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In this paper, we develop a prototype of a new snake-like robot using the screw drive mechanism, and design a control system design for trajectory tracking. First, we explain the outline of the snake-like robot using the screw drive units and joint units, as well as the principle of the screw drive mechanism. Next, in order to achieve the trajectory tracking of the robot, we derive a kinematic model. Finally, the validity of the derived model and the effectiveness of the proposed control system are demonstrated by simulations and experiments.