## 3 Spiral Rolling-Gait Generation
### Modeling of the Normalized Asymmetric CPG Network  
When the snake-like robot moves forward with a spiral rolling gait, the cooperative coordination between the rotary joints and deflection joints achieves overall rolling propulsion. During obstacle traversal, to better adapt the snake-like robot to the shape and position of the obstacle, asymmetric waveform signals are generated to drive the deflection joints between adjacent rotary joints, thereby adjusting the winding radius of the body units and enabling smooth passage over obstacles along the path. In the non-obstacle traversal phase, the stable symmetric periodic oscillatory signals generated by the CPG (Central Pattern Generator) neural model effectively support the snake-like robot's periodic and regular motion along the power transmission cable. The deflection joints oscillate in a periodic sine wave pattern, ensuring the robot maintains its normal locomotion mode. The traditional Hopf-CPGs oscillator model (see Equation (18)), benefiting from the stability of its limit cycle, allows the network signals to converge quickly under an appropriate learning rate [26]. However, during the winding operation of the snake-like robot, actions such as crossing obstacles can trigger asymmetric movements, requiring acceleration and deceleration during different phases of the cycle. In particular, during the stages of winding over and under the obstacle, it is necessary to reduce the input parameters of the oscillator, such as amplitude \(A\) and angular frequency \(\omega\), at certain moments to generate smooth motion gaits, enabling the robot to pass through the target obstacle safely and smoothly. Additionally, when the obstacle volume is large, the original amplitude range may no longer be suitable for the snake-like robot to wind successfully, necessitating a smooth transition of the output signal to a new amplitude range. However, when reducing input parameters such as amplitude \(A\) (as shown in Figure 7a), angular frequency \(\omega\) (as shown in Figure 7b), and amplitude range (as shown in Figure 7c), it is observed that the output signals of the model become excessively sensitive, leading to signal discontinuities and causing spike-like signals. \[\left[ \begin{array}{c}\dot{x}_{i}\\ \dot{y}_{i}\end{array} \right]=\left[ \begin{array}{cc}\alpha(A^{2}-r_{i}^{2})& -\omega\\ \omega_{i}&\alpha(A^{2}-r_{i}^{2})\end{array} \right]\left[ \begin{array}{c}x_ {i}\\ y_{i}\end{array} \right] \tag{18}\]  
Figure 7: Abnormal output in traditional Hopf-CPG models. (**a**) Output signal caused by \(A\) mutation. (**b**) Output signal caused by \(\omega\) mutation. (**c**) Output signal caused by amplitude range mutation. To address this issue, this paper improves the traditional Hopf-CPG oscillator model. First, the desired limit cycle amplitude \(A\) (referred to as \"amplitude\" in the text) is uniformly standardized to 1, with the limit cycle compensation value incorporated into subsequent calculations, as shown in Equations (19) and (20). This approach prevents the system state variables \(x_{i}\) and \(y_{i}\) from being divided by the old amplitude \(A_{old}\) when changing the winding radius, thereby avoiding an initial position offset of the limit cycle. It also prevents the deflection joint output signal from experiencing abrupt changes, such as those shown in Figure 7a,c, when adjusting the input angular frequency and amplitude during operation. \[\left[ \begin{array}{c}\dot{x}_{i}\\ \dot{y}_{i}\end{array} \right]=\left[ \begin{array}{cc}\alpha(1-r_{i}^{2})& \omega_{i}\\ \omega_{i}&\alpha(1-r_{i}^{2})\end{array} \right]\left[ \begin{array}{c}x_{i} \\ y_{i}\end{array} \right] \tag{19}\]  
\[\theta_{i}=A_{i}\times y_{i} \tag{20}\]  
Secondly, due to the periodicity of the limit cycle [27], its acceleration and deceleration are symmetric. The periodic oscillatory signal is only suitable for segment \(V_{1}\) during cable winding and segment \(V_{3}\) during the ascent onto the vibration damper.