# MViDO: A High Performance Monocular Vision-Based System for Docking A Hovering AUV
## 4 Mvido: Guidance Law  
It is a challenge to define guidance laws that conduct the AUV to the dock, while keeping visual markers in sight. For our point of view, the solution can be based on basic motion primitives that are already implemented by the vehicle on-board control system. This section describes the implementation of a guidance system for docking maneuvers for a hovering AUV. We address the problem of driving the AUV to a docking station with the help of visual markers detected by the vehicle on-board camera, and show that by adequately choosing the references for the linear degrees of freedom of the AUV it is possible to conduct it to the dock while keeping those markers in the field of view of the on-board camera. We address the problem of defining a guidance law for a hovering AUV when performing a docking maneuver. We used as a case study the docking of AUV MARES, but our approach is pretty general and can be applied to any hovering vehicle. During the docking, it is of utmost importance that the visual markers are within the field of view of the camera. Considering a  
Figure 19: Sensitivity analysis.  
Figure 18: Linearize around a certain point:the red marker was centered at zero.  
local frame with origin in the center of the markers, with the usual North-East-Down convention, there should be a compromise between the motion on the vertical coordinate \(z\) and the motion on the horizontal plane \(\rho=\sqrt{x^{2}+y^{2}}\) (where \(x\) and \(y\) are Cartesian coordinates) from the center of the markers to the center of the AUV's camera. We want to ensure that the ratio \(\frac{z}{\rho}\) gives a slope greater than the slope of the cone, which is defined by the angle of view of the camera \(AOV\). This means that we define the condition:  
\[\rho\leqslant-z.tan(\frac{AOV}{2}) \tag{54}\]  
As is commonly the case with AUVs, we consider here that its on-board control system is able to independently control its DOFs. Relevant for this work is the ability to keep a fixed position on the horizontal plane and of keeping a given depth. Such controllers are part of the Mares on-board control system as described in [1]. For the purpose of defining the guidance law and testing its performance, we model here the closed-loop behavior of the AUV as a first-order system, both in the horizontal plane and in the vertical coordinate [30; 31]. Two decoupled models were considered: a model for the behavior in the horizontal plane \(\rho\) and a model for the behavior in the vertical coordinate \(z\). In this way:  
\[\frac{Z(s)}{Z_{ref}(s)}=\frac{p_{z}}{s+p_{z}} \tag{55}\]  
\[\frac{P(s)}{P_{ref}(s)}=\frac{p_{h}}{s+p_{h}}, \tag{56}\]  
where \(P(s)\) and \(Z(s)\) are respectively the Laplace transforms of \(\rho\) and of the vertical coordinate \(z\). In order to drive the AUV to the cradle while keeping the markers within the camera field of view, we propose a guidance law that can ensures a much faster convergence of \(\rho\) to zero that of \(z\) to zero. This law is defined by:  
\[\rho_{ref}(t)=0 \tag{57}\]  
and,  
\[z_{ref}(t)=-\rho(t)\epsilon^{(|T_{sZ}-T_{sP}|)\epsilon_{\rho}t}, \tag{58}\]  
where \((|T_{sZ}-T_{sP}|)\) is the difference between the settling times of the closed loop models (vertical and horizontal) for a unit step input, and \(\epsilon_{\rho}\) is the actual error to the reference of \(\rho\).  
For this assumptions it is possible to conclude that  
\[\lim_{t\rightarrow\infty}\frac{\rho(t)}{z(t)}=0. \tag{59}\]  
Observation: in our law, we force a restriction on the evolution generated for the reference \(z_{ref}(t)\) by a condition: reference values below to \(z_{0}\) are not considered, keeping the actual \(z\).