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Near-Earth Objects (NEOs) are asteroids, comets and meteoroids in heliocentric orbits with perihelion below 1.3 au. Similarly to the population of the Main Asteroid Belt, NEOs are primordial bodies and their study can improve our understanding of the origins of the Solar System. With a catalog of over 30~000 known asteroids and approximately 100 listed short-period comets, the NEO population represents an inventory of exploration targets reachable with significantly lower cost than the objects of the Main Asteroid Belt. In addition, the materials present in these bodies could be used to resupply spacecraft en route to other destinations. The trajectories of past missions to NEOs have been designed with the patched-conics technique supplemented by impulsive and/or low-thrust maneuvers and planetary gravity assist. The transfer times range from some months to a few years, and the close-approach speeds relative to the target have been as high as 10 km/s. The design technique described in this work leverages the invariant structures of the circular restricted three-body problem (CR3BP) to connect the vicinity of the Earth with NEOs in low-eccentricity, low-inclination orbits. The fundamental building blocks are periodic orbits around the collinear points L$_1$ and L$_2$ of the Sun-Earth CR3BP. These orbits are used to generate paths that follow the associated hyperbolic invariant manifolds, exit the sphere of influence of the Earth and reach NEOs on nearby orbits. The strategy is simple, can be applied to depart either a libration point orbit or the vicinity of the Earth, and offers attractive performance features.

This contribution focuses on the design of low-energy transfers between planetary moons and presents an efficient technique to compute trajectories characterized by desirable behaviors in the vicinities of the departure and destination bodies. The method utilizes finite-time Lyapunov exponent maps in combination with the Moon-to-Moon Analytical Transfer (MMAT) method previously proposed by the authors. The integration of these two components facilitates the design of direct transfers between moons within the context of the circular restricted three-body problem, and allows the inclusion of a variety of trajectory patterns, such as captures, landings, transits and takeoffs, at the two ends of a transfer. The foundations and properties of the technique are illustrated through an application based on impulsive direct transfers between Ganymede and Europa. However, the methodology can be employed to assist in the design of more complex mission scenarios, such as moon tours.

Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to efficiently determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet. In particular, connections between the periodic orbits of such two different moons are achieved. The strategy is applicable for any type of direct transfers that satisfy the analytical constraints. Case studies are presented for the Jovian and Uranian systems. The transition of the transfers into higher-fidelity ephemeris models confirms the validity of the MMAT method as a fast tool to provide possible transfer options between two consecutive moons.

Mission designers must study many dynamical models to plan a low-cost spacecraft trajectory that satisfies mission constraints. They routinely use Poincar\'e maps to search for a suitable path through the interconnected web of periodic orbits and invariant manifolds found in multi-body gravitational systems. This paper is concerned with the extraction and interactive visual exploration of this structural landscape to assist spacecraft trajectory planning. We propose algorithmic solutions that address the specific challenges posed by the characterization of the topology in astrodynamics problems and allow for an effective visual analysis of the resulting information. This visualization framework is applied to the circular restricted three-body problem (CR3BP), where it reveals novel periodic orbits with their relevant invariant manifolds in a suitable format for interactive transfer selection. Representative design problems illustrate how spacecraft path planners can leverage our topology visualization to fully exploit the natural dynamics pathways for energy-efficient trajectory designs., Comment: 11 pages, 13 figures, to appear at IEEE VIS 2020

The baseline trajectory proposed for the Gateway is a southern Earth-Moon L2 Near Rectilinear Halo Orbit (NRHO). Designed to avoid eclipses, the NRHO exhibits a resonance with the lunar synodic period. The current investigation details the eclipse behavior in the baseline NRHO. Then, phase control is added to the orbit maintenance algorithm to regulate perilune passage time and maintain the eclipse-free characteristics of the Gateway reference orbit. A targeting strategy is designed to periodically target back to the long-horizon virtual reference if the orbit diverges over time in the presence of additional perturbations.

The upcoming Lunar IceCube (LIC) mission will deliver a 6U CubeSat to a low lunar orbit via a ride-share opportunity during NASAs Artemis 1 mission. This presents a challenging trajectory design scenario, as the vast change in energy required to transfer from the initial deployment state to the destination orbit is compounded by the limitations of the LICs low-thrust engine. This investigation addresses these challenges by developing a trajectory design framework that utilizes dynamical structures available in the Bicircular Restricted Four-Body Problem (BCR4BP) along with a robust direct collocation algorithm. Maps are created that expedite the selection of invariant manifold paths from a periodic staging orbit in the BCR4BP that offer favorable connections between the LIC transfer phases. Initial guesses assembled from these maps are passed to a direct collocation algorithm that corrects them in the BCR4BP while including the variable low-thrust acceleration of the spacecraft engine. Results indicate that the ordered motion provided by the BCR4BP and the robustness of direct collocation combine to offer an efficient and adaptable framework for designing a baseline trajectory for the LIC mission.

