Hung Lam

Nguyen

Minh Duc Hua

Professeur Tarek Hamel

Devant Le

Professeur Pascal Morin

Professeur Carlos J F Silvestre

MCF Guillaume Allibert

Alain Fidani

Thèse De Doctorat

Université Côte D'azur

MCF Vicent Creuze

Contrôle non-linéaire et guidage visuel de véhicules sous-marins

Keywords: AUV, Visual servoing, Homography-based control, Station keeping, Trajectory tracking, Underactuated vehicles, Nonlinear control AUV, asservissement visuel, contrôle basé sur l'homographie, maintien en position, suivi de trajectoire, véhicules sous-actionnés, contrôle non linéaire

Recent advances in technology have led to a growing interest in underwater robotic applications. Historically, payload constraints have severely limited the development of scale-model underwater vehicles. They are small; their dynamics are highly nonlinear, subjected to complex and imprecise hydrodynamic effects and to disturbances commensurable with the control actions. Consequently, they provide an ideal testing ground for sophisticated nonlinear control techniques. Classical control techniques applied to these vehicles seem to be inadequate in dealing with their highly nonlinear dynamics, model uncertainties and external disturbances. Another key arising issue is the difficulty of navigation in cluttered environments and close to obstacles without precise positioning with respect to the environment and without velocity measurements.

In the present thesis, two research topics have been considered. Firstly, two novel dynamic visual servoing controllers exploiting the so-called homography matrix for fullyactuated vehicles (without measuring the linear velocity) have been proposed. These two approaches are derived for two different camera configurations: downward-looking and forward-looking. The key contribution here is the non-requirement of costly Doppler Velocity Log (DVL) velocity sensor. The control performance has been validated over simulations and via practical trials in a real challenging environment. The experimental platform as well as the software was built over the period of the thesis. Secondly, a novel trajectory tracking algorithm for a class of underactuated vehicles with axisymmetric slender body has been developed. The proposed approach makes use of a new nonlinear dynamic model of the vehicle, and exploits the symmetry of the vehicle in control design. The control performance and its robustness have been validated via simulations using a realistic model of an underwater vehicle.

Résumé

Les récents progrès technologiques ont suscité un intérêt croissant pour les applications robotiques sous-marines. Historiquement, les contraintes liées à la charge utile ont fortement limité le développement des véhicules sous-marins à modèle réduit. Ils sont petits, leur dynamique est très non linéaire, soumis à des effets hydrodynamiques complexes et imprécis et à des perturbations proportionnables aux actions de contrôle. Par conséquent, ils constituent un terrain d'essai idéal pour les techniques de contrôle nonlinéaire sophistiquées. Les techniques de contrôle classiques appliquées à ces véhicules semblent inadéquates pour faire face à leur dynamique hautement non-linéaire, aux incertitudes des modèles et aux perturbations externes. Une autre question clé qui se pose est la difficulté de la navigation dans des environnements encombrés et à proximité d'obstacles sans positionnement précis par rapport à l'environnement et sans mesure de vitesse.

Dans la présente thèse, deux sujets de recherche ont été considérés:

1) Deux nouveaux schémas de contrôle par asservissement visuel dynamique (sans mesure de la vitesse linéaire) pour les véhicules sous-marins complètement actionnés ont été proposés. Ces deux contrôleurs ont été développés pour deux configurations différentes de la caméra : pointant vers le bas et vers l'avant. La principale contribution ici est de s'en passer du coûteux capteur de vitesse (DVL) utilisé pour mesurer la vitesse linéaire. Les performances des deux algorithmes de contrôle ont été validées par des simulations et via des essais pratiques dans un environnement très difficile. La plateforme expérimentale ainsi que le logiciel de développement a été conçue sur la période la thèse.

2) Nous avons proposé un nouvel algorithme de poursuite de trajectoire pour une classe de véhicules sous-déclenchés ayant une forme axisymétrique allongée. La méthode proposée utilise un nouveau modèle dynamique non linéaire du véhicule et exploite la symétrie du véhicule pour la synthèse de la conception des commandes. La performance de la loi de contrôle élaborée et sa robustesse ont été validées via des simulations en utilisant un modèle réaliste d'un véhicule sous-marin.

Summary in French

• Chapitre 1: Motivations, contributions et structure de la thèse.

Comme indiqué sur le titre, ce chapitre de Partie I: Introduction tout d'abord brièvement présente les motivations et les objectifs de ce projet de thèse. Ensuite, les contributions principales sont fourniées. En fin, la section de structure de la thèse présente brièvement le contenu de tous les chapitres.

• Chapitre 2: Une nouvelle tendance dans la robotique sous-marine.

Tout d'abord, ce chapitre discute sur le développement de la communauté mondiale de petits robots sous-marins, puis sur la technologie des petits ROV (véhicule télécommandé) et AUV (véhicule sous-marin autonome) de pointe. Enfin, certaines technologies récentes utilisées pour les véhicules sous-marins de petite taille et à faible coût sont présentées.

• Chapitre 3: Modèle mathématique des véhicules sous-marins.

Ce chapitre rappelle la modélisation de la dynamique complexe des véhicules sous-marins selon l'approche lagrangienne. Divers phénomènes hydrodynamiques et hydrostatiques affectant la dynamique des véhicules sous-marins sont discutés.

La formulation finale du modèle non linéaire 6 degrés de liberté est dérivée.

• Chapitre 4: Préliminaires sur l'estimation de l'homographie et les techniques de contrôle basées sur l'homographie.

Ceci est le premier chapitre de Partie II: Contrôle asservissement visuel basé sur l'homographie (HBVS) d'AUVs completement actionnés sans mesures de vitesse linéaire. Tout d'abord, ce chapitre rappelle la définition de l'homographie, puis discute de certaines techniques d'estimation de l'homographie et de méthodes existantes de contrôle du HBVS.

• Chapitre 5: Contrôle dynamique HBVS d'AUVs completement actionnés sans mesures de vitesse linéaire: le cas d'une caméra orientée en bas.

Une nouvelle approche de contrôle dynamique du HBVS développée pour les véhicules sous-marins completement actionnés équipés d'une caméra observant une cible visuelle (presque) plane et texturée. Tout d'abord, la formulation du problème et les idées de base de la conception des contrôles sont présentées. En suite, la conception du contrôle sous forme d'une architecture de contrôle en cascade est fourniée avec des analyses rigoureuses de la stabilité et de la convergence. Les performances et la robustesse du contrôleur proposé sont d'abord vérifiées par simulation basée sur un modèle réaliste de véhicule sous-marin, puis validées expérimentalement par des essais dans un environnement réel de lac en utilisant la plate-forme I3S-UV. Enfin, des extensions potentielles sont proposées pour élargir les fonctionnalités de l'application de maintien de poste.

• Chapitre 6: Contrôle dynamique HBVS d'AUVs completement actionnés sans mesures de vitesse linéaire: le cas d'une caméra orientée ver l'avant.

Une nouvelle approche de contrôle dynamique du HBVS est développée pour le cas plus difficile dans lequel le sous-marin considéré est completement actionné. Le robot est équipé d'une caméra qui est orientée vers l'avant en observant une cible visuelle plane et (presque) verticale. Les performances du contrôleur proposé sont d'abord validées par simulation, puis par des expériences préliminaires dans une cuve d'eau.

• Chapitre 7: Développement de la plate-forme expérimentale I3S-UV.

Ce chapitre explique la nécessité pour l'équipe I3S-OSCAR d'avoir d'une plateforme sous-marine pour valider expérimentalement les algorithmes de contrôle et d'estimation développés. Tous les processus de développement en termes de la conception électro-mécanique et de logiciel de contrôle sont présentés en détail.

• Chapitre 8: Suivi de trajectoire de véhicules sous-marins sous-actionnés et axisymétriques à forme allongée.

Ce chapitre se trouve dans Partie III: Contrôle des véhicules sous-marins sousactionnés. Un nouveau contrôleur est développé pour le problème de suivi de trajectoire de véhicules sous-marins à forme allongée et (presque) axisymétrique par rapport à leur axe longitudinal. La conception des commandes exploite la dynamique 3D des véhicules et certaines propriétés particulières héritées de leur forme en termes de résistance et de masse ajoutée. Les performances et la robustesse du contrôleur proposé sont vérifiées par simulation basée sur un modèle réaliste d'un véhicule sous-marin.

To my whole beloved family thanks are also extended to DR. Claude SAMSON, who together with PR. Tarek HAMEL has providing me opportunities to work in the OSCAR team of the I3S laboratory, starting by supervising me within my masters' internship. They all bring me to a new horizon of knowledge: nonlinear control and nonlinear observers. These tools with powerful mathematical insight make me astonishing by efficient solutions for many practical problems.

I would like to thank Guillaume ALLIBERT, Andé ANGLADE, Ninad MANERIKAR, Simeone De MARCO and Jean-Marie KAI for all their supports to complete my work successfully. I am also grateful to other friends who are or are used to be at the I3S: Fernando IRETA MUNOZ, José Henrique GOULART, Myriana RIFAI, Yousef ALF, Arnaud TANGUY. Thank you all for the time we spend together.

I would like to particularly thank my wife and my daughters for being with me in such a long journey. I really appreciate how patient you were while I worked on this PhD project. I am grateful for my parents and parents-in-law for continuously supporting my life and study. I n the identity of R n×n , n ∈ Z + 0 n the zero matrix of R n×n , n ∈ Z + {A} the inertial reference system {B} the reference system attached to the center of buoyancy of the mobile robot (non-inertial) {C} the reference system attached to the on-board camera (non-inertial) m mass J 0 moment of inertia tensor around the center of gravity V body-fixed linear velocity vector,

V = [V 1 , V 2 , V 3 ] Ω
angular velocity vector SL [START_REF] Agarwal | A survey of planar homography estimation techniques[END_REF] the Special Linear group, the set of all real valued 3 × 3 matrices with unit determinant sl [START_REF] Agarwal | A survey of planar homography estimation techniques[END_REF] Lie-algebra of SL [START_REF] Agarwal | A survey of planar homography estimation techniques[END_REF], the set of matrices with trace equal to zero SO [START_REF] Agarwal | A survey of planar homography estimation techniques[END_REF] the Special Orthogonal group of the orthogonal 3 × 3 matrices with

Motivations and contributions

Offshore operations are always extremely challenging and classically addressed by Remotely Operated Vehicles (ROVs). However, the use of ROVs is very costly, complicated and inefficient in deep sea and in presence of obstacles. Automatization of surveillance and inspection tasks relying on Autonomous Underwater Vehicles (AUVs) is thus highly relevant. Nevertheless, for such systems to be reliable and used in daily-routine applications, several fundamental and technological challenges must be overcome. For instance, the dynamics of AUVs are highly nonlinear and the translational and rotational dynamics are highly coupled, essentially due to added mass effects. The complexity of hydrodynamic effects often impedes to obtain a precise dynamic model, and the magnitude of external perturbations can become commensurable with the available actuation power. Thus, robust nonlinear control design is of prime importance. Existing control approaches, generally based on the modeling framework proposed by Fossen [START_REF] Fossen | Guidance and Control of Ocean Vehicles[END_REF], often make use of a minimal parametrization of the system (Euler angles for example), 1.1. Motivations and contributions leading to very complex equations involving mathematical singularities in the dynamic representation of the system. The resulting control laws are either too simplistic (only dealing with linear tangent behaviors) or too complex (not allowing for taking into account the natural physics of the system such as passivity and dissipativity of external forces). Typically, the 6-degree of freedom (d.o.f) vehicle model is decoupled into two reduced dynamical systems that are a "depth-pitch" model for the motion in the vertical plane and a "plane-yaw" model for the motion in the horizontal plane, by neglecting the interactions between the two types of motion and/or the Coriolis effects. Such a simplification may be meaningful for the class of box-shaped AUVs moving at low speed, since the hydrodynamic Coriolis forces can be dominated by the damping forces. However, for slender-body AUVs moving with some forward speed, this assumption is far from realistic. Stability and performance then suffer significantly when strong sea currents or aggressive maneuvers excite the effects of complex hydrodynamics and strong dynamic couplings. Moreover, control design for underactuated AUVs (by conception) as in the case of torpedo-shaped AUVs propelled by a single thruster (and 2 or 3 control torques) is more involved than the case of (almost) fully-actuated AUVs. For instance, there does not exist any time-invariant feedback controllers capable of stabilizing a fixed reference pose (i.e. position and orientation) asymptotically -a fact known to the automatic control community as a consequence of a theorem due to Brockett [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF]. On the other hand, AUVs may navigate in cluttered areas where global acoustic positioning systems are unusable or insufficiently precise for safe navigation. These vehicles may also be required to navigate relatively to their environment so as to carry out inspection and surveillance tasks. In these situations, their perception and navigation rely heavily on embarked exteroceptive sensors and sensor-based control techniques. However, sensor-based control (and in particular vision-based control) of highly nonlinear dynamical systems is usually difficult due to the fact that the vehicle's pose estimate may be unavailable for feedback control (as in the case of image-based visual servoing or homography-based visual servoing), and that indirect and coupled information about the state and the environment provided by sensor data has to be exploited in the control design process. The fact that linear velocity sensors such as Doppler Velocity Log (DVL) may not be available in small and/or low-cost AUVs due to their high price and weight even amplifies the degree of complexity of sensor-based control design, especially for highly nonlinear and dynamically complex systems like AUVs. In addition, multiple factors specific to underwater environment cause most of sensors and communication means that are widely used in ground and aerial robotics not to work under water. Last but not least, AUV platform and sensor costs, together with requirements for experimental setups such as test area, logistic support, etc., have never been easily accessible for any robotics laboratory.

