Underactuated systems are mechanical systems for which one or several degrees of freedom are not actuated, alike most standard aerial and marine vehicles, and also a few ground vehicles (hovercraft, for example). By contrast with fully actuated nonholomic systems their equations of motion cannot be reduced to (driftless) kinematic equations. They also involve dynamic equations that express how displacements along directions associated with non-actuated degrees of freedom are coupled with displacements that are directly controlled with the available control input forces and/or torques. Alike nonholonomic systems, eventhough the original nonlinear system may be (locally) controllable everywhere, the linearization of these equations at certain equilibria, or along certain solution trajectories, may not be controllable, and Brockett’s obstruction concerning the non-existence of pure-state asymptotic stabilizers may also apply. For instance, in the case of conventional motor boats –in the absence of sea currents and wind– the problem occurs when it comes to asymptotically stabilize a desired fixed position/orientation, i.e. a configuration for which the sum of all external (environmental) forces acting on the system vanishes. The problem is less critical for aerial vehicles due to the action of the gravitational force which prevents the sum of external forces to vanish along most trajectories.

With respect to the case of kinematic nonholonomic systems, the application of the transverse function approach to the design of practical stabilizers for underactuated systems is less direct and involves a complementary backstepping stage.