http://hdl.handle.net/2117/3230 Sun, 26 May 2013 05:11:17 GMT 2013-05-26T05:11:17Z webmaster.bupc@upc.edu Universitat Politècnica de Catalunya. Servei de Biblioteques i Documentació no http://hdl.handle.net/2117/17854 Title: Fast numerical computation of Lissajous and quasi-halo libration point trajectories Authors: Mondelo, Josep Maria; Barrabes, E.; Gómez Muntané, Gerard; Ollé Torner, Mercè Abstract: In this paper we present a methodology for the automatic generation of quasi–periodic libration point trajectories (Lissajous and quasi–halo) of the Spatial, Circular Restricted Three–Body Problem. This methodology is based on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate quantitative representation of the full set of libration point orbits. This representation, when combined with the one obtained using Poincar´e maps, provides a useful tool for the design of missions to libration points fulfilling specific requirements. The same methodology applies to stable and unstable manifolds as well. This paper extends and improves results presented in [10]. Mon, 18 Feb 2013 15:32:39 GMT http://hdl.handle.net/2117/17854 2013-02-18T15:32:39Z Mondelo, Josep Maria; Barrabes, E.; Gómez Muntané, Gerard; Ollé Torner, Mercè no In this paper we present a methodology for the automatic generation of quasi–periodic libration point trajectories (Lissajous and quasi–halo) of the Spatial, Circular Restricted Three–Body Problem. This methodology is based on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate quantitative representation of the full set of libration point orbits. This representation, when combined with the one obtained using Poincar´e maps, provides a useful tool for the design of missions to libration points fulfilling specific requirements. The same methodology applies to stable and unstable manifolds as well. This paper extends and improves results presented in [10]. http://hdl.handle.net/2117/17850 Title: Detecting invariant manifolds using Lagrangian coherent structures Authors: Pérez, Daniel; Gómez Muntané, Gerard; Masdemont Soler, Josep Abstract: Using as reference test model the Planar Circular Restricted Three Body Problem, this paper explores its Lagrangian Coherent Structures, as well as its Hyperbolic Lagrangian Coherent Structures. The purpose is to identify stable and unstable manifolds acting as separatrices between orbits with different qualitative behaviour and, therefore, relevant to the dynamics of the problem. Particular attention is given to the manifolds associated to the collinear libration points and to the practical stability regions around the triangular equilibrium points Mon, 18 Feb 2013 15:16:28 GMT http://hdl.handle.net/2117/17850 2013-02-18T15:16:28Z Pérez, Daniel; Gómez Muntané, Gerard; Masdemont Soler, Josep no Using as reference test model the Planar Circular Restricted Three Body Problem, this paper explores its Lagrangian Coherent Structures, as well as its Hyperbolic Lagrangian Coherent Structures. The purpose is to identify stable and unstable manifolds acting as separatrices between orbits with different qualitative behaviour and, therefore, relevant to the dynamics of the problem. Particular attention is given to the manifolds associated to the collinear libration points and to the practical stability regions around the triangular equilibrium points http://hdl.handle.net/2117/17849 Title: A note on the dynamics around the L1,2 Lagrange points of the Earth-Moon system in a complete solar system model Authors: Tang, Guojian; Gomez Muntaner, Gerard; Masdemont Soler, Josep; Yijun, L. Abstract: The purpose of this paper is the study of the phase space around the collinear libration points L1 and L2 of the Earth–Moon system when the gravitational effects of the remaining bodies of the Solar System are taken into account. In the simplified model defined by the Circular Restricted Three Body Problem (CR3BP), the description of the phase space around these two points has already been done in the past, either using semi-analytical techniques1 or numerical ones.2 When using more realistic models of motion, the refinement of the different kinds of libration point orbits around both points (see G´omez et al.3) has some problems when the time interval used is large. These problems are more evident for the Earth–Moon L2 point due to a 1:2 resonance between the natural frequency of some halo orbits (!h) and the external frequency due to the perturbation of the Sun (!s); in fact !h ' 2!s for some orbits of the halo family of periodic orbits around this point. To analyse more closely this fact, Andreu4 introduced an intermediate model between the CR3BP and the restricted n-body problem, which is the so called Quasi-Bicircular Problem. It is a restricted four body problem in which the three primaries move following a “true” solution of the three body problem along orbits close to circular (the Earth and the Moon around their barycenter, and the Earth-Moon barycenter around the Sun). The description of the different types of orbits in the neighbourhood of L2 is also done in Reference 4 by means of the reduction of the Hamiltonian of the problem to the central manifold. The reduced Hamiltonian is then studied by means of Poincar´e maps at different energy levels. Mon, 18 Feb 2013 15:11:57 GMT http://hdl.handle.net/2117/17849 2013-02-18T15:11:57Z Tang, Guojian; Gomez Muntaner, Gerard; Masdemont Soler, Josep; Yijun, L. no The purpose of this paper is the study of the phase space around the collinear libration points L1 and L2 of the Earth–Moon system when the gravitational effects of the remaining bodies of the Solar System are taken into account. In the simplified model defined by the Circular Restricted Three Body Problem (CR3BP), the description of the phase space around these two points has already been done in the past, either using semi-analytical techniques1 or numerical ones.2 When using more realistic models of motion, the refinement of the different kinds of libration point orbits around both points (see G´omez et al.3) has some problems when the time interval used is large. These problems are more evident for the Earth–Moon L2 point due to a 1:2 resonance between the natural frequency of some halo orbits (!h) and the external frequency due to the perturbation of the Sun (!s); in fact !h ' 2!s for some orbits of the halo family of periodic orbits around this point. To analyse more closely this fact, Andreu4 introduced an intermediate model between the CR3BP and the restricted n-body problem, which is the so called Quasi-Bicircular Problem. It is a restricted four body problem in which the three primaries move following a “true” solution of the three body problem along orbits close to circular (the Earth and the Moon around their barycenter, and the Earth-Moon barycenter around the Sun). The description of the different types of orbits in the neighbourhood of L2 is also done in Reference 4 by means of the reduction of the Hamiltonian of the problem to the central manifold. The reduced Hamiltonian is then studied by means of Poincar´e maps at different energy levels. http://hdl.handle.net/2117/17714 Title: Capturing small asteroids into Sun-Earth Lagrangian points for mining purposes Authors: Lladó, Neus; Ren, Yuan; Masdemont Soler, Josep; Gomez Muntaner, Gerard Abstract: The aim of this paper is to study the capture of small Near Earth Objects (NEOs) into the Sun-Earth L2 using low-thrust propulsion for mining or science purposes. As it is well known, the vicinity of these points is inside a net of dynamical channels suitable for the transport in the Earth-Moon neighborhood, so different final destinations from here could be easily considered. Asteroids with very small mass and not representing a potential hazard are analyzed. An initial pruning of asteroids is made, considering NEOs with stellar magnitude bigger than 28, which are the smallest available, and NEOs close to the Earth orbit with semi-major axis between 0.85-1.15. Due to the difficult determination of their physical properties, two methods to estimate the asteroid masses are conducted. A procedure to find the low-thrust optimization trajectories has been implemented. The initial seed is obtained integrating forward the equations of motion plus its conjugated equations expressed in cartesian coordinates and applying the Pontryagins maximum principle to obtain the optimal control with a switching function for the thrust. To refine the trajectory a 4 order Runge-Kutta shooting method has been used. The objective function in this study is the fuel consumption. Finally, the capable asteroids to get captured by a low-thrust engine have been listed indicating the main parameters. Wed, 13 Feb 2013 11:39:10 GMT http://hdl.handle.net/2117/17714 2013-02-13T11:39:10Z Lladó, Neus; Ren, Yuan; Masdemont Soler, Josep; Gomez Muntaner, Gerard no The aim of this paper is to study the capture of small Near Earth Objects (NEOs) into