We investigate the existence of a symmetric singular periodic brake orbit in the equal mass, fully symmetric planar four body problem. Using regularized coordinates, we remove the singularity of binary collision for each symmetric pair. We use topological and symmetry tools in our investigation.

Orientador: Ernesto Vieira Neto / Banca: Othon Cabo Winter / Banca: Helio Koiti Kuga / Resumo: Com o lançamento do primeiro satélite artificial da Terra, Sputnik I, surgiu a necessidade do desenvolvimento de satélites mais eficientes e mais econômicos. Um dos mecanismos utilizados para economizar combustível numa transferência completa de um veículo espacial em órbita da Terra para uma órbita em torno da Lua, é o fenômeno de captura gravitacional temporária. Nesse trabalho é feita a análise numérica de diversas trajetórias em torno da Lua, considerando-se as dinâmicas de três e quatro corpos, com o objetivo de estudar o fenômeno da captura gravitacional temporária, através do monitoramento do sinal da energia relativa de dois corpos partícula-Lua e das componentes radiais das forças gravitacionais da Terra, da Lua e do Sol. Através desses estudos também foram obtidos diversos mapas de escape e colisão, considerando-se os movimentos prógrado e retrógrado. / Abstract: With the launch of the first artificial satellite of the Earth, Sputnik I, arose the necessity of the development of the satellites more efficient and more economic. One of the mechanisms used to save fuel in a complete transference of one spacecraft in orbit of the Earth to an orbit around the Moon, is the phenomena of the temporary gravitational capture. In this paper is made the numerical analysis of the several trajectories around the Moon, considering the dynamics of the three and four-bodies, with the objective of studying the phenomena of temporary gravitational capture, through monitoring the sign of the relative two-body energy particle-Moon and the radial component of the force of attraction, gravitational of the Earth, of the Moon and of the Sun. Though of these studies also were obtained several maps of the escape and collision, considering the prograde and retrograde movements. / Mestre

Spacecraft trajectory optimization is a topic of crucial importance to space missions design. The less fuel required to accomplish the mission, the more payload that can be transported, and the higher the opportunity to lower the cost of the space mission. The objective is to find the optimal trajectory through space that will minimize the fuel used, and still achieve all mission constraints. Most space trajectories are designed using the simplified relative two-body problem as the base model. Using this patched conics approximation, however, constrains the solution space and fails to produce accurate initial guesses for trajectories in sensitive dynamics. This dissertation uses the Circular Restricted Three-Body Problem (CR3BP) as the base model for designing transfer trajectories in the Circular Restricted Four-Body Problem (CR4BP). The dynamical behavior of the CR3BP guides the search for useful low-energy trajectory arcs. Two distinct models of the CR4BP are considered in this research: the Concentric model, and the Bi-Circular model. Transfers are broken down into trajectory arcs in two separate CR3BPs and the stable and unstable manifold structures of both systems are utilized to produce low-energy transfer arcs that are later patched together to form the orbit-to-orbit transfer. The patched solution is then used as an initial guess in the CR4BP model. A vital contribution of this dissertation is the sequential process for initial guess generation for transfers in the CR4BP. The techniques discussed in this dissertation overcome many of the difficulties in the trajectory design process presented by the complicated dynamics of the CR4BP. Indirect optimization techniques are also used to derive the first order necessary conditions for optimality to assure the optimality of the transfers and determine whether additional impulses might further lower the total cost of the mission. / text

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-12Bitstream added on 2014-06-13T20:33:10Z : No. of bitstreams: 1 peixoto_ln_me_guara.pdf: 13781791 bytes, checksum: 7047ea962d175dfc7039c195697ef84d (MD5) / Com o lançamento do primeiro satélite artificial da Terra, Sputnik I, surgiu a necessidade do desenvolvimento de satélites mais eficientes e mais econômicos. Um dos mecanismos utilizados para economizar combustível numa transferência completa de um veículo espacial em órbita da Terra para uma órbita em torno da Lua, é o fenômeno de captura gravitacional temporária. Nesse trabalho é feita a análise numérica de diversas trajetórias em torno da Lua, considerando-se as dinâmicas de três e quatro corpos, com o objetivo de estudar o fenômeno da captura gravitacional temporária, através do monitoramento do sinal da energia relativa de dois corpos partícula-Lua e das componentes radiais das forças gravitacionais da Terra, da Lua e do Sol. Através desses estudos também foram obtidos diversos mapas de escape e colisão, considerando-se os movimentos prógrado e retrógrado. / With the launch of the first artificial satellite of the Earth, Sputnik I, arose the necessity of the development of the satellites more efficient and more economic. One of the mechanisms used to save fuel in a complete transference of one spacecraft in orbit of the Earth to an orbit around the Moon, is the phenomena of the temporary gravitational capture. In this paper is made the numerical analysis of the several trajectories around the Moon, considering the dynamics of the three and four-bodies, with the objective of studying the phenomena of temporary gravitational capture, through monitoring the sign of the relative two-body energy particle-Moon and the radial component of the force of attraction, gravitational of the Earth, of the Moon and of the Sun. Though of these studies also were obtained several maps of the escape and collision, considering the prograde and retrograde movements.

