# Quadrotor UAV Dynamic Visual Servoing Based on Differential Flatness Theory
## 1 Introduction
Several methods have been employed to track humans, including the use of bounding boxes and minutiae [24; 25]. Nevertheless, the targets tracked using these methods moved at slow speeds, which could result in their movements being ignored. Furthermore, it has been reported that target orientation is frequently unavailable [26]. In [27], the utilization of model-based predictive control was demonstrated for the purpose of tracking a periodically moving target. This approach resulted in a reduction in the complexity associated with controller design. The aforementioned methods failed to consider the interaction between the unmanned aerial vehicle (UAV) and the intended target. Additionally, the angle between the camera and the target was neither modeled nor quantified. Quadrotor dynamics are typically fast and unpredictable. To control this type of system, it is necessary to develop a high-frequency controller [28]. On the other hand, visual servoing goes through an image processing step that aims to extract the characteristics of the object. This can have a detrimental effect on the frequency of control law development. Furthermore, the complexity of image processing can cause delays in motion control, which can lead to the loss of a target. In order to solve these problems, a new 2D dynamic visual servoing (Dynamic IBVS) method is proposed. Its objective is to generate the necessary movements of the quadrotor to keep the target centered in the image plane. The proposed method transforms the problem into an asymptotic tracking process of a desired trajectory in the image plane, using the inverse dynamic of the estimated model of the vehicle to be followed. Since the proposed method allows the altitude of the quadrotor to be controlled independently of other variables, it is possible to set the altitude to a high level in order to reduce the risk of losing the target out of sight of the on-board camera, even during discontinuous and significant movements of the target. Moreover, this method only uses a single point on the target as a visual primitive. To increase the robustness and flexibility of the detection, this point can represent either the center of gravity of the target to be tracked or a specific part of the target. The flatness property of a system is a relatively recent concept in automatic control that was proposed and developed in 1992 by [PERSON] et al. [29]. This property, which makes it possible to parameterize in a very simple way the dynamic behavior of a system, is based on the highlighting of a set of fundamental variables of the system: its flat outputs. This point of view, as we demonstrate, has multiple and interesting consequences relative to the control of systems. First of all, this makes it possible to return to the center of the control of a process the notion of trajectory that the system must execute; that is to say, the movement requested from a system must above all be achievable by this system. This avoids many of the problems faced by automation engineers. One of the first steps in flatness control is to generate an adequate desired trajectory that implicitly takes into account the system model. In this work, we consider as an application the tracking of a car-type vehicle (Dynamic IBVS) by a quadrotor UAV equipped with a single facing-down perspective camera. In our case, the information concerning the altitude and the lateral angles (the roll angle and the pitch angle) are of no importance to the visual system. Indeed, to perform the tracking, we only need to know the longitudinal displacements (along the \(x\)- and \(y\)-axes) and the orientation along the \(z\)-axis. Those details are necessary for the problem of guiding the quadrotor. In [30; 31; 32; 33], the authors proposed to use a rotating image plane, called a \"virtual image plane\", thus making it possible to obtain a dynamic of decoupled image characteristics. This method is applied to a fixed target and requires the detection of at least three points on the target.