The work presented here is a continuation of Spacecraft Trajectory Optimization Suite (STOpS), a master’s thesis written by Timothy Fitzgerald at California Polytechnic State University, San Luis Obispo. Low-thrust spacecraft engines are becoming much more common due to their high efficiency, especially for interplanetary trajectories. The version of STOpS presented here optimizes low-thrust trajectories using the Island Model Paradigm with three stochastic evolutionary algorithms: the genetic algorithm, differential evolution, and particle swarm optimization. While the algorithms used here were designed for the original STOpS, they were modified for this work. The low-thrust STOpS was successfully validated with two trajectory problems and their known near-optimal solutions. The first verification case was a constant-thrust, variable-time Earth orbit to Mars orbit transfer where the thrust was 3.787 Newtons and the time was approximately 195 days. The second verification case was a variable-thrust, constant-time Earth orbit to Mercury orbit transfer with the thrust coming from a solar electric propulsion model equation and the time being 355 days. Low-thrust STOpS found similar near-optimal solutions in each case. The final result of this work is a versatile MATLAB tool for optimizing low-thrust interplanetary trajectories.

While the interest in the Martian moons increases, the low-thrust propulsion technology is expected to enable novel mission scenarios but is associated with unique trajectory design challenges. Accordingly, the current investigation introduces a multi-phase low-thrust design framework. The trajectory of a potential spacecraft that departs from the Earth vicinity to reach both of the Martian moons, is divided into four phases. To describe the motion of the spacecraft under the influence of gravitational bodies, the two-body problem (2BP) and the Circular-Restricted Three Body Problem (CR3BP) are employed as lower-fidelity models, from which the results are validated in a higher-fidelity ephemeris model. For the computation and optimization of low-thrust trajectories, direct collocation algorithm is introduced. Utilizing the dynamical models and the numerical scheme, the low-thrust trajectory design challenge associated each phase is located and tackled separately. For the heliocentric leg, multiple optimal control problems are formulated between the planets in heliocentric space over different departure and arrival epochs. A contour plot is then generated to illustrate the trade-off between the propellant consumption and the time of flight. For the tour of the Martian moons, the science orbits for both moons are defined. Then, a new algorithm that interfaces the Q-law guidance scheme and direct collocation algorithm is introduced to generate low-thrust transfer trajectories between the science orbits. Finally, an end-to-end trajectory is produced by merging the piece-wise solutions from each phase. The validity of the introduced multi-phase formulation is confirmed by converging the trajectories in a higher-fidelity ephemeris model.<br>

Tradition spacecraft proximity operations require large and expensive on-board sensors and significant ground support. Relative angle measurements can be obtained from small, simple, and inexpensive on-board sensors, but have not traditionally been used for proximity operation because of difficulty generating rang information. In this thesis it is shown that useful relative range data can be generated provided that the spacecraft is experiencing a small continuous thrust such as would be provided by a low-thrust propulsion system.

