Saturating-proportional-derivative-type global continuous control for the finite-time or (local) exponential stabilisation of mechanical systems with bounded inputs is achieved involving the desired conservative-force compensation. Far from what one could expect, the proposed controller is not a simple extension of the on-line compensation case but it rather proves to entail a closed-loop analysis with a considerably higher degree of complexity. This gives rise to more involved requirements to guarantee its successful performance and implementability. Interesting enough, the proposal even shows that actuators with higher power-supply capabilities than in the on-line compensation case are required. Other important analytical limitations are further overcome through the developed algorithm. Experimental tests on a two-degree-of-freedom robotic arm corroborate the efficiency of the proposed scheme.