<p dir="ltr"> With NASA's Artemis program and international collaborations focused on building a sustainable infrastructure for human exploration of the Moon, there is a growing demand for lunar exploration and complex spaceflight operations in cislunar space. However, designing efficient transfer trajectories between the Earth and the Moon remains complex and challenging. This investigation focuses on developing a dynamically informed framework for constructing low-energy transfers in the Earth-Moon-Sun Bicircular Restricted Four-body Problem (BCR4BP). Techniques within dynamical systems theory and numerical methods are exploited to construct transfers to various cislunar orbits. The analysis aims to contribute to a deeper understanding of the dynamical structures governing spacecraft motion. It addresses the characteristics of dynamical structures that facilitate the construction of propellant-efficient pathways between the Earth and the Moon, exploring periodic structures and energy properties from the Circular Restricted Three-body Problem (CR3BP) and BCR4BP. The investigation also focuses on constructing families of low-energy transfers by incorporating electric propulsion, i.e., low thrust, in an effort to reduce the time of flight and offer alternative transfer geometries. Additionally, the investigation introduces a process to transition solutions to the higher fidelity ephemeris force model to accurately model spacecraft motion through the Earth-Moon-Sun system. This research provides insights into constructing families of ballistic lunar transfers (BLTs) and cislunar low-energy flight paths (CLEFs), offering a foundation for future mission design and exploration of the Earth-Moon system.</p>

<p>An increasing interest in lunar exploration calls for low-cost techniques of reaching the Moon. Ballistic lunar transfers are long duration trajectories that leverage solar perturbations to reduce the multi-body energy of a spacecraft upon arrival into cislunar space. An investigation is conducted to explore methods of constructing ballistic lunar transfers. The techniques employ dynamical systems theory to leverage the underlying dynamical flow of the multi-body regime. Ballistic lunar transfers are governed by the gravitational influence of the Earth-Moon-Sun system; thus, multi-body gravity models are employed, i.e., the circular restricted three-body problem (CR3BP) and the bicircular restricted four-body problem (BCR4BP). The Sun-Earth CR3BP provides insight into the Sun’s effect on transfers near the Earth. The BCR4BP offers a coherent model for constructing end-to-end ballistic lunar transfers. Multiple techniques are employed to uncover ballistic transfers to conic and multi-body orbits in cislunar space. Initial conditions to deliver the spacecraft into various orbits emerge from Periapse Poincaré maps. From a chosen geometry, families of transfers from the Earth to conic orbits about the Moon are developed. Instantaneous equilibrium solutions in the BCR4BP provide an approximate for the theoretical minimum lunar orbit insertion costs, and are leveraged to create low-cost solutions. Trajectories to the <i>L</i>2 2:1 synodic resonant Lyapunov orbit, <i>L</i>2 2:1 synodic resonant Halo orbit, and the 3:1 synodic resonant Distant Retrograde Orbit (DRO) are investigated.</p>

