A Methodology for the Computation of Constrained Orbits and its Application to the Design of Solar System Trajectories

Abstract

The purpose of this paper is the development of a procedure for the determination of constrained trajectories of spacecraft in the Solar System and its application to the computation of orbits, such as heteroclinic connections in the Sun-Earth-Moon system, that cannot be computed using only a two or three-body problem approach. Starting from a nominal trajectory computed in a simplified force model, the aim is to compute a new one, close to the original, but satisfying more realistic equations of motion. The formulation adopted allows for the existence of discontinuities in velocity and specific constraints for the states of the orbit, including boundary conditions. The continuity requirement of the orbit is accomplished in position by a differential correction process. The discontinuities in velocity are minimized using an optimization procedure. Some significant applications are presented, mainly regarding libration point orbit missions. It will be shown how using the initial input, the constraints and weights, an orbit, initially determined in the Circular Restricted Three-Body Problem, can be refined in order to fulfill the equations of motion of the Restricted n-Body Problem and specific scopes. In all the cases, an accurate initial guess is a key point.

Rendezvous design in a cislunar near rectilinear Halo orbit

“…This is further simplified by symmetry properties of Halo orbits that impose perilune and apoapsis states witḣ x =ż = 0 at the (O, x, z) plane crossing. The CR3BP reference orbit provides a good initial guess for a correction procedure in the Ephemeris (14,15) .…”

The reentry to Earth as a valuable option at the end-of-life of Libration Point Orbit missions

“…We remark that in this framework an invariant manifold cannot be constructed analytically, but still there exist trajectories shadowing it. To exploit the corresponding dynamics, we could either take a given reentry solution designed in the CR3BP, and refine it taking into account all the relevant perturbations (as explained, for instance, in Alessi et al (2012)), or consider the initial conditions on the nominal LPO provided by JPL HORIZONS, and design the suitable maneuver to drive the spacecraft along the proper direction. The latter approach is the one adopted in this work, because the JPL HORIZONS ephemeris are computed with the highest level of accuracy, while the analysis performed in the previous section represents a good background to interpret the outcome.…”

High-Fidelity Trajectory Optimization with Application to Saddle-Point Transfers

“…Multiple shooting methods with propagation of the variational equations in n-body dynamics are used in [13,14] for trajectory optimization, and in [15] to analyze the evolution of quasi-periodic orbits in the Earth-Moon system. Multiple shooting techniques are used to compute constrained transfers to LPO with a sequence of impulsive control maneuvers [16], as well as for the solution of low-thrust transfers in the Earth-Moon system, including applications to LPO [17]. Direct transcription of the dynamics is also employed in low-thrust trajectories to boost numerical efficiency [18].…”