










































  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. 



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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 



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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 



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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. 

[4] Rathbun, David, Kragelund Sean, and Pongpunwattana, An-
awat, “An Evolution Based Path Planning Algorithm for Au-
tonomous Motion of a UAV Through Uncertain Environ-
ments,” University of Washington, Seattle, WA, 2002. 

[5] Zhao, Yiyuan J. and Celia Qi, Ying, “Minimum Fuel Powered 
Dynamic Soaring of Unmanned Aerial Vehicles Utilizing 
Wind Gradients,” University of Minneapolis, MN, 2004. 

[6] Morgan, Daniel, Chong, Soon-Jo, and Hadaegh, Fred Y., 
“Model Predictive Control of Swarms of Spacecraft Using Se-
quential Convex Programming,” Journal of Guidance, Con-
trol, and Dynamics, Vol. 37, No 6, November-December 
2014, 1725-1740. 

[7] “Cassini-Solstice Mission FAQs,” Jet Propulsion Laboratory. 
Retrieved January 24, NASA, 2014. 

[8] Blocher, A., “Alternative Mission Concepts for the Explora-
tion of Outer Planets,” California Polytech State University, 
Master’s Thesis 2017.  

[9] Hadaegh, F. Y., Chung, S., Manohara, H. M., “On Develop-
ment of 100-Gram Class Spacecraft for Swarm Applications,” 
IEEE Systems Journal, Vol. 10, No. 2, Pages 673-684, June 
2016, 2016. 

[10] Hinchley, M. G., Rash, J. L., Truszkowski, W. F.,“Autonomous 
and Autonomic Swarms,” Information Systems Division, 
NASA Goddard Space Flight Center, Technical Report, 
2006.  

[11] Nallapu, R., Thangavelautham, J., “Design of Spacecraft 
Swarm Flybys for Planetary Moon Exploration,” Structures, 
Structural Design and Materials Conference, Technical Re-
port, 2019. 

[12] Blocher, A., Atkinson, D., Freeman, T., “Saturn Swarm Study 
Small Probe and CubeSat Architectures to accompany New 
Frontiers missions at Saturn” Low Cost Planetary Mission 
Conference, Presentation, 2017. 

[13] MATLAB Simulink, math and computer algebra system, cod-
ing interface, and simulation environment, Mathworks, 2019.  

[14] Hibbeler, Russel C., Engineering Mechanics: Dynamics, 14th 
Edition, 2016. 

 


