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This work demonstrates that the underactuated mechanical systems such as the acrobot, the pendubot, the cart-pendulum system, the Furuta pendulum, and the inertia wheel pendulum can be swung up with a common control principle, and derives a generic control method that uses the principles of the energy pumping and the control of the global integration error. The main contribution of this work is using the numerical solutions of the system equations in the control law with a reasonable computational effort. An integration tool, which is capable of controlling the global integration error, is required to determine the controller parameters. The control aim is achieved in two steps. First, a control law, which steers the system to an arbitrarily close neighborhood of the target equilibrium is presented. Afterwards, the system is asymptotically steered to the target equilibrium state by activating a second control law that is linear, and locally stabilizes the target equilibrium.