During the last decade, we have seen the presence of unmanned aerial vehicles (UAVs) in different applications, whether in the commercial industry or high-end application fields such as the military, avionics, and artificial intelligence [1]. A UAV is a pilotless aircraft with the ability to fly and stay in a hover state without any human interaction. They are piloted remotely through control commands from a ground station [2].

In general, depending on the characteristics of the UAV, its application, speed, weight, and operation vary. UAVs are generally divided into four categories: fixed wing, fixed-wing hybrid, multirotor, and single rotor. Fixed-wing UAVs are vehicles based on wings, a main body, a motor, and a propeller. This type of UAV requires unique and extensive training to pilot them. One of its main advantages is the flight time of several hours. However, this vehicle is not able to perform a backward flight, hover, or rotate. The next type is the fixed-wing hybrid, which combines automated and manual gliding. Although that is an advantage, this type of UAV could be better at forward flight and hovering. Multirotor vehicles are the most common in the industry. There are different types of multirotors, and they are generally classified by the number of rotors or propellers they have. There are tricopters, quadcopters, hexacopters, and octocopters. The most used are quadcopters since they have a vertical landing, small size, and simple structure [2,3]. The last one is the single-rotor UAV, which has a single rotor as its name mentions. This type of UAV is less developed than the others since it is complex to design. Nevertheless, they have distinct advantages, like a heavy payload, a mixture of hovering with long endurance, and fast forward flight [4].

In recent years, it has been concluded that UAVs are no longer sufficient to fly autonomously, and more advanced technologies are needed to bring innovations and improvements to achieve more complex goals. While UAVs excel in speed and flight duration, innovative solutions are imperative to address their limited robustness, a pivotal consideration for large-scale deployment and diverse task completion [2].

It is common to assume that a single-rotor system resembles a conventional helicopter; however, this assumption only holds partially. The fundamental distinction lies in their dynamic configuration and weight-handling approaches [5]. Unlike helicopters, which employ a primary rotor and tail rotor to counteract asymmetry and reaction torque, single-rotor systems rely on fins or control surfaces to achieve stability and direction [4].

Helicopter dynamics revolve around the lift and control generated by the main rotor, counteracted by tail-rotor reaction torque [6]. Conversely, single-rotor systems employ fins and control surfaces to manipulate airflow, enabling precise aircraft control. Furthermore, single-rotor systems, generally lighter than traditional helicopters, leverage more efficient configurations and sophisticated control systems, compensating for dynamic and weight differences [4].

Single-rotor UAVs only have one propeller, and although it could be seen as a disadvantage since it would lose maneuverability, in reality, the energy use also decreases. Using only one rotor increases the flight efficiency, reduces energy consumption, and allows for an important feature: the VTOL (vertical take-off and landing) [7]. Different single-rotor UAV designs have been introduced over the years. A single rotor composed of two fixed-wing airplanes was analyzed in [8], in which two attached rotating airplanes can be modeled and controlled. Another variation studied is the single-rotor aerial vehicle, which features a model with a single rotor and a tail rotor [9]. In [4], a different approach based on the design of a single-rotor VTOL is presented. This design is based on thrust vectoring through a single propeller and four deflecting fins actuated by small motors. In that work, the process design is simplified to obtain lower costs and an increase in the number of applications.

Path following and trajectory tracking constitute primary control challenges for UAVs [10]. The PID (Proportional Integral Derivative) controller is commonly used in UAVs [11] due to its versatility and applicability to various UAV variants. However, the PID has limitations in handling complex systems, uncertainties, disturbances, and nonlinearities, making more robust control techniques desirable. The PID does not require a mathematical model of the system but has limitations when systems are complex. Changes in the system parameters can affect the stability and performance of the control if the parameters are not tuned properly. In addition, the PID is not efficient for handling uncertainties and disturbances compared to other more robust controllers. Because the PID is designed on linear assumptions, it has difficulties controlling nonlinear systems. The elimination of steady-state errors is also a limitation that reduces the control performance [12]. For this reason, more robust nonlinear control techniques are considered in this work to deal with such issues.

