ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. MINIMIZATION OF FUEL CONSUMPTION OF A SWARM OF SPACECRAFT THROUGH A GENETIC ALGORITHM APPROACH SHANE LECOMPTE, DR. ANNALISA SCACCHIOLI (FACULTY ADVISOR) ✵ ABSTRACT As humanity moves closer to forming realis- tic paths toward space exploration beyond that what we have already accomplished, multiple new chal- lenges have presented themselves. Traditional large spacecraft prove to be unfeasible both logistically and economically for missions where a single prob- lem can completely halt operations, especially given that higher reward missions are also of higher risk. A possible alternative to large craft is using a swarm of smaller craft made to accomplish the same goals while mitigating some of the drawbacks large craft face. Rockets, space shuttles, and satellites all prove to be too large to navigate areas of space dense with obstacles. Smaller craft on the scale of one meter in a large swarm would navigate these regions. Due to the decentralized nature of a swarm, any problems faced by one craft do not necessarily affect the oth- ers, allowing the swarm to stay operational despite some crafts becoming compromised. This feature means that a problem or miscalculation that could completely derail an entire mission in the context of a large spacecraft would not do the same to a swarm. In the context of exploring dense and/or extreme en- vironments in space, many logistic and economic problems faced by large craft due to their size and centralized nature will not affect a swarm. With an ac- curate mathematical model of the swarm dynamics from Benet et al.[1], a genetic algorithm’s metaheuristic method is utilized[2] to find optimal pa- rameters that yield a minimal fuel consumption value for a given trajectory/mission objective. From this approach, the total fuel consumption was cut in half while retaining desirable characteristics of the trajec- tory such as collision avoidance and final formation constraints, giving us a similar course that accom- plishes the same goal of transporting craft around objects and disturbances while also minimizing eco- nomic losses. 1 INTRODUCTION Sending a large spacecraft to a destination far from Earth costs more money and has higher risk factors and implications of failure than the alternative presented in this paper. The materials fees, fuel cost for propelling the craft, and opportunity cost associ- ated with large craft failure all heavily outweigh that of a swarm. Failure of any portion of the large craft can result in failure of the entire mission in a domino effect, while the swarm approach mitigates this. Sending a swarm through a field of densely packed asteroids or ice rocks in planetary rings involves a co- operative intelligence between individual agents that may not be as sophisticated on their own, but as a whole are comparable to any individual many or- ders of magnitude larger/smarter on its own. This idea is mimicked in nature with swarms of bees to- gether to accomplish great feats of engineering comparable to humans despite their individual intel- ligence paling in comparison. Despite the hackneyed image that space travel is an incredibly advanced and technical area for humankind, controlling a swarm of autonomous agents moving as a single unit was already solved by nature millions of years before humans ever existed. This trend can be seen in flocks of birds, schools of fish, and colonies of ants. They react to disturbances as a whole and can continue to survive even if indi- vidual swarm members are compromised. Following the biological inspiration for translating of this prob- lem into a mathematical equation/model, a biologi- cally inspired method of solving it seemed to be in order. The genetic algorithm is based on Darwin’s theory of evolution. The idea of “survival of the ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II fittest” as it is described in natural selection is utilized in this algorithm to only allow the most optimal members of each generation to survive to the next, thus giving rise to a criterion for convergence to a global optimum if this algorithm is run consecutively and recursively. These “members” are the parame- ters used in the differential equation model pro- vided by Benet et al.[1] and tweaked in this paper. Dif- ferent values of certain parameters give rise to other solutions (trajectories). Thus, the genetic algorithm is utilized to find which parameters result in the most optimal trajectory. In their paper, Conn et al. detail NASA’s lat- est developments with autonomous spacecraft swarms.