# Trajectory Tracking with Adaptive Robust Control for Quadrotor
## 1 Introduction
Otherwise, MRAC modification in [20] is used to catch possibly dangerous UAVs mid-air using a net carried by two drones. The modification proposed in [20] is adding a compensation with a proportional-derivative-integral action to avoid instability of the system due to abrupt changes in altitude. Besides, a comparative between Proportional-Integral-Derivative and MRAC control for a quadrotor to stabilize the Euler angles roll, pitch, and yaw was analyzed in [21]. On the other hand, in [22] neural networks were employed to design an indirect MRAC to work for any quadrotor system. The neural networks have to learn the system parameters online (during the flight), and the challenge is to keep the quadrotor stable during online learning. In [23], it is shown how to apply the sliding mode control theory, MRAC, and the adaptive sliding mode control to design flight controllers for quadrotors; these proposals are evaluated and compared in numerical simulations. In [24], a study of multivariable output feedback MRAC applied to a quadrotor with parameter uncertainties is presented. The feedback solution requires one-third the number of measurements as full state feedback. In [25], a baseline MRAC controller applied to a quadrotor is shown with parameter changes or uncertainty parameters during the quadrotor flight, and the recursive least squares method is used with a forgetting factor to identify or estimate the unknown parameters. Otherwise, to solve the problem of parametric and nonparametric uncertainties in quadrotors, a decentralized adaptive controller with the methodology of Lyapunov and with an MRAC structure is presented in [26]. The controller is applied to achieve the desired angles (yaw, pitch, roll) and the desired altitude. Even in [27], the MRAC is used to try with the uncertainties, but it involves the adaptive control only in the Euler angles defined by the approximation of the quadrotor dynamics. Then the general problem is the trajectory tracking with a quadrotor, and this implies, based on the research presented before, that a controller is necessary for attitude, position, and with enough robustness for the quadrotor to achieve the desired trajectory for some applications as mentioned at the beginning of this introduction. Thus, we proposed resolving this problem with a robust mechanism for an adaptive control law to steer a quadrotor UAV in a predefined trajectory by equations to obtain an adequate trajectory and avoid some physics singularity of the system in the position and the quadrotor attitude. Then, the controller in this work is an adaptive proportional-derivative control, starting from the methodologies MIT rule and sliding modes. The MIT rule is the original approach to model-reference adaptive control; the name is from the Instrumentation Laboratory (now the Draper Laboratory) at MIT [28]. Specifically, the main contribution of this work is the robustness in the design of the adjustment mechanism that has to adjust the adaptive gains. With this, we can obtain better results in time convergence, reduce the oscillations in the steering trajectory, and obtain a stable flight during some applications. Several simulations are presented to show that the adaptive control law with a robust adjustment mechanism is effective. Other contributions of this work are  
* The obtention of robust adaptive controller considering a Proportional-Derivative structure. * Design only two adjustment mechanisms for two gains and achieve the control objective. * The obtention of a robust adjustment mechanism with the use of the sliding mode theory. * Achieve the desired trajectory with a reduced aerodynamic model. * The modification of the MIT rule is different that that presented in [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28].