To assist patients with upper limb motor dysfunction for rehabilitation training, an upper limb rehabilitation exoskeleton robot system is established and an augmented neural network adaptive admittance control strategy based on the barrier Lyapunov function is proposed. Firstly, the mechanical mechanism and control system of upper limb rehabilitation exoskeleton are introduced. Then, the design process of the controller is illustrated and Lyapunov stability is demonstrated. Finally, the passive training experiment of trajectory tracking with different inner control loops and the active interaction training experiment based on human-robot interaction force under different admittance parameters are carried out. Experimental results of passive training show that the effectiveness of the augmented neural network is close to human-robot dynamics, and the maximum trajectory tracking error is only 53% of that of fuzzy PID controller. The active interaction training experiment proves that different intensities of rehabilitation training can be achieved under the same training task by adjusting the admittance parameters to match patients with different levels of recovery.