[3] SODA, or Swarm Orbital Dynamics Advi- sor, is a computational framework for a control net- work that governs communications between space- craft in a swarm. However, a fuel optimization ap- proach is still yet to be fully explored in this context. Optimal path planning and fuel minimization ap- proaches for UAVs (unmanned aerial vehicles) have been researched for terrestrial environments[4,5], and in space in the context of transferring between mul- tiple orbits[6], but a combination of these approaches has yet to be extensively developed for space explo- ration in dynamically dense environments. In this pa- per, calculating the optimal trajectory of a swarm of spacecraft is investigated, minimizing the total fuel consumption in this specific context, thus finding both a logistic and economically viable alternative to that of traditional missions involving a singular large craft. A specific area of space exploration, as men- tioned earlier, in which the results of this work may be applicable is in the exploration of Saturn’s rings. While we have been able to visit many moons of other planets in our solar system, we have only been able to visualize Saturn’s rings from a distance due to the density and unpredictable size (meters to kil- ometers) of the ice rocks present in them. This envi- ronment provides an unacceptable risk for a multi- million-dollar spacecraft, so such missions have been avoided. A swarm of spacecraft with the same instrumental capabilities of a large vessel and collec- tive intelligence equivalent to that of a larger com- puter system on a traditional craft that can morph around complex obstacles and dynamic obstacle patterns can provide a solution. The total cost of the Cassini mission, a probe that orbited Saturn retrieving much of the data we have today about Saturn’s rings, was about $3.26 bil- lion, with operations, fuel, and communications cost- ing about $760 million.[7] This last figure is for the costs mostly associated with the Cassini mission only after it had entered Saturn’s orbit and not the costs from the launch from Earth. With this being said, costs associated with launch and leaving Earth for both the traditional and swarm approach are the same considering the costs of operations in orbit are lessened by the swarm approach, not necessarily launch and manufacturing costs on Earth. The pur- pose of this paper is to highlight the economic via- bility of the swarm approach in the context of opera- tions post-launch and during the actual mission. The swarm itself will most likely leave Earth on a larger craft and then be deployed from the mothercraft once the destination has been reached. Only then will the total cost of the entire project start to be- come much less than that of a traditional mission where a singular large craft leaves Earth and carries out the mission at the destination. Not only will a swarm approach to this spe- cific problem have the potential to cost much less, but it also has a much greater margin of error, result- ing in less risk. There have been multiple studies showing both the economic and lower risk ad- vantages of the swarm approach instead of the tra- ditional large craft approach.[8,9] Various work on the validity of swarm approaches to drone applications and general robotics applications has already been explored.[8,10,11] In addition, NASA has considered projects involving “cubesats” that would accomplish similar goals using a swarm guidance framework within Earth’s orbit, also including ideas for swarm missions to Saturn in particular to individual moons and the upper atmosphere, but never within the dense parts of the rings.[12] In essence, a swarm ap- proach not only can provide a method of getting more data about Saturn’s rings to advance the sci- ence of planetary rings but can provide an econom- ically viable and less risk-prone alternative to tradi- tional mission outlines. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II List of Relevant Terms ARTIFICIAL POTENTIAL FIELD – The assignment of values to every point in space that corresponds to the willing- ness of an object to move toward or away from that point in space while not being tied to any actual physical phenomena or interactions such as gravita- tion. A standard potential field encompasses these fundamental forces.[1] BIFURCATION – The ability of a dynamic system to dras- tically change its behavior with the tuning of a single parameter.[1] CONVERGENCE – When a system tends toward a certain steady value or state after a certain period of time. LINE INTEGRAL – The integral of a function evaluated along a path in space.[14] WORK – The amount of energy transfer associated with a force acting on an object, as calculated via the line integral of the force along a specified path in space.[14] METAHEURISTIC – A computer search algorithm that is more sophisticated than simply checking all possible solutions in order to solve an optimization prob- lem.[2] GENETIC ALGORITHM – A metaheuristic method of opti- mizing a function that is otherwise undefinable ana- lytically or extremely difficult to do using methods of differential calculus. A global optimum is approxi- mated numerically by mimicking Darwin’s theory of evolution. With each successive generation, a more “fit” (lower fuel consumption) solution is created by taking the desirable characteristics of the previous generation and using them to pseudo-randomly generate the next. Each iteration (or generation) has “genes” associated with it that combine in a way with the possibility of mutation to produce the next gen- eration.[2] 2 METHODS i. Mathematical Model In this paper, a swarm of twenty spacecraft is considered with dynamics modeled by a system of differential equations provided by Bennet et al.