Two typical relative motion control problems of Lorentz-augmented spacecraft implemented in the Earth's magnetic field are studied and further discussed. The Lorentz force acts on a charged spacecraft that could actively generate and modulate when it is flying through a magnetic field, and provides a new concept of propellantless propulsion strategy for spacecraft formation flying and hovering system control. It is a fact that the directions of Lorentz force are limited by the local magnetic field. In view of this reason, it does not provide or satisfy the required control acceleration for spacecraft formation flying and hovering timely; therefore, it always works as an auxiliary strategy to reduce the fuel consumptions. Based on the above considerations, a dynamical model for relative motion of charged spacecraft, including the effects of the J2 perturbation and the Lorentz force, is derived and its application to spacecraft formation flying and hovering control problems are discussed. Then, the optimal sliding model error feedback control method is derived based on the novel dynamical model, which is proposed by theory integrating between optimal sliding model control theory and the principle of minimum sliding mode error. Moreover, the optimal design of the required charge for the Lorentz spacecraft and the thruster-output control acceleration has been developed with details. It is shown that the proposed controller owns the advantages of the optimal control theory and has the ability to estimate and offset the unknown disturbances. The numerical simulations are performed to illustrate the efficacy of the proposed dynamical model and controller to maintain the spacecraft formation flying and hovering system with optimal fuel consumptions and high precision in the presence of the unknown disturbances.
