost decreases. On the other side, an ambiguity on the choice of the orbit has been introduced. A criterion to solve the ambiguity needs to be defined. The idea here introduced is the following: since the value of a does not solve for the ambiguity, a parameter η ∈ [0, 1] is used as gene or optimization parameter. The value of the semimajor axes is calculated from η using a parabolic law. a=

amin 4η (1 − η)

(6)

The vertex of the parabola represents amin , i.e. the minimum energy ellipse solution. Besides providing a, the parameter η can be used to solve the ambiguity. Indeed, fixed the point P1 and P2 and the semimajor axes value, two possible elliptical orbits connecting the points exist. The two orbits are characterized by different flight times, one shorter, the other longer than tm previously defined. Defining the two solutions the fast and the slow solution, one chooses the fast solution for η < 0.5, otherwise the slow one. The ambiguity solution criterion becomes:   α = α0 if η < 0.5 β = β0 if ∆ν ∈ [0, π] α = 2π − α0 if η > 0.5 β = −β0 if ∆ν ∈ [π, 2π] 4.2

Algorithm of the problem

The genes composing the solution of the problem are represented graphically in Fig. 1. Using these genes the first and third branches can be computed, followed by the second branch. The genes η1 and η3 are used as inputs for the direct determination of the semimajor axes of the corresponding orbital arcs, using the algorithm proposed in section section 4.1. Knowing the times of flight of the first and third branches, the time of flight on the second arc is easily obtained. A classical Lambert problem must now be solved to determine the second orbital arc. Once the computation of the three orbital arcs has been completed (the waiting orbit is known as coincides with the departure orbit), the velocities at the positions corresponding

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Figure 1: Genes representation

to the burns can be calculated, and all the single ∆Vi are obtained. The total velocity impulse is then passed to the genetic optimizer, that uses it as fitness function. The physical and mathematical model used to describe the problem to be optimized is totally independent of the genetic algorithm, since the routine used to compute the fitness of the individuals is the only connection between the optimizer and the physical problem treated. 5

Results

The aim of this section is to present the results obtained with the optimization program. The cases of major interest are shown here. All the results have been calculated using the tournament selection, with victory probability r◦ = 0.75. The mutation operator with polynomial probability distribution and the SBR crossover operator with a crossover probability of 90% have been chosen. 5.1

Rendez-vous between non-coplanar orbits

The example here described is a classical example of transfer mission between circular non-coplanar orbits. Impulses ∆V0 ∆V1 ∆V2 ∆V3

= = = =

0.2265 0 0.2378 0.0799

∆VT = 0.5442

The departure orbit has a null value of inclination, while for the arrival orbit (r2 = 1.2r1 ) a value of i = 20◦ has been considered. The optimal solution is shown in Fig. 2. It is worth to notice in Fig. 2(b) that the first and the second branches of the trajectory belong to the same orbit. Indeed, analyzing the obtained results, the value of the impulse in thrust point 1 results to be null (see the corresponding table). This result is very interesting: even if the structure of the mission is a priori imposed, (i.e. composed of three orbital arcs), the optimization program is able to find optimal trajectories with a lower number of branches, and so a lower number of impulses. In this case the second branch is a waiting orbit and the inclination variation is split in two different burns. This happens because the change of inclination is a very expensive mission, so it is convenient to carry out a small i change together with an energy increase, instead of separating the two manoeuvres.

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5.2

Optimization of orbital trajectories using genetic algorithms

Transfer mission between coplanar orbit

As an interesting case, one can consider a satellite rephasing mission. The result obtained is shown in Fig. 3. The optimal trajectory is composed by three orbital arcs. The characteristic of the mission is its symmetry with respect to a symmetry axes, as shown in Fig. 3(b). For coplanar orbit, obviously, the optimal mission lies on the orbital plane. Therefore, for this specific case, a 2D case can be considered, since the two-dimensional and the three-dimensional model give the same results. The 2D approach is characterized by a different mathematical problem, so for a two-dimensional model a new and different program has to be used. Thanks to the great flexibility of GAs, the optimization algorithm remains the same, but the optimization parameters and the algorithm change. In particular, the number of parameters to be optimized is reduced from 9 to 6, so a decrease of computational cost is obtained. In Table 1 it is shown the comparison of the results obtained with the two models. This comparison provides also a measure of the accuracy of this genetic algorithm. Table 1: Comparison between three and two-dimensional models ∆V 3D model 2D model ∆V0 ∆V1 ∆V2 ∆V3

