The snake-like robot has superior flexibility in complex and extreme environments due to its long and narrow robotic arm and redundant degrees of freedom. In this paper, based on the structure of the underactuated snake-arm robot, two trajectory tracking algorithms based on equivalent rod length and equivalent virtual joint are proposed to realize the tracking of the target curve by the snake-arm robot and meet the motion control requirements of the snake-arm robot. Finally, a snake-like robot prototype with an arm length of 1075 mm and 7 joint units is designed and the trajectory tracking experiment is carried out. The results show that the snake-like robot can easily pass through a narrow pipe with a diameter of more than 90 mm. The smooth and vibration-free motion process verifies the correctness of the trajectory control algorithm and the rationality of the structural design.