Bernd Dachwald Low-Thrust Trajectory Optimization and Interplanetary Mission Analysis Using Evolutionary Neurocontrol Universität der Bundeswehr München Fakultät für Luft- und Raumfahrttechnik Institut für Raumfahrttechnik Low-Thrust Trajectory Optimization and Interplanetary Mission Analysis Using Evolutionary Neurocontrol Dipl.-Ing. Dipl.-Wirt.Ing. Bernd Dachwald Vollständiger Abdruck der bei der Fakultät für Luft- und Raumfahrttechnik der Universität der Bundeswehr München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) eingereichten Dissertation Vorsitzender : Prof. Dr.-Ing. Axel Schulte 1. Berichterstatter : Prof. Dr. rer. nat. Bernd Häusler 2. Berichterstatter : Prof. Dr. sc. math. Joachim Gwinner Diese Dissertation wurde am 5. Juni 2003 bei der Universität der Bundeswehr München, 85577 Neubiberg eingereicht und durch die Fakultät für Luft- und Raumfahrttechnik am 5. Juni 2003 angenommen. Tag der Prüfung : 2. April 2004 to my family If the human brain were so simple that we could understand it, we would be so simple that we couldn't. Emerson Pugh Preface and Acknowledgements This work originates from my research at the "Deutsches Zentrum für Luft- und Raumfahrt" (DLR), "Institut für Weltraumsensorik und Planetenerkundung" (until December 2002), and "Institut für Raumsimulation" (since January 2003), Cologne, in cooperation with the "Institut für Raumfahrttechnik" at the "Universität der Bundeswehr München", Neubiberg. Parts of this work have been presented in "Performance Requirements for Near-Term Interplanetary Solar Sailcraft Missions" at the 6th International AAAF Symposium on Space Propulsion (2002, Versailles, France) [28], in "Optimization of Interplanetary Rendezvous Trajectories for Solar Sailcraft Using a Neurocontroller" at the AIAA/AAS Astrodynamics Specialist Conference 2002 (Monterey, USA) [26], and in "Verwendung eines neuronalen Reglers und evolutionärer Algorithmen zur Berechnung optimaler interplanetarer Sonnenseglerbahnen" at the Deutscher Luft- und Raumfahrtkongress 2002 (Stuttgart, Germany) [24]. Additionally, parts of this work have been submitted as "Optimization of Interplanetary Solar Sailcraft Trajectories Using Evolutionary Neurocontrol" to the AIAA Journal of Guidance, Control, and Dynamics [22]. Furthermore, parts of this work are intended for presentation in "Interplanetary Mission Analysis for Non-Perfectly Reflecting Solar Sailcraft Using Evolutionary Neurocontrol" at the AAS/AIAA Astrodynamics Specialist Conference 2003 (Big Sky, USA) [25], in "Multiple Rendezvous and Sample Return Missions to Near Earth Asteroids Using Solar Sailcraft" at the 5th IAA International Conference on Low-Cost Planetary Missions (2003, Noordwijk, The Netherlands) [29], and in "Solar Sailcraft of the First Generation" at the 54th International Astronautical Congress (2003, Bremen, Germany) [27, 85]. I especially thank Prof. Dr. Bernd Häusler for his support and time dedication on this work. Moreover, I would like to express my gratefulness towards Prof. Dr. Joachim Gwinner for assuming the co-chair and for providing comments and recommendations throughout this project. My special thanks are directed to Dr. Wolfgang Seboldt who strongly supported my work from the beginning and who always took the time for in-detail discussions of various aspects of this work. I also give thanks to all colleagues who are not named here, but have contributed to this work through discussions, suggestions, and comments. A special thank is due to my wife Bettina. Without her support this research would not have been possible. 1 German Aerospace Center 2 Institute of Space Sensor Technology and Planetary Exploration 3 Institute of Space Simulation Abstract Innovative solar system exploration missions require ever larger velocity increments and thus ever more demanding propulsion capabilities. Using for those high-energy missions the state-of-the-art technique of chemical propulsion in combination with (eventually multiple) gravity assist maneuvers results in long, complicated, and inflexible mission profiles. Low-thrust propulsions systems can significantly enhance or even enable those high-energy missions, since they utilize the propellant more efficiently – like electric propulsion systems – or do not consume any propellant at all – like solar sails, that utilize solely the freely available solar radiation pressure for propulsion. Consequently, low-thrust propulsion systems permit significantly larger velocity increments and/or larger payload ratios and/or smaller launch vehicles, while at the same time allowing direct trajectories with reduced flight times, simpler mission profiles, and extended launch windows. One of the most important tasks during the feasibility analysis and the preliminary design of a deep space mission is the design and the optimization of the interplanetary transfer trajectory. Searching trajectories for low-thrust spacecraft, that are optimal with respect to transfer time or propellant consumption, is usually a difficult and time-consuming task that involves a lot of experience and expert knowledge, since