Feedback Control Methods on Short-Period Orbits Of the Earth-Moon Equilateral Libration Points

Recent research by the authors suggests a unique approach to perform Lunar occultations for a diverse set of scientific applications. Under the circular restricted three-body problem assumptions, short-period orbits (SPOs) near the Earth-Moon equilateral Libration points have been suggested for optimal eclipse time and minimal fuel consumption requirements to stay in orbit. Nevertheless, under the presence of orbital perturbations, SPOs are no longer stable as gravitational effects from neighboring celestial bodies continuously perturb these orbits. In this sense, the current study compares a wide range of control methods, including Lyapunov-based adaptive control schemes and fuel-optimal control policies, to address the fuel consumption and tracking issues of the perturbed system. This inquiry attests that perturbations are effectively cancelled out to achieve the proposed scientific objectives with minimal station-keeping requirements

Computation of libration point orbits and manifolds using collocation methods

This thesis contains a methodology whose aim is to compute trajectories describing natural motion of the phase space in a neighborhood of Libtation points and stable/unstable manifolds which correspond to these orbits in the Restricted Three Body Problem. There are two models the Circular Restricted Three Body Problem and Elliptic Restricted Three Body Problem which are special cases of RTBP . In this paper we pay attention to CRTBP which is autonomous (depending on time). The CRTBP is the most easily understood and well-analysed in a coordinate system rotating with two large bodies. The method is based on the collocation method implemented in AUTO - 07p software and must provide an isolated periodic solution. The paper includes explanation of the collocation method, its application in case of CRTBP, numerical and graphical results of its implementation

Tadpole orbits in the L4/L5 region: Construction and links to other families of periodic orbits

The equations of motion in the Circular Restricted Three-Body Problem (CR3BP) allow five equilibrium solutions, that is, the Lagrange or libration points. Two of the five equilibrium solutions are the triangular or equilateral libration points, L4 and L 5. As the secondary gravitational body moves in its orbit about the larger mass, L4 and L5 lead and trail the secondary by 60 degrees, respectively. This investigation focuses on periodic solutions in the vicinity of the triangular libration points, specifically horseshoe and tadpole orbits. Horseshoe orbits are symmetric periodic solutions in the plane of primary motion encompassing both triangular points, as well as one of the collinear libration points, L 3. As a result of these known properties, it is possible to identify regions bounding the motion of horseshoe orbits. Also planar, tadpole orbits represent stable oscillations about the triangular points, combining a long-period librational motion and a short-period epicyclic motion reflecting the period of the two large gravitational bodies about their barycenter. Different strategies are developed to effectively construct tadpole orbits numerically, since the motion is not symmetrical and cannot be bounded to a limiting region as accomplished with horseshoe orbits. The relationship between tadpole orbits and other periodic orbits in the vicinity of L4 and L 5 is examined to explore the natural dynamical evolution of motion and produce useful insight for applications

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

A Heuristic Strategy to Compute Ensemble of Trajectories for 3D Low Cost Earth-Moon Transfers

The problem of finding optimal trajectories is essential for modern space mission design. When considering multibody gravitational dynamics and exploiting both low-thrust and high-thrust and alternative forms of propulsion such as solar sailing, sets of good initial guesses are fundamental for the convergence to local or global optimal solutions, using both direct or indirect methods available to solve the optimal control problem. This paper deals with obtaining preliminary trajectories that are designed to be good initial guesses as input to search optimal low-energy short-time Earth-Moon transfers with ballistic capture. A more realistic modelling is introduced, in which the restricted four-body system Sun-Earth-Moon-Spacecraft is decoupled in two patched planar Circular Restricted Three-Body Problems, taking into account the inclination of the orbital plane of the Moon with respect to the ecliptic. We present a heuristic strategy based on the hyperbolic invariant manifolds of the Lyapunov orbits around the Lagrangian points of the Earth- Moon system to obtain ballistic capture orbits around the Moon that fulfill specific mission requirements. Moreover, quasi-periodic orbits of the Sun-Earth system are exploited using a genetic algorithm to find optimal solutions with respect to total Dv, time of flight and altitude at departure. Finally, the procedure is illustrated and the full transfer trajectories assessed in view of relevant properties. The proposed methodology provides sets of low-cost and shorttime initial guesses to serve as inputs to compute fully optimized three-dimensional solutions considering different propulsion technologies, such as low, high, and hybrid thrust, and/or using more realistic models

