A new, novel numerical optimization algorithm is developed, tested, and used to solve difficult numerical problems from the field of astrodynamics. First, a brief review of optimization theory is presented and common numerical optimization techniques are discussed. Then, the new method, called the Learning Approach to Sampling Optimization (LA) is presented. Simple, illustrative examples are given to further emphasize the simplicity and accuracy of the LA method. Benchmark functions in lower dimensions are studied and the LA is compared, in terms of performance, to widely used methods. Three classes of problems from astrodynamics are then solved. First, the N - impulse orbit transfer and rendezvous problems are solved by using the LA optimization technique along with derived bounds that make the problem computationally feasible. This marriage between analytical and numerical methods allows an answer to be found for an order of magnitude greater number of impulses than are currently published. Next, the N -impulse work is applied to design periodic close encounters (PCE) in space. The encounters are defined as an open rendezvous, meaning that two spacecraft must be at the same position at the same time, but their velocities are not necessarily equal. The PCE work is extended to include N -impulses and other constraints, and new examples are given. Finally, a trajectory optimization problem is solved using the LA algorithm and comparing performance with other methods based on two models-with varying complexity-of the Cassini-Huygens mission to Saturn. The results show that the LA consistently outperforms commonly used numerical optimization algorithms.

This thesis addresses the orbit transfer optimization problem of a spacecraft. The optimal orbit transfer is the process of altering the orbit of a spacecraft with minimum propellant consumption. The spacecrafts are needed to realize orbit transfer to reach, change or keep its orbit. The spacecraft may be a satellite or the last stage of a launch vehicle that is operated at the exo-atmospheric region. In this study, a genetic algorithm based orbit transfer method has been developed. The applicability of genetic algorithm based orbit transfer method has been verified using orbit transfers which are optimal at specific cases. The solution to orbit transfer problem is also searched using steepest descent algorithm.While genetic algorithm can reach the optimal solution, steepest descent algorithm can reach optimal solution when a good initial prediction is provided. The effects of the initial orbital values on the orbit transfer solutions are also studied.

Analytical solutions to complex trajectory design problems are scarce, since only a few specific cases allow for closed-form solutions. The main purpose of this dissertation is to design simple algorithms for trajectory design using continuous thrust, with a focus on low-thrust applications. By “simple” here we seek to achieve algorithms that either admit an analytical solution, or require minimal input by the user and minimal computation time. The three main contributions of this dissertation are: designing Lyapunov-based closed-loop guidance laws for orbit transfers, finding semi-analytical solutions using a constant magnitude thrust, and perturbation theory for approximate solutions to low-thrust problems. The technical aspect that these problems share in common is that they all use, fully or partially, a thrusting model in which the energy of the system is kept constant. Many orbit transfer problems are shown to be solved with this thrusting protocol. / text

An analysis of the use of complex-step differentiation (CSD) in optimization problems is presented. Complex-step differentiation is a numerical approximation of the derivative of a function valid for any real-valued analytic function. The primary benefit of this method is that the approximation does not depend on a difference term; therefore round-off error is reduced to the machine word-length. A suitably small choice of the perturbation length, h, then results in the virtual elimination of truncation error in the series approximation. The theoretical basis for this method is derived highlighting its merits and limitations. The Lunar Ascent Problem is used to compare CSD to traditional forward differencing in applications useful to the solution of optimization problems. Complex-step derivatives are shown to sufficiently apply in various interpolation and integration methods, and, in fact, consistently outperform traditional methods. Further, the Optimal Orbit Transfer Problem is used to test the accuracy, robustness, and runtime of CSD in comparison to central differencing. It is shown that CSD is a considerably more accurate derivative approximation which results in an increased robustness and decreased optimization time. Also, it is shown that each approximation is computed in less time using CSD than central differences. Overall, complex-step derivatives are shown to be a fast, accurate, and easy to implement differentiation method ideally suited for most optimization problems. / text

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.