Pour mener à bien leur mission, les satellites de télécommunications doivent rester à la verticale d'un même point de la Terre, sur une orbite dite géostationnaire, pour laquelle la période de révolution des satellites sur leur orbite est identique à la période de rotation de la Terre sur elle-même. Cependant, à cause des perturbations orbitales, les satellites tendent à s'en éloigner, et il est alors nécessaire de concevoir des stratégies de commande pour les maintenir dans un voisinage de cette position de référence. Du fait de leur grande valeur de poussée, les systèmes à propulsion chimique ont largement été utilisés, mais aujourd'hui les systèmes à propulsion électrique avec leur grande impulsion spécifique sont des alternatives viables pour réduire la masse d'ergols du satellite, et ainsi le coût au lancement, ou allonger la durée de vie du satellite, ce qui permettrait de limiter l'encombrement dans l'espace. Cependant, l'utilisation d'un tel système propulsif induit des contraintes opérationnelles issues en partie du caractère limité de la puissance électrique disponible à bord. Ces contraintes sont difficiles à prendre en compte dans la transcription du problème de maintien à poste en un problème de contrôle optimal à consommation minimale avec contraintes sur l'état et le contrôle. Ce manuscrit propose deux approches pour résoudre ce problème de commande optimale. La première, basée sur le développement et l'exploitation de conditions nécessaires d'optimalité, consiste à découper le problème initial en trois sous-problèmes pour former une méthode de résolution à trois étapes. La première étape permet de résoudre un problème de maintien à poste expurgé des contraintes opérationnelles, tandis que la deuxième, initialisée par le résultat de la première, produit une solution assurant le respect de ces dernières contraintes. La troisième étape permet d'optimiser la valeur des instants d'allumage et d'extinction des propulseurs dans le cadre du formalisme des systèmes à commutation. La seconde approche, dite " directe ", consiste à paramétrer le profil de commande par une fonction binaire et à le discrétiser sur l'horizon temporel de résolution. Les contraintes opérationnelles sont ainsi facilement transcrites en contraintes linéaires en nombres entiers. Après l'intégration numérique de la dynamique, le problème de contrôle optimal se résume à un problème linéaire en nombres entiers. Après la résolution du problème de maintien à poste sur un horizon court d'une semaine, le problème est résolu sur un horizon long d'un an par résolutions successives sur des horizons courts d'une durée de l'ordre de la semaine. Des contraintes de fin d'horizon court doivent alors être ajoutées afin d'assurer la faisabilité de l'enchaînement des problèmes sur l'horizon court constituant le problème sur l'horizon long. / Geostationary spacecraft have to stay above a fixed point of the Earth, on a so-called geostationary Earth orbit. For this orbit, the orbital period of the spacecraft is equal to the rotation period of the Earth. Because of orbital disturbances, spacecraft drift away their station keeping position. It is therefore mandatory to create control strategies in order to make the spacecraft stay in the vicinity of the station keeping position. Due to their high thrust capabilities, chemical thrusters have been widely used. However nowadays electric propulsion based thrusters with their high specific impulse are viable alternative in order to decrease the spacecraft mass or increase its longevity. The use of such a system induce the necessity to handle operational constraints because of the limited on-board power. These operational constraints are difficult to take into account in the mathematical transcription of the station keeping problem in an optimal control problem with control and state constraints. This thesis proposed two techniques in order to solve this optimal control problem. The first one is based on the computation of first order necessary conditions and consists in decomposing the overall problem in three sub-problems, leading to a three-step decomposition method. The first step solves an optimal control problem without the operational constraints. The second steps enforces these operational constraints thanks to dedicated equivalence schemes and the third one optimises the switching times of the control profile thanks to a method borrowed from the switched systems theory. The second proposed method consists in parametrising the on-off control profile with binary functions. After a time discretisation of the station keeping horizons, the operational constraints are easily recast as linear constraints on integer variables, the dynamics is numerically integrated and the station keeping problem is recast as a mixed integer linear programming problem. After the resolution of the problem over a short time horizon of one week, the station keeping problem is solved over a long time horizon of one year. To this end, the long time horizon is split in shorter horizons over which the problem is successively solved. End-of-cycle constraints have been set up in order to ensure the feasibility of the solution one short horizon after another.

Low-thrust interplanetary spacecraft trajectory optimization poses a uniquely difficult problem to solve because of the inherent nonlinearities of the dynamics and constraints as well as the large size of the search space of possible solutions. Tools currently exist that optimize low-thrust interplanetary trajectories, but these tools are rarely openly available to the public, and when they are available they require multiple interfaces between multiple different packages. The goal of this work is to present a new piece of low-thrust interplanetary spacecraft trajectory optimization software that is open-source and entirely self-contained so that more people can have access to the ability to design interplanetary trajectories. To achieve this goal, a gradient-descent based nonlinear programming method, called the interior point method, was used. The nonlinear programming method was chosen so that results from this work could be compared and contrasted with results from Spacecraft Trajectory Optimization Suite (STOpS), which uses heuristics to iterate towards a solution. Interior point methods are popular because of their ability to handle large amounts of equality and inequality constraints, which is a characteristic that is valuable for low-thrust interplanetary spacecraft trajectories. The software developed, Interior Point Optimizer (IP Optimizer), was then validated against test cases with known solutions to ensure that the software delivered the intended results. Lastly, a constraint satisfaction, a minimum-time, and a maximum-final-mass optimization problem were solved and compared with literature to illustrate the advantages of IP Optimizer and the methods it employs. For the constraint satisfaction problem, IP Optimizer was able to find a solution that exactly satisfied the desired terminal constraints whereas STOpS had an error of 2.29 percent. In this case, IP Optimizer had a reduced runtime of 15 percent compared to STOpS as well. When minimizing time for a spacecraft transfer, IP Optimizer improved upon the solution found by STOpS by 5.3 percent. The speed of convergence for IP Optimizer was almost twice as fast as STOpS for this case. These results show that IP Optimizer is faster than STOpS at converging on a solution and the solution it converges to has a better objective value and more accurately satisfies the terminal constraints than STOpS. Lastly, the maximum-final-mass problem resulted in an objective value that was only 0.5 percent lower than the value found in literature.