Spacecraft departing from the Gateway in a Near Rectilinear Halo Orbit (NRHO) experience gravitational forces from the Moon, the Earth, and the Sun, all of which can be simultaneously significant. These complex dynamics influence the post-separation risk of recontact with the Gateway and the eventual destinations of the departing spacecraft. The current investigation examines the flow of objects leaving NRHOs in the Bicircular Restricted Four-Body Problem, and results are applied to heliocentric escape and lunar impact trajectories in a higher-fidelity ephemeris model. Separation maneuver magnitude, direction, and location are correlated with risk of recontact with the Gateway and successful departure to various destinations.

A proposed Gateway facility in a lunar Near Rectilinear Halo Orbit (NRHO) will serve as an outpost in deep space, with spacecraft periodically arriving and departing. Departing objects will include logistics modules, requiring safe disposal, cubesats, deployed to various destinations, and debris objects, whose precise paths may be unknown. Escape dynamics from NRHOs are complex; primarily influenced by the Earth and Moon within the orbit, spacecraft are significantly impacted by solar gravity upon departure. The current investigation explores the dynamics of departure from the NRHO, including the risk of debris recontact, safe heliocentric disposal, and deployment to select destinations.

A primary challenge of low-thrust mission design is the development of an initial guess for the state and control history of a trajectory. To address this challenge, one technique assembles dynamical structures, such as periodic orbits and their associated manifolds, into discontinuous chains that are corrected to locally optimal transfer solutions via direct transcription. In this investigation, dynamical structures that leverage low-thrust in a multi-body regime are incorporated into the orbit chain approach to expand the options available for construction of an initial guess and to guide the direct transcription algorithm toward different categories of locally optimal solutions. The properties and structures of the low-thrust model which facilitate orbit chain construction are demonstrated in representative transfer scenarios. Direct transcription is applied to converge upon locally optimal transfer trajectories in both simplified and ephemeris models. Results indicate that low-thrust dynamical structures offer a promising new catalog of options for use in the orbit chain approach to designing optimal low-thrust trajectories.

A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously specify the spacecraft position and velocity vectors, as well as the thrust history. To mitigate this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP) are nvestigated as candidate solutions to seed initial low-thrust trajectory designs. The addition of low-thrust to the CR3BP modifies the locations and stability of the equilibria, offering novel geometries for mission applications. Transfers between these novel equilibria are constructed by leveraging the associated stable and unstable manifolds and insights from the low-thrust CR3BP.

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 Near Rectilinear Halo Orbit (NRHO) and applies these insights to the design of a low-cost heliocentric Logistics Module disposal option. The effects of the solar gravitational perturbations are assessed in both the bicircular restricted 4-body problem and in an ephemeris force model.

A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously specify the spacecraft position and velocity vectors, as well as the thrust history. To mitigate this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP) are investigated as candidate solutions to seed initial low-thrust trajectory designs. The addition of low-thrust to the CR3BP modifies the locations and stability of the equilibria, offering novel geometries for mission applications. Transfers between these novel equilibria are constructed by leveraging the associated stable and unstable manifolds and insights from the low-thrust CR3BP.

Lunar IceCube is a 6U CubeSat that is designed to detect and observe lunar volatiles from a highly inclined orbit. This spacecraft, equipped with a low-thrust engine, will be deployed from the upcoming Exploration Mission-1 vehicle in late 2018. However, significant uncertainty in the deployment conditions for secondary payloads impacts both the availability and geometry of transfers that deliver the spacecraft to the lunar vicinity. A framework that leverages dynamical systems techniques is applied to a recently updated set of deployment conditions and spacecraft parameter values for the Lunar IceCube mission, demonstrating the capability for rapid trajectory design.

The challenges of targeting specific lunar science orbit parameters from a concomitant Sun-Earth/Moon system trajectory are examined. While the concept of ballistic lunar capture is well-studied, achieving and controlling the time evolution of the orbital elements to satisfy mission constraints is especially problematic when the spacecraft is equipped with a low-thrust propulsion system. Satisfying these requirements on the lunar approach and capture segments is critical to the success of the Lunar IceCube mission, a 6U CubeSat that will prospect for water in solid (ice), liquid, and vapor forms and other lunar volatiles from a low-periapsis, highly inclined elliptical lunar orbit.

