## 1 Introduction  
There are two main approaches in visual servoing: Position-Based Control (PBC) (or 3D visual servoing) [1, 2] and Image-Based Control (IBC) (or 2D visual servoing) [3]. The error function in PBC is computed in the Cartesian coordinate system, while in IBC, this error is measured in the image coordinate system. In this paper, we are interested in 3D visual servoing in the case of a mobile robot equipped with a perspective camera. The generated control law must minimize the error between the instantaneous position of the robot and a desired position. These two positions are calculated (estimated) from the observations of the camera. Features are extracted from images. An ideal model of the object (target) is generated and used to determine its position. The camera/robot position is deduced according to the intrinsic and extrinsic parameters of the camera, namely, through pose estimation. Several works have attempted to solve the pose estimation problem [1, 2, 3, 4, 5]. In general, proposed methods exploit the 2D or 3D object (target) model and the camera's intrinsic parameters. They exploit various types of visual information such as points [6], rights [7], etc. We can consider these methods as static methods or approaches. Other attempts of pose estimation explore the dynamic aspect of the whole servoing problem. They use 3D reconstruction by dynamic vision techniques; they allow one to estimate the object model or to determine the camera's location based on the 2D or 3D movement of the robot [8; 9]. However, regardless of the pose reconstruction method used, pose estimation is very sensitive to measurement noise and visual sensor calibration errors. This generates, after the convergence of servo control (when the steady state is reached), a bias between the position of the robot and the object of interest to be reached. It is, in fact, the major problem of 3D visual servoing. To deal with this problem, we perform the pose estimation by using neural networks. Generally, to solve the problem of pose estimation, we need at least three points in the image [1] (six neurons in the input layer), which can generate an excessive number of neurons in the hidden layer. To reduce this number, the concept of differential flatness is used: we proved in [10] that, in the case of a two-degrees-of-freedom mobile robot, the variation in a single point in the picture coordinate space can be used to ensure the robot's position and orientation in relation to the target. In PBC, the camera trajectory is directly controlled in Cartesian coordinate space. There are, however, some issues, including the lack of control over the image space, which means that the object may leave the field of view of the camera during navigation, as well as the pose estimation algorithm generally causing errors [11]. Using the flatness concept [12; 13; 14; 15; 16; 17], we propose a new approach to solve these problems. While servoing, the object leaves the camera's field of view. At this moment, we propose implementing a blind motion of the robot by using an open-loop control strategy that relies on the flatness principle. Robust feedback control is required to reject the error pose estimation in the input. The path planning problem is, in general, difficult to solve due to the need to solve the differential system equation. The flatness principle can be used to fix this problem without approximation or to solve the differential system equation. In fact, the flatness property guarantees the existence of a flat output, allowing all system variables to be parameterized as a function of a finite number of derivatives [18]. We use the RVCTOOLS MATLAB toolbox to demonstrate the efficiency of the proposed algorithm. This paper is organized as follows: Section 2 focuses on the 3D visual servoing problem. Section 3 introduces the flatness control theory. The neural network architecture used to estimate the robot's pose is described in Section 4.