# Curved Path Following Control for a Small Fixed-Wing UAV with Parameters Adaptation
## 5 Conclusions  
This paper has considered the problem of horizontal curved-path following using fixed-wing UAV. The guidance strategies are derived using a kinematic model of the aircraft and using the Lyapunov stability theory. The control parameters in the guidance law are time-varying and adaptive to the initial states and dynamics of the corresponding nonlinear system. The rules of the time-varying P-like parameter are designed by using an exponential equation shown as (15), where its corresponding parameter \(\tau\) is determined by using the fourth order Runge-Kutta algorithm and the bilinear interpolation method. The rules of the time-varying D-like parameter are designed based on the fuzzy logic technique. The stability of the corresponding nonautonomous nonlinear system is also guaranteed. Finally, we have demonstrated the effectiveness of the proposed strategies in a Matlab/Simulink simulation environment with an Aerosonde model. For the given square and circular paths, the method using the adaptive parameters can make the UAV fly more accurately on the target path based on the performance indexes LOPPF and LOPCPF. In the sense of path following, the proposed method provides a methodology that will be applied to realize the better path following of unmanned systems, including aerial, underwater, and ground-based robots. In the actual flight process, the roll angle command is bounded, which should be considered in the parameter design of the controller. At the same time, if the bank angle command tracking control response of the UAV is very fast, the method described in this paper will be effective. If its response takes a certain time, the closed-loop roll dynamics must be considered in the design of controller. The closed-loop roll dynamics can be expressed as a first-order system. The follow-up research work will further design the path following controller by using the backstepping technique. In the future, the closed-loop roll dynamics of the bank angle presented by a first-order system with the unmodeled parts will be used to generate the bank angle command in order to make the vehicle follow the predefined path more precisely. The method to estimate the disturbance described in [28] will also be considered to design a robust controller. [PERSON] has contributed to this paper by carrying out the methodology and formulation, the supporting algorithms, and original draft preparation. [PERSON] has also contributed to this research by performing the simulations. [PERSON] has contributed to this paper by analyzing how the parameters will influence the path-following performance. [PERSON] was also involved in the aspect of the revision of the manuscript and in the literature review. His expertise in UAV has contributed towards optimizing the control parameters to improve control performance. [PERSON], [PERSON], [PERSON] and [PERSON] discussed the results. All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Circle**} & \multicolumn{4}{c}{**LOPCPF (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 A & 3464.27 & 3598.88 & 3585.89 & 3725.95 & 28.40 & 22.65 & 22.28 & 14.98 & 22.65 & 11.07 & 12.52 & \textless{}0.1 \\ B & 1892.61 & 2063.49 & 2156.22 & 2153.20 & 20.95 & 14.83 & 10.60 & 11.00 & 14.59 & 2.25 & 3.03 & \textless{}1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The performance comparison of the circular path following using the four methods. **Funding:** This work was supported by the National Natural Science Foundation of China (Grant Nos. 61503172 and 61773194), the Natural Science Foundation of Fujian Province (Grant No. 2020J05197), and the Shenzhen Special Funds for the Construction of High-Level Universities (Grant No.