Innovative trajectory design tools are required to support challenging multi-body regimes with complex dynamics, uncertain perturbations, and the integration of propulsion influences. Two distinctive tools, Adaptive Trajectory Design and the General Mission Analysis Tool have been developed and certified to provide the astrodynamics community with the ability to design multi-body trajectories. In this paper we discuss the multi-body design process and the capabilities of both tools. Demonstrable applications to confirmed missions, the Lunar IceCube Cubesat lunar mission and the Wide-Field Infrared Survey Telescope (WFIRST) Sun-Earth L2 mission, are presented.

Lunar IceCube, a 6U CubeSat, will prospect for water and other volatiles from a low-periapsis, highly inclined elliptical lunar orbit. Injected from Exploration Mission-1, a lunar gravity assisted multi-body transfer trajectory will capture into a lunar science orbit. The constrained departure asymptote and value of trans-lunar energy limit transfer trajectory types that re-encounter the Moon with the necessary energy and flight duration. Purdue University and Goddard Space Flight Center's Adaptive Trajectory Design tool and dynamical system research is applied to uncover cislunar spatial regions permitting viable transfer arcs. Numerically integrated transfer designs applying low-thrust and a design framework are described.

This investigation is focused specifically on transfers from Earth-Moon L(sub 1)/L(sub 2) libration point orbits to Mars. Initially, the analysis is based in the circular restricted three-body problem to utilize the framework of the invariant manifolds. Various departure scenarios are compared, including arcs that leverage manifolds associated with the Sun-Earth L(sub 2) orbits as well as non-manifold trajectories. For the manifold options, ballistic transfers from Earth-Moon L(sub 2) libration point orbits to Sun-Earth L(sub 1)/L(sub 2) halo orbits are first computed. This autonomous procedure applies to both departure and arrival between the Earth-Moon and Sun-Earth systems. Departure times in the lunar cycle, amplitudes and types of libration point orbits, manifold selection, and the orientation/location of the surface of section all contribute to produce a variety of options. As the destination planet, the ephemeris position for Mars is employed throughout the analysis. The complete transfer is transitioned to the ephemeris model after the initial design phase. Results for multiple departure/arrival scenarios are compared.

As demonstrated by ongoing concept designs and the recent ARTEMIS mission, there is, currently, significant interest in exploiting three-body dynamics in the design of trajectories for both robotic and human missions within the Earth-Moon system. The concept of an interactive and 'dynamic' catalog of potential solutions in the Earth-Moon system is explored within this paper and analyzed as a framework to guide trajectory design. Characterizing and compiling periodic and quasi-periodic solutions that exist in the circular restricted three-body problem may offer faster and more efficient strategies for orbit design, while also delivering innovative mission design parameters for further examination.

Within the context of manned spaceflight activities, Earth-Moon libration point orbits could support lunar surface operations and serve as staging areas for future missions to near-Earth asteroids and Mars. This investigation examines preliminary design considerations including Earth-Moon L1/L2 libration point orbit selection, transfers, and stationkeeping costs associated with maintaining a spacecraft in the vicinity of L1 or L2 for a specified duration. Existing tools in multi-body trajectory design, dynamical systems theory, and orbit maintenance are leveraged in this analysis to explore end-to-end concepts for manned missions to Earth-Moon libration points.

Collinear Earth-Moon libration points have emerged as locations with immediate applications. These libration point orbits are inherently unstable and must be maintained regularly which constrains operations and maneuver locations. Stationkeeping is challenging due to relatively short time scales for divergence effects of large orbital eccentricity of the secondary body, and third-body perturbations. Using the Acceleration Reconnection and Turbulence and Electrodynamics of the Moon's Interaction with the Sun (ARTEMIS) mission orbit as a platform, the fundamental behavior of the trajectories is explored using Poincare maps in the circular restricted three-body problem. Operational stationkeeping results obtained using the Optimal Continuation Strategy are presented and compared to orbit stability information generated from mode analysis based in dynamical systems theory.

The purpose of this study is to investigate alternative return strategies for the Orion trans-Earth injection (TEI) phase. A dynamical systems analysis approach considers the structure of the stable and unstable Sun perturbed Earth-Moon manifolds near the Earth-Moon interface region. A hybrid approach, then, combines the results from this analysis with classical two-body methods in a targeting process that seeks to expand the window of return opportunities in a precision entry scenario. The resulting startup arcs can be used, for instance, to enhance the block set of solutions available onboard during an autonomous targeting process.

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