3D CR3BD Halo Orbit Modeling
The Pennsylvania State University - Dr. Puneet Singla
Analyzing Halo Orbits near the Moon
May 2022
The objective of this project was to analyze the Three-Dimensional Circular Restricted Three-Body Problem (3D CR3BP). The 2D version was completed in an earlier project, but was built upon to develop a 3D solution by adding another dimension. Matlab was then used to model and graph Halo Orbits around the Moon.
Background
AERSP 304 was a junior-level System Dynamics and Control course that covered analytical and computational techniques for solving mathematical models of multiple degree of freedom engineering systems. This is achieved by exposure to topics such as model development, approximating systems with linear ODEs, linear feedback control systems, and stability analysis.
This project added to a project completed earlier in the semester for honors credit.
My Involvement
A mathematical dynamics model was created from a dimensionless graph of the 3BP. Three Lagrange equations were found using non-linear equations. The five Lagrange points were found by finding where velocity and acceleration were zero. It was also found that the z coordinates were always zero, showing that Lagrange points occur on a 2D plane. This further led into the discovery that the x and y equations of motion were decoupled from the z motion. A state space was created to allow for easy implementation into Matlab.
From there, a script was written to graph the resulting motion, using ODE45, over a period of time in the Earth-Moon system. Lagrange Points 2 and 4 were run through the program. A guess and check method with the position and velocity initial conditions was used to find different halo orbits, to experiment with stability. These initial conditions were also used to find the frequency of the x and y motion, as well as the z motion.