Alternative approach to the solution of Lambert's problem

Abstract

A method for solving the two-point, two-body orbital transfer boundary-value problem, commonly referred to as Lambert's problem, is presented. Previous algorithms have depended heavily on the geometric properties of conic sections to obtain an iterative solution. An alternative approach is offered, making use of velocity and time functions of the flyout angle that have been derived directly from the equations of motion. A procedure is presented that rapidly iterates directly on the flyout angle until the desired initial velocity vector can be obtained. Statement of Lambert's Problem T HE boundary-valu e problem for a two-point, two-body transfer in a Newtonian central gravity field (often referred to as Lambert's problem) can be stated as follows: given an initial position vector r^ and a final position vector r2, find the initial velocity vector K, which will allow a transfer from r to r2 in time tf. Originally, this problem concerned tracking bodies that were orbiting the sun, but today it has application in problems such as spacecraft navigation and strategic missile guidance. Ideally, given the parameters rt, r2, and /i (the gravitational parameter), the solution to Lambert's problem would be achieved with an equation that gives the velocity vector Fas a function of the transfer time: