## 4 Simulation Experiments and Analysis
### Experiments on Heading Control Strategy  
To evaluate the effectiveness of the improved adaptive heading control strategy on path following, we compare its performance with a state feedback control strategy. The comparison is performed using a fixed acceptance circle radius: \(R_{0}=0.5\) m. The results of this experiment show that the new strategy improves the path-following accuracy. The flying speed of the robot was maintained at \(v=0.2\) m/s during the simulation experiment.  
The experimental results are presented in the following figures: Figure 9 shows the 3D path-following results, Figure 9a shows the following trajectory of the 3D path following, Figure 9b shows the _xoy_ plane projection of the 3D path-following trajectory, and Figure 9c shows the _xoz_ plane projection of the 3D path-following trajectory. The simulation results reveal the robot's ability to execute path following on the reference path under different following methods. Figure 9d shows the following error versus time curve between the robot's real-time position and the reference path. The abrupt change in the error curve corresponds to the moment it reaches the corner of the reference path. It is worth noting that the oscillations of the following error in Figure 9d are caused by the earlier updating of the error statistics with the \(P_{LOS}\) update. The variations in Figure 9d correspond to the path points \(a,b,\) and \(c\). As depicted in Figure 9, the following trajectory with the improved LOS is closer to the reference path.  
In order to further verify the effect of adaptive heading control on improving path-following accuracy and convergence speed, the details of the horizontal projection, vertical projection, and following error curve of the robot's following trajectory at points \(a,b,\) and \(c\) are shown in Figures 10-12. The following conclusions can be obtained by analyzing Figures 10-12.  
Figure 9: Results of 3D path following. (**a**) 3D path following trajectory; (**b**) projection of 3D path-following trajectory onto the _xoy_ plane; (**c**) projection of 3D path-following trajectory onto the _xoz_ plane; (**d**) comparison of following errors in 3D path-following trajectories.  
point \(c\), the maximum following error decreases from 0.152 m to 0.148 m, indicating a reduction in the following error of 0.004 m. This improvement also enhances the oscillatory convergence phenomenon.
2. The convergence time of the adaptive heading control is faster. At point \(a\), the convergence time is reduced from 8.1 s (from 33.7 s to 41.8 s) to 5.1 s (from 33.7 s to 38.8 s), and the time taken to converge is reduced by 3 s. At point \(b\), the convergence time is reduced from 9.3 s (from 67.9 s to 77.2 s) to 2.8 s (from 66.6 s to 69.4 s), and the time consumed for convergence is reduced by 6.5 s. Similarly, at point \(c\), the convergence time is reduced from 5.8 s (from 102.1 s to 107.9 s) to 4.7 s (from 100.0 s to 104.7 s), and the time taken for convergence is reduced by 1.1 s. Convergence time is determined when the following curve intersects the 5% error margin.  
Similarly, an adaptive heading control strategy will provide a much higher approach speed and lower following error, resulting in better control results.