Chapter 1. Motivations, contributions and thesis structure This PhD project has been carried out within a more global research activity in Underwater Robotics of the I3S-OSCAR team that aims to develop theoretical fundamentals of robust nonlinear control, sensor-based control and nonlinear observers in order to provide the AUVs with a significantly improved overall performance, enabling the offshore automatic inspection and surveillance operations. The different dynamics of underwater vehicles with respect to the ones of aerial vehicles (due to their different ambient fluids) and the differences in capacity of exteroceptive sensors in an underwater environment compared to an aerial environment (e.g. the range of cameras under water is much more limited than that in air) represent new challenges, opportunities and sources of inspiration for the team to apply our knowledge (theoretical and practical) in nonlinear and sensor-based control and nonlinear observer design (that has been developed for two decades in Aerial Robotics) to the field of Underwater Robotics. Within this global research panorama in Underwater Robotics of the I3S-OSCAR team, my PhD project focuses on the two following research topics:

1. Dynamic homography-based visual servoing of fully-actuated underwater vehicles without relying on linear velocity measurements: Vision-based stabilization and positioning functionality is useful for AUV navigation in close proximity to underwater infrastructures or a seafloor with many potential applications such as high-resolution imaging, monitoring, inspection, manipulation, station keeping, and docking, etc. In the case of monocular camera, this problem has been mostly addressed in the literature by relying on the assumption of (local) planarity of the visual target and by exploiting the so-called homography -an invertible mapping relating two camera views of the same planar scene. For instance, the 2 1 2 -D visual servoing control approach proposed by Malis et al. [START_REF] Malis | 2 1/2d visual servoing[END_REF] is one of the most successful visual servoing control paradigms. That approach, however, requires a homography decomposition process which is often computationally expensive and highly sensitive to measurement noise. To deplete the need of homography decomposition, Benhimane and Malis [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF] proposed a more advanced kinematic Homographybased Visual Servoing (HBVS) control algorithm. That work has inspired the members of I3S-OSCAR team to develop a dynamic HBVS control approach which takes the full dynamics of fully-actuated AUVs into account and, consequently, ensures a large provable domain of stability [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]. In this thesis work, we extended the prior work of the team [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] to the case where linear velocity measurements are not available. One of the main motivations behind this effort is related to the development of a low-cost but efficient solution for the stabilization of AUVs without the need of a costly DVL velocity sensor. More precisely, the proposed solutions make use of a minimal and inexpensive sensor suite consisting of an Inertial Measurement Unit 1.1. Motivations and contributions (IMU) and an embedded video camera. Two algorithms have been successfully developed, with the support of rigorous Lyapunov-based stability analyses, for two different camera configurations: downward-looking and forward-looking. In particular, the novel visual error used for the second case is also an original contribution. The development of a low-cost and man-portable underwater vehicle called I3S-UV during my PhD project, together with the development of the homography estimation library called HomographyLab © by the I3S-OSCAR team, is a defining factor to the successful experimental validations of these algorithms.

2. Trajectory tracking of slender body underactuated underwater vehicles: Slender body underwater vehicles are often used for data acquisition applications that require them to move at high speed. The slender shape is conceived for reducing hydrodynamic drag along a nominal axis and thus energy consumption. However, this design leads to highly nonlinear dynamics of the system due to added mass and hydrodynamic effects since the total mass can no longer be considered as proportional to identity and the resulting hydrodynamic force also strongly depends on the vehicle's orientation. On the other hand, when traveling forward at high speed, the lateral thrusters in sway and heave directions can be ineffective, leading any fully actuated vehicle to behave like an underactuated one. By considering this fact together with cost effective and weight/volume, slender body underwater robots are often underactuated by conception with the main thruster mounted at the back and control surfaces employed for orientation control. Control design for underactuated AUVs is challenging and still remains an everlasting source of inspiration for applying new control techniques. When the control objective concerns the tracking of a reference position trajectory, whose velocity does not vanish for all time, various classical control design techniques can be applied. For instance, classical methodologies, linear and nonlinear, have been applied on the basis of linear approximations of the two simplified subsystems "depth-pitch" and "planeyaw" about nominal operating points. The main limitation of these approaches is the local nature of the control design and analysis. Moreover, stability and performance can suffer significantly when strong sea currents or aggressive manoeuvres excite the complex hydrodynamic and added-mass effects. To overcome some of these limitations, nonlinear Lyapunov-based control designs have been recently investigated first for the trajectory tracking problem in a horizontal plane using a simplified and reduced "plane-yaw" 3-d.o.f model and then for the motion in 3D space. A nonlinear high-gain backstepping-based controller proposed in [START_REF] Aguiar | Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty[END_REF] allows for exponential convergence of the position error to a small neighborhood of the origin, which means that asymptotic stabilization to zero is not fulfilled. Moreover, Chapter 1. Motivations, contributions and thesis structure the attitude is not explicitly controlled but guided by the closed-loop system's zero dynamics, thus possibly resulting in undesirable attitude dynamics. High-gain controllers are also known to be sensitive to measurement noise and time delays of control inputs. Nonlinear robust control design for underactuated AUVs thus remains an active research topic. My work on this topic has been motivated by the extension of the thrust direction control paradigm initially developed for underactuated aerial vehicles by the members of the I3S-OSCAR team [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to VTOL drones[END_REF][START_REF] Pucci | Nonlinear feedback control of axisymmetric aerial vehicles[END_REF][START_REF] Kai | A nonlinear approach to the control of a discshaped aircraft[END_REF] to underactuated underwater vehicles, making a step towards a unified control approach for both aerial and underwater vehicles. This work addresses the trajectory tracking control design for slender-body underactuated AUVs, whose body shape is symmetric with respect to the longitudinal axis, using a full 6-d.o.f model. The main objective of this study is to find an appropriate methodology to deal with added mass and hydrodynamic effects. In this work, we have proposed various modifications and adaptations, resulting in a modified apparent force independent of the vehicle's orientation and subsequently a nonlinear system with a triangular control structure. To compensate for unavoidable model uncertainties and external disturbances, the proposed controller is also complemented with bounded integral correction actions via a novel formalism. Future extensions to other control objectives and applications such as path following, vision-based pipeline following, etc. should benefit from this study. '20), St Petersburg, Russia. [START_REF] Nguyen | A homography-based dynamic control approach of autonomous underwater vehicles observing a (near) vertical target without linear velocity measurements[END_REF] In addition, I am the main contributor to the development (construction from scratch) of the I3S-UV underwater platform and of the control software system that includes a modified open-source PX4 autopilot and a high level control package in ROS. The robot can be easily upgraded for future needs.

Most

Thesis structure

The present thesis is organized in three parts and partitioned in eight chapters.

• Chapter 1 -Motivations, contributions and thesis structure. However, because of these complexities and difficulties, and of practical needs, the research community always showed their interest to this specific domain. Fortunately, in recent years the same trend that appeared in Aerial Robotics has also occurred in the field of Underwater Robotics. It is partly because many researchers have changed their focus onto underwater environment after aerial robotics has gained considerable achievements. In addition, many electronic components and technologies previously used for aerial robots or robots in a general sense could now be reused for underwater robots such as autopilots, flight controllers, Robot Operating System (ROS), etc. Moreover, there are more companies providing essential components required for building underwater robots. One can consider Blue Robotics2 as one of the pioneers. These companies not only sell their products, but also provide support to their customers to build and operate their own robots through a large number of free tutorials with open forums and free open-source autopilot firmware.

A more "romantic" reason for having stimulated the development of the DIY community in underwater robotics may be linked to the beautiful underwater world that has often been accessed only by a small group of people possessing enough financial support and diving skills. Small and low-cost underwater robots equipped with a tethered link to the operator (i.e. ROVs) in this case would be an ideal solution for those without these resources.

Here again, the DIY community, affordable components and open source autopilot firmwares ease the involvement of newcomers to the underwater robotics community.

The capability of owning small experimental platforms allows researchers to easily and proactively test their theoretical works since less support is required in terms of trans-Chapter 2. A new trend in underwater robotics portation and operational cost. In the past, research groups which did not come from an underwater robotics background often needed collaboration with the ones from the field in order to have access to their vehicles. But nowadays, due to the availability of low cost experimental platforms which are easy to build and assemble, a large number of research groups and individuals (even though not in the field) are able to apply their knowledge to develop new applications. This stimulates the development of research works in the field of underwater robotics, and allows to create more startup companies, especially spin-off ones.

In recent years, there has been a significant increase in the number of small underwater robots developed by worldwide companies or start-ups. These systems can perform a wide range of applications such as underwater photography, hull inspection, treasure hunting, etc. Giants in underwater systems such as Konsberg Maritime with its subsidiary Hydroid3 or General Dynamics4 provide to the market a large choice of small underwater robotic vehicles. While these small systems with simple deployment and retrieval can significantly reduce the budget needed for their operation, they are still relatively expensive. It seems that the market of professional applications is monopolized by a limited number of companies in the field. However, starting by ROV products for hobby/entertainment purposes, start-up companies now show their endeavor to gain market share of ROVs in some specific industrial applications such as underwater surveillance, hull inspection, etc. in areas of shallow water because of less technological barriers. We believe that with the continuous development of many related technologies, the modern underwater robots with smaller dimensions and more intelligence will play an extremely significant role in military, industrial and civil activities in the near future.

In this thesis we focus our attention to the class of small underwater robots, which can be man-portable and are easily to be deployed and retrieved. We will provide detailed insights on the state of the art of small underwater robots in the next section. Here the term small underwater robots stands for Remotedly Operated Vehicles (ROVs) and Autonomous Underwater Vehicles (AUVs) in mini (around 16 kg) or micro (around 3 kg) sizes.

Recent small underwater robots

In this section, we present some recent small underwater robots with discussions on their innovative features. The Bluefin-9 AUV requires at least a group of 3 persons to deploy and retrieve.

Recent small underwater robots

Small ROVs

However, for a smaller one, for instance one-man portable Riptide µU U V , these tasks can be significantly simplified. As can be seen in Fig. 2 or remotely control. Nowadays, hybrid vehicles are becoming more popular since their exceptional performance allows for implementing many specific tasks. 2.2 Review on the technologies used for small underwater robots

Brushless thruster

Thrusters always remain troublesome for ones who desired to build their own robots.

While the in-house thrusters (e.g. hack-pump thruster) are not reliable, the industrial (e.g. SeaBotix thrusters) are significant expensive (at least US$550 per item). Moreover, they are required periodical maintenance. Many people tried to overcome these issues but were not successful. It was the founder of Blue Robotics, Rustom Jehangir with his friends had noticed these facts when intended to build a small GPS-guided solarpowered boat that can travel autonomously from Los Angeles to Hawaii. They then decided to design new thruster which resists well to saltwater, operating continuously and reliably. However, the thruster must be affordable for everyone who would like to explore the ocean. After introducing the prototype of their first thruster (namely T100)

Camera

Small underwater robots are mainly used for observation applications. Thus, camera is one of the most important sensor. From diving sport, goPro 6

Depth sensor and single-beam echo sounder

For underwater vehicle, depth sensor (c.f. 2.13a) is important to measure submerged depth while single beam echo sounder (c.f. 2.13b) is useful for measuring distance from the vehicle to the seabed or obstacle. For vehicles working in low-visibility water conditions, the mechanical scanning imaging sonar (Ping360 scanning sonar, c.f. 2.14) can be employed for navigation. This sensor is also useful for applications such as inspection, obstacle avoidance, target location and tracking, autonomous systems development, etc.

These sensors with reasonable prices allow developers to implement many interesting functionalities for small underwater robots.

2.2. Review on the technologies used for small underwater robots 3) is adopted for attitude parametrization so that singularity issue can be avoided. As for the dynamic equations, the dynamic formulation proposed lately by N. E. Leonard [START_REF] Leonard | Stability of a bottom-heavy underwater vehicle[END_REF] is preferred due to its compact form and physical interpretation related to the translational and rotational momentums of the body-fluid system. However, the interested reader can find the equivalent transformation between the Leonard model and the Fossen model in [START_REF] Krupínski | Offshore structure following by means of sensor servoing and sensor fusion[END_REF].

In this chapter the basics of AUV modeling are recalled. First, reference frames used for representing the vehicle's motion are introduced. Then, kinematic and dynamic equations are provided. Since the AUV is submersed in a fluid, hydrodynamic effects in terms of added mass and drag are carefully investigated. The effect of sea current can be considered as external perturbation.

Reference frames

Before deriving the equations of motion, it is important to define the reference frames used for that purpose. Classically, the dynamic equations of the vehicle are derived in the body-fixed coordinates because the expressions of hydrodynamic and control terms are much simple. It is well-known that the principal axes of symmetry of the vehicle are often chosen as the coordinate axes since the resulting inertia matrix can be relatively simplified. The body-fixed, mobile reference frame {B} is chosen such that -→ e b 3 pointing towards the design "bottom" of the vehicle, while -→ e b 1 points towards its bow and -→ e b 2 towards its starboard. The coordinate vector representing the position of the Center of

Gravity (CG) in {B} is denoted as r G ∈ R 3 .
For the one observing the AUV's motion on Earth, it is convenient to define an Earthfixed reference frame. Following SNAME convention, the Earth-fixed coordinate system {A} is a North-East-Down (earth fixed reference frame convention) (NED) reference frame, with its base vector -→ e a 3 aligning the gravitational direction. The notation of two coordinate systems as depicted in Fig. 3.1 are summarized in Table

Kinematics

Since the AUV is considered as a rigid body moving in a 3 dimensional space, the method of attitude parameterization should be carefully noticed. In this thesis the notion of rotation matrix is used in model formulation. Unit quaternions are used for efficient computational calculations since they have less parameters and require less computation cost to renormalize after each integration step due to rounding errors than the rotation matrix. Euler angles, despite their singularity issue, are still useful for visualization purposes due to their intuitive interpretation. The transformation between different attitude parametrizations including rotation matrix, Euler angles and unit quaternions can be found in [START_REF] Blanco | A tutorial on SE(3) transformation parameterizations and on-manifold optimization[END_REF][START_REF] Hua | Contributions to the automatic control of aerial vehicles[END_REF].

The orientation of the body-fixed reference frame {B} with respect to the inertial {A} is represented by a rotation matrix R ∈ SO(3). Let us denote the coordinate vector representing the position of the CB in the inertial frame {A} as ξ ∈ R 3 .