Two-Craft Coulomb Formation Study about Circular Orbits and Libration Points

This dissertation investigates the dynamics and control of a two-craft Coulomb formation in circular orbits and at libration points; it addresses relative equilibria, stability and optimal reconfigurations of such formations. The relative equilibria of a two-craft tether formation connected by line-of-sight elastic forces moving in circular orbits and at libration points are investigated. In circular Earth orbits and Earth-Moon libration points, the radial, along-track, and orbit normal great circle equilibria conditions are found. An example of modeling the tether force using Coulomb force is discussed. Furthermore, the non-great-circle equilibria conditions for a two-spacecraft tether structure in circular Earth orbit and at collinear libration points are developed. Then the linearized dynamics and stability analysis of a 2-craft Coulomb formation at Earth- Moon libration points are studied. For orbit-radial equilibrium, Coulomb forces control the relative distance between the two satellites. The gravity gradient torques on the formation due to the two planets help stabilize the formation. Similar analysis is performed for alongtrack and orbit-normal relative equilibrium configurations. Where necessary, the craft use a hybrid thrusting-electrostatic actuation system. The two-craft dynamics at the libration points provide a general framework with circular Earth orbit dynamics forming a special case. In the presence of differential solar drag perturbations, a Lyapunov feedback controller is designed to stabilize a radial equilibrium, two-craft Coulomb formation at collinear libration points. The second part of the thesis investigates optimal reconfigurations of two-craft Coulomb formations in circular Earth orbits by applying nonlinear optimal control techniques. The objective of these reconfigurations is to maneuver the two-craft formation between two charged equilibria configurations. The reconfiguration of spacecraft is posed as an optimization problem using the calculus of variations approach. The optimality criteria are minimum time, minimum acceleration of the separation distance, minimum Coulomb and electric propulsion fuel usage, and minimum electrical power consumption. The continuous time problem is discretized using a pseudospectral method, and the resulting finite dimensional problem is solved using a sequential quadratic programming algorithm. The software package, DIDO, implements this approach. This second part illustrates how pseudospectral methods significantly simplify the solution-finding process

Minimum-Fuel Trajectory Design in Multiple Dynamical Environments Utilizing Direct Transcription Methods and Particle Swarm Optimization

Particle swarm optimization is used to generate an initial guess for designing fuel-optimal trajectories in multiple dynamical environments. Trajectories designed in the vicinity of Earth use continuous or finite low-thrust burning and transfer from an inclined or equatorial circular low-Earth-orbit to a geostationary orbit. In addition, a trajectory from near-Earth to a periodic orbit about the cislunar Lagrange point with minimized impulsive burn costs is designed within a multi-body dynamical environment. Direct transcription is used in conjunction with a nonlinear optimizer to find locally-optimal trajectories given the initial guess. The near-Earth transfers are propagated at low-level thrust where neither the very-low-thrust spiral solution nor the impulsive transfer is an acceptable starting point. The very-high-altitude transfer is designed in a multi-body dynamical environment lacking a closed-form analytical solution. Swarming algorithms excel given a small number of design parameters.When continuous control time histories are needed, employing a polynomial parameterization facilitates the generation of feasible solutions. For design in a circular restricted three-body system, particle swarm optimization gains utility due to a more global search for the solution, but may be more sensitive to boundary constraints. Computation time and constraint weighting are areas where a swarming algorithm is weaker than other approaches