An N-Observation Modification to Gooding's Method for Initial Orbit Determination

This report tests an extension of Gooding's Method for Initial Orbit Determination (IOD) to process N line of sight (LOS) measurements for space-based observations of a target taken in a near co-planar orbit relative to the observer. Additional error frames are added for measurements 2 through N-1 to extend the gradient used in a Newton-Raphson iterative refinement formula from 2 target functions to 2*(N-2). The N-Gooding algorithm utilizes a least-squares approach to perform the iterative refinement on a non-square matrix. Measurement errors and process noise are not considered in this study, and dynamics are represented from simple Keplerian orbits. Line of sight observations are processed as unit vectors and not converted to error angles as seen in other N-Gooding approaches. A sample of 100 orbits were tested for N = 3 through N = 24 observations. The accuracy of the N-Gooding algorithm significantly improves on the original Gooding's algorithm in consistently fewer steps. The resulting N-Gooding estimate shows an average of 80% improvement in position error as compared to the traditional Gooding algorithm, with average errors of 0.0467 km and 0.2378 km respectively. Both accuracy and precision improved with more observations for a majority of target orbits, with standard deviations reduced by more than 60%. The number of steps to converge to within an error frame tolerance of +/- 1E-8 km was reduced by an average of 99%. The resulting convergence plots show that the estimate improves according to a logarithmic trend. Future work includes testing the effects of process and measurement noise on the accuracy and convergence of the N-Gooding algorithm. Classification of near co-planar orbits that yield optimal and predictable results against target that are less likely to converge will also be studied. Last, comparison of N-Gooding to other IOD procedures such as those based on Physics Informed Neural Networks is of particular interest.