Design of natural Lissajous formations with nearly constant baselines

A semi-analytical approach is presented for the design of spacecraft formations in quasi-periodic Lissajous orbits. The approach utilizes Lindstedt–Poincaré asymptotic series to approximate the center manifold tori around a collinear libration point. For several performance metrics based on the intersatellite distance, the formation design parameters are analytically optimized in the linearized dynamics, which is then used as an initial guess for a numerical optimization procedure in the 15th-order approximation model. The proposed approach avoids the need for numerical integration in a highly unstable environment by leveraging the full range of center manifold approximations. The formation design obtained can be easily adapted to the high-fidelity ephemeris model by the multiple-shooting technique. The efficiency of the developed methodology is demonstrated on the problem of designing two- or three-spacecraft formations with the nearly constant baseline property in Lissajous orbits around the L₂ libration point of the Sun–Earth system.