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This study addresses the trajectory tracking control problem of dynamic non-holonomic robotic systems in chained form in presence of parametric and non-prarametric uncertainties. At first, a global adaptive tracking controller is designed by exploiting the full use of the backstepping technique. Then, the proposed controller is modified to preserve its robustness to non-parametric uncertainties. In contrast to most of the previously developed kinematic and dynamic tracking controllers, the proposed controller makes all tracking errors and parameter estimation errors be at least globally uniformly ultimately bounded (GUUB) and exponentially converge to a small ball containing the origin. A Lyapunov-based stability analysis is presented to guarantee the GUUB stability of the tracking errors. The controller is applied to a non-holonomic wheeled mobile robot and simulation results are presented to illustrate the effectiveness of the proposed control law.