# Curved Path Following Control for a Small Fixed-Wing UAV with Parameters Adaptation
## 4 Simulation
### Circular Path Following  
Figure 11a shows the position comparison while following the circular path using the three methods. The desired circular path is defined as two concentric circles with circle A: \(\sqrt{x^{2}+\left(y-350\right)^{2}}-250=0\), and circle B \(\sqrt{x^{2}+\left(y-350\right)^{2}}-350=0\). The UAV first flew toward circle A. About 180 s later, the UAV started to switch from the position (0, 600) and flew toward circle B with a radius of 350 m.  
The distance \(f(x,y)\) presented in Figure 11b shows that, with the parameter \(k_{1}\) adaptive, the method PFC_\(k_{1,adaptive}\) inherits the advantages of the rapidity of the method PFC_\(k_{1,max}\) and the small overshoot of the method PFC_\(k_{1,min}\). The error overshoots of \(f(x,y)\) for the first circular path following using the three methods PFC_\(k_{1,max}\), PFC_\(k_{1,min}\), and PFC_\(k_{1,adaptive}\) are 22.65 m, 11.07 m and 12.52 m, respectively. When switching to fly toward circle B, the error overshoots of \(f(x,y)\) using the methods PFC_\(k_{1,max}\), PFC_\(k_{1,min}\), and PFC_\(k_{1,adaptive}\) are 14.59 m, 2.25, and 3.03 m, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Segment**} & \multicolumn{4}{c}{**LOPPF (m)**} & \multicolumn{4}{c}{**Convergence Time (s)**} & \multicolumn{4}{c}{**Error Overshoot (m)**} \\ \cline{2-13}  & \(k_{1,max}\) & \(k_{1,min}\) & \(k_{1,adaptive}\) & **VF** & \(k_{1,max}\) & \(k_{1,min}\) & \(k_{1,adaptive}\) & **VF** & \(k_{1,max}\) & \(k_{1,min}\) & \(k_{1,adaptive}\) & **VF** \\ \hline AB & 2457.48 & 2236.19 & 2308.61 & 2193.64 & 11.68 & 18.70 & 16.38 & 20.35 & – & – & – & – \\ BC & 1462.99 & 1537.41 & 1578.74 & 1468.67 & 28.13 & 23.53 & 21.85 & 25.83 & 19.15 & 4.46 & 6.46 & \(<\)0.2 \\ CD & 2931.19 & 3057.74 & 3080.79 & 2991.44 & 28.40 & 20.28 & 19.00 & 24.03 & 15.39 & 6.02 & 8.48 & \(<\)0.1 \\ DE & 1563.01 & 1685.46 & 1717.57 & 1999.72 & 29.13 & 20.93 & 18.93 & 21.60 & 13.04 & 6.12 & 8.01 & \(<\)0.1 \\ EF & 5339.77 & 5364.65 & 5438.53 & 5258.51 & 23.58 & 21.78 & 18.63 & 26.80 & 8.38 & 1.44 & 2.29 & \(<\)0.1 \\ FA & 1414.37 & 1195.11 & 1316.10 & 1353.38 & 23.18 & 29.40 & 25.73 & 25.18 & 6.12 & \(<\)0.1 & \(<\)0.1 & \(<\)0.1 \\ AB\({}_{900s}\) & 1054.74 & 1431.50 & 1369.55 & 1232.40 & 14.80 & 6.90 & 8.33 & 12.18 & 10.82 & \(<\)0.4 & 3.17 & \(<\)0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The performance comparison of the polygon path following using the four methods.  
The length of the predefined circular path followed (LOPCPF) is defined as the length of the actual flight path after the time that the UAV has converged to the desired circular path. Table 4 lists the LOPCPF, the convergence time, and the error overshoot for the  
Figure 11: Comparison of the circular path following using the four methods PFC_\(k_{1,max}\), PFC_\(k_{1,min}\), PFC_\(k_{1,adaptive}\) and VF. (**a**) The position; (**b**) the distance \(f(x,y)\); (**c**) the parameter \(k_{1}\); (**d**) the parameter \(k_{2}\); (**e**) the course rate command \(u_{cmd}\); (**f**) the bank angle command \(\phi_{cmd}\).  
UAV to follow the two circles A and B. It is determined that it takes only 10.60 s for the UAV to converge to circle B using the method PFC_\(k_{1,adaptive}\). While using the methods PFC_\(k_{1,max}\) and PFC_\(k_{1,min}\), the convergence times are 20.95 s and 14.83 s, respectively. During the 300 seconds' flight, the LOPCPF by using the three methods PFC_\(k_{1,max}\), PFC_\(k_{1,min}\), PFC_\(k_{1,adaptive}\), and VF are 5356.88 m, 5662.37 m, 5742.11 m, and 5879.15 m, respectively. The above indicates that the method PFC_\(k_{1,adaptive}\) will still cause the UAV to fly the second most accurately on the desired circular path, with moderate convergence time and error overshoot.