the Sun-Earth L2 using low-thrust propulsion for mining or science purposes. As it is well known, the vicinity of these points is inside a net of dynamical channels suitable for the transport in the Earth-Moon neighborhood, so different final destinations from here could be easily considered. Asteroids with very small mass and not representing a potential hazard are analyzed. An initial pruning of asteroids is made, considering NEOs with stellar magnitude bigger than 28, which are the smallest available, and NEOs close to the Earth orbit with semi-major axis between 0.85-1.15. Due to the difficult determination of their physical properties, two methods to estimate the asteroid masses are conducted. A procedure to find the low-thrust optimization trajectories has been implemented. The initial seed is obtained integrating forward the equations of motion plus its conjugated equations expressed in cartesian coordinates and applying the Pontryagins maximum principle to obtain the optimal control with a switching function for the thrust. To refine the trajectory a 4 order Runge-Kutta shooting method has been used. The objective function in this study is the fuel consumption. Finally, the capable asteroids to get captured by a low-thrust engine have been listed indicating the main parameters. http://hdl.handle.net/2117/17711 Title: Methodology and estimation costs for proximity maneuvering spacecraft formations in the vicinity of libration points Authors: Garcia Taberner, Laura; Masdemont Soler, Josep Abstract: The aim of this paper is to study and provide estimations of the cost of reconfiguration maneuvers in the neighbor- hood of L1 and L2 libration points of the Sun-Earth system. The problem is considered as a function of the parameters of the formation (baseline length, security distances, orientation, reconfiguration time) and simple formulae to evaluate the costs are given. Wed, 13 Feb 2013 11:29:45 GMT http://hdl.handle.net/2117/17711 2013-02-13T11:29:45Z Garcia Taberner, Laura; Masdemont Soler, Josep no The aim of this paper is to study and provide estimations of the cost of reconfiguration maneuvers in the neighbor- hood of L1 and L2 libration points of the Sun-Earth system. The problem is considered as a function of the parameters of the formation (baseline length, security distances, orientation, reconfiguration time) and simple formulae to evaluate the costs are given. http://hdl.handle.net/2117/17710 Title: The zero, minimum and maximum relative radial accelerations along a family of periodic orbits around L4 in the Earth-Moon system Authors: Salazar, F.J.T.; Masdemont Soler, Josep; Gómez Muntané, Gerard; Macau, E.E.N.; Winter, O.C. Abstract: Assume a constellations of satellites flighting close a given nominal trajectory around L4 or L5 in the Earth-Moon system in such a way that there is freedom in the selection of the geometry of the constellation. We are interested in to avoid large variations of the mutual distance between the spacecrafts. In this case, the possible existence of regions of zero relative radial acceleration with respect to the nominal trajectory will prevent from the expansion or contraction of the constellation. The goal of this paper is the study of theses regions. Description: Constellations of satellites, Zero relative radial acceleration, Earth-Moon. Wed, 13 Feb 2013 11:14:00 GMT http://hdl.handle.net/2117/17710 2013-02-13T11:14:00Z Salazar, F.J.T.; Masdemont Soler, Josep; Gómez Muntané, Gerard; Macau, E.E.N.; Winter, O.C. no Assume a constellations of satellites flighting close a given nominal trajectory around L4 or L5 in the Earth-Moon system in such a way that there is freedom in the selection of the geometry of the constellation. We are interested in to avoid large variations of the mutual distance between the spacecrafts. In this case, the possible existence of regions of zero relative radial acceleration with respect to the nominal trajectory will prevent from the expansion or contraction of the constellation. The goal of this paper is the study of theses regions. http://hdl.handle.net/2117/16639 Title: From b-Poisson manifolds to symplectic mapping tori and back Authors: Miranda Galcerán, Eva; Guillemin, Victor; Pissarra Pires, Ana Rita Thu, 04 Oct 2012 11:14:26 GMT http://hdl.handle.net/2117/16639 2012-10-04T11:14:26Z Miranda Galcerán, Eva; Guillemin, Victor; Pissarra Pires, Ana Rita no http://hdl.handle.net/2117/15892 Title: Input observability analysis of Fixed