<p>Secondary payloads have a rich and successful history of utilizing cheap rides to orbit to perform outstanding missions in Earth orbit, and more recently, in cislunar space and beyond. New launch vehicles, namely the Space Launch System (SLS), are increasing the science opportunity for rideshare class missions by providing regular service to the lunar vicinity. However, trajectory design in a multi-body regime brings a host of novel challenges, further exacerbated by constraints generated from the primary payload’s mission. Often, secondary payloads do not possess the fuel required to directly insert into lunar orbit and must instead perform a lunar flyby, traverse the Earth-Moon-Sun system, and later return to the lunar vicinity. This investigation develops a novel framework to construct low-cost, end-to-end lunar transfer trajectories for secondary payload missions. The proposed threephase approach provides unique insights into potential lunar transfer geometries. The phases consist of an arc from launch to initial perilune, an exterior transfer arc, and a lunar approach arc. The space of feasible transfers within each phase is determined through low-dimension grid searches and informed filtering techniques, while the problem of recombining the phases through differential corrections is kept tractable by reducing the dimensionality at each phase transition boundary. A sample mission demonstrates the trajectory design approach and example solutions are generated and discussed. Finally, alternate strategies are developed to both augment the analysis and for scenarios where the proposed three-phase technique does not deliver adequate solutions. The trajectory design methods described in this document are applicable to many upcoming secondary payload missions headed to lunar orbit, including spacecraft with only low-thrust, only high-thrust, or a combination of both. </p>

Στην παρούσα διατριβή ασχολούμαστε με την μελέτη περιοδικών και ασυμπτωτικών λύσεων στο περιορισμένο πρόβλημα των τεσσάρων σωμάτων. Πιο συγκεκριμένα: Στο κεφάλαιο 1 περιγράφουμε το πρόβλημα των τριών και των τεσσάρων σωμάτων, κάνοντας μια ιστορική αναδρομή και παραθέτουμε τις αρχικές εξισώσεις της κίνησης. Στο κεφάλαιο 2 μελετάμε αριθμητικά το περιορισμένο πρόβλημα των τεσσάρων σωμάτων, στην Lagrangian διαμόρφωση. Υπολογίζουμε τα σημεία ισορροπίας, καθώς και τις επιτρεπτές περιοχές κίνησης του τέταρτου σώματος. Στο κεφάλαιο 3 μελετάμε την ευστάθεια των σημείων ισορροπίας. Επίσης υπολογίζουμε και παρουσιάζουμε τις περιοχές έλξης, για το δυναμικό σύστημα των τεσσάρων σωμάτων. Στο κεφάλαιο 4 μελετάμε οικογένειες απλών συμμετρικών και μη συμμετρικών περιοδικών τροχιών του περιορισμένου προβλήματος των τεσσάρων σωμάτων. Υπολογίζουμε για κάθε περίπτωση τιμών των μαζών, σειρές κρίσιμων περιοδικών τροχιών κάθε οικογένειας ξεχωριστά. Τέλος στο κεφάλαιο 5 μελετάμε αριθμητικά οικογένειες απλών ασύμμετρων περιοδικών τροχιών στο περιορισμένο πρόβλημα των τεσσάρων σωμάτων, έχοντας θέσει ως πρωτεύοντα σώματα τους ΄Ηλιο - Δία και έναν Τρωικό Αστεροειδή και θεωρώντας ως τέταρτο αμελητέας μάζας σώμα ένα διαστημόπλοιο. Τα πρωτεύοντα σώματα υπακούουν στην ευσταθή Lagrangian τριγωνική διαμόρφωση. Μελετήσαμε επίσης αναλυτικά και αριθμητικά τις λύσεις στην περιοχή των ευσταθών σημείων ισορροπίας του συστήματος, βρήκαmε οικογένειες περιοδικών λύσεων και μελετήσαμε την γραμμική ευστάθεια τους. Τα αποτελέσματα των κεφαλαίων 2,3,4 και 5 έχουν δημοσιευτεί σε τρία διεθνή περιοδικά και ένα κομμάτι του κεφαλαίου 5 παρουσιάστηκε σε διεθνές συνέδριο (με συγγραφείς τους Μπαλταγιάννη Α. και Παπαδάκη Κ.). Πιο συγκεκριμένα η μελέτη των κεφαλαίων 2 και 3 έχει δημοσιευτεί στο περιοδικό “International Journal of Bifurcation and Chaos, 21, 2011, pp. 2179-2193” με τον τίτλο: “Equilibrium Points and their stability in the restricted four-body problem”. Τα αποτελέσματα του κεφαλαίου 4 δημοσιεύτηκαν mε τον τίτλο: “Families of periodic orbits in the restricted four-body problem” στο περιοδικό “Astrophysics and Space Science, 336, 2011, pp. 357-367”. Επίσης το κεφάλαιο 5 υπό τον τίτλο “Periodic solutions in the Sun - Jupiter - Trojan Asteroid - Spacecraft system”, δημοσιεύτηκε στο περιοδικό ”Planetary and Space Science, 75, 2013, pp. 148-157”. Το διεθνές συνέδριο στο οποίο παρουσιάστηκε τμήμα του κεφαλαίου 5 ήταν το : “10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina, Greece, 5-8 September 2011, pp. 23-24” και η εργασία είχε τίτλο: “Families of periodic orbits in the Sun - Jupiter - Trojan Asteroid system”. Η παρούσα διατριβή εκπονήθηκε με την οικονομική υποστήριξη του ερευνητικού προγράμματος του Πανεπιστημίου Πατρών: Κ. Καραθεοδωρή. / In this thesis we are concerned with the periodic and asymptotic solutions of the restricted four - body problem. In chapter 1 we describe the three - body and four - body problem, starting with historical information. We also present the needed equations of motion and integrals of the problem. In chapter 2 we study numerically the problem of four - bodies, according to the Lagrangian equilateral triangle configuration. We find the equilibrium points and the allowed regions of motion. In chapter 3 we study the stability of the relative equibrium solutions. We also illustrate the regions of the basins of attraction for the equilibrium points of the present dynamical model. In chapter 4 we present families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem. Series of critical periodic orbits of each family and in any case of the mass parameters are also calculated. In chapter 5 we study, numerically, families of simple non-symmetric periodic orbits of the restricted four-body problem, where we consider the three primary bodies as Sun, Jupiter and a Trojan Asteroid and as a massless fourth body, a spacecraft. The primary bodies are set in the stable Lagrangian equilateral triangle configuration. We also study analytically the solutions in the neighborhood of the stable equilibrium points and the linear stability of each periodic solution. The results of the chapters 2,3,4 and 5 have been published in three journals and a part of chapter 5 has been presented in an international conference. Chapters 2 and 3 have been published in “International Journal of Bifurcation and Chaos, 21, 2011, pp. 2179-2193” under the title of “Equilibrium Points and their stability in the restricted four-body problem”. Chapter 4 has been titled “Families of periodic orbits in the restricted four- body problem” and published in “Astrophysics and Space Science, 336, 2011, pp. 357-367”. Chapter 5 has been titled “Periodic solutions in the Sun - Jupiter - Trojan Asteroid - Spacecraft system,” and published in “Planetary and Space Science, 75, 2013, pp. 148-157”. The conference was the “10th Hellenic Astronomical Conference, Proceedings of the conference held at Ioannina, Greece, 5-8 September 2011, pp. 23-24” and part of the chapter 5 was presented under the title of “Families of periodic orbits in the Sun - Jupiter - Trojan Asteroid system”. This thesis was compiled while the author was in receipt of “K.Karatheodory” research grant.