the convergence behavior of traditional optimizers, that are based on numerical optimal control methods, depends strongly on an adequate initial guess, which is often hard to find. Even if the optimizer converges to an "optimal" trajectory, this trajectory is typically close to the initial guess that is rarely close to the (unknown) global optimum. Within this work, trajectory optimization problems are attacked from the perspective of artificial intelligence and machine learning, which is quite different from that of optimal control theory. Inspired by natural archetypes, a smart method for spacecraft trajectory optimization – that fuses artificial neural networks and evolutionary algorithms to evolutionary neurocontrollers – is developed. Before the novel method is employed for the trajectory optimization and mission analysis of some exemplary deep space missions, its convergence behavior is evaluated and the quality of the obtained solutions is assessed. It is demonstrated, by re-calculating trajectories for several existing low-thrust problems, that this novel method can be applied successfully for near-globally optimal spacecraft steering. Since evolutionary neurocontrollers explore the trajectory search space more exhaustively than a human expert can do by using traditional optimal control methods, they are able to find spacecraft steering strategies that generate better trajectories, which are closer to the global optimum. Using evolutionary neurocontrollers, low-thrust trajectories can be optimized without an initial guess and without the permanent attendance of an expert in astrodynamics and optimal control theory. Their field of application may be extended to a variety of optimal control problems. Zusammenfassung Innovative Missionen für die Erkundung des Sonnensystems erfordern immer höhere Geschwindigkeitsänderungen und somit immer bessere Antriebssysteme. Für solch hochenergetische Missionen führt die gegenwärtig übliche Methode – Verwendung chemischer Antriebe in Verbindung mit (eventuell mehrfachen) Gravity-Assist-Manövern – zu langen Missionsdauern und zu komplizierten unflexiblen Missionsprofilen. Durch Niedrigschubantriebssysteme kann die Durchführung von hochenergetischen Missionen erleichtert oder teilweise sogar erst ermöglicht werden, da diese Antriebssysteme den Treibstoff effizienter nutzen – wie z.B. elektrische Antriebssysteme – oder sogar überhaupt keinen Treibstoff benötigen – wie Sonnensegler, die den „kostenlosen“ solaren Strahlungsdruck als Antriebsquelle nutzen. Infolgedessen erlauben Niedrigschubsysteme höhere Geschwindigkeitsänderungen und/oder größere Nutzlastverhältnisse und/oder leichtere Trägerraketen. Gleichzeitig erlauben sie schnelle Direktbahnen und einfache Missionsprofile mit ausgedehnten Startfenstern. Eine der wichtigsten Aufgaben während der Missionsdurchführbarkeitsanalyse und des Grobentwurfs einer interplanetaren Mission ist der Entwurf und die Optimierung der Transferbahn(en). Das Konvergenzverhalten von traditionellen Niedrigschub-Bahnoptimierungsverfahren, die auf numerischen Methoden der optimalen Steuerung basieren, hängt entscheidend von einer guten Startschätzung der Lösung ab, wie sie oft schwer zu finden ist. Selbst wenn der Optimierer gegen eine „optimale“ Bahn konvergiert, so befindet sich diese meist nahe an der Startschätzung, welche sich wiederum selten nahe am (unbekannten) globalen Optimum befindet. Dies macht die Suche nach zeit- oder treibstoffoptimalen Niedrigschubbahnen für gewöhnlich zu einer schwierigen und zeitraubenden Arbeit, die viel Erfahrung und Expertenwissen erfordert. Im Rahmen dieser Arbeit werden Bahnoptimierungsprobleme anders angepackt: aus dem Blickwinkel der Künstlichen Intelligenz und des Maschinenlernens. Inspiriert von natürlichen Vorbildern, wird eine „intelligente“ Methode zur Bahnoptimierung von Raumfahrzeugen entwickelt, welche Neuronale Netze und Evolutionäre Algorithmen zu Evolutionären Neuronalen Reglern verbindet. Bevor diese neuartige Methode zur Bahnoptimierung und Missionsanalyse einiger beispielhafter interplanetarer Missionen verwendet wird, wird sie hinsichtlich ihres Konvergenzverhaltens und der Güte der Lösungen geprüft und bewertet. Anhand der Neuberechnung von Bahnen für Niedrigschubprobleme, für die bereits Bahnen in der Literatur vorhanden sind, wird gezeigt, dass die neue Methode erfolgreich für eine beinahe-global optimale Steuerung von Raumfahrzeugen eingesetzt werden kann. Da Evolutionäre Neuronale Regler den Lösungsraum möglicher Bahnen erschöpfender durchsuchen als dies ein menschlicher Experte unter Verwendung traditioneller Methoden der optimalen Steuerung kann, finden sie Steuerstrategien, die bessere – näher am globalen Optimum liegende – Bahnen generieren. Mit Evolutionären Neuronalen Reglern können Niedrigschubbahnen ohne Anfangsschätzung und ohne die ständige Anwesenheit eines Astrodynamik- und Bahnoptimierungsexperten optimiert werden. Ihr Anwendungsbereich ist auf eine Vielzahl von optimalen Steuerungsproblemen erweiterbar. Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx List of Symbols, Constants, and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1 Introduction 1 1.1 Low-Thrust Trajectory Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation for Evolutionary Neurocontrol . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Work Objectives and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Simulation and Propulsion System Models 4 2.1 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Solar Sail Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Solar Sail Mission Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 DLR Solar Sailcraft Baseline Design . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 Solar Radiation Pressure Force Models . . . . . . . . . . . . . . . . . . . . . 7 2.2.4 Sail Performance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.5 Equations of Motion for Solar Sailcraft . . . . . . . . . . . . . . . . . . . . 16 2.3 Electric Propulsion System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1 EP Mission Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 NEP System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.3 SEP System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Equations of Motion for EP Spacecraft . . . . . . . . . . . . . . . . . . . . 24 3 Traditional Trajectory Optimization 26 3.1 The Low-Thrust Trajectory Optimization Problem . . . . . . . . . . . . . . . . . . 26 3.1.1 Objectives for Trajectory Optimization . . . . . . . . . . . . . . . . . . . . 26 3.1.2 High-Thrust and Low-Thrust Trajectory Optimization . . . . . . . . . . . . 27 3.1.3 Low-Thrust Trajectory Optimization from the Perspective of Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Spacecraft Steering Using Local Steering Laws . . . . . . . . . . . . . . . . . . . . 30 3.2.1 Lagrange’s Planetary Equations . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 Pure Local Steering Laws for Spacecraft . . . . . . . . . . . . . . . . . . . . 31 3.2.3 Blended Local Steering Laws for Spacecraft . . . . . . . . . . . . . . . . . . 31 3.2.4 Locally Optimal Spacecraft Steering . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Traditional Trajectory Optimization Methods . . . . . . . . . . . . . . . . . . . . 34 3.4 Concept of a Smart Global Trajectory Optimization Method . . . . . . . . . . . . 36 4 Trajectory Optimization Using Evolutionary Neurocontrol 37 4.1 Spacecraft Steering Using Steering Strategies . . . . . . . . . . . . . . . . . . . . . 38 4.1.1 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1.2 Low-Thrust Trajectory Optimization from the Perspective of Machine Learning 38 4.2 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2.1 What are Artificial Neural Networks? . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 How do Artificial Neural Networks Work? . . . . . . . . . . . . . . . . . . . 40 4.2.3 Learning in Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . 41 4.3 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.1 What are Evolutionary Algorithms? . . . . . . . . . . . . . . . . . . . . . . 42 4.3.2 How do Evolutionary Algorithms Work? . . . . . . . . . . . . . . . . . . . . 43 4.3.3 Why do Evolutionary Algorithms Work? . . . . . . . . . . . . . . . . . . . . 46 4.3.4 Advantages and Disadvantages of Evolutionary Algorithms . . . . . . . . . 48 4.4 Evolutionary Neurocontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Spacecraft Steering Using Evolutionary Neurocontrol . . . . . . . . . . . . . . . . . 50 4.5.1 Neurocontroller Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.2 Neurocontroller Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.3 Neurocontroller Fitness Assignment . . . . . . . . . . . . . . . . . . . . . . 55 4.5.4 Evolutionary Neurocontroller Design . . . . . . . . . . . . . . . . . . . . . . 57 5 InTrance Implementation 59 5.1 Precision and Real-valued Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Premature Convergence and Selection Methods . . . . . . . . . . . . . . . . . . . . 60 5.3 Delta Coding and Real Delta Coding . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4 Evolutionary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4.1 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4.2 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.5 Additionally Encoded Problem Parameters . . . . . . . . . . . . . . . . . . . . . . 66 6 InTrance Evaluation 67 6.1 Mercury Rendezvous Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.1.1 Convergence Behavior and Stability . . . . . . . . . . . . . . . . . . . . . . 68 6.1.2 Different Population Sizes and Accuracy Requirements . . . . . . . . . . . . 70 6.1.3 Different Neurocontrollers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.1.4 Noise and Disturbing Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.1.5 Optimization of the Launch Date . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 Near-Earth Asteroid Rendezvous Mission . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2.1 Convergence Behavior and Stability . . . . . . . . . . . . . . . . . . . . . . 77 6.2.2 Optimization of the Launch Date . