Novel Method of Optimizing L1 to L2 Lyapunov Transfer Trajectories

This thesis utilizes the Planar Circular Restricted Three-Body Problem (PCRTBP) to develop a novel method of optimizing transfer trajectories between L1 and L2 Lyapunov orbits. As future space projects progress, the desire for placing spaceports and cargo storage systems in orbits around the Moon or other planets increases, and trajectory optimization will be a vital instrument. The method outlined in this thesis utilizes differential correctors, and a gradient descent method iterates a given trajectory to produce a more optimized trajectory. Although free transfers can be achieved between Lyapunov orbits sharing equivalent Jacobi constants, this thesis aimed to produce an optimized trajectory for Lyapunov orbits of varying Jacobi constants. To accomplish this, a free transfer was utilized as the initial transfer orbit between L1 and L2. Multiple orbits were selected between the free transfer and a final selected orbit. These intermediary orbits would provide steps for the iteration to be conducted within. Each trajectory being inputted would include the position vectors, velocity vectors, and time. The initial trajectory would be scaled to a resolution, which would be defined by the number of points (segments) within the stable or unstable manifolds. As iterations are run, a cost value is outputted, defining the optimization of the trajectory.

Analysis results found a direct relationship between the cost and trajectory resolution. As the resolution of the trajectory increases, as more segments are added, the finer the control of the trajectory becomes. This contrasts with a lower-resolution trajectory, allowing for more significant shifts in the trajectory's position with fewer iterations. Although a higher resolution trajectory could achieve the same cost value, the compute time would be much longer than that of the latter. A hybrid method was designed to utilize the lower resolution optimization method for a majority of the trajectory optimization, then to convert the final trajectory to a higher resolution trajectory and run it through another larger iteration batch for finer modifications. This method was found to be the ideal method as it provided a much lower cost value than running the same number of iterations on a constant resolution optimization. Further improvements can be made to the optimization technique, such as running higher iterations and increasing the intermediary orbits to provide a much smoother transition between the orbits allowing higher optimization to occur. Overall, the final result of the thesis achieved the goal of developing a method for optimizing the trajectories between L1 and L2 Lyapunov orbits with excellent accuracy.