Au cours des dernières décennies, l’étude de la dynamique autour des points de Lagrange des systèmes Terre-Lune (EMLi) et Terre-Soleil (SELi) a ouvert de nouvelles possibilités pour les orbites et les transferts spatiaux. Souvent modélisés comme des Problèmes à Trois Corps (CR3BP) distincts, ces deux systèmes ont également été combinés pour produire des trajectoiresà faible coût dans le système Terre-Lune-Soleil étendu. Cette approximation (PACR3BP) a permis de mettre en évidence un réseau à faible énergie de trajectoires (LEN) qui relie la Terre, la Lune, EML1,2 et SEL1,2. Cependant, pour chaque trajectoire calculée, le PACR3BP nécessite une connexion arbitraire entre les CR3BPs, ce qui complique son utilisation systématique. Cette thèse vise à mettre en place une modélisation à quatre corps non autonome pour l’étude du LEN basé sur un système Hamiltonien périodique cohérent, le Problème Quasi-Bicirculaire (QBCP). Tout d’abord, la Méthode de Paramétrisation est appliquée afin d’obtenir une représentation semi-analytique des variétés invariantes autour de chaque point de Lagrange. Une recherche systématique de connexions EML1,2-SEL1,2 peut alors être effectuée dans l’espace des paramètres : les conditions initiales sur la variété centrale-instable de EML1,2 sont propagées et les trajectoires résultantes sont projetées sur la variété centrale de SEL1,2 . Un transfert est détecté lorsque la distance de projection est proche de zéro. Les familles de transfert obtenues sont corrigées dans un modèle newtonien haute-fidélité du système solaire. La structure globale des connections est largement préservée et valide l’utilisation du QBCP comme modèle de base du LEN. / In recent decades, the dynamics about the libration points of the Sun-Earth (SELi) and Earth-Moon (EMLi ) systems have been increasingly studied and used, both in terms of transfer trajectory computation and nominal orbit design. Often seen as two distinct Circular Restricted Three Body Problems (CR3BP), both systems have also been combined to produce efficient transfers in the Sun-Earth-Moon system. This patched CR3BP approximation (PACR3BP) allowed to uncover a low-energy network (LEN) of trajectories that interconnect the Earth, the Moon, EML1,2 and SEL1,2 . However, for every computed trajectory, the PACR3BP requires an arbitrary connection between the CR3BPs, which limits its use in a systematic tool. This thesis introduces a single non-autonomous four-body framework for the study of the LEN based on a coherent periodically-forced Hamiltonian system, the Quasi-Bicircular Problem (QBCP). First, the Parameterization Method is applied in order to obtain high-order, periodic, semi-analytical parameterizations of the invariant manifolds about each libration point. A systematic search for EML1,2 -SEL1,2 connections can then be performed in the parameterization space: initial conditions on the center-unstable manifold at EML1,2 are propagated and projected on the center manifold at SEL1,2. A transfer is found each time that the distance of projection is close to zero. These trajectories are refined as solutions of a Boundary Value Problem, which uncover families of natural transfers, later transitioned into a higher-fidelity model. The global structure of the connecting orbits is largely preserved, which validates the QBCP as a relevant model for the LEN.

