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

Abstract

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.

Peer-to-Peer Refueling for Circular Satellite Constellations

“…As shown in Ref. 25, allowing multi-revolution orbital transfers may reduce the fuel significantly. Allowing more than two impulses, on the other hand, offers little improvement.…”

“…The velocity change for each rendezvous is calculated according to the method presented in Ref. 25. Only multi-revolution rendezvous trajectories whose perigees are higher than the radius of the earth are considered valid.…”

“…It is shown in Ref. 25 that the rendezvous cost between two satellites in the same circular orbit decreases monotonically as the total time to conduct the rendezvous increases. As a result, the number of edges involved in the optimal matching changes, depending on the refueling period.…”

“…This assumption is valid only if the times of transfer between the satellites are long. 25,26 Although in practice binding time constraints will invalidate this assumption, nevertheless by neglecting the delivery cost of the transfers, one allows to establish a market between the satellites in the constellation. In this market, satellites having an amount of fuel above the constellation average are designated as the sellers, whereas satellites in the constellation having an amount of fuel below the constellation average are designated as the buyers.…”

Optimal impulsive ellipse-to-circle coplanar rendezvous

“…During the early stage of rendezvous study, Lambert's problem was used to solve the two-body problem of fixed-time rendezvous between conic orbits. In recent years, Lambert rendezvous has become a new topic that deals with two-impulse rendezvous using multiplerevolution Lambert solutions [10][11][12] . For optimal impulsive rendezvous based on numerical methods, the performance index is regarded as optimization function, the boundary problems and other constraints of rendezvous as optimization constraints, then the optimal rendezvous turns into the problem of optimization so that numerical methods can be used to get the solutions of rendezvous [13] .…”