Many observation satellites are subject to attitude constrains arising from peculiar mission requirements or environmental conditions. These obstructions often constrain the direction of the thrust vector to remain within a cone. In this study, we investigate the local controllability of station-keeping maneuvers of satellites with low thrust capabilities or small chemical impulsions on nominal periodic orbits subject to such constraints. We offer a numerical methodology based on convex optimization to identify the minimum cone angle guaranteeing local controllability for a specific orbit. We compare the results with a dynamical approach using Floquet Modes. An illustrative example inspired by the James Webb Space Telescope is proposed. Specifically, we consider a satellite is on a Halo orbit around L 2 in the Sun-Earth circular restricted three-body problem.