## 1 Introduction  
The power industry plays a vital role in socioeconomic development. However, because of factors such as elevated power lines and challenging terrain, power line inspection has become a dangerous and labor-intensive task. To improve efficiency and safety, the advent of power line inspection robots has been instrumental in replacing manual inspection. Over the past decade, power line inspection robots have been developed mainly in three different modes: aerial mode, climbing mode, and hybrid mode. Aerial-mode robots, equipped with lightweight and highly accurate remote sensing devices, have enabled efficient and accurate inspections. However, their stability and endurance are severely limited by various factors such as weather and wind directions. Climbing-mode robots offer longer endurance and greater stability, but their complicated structural designs and inability to autonomously ascend and descend power lines have become limiting factors in their application to power line inspection [1; 2; 3]. Hybrid-mode robots combine the advantages of aerial-mode robots and climbing-mode robots. They use the flying mode to suspend themselves on the power line and the walking mode to control their movement along the power line. In addition, when they encounter obstacles, they activate the flying mode to bypass the obstacles and then reattach to the power line and resume the inspection [4; 5; 6]. This design overcomes some of the limitations of climbing-mode robots and provides greater stability and precision compared with aerial-mode robots. Power lines are often installed in varied and harsh terrain such as mountains, valleys, and deserts. These geographical features, coupled with the segmentation of the aerial flight space by the presence of power lines, impose limitations on the flight space for aerial-mode robots. As a result, viable path generation becomes notably intricate [7; 8; 9]. Within the confined space constrained by these power lines, the ascent and descent process for hybrid-mode robots requires careful attention to both path-following accuracy and maintaining flight safety. This is essential in order to avoid any potential damage to the surrounding environment and equipment. Currently, unmanned aerial vehicles (UAVs) are being applied in various fields, including the military and civilian sectors. Achieving accurate path-following technology is important for improving the performance and applications of UAVs [10; 11; 12]. Consequently, path-following research is of great value. At present, both domestic and international studies of path following can be categorized into two approaches: those based on control theory and those based on geometry [13]. Among the cybernetic-based path-following methods, [PERSON] et al. [14] address the UAV trajectory-tracking problem by deriving explicit parameter uncertainty equations and applying neural networks for uncertainty compensation. Meanwhile, a model reference based on the adaptive control framework has been adopted to improve tracking performance. [PERSON], [PERSON] et al. [15] used an optimization strategy combining the genetic algorithm with sequential quadratic programming for offline optimal trajectory computation and online path updating. A trajectory-tracking algorithm based on nonlinear model predictive control regulates air braking to ensure trajectory rationality. [PERSON], M et al. [16] proposed an estimation-based propulsion class control strategy for the UAV tracking of a moving target in 3D space. The target state is estimated using an extended Kalman filter and combined with the estimation and the UAV state to achieve velocity, trajectory, and heading angle control. That study considered measurement uncertainty and simulated tracking performance in different scenarios. This approach is expected to improve the practical effectiveness of UAV target tracking. [PERSON], P et al.