One idea to stop the increasing threat of global warming is to build a space sunshade, made up of a constellation of satellites to reflect a portion of the sunlight. These satellites need to reach an equilibrium point where they can stay in orbit with minimal adjustments. A few feasibility studies have been made in this area and the location of the equilibrium point, L1' has been found. An area that is still unclear is the best way of getting the satellites to L1'. One idea is to use electric propulsion engines , and utilizing a gravity assist around the Moon on the way to L1'. Gravity assists have mostly been performed with chemical rocket engines, so doing it with electric propulsion causes some unique issues. The goal was to find a trajectory including the gravity assist showing that this could be an effective way of transporting the satellites. To measure how effective this would be, a trajectory without a gravity assist was also created as a means of comparison. The trajectory and orbit simulations were done in the program General Missions Analysis Tool (GMAT), and the resulting trajectory took 51 days and used 83.4kg of fuel for a fuel-to-mass-ratio of 21.5%. This was a worse result compared to the trajectory without a gravity assist, which only used 80.4kg of fuel. Finally a discussion around a potential trajectory which was shown to have a much greater velocity increase was had, which would indicate that a gravity assist maneuver could provide a trajectory that does save on fuel compared to using no gravity assist.

The information presented in the thesis is a continuation of the Spacecraft Trajectory Optimization Suite (STOpS). This suite was originally designed and developed by Timothy Fitzgerald and further developed by Shane Sheehan, both graduate students at California Polytechnic State University, San Luis Obispo. Spacecraft utilizing low-thrust transfers are becoming more and more common due to their efficiency on interplanetary trajectories, and as such, finding the most optimal trajectory between two planets is something of interest. The version of STOpS presented in this thesis uses Multiple Gravity-Assist Low-Thrust (MGALT) trajectories paired with the island model paradigm to accomplish this goal. The island model utilizes four different global search algorithms: a Genetic Algorithm, Differential Evolution, Particle Swarm Optimization, and Monotonic Basin Hopping. The first three algorithms were featured in the initial version of STOpS written by Fitzgerald [1], and were subsequently modified by Sheehan [2] to work with a low-thrust adaptation of STOpS. For this work, Monotonic Basin Hopping was added to aid the suite with the MGALT trajectory search. Monotonic Basin Hopping was successfully validated against four different test functions which had been used to validate the other three algorithms. The purpose of this validation was to ensure Monotonic Basin Hopping would work as intended, ensuring it would work in cooperation with the other three algorithms to produce a near optimal solution. After verifying the addition of Monotonic Basin Hopping, all four algorithms were used in the island model paradigm to verify MGALT STOpS’ ability to solve two known orbital transfer problem. The first verification case involved an Earth to Mars transfer with fixed thruster parameters and a predetermined time of flight. The second verification case involved a transfer from Earth to Jupiter via a Mars gravity assist; two different versions of the verification case were solved against trajectories produced by industry optimization software, the Satellite Tour Design Program Low-Thrust Gravity Assist and the Gravity Assisted Low-thrust Local Optimization Program. In the first verification case, MGALT STOpS successfully validated the Earth to Mars trajectory problem and found results agreeable to literature. In the second verification case, MGALT STOpS was partially successful in validating the Earth to Mars to Jupiter trajectory problems, and found results similar to literature. The final software produced for this work is a trajectory optimization suite implemented in MATLAB, which can solve interplanetary low-thrust trajectories with or without the inclusion of gravity assists.

One of the most important aspects of preliminary interplanetary mission planning entails designing a trajectory that delivers a spacecraft to the required destinations and accomplishes all the objectives. The design tool described in this thesis allows an investigator to explore various interplanetary trajectories quickly and easily. The design tool employs the patched conic method to determine heliocentric and planetocentric trajectory information. An existing Lambert Targeting routine and other common algorithms are utilized in conjunction with the design tool’s specialized code to formulate an entire trajectory from Earth departure to arrival at the destination. The tool includes many options for the investigator to accurately configure the desired trajectory, including planetary gravity assists, deep space maneuvers, and various departure and arrival conditions. The trajectory design tool is coded in MATLAB, which provides access to three dimensional plotting options and user adaptability. The design tool also incorporates powerful MATLAB optimization functions that adjust trajectory characteristics to find a configuration that yields the minimum spacecraft propellant in the form of change in velocity. / text

