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In this paper, a novel biologically inspired tracking controller is proposed for real-time navigation of nonholonomic mobile robots, which is inspired by the agonist/antagonist effects in muscular sensory motor reaction of a gated dipole neural model. Through the incorporation of the biological computation element, the proposed controller is capable of generating smooth, bounded acceleration command signals for the mobile robot to track reference paths. It resolves the velocity jump problem in the conventional backstepping controllers that result from the initial tracking errors or the discontinuities in discrete paths. In addition, the proposed controller removes the unpractical assumption of "perfect velocity tracking" in some existing approaches. The effectiveness and efficiency of the proposed approach are demonstrated by simulation and comparison studies.