[1] The model works for any number of craft, but twenty are used throughout this paper. The swarm is as- sumed to start stationed at a larger craft or satellite within space. The swarm will then be launched hori- zontally into space from this dock. This system of equations is derived from the basic principle of a particle’s behavior in a potential field. In this model, at every point in three-dimensional space, a value for potential is defined, and in this case, the particles are individual spacecraft moving through this space. A potential field can be thought of as either a hill or a well where objects (particles) tend to either roll down the hill (move away from a higher potential) or roll into the well (move closer to a lower potential) wherein this analogy the potential is gravity. We can extend this same concept to a more abstract mathe- matical model by defining points in three-dimen- sional space that the spacecraft tend to move away from and others where the spacecraft tend to con- verge to. These artificial potentials depend on both the position of a craft in space as well as its relative positions to all others in the swarm. Suppose there are preset positions in space where low potentials occur in addition to high potentials being defined at the positions of each spacecraft and any obstacles. In that case, a mathematical scheme can be created where both external obstacle and inter-craft collision avoidance and convergence to desired final for- mations are well defined. The following model (FIGURE 1) was based on the theory of bifurcating potential fields with a more rigorous mathematical derivation in the paper by Bennet et al.[1] They were able to derive a relation be- tween the velocity of each craft in a conveniently de- fined (in the sense that it is artificially constructed with the final product in mind during its inception) potential field to get desired final formation and tra- jectory characteristics. Mathematically, the velocity is equal to the negative gradient of the potential field. The effects of these potential fields on the spacecraft ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II only initiate the thrusters to force the spacecraft in a certain direction. In other words, these potentials are not real and are only defined within the computer- sensor framework of the swarm. The equation above is a modified version of the model from Bennet et al.,[1] taking this into consideration. Above in FIGURE 1 the system of differential equations modeling the dynamics of the swarm is given. Without going into mathematical detail, these are a system of equations relating the velocities, po- sitions, and relative positions of all swarm members to one another. The solution to this equation is the set of all the position functions of time of each mem- ber of the swarm, describing the path each craft fol- lows through space. The diagram next to the model is the representation of the position vectors of two arbitrary crafts with respect to the origin and their rel- ative position vector. This model contains four parts: the exponential potential, the hyperbolic potential, the inter-craft repulsive potential, and the modifica- tion, which we introduce as the craft-obstacle avoid- ance potential. The exponential and hyperbolic po- tentials control the final formation and trajectory, while the repulsive potential experienced by each craft governs obstacle and inter-craft avoidance. From this, a system of 20 first order, nonlinear, highly coupled, vector differential equations governing the motion of the swarm is obtained. Relevant variables and parameters are summarized in the following ta- ble (TABLE 1). ii. Formulation of Optimization Problem The main goal in this paper is to determine the values of certain parameters in the model that yield a trajectory that minimizes the total fuel con- sumption of the swarm. Fuel consumption in a craft is defined as being equal to the energy loss associ- ated with a trajectory. Under the assumption that there is a direct proportion between fuel used and energy lost, the following calculations are all in joules per kilogram of fuel used. In other words, one kilo- gram of fuel used is equal to one joule of energy. This fact can be changed for any fuel/energy ratio for a real fuel with a simple multiplicative factor. In this case, energy loss is the work done on the craft by the thruster force along its trajectory. Since we neglect gravitation and any other forces acting on each craft, the only force acting on one craft is the force of its thrusters acting to either accelerate or decelerate it. Thus, the total work done by the thrusters on the craft will be equal to the amount of fuel used under these assumptions. The expression for work then that we are interested in is the following line integral evaluated along the path defined by the trajectory (FIGURE 2). FIGURE 1: Mathematical model of swarm dynamics and reference system FIGURE 2: Work line integral ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II Where 𝒓𝒓 is the displacement of the spacecraft and 𝑭𝑭 is the net force experienced by the craft due to the thrusters along the path it follows to the end goal. We can then see that 𝑑𝑑𝒓𝒓/𝑑𝑑𝑑𝑑 is equal to the ve- locity of the craft traveling on this trajectory. We then arrive at 𝑑𝑑𝒓𝒓 = 𝒗𝒗𝑑𝑑𝑑𝑑. It is common to use in the field of dynamical systems to denote time derivatives with a small dot above the variable of interest[13]. The only forces exerted on the crafts being considered are the forces due to the thrusters. Gravitation from Sat- urn is neglected due to all particles and agents be- ing within Saturn’s orbital reference frame, where the force of gravity is already