Piecewise Sundman Transformation for Spacecraft Trajectory Optimization Using Many Embedded Lambert Problems

Abstract

Direct optimization of many-revolution spacecraft trajectories requires path discretization into segments. The well-known Sundman transformation guides the development of a piecewise function that regularly varies flight time for each segment. This flight-time function approximates the regularization effect on the dynamics for a spatially even discretization, conveniently not changing the independent variable from time. The flight-time functions are analyzed in the following two ways: first, in a trajectory propagation problem, and second, in several optimization problems. First, an eccentric Keplerian orbit is propagated to reveal similar behavior between flight-time functions and dynamic regularization, increasingly so for more segments per orbit. Second, the piecewise transformation is applied to trajectory optimization problems that embed the Lambert boundary value problem for every segment, defined here as the embedded boundary value problem (EBVP) technique. This EBVP technique performs favorably to a similar, state-of-the-art, optimization technique using a Kepler solver. The improvement is a result of eliminating explicit enforcement of position and velocity constraints by the optimizer. The EBVP technique is further enhanced by the approximate Sundman function by using an exponent of , enabling otherwise infeasible fuel- and energy-optimal problems. The largest solution contains up to 12,287 segments and 512 revolutions, and significantly varies semimajor axis and eccentricity.