# Dr Nicola Baresi

## About

### Biography

Nicola Baresi started his Astrodynamics career with a MSc thesis on “Optimal Control of Formation Flying Satellites”. After graduating full marks in Physics from the University of Padova in 2011, he moved to Israel where he worked as a postgraduate researcher for the Distributed Space Systems Laboratory of the Technion, Israeli Institute of Technology. Starting from 2013, Nicola moved to the United States of America as a US-Italy Fulbright scholarship awardee and pursued his PhD studies on spacecraft formation flying and dynamical systems theory. He was later awarded with a PhD in Astrodynamics and Satellite Navigation Systems from the University of Colorado Boulder, as well as with an outstanding graduate research award from the department of aerospace engineering sciences of the same university. Following graduation, Nicola was employed at the Japanese Aerospace eXploration Agency (JAXA), working on small and large scale satellite missions to the Moon and Mars. He eventually joined the University of Surrey in 2019, first as a Surrey Research Fellow and now as a lecturer in Orbital Mechanics at Surrey Space Centre.

## News

## Research

### Research interests

- Astrodynamics
- Dynamical Systems Theory
- Spacecraft Formation Flying
- Autonomous Guidance, Navigation & Control
- Trajectory Optimization
- Orbit Determination
- Asteroids & Comets

## Teaching

- EEE3039: SPACE DYNAMICS AND MISSIONS

- EEEM009: ADVANCED GUIDANCE, NAVIGATION AND CONTROL

## Publications

Air-breathing electric propulsion (ABEP) enables long duration missions at very low orbital altitudes through the use of drag compensation. A system-level spacecraft model is developed, using the interaction between thruster, intake and solar arrays, and coupled to a calculation of the drag. A quadratic solution is found for specific impulse and evaluated to identify the thruster performance required for drag-compensation at varying altitudes. An upper altitude limit around 190 km is based on a minimum thruster propellant density, resulting in required thruster performance values of 𝐼𝑠𝑝 > 3000 s and 𝑇 ∕ 𝑃 > 8 mN/kW for a realistic ABEP spacecraft. The orbit of an air-breathing spacecraft is propagated with time, which highlights the prescribed orbit eccentricity due to non-spherical gravity and therefore an increased variability in the atmospheric conditions. A thruster control law is introduced which avoids a divergent altitude behaviour by preventing thruster firings around the orbit periapsis, as well as adding robustness against atmospheric changes due to season and solar activity. Through the use of an initial frozen orbit, thruster control and an augmented 𝑇 ∕ 𝑃 , a stable long-term profile is demonstrated based on the performance data of a gridded-ion thruster tested with atmospheric propellants. An initial mean semi-major axis altitude of 200 km relative to the equatorial Earth radius, a spacecraft mass of 200 kg, 𝐼𝑠𝑝 = 5455 s and 𝑇 ∕ 𝑃 = 23 mN/kW, results in an altitude range of around 10 km at altitudes of 160–183 km during a period of medium to high solar activity.

Understanding the solar corona and its structure, evolution and composition can provide new insights regarding the processes that control the transport of energy throughout the solar atmosphere and out into the heliosphere. However, the visible emission coming from the corona is more than a million times weaker than the emission from the photosphere, implying that direct corona observations are only possible when the disk of the Sun is fully obscured. In this paper we perform a feasibility study of a Sun occultation mission using the Earth as a natural occulter. The challenge is that the occultation zone created by the Earth does not follow a Keplerian trajectory, causing satellites placed in this region to quickly drift away from eclipse conditions. To increase the number of revisits while optimizing the propellant budget, we propose optimal trajectories in the Sun–Earth-Spacecraft circular restricted three body problem that account for scientific and engineering constraints such as limited power budget and mission duration. Chemical propulsion, electric propulsion and solar sailing configurations are compared in terms of performance and mission feasibility, revealing how 24 h of corona observations would be possible every 39 days with as little as 199 m/s of í µí»¥í µí±. The feasibility of the solar sail approach is hereby demonstrated, making it a challenging engineering alternative to currently available technologies.

This paper explores the application of stochastic continuation methods in the context of mission analysis for spacecraft trajectories around libration points in the planar circular restricted three-body problem. Traditional deterministic approaches have limitations in accounting for uncertainties, requiring a two-step process involving Monte Carlo techniques for assessing the robustness of the deterministic design. This might lead to suboptimal solutions and to a long and time-consuming design process. Stochastic continuation methods, which extend numerical continuation techniques to moments of probability density functions, offer a promising alternative. This paper aims to pioneer the application of stochastic continuation procedures in mission analysis, incorporating and acknowledging the stochastic nature of spacecraft missions from the early design phases. By extending existing frameworks to handle fixed points of stroboscopic or Poincaré mappings, the study focuses on robustifying and enhancing trajectory design by considering uncertainties in the determination of periodic orbits. The proposed approach has the potential to discover new solutions that may remain hidden in deterministic analyses, offering improved mission design outcomes. Specifically, this work concentrates on the planar circular restricted three-body problem, assuming uncertainties in both initial conditions and the mass ratio parameter. Stochastic continuation is employed to identify equilibrium points and periodic orbits in this uncertain dynamical system. The generalization of steady states and periodic orbits in uncertain environments is discussed, demonstrating the effectiveness of stochastic continuation in identifying safe operational regions in uncertain astrodynamics problems.

Space exploration has often benefitted from the qualitative analyses of non integrable problems enabled by numerical continuation procedures. Yet, standard approaches based on Newton’s method typically end with discrete representations of family branches that may be subject to misinterpretation and overlook important dynamical features. In this research, we introduce novel continuation procedures based on the differential algebra of Taylor polynomials. Our algorithms aim at generating dense family branches as an atlas of polynomial charts that are locally valid for a range of system and continuation parameters. Examples of particular solutions will be shown within the framework of the Circular Restricted Three-Body Problem, along with fold and period-doubling bifurcations that are efficiently detected using automatic domain splitting and map inversion techniques.

This paper investigates the dynamical substitutes of the Moon's synodic and side-real resonant Near-Rectilinear Halo Orbits (NRHOs) under the Circular-Elliptic Restricted Four-Body Problem formulation. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. The resonant periodic NRHOs are replaced by two-dimensional quasi-periodic tori in this model, which better approximate the real dynamics. We present the steps and algorithms needed to compute these dynamical structures as well as their geometry in the Circular-Elliptic model.

Closed-loop feedback-driven control laws can be used to solve low-thrust many-revolution trajectory design and guidance problems with minimal computational cost. They treat the problem from a targeting perspective and hence value stability over optimality. The optimality can be increased by making the parameters state-dependent at the cost of reduced stability. In this paper, an actor-critic reinforcement learning framework is used to make the parameters of the Lyapunov-based Q-law state-dependent. A single-layer neural network ensures the Jacobian of these state-dependent parameters can be calculated and used to enforce stability throughout the transfer. The current results focus on GTO-GEO and LEO-GEO transfers in Keplerian dynamics. A trade-off between optimality and stability is observed for the first, but the added stability increases optimality for the later. Robustness to uncertainties in position and velocity are also investigated, along with the effects of eclipses and dynamical perturbations such as J2, Sun and Moon third body attractions.

Proximity operations around Phobos are critical for the Martian Moons eXploration (MMX) mission, and accurate knowledge of the spacecraft's trajectory is required to perform scientific observations and maximise the chances of understanding the moon's internal structure. Possible solutions to implement autonomous navigation around Phobos while orbiting at lower altitudes are presented in this paper and compared with standard navigation techniques. For this purpose, our analyses will compare the level of knowledge achievable once the connection with the Deep Space Network is cut off, showing what can be achieved without ground support, using only information that can be collected and processed onboard, mainly lidar and optical data. Different strategies to build a map of the moon's surface's landmarks are also discussed, demonstrating how this operation can be performed autonomously by the spacecraft and how landmark-based navigation is paramount in estimating the spacecraft's state. Finally, the capability of this setup to observe the moon's gravity field will be assessed, returning precious information for the future geodetic investigations of the Martian Moons eXploration mission around Phobos.

