The direct approach for trajectory optimization was found to be very robust. For most problems, solutions were obtained even with poor initial guesses for the controls. The direct approach was also found to be only slightly less accurate than the indirect methods found in the literature (within 0.7%). The present study investigates minimum-time and minimum-fuel low-thrust trajectory problems via a single shooting direct method. Various Earth-based and interplanetary case studies have been examined and have yielded good agreement with similar cases in the literature. Furthermore, new near-optimal trajectory problems have been successfully solved. A multiple-orbit thrust parameterization strategy was also developed to solve near-optimal very-low-thrust Earth-based transfers. Lastly, this thesis examines the use of the high-accuracy complex-step derivative approximation method for solving low-thrust transfer problems. For certain very nonlinear transfer problems, the complex-step derivative approximation was found to increase the robustness of the single shooting direct method.
The dissertation presents advances in trajectory optimization and flight mechanics of low-thrust spacecraft. With the aid of the extended multiple-shooting techniques with state and costate nodes, the hybrid method and the direct-shooting method are systematically described and used to solve a variety of optimal orbit transfer problems. The optimization methods are demonstrated by presenting solutions for optimal Earth-orbit and interplanetary trajectory examples, and complex interplanetary missions using solar electric propulsion (such as Eros sample return and Pluto-flyby missions). Alternative formulations of equations of motion are discussed, which include inertial frame transformation in terms of three Euler angles and a modified set of equinoctial elements using non-dimensional angular momentum. Finally, a low-thrust Earth-capture guidance scheme is developed and presented, which makes novel use of Perkins' low-thrust universal solution and doesn't require a stored reference trajectory. The simplicity and performance of this new guidance design makes it a viable candidate for onboard implementation.
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.
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.
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.
The purpose of this study is to find optimal low thrust capture trajectories in the restricted three body problem. The region of phase space which corresponds to capture by a body is bounded by the curve of zero velocity passing through the equilibrium point (Lagrange point) L2. If the spacecraft is driven to rest at L2, capture has been achieved by adding the least amount of energy. An optimal control law is developed to achieve this based on maximizing the Jacobi integral. The dynamics are then linearized around L2 for the Earth Moon system and the shooting method is used to solve the two point boundary value problem. This problem was found to be singular so the shooting method was modified to avoid making corrections along the null eigenvector. Four trajectories were found which demonstrate the optimal control law successfully caused the spacecraft to be captured by the Moon. Theses. (mjm).
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.
