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Formation flying in elliptic orbits with the J2 perturbation

Relative dynamics between the chief satellite and the deputy ones in formation flying is crucial to maintaining the formation. A good choice of the formation usually requires a lower control frequency or less control energy. For formation flying missions in highly elliptic orbits, the well-known C-W equation is not accurate enough. Instead, Lawden's equation is often used. First, the solution to Lawden's equation with a very simple form is deduced. Then the J2 perturbation is added. It is found that Lawden's solution is not necessarily valid when the J2 perturbation is considered. Completely discarding Lawden's solution and borrowing the idea of mean orbit elements, two rules to initialize the formation are proposed. The deviation speed is greatly reduced. Different from previous studies on the J2 perturbation, except for the relatively simple expression for the semi-major axis, the tedious formulae of the long period terms and the short period terms of other orbital elements are not used. In addition, the deviation speed is further reduced by compensation of the nonlinear effects. Finally, a loose control strategy of the formation is proposed. To test the robustness of this strategy, a third body perturbation is added in numerical simulations.