## 2. Serpentine Robot Configuration and Kinematic Modeling  
The serpentine inspection robot designed for high-voltage transmission lines consists of ten P-R-S module units connected in series. Each unit has three degrees of freedom: deflection, rotation, and translation, with the head unit requiring no deflection and the tail unit requiring no rotation [23]. Figure 1 shows the model of a single unit of the serpentine robot. The deflection servo is connected to the inner sleeve via a rudder disk, linked in series to the next unit. It achieves deflection motion through the rotation of the servo. The rotary servo drives the internal gear via a small gear to complete the rotational motion between units. The translation servo, on the other hand, is connected to the inner and outer sleeves through a bolt and nut, enabling relative sliding motion. Given that the spiral rolling gait and obstacle-crossing movement mainly rely on the deflection and rotation of the joints to achieve complex trajectories and posture adjustments, this paper simplifies the model by focusing solely on the signal processing of the deflection and rotary servos. Figure 2 shows the coordinate system describing the kinematic model of the snake-like robot. {O} represents the global coordinate system, {B} represents the base coordinate system, and the coordinate system {K} represents the rotary joint, which rotates around the Z-axis. {2K} represents the deflection joint, which rotates around the Y-axis. The adjacent units of the snake-like robot (i.e., the deflection joint of the previous unit and the rotary joint of the next unit) are connected at node \(n\), with the node position denoted as \(L_{n}\). The position of the node is expressed as follows:  
\[L_{n}=\frac{nL_{T}}{N_{T}} \tag{1}\]  
Figure 1. Serpentine robot unit. 1. Check plate sleeve 2. Deflection servo 3. Translation servo 4. Rotary servo 5. Inner sleeve. where \(L_{T}\) represents the total length of the robot, and \(N_{T}\) represents the number of units. For the convenience of calculation, a base is added to the head joint. Figure 3 describes the geometric relationship between the joints of the snake-like robot. The joint angle \(\theta_{pi}\) is the angle between the central axes of adjacent units \(i\) and \(i-1\), representing the rotation angle of the deflection joint. The cable is approximated as a cylindrical shape, and its winding is approximated as an equidistant helix. Based on the coordinate system shown in Figure 2, the rotation matrix of the first joint in the base coordinate system B is as follows:  
\[ \begin{array}{ll}\mathbb{\ }_{1}^{B}R=\begin{bmatrix}\cos\theta_{1}&-\sin\theta_{1}& 0\\ \sin\theta_{1}&\cos\theta_{1}&0\\ 0&0&1\end{bmatrix}\end{array}  \tag{2}\]  
The rotation matrix of the \(2K\)th joint module is as follows:  
\[ \begin{array}{ll}\mathbb{\ }_{2K}^{K-1}R=\begin{bmatrix}\cos\theta_{2K}&0&\sin \theta_{2K}\\ 0&1&0\\ -\sin\theta_{2K}&0&\cos\theta_{2K}\end{bmatrix}\end{array}  \tag{3}\]  
The rotation matrix of the \((2K+1)\)th joint module is as follows:  
\[ \begin{array}{ll}\mathbb{\ }_{2K+1}^{K}R=\begin{bmatrix}\cos\theta_{2K+1}&-\sin \theta_{2K+1}&0\\ \sin\theta_{2K+1}&\cos\theta_{2K+1}&0\\ 0&0&1\end{bmatrix}\end{array}  \tag{4}\]  
Figure 3: Schematic diagram of the snake-like robot joints. Figure 2: Schematic diagram of the coordinate system of the snake-like robot. where \(\theta_{k}\) represents the angle between the current joint and the extended line of the previous joint's central axis. Since each unit of the snake-like robot has both a rotary joint and a deflection joint, the position \(P^{B}\) of a point \(P^{i}\) (i is even) on the deflection joint is used to represent the pose coordinates of the snake-like robot unit relative to the reference coordinate system.