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.3 Interplanetary Insertion with Hyperbolic Excess Energy . . . . . . . . . . . 79 6.3 Fast Pluto Fly-By Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3.1 Convergence Behavior and Stability . . . . . . . . . . . . . . . . . . . . . . 81 6.3.2 Close Solar Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.4 Minimal Orbit Transfer Times for Ideal Solar Sailcraft . . . . . . . . . . . . . . . . 83 6.5 Multiple Near-Earth Asteroid Rendezvous Using SEP . . . . . . . . . . . . . . . . . 84 7 Mission Analysis Using InTrance 87 7.1 Minimal Orbit Transfer Times for Non-Ideal Solar Sailcraft . . . . . . . . . . . . . . 87 7.2 Near-Earth Asteroid Rendezvous and Sample Return Missions . . . . . . . . . . . . 89 7.2.1 Mission Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.2.2 Near-Earth Asteroid Rendezvous Mission . . . . . . . . . . . . . . . . . . . . 90 7.2.3 Near-Earth Asteroid Sample Return Mission . . . . . . . . . . . . . . . . . . 94 7.2.4 Multiple Near-Earth Asteroid Rendezvous and Sample Return Mission . . . . 100 7.3 Mercury Rendezvous Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.3.1 Mission Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.3.2 Propulsion Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.3.3 Mission Analysis for Solar Sailcraft . . . . . . . . . . . . . . . . . . . . . . . 103 7.4 Piloted Mars Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.4.1 Mission Objectives and Propulsion Options . . . . . . . . . . . . . . . . . . 106 7.4.2 Mission Analysis for NEP Spacecraft . . . . . . . . . . . . . . . . . . . . . . 106 8 Summary and Conclusions 109 Bibliography 112 Appendices 119 Appendix A Reference Frames 121 A.1 Inertial Cartesian Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.2 Ecliptic Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.3 Orbit Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.4 Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Appendix B NSTAR Cluster Control Strategies 125 Appendix C Locally Optimal Spacecraft Steering 127 Appendix D InTrance Program Overview 129 D.1 Functionalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 D.2 Input and Output Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 D.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 D.2.2 The InTrance Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 D.2.3 The Simulation Parameter File . . . . . . . . . . . . . . . . . . . . . . . . . . 131 D.2.4 The Spacecraft Parameter File . . . . . . . . . . . . . . . . . . . . . . . . . . 132 D.2.5 The NC Definition File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 D.2.6 The EA Parameter File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.2.7 The Simulation Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.2.8 The Trajectory Data File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.2.9 The VRML File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 D.2.10 The Control Vector History File . . . . . . . . . . . . . . . . . . . . . . . . . 135 D.2.11 The (Best) Chromosome File . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 D.2.12 The InTrance Report File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 1 Introduction Innovative solar system exploration missions require ever larger velocity increments and thus ever more demanding propulsion capabilities. Using the state-of-the-art technique of chemical propulsion in combination with (eventually multiple) gravity assist maneuvers for those high-energy missions results in long, complicated, and inflexible mission profiles. Low-thrust propulsions systems can significantly enhance or even enable those high-energy missions, since they utilize the propellant more efficiently – like electric propulsion systems – or do not consume any propellant at all – like solar sails, that utilize solely the freely available solar radiation pressure for propulsion. Consequently, low-thrust propulsion systems permit significantly larger velocity increments and/or larger payload ratios and/or smaller launch vehicles, while at the same time allowing direct trajectories with reduced flight times, simpler mission profiles, and extended launch windows, providing more mission flexibility. 1.1 Low-Thrust Trajectory Optimization One of the most important tasks during the analysis and the design of a deep space mission is the design and the optimization of the interplanetary transfer trajectory. This work deals with the problem of searching optimal interplanetary trajectories for low-thrust spacecraft, where a low but continuous thrust is applied to modify the spacecraft’s orbit over an extended period of time. In simple words, a spacecraft trajectory is the spacecraft’s path from A (the initial body or orbit) to B (the target body or orbit). In general, optimality can be defined according to several objectives like transfer time or propellant consumption. Since solar sailcraft do not consume any propellant, their trajectories are typically optimized with respect to transfer time alone. Trajectory optimization for electric propelled spacecraft is less straightforward, since transfer