Two Periodic Models for the Earth-Moon System

Jorba-Cuscó, Marc; Farrés, Ariadna; Jorba, Àngel

doi:10.3389/fams.2018.00032

Cited by 19 publications

(6 citation statements)

References 39 publications

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“…Within the pulsatingrotating frame, it is trivially demonstrated that the ER3BP also admits an equilibrium L 2 point (to observe this, refer to Park and Howell (2024b)). Within the QBCP, the L 2 point evolves to a PO with three distinct lobes as determined within the pulsatingrotating frame and discussed in Andreu (1998); Jorba-Cuscó et al (2018); Rosales et al (2023). The HR4BP's L 2 counterpart also displays a periodic motion, but represented within the uniform-rotating frame as discussed in Olikara et al (2016); Henry et al (2023); Peterson et al (2023).…”

Section: Lagrange Pointmentioning

confidence: 95%

A Frequency-Based Hierarchy of Dynamical Models in Cislunar Space: Leveraging Periodically and Quasi-Periodically Perturbed Models

Park,

Sanaga,

Howell

2024

Preprint

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Understanding the applicability of different dynamical models within cislunar space is a useful yet challenging problem. The current work constructs a hierarchy of dynamical models leveraging a common rotating reference frame that accommodates multiple dynamical models. In such a formulation, the Higher-Fidelity Ephemeris Model (HFEM) is approximated as quasi-periodically perturbed model with multiple frequencies in addition to the autonomous Circular Restricted Three-Body Problem (CR3BP) dynamics. Then, different intermediate models are evaluated within the hierarchy in terms of the types and numbers of the incorporated frequencies. The analysis focuses on three main topics. Firstly, coherent solar gravity modeling is properly defined with analytical and visual criteria that serve as useful metrics in assessing intermediate models. Secondly, two periodically perturbed models, i.e., Quasi Bi-Circular Problem (QBCP) and Hill Restricted Four-Body Problem (HR4BP), are investigated within the common rotating reference frame. The capabilities and limitations of these two models within the hierarchy between the CR3BP and HFEM are examined, focusing on the similarities and differences in the models. Thirdly, novel quasi-periodically perturbed models are introduced. Leveraging the insight from Hill's lunar theory, two models are designed denoted as In-plane Quasi-Hill Restricted Four-Body Problem (I-QHR4BP) and Out-of-plane Quasi-Hill Restricted Four-Body Problem (O-QHR4BP). These two models selectively introduce the eccentricity between the Earth-Moon system and the inclination of the lunar orbit with respect to the Earth ecliptic plane, respectively, in addition to the HR4BP. The global characteristics of these new models are analyzed within the common rotating reference frame. The capabilities of multiple models within the model hierarchy are analyzed for two sample transition cases from the CR3BP to HFEM: L2 Lagrange point and the L2 9:2 synodic halo orbit.

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Section: Lagrange Pointmentioning

confidence: 95%

A Frequency-Based Hierarchy of Dynamical Models in Cislunar Space: Leveraging Periodically and Quasi-Periodically Perturbed Models

Park,

Sanaga,

Howell

2024

Preprint

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“…Focusing on the BCP and QBCP, it is interesting to mention that despite modeling the same system, there are qualitatively differences between these two models around L 2 . (See Jorba-Cuscó et al [2018] for a discussion.) Thus, it is important to address the question of why we chose the BCP over the QBCP to study the dynamics around L 2 .…”

Section: Introductionmentioning

confidence: 99%

“…In the QBCP the L 2 is replaced by a periodic orbit that is small in the sense that its maximal distance to L 2 is of the order of 10 −6 , and it has the same stability type of the L 2 point. See Andreu [2002], Jorba-Cuscó et al [2018] and references therein for the details.…”

Section: Introductionmentioning

confidence: 99%

Families of Halo-like invariant tori around $$L_2$$ in the Earth-Moon Bicircular Problem

Rosales

Jorba

Jorba-Cuscó

2021

Celest Mech Dyn Astr

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The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around L 2 in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that there are at least two different families of Halo-like quasi-periodic orbits around L 2 .