The rate of change of the vehicle's position represents the velocity vector that relates to the body-referenced velocity V ∈ R 3 by ξ = RV (3.1)

Dynamics

The kinematics of the rotation matrix of the vehicle [START_REF] Hamano | Derivative of rotation matrix -direct matrix derivation of well known formula[END_REF] satisfy:

Ṙ = RΩ × (3.2)

Dynamics

In comparison with any heavier-than-air aerial vehicle, the dynamics of an underwater vehicle are remarkably different because the surrounding environment of the vehicle is more dense and viscous. In the sequel the vehicle's dynamic equations are linear superposed of the rigid body and of the surrounding liquid.

Rigid body dynamics

Let us denote J 0 ∈ R 3×3 the inertia matrix of the rigid body in a body-fixed frame whose axes are aligned with the ones of {B} and whose origin coincides with the CG. In most of the cases, J 0 can be approximately described by a diagonal positive-definite matrix:

J 0 = J x 0 0 0 J y 0 0 0 J z (3.3)
For the inertia matrix J ∈ R 3×3 expressed in {B}, it can be deduced based on J 0 by applying the parallel axis theorem [START_REF] Kane | Dynamics: Theory and Applicationss[END_REF]:

J B = J 0 -mr 2 G× (3.4)
Denote P ∈ R 3 and Π ∈ R 3 the translational and rotational momentums expressed in {B}, respectively. In view of (3.4), the expressions of the quantity of momentum of the rigid body in the frame {B} are given by

P = mV -mr G× Ω Π = J B Ω + mr G× V (3.5)
Let us denote f ∈ R 3 the coordinate vector expressed in {A} of the total force acting on the body at a certain point, and τ ∈ R 3 the coordinate vector expressed in {A} of the total torque acting around an axis through that point. Assume that the acted point is the CB, then one can neglect the rotational momentum terms due to eccentric forces. The dynamics of the rigid body is thus given by [START_REF] Leonard | Stability of a bottom-heavy underwater vehicle[END_REF]:

Ṗ = P × Ω + F (3.6a) Π = Π×Ω + P×V + Γ (3.6b)
where F = R f and Γ = R τ are the coordinate vectors expressed in {B} of the total force and torque, respectively.

The kinetic energy of the rigid body including translational and rotational kinetic energies has the following form

E B = 1 2 W M B W (3.7)
with the abstract rigid body mass matrix

M B mI 3 -mr G× mr G× J B (3.8)
and

W [V , Ω ] ∈ R 6 .

Added mass

In 1828, after accounting for buoyancy effect F. Bessel observed that period of a pendulum submersed in a fluid relatively increase in comparison with in vacuum. He proposed the concept of added mass [START_REF] Stokes | On the effect of the internal friction of fluids on the motion of pendulums[END_REF] to indicate that the surrounding fluid must be accelerated that increases the effective mass of accelerating submersed body.

According to Kirchhoff and Lamb theory [START_REF] Lamb | Hydrodynamics[END_REF], the liquid surrounding the body has the following kinetic energy

E F = 1 2 W M A W (3.9)
where the added mass matrix is denoted as

M A M 11 A M 12 A M 21 A M 22 A ∈ R 6×6 (3.10)
For a fully submerged body, M A is positive definite and all its diagonal components are positive. In [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF] it is shown that

M A = M A is a good approximation, thereby result- ing in M 21 A = M 12 A .
Identification experiments in testing pool only allow for estimating the diagonal terms -the most significant ones of the added mass matrix. Alternatively to this time-consuming and high-cost method, the identification can be numerically carried out using WAMIT1 , a panel program designed to solve the boundary-value problem for the interaction of water-waves with submersed body. However, this costly commercial software is only affordable by a limited number of companies and institutions. In addition, the result of identification strongly depends on mesh generation.

In practice for low-speed maneuvering underwater vehicle with three planes of sym-3.3. Dynamics metry, the effect of off-diagonal components of M A can be neglected [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF]. This fact thus allows one to approximate M A by a diagonal matrix, which in turn leads to the approximation M 21 A = M 12 A ≈ 0. In this thesis, due to the unavailability of testing pool and identification commercial program, the diagonal components of M A are identified based on empirical method. By approximating the AUV's main geometric components by corresponding symmetrical objects which have added mass expressed in analytical forms [START_REF] Newman | Marine Hydrodynamics[END_REF], these diagonal components can be roughly estimated.

Body-fluid dynamics

The vehicle's dynamic equations are derived based on the translational and rotational momentums of the body-fluid system. The total kinetic energy of the body-fluid system

is E T = E B + E F . One thus deduces E T = 1 2 W M T W, with M T = M Ξ Ξ J (3.11)
where

M mI 3 + M 11 A , J J B + M 22
A , and

Ξ mr G× + M 21 A (3.12)
In the particular case where M 21 A = M 12 A ≈ 0, one has Ξ = -Ξ ≈ -mr G× . The translational and rotational momentums are computed as

   P = ∂E T ∂V = MV -ΞΩ Π = ∂E T ∂Ω = JΩ + ΞV (3.13)
In view of (3.6) and (3.13) and due to the fact that (r G× V) × = r G× V × -V × r G× , the dynamics of the AUV can be expressed as follows

M V -Ξ Ω = (MV -ΞΩ) × Ω + F (3.14a) J Ω + Ξ V = (JΩ)×Ω + (MV)×V + Ξ(V × Ω) + Γ (3.14b)
So far, the dynamics of the AUV is derived using an abstract form of the total force F and total torque Γ acting on the vehicle. In the next sections, details on F and Γ will be discussed.

Chapter 3. Mathematical model of underwater vehicles

External forces and torques

Restoring forces

Underwater vehicles are constrained by the gravity and buoyancy forces which are aligned with -→ e a 3 . While the former acts on the body at the CG, the acted point of the latter is at the CB, which is not usually coincident with the CG. The torque generated by the two forces statically stabilizes the vehicle to its orientation of equilibrium. These forces are thus called restoring forces in naval architecture.

The resulting restoring forces expressed in {B} is defined as

F gb β gb R e 3 (3.15)
where β gb mg -F b is the sum of the gravity and the buoyancy force F b which is equal to the gravitational weight of the liquid displaced by the vehicle.

Since the CB is the origin of the frame {B}, the restoring torque expressed in {B} is the gravity torque which is defined as

Γ g mgr G× R e 3 (3.16)
In practice, marine vehicles are often intendedly designed as "heavy bottom" to benefit from restoring forces. To keep the vehicle safe from sinking, F B is slightly larger than its weight. Additional masses (payloads or/and ballasts) and floats are employed for adjusting zero-pitch and zero-roll orientation.

Drag

A marine vehicle submersed in a fluid environment is constrained by drag (also called as fluid resistance). It is a force preventing the relative motion of the vehicle moving with respect to a surrounding fluid. In Section 3.3.2, the term of added mass is also proposed for explaining the interaction between the vehicle and the surrounding fluid.

The drag and the added mass (or added inertia) thus must be carefully differentiate.

Since the former describes the effect of the surrounding liquid on a body moving at certain relative speed, the latter is only applied in the context of acceleration.

For underwater vehicles, the wavemaking drag is ignored because of lacking of interaction between vehicles and the free water surface. Therefore, only three other kinds of drag are considered including potential, skin friction and vortex shedding. However, it is difficult to separate these terms in the total drag of underwater vehicles in real fluid.

In practice, it is often assumed that high order drag effects can be neglected, and different drag terms contribute to either linear or quadratic drag. The drag force and 3.4. External forces and torques torque are thus modeled as the sum of linear and quadratic terms as follows or open-source Code_Saturne3 . However, the identification results depend strongly on choosing and setting parameters of the calculation modules, more importantly the form and size of computational grid.

F d (V) = -(D Vl + |V|D Vq )V Γ d (Ω) = -(D Ωl + |Ω|D Ωq )Ω (3.
For a certain number of basic two-or three-dimensional bodies like flat plate, sphere, half-sphere, cone, circular cylinder with different noses, etc. in a certain range of Reynolds number, the value of drag coefficients are almost constant [START_REF] Hoerner | Fluid dynamic drag[END_REF]. Therefore, the values of damping matrices of underwater vehicles in this thesis are identified by empirical method. The vehicles are decomposed into basic bodies with defined drag coefficients.

By superposition, the terms of damping matrices D Vl , D Vq of the whole vehicle can be calculated. The terms of rotational damping matrices D Ωl , D Ωq then are calculated by considering decomposed bodies rotating around the vehicle's principal axis. Also, the terms of damping matrices are verified by comparison with published data sets, for instance in [START_REF] Eidsvik | Identification of hydrodynamic parameters for remotely operated vehicles[END_REF].

Propulsion system

The propulsion system gives underwater vehicle movement and maneuverability against water resistance. In view of dynamical model of the vehicle, it provides control force F c and torque Γ c . There exist different kinds of propulsion for underwater vehicles like variable-buoyancy propulsion system [START_REF] Rudnick | Underwater gliders for ocean research[END_REF] for underwater gliders, hydraulic thrusters 4typically for large work class ROVs, or wide range of thermal propulsion systems [START_REF] Joseph | Modern Torpedoes And Countermeasures[END_REF] for Chapter 3. Mathematical model of underwater vehicles torpedoes. However, only electric thrusters employed in experimental platforms used in this work are considered.

For slender body vehicles traveling at high speed, the thrusters whose axes are perpendicular to the vehicle longitudinal axis can be ineffective. Therefore, control surfaces (i.e. elevator or ruder) are often employed for orientation control.

Current

For underwater vehicles, current is an external factor that effects the vehicle operation.

It is thus must be taken into account in the modeling. For simplification, it is assumed that the current velocity v f expressed in inertial frame A is constant and irrotational.

Let V f be the vector of coordinates of the current velocity expressed in B. Then the CoB's velocity relative to the current expressed in B is denoted as

V h V -V f . Let us denote W h [V h , Ω ] ,
then the total kinetic energy of the body-fluid system yields

E T = 1 2 W h M T W h (3.18)
The translational and rotational momentums with current taken into account are computed as

   P h = ∂E T ∂V h = MV h -ΞΩ Π h = ∂E T ∂Ω = JΩ + ΞV h (3.19)
In view of (3.6) and (3.13), the dynamics of the AUV can be expressed as follows

M Vh -Ξ Ω = (MV h -ΞΩ) × Ω + F (3.20a) J Ω + Ξ Vh = (JΩ)×Ω + (MV h )×V h + Ξ(V h × Ω) + Γ (3.20b)
Current also has effect on the drag of the vehicle. The first equation of (3.17) representing the drag force is thus written as follows

F d (V h ) = -(D Vl + |V h |D Vq )V h (3.21)
In control design, the terms relating to current are often considered as disturbances.

With considering the fact that Vf = V f × Ω, equations (3.20a) and (3.20b) are thus rewritten as

M V -Ξ Ω = (MV -ΞΩ)×Ω + F + ∆ F (3.22a) J Ω + Ξ V = (JΩ)×Ω + (MV)×V + Ξ(V × Ω) + Γ + ∆ Γ (3.22b)

Complete nonlinear model

with the "disturbance" terms

∆ F -(MV f ) × Ω -MΩ × V f ∆ Γ -(MV) × V f -(MV f ) × (V -V f )

Complete nonlinear model

So far in this chapter, the mathematical modeling for AUV is briefly recalled. The model is foundation for identification, control design and simulation in the next chapters. In summary, the following mathematical model is considered:

M V -Ξ Ω = (MV -ΞΩ) × Ω + F c + F gb + F d (3.23a) J Ω + Ξ V = (JΩ)×Ω + (MV)×V + Ξ(V × Ω) + Γ c + Γ g + Γ d (3.23b)
Depending on the quantity of thrusters and how they are distributed, underwater vehicles can be fully actuated or underactuated. The vehicles considered in Chapters 5 and 6

is fully actuated where F c together with Γ c provides full six control variables for controlling vehicle's position and orientation. In contrast, the vehicles considered in Chapter 8 are underactuated. Since F c is fixed to be parallel to the -→ e b 1 longitudinal axis of the vehicle, there is only one variable for controlling the vehicle's translation (i.e. F c = T e 1 ) whereas its rotational dynamics are fully actuated. In this work, for simplification, we assume that the available components of the control force F c and control torque Γ c are not bounded.

Remark 1. If the vehicle shape is cubic or a spherical, then system model (3.23) can be more simplified. These particular shapes result in the approximations

M 21 A ≈ M 12 A ≈ M 22 A ≈ 0 and M 11 A ≈ m A I 3
, where m A = αρl 3 with α ≈ 0.7 for a cube with length of edge l submerged in water with density ρ, or m A = 2 3 πρr 3 for a sphere with radius r. The term M thus has the form mI 3 with m m + m A , and consequently the Munk moment term (MV) × V is null.