time minimization and propellant minimization are sometimes competing objectives, so that one objective can only be optimized at the cost of the other objective. Spacecraft trajectories are obtained from the (numerical) integration of the spacecraft’s equations of motion, which contain terms for the external forces that are acting on the spacecraft (gravitational forces and "disturbing" forces like solar radiation and solar wind) and for the thrust force. Besides the inalterable external forces, the trajectory is determined by the variation of the thrust vector, which is typically described by a (spacecraft) control function. Therefore, the actual optimization problem is to find the optimal spacecraft control function that yields the optimal trajectory. For spacecraft with high thrust like chemical rockets, optimal interplanetary trajectories can be found relatively easily, since only a few thrust phases are necessary. These thrust phases are very short compared to the transfer time, so that they can be approximated by singular events that change the spacecraft’s velocity instantaneously while its position remains fixed. In contrast to those high-thrust propulsion systems, low-thrust propulsion systems are required to operate for a significant part of the transfer to generate the necessary velocity increment ΔV. Consequently, the spacecraft control function is a continuous function of time and the dimension of the solution space is infinite. This renders low-thrust trajectory optimization a very difficult problem. Traditionally, low-thrust trajectories are optimized by the application of numerical optimal control methods that are based on the calculus of variations. All these methods can be generally classified as local trajectory optimization methods, where the term optimization does not mean "finding the best solution" but rather "finding a solution". The convergence behavior of local trajectory optimization methods depends on an adequate initial guess of the solution, which is often hard to find. Mostly, many initial guesses have to be conceived until an adequate one is found and convergence is achieved. Since all tasks require frequent manual interactions and thus the permanent attendance of an expert in astrodynamics and optimal control theory, the search for a good trajectory can become very time-consuming and thus expensive. Even if convergence is achieved, a local optimum is typically found, which is close to the initial guess that is rarely close to the (unknown) global optimum. Emanating from the drawbacks of the traditional local trajectory optimization methods, a smart global trajectory optimization method is sought that runs without an initial guess and without the permanent attendance of a trajectory optimization expert. 1.2 Motivation for Evolutionary Neurocontrol Evolutionary neurocontrollers fuse artificial neural networks with evolutionary algorithms. Like the underlying constructs, they are inspired by the natural processes of information processing and optimization. Animal nervous systems incorporate natural evolutionary neurocontrollers to control their actions, giving them marvellous capabilities. One brilliant example for this proposition is the smart flight control system of the housefly. The nervous system of the housefly comprises about 100 000 neurons. This natural neural network manages the flight control of the fly as well as many even more difficult tasks like finding food, finding a mate, producing offspring, etc. As a matter of fact, this flight control system is more ingenious than the one of any aircraft man has ever built. Nature has optimized the fly’s neurocontroller with respect to one single objective: survive to produce offspring. This optimization problem is really very difficult, much more than finding some optimal spacecraft trajectory. Nature has solved this problem through the recombination and mutation of the fly’s genetic material and through natural selection, the famous so-called "survival of the fittest". Fitter flies produce more offspring and there is a high probability that some of them are still smarter than their parents. This very elegant optimization process runs without an initial guess and without employing the calculus of variations! So, if a natural evolutionary neurocontroller can steer a housefly optimally from A to B, why should an artificial evolutionary neurocontroller not be able to steer a spacecraft optimally from A to B? 1.3 Work Objectives and Outline The primary objective of this work is to develop a smart global trajectory optimization method that does not have the drawbacks of the traditional local trajectory optimization methods, as stated above. This novel method, termed "InTrance" (which stands for "Intelligent Trajectory optimization using neurocontroller evolution") employs evolutionary neurocontrol to search near-globally optimal spacecraft trajectories without an initial guess and without the permanent attendance of a trajectory optimization expert. Within chapter 2, the simulation model and the propulsion system models are established that are used