This paper investigates the finite-time robust tracking problem for a 5-DOF (degrees of freedom) upper-limb exoskeleton robot subjected to parametric uncertainties, unmodelled dynamics, and unknown human efforts. By developing the non-singular terminal sliding mode control approach, three innovative schemes of robust torques are proposed to steer configuration variables (angular displacements of joints) of the 5-DOF robotic exoskeleton to the reference trajectories within adjustable finite times. Based on mathematical analysis, it is proven that all suggested schemes of control inputs (input torques) accomplish and provide the mentioned tracking objective accurately. In addition, several new formulas (in the form of inequalities) are derived to determine and tune the needed finite times for achieving the tracking objective. Finally, three numerical examples are given to show the appropriate performance and the acceptable effectiveness of the proposed finite-time robust control schemes.