| Considering the special configuration with respect to the primaries and complicat-ed dynamics of libration points, they not only provide the ideal locations for observing the Sun and the space environment, but also they provide the perfect gateways for inter-planetary explorations. In recent years, space manifold dynamics theory has attracted a great deal of attention in the fields of scientific research and engineering application, and provides the theoretical foundation for the trajectory design of complicated mis-sions, for which the transfer trajectory cannot be approximated by two-body problem. In particular, space manifold dynamics can be applied to design low-energy transfer trajectory which can save a great deal of fuel consumption, compared to the traditional transfer.In the research of this dissertation, the basic dynamical models include the re-stricted N-body problem where the primaries are considered as particles, and the poly-hedron model of irregular asteroid. According to the value of N, there are various kinds of restricted N-body problems, such as the dynamical model defined by the JPL planetary ephemeris, restricted four-body problem, circular and/or elliptic restricted three-body problems, and the model of relative motion with circular and/or elliptic ref-erence orbits. Circular restricted three-body problem and elliptic restricted three-body problem are together called restricted three-body problem. In particular, when the ec-centricities of primaries are zero, the elliptic restricted three-body can be reduced to circular restricted three-body problem; and when the mass parameter of system is ze-ro, one can get that the elliptic restricted three-body problem is reduced to the model of relative motion with elliptic reference orbit and the circular restricted three-body problem is reduced to the model of relative motion with circular reference orbit. In ad-dition, if we consider the gravitational effect of the bodies except for the two primary bodies as the perturbation, we could also called restricted four-body problem and the real system defined by JPL ephemeris as perturbed restricted three-body problem.In the frameworks of the dynamical models mentioned above, we investigated the following topics:Dynamics in the vicinity of equilibrium points, theory of invariant manifold transfer, low-energy trajectory design and dynamics of libration points in the vicinity of irregular astroid. Some novel results have been obtained. The investiga-tion within this dissertation riches the theory of dynamical system, and provides the theoretical groundwork for the application of space manifold dynamics to deep space exploration. The main innovations of this dissertation are briefly summarized as fol-lows:The series expansions of the configuration of relative motion with elliptic refer-ence orbit are studied.the solution is expanded as formal series of the eccentricity of the reference orbit, in-plane amplitude and out-of-plane amplitude, then, taking the Lawden periodic solution as starting point, the high-order analytical solution is constructed by means of the Lindstedt-Poincare method. The series expansions con-structed provide an accurate mathematical expression for the formation configuration with large amplitude with elliptic reference orbit, and can be applied to the studied on capture, station-keeping and reconstruction of configurations.Based on the series solutions of the equations of relative motion, we propose a novel approach to compute the periodic and/or quasi-periodic orbits around equilibri-um points of circular and/or elliptic restricted three-body problems. Firstly, we studied high-order series solutions around any equilibrium points of the model of relative mo-tion with elliptic reference orbit, then taking the series solution as initial solution, the periodic and/or quasi-periodic orbits around equilibrium points of circular and/or el-liptic restricted three-body problem are obtained by combing numerical continuous technique with multiple shooting method.The high-order series solutions of the motions around triangular libration points in the circular restricted three-body problem are constructed. We know that when μ<μc, triangular libration points in circular restricted three-body problem are linearly stable, and according to Lyapunov center manifold theory, there are three kinds of periodic motions around them:long, short and vertical periodic motions. The general motions are quasi-periodic and composed of the three kinds of basic motions. Based on the non-linear equations of motion, we expand the motions around triangular libration points as formal series of long periodic amplitude, short periodic amplitude and vertical periodic amplitude, and with the aid of computer, high-order series solutions are constructed. The advantages of series expansions lie in that the state of the point on the triangular point orbit can be parameterized by several related parameters, which are of benefit in the process of trajectory optimization.The motions around artificial equilibrium points of the low-thrust circular restrict-ed three-body problem are analytically studied. Different from the classical circular re-stricted three-body problem, the locations of artificial equilibrium points can be fixed at some appropriate positions in space by adding the low-thrust propulsion, therefore artificial equilibrium points could increase the flexibility of mission design to satisfy some special requirements, such as continuous observation of the polar region, Earlier prediction of solar activities (compared to the classical L1 point).We constructed high-order series expansions of invariant manifolds associated with Lissajous and Halo orbits around the collinear libration points in the elliptic re-stricted