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In this paper, we develop a new type of snake-like robot using screw-drive units that are connected by active joints. The screw-drive units enable the robot to generate propulsion on any side of the body in contact with environments. Another feature of this robot is the omnidirectional mobility by combinations of screws' angular velocities. We also derive a kinematic model and apply it to trajectory tracking control. Furthermore, we design a front-unit-following controller, which is suitable for manual operations. In this control system, operators are required to command only one unit in the front; then, commands for the rest of the units are automatically calculated to track the path of the preceding units. Asymptotic convergence of the tracking error of the front-unit-following controller is analyzed based on a Lyapunov approach for the case of constant curvature. The effectiveness of the control method is demonstrated by numerical examples and experiments.