Comparing Radius of Convergence in Solving the Nonlinear Least Squares Problem for Precision Orbit Determination of Geodetic Satellites

摘要

Precision orbit determination methods often rely on nonlinear least squares estimation. A large body of literature is dedicated to various methods for effectively solving the matrix equations involved. Test cases were constructed using satellite laser ranging data for LAGEOS-1 to examine the effects of state update methods, the treatment of the normal equations, and the matrix decomposition method. Cases are run with varying initial conditions to characterize the radius of convergence for each method. The Levenberg-Marquardt and Gauss-Newton state update methods are compared. It is found that the Levenberg-Marquardt state update method increases the radius of convergence by a factor of eighteen over the Gauss-Newton state update method. The Lower-Upper (LU) and Choleksy matrix decomposition methods, where the least squares normal equations are explicitly formed, are compared to the Singular Value Decomposition (SVD) and QR matrix decomposition methods, where the least squares normal equations are not explicitly formed. It is found that there is only a minor difference in the radius of convergence regardless of whether the least squares normal equations are explicitly formed, or which matrix decomposition method is used. Ultimately, more data is needed to draw generalized conclusion; however, it is clear that using the Levenberg-Marquardt state update method can greatly increase the radius of convergence, which is beneficial for orbit determination applications.