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In this paper, a novel biologically inspired approach to real-time tracking control of a nonholonomic mobile robot is proposed. The proposed algorithm incorporates a neural dynamics model derived from a biological membrane equation with the conventional full-state tracking control technique. It is capable of generating real-time smooth and continuous velocity control signals that drive the mobile robot to follow desired trajectories. The proposed approach resolves the speed jump problem existing in some previous tracking controllers. In addition, it can track both continuous and discrete paths. The practicality and effectiveness of the proposed tracking controller were demonstrated by simulation and comparison results.