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A backstepping-based tracking control design for uncertain mobile robot systems with non-holonomic constraints is presented. For avoiding the singularity and the necessity of the repeated differentiation of the virtual controller, high-degree polynomials of the affine functions are generally included in many existing kinematic controllers. That unfortunately would cause the possible blowup of the actuators for high-order kinematic systems (e.g. a trailer-type mobile robot) in high-speed motions. Regarding this, an exponentially modulated linear stabilising function is included in this design to alleviate such a difficulty. Next at the dynamic design level, an adaptive control algorithm is developed for attaining the global asymptotic tracking stability of the overall closed-loop system. Two case studies of a unicycle-like and a trailer-type wheeled mobile robots are conducted in the final to demonstrate the effectiveness of the proposed design.