Multiobjective Optimization of Low-Thrust Trajectories using a Genetic ...

AAS 09-151

MULTIOBJECTIVE OPTIMIZATION OF LOW-THRUST TRAJECTORIES USING A GENETIC ALGORITHM HYBRID Matthew A. Vavrina* and Kathleen C. Howell† In low-thrust, gravity-assist trajectory design, two objectives are often equally important: maximization of final spacecraft mass and minimization of time-of-flight. Generally, these objectives are coupled and competing. Designing the trajectory that is best-suited for a mission typically requires a compromise between the objectives. However, optimizing even a single objective in the complex design space of low-thrust, gravity-assist trajectories is difficult. The technique in this development hybridizes a multiobjective genetic algorithm (NSGA-II) and an efficient, calculus-based direct method (GALLOP). The hybrid algorithm capitalizes on the benefits of both methods to generate a representation of the Pareto front of near-globally optimal solutions. INTRODUCTION It is well established that propellant-efficient, low-thrust electric propulsion can improve spacecraft delivered mass and other interplanetary trajectory characteristics when compared to chemical propulsion. Typically, only a single objective is considered when generating optimal solutions. However, in low-thrust trajectory design, the maximization of final spacecraft mass and the minimization of time-of-flight (TOF) are both critical objectives. In general, these two objectives are coupled and competing, creating a complex multiobjective optimization problem. Designing the trajectory that is best-suited for a mission typically requires a compromise between these objectives. Thus, a multiobjective technique that can simultaneously optimize both TOF and final mass significantly aids the decision process, enabling the development of the best design. In contrast to single objective optimization, where the goal is a single optimal result, multiobjective optimization techniques attempt to generate an entire set of equally optimal solutions, the Pareto-optimal set. Traditionally, multiobjective optimization has been accomplished by modifying single objective schemes and executing an optimization routine many times to produce a set of solutions. However, the nontraditional, population-based formulation of evolutionary algorithms (EAs),1 which include genetic algorithms (GAs),2 allows the evaluation of an entire set of possible solutions. Evolutionary algorithms are stochastic optimization routines that simulate biological evolution by mimicking natural selection and reproduction. The population ‘evolves’ through application of genetic operators to generate design(s) that are best adapted to a fitness landscape. Therefore, by adapting the function that defines the fitness landscape to incorporate all objectives, a multiobjective optimization can be executed in a single run of an EA, a distinct advantage over traditional methods. Additionally, a multiobjective evolutionary algorithm (MOEA)3 does not require a userdefined initial guess. Hence, for a trajectory design problem, a representation of the Pareto front can potentially be generated, where each solution is globally optimal in terms of both final mass and TOF. The development of Pareto-optimal, low-thrust trajectories has been the focus of several recent investigations. Work by Hartmann4 and Hartmann et al.5 has demonstrated the benefits of pairing a MOEA with a calculus-based indirect method for low-thrust trajectory optimization. Specifically, Hartmann et al. investigated direct SEP missions by coupling the Non-dominated Sorting Genetic Algorithm (NSGA)6 with the software SEPTOP7 to generate Pareto-optimal solutions to Mars and Mercury for a fixed launch date. In other work on multiobjective low-thrust trajectory optimization, Lee and Russell applied a specialized mul* Graduate Student, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall, 701 W. Stadium Ave.; Student Member AIAA; Currently: Research Engineer, Boeing Research and Technology, PO Box 3707, Seattle, Washington 98124. † Hsu Lo Professor of Aeronautical and Astronautical Engineering, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall, 701 W. Stadium Ave., West Lafayette, Indiana 47907; Fellow AAS; Associate Fellow AIAA.

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tiobjective GA to generate initial costates that were propagated with the primer vector control law for orbit transfers about Earth and Europa.8 Additionally, Lee et al.9 used a multiobjective EA to develop initial conditions for propagation in Petropoulos’ Q-Law.10 Russell,11 on the other hand, used a randomized global search strategy with Primer Vector theory to generate Pareto-optimal solutions. Evolutionary algorithms, while capable of developing a globally optimal set of multiobjective solutions, do not exploit gradient information and, thus, can struggle with tightly constrained problems. Such is the case for low-thrust, gravity-assist (LTGA) trajectory optimization, where the design space is mostly infeasible with exceedingly small feasible regions. However, developing simply a locally optimal solution in problems formulated with a single objective poses significant challenges for even calculus-based techniques because of the expansive, multimodal design space associated with LTGA trajectories. Furthermore, both conventional direct and indirect implementations12 require an adequate initial guess for convergence. Purdue’s GALLOP software,13,14 through a direct method devised by Sims and Flanagan,15 has been demonstrated as an efficient and robust local optimization technique. GALLOP is a preliminary design tool capable of generating locally optimal LTGA trajectories in terms of a single objective. To capitalize on the multiobjective capabilities of EAs as well as the efficient constraint handling and robustness of a calculus-based, direct method, the technique in this development hybridizes a MOEA, specifically the NSGA-II algorithm,16 and GALLOP. For every generation in the hybrid algorithm, each member of the MOEA population is sent to GALLOP to be locally optimized. The MOEA and GALLOP operate synergistically, where the NSGA-II explores globally, aiming to uncover regions of the design space containing Pareto-optimal solutions, and GALLOP searches locally with the aid of gradient information, to refine each design. The hybrid algorithm optimizes both final mass and TOF for LTGA trajectories, to generate a set of near-globally optimal solutions that form a representation of the Pareto front. The automated algorithm does not require a user-defined initial guess and, as a result, is not biased to any preconceived notions of optimal solutions. This development is an extension of previous work on single objective global optimization through hybridization of a simple GA and GALLOP.17,18 The multiobjective GA-GALLOP hybrid algorithm is applied to three design scenarios: a solar electric propulsion (SEP) Earth-Mars (EM) rendezvous mission, a nuclear electric propulsion (NEP) Earth-Jupiter (EJ) rendezvous mission, and a nuclear electric propulsion Earth-Earth-Jupiter (EEJ) rendezvous. In each example, the goal of the optimization process is a representation of the Pareto front. Previously published optimal trajectories (in terms of final mass) are used as test points to verify that the front produced by the multiobjective GA-GALLOP algorithm is near the true, globally-optimal Pareto front. MULTIOBJECTIVE OPTIMIZATION In multiobjective optimization, the problem is the minimization of the components in a vector-valued function. The multiobjective optimization problem is formally stated in a general form as

minimize : subject to :

f ( x) g j ( x) ≤ 0 hk ( x) = 0 x iL ≤ x i ≤ x iU

where f(x) is a vector of objectives

[

f (x) = f 1 (x)

f 2 (x) L

j = 1, m (1)

k = 1, l i = 1, n

f nob ( x )

]

T

.