The model of the vehicle can thus be more simplified as

m V -Ξ Ω = ( mV -ΞΩ) × Ω + F c + F gb + F d J B Ω + Ξ V = (J B Ω)×Ω + Ξ(V × Ω) + Γ c + Γ g + Γ d
Part II Dynamic Homography-Based Visual Servo (HBVS) control of fully-actuated AUVs without linear velocity measurements

S

afe and efficient navigation of AUVs in cluttered environments remains a challenging task. Scientific issues are particularly related to the fact that global acoustic positioning systems become unusable or insufficiently precise in such situations, leading to an obvious interest in developing advanced sensor-based control strategies for AUV applications in close proximity to a complex sea bottom or submarine structures. While acoustic systems have been widely used for sensing underwater environments, cameras offer an appealing alternative due to the rich information captured by images and their high update rate. By using vision sensors as a sensor modality for relative (scaled) position and orientation, the control problems can be cast into Position-Based Visual Servoing (PBVS) or Image-Based Visual Servoing (IBVS) [START_REF] Chaumette | Visual servo control, part ii: Advanced approaches[END_REF]. Classical visual servo control techniques have been initially developed for robotic manipulators and mobile ground vehicles [START_REF] Hutchinson | A tutorial on visual servo control[END_REF][START_REF] Ma | Vision guided navigation for a nonholonomic mobile robot[END_REF] and then for aerial drones [START_REF] Hamel | Visual servoing of an under-actuated dynamic rigid-body system: an image-based approach[END_REF][START_REF] Gonçalves | Vision-based autonomous approach and landing for an aircraft using a direct visual tracking method[END_REF][START_REF] Plinval | Visual servoing for underactuated VTOL UAVs: a linear, homography-based framework[END_REF]. In underwater robotics, vision sensors have been used to perform station keeping or positioning [START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF][START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF][START_REF] Garcia | Positioning an underwater vehicle through image mosaicking[END_REF][START_REF] Cufi | An approach to vision-based station keeping for an unmanned underwater vehicle[END_REF], docking [START_REF] Brignone | A fully autonomous docking strategy for intervention auvs[END_REF][START_REF] Maire | A vision based target detection system for docking of an autonomous underwater vehicle[END_REF][START_REF] Wirtz | Design and test of a robust docking system for hovering AUVs[END_REF][START_REF] Lwin | Visual docking against bubble noise with 3-D perception using dual-eye cameras[END_REF], and pipeline following [START_REF] Matsumoto | Real-time vision-based tracking of submarine-cables for AUV/ROV[END_REF][START_REF] Rives | Underwater pipe inspection task using visual servoing techniques[END_REF][START_REF] Antich | Development of the control architecture of an underwater cable tracker[END_REF][START_REF] Allibert | Pipeline following by visual servoing for autonomous underwater vehicles[END_REF], etc.

Both stereo and monocular cameras have been exploited for stabilization and positioning of AUVs. When the vehicle's pose (i.e. position and orientation) can be estimated, existing PBVS controllers can be directly applied [START_REF] Palomeras | Vision-based localization and mapping system for AUV intervention[END_REF]. In contrast, the case of monocular vision without the assumption of planarity of the visual target and the prior knowledge of its geometry is more involved since full pose reconstruction from visual data is not possible. However, monocular vision can be sufficient to achieve stabilization of an AUV in front of a planar target [START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF][START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF][START_REF] Brignone | A fully autonomous docking strategy for intervention auvs[END_REF]. Recently, an advanced kinematic IBVS control scheme was proposed in [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF] by exploiting the so-called homography that is an invertible mapping relating two camera views of the same planar scene by encoding in a single matrix the camera pose, the distance between the camera and the scene, and the normal direction to the scene [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF][START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF]. A noteworthy feature of this approach is the nonrequirement of homography decomposition which is often computationally expensive and sensitive to measurement noise (see e.g. [START_REF] Malis | Deeper understanding of the homography decomposition for vision-based control[END_REF]), as opposed to other homographybased visual servo (HBVS) controllers [START_REF] Malis | 2 1/2d visual servoing[END_REF][START_REF] Brignone | A fully autonomous docking strategy for intervention auvs[END_REF]. More recently, this kinematic HBVS control approach has been extended in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] in order to account for the full dynamics of fully-actuated AUVs and to obtain an enlarged provable domain of stability. The research work presented in this part is an extension of [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] to the case where linear velocity measurements are unavailable. One of the main motivations behind our efforts are related to the development of a low-cost but efficient solution for station keeping and positioning of AUVs without the need of a costly DVL velocity sensor. More precisely, the proposed solution makes use of a minimal and inexpensive sensor suite consisting of an IMU and an embedded video camera.

HBVS control can be applied to numerous AUV applications once a locally planar visual target is available. For instance, station keeping using a downward-looking camera to observe the ocean floor is a classical application. One can also mention stabilization or positioning in front of a man-made subsea manifold for high-resolution imaging, monitoring, or inspection, or for manipulation like valve-turning, or for maintenance like cleaning, repairing, or changing underwater structures. Finally, docking on a planar docking station is also a relevant application for HBVS control.

This part is structured into four chapters. Homography definition, some featurebased homography estimation methods and a review of existing homography-based visual servo control techniques are provided in Chapter 4. In Chapter 5, a dynamic HBVS control approach for AUV equipped with a downward-looking camera observing a (near) horizontal target is proposed. Then, another dynamic HBVS control approach for AUV equipped with a forward-looking camera observing a (near) vertical target is developed in Chapter 6. The control designs for both cases have a cascade inner-outer loop control architecture. While the inner-loop designs are identical, the outer-loop designs for these cases are significantly different. All the proposed control approaches are first validated by simulation using Matlab/Simulink and then by experiment using an in-house platform named I3S-UV. In Chapter 7, practical issues relating to the development of the I3S-UV platform are presented.

4

Preliminaries on homography estimation and homography-based control techniques

P

art II is dedicated to the control design for the stabilization or positioning problem of fully-actuated AUVs equipped with a monocular camera and an IMU as sensor modalities. By focussing on the case of a camera observing a textured (near) planar visual target such as a sea floor, it is natural to exploit the so-called homography as feedback information for control design. Robust and precise homography estimation is one of the essential requirements to guarantee a high performance of the designed homography-based visual servo (HBVS) controllers. Aiming to provide preliminary materials about homography, the organization of this chapter is thereby sequential.

First, some basic notion of homography is recalled. Then we discuss about some relevant feature-based methods for homography estimation with a particular focus on a nonlinear homography observer on SL(3) that has been developed by the I3S-OSCAR team [START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF]. Note that the library HomographyLab 1 implementing this observer for real-time, 4.1. Homography definition highly robust and efficient homography estimation has been used for the experimental validations of our proposed HBVS controllers. Finally, we briefly recall and discuss about some state-of-the-art HBVS controllers that have inspired our works on this topic.

Homography definition

Originated from the field of Computer Vision, the so-called homography is an invertible mapping that relates two camera views of the same planar scene by encoding in a single matrix the camera pose, the distance between the camera and the scene, along with the normal direction to the scene (e.g., [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]). For further understanding, more details about the homography (by borrowing some elements of [START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF]) are developed next. to the observed planar scene and the coordinate normal vector pointing towards the scene expressed in {A} (resp. {C}), respectively. One easily verifies that

n = R n d = d -n ξ C (4.1)
The coordinate vectors P ∈ {A} and P ∈ {C} of the same point P on the scene are related by

P = R (P -ξ C ) (4.2)
Since the considered points belong to the observed planar scene

Π := {∀P ∈ R 3 : n P -d = 0} = {∀P ∈ R 3 : n P -d = 0}
one derives from the plane constraint n P d = 1 and Eq. ( 4.2) that

P = R I - ξ C n d P (4.3)
Let p img ∈ A (resp. p img ∈ C) denote the image of the considered point when the camera is aligned with the frame {A} (resp. frame {C}). Note that p img and p img have the form (u, v, 1) using the homogeneous coordinate representation and they are related to the 3D coordinates of that point by2 :

p img ∼ = KP , p img ∼ = KP (4.4)
with K ∈ R 3×3 denoting the camera calibration matrix that contains the intrinsic parameters of the camera such as the focal length, the pixel aspect ratio, the principal point, etc [START_REF] Ma | An Invitation to 3-D Vision: From Images to Geometric Models[END_REF]. If the camera is well calibrated (i.e. K is known) then all quantities can be re-normalized onto the unit 2-sphere S 2 as

p := P |P | = K -1 p img |K -1 p img | , p := P |P| = K -1 p img |K -1 p img | (4.5)
Using Eqs. (4.3) and (4.5), the projected points satisfies

p ∼ = R I - ξ C n d p ∼ = Hp (4.6)
where the projective mapping

H : ∼ = R -R ξ C n d (4.7)
is defined as the Euclidean homography that maps Euclidean coordinates of the scene's points from {A} to {C}. Using (4.1) one verifies that H such that det( H) = 1, without loss of generality it can be assumed that H is an element of SL(3) as originally proposed in [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF].

H -1 ∼ = R + ξ C n d (4.
Recall that the scale factor γ such that The so-called "image" homography matrix H img ∈ SL(3) that maps pixel coordinates from A to C (i.e. p img ∼ = H img p img ) then satisfies H img = KHK -1 . Expression (4.6) provides the transformation by the homography H of point-feature correspondences between two image frames. Analogously, one can find the transformation by H of the correspondences of line features in [START_REF] Hua | Nonlinear observer design on sl(3) for homography estimation by exploiting point and line correspondences with application to image stabilization[END_REF] and conic features (i.e ellipses, hyperbolas) [START_REF] Hua | Explicit Complementary Observer Design on Special Linear Group SL(3) for Homography Estimation using Conic Correspondences[END_REF].

H = γR I -ξ C n d is equal (d /d)

Existing feature-based homography estimation techniques

Classical algorithms for homography estimation taken from the computer vision community consist of computing the homography on a frame-by-frame basis by solving algebraic constraints related to correspondences of image features (points, lines, conics, contours, etc.) [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF][START_REF] Agarwal | A survey of planar homography estimation techniques[END_REF][START_REF] Jawahar | Homography estimation from planar contours[END_REF][START_REF] Kaminski | Multiple view geometry of general algebraic curves[END_REF][START_REF] Conomis | Conics-based homography estimation from invariant points and polepolar relationships[END_REF]. These algorithms only considered the homography as an incidental variable and were not focused on improving (or filtering) the homography over time. In recent years, advances have been made in homography estimation algorithms by exploiting the temporal correlation of data across a video sequence rather than Chapter 4. Preliminaries on homography estimation and homography-based control techniques computing algebraically individual raw homography for each image. Powerful methodologies for nonlinear observer design on Lie groups (e.g. [START_REF] Mahony | Observers for kinematic systems with symmetry[END_REF]) have been instrumental

for the derivation of these algorithms.

A nonlinear observer was proposed in [START_REF] Mahony | Nonlinear complementary filters on the special linear group[END_REF] based on the underlying structure of the Special Linear group SL(3), which is isomorphic to the group of homographies [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF]. Velocity information was exploited to interpolate across a sequence of images and improve the individual homography estimates. The observer, however, still requires individual image homographies (previously computed using an algebraic technique) as the feedback information. Thus, it needed both a classical homography algorithm and a temporal filter algorithm, and only functions if each pair of images provides sufficient features to algebraically compute a raw homography.

In order to overcome these drawbacks, the question of deriving an observer for a sequence of image homographies, which takes image point-feature correspondences directly as input has been considered [START_REF] Hamel | Homography estimation on the Special Linear group based on direct point correspondence[END_REF][START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF]. The previous observer is extended by also incorporating image line-feature correspondences (in addition to point-feature correspondences) directly as input in the design of observer innovation [START_REF] Hua | Point and line featurebased observer design on SL(3) for Homography estimation and its application to image stabilization[END_REF]. In line with this effort, conic-feature correspondences (i.e. non-degenerate second-order features such as ellipses and hyperbolas) are considered for the construction of observer innovation [START_REF] Hua | Explicit Complementary Observer Design on Special Linear Group SL(3) for Homography Estimation using Conic Correspondences[END_REF]. Without requiring any prior step for reconstruction of individual homographies for feeding the observer innovation, these algorithms are suitable for real-time applications using an embedded computer. In contrast with algebraic techniques, these observers are also well posed even when there is insufficient data for full reconstruction of a homography. In such situations, these algorithms continue to operate by incorporating available information and relying on propagation of prior estimates.

In this work, the experimental validations of homography-based visual servo control algorithms proposed in Chapters 5 and 6 have been performed using HomographyLab library 3 that implements in C++ the homography estimation observer proposed in [START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF].

This library allows for running in real time and sufficiently fast the homography estimation for control applications with a modest companion computer (for instance 10 Hz with a Hardkernel XU-4 or 20 Hz with an Nvidia Jetson Nano).

Thereafter, a classical algebraic algorithm and a state-of-the-art nonlinear observer on SL(3) for homography estimation that exploit the simplest feature correspondences -the point correspondences -are recalled for the purpose of understanding.

A classical algebraic algorithm of homography estimation

Homography estimation is a topic well developed and discussed in classical computer vision books [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF][START_REF] Prince | Computer Vision: Models Learning and Inference[END_REF]. In this section, the so-called Direct Linear Transformation (DLT) method which employs point correspondences for homography estimation is briefly revised.

Given a set of four 2D to 2D point correspondences, p i ↔ p i , where p i (resp. p i ) is the re-normalized point of P i (resp. P i ), as shown in (4.5). Denote

[u i , v i , w i ] coordi- nates of p i . Equation (4.6) implies that p i × (Hp i ) = 0 which in turn yields    0 -w i p i v i p i w i p i 0 -u i p i -v i p i u i p i 0       h 1 h 2 h 3    = 0 (4.9)
with h j (j = 1, 2, 3) the j th column of H. Equation (4.9) contains three equations, however only two of them are linearly independent. By omitting, for instance, the third equation, each point correspondence p i ↔ p i gives two equations in the entries of H as

0 -w i p i v i p i w i p i 0 -u i p i    h 1 h 2 h 3    = 0
These equations have the form

L i h = 0 where L i is a 2×9 matrix and h = [h 1 h 2 h 3 ]
the vector of 9 unknown entries of H. From a set of four point correspondences on the observed plane, a set of 8 equations in form of Lh = 0 is obtained, where L is the matrix of dimension 8 × 9 obtained by stacking the rows of L i contributed from each correspondence. One observes that h = 0 is an obvious solution.

For a set of four consistent points (in the sense that all triplets of these four points are linearly independent), L has rank 8, and thus with an additional constraint of the norm |h| > 0, the obvious solution is avoided and h is defined up to scale. For simplification, one can choose |h| = 1 which is equivalent to having the Frobenius norm of H equal to 1.

Solving these algebraic equations on a frame-by-frame basis requires computation

p i ∼ = Hp i (4.10)
The re-normalized point p i is thus given by

p i = H-1 p i | H-1 p i | (4.11)
Since the determinant of H ∈ SL( 3) is equal to 1, the determinant of H is also equal to 1 and H thus also belongs to SL(3) -the set of all real valued 3 × 3 matrices with unit determinant. The Lie-algebra sl(3) of SL( 3) is the set of all real valued 3 × 3 matrices with trace equal to zero. The adjoint operator is a mapping Ad :

SL(3) × sl(3) → sl(3)
defined by:

Ad H X := HX H-1 , H ∈ SL(3), X ∈ sl(3).
In the sequel, basic ideas of observer design proposed in [START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF] for H on SL(3) based on direct point correspondence are recalled.

