The welding robot in nuclear power plant is used for repairing tasks of control rod guide tubes, thermocouple guide tubes, and baffle screws to prevent loosening welds. It can effectively replace manual operations in high radiation underwater environments. Due to the model uncertainty of the robot system itself and external disturbances, it is difficult for the robot to accurately track operational trajectories. This paper proposes an Adaptive Neural Network Sliding Mode Controller (ANNSMC) applied to welding robots in nuclear power plants, which includes three parts: sliding mode controller (SMC), neural network (NN)controller, and barrier Lyapunov function (BLF) output-constrained controller. The controller uses NN to improve adaptability to uncertain parts of the system and utilizes BLF to ensure tracking capability of trajectories when subjected to external disturbances. A simulation was conducted on a 2-link robot to evaluate the performance of this controller. The steady-state position error at the end of the robot is better than 0.01 rad.