(2)

Then, x is a vector of design variables, and nob is the scalar number of objectives, i.e., the number of rows in f(x). The objective space, the space in which the objective vector belongs, is nob-dimensional. Often in multiobjective optimization, the objectives are not only competing but also coupled, where variables in one objective function appear in the other functions. In general, because there are multiple, different design spaces, multiobjective optimization is more complex than single objective optimization. Classical methods, such as the weighted sum approach3 and the εconstraint method,19 incorporate single objective strategies to build up a set of optimal solutions. These techniques, however, require many simulations of a single objective method and often necessitate the specification of parameter values that can be difficult to determine a priori. Furthermore, these classic techniques are often unable to identify all areas of the optimal objective space, warranting the development of other multiobjective optimization strategies such as multiobjective evolutionary algorithms.

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f1: final mass

Pareto Optimality The optimal set of solutions in multiobjective opfront 1 (Pareto front) timization is termed the Pareto-optimal set.20 A solu5 tion is Pareto optimal if no improvement in one objecfront 2 tive can be accomplished without adversely affecting front 3 2 at least one other objective. In the objective space, the 1 7 hypersurface (of lower dimension than nob) that repre4 3 sents all possible Pareto-optimal solutions is defined as the Pareto front (or frontier). A design that is located along the Pareto front is neither better nor worse than 6 any other solution along the Pareto front. That is, the solutions that compose the Pareto-optimal set are equivalently optimal. Thus, the goal of multiobjective f2: time of flight optimization is to generate as many Pareto-optimal Figure 1. Non-dominated fronts of multiobjecsolutions as possible to adequately represent the Pareto tive trajectory optimization example. front, so that sufficient information for a tradeoff decision is available. The Pareto front can be discontinuous as well as concave or convex, and, in general, is not known a priori. As an example, consider the optimization of the competing objectives for an interplanetary trajectory: final mass and time-of-flight. The feasible objective space along with seven designs is illustrated in Figure 1. Because final mass is to be maximized and flight time minimized, the Pareto front is located in the upper left region of the feasible objective space. Design 1, x1, and design 5, x5, are along the Pareto front and compose the Pareto-optimal set for this group of designs; all other designs are non-optimal. Though x5 has the highest final mass, x1 has shorter time of flight; neither x1 nor x5 is dominated by any other design. Domination The concept of domination allows for the comparison of a set of designs with multiple objectives. Such a concept is not required for single objective optimization because the value of the objective function is the sole measure of the quality of the design. When comparing two multiobjective designs (with all objectives to be minimized), the design x1 dominates design x2 if:

∀p : f p (x1 ) ≤ f p (x 2 )

p = 1,2,..., nobj

and

(3)

∃p : f p (x1 ) < f p (x 2 )

p = 1,2,..., nobj

That is, x1 dominates design x2 if, for all objectives, x1 is better than or equal to x2, and x1 outperforms x2 for at least one objective. Thus, in a direct comparison of two designs, if one design dominates another, the dominating design is superior and ‘nearer’ to the Pareto front. If neither design dominates the other, the designs are non-dominant to each other. Hence, the best designs (with equally good objective vectors) in an arbitrary set of solutions can be distinguished because they are not dominated by any other design in the set; they compose the non-dominated subset. Similarly, the designs that compose the Pareto-optimal set are the non-dominated set associated with the entire feasible space and are located along the Pareto front. Non-Dominated Sorting Domination can be used to categorize each design within a set into a hierarchy of non-dominated levels or fronts. Each different level of non-domination represents a relative ‘distance’ from the Pareto front. The best non-dominated front is closest to the Pareto front and each subsequent front lags further and is, thus, increasingly inferior. Through this sorting, each design is associated with a front that defines the quality of the design relative to the rest of the group. To isolate the various fronts, the designs that belong to the nondominated subset of the entire group are first identified. These designs are the best in the group, the closest to (or members of) the Pareto front, and are classified as front 1, or F1. Any design belonging to F1 is then temporarily set aside and another comparison process determines the next level of non-dominated designs from the remaining population. This non-dominated subset is front 2, or F2, and the procedure is repeated until the entire population has been sorted into the appropriate level. For example, return to the multiobjective trajectory optimization example in Figure 1. Designs x1 and x5 are clearly members of F1 because they lie along the Pareto front. Designs x1 and x5 are then discarded and a second sorting reveals that x2, x4, and x6 are non-dominated and belong to F2. A final comparison demonstrates that x3 and x7 are non-dominant to each other and constitute F3. Note that the members of the best front, F1, are not necessarily Pareto optimal.