0.107377 0.130670 0.130674 0.107373

0.107356 0.130693 0.130678 0.107369

∆VT

0.476094

0.476096

(a) Three-dimensional view

(b) Normal to orbital plane view

(c) Frontal view

Figure 2: Non-coplanar orbit transfer (r2 /r1 = 1.2 and ∆i = 20◦ )

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Target at final time

Axis of symmetry

(a) Frontal view

(b) Normal to orbital plane view

(c) Line of nodes view

Figure 3: Rephasing manoeuvre 5.3 5.3.1

Algorithm Behavior and Performances Genes Distribution

The analysis of the distributions of the gene values during the computation provides interesting results. As already stated in previous sections, the evolution starts with a random first population, and thus a uniform (white noise shaped) distribution of the gene values can be expected. The operators will modify this distribution: the population is moved towards more convenient zones of the search space, and thus the distributions will tend to drift towards the gene values corresponding to the highest fitness areas of the problem domain. Ideally, the gene distribution will change assuming a gaussian-like shape, centered in the variable values corresponding to the global optimum. With the progression of the generations the variance of the individual genome will decrease, and all the solutions of the population will tend to become identical to the global optimum individual. The gene distribution therefore changes from an initial white-noise shape, to a gaussian-like shape, and finally to a Dirac’s delta. The effect of the selective pressure on the distributions is to rapidly make them concentrated around the above average individuals first found, despite their sub-optimality. The individuals will become almost identical from the first generations of the evolution. The GA will operate on a unique individual continuously repeated in the population, moving it around the search space until the solution improvements get to an undue end because of the lack of variety in the population. It is evident that in this case the solution parallelism given by the use of a population does not exist anymore, and to all purposes we are handling only one solution per iteration. The ideal behavior is observed if a convenient selective pressure is obtained: the correct equilibrium between exploration and exploitation is reached, and the distributions of the gene values will in the first generations drift towards the zones of higher fitness found, anyway maintaining a good scattering of the individuals. Once the global optimum area is found, the population will concentrate around the values corresponding to it, and will assume the afore mentioned Dirac’s delta shape in the final steps of the computation. Figures 4 and 5 present an example of the different behavior of the distributions in relation to the selection pressure given by the implemented GA operators. The shape of the distribution of the gene ∆u1 is drawn

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Optimization of orbital trajectories using genetic algorithms

(a) Tournament Selection

(b) Sigma Scaling with Roulette Wheel

Figure 4: Distribution of the values of the gene ∆u1 for different generations along the computation (the choice of the gene is irrelevant). Figure 4(a) refers to the gene distributions of an algorithm implementing Tournament Selection: the progressive thickening of the distribution around the optimum is clearly visible. Figure 4(b) is relative to the same algorithm but with Sigma Scale Selection and Roulette Wheel Sampling: the behavior previously described for the high selective pressure cases is evident. The sampling algorithm is necessary to convert the theoretical expected value provided by the sigma scaling for each selected individual (see section 2.1.1) in the effective number of times that the considered individual is selected. The selection pressure of the sigma scaling and the sampling errors due to the roulette wheel make the individuals identical even from the first stages of the calculation. Due to the sampling errors the individuals are not selected the correct number of times corresponding to their expected value, and it’s reasonable to think that higher fitness individuals will get even more chances to be selected than the lower fitness ones; as a consequence, the selective pressure will experiment a further increase. 5.3.2

GA Performances

The different performances obtained using the selection methods described in the section presenting the working principles of the genetic algorithms are reported in this section. Due to the high number of relevant parameters and considered cases, many are the possible comparisons. Even so, the restricted study carried out here is useful to give an impression of the practical effects of different selection operators. For the same optimization problem five selection methods are compared: Tournament, Fitness Proportionate Selection (with Roulette Wheel Extraction or Baker’s Stochastic Universal Sampling -SUS-), Sigma Scaling (with Roulette Wheel) and Linear Ranking (with SUS). The comparison results are presented in Fig. 5. The optimization has been executed ten times for each selection method. The results outline the poor performances obtained with Roulette Wheel caused by its sampling errors: the low number of computed generations is due to the excessive selective pressure of the GA, that causes the algorithm to rapidly converge to a local optimum: the optimizer stops early due to the premature convergence. Sigma Scaling provides better results than the simple FPS: the selective pressure modulation has a positive influence of the performances of the operator. A remarkable improvement is obtained also using FPS with Baker’s SUS instead of Roulette Wheel. In this last case however, some of the calculations stopped after a big