This research explores robust nonlinear control techniques such as the Adaptive Sliding Mode (ASM) and Super Twisting (ST) for single-rotor UAVs. ST is a control technique used for nonlinear dynamic systems based on the theory of sliding mode and trajectory tracking to zero. Several approaches have been used in the literature to quantify the position control of small-rotor UAVs. In [13], it is demonstrated that ST control is one of the most powerful second-order sliding mode controllers for systems with a relative degree equal to one. The main contribution of this work is the robust ST control for solving the attitude-tracking problem of a quadrotor subject to perturbations. Limitations in the controller are also observed. The main one is the presence of chattering [14]. Although the controller is shown to be robust, it has limitations with regard to working with uncertainties, requiring extensive knowledge for tuning [15].

Sliding mode techniques have also been applied in the control of UAVs. For instance, [16] presents a bond graph (BG) model and a robust cascaded controller for a twin-rotor system. The BG model accounts for all the energetic and dynamical couplings. An adaptive integral backstepping sliding mode controller is used in an outer loop to control the yaw and pitch dynamics. Although the proposed controller was able to obtain less chatter and lower the following error than the others during experiments, it did so at the expense of an increased response time, which implies a trade-off between accuracy and speed. In [17], a robust and fault-tolerant controller is designed to determine the position and attitude dynamics by using a higher-order integral dynamic sliding mode controller. The proposed method requires redundant actuators, which increases the cost of implementation. Here, the performance of finite-time disturbance observers used to estimate the combined effects of parametric uncertainty, external disturbances, and actuator faults is satisfactory, but the estimators start deteriorating after intermittent faults. Additionally, in [18], the attitude-position tracking control of fully actuated multirotor aerial vehicles equipped with fixed rotors and subject to matched model uncertainties and disturbances is presented. Here, a joint geometric attitude-position control law is designed by using a multi-input smooth second-order sliding mode strategy. The simulations show that the proposed method reduces the chattering and improves the tracking performance in comparison with Super Twisting and second-order sliding mode controllers, but it still presents oscillatory behavior, which is mainly caused by the unmodeled rotor dynamics. Finally, ref. [19] presents the development of a terminal sliding mode attitude-position quaternion-based control of a quadrotor unmanned aerial vehicle (UAV). The dynamics of the UAV are split into an attitude and position loop to design independent control laws. The control law implemented for each loop is the hybrid terminal sliding mode and quaternion controller. The results obtained in simulations showed better performance when compared with a higher-order sliding mode controller, but like the other previously mentioned works, they did not test the method with sustained wind gust disturbances as we did in our work.

Another nonlinear control technique for complex dynamic systems is the Adaptive Sliding Mode [20]. It also has the ability to handle uncertainties in the system model. It is based on the theory of bringing the system’s state to a sliding surface controlled by a linear controller [21]. The ASM has the ability to provide smooth and precise control, but the outstanding feature of the ASM is its ability to reduce the chattering effect. This effect is an unwanted vibration in the ST controller under certain conditions. The ASM has an inherent disturbance rejection capability to compensate for disturbances without requiring additional control action [22].

Regarding the implementation of the ASM in the SR-UAV, it is beneficial due to the nonlinear complexity of the system and the variable perturbations it faces in different flight conditions. The adaptive capability of the ASM is presented as a prosperous solution for controlling a nonlinear system such as the SR-UAV [23]. However, the limitation of the complexity of the ASM implementation must be kept in mind, as well as the requirement for the selection of the controller parameters and the detailed knowledge of the system model [24].

To deal with uncertainty and external disturbances in UAVs, there have been previous works; for instance, ref. [25] addresses the design of a robust controller based on nonlinear estimators for a quadrotor. This controller ensures the tracking of desired references even with parameter variations and external disturbances. It uses high-order sliding mode estimators to estimate the disturbances, which can then be canceled by the controller, thus improving the dynamic behavior of the controlled system. However, experiments to show the performance of the proposed controller in more complex trajectories, as the helical route we use in our testing, are missing. Another type of perturbation implementation is based on the Von Karman wind model [26]. This model allows one to understand and characterize different types of wind, such as steady wind, turbulent flow, wind speed variations, and a propulsor vortex. By combining this model with the mathematical representation of the UAV, the motion of the UAV in a wind field can be illustrated from three different perspectives: speed, force, and energy.