speed wind turbine Authors: García Planas, María Isabel Abstract: This paper deals with the concept of input observability in a fixed speed wind turbine. A linear system has been calculated from the nonlinear equations of the squirrel cage induction generator, supposing it connected directly to the grid and assuming a steady state operating point. The observability of the input from the output of the system could be an interesting way to know if its possible to develop some new controls without introduce special sensors in the system. Furthermore, it is interesting to analyse which is the parameter variation margin of the wind turbine from input-observable state to non-input observable, in order to obtain some restrictions to design future controllers, or limit the operating points Fri, 18 May 2012 10:24:39 GMT http://hdl.handle.net/2117/15892 2012-05-18T10:24:39Z García Planas, María Isabel no This paper deals with the concept of input observability in a fixed speed wind turbine. A linear system has been calculated from the nonlinear equations of the squirrel cage induction generator, supposing it connected directly to the grid and assuming a steady state operating point. The observability of the input from the output of the system could be an interesting way to know if its possible to develop some new controls without introduce special sensors in the system. Furthermore, it is interesting to analyse which is the parameter variation margin of the wind turbine from input-observable state to non-input observable, in order to obtain some restrictions to design future controllers, or limit the operating points http://hdl.handle.net/2117/15696 Title: Output controllability analysis of fixed speed wind turbine Authors: Domínguez García, José Luís; García Planas, María Isabel Abstract: This paper deals with the concepts of output controllability and steady output controllability, it demonstrates that they are not equivalent of concepts. A linear system has been calculated from the nonlinear equations of the squirrel cage induction generator, supposing it connected directly to the grid and assuming a steady state operating point. The study of output controllability and steady-output controllability concepts of the introduced system is done. Mon, 02 Apr 2012 10:31:41 GMT http://hdl.handle.net/2117/15696 2012-04-02T10:31:41Z Domínguez García, José Luís; García Planas, María Isabel no This paper deals with the concepts of output controllability and steady output controllability, it demonstrates that they are not equivalent of concepts. A linear system has been calculated from the nonlinear equations of the squirrel cage induction generator, supposing it connected directly to the grid and assuming a steady state operating point. The study of output controllability and steady-output controllability concepts of the introduced system is done. http://hdl.handle.net/2117/15612 Title: Learning engineering to teach mathematics Authors: Ferrer Llop, Josep; Peña Carrera, Marta; Ortiz, Carmen Abstract: The Bologna process is a good opportunity to bring together first-year mathematics courses of engineering degrees and technology courses offered in subsequent years. In fact, the Faculty Council has decided that 20% of the credits from basic courses must be related to technological applications. To this end, during the past academic year a mathematical engineering seminar was held with each session dealing with one technological discipline. The main goal of the seminar, which relied on the presence of speakers from both mathematics and engineering departments, was to identify the most commonly used mathematical tools. Furthermore, a set of exercises and some guidelines addressed to faculty lacking an engineering background were created. Here, we present some of this material: first, a summary of the collection of exercises illustrating the use of Linear Algebra in different engineering areas such as Mechanical Engineering, Control and Automation, and second, some exercises Mon, 19 Mar 2012 11:27:02 GMT http://hdl.handle.net/2117/15612 2012-03-19T11:27:02Z Ferrer Llop, Josep; Peña Carrera, Marta; Ortiz, Carmen no The Bologna process is a good opportunity to bring together first-year mathematics courses of engineering degrees and technology courses offered in subsequent years. In fact, the Faculty Council has decided that 20% of the credits from basic courses must be related to technological applications. To this end, during the