## 6 Conclusions  
This paper analyzes the robot flight environment and different working conditions of the FPLIR. It investigates the problem of 3D path-following control in the restricted space of power lines. In addition, this paper proposes the adaptive acceptance circle strategy and the adaptive heading control strategy. The numerical simulation experiments are carried out, and the following conclusions are drawn:  
1. Under the condition of a fixed acceptance circle, compared with the traditional LOS method, the average path-following error of the improved LOS method with adaptive heading control is reduced by 0.167 m, and the average convergence time is reduced by 2.05 s for a path angle of \(\theta\in(0,\pi/2)\). However, when \(\theta\in(\pi/2,\pi)\), the following error remains almost constant, and the convergence time is reduced by 0.6 s. The proposed adaptive heading control strategy significantly improves following accuracy and the convergence speed of the path following, reducing following errors caused by large switching angles and effectively following the reference path.
2. When keeping the reference path and the adaptive heading control scale coefficient unchanged, the adaptive acceptance circle strategy improves the problem of switching the reference path too early or lagging behind. This strategy fully follows the reference path and reduces the following error. When the path angle is between \(\theta\in(0,\pi/2)\), the following trajectory based on the adaptive acceptance circle switches the reference path segment earlier due to the acceptance circle's larger radius. However, this increases the following error while maintaining the controllable state and reduces the convergence time by 0.8 s. When the path angle is between \(\theta\in(\pi/2,\pi)\), the following trajectory based on the adaptive acceptance circle is closer to the reference path because of the smaller radius. This approach reduces the maximum following error by 0.3 m and the convergence time by approximately 0.4 s, leading to shorter convergence times and smaller following errors.
3. Using the improved LOS method, the state feedback path-following control method satisfies the following requirements of convergence, reliability, and accuracy and also has advantages in following accuracy and convergence speed.
4. The proposed method is highly reliant on the selection of the scaling coefficient, \(m\), which must be determined via testing under various conditions. The method is effective for following a path consisting of straight lines, but further research is needed to solve the problem of following a path under different types of paths.  
Conceptualization, T.F. and X.Q.; methodology, T.F., J.L. and X.Q.; software, T.F., J.L. and Y.Z.; validation, T.F., J.L. and Y.Z.; formal analysis, T.F., J.L. and X.Q.; investigation, T.F., J.L. and Y.Z.; resources, Y.W., D.W. and W.J.; data curation, Y.W., D.W. and W.J.; writing--original draft preparation, T.F. and J.L.; writing--review and editing, T.F., J.L. and Y.Z.; visualization, T.F., J.L. and Y.Z.; supervision, X.Q.; project administration, J.L. and X.Q.; funding acquisition, J.L. and X.Q. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the National Natural Science Foundation of China (grant nos. 62163032 and 62063030), the Financial Science and Technology Program of the XPCC (grant nos. 2021 DB003, 2022 CB011 and 2022 CB002-07), and the Science and Technology Special of the Ninth Division (grant no. 2021 JS008).  
Not applicable.  
The authors declare no conflict of interest.