In order to improve the energy efficiency of the underwater snake-like robot, the gliding motion of the underwater glider is introduced into the underwater snake-like robot. The novel underwater glide snake-like robot owns good properties of the underwater snake-like robot with high mobility and the underwater glider with long endurance. In this paper, the gliding gait is mainly studied. The telescopic joint and pitch-yaw joint of the underwater glide snake-like robot are presented firstly. The designed telescopic joint has the characteristic of simultaneously changing the net buoyancy, and the relative position between the center of gravity(CG) and the center of buoyancy(CB), which simplifies the structure of the robot. Then, a detailed three-dimensional mathematical model is derived based on the theorem of momentum and the theorem of moment of momentum, and a steady motion model in two-dimensional plane is also obtained for the analysis of the equilibrium gliding. Further, the values of parameters in the equilibrium state are computed through simulation. According to the developed prototype, pool experiments are carried out and the results verify the effectiveness of the proposed joint structure.