past academic year a mathematical engineering seminar was held with each session dealing with one technological discipline. The main goal of the seminar, which relied on the presence of speakers from both mathematics and engineering departments, was to identify the most commonly used mathematical tools. Furthermore, a set of exercises and some guidelines addressed to faculty lacking an engineering background were created. Here, we present some of this material: first, a summary of the collection of exercises illustrating the use of Linear Algebra in different engineering areas such as Mechanical Engineering, Control and Automation, and second, some exercises http://hdl.handle.net/2117/14651 Title: Maximum IR-drop in On-Chip Power Distribution Networks of Wire-Bonded Integrated Circuits Authors: Rius Vázquez, José; Aguareles Carrero, María Abstract: A compact IR-drop model for on-chip power distribution networks in wire-bonded ICs is presented. Chip dimensions, metal coverage and piecewise distribution of the IC consumption are taken into account to obtain closed form expressions for the maximum IR-drop as well as its place. Comparison with simulations shows an error as small as 2% in most the cases. Wed, 18 Jan 2012 12:46:01 GMT http://hdl.handle.net/2117/14651 2012-01-18T12:46:01Z Rius Vázquez, José; Aguareles Carrero, María no A compact IR-drop model for on-chip power distribution networks in wire-bonded ICs is presented. Chip dimensions, metal coverage and piecewise distribution of the IC consumption are taken into account to obtain closed form expressions for the maximum IR-drop as well as its place. Comparison with simulations shows an error as small as 2% in most the cases. http://hdl.handle.net/2117/12453 Title: Aproximación del problema de Stokes mediante elementos finitos mixtos de tipo cross-grid Authors: Blasco Lorente, Jorge; Armentano, María G. Abstract: En esta comunicación introducimos una familia de métodos de elementos finitos mixtos para la resolución numérica del problema de Stokes en dimensión 2. En estos métodos, la presión se interpola en una malla de elementos cuadriláteros, mientras que la velocidad se interpola en una malla de elementos triangulares obtenida subdividiendo cada cuadrilátero en cuatro triángulos por sus diagonales. Se consideran entonces interpolaciones de grado k para las velocidades y grado l para la presión, siendo k ≥ l ≥ 1. Por todo ello, estos elementos se han denominado de tipo cross-grid PkQl (ver [3]). Se presenta un análisis numérico de la estabilidad de estos métodos para elementos rectangulares, basado en la técnica de los macroelementos de Stenberg (ver [4], [5], [6]), y se analizan en particular los casos de orden menor, P1Q1 y P2Q1. En el primer caso, se demuestra la existencia de un modo espurio global para la presión, de manera que este elemento no es estable. En el segundo caso se demuestra la estabilidad del método, y por tanto su convergencia óptima. Se presentan también resultados numéricos obtenidos con estos elementos en varios casos test, tanto con mallas de elementos rectangulares como de cuadriláteros generales. Dichos resultados confirman la existencia del modo espurio de presión para el elemento P1Q1 y la estabilidad del elemento P2Q1. Tue, 03 May 2011 11:53:31 GMT http://hdl.handle.net/2117/12453 2011-05-03T11:53:31Z Blasco Lorente, Jorge; Armentano, María G. no Stokes, Metodos Mixtos En esta comunicación introducimos una familia de métodos de elementos finitos mixtos para la resolución numérica del problema de Stokes en dimensión 2. En estos métodos, la presión se interpola en una malla de elementos cuadriláteros, mientras que la velocidad se interpola en una malla de elementos triangulares obtenida subdividiendo cada cuadrilátero en cuatro triángulos por sus diagonales. Se consideran entonces interpolaciones de grado k para las velocidades y grado l para la presión, siendo k ≥ l ≥ 1. Por todo ello, estos elementos se han denominado de tipo cross-grid PkQl (ver [3]). Se presenta un análisis numérico de la estabilidad de estos métodos para elementos rectangulares, basado en la técnica de los macroelementos de Stenberg (ver [4], [5], [6]), y se analizan en particular los casos de orden menor, P1Q1 y P2Q1. En el primer caso, se demuestra la existencia de un modo espurio global para la presión, de manera que este elemento no es estable. En el segundo caso se demuestra la estabilidad del método, y por tanto su convergencia óptima. Se