One of the main problems found in UAV controllers is the tuning process over which the controller gains are adjusted by trial-and-error to minimize the settling time and the corresponding oscillations of the closed-loop system. Many gains need to be adjusted depending on the number and the type of controllers in the system. Each set of gains represents a new possible scenario to obtain the best conditions for the experiment, which can make tuning rather long, tedious, and costly. Particle Swarm Optimization is a global metaheuristic optimization method that belongs to the family of algorithms based on the swarm intelligence concept introduced by Kennedy and Eberhart [27]. This method is based on the concept of swarming, which is inspired by the collective behavior of social animals, especially bird flocks and fish schools. A swarm is a population of agents which perform tasks by interacting locally with each other and with an environment. There is no central control, and their behavior arises from stochastic local self-organization and interaction. These swarm individuals could be more satisfied with limited capabilities; the goal is achieved by exchanging information obtained through the behavioral and interaction parameters [28]. Specifically, this type of optimization is suitable for solving problems where the optimal solution is a point in a multidimensional parameter space. Each particle is evaluated to determine its fitness value. At each iteration, a fitness value is calculated and compared with the previous value to obtain the best one. This social interaction is direct, which means that each particle is influenced by its memory (the best solution so far) and by those of the other agents. Individual and global best positions are updated, where if a stop criterion is not met, the velocity and position are updated to create a new swarm. The two fundamental operations are the velocity and position update. The actual velocity depends on the old velocity, the individual particle’s experience, and the whole swarm’s experience. Each term has an associated weight constant [29].

PSO has been used in the control of UAVs due to the difficulty of adjusting the parameters of the controllers. For instance, the camera-position control on a UAV is presented in [30]. The designed PID controller keeps the camera stable. Traditional fitting methods and evolutionary/bioinspired algorithms such as PSO are discussed in this article. It is highlighted that PSO is used due to its stable convergence, both in terms of its dynamic and static performance and its good computational efficiency, which allows for improving the system performance through error minimization. The research concluded that the PSO-tuned PID controller offers accuracy and convergence stability, which makes it the preferred option. In [30], PSO is applied to camera-position control, offering stable convergence and improved system performance compared to manual tuning. The approach proposed in [31] is the modified PSO (MOPSO), which optimizes the gains of a PID controller for an AR drone quadcopter. The MOPSO algorithm demonstrates the efficient adjustment of the PID parameters, resulting in no overshoot, zero steady-state errors, and a short rise time. Similarly, in [32], PSO is used to optimize parameters in a four-rotor PID controller, achieving a shorter tuning time and reduced overshoot. However, despite these achievements, there is limited research on using PSO in UAV control with nonlinear methods. For instance, in [33], PSO is employed to tune the gains of three nonlinear control strategies for quadrotor attitude control and trajectory tracking. The GTNC controller stands out as it consistently converges to the desired attitude and effectively reduces the position error and basin of attraction to zero, making it more reliable at high speeds. However, it requires more processing time compared to other controllers. In [34], a control strategy is proposed for a quadrotor with a sliding mode controller (SMC) that will use the gains obtained from Lyapunov stability analysis and PSO. The controller is shown to be stable with the gains obtained from PSO. A PID controller is also added to the SMC control law when a fault is detected to stabilize it. The results show that this control technique meets the desired requirements given the gains obtained by the PSO algorithm.

The main contributions of this paper strengthen our existing knowledge in two aspects:

• First, in this research, robust nonlinear controllers are used. To the best of our knowledge, none of these approaches have been applied before with single-rotor UAVs. We show that these controllers are effective in improving SR-UAV performance, especially under wind gusts disturbances.
• Second, we prove that the gain values obtained with the PSO algorithm satisfy Lyapunov stability conditions for Super Twisting and Adaptive Sliding Mode controllers.

In addition, to address the challenging and laborious task of tuning the multiple controller parameters, we used an evolutionary PSO method. This automated tuning approach has allowed us to find the optimal parameter values for the PID and the other two nonlinear controllers. The results obtained in simulation scenarios, with the presence of significant wind disturbances, have demonstrated the superiority of the PSO-tuned controllers compared to the controllers tuned by the traditional trial-and-error method. Also, we have to mention that corrections were made to the original mathematical model obtained in [4].

It is important to note that a simulation is an initial and fundamental stage in the process of developing and evaluating the proposed method. At this initial stage of the project, simulations provide an opportunity to explore and validate the operation of the controllers virtually. The validation of a method is an iterative and gradual process [35]. Once the controllers have been validated and optimized through simulation, the next stage may involve testing in a controlled laboratory environment or even on a physical prototype if the necessary resources are available at later stages of the project. In this way, the proposed approach is strengthened as validation progresses and more experimental evidence is acquired.