Investigation of Initial Orbit Determination in the CR3BP with Multi-Order Differential Corrections

open_access
Open Access
Author:
Hope, Jonathon
Area of Honors:
Aerospace Engineering
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
Puneet Singla, Thesis Supervisor
Robert G. Melton, Thesis Honors Advisor
Roshan Thomas Eapen, Thesis Supervisor
Keywords:
Initial Orbit Determination
CR3BP
Space Domain Awareness
Abstract:
The application of methods from dynamical systems theory (DST)—such as bifurcation analysis, periodic orbit stability, and invariant manifolds associated with periodic orbits—has been fundamental in advancing the study of orbits and transfers in cislunar space. The dynamics in cislunar space is largely governed by the multi-body gravitational influence of both the Earth and the Moon. The dynamical model of the circular restricted three-body problem (CR3BP) is employed in this work to design periodic solutions about the system’s equilibrium points, the Lagrange points. The CR3BP admits planar solutions – the Lyapunov orbit family, and spatial solutions – the Halo, Axial and Vertical orbit families. Additionally, in-plane and spatial resonant orbits – orbits that exhibit a near rational ratio of their periods with that of the moon about the Earth - are also studied in this work leveraging preliminary design in Keplerian dynamics. This work also develops a design methodology for constructing transfers from the Earth to these periodic orbits in cislunar space leveraging tools from DST such as the Poincaré section. The orbits and transfers examined in this thesis serve as simulated data for initial orbit determination (IOD) algorithms, as they closely reflect observations from past mission profiles. Optical measurements are typically the azimuth and elevation of a space object from an observation site on the Earth. IOD in cislunar space presents unique challenges compared to near-Earth IOD. One key difficulty arises from the underlying system dynamics: the chaotic motion inherent in the CR3BP complicates the estimation of initial conditions, given the vast expanse of cislunar space. Additionally, sensor limitations result in sparse observational datasets, further complicating the problem. A commonly used approach for IOD is the nonlinear least squares (NLS) method. By linearizing the measurement residuals, this method approximates the problem and applies Newton’s method to iteratively minimize errors and converge to a solution. However, linearization leads to a loss of information and increased sensitivity to initial conditions. Since little guidance is available for selecting suitable initial guesses in IOD, NLS often struggles to converge effectively. To address these limitations, a multi-order shooting scheme (MOSS) is introduced which integrates high-order sensitivities into the correction process, enhancing its ability to navigate the large solution space and reducing its dependence on precise initial guesses. MOSS achieves this by employing a derivative-free quadrature scheme known as the conjugate unscented transform to compute higher-order sensitivities of the spacecraft location with respect to an initial condition in the CR3BP. To compare the performance of MOSS and NLS, two test cases are examined: a 3:1 resonant orbit and a set of Northern L1 halo orbits. The first case assesses MOSS’s robustness by analyzing a distribution of initial conditions centered around the true initial state, while the second case aims to determine the orbit characteristics influencing convergence behavior in different sections of halo orbits. When testing 500 initial condition samples for the resonant orbit, MOSS successfully converged to the correct solution 60% of the time, compared to only 13% for NLS. For the halo orbits, MOSS demonstrated an average convergence rate 72% higher than NLS. Further analysis of the convergence history reveals that the target spacecraft’s location relative to the initial conditions plays a crucial role in achieving a correct solution. These findings validate the initial hypothesis that MOSS achieves a higher convergence rate despite poor initial guesses. Additionally, for cases where both methods converged, MOSS yielded errors an order of magnitude lower than those of NLLS relative to the true initial conditions.