presentan también resultados numéricos obtenidos con estos elementos en varios casos test, tanto con mallas de elementos rectangulares como de cuadriláteros generales. Dichos resultados confirman la existencia del modo espurio de presión para el elemento P1Q1 y la estabilidad del elemento P2Q1. http://hdl.handle.net/2117/11562 Title: Spacecraft formations reconfiguration using finite element methodology with adaptive remeshing Authors: Masdemont Soler, Josep; Garcia Taberner, Laura Abstract: We present a methodology to compute trajectories for formation °ying recon¯g- uration problems using a methodology based on ¯nite elements: FEFF (Finite Elements for Formation Flying). In this paper we center ourselves on the obtention of optimal meshes using adaptive remeshing. As an application we show how adaptive remeshing is applied to two kinds of recon¯gurations: a class where is known that a bang-bang control is the optimal solution and another class where bang-bang would end up in collision. In the ¯rst case the methodology tends to the optimal bang-bang, while in the second one, tends to a low thrust control. Finally, we update the trajectories obtained with FEFF and adaptive remeshing to the JPL-ephemeris model to test the reliability of the methodology. Mon, 28 Feb 2011 10:36:03 GMT http://hdl.handle.net/2117/11562 2011-02-28T10:36:03Z Masdemont Soler, Josep; Garcia Taberner, Laura no We present a methodology to compute trajectories for formation °ying recon¯g- uration problems using a methodology based on ¯nite elements: FEFF (Finite Elements for Formation Flying). In this paper we center ourselves on the obtention of optimal meshes using adaptive remeshing. As an application we show how adaptive remeshing is applied to two kinds of recon¯gurations: a class where is known that a bang-bang control is the optimal solution and another class where bang-bang would end up in collision. In the ¯rst case the methodology tends to the optimal bang-bang, while in the second one, tends to a low thrust control. Finally, we update the trajectories obtained with FEFF and adaptive remeshing to the JPL-ephemeris model to test the reliability of the methodology. http://hdl.handle.net/2117/11560 Title: On the mechanisms of natural transport in the solar system Authors: Masdemont Soler, Josep; Ren, Yuan; Gómez Muntané, Gerard; Fantino, Elena Abstract: Some minor bodies in the solar system (i.e., comets, asteroids and planetary ejecta) are capable of performing transfers from their original location to very distant places, provided they possess sufficiently large energies. Some of them can reach the surface of a planet. This phenomenon is called natural transport. Within the planar circular restricted three-body problem (PCR3BP) with the Sun and a planet as primaries, and also within two coupled PCR3BPs, the gravity of the secondary causes long-term perturbations and the minor body performs swingbys at the secondary, thus resulting in different behaviours with respect to those observed in the heliocentric two-body model. In this contribution, two natural transport mechanisms in the PCR3BP framework are considered. The first is a short-time transport, consisting in heteroclinic connections between libration point orbits of pairs of Sun-planet PCR3BPs: by varying the relative orbital phase of the involved planets at the start of the transfer, the location of the Poincar´e section at which the connection is sought and the size of the departure and arrival periodic orbits, the intersection between the associated unstable and stable manifolds (respectively in the departure and arrival PCR3BP) is computed. The second mechanism corresponds to a long-time transport, the result of the strongly chaotic motion of the minor body in the PCR3BP: the heliocentric orbit changes significantly due to the gravitational interactions with the Sun and the planets (especially the giant planets), and this eventually allows the minor body to reach the vicinity of some planet. In this contribution we provide an analysis of the natural transport in solar system by these two mechanisms. In particular we discuss the key properties of the natural transport, such as the possibility of transferring between two specified celestial bodies, the type of transport and the time of flight. The final aim is to get a deeper insight into the motion of the minor bodies and the exchange of natural material in the solar system. Mon, 28 Feb 2011 10:28:56 GMT