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This article proposes an adaptive trajectory tracking control scheme for low-speed car-like vehicles with less efforts in tuning of the control gains. An interesting way of integrating adaptive control gains with consideration of steering saturation by using the backstepping technique is designed to enhance trajectory tracking while ensuring the commanded inputs within the input boundaries. The design of such adaptive control gains is also based on enhancing the convergence rate of tracking errors, especially for lateral deviation from the reference trajectory. It is further theoretically proven that, even under the influence of steering saturation, the proposed controller can make the closed-loop system approximately globally asymptotically stable at zero errors. Comparative MATLAB/Simulink simulations and experimental tests based on Quanser latest self-driving car have been conducted to verify the effectiveness of the proposed control scheme in accurate tracking without violating the input constraints. [ABSTRACT FROM AUTHOR]