This article presents a method to design controllers that force an underactuated omni-directional intelligent navigator (ODIN), a spherical underwater vehicle, to track a reference trajectory in three-dimensional space. The vehicle under consideration has only four thrusters but six degrees of freedom to be controlled. Motivated by the vehicle's steering practice, the roll and pitch angles regarded as virtual controls plus four actual control forces provided by the thrusters are used to force the position and yaw angle of the vehicle to globally and asymptotically track their reference trajectories. The control design is based on the one-step ahead backstepping method and Lyapunov's direct method. A combination of Euler angles and unit-quaternion for the attitude representation of the vehicle is used to obtain global tracking control results. Simulations illustrate the results.