within this work to simulate the motion of low-thrust spacecraft in interplanetary space. Within chapter 3, a formal framework for trajectory optimization is given from the perspective of optimal control theory, and it is exemplified how – using local spacecraft steering laws and numerical methods of optimal control – low-thrust trajectories are generally optimized. Special emphasis is given to the problems that local trajectory optimization methods typically encounter. Within chapter 4, it is demonstrated that trajectory optimization problems can also be attacked from the perspective of artificial intelligence and machine learning. Within this context, the problem of searching an optimal trajectory is equivalent to the problem of searching an optimal spacecraft steering strategy. An artificial neural network can be employed as a neurocontroller to implement such a steering strategy, and an evolutionary algorithm can be used to find the neurocontroller that represents the optimal steering strategy that in turn yields the optimal trajectory. Within chapter 5, the implementation of InTrance is described. Before InTrance can be considered as a viable low-thrust trajectory optimization tool, its convergence behavior has to evaluated and the quality of the obtained solutions has to be assessed. Within chapter 6, InTrance is applied to re-calculate existing trajectories for several interplanetary low-thrust trajectory optimization problems (rendezvous problems, fly-by problems, orbit transfer problems, solar sails, solar electric propulsion systems). By comparing the obtained InTrance-results with the existing results, it is shown that InTrance is able to find trajectories that are better (faster and fairly accurate for mission feasibility analysis) than the original trajectories, which have been calculated using traditional local trajectory optimization methods. The secondary objective of this work is to assess the near-term applicability of solar sail propulsion for deep space missions. Within chapter 7, the suitability of InTrance as a tool for mission feasibility analysis and design is assessed. Therefore, InTrance is applied to analyze and design some innovative solar sail missions, and to compare the solar sail mission parameters with those of electric propulsion systems. This is first done for a near-Earth asteroid rendezvous mission, for a near-Earth asteroid sample return mission, and for a mission to rendezvous three different near-Earth asteroids (including a sample return option). Then, two missions are investigated that employ a solar sail for a MESSENGER- and BepiColombo-like mission to Mercury. Finally, it is shown how InTrance can be used to facilitate the feasibility analysis for a piloted Mars mission. 8 Summary and Conclusions Low-thrust propulsion systems provide a small thrust force over an extended period of time to modify the orbit of spacecraft and to generate the required velocity increment. Unlike for high-thrust propulsion systems – like chemical rockets – the trajectory can not be approximated by ballistic flight arcs combined with singular thrusting events that change the spacecraft’s velocity instantaneously while its position remains fixed. Low-thrust trajectory optimization is equivalent to the search for the optimal thrust vector variation along the entire trajectory from a given initial state A to a desired final state B. The optimality of the trajectory can be defined with respect to several (possibly competing) objectives, e.g. transfer time and propellant consumption. Traditionally, low-thrust trajectory optimization problems are solved using various numerical optimal control methods that are based on the calculus of variations. All these methods can be generally classified as local trajectory optimization methods. Their convergence behavior depends on an adequate initial guess of the solution, which is often hard to find, so that the search for a good trajectory can become very time-consuming. Even if convergence is achieved, a local optimum is typically found, which is close to the initial guess that is rarely close to the global optimum. Within this work, low-thrust trajectory optimization problems have been attacked from a different perspective: the perspective of artificial intelligence and machine learning. Inspired by natural archetypes, a smart global method for spacecraft trajectory optimization was developed that fuses artificial neural networks and evolutionary algorithms to evolutionary neurocontrollers (this method was termed InTrance, which stands for "Intelligent Trajectory optimization using neurocontroller evolution"). From the perspective of machine learning, a trajectory is regarded as the result of an explicitly not time-dependent steering strategy that manipulates the spacecraft’s thrust vector according to the actually perceived state of the environment. A standard feedforward artificial neural network is used as a so-called neurocontroller to implement such a spacecraft steering strategy. This way, the trajectory is defined by the internal parameters of the neurocontroller. An