Tracking control is a fundamentally important issue for robot and motor systems, where smooth velocity commands are desirable for safe and effective operation. In this paper, a novel biologically inspired tracking control approach to real-time navigation of a nonholonomic mobile robot is proposed by integrating a backstepping technique and a neurodynamics model. The tracking control algorithm is derived from the error dynamics analysis of the mobile robot and the stability analysis of the closed-loop control system. The stability of the robot control system and the convergence of tracking errors to zeros are guaranteed by a Lyapunov stability theory. Unlike some existing tracking control methods for mobile robots whose control velocities suffer from velocity jumps, the proposed neurodynamics-based approach is capable of generating smooth continuous robot control signals with zero initial velocities. In addition, it can deal with situations with a very large tracking error. The effectiveness and efficiency of the proposed neurodynamics-based tracking control of mobile robots are demonstrated by experimental and comparison studies.