Singular value decomposition and least squares orbit determination

Convergence using least squares for orbit determination is often difficult to assure when a) there are only a small number of measurements with large time intervals between the measurements (sparse measurements), b) the measurement accuracy is poor, c) there are crude or abnormal measurements present, or d) no good initial guess is available. The first case, accurate but sparse measurements, but with a good initial guess represents a linear least-squares problem. It was shown long ago that such problems are successfully solved by applying singular value decomposition (SVD) to the matrix of the conditional equations. In the nonlinear case, the poor determination of the conditional equations matrix is further aggravated by the difficulties of obtaining a good Hessian approximation in regions which are remote from the solution. In this paper we present an algorithm using SVD in which both of these problems are resolved by choosing the dimension of the minimization subspace for each minimization step. If necessary, minimization in a low dimensionality subspace (2-D or 3-D) can be carried out with improved accuracy Hessian matrices. Results of the application of the algorithm are discussed for typical measurement-based orbit determination. We compare these results with those produced using the Gauss-Newton method. Areas of convergence of the proposed technique are investigated as well.

Singular value decomposition and least squares orbit determination Conference Paper