Optimal Projected Circular Orbit for Spacecraft Formation Flying Near a Slowly Rotating Asteroid

In spacecraft formation flying missions, the projected circular (relative) orbit (PCO), which appears circular when viewed along the zenith-nadir direction of the nominal chief spacecraft, is suitable for constructing a synthetic aperture radar. In this paper, the analysis of a PCO for spacecraft formation flying near a slowly rotating asteroid is conducted. By matching the in-plane and out-of-plane relative motion frequency, the critical values of the initial phase angle that characterizes the PCO are identified, resulting in the minimum precession of relative orbit. The analysis is then utilized to develop the fuel-optimal control law for establishing a target PCO. In particular, the fuel-optimal problem is addressed with an indirect method, where a homotopy approach is adopted, and the initial (unknown) costate vector is calculated with a scaling technique, so that the sensitivity to the initial guess problem is mitigated.