To maintain the periodic orbits in a three-body regime, a high-order Target Phase Approach (TPhA) is proposed in this work. Two types of polynomial maps, the phase-angle Poincaré map and high-order maneuver map, are established respectively for the determination of stationkeeping epochs and calculation of correction maneuvers. A stochastic optimization framework tailored for the TPhA-based stationkeeping process is leveraged in search of fuel-optimal and error-robust TPhA parameters. Quasi-Satellite Orbits (QSOs) around Phobos are investigated to demonstrate the efficacy of this approach in both low-and high-fidelity models. Monte-Carlo simulations demonstrate that the baseline QSO of JAXA's Martian Moons eXploration (MMX) mission can be maintained with a monthly manuever budget of around 1.13m/s.

Near Rectilinear Halo Orbits (NRHOs) are orbits of great interest for the upcom-ing lunar missions. To maintain NRHOs in a three-body regime, a stationkeeping strategy based on a high-order Target Point Approach (TPA) is proposed, where fuel-optimal and error-robust TPA parameters are acquired from stochastic global optimization. Accurate TPA manevuers are calculated in a high-order fashion enabled by Differential Algebra (DA) techniques. Stochasticity is handled by incorporating Monte Carlo simulations in the process of optimization and the evaluation of high-order ODE expansions is employed to supplant the time-consuming numerical integration. Multiple candidate NRHOs with different stability properties are investigated.

Fifty years after the first images of the Martian moons were downlinked from the Mariner 7 and 9, Phobos and Deimos remain mysterious objects of the Solar System. On the one hand, the near-equatorial near-circular orbits of these remote bodies suggest that Phobos and Deimos are likely the product of a giant impact between a protoplanetary object and Mars' ancestor. On the other hand, the spectral and geophysical features observed by several spacecraft missions indicate that both of the Martian moons are extremely dark asteroids captured by the gravity well of the Red planet. The goal of the Martian Moons eXploration (MMX) mission–currently under development by JAXA and international collaborators–is to finally settle the debate on the moons' origin by retrieving pristine samples from the surface of Phobos. The samples will be collected in 2027 after extensive observation campaigns carried out from low-altitude retrograde orbits around the Martian satellite. This paper presents the orbit design and maintenance strategy for the proximity phase of MMX. We will start by reviewing the equations of the elliptical Hill problem in order to account for irregular gravity field of Phobos. The resulting non-autonomous system is populated by families of quasi-periodic orbit invariant tori that are generated through numerical continuation procedures on stroboscopic mappings. Five baseline trajectories are selected with different altitude profiles to meet the scientific instrument requirements of the MMX mission. Each of the candidate trajectories is tested under navigation errors and mismodeled dynamics to assess the operational feasibility of the newly found solutions and identify points of minimum and maximum sensitivity. A two-burn impulse per day strategy is implemented in order to track the reference retrograde orbits. Preliminary results show that the proposed relative trajectories can be maintained around Phobos for more than 30 days with less than 10 m/s.

The Martian Moons eXploration mission will be the first of its kind to sample and study Mars's moon Phobos for a prolonged period of time. The aim of this work is to show that the adoption of periodic and quasi-periodic retrograde trajectories would be beneficial for the scientific return of MMX. A consider covariance analysis is hereby implemented in order to compare the estimation of high-order gravitational field coefficients from different orbital geometries and processing different sets of observables. It is shown that low-altitude non-planar quasi-satellite orbits would refine the knowledge of the moon's gravity field.

Stable Quasi-Satellite Orbits (QSOs) have gained a lot of attention as suitable candidate orbits to explore remote planetary moons. Despite many studies on QSO and its orbital stability are found in literature, the problem of efficient transfer between two QSO is still unsolved. Previous works on transfers between two QSOs includes transfers between planar QSOs using impulsive maneuvers and utilizing the bifurcated multiple-revolutional periodic QSOs (MP-QSOs). Purpose of this paper is to explore the three-dimensional case and design transfer trajectories from lower altitude QSOs to three-dimensional QSOs being considered for upcoming sample return missions like Martian Moons eXploration (MMX) and PHOOTPRINT. Specifically, we utilize the unstable 3D QSO to generate manifolds and connect them to the planar QSO in the framework of Mars-Phobos Circular Hill Problem with Ellipsoidal secondary (HPE). Numerical simulations suggest that transfer between a planar and spatial QSO around Phobos is possible with ΔV as low as 8.277 m/s with TOF of 4.19 days and similarly transfer between a spatial and planar QSO with ΔV of 8.286 m/s with TOF of 6.75 days. As a result, transfer design space between planar and spatial QSOs is established using the invariant manifolds.

Guidance and navigation algorithms play a crucial role in ensuring a successful spacecraft mission. This work proposes a full guidance and navigation algorithm based on differential algebra successive convex programming technique (SCVX). By leveraging the high-order expansions around the reference trajectory it is possible to enhance the computational efficiency of convex-based guidance and navigation algorithms. The high-order expansion enables to capture of the non-linearities in the estimation and guidance problems without sacrificing the robustness of the algorithms. Monte Carlo analyses are carried out to assess the benefits of recom-puting the guidance from the estimated state with this new high-order approach while being robust to uncertainties and errors.

Methods of generating heteroclinic connections between quasi-periodic orbits typically rely on human-in-the-loop or machine learning techniques to find intersections in sets of data in more than three dimensions. We propose a fully systematic method of generating these connections using an invariant property found in knot theory: the linking number. This method proves to be robust in detecting heteroclinic connections between isoenergetic invariant tori in the circular restricted three-body problem.

Many Earth-Moon libration point orbits are being evaluated as candidates for future space missions, ranging from lunar gateways and possibly inhabited assets in the vicinity of the Moon to science missions in resonant synodic orbits, as in the cases of EQUULEUS and LUMIO CubeSat missions. The problem of libration point orbit maintenance is investigated here using the Target Point Approach, which leads to a closed LQR formulation of the station-keeping Delta-v once a set of parameters is specified. These parameters strongly depend on the orbit type. In view of the dramatic variety of stability property featured by halo orbits, these parameters must be fine-tuned with ad-hoc Monte-Carlo simulations or via a trial and error procedure. In this paper, a genetic algorithm is used to optimize the parameters of the target point approach for halo orbits with period ranging from 7 to 14 days and to provide a unified framework for orbit maintenance analysis of halo orbits. Numerical simulations in a high-fidelity ephemeris model show reductions in the station-keeping Delta-v budget for libration point orbits compared to previous results found in the literature.

CubeSat missions that take advantage of ride-sharing opportunities to the Moon are typically limited by a small v budget. In this paper, a patching point method is applied to design a low-energy lunar transfer for a spacecraft initially placed into a circumlunar free-return trajectory. Initial conditions generated within weak stability boundaries further guarantee ballistic lunar capture upon arrival. A large number of optimised trajectories are produced, with a minimum mission v cost of 32.51 m/s. Additionally, the B-plane values of the spacecraft during its initial flyby of the Moon are investigated. A relationship between the initial position of the Sun in the synodic frame and the B-plane values is observed, in which successful trajectories appear to favour entering into certain orbital resonances with the Moon.

