This thesis proposes a continuous low-thrust guidance and control strategy for satellite formation-flying. Stabilizing feedback based on mean relative orbit elements and Lyapunov theory is used. A novel feedback gain matrix inspired by the fuel-optimal impulsive solution is designed to achieve near-optimal fuel consumption. A reference governor is developed to autonomously guide the spacecraft through the relative state-space in order to allow for arbitrarily constrained satellite formations. Constraints include desired  thrust levels, time constraints, passive collision avoidance and locally constrained state-space areas. Keplerian dynamics are leveraged to further decrease fuel consumption. Simulations show fuel consumptions of only 4% higher delta-v than the fuel-optimal impulsive solution. The proposed control and guidance strategy is tested in a high-fidelity orbit propagation simulation using MATLAB/Simulink. Numerical simulations include orbit perturbations such as atmospheric drag, high-order geopotential, solar radiation pressure and third-body (Moon and Sun) effects. Test cases include reconfiguration scenarios with imposed wall, thrust and time constraints and a formation maintenance experiment as flown by TanDEM-X, the TanDEM-X Autonomous Formation-Flying (TAFF) experiment.

Trajectory optimization is a powerful technique to analyze mission feasibility during mission design. High-thrust trajectory optimization problems are typically formulated as discrete optimization problems and are numerically well-behaved. Low-thrust systems, on the other hand, operate for significant periods of the mission time. As a result, the solution approach requires continuous optimization; the associated optimal control problems are in general numerically ill-conditioned. In addition, case studies comparing the performance of low-thrust technologies for space travel have not received adequate attention in the literature and are in most instances incomplete. The objective of this dissertation is therefore to design an efficient optimal control algorithm and to apply it to the minimum-time transfer problem of low-thrust spacecraft. We devise a cascaded computational scheme based on numerical and analytical methods. Whereas other conventional optimization packages rely on numerical solution approaches, we employ analytical and semi-analytical techniques such as symmetry and homotopy methods to assist in the solution-finding process. The first objective is to obtain a single optimized trajectory that satisfies some given boundary conditions. The initialization phase for this first trajectory includes a global, stochastic search based on Adaptive Simulated Annealing; the fine tuning of optimization parameters — the local search — is accomplished by Quasi-Newton and Newton methods. Once an optimized trajectory has been obtained, we use system symmetry and homotopy techniques to generate additional optimal control solutions efficiently. We obtain optimal trajectories for several interrelated problem families that are described as Multi-Point Boundary Value Problems. We present and prove two theorems describing system symmetries for solar sail spacecraft and discuss symmetry properties and symmetry breaking for electric spacecraft systems models. We demonstrate how these symmetry properties can be used to significantly simplify the solution-finding process. / Ph. D.

A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously detail the evolution of the spacecraft position and velocity vectors, as well as the thrust history. To address this difficulty, a dynamical model that incorporates a low-thrust force into the circular restricted 3-body problem (CR3BP), i.e., the CR3BP+LT, is constructed and analyzed. Control strategies that deliver specific energy changes (including zero energy change to deliver a conservative system) are derived and investigated, and dynamical structures within the CR3BP+LT are explored as candidate solutions to seed initial low-thrust trajectory designs. Furthermore, insights from dynamical systems theory are leveraged to inform the design process. In the combined model, the addition of a low-thrust force modifies the locations and stability of the equilibrium solutions, resulting in flow configurations that differ from the natural behavior in the CR3BP. The application of simplifying assumptions yields a conservative, autonomous system with properties that supply useful insights. For instance, "forbidden regions" at fixed energy levels bound low-thrust motion, and analytical equations are available to guide the navigation through energy space. Linearized dynamics about the equilibria supply hyperbolic and center manifold structures, similar to the ballistic CR3BP. Low-thrust periodic orbits in the vicinity of the equilibrium solutions also admit hyperbolic and center manifolds, providing an even greater number of dynamical structures to be employed in preliminary trajectory designs. Several applications of the structures and insights derived from the CR3BP+LT are presented, including several strategies for transit and capture near the smaller CR3BP primary body. Finally, an interactive trajectory design framework is presented to explore and utilize the structures and insights delivered by this investigation.

