Interconnection and damping assignment passivity-based control (IDA-PBC) is a method that has been developed to (asymptotically) stabilize nonlinear systems formulated in the port-controlled Hamiltonian (PCH) structure. This method has gained increasing popularity and has been successfully applied to a wide range of dynamical systems. However, little is known about the robustness of this method in response to the effects of uncertainty which could result from disturbances, noises, and modeling errors. This paper explores the possibility of extending some energy shaping methods, taking into account the robustness aspects, with the aim of maintaining (asymptotic) stability of the system in the presence of perturbations which inevitably exist in any realistic applications. We propose constructive results on robust IDA-PBC controllers for underactuated mechanical systems that are quite commonly found in practice and have the most challenging control problems within this context. The proposed results extend some existing methods and provide a new framework that allows the implementation of integral and input-to-state stability controllers to underactuated mechanical systems. The results are applied to two physical systems: an inertia wheel pendulum and a rotary inverted pendulum that represent separable and nonseparable PCH systems, respectively.