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Dr Nicola Baresi

Lecturer in Orbital Mechanics
+44 (0)1483 688272
30 BA 01

Academic and research departments

Surrey Space Centre.



Research interests

My publications


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.

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.

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.

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.

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.

Nicola Baresi, Daniel J Scheeres (2017)Bounded relative motion under zonal harmonics perturbations, In: Celestial mechanics and dynamical astronomy127(4)pp. 527-548 Springer
Nishanth Pushparaj, Nicola Baresi, Kento Ichinomiya, Yasuhiro Kawakatsu (2021)Transfers around Phobos via bifurcated retrograde orbits: Applications to Martian Moons eXploration mission, In: Acta Astronautica181pp. 70-80 Elsevier Ltd

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(

Kenshiro Oguri, Kenta Oshima, Stefano Campagnola, Kota Kakihara, Naoya Ozaki, Nicola Baresi, Yasuhiro Kawakatsu, Ryu Funase (2020)EQUULEUS Trajectory Design, In: The Journal of the astronautical sciences67(3)pp. 950-976 Springer

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.

Nicola Baresi, Daniel J Scheeres, Hanspeter Schaub (2016)Bounded relative orbits about asteroids for formation flying and applications, In: Acta astronautica123pp. 364-375 Elsevier Ltd

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.

Onur Çelik, Nicola Baresi, Ronald-Louis Ballouz, Kazunori Ogawa, Koji Wada, Yasuhiro Kawakatsu (2019)Ballistic deployment from quasi-satellite orbits around Phobos under realistic dynamical and surface environment constraints, In: Planetary and space science178104693 Elsevier Ltd

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.

Nicola Baresi, Daniel J Scheeres (2017)Design of Bounded Relative Trajectories in the Earth Zonal Problem, In: Journal of guidance, control, and dynamics40(12)pp. 3075-3087 American Institute of Aeronautics and Astronautics

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.

Nicola Baresi, Zubin Olikara, Daniel Scheeres (2018)Fully Numerical Methods for Continuing Families of Quasi-Periodic Invariant Tori in Astrodynamics, In: The Journal of the astronautical sciences65(2)pp. 157-182 Springer US

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.

Nicola Baresi, Daniel J Scheeres (2018)Drag-perturbed bounded relative trajectories in low Earth orbit: A semi-analytical approach, In: Acta astronautica153pp. 229-239 Elsevier Ltd

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.

NICOLÒ BERNARDINI, NICOLA BARESI, Roberto Armellin, Steve Eckersley (2020)Feasibility Study of Sun Occultation Missions Using Natural Bodies

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.

L Riu, R.-L Ballouz, S Van wal, O Çelik, N Baresi, R Boden, S Crites (2020)MARAUDERS: A mission concept to probe volatile distribution and properties at the lunar poles with miniature impactors, In: Planetary and space science189104969 Elsevier Ltd

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 ​

HARRY JAMES HOLT, Roberto Armellin, NICOLA BARESI, Yoshi Hashida, Andrea Turconi, Andrea Scorsoglio, Roberto Furfaro (2021)Optimal Q-laws via reinforcement learning with guaranteed stability, In: Acta astronautica187pp. 511-528 Elsevier

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.


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.

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.

Takuya Chikazawa, Nicola Baresi, Stefano Campagnola, Naoya Ozaki, Yasuhiro Kawakatsu (2021)Minimizing eclipses via synodic resonant orbits with applications to EQUULEUS and MMX, In: Acta Astronautica180pp. 679-692 Elsevier Ltd

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.

Nishanth Pushparaj, Nicola Baresi, Yasuhiro Kawakatsu (2020)Design of Transfer Trajectories Between Planar and Spatial Quasi-Satellite Orbits, In: AIAA Scitech 2020 Forum

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.


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.

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.

Diogene Alessandro Dei Tos, Nicola Baresi (2020)Genetic optimization for the orbit maintenance of libration point orbits with applications to EQUULEUS and LUMIO, In: AIAA Scitech 2020 Forum

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.

Nicola Baresi, Diogene A Dei Tos, Hitoshi Ikeda, Yasuhiro Kawakatsu (2020)Trajectory Design and Maintenance of the Martian Moons eXploration Mission Around Phobos, In: Journal of guidance, control, and dynamicspp. 1-12 American Institute of Aeronautics and Astronautics

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.

Nicola Baresi, Lamberto Dell'Elce, Josué Cardoso dos Santos, Yasuhiro Kawakatsu (2020)Long-term Evolution of Mid-altitude Quasi-satellite Orbits, In: Nonlinear dynamics99(4)pp. 2743-2763 Springer Verlag