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

Abstract

A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheimo–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.