In this paper, an adaptive neural network approach is developed based on the integral nonsingular terminal sliding mode control method, with the aim of fixed-time position tracking control of a wheelchair upper-limb exoskeleton robot system under external disturbance. The dynamical equation of the upper-limb exoskeleton robot system is obtained using a free and typical model of the robotic manipulator. Afterward, the position tracking error between the actual and desired values of the upper-limb exoskeleton robot system is defined. Then, the integral nonsingular terminal sliding surface based on tracking error is proposed for fixed-time convergence of the tracking error. Furthermore, the adaptive neural network procedure is proposed to compensate for the external disturbance which exists in the upper-limb exoskeleton robotic system. Finally, to demonstrate the effectiveness of the proposed method, simulation results using MATLAB/Simulink are provided.