This paper concerns the swing-up control of an underactuated two-link robot with a linear torsional spring-attached ﬁrst joint and an actuated second joint (called the FA robot below) moving in a vertical plane. First, we present a necessary and sufficient condition such the FA robot is linearly controllable at the upright equilibrium point (UEP, where two links are both upright). Second, we prove without any assumption that the FA robot is at an equilibrium point provided that its actuated joint angle is constant under a constant torque. Third, for the FA robot with its torsional stiﬀness of the spring being smaller than a value determined by the coefficients of its gravitational terms, we propose an energy-based controller without singular points to swing it up. We conduct a global motion analysis for the FA robot under the proposed controller. For the case that the total mechanical energy of the FA robot converges to its desired value, we present its phase portrait. For the case that the convergence is not achieved, we show that the FA robot approaches an equilibrium point belonging to a set of equilibrium points, and propose a sufficient condition to check its stability. From the motion analysis, we present a sufficient condition such that the FA robot can be swung-up close to the UEP under the proposed swing-up controller. Finally, we verify our theoretical results through a numerical simulation.