A closed-form solution to the minimum ${\Delta V_{\rm tot}^2}$ Lambert’s problem

Abstract

A closed form solution to the minimum ${\Delta V_{\rm tot}^2}$ Lambert problem between two assigned positions in two distinct orbits is presented. Motivation comes from the need of computing optimal orbit transfer matrices to solve re-configuration problems of satellite constellations and the complexity associated in facing these problems with the minimization of ${\Delta V_{\rm tot}}$. Extensive numerical tests show that the difference in fuel consumption between the solutions obtained by minimizing ${\Delta V_{\rm tot}^2}$ and ${\Delta V_{\rm tot}}$ does not exceed 17%. The ${\Delta V_{\rm tot}^2}$ solution can be adopted as starting point to find the minimum ${\Delta V_{\rm tot}}$. The solving equation for minimum ${\Delta V_{\rm tot}^2}$ Lambert problem is a quartic polynomial in term of the angular momentum modulus of the optimal transfer orbit. The root selection is discussed and the singular case, occurring when the initial and final radii are parallel, is analytically solved. A numerical example for the general case (orbit transfer “pork-chop” between two non-coplanar elliptical orbits) and two examples for the singular case (Hohmann and GTO transfers) are provided.