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NSGA-II: A MULTIOBJECTIVE GENETIC ALGORITHM The utilization of a population in a genetic algorithm for search and optimization is computationally expensive for single objective optimization, but the population-based approach also affords the algorithm the unique ability to operate on many designs simultaneously. Such an approach is advantageous for multiobjective optimization because a sufficient global Pareto front representation requires many solutions. Thus, rather than many repetitions of a single objective method, one simulation run can develop an entire population of Pareto-optimal solutions. Goldberg realized the potential of a GA for successful application to multiobjective optimization problems; and he outlined a procedure that assigns fitness based on the rank of a design’s non-dominated front, determined through non-dominated sorting.2 Srinivas and Deb first implemented this technique with the development of the NSGA. Deb et al. followed with a second-generation non-dominated sorting genetic algorithm, the NSGA-II, that employs elitism and demonstrates improved efficiency. The NSGA-II is especially effective for a large range of difficult multiobjective optimization problems3,21 and is the technique implemented for this development. Genetic Algorithms A genetic algorithm is a stochastic search and optimization technique that simulates natural selection and reproduction to evolve ‘highly adapted’ (optimized) designs.2 Unlike calculus-based methods, GAs do not use gradient information in a deterministic, point-to-point search. They operate with an entire population of designs and incorporate probabilistic transition rules executed by three key genetic operators: selection, crossover, and mutation. These operators imitate biological evolution, iteratively constructing a new, and hopefully better ‘adapted’, population with each successive generation. A simple genetic algorithm is aesthetically straightforward, but produces powerful results. An initial population of designs is first created using a random number generator. Each individual design in the population is assigned a fitness value based on its cost function. The best performing designs, those with lower fitness values, are selected to form a parent pool. This process is analogous to natural selection in biological populations. Next, the crossover operator generates an offspring population from designs in the parent pool. In uniform crossover,22 the scheme applied in this implementation, two parents are selected from the pool at random and are “mated”, or combined, to create two new offspring designs. The offspring possess a combination of characteristics from both parents and form the next generation. This reproduction step enables “genes” from the parents to be passed on to the next generation while simultaneously generating new genetic patterns to search the design space. Mutations are then imposed on a specified percentage of the children to further explore the design space, finalizing the new offspring population. This process is repeated until the population has become homogenous, the same best design has occurred for several consecutive generations, or a maximum generation count is exceeded. The advantages of GAs include the fact that no initial guesses are required to start the optimization process and, unlike gradient based methods, GAs are not confined to local searches. The algorithms are capable of searching expansive and complex design spaces and can generate unbiased, nonintuitive solutions. Furthermore, slight modifications to the simple GA can transform the algorithm into an effective multiobjective optimization scheme. GAs, however, are computationally expensive, and without employing gradient information, it is often difficult to meet constraints. NSGA-II Formulation One of the most effective multiobjective evolutionary algorithms is the Non-dominated Sorting Genetic Algorithm II (NSGA-II).3 The NSGA-II attempts to achieve broad coverage of the Pareto front by emphasizing the designs that are closest to the Pareto front, while maintaining diversity and incorporating elitism. The fitness value of an individual in the population is based on the rank of the non-dominated front with which it is associated. Because the designs that are closer to the Pareto front are associated with a lower ranking and, thus, a better fitness, evolution is biased toward the Pareto front. The NSGA-II employs elite preservation, which yields improved convergence properties. Furthermore, the NSGA-II includes a strategy to develop a wide set of solutions along the Pareto front by affording preference to less crowded designs in two different locations within the algorithm. The NSGA-II algorithm diverges from the original NSGA by retaining two populations: a parent population, Pt, and an offspring population, Qt, both of size N. Initially, the parent population is created randomly, consistent with the standard GA, and fitness is assigned through non-dominated sorting. The parent population then produces the first offspring population with the three genetic operators. However, Pt is not discarded; it is combined with Qt to form a new population, Rt, with size 2N. Non-dominated sorting then determines the rank of each design in Rt according to its non-dominated front level. After the ranking, a new 4

parent population, Pt+1, is created from the best N designs in Rt. The designs in the best non-dominated front receive first priority in filling the available slots in Pt+1. Then, if the size of the first non-dominated front is less than N, the designs in the second non-dominated front start filling the remaining available slots. This filling continues with the subsequent non-dominated fronts until Pt+1 can no longer accommodate an entire non-dominated front from Rt. However, it is unlikely that the last allowed front will fit exactly into the remaining open slots in Pt+1. Thus, some mechanism must be employed to determine which designs from the last allowed non-dominated front should be placed into the new parent population. To encourage diversity, preference is given to designs that are less “crowded” in the objective space along the non-dominated front. A sorting based on crowding distance differentiates designs in a non-dominated set by favoring designs in less congested areas of the front. To define a crowding distance it is useful to impose some geometric structure onto a front. The crowding distance, di, associated with a design i is the sum of the average distance (in objective space) between the two designs immediately surrounding design i for each objective along a non-dominated front, F. The crowding distance provides a measure of the density of the designs along the front at the location of the ith design by estimating the perimeter of the hyperrectangle formed with the two adjacent designs of each objective as vertices. A crowding distance is assigned for each member of the non-dominated front. The smaller the value of di, the less distance separates the neighboring designs from design i along the front. A high di value indicates that the individual is far from the other designs in the front and is preferred to encourage diversity. Because the designs on the boundaries of the non-dominated front represent extremes in terms of the objectives, and only one adjacent neighbor exists, they receive a large crowding distance to ensure precedence. The crowding mechanism for completing the last slots in the population Pt+1 becomes critical to developing a sufficient spread of designs along the Pareto front in the later generations when many designs are already Pareto optimal. In early generations, there are many smaller groups of non-dominated sets and only a few designs in lower-ranking sets require sorting in terms of their crowding distance. In the final generations, however, the best non-dominated front in Rt may include more than the allotted N designs and only the least crowded individuals in the set are included in the new parent population. Once created, Pt+1 moves to the genetic operators, which generate a new, and hopefully more optimal, offspring population, Qt+1. A second diversity preservation mechanism in the NSGA-II occurs in the selection operator. Before selection, each design in Pt+1 is assigned a fitness value equal to the rank of the corresponding non-dominated front. That is, a design along the best non-dominated front is assigned a fitness of 1. Then, the fitness of any design along the second non-dominated front is 2, and so on. However, the NSGA-II algorithm also distinguishes between designs with the same fitness by sorting each front in the population Pt+1 in terms of crowding distance. Differentiating designs via the crowding distance metric allows for application of the ‘crowded tournament’ selection operator and avoids the necessity of specifying a sharing parameter as required in the NSGA. The crowded tournament operator is structured in a manner that is similar to standard tournament selection, but compensates for the possibility of competition between two designs with the same fitness value. The operator compares two designs at a time and, first, evaluates the fitness values, which are equivalent to the ranking of the non-dominated front associated with the designs. If the designs belong to different fronts and, thus, have a different fitness, the design belonging to the better front wins the competition. However, when the fitness values of two competing designs are equal, the design with a larger di is selected for promotion to the parent pool. In this way, the crowded tournament selection operator further promotes diversity. After selection, the members of the parent pool are mated in the crossover operator and the resulting children are passed to the mutation operator, which completes the creation of the offspring population. GALLOP For the local improvement component of the hybrid algorithm, an efficient and robust calculus-based technique is desired. A direct method fitting these qualifications was developed by Sims and Flanagan15 and has been incorporated in the software package Gravity-Assist Low-thrust Local Optimization Program (GALLOP)13,14 developed by McConaghy et al. at Purdue University (as well as the program MALTO23 constructed at the Jet Propulsion Laboratory). GALLOP has been used in a number of preliminary SEP and NEP trajectory design studies, demonstrating its capability to locally optimize complex LTGA trajectories.24-27 For these reasons, GALLOP is incorporated as the local optimization component of the hybrid approach. The structure of the Sims-Flanagan trajectory model affords GALLOP a reduced sensitivity to the initial guess when compared to many indirect methods. To accommodate the complexities of multiple gravity assists, trajectories in GALLOP are partitioned into independent legs, which are bounded by gravitational bodies, or control nodes. Along each leg, a match point is defined at a particular fraction of the leg’s duration.

