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

In this work, we present 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 problem of computing a set of geodesics for a suitable Riemannian metric. This transformation is achieved by using the Jacobi metric, whose geodesics are dynamically feasible trajectories for the small body in a restricted two-body system. The proposed method finds the fuel-optimal transfer trajectory by sampling candidate energy changes (ΔV) between 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 initial guesses, and can be easily applied to two-body systems with perturbations. First, the approach is demonstrated on the minimum-ΔV two-impulse phase-free orbit transfer problem in a restricted two-body system, and second on the same system with an added J2 perturbation. The proposed method is shown to meet or exceed state-of-the-art methods of finding minimum-ΔV transfer orbits in the restricted two-body system. The generality and versatility of the approach is demonstrated by seamlessly including the J2 perturbation, a case that many existing methods cannot handle. Numerical simulations and performance comparisons showcase the effectiveness of this new approach.