This thesis develops and validates a satellite trajectory optimization model. A summary is given of the general mathematical principles of dynamic optimal control to minimize fuel consumed or transfer time. The dynamic equations of motion for a satellite are based upon equinoctial orbital elements in order to avoid singularities for circular or equatorial orbits. The study is restricted to the two-body problem, with engine thrust as the only possible perturbation. The optimal control problems are solved using the general purpose dynamic optimization software, DIDO. The dynamical model together with the fuel optimal control problem is validated by simulating several well known orbit transfers. By replicating the single satellite model, this thesis shows that a multi-satellite model which optimizes all vehicles concurrently can be easily built. The specific scenario under study involves the injection of multiple satellites from a common launch vehicle; however, the methods and model are applicable to spacecraft formation problems as well.
This is a long-overdue volume dedicated to space trajectory optimization. Interest in the subject has grown, as space missions of increasing levels of sophistication, complexity, and scientific return - hardly imaginable in the 1960s - have been designed and flown. Although the basic tools of optimization theory remain an accepted canon, there has been a revolution in the manner in which they are applied and in the development of numerical optimization. This volume purposely includes a variety of both analytical and numerical approaches to trajectory optimization. The choice of authors has been guided by the editor's intention to assemble the most expert and active researchers in the various specialities presented. The authors were given considerable freedom to choose their subjects, and although this may yield a somewhat eclectic volume, it also yields chapters written with palpable enthusiasm and relevance to contemporary problems.
This book explores the design of optimal trajectories for space maneuver vehicles (SMVs) using optimal control-based techniques. It begins with a comprehensive introduction to and overview of three main approaches to trajectory optimization, and subsequently focuses on the design of a novel hybrid optimization strategy that combines an initial guess generator with an improved gradient-based inner optimizer. Further, it highlights the development of multi-objective spacecraft trajectory optimization problems, with a particular focus on multi-objective transcription methods and multi-objective evolutionary algorithms. In its final sections, the book studies spacecraft flight scenarios with noise-perturbed dynamics and probabilistic constraints, and designs and validates new chance-constrained optimal control frameworks. The comprehensive and systematic treatment of practical issues in spacecraft trajectory optimization is one of the book’s major features, making it particularly suited for readers who are seeking practical solutions in spacecraft trajectory optimization. It offers a valuable asset for researchers, engineers, and graduate students in GNC systems, engineering optimization, applied optimal control theory, etc.
Reflecting the results of twenty years; experience in the field of multipurpose flights, this monograph includes the complex routes of the trajectories of a number of bodies (e.g., space vehicles, comets) in the solar system. A general methodological approach to the research of flight schemes and the choice of optimal performances is developed. Additionally, a number of interconnected methods and algorithms used at sequential stages of such development are introduced, which allow the selection of a rational multipurpose route for a space vehicle, the design of multipurpose orbits, the determination of optimal space vehicle design, and ballistic performances for carrying out the routes chosen. Other topics include the practical results obtained from using these methods, navigation problems, near-to-planet orbits, and an overview of proven and new flight schemes.
STOpS successfully found optimal trajectories for the Mariner 10 mission and the Voyager 2 mission that were similar to the actual missions flown. STOpS did not necessarily find better trajectories than those actually flown, but instead demonstrated the capability to quickly and successfully analyze/plan trajectories. The analysis for each of these missions took 2-3 days each. The final program is a robust tool that has taken existing techniques and applied them to the specific problem of trajectory optimization, so it can repeatedly and reliably solve these types of problems.
All-electric satellites are gaining favor among the manufacturers and operators of satellites in Geostationary Earth Orbit (GEO) due to cost saving potential. These satellites have the capability of performing all propulsive tasks with electric propulsion including transfer to GEO. Although fuel-efficient, electric thrusters lead to long transfer, during which the health and the usability of spacecraft is affected due to its exposure to hazardous space radiation in the Van Allen belts. Hence, determining electric orbit-raising trajectory that minimize transfer time is crucial for all-electric satellite operation. This thesis proposes a novel method to determine minimum-time orbit-raising trajectory by blending the ideas of direct optimization and guidance-like trajectory optimization schemes. The proposed methodology is applicable for both planar and non-planar transfers and for transfers starting from arbitrary circular and elliptic orbits. Therefore, it can be used for rapidly analyzing various orbit-raising mission scenarios. The methodology utilizes the variational equations of motion of the satellite in the context of the two-body problem by considering the low-thrust of an electric engine as a perturbing force. The no-thrust condition due to Earth's shadow is also considered. The proposed methodology breaks the overall optimization problem into multiple sub-problems and each sub-problem minimizes a desired objective over the sun-lit part of the trajectory. Two different objective types are considered. Type I transfers minimize the deviation of the total energy and eccentricity of final position from the GEO, while type II transfers minimize the deviation of total energy and angular momentum. Using the developed tool, several mission scenarios are analyzed including, a new type of mission scenarios, in which more than one thruster type are used for the transfer. The thesis presents the result for all studied scenarios and compares the performance of Type I and Type II transfers.