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The trajectory is propagated forward in time from the originating control node of the leg to the match point, and backward in time from the terminating control node to the match point. If a mass, position, or velocity discontinuity exists at the match point, the trajectory arc is infeasible and the optimizer attempts to reduce these discrepancies to produce a feasible trajectory. This multiple shooting strategy decreases the sensitivities that are incurred by propagation only forward in time from the start of the trajectory. A fundamental simplifying assumption is incorporated in GALLOP’s modeling of the spacecraft thrust. In contrast to indirect methods, which generate a continuous thrust profile, GALLOP discretizes the continuous thrust arc to create a finite-dimensional problem. Each leg is divided into a specified number of segments and the continuous thrust is approximated by an impulsive ∆V at the midpoint, in time, of each segment. Thus, continuous thrusting is assumed along the entire segment if the ∆V magnitude is greater than zero. Accordingly, the more segments that are specified per leg, the better the thrust approximation. The benefits of robustness and efficiency come at the expense of a high dimensionality and limited accuracy due to a simplified trajectory model. Therefore, the trajectory produced from GALLOP, albeit a generally good approximation,25 must be refined in a higher fidelity model for use on an actual mission. To accomplish the optimization, GALLOP employs the nonlinear optimization software SNOPT (Sparse Nonlinear OPTimizer).28 SNOPT is based on a sequential quadratic programming algorithm, and is wellsuited for large-scale, nonlinear problems. To improve execution time and robustness, analytic derivatives are provided to SNOPT. Note that, convergence to a feasible, locally optimal trajectory depends critically on the initial guess for the independent variables. Thus, an initial guess that is sufficiently “close” to the optimal solution is required, often a challenging task. Several user options determine the exit conditions for SNOPT, including the “major feasibility” tolerance, the “major optimality” tolerance, and the “major iteration” limit. The SNOPT output “major feasibility” indicates the degree to which the constraints and bounds are satisfied. When the “major feasibility” value is below the tolerance setting, the trajectory is considered feasible. Similarly, the lower the “major optimality” output value, the better the optimality conditions are met. For convergence, both the major feasibility and optimality values must be within the tolerance. The “major iteration” limit regulates the number of iterations that SNOPT is allowed before the run is terminated. Further details on GALLOP (including the constraints and independent variables) are available in McConaghy’s dissertation.13 MULTIOBJECTIVE GA-GALLOP HYBRID ALGORITHM To exploit the advantages of both a multiobjective evolutionary algorithm and a calculus-based direct method, the NSGA-II algorithm and GALLOP software are combined to form a multiobjective GAGALLOP hybrid algorithm. In hybridizing the two techniques, a formulation is achieved that allows for a global search of the entire design space without the necessity of a user-defined initial guess. The scheme evolves a distribution of globally optimal solutions that form a representation of the Pareto front. The hybrid algorithm is structured so that the NSGA-II acts as a ‘wrapper’ around GALLOP (Figure 2). In every generation, each individual in the NSGA-II offspring population is sent to GALLOP. GALLOP refines and repairs each individual, attempting to shift the design variables so that the trajectory is feasible and locally optimal. Through this formulation, the beneficial characteristics of the two methods offset the disadvantages intrinsic to each technique applied alone. Inclusion of GALLOP into the main loop of the NSGA-II permits constraint handling to be delegated to GALLOP. GALLOP can then exploit gradient information and efficiently search for local optimums within the design space neighborhood of the individual. As the global component, the NSGA-II explores the entire design space for solutions along the Pareto front that are globally optimal in terms of final mass and TOF, without the requirement of an initial guess. This type of hybrid formulation, where an evolutionary algorithm is coupled with a local improvement scheme, is sometimes designated a memetic algorithm (MA).29,30 MAs are often significantly more efficient and effective than a genetic algorithm alone when applied to complex optimization problems.31-34 In a MA, two inheritance schemes are available for the design variables after local refinement, or adaptation.17,32,34 One choice is Lamarckian inheritance, where the adapted design variables resulting from the local search replace an individual’s original design variables. The other choice is Baldwinian inheritance, where the original design variables that define an individual are retained after the local improvement execution (i.e., GALLOP’s optimization does not affect the design variables). In both strategies, the fitness of the individual design is based on the result of the local improvement step. Hybrid Algorithm Structure The multiobjective implementation of the hybrid is similar in structure to the single objective version of the GA-GALLOP hybrid algorithm.18 The fundamental difference is that the dual objective nature of the 6

