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This paper presents the dynamic modelling and tracking control of a holonomic 3-wheel omni-directional mobile robot. A proportional integral derivative (PID) controller is used, and its performance depends on the dynamic model's accuracy. The controller design for the switched non-linear system is based on both a dynamic model and a kinematic design. A proposed dynamic model averages the contact radius of the 3-wheels. The resultant dynamic model, on average, is non-linear and smooth, serving as a potential solution for model-based control design. The closed-loop simulation results clearly illustrate the dynamic characteristics of the 3-wheels omni-directional mobile robot, offering valuable insights into its behaviour for further analysis and optimisation. Comparative simulation results using MATLAB are presented and demonstrate the effectiveness and legitimacy of the control mechanism.