Improvement of spacecraft trajectory optimization approaches, methods, and techniques is critical for better mission design. Preliminary low-fidelity analysis precedes high-fidelity analysis to efficiently explore the space of a problem. The work of this dissertation extends an embedded boundary value problem (EBVP) technique for preliminary design in the two-body problem. The EBVP technique is designed for direct, unconstrained optimization using many, short-arc, embedded Lambert problems that discretize the trajectory. The short arcs share terminal positions to implicitly enforce position continuity and the instantaneous velocity discontinuities in between segments are the control. These coasting arcs and impulsive maneuvers in between segments are defined collectively as a coast-impulse model, similar to the well-known Sims-Flanagan model. Use of EBVPs is not new to spacecraft trajectory optimization, extensively used in primer vector theory, flyby-tour design, direct impulsive-maneuver optimization, and more. Lack of fast and accurate BVP solvers has prevented the use of the EBVP technique on problems with more than dozens of segments. For the two-body problem, a recently-developed Lambert solver, complete with the necessary partials, enables the extension of the EBVP technique to many hundreds to thousands of segments and hundreds of revolutions. The use of many short arcs guarantees existence and uniqueness for the Lambert problem of each segment. Furthermore, short arcs simultaneously approximate low thrust and eliminates the need to know the structure of a high-thrust impulsive-maneuver solution. A set of examples show the EBVP technique to be efficient, robust, and useful. In particular, an example using 256 revolutions, 6143 segments, and a constant flight time per segment, optimizes in 5.5 hours using a single processor. After this initial demonstration, the EBVP technique is improved by a function which enables variable flight time per segment. Guided by the well-known Sundman transformation, these piecewise Sundman transformation (PST) functions divide the total flight time of the trajectory into spatially-even arcs, importantly not modifying the dynamics. Flight-time functions and their dynamical regularization counterpart are shown to share similar behavior for Keplerian orbit propagation. The PST functions are also shown to extend the EBVP technique to a large design space, where a runtime-feasible transfer with 512 revs and 12287 segments is presented that significantly changes semimajor axis, eccentricity, and inclination. Moreover, another example is presented that transfers through the numerically challenging parabolic boundary, i.e. a transfer from a circular to hyperbolic orbit. Both these examples use an exponent of 3/2 for the PST to enforce the spatially-even arcs or equal steps in eccentric anomaly. Lastly, an optimal control problem is formulated to solve a class of many-revolution trajectories relevant to the EBVP technique. For transfers that minimize thrust-acceleration-squared, primer vector theory enables the mapping of direct, many-impulsive-maneuver trajectories to the indirect, continuous-thrust-acceleration equivalent. The mapping algorithm is independent of how the direct solution is obtained and the mapping computations only require a solver for a BVP and its partial derivatives. For the two-body problem, a Lambert solver is used. The mapping is simple because the impulsive maneuvers and co-states share the same linear space around an optimal trajectory. For numerical results, the direct coast-impulse solutions are demonstrated to converge to the indirect continuous solutions as the number of impulses and segments increase. The two-body design space is explored with a set of three many-revolution, many-segment examples changing semimajor axis, eccentricity, and inclination. The first two examples change either a small amount of semimajor axis or eccentricity, and the third example is a transfer to geosynchronous orbit. Using a single processor, the optimization runtime is seconds to minutes for revolution counts of 10 to 100, while on the order of one hour for examples with up to 500 revolutions. Any of these thrust-acceleration-squared solutions are good candidates to start a homotopy to a higher-fidelity minimization problem with practical constraints
The optimization of spacecraft trajectories has been, and continues to be, critical for the development of modern space missions. Longer flight times, continuous low-thrust propulsion, and multiple flybys are just a few of the modern features resulting in increasingly complex optimal control problems for trajectory designers to solve. In order to efficiently tackle such challenging problems, a variety of methods and algorithms have been developed over the last decades. The work presented in this dissertation aims at improving the solutions and the robustness of the optimal control algorithms, in addition to reducing their computational load and the amount of necessary human involvement. Several areas of improvement are examined in the dissertation. First, the general formulation of a Differential Dynamic Programming (DDP) algorithm is examined, and new theoretical developments are made in order to achieve a multiple-shooting formulation of the method. Multiple-shooting transcriptions have been demonstrated to be beneficial to both direct and indirect optimal control methods, as they help decrease the large sensitivities present in highly nonlinear problems (thus improving the algorithms' robustness), and increase the potential for a parallel implementation. The new Multiple-Shooting Differential Dynamic Programming algorithm (MDDP) is the first application of the well-known multiple-shooting principles to DDP. The algorithm uses a null-space trust-region method for the optimization of quadratic subproblems subject to simple bounds, which permits to control the quality of the quadratic approximations of the objective function. Equality and inequality path and terminal constraints are treated with a general Augmented Lagrangian approach. The choice of a direct transcription and of an Augmented Lagrangian merit function, associated with automated partial computations, make the MDDP implementation flexible, requiring minimal user effort for changes in the dynamics, cost and constraint functions. The algorithm is implemented in a general, modular optimal control software, and the performance of the multiple-shooting formulation is analyzed. The use of quasi-Newton approximations in the context of DDP is examined, and numerically demonstrated to improve computational efficiency while retaining attractive convergence properties. The computational performance of an optimal control algorithm is closely related to that of the integrator chosen for the propagation of the equation of motion. In an effort to improve the efficiency of the MDDP algorithm, a new numerical propagation method is developed for the Kepler, Stark, and three-body problems, three of the most commonly used dynamical models for spacecraft trajectory optimization. The method uses a time regularization technique, the generalized Sundman transformation, and Taylor Series developments of equivalents to the f and g functions for each problem. The performance of the new method is examined, and specific domains where the series solution outperforms existing propagation methods are identified. Finally, because the robustness and computational efficiency of the MDDP algorithm depend on the quality of the first- and second-order State Transition Matrices, the three most common techniques for their computation are analyzed, in particular for low-fidelity propagation. The propagation of variational equations is compared to the complex step derivative approximation and finite differences methods, for a variety of problems and integration techniques. The subtle differences between variable- and fixed-step integration for partial computation are revealed, common pitfalls are observed, and recommendations are made for the practitioner to enhance the quality of state transition matrices.