<div> <div> <div> <p>As cislunar and outer space exploration regains worldwide popularity, the low-thrust spacecraft technology, whether in the form of solar sails, electric propulsion or nuclear propulsion, has seen a major increase in the last two decades, as new technologies arise that not only seek for a reduction of the size of the spacecraft —and/or the payloads— but also to minimize the cost of spaceflights, while trying to approach further destinations in our solar system. Mission designers are being challenged with the need to develop new strategies to generate rapid and informed initial guesses for low-thrust spacecraft trajectory design, that are easily converged into fully continuous solutions in position, velocity and mass states, in a high-fidelity dynamical model that incorporates the true ephemerides and perturbations of the gravitational attracting bodies acting on the spacecraft as it navigates through space. </p> <p>In an effort to explore further mission options for spacecraft traveling in the lunar vicinity, new interest arises into the problem of constructing a general framework for the initial guess generation of low-thrust trajectories in cislunar space, that is independent of the force models in which the orbits of interest are de ned. Given the efficiency of the low-thrust engines, most vehicles are equipped to perform further exploration of the cislunar space after completion of their primary science and technology demonstrations in orbits around the Moon. In this investigation, a generalized strategy for constructing initial guesses for low-thrust spacecraft traveling between lunar orbits that exist within the context of multiple dynamical models is presented. These trajectories are converged as mass-optimal solutions in lower fidelity model, that are easily transitioned and validated in the higher-fidelity ephemeris model, and, achieve large orbital plane changes while evolving entirely within the cislunar region. </p> <p>The robustness of the initial guess generation of the spacecraft’s path, depends highly on the fidelity of the dynamical model utilized to construct such trajectories, as well as on the numerical techniques employed to converge and propagate them into continuous solutions. Other researchers have extensively investigated novel techniques for the generation of initial guesses for the low-thrust spacecraft trajectory design problem including, but not limited to, patched conics strategies, methodologies for the transformation of impulsive burns into nite burns, the orbit chaining framework and, more recently, artificial intelligence schemes. This investigation develops an adaptive orbit chaining type approach that relies on the energy parametrization of periodic orbits that exist within the context of the circular restricted three-body problem, to construct informed initial guess for the low-thrust spacecraft trajectory.</p> <div> <div> <div> <p>A variety of multiple transfer applications for vehicles traveling between orbits in the cislunar region is explored for a wide range of low-thrust spacecraft with varying thrust acceleration magnitude. The examples presented in this investigation are consistent with the low-thrust parameters of previously own missions that utilized the same propulsion capabilities, such as, the DAWN mission and the Japanese Hayabusa missions 1 and 2. The trajectories presented in this work are optimized for either propellant consumption or time- of-flight in the lower-fidelity model, and later transitioned into a higher-fidelity ephemeris model that includes the gravitational attraction of the Sun, the Earth and the Moon. </p> <p>Two strategies are explored for the transition of trajectories from a lower-fidelity model to the higher-fidelity ephemeris model, both of which are successful in retaining the transfer geometry. The framework presented in this investigation is further applied to the upcoming NASA Lunar IceCube (LIC) mission to explore possible extended mission options once its primary science and technology demonstration objectives are achieved. It is demonstrated in this investigation that the strategies developed and presented in this work are not only applicable to the specific low-thrust vehicles explored, but it is applicable to any spacecraft with any type of propulsion technology. Furthermore, the energy-informed adaptive algorithm is easily transition to generate trajectories in a range of varying dynamical models. </p> </div> </div> </div> </div> </div> </div>

<div>In recent decades the revolutionary possibilities of low-thrust electric propulsion have been demonstrated by the success of missions such as Dawn and Hayabusa 1 and 2. The efficiency of low-thrust engines reduces the propellant mass required to achieve mission objectives and this benefit is frequently worth the additional time of flight incurred, particularly for robotic spacecraft. However, low-thrust trajectory design poses a challenging optimal control problem. At each instant in time, spacecraft control parameters that minimize an objective, typically propellant consumption or time of flight, must be determined. The characteristics of low-thrust optimal solutions are often unintuitive, making it difficult to develop an <i>a priori</i> estimate for the state and control history of a spacecraft that can be used to initialize an optimization algorithm. This investigation seeks to develop a low-thrust trajectory design framework to address this challenge by combining the existing techniques of orbit chaining and direct collocation. Together, these two methods offer a novel approach for low-thrust trajectory design that is intuitive, flexible, and robust.</div><div><br></div><div>This investigation presents a framework for the construction of orbit chains and the convergence of these initial guesses to optimal low-thrust solutions via direct collocation. The general procedure is first demonstrated with simple trajectory design problems which show how dynamical structures, such as periodic orbits and invariant manifolds, are employed to assemble orbits chains. Following this, two practical mission design problems demonstrate the applicability of this framework to real world scenarios. An orbit chain and direct collocation approach is utilized to develop low-thrust transfers for the planned Gateway spacecraft between a variety of lunar and libration point orbits (LPOs). Additionally, the proposed framework is applied to create a systematic method for the construction of transfers for the Lunar IceCube spacecraft from deployment to insertion upon its destination orbit near the Moon. Three and four-body dynamical models are leveraged for preliminary trajectory design in the first and second mission design applications, respectively, before transfers are transitioned to an ephemeris model for validation. Together, these realistic sample applications, along with the early examples, demonstrate that orbit chaining and direct collocation constitute an intuitive, flexible, and robust framework for low-thrust trajectory design. </div>

