This paper presents a backstepping control method using sum of squares (SOS) technique to synthesize a nonlinear controller for a three wheels omni-directional mobile robot. The characteristic of strict-feedback systems is also applied to construct the backstepping controller. The controller design satisfies the Lyapunov direct method and LaSalle-Yoshizawa theorem so that the stability of control systems can be assured. Subsequently, the sum of squares conditions are developed to constraint the performances such that the backstepping controller is optimal. Computer simulations and experimental results for mobile robots demonstrate the effectiveness of the proposed backstepping controller.