Two-Point Boundary Value Problem Solutions to Spacecraft Formation Flying

Abstract

The two-point boundary value problem of a leader-follower spacecraft formation flying in unperturbed elliptical reference orbits is studied. The initial and final relative positions and times and the orbit of the leader are known, and the orbit of the follower must be determined. This problem will be called the relative Lambert's problem. First, we will show this relative Lambert's problem can be solved like the classical Lambert's problem. Then, a set of approximate analytic solutions are obtained through linearizing the Lagrange's time equation. Meanwhile, a simplified Newton-Raphson algorithm is applied to obtain numerical solutions, and the relevant quantities of the leader are used as initial conditions. Here, our special efforts are dedicated to the study of the periodic relative orbits of the follower. In particular, a set of first-order analytic solutions to the relative Lambert's problem are derived from the periodic solutions of Lawden's equations, and a constraint on the leader's true anomalies (implicitly in time) and relative positions is obtained. From that constraint on the radial/in-track plane of the leader local-vertical-local- horizontal frame, we found that, for specified initial and final times, the locus of final positions of the follower with fixed initial position is a straight line, and so is the locus of initial positions with fixed final position. Furthermore, for fixed initial and final positions, the transfer times with either specified initial time or final time can be expressed as the real roots of a cubic equation, for which there are at most three solutions. Several examples will be given to support these conclusions.