evolutionary algorithm is used for finding the optimal network parameters. The trajectory optimization problem is solved, if the optimal parameter vector is found, which defines the optimal spacecraft steering strategy, which in turn generates the optimal trajectory. Using an evolutionary algorithm for the optimization of the neurocontroller, this algorithm may be additionally used for finding good initial conditions. InTrance has been applied to a variety of interplanetary low-thrust trajectory optimization problems, for which reference trajectories have been found in the literature. In accordance with those reference problems, a one-body simulation model without disturbing forces was used. The re-calculation of those problems has revealed that many of the trajectories, which have been generated using traditional local trajectory optimization methods, are quite far from the global optimum. Using InTrance, improvements in transfer time of up to 74% have been achieved. For mission feasibility analysis, the obtained InTrance-trajectories are usually sufficiently accurate with respect to the terminal constraint. However, especially for very difficult problems, demanding accuracy requirements, and indirect steering strategies, InTrance does not in all instances converge against the near-globally optimal solution, or does not achieve the required accuracy, so that more than one InTrance-run should be performed to assess the quality of the generated solution(s) and to raise the likelihood for obtaining a near-globally optimal solution. If a more accurate solution than the InTrance-generated one is required, the InTrance-solution might be used as an initial guess for some numerical optimal control method. The obtained results indicate that – using an evolutionary neurocontroller for spacecraft steering – InTrance is a suitable method for finding near-globally optimal low-thrust trajectories. Since evolutionary neurocontrollers explore the trajectory search space more exhaustively than a human expert can do by using traditional optimal control methods, they are able to find spacecraft steering strategies that generate better trajectories, which are closer to the global optimum. Unlike the traditional methods, InTrance runs without an initial guess and without the permanent attendance of an expert in astrodynamics and optimal control theory. The influence of various InTrance-parameters – like neurocontroller type, population size, and accuracy requirements – on the convergence behavior of the algorithm and on the quality of the obtained solutions has been investigated. Both, quality and convergence behavior have been found to depend considerably only on the chosen neurocontroller input and output set, and on an adequate choice of the fitness function. Thereby, direct steering strategies have been found to perform better than indirect steering strategies. Concerning the other varied parameters, the convergence behavior of InTrance has found to be quite robust. Many InTrance-parameters have not been varied, since reasonably robust settings had been found during the development of InTrance, and since the primary objective of this work was not to find the optimal neurocontroller and evolutionary algorithm parameters for each particular problem, but to demonstrate that evolutionary neurocontrol can be successfully applied to the near-globally optimal steering of low-thrust spacecraft. The optimization of the neurocontroller and the evolutionary algorithm as well as investigations about the problem-dependency of the parameters remain a wide field of research. The InTrance-generated steering strategies have been found to be quite insensitive to neurocontroller input noise, as it is expected for a real mission due to errors in the spacecraft’s autonomous sensors and/or errors in the measurement of the spacecraft’s position and velocity from Earth. However, the InTrance-generated steering strategies have been found to be sensitive to disturbing accelerations acting on the spacecraft, so that a thorough mission design should employ a more sophisticated simulation and spacecraft model than the one that has been used within this work. Many promising directions concerning the application of evolutionary neurocontrol for spacecraft trajectory optimization are proposed for future research: • Evolutionary neurocontrol might be applied to a wide variety of trajectory optimization problems, including different propulsion systems and planetocentric problems. To solve combined low-thrust planetary escape -> interplanetary transfer -> planetary capture problems, three neurocontrollers might be encoded on a single chromosome, so that they can be co-evolved by the evolutionary algorithm. To find still faster transfer trajectories, the inclusion of gravity-assist maneuvers would also be desirable. The number of intermediate fly-bys – e.g. at near-Earth objects – along a trajectory would be another interesting objective for optimization, if longer transfer times can be tolerated. Evolutionary neurocontrol may also be successfully applied to autonomous multiple