Optimal reconfigurations between equilibria of two-craft electromagnetic formations along manifolds

As a novel approach to control the relative motion of a spacecraft formation, electromagnetic formation flight (EMFF) has some prominent advantages, such as no propellant consumption and no plume contamination. The relative motion actuated by inter-craft electromagnetic force/torque is characterized by equilibrium states which could be utilized to accomplish particular missions, such as close formation, interferometer, etc. However, to maneuver among and maintain these equilibrium configurations, the control capability would be insufficient with only electromagnetic actuation and the inertial thrust is required, so an optimal method which exploits the invariant manifold theory is investigated in this paper. Firstly, the nonlinear translational dynamic model of two-craft electromagnetic formations is derived, and then relative equilibrium configurations and their stabilities are analytically studied. Secondly, the manifolds displaying characteristics of equilibria for the unstable equilibrium configurations are analyzed for the first time. Based on these manifolds, a novel and generalized method is proposed to formulate and parameterize optimal reconfigurations from one electromagnetic equilibrium configuration to another with the electromagnetic control inputs from the initial value to the target value. Specifically, the system's uncontrolled and discontinuous flows along manifolds compose majority of the transfer trajectories, and then optimal controls are applied to a portion of the discontinuous trajectories to differentially correct them to match continuity. By defining the control histories as a finite set of time independent variables, the optimal control problem is converted to a parameter optimization problem and solved by Particle Swarm Optimization. The method can attain the shape-changing ability and consume as little inertial thrust as possible, in exchange for added electromagnetic control effort. Lastly, numerical simulations are presented to verify the feasibility and optimality of the designed method.