| By assuming the motion of a satellite about the earth's geopotential mimics the known Kolmogorov-Arnold-Moser (KAM) solution of a lightly perturbed integrable Hamiltonian system, this research focused on applying trajectory following spectral methods to estimate orbital tori from sampled orbital data. From an estimated basis frequency set, orbital data was decomposed into multi-periodic Fourier series, essentially compressing ephemerides for long-term use. Real-world Global Positioning System (GPS) orbital tracks were decomposed and reconstructed with error from as low as few kilometers per coordinate axis over a 10-week span to tens of kilometers per coordinate axis over the same time period, depending on the method chosen. These less-than-precision-level results were due primarily to the resonant orbits of the GPS constellation. Additionally, the trajectory following spectral methods chosen experienced difficulties converging on a complete basis set when using data time spans much smaller than the period of the slowest system frequency. However, the lessons learned from GPS led to a new orbital tori construction method. This approach focused on fitting local spectral structures, denoted as frequency clusters, within the sampled orbital data to the analytical form of the windowed, truncated, continuous Fourier transform. Methods employing direct use of the observed spectrum as well as least squares fitting techniques were developed with considerable success. For portions of the low-earth-orbit regime, maximum errors per coordinate axis in orbital tori fits were kept below 5 meters over a time period of 1 year. Simulations using the Hubble Space Telescope yielded 1-dimensional root mean square errors of less than 2 meters in each coordinate axis in the initial and predicted ephemeris fits, both of which used 1-year-long tracks of numerically integrated data. |
