Available at: https://digitalcommons.calpoly.edu/theses/2977
Date of Award
3-2025
Degree Name
MS in Aerospace Engineering
Department/Program
Aerospace Engineering
College
College of Engineering
Advisor
Kira Jorgensen Abercromby
Advisor Department
Aerospace Engineering
Advisor College
College of Engineering
Abstract
There are a multitude of computational techniques capable of modeling the behavior of nonlinear dynamic systems over time. In the field of astrodynamics, the most frequently used tools are numerical integrators, specifically Runge-Kutta algorithms. These algorithms have successfully produced numerous two-body trajectories, as well as an increasing number of three-body trajectories when two-body dynamic models fail to produce Δv values within the realm of mission feasibility. Koopman operator theory is an alternative method for modeling nonlinear dynamic behavior that captures an approximation of a nonlinear dynamic system on a defined basis in the Hilbert space. The Koopman operator is an infinite dimensional linear operator capable of evolving all observable functions of the state, and when the observable is chosen to be the identity observable, can evolve the states themselves. In this thesis, Koopman operator theory is applied to the circular restricted three body problem to solve for halo orbits in the Sun-Earth system, as well as their resulting manifold trajectories. The goal is to possibly expand the initial trajectory design tools available to mission designers as the need for nonlinear three body solutions increases. A complex-normal Hamiltonian dynamic model and Hamiltonian dynamic model are used with a third order Richardson polynomial for the nonlinear dynamic representation. The implementation of the complex-normal Hamiltonian is found to be highly susceptible to error and ill suited to an initial trajectory design application. The implementation of the Hamiltonian is much simpler and using this dynamic model, the Koopman operator produces partial halo orbits, but diverges before a full period is achieved. The solutions become less accurate as the halo amplitude increases, and as the solution order decreases. Manifolds are successfully computed along the converged portion of the trajectories, but their trajectories diverge as well before long term behavior can be identified. The lack of converging solutions from the Hamiltonian model is attributed to an insufficient basis. The computational burden required to form a sufficiently large basis that produces a converging solution exceeds the capacity of the computer used for this analysis and what is desirable for low-cost initial trajectory design. Thus, Koopman Operator Theory is deemed an ineffective initial trajectory design tool.
