## 3 Path-following Control Based on Improved LOS
### Improved LOS  
To reduce path following error and improve convergence speed, while taking into account the effects of the minimum turning radius, the flight speed, and the angle between the path segments, the acceptance circle in the method must be adaptively adjusted based on the angle of the path segments and the following error. This avoids situations of switching lagging or premature switching when the reference path is changed, which can cause an increase in the path-following error and a slower convergence speed. Therefore, a turning radius threshold is set based on the segment angle and the flight speed of the robot during switching to establish a correspondence between the acceptance circle radius, \(R_{0,i}\); the path segment angle, \(\theta\); and the flight speed \(V\). This reduces the path-following error and improves the convergence speed. The principle of the adjustment is as follows:  
When \(\theta\in(0,\pi/2)\), the robot switches the path segment with a larger acceptance circle radius. The robot tracks the position of the reference point in the next path segment beforehand and adjusts the heading angle and trajectory inclination to avoid the increased  
Figure 4: The following strategy of the LOS method. following error due to the delay in switching the reference path segment. When \(\theta\in(\pi/2,\pi)\), a smaller acceptance circle radius is required to follow the reference path to avoid the incomplete following of the previous path segment due to switching the reference path too early. The path-following method includes a navigation subsystem and a control subsystem. The navigation subsystem calculates the path-following error based on the improved LOS, while the control subsystem computes the control command to implement the path-following control based on the output of the navigation subsystem. Scaling coefficients are introduced to adjust the following priorities of parallel and vertical paths for the control commands of the navigation subsystem. The following error is quickly eliminated, and a high-precision path-following effect is obtained. This paper investigates the adaptive admittance circle and adaptive heading control to reduce the path-following error and accelerate the convergence speed, taking into account the UAV's maneuvering performance constraints and following path information. During path following, the following error changes, while the reference point is determined by the forward-looking distance. When the reference point is in the next segment of the reference path, the radius of the admittance circle at the current corner is calculated based on the size of the corner and the current speed. When the UAV is inside the acceptance circle, the reference path segment is switched, and the next closed-loop control begins. Algorithm 1 shows the operation logic of the improved LOS method:  
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#The forward control vector, \(n^{t}\), is continuously adjusted based on the distance from the robot's position, \(P\), to the path in order to achieve the fast convergence of the robot's trajectory. functionPathFollowing #Calculate the path segment on which the reference point, \(P_{LOS}\), is located: \(P_{l\to 1}P_{l}/P_{l}P_{l+1}\) Whilei<ndo \((n^{t}_{L},n^{t}_{l})\leftarrow\) ConstructVector(); \(\theta_{d}\leftarrow\) TheoryForwardDirectionAngle(\(n^{t}\)) ; (\(\theta_{d}\) is theoretical angle) \(\theta\leftarrow\) ActualAngle(); (\(\theta\) Real angle) \(u\leftarrow\) StateFeedbackControl(\(\theta_{d},\theta\)) ; #The mass center arrives at the end neighborhood and then moves to the next control segment. The next control phase starts when \(P_{LOS}\in P_{l}P_{l+1}\).