# Quadrotor UAV Dynamic Visual Servoing Based on Differential Flatness Theory
## 1 Introduction
Thanks to the concept of differential flatness, we demonstrate that if we manage to extract the displacements according to the three axes and the orientation according to the yaw angle (the vertical axis) of the quadrotor, we can control all the other variables of the system. For this, we consider the following conditions. The camera is equipped with a vertical stabilizer, which keeps the camera in a vertical position during its movement; in other words, we neglect the lateral angles. It is also assumed that the quadrotor flies over at a given altitude. This altitude is not necessarily constant, but it must be known a priori. It should be noted here that these hypotheses only concern the visual system, which makes it possible to generate the movements necessary for the quadrotor in order to ensure the tracking of the vehicle. We use additional sensors to measure its magnitudes in order to achieve the trajectory that the visual system has thus generated. In the case of traditional 2D visual servoing, the development of the quadrotor's kinematic torsor does not guarantee the physical realization of control instructions (a controllability issue compounded by an under-actuated system). In fact, the kinematic torsor has six dimensions, whereas the quadrotor has only four inputs. With only four control inputs, it is nearly impossible to implement the six instructions generated by the visual servoing algorithm. To solve this problem, ref. [34] proposed a linear model predictive control (MPC), but this method uses linear approximations and is not generally suitable for systems with very fast dynamics. [PERSON] et al. [31] offered a command by backstepping; it was necessary to make many modifications to the model to render it in a particular form. The proposed control by flatness takes into account all the variables of the system, guarantees its controllability, and ensures the asymptotic convergence of the resulting trajectory. The contributions of this study can be summarized as follows:Using the concept of differential flatness, we have developed a new method of dynamic visual servoing for quadrotors. This method generates the necessary movements (translation and orientation) in order to keep the target centered in the image plane. Since quadrotors are fast systems working in outdoor environments, we have simplified the image processing and ensured the robustness of the visual primitive by using only one point of the target. Since quadrotors are under-actuated and strongly coupled systems, the realization of the kinematic tensor generated by the visual servoing algorithm becomes a problem. To solve this, we have proposed a control by flatness that ensures controllability and asymptotic tracking of the generated trajectory. In order to ensure robustness against climatic conditions, such as wind, we have added a PD-type correction term to the open-loop flatness control. This paper is organized as follows: Section 2 presents the dynamic model of the quadrotor. The tracking strategy of a vehicle is detailed in Section 3. This strategy includes three loops: the first controls the altitude of the quadrotor; the second is dedicated to the generation of the trajectory; and finally, the third loop ensures tracking by flatness. Section 4 displays the simulation results validating the proposed approach.