Observer on SL(3) based on direct point correspondences

Consider the kinematics of SL(3) given by

Ḣ = F ( H, U) := HU, H(0) ∈ SL(3) (4.12)
with U ∈ sl(3) the group velocity. To expose the underlying ideas of observer design, in this section we consider the simplified case where the group velocity U is known.

Assume that a set of n measurements

p i = h( H, p i ) ∈ P 2 , i = {1 . . . n} in form of (4.11)
in the camera frame {C} is available, where p i ∈ P 2 are constant and known.

In [START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF] it is verified that the kinematics (4.12) are right equivariant, the observer design framework proposed in [START_REF] Mahony | Observers for kinematic systems with symmetry[END_REF] thus can be applied.

Assume that a set M n of n ≥ 4 vector directions p i ∈ P 2 , with i = {1 . . . n} contains a subset M 4 ⊂ M n of 4 constant vector directions such that all vector triplets in M 4 are linearly independent. In this case , M n is called consistent.

e i := ρ( Ĥ-1 , p i ) = Ĥp i | Ĥp i | = Ep i |Ep i | .
The proposed observer takes the form

Ḣ = ĤU -∆( Ĥ, p) Ĥ, Ĥ(0) ∈ SL(3 (4.13)
where ∆( Ĥ, p) ∈ sl(3) is the innovation term designed as

∆( Ĥ, p) = grad 1 C p ( Ĥ, p) Ĥ-1
where grad 1 is the gradient using a right-invariant Riemannian metric on SL(3) and the right-invariant cost function C p( Ĥ, p) exploiting the point correspondence is defined as

C p( Ĥ, p) n i=1 k i 2 Ĥp i | Ĥp i | -p i 2 Straightforward computations then yield ∆( Ĥ, p) = - n i=1 k i π e i p i e i (4.14) with π x := (I -xx ), ∀x ∈ S 2 .
It is verified that ∆( Ĥ, p) is right equivariant and according to [START_REF] Mahony | Observers for kinematic systems with symmetry[END_REF] the dynamics of the group error E are autonomous and given by

Ė = -∆(E, p )E (4.15)
Furthermore, provided that the measurement set M n is consistent, the estimate Ĥ converges locally exponentially to H [34, Th. 1].

Observers with partially known group velocity

In previous section, it is assumed that the group velocity U is known for the purpose of observer design. However, that assumption is not fully true in practice. The observer design with partial knowledge of the group velocity is thus recalled thereafter.

U = Ω × + V C n d - n V C 3d I
Note that the group velocity U induced by camera motion depends on the additional variables n and d that define the scene geometry at time t as well as the scale factor γ. Since these variables are unmeasurable and cannot be extracted directly from the measurements, one rewrites

U = Ω × + Γ = Ω × + Γ 1 - 1 3 tr(Γ 1 )I (4.16) with Γ V C n d -n V C 3d I and Γ 1 V C n d .
Since {A} is stationary by assumption, the vector Ω can be directly obtained from the set of embedded gyroscopes. Assume that the unknown part Γ (resp. Γ 1 ) is constant or slowly time varying. In practice, instead of the group velocity U, only the angular velocity Ω is measured, then an additional observer for estimating the unknown part Γ (resp. Γ 1 ) of the group velocity U is employed.

The observer when ξC d is constant (e.g., the situation in which the camera moves with a constant velocity parallel to the scene or converges exponentially towards it) is chosen as follows (compare to (4.13))

   Ḣ = Ĥ(Ω × + Γ) -∆( Ĥ, p) Ĥ Γ = ΓΩ × -Ω × Γ -k I Ad Ĥ ∆( Ĥ, p) (4.17)
and the observer when V d is constant (e.g., the situation in which the camera follows a circular trajectory over the scene or performs an exponential convergence towards it) is defined as follows

   Ḣ = Ĥ Ω × + Γ1 -1 3 tr( Γ1 )I -∆( Ĥ, p) Ĥ Γ1 = Γ1 Ω × -k I Ad Ĥ ∆( Ĥ, p) (4.18)
with some positive gain k I and ∆( Ĥ, p) given by (4.14).

The assumption of the consistency of M n with n ≥ 4 guarantees the convergence of the estimate Ĥ to H. This condition is equivalent to the geometric constraint that requires at least 4 point-correspondences to fully reconstruct the 8 degrees of freedom of the homography using algebraic techniques as discussed in Section 4.2.1.

The homography observers (4.15), (4.17 sical algebraic techniques is that the proposed algorithm is well posed even when there is insufficient data (i.e. less than 4 point correspondences) for full reconstruction of homography. In such situations while algebraic algorithms stop working, the observers proposed in [START_REF] Hua | Feature-based recursive observer design for homography estimation and its application to image stabilization[END_REF] continue to operate by incorporating available information and relying on propagation of prior estimates.

Discussions on existing HBVS controllers

With the assumption of planar visual target and by using monocular camera as sensor modality, one can estimate homography and use it for control design purposes.

HBVS kinematic controllers based on homography decomposition

In this section we first briefly recall the well known 2 1 2 D visual servoing technique proposed by Malis et al. [START_REF] Malis | 2 1/2d visual servoing[END_REF].

• Assume that the planar stationary target is textured enough so that the "image" homography matrix H img can be estimated using either feature-based or densebased algorithms. Assume also that the camera calibration matrix K is precisely calibrated so that the Euclidean homography matrix can be determined as follows

H (= R + R ξ C n d ) = K -1 H img K γ (4.19)
where the scale factor γ is equal to the second singular value of K -1 H img K.

• Using homography decomposition techniques [START_REF] Malis | Deeper understanding of the homography decomposition for vision-based control[END_REF], H can be decomposed into

ṁe = 1 d L v L v,w V C Ω (4.22)
with • The control of the orientation of the camera w.r.t the target image is more trivial since the rotation matrix R can be calculated by homography decomposition.

L v 1 ρ 1    -1 0 x 0 -1 y 0 0 -1    , L v,w    xy -(1 + x 2 ) y 1 + y 2 -xy -x -y x 0    ,
Denote u and θ are rotation axis and rotation angle obtained from R, respectively. It is proved that

d(uθ) dt = 0 L w V C Ω (4.23)
with the Jacobian matrix L w defined by The first two components of m e -m e are directly calculated from the current and reference images whereas the third one is estimated using (4.20).

L w I 3 - θ 2 u × + 1 - sinc(θ) sinc 2 ( θ 2 ) (u × ) 2 , with sinc(θ) sin(θ) θ (4.
In view of (4.22) and (4.23), the time derivative of the task function (4.25) is related to the camera velocities by

ė = L V C Ω , with L 1 d L v L v,w 0 L w (4.26)
• The kinematic control law for the objective of exponential stabilization of e to zero has been proposed

V C Ω = -λ L-1 e (4.27)
where L-1 is an approximation of L -1 , λ is a positive number used for tuning the convergence rate. Denote d the estimate of d and from the fact that L -1 w uθ = uθ, the control law (4.27) is finally given by

V C Ω = -λ d L -1 v -d L -1 v L u,w 0 I 3 e (4.28)
Some significant attributes of this 2 1 2 D visual servoing technique are discussed next.

• The advantage of the extended image coordinates m e expressed in (4.21) is that it links the coordinates in the 3D space with the image space (2D), and because of this the visual servoing technique is called 2 1 2 D.

• The rotational control is decoupled from translational one. The equilibrium e = 0 of System (4.29) is exponentially stable even when d is poorly estimated, showing a certain robustness of this kinematic 2 1 2 D visual servo controller.

• The estimation of ρ 1 (= ρ n m ) and ln( Z Z ) = ln(ρ n m n m ) can be carried out only if the knowledge of the normal vector n and n is available, for instance, from homography decomposition. However, precisely estimating n and n from homography decomposition is quite challenging in practice due to noise influence. For instance, in case of station keeping it is desirable that (R, ξ C ) is stabilized about (I, 0). However, when the camera approaches the desired position, the normal vector n (and n) from homography decomposition can be poorly estimated due to measurement noise and is degenerate when ξ C = 0. The dependence of the Jacobian matrix L v and the visual error ln( Z Z ) on the normal vector estimates thus renders this kinematic 2 1 2 D visual servo controller extremely sensitive to measurement noise especially in a close neighborhood of the desired position. This drawback yields an evident interest of developing HBVS controllers without relying on homography decomposition as initially leaded by Malis and his former student [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF].

The 2 1 2 D technique of Malis et al. [START_REF] Malis | 2 1/2d visual servoing[END_REF] is a milestone in the history of visual servo control. Hereafter, several control works for underwater vehicles based on that technique are briefly recalled and discussed.

In [START_REF] Lots | Application of 2 1/2 D visual servoing to underwater vehicle station-keeping[END_REF], Lots et al. applied

ė = L V C Ω 3 , where L            -1 Z 0 x Z y 0 -1 Z y Z -x 0 0 -1 Z 0 0 0 0 0 0 0 0 0 0 0 0 1            6×4 (4.30)
In order to render the scheme robust to disturbances, Lots et al. replaced the original error variable by a composition of proportional, derivative and integral terms, resulting in the following kinematic control law

V C Ω 3 = -Λ L+ K p e + K d ė + K i ˆt 0 edt (4.31) with Λ ∈ R 4×4 , K p , K d , K i ∈ R 6×6
positive diagonal matrices and L+ ∈ R 4×6 the pseudo-inverse of the estimated Jacobian L which requires the knowledge of the initial depth Z . However, the authors did not provide any stability analysis for that control law.

In the later work [START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF] also with the assumption that the target plane is horizontal and the vehicle's -→ e b 3 -axis is always parallel with the inertial -→ e a 3 -axis, Lots et al. proposed a more simplified control solution. In more detail, the depth Z is estimated as Z = det(H)Z with the initial depth Z assumed to be known. However, the depth control is not explicitly explained and presented. While discarding roll and pitch control, and considering that the heading (i.e yaw) can be independently and directly controlled in view of (4.30), the authors argue that the control problem can be restricted to two degrees of freedom, equivalently to a 2D visual servoing. By denoting ē a simplified visual error vector containing only the first two components of e defined in (4.25), it is verified

that ė = -1 Z 0 0 -1 Z V C1 V C2 (4.32)
The authors consider the simplified dynamics of the vehicle on the horizontal plane Similarly to the previously cited work [START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF], Van der Zwaan et al. [START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF] proposed a decoupled control scheme between station keeping in the horizontal plane and depth control. While the former task is almost identical to 2D visual servoing in [START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF], the later one is explicitly introduced. The depth controller is designed based on the partial information obtained from homography matrix. In fact the depth scale factor ρ (= d/d )

M 11 VC1 = B 11 (V C1 -V f 1 )(|V C1 -V f 1 | + D 1 ) + α 1 β + α 2 |β| (4.33a) M 22 VC2 = B 22 (V C2 -V f 2 )(|V C2 -V f 2 | + D 2 ) + α 3 γ (4.
β γ = -Z 0 0 -Z K p ē + K d ė + K i ˆt 0 ēdt (4.34) with K p , K d , K i ∈ R 2×2 positive
satisfies ρ = det h 11 h 12 h 21 h 22 (4.35)
By using ρ as feedback information for depth control (in fact, the authors have used s = √ ρ maybe by error), the desired vertical velocity in form of PID terms is given analogously to (4.31). In addition, the authors propose a method to enhance tracker system by using optical flow information in prediction phase. The experiments were performed with positive results, first on an unmanned blimp in laboratory environment and then on a hover-capable ROV in the open sea.

In [START_REF] Brignone | A fully autonomous docking strategy for intervention auvs[END_REF], Brignone et al. proposed a fully autonomous docking technique for an underwater vehicle [START_REF]Véhicule Ouvert et Reconfigurable pour les Techniques EXpérimentales[END_REF]. The task includes two phases. In the first one, a passive acoustic marker and sonar are used for identifying the docking station on the seabed from dis-57 4.3. Discussions on existing HBVS controllers tance up to 50 m. In the second one when the vehicle approaches the docking station at a visual range (about 2 m to 3 m), a controller based directly on 2 1 2 D visual servoing technique [START_REF] Malis | 2 1/2d visual servoing[END_REF] is employed. A downward-looking camera with optical axis parallel to vehicle's body -→ e b 3 -axis is used for that purpose. Even the thrusters' configuration is not explicitly presented, it can be guessed that the vehicle is fully actuated. For guiding the vehicle to vertically align with the docking structure, the simplified visual error ē that contains only the first two components of e (defined in (4.25)) is used. The expression (4.26) can be simplified as

ė = L V C Ω , with L(x, y, Z) -1 Z 0 x Z xy -(1 + x 2 ) y 0 -1 Z y Z 1 + y 2 -xy -x
The kinematic control law for the objective of exponential stabilization of ē to zero was propose as follows

V C Ω = -λ L + (x , y , Z ) ē (4.36)
Here, instead of choosing the pseudo-inverse of the Jacobian L for (x, y, Z) (i.e. L + (x, y, Z)), a simplified choice L + (x , y , Z ) is adopted in (4.36). The authors argue that even though this convenient simplification may have implications, the stability of the equilibrium ē = 0 can still be successfully achieved because of the slow dynamic response of the hovering vehicle. The controller has been tested in a pool with checkerboards placed in the bottom. The overall visual servoing algorithm was performed on a embedded computer at 12 Hz.

