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The purpose of this paper is to design an intelligent controller and system experimental implementation for nonlinear translational oscillations with a rotational actuator (TORA) system. In this paper, an adaptive backstepping control scheme using wavelet-based neural networks (WNNs), called ABCWNN, is proposed for TORA system control. According to the estimations of the WNNs, the ABCWNN control is developed by backstepping design procedure such that the system output follows the desired trajectory. Based on the universal approximation ability, we use the WNN to estimate the system uncertainty including fractional force and parameter variance. For system development, the effect of frictional forces is discussed and solved by using the estimation of WNN. Based on the Lyapunov approach, the adaptive laws of WNNs' parameters are obtained. The experimental results are presented to demonstrate the effectiveness of our approach.