Application of Floquet theory to formation flying in elliptic orbits via a Hamiltonian structure-preserving control

In this study, an improved Hamiltonian structure-preserving controller is applied for formation flying in elliptic orbits. The Lyapunov-Floquet theory is applied to the Tschauner-Hempel (TH) equation in order to generate its transformed form with two zero eigenvalues and two pairs of imaginary eigenvalues. Then, an improved Hamiltonian structure-preserving (HSP) control is introduced to the transformed system to eliminate the instability of double zeros. All bounded trajectories can be generated by basic solutions, which effectively reveal the following performances of the controller, such as the relationship between the frequencies of basic motions and control gains are detected and the wide selection of gains for stability. Local and global optimizations on gains are discussed based on different performances of the controller. One focuses on the sensitivity of output with respect to input of controller and the other aims at a minimal fuel consumption for the maximum likelihood. The ergodic relationship between the fuel consumption and initial condition is established for the minimal consumption. To avoid in-orbit collision, a collision-free strategy based on the artificial potential function is proposed to enhance the proposed controller, which make sure a minimum safety distance for the formation.