Remark 2. The 2 1 2 D visual servoing technique in [START_REF] Malis | 2 1/2d visual servoing[END_REF] that is proposed for a general case of a visual target requires the estimates of the normal vector (i.e. n and n) for establishing the Jacobian matrix term L v and the visual error ln( Z Z ). As discussed above, this can be problematic for station keeping task since n (and n) calculated from homography decomposition can be strongly effected by noise or even degenerate. However, in underwater works [START_REF] Lots | Application of 2 1/2 D visual servoing to underwater vehicle station-keeping[END_REF][START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF][START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF][START_REF] Brignone | A fully autonomous docking strategy for intervention auvs[END_REF], only a particular case is considered when the textured target plane is horizontal, the vehicle's 2 D visual servoing technique thus can be avoided. However, it can be seen from this scenario that the proposed technique has considerable limitation in practical application. Above all, the requirement of the visual target to be horizontal seems restrictive in reality.

HBVS kinematic controllers without homography decomposition

All the aforementioned works [START_REF] Malis | 2 1/2d visual servoing[END_REF][START_REF] Lots | Application of 2 1/2 D visual servoing to underwater vehicle station-keeping[END_REF][START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF][START_REF] Van Der Zwaan | Visual station keeping for floating robots in unstructured environments[END_REF][START_REF] Brignone | A fully autonomous docking strategy for intervention auvs[END_REF] in the previous section are based on homography decomposition that often requires much computational resource and is sensitive to measurement noise [START_REF] Malis | Deeper understanding of the homography decomposition for vision-based control[END_REF]. For avoiding these practical issues, an advanced HBVS approach without relying on homography decomposition has been proposed by Benhimane and Malis in [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF]. This kinematic control approach consists in using (V C , Ω) as control inputs to stabilize the visual errors e p , e Θ ∈ R 3 to zero, where these errors are directly computed from the homography matrix

H = R + R ξ C n d as follows e p (I 3 -H)m , e Θ vex(H -H), (4.37)
with m ∈ S 2 an arbitrary unit vector satisfying n m > 0. Denote V C the vector of coordinates (expressed in {C}) of the linear velocity of the camera C. The derivatives of

H satisfies Ḣ = -Ω × H - 1 d V C n (4.38)
It is shown in [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF] that the following kinematic controller

V C = -k p e p , Ω = -k Θ e Θ (4.39)
with positive gains k p and k Θ ensures the local exponential stability of the equilibrium (R, ξ C ) = (I 3 , 0), i.e. H = I 3 .

HBVS dynamical controllers without homography decomposition

The presented controllers in Section 4.3.1 and 4.3.2 (except for [START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF]) are kinematic based.

However, for mechanical systems such as AUVs, dynamical model should be taken into account while forces and torques should be used as control inputs instead of the linear and angular velocities. In this section we recall a prior work on this topic of the I3S-OSCAR team [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]. For discussion purposes, the following simplified dynamical system

is considered VC = F c Γ = Γ c (4.40)
with F c , Γ c ∈ R 3 control variables. This dynamical system can be controlled to stabilize V C and Ω at any smooth reference values V Cr and Ω r as long as the derivatives VCr

and Ωr are at our disposal, by the following proportional controller

F c = -k V (V C -V Cr ) + VCr Γ c = -k Ω (Ω -Ω r ) + Ωr (4.41)
with positive gains k V and k Ω . It is not difficult to verify that the equilibrium (V C , Ω) = (V Cr , Ω r ) of the controlled system is globally exponential stable.

Continuing to exploit the visual errors e p , e Θ defined in (4.37), in view of (4.39), it is logical to define the reference velocities as with the "hierarchical" PD(proportional-derivative)-controller is developed. The proposed hierarchical PD-controller

V Cr = -k p e p Ω r = -k Θ e Θ ( 4
F c = -k V (V C -V Cr ) , Γ c = -k Ω (Ω -Ω r ) (4.43)
with V Cr and Ω r defined by (4.42) and positive gains k p , k Θ , k V , k Ω ensures the local exponential stability (LES) of the equilibrium (H, V C , Ω) = (I, 0, 0) of the controlled system.

The control design and associated stability analysis of the hierarchical PD-controller (4.43) are carried out on a local basis. It is thus difficult to characterize its domain of attraction. For underwater vehicles whose translational and rotational dynamics are nonlinear and strongly coupled due to added mass effects (see Chapter 3), this controller may not guarantee an acceptable performance.

In order to overcome such local stability property, in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] the authors proposed a HBVS dynamical control approach relying on the classical inner-outer loop strategy, such that the inner loop stabilizes the linear and angular velocities V C , Ω to any smooth reference values V Cr , Ω r , provided that VCr , Ωr are computable, while the outer loop makes use of V Cr , Ω r as intermediate control variables to stabilize H about I 3 (or equivalently, stabilize (e p , e Θ ) about zero) to fulfill the visual servoing control objective. Since the vehicle dynamics is fully actuated, the inner loop control objective can be achieved without too much difficulty. The interested readers are recommended to read the details of inner-loop control design in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]. The design of the outer loop which is more important for our discussion will thus be recalled thereafter.

First, the design of reference linear velocity V Cr is presented. It is verified that the and the following augmented dynamics

ėp = -Ω × êp -k p2 (ê p -e p ), êp (0) ∈ R 3 , k p2 > 0 (4.46)
It is assumed that the reference angular velocity Ω r and its derivative are bounded and computable by the inner loop controller, and the inner loop controller ensures the global asymptotical stability and local exponential stability of the equilibrium

(V C , Ω) = (V Cr , Ω r ).
Then, by considering visual error dynamics (4.44), employing the augmented system (4.45)+(4.46), and introducing the following reference linear velocity used by the inner loop controller:

V Cr = -k p êp -z p Ω r × m (4.47)
it is shown that there exists a positive number κ p such that if

k p2
k p1 > κ p so that e p is globally stabilized about zero [44, Propo. 1].

Remark 3. In the particular case where m is chosen parallel to u (for example in the case where u = e 3 (gravity direction) and m = e 3 (downward-looking camera observing a seafloor)), the adaptive term z p given in (4.45) can be neglected in the outer-loop controller V Cr defined by (4.47) so that the latter can be simplified to

V Cr = -k p1 êp , k p1 > 0
and êp is the solution of the following equation (instead of (4.46))

ėp = -ω r u × êp -k p2 (ê p -e p ), k p2 > 0
It can be proved that the global asymptotic stability of e p = 0 no longer requires any condition on k p1 and k p2 [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]Rem. 7].

Second, the design of reference angular velocity Ω r to ensure the convergence of e Θ is recalled. By exploiting visual error e Θ while introducing the following dynamics

Ωr = -k Θ2 Ω r -k Θ1 sat ∆ω (e Θ ), Ω r (0) ∈ R 3 , (4.48)
with positive numbers k Θ1 , k Θ2 and ∆ ω , it is stated that the equilibrium (H, V C , Ω) = (I 3 , 0, 0) of the controlled system is locally asymptotically stable (LAS). Furthermore, there exists a positive number

k Θ1 such that if k Θ1 ≤ k Θ1 then the equilibrium (H, V C , Ω) = (I 3 , 0, 0) is LES [44, Propo. 2].
Although convergence of the visual error e p to zero is global, the visual error variable e Θ is only locally stabilized about zero. For obtaining a stronger stability result, the authors [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] proposed an enhanced design of the reference angular velocity (instead of (4.48))

Ω r k u u × R u + ω r u (4.49)
where u ∈ S 2 a known inertial unit vector, which is assumed to be measured in the vehicle body-fixed frame {B}, is employed for control design. In practice, u can be the gravity direction that can be obtained from accelerometer measurements. The innerloop control is thus re-designed correspondingly to ensure not only the convergence of (V C , Ω) to (V Cr , Ω r ) but also the convergence of R u to u.

It is verified that for any u ∈ S 2 , there exists a well-defined rotation matrix So far, [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] establishes a dynamical control approach in the form of inner-outer loop control scheme for fully-actuated AUVs. In the outer loop level, visual errors e p , e Θ computed directly from homography matrix without the need of homography decomposition are employed to stabilize H about I 3 . In the inner loop level, the vehicle nonlinear dynamics are taken into account. In addition to the estimated homography, implementing these controllers requires measurements of the linear and angular velocities which can be obtained from a DVL and an IMU, respectively. In practice, while IMUs are relatively inexpensive, the cost of the cheapest DVL exceeds €15,000 per item. DVLs are thus unaffordable for underwater vehicle projects with a limited budget. Their large dimensions and high weight also limit the possibility of mounting them into small size 62 Chapter 4. Preliminaries on homography estimation and homography-based control techniques underwater vehicles, for instance man-portable ones. Moreover, many applications such as intervention and high-resolution imaging for inspection require the vehicle to operate in close proximity to man-made structure. However, at a close distance less than 1 m DVL measurements are very imprecise. For these reasons, there is an obvious need of advanced vision-based control techniques without relying on linear velocity measurements. This constitutes the main motivation of our works presented in Chapters 5 and 6.

5

Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera T his chapter establishes a novel inertial-aided homography-based dynamic control approach of fully-actuated AUVs without relying on linear velocity measurements. A particular application of the proposed approach concerns the capability of maintaining the AUV at a desired pose during a long period (i.e. stationkeeping).

This chapter is structured into seven sections. Problem formulation of HBVS control of fully-actuated AUVs is provided in Section 5. Section ?? presents prospective extensions of the proposed control approach.

Problem formulation

Assume that the AUV, equipped with a downward-looking monocular camera, operates

= R + R ξ C n d ∈ R 3×3
between the current image and the reference image is available for control design. Such a homography matrix contains coupled information about the rotation and translation between the current camera frame {C} and the inertial {A}. Recall that the derivative of H satisfies (4.38).

In addition to the estimate of H, (approximate) measurements of the angular velocity Ω and of the gravity direction R e 3 provided by an IMU are also available for control design.

The considered control objective consists in asymptotically stabilizing H about I 3 , which is equivalent to the stabilization of (R, ξ C ) about (I 3 , 0). Main difficulties for con-

Simplified model for control design

The translational and rotational dynamics (3.23a)-(3.23b) are tightly coupled due to the coupling matrix Ξ involved in the definition (3.5) of the momentum terms. These complex dynamic couplings are often neglected in the literature by neglecting all terms involving Ξ using the fact that the considered AUV is compact and the distance between the CB and CG is relatively small. Moreover, since the linear velocity is not measured, the "Munk moment" (MV h ) × V h is here considered as a disturbance. Finally, all terms involving unknown current velocity V f , together with the damping force and torque, are also considered as disturbances. These considerations result in the following simpler dynamic equations that decouple the translational and rotational dynamics:

M V = (MV)×Ω + F c + F gb + ∆ F (5.3a) J Ω = (JΩ)×Ω + Γ c + Γ g + ∆ Γ (5.3b)
with the "disturbance" terms

∆ F -(MV f ) × Ω -MΩ × V f +(Ξ Ω) × Ω -Ξ Ω + F d ∆ Γ (ΞV h )×Ω + P h ×V h -Ξ Vh + Γ d
In the case of station keeping, ∆ F and ∆ Γ would eventually converge to constant vectors. In the sequel, these terms will be first neglected in the derivation of a basic controller, which later on will be robustified via both integral correction actions and a high-gain observer of ∆ Γ .

Remark 4. Assumption that ∆ F and ∆ Γ are constant seems to be restrictive but we have demonstrated via simulation and experimental results that such an assumption is reasonably 5.3. Basic ideas of control design acceptable.

Basic ideas of control design

For illustrating the main idea of control design, let us consider the following simple system:

ėp = a V V = u (5.4)
with e p ∈ R 3 available for control design, a a positive unknown constant, and u ∈ R 3 control input.

Let us first assume that V is available for control design. Since a is positive, the

following controller u = -k 1 e p -k 2 V; k 1 , k 2 > 0
results in a stable closed-loop system ëp + k 2 ėp + a k 1 e p = 0.

Another possible solution is reminiscent of the one discussed in Remark 3 in Chapter 4. Since System (5.4) is in cascade form, the control design can be carried out with an inner-outer loop control architecture.

• The inner-loop controller, governing the dynamics V = u, is given by u = -k 3 (V-V r ) + Vr with k 3 > 0 provided that Vr is computable. Global exponential stability of V about any smooth reference velocity V r is ensured in view of the resulting closed-loop system V -Vr = -k 3 (V -V r ).

• The outer-loop controller, governing the kinematic ėp = a V, defines the reference velocity V r = -k 2 êp with k 2 > 0 and an augmented system

ėp = -k 1 (ê p -e p ); êp (0) ∈ R 3 ; k 1 > 0 (5.5)
The resulting outer-loop system can be written as

ėp = -a k 2 êp + ε(t) ėp = k 1 e p -k 1 êp (5.6)
with ε(t) a (V -V r ) an exponentially vanishing term as a result of the innerloop controller. System (5.6) can be rewritten in the following particular timevarying cascaded interconnection studied by Panteley and Loria [START_REF] Panteley | Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems[END_REF]:

ẋ1 = f 1 (t, x 1 ) + Gx 2 (5.7)
where x 1 = [e p , ê p ] , x 2 = ε, the function f 1 (t, x 1 ) can be easily deduced, and measurements: the case of downward-looking camera G is a constant matrix composed of 0 and 1 in our case. System (5.7) can be seen as a nominal system ẋ1 = f 1 (t, x 1 ) perturbed by the output x 2 of a exponentially stable system. In view of [START_REF] Panteley | Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems[END_REF]Theorem 3] corresponding to the case where the function f 1 (t, x 1 ) grows faster than G, it suffices to prove that the nominal system ẋ1 = f 1 (t, x 1 ) (i.e. setting ε(t) ≡ 0 in (5.6)) is uniformly globally asymptotically stable. This system is clearly uniformly globally asymptotically stable by studying its characteristic polynomial given by

P (λ) = (λ 2 + k 1 λ + a k 1 k 2 ) 3 .
However, in the case when V is unavailable for control design, the aforementioned inner-outer loop control architecture is not applicable. In order to overcome this issue, let us keep using the augmented variable êp specified in (5.5) and propose the following

controller u = k 2 (ê p -e p ) -k 3 e p , k 2 , k 3 > 0 (5.8)
The global asymptotical stability of the equilibrium (ê p , e p , V) = (0, 0, 0) of System (5.4)+(5.5) can be pointed out by examining the following Lyapunov function

L = 1 2 |ê p -e p | 2 + k 3 2k 2 |e p | 2 + a 2k 2 |V| 2 whose derivative satisfies L = -k 1 |ê p -e p | 2 ≤ 0
It is interesting that even though a is an unknown constant and the measurement of V is not available, one can still stabilize e p about zero by employing the augmented variable êp (5.5) and the controller (5.8).