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“…The work done by Jorba-Cuscó et al [8,[72][73][74] for this problem, considers two more realistic dynamical models that take into account the gravitational effect of the Sun: the BCP and the QBCP models that include in both cases the SRP acceleration. In fact, including the Sun's gravity into the Earth-Moon-sail system does not increase the dynamical complexity of the model, since the SRP depends periodically on time with the same period as the Sun around the Earth-Moon system.…”

Section: State Of the Art 121 Solar Sailing And Hybrid Sail Propulsionmentioning

confidence: 99%

“…The QBCP is a coherent version of the BCP. According to [31,112], the BCP is more adequate for the study of the dynamics near the triangular libration points, while the QBCP is more suitable for the study of the dynamics near the collinear libration points. In both models, the two libration points of the RTBP under consideration (the L 1 and L 2 points) are substituted by small periodic orbits, usually called their dynamical substitutes.…”

Section: Remarks and Conclusionmentioning

confidence: 99%

Dynamics and control for continuous low-thrust spacecraft near collinear libration points

Gao

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Due to the rich dynamic properties of the equilibria of the Restricted Three Body Problem (RTBP), the libration point orbits (LPOs) around them are ideal locations for scientific missions. Fortunately, these invariant dynamical objects persist when adding an appropriate low-thrust propulsion, which opens additional opportunities for observation or communication missions. Moreover, the low-thrust propulsion can help to achieve active orbit control which are of great importance for advanced space missions. Focusing on the realm of collinear libration points, the main results of the dissertation are the following: Determination and analysis of displaced orbits of a hybrid-sail spacecraft around the L2 point in the Earth-Moon Elliptic RTBP. This dynamical model includes the eccentricities of the Earth-Moon and Sun-Earth/Moon systems, as well as the solar radiation pressure (SRP) acceleration. Using first-order approximations, the problem of generating displaced orbits is transformed into that of computing the particular solutions of some algebraic equations. The approximate analytical solutions are essentially quasi-periodic displaced orbits. Due to model errors and the inherent instability of the orbits, a backstepping station-keeping controller is proposed, while the reflectivity rate and orientation of the hybrid-sail spacecraft are optimized in such a way that the overall propellant consumption of the solar electric propulsion (SEP) system is minimized. Computation and station-keeping of resonant LPOs in the Earth-Moon Circular RTBP. The orbit and attitude dynamics of a high-fidelity hybrid-sail spacecraft are established, where the reflectivity control devices are taken as actuators. By using a parallel shooting method, combined with a continuation on the SRP acceleration, the families of resonant Lyapunov and halo orbits are investigated. To achieve high-precision station keeping and to minimize propellant consumption, a coupled orbit-attitude control strategy scheme is proposed. It is composed of three parts: an optimal periodic orbital controller, a SEP acceleration optimization and a robust backstepping attitude controller. In particular, nonlinear disturbance observers are incorporated into the orbit and the attitude control to enhance the robustness of the procedures in front of force and torque perturbations. Study of the influence of the SRP on the dynamics around the L1 and L2 dynamical substitutes in the Earth-Moon Quasi-Bicircular Problem (QBCP). Starting from the simplest invariant objects in the QBCP, i.e. the L1 and L2 dynamical substitutes, as well as the low order resonant orbits with the synodic period of the Sun, the evolution of the families of resonant periodic orbits is investigated when two sail parameters, defining its orientation and efficiency, are varied. The study shows an intricate web of connections between the families. As an important remark, it has been found that for some particular values of the parameters there exists periodic orbits that become stable under the influence of the SRP acceleration. Station-keeping of LPOs using low-thrust propulsion. Based on the dynamical properties of the phase space near LPOs, families of limit impulsive and dynamical reshaping control laws are derived. Despite following different approaches, the geometrical analysis shows that both strategies can reshape the dynamical structures about the LPOs. With the help of the Jet Transport technique, the control laws can be explicitly written as polynomials in terms of the deviation between the state of the spacecraft and the one of a nominal point, which opens the possibility of compact on-board implementation. The proposed control laws are geometrically intuitive, suitable to be implemented using a low-thrust propulsion device, and the fuel requirements of the laws are reasonable. The numerical results