Quasi-satellite orbits (QSO) are stable retrograde parking orbits around Phobos that are currently being considered for JAXA's upcoming robotic sample return mission MMX. During the proximity operations of MMX, the spacecraft inserted in a high altitude QSO will gradually descend to lower altitude QSOs with suitable transfer and station-keeping techniques between different relative QSOs. Preliminary analysis of two-impulsive planar transfers between relative retrograde orbits utilizing the bifurcated QSOs families is studied to estimate the ∆V costs and time of flights of the transfers. In spatial transfer problem, trajectories utilizing the invariant manifolds of unstable 3D-QSOs are weakly to highly unstable and require additional station-keeping strategies to perform MMX scientific observations. These transfer trajectories have a longer flight time and might need minor correction maneuvers along the transfer paths. In this paper, an orbital maintenance strategy that suppresses and eliminates linear dynamical instability of the unstable 3D-QSOs has been considered for shortlisting feasible 3D-QSOs for high-latitude observations. Differently from previous works, we utilize the initial guesses found through the preliminary results that provide two-impulsive transfer ∆V execution points and optimize the transfers between relative QSOs around Phobos. Primer vector theory is applied to investigate the primer vector of the transfer trajectories to evaluate whether intermediate maneuver or initial/final coasting times along the trajectories can minimize the total ∆V cost between the transfers.

Missions around small bodies present many challenges from their design to the operations, due to the highly non-linear and uncertain dynamics, the limited ∆v budget and constraints coming from orbit determination and mission design. Within this context, mathematical tools to enhance the understanding of the dynamics behavior can be proven useful to support the mission design process. Chaos indicators are adopted to reveal patterns of time-dependent dynamical systems and to enable the identification of practical stability regions, which are then exploited to design bounded orbits in the proximity of small bodies. The methodology is applied to study the MMX and Hera missions. In the MMX context, the final goal is to obtain bounded orbits useful for the global surface mapping and gravity potential determination of Phobos. On the other hand, concerning the Hera mission, a qualitative analysis of the natural motion about the Didymos binary asteroid system is carried out to compute bounded orbits convenient for the global characterization of the two asteroids and to investigate potential landing trajectories. Sensitivity analyses via Monte Carlo simulations are performed to prove the robustness of the different bounded orbits.

The application of dynamical systems theory in astrodynamics has enabled mission designers to construct innovative fuel-efficient trajectories within chaotic systems. While these solutions often take the form of periodic orbits and two-dimensional invariant manifolds, we can extend these solutions to higher dimensional objects such as quasi-periodic trajectories and their manifolds. Quasi-periodic trajectories allow us to better explore the dynamical environment and expand the design domain of spacecraft missions. By increasing the dimensional space of our solutions, we can obtain trajectories that better fulfill scientific objectives in space missions and that offer significant advantages in orbit maintenance and control. Additionally, these solutions are closer to what we would find in high-fidelity or full-ephemeris models, where perfect periodic solutions do not exist. This is particularly true for space missions flying towards chaotic environments, such as JAXA's Martian Moons eXploration (MMX) mission, which aims to retrieve samples from the largest moon of Mars, Phobos. This paper explores advances in the numerical computation of quasi-periodic tori families and possible applications to the MMX mission around the Mars-Phobos system under the formulation of a restricted three-body model with an ellipsoidal secondary.

The risk of collisions in Earth’s orbit is growing markedly. In January 2021, SpaceX and OneWeb released an operator-to-operator fact sheet that highlights the critical reliance on conjunction data messages (CDMs) and observations, demonstrating the need for a diverse sensing environment for orbital objects. Recently, the University of Oxford and the University of Surrey developed, in collaboration with Trillium Technologies and the European Space Operations Center, an opensource Python package for modeling the spacecraft collision avoidance process, called Kessler. Such tools can be used for importing/exporting CDMs in their standard format, modeling the current low-Earth orbit (LEO) population and its short-term propagation from a given catalog file, as well as modeling the evolution of conjunction events based on the current population and observation scenarios, hence emulating the CDMs generation process of the Combined Space Operations Center (CSpOC). The model also provides probabilistic programming and ML tools to predict future collision events and to perform Bayesian inference (i.e., optimal use of all available observations). In the framework of a United Kingdom Space Agency-funded project, we analyze and study the impact of megaconstellations and observation models in the collision avoidance process. First, we monitor and report how the estimated collision risk and other quantities at the time of closest approach (e.g. miss distance, uncertainties, etc.) vary, according to different observation models, which emulate different radar observation accuracy. Then, we analyze the impact of future megaconstellations on the number of warnings generated from the increase in the number of conjunctions leading to an increased burden on space operators. FCC licenses were used to identify credible megaconstellation sources to understand how a potential consistent increase in active satellites will impact LEO situational safety. We finally present how our simulations help understand the impact of these future megaconstellations on the current population, and how we can devise better ground observation strategies to quantify future observation needs and reduce the burden on operators.

PLATOR is a new electrothermal thruster for space logistics applications, developed by the University of Surrey and the University of Leicester. This paper describes the technology behind the development of the thruster and presents a mission scenario where a PLATOR-propelled spacecraft is used to capture and de-orbit the European Space Agency (ESA)'s Envisat satellite. The orbital transfer trajectory is designed using a time-optimal control approach, and the spacecraft's state vector's uncertainties are assessed through a covariance analysis. A navigation analysis is then performed to evaluate the spacecraft's capability to autonomously track its motion during the transfer using GPS measurements. Finally, a target proximity phase is then simulated to demonstrate the spacecraft's capability to rendezvous and dock with Envisat, using the uncertainties obtained from the covariance analysis, showing the potential of the PLATOR thruster for in-orbit servicing and active debris removal applications.

Successive convex programming is a promising technique for onboard applications thanks to its speed and guaranteed convergence. Hence it can be an enabler for future missions where spacecraft autonomy plays a key role. The definition of a good value of the trust region plays a vital role in the successful convergence of SCVX algorithms. This work presents an improved trust region algorithm based on a differential algebra technique that relies on the information given by the nonlinearities of the constraints and does not depend on the user for the initialization of the trust region.

Water ice and other volatile compounds found in permanently shadowed regions near the lunar poles have attracted the interests of space agencies and private companies due to their great potential for in-situ resource utilization and scientific breakthroughs. This paper presents the mission design and trade-off analyses of the Volatile Mineralogy Mapping Orbiter, a 12U CubeSat to be launched in 2023 with the goal of understanding the composition and distribution of water ice near the lunar South pole. Spacecraft configurations based on chemical and electric propulsion systems are investigated and compared for different candidate science orbits and rideshare opportunities.

The JUICE and Europa Clipper missions will soon be launched to investigate the presence of liquid subsurface oceans underneath the icy moons of Jupiter. Both missions rely on complex moon tour trajectories whose design can benefit from a preliminary understanding of the moons' relative geometry when candidate flyby sequences are researched. To aid with this objective, a new time-periodic system of equation of motion is introduced based on the 4:2:1 Laplace resonance of the Io-Europa-Ganymede system. The dynamical system is investigated with numerical continuation techniques that seek to grow equilibria and periodic orbits from the circular restricted three-body problem into periodic and quasi-periodic solutions that populate the phase space when fourth-and fifth-body perturbations are taken into account. The stability of these solutions is further analyzed in the new time-periodic model, searching for stable and unstable manifolds that can help identify fuel-efficient transfer opportunities between the Galilean moons of Jupiter.

Current and future missions such as JUICE and Europa Clipper will attempt to answer long-standing questions regarding the presence and composition of liquid subsurface oceans below the crusts of Jupiter's icy moons. To achieve their scientific objectives, such missions require complex trajectories in the vicinity of multiple large bodies. In support of moon tour design, the tri-circular restricted five-body problem has been introduced in previous work to model the Jupiter-Io-Europa-Ganymede system while exploiting the Laplace resonance of these Galilean moons, allowing for a non-autonomous periodic dynamical system to be defined in which periodic and quasi-periodic orbits, and their associated invariant manifolds, have been shown to exist. In this paper we demonstrate the trajectory design benefits of the tri-circular problem via the generation of a variety of low-energy transfers between libration orbits at Europa and Ganymede. This is achieved by investigating regions of overlap in the stable and unstable manifolds of the desired departure and arrival orbits in the planar case of the tri-circular problem. Transfers were found with ∆V costs as low as 500 m/s and times of flight as low as 30 days.

