Spherical robot is a special kind of nonholonomic system that cannot be converted to chained form, which means most of the well-known control methodologies are not suitable for this system. For the trajectory tracking of BHQ-1, a spherical robot designed by our lab, a two-state trajectory tracking controller is proposed in this paper. First, the kinematic model of the robot is built using screw theory and exponential method and the controllability is proved based on the differential geometric control theory. Then to solve the two-state trajectory tracking problem of BHQ-1, a shunting model of neurodynamics and Lyapunov's direct method are combined to design a two-state trajectory tracking controller, of which the Lyapunov stability is validated. Finally, typical simulation examples, such as tracking linear, circular, and sinusoidal trajectories, are introduced to verify the effectiveness of the proposed controller. In this paper the proposed method can also be applied to the control of other spherical robots.