Two-Impulse Trajectory Design in Two-Body Systems With Riemannian Geometry

This work presents a new method for generating impulsive trajectories in restricted two-body systems by leveraging Riemannian geometry. The proposed method transforms the standard trajectory optimization problem into a purely geometric one that involves computing a set of geodesics for a suitable Riemannian metric. This transformation is achieved by defining a metric, specifically the Jacobi metric, that embeds the dynamics directly into the metric, so any geodesic of the metric is also a dynamically feasible trajectory. The method finds the fuel-optimal transfer trajectory by sampling candidate energy ($\Delta V$) changes for different points on the current and desired orbit, and efficiently computing and evaluating each candidate geodesic, which are equivalent to candidate orbit transfer trajectories via the Jacobi metric. The method bypasses the known issues of optimization-based methods, e.g., sensitivity to the initial guess, and can be applied to more complex two-body systems. The approach is demonstrated on the minimum-$\Delta V$ two-impulse phase-free orbit transfer problem, first on a Keplerian system and second on a system with a modeled $J_2$ perturbation. The proposed method is shown to meet or exceed the state-of-the-art methods in the minimum-$\Delta V$ problem in the Keplerian system. The generality and versatility of the approach is demonstrated by seamlessly including the $J_2$ perturbation, a case that many existing methods cannot handle. Numerical simulations and performance comparisons showcase the effectiveness of the approach.