One-Dimensional Constrained Coulomb Structure Control with Charge Saturations

A Coulomb structure is a cluster of free-flying satellites which maintains its shape through inter-vehicle electrostatic forces. These Coulomb forces are generated using on-board charge emission devices. This paper investigates the 1-D restricted motion of a 3-craft cluster. Two charge feedback strategies are discussed where the charge saturation limitation is considered. First a continuous formation shape feedback control strategy is presented. Next a saturated control strategy is developed to arrest any relative velocities of the Coulomb structure. If the structure can be brought to rest, then the continuous charge control can be engaged to achieve the desired virtual structure. The saturated feedback control is developed using Lyapunov´s direct method and can control the separation rates between the satellites by changing the signs of the three saturated charge products. Implementable real-charge solutions are ensured through scaling the Lyapunov function rate. The control is shown to be Lyapunov stable. Because of the limitations of the control charge magnitudes, certain initial conditions will not lead to the desired zero relative motion rates. Conditions under which the relative motion of the Coulomb structure can be stabilized are analyzed through investigating the total energy of the system in the symmetric motion assumption. The general convergence areas are illustrated numerically in various state planes. Simulations demonstrate the performance of the control.