Rehabilitation robotics has seen growing popularity in recent years due to its immense potential for improving the lives of people with disabilities. However, the complex, uncertain dynamics of these systems present significant control challenges, requiring advanced techniques. This paper introduces a novel adaptive control framework integrating modified function approximation (MFAT) and double-integral non-singular terminal sliding mode control (DINTSMC). The goal is to achieve precise tracking performance, high robustness, a fast response, a finite convergence time, reduced chattering, and effective handling of unknown system dynamics. A key feature is the incorporation of a higher-order sliding mode observer, eliminating the need for velocity feedback. This provides a new solution for overcoming the inherent variations and uncertainties in robot manipulators, enabling improved accuracy within fixed convergence times. The efficacy of the proposed approach was validated through simulations and experiments on an exoskeleton robot. The results successfully demonstrated the controller’s effectiveness. Stability analysis using Lyapunov theory proved the closed-loop system’s uniform ultimate boundedness. This contribution is expected to enable enhanced control for rehabilitation robots and improved patient outcomes.