Being able to identify fuel efficient transfers between orbits is critical to planning and executing missions involving spacecraft. With a renewed focus on missions in cislunar space, identifying efficient transfers in the dynamical environment characterized by the Circular Restricted Three-Body Problem (CR3BP) will be especially important, both now and in the immediate future. The focus of this thesis is to develop a methodology that can be used to identify a valid low-cost transfer between a variety of orbits in the CR3BP. The approach consists of two distinct parts. First, tools related to dynamical systems theory and manifolds are used to create an initial set of possible transfers. An optimization scheme is then applied to the initial transfers to obtain an optimized set of transfers. Code was developed in MATLAB to implement and test this approach. The methodology and its implementation were evaluated by using the code to identify a low-cost transfer in three different transfer cases. For each transfer case, the best transfers from each set were compared, and important characteristics of the transfers in the first and final sets were examined. The results from those transfer cases were analyzed to determine the overall efficacy of the approach and effectiveness of the implementation code. In all three cases, in terms of cost and continuity characteristics, the best optimized transfers were noticeably different compared to the best manifold transfers. In terms of the transfer path identified, the best optimized and best manifold transfers were noticeably different for two of the three cases. Suggestions for improvements and other possible applications for the developed methodology were then identified and presented. / Master of Science / Being able to identify fuel efficient transfers between orbits is critical to planning and executing missions involving spacecraft. With a renewed focus on lunar missions, identifying efficient transfers between orbits in the space around the Moon will be especially important, both now and in the immediate future. The focus of this thesis is to develop a methodology that can be used to identify a valid low-cost transfer between a variety of orbits in the space around the Moon. The approach was evaluated by using the code to identify a low-cost transfer in three different transfer cases. The results from those transfer cases were analyzed to determine the overall efficacy of the approach and effectiveness of the implementation code. Suggestions for improvements and other possible applications for the developed methodology were then identified and presented.

<div>After completion of a resupply mission to NASA’s proposed Lunar Orbital Platform - Gateway, safe disposal of the Logistics Module is required. One potential option is disposal to heliocentric space. This investigation includes an exploration of the trajectory escape dynamics from an Earth-Moon L2 Near Rectilinear Halo Orbit (NRHO). The effects of the solar gravitational perturbations are assessed in the Bicircular Restricted 4-Body Problem (BCR4BP), as defined in the Earth-Moon rotating frame and in the Sun-B1 rotating frame, where B1 is the Earth-Moon barycenter. Disposal trajectories candidates are classified in three outcomes: direct escape, in direct escapes and captures.</div><div>Characteristics of each outcome is defined in terms of three parameters: the location of the apoapses within to the Sun-B1 rotating frame, a characteristic Hamiltonian value, and the osculating eccentricity with respect to the Earth-Moon barycenter. Sample trajectories are presented for each outcome. Low-cost disposal options are introduced.</div>

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>

<p dir="ltr">This research considers orbit maintenance strategies for multi-body orbits in the context of the Earth-Moon Circular Restricted Three Body Problem (CR3BP). Dynamical requirements for successful long-term orbit maintenance strategies are highlighted.</p>

<p dir="ltr">At the cusp of scientific discovery and innovation, mankind's next greatest challenge lies in developing capabilities to enable human presence in deep space. This entails setting up space infrastructure, travel pathways, managing spacecraft traffic, and building up deep space operation logistics. Spacecrafts that are a part of the infrastructure must be able to perform myriad of operations and transfers such as rendezvous and docking, station-keeping, loitering, collision avoidance etc. In support of this endeavour, an investigation is done to analyze and recreate the solution space for fuel-optimal trajectories and control histories required for onboard trajectory design of inexpensive spacecraft transfers and operations. This study investigates close range rendezvous (CRR), nearby orbital transfer, collision avoidance, and long range transfer maneuvers for spacecrafts whose highly complex and nonlinear behavior is modelled using the circular restricted three-body problem (CR3BP) dynamics and to which a finite-burn maneuver is augmented to model low-propulsion maneuvers. In order to study the nonlinear solution space for such maneuvers, this investigation contributes new formulations of nonlinear programming (NLP) optimal control problems solved to minimize fuel consumption, and validated by traditional methods already in use. This investigation proposes a Feature Learning based Optimal Control Method (L-OCM) to learn the solution space and recreate results in real-time. The NLP problem is solved off-line for a range of initial conditions. The set of solutions is used to generate datasets with initial conditions as inputs and the identified features of the optimal control solution as outputs. These features are inherent to reconstructing the optimal control histories of the solution and are selected keeping onboard computational capabilities in mind. Deep Neural Networks (DNNs) are trained to map the complex, nonlinear relationship between the inputs and outputs, and then implemented to find on-line solutions to any initial condition. The L-OCM method provides fuel-optimal, real-time solutions that can be implemented by a spacecraft performing operations in cislunar space.</p>