Understanding the solar corona and its composition can provide new insights regarding the temperature and the magnetic field of the Sun. The light coming from the corona is more than a million of times weaker than the direct light from the Sun; consequently observing the corona is only possible when the Sun is obscured. From ground, total solar eclipses offer a good opportunity to observe the corona; however, these events only occur every 18 months on average, lasting typically only for a few minutes. The goal of this paper is to perform a feasibility analysis of a Sun occultation mission using Earth as an occulter. However, the occultation zone created by the Earth does not follow a Keplerian trajectory, causing satellites placed in this region to quickly drift apart from the target area. To increase the number of revisits while optimizing the propellant budget, we propose optimal trajectories in the Sun-Earth-Spacecraft circular restricted three body problem that account for scientific and engineering constraints such as limited power budget and mission duration. Chemical Propulsion, Electric Propulsion and Solar Sailing configurations are compared in terms of performance and mission feasibility, revealing how 20 hours of corona observations per cycle would be possible with 0.25 km/s with a revisit of the occultation zone every 35 days. In addition to that the solar sail was proven to be an interesting alternative to chemical and low-thrust propulsion systems.

Upcoming missions towards remote planetary moons will fly towards chaotic dynamical environments that are significantly perturbed by the oblateness of the host planet. This paper introduces a new time-periodic set of equations of motion that is based on the analytical solution of the zonal equatorial problem. Such a system, hereby referred to as the Zonal Hill Problem, remains populated by resonant periodic orbits and families of two-dimensional quasi-periodic invariant tori that are calculated by means of homotopy continuation procedures. The resulting periodic and quasi-periodic trajectories are investigated for the mission analysis and design of future planetary moons explorers.

Future missions to the Moon and beyond are likely to involve low-thrust propulsion technologies due to their propellant efficiency. However, these still present a difficult trajectory design problem. Lyapunov control laws can generate sub-optimal trajectories with minimal computational cost and are suitable for feasibility studies and as initial guesses for optimisation methods. In this work we combine Lyapunov control laws with state-dependent weights trained via reinforcement learning to design low-thrust transfers from GTO towards low-altitude Lunar orbits. The agent is able to explore third-body effects during training and learn to remain stable to perturbations during the different transfer phases. Three different approaches are investigated: backwards propagation, backwards propagation with freed geometry, and forwards propagation including rendezvous capability with the Lunar SOI. The last of these proves to be the most successful, coming within 6.6% of the optimal solution.

Despite the advantages of very-low altitude retrograde orbits around Phobos, questions remain about the efficacy of conventional station-keeping strategies in preventing spacecraft such as the Martian Moons eXploration from escaping or impacting against the surface of the small irregular moon. This paper introduces new high-fidelity simulations in which the output of a sequential Square-Root Information Filter is combined with recently developed orbit maintenance strategies based on differential algebra and convex optimization methods. The position and velocity vector of the spacecraft are first estimated using range, range-rate, and additional onboard data types such as LIDAR and camera images. This information is later processed to assess the necessity of an orbit maintenance maneuver based on the estimated relative altitude of MMX about Phobos. If a maneuver is deemed necessary, the state of the spacecraft is fed to either a successive convex optimization procedure or a high-order target phase approach capable of providing sub-optimal station-keeping maneuvers. The performance of the two orbit maintenance approaches is assessed via Monte Carlo simulations and compared against work in the literature so as to identify points of strength and weaknesses.

Numerical continuation techniques are powerful tools that have been extensively used to identify particular solutions of nonlinear dynamical systems and enable trajectory design in chaotic astrodynamics problems such as the Circular Restricted Three-Body Problem. However , the applicability of equilibrium points and periodic orbits may be questionable in real-world applications where the uncertainties of the initial conditions of the spacecraft and dynamical parameters of the problem (e.g., mass ratio parameter) are taken into consideration. Usually, the robustness of a candidate trajectory is tested via a two-step approach, whereby trajectories are first designed in a deterministic scenario, and then Monte Carlo methods are a posteriori used to check their robustness. While this strategy is ubiquitous in preliminary mission design, it can however lead to time-consuming and potentially not robust solutions, meaning that the found tra-jectories are not designed to account for uncertainties. Instead, the robustness of the determin-istic optimal solutions is usually ensured. Due to uncertain parameters and initial conditions, the spacecraft might not follow the reference periodic orbit owing to growing uncertainties that cause the satellite to deviate from its nominal path. Hence, it is crucial to keep track of the probability of finding the spacecraft in a given region. Building on previous work, we extend numerical continuation to moments of the distribution (i.e., stochastic continuation) by directly continuing moments of the probability density function of the spacecraft state. Only assuming normality of the initial conditions, and leveraging moment-generating functions, Isserlis' theorem, and the algebra of truncated polynomials, we propagate the distribution of the spacecraft state at consecutive surface of section crossings while retaining a symbolic map of the final moments of the distribution that depend on the initial mean and covariance matrix only. While the technique is only valid for initial Gaussian distributions, it does not assume that the distribution maintains its normality throughout the integration. The symbolic Poincaré map can then be directly used to evaluate the final moments of an initial distribution , as a function of the initial mean and covariance. This can therefore be used to accelerate the evolving step in the stochastic continuation procedure. The goal of the work is to offer a differential algebra-based general framework to continue 3D periodic orbits in the presence of uncertain dynamical systems. The proposed approach is compared against traditional Monte Carlo simulations to validate the uncertainty propagation approach and demonstrate the advantages of the proposed in terms of uncertainty propagation computational burden and access to higher-dimensional problems.

An algorithm for the second-order approximation of fixed-time, low-thrust orbital transfers and generation of Bacon plots is proposed. After averaging the extremal flow of the optimal-control Hamiltonian, a one-parameter family of solutions of a reduced-order, two-point boundary value problems is computed by means of a differential-continuation scheme. Sensitivities of the shooting function are then used in conjunction with an ad hoc near-identity transformation between averaged and osculating variables to achieve an accurate solution for all longitudes of the departure and arrival orbits. Hence, a single simplified shooting problem has to be solved to approximate the solution for any combination of departure and arrival dates and to draw a Bacon plot for an arbitrary launch window. Both the averaged flow and the near-identity transformation are efficiently evaluated via the fast-Fourier transform algorithm, yielding a fully numerical procedure.

Heteroclinic connections represent unique opportunities for spacecraft to transfer between isoenergetic libration point orbits for zero deterministic Delta V expenditure. However, methods of detecting them can be limited, typically relying on human-in-the-loop or computationally intensive processes. In this paper we present a rapid and fully systematic method of detecting heteroclinic connections between quasi-periodic invariant tori by exploiting topological invariants found in knot theory. The approach is applied to the Earth-Moon, Sun-Earth, and Jupiter-Ganymede circular restricted three-body problems to demonstrate the robustness of this method in detecting heteroclinic connections between various quasi-periodic orbit families in restricted astrodynamical problems.

Spacecraft trajectory optimization is essential for all the different phases of a space mission, from its launch to end-of-life disposal. Due to the increase in the number of satellites and future space missions beyond our planet, increasing the level of autonomy of spacecraft is a key technical challenge. In this context, traditional trajectory optimization methods, like direct and indirect methods are not suited for autonomous or on-board operations due to the lack of guaranteed convergence or the high demand for computational power. Heuristic control laws represent an alternative in terms of computational power and convergence but they usually result in sub-optimal solutions. Successive convex programming (SCVX) enables to extend the application of convex optimization to non-linear optimal control problems. The definition of a good value of the trust region size plays a key role in the convergence of SCVX algorithms, and there is no systematic procedure to define it. This work presents an improved trust region based on the information given by the nonlinearities of the constraints which is unique for each optimization variable. In addition, differential algebra is adopted to automatize the transcription process required for SCVX algorithms. This new technique is first tested on a simple 2D problem as a benchmark of its performance and then applied to solve complex astrodynamics problems while providing a comparison with indirect, direct, and standard SCVX solutions.

