In this paper, we revisit the energy-based swing-up control problem for a rotational pendulum. Different from the existing energy-based control solution, first, we present a necessary and sufficient condition such that the control law has no singularities for all states of the rotational pendulum. Next, we carry out a global motion analysis of the pendulum, and we remove the previous required constraint on the initial state of the pendulum and the control parameters for preventing the pendulum getting stuck at the downward equilibrium point by revealing that the point is a saddle. Specifically, we show that the Jacobian matrix evaluated at the point has two and two eigenvalues in the open left-and right-half planes, respectively. We prove that the pendulum will eventually be swung up into the basin of attraction of any (locally) stabilizing controller for all initial conditions with the exception of a set of Lebesgue measure zero. Finally, we validate the presented theoretical results via numerical simulation. Our simulation results show that the swing-up control can be achieved quickly under the improved conditions on the control parameters.