<p> </p> <p>In the coming decades, numerous missions plan to exploit multi-body orbits for operations. Given the complex nature of multi-body systems, trajectory designers must possess effective tools that leverage aspects of the dynamical environment to streamline the design process and enable these missions. In this investigation, a particular class of dynamical structures, quasi-periodic orbits, are examined. This work summarizes a computational framework to construct quasi-periodic orbits and a design framework to leverage quasi-periodic motion within the path planning process. First, quasi-periodic orbit computation in the Circular Restricted Three-Body Problem (CR3BP) and the Bicircular Restricted Four-Body Problem (BCR4BP) is summarized. The CR3BP and BCR4BP serve as preliminary models to capture fundamental motion that is leveraged for end-to-end designs. Additionally, the relationship between the Earth-Moon CR3BP and the BCR4BP is explored to provide insight into the effect of solar acceleration on multi-body structures in the lunar vicinity. Characterization of families of quasi-periodic orbits in the CR3BP and BCR4BP is also summarized. Families of quasi-periodic orbits prove to be particularly insightful in the BCR4BP, where periodic orbits only exist as isolated solutions. Computation of three-dimensional quasi-periodic tori is also summarized to demonstrate the extensibility of the computational framework to higher-dimensional quasi-periodic orbits. Lastly, a design framework to incorporate quasi-periodic orbits into the trajectory design process is demonstrated through a series of applications. First, several applications were examined for transfer design in the vicinity of the Moon. The first application leverages a single quasi-periodic trajectory arc as an initial guess to transfer between two periodic orbits. Next, several quasi-periodic arcs are leveraged to construct transfer between a planar periodic orbit and a spatial periodic orbit. Lastly, transfers between two quasi-periodic orbits are demonstrated by leveraging heteroclinic connections between orbits at the same energy. These transfer applications are all constructed in the CR3BP and validated in a higher-fidelity ephemeris model to ensure the geometry persists. Applications to ballistic lunar transfers are also constructed by leveraging quasi-periodic motion in the BCR4BP. Stable manifold trajectories of four-body quasi-periodic orbits supply an initial guess to generate families of ballistic lunar transfers to a single quasi-periodic orbit. Poincare mapping techniques are used to isolate transfer solutions that possess a low time of flight or an outbound lunar flyby. Additionally, impulsive maneuvers are introduced to expand the solution space. This strategy is extended to additional orbits in a single family to demonstrate "corridors" of transfers exist to reach a type of destination motion. To ensure these transfers exist in a higher fidelity model, several solutions are transitioned to a Sun-Earth-Moon ephemeris model using a differential corrections process to show that the geometries persist.</p>