# Curved Path Following Control for a Small Fixed-Wing UAV with Parameters Adaptation
## 1 Introduction
applied the pure pursuit and line-of-sight (PLOS) method to realize following in a straight-line [18]. By using the virtual target point, [PERSON] et al. [19] developed a nonlinear path-following guidance law, which can follow both straight-line and circular paths. The use of heading guidance generated from vector fields is a candidate for UAVs to implement path following [2]. [PERSON] et al. combined the techniques of vector field and sliding mode to generate the straight-line and circular paths following control law [20]. [PERSON] et al. presented an adaptive vector field control law which can compensate for the lack of knowledge of the wind vector and for the presence of unmodelled course angle dynamics [21]. [PERSON] et al. developed a curved-path following controller approach with the combination of the vector field and the input-to-state stability theorem [22]. [PERSON] et al. designed a saturated course rate controller based on a combined vector field, which combined a conservative vector field and a solenoidal vector field [23; 24]. The theory of nested saturations is an alternative approach which can be used to deal with the input constraints [3]. [PERSON] et al. developed a curved-path following control scheme with the control constraints by using the theory of nested saturations [3]. [PERSON] et al. designed the guidance strategies for following the straight-lines and orbits in wind by using the nested saturations technique. The strategies have considered the roll angle and flight path constraints [25]. While designing the guidance law for path following by using the kinematic and dynamic models of the UAV, the backstepping technique can be applied to design the virtual control to meet the requirements step by step, finally designing the real control law. By using the backstepping and parameter adaptation techniques, [PERSON] et al. have developed the roll angle command, which can follow the flyable smooth path [26]. [PERSON] et al. have designed a path following controller by introducing a virtual particle moving along the geometric path that is represented by parameterized smooth functions. The error kinematic model in the Frenet-Serret frame, together with the roll dynamics, is used to generate the guidance law [27]. Furthermore, the wind disturbances, which are estimated by using the arbitrary-order exact robust differentiator, are considered to improve robustness [28]. The nonlinear controllers designed using the Lyapunov stability theory are very attractive, both in theory and experiment. Nevertheless, in the process of designing this kind of controller, the control parameters are generally fixed to ensure the stability of the corresponding closed-loop nonlinear system. In fact, the values of the control parameters have a very important impact on the path-following performance, which is mainly reflected in the convergence speed, overshoot, following error, and so on. If the above controller parameters change within a certain range, the corresponding stability proof process will no longer be applicable. Therefore, designing a stable path following controller based on the Lyapunov stability theory, where its corresponding control parameters can be adjusted in real time to adapt to different initial conditions and dynamics, is vital to improve the path-following performance of a UAV. In our previous works [29], a PD-like nonlinear guidance law was presented to guide the fixed-wing UAV toward the predefined straight-line path. To achieve better path-following performance, one of the control parameters was tuned by fuzzy logic. However, the other control parameter is still fixed; therefore, it cannot give the UAV a better path-following performance. In this study, the PD-like nonlinear guidance law is extended to follow a class of horizontal paths. Both of the control parameters are designed adaptively to improve the path-following performance.