A Low-Complexity Algorithm for Trajectory Generation in the Three-Body Problem

The Cislunar realm holds intrinsic value for scientific, commercial, and military applications as numerous entities have begun to invest resources into the expansion and utilization of the region. As the Cislunar region continues to grow, there is an escalating need for efficient methods of trajectory generation in this multi-body dynamical system for computationally limited systems. Thus, a low-complexity classical algorithm is proposed to achieve accurate orbital trajectories in the three-body problem. The proposed algorithm solves a polynomial interpolation problem, formulated using state measurements, through the unique decomposition of a dense system into sparse matrices. Several relevant Cislunar trajectories are simulated using the algorithm, yielding favorable results in terms of arithmetic and time complexities over existing iterative techniques at the cost of accuracy. The algorithm does not directly consider the dynamics of the system, allowing for fast solutions to be determined, but at the cost of the algorithm struggling to propagate trajectories forward past known measurements for long periods of time. Furthermore, the algorithm remains stable under perturbations and an analytical boundary of perturbed trajectories is determined. In general, the LCA offers a computationally efficient method for trajectory generation in the three-body problem that may be used to supplement traditional iterative techniques in computationally stringent scenarios.