<p dir="ltr">A dramatic increase in the number of missions for inserting both large satellites as well as rideshare spacecraft into cislunar trajectories has been noted as of recently. While bal- listic lunar transfers (BLTs) have proven a reliable means for sending primary missions to their destination orbits, the inflexible jettison conditions imposed upon secondary payloads may significantly limit viable pathways. This investigation is centered about designing a framework to construct lunar transfers for secondary payloads from various commissioning maneuver (CM) states to select periodic orbits near the Moon. These continuous passage- ways are modelled in the Bi-Circular Restricted 4-Body Problem (BCR4BP), while necessary dynamical insights are recovered from the application of dynamical systems theory to both the BCR4BP and the Circular Restricted 3-Body Problem (CR3BP). To understand the impact of a Moon encounter on an outbound lunar transfer, families of BLTs are generated for primary payloads, where select members that have close flybys are isolated an examined. A modular, multi-phase framework is then developed, stemming from the lunar encounter. With this, transfers from a variety of sample CM states to Halo, Butterfly and Lyapunov orbits are presented. The versatility of the design framework is highlighted through a case study for a double-flyby transfer to a select Lyapunov orbit. The presented analysis provides an intuitive strategy for diversifying the otherwise limited pool of viable transfers that send secondary payloads to cislunar orbits.</p>

Engineering and technical challenges exist with the material transport of natural resources in space. One aspect of this transport problem is the design of an orbit architecture in the Earth-Moon system (EMS) that facilitates these resources through the mining cycle. In this thesis, it is proposed to use the Circular Restricted 3-Body Problem (CR3BP) to design an orbit architecture composed of L3 Lyapunov orbits, hyperbolic invariant stable and unstable manifolds, and geosynchronous (GEO) orbits. A single shooting method (SSM) and natural parameter continuation (NPC) numerical algorithm is used to compute a family of L3 Lyapunov orbits. Invariant Manifold Theory (IMT) is leveraged to find the set of feasible hyperbolic invariant stable and unstable manifolds associated with a L3 Lyapunov orbit. Ideal L3 Lyapunov orbits are chosen to construct an orbit architecture based off favorable metrics like orbital period, Jacobi Constant, and stability index. Manifolds that enter the GEO and xGEO (beyond GEO) volumes are identified. Finally, a ∆V analysis for GEO to manifold transfer is conducted. An achievement of this study is the computation of stable L3 Lyapunov orbits. The primary contribution of this paper lies in its modeling of a L3 Lyapunov orbit architecture using the CR3BP. / Master of Science / Engineering and technical challenges exist with the material transport of natural resources in space. One aspect of this transport problem is the design of an orbit architecture in the Earth-Moon system (EMS) that facilitates these resources through the mining cycle. In this thesis, it is proposed to use the Circular Restricted 3-Body Problem (CR3BP) to design an orbit architecture composed of L3 Lyapunov orbits, hyperbolic invariant stable and unstable manifolds, and geosynchronous (GEO) orbits. L3 is a unique point in space in a rotating frame of reference where the gravity of the Earth and Moon create a dynamical equilibrium point. Due to its location in a rotating frame of reference relative to the Earth and the Moon, orbits around L3 tend to greater stability than L1 or L2. A single shooting method (SSM) and natural parameter continuation (NPC), which are computational methods for finding solutions that connect discrete boundary conditions, numerical algorithm is used to compute a family of L3 Lyapunov orbits. Invariant Manifold Theory (IMT), which is a dynamical system structure that is invariant throughout the action of the system, is leveraged to find the set of feasible hyperbolic invariant stable and unstable manifolds associated with L3 Lyapunov orbits. Ideal L3 Lyapunov orbits and manifolds are chosen to construct an orbit architecture based off favorable metrics like orbital period, Jacobi Constant, and stability index. Manifolds that enter the GEO and xGEO (beyond GEO) volumes are identified. Finally, a ∆V analysis for GEO to manifold transfer is conducted. An achievement of this study is the computation of stable L3 Lyapunov orbits. The primary contribution of this paper lies in its modeling of a L3 Lyapunov orbit architecture using the CR3BP.