accounted for in defining this frame. Gravitation between spacecraft and ring particles is also neglected due to their extremely small masses that would result in small forces com- pared to the thruster forces. Since the variable “𝑥𝑥” denotes the position of a craft, “𝑥𝑥” with one dot de- notes the velocity, while “𝑥𝑥” with two dots denotes acceleration. From this, and the fact that force is equal to mass times acceleration (𝑭𝑭 = 𝑚𝑚𝑚𝑚) by Newton’s second law, we obtain the total work done by the thrusters in moving the craft along its trajec- tory as an integral from the starting time (0) to the final time of the force vector. This pattern is ex- pressed as the product of mass and the second time derivative (acceleration) of the i’th craft’s displace- ment and 𝑑𝑑𝒓𝒓 being equal to 𝒗𝒗𝑑𝑑𝑑𝑑, which is equiva- lently expressed as the product of 𝑑𝑑𝑑𝑑 and the first time derivative of the i’th craft’s displacement. Fi- nally, we sum over all craft to get the total energy lost (fuel consumed) of the swarm over the course of the trajectory (FIGURE 3). a,b,c PARAMETERS THAT ARE MEMBERS OF THE VECTOR K – control the shape of the final formation (circles, spheres, etc.) Ch MAGNITUDE OF THE HYPERBOLIC POTENTIAL TERM – determines speed of convergence and works together with expo- nential potential to determine general trajectory shape Ce,Le MAGNITUDE AND LENGTH SCALE OF EXPONENTIAL POTENTIAL TERM – work together with hyperbolic potential to deter- mine general trajectory shapes and provide an upper bound on velocity Cr,Lr MAGNITUDE AND LENGTH SCALE OF INTER-CRAFT REPULSIVE POTENTIALS – determine how strong the repulsive force be- tween crafts is. Constants of craft geometry and material Cdm,Ldm MAGNITUDE AND LENGTH SCALE OF CRAFT-OBSTACLE REPULSIVE POTENTIALS – determine how strong the repulsive force between crafts and obstacles is. Constants of obstacle geometry μ BIFURCATION PARAMETER – controls how many steady state formations swarm converges to r SCALAR – determines how spread out the swarm is as a whole by controlling locations of hyperbolic and expo- tential potentials xi POSITION VECTOR OF THE I’TH CRAFT xij RELATIVE POSITION VECTOR BETWEEN I’TH AND J’TH CRAFTS ux AVERAGE VELOCITY WITH WHICH SWARM TRAVELS IN POSITIVE X-DIRECTION xim RELATIVE POSITION VECTOR BETWEEN I’TH CRAFT AND M’TH OBSTACLE TABLE 1: Relevant parameters summary FIGURE 3: Total energy expression of swarm along a trajectory ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II Where 𝑑𝑑𝑓𝑓 is the final time during the simula- tion. The absolute value is necessary considering that if the thruster force is ever acting to decelerate a craft (it is acting opposite the direction of the craft’s velocity), then the mathematical expression for work will yield a negative value, representing an energy gain in the system. Fuel is not generated in this in- stance, so the absolute value bars restrict this work to always represent an energy loss in the system. The above expression is equal to the total en- ergy lost during a swarm maneuver, representing the objective function we wish to minimize. Since this function is locked within a highly coupled system of differential equations and an integral that has no closed-form expression (an expression involving a fi- nite amount of known algebraic operations and var- iables without derivatives and integrals), we must use a more advanced method of optimization which in this case is the genetic algorithm. iii. Genetic Algorithm In the genetic algorithm, a single member of a population is the set of parameters and its associ- ated final fuel consumption value. We only consider the parameters 𝑚𝑚, 𝑏𝑏, 𝑐𝑐, 𝐶𝐶𝑒𝑒, 𝐿𝐿𝑒𝑒, 𝐶𝐶ℎ, and 𝑟𝑟 as the rest are either constants associated with the spacecraft or obstacles, which cannot be changed or are con- stants associated with a desired final formation type that we wish to keep constant. The algorithm (refer to FIGURE 4) works by tak- ing a set of random initial parameters, running them through a MATLAB Simulink[14] simulation to obtain the fuel consumed in that trajectory, and then select- ing the two sets of parameters that result in the low- est fuel consumed out of all of the initial sets. These two sets are chosen as the parents for the next gen- eration. The parameters associated with these are then converted to a type of binary vector of ones and zeroes. The reason for this is because the next step FIGURE 4: Genetic algorithm visualization[2] ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II is the exchanging of the “genetic material” of the two parents to produce offspring. The binary represen- tations of each parent’s parameter are lined up, and a random stopping point is then selected. This ran- dom stopping point is selected through a random number generator in the MATLAB code. Every value before the stopping point is swapped amongst the two parents, along with a random chance for one of the values to be inverted (1 to 0 or 0 to 1) at a low probability, representing random mutations. This process is continued a number of times until multi- ple “child” binary vectors are created, which are then converted back to their decimal representation. To- gether with this genetic crossover process and ran- dom mutations, there is a chance for every genera- tion for a more optimal set of parameters to arise while not converging prematurely. After the