## 3 Spiral Rolling-Gait Generation
### Generation of Helical Rolling Gait
From the 7 th to the 23 nd second, due to the need to navigate around a vibration damper with a diameter of 127 mm, the deflection joint's movement speed gradually decreases, and the maximum pitch angle reduces to accommodate the larger diameter. Once the robot successfully navigates the vibration damper, the deflection joint's movement returns to its normal oscillatory form. From the 26 nd to the 32 th second, as the robot's tail gradually detaches from the vibration damper and continues winding along the cable, both the movement speed and maximum pitch angle of the deflection joints gradually return to the winding cable mode. Figure 11: Snake robot spiral winding rolling crawling process. Figure 12: The variation in joint angles of the snake-like robot over the 0–28 s period. (**a**) Signal curves of the pitch joints from 0 to 28 s. (**b**) Signal curves of the yaw joints from 0 to 28 s.  
Throughout the motion, the yaw joints' motion period remains synchronized with that of the pitch joints, and their acceleration and deceleration patterns are consistent. The signal variation curves of the pitch and yaw joints during the winding motion indicate that, when using the UA-CPGs oscillator for helical winding and rolling motion, the system naturally generates periodic oscillatory signals. It is also capable of smoothly adjusting the output amplitude, velocity during the period, and the resulting oscillation signals transition smoothly. Additionally, the joint angle signals do not exhibit noticeable sudden changes, and the joint noise is relatively low, suggesting that the gait demonstrates a certain level of stability during the snake-like robot's winding and inspection process along the cable. To validate the capability of the UA-CPGs oscillator-generated spiral rolling gait in traversing obstacles of different diameters, motion simulations were performed on cables with cylindrical obstacles of 75 mm, 127 mm, and 250 mm in diameter. By incorporating the varying obstacle diameters into Equations (6)-(17), the state parameters of the UA-CPGs at each stage were calculated, followed by simulation analyses of the snake-like robot's obstacle-crossing performance in CoppeliaSim Edu (v.4.1.0). Figure 13 shows the motion simulation of the snake robot on the cable of a cylindrical obstacle with a diameter of 75 mm, 127 mm and 250 mm. The results demonstrate that the spiral rolling gait designed in this study enables the robot to smoothly climb and traverse obstacles of different sizes without any occurrences of jamming, detachment, or instability during motion. Hence, the proposed spiral rolling gait exhibits considerable reliability. To evaluate the effectiveness of the UA-CPGs-generated spiral rolling gait in improving speed, a comparative analysis was conducted in a simulation environment between the gait speed curves generated by the dual-chain Hopf-CPGs oscillator and the UA-CPGs oscillator, as shown in Figure 14. Considering weather conditions such as rain and ice that may affect the actual friction between the snake-like robot and the cable surface, the cable friction coefficient was set to 0.3, 0.4, 0.5, and 0.6 in the simulations, while other conditions remained unchanged. For the dual-chain Hopf-CPGs model, the maximum crawling speeds around the cable at different friction coefficients are 0.90 cm/s, 0.88 cm/s, 0.87 cm/s, and 0.80 cm/s, respectively. For the UA-CPGs model, the maximum speeds are 1.28 cm/s, 1.43 cm/s, 1.35 cm/s, and 1.32 cm/s, respectively. Compared to the dual-chain Hopf-CPGs model, the UA-CPGs model shows an average speed increase of 55.70%. This demonstrates that the helical rolling gait generated by this method offers a significant advantage in improving the robot's motion speed. Figure 13: Simulation of the snake-like robot’s motion with obstacle diameters of 75 mm, 127 mm, and 250 mm.