<p dir="ltr">With increasing interest in cislunar operations and exploration of deep space destinations like Mars, a foundational understanding of cislunar dynamics and their potential for facilitating departure from the vicinity of Earth is essential. This investigation addresses this need by analyzing system departure characteristics from a variety of periodic orbit families with unstable members in the Earth-Moon Circular-Restricted 3-Body Problem (CR3BP).</p><p dir="ltr">Specifically, a cislunar-to-Mars transfer methodology is developed, leveraging multi-body dynamical systems theory, especially invariant manifolds of periodic orbits, to design lower energy deep space transfers in comparison to traditional methods. The proposed approach generates families of end-to-end transfers that vary in total maneuver delta-v cost and time-of flight, originating from different unstable cislunar orbits. The tradespaces of these transfer families are then analyzed and compared across various departure orbits to identify departure characteristics across orbit families and energy levels (Jacobi constants). The analysis reveals certain unstable Earth-Moon CR3BP orbit families with more favorable departure characteristics.</p><p dir="ltr">Additionally, this investigation compares the computed deep space transfer costs with those of traditional interplanetary transfers and others from existing literature. Although this transfer design strategy is specifically applied to Mars transfers in this investigation, the methodology is broadly applicable to other deep space destinations. Furthermore, the general findings on cislunar departure characteristics have implications for mission designs to destinations beyond the Earth-Moon region.</p>

<p>The 9:2 lunar synodic resonant near rectilinear halo orbit (NRHO) is the chosen orbit for the Gateway, a future lunar space station constructed by the National Aeronautics and Space Administration (NASA) as well as several commercial and international partners. Designing trajectories in this sensitive lunar region combined with the absence of a singular systematic methodology to approach mission design poses challenges as researchers attempt to design transfers to and from this nearly stable orbit. This investigation builds on previous research in Poincar\'e mapping strategies to design transfers from the 9:2 NRHO using higher-dimensional maps and maps with non-state variables. First, Poincar\'e maps are applied to planar transfers to demonstrate the utility of hyperplanes and establish that maps with only two or three dimensions are required in the planar problem. However, with the addition of two state variables, the spatial problem presents challenges in visualizing the full state. Higher-dimensional maps utilizing glyphs and color are employed for spatial transfer design involving the 9:2 NRHO. The visualization of all required dimensions on one plot accurately reveals low cost transfers into both a 3:2 planar resonant orbit and an L2 vertical orbit. Next, the application of higher-dimensional maps is extended beyond state variables. Visualizing time-of-flight on a map axis enables the selection of faster transfers. Additionally, glyphs and color depicting angular momentum rather than velocity lead to transfers with nearly tangential maneuvers. Theoretical minimum maneuvers occur at tangential intersections, so these transfers are low cost. Finally, a map displaying several initial and final orbit options, discerned through the inclusion of Jacobi constant on an axis, eliminates the need to recompute a map for each initial and final orbit pair. Thus, computation time is greatly reduced in addition to visualizing more of the design space in one plot. The higher-dimensional mapping strategies investigated are relevant for transfer design or other applications requiring the visualization of several dimensions simultaneously. Overall, this investigation outlines Poincar\'e mapping strategies for transfer scenarios of different design space dimensions and represents initial research into non-state variable mapping methods.</p>

<p dir="ltr">In the pursuit of establishing a sustainable space economy within the cislunar region, it is vital to formulate transfer design strategies that uncover economically viable highways between different regions of the space domain. The inherent complexity of spacecraft dynamics in the cislunar space poses challenges in determining feasible transfer options. However, the motion characterized by known dynamical structures modeled through the circular restricted three-body problem (CR3BP) aids in the identification of pathways with reasonable maneuver costs and flight times. A framework is proposed that incorporates a quasi-periodic orbit (QPOs) as an option to design transfer scenarios. This investigation focuses on the construction of transfers between periodic orbits. The framework is exemplified by the construction of pathways between an L2 9:2 synodic resonant Near-Rectilinear Halo Orbit (NRHO) and a planar Moon-centered Distant Retrograde Orbit (DRO). The innate difference in the geometries of the departure and arrival orbits of the sample case, along with the lack of natural flows towards and away from them, imply that links between these orbits may necessitate costly maneuvers. A strategy is formulated that leverages the stable and unstable manifolds associated with intermediate periodic orbits and quasi-periodic orbits to construct end-toend trajectories. As part of this strategy, a systematic methodology is outlined to streamline the determination of transfer options provided by the 5-dimensional manifolds associated with a QPO family. This approach reveals multiple local basins of solutions, both interior and exterior-types, characterized by selected intermediate orbits. The construction of transfers informed by the manifolds associated with QPOs is more intricate than those based on periodic orbits. However, QPO-derived solutions allow for the recognition of alternative local basins of solutions and often offer more cost-effective transfer options when compared to trajectories designed using periodic orbits that underlie the QPOs.</p>