Thesis (MScEng (Mechanical and Mechatronic Engineering))--University of Stellenbosch, 2009. / This thesis describes the development of a vacuum arc thruster (VAT) to be used as a potential low mass (< 500 g), low power (< 5–10W) propulsion system for nanosatellites. The thruster uses a high voltage capacitive circuit to initiate and power the arc process with a 400 ns high current (150–800A) pulse. A one-dimensional steady state analyticalmodel describing the cathode region of the vacuum arc was developed. The model made use of mass and energy balances at the sheath region and cathode surface respectively to predict key quantities such as thrust, ion velocity, ion-to-arc current ratio and erosion rate. Predicted results were shown to be within the limits of reported literature (∼63 μN/A, 26.12 km/s, 0.077 and 110 μg/C respectively). A sensitivity analysis of the analytical model found that a high electric field in the cathode region impedes and decelerates ion flow, which is used for thrust. This was confirmed experimentally for thrust values at arc voltages greater than 2000 V. Both direct and indirect means of measuring thrust were achieved by using a deflecting cantilever beam and an ion collector system, respectively. The transient response of the cantilever beam to impulsive thrust was analytically modeled, whilst the ion current was found by measuring the current induced on a plate subject to ion bombardment. Knowledge of the ion current density distribution was successfully used to approximate the effective normal thrust vector. Direct and indirect thrust levels were roughly 140 and 82 μN/A of average arc current, respectively. Measured thrust was found to be higher than predicted thrust due to thrust contributions fromthe ablation of Teflon insulation. The discrepancy is also due to the uncertainty in quantifying free parameters in the analytical model such as the fraction of generated ions flowing away from the cathode region. The thrust-topower ratio, specific impulse and efficiency of the vacuum arc thruster at an average arc current of 200 A was measured to be 0.6 μN/W, 160 s and 0.05 %, respectively. A thruster performance analysis and specification showed that the VAT is capable of achieving specific orbital and slew manoeuvres within a constant 5–10 W average power. It was concluded that thruster performance could be improved by using a two-stage arc circuit consisting of a high voltage, low current, short pulse trigger and a low voltage, high current, long pulse driver.

Future ambitious solar system exploration missions are likely to require ever larger propulsion capabilities and involve innovative interplanetary trajectories in order to accommodate the increasingly complex mission scenarios. Two recent advances in trajectory design can be exploited to meet those new requirements: the use of low-thrust propulsion which enables larger cumulative momentum exchange relative to chemical propulsion; and the consideration of low-energy transfers relying on full multi-body dynamics. Yet the resulting optimal control problems are hypersensitive, time-consuming and extremely difficult to tackle with current optimization tools. Therefore, the goal of the thesis is to develop a methodology that facilitates and simplifies the solution finding process of low-thrust optimization problems in multi-body environments. Emphasis is placed on robust techniques to produce good solutions for a wide range of cases despite the strong nonlinearities of the problems. The complete trajectory is broken down into different component phases, which facilitates the modeling of the effects of multiple bodies and makes the process less sensitive to the initial guess. A unified optimization framework is created to solve the resulting multi-phase optimal control problems. Interfaces to state-of-the-art solvers SNOPT and IPOPT are included. In addition, a new, robust Hybrid Differential Dynamic Programming (HDDP) algorithm is developed. HDDP is based on differential dynamic programming, a proven robust second-order technique that relies on Bellman's Principle of Optimality and successive minimization of quadratic approximations. HDDP also incorporates nonlinear mathematical programming techniques to increase efficiency, and decouples the optimization from the dynamics using first- and second-order state transition matrices. Crucial to this optimization procedure is the generation of the sensitivities with respect to the variables of the system. In the context of trajectory optimization, these derivatives are often tedious and cumbersome to estimate analytically, especially when complex multi-body dynamics are considered. To produce a solution with minimal effort, an new approach is derived that computes automatically first- and high-order derivatives via multicomplex numbers. Another important aspect of the methodology is the representation of low-thrust trajectories by different dynamical models with varying degrees of fidelity. Emphasis is given on analytical expressions to speed up the optimization process. In particular, one novelty of the framework is the derivation and implementation of analytic expressions for motion subjected to Newtonian gravitation plus an additional constant inertial force. Example applications include low-thrust asteroid tour design, multiple flyby trajectories, and planetary inter-moon transfers. In the latter case, we generate good initial guesses using dynamical systems theory to exploit the chaotic nature of these multi-body systems. The developed optimization framework is then used to generate low-energy, inter-moon trajectories with multiple resonant gravity assists.

