In this presented work, a new tracking control strategy is designed for a kind of nonholonomic system under external disturbances. First, the whole system is divided into two design stages by using the relay switching technique. Then, using state transformations, the two primitive subsystems are converted into unconstrained systems for which tracking controllers are designed based on dynamic surface control. Different from most existing finite-time control strategies and fixed-time control strategies, our controllers are able to control the tracking error within a specified accuracy within a specified time, starting from anywhere, without changing the control structure/parameters. Meanwhile, all closed-loop signals keep bounded in the whole control process. Extensive computer simulation and physical experiments have been given to validate good performance of the designed controller.