Addressing the ill-conditioned problem in initial orbit determination via the Gooding algorithm

The accuracy of angles-only initial orbit determination (IOD) is significantly compromised when only a short-arc orbit is observed. The ill-conditioned problem in matrices due to weak geometric constraints caused by short arcs and observation errors typically causes significant errors in the estimated ranges and thus unsatisfactory IOD. This paper presents a critical analysis of the ill-conditioned problem using the Gooding algorithm and proposes several techniques to improve it. On the basis of multiple observations, a least-squares method is proposed to solve the ranges at the first and last epochs. For the short-arc case, the ridge estimation technique is applied to mitigate the ill-conditioned problem. To determine whether an orbit is eccentric, a procedure to assess orbit eccentricity is developed via the range-search method, which aims to provide reasonably accurate initial ranges to the Gooding algorithm. Finally, an eccentricity-constraint technique for the Gooding algorithm is proposed for cases where the orbit is determined to be nearly circular. The performances of these techniques on space-based simulation data are assessed, and an improved Gooding algorithm (I-Gooding) suitable for various observation conditions is proposed. The I-Gooding algorithm is subsequently applied to process actual ground-based observations. The results show that its accuracy in estimating the semimajor axis is 47% higher than that afforded by the standard Gooding algorithm.