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A position tracking control system is developed for a rotorcraft-based unmanned aerial vehicle (RUAV) using robust integral of the signum of the error (RISE) feedback and neural network (NN) feedforward terms. While the typical NN-based adaptive controller guarantees uniformly ultimately bounded stability, the proposed NN-based adaptive control system guarantees semi-global asymptotic tracking of the RUAV using the RISE feedback control. The developed control system consists of an inner-loop and outer-loop. The inner-loop control system determines the attitude of the RUAV based on an adaptive NN-based linear dynamic model inversion (LDI) method with the RISE feedback. The outer-loop control system generates the attitude reference corresponding to the given position, velocity, and heading references, and controls the altitude of the RUAV by the LDI method with the RISE feedback. The linear model for the LDI is obtained by a linearization of the nonlinear RUAV dynamics during hover flight. Asymptotic tracking of the attitude and altitude states is proven by a Lyapunov-based stability analysis, and a numerical simulation is performed on the nonlinear RUAV model to validate the effectiveness of the controller.