The sim-to-real gap represent a very important issue in the robotics research field. Challenging problems like the stabilization of highly chaotic underactuated systems are very sensitive to changes with respect to the simulated environment. Region of attraction (RoA) estimation gives an intuitive point of view about the controller robustness to uncertainty in the environment, in the parametric model and to disturbances. Moreover, the time-varying version of this analysis can provide a stability guarantee for the whole of a nominal trajectory that brings the initial state to the desired fixed point. This powerful tool permits to express the controller robustness as an objective in optimization problems. An optimal behaviour of a system to achieve a specific task is the result of an interplay between trajectory optimization, stabilization and design optimization. This coupling renders itself even stronger while dealing with underactuated robots where the controller necessitates the use of passive dynamics to fulfill control objectives. To exploit this shared optimality by taking into account the robust feature is the aim of my thesis work. In this first presentation an overview of previous results as well as future plans will be shown.