prove the robustness of the controls in front of measurement uncertainties. La memòria està dedicada a l'estudi de la dinàmica i el control de satèl·lits artificials en entorns de libració colineal considerant impuls feble. Els resultats més importants són els següents: 1. Determinació i anàlisi del desplaçament de l'òrbita d'un satèl·lit híbrid amb vela situat al voltant del punt L2 en el model Terra-Lluna el·líptic. Aquest model dinàmic inclou les excentricitats orbitals dels sistemes Terra-Lluna i Sol-Terra/Lluna, així com l'acceleració deguda a la pressió de radiació solar on la inclinació del pla orbital de la Lluna respecte de l'eclíptica es té en compte. El problema es formula usant la dinàmica lineal al voltant del punt L2, mentre que la propulsió elèctrica auxiliar s'utilitza per a estabilitzar l'òrbita. En el treball, el problema de la generació d'òrbites desplaçades es transforma en el càlcul de solucions d'unes equacions algebraiques. Les solucions analítiques aproximades de les equacions són bàsicament òrbites quasi periòdiques desplaçades, amb la distància fora del pla determinada per la reflectivitat de la vela i l'actitud. Es proposa també un controlador backstep que assegura un seguiment precís, mentre que la reflectivitat i orientació de la vela híbrida s'optimitza minimitzant la part de propulsió elèctrica. 2. Càlcul, manteniment en estació i control d'actitud d'òrbites periòdiques de libració ressonant en el RTBP circular Terra-Lluna. S'estableix l'òrbita i la dinàmica d'actitud d'un satèl·lit amb vela híbrida d'alta fidelitat, amb control per reflectivitat. Mitjançant tir paral·lel i continuació respecte l'acceleració induïda a la vela solar, s'investiguen famílies d'òrbites Lyapunov i halo ressonants. Degut a l'entorn tan pertorbat del sistema Terra-Lluna, cal un sistema auxiliar de propulsió elèctrica pel manteniment en estació. Per a aconseguir-hi precisió i per a minimitzar el consum de propulsor es proposa una estratègia acoblada de control orbital i d'actitud. Té tres parts: un controlador orbital periòdic optimal, una optimització d'acceleració provinent de la propulsió elèctrica i un controlador backstep robust per a l'actitud. En particular, a les simulacions s'hi han incorporat pertorbacions en les observacions i en les forces i moments pel control d'actitud, la qual cosa ressalta la robustesa del mètode. 3. Estudi de la influència de la pressió de radiació solar en la dinàmica dels substituts dinàmics de L1 i L2 en el problema quasi-bicircular Terra-Lluna. Aquest model és una pertorbació periòdica del RTBP que inclou l'efecte gravitacional del Sol i l'acceleració de la pressió de radiació solar en la vela del satèl·lit. Començant pells substituts dinàmics dels dos punts d'equilibri, així com òrbites periòdiques ressonants d'orbre baix amb el període sinòdic del Sol, s'investiga l'evolució d'aquestes òrbites variant dos paràmetres de la vela que defineixen la seva orientació i eficiència. L'estudi mostra una intricada teranyina de connexions entre les famílies. També s'ha trobat que, per certs valors dels paràmetres existeixen òrbites periòdiques estables amb la pressió de radiació. 4. Manteniment en estació d'òrbites de libració usant propulsió feble contínua. Usant propietats dinàmiques de l'espai de fases prop de les òrbites de libració es desenvolupen dos procediments de control per aquesta propulsió. La primera família s'obté com a límit de maniobres impulsives i la segona mitjançant la remodelació dels modes de Floquet. Tot i seguir diferents idees, l'anàlisi geomètrica dels dos procediments mostra que les estratègies poden remodelar les estructures dinàmiques al voltant de les òrbites de libració i estabilitzar el moviment. Usant tècniques de transport del jet, les lleis de control es poden escriure explícitament com a polinomis en termes de les desviacions entre l’estat del satèl·lit i el seu punt nominal a l’òrbita de referència, la qual cosa obra la possibilitat d’implementacions compactes que poden ser posades a bord del mateix satèl·lit. Les lleis de control que es proposen són geomètricament intuïtives, apropiades per a ser implementades amb dispositius d’impuls feble i els requeriments de propulsor són raonables. Els resultats numèrics proven la robustesa del mètode en front de determinació orbital afectada d’incertesa i d’un gran error d'injecció inicial. Aquesta metodologia proveeix també un marc general per a analitzar el comportament geomètric de qualsevol llei de control basada en impuls feble.

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Two Periodic Models for the Earth-Moon System

Cited by 19 publications

References 39 publications

A Frequency-Based Hierarchy of Dynamical Models in Cislunar Space: Leveraging Periodically and Quasi-Periodically Perturbed Models

A Frequency-Based Hierarchy of Dynamical Models in Cislunar Space: Leveraging Periodically and Quasi-Periodically Perturbed Models

Families of Halo-like invariant tori around $$L_2$$ in the Earth-Moon Bicircular Problem

Dynamics and control for continuous low-thrust spacecraft near collinear libration points

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