Initial Orbit Determination of the Earth Orbiters Using a Single Ground Optical Tracking Station Based on the Tikhonov Regularized Total Least Squares Estimation

Increasing demand on the launch of the Earth orbiters with a variety of applications makes the problem of the initial orbit determination problem more interesting. The problem is to determine the Keplerian orbital elements of any orbiters using minimum number of observations.  Observing the orbiters in their initial phase of orbital launch using the ground optical tracking stations is one of the most reliable and frequently used methods for the problem solution. Slant distance, horizontal and vertical angles of satellite with respect to the local north and zenith are the observed quantities in the optical tracking systems. Short-arc of the observations in particular for the Low Earth Orbiters (LEO) and modeling the initial orbit determination as a two-body problem and ignoring perturbing forces are the main challenging issues in orbital mechanics. Neglecting the perturbation force contribution in the mathematical models of the initial orbit determination sets up of the observation equations with some errors or the so called Error In Variable (EIV) model. Moreover, fitting an ellipse to the observed three dimension position time series of the orbiter and determination of the Kepler elements is an ill-posed problem. It is due to fact of fitting an ellipse to a few closely distributed data points along the orbit in three dimensional space. Therefore, one has to implement the method of Total Least Squares (TLS) with an appropriate regularization technique for the orbital parameter estimation. Different regularization techniques have been already introduced for solution of ill-posed problems. Herein, Tikhonov regularization method with the aim of minimization of bias term along with the error in measurements is applied and the orbital elements are estimated. Implementation of regularization method significantly improves the results and in particular the LEO Kepler elements. Numerically, the proposed method is implemented on the estimation of the parameters with different orbital geometry type; including the LEO and Medium Earth Orbiter (MEO) satellite in the polar and non-polar orbits. In all cases, the orbital parameters and their variances are estimated and statistically tested. Moreover, relative errors of the estimated parameters and their meaningfulness are checked in different scenarios.     Theoretically, the problem of orbital parameter estimation is a nonlinear problem by its nature. We are implemented iterative gradient-based solution and therefore linearization of the nonlinear equations is required. For quarantined convergence of the linearized model, initial value of the unknowns with acceptable accuracy is needed. The classical methods, e.g., Gauss, Gibbs and Lambert method of the initial orbit determination problem provide an approximate solution with enough accuracy for initialization of the estimation problem. The Picard condition as an indicator of ill-posedness on the inverse problem is used to demonstrate in the orbital parameters estimation. The L-curve method is implemented to get the solution with minimum bias value. The method of Ordinary Least Squares (OLS) is simultaneously implemented on the problem to show how TLS can efficiently be used.