**1. Introduction**

It is known that, in motion control systems, it is required that the system move to match some desired features of acceleration, velocity, position, or a combination of them. Unmanned aerial vehicles (UAVs) are dynamic systems where the controlled motion is

**Citation:** Yañez-Badillo, H.; Beltran-Carbajal, F.; Tapia-Olvera, R.; Favela-Contreras, A.; Sotelo, C.; Sotelo, D. Adaptive Robust Motion Control of Quadrotor Systems Using Artificial Neural Networks and Particle Swarm Optimization. *Mathematics* **2021**, *9*, 2367. https:// doi.org/10.3390/math9192367

Academic Editor: Alfonso Baños

Received: 3 August 2021 Accepted: 10 September 2021 Published: 24 September 2021

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fundamental to complete specific applications. Recently, diverse types of UAVs vehicles have been developed, with fixed-wing unmanned aerial vehicles (FW-UAVs) being the most common and most developed. These aircraft are similar to passenger aircraft, with a pair of wings to provide lift, a propellant system to provide thrust, and aerodynamic surfaces to control the motion. Their efficiency is higher compared to other UAVs, allowing it to perform long flights. Nevertheless, their indoors use is exclude since they do not have the ability to hovering and can not turn at reduced distances [1]. For their part, rotary-wing unmanned aerial vehicles (RW-UAVs) have various configurations including the conventional helicopter, the coaxial helicopter, and multi-rotors, which can sustain hover flight and take-off-landing vertically (VTOL). The FW-UAVs and RW-UAVs are the classic configurations most used in the applications assigned to unmanned aerial vehicles. Among the main ones are surveillance, monitoring, photography, inspection, and cargo transportation [2], with RW-UAVs having more civil applications than FW-UAVs [3]. On the other hand, technological advances have also allowed the development of new UAV configurations, such as bio-inspired flapping-wing unmanned aerial vehicles (Fl-UAV) [4] and lighter-than-air unmanned aerial vehicles, (LtA-UAVs) [5]. The four rotor helicopter or quadrotor is the most common rotorcraft platform in the research community due to its properties of under-actuation, low construction cost, symmetrical structure, high coupling non-linear dynamics, and capabilities of VTOL and hovering.

In the literature, several important contributions have been reported for controlling the quadrotor dynamics. Motion controllers based on theories, such as sliding modes [6], active disturbance rejection [7], backstepping [8], Lyapunov functions [9], *H*∞ [10], adaptive controllers based on L<sup>1</sup> [11,12], fuzzy logic [13], neural networks [14], model predictive control [15], or combination of them. Since, to some, drawbacks are inherent to each control strategy, such as high-frequency control actions, unmeasurable system information required, high dependency of mathematical models, high-gain feedback, and high sensibility against exogenous disturbances, some researchers have been properly exploited the properties of adaptive and robust control for designing advanced control methodologies.

In contrast with conventional control, intelligent control techniques are able to efficiently deal with incomplete information of many dynamic systems and its environment within a wide range of operational conditions. Then, adaptive control strategies represent a potential alternative for improving the performance of robust motion control schemes. In the literature, adaptive control stands for a class of control techniques used for compensating parameter changes, disturbances, and unknown changes in the system, by adaptations based on observations [16]. Relevant and recently research have been inspired by the qualities of adaptive and robust control schemes for quadrotor motion control. Authors in [17] introduce a model reference adaptive control scheme for a four-rotor helicopter in order to increase robustness against parametric uncertainty. A baseline controller is proposed for trajectory tracking task which is further improved by including adaptive capabilities. Similarly, switched adaptive controller are properly introduced in [18,19]. Here, controllers are suitably designed for controlling a quadrotor in the presence of unknown external disturbances and variations in the mass and inertia of the quadrotor due to unknown payload. Strict simulation scenarios are brought out to validate their proposal.

On the other hand, another adaptive control scheme is presented by authors in [20], where the quadrotor attitude is stabilized by an adaptive multi-variable finite-time algorithm. The controller design is carried out by using an improved super-twisting technique. Control methodologies designed based on the central ideas of adaptive sliding mode control are presented in [21,22]. In [21], a disturbance observer (DO) is integrated in control design to compensate external disturbances. The tune of the gain of sliding surface is accomplished via neural networks. In contrast, authors in [22] implement an adaptive scheme by proposing a super twisting controller along with Lyapunov-based function methodology and discontinuous projection operators. The research in [23] presents a fuzzy adaptive linear active disturbance rejection controller. The fuzzy framework is setup

successfully to compute the observer bandwidth, controller bandwidth, as well as the control compensation factor.

Considering the aforementioned information, in this paper, authors introduce a novel and efficient adaptive robust motion tracking control for quadrotor non-linear systems. The main differences with others proposals reported in the literature are enlisted below:


The content of this paper is summarized as follows: the quadrotor non-linear and high coupling mathematical model is presented in Section 2. In Section 3, the design procedure of the novel robust and adaptive controller is introduced. Subsequently, some simulation experiments are presented in Section 4 in order to highlight the performance of the introduced methodology. Finally, some conclusions, remarks, and future work are mentioned in conclusions section.