problem requires a global search scheme that Gen=1 Generate random initial parent can simultaneously maximize final mass and population of trajectories, P1 minimize time-of-flight. Thus, instead of employing a simple genetic algorithm, the NSGALocally optimize all of P1 in GALLOP II is utilized to accomplish the global search. Further details such as the motivation and charNon-dominated sort P1 based acteristics of the GA-GALLOP hybrid, as well on final mass and TOF as an in-depth discussion on the inheritance schemes, can be found in Reference 17. Crowded tournament selection The initial step in the multiobjective algoGenetic operators to rithm is similar to the single objective impleUniform crossover creates create initial mentation. The initial parent population, P1, of offspring population, Q1 offspring size N is created with a uniform random numpopulation ber generator. However, in the multiobjective Clock mutation on Q1 version, there are two objectives and the fitness value is developed using a non-dominated sortGen=Gen+1 Locally optimize Q in GALLOP ing routine. An attempt to locally optimize P1 is accomplished in GALLOP; then, a user-defined penalty value is added to both the final mass and the time-of-flight if the individual solution No Maximum Combine Pt and Qt to form Rt is not feasible. The penalized objectives are generation? employed in the sorting routine to develop the Non-dominated sort Rt to various non-dominated fronts. The fitness of Yes generate Pt+1 an individual is assigned according to the rank of the non-dominated front to which it belongs. Stop Crowded tournament selection Recall that the closer the front is to the Pareto on Pt+1 front, the lower the fitness value of individual. Genetic operators At this stage, the inheritance of the design variUniform crossover creates ables from GALLOP are also determined. The offspring population, Qt+1 inheritance options are the same in the multiobjective hybrid as in the single objective hybrid Clock mutation on Qt+1 algorithm. Thus, each individual in the population can use Lamarckian or Baldwinian inheri- Figure 2. Multiobjective GA-GALLOP hybrid structure. tance based on user-defined parameters. A significant difference between the multiobjective and single objective versions of the hybrid is the development of an offspring population of trajectories in the multiobjective GA-GALLOP hybrid. The initial offspring population, Q1, of size N is generated by application of the genetic operators: crowded tournament selection, uniform selection, and clock mutation. Once Q1 is created, the procedure enters the main loop of the hybrid algorithm, and each member in the offspring population is locally optimized with GALLOP. Consistent with the initial parent population, the inheritance in the offspring population is determined via userdefined parameters. After the final mass and the time-of-flight corresponding to the infeasible solutions are penalized according to the “major feasibility” value returned from GALLOP, the stopping criterion is checked. If the number of generations is greater than a specified maximum, the process is terminated. Otherwise, the offspring solution is then combined with the parent population, creating a combined population, Rt, of size 2N. The use of both a parent and offspring population allows for the incorporation of elitism by maintaining the elite individuals from a generation in a new parent population. In the main loop of the hybrid algorithm, the combined population is ranked with a non-dominated sorting. The new parent population, Pt+1, is generated from the individuals in the best non-dominated fronts, where the last available slots in the new population are filled based on the crowding distance. Thus, this parent population is composed of the elites of the combined population and generates the next offspring population. The new offspring population, Qt+1, is developed in the same way as the initial offspring through crowded tournament selection, uniform crossover, and clock mutation. The loop then restarts, with each individual in the new offspring population passing to GALLOP. As in the single objective implementation of the GA-GALLOP hybrid, a reduced thrust parameterization is incorporated in the NSGA-II component of the hybrid (GALLOP still operates with the standard thrust representation). The thrust is represented with a Chebyshev series to reduce the number of design variables

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that the NSGA-II is required to manage, decreasing the length of the binary string encoding each design variable and, thus, lessening computational complexity. All other design variables associated with the hybrid algorithm are the same as the independent variables from GALLOP. Except for the elitism percentage, the same user controls as in the single objective implementation are required for the multiobjective hybrid. These parameters include the population size, maximum number of generations, and mutation probability for the NSGA-II component; along with the parameters “major feasibility”, “major optimality” and “major iteration limit” for the GALLOP component. Additionally, the user must define the degree of the Chebyshev series to parameterize the thrust for each leg and a penalty parameter. The user must also specify the two parameters for the inheritance strategy associated with the design variables that are output from GALLOP: the Lamarckian probability and the Lamarckian buffer.17 RESULTS Direct Earth-Mars Rendezvous Table 1. Multiobjective GA-GALLOP hyThe multiobjective GA-GALLOP hybrid algorithm is brid parameters for EM rendezvous case applied to a direct Earth-to-Mars rendezvous mission sceParameter Value nario with the goal of generating a Pareto front comprising Population size 200 trajectories that are optimal in terms of final mass and timeof-flight. To verify the capability of the hybrid algorithm to Mutation probability 3% develop a representative front of globally optimal solutions Lamarckian probability 100% in terms of both objectives, the same solar electric propulLamarckian buffer 1E-4 sion powered spacecraft and launch vehicle as applied in a 35 Maximum generation 800 study by Williams and Coverstone-Carroll is incorpoPenalty parameter 1E8 rated. The same spacecraft and launch vehicle parameters are also used in Reference 18 in a single objective global Major iterations limit 300 optimization of an Earth-Mars (EM) trajectory for verificaMajor feasibility tolerance 1E-4 tion of the single objective implementation of the GAMajor optimality tolerance 1E-4 GALLOP hybrid. The ranges for some of the key design Degree of Chebyshev series 4 variables, such as the launch and arrival dates, are listed in Table 2 (the same ranges are applied in the single Table 2. GA-GALLOP hybrid variable ranges objective EM example of Reference 18), where the for Earth-Mars rendezvous example date ranges are based on Reference 35 for compariDesign Variable Lower Bound Upper Bound son purposes. All other variables are ‘free’, such 1 May 2004 31 Dec. 2009 that the ranges are sufficiently broad to encompass Launch date 0 km/s 10 km/s all physically feasible trajectories. Globally optimal Launch v∞ solutions in terms of final mass are available for Mars arrival date 1 May 2004 29 Apr. 2014 flight times of 1.5, 2.0, 2.5, 3.0, and 3.5 years from Mars arrival v∞ 0 km/s 0 km/s Reference 35 and were corroborated with the single Mars arrival mass 200 kg 1300 kg objective implementation of the GA-GALLOP hybrid algorithm. Thus, these five points can be safely assumed to be located along the Pareto front and serve as test points to validate the hybrid algorithm. The launch and arrival dates (as with the rest of the design variables) are not fixed, and are allowed to vary within the specified ranges. The time-of-flight is constrained to be less than 3.5 years, and the population size is 200. All of the hybrid parameters are listed in Table 1. To demonstrate the evolution of the population in the multiobjective GA-GALLOP hybrid algorithm, both objectives are plotted against each other for generations 3, 25, 100, and 800 in Figure 3. By the third generation, all of the trajectories in the population are feasible, though many are clustered near the TOF upper limit (3.5 years). By the 25th generation, the hybrid algorithm has discovered several more trajectories with lower flight times that are non-dominated. At generation 100, most of the population lies along a distinct non-dominated front. From generation 100 to 800, this front is progressively shifted up (higher final mass) and to the left (lower flight time), towards the Pareto front. By the 800th generation, all members of the population are non-dominated and distributed uniformly along the front. The final mass versus the TOF for all of the feasible designs throughout the evolution (800 generations) are plotted in Figure 4. The trajectories corresponding to the final generation are identified in red, whereas the rest of the designs are blue. Note that four clusters of trajectories have emerged in different TOF regions. One family resides in the TOF region from 300 days to roughly 510 days, where relatively dramatic improvements in final mass can be realized with only small compromises in TOF. The next family stems from the disjointed shoulder of the non-dominated front from ~510 days to ~570 days. The third family is the 8

grouping of trajectories with flight times between 700 days and about 950 days, where the front begins to level out, and only small improvements in final mass are attained with increasing TOF. The final grouping is located in the upper right hand corner, with TOFs ranging from ~1050 days to 1278 days. The trajectories in each family possess similar characteristics and thrusting structures (e.g., coast-thrust or thrust-coast-thrust), where the trajectories with longer flight times and higher mass within each family, generally, launch earlier, with a lower v∞, and arrive later.