Preliminary interplanetary trajectory designs frequently use simplified two-body orbital mechanics and linked conics methodology to model the complex trajectories in multi-body systems. Incorporating gravity assists provides highly efficient interplanetary trajectories, enabling otherwise infeasible spacecraft missions. Future missions may employ powered gravity assists, using a propulsive maneuver during the flyby, improving the overall trajectory performance. This dissertation provides a complete description and analysis of a new interplanetary trajectory design tool known as TRACT (TRAjectory Configuration Tool). TRACT is capable of modeling complex interplanetary trajectories, including multiple ballistic and/or powered gravity assists, deep space maneuvers, parking orbits, and other common maneuvers. TRACT utilizes an adaptable architecture of modular boundary value problem (BVP) algorithms for all trajectory segments. A bi-level optimization scheme is employed to reduce the number of optimization variables, simplifying the user provided trajectory information. The standardized optimization parameter set allows for easy use of TRACT with a variety of optimization algorithms and mission constraints. The dissertation also details new research in powered gravity assists. A review of literature on optimal powered gravity assists is presented, where many optimal solutions found are infeasible for realistic spacecraft missions. The need was identified for a mission feasible optimal powered gravity assist algorithm using only a single impulsive maneuver. The solution space was analyzed and a complete characterization was developed for solution types of the optimal single-impulse powered gravity assist. Using newfound solution space characteristics, an efficient and reliable optimal single-impulse powered gravity assist BVP algorithm was formulated. The mission constraints were strictly enforced, such as maintaining the closest approach above a minimum radius and below a maximum radius. An extension of the optimal powered gravity assist research is the development of a gravity assist BVP algorithm that utilizes an asymptote ΔV correction maneuver to produce ballistic gravity assist trajectory solutions. The efficient algorithm is tested with real interplanetary mission trajectory parameters and successfully converges upon ballistic gravity assists with improved performance compared to traditional methods. A hybrid approach is also presented, using the asymptote maneuver algorithm together with traditional gravity assist constraints to reach ballistic trajectory solutions more reliably, while improving computational performance.

Interplanetary multiple gravity assist (MGA) trajectory optimization has long been a field of interest to space scientists and engineers. Gravity assist maneuvers alter a spacecraft's velocity vector and potentially allow spacecraft to achieve changes in velocity which would otherwise be unfeasible given our current technological limitations. Unfortunately, designing MGA trajectories is difficult and in order to find good solutions, deep space maneuvers (DSM) are often required which further increase the complexity of the problem. In addition, despite the active research in the field over the last 50 years, software for MGA trajectory optimization is scarce. A few good commercial, and even fewer open-source, options exist, but a majority of quality software remains proprietary. The intent of this thesis is twofold. The first part of this work explores the realm of global optimization applied to multiple gravity assist trajectories with deep space maneuvers (MGA-DSM). With the constant influx of new global optimization algorithms and heuristics being developed in the global optimization community, this work aims to be a high level optimization approach which makes use of those algorithms instead of trying to be one itself. Central to this approach is PaGMO, which is the open-source Parallel Multiobjective Global Optimizer created by ESA's Advanced Concepts Team (ACT). PaGMO is an implementation of the Island Model Paradigm which allows the parallelization of different global optimizers. The second part of this work introduces the IGATO software which improves PaGMO by complementing it with dynamic restart capabilities, a pruning algorithm which learns over time, subdomain decomposition, and other techniques to create a powerful optimization tool. IGATO aims to be an open-source platform independent C++ application with a robust graphical user interface (GUI). The application is equipped with 2D plotting and simulations, real time Porkchop Plot generation, and other useful features for analyzing various problems. The optimizer is tested on several challenging MGA-DSM problems and performs well: consistently performing as well or better than PaGMO on its own.

Spacecraft trajectory optimization is a near-infinite problem space with a wide variety of models and optimizers. As trajectory complexity increases, so too must the capabilities of modern optimizers. Common objective cost functions for these optimizers include the propellant utilized by the spacecraft and the time the spacecraft spends in flight. One effective method of minimizing these costs is the utilization of one or multiple gravity assists. Due to the phenomenon known as the Oberth effect, fuel burned at a high velocity results in a larger change in orbital energy than fuel burned at a low velocity. Since a spacecraft is flying fastest at the periapsis of its orbit, application of impulsive thrust at this closest approach is demonstrably capable of generating a greater change in orbital energy than at any other location in a trajectory. Harnessing this extra energy in order to lower relevant cost functions requires the modeling of these “powered flybys” or “powered gravity assists” (PGAs) within an interplanetary trajectory optimizer. This paper will discuss the use and modification of the Spacecraft Trajectory Optimization Suite, an optimizer built on evolutionary algorithms and the island model paradigm from the Parallel Global Multi-Objective Optimizer (PaGMO). This variant of STOpS enhances the STOpS library of tools with the capability of modeling and optimizing single and multiple powered gravity assist trajectories. Due to its functionality as a tool to optimize powered flybys, this variant of STOpS is named the Spacecraft Trajectory Optimization Suite - Flybys with Impulsive Thrust Engines (STOpS-FLITE). In three test scenarios, the PGA algorithm was able to converge to comparable or superior solutions to the unpowered gravity assist (uPGA) modeling used in previous STOpS versions, while providing extra options of trades between time of flight and propellant burned. Further, the PGA algorithm was able to find trajectories utilizing a PGA where uPGA trajectories were impossible due to limitations on time of flight and flyby altitude. Finally, STOpS-FLITE was able to converge to a uPGA trajectory when it was the most optimal solution, suggesting the algorithm does include and properly considers the uPGA case within its search space.