A novel high-order target phase approach (TPhA) for the station-keeping of periodic orbits is proposed in this work. The key elements of the TPhA method, the phase-angle Poincare map and high-order maneuver map, are constructed using differential algebra (DA) techniques to determine station-keeping epochs and calculate correction maneuvers. A stochastic optimization framework tailored for the TPhA-based station-keeping process is leveraged to search for fuel-optimal and error-robust TPhA parameters. Quasi-satellite orbits (QSOs) around Phobos are investigated to demonstrate the efficacy of TPhA in mutli-fidelity dynamical models. Monte Carlo simulations demonstrated that the baseline QSO of JAXA's Martian Moons eXploration (MMX) mission could be maintained with a monthly maneuver budget of approximately 1 m/s.

Understanding the internal composition of a celestial body is fundamental for formulating theories regarding its origin. Deep knowledge of the distribution of mass under the body's crust can be achieved by analyzing its moments of inertia and gravity field. In this regard, the two moons of the Martian system have not yet been closely studied and continue to pose questions regarding their origin to the space community; thus, they deserve further characterization. The Martian Moons eXploration mission will be the first of its kind to sample and study Phobos over a prolonged period. This study aims to demonstrate that the adoption of periodic and quasi-periodic retrograde trajectories would be beneficial for the scientific value of the mission. Here, a covariance analysis was implemented to compare the estimation of high-order gravitational field coefficients from different orbital geometries and for different sets of processed observables. It was shown that the adoption of low-altitude non-planar quasi-satellite orbits would help to refine the knowledge of the moon's libration angle and gravitational field.

This paper develops a high-order guidance scheme based on the expansion of the solution to a time-optimal low-thrust optimal control problem by using differential algebraic techniques. The result allows new optimal trajectories to be autonomously computed from the simple evaluation of polynomials, including updated times of flight. Furthermore, a differential algebraic technique known as automatic domain splitting is implemented to improve the guidance scheme by adaptively and automatically splitting the uncertainty domain into smaller subdomains, in which new guidance polynomials are then generated which are better equipped to handle deviations in their respective subdomains. The result is not only a more effective handling of deviations in the trajectory compared to previous algorithms which use just a single set of polynomials but one that can provide the control and time-of-flight updates to a desired accuracy across the entire domain. Combined with its high-order nature, the guidance scheme is capable of dealing with nonlinearity and large deviations with respect to the reference trajectory, thus being robust to mismodeling and unplanned events. The guidance scheme is evaluated in three scenarios: simplified rendezvous using Clohessy–Wiltshire equations to demonstrate automatic domain splitting in the guidance problem, an Earth– Mars transfer, and an asteroid landing trajectory. The results show the ability of the differential algebraic-based guidance to provide optimal control and time-of-flight updates and improvements of several orders of magnitude over previous differential algebraic-based schemes in handling large uncertainty experienced in spaceflight by leveraging automatic domain splitting.

Enhancements in evolutionary optimization techniques are rapidly growing in many aspects of engineering, specifically in astrodynamics and space trajectory optimization and design. In this chapter, the problem of optimal design of space trajectories is tackled via an enhanced optimization algorithm within the framework of Estimation of Distribution Algorithms (EDAs), incorporated with Lyapunov and Q-law feedback control methods. First, both a simple Lyapunov function and a Q-law are formulated in Classical Orbital Elements (COEs) to provide a closed-loop low-thrust trajectory profile. The weighting coefficients of these controllers are approximated with various degrees of Hermite interpolation splines. Following this model, the unknown time series of weighting coefficients are converted to unknown interpolation points. Considering the interpolation points as the decision variables, a black-box optimization problem is formed with transfer time and fuel mass as the objective functions. An enhanced EDA is proposed and utilized to find the optimal variation of weighting coefficients for minimum-time and minimum-fuel transfer trajectories. The proposed approach is applied in some trajectory optimization problems of Earth-orbiting satellites. Results show the efficiency and the effectiveness of the proposed approach in finding optimal transfer trajectories. A comparison between the Q-law and a simple Lyapunov controller is done to show the potential of the EEDA in enabling the simple Lyapunov controller to recover the finer nuances explicitly given within the analytical expressions in the Q-law.

Quasi-periodic invariant tori are emerging as a powerful tool to populate the phase space and enable a better understanding of key astrodynamics problems. This chapter introduces modern numerical continuation techniques for generating two-dimensional invariant tori in the elliptical restricted three-body problem of the Earth–Moon system. As a test case and baseline trajectories for the upcoming Lunar Orbital Platform Gateway, the (2:9) and (1:4) synodic resonant near rectilinear halo orbits are hereby considered. Their dynamical substitutes are first calculated and later analyzed using two-dimensional torus maps that facilitate the creation of accurate initial guesses for real-ephemeris orbits. The transition process is demonstrated with a trajectory optimization procedure that successfully generates continuous ballistic arcs around the Moon for more than one year.

Modern space missions are frequently targeted towards new and unexplored regions of space, such as the region between the Earth and the Moon, which is denoted as the Cislunar space (NASA, 2020; International Space Exploration Coordination Group – ISECG, 2018); binary asteroid systems (Rivkin et al., 2021); comets and other irregularly shaped celestial objects; satellites of other Solar system planets. In all of these mission scenarios, the spacecraft dynamics is governed by an intriguing, yet complex and chaotic dynamical environment that is driven by the presence of multiple and/or non-spherical massive bodies. The gravitational influence of these objects shall be addressed with methods and techniques that are different from the standard Keplerian tools available in the classic Two-Body Problem. In recent years, the space community has shown a renovated scientific and technological interest in mastering the multi-body non-Keplerian astrodynamics for practical applications. Immediately, new technological and engineering challenges emerged in order to cope with this uninvestigated portion of outer space. In particular, the Guidance, Navigation and Control (GNC) and the Propulsive subsystems developments have been strongly supportive of this endeavor.

The Lunar Volatile and Mineralogy Mapping Orbiter (VMMO) comprises a low-cost 12U Cubesat with deployable solar arrays, X-Band/UHF communications, option of electric or chemical propulsion, the Lunar Volatiles and Mineralogy Mapper (LVMM) payload, and an optional GPS receiver technology demonstrator. The LVMM facilitates three operational modes: Active mode using illumination of the lunar surface at 532nm, 1064nm and 1560nm to enable volatiles mapping during the lunar night and within Permanently Shadowed Regions (PSRs); Passive mode during the lunar day with spectral channels at 300nm, 532nm, 690nm, 1064nm and 1560nm for mapping lunar surficial ilmenite (FeTiO3); and a Communications mode for an optical data downlink demonstration at 1560nm. Previous lunar missions have detected the presence of water-ice in the lunar South Pole region. However, there is considerable uncertainty with regards to its distribution within and across the lunar surface. A number of planned future missions will further map water ice deposits, but the spatial resolution of these observations is expected to be on the order of kilometers. The LVMM using single-mode fiber lasers can improve the special resolution of the mapping to 10s of meters. VMMO has completed the Phase A study with ESA. This paper discusses the baseline LVMM payload design and its dual-use applications for both the stand-off mapping of lunar volatiles and a high-speed optical data link demonstration. In particular, the supporting fiber-laser technology readiness was advanced through ground qualification.

