Kinematic inversion via closed-loop schemes is at the core of robotic manipulators' success. These algorithms robustly and efficiently compute a reference for low-level controllers to position the end effector or other selected parts of the robot's body in space. This work concerns the extension of kinematic inversion algorithms beyond fully actuated systems into the realm of underactuated robotics. We revise a well-known kinematic inversion algorithm in continuum (soft) robotics, the direct task-to-actuators inversion, and generalize it beyond this field and its current strong assumptions. More precisely, in its existing formulation, the algorithm relies on geometrical assumptions that hold valid only if the robot is fully actuated. We leverage recent advancements in system dynamics of mechanical systems to generalize the method beyond soft robotics and provide a constructive strategy that does not rely on any full actuation assumption. After that, we propose a second closed-loop inversion algorithm that works in even more general conditions, not requiring the explicit derivation of end-to-end actuators to task space mapping.