State transition matrix for long-distance formation with J2 in eccentric orbits

• Long-distance formation flying in eccentric orbits with J2 is studied.
• A state transition matrix applicable to the long-distance formation is derived.
• Nonlinear terms are calculated by using the difference between semi-major axes.
• The position error is largely reduced in the case of a large-size formation.

The relative state transition of spacecraft formation flying is considered in this paper. In eccentric orbits, the Tschauner–Hempel (TH) equations are used to express the relative motion between a deputy spacecraft and a chief spacecraft. Perturbation forces are not considered in the TH equations, and the relative distance between two spacecraft is limited to a short range. In this paper, the effects of the J2 perturbation forces are considered, and the case of relatively long distance between two spacecraft is focused. A state transition matrix applicable to such cases is derived. The state transition matrix is expressed by adding some compensating terms to the state transition matrix of the TH equations. The usefulness of the proposed state transition matrix relative to the state transition matrix of the TH equations is shown through numerical simulations from the viewpoint of position error.