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Formation flying in elliptical orbits

We present a general approach to the analysis of the formation flying problem, based on FreeFlyer(R) and MATLAB(R), that is designed with spacecraft formations in mind. With this approach, we are able to quickly combine new algorithms for the initialization and control of a formation within the context of a high-fidelity operational tool. As an example, we can use a Hill-Clohessy-Wiltshire (HCW) approximation to obtain a first-guess of the initial conditions that result in a desired relative motion and subsequently use these initial conditions in the full equations of motion. By exploiting the object-oriented nature of FreeFlyer/sup R/, the geometric and dynamic requirements, constraints, and interactions among the member spacecraft are easily monitored, controlled, and visualized. Immediate feedback yields results that can then be used to refine the first-guess. Due to the lack of a closed-form analytical solution to the variational equations of relative motion about highly elliptical orbits, we find that our approach provides valuable insight into the establishment and control of formations in this context. In particular, we have applied this approach to the problem of generating initial conditions for both simple synthetic aperture radar and 'tetrahedron' formations in elliptical orbit. We show that a first-guess for the initial conditions obtained from the Hill-Clohessy-Wiltshire equations works well over a fairly broad range of eccentricities (0 to /spl sim/0.4) but is dependent on the true anomaly in the orbit where the approximation is applied. For highly elliptical orbits (eccentricity >0.8) the resulting relative motion is coupled strongly to true anomaly. In particular, finding good initial conditions for formations with requirements defined at apogee, such as those desiring a tetrahedron, becomes difficult. As a result we developed a new algorithm for setting the initial conditions based on the desired geometry. We compare our HCW results to those obtained from the purely geometrical first-guess algorithm. Analysis of these results will provide vital clues in our effort to construct control strategies for formation flying in elliptical orbits.