Figure 3. Evolution of the population for the EM rendezvous example (gens. 3, 25, 100, and 800). After 800 generations, each member of the final population is sent to GALLOP to be reoptimized with “major feasibility” and “major optimality” tolerances both decreased from 1E-4 to 1E-6, and the flight time constrained to its current value. Recall that GALLOP optimizes for final mass only, and in the nominal execution of the multiobjective GAGALLOP hybrid, the TOF is not constrained to a particular length; however, an upper limit of 3.5 years is in effect. The refined population is plotted in Figure 5. Several of the individuals in the first family, those with lower flight times, shift to solutions with a higher final mass. Correspondingly, the non-dominated front is

Figure 4. All feasible designs through 800 generations in the EM rendezvous example (designs from generation 800 in red). 9

also shifted, and as a result, the g h population no longer comprises a f e single set of non-dominated individuals. However, the nond 3.5 years 2.5 years 3.0 years dominated front associated with the 2.0 years re-optimized population should be 1.5 years more representative of the true c Pareto front in the shorter TOF region. The five test points from Coverstone-Carroll,35 validated using the single objective GAGALLOP hybrid, are: (1.5 years, 943 kg), (2.0 years, 1008 kg), (2.5 b years, 1024 kg), (3.0 years, 1038 kg), (3.5 years, 1039 kg). The test points successfully lie along the non-dominated front generated by a the multiobjective GA-GALLOP hybrid. However, no test points Figure 5. Re-optimized designs from generation 800 for the EM exist to confirm that the individuals rendezvous example. in the early TOF family, those that shifted after re-optimization, are representative of the Pareto front. To illustrate the characteristics of the four families, several representative trajectories from the nondominated front are selected for plotting. The location of these trajectories along the front is indicated by the green dots and a corresponding identification letter (a – h) in Figure 5. The eight selected trajectories are plotted in Figure 6 (where the red lines identify the thrust direction), and the characteristics of each trajectory are listed in Table 3. Trajectories in the short TOF family (trajectories a - c) possess similar launch and arrival dates, as well as a similar thrusting structure, that is, thrust-coast-thrust. However, as the TOF increases, the launch v∞ decreases: from 317 days and 8.69 km/s in trajectory a to 427 days and 4.6 km/s in trajectory c. The trajectories with a higher TOF are able to thrust longer, allowing the propellant-efficient low-thrust engine to accumulate enough ∆V to compensate for the reduced launch v∞, which, otherwise, must be generated by the less efficient launch vehicle. In trajectory a, the spacecraft is thrusting along approximately 50% of the trajectory, whereas in trajectory c, the spacecraft is thrusting for all but one segment. In the ‘shoulder’ family, which includes trajectory d, the launch date is in late-August 2007, and the arrival date is in early2009. In this trajectory, and the family in general, the spacecraft thrusts for nearly the entire trajectory. However, the trajectories in the family with longer flight times allow for a reduced launch v∞ and, thus, a higher injection mass and final mass. Trajectories e and f are members of the third family, in which the spacecraft completes at least one solar revolution, launching in mid-2009 and then spiraling out to Mars by thrusting in a direction that is nearly tangential to the velocity. While trajectories e and f possess a similar launch v∞, the longer TOF associated with trajectory f allows for the elimination of thrusting in inefficient locations along the trajectory (i.e., when the spacecraft is further from the Sun). The fourth, and final family, includes trajectories g and h. These solutions are also spiral trajectories; however, they launch in 2008, and complete more solar revolutions than the trajectories in the third family. Table 3. Characteristics of selected trajectories (from Figure 5) for the EM rendezvous example Trajectory

Launch Date

a b c d e f g h

19 Dec. 2009 27 Nov. 2009 30 Oct. 2009 27 Aug. 2007 25 July 2009 15 June 2009 18 Oct. 2008 15 June 2008

Launch v∞ (km/s) 8.69 6.86 4.66 2.90 1.92 1.73 0.75 0.75

Arrival Date 1 Nov. 2010 7 Nov. 2010 31 Dec. 2010 12 Feb. 2009 4 July 2011 15 Aug. 2011 14 June 2011 14 Dec. 2011

10

TOF (days) 317 345 427 536 709 792 969 1278

Final Mass (kg) 211 412 706 933 1003 1016 1037 1039

mf = 211 kg, TOF = 317 days


a

b a

b


c

d mf = 1016 kg, TOF = 792 days

mf = 1003 kg, TOF = 709 days c

d

e

f mf = 1037 kg, TOF = 969 days

h

f

e

g


mf = 1039 kg, TOF = 1278 days

h

Figure 6. Ecliptic projections of selected trajectories from the Pareto front for the EM example.

