Convergence Basin Analysis in Perturbed Trajectory Targeting Problems

Increasingly, space flight missions are planned to traverse regions of space with complex dynamical environments influenced by multiple gravitational bodies. The nature of these systems produces motion and regions of sensitivity that are, at times, unintuitive,

and the accumulation of trajectory dispersions from a variety of sources guarantees that spacecraft will deviate from their pre-planned trajectories in this complex environment, necessitating the use of a targeting process to generate a new feasible reference path. To ensure mission success and a robust path planning process, trajectory designers require insight into the interaction between the targeting process, the baseline trajectory, and the dynamical environment. In this investigation, the convergence behavior of these targeting processes is examined. This work summarizes a framework for characterizing and predicting the convergence behavior of perturbed targeting problems, consisting of a set of constraints, design variables, perturbation variables, and a reference solution within a dynamical system. First, this work identifies the typical features of a convergence basin and identifies a measure of worst-case performance. In the absence of an analytical method, efficient numerical discretization procedures are proposed based on the evaluation of partial derivatives at the reference solution to the perturbed targeting problem. A method is also proposed for approximating the tradespace of position and velocity perturbations that achieve reliable

convergence toward the baseline solution. Additionally, evaluated scalar quantities are introduced to serve as predictors of the simulation-measured worst-case convergence behavior based on the local rate of growth in the constraints as well as the local relative change in the targeting-employed partial derivatives with respect to perturbations.

A variety of applications in different dynamical regions and force models are introduced to evaluate the improved discretization techniques and their correlation to the predictive metrics of convergence behavior. Segments of periodic orbits and transfer trajectories from past and planned missions are employed to evaluate the relative convergence performance across sets of candidate solutions. In the circular restricted three-body problem (CRTBP), perturbed targeting problems are formulated along a distant retrograde orbit and a near-rectilinear halo orbit (NRHO) in the Earth-Moon system. To investigate the persistence of results from the CRTBP in an ephemeris force model, a targeting problem applied to an NRHO is analyzed in both force models. Next, an L1 -to-L2 transit trajectory in the Sun-Earth system is studied to explore the effect of moving a maneuver downstream along

a trajectory and altering the orientations of the gravitational bodies. Finally, a trans-lunar return trajectory is explored, and the convergence behavior is analyzed as the final maneuver time is varied.