## 1 Introduction
[17] used state correlation coefficients in the form of the nonlinear properties of the system embedded in a pseudo-linear system matrix and obtained nonlinear equivalents of model predictive control. The nonlinear model predictive control law was derived by converting the continuous system into a sampled data form and using a sequential quadratic programming solver to handle input, output, and state constraints. Among the geometry-based methods, pursuit algorithms [18; 19], the LOS method [20; 21; 22; 23], and nonlinear guidance techniques [24; 25; 26] have been widely employed for UAV path following. Among these, the LOS method is distinguished by its real-time capability, high accuracy, and ease of implementation. Given these advantages, it has been widely used in UAV path-following applications. Consequently, this study uses the LOS method to investigate 3D path following. The LOS method uses the geometric relationship between the UAV and its target to facilitate path following. This is achieved by manipulating the relative angular disparity. The system utilizes inputs such as the UAV's current position and waypoints along the target path. It employs a pure following control system to calculate the desired LOS direction for the UAV. This calculated direction is then compared with the actual LOS direction. The resulting control deviation is processed through a control algorithm to generate control commands that ultimately cause the UAV to follow the target path. In practical applications, this method needs increased accuracy and real-time reliability. Within 3D path following, the LOS method has to consider factors such as the UAV's distance from the target points and their heights. To facilitate 3D path following, the traditional LOS method needs to be adapted to address its limitations in this context. The traditional LOS method uses a fixed acceptance radius, which is characterized by its simplicity but is accompanied by significant following errors and convergence problems. In an effort to address these limitations, [PERSON] et al. [20] introduced an integral term compensation approach for 3D path following in underwater vehicles. This approach counteracts the drift effects caused by environmental disturbances, thereby enabling improved straight-line navigation. [PERSON] et al. [27] and [PERSON] et al. [28] introduced an adaptive acceptance radius strategy where the radius is inversely proportional to the angle of the path, providing favorable following results. [PERSON] et al. [29] proposed an exponential acceptance radius strategy that significantly improves convergence. However, the introduction of the exponential function results in a lengthy LOS angle calculation. While these methods have collectively reduced the average following error, challenges remain for small angles where significant errors and slow convergence problems exist. The traditional LOS method has certain shortcomings in 3D path following for UAVs. Adjusting the acceptance radius is one way to improve following accuracy and convergence speed. Future research will focus on improving following accuracy, reducing computation time, and overcoming following errors and slow convergence associated with small path angles. Heading control for robots is an important issue in 3D path following. The chosen heading control strategy has a direct impact on the convergence and performance of path-following algorithms. An appropriate heading control strategy can significantly improve both the accuracy and efficiency of path following. Currently, numerous heading control strategies have been extensively researched and applied in the field of path following. [PERSON] et al. [30] augmented the LOS method with an extended state observer to achieve more accurate heading angle estimation errors, resulting in remarkable effectiveness. [PERSON] et al. [31] introduced two extended state observers to estimate lateral and vertical velocities, indirectly calculating the sideslip angle. This information was used to perform accurate compensation within the LOS method.