Initial orbit determination using the generalized Laplacian method with differential correction and circular-orbit constraint

Initial orbit determination (IOD) for spacecraft is crucial in various scenarios, including telemetry, tracking, and space debris surveillance. The generalized Laplacian method, based on the classic Laplacian method, supports various perturbations and measurement types. It is simple to comprehend, implement, and convenient to use and has been proven effective in practical applications. However, the current implementation of the generalized Laplacian method can fail when dealing with certain medium earth orbit (MEO) or geosynchronous orbit (GEO) objects. IOD solutions suffer from reduced observation geometry due to the short tracking length of MEO and GEO. The measurement error can cause in plane elements in IOD solutions, such as the semi-major axis and eccentricity, to be significantly biased even if IOD converges and the measurement residual RMS (root mean square) appears small. In this paper, differential correction (Newton-Raphson iteration) is used to improve the convergence of the generalized Laplacian method. We show that using the first-order derivative (Jacobian matrix) reduces the number of iterations significantly. It also helps in the convergence of iteration procedures when simple iteration techniques fail to solve the condition equation iteratively. Tracking data can be too short or sparse in some cases to effectively constrain the IOD solution to a reasonable level. An eccentric orbit solution can result from the measurements of a circular-orbit object. To overcome the inappropriate solution induced by insufficient tracking data, we introduce an external circular-orbit constraint in the condition equation when dealing with tracking data whose orbit is known to be circular. This constraint is effective in producing a suitable IOD solution when used with the measurements in the generalized Laplacian method. Our results indicate that, with a slight adjustment, the circular-orbit constraint can perform effectively in the generalized Laplacian method even when the tracking data are extremely sparse.