A homotopy approach to the solutions of minimum-fuel space-flight rendezvous problems

Abstract

A homotopy approach for solving constrained parameter optimization problems is examined. The first order necessary conditions, with the complementarity conditions represented using a technique due to Mangasarian, are solved. The equations are augmented to avoid singularities which occur when the active constraint set changes. The Chow-Yorke algorithm is used to track the homotopy path leading to the solution to the desired problem at the terminal point. Since the Chow-Yorke algorithm requires a fairly accurate computation of the Jacobian matrix, analytical representation of the system of equations is desired. Consequently, equations obtained using the true anomaly regularization of the governing equations were employed for the above purpose. A homotopy map suited for the space-flight rendezvous problem including a minimum radius constraint is developed, which can naturally deform any initial problem into some other valid desired problem. Several coplanar and non-coplanar solutions for circular and elliptic cases have been presented for the restricted time problem with a minimum radius constraint.