Two-Impulse Cotangent Rendezvous Between Coplanar Elliptic and Hyperbolic Orbits

Abstract

T HE cycler orbit is an important trajectory for the human exploration of Mars because it repeatedly encounters Earth and Mars on a regular schedulewithout stopping. The Aldrin cycler [1] is the simplest circulating orbit between Earth and Mars. There are two types of spacecraft in the cycler architecture. One is the interplanetary transfer vehicle; the other is the “taxi” vehicle, which travels between the surface of Earth or Mars and the transfer vehicle. Thus, the taxi must rendezvous with the transfer vehicle, which is on a hyperbolic trajectory during the planetary flyby [2]. Assume that the taxi vehicle is in a coplanar elliptic parking orbit, and then the rendezvous problem between the elliptic and hyperbolic orbits needs to be solved in the cycler architecture. The two-impulse method is the simplest approach to fulfill rendezvous mission. If two endpoints and the rendezvous time are assigned, the two-body two-impulse rendezvous problem can be obtained by solving Lambert’s problem [3–5]. There exist many elliptic transfers when multiple revolutions are allowed [6–8]. By considering the coasting time, the L-norm minimum-fuel solution can be obtained by numerical optimization techniques [9,10]. For the two-impulse rendezvous between two coplanar orbits, the L-norm optimal solution was obtained in closed form [11]. However, no constraints are imposed on the impulse direction for the Lambert approach.Among all constraints on the impulse direction, the tangent impulse is the most interesting because the tangent orbit has many merits; for example, simpler operation, less energy consumption, and significant improvement in rendezvous safety [12]. For the transfer between two coplanar circular orbits or two coplanar coaxial elliptic orbits, the cotangent transfer, known as the classic Hohmann transfer, is the minimum-energy one among all the two-impulse transfers [3]. For two coplanar noncoaxial elliptic orbits, the cotangent transfer problem has been solved by many methods, including the orbital hodograph theory [12], the geometrical method [13], and the method based on the flight-direction angle [14]. Different from the transfer problem, the rendezvous problem requires the same flight time for the chaser and the target. For this propose, Zhang et al. [15] solved a tangent orbit rendezvous problem with the same velocity direction at an assigned rendezvous point. In addition, Zhang et al. [16] provided a method for all solutions and the minimum-fuel solution to the two-impulse cotangent rendezvous problemwith bounded revolutionnumbers.However, thismethod for the cotangent rendezvous problem is not valid for hyperbolic orbits. Themain contribution of this work is an extension of themethod in [16], which presented cotangent rendezvous between two coplanar elliptic orbits, to cotangent rendezvous between coplanar elliptic and hyperbolic orbits. The solutions will be obtained by a numerical iterative algorithm for a piecewise function. The minimum-time solution and the L-norm minimum-fuel solution will also be determined.