In the following section, the control design for performing station keeping of underwater vehicle equipped with a downward-looking camera observing (nearly) planar textured seabed will be presented. The outer-loop controller for the stabilization of the visual error e p about zero will be established based on the above-presented basic idea.

Control design

By analogy to the cascade inner-outer loop control architecture proposed in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF], the following modified version (illustrated by Fig. 5.2) handling the unavailability of linear velocity measurements is adopted:

• An inner-loop controller, governing the rotation dynamics (3.2) and (5.3b), defines the torque control vector Γ c to ensure the asymptotic stabilization of (Ω, R e 3 ) about (Ω r , e 3 ), where the reference angular velocity Ω r is defined by Ω r k g e 3 × R e 3 + ω 3r e 3 (5.9) with k g > 0 positive gain and ω 3r ∈ R the reference yaw angular velocity to be specified by the outer-loop controller. Note that ω3r must be computable so that the feedforward term Ωr is also computable by the torque controller.

• An outer-loop controller defines the force control vector F c together with the reference yaw angular velocity ω 3r (for the inner-loop controller) to fulfill the main objective of stabilizing H about I 3 , provided that the inner-loop controller ensures the almost global asymptotic stability (almost-GAS) and local exponential stability (LES) of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ).

The inner-loop control design is less involved and is postponed after the outer-loop control design, which is the main contribution of this thesis.

Outer-loop control design

Force control design

Analogously to the control approach proposed in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF], the control design for stabilizing the visual error e p defined in (4.37) about zero is first carried out, but here without linear velocity measurements.

The angle between the target plane's normal vector and the camera axis at the desired pose should be smaller than 90 • , which implies that n e 3 > 0. It follows that the simple choice m = e 3 ensures the validity of the condition n m > 0 without any prior knowledge of n .

In view of (4.38) and (4.37) and using the choice m = e 3 , the dynamics of e p satisfy ėp = -Ω × (e p -e 3 ) + a V C (5.10) with a

(n e 3 ) d an unknown number. However, we know that it is positive using the condition n e 3 > 0.

Chapter 5. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera

In order to handle the case with an arbitrary position of the camera w.r.t the CB in the control design, instead of V, a modified velocity variable is employed:

V V + ω 3r e 3 × r C (5.11)
As a result of the inner-loop controller (to be designed thereafter) that ensures the exponential convergence of (Ω, R e 3 ) to (Ω r , e 3 ), one ensures that Ω converges to ω 3r e 3 , which in turn implies that V C converges to V and Ω × e 3 converges to zero. Therefore,

(5.10) can be rewritten as ėp = -Ω × e p + a V + ε(t) (5.12)

with exponentially vanishing term ε(t) Ω × m + a (V C -V). On the other hand, using

(5.3a) the dynamics of V verify

M V = (M V) × Ω + F c + F gb + ∆ F + ω3r M(e 3 × r C ) -ω 3r (M(e 3 × r C )) × Ω = (M V) × Ω + Fc + F gb + ∆ F + ε 1 (t) (5.13)
with new control force variable

Fc F c + ω3r M(e 3 × r C ) -ω 2 3r (M(e 3 × r C )) × e 3 = F c + ( ω3r I 3 + ω 2 3r [e 3 ] × )M(e 3 × r C ) (5.14)
and exponentially vanishing term

ε 1 (t) -ω 3r (M(e 3 × r C )) × (Ω -ω 3r e 3 )
To expose the main ideas of control design, the outer-loop control design will be first carried out for the case where the disturbance term ∆ F involved in (5.13) is considered null, i.e. ∆ F ≡ 0. Then, we will show later on how to cope with external disturbances and model uncertainties via integral correction actions.

Proposition 1. Consider system including the dynamics (5.12) of e p and translation dynamics given by (5.13) with ∆ F ≡ 0. Assume that the disturbance terms ε(t) and ε 1 (t) remain bounded for all time and converge exponentially to zero. Introduce the augmented system ėp = -Ω × êp -K 1 êp + K 1 e p , êp (0) = e p (0) (5.15)

with K 1 ∈ R 3×3 positive gain matrix. Assume that Ω remains bounded for all time. Apply the control force Fc = mM -1 sat η 1 (k 2 ẽp ) -sat η 2 (k 3 e p ) -F gb (5.16)

Control design

with m, k 2 , k 3 , η 1 , η 2 positive numbers, and ẽp êp -e p . Then, the equilibrium (ê p , e p , V) = (0, 0, 0) is globally asymptotically stable (GAS). Moreover, Fc remains bounded by

| Fc | ≤ mλ -1 M (η 1 + η 2 ) + |mg -F b | (5.17)
with λ M the smallest eigenvalue of M.

Proof. The closed-loop system of system (5.12)+(5.13)+(5.15) can be rewritten in the following particular time-varying cascaded interconnection studied by Panteley

and Loria [START_REF] Panteley | Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems[END_REF]:

ẋ1 = f 1 (t, x 1 ) + Gx 2 (5.18)
where

x 1 = [ê p , e p , V ] , x 2 = [ε , ε 1 ] , the function f 1 (t, x 1
) can be easily deduced, and G is a constant matrix composed of 0 and 1 in our case. System (5.18) can be seen as a nominal system ẋ1 = f 1 (t, x 1 ) perturbed by the output x 2 of a exponentially stable system. In view of [67, Theorem 3] corresponding to the case where the function f 1 (t, x 1 ) grows faster than G, it suffices to prove that the nominal system ẋ1 = f 1 (t, x 1 ) (i.e. setting ε(t) ≡ 0 in (5.12) and ε 1 (t) ≡ 0 in (5.13)) is uniformly globally asymptotically stable. This allows us to avoid using input-to-state stability (ISS) argument and, subsequently, the need of constructing of a strict Lyapunov function for the nominal system.

From (5.15) and (5.12) (with ε(t) ≡ 0) one obtains

ėp = -Ω × ẽp -K 1 ẽp -a V (5.19)
Consider the following Lyapunov function candidate

L ˆ|ẽ p| 0 sat η 1 (k 2 s)ds + ˆ|ep| 0 sat η 2 (k 3 s)ds + a 2 m |M V| 2 (5.20)
One verifies that the time-derivative of L along any solution to the controlled unperturbed system is

L = -sat η 1 (k 2 ẽp ) K 1 ẽp -a sat η 1 (k 2 ẽp ) V + a sat η 2 (k 3 e p ) V + a m-1 M V) ( Fc + F gb = -sat η 1 (k 2 ẽp ) K 1 ẽp (5.21)
From (5.21) one deduces that L is semi-negative definite and, thus, ẽp , e p , and V remain bounded by initial conditions. One then easily verifies that ėp and L also Chapter 5. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera remain bounded, implying the uniform continuity of L. Then, application of Barbalat's lemma [START_REF] Khalil | Nonlinear systems (3rd Edition)[END_REF] ensures the convergence of L and, thus, of ẽp to zero. Then, from (5.19) and by application of the extended Barbalat's lemma [START_REF] Micaelli | Trajectory tracking for unicycle-type and two-steeringwheels mobile robots[END_REF] one also ensures the convergence of ėp to zero, which in turn implies that V converges to zero. Similar arguments can be used to show the convergence of V to zero. In view of (5.13) (with ∆ F ≡ 0), ε 1 (t) ≡ 0 and Fc given by (5.16), one ensures that e p converges to zero. The stability of the equilibrium (e p , êp , V) = (0, 0, 0) is a direct consequence of (5.20) and (5.21). Finally, the bound of the force control vector Fc given by (5.17) is directly deduced from (5.16).

Remark 5. In first order approximations, the force control Fc given by (5. [START_REF] De Plinval | Stabilization of a class of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF]) is given by

Fc = k 2 mM -1 êp -(k 2 + k 3 ) mM -1 e p -F gb (5.22)
The proof of global asymptotic stability of the equilibrium (ê p , e p , V) = (0, 0, 0), with Fc given by (5.22), proceeds analogously to the proof of Proposition 1. The linear approximation (5.22) of Fc is useful for gain tuning using, for instance, pole placement technique, while its nonlinear expression (5.16) that involves saturation functions allows us to define explicitly the bound of the force control input Fc as given by (5.17). The latter property is of particular importance in practice since it is often desirable to take explicitly actuation limitations into account. For instance, (5.17) implies that the desired bound of | Fc | can be set to any value

µ (> |mg -F b |) if η 1 + η 2 ≤ m-1 λ M (µ -|mg -F b |)
Note that Proposition 1 applies to the case where the perturbation term ∆ F is negligible. While this basic controller would be able to handle small currents, in practice it is often desirable to enhance control robustness by incorporating integral correction actions. However, in our case the system considered in Proposition 1 (i.e. (5.15)+(5.12)+(5.13))

is already a third-order time-varying system. Thus, adding an integrator would lead to a fourth-order time-varying system. Too high order system, together with the presence of an unknown multiplicative factor a in (5.12) and the unavailability of linear velocity measurements, excludes the possibility of establishing global (or semi-global) stability results similar to Proposition 1. However, it is still possible to state local exponential stability. For simplicity, let us consider the case where M can be roughly approximated by a positive diagonal matrix, i.e. M ≈ diag(m 1,1 , m 2,2 , m 3,3 ).

Proposition 2. Consider system including dynamics of e p given by (5.12) and translation dynamics given by (5.13) with constant disturbance ∆ F and diagonal total mass matrix M = diag(m 1,1 , m 2,2 , m 3,3 ). Assume that the disturbance terms ε(t) and ε 1 (t) remaining bounded Apply the control force

Fc = mM -1 sat η 1 (k 2 ẽp ) -sat η 2 (k 3 ēp ) -F gb (5.25)
with positive numbers m, k 2 , k 3 , η 1 , η 2 , and ẽp êpēp . Choose η 2 high enough such that

η 2 > m-1 |M∆ F | (5.26)
and choose k I satisfying

k I < k 2 λ K 1 k 2 + k 3 (5.27) with λ K 1 the smallest diagonal component of K 1 .
Assume that the outer-loop control ω 3r together with the inner-loop control Γ c ensures that Ω can be considered as a first order term in first order approximations. Then, the equilibrium (ê p , e p , V, z) = (k I z , 0, 0, z ), with z ( mk 3 k I ) -1 M∆ F , of the controlled system is locally exponentially stable (LES).

Proof. One verifies that the linearized system of Eqs. (5.12), (5.3a), (5.24) augmented with integrator (5.23) around the equilibrium (ê p , e p , V, z) = (k I z , 0, 0, z ) is Ẋ = AX with X ∈ R 12 and A ∈ R 12×12 given by

A      -K 1 K 1 0 k I K 1 0 0 a M -1 0 k 2 mM -1 -(k 2 +k 3 ) mM -1 0 -k I (k 2 +k 3 ) mM -1 0 I 3 0 0      X [ê p e p (M V) z ]
One verifies from (5.3a) and (5.25) that at equilibrium configuration z ( mk

3 k I ) -1 M∆ F .
Also from these equations, when ẽp = 0, if |k 2 ēp | ≥ η 2 , one deduces that η 2 = m-1 |M∆ F |. Therefore, condition (5.26) is necessary so that the integral action can compensate for the disturbance ∆ F .

The 12th-order characteristic polynomial of the linearized system is

Q(λ) = Q 1 (λ)Q 2 (λ)Q 3 (λ), with
Chapter 5. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera

Q i (λ) = λ 4 + k 1i λ 3 + (k 2 + k 3 )a m m 2 i,i λ 2 + k 1i k 3 + (k 2 + k 3 )k I a m m 2 i,i λ + k 1i k 3 k I a m m 2 i,i
with k I satisfying (5.27). Application of Routh-Hurwitz criterion ensures the exponential stability of the linearized system.

Control design of the reference yaw angular velocity

The previous part of outer-loop control design ensures the convergence of e p to zero.

From there, the design of ω 3r can proceed identically to our prior work [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] and is, thus, recalled here for the sake of completeness.

Let ψ denote the AUV's yaw angle and R ψ the corresponding rotation matrix around AUV's axis -→ e b 3 . In view of (5.9), the designed angular velocity Ω r includes two components. The first one (i.e k g e 3 × R e 3 ) is used for the convergence of R e 3 about e 3 , or equivalently roll and pitch angles about zero, in the inner-loop. Consequently, the designed control force F c in form of (5.16) or (5.25) stabilizes e p to zero or equivalently ξ C to zero, in the outer-loop. In view of the definition of homography matrix H in (4.7), H converges to R , which in turn converges to R ψ as a consequence of the inner-loop controller. In view of (5.1) and (5.2), the component h 12 of H converges to sin ψ. The second component (i.e. ω 3r e 3 ) with ω 3r is thus designed for guiding the yaw angle to zero using the information embedded in h 12 as follows: Proposition 3. (see [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] for proof) Assume that the inner-loop torque controller Γ c ensures the almost-GAS and LES of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ), with Ω r defined by (5.9) combined with ω 3r (involved in (5.9)) solution to the following system

ω3r = -k Θ2 ω 3r -k Θ1 sat ∆ Θ (h 12 ), ω 3r (0) ∈ R (5.28)
with k Θ1 , k Θ2 , ∆ Θ positive numbers and h 1,2 the element at the first row and second column of H. Apply the outer-loop force controller F c given either by Proposition 1 (when

∆ F ≡ 0)
or Proposition 2 (when ∆ F is constant). Then, the equilibrium H = I 3 is LES. Moreover, this equilibrium is almost-GAS in the case where Fc is given by Proposition 1 and ∆ F ≡ 0.