conver- sion back to decimal form, the children are run through the same fitness test the previous popula- tion went through, and two new parents are selected while the other sets of parameters are scrapped. This process continues until a population converges to a global optimum where the fuel consumption values are all equally as fit as each other. Of course, other convergence criteria could be used (energy con- sumption below a certain value across the swarm, to- tal time spent, etc.), but defining convergence in this way where every craft is equally optimal tends to pri- oritize the crafts expending the most fuel first. This process was implemented in MATLAB[14] with multi- ple Simulink simulations running per generation to get the final fuel consumption value for each new set of parameters. Information and methods followed that were used to code this algorithm are presented in the article by Mallawaarachchi.[2] A similar evolu- tion-based optimization method was successfully used in path planning a single UAV in a terrestrial en- vironment in the paper by Rathbun et al.[4] 3 NUMERICAL RESULTS In the following simulations, there is a swarm of twenty spacecraft initially arranged in a circle at the vertical launch pad and three static disturbances (ice rocks/asteroids) in space along the trajectory. In the context of Saturn’s rings, this simulation repre- sents a swarm being deployed from a mothercraft near Saturn into a region of Saturn’s rings with ring particles acting as obstacles. It should be noted that this simulation and approach will work for any num- ber of spacecraft and disturbances that may or may not be static. Still, for this work, a simpler scenario was chosen, primarily to demonstrate the efficacy of the chosen genetic algorithm in reducing fuel con- sumption. As described before, the fuel consump- tion depends on the force exerted on a craft by the thrusters, which is proportional to the acceleration of the craft. Whenever crafts are accelerating to form a final formation, avoid an obstacle, or avoid another craft, the thrusters must exert force on the craft to move it, thus spending fuel. Smoother trajectories generally use less fuel than trajectories where sharp turns and abrupt accelerations are required to avoid collisions. Within this preliminary model, collisions may happen and can be checked for but will not be physically accurate. The agents will move through obstacles or other craft upon collision. This simula- tion’s main goal is to validate the genetic algorithm before making the mathematical model and envi- ronment too complicated. Future work on this pro- ject will involve the implementation of physically ac- curate collision physics and the event of a craft col- liding and being compromised, leaving the swarm. First, we start with an initial population of five sets of parameters, listed in the chart to the left of FIGURE 5. The trajectory and fuel consumption analysis for one craft is shown below in FIGURE 7. In FIGURE 5 we show the parameters used for this simulation as well as the swarm trajectory. FIGURE 6 presents an alternate view. As seen from FIGURE 5 and FIGURE 6, the swarm starts in an initial circular launch apparatus. It then moves through space approaching the final for- mation while avoiding the obstacles at locations de- noted by large open circles and the other craft. The blue craft on the right provides an example of such aggressive avoidance behavior with an aggressive FUEL CONSUMPTION DEPENDS ON THE FORCE EXERTED ON A CRAFT BY THE THRUSTERS, WHICH IS PROPORTIONAL TO THE ACCELERATION OF THE CRAFT ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II maneuver at around 10 meters. The red trajectory next to it also appears to move out of the way of the blue one when it moves in the red trajectory’s path to avoid the obstacle as well. This trend is better visible in FIGURE 5. As the blue craft dips inward toward the left, the red craft above it speeds up (more space in between consecutive dots represents faster move- ment) to avoid the blue craft. Although they are not necessarily on a collision course, they could be as these dots only represent the centers of spherical crafts, which could have varying sizes. The magni- tudes of the inter-craft repulsive potentials responsi- ble for this collision avoidance behavior would factor in the craft’s radius when being chosen. In this case, the red and blue crafts became a bit too close to each other, so the avoidance mechanism governed by the potentials was initiated. In FIGURE 7 we show the fuel consumption against the time of this same blue craft described above. A graph of a singular craft’s fuel consump- tion is presented as it is still indicative of the swarm’s consumption as a whole. This blue craft was the most responsible for increased fuel consumption in this trajectory due to its harsh avoidance maneuver. The fuel consumption spikes during periods of high ac- celeration and saturates once the swarm converges to the final formation as expected since the craft is no longer accelerating at that point. Also noteworthy is that fuel consumption spikes around when the blue trajectory aggressively avoids the obstacle (as- teroid) bound for a head-on collision if it did not move out of the way. All of the fuel consumptions summed for each craft for this simulation was 3462.7 J/kg. This value