Benchmark problems continue to represent an actively studied domain, focusing on application-based situations, where controllers have to deal with typical real environments. In this paper, a Robust Fixed Point Transformations (RFPT)-based Model Reference Adaptive Controller (MRAC) is designed for a modified Translational Oscillations by a Rotational Actuator (TORA) system, which is an indirectly driven, underactuated classical mechanical system with peculiar properties. The RFPT-based design has the advantage of working only with three free parameters, and does not need complex a priori calculations. It is founded on the idea that at the cost of replacing the requirement for global stability with local stability, a mathematically very simple and geometrically lucid, well interpreted methodology can be developed. The resulting structure directly concentrates on the primary design intent, i.e., on the realization of a purely kinematically prescribed trajectory tracking. Examples and simulation results are presented in this paper, demonstrating that the RFPT-based design can provide an efficient MRAC controller for a very special physical system.