At first, the non-holonomic and holonomic constraints of the three-wheeled mobile robot were clarified by means of its geometric relation. Then, its dynamical equation was deduced by using the Euler-Lagrange equation of non-holonomic mechanical systems. In order to make the three-wheeled mobile robot move along a given trajectory curve accurately, the target trajectory curve was transformed into a speed form. After that, the speed target was converted to a form approximating to the actual initial speed by introducing a differential homeomorphic transformation. In this case, the initial speed error and cumulative position error of the controlled systems can be minimized. Finally, the trajectory tracking control of the three-wheeled mobile robot was designed by using optimal control and integral sliding mode control. Simulation results show that the proposed control method can compel the three-wheeled mobile robot to move along a given trajectory curve accurately with a certainty of robustness.