In this work, an adaptive robust control method based on Radial Basis Function Neural Network (RBFNN) is proposed. Inspired by the local response characteristics of biological neurons, this method can reduce the influence of nonlinear errors and unknown perturbations in the extreme working conditions of the aircraft, such as high dynamics and strong vibration, so as to achieve high tracking accuracy. In this method, the dynamic model of the nonlinear error of the fiber optic gyroscope is proposed, and then the unknown external interference observer is designed for the system to realize the estimation of the unknown disturbances. The controller design method combines the design of the adaptive law outside the finite approximation domain of the achievable condition design of the sliding mode surface, and adjusts the controller parameters online according to the conditions satisfied by the real-time error state, breaking through the limitation of the finite approximation domain of the traditional neural network. In the finite approximation domain, an online adaptive controller is constructed by using the universal approximation ability of RBFNN, so as to enhance the robustness to nonlinear errors and external disturbances. By designing the output constraint mechanism, the dynamic stability of the system is further guaranteed under the constraints, and finally its effectiveness is verified by simulation analysis, which provides a new solution for high-precision inertial navigation.