# Curved Path Following Control for a Small Fixed-Wing UAV with Parameters Adaptation
## 1 Introduction  
In the past two decades, small fixed-wing UAVs have been widely applied in military and civil fields, such as anti-terrorism reconnaissance, communication relay, land surveying and mapping, forest fire prevention, power inspection, and so on [1]. One important function of the autopilot is that it can give the UAV the ability to fly over the target area according to the predefined path [2]. The methods for realizing the objective can be classified into two categories: trajectory tracking and path following. Path following is to make the UAV fly along the geometric path at any feasible speed. Trajectory tracking requires the UAV to converge and follow a time-parameterized path [3]. It requires the autopilot to produce the velocity command, which is critical for the small fixed-wing UAV [4]. Various approaches have been proposed in the literature for path following. Two main categories are linear-control-based approaches and nonlinear-control-based approaches. The linear path following control methods mainly include PID and LQR techniques. The structure of a PID-based path following controller is simple and easy to implement [5], but the selection of control parameters mostly depends on the experience of the designers [6], the controller's ability to resist unmodeled parts and external disturbances is poor [7]. The LQR-based method is usually applied to follow a straight-line and circular paths. The LQR-based path following approach requires a suitable linear model, which includes the dynamic characteristics of the system and the differential equation of the headingerror [8]. It should be pointed out that the heading error must be small enough to meet the linearization conditions [9]. Otherwise, the linear model will not match the actual characteristics, and the controller will fail. The nonlinear path following control methods mainly include the intelligent control techniques, nonlinear model predictive control (NMPC), and the Lyapunov-stability-theory-based methods. In the aspect of intelligent control techniques, [PERSON] et al. applied the Takagi-Sugeno fuzzy technique to produce the desired heading to follow the predefined waypoints [10]. [PERSON] et al. applied the convolutional neural networks to produce the head direction control and lateral offset [11]. By combining the above two control quantities linearly, the yaw rate command was generated to guide the UAV to follow the desired path. [PERSON] et al. [12] used the deep reinforcement learning approach to generate the desired heading command, where a specific reward function was designed for minimizing the cross-track error of the path following problem. The simulations with the above intelligent techniques have been presented to show their effectiveness. However, the stability of the relative controllers has not been proved. Considering the input and state constraints of the UAV, the NMPC technique is usually applied to solve the path following problems by combining the dynamics of the UAV and the path following error. In the early stage of applying NMPC, the methods have satisfactory path-following performance [13; 14], but they do not consider the stability of the relative closed-loop system. [PERSON] et al. applied the NMPC technique to produce the lateral guidance law by using the continuation/generalized minimum residual method with an extended Kalman filter [15]. In [16], [PERSON] et al. applied the NMPC technique to generate the heading rate command to first follow the straight-line path. By varying the control horizon depending upon the path curvature profile, an adaptive NMPC algorithm was presented to follow the continuous curvature path [17]. The conditions that can assure the closed-loop stability of the corresponding system were obtained. However, this method presents higher requirements for the calculation of the optimization problems, especially when the control horizon increases. Other nonlinear path following controllers are mainly based on the Lyapunov stability theory. [PERSON] et al.