Trajectory tracking is a fundamental requirement for wheeled mobile robots, particularly in industrial applications that demand precise motion control. This experimental study presents a trajectory-tracking control strategy for a four-wheeled mecanum robot. The work begins with the derivation of the robot kinematic model. Following this, a system identification approach is employed to capture the robot's dynamic behavior accurately. This involves collecting and analyzing input-output data to develop a comprehensive dynamic model of the robot. A deep neural network (DNN) is utilized as a black-box model to effectively learn and represent the complex nonlinear behavior of the system. The DNN is trained on extensive input-output data, ensuring it can generalize well to various operational scenarios. Once the dynamic model is established, a separate DNN-based controller is designed. This controller leverages the insights gained from the dynamic model to generate precise control signals, enabling the robot to follow the desired trajectory accurately. The proposed system identification method demonstrates remarkable accuracy, achieving a 99.98% fit to the training data, which is indicative of the model's robustness and reliability. To validate the effectiveness of the approach, experimental tests are conducted using an infinity-shaped trajectory. The results are highly promising, with the controller achieving precise tracking marked by a mean squared error of 0.0005 meter. This level of precision highlights the potential of deep learning techniques in addressing complex control challenges in wheeled mobile robots. The combination of system identification and deep learning offers a powerful toolset for developing advanced control systems.