rendezvous trajectory optimization (e.g. rendezvous three arbitrary near-Earth objects from a set of some hundred objects in minimal time, or low-thrust Earth-Mars-cycler trajectories, etc.) • The found neurocontroller steering strategies are not universal steering strategies, but achieve acceptable results only for a launch within a small interval around the launch date, for which they are optimized. For a given spacecraft, a universal steering strategy would achieve the required accuracy for an entire class of transfer problems (e.g. rendezvous with any arbitrary solar system body for all launch dates). Such universal steering strategies have not been found. It remains questionable, whether universal steering strategies exist and whether they can be represented using a feedforward artificial neural network. Universal steering strategies would be very interesting, since they would suggest that low-thrust trajectory optimization problems might have a (not analytically obtainable and yet unknown) closed-form solution. It may also be beneficial to utilize more general artificial neural networks, where the neural transfer function and the network topology are also optimized by the evolutionary algorithm. (Truncated) Fourier series provide another option to represent the variation of control variables. They might be used instead of artificial neural networks to represent a (non-universal) steering law. In this case, the evolutionary algorithm could be used to determine the optimal coefficients of the series. For a universal steering law, the coefficients can not be fixed but are expected to be a function of the problem parameters. • To perform more thorough mission analyses, the implementation of a more sophisticated simulation model, that includes the gravitational influence of all major solar system bodies is desirable. Additionally, more realistic spacecraft models must be used in this case. Especially for solar sailcraft, the translational motion is highly interrelated with the rotational motion, so that a model for all six degrees of freedom should be developed. Also the billowing, wrinkling and aging of the sail must be studied through FEM simulations and in-space tests. The secondary objective of this work was to assess the performance of solar sail propulsion with respect to chemical and solar electric propulsion (SEP) for interplanetary missions. This has been done exemplary for missions to near-Earth asteroids and to Mercury. For the near-Earth asteroid missions, a near-term sail engineering technology was assumed. The obtained InTrance-results indicate that for this kind of missions (moderate velocity increments, relatively far from the sun), near-term solar sailcraft is clearly outperformed by the SEP option, if only transfer time is considered. However, the launch mass for the solar sail option is considerably smaller in all cases, thus allowing smaller and cheaper launch vehicles. If ground operation costs can be kept low and if the transfer time plays a subordinate role with respect to cost, the solar sail might be the favorable option for such near-Earth asteroid rendezvous and sample return missions. In any case, the development of near-term sails is an indispensable first stepping stone on the way to more advanced solar sails. For the Mercury rendezvous, it has been found that medium-term solar sails are able to deliver the same payload to Mercury than chemical propulsion systems and SEP systems within the same transfer time, allowing at the same time smaller launch vehicles and direct trajectories with simpler mission profiles and practically ever-existent launch windows. Another important result, that has been obtained using InTrance, is the fact, that for a thorough mission analysis the non-perfect reflectivity of the solar sail must be considered through an appropriate solar radiation pressure force model. The widely used simplification, that the non-perfect reflectivity of the sail can be taken into account by using an overall efficiency factor, should only be made for very preliminary mission feasibility analyses. Finally, to provide another innovative example of how InTrance can be applied to support space mission analysis for low-thrust propulsion systems, it has been used to analyze the feasibility of a piloted Mars mission for spacecraft using a nuclear electric propulsion system. In conclusion, it is astonishing that small and primitive artificial neural networks, bred by a crudely simulated biological process, are able to perform tasks that – after some million years of intellectual and cultural evolution and some thousand years of scientific evolution – we are not able to solve with our bare mind. Luckily, although we do not understand their solution method, we are able to replicate their solutions. The citation by Emerson Pugh at the beginning of this work states that we might probably never be able to understand the human brain, since it is as complex as we are. The results that have been obtained within this work support this statement, although we might probably be more simple than Pugh had ever imagined.