Reaction wheel assemblies have been applied for the attitude control of satellites since the dawn of the space exploration era. This presentation demonstrates how reaction wheel assemblies can be simulated in GODOT while taking advantage of its automatic differentiation and time evaluable features. In particular, time evaluables for the reaction wheel angular momentum and gravity gradient of a planet will be added so as to expand upon the classic form of Euler’s equations available in GODOT, thereby increasing the fidelity of the attitude simulations. Time evaluables and automatic differentiation features will be also applied for generating minimum-effort slew manoevres of a rigid spacecraft equipped with a tetrahedral reaction wheel assembly around Jupiter.

Upcoming missions towards remote planetary moons will fly in chaotic dynamic environments that are significantly perturbed by the oblateness of the host planet. In this presentation, we will be overviewing the equations of the Zonal Hill Problem that govern the dynamic evolution of a satellite subject to the gravitational attraction of a moon and its oblate host planet. Despite time-periodic effects, the system remains populated by families of quasi-periodic trajectories that can be used as a baseline for the proximity operations of future planetary moons explorers such as MMX.

Despite several missions, the origin of the two Martian moons, Phobos and Deimos, remains an open question. The goal of the next JAXA’s flagship mission Martian Moons eXploration will be to explore the two Martian moons. The satellite will be injected into a quasi-satellite orbit and it will require some station-keeping maneuvers to maintain the satellite on these orbits. Traditional methods for station-keeping around libration points are not applicable for these orbits due to their rapid evolution. In this paper we propose a new approach to perform station-keeping on periodic and quasi-periodic orbits based on convex optimization. Successive convex optimization is used to solve the time free fuel optimal problem to drive the satellite back to a reference trajectory. The latter is updated every GNC (Guidance Navigation and Control) loop by means of an innovative Discrete Fourier Transform approach that exploits the periodicity and quasi-periodicity of quasi-satellite orbits. To assess the robustness of the methodology the control and the references are computed in the autonomous dynamical model while the propagation is performed in the non-autonomous model while adding injection, orbit determination and executions errors. Monte Carlo analysis demonstrate that quasi-satellite orbits can be maintained using less than 6 m/s per month.

The dynamical environment on and about the Martian moon Phobos is explored. This planetary moon provides a unique dynamical environment in the solar system, being subject to extreme tidal forces and having a characteristically non-spherical shape. Further, it is not in a fully circular orbit, meaning that it has librations that arise from its eccentricity, contributing to a periodic forcing environment. Thus, to plan and implement missions in the vicinity of and on Phobos will require these considerations be taken into account. In this paper the latest published models of the Phobos shape and dynamics are used to characterize its dynamical environment in close proximity orbit about the body, for motion across its surface and for controlled hovering motion in its vicinity. It is found that surface motion is subject to a number of "speed limits" that can cause a moving vehicle to leave the surface and to possibly escape the moon and enter orbit about Mars. In terms of orbital stability, the existence of libration orbit families are characterized down to the surface using an exact potential, and the known stable QSO orbits are shown to be associated with families of stable quasi-periodic orbits. (C) 2018 COSPAR. Published by Elsevier Ltd. All rights reserved.

The Asteroid Impact & Deflection Assessment (AIDA) mission is planning to visit the Didymos binary system in 2022 in order to perform the first demonstration ever of the kinetic impact technique. Binary asteroids are an ideal target for this since the deflection of the secondary body can be accurately measured by a satellite orbiting in the system. However, these binaries offer an extremely rich dynamical environment whose accurate investigation through analytical approaches is challenging at best and requires a significant number of restrictive assumptions. For this reason, a numerical investigation of the dynamical environment in the vicinity of the Didymos system is offered in this paper. After computing various families of periodic orbits, their robustness is assessed in a high-fidelity environment consisting of the perturbed restricted full three-body problem. The results of this study suggest that several nominally stable trajectories, including the triangular libration points, should not be considered as safe as a state vector perturbation may cause the spacecraft to drift from the nominal orbit and possibly impact one of the primary bodies within a few days. Nonetheless, there exist two safe solutions, namely terminator and interior retrograde orbits. The first one is adequate for observation purposes of the entire system and for communications. The second one is more suitable to perform close investigations of the primary body.

The science operations of the spacecraft and remote sensing instruments for the Martian Moon eXploration (MMX) mission are discussed by the mission operation working team. In this paper, we describe the Phobos observations during the first 1.5 years of the spacecraft's stay around Mars, and the Deimos observations before leaving the Martian system. In the Phobos observation, the spacecraft will be placed in low-altitude quasi-satellite orbits on the equatorial plane of Phobos and will make high-resolution topographic and spectroscopic observations of the Phobos surface from five different altitudes orbits. The spacecraft will also attempt to observe polar regions of Phobos from a three-dimensional quasi-satellite orbit moving out of the equatorial plane of Phobos. From these observations, we will constrain the origin of Phobos and Deimos and select places for landing site candidates for sample collection. For the Deimos observations, the spacecraft will be injected into two resonant orbits and will perform many flybys to observe the surface of Deimos over as large an area as possible.

The Near-Earth Asteroid Characterization and Observation (NEACO) mission is a concept study proposing to explore the fast-rotating asteroid (469219) 2016 HO3, one of Earth's few quasi-satellites. In this study, a SmallSat spacecraft performs a scientific investigation that characterizes the asteroid at a sufficient degree to enable future, more in-depth missions. The 166 kg NEACO spacecraft uses a low-thrust, solar electric propulsion system to reach HO3 within 22 months from launch. Its instrument suite consists of two optical cameras, two spectrometers, an altimeter, and a low-velocity impactor. Upon arrival at HO3, NEACO uses pulsed plasma thrusters to hover at varying altitudes to enable lit surface mapping, shape modeling, and surface spectroscopy. The spacecraft will then perform several flybys to estimate the asteroid's mass. Finally, NEACO releases a low-velocity impactor during very low-altitude hovering to validate the existence of regolith and estimate the magnitude of surface cohesion. The science operations are completed within 8 months and the total mission is completed in less than 3 years. The NEACO mission concept integrates novel small-body analyses and proximity operation techniques with high-technology-readiness-level spacecraft components to achieve its science objectives within a reasonable mission timeline.

The surface of the Martian moon Phobos exhibits two distinct geologic units, red and blue, characterized by their spectral slopes. The provenance of these units is uncertain yet crucial to understanding the origin of the Martian moon and its interaction with the space environment. Here we present a combination of dynamical analyses and numerical simulations of particle dynamics to show that periodic variations in dynamic slopes, driven by orbital eccentricity, can cause surface grain motion. For regions with steep slopes that vary substantially over one Phobos orbit, the surface is excavated at a faster rate than the space weathering timescale. Our model predicts that this new mechanism is most effective in regions that coincide with blue units. Therefore, space weathering is the likely driver of the dichotomy on the moon's surface, reddening blue units that represent pristine endogenic material.

In order to investigate the origin of Phobos and Deimos, the Japanese Martian Moons eXploration (MMX) mission is scheduled for launch in 2024. MMX will make comprehensive remote-sensing measurements of both moons and return regolith samples from Phobos to Earth. Geodetic measurements of gravity, shape, and rotation parameter of a body provides constraints on its internal structure reflecting its origin and evolution. Moments of inertia are important parameters to constrain the internal mass distribution, but they have not been well determined for the Martian moons yet. We discuss the mission requirements related to the moments of inertia to detect a potential heterogeneity of the mass distribution inside Phobos. We introduce mission instruments and operational strategies to meet the mission requirements. We present a preliminary imaging strategy from a quasi-satellite orbit for a base shape model that is expected to be created at the early stage of the mission. Geodetic products including ephemeris, gravity field, rotation parameter of Phobos, and spacecraft orbit are of importance not only for the geodetic study, but also for interpreting data from various mission instruments and selecting possible landing sites. Graphical Abstract

Periodic orbits in the Restricted Three-Body Problem are widely adopted as nominal trajectories by di↵erent missions. To maintain periodic orbits in a three-body regime, a stationkeeping strategy based on a high-order Target Point Approach (TPA) is proposed, where fuel-optimal and error-robust TPA parameters are acquired from stochastic global optimization. Accurate TPA maneuvers are calculated in a high-order fashion enabled by Di↵erential Algebra techniques. Orbit determination epoch is selected using a sensitivity analysis based on the convergence radius of a stroboscopic map. Stochasticity is handled by incorporating Monte Carlo simulations in the process of optimization and the evaluation of high-order ODE expansions is employed to supplant the time-consuming numerical integration. Two specific types of periodic orbits, Near Rectilinear Halo Orbits and Quasi-Satellite Orbits, are investigated to demonstrate the validity and eciency of the strategy.

