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

For satellite formation with separated orbit planes that differ by around 1° at the right ascension of ascending node (RAAN), the fuel expenditure of configuration establishment will directly affect the mission's success. This paper 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. The symplectic manifold approach is applied in construction of Kepler dynamical system Σelliptic, by which the angular variables are identified as a symplectic S¹ action on the union of non-degenerate elliptical orbits. Furthermore, the Delaunay variables consisting of S¹ actions and accompanied momentum mappings are used to derive the geometric relationship of the two torus T²= S¹*S¹, to obtain the torus distribution 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. As an application of the new derivation with Ω: 0→2π and u: 0→2π, the phase flows specified by the (μ, Σ_elliptic) dynamical system succeed in establishing the formation reconfiguration after several equivalent substitutions of u, through arranging the semi-major axis with in-plane maneuver. The maneuver reaches 132m/s if the conventional normal maneuver is applied, exceeding the control capability of small and micro satellites or the total fuel constraints. Contrarily, through the orbital verification of Sat 36413, 36414 and 36415, the topological equivalence of the phase flows on the torus T² is utilized to achieve “time for fuel”, which only consumes 49m/s of maneuver, saving fuel consumption by 22.2kg accounting for 60~70%, and greatly prolongs the in-orbit life of long-term constellations. The topological equivalence method of accumulating SCV and coupling RCV proposed in this paper contributes to tasks where the formation mission has no time constraints on the configuration establishment, or where the scale of reconfiguration is large with RAAN as one of the targeted orbital elements.