Koopman Operator Theory Applied to Lambert’s Problem with a Spectral Behavior Analysis

Astrodynamics is abundant with nonlinear dynamical systems, such as satellites operating within Earth’s gravitational field. With the increase in the number of satellite constellations, making accurate predictions of the motion of satellites throughout space is becoming more relevant than ever. Influences from gravitational forces, atmospheric drag, and solar radiation pressure introduce highly nonlinear terms in the equations that model these dynamical systems. The predictions of these effects are essential for planning future space missions. Intrinsically tied to this is Lambert’s problem, which concerns finding an optimal transfer orbit that connects two position vectors within a specified time of flight. Furthermore, solving Lambert’s problem in the context of these nonlinear dynamical systems is crucial for identifying optimal or- bit trajectories of spacecraft in Earth orbit and beyond. Traditional Lambert solvers often involve iterative methods that are computationally intensive, which may not be able to capture the nonlinearities of the dynamical systems accurately, and might have constraints in their applications. Using operator theory to simplify a system’s nonlinear dynamics presents a promising avenue for research. This Thesis bridges the gap in implementing operator theory to effectively solve Lambert’s problem. The Koopman Operator is used to embed the nonlinear dynamics involved in Lambert’s problem into a global linear representation, enabling the study of the nonlinear dynamical systems from a global perspective for future state prediction away from fixed points. The Koopman Operator is applied to solve variants of Lambert’s problem including solving for the minimum energy and minimum Δv solutions, the single and multi-revolutions solutions, and the multi-impulse solution. Furthermore, the Koopman Operator enables the computation of these solutions with low computational complexity. A variety of initial conditions are considered, proving the range of applicability of the Koopman Operator to Lambert’s problem. Comparisons made with numerical methods and another Lambert solver demonstrate the robustness and accuracy of the Koopman Operator solutions. Finally, an analysis of the spectral behaviors of the dynamics considered is provided, with insights into the stability of the dynamical systems and accuracy of the solutions found.

Name thesis:degree_name

Master

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Aeronautics and Astronautics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2023

Author dc:creator

• Pasiecznik, Julia

Advisor dc:contributor.advisor

• Linares, Richard

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/151603

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/151603