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(a) Results obtained with different selection algorithms

(b) Zoom in the lower impulse area

Figure 5: Comparison of Selection Operators number of generations. The analysis shows that Linear Ranking with SUS is comparable to Tournament Selection: similar fitness function values are obtained, but using the linear ranking operator the calculation requires a bigger number of generations. Tournament selection results to be the best choice considering also the computation time: linear ranking provides similar results, but with the highest calculation time. 6

Conclusion

In this work the optimization process for impulsive transfers has been presented. An alternative method to the classical one has been here proposed: the choice of the genes to be optimized does not include the mission times of the arcs and this let the algorithm to solve only one (iterative) Lambert’s problem, in order to get a decrease of the computational cost. GAs have been chosen as optimization algorithm. Particular attention has been payed to the optimization method and the different selection operators have been compared, in order to obtain new information about them. The numerical results let the authors say that a convenient selective pressure has to be selected to reach a convenient equilibrium between exploration of the search space and exploitation. The good scattering of the individuals in all the areas of the definition domain has been found to be a very important feature of the optimization process, in order to let the solution converge towards the global optimum and not a suboptimal solution. The algorithm described in the paper drives the solution to the global optimum area where the population concentrates around the values corresponding to the optimal solution. It is worth to notice that the algorithm here described provides a great flexibility since the unique link between the optimization process and the physical problem is represented by the calculation of the fitness function.

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Cacciatore, F., Toglia, C.

Optimization of orbital trajectories using genetic algorithms

References Bate, R.R., Mueller, D.D. and White, J.E., 1971, Fundamentals of astodynamics, Dover Publications, New York. Battin, R.H. and Vaughan, R.M, 1984, An Elegant Lambert Algorithm, Journal of Guidance, Vol.7, No.6, pp.662-670. Casalino, L., Colasurdo, G. and Pastrone, D., 1999, Optimal Low-Thrust Escape Trajectories Using Gravity Assist, Journal of Guidance, COntrol and Dynamics, Vol. 22, No. 5, pp 637-642, SeptemberOctober. Chambers, L., 2000, The Practical Handbook of Genetic Algorithms, Chapman & Hall/CRC, second edition. Colasurdo, G. and Pastrone, D., 1994, Indirect Optimization Method for Impulsive Transfers, Paper AIAA 94-3762. Coley, D.A., 1999, An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific. Deb, K., 1998, Non-linear Goal Programming Using Multi-Objective Genetic Algorithms, Technical Report CI-60/98, Department of Computer Science/XI, University of Dortmund. Deb, K., 1999, An Introduction to Genetic Algorithms, Sadhana Journal, Vol. 24, No. 4, pp. 293-315. Deb, K., 2005, A Population-Based Algorithm-Generator for Real-Parameter Optimization, Department of Mechanical Engineering, Indian Institute of Technology Kampur. Lankhorst, M.M., 1996, Genetic Algorithms in data analysis, Printed by Universiteitsdrukkerij Groningen. Mengali, G., 2001, Meccanica del volo spaziale, Edizioni Plus, Cap.6. Prussing, J.E., 1979, Geometrical Interpretation of the Angles α and β in Lambert’s Problem, Journal of Guidance and Control, Vol.5, No.2, pp. 442-443. Prussing, J.E., 1993, Equation for Optimal Power-Limited Spacecraft Trajectories, Journal of Guidance, Control, and Dynamics, Vol.16, No.2, pp. 391393. Rosa Sentinella, M. and Casalino, L., 2006, Genetic Algorithm and Indirect Method Coupling for LowThrust Trajectory Optimization, AIAA No. 064468.

Journal of Aerospace Engineering, Sciences and Applications, Jan–April 2008, Vol. I, No 1

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OPTIMIZATION OF ORBITAL TRAJECTORIES USING ...

a big number of times, reducing the exploration of the search space; at the end of .... (1971) directly calculate the velocity vector in the extreme points of the orbital arc ..... Algorithms in data analysis, Printed by Universiteitsdrukkerij Groningen.

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