For deep-space missions that remain in the vicinity of a target body, solar eclipses might become a source of major concern due to thermal and power constraints. This paper presents a method for minimizing the eclipse duration of periodic trajectories in three-body systems using synodic resonant periodic orbits. The proposed methodology successfully captures the global eclipse structure of synodic resonant periodic orbits in terms of driving parameters such as elongation and phase angles. Two-dimensional Eclipse maps are introduced to identify optimal orbit insertion conditions that avoid or minimize eclipse intervals. The validity and applicability of the proposed method is tested in the full-ephemeris model (DE430) of the Earth–Moon and Mars–Phobos systems, respectively. As a result of these investigations, we propose new science minimum-eclipse trajectories that are under consideration for the upcoming JAXA missions EQUULEUS and MMX. •Eclipses have the potential to endanger the safety and operations of space missions.•We derive relation between eclipse of periodic orbits and mission design parameters.•Minimum eclipse orbits can be achieved through the proposed graphical tool.•The validity of developments has been investigated and applied for two JAXA missions.

We present a small-scale mission concept to characterize the permanently shadowed regions of the lunar south pole. MARAUDERS aims to measure in situ for the first time the presence, distribution, and state of volatiles in one permanently shaded crater at a greater resolution than existing orbital measurements using up to 12 deployed impactors. A total of 15 permanently shadowed regions have been characterized as potential landing sites candidates for the probes. The science principle is based on penetrometry, that has proven in the past to be an efficient technique to estimate regolith properties from acceleration profiles. We demonstrate this concept by numerically simulating the surface interaction between our probes and the lunar regolith, thereby demonstrating how deceleration profiles can elucidate information on key regolith properties and help discriminate between two ice-regolith end-members. The preliminary payload design indicates that a good baseline for the impactors would be a spherical shell of 30–40 mm in size and ~90 g in mass per impactor, including electronics and the communication system. This would sum up to an overall payload of ~1 kg contained in a volume of ~15.10−4 m3, which is in agreement with a small-scale payload. Preliminary landing trajectory design enabled the computing of a nominal deployment scenario (with constraint on altitude, ejection velocity and spin rate) that would provide dispersions of the probes from ~250 m down to ~20 m if deployed from orbit, and down to ~10 m if deployed from a carrier lander/rover. Both scenarios will be able to comply with the MARAUDERS’ objectives to assess: (1) the presence (2) the distribution and (3) the surface strength heterogeneity (that can be traced back to the state of volatiles through lab experiments) of water-ice volatiles in permanently shadowed regions at a resolution

Numerical methods have proven to be reliable tools in generating long-term bounded relative trajectories in nontrivial dynamical environments. Yet, none of the existing procedures have a systematic approach to come up with all of the bounded relative trajectories associated with a variety of satellites in Earth orbit. In this paper, such a systematic procedure is developed based on the assumption that any trajectory, except at critical inclinations, is either periodic or quasi periodic in the four-dimensional Routh reduced system describing the motion of a mass particle about an axisymmetric body. This allows key design parameters to be identified, such as the nodal period and the node drift per nodal period, which are later used to constrain a differential corrector scheme looking for families of quasi-periodic invariant tori that yield bounded relative motion in the Earth-centered inertial frame. Numerical simulations show that the computed solutions are good candidates for cluster flight missions at a large range of altitude and inclination values.

Quasi-periodic invariant tori are of great interest in astrodynamics because of their capability to further expand the design space of satellite missions. However, there is no general consent on what is the best methodology for computing these dynamical structures. This paper compares the performance of four different approaches available in the literature. The first two methods compute invariant tori of flows by solving a system of partial differential equations via either central differences or Fourier techniques. In contrast, the other two strategies calculate invariant curves of maps via shooting algorithms: one using surfaces of section, and one using a stroboscopic map. All of the numerical procedures are tested in the co-rotating frame of the Earth as well as in the planar circular restricted three-body problem. The results of our numerical simulations show which of the reviewed procedures should be preferred for future studies of astrodynamics systems.

Upcoming missions towards remote planetary moons will fly in chaotic dynamical environments that are significantly perturbed by the oblateness of the host planet. Such a dominant perturbation is often neglected when designing spacecraft trajectories in planetary moon systems. This paper introduces a new time-periodic set of equations of motion that is based on the analytical solution of the zonal equatorial problem and better describes the dynamical evolution of a spacecraft subject to the gravitational attraction of a moon and its oblate host planet. Such a system, hereby referred to as the Zonal Hill Problem, remains populated by resonant periodic orbits and families of two-dimensional quasi-periodic invariant tori that are calculated by means of numerical continuation procedures. The resulting periodic and quasi-periodic trajectories are investigated for the trajectory design of future planetary moons explorers.

This paper presents the trajectory design for EQUilibriUm Lunar-Earth point 6U Spacecraft (EQUULEUS), which aims to demonstrate orbit control capability of CubeSats in the cislunar space. The mission plans to observe the far side of the Moon from an Earth-Moon L2 (EML2) libration point orbit. The EQUULEUS trajectory design needs to react to uncertainties of mission design parameters such as the launch conditions, errors, and thrust levels. The main challenge is to quickly design science orbits at EML2 and low-energy transfers from the post-deployment trajectory to the science orbits within the CubeSat’s limited propulsion capabilities. To overcome this challenge, we develop a systematic trajectory design approach that 1) designs over 13,000 EML2 quasi-halo orbits in a full-ephemeris model with a statistical stationkeeping cost evaluation, and 2) identifies families of low-energy transfers to the science orbits using lunar flybys and solar perturbations. The approach is successfully applied for the trajectory design of EQUULEUS.

Quasi-satellite orbits (QSOs) are stable retrograde orbits in the restricted three-body problem that have gained attention as a viable candidate for future deep-space missions towards remote planetary satellites. JAXA's robotic sample return mission MMX will utilize QSOs to perform scientific observations of the Martian moon Phobos before landing on its surface and attempt sample retrieval. The complex dynamical environment around Phobos makes the proximity operations of MMX quite challenging and requires novel and sophisticated techniques for maintaining and transferring between different quasi-satellite orbits. The present paper explores the application of invariant manifolds of unstable retrograde orbits to design transfer trajectories around Phobos. Starting from the equations of the Circular Hill Problem with ellipsoidal Phobos, we first compute families of three-dimensional QSOs using out-of-plane bifurca-tions near planar orbits. The feasibility of using unstable family members as staging orbits between high-altitude and low-altitude QSOs is later assessed. The final candidates are ranked based on MMX scientific requirements, transfer analyses, and station-keeping costs. It is found that intermediate 3D-QSOs can be maintained with as little as 1 m/s per month. Furthermore, it is discovered that transfer from high-altitude QSOs to low-altitude QSOs can be executed with a total ∆V of less than 40 m/s and total time of flight of less than 5 days.

