In this study, the desired tracking control of the upper-limb exoskeleton robot system under model uncertainty and external disturbance is investigated. For this reason, an adaptive neural network using a backstepping control strategy is designed. The difference between the actual values of the upper-limb exoskeleton robot system and the desired values is considered as the tracking error. Afterward, the auxiliary variable based on the tracking error is defined and the virtual control input is obtained. Then, by using the backstepping control procedure and Lyapunov stability concept, the convergence of the position tracking error is proved. Moreover, for the compensation of the model uncertainty and the external disturbance that exist in the upper-limb exoskeleton robot system, an adaptive neural-network procedure is adopted. Furthermore, for the estimation of the unknown coefficient related to the parameters of the neural network, the adaptive law is designed. Finally, the simulation results are prepared for demonstration of the effectiveness of the suggested method on the upper-limb exoskeleton robot system.