Distributed situational observer in a displaced orbit: Relative dynamics and control

We design a distributed situational observer using formation flying in a displaced orbit. The main focus of our investigation is the relative dynamics and control of displaced orbits obtained by low-thrust propulsion. The spatial dynamics in Newtonian form are used to derive the numerical relative motions, and their natural frequencies discovered by eigenvalue decomposition separate from each other at a critical height that differentiates the structural stability, bifurcation, and instability. Using the Jordan decomposition, six fundamental motions are achieved, including the stationary multiequilibria, the periodic oscillations that correspond to the natural frequencies, and the maximum leaving or approaching velocity caused by the different geometric and algebraic multiplicities. Off-axis equilibrium is obtained by a proposed open-loop control, and the motions nearby are proven to be equivalent to the numerical relative motions. The reduced dynamics in Hamiltonian form are used to derive the analytical solutions for linearized relative motions. Bounded relative trajectories with arbitrary initial values are achieved by two extraclosed-loop controls. Using the off-axis equilibrium and resonance of natural frequencies, the applications of a fixed relative baseline vector for interferometric SAR or Fresnel zone lens missions and repeating relative ground tracks for a phased array antenna mission are addressed in terms of the trajectory design.