Landing on Phobos and bringing samples from its surface would settle the debate on the origin of the Martian moons and support future manned exploration to Mars. To fulfill these scientific objectives, JAXA is planning to send a sample return probe to Phobos by the first half of the next decade, named the Martian Moons eXploration (MMX) mission. In order to explore scientifically interesting regions of Phobos, as well as to support the sampling operations of MMX, a number of Deployable CAMera 5 payloads are proposed to be deployed from quasi-satellite orbits (QSOs) around the Martian moon. This paper explores the feasibility of ballistic deployments from QSOs under realistic dynamical environment and surface constraints in order to guarantee surface settlement within the lifespan of DCAM5. First, we analyze the dynamical environment and escape speeds from Phobos by means of the Circular Hill Problem. Then, the surface coefficient of restitution is estimated by generic impacts onto Phobos regolith via discrete element method simulations. By combining these two analyses, maximum allowable impact velocities for surface settling are calculated and applied to downselect the number of feasible ballistic landings from QSOs. It is found that access to Phobos surface is possible especially along the leading and trailing sides of the Martian moon and in agreement with the engineering requirements of DCAM5.

The relative motion about 4179 Toutatis is studied in order to investigate the feasibility of formation flying as an alternative concept for future asteroid exploration missions. In particular, the existence of quasi-frozen orbits about slowly rotating bodies allows us to compute families of periodic orbits in the body-fixed frame of the asteroid. Since these periodic orbits are of the center×center type, quasi-periodic invariant tori are calculated via fully numerical procedures and used to initialize spacecraft formations about the central body. Numerical simulations show that the resulting in-plane and out-of-plane relative trajectories remain bounded over long time spans; i.e., more than 30 days. •We consider two spacecraft flying in a formation about the asteroid 4179 Toutatis.•We derive first order conditions that minimize the drift between the satellites.•We find a full family of stable periodic orbits about the target asteroid.•We compute quasi-periodic invariant tori and use them to initialize the formation.•Numerical simulations prove that the satellites stay bounded for long time spans.

The Martian Moons eXploration (MMX) mission will study the Martian moons Phobos and Deimos, Mars, and their environments. The mission scenario includes both landing on the surface of Phobos to collect samples and deploying a small rover for in situ observations. Engineering safeties and scientific planning for these operations require appropriate evaluations of the surface environment of Phobos. Thus, the mission team organized the Landing Operation Working Team (LOWT) and Surface Science and Geology Sub-Science Team (SSG-SST), whose view of the Phobos environment is summarized in this paper. While orbital and large-scale characteristics of Phobos are relatively well known, characteristics of the surface regolith, including the particle size-distributions, the packing density, and the mechanical properties, are difficult to constrain. Therefore, we developed several types of simulated soil materials (simulant), such as UTPS-TB (University of Tokyo Phobos Simulant, Tagish Lake based), UTPS-IB (Impact-hypothesis based), and UTPS-S (Simpler version) for engineering and scientific evaluation experiments.

Quasi-satellite or distant retrograde orbits are stable orbits in the restricted three-body problem will be used in remote planetary satellite missions like JAXA’s Martian Moons eXploration (MMX). Due to Phobos’ irregular gravity field, the proximity operations of the MMX mission require sophisticated techniques for transfers and maintenance between different quasi-satellite orbits. This paper focuses on the design of transfer trajectories between Quasi-Satellite Orbits (QSO) around Phobos. We utilize horizontal bifurcated Multi-Revolution Periodic QSOs (MP-QSOs) that are found in the vicinity of an ellipsoidal Phobos under the assumptions of the circular Hill Problem. Transfer strategy via MP-QSOs is explored through transfer maps that illustrate transfer design space between different altitudes QSOs. It is found that transfers via MP-QSOs provide insights on minimum ΔV transfers and the parameters determining the transfer cost. The proposed transfer method is explicitly applied to MMX baseline QSOs. This research has identified that transfers between high(100 × 200 km)-to-mid(50 × 100 km), mid-to-low(30 × 50 km), and lower altitude(

Motivated by the near-future re-exploration of 1 the cislunar space, this paper investigates dynamical substitutes of the Earth-Moon’s resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation of the Earth-Moon-Sun system. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. By making use of this higher-fidelity dynamical model, we are able to incorporate the Sun’s influence and the Moon’s eccentricity, two of the most significant perturbations of the cislunar environment. As a result of these perturbations, resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) are hereby replaced by two-dimensional quasi-periodic tori that better represent the dynamical evolution of satellites near the vicinity of the Moon. We present the steps and algorithms needed to compute these dynamical structures in the Elliptic-Circular model and subsequently assess their utility for spacecraft missions. We focus on the planned orbit for the NASA-led Lunar Gateway mission, a 9:2 synodic resonant L2 southern NRHO, as well as on the 4:1 synodic and 4:1 sidereal resonances, due to the proximity to the nominal orbit and their advantageous dynamical properties. We verify that the dynamical equivalents of these orbits preserve key dynamical attributes such as eclipse avoidance and near-linear stability. Furthermore, we find that the higher dimensionality of quasi-periodic solutions offers interesting alternatives to mission designers in terms of phasing maneuvers and low-altitude scientific observations.

The Martian Moons eXploration mission, currently under development by the Japan Aerospace Exploration Agency (JAXA), will be launched in 2024 with the goal of retrieving pristine samples from the surface of Phobos. Soon after arrival, the spacecraft will inject into retrograde relative trajectories known as quasi-satellite orbits and study the geophysical environment of the Martian moon for more than three years. This paper presents the orbit design and maintenance strategy of the Martian Moons eXploration mission in the framework of the elliptic Hill problem with ellipsoidal secondary. This paper first introduces a numerical continuation procedure on the eccentricity of Phobos to replace purely periodic solutions with families of quasi-periodic invariant tori. Two-dimensional torus maps can be then constructed and used to represent physical quantities of interest, as well as to generate reference trajectories at arbitrary epochs. Sensitivity and stability analyses are carried out to investigate the dynamic properties of retrograde relative trajectories in the elliptic case. Finally, a linear quadratic regulator is implemented in order to assess the robustness of the computed trajectories under injection, navigation, and execution errors. Monte Carlo simulations demonstrate that the baseline quasi-satellite orbits of the Martian Moons eXploration mission can be maintained with as low as 6.265 m/s per month.

Closed-loop feedback-driven control laws can be used to solve low-thrust many-revolution trajectory design and guidance problems with minimal computational cost. Lyapunov-based control laws offer the benefits of increased stability whilst their optimality can be increased by tuning their parameters. In this paper, a reinforcement learning framework is used to make the parameters of the Lyapunov-based Q-law state-dependent, increasing its optimality. The Jacobian of these state-dependent parameters is available analytically and, unlike in other optimisation approaches, can be used to enforce stability throughout the transfer. The results focus on GTO–GEO and LEO–GEO transfers in Keplerian dynamics, including the effects of eclipses. The impact of the network architecture on the behaviour is investigated for both time- and mass-optimal transfers. Robustness to navigation errors and thruster misalignment is demonstrated using Monte Carlo analyses. The resulting approach offers potential for on-board autonomous transfers and orbit reconfiguration.

Although recent numerical studies have demonstrated the possibility to obtain entire families of bounded relative trajectories in the Earth zonal problem, it is still unclear whether such initial conditions can cope with mismodeled dynamics such as atmospheric drag, luni-solar attraction, and solar radiation pressure. In the attempt to deal with the dynamical perturbations caused by the Earth's atmosphere, this paper offers a semi-analytical approach that combines previous work developed by the authors with newly found analytical relationships describing the effects of drag on the nodal and sidereal periods of a satellite. It is shown that both of these quantities change in a secular fashion due to the variations in the semi-major axis of spacecraft. Nevertheless, bounded relative motion can be guaranteed over the course of several days by either initializing the satellites on the same invariant curve of a stroboscopic mapping or by design of appropriate ballistic coefficients. These conclusions are supported by numerical simulations that prove the feasibility of our approach for spacecraft formations in low Earth orbit.