An Establishment and Transfer Strategy for Formation Configurations Based on Ω–u Torus Topological Equivalence

For satellite formations with orbit planes differing by the right ascension of ascending node (RAAN), the success of the configuration establishment is directly influenced by the fuel expenditure. This article suggests a topological equivalent formation configuration establishment strategy, for RAAN, Ω and argument of latitude u deployment problems, based on the coupling relationship between rapidly changing variable (RCV) time accumulation and slowly changing variable (SCV) fuel consumption. First, a symplectic manifold approach is applied in the construction of the Kepler dynamical system Σelliptic, by which the angular variables are identified as a symplectic S1 action on the union of nondegenerate elliptical orbits. Then, the Delaunay variables consisting of S1 actions and accompanied momentum mappings are used to derive the geometric relationship of the two torus ${\mathbb{T}^2}$ = S1 × S1 spanned by Ω and u. Through the deduced one-parameter transformation groups, a topological equivalent relation of phase flow (Ω, u) is illustrated on the 2-torus. Finally, as an application of the topological equivalence with Ω: 0→2π and u: 0→2π, the phase flows specified by the dynamical system (μ, Σelliptic) succeed in establishing the formation reconfiguration of Ω after several equivalent substitutions of u with in-plane maneuvers. The topological equivalence of the phase flows on the torus ${\mathbb{T}^2}$ is utilized to achieve the “time for fuel” strategy, which is verified by a three-satellite formation. In comparison with the conventional plane-changing with normal maneuvers, the “time for fuel” strategy realizes a noteworthy fuel saving of 60%–70%, total 26.26 kg, which contributes to a substantial extension of the satellite's in-orbit operational lifespan. The topological equivalence method of accumulating SCV and coupling RCV is beneficial in scenarios where the formation has no time constraints on configuration establishment or involves large-scale reconfiguration with RAAN as one of the targeted orbital elements.