three-body problem. In almost all the three-body systems in the Solar system, such as Sun-planet system and planet-satellite system, the eccentricities of the pri-maries are different from zero, therefore elliptic restricted three-body problem could approximate the restricted three-body system in Solar system more well, compared to circular restricted three-body problem. In the research, we expand the invariant manifolds as formal series of five parameters:eccentricity of primaries, amplitude cor-responding to unstable manifold, amplitude corresponding to stable manifold, in-plane amplitude and out-of-plane amplitude. By taking advantage of the series expansions constructed, the motions around collinear libration points, such as stable and/or unsta-ble manifolds, transit and/or non-transit trajectories, Lissajous and Halo orbits, can all be parameterized. In particular, when the eccentricity of primaries is equal to zero, the series expansions constructed can be reduced to describe the center and hyperbolic manifolds in the circular restricted three-body problem.High-order series solutions are constructed for describing the motions around tri-angular libration points in the elliptic restricted three-body problem. In the research, the motions around triangular points are expressed as formal series of eccentricity of primaries, long periodic amplitude, short periodic amplitude and vertical periodic am-plitude. In order to check the validity of the series solutions constructed, the practical convergence is computed.Similar to the concept of weak stability boundary (WSB) transfers to the Moon, we investigated two-impulse and/or low-thrust low-energy transfers to the vicinity of triangular points of Earth-Moon system. Compared to the traditional Hohmann trans-fer, the low-energy transfers could save a great deal of fuel consumption.In the frame work of circular restricted three-body problem, we proposed a method for designing transfers from the Earth to Moon based on Jacobi constants. Combining with the numerical corrector, we obtained the relations among trajectory parameters, such as transfer time, velocity impulse and energy of trajectory, where the relationship between transfer time and velocity impulse is the most important and could provide reference to the practical trajectory design in the process of Moon exploration. Con-sidering the advantages and drawbacks of particle swarm optimization and differential evolution algorithm when they are dealing with the optimization problems solely, we proposed an improved cooperative evolutionary algorithm, which is applied to solve the global optimization problems and performs very well. In the real system defined by JPL ephemeris, the optimization problems established for solving the low-energy transfers are solved by combining the improved cooperative evolutionary algorithm with sequential quadratic programming algorithm, with the initial information provid-ed by the low-energy transfers designed in circular restricted three-body problem, and then several optimal low-energy transfers are obtained. Conclusions can be drawn as follows:by taking advantage of the perturbation of the Sun and other planets and the perturbation of the elliptic motion of the Moon, the low energy trajectories in the real system are more fuel-efficient than the ones in the model of circular restricted three-body problem.By using invariant manifolds, the single-impulse and low-thrust transfers between Li(i=1,2) points of Sun-Earth system and Li(i=3,4,5) points of Earth-Moon system are designed. This investigation on one hand demonstrates the potential ap-plications of Li(i=1,2) points of Sun-Earth system to providing the gateways for deep space exploration, and is of benefit to the extended mission design for Sun-Earth Li(i=1,2) missions, on the other hand, the investigation provides an alternative trans-fer approach to the vicinity of L,(i=3,4,5) of Earth-Moon system:The spacecraft is forced to the vicinity of Li(i=1,2) points of Sun-Earth system first, and then fol-lowing the unstable manifold, the spacecraft can arrive at the vicinity of Li(i=3,4,5) of Earth-Moon system, at last the spacecraft is inserted into the nominal orbits by performing an maneuver.Taking advantage of the series expansions of invariant manifolds, the state of the point on the invariant manifolds can be parameterized. Then the optimization prob- lem for low-thrust transfer to the libration point orbits (Lissajous and Halo orbits) of Sun-Mars system is established and solved by using global and local optimization al-gorithms.Taking Sun-Earth+Moon system as an example, we studied two strategies of station-keeping for the triangular libration point mission:i) Multiple shooting station-keeping strategy; and ii) the reconstruction of the nominal orbit in the real system. In the research, the problem of station-keeping is transcribed into a nonlinear program-ming problem, which is solved by sequential quadratic programming method. Simula-tion results indicate that optimization techniques perform efficiently in the process of solving the station-keeping problem.At last, the dynamics of equilibrium points in the vicinity of bar-shaped asteroid are investigated. Firstly, the gravitational model of the bar-shaped asteroid is estab-lished by means of polyhedron model, then dependence of the locations, linear dynam-ics of equilibrium points on the system parameters are studied. By means of numerical methods, the families of horizontal Lyapunov orbits and vertical Lyapunov orbits are computed. For the unstable equilibrium points, the invariant manifolds around them are computed and their applications to landing and escaping missions for asteroid ex-ploration are discussed. |
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