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A robust adaptive second-order sliding-mode control scheme with finite reaching time is proposed for a class of uncertain non-linear systems. It is shown that the problem is equivalent to the finite-time stabilisation of a second-order input-output dynamics with bounded uncertainties. The controller is designed by using geometric homogeneity and adaptive sliding-mode concept. It contains two parts: one part achieves finite-time stabilisation of the second-order input-output dynamics without uncertainties; the other part rejects bounded uncertainties, and the upper bounds of uncertainties are not required to be known in advance. As a result, a finite-time convergent second-order sliding mode is established. The convergence rate of the second-order sliding mode can be hastened through tuning the controller parameters, and the robustness is ensured. The method is evaluated in simulations on an academic example.