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 approximated 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 decreased as far as possible. 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.