This thesis takes a closer look at the complex maneuver known as gravity assist, a popular method of interplanetary travel. The maneuver is used to gain or lose momentum by flying by planets, which induces a speed and direction change. A simulation model is created using the General Mission Analysis Tool (GMAT), which is intended to be easily reproduced and altered to match any desired gravity assist maneuver. The validity of its results is analyzed, comparing them to available data from real missions. Some parameters, including speed and trajectory, are found to be extremely reliable. The model is then used as a tool to investigate the way that different parameters impact this complex environment, and the advantages of performing thrusting burns at different points during the maneuver are explored. According to theory, thrusting at the point of closest approach to the planet is thought to be the most efficient method for changing speed and direction of flight. However, the results from this study show that thrusting before this point can have some major advantages, depending on the desired outcome. The reason behind this is concluded to be the high sensitivity of the gravity assist maneuver to the altitude and location of the point of closest approach. / Detta examensarbete tittar närmare på den komplexa manöver inom banmekanik som kallas gravitationsassisterad manöver, vilken är vanligt förekommande vid interplanetära rymduppdrag. Manövern används för att öka eller minska farkostens rörelsemängd genom att flyga förbi nära planeter, vilket ger upphov till en förändring i fart och riktning. En simuleringsmodell är skapad i NASAs mjukvara GMAT med syftena att den ska vara reproducerbar samt möjlig att ändra för olika gravitationsassisterade manövrar. Resultaten från simuleringarna är validerade mot tillgängliga data från riktigt rymduppdrag. Vissa parametrar, som fart och position, har en väldigt bra överenstämmelse. Modellen används sedan för att noggrannare undersöka hur olika parametrar påverkar det komplexa beteendet vid en graviationsassisterad manöver, genom att specifikt titta på effekterna av en pålagd dragkraft från motorn under den gravitationsassisterade manövern. Teoretiskt fås mest effekt på fart och riktning om dragkraften från motorn läggs på vid punkten närmast planeten. Resultaten från denna studie visar att beroende på vilken parameter man vill ändra så kan man erhålla mer effekt genom att lägga på dragkraften innan den närmsta punkten. Förklaringen till detta är att den gravitationsassisterade manövern är väldigt icke-linjär, så en tidigare pålagd dragkraft kan kraftigt förändra farkostens bana nära planeten, så att farkosten t.ex. kommer närmare och då påverkas mer.

Multiple gravity assist (MGA) spacecraft trajectories can be difficult to find, an intractable problem to solve completely. However, these trajectories have enormous benefits for missions to challenging destinations such as outer planets and primitive bodies. Techniques are presented to aid in solving this problem with a global search tool and additional investigation into one particular proximity operations option is discussed. Explore is a global grid-search MGA trajectory pathsolving tool. An efficient sequential tree search eliminates v∞ discontinuities and prunes trajectories. Performance indices may be applied to further prune the search, with multiple objectives handled by allowing these indices to change between trajectory segments and by pruning with a Pareto-optimality ranking. The MGA search is extended to include deep space maneuvers (DSM), v∞ leveraging transfers (VILT) and low-thrust (LT) transfers. In addition, rendezvous or nπ sequences can patch the transfers together, enabling automatic augmentation of the MGA sequence. Details of VILT segments and nπ sequences are presented: A boundaryvalue problem (BVP) VILT formulation using a one-dimensional root-solve enables inclusion of an efficient class of maneuvers with runtime comparable to solving ballistic transfers. Importantly, the BVP VILT also allows the calculation of velocity-aligned apsidal maneuvers (VAM), including inter-body transfers and orbit insertion maneuvers. A method for automated inclusion of nπ transfers such as resonant returns and back-flip trajectories is introduced: a BVP is posed on the v∞ sphere and solved with one or more nπ transfers – which may additionally fulfill specified science objectives. The nπ sequence BVP is implemented within the broader search, combining nπ and other transfers in the same trajectory. To aid proximity operations around small bodies, analytical methods are used to investigate stability regions in the presence of significant solar radiation pressure (SRP) and body oblateness perturbations. The interactions of these perturbations allow for heliotropic orbits, a stable family of low-altitude orbits investigated in detail. A novel constrained double-averaging technique analytically determines inclined heliotropic orbits. This type of knowledge is uniquely valuable for small body missions where SRP and irregular body shape are very important and where target selection is often a part of the mission design.