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

For the upcoming formation flying technology demonstration mission NetSat a relative orbit propagator as well as a relative orbit controller were developed. The formation will consist of four equal nano-satellites with an electric propulsion system for orbit correction manoeuvres. This demands the use of continuous low-thrust models for relative orbit control, which is a novel field. A software framework was developed which allows orbit simulations of the whole fleet in a fully non-linear environment. The final on-board relative propagator is based on the Gim-Alfriend STM and incorporates eccentricity and the non-spherical shape of the Earth. The controller uses control Lyapunov function-based design and model predictive control, depending on the task. The guidance and control system is able to safely govern the relative motion for one-, two and three-dimensional formation configurations with inter-satellite distances as low as 50m. Based on these results, a complete mission plan is proposed.

Approved for public release, distribution is unlimited / This thesis explores using a direct pseudospectral method for the solution of optimal control problems with mixed dynamics. An easy to use MATLAB optimization package known as DIDO is used to obtain the solutions. The modeling of both low thrust interplanetary trajectories as well as aerocapture trajectories is detailed and the solutions for low thrust minimum time and minimum fuel trajectories are explored with particular emphasis on verification of the optimality of the obtained solution. Optimal aerocpature trajectories are solved for rotating atmospheres over a range of arrival Vinfinities. Solutions are obtained using various performance indexes including minimum fuel, minimum heat load, and minimum total aerocapture mass. Finally, the problem formulation and solutions for the mixed dynamic problem of low thrust trajectories with a terminal aerocapture maneuver is addressed yielding new trajectories maximizing the total scientific mass at arrival. / Lieutenant, United States Navy

<div>Applications of electric propulsion to spaceflight in multi-body environments require a targeting algorithm to produce suitable trajectories on the ground and on board spacecraft. The two-level targeter with low thrust (TLT-LT) provides a framework to implement differential corrections in computationally-limited autonomous spacecraft applications as well as the larger design space of pre-mission planning. Extending existing two-level corrections algorithms, applications of the TLT-LT to spacecraft with a range of propulsive capabilities, from nearly-impulsive to low-thrust, are explored. The process of determining partial derivatives is generalized, allowing reduced logical complexity and increased flexibility in designing sequences of thrusting and ballistic segments. Various implementation strategies are explored to enforce constraints on time and other design variables as well as to improve convergence behavior through the use of dynamical systems theory and attenuation factors. The TLT-LT is applied to both nearly-impulsive and low-thrust spacecraft applications in the circular restricted three-body problem to demonstrate the flexibility of the framework to correct trajectories across the spectrum of thrust magnitude. Finally, parameter continuation is employed to extend a family of trajectories from a solution with nearly-impulsive thrust events to the low-thrust regime, and the characteristics of this transition are investigated.</div>

Low-thrust spacecraft trajectory optimization is often a difficult and time-consuming process. One alternative is to instead use a closed-loop, feedback-driven control law, which calculates the control using knowledge of only the current state and target state, and does not require the solution of a nonlinear optimization problem or system of nonlinear equations. Though generally suboptimal, such control laws are attractive because of the ease and speed with which they may be implemented and used to calculate feasible low-thrust maneuvers. This thesis presents the theoretical foundations for seven modern low-thrust control laws based on control law "blending" and Lyapunov control theory for a particle spacecraft operating in an inverse-square gravitational field. The control laws are evaluated critically to determine those that present the best combinations of thoroughness of method and minimization of user input required. The three control laws judged to exhibit the most favorable characteristics are then compared quantitatively through three numerical simulations. The simulations demonstrate the effectiveness of feedback-driven control laws, but also reveal several situations in which the control laws may perform poorly or break down altogether due to either theoretical shortcomings or numerical difficulties. The causes and effects of these issues are explained, and methods of handling them are proposed, implemented, and evaluated. Various opportunities for further work in the area are also described. / text