Analysis an‎d design of optimal trajectories in the restricted three-body problem using stable an‎d unstable invariant manifolds

1/14/2027 12:00:00 AM

Given the growing interest of space agencies in the exploration an‎d utilization of deep space, an‎d the considerable potential of these regions for enabling long-duration missions, this research investigates the design of optimal transfer trajectories for deploying spacecraft in dynamically favorable zones suitable for imaging an‎d establishing stable communications. Lagrange points within the framework of the Circular Restricted Three-Body Problem (CR3BP), along with their surrounding regions, are recognized as promising can‎didates for such missions due to their unique dynamical properties. However, direct inser‎tion of spacecraft into these points is typically accompanied by significant challenges, particularly severe fuel consumption constraints. To address this issue, the present thesis follows a two-step methodology. In the first step, considering the high sensitivity of the system’s dynamics to initial conditions, periodic orbits around the Lagrange points are identified using Poincaré mapping. The precise initial conditions of these orbits are then extracted through the application of a differential correction algorithm. In the second step, the stable an‎d unstable manifolds associated with these periodic orbits are computed based on the analysis of the monodromy matrix an‎d its eigenvalues. These manifolds, which represent the intrinsic pathways of the system in phase space, define the dynamical structure of the problem an‎d enable the design of low-energy transfer trajectories. Initial transfer paths are extracted from the intersection of the unstable manifold of the departure orbit with the stable manifold of the destination orbit. Subsequently, in order to determine the most optimal intersection point,taking into account the significance of the mission’s departure an‎d arrival locations—a hybrid metaheuristic algorithm combining Particle Swarm Optimization (PSO) an‎d Invasive Weed Optimization (IWO) is employed to design transfer trajectories with minimal fuel consumption. Simulation results demonstrate that the proposed method performs efficiently in designing low-energy transfer paths with high fuel efficiency, an‎d can play a significant role in the planning of future space missions, particularly in regions adjacent to Lagrange points. Keywords: trajectory design, Circular Restricted Three-Body Problem, Poincaré mapping, periodic orbits, Lagrange points, manifolds, hybrid metaheuristic algorithm

طراحي مسير , مسئله‌ي سه جسم محدودشده‌ي دايروي , نگاشت پوانكاره , نقاط لاگرانژي , مدارهاي متناوب , منيفولد , الگوريتم فراابتكاري تركيبي

trajectory design , Circular Restricted Three-Body Problem , Poincaré mapping , periodic orbits , Lagrange points , hybrid metaheuristic algorithm , manifolds

Arghavan Kamali

dr. Majid Bakhtiari