New Solutions for the Perturbed Lambert Problem Using Regularization and Picard Iteration

摘要

A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheuno-Stiefel transformation and Modified Chebyshev-Picard iteration is presented. The first contribution is the development of an analytical solution to the elliptic Keplerian Lambert problem based on Kustaanheimo-Stiefel regularization. This transforms the nonlinear three-dimensional orbit equations of motion into four linear oscillators. The second contribution solves the elliptic Keplerian two-point boundary value problem and initial value problem using the Kustaanheimo-Stiefel transformation and Picard iteration. The Picard sequence of trajectories represents a contraction mapping that converges to a unique solution over a finite domain. Solving the Keplerian two-point boundary value problem in Kustaanheimo-Stiefel variables increases the Picard domain of convergence from about one-third of an orbit (Cartesian variables) to over 95% of an orbit (Kustaanheimo-Stiefel variables). These increases in the domain of Picard iteration convergence are independent of eccentricity. The third contribution solves the general spherical harmonic gravity perturbed elliptic two-point boundary value problem using the Kustaanheimo-Stiefel transformation and Picard iteration, and it does not require a Newton-like shooting method for fractional orbit transfers. For multiple revolution transfers, however, a shooting method can make use of the Modified Chebyshev-Picard iteration/ Kustaanheimo-Stiefel/initial value problem and the Method of Particular Solutions to obtain solutions given a Keplerian Lambert solution as the starting iterative. The Kustaanheimo-Stiefel perturbed solution is illustrated using a (40,40) degree and order spherical harmonic gravity model. A general three-dimensional recipe is introduced for solving the perturbed Lambert Problem via Modified Chebyshev-Picard iteration without a Newton-like shooting method for the fractional orbit case. The increase in the domain of convergence of the Kustaanheimo-Stiefel transformed, perturbed Lambert problem via Modified Chebyshev-Picard iteration versus the Cartesian Modified Chebyshev-Picard iteration Lambert solution is analogous to the results for the Keplerian case. The three-dimensional two-impulse perturbed Lambert problem is efficiently convergent up to about 85% of the Keplerian orbit period with a (40,40) spherical harmonic gravity model. The efficiency of the current two-point boundary value problem solver is compared with MATLAB's fsolve, where Runge-Kutta-Nystrom 12(10) and Gauss-Jackson are the integrators.