11

Direct Earth-Jupiter Rendezvous As another example, a direct Earth-Jupiter rendezvous Table 4. GA-GALLOP hybrid parameexample is considered for a spacecraft with a NEP systers for the EJ rendezvous example tem. The same spacecraft and launch vehicle as in a Parameter Value comprehensive study of trajectories to the outer planets by 26,27 Population size 200 is applied. Yam et al. employed a two-step Yam et al. 36 Mutation probability 3% global approach (1. STOUR-LTGA ; 2. GALLOP) over a wide range of launch dates to generate optimal trajectoLamarckian probability 100% ries in terms of final mass. This same study was used for Lamarckian buffer 1E-4 comparison with results from the single objective GAMaximum generation 1300 GALLOP hybrid.18 In this example, 48 segments are used Penalty parameter 1E8 for the trajectory (a single GALLOP leg), and the TOF is Major iterations limit 400 constrained to be less than 9 years. The hybrid parameMajor feasibility tolerance 1E-4 ters and the ranges on key design variables are listed in Tables 4 and 5, respectively. The range on the launch Major optimality tolerance 1E-4 date is the same as that used in Reference 26. Both the Degree of Chebyshev series 5 launch date range (11 years) and the arrival date range (14 years) are broad, and the other design variTable 5. GA-GALLOP hybrid variable ranges for ables are effectively free, defining a large EJ rendezvous example design space. Thus, a wider variety of optiDesign Variable Lower Bound Upper Bound mal trajectory types along the Pareto front is expected as compared to the EM example, Launch date 1 Jan. 2014 1 Jan. 2025 creating a challenging multiobjective optimiLaunch v∞ 0 km/s 7 km/s zation problem. Jupiter arrival date 1 Jan. 2016 1 Jan. 2030 The evolution of the population is illus0 km/s 0 km/s Jupiter arrival v ∞ trated in Figure 7 with plots of the final mass Jupiter arrival mass 4,000 kg 20,000 kg versus TOF for generations 13, 50, 200, and 1300. By generation 13, all of the members of the population are feasible, but many designs are clustered towards the longer TOF regions of the feasible objective space. This clustering behavior is expected in the first several generations because GALLOP only optimizes for final mass, often increasing the TOF of designs to improve final mass. However, as the generations increase, diversity mechanisms in the NSGA-II algorithm attempt to spread out the designs in objective space, countering the pressure GALLOP places on the final mass objective in the hybrid algorithm. The population of the 50th generation is distributed more evenly across time-of-flight and includes more designs in the best non-dominated front. The 200th generation comprises even more trajectories in the non-dominated front. By the 1300th generation, the maximum generation, the members of non-dominated front are apparently on or near the Pareto front and globally optimal in terms of final mass and TOF. As in the EM example, the entire population of the last generation is re-optimized in GALLOP with the tolerances on “major feasibility” and “major optimality” decreased from 1E-4 to 1E-6; the TOF is also constrained to its current value. After ‘re-optimization’ there are only slight increases in final mass for a few designs, and 81 designs out of the 200 members of the population are non-dominated. The changes in final mass are not as large as in the EM example, however, and only a minor shifting of the non-dominated front occurs in the lower TOF regions. All feasible trajectories through 1300 generations, in addition to the re-optimized final generation, are plotted in Figure 8. The red dots indicate the non-dominated designs for the re-optimized final trajectory and represent the apparent globally-optimal Pareto front for this example. The green dot locates an EJ trajectory from Yam et al.26 (mf = 13,709, TOF = 2,020 days). This test point was not designed to be globally optimal, but serves as a reference for the EJ trajectory from the study by Yam et al. Eight representative trajectories (a – h) from the apparent Pareto front are selected for plotting (orange “x”) in Figure 9, and the trajectory characteristics are listed in Table 6. From the trajectory plots, it is clear that the longer the TOF associated with the trajectory, the higher the degree of spiraling (the more solar revolutions). The flight times corresponding to the trajectories associated with the highest final mass values (e.g. trajectories g and h) are sufficiently long to accommodate significant periods of coasting, allowing the spacecraft to thrust at more efficient locations. As expected, the launch v∞ of trajectories decreases with increasing TOF and final mass.

12

Figure 7. Evolution of the population for the EJ rendezvous example (gens. 16, 50, 200, & 1300).

f

EJ test point

g

h

e d c

b

a i

Figure 8. All feasible designs through 1300 generations for the EJ example (nondominated front of the re-optimized last generation in red).

13

mf = 5,754 kg, TOF = 760 days

a

mf = 8,811 kg, TOF = 1,034 days

b b m = 12,417 kg, TOF = 1,619 days f

mf = 11,334 kg, TOF = 1,272 days

c

d mf = 15,132 kg, TOF = 2,124 days

mf = 13,937 kg, TOF = 1,866 days c

d

e

f h mf = 15,428 kg, TOF = 3,216 days

mf = 15,320 kg, TOF = 2,600 days

f

e

g

h

Figure 9. Ecliptic projections of trajectories from the non-dominated front for the EJ example. example 14

Table 6. Characteristics of selected trajectories (from Figure 9) fom the EJ rendezvous example Trajectory

Launch Date

Launch v∞ (km/s)

Arrival Date

TOF (years)

Final Mass (kg)

a

15 Apr. 2021

6.97

15 May 2023

2.08

5,754

b

30 Apr. 2022

5.66

28 Feb. 2025

2.83

8,810

c

22 Mar. 2022

4.30

14 Sept. 2025

3.48

11,334

d

14 Dec. 2020

3.60

22 May 2025

4.43

12,417

e

31 Mar. 2020

2.46

10 May 2025

5.10

13,937

f

23 Dec. 2018

0.58

16 Oct. 2024

5.81

15,132

g

27 Apr. 2019

0.55

10 June 2026

7.11

15,320

h

4 May 2017

0.50

22 Feb. 2026

8.81

15,428

Earth-Earth-Jupiter Rendezvous The final example is also a rendezvous Table 7. GA-GALLOP hybrid variable ranges for mission to Jupiter, but includes a gravityEarth-Jupiter rendezvous example assist from Earth. This Earth-Earth-Jupiter Design Variable Lower Bound Upper Bound trajectory scenario is the most challenging of Launch date 1 Jan. 2014 1 Jan. 2025 the three cases and, thus, a larger population Launch v∞ 0 km/s 7 km/s size of 300 is employed. All other GAEarth arrival date 30 June 2014 31 Dec. 2028 GALLOP hybrid parameters are the same as in the EJ example (Table 4) except for the Earth arrival flyby alt. 500 km 10,000 km “major iterations” limit, which is increased Jupiter arrival date 31 Dec. 2016 1 Jan. 2030 from 400 to 500, and the maximum number Jupiter arrival v∞ 0 km/s 0 km/s of generations, which is decreased to 1200. Jupiter arrival mass 6,000 kg 20,000 kg The ranges on the key design variables are specified in Table 7 and the TOF is constrained to less than 7 years. Additionally, a 50-day minimum flight time on the Earth-Earth leg is imposed to keep GALLOP from searching for physically infeasible trajectories. Note that the launch, encounter, and arrival date ranges are all wide, establishing not only a large design space, but also a more complex objective space than the other examples. The number of segments on both legs is 24. The evolution of the population is demonstrated with plots of the final mass versus time-of-flight for generations 16, 60, 400, and 1200 in Figure 10. Similar trends as compared to the evolution of EM and EJ examples are observed. After 1200 generations, the final population is re-optimized in the same fashion as the previous examples to generate an apparent Pareto front of globally-optimal trajectories, as illustrated in Figure 11 in red. Out of the 300 designs in the population, 29 are non-dominated after 1200 generations. Allowing more generations should increase the number of non-dominated designs to produce a better representation of the Pareto front. The final mass and TOF of all designs through 1200 generations are also plotted in Figure 11 in blue. To verify the GA-GALLOP hybrid algorithm’s capability to generate known results, two EEJ trajectories from Yam et al.26 are used as test points and are indicated by green dots in Figure 11: (1,651 days, 16,181 kg), (1,801 days, 16,260 kg). Both test points lie along the apparent Pareto front evolved by the hybrid algorithm. From the 29 trajectories in the non-dominated front, eight are selected as representative examples and plotted in Figure 12. The three trajectories with the lowest TOF (a – c) possess similar, and rather unique, geometry on the Earth-Earth leg. In all three trajectories, the spacecraft initially thrusts in the opposite direction of the velocity to place it on a trajectory that passes inside of Earth’s orbit. It then reverses the thrust direction suddenly so that the thrust is in the same direction as the spacecraft velocity, allowing the spacecraft to flyby Earth almost exactly six months after launch. The flight time corresponding to the Earth-Earth leg for the other representative trajectories (d – h) is greater than one year, allowing for more optimal geometries in terms of final mass. The trajectory characteristics of the eight representative trajectories are listed in Table 8. As final mass increases along the apparent Pareto front, the flight time of Earth-Earth leg also increases.