La technique de moyennation est un moyen efficace pour simplifier les transferts optimaux pour un satellite à faible poussée dans un problème à deux corps contrôlé. Cette thèse est une étude analytique et numérique du transferts orbital à poussée faible en temps optimal qui généralise l'application de la moyennation du problème à deux corps à des transferts dans le problème à deux corps perturbés et aux transfert d'une orbite proche de la Terre au point de Lagrange L1, dans le cadre du problème à quatre corps bi-circulaire où l’effet perturbatif de la Lune et du Soleil est modélisé. Dans le transfert à faible poussée à deux corps, nous comparons le cas du temps minimal et de l'énergie. Nous déterminons que le domaine elliptique pour les transferts orbitaux temps-minimal est géodésiquement convexe pour un transfert coplanaire et vers une orbite circulaire, contrairement au cas de l’énergie. Nous examinons ensuite l’effet la perturbation lunaire, nous montrons que dans ce cas le Hamiltonien moyenné se trouve être celui associé à un problème de navigation de Zermelo. Nous étudions numériquement à l’aide du code Hampath, les points conjugués pour caractériser l’optimalité globale des trajectoires. Enfin, nous construisons et réalisons numériquement un transfert d'une orbite terrestre au point de Lagrange L1, qui utilise la moyennation sur un arc (proche de la Terre) pour simplifier les calculs numériques. Dans ce dernier résultat nous voyons qu'un transfert concaténant une trajectoire moyennée avec une trajectoire temps minimal au voisinage du point de Lagrange est en effet proche d’un transfert de temps optimal calculé avec une méthode numérique de tir. / The technique of averaging is an effective way to simplify optimal low-thrust satellite transfers in a controlled two-body Kepler problem. This study takes the form of both an analytical and numerical investigation of low-thrust time-optimal transfers, extending the application of averaging from the two-body problem to transfers in the perturbed low-thrust two body problem and a low-thrust transfer from Earth orbit to the L1 Lagrange point in the bicircular four-body setting. In the low-thrust two-body transfer, we compare the time-minimal case with the energy-minimal case, and determine that the elliptic domain under time-minimal orbital transfers (reduced in some sense) is geodesically convex. We then consider the Lunar perturbation of an energy-minimal low-thrust satellite transfer, finding a representation of the optimal Hamiltonian that relates the problem to a Zermelo navigation problem and making a numerical study of the conjugate points. Finally, we construct and implement numerically a transfer from an Earth orbit to the L1 Lagrange point, using averaging on one (near-Earth) arc in order to simplify analytic and numerical computations. In this last result we see that such a `time-optimal' transfer is indeed comparable to a true time-optimal transfer (without averaging) in these coordinates.

In public opinion, protection against asteroids impact has always been on the agenda of space engineering. Actually it started from 1994 when Shoemaker Levy stroke Jupiter. This protection works in two steps: detection of threat and deflection. Some space agencies and foundations monitor the sky and set up scenario. Although the sky is nowadays well monitored and mapped, there is no global plan nowadays against this threat. This paper focuses on the deflection step, and aims at forecasting which variables are involved and their consequences on the deflection mission. In fact the result depends on several factors, like the time before hazardous moment, the accuracy of detection tools, the choice of deflection method, but the most unpredictable are human factors. This study shows a strategy and so tries to give some new response parts to the global deflection problem.

<p>The missed-thrust problem is a modern challenge in the field of mission design. While some methods exist to quantify its effects, there still exists room for improvement for algorithms which can fully anticipate and plan for a realistic set of missed-thrust events. The present work investigates the use of machine learning techniques to provide a robust controller for a low-thrust spacecraft. The spacecraft’s thrust vector is provided by a neural network controller which guides the spacecraft to the target along a trajectory that is robust to missed thrust, and the controller does not need to re-optimize any trajectories if it veers off its nominal course. The algorithms used to train the controller to account for missed thrust are supervised learning and neuroevolution. Supervised learning entails showing a neural network many examples of what inputs and outputs should look like, with the network learning over time to duplicate the patterns it has seen. Neuroevolution involves testing many neural networks on a problem, and using the principles of biological evolution and survival of the fittest to produce increasingly competitive networks. Preliminary results show that a controller designed with these methods provides mixed results, but performance can be greatly boosted if the controller’s output is used as an initial guess for an optimizer. With an optimizer, the success rate ranges from around 60% to 96% depending on the problem.</p> <p><br></p> <p>Additionally, this work conducts an analysis of a novel hyperbolic rendezvous strategy which was originally conceived by Dr. Buzz Aldrin. Instead of rendezvousing on the outbound leg of a hyperbolic orbit (traveling away from Earth), the spacecraft performs a rendezvous while on the inbound leg (traveling towards Earth). This allows for a relatively low Delta-v abort option for the spacecraft to return to Earth if a problem arose during rendezvous. Previous work that studied hyperbolic rendezvous has always assumed rendezvous on the outbound leg because the total Delta-v required (total propellant required) for the insertion alone is minimal with this strategy. However, I show that when an abort maneuver is taken into consideration, inserting on the inbound leg is both lower Delta-v overall, and also provides an abort window which is up to a full day longer.</p>