http://hdl.handle.net/2117/11560 2011-02-28T10:28:56Z Masdemont Soler, Josep; Ren, Yuan; Gómez Muntané, Gerard; Fantino, Elena no Some minor bodies in the solar system (i.e., comets, asteroids and planetary ejecta) are capable of performing transfers from their original location to very distant places, provided they possess sufficiently large energies. Some of them can reach the surface of a planet. This phenomenon is called natural transport. Within the planar circular restricted three-body problem (PCR3BP) with the Sun and a planet as primaries, and also within two coupled PCR3BPs, the gravity of the secondary causes long-term perturbations and the minor body performs swingbys at the secondary, thus resulting in different behaviours with respect to those observed in the heliocentric two-body model. In this contribution, two natural transport mechanisms in the PCR3BP framework are considered. The first is a short-time transport, consisting in heteroclinic connections between libration point orbits of pairs of Sun-planet PCR3BPs: by varying the relative orbital phase of the involved planets at the start of the transfer, the location of the Poincar´e section at which the connection is sought and the size of the departure and arrival periodic orbits, the intersection between the associated unstable and stable manifolds (respectively in the departure and arrival PCR3BP) is computed. The second mechanism corresponds to a long-time transport, the result of the strongly chaotic motion of the minor body in the PCR3BP: the heliocentric orbit changes significantly due to the gravitational interactions with the Sun and the planets (especially the giant planets), and this eventually allows the minor body to reach the vicinity of some planet. In this contribution we provide an analysis of the natural transport in solar system by these two mechanisms. In particular we discuss the key properties of the natural transport, such as the possibility of transferring between two specified celestial bodies, the type of transport and the time of flight. The final aim is to get a deeper insight into the motion of the minor bodies and the exchange of natural material in the solar system. http://hdl.handle.net/2117/11559 Title: Low-energy impact dynamics in the Earth – Moon system Authors: Masdemont Soler, Josep; Gómez Muntané, Gerard; Alessi, Elisa Maria Abstract: Most of the craters on the surface of the Moon were created by the collision of minor bodies of the Solar System, in particular asteroids coming from the Main Belt as a consequence of different types of resonance. Our aim is to investigate the dynamics of such asteroids, paying special attention on the hyperbolic invariant manifolds associated with the equilibrium point L2 of the Earth – Moon system within the framework of the Circular Restricted Three – Body Problem. We analyze how different distributions of initial conditions for transit trajectories and the value considered for the relative Earth – Moon distance can vary the probability of a lunar impact. Then, we add the gravitational effect of the Sun by means of the Bicircular Restricted Four – Body Problem, showing that the initial phase associated with the Sun and the ratio between the Earth – Moon – Sun distance and the Earth – Moon one can affect the collision pattern in terms of lunar longitude and latitude. Mon, 28 Feb 2011 10:19:36 GMT http://hdl.handle.net/2117/11559 2011-02-28T10:19:36Z Masdemont Soler, Josep; Gómez Muntané, Gerard; Alessi, Elisa Maria no Most of the craters on the surface of the Moon were created by the collision of minor bodies of the Solar System, in particular asteroids coming from the Main Belt as a consequence of different types of resonance. Our aim is to investigate the dynamics of such asteroids, paying special attention on the hyperbolic invariant manifolds associated with the equilibrium point L2 of the Earth – Moon system within the framework of the Circular Restricted Three – Body Problem. We analyze how different distributions of initial conditions for transit trajectories and the value considered for the relative Earth – Moon distance can vary the probability of a lunar impact. Then, we add the gravitational effect of the Sun by means of the Bicircular Restricted Four – Body Problem, showing that the initial phase associated with the Sun and the ratio between the Earth – Moon – Sun distance and the Earth – Moon one can affect the collision pattern in terms of lunar longitude and latitude.