is accurate within the model itself as it is a result of pure mathematical calculations with- out any data being taken. After running this data set (set of the seven parameters being considered) and four others through the genetic algorithm, it converged to a possible optimal data set after 71 generations (or 355 individual simulations in Simulink). This process worked by taking the original population of five sets of the seven parameters being considered and run- ning the Simulink simulation and corresponding to- tal fuel usage for a swarm. With the five values of fuel consumption, the two smallest are considered and FIGURE 5: Parameters and associated trajectories of swarm traveling in x-direction. Different colored trajectories repre- sent different spacecraft. FIGURE 6: Top-Down View of Same Trajectory. Different col- ored trajectories represent different spacecraft. FIGURE 7: Fuel consumption (J/kg) of one craft against time (s). ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II used as parents for the next generation. Using the technique described in the section on the genetic al- gorithm, five more sets of parameters (that have a higher chance of being optimal) are generated from the parents. One generation consists of the five sim- ulations and calculations of total fuel consumption. So in total, the algorithm converged after 71 of these or 355 individual instances of numerically solving the system of equations and calculating integrals. The al- gorithm’s optimal parameters output and their cor- responding trajectories when run through the same simulation are described below in FIGURES 8 and 9. One of the first things immediately made ap- parent is that the blue trajectories now no longer need to avoid the obstacle so aggressively, and the swarm as a whole being generally smoother as it converges to a final formation. With the use of the genetic algorithm, the swarm could navigate through the field of obstacles and arrive at the same destination while minimizing the fuel used to arrive there. In FIGURE 10 we display the fuel consumption versus time of the same craft as before, only now af- ter the algorithm has run and the new parameters put in place. The fuel consumption saturates in about half the time as the original simulation reaching a maxi- mum value of a little less than 60 J/kg, which is much smaller than the original in FIGURE 7, being at about 170 J/kg. Another notable aspect of this new trajec- tory is that the fuel consumption curve in FIGURE 10 in- creases much more smoothly than that of the unop- timized swarm, which has many abrupt spikes. In the context of a real-world scenario, the same mission could be completed for half the fuel cost while also putting less strain on the thrusters. 4 CONCLUSIONS Although these are preliminary results, they show that the genetic algorithm approach can be successfully optimize the fuel consumption of a swarm trajectory of spacecraft defined by this com- plex nonlinear model. The solution to this problem provides a method of exploring and gaining new in- formation on the rings of Saturn that otherwise would not be possible. Future work will include mak- ing the model more realistic and transitioning the FIGURE 8: Parameters and associated trajectories of swarm af- ter genetic algorithm. Different colored trajectories repre- sent different spacecraft. FIGURE 9: Side view of same trajectory. Different colored tra- jectories represent different spacecraft. FIGURE 10: Fuel consumption (J/kg) of one craft against time (s) after genetic algorithm. ARESTY RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II deterministic problem to a stochastic one; denser obstacle distributions, and obstacles that are mov- ing, will also be considered in future simulations more akin to the real-world density of ice rocks in the rings of Saturn. Applying the genetic algorithm method to the same spacecraft swarm system de- scribed by more accurate stochastic models can lead to future innovative developments in this field. New methods of optimization will also be investi- gated∎ 5 ACKNOWLEDGEMENTS I would first like to thank Dr. Annalisa Scac- chioli for guiding me through this process and open- ing up this wonderful opportunity to learn as well as create something meaningful. Without her, this pro- ject would not have come to fruition. I would also like to thank the NASA New Jersey Space Grant Consor- tium for funding and facilitating this research. Also, I would like to thank Dr. Jingang Yi as well as Dr. Enver Koray Akdogan for providing insight into the inner workings of the mathematical model used in this pa- per as well as possible avenues to best utilize it for this project. 6 REFERENCES [1] Bennet, Derek J., McInnes, C.R., Suzuki, M., and Uchiyama K., “Autonomous Three-Dimensional Formation Flight for a Swarm of Unmanned Aerial Vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 6, November-December 2011, 1899-1908. [2] Mallawaarachchi, Vijini, “Introduction to Genetic Algorithms – Including Example Code,” Medium Towards Data Science, 7 July 2017. [3] Conn, T., Plice L., Dono Perez, A., and Ho, M., “Operating Small Sat Swarms as a Single Entity: Introducing SODA,” NASA Ames Research Center / Mission Design Division, 31st Annual AIAA/USU Conference on Small Satellites, 2017. 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