# Trajectory Tracking with Adaptive Robust Control for Quadrotor
## 1 Introduction  
Unmanned aerial vehicles (UAVs) have a lot of acceptance in the control theory research due to the challenge of getting a stable flight and finding some application to solve some science and engineering problems. Some applications with these unmanned aerial vehicles are: forest fire detection, in civil engineering (topography, analysis structural and others) [1], photogrammetry, and military applications [2], car detection [3] or for landing [4]. We can find different strategies to resolve the trajectory tracking with a quadrotor; for example, ref. [5] was proposed a flight controller with a hierarchical structure designed to control the altitude and the position. The controllers used in [5] are proportional-derivative (PD) and proportional-integral-derivative (PID) with a genetic algorithm method. In [6], a nonlinear system error was proposed for trajectory tracking of a six-degree of freedom quadrotor, and to stabilize the quadrotor attitude a backstepping approach and a nonlinear disturbance observer/sliding mode control approach are used. Likewise, a nonlinear adaptive state feedback technique as presented in [7, 8] was applied to an adaptive algorithm for steering a quadrotor in a trajectory with parametric uncertainty. Then, trajectory tracking with a quadrotor is a problem of accuracy in the position, and in [9] a solution based on trajectory planning algorithm with a genetic algorithm (GA) and A\({}^{*}\) algorithm is presented. On the other hand, in [10] a novel 3D path planning technique to steer a sinusoidal path for a quadrotor is presented. Some scientific research, before resolving the problem of trajectory tracking with a quadrotor, first resolves the problem of the disturbances acting in unmanned aerial vehicles such as in [11], where an estimator considers the sliding mode techniques to attenuate the translational disturbance. In [12], an optimal controller in a quadrotor is applied considering external perturbations in the system and a quadrotor linear mathematical model is considered; that is, the control law is working in some quadrotor operating pwith input constraints to avoid the external disturbances and even is demonstrated finite-time convergence. In [13], a backstepping control and integral sliding mode is proposed to eliminate external perturbations. A novel multivariable super-twisting algorithm to avoid the chattering effect generated by the integral sliding mode is presented in [13]. On the other hand, in [14], a robust controller to stabilize a quadrotor is proposed; the control law was based on the proportional-derivative control, and to obtain a better flight performance a disturbance compensation was introduced. The mathematical model that defines the UAV in [14] is rotation in SO(3). It is worth mentioning that in [11; 12; 13; 14; 15] do not applying any adaptive controller. Other authors suggest that adaptable techniques are better to stabilize a quadrotor. For instance, in [16] an adaptive controller applied to attitude tracking is presented for an unmanned aerial vehicle to eliminate the adverse effect of measurement noises. Even in [17], a robust adaptive neural network controller is proposed, and is applied to a quadrotor UAV. This control law is not necessary for the prior information of disturbances to stabilize the quadrotor. Meanwhile, in [18] an adaptive backstepping is presented with an extended state observer applied to a quadrotor to follow and maintain trajectory, even if it is subject to external disturbances. The perturbations in [16; 17; 18] are considered as a noise in the measurement from the sensors that feedback the control law. On the other hand, Ref [19] presented the adaptive tracking control for a quadrotor in different or multiple operating conditions. These conditions are when the quadrotor deals with system uncertainties or unknown parameters; the control objective is to obtain a uniform update law for the controller parameters based on the Model Reference Adaptive Control (MRAC).