Abstract: Parametric stability of liquid sloshing under verticalexcitations is widely concerned in spacecraft dynamics. However, theinfluence of the coupling between the liquid sloshing and the spacecraftmotion on the parametric stability has not been investigated in the previousstudies. In this paper, the Mathieu's equation of a translationalrigid-liquid coupled system subject to vertical excitations is presentedwith the equivalent pendulum model of liquid sloshing. The one-halfsubharmonic and harmonic stability boundaries of the coupled system areanalyzed using the perturbation method. It is illustrated that, the lateralcoupling between the liquid sloshing and the rigid body motion enlarges theparametric unstable regions and enhances the resonance frequencies. Theeffects decline with the decrease of equivalent sloshing mass. Due to thelinear slosh damping, the reduction of the one-half subharmonic unstableregion is much less than that of the harmonic unstable region. The one-halfsubharmonic responses associated with the first sloshing mode of thecoupled system show that, when the parameters of vertical excitations dropin the unstable region, the rigid body may experience a vertical-lateralcoupling vibration.