15

Figure 10. Evolution of the population for the EEJ example (gens. 16, 60, 400, and 1200).

d i

EEJ test points e i

f i

g i

h i

c i b i

a i

Figure 11. All feasible designs through 1200 generations for the EEJ rendezvous example (nondominated front from the re-optimized final generation in red). 16

mf = 10,082 kg, TOF = 1,182 days

mf = 7,148 kg, TOF = 1,036 days

a

b mf = 12,045 kg, TOF = 1,279 days

c

b mf = 15,897 kg, TOF = 1,526 days

d mf = 16,201 kg, TOF = 1,708 days

mf = 16,303 kg, TOF = 2,020 days

c

d

e

f mf = 16,320 kg, TOF = 2,303 days

h mf = 16,396 kg, TOF = 2,555 days

f

e

g

h

Figure 12. Ecliptic projections of trajectories from the non-dominated front for the EEJ example. example 17

Table 8. Characteristics of selected trajectories (from Figure 12) for the EEJ rendezvous example Trajectory

Launch Date

Launch v∞ (km/s)

Earth Arrival Date

Earth Arrival v∞ (km/s)

Jupiter Arrival Date

TOF (years)

Final Mass (kg)

a

5 June 2016

6.17

6 Dec. 2016

8.06

7 Apr. 2019

2.84

7,148

b

2 Nov. 2021

4.87

2 May 2022

6.23

27 Jan. 2025

3.24

10,082

c

4 July 2018

3.81

5 Jan. 2019

5.25

2 Jan. 2022

3.50

12,045

d

28 Mar. 2021

1.10

2 June 2022

7.55

1 June 2025

4.18

15,897

e

13 Dec. 2018

0.56

16 Mar. 2020

6.94

18 Aug. 2023

4.68

16,201

f

9 Oct. 2014

0.50

19 Dec. 2016

7.85

21 Apr. 2020

5.53

16,303

g

30 Sept. 2019

0.51

9 June 2022

7.70

19 Jan. 2026

6.30

16,320

h

12 Oct. 2020

0.68

28 July 2024

8.77

11 Oct. 2027

7.00

16,396

To compare the two scenarios for transfer from Earth to Jupiter, both the EJ and EEJ Pareto fronts are plotted against each other in Figure 13. It is clear that for flight times from ~760 days to ~1264 days (2.08 years to 3.46 years) the best options for a mission designer are direct EJ trajectories. For these low TOF trajectories, the additional Earth flyby is detrimental to the final mass (as compared to a direct trajectory) because the spacecraft is forced into an inefficient thrusting profile to return to Earth without completing an entire solar revolution. After 1264 days, however, the disparity between the two Pareto fronts exemplifies the benefits of a gravity assist. Dramatic advantages in either TOF or final mass are attained in selecting an EEJ trajectory over an EJ trajectory. At a TOF of 1561 days (4.27 years), the final mass associated with an EEJ trajectory along the Pareto front is 16,108 kg, a 31% increase in final mass (3,844 kg) over an optimal EJ trajectory with the same TOF.

Figure 13. Comparison of the globally-optimal Pareto fronts from the EJ and EEJ examples.

18

CONCLUSION An algorithm is developed for global optimization of low-thrust, gravity-assist trajectories in terms of both final mass and TOF by hybridizing a multiobjective genetic algorithm and the direct-method-based software GALLOP. The global component of the hybrid algorithm, the NSGA-II, incorporates a broad search of the design space and is able to place evolutionary pressure towards designs that are closest to the Pareto front by assigning fitness based on a non-dominated sorting. As the local improvement component, GALLOP employs a gradient-based search to refine and repair individuals in the population created by the NSGA-II. The GA-GALLOP hybrid algorithm exploits the strengths of each individual method to counter the disadvantages of the other method when applied alone. The technique is automated and can generate a representation of the Pareto front for LTGA trajectories in a single execution, even for complex, expansive design scenarios. Additionally, the algorithm does not require a user-defined initial guess and, therefore, is free of any associated biases allowing for the generation of nonintuitive solutions. The multiobjective GA-GALLOP hybrid algorithm is applied to a SEP Earth-Mars rendezvous scenario, a NEP Earth-Jupiter rendezvous mission, and a gravity-assist scenario for a NEP Earth-Earth-Jupiter rendezvous. In each example, a representation of the apparent Pareto front of globally optimal trajectories in terms of final mass and time-of-flight is generated. These apparent Pareto fronts are successfully verified with previously published optimal trajectories, demonstrating the efficacy of the technique. The entire set of nondominated solutions forming the Pareto front can be utilized to perform a trade study on the two objectives, significantly aiding the mission designer in the selection of a final trajectory design. ACKNOWLEDGMENTS The authors thank Dr. Chit Hong Yam for his assistance with GALLOP. Portions of this work were supported by Purdue University. REFERENCES 1

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