Optimal Earth Return Transfers from Lagrange Point Orbits Using Particle Swarm Optimization

Open Access
Nayyar, Mollik
Graduate Program:
Aerospace Engineering
Master of Science
Document Type:
Master Thesis
Date of Defense:
November 10, 2016
Committee Members:
  • Dr David Spencer, Thesis Advisor
  • Three body system
  • Lagrange points
  • manifolds
  • optimization
  • particle swarm
  • PSO
  • Dynamical Systems
  • Optimal Transfers
  • Astrodynamics
  • Lagrange Point Orbits
  • LPO
  • Lambert's Problem
A future where space mining and cargo transportation is fast approaching reality, for such a future, one needs a way to efficiently transport the materials gathered in space to Earth or other space based depots. The focus of this thesis is on such return trajectories from Lagrange point orbits to Earth. Since travel in space is currently very expensive, it is desired to utilize more efficient methods to travel through space. Current trajectory design methods require a lot of computational resource which contributes to mission cost. This thesis presents an approximate, computationally inexpensive method to scan the available search space and obtain the locations and trajectories associated the with lowest ΔV, which can be used as an initial guess that can be later used in advanced methods to obtain ΔV optimal trajectories. The use of invariant manifolds in the Earth-Moon system is proposed for this purpose. The three-body system is studied and a simplified methodology for the design of trajectories from the Lagrange point orbits is presented. The techniques to generate the invariant manifold associated with Lagrange point orbits is presented. To obtain the best possible trajectory based on ΔV requirements, an optimization scheme is introduced to select the best location on the manifold to initiate transfer. While gradient based methods have been previously used to study optimal trajectories in the three-body system, these are computationally very expensive. Therefore, this thesis tackles the computational requirements associated with design of the space trajectories in the Earth-Moon-spacecraft three-body system for space transportation. The solver used to survey the search space of the invariant manifold is presented. The two-body approximation is utilized to provide quick, preliminary study of the transfers in space, allowing the mission designer to be able to devise a suitable trajectory in the three-body system using the insights from the two-body approximation. Essentially, the burden of finding the location of the transfer point with the minimum ΔV transfer is offloaded to the particle swarm optimization algorithm that uses a simplified Lambert's solver to scan the search space on the manifold and provide the patch point with the lowest ΔV. The solver used here is validated using the results obtained from literature for transfers from Earth to a Lagrange point orbit. Validation results show that the technique used is able to provide fairly accurate results for a two-impulse scenario at a very significant computational advantage. Then, the method is used to generate results for Earth return trajectories. The ΔV requirements from Lambert's solver for halo and Lyapunov orbits of L_1,L_2 points are presented. The trends obtained using this method shows the location of orbit insertion in the LEO that has the potential of providing the lowest ΔV. The present analysis suggests that it is possible to reach almost any inclination from the manifold of a halo LPO without incurring huge ΔV costs if the target location for insertion along the final orbit's true anomaly is either at the ascending or the descending node. ΔV was found to increase as the point of insertion approached a true anomaly corresponding to maximum z-component displacement from the equator. These trends seem to hold for higher inclinations of the final orbit. Similar trends seem to hold for transfers from Lyapunov orbit as well. The results and initial and final ΔV vectors obtained by this method has the potential to be used as an initial guess for gradient-based methods. Finally, ways to improve upon the solver and the optimization algorithm are suggested. The technique maybe combined with gradient-based methods to quickly and with reasonable accuracy locate the point on the manifold in the vicinity of the optimal point and then a detailed analysis may be pursued for better and more ΔV optimal results.