Optimal Two-Impulse Rendezvous Between Two Circular Orbits Using Multiple-Revolution Lambert’s Solutions

In this paper, we study the optimal fixed-time, two-impulse rendezvous between two spacecraft orbiting along two coplanar circular orbits in the same direction. The fixed-time two-impulse transfer problem between two fixed points on two circular orbits, called a fixed-time fixed-endpoint transfer problem, is solved first. Our solution scheme involves first the solution to the related multiple-revolution Lambert problem. A solution procedure is proposed to reduce the calculation of an existing algorithm, thanks to the introduction of an auxiliary transfer problem. Unlike the fixed-endpoint transfer problem with a fixed time of flight, the auxiliary transfer problem studies the relationship between the transfer cost and the transfer orbit semimajor axis with the transfer time being free. The characteristics of the auxiliary problem are thoroughly explored and then applied to the fixed-time fixed-endpoint problem. As a result, the solution candidates are narrowed down from 2N max + 1 (N max is the maximum number of revolutions permitted) to at most two. Using this solution procedure, the minimum cost of the fixed-time transfer problem is easily obtained for all cases of different separation angles and times of flight. Thus a contour plot of the cost is obtained as a function of the separation angle and the transfer time. This contour plot along with a sliding rule facilitates the task of finding the optimal initial and terminal coasting periods , and thus obtaining the solutions for the original problem. Numerical examples demonstrate the application of the methodology to multiple rendezvous of satellite constellations on circular orbits.