In this paper a strong Lyapunov function is obtained, for the first time, for the supertwisting algorithm, an important class of second order sliding modes (SOSM). This algorithm is widely used in the sliding modes literature to design controllers, observers and exact differentiators. The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm. These modification allows the system to deal with linearly growing perturbations, that are not endured by the basic supertwisting algorithm. Moreover, the introduction of Lyapunov functions opens many new analysis and design tools to the Higher Order Sliding Modes research area.