Comparison of optimization methods for computing bi-impulsive transfer trajectories in the bi-circular restricted four-body problem

Optimization is a crucial process in astrodynamics, focused on finding the most efficient trajectories with minimum cost, especially in terms of total velocity change ($\Delta V$). The inherent complexity of orbital mechanics, particularly in multi-body systems and for long-duration transfers, often limits the effectiveness of traditional gradient-based methods, such as Non-Linear Programming (NLP), which are highly dependent on a high-quality initial guess. To overcome this limitation, meta-heuristic methods, which are robust and gradient-independent, are presented as a viable alternative. The present work evaluates the performance and efficiency of several single-objective meta-heuristic algorithms in the optimization of bi-impulsive transfer trajectories, starting from a Low Earth Orbit (LEO) at 200 km altitude toward retrograde co-orbital resonance using the Planar Restricted Bi-Circular Four-Body Problem (PBCR4BP). A comprehensive comparison was conducted using methods such as variants of Differential Evolution (DE), the Evolutionary Centers Algorithm (ECA), Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Re-sampled Inheritance Search (RIS) and others, comparing them against a Random Search (RAS) baseline. The results demonstrated that Differential Evolution (DE) based methods, notably the DE/rand/1/bin RL and AD-DE/rand/1/bin RL variants, achieved the best overall performance, effectively minimizing both the required $\Delta V$ and the computational time. Furthermore, the robust optimization strategy successfully identified more efficient solutions than previously reported in the literature, achieving a minimum $\Delta V$ of 0.1478 km/s. We conclude that meta-heuristic methods, particularly the advanced variants of Differential Evolution, are powerful, reliable, and efficient tools for optimizing transfer trajectories in the complex dynamic environment of the PBCR4BP. However, the sensitivity analysis indicated that long-duration transfers can be highly sensitive to small perturbations, especially for eccentric target orbits near Earth.