Inner-loop control design

The more involved part concerning the outer-loop control design has been presented. It remains to design an effective inner-loop torque controller that ensures the stability of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ), with Ω r defined by (5.9) combined with (5.28).

In view of the rotation dynamics (i.e. (3.2) and (5.3b)), it is not too difficult to carry 5.4. Control design out the above-mentioned objective since the sub-system under consideration is fullyactuated and the measurements of both Ω and R e 3 are at our disposal. However, the troublesome term ∆ Γ involved in (5.3b) should be carefully addressed, especially when the vehicle is subjected to strong currents that excite the Munk moment effects. Since the angular velocity can be measured at high frequency and with good precision, we propose to estimate ∆ Γ using a high-gain observer similarly to the idea proposed in [START_REF] Hua | Contributions to the automatic control of aerial vehicles[END_REF]Proposition 8].

Lemma 1. Consider the following observer of ∆ Γ :

J Ω = (JΩ)× Ω + Γ c +Γ g + ∆Γ + k 0 J(Ω -Ω) ∆Γ = a 2 0 k 2 0 J(Ω -Ω) (5.29)
with Ω and ∆Γ the estimates of Ω and ∆ Γ , respectively; Ω(0) ∈ R 3 , ∆Γ (0) ∈ R 3 ; a 0 , k 0 some positive gains. Assume that ∆Γ is uniformly ultimately bounded (u.u.b.). Then for any

a 0 ∈ (1 - √ 2/2, 1 + √ 2/2),
1. The errors Ω -Ω and ∆Γ -∆ Γ are u.u.b. by a positive constant ε(k 0 ) that tends to zero when k 0 tends to +∞. Moreover, these terms converge exponentially to zero for any

k 0 > 0 if ∆ Γ is constant.
2. ∆Γ is u.u.b. by a constant independent of k 0 .

The proof proceeds identically to the proof of [START_REF] Hua | Contributions to the automatic control of aerial vehicles[END_REF]Proposition 8]. Now, we can use the estimate ∆Γ as a feedforward term for the inner-loop torque control design.

Define the angular velocity error variable Ω Ω -Ω r . From (5.3b), one obtains the following error equation

J Ω = (JΩ) × Ω + Γ c + Γ g + Γ + ∆Γ + ∆Γ (5.30)
with Γ (JΩ) × Ω r -J Ωr and ∆Γ ∆ Γ -∆Γ .

Proposition 4. Consider error equation given by (5.30). Assume that the unknown perturbation term ∆Γ is constant and bounded by a known value ¯ . Define an anti-windup integrator I Ω solution to the following differential equation

İΩ = -kI Ω + sat ∆ 1 kI Ω + sat ∆ 2 (Q Ω) , I Ω (0) ∈ R 3 (5.31)
with k a positive gain, ∆ 1 , ∆ 2 some positive constants, and Q ∈ R 3×3 satisfying Q Q = J. measurements: the case of downward-looking camera

Apply the control torque

Γ c = -sat η 3 (K Ω Ω) -k iΩ Q I Ω + max 0, |Q Ω| ∆ 2 -1 Γ g -Γ -∆Γ (5.32)
with K Ω ∈ R 3×3 positive diagonal gain matrix, k iΩ positive gain, η 3 a positive number, and Ω r defined by (5.9) combined with (5.28).

If k k iΩ ||Q -||¯ + ∆ 2 ≤ ∆ 1 (5.33)
then, the following properties hold.

1. The error state ( Ω, I Ω , R e 3 ) converges either to (0, I Ω , e 3 ) or (0, I Ω , -e 3 ) for all initial conditions, with

I Ω k -1 iΩ Q - ∆Γ .
2. The "desired" equilibrium ( Ω, I Ω , R e 3 ) = (0, I Ω , e 3 ) is almost-GAS and LES. The "undesired" equilibrium ( Ω, I Ω , R e 3 ) = (0, I Ω , -e 3 ) is unstable.

Proof. Consider the positive storage function

V      1 2 |Q Ω| 2 if |Q Ω| ≤ ∆ 2 1 2 (2|Q Ω| -∆ 2 )∆ 2 otherwise
Using the fact that J = Q Q one verifies that

V = min 1, ∆ 2 |Q Ω| Ω J Ω (5.34)
From (5.30), (5.32) and the definition of Ω one obtains

J Ω = (JΩ) × Ω -sat η 3 (K Ω Ω) + max 1, |Q Ω| ∆ 2 Γ g -k iΩ Q I Ω + ∆Γ (5.35)
Consider the Lyapunov function candidate

L 1 V + mgl(1 -e 3 R e 3 ) + 1 2 k iΩ | ĨΩ | 2 (5.36)
with ĨΩ I Ω -I Ω . Calculating the time-derivative of L 1 using the expression (5.34) of V, equation (5.30), the torque control expression (5.32), the definition (5.9) of Ω r , 5.4. Control design the expression (5.31) of İΩ and the fact that

sat ∆ 2 (Q Ω) = min(1, ∆ 2 |Q Ω| )Q Ω, one deduces L1 = -min(1, ∆ 2 |Q Ω| ) Ω sat η 3 (K Ω Ω) -k iΩ Ĩ Ω sat ∆ 2 (Q Ω) -mglΩ r e 3 × R e 3 + k iΩ Ĩ Ω İΩ = -min(1, ∆ 2 |Q Ω| ) Ω sat η 3 (K Ω Ω) -mglk g |e 3 × R e 3 | 2 -kk iΩ | ĨΩ | 2 + k iΩ Ĩ Ω -kI iΩ -sat ∆ 2 (Q Ω) + sat ∆ 1 k ĨΩ + kI Ω + sat ∆ 2 (Q Ω) ≤ -min(1, ∆ 2 |Q Ω| ) Ω sat η 3 (K Ω Ω) -mglk g |e 3 × R e 3 | 2 (5.37)
where the last inequality is obtained using condition (5.33) and the fact that ∀a, b ∈ [START_REF] Hua | Contributions to the automatic control of aerial vehicles[END_REF] for the proof). Clearly L1 is negative semi-definite. Remark that system (5.35) and (5.31) is not autonomous due to the time-varying term Ω and consequently La Salle's principle does not apply. However, Ω and I Ω are bounded with respect to initial conditions.

R 3 , ∆ ∈ R + one has | -a + sat ∆ (b + a) ≤ |b| if |a| ≤ ∆ (see
Since Ω r and its derivative are bounded thanks to the expressions (5.9) and (5.28), one deduces from (5.35) that Ω is also bounded. Then it is straightforward to verify that L is also bounded, implying the uniform continuity of L. Then, application of Barbalat's lemma ensures that L and, thus, Ω and e 3 × R e 3 converge to zero.

Next, using Barbalat-like arguments it can be shown that Ω also converges to zero, implying the convergence of I Ω to I Ω . The convergence of e 3 × R e 3 to zero implies that Re 3 converges to either e 3 or -e 3 . So far we have proved that ( Ω, I Ω , Re 3 ) converges either to (0, I Ω , e 3 ) or (0, I Ω , -e 3 ).

It remains to show that the "desired" equilibrium ( Ω, I Ω , Re 3 ) = (0, I Ω , e 3 ) is LES and the "undesired" equilibrium ( Ω, z Ω , Re 3 ) = (0, I Ω , -e 3 ) is unstable. Note that the almost-GAS of the "desired" equilibrium then directly follows. In the firstorder approximations, one has R ≈ I + Θ × with Θ = [φ, θ, ψ] and, subsequently,

e 3× R e 3 ≈ [-φ, -θ, 0] . Denoting [ω 1 , ω2 , ω3 ]
Ω and using the approximation Θ ≈ Ω, one obtains the following linearized system of (5.30) and (5.31)

           φ ≈ ω1 -k g φ θ ≈ ω2 -k g θ Ω ≈ -J -1 K Ω Ω-J -1 k iΩ Q ĨΩ +mglJ -1 [-φ,-θ, 0] İΩ = Q Ω (5.
L Ω = 1 2 Ω J Ω + 1 2 k iΩ | ĨΩ | 2 + 1 2 mgl φ 2 + θ 2 (5.39) One verifies that LΩ = -Ω K Ω Ω -k g mgl(φ 2 + θ 2 ) ≤ 0 (5.40)
From here, LaSalle's principle ensures that Ω, φ and θ and, thus, Ω converge to zero, which implies the convergence of z Ω to zero. The convergence of φ and θ to zero is equivalent to the convergence of R e 3 to e 3 . Since the equilibrium ( Ω, z Ω , R e 3 ) = (0, 0, e 3 ) of the linearized system (5.38) is asymptotically stable, it is also exponentially stable. Now, the Chetaev's theorem is used to prove the instability of the equilibrium ( Ω, z Ω , R e 3 ) = (0, 0, -e 3 ). Define y = e 3 + R e 3 . Consider the positive function S 1 (y) y e 3 = 1 + e 3 R e 3 , satisfying S 1 (0) = 0. Define U r {y|S 1 (y) > 0, |y| < r} for some number 0 < r < 1. Note that U r is non-empty. By neglecting all highorder terms, one verifies that

Ṡ1 ≈ e 3 RΩ r× e 3 = k g |e 3× R e 3 | 2 = k g |e 3× y| 2
For all y ∈ U r , the fact that y e 3 > 0 is equivalent to |e 3× y| 2 > 0, which implies that Ṡ1 > 0. Since all conditions of Chetaev's theorem are now united [START_REF] Khalil | Nonlinear systems (3rd Edition)[END_REF], the origin of the linearized system about the undesired equilibrium (so that y = 0) is unstable.

Comparative simulation results

The proposed control approach applied to station keeping of fully-actuated AUVs without the need of linear velocity measurements is, in fact, inspired by the one proposed in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]Remark 7] which corresponds to the particular case of using a downward-looking camera and which makes use of linear velocity measurements. Although the novel approach already has a practical advantage by depleting the need of a costly DVL, it is even more desirable if its performance is also comparable to the previous approach. Therefore, comparative simulation results of the two approaches using a realistic model of a fully-actuated AUV and in presence of a constant horizontal current

v f = [ 1 2 √ 2 , 1 2 √ 2 , 0] [m/
s] will be reported thereafter. For convenience, let us call the proposed controller and the one proposed in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] as Controller 1 and Controller 2, respectively. The simulated vehicle is the BlueROV1 . Its physical parameters are provided in Tab. 5.1, where the added-mass, added-inertia and damping coefficients are roughly identified from the given shape. To test the robustness of the controllers with respect to model uncertainties, the following "erroneous" estimates of M and J are used: One notes that these estimated parameters are quite different from the corresponding "real" ones.

M = mI 3 + M11 A = diag(8.
The homography H is directly computed using (5.1)

with d = 1 [m] and n = R { π 18 , π 6 ,0} e 3 = [0.4924, -0.1736, 0.8529] . The initial conditions are p C (0) = [-2, -1.5, -1] [m], R(0) = R { π 18 ,-π 18 ,π} , V(0) = Ω(0) = 0.
The initial roll and pitch errors are chosen small to guarantee that the target scene stays in the field of view of the downwardlooking camera. In contrast, a very large initial yaw error is chosen (i.e. ψ = π) to verify the large stability domain of the control algorithms.

• Simulation with Controller 1 (i.e proposed approach): The proposed control approach including the outer-loop controller given in Propositions 2-3 and the inner-measurements: the case of downward-looking camera

Specification

Numerical value m [kg] 7.6

F b [N ] 1.01mg l [m] 0.025 r C [m] [0 0 0.15] J 0 [kg.m 2 ]   0.

Controller

Gains and other parameters Outer-loop K 1 = diag(3s, 3s, 3s) • Simulation with Controller 2 (i.e. approach proposed in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF]): To make a fair comparison, the inner-loop controller of [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] has been revised by also incorporating a highgain observer of the perturbation torque induced by sea current, similarly to the one visual error e p are quite reminiscent of the corresponding ones resulted by Controller 1. This implies that the performance of the proposed controller is comparable to our prior control approach [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] that in contrast relies on a costly DVL for linear velocity measurements.

k 2 = 8 3 s 2 a , k 3 = 1 3 s 2 a , s = √ 2 k I = 0.7, η 1 = 1.8, η 2 = 2.3 k g = 1, k Θ1 = 1, k Θ2 = √ 2, ∆ Θ = 1 Inner-loop K Ω = diag(3, 3, 3), k iΩ = 2 k = 10, ∆ 1 = 6.25, ∆ 2 = 2 a 0 = 0.5, k 0 = 20, η 3 = 8
k p2 = √ 2 k u = 1, k Θ1 = 1, k Θ2 = √ 2, ∆ Θ = 1 Inner-loop K V = diag(5, 5, 5), K iV = 0.4K V K Ω = diag(3, 3, 3), k iΩ = 2 k = 10, ∆ 1 = 6.25, ∆ 2 = 2 a 0 = 0.5, k 0 = 20, η 3 = 8

Experimental validations

Experimental setup

The implementation of the proposed algorithm with real-time homography estimation has been performed on an in-house AUV experimental platform (see Fig. To perform the station-keeping task, the UAV is initially in the teleoperation mode and is manually positioned at a certain depth above the lakebed so that the latter is visible by the camera ensuring a sufficient number of detected features for good homography estimation. Such unknown depth thus varies according to lighting conditions and water turbidity. In the reported experiment, the vehicle was positioned at about 0.

5[m]

above the lakebed.

The parameters and control gains involved in the computation of the control inputs are given in Tab. 5.4 and Tab. 5.5.

Experimental results

Experimental results carried out in lake Saint-Cassien (France) are reported next (see Fig. 5.9). Due to space limitation, only brief but most representative parts of total results • https://youtu.be/mkAAjX3mgVk

• https://youtu.be/KjAfYu1jG18

Experimental validations

Controller

Gains and other parameters Proposition 2 K 1 = diag(3s, 3s, 3s)

k 2 = 8 3 s 2 a , k 3 = 1 3
s 2 a , s = 0.9 As can be observed from the reported video, the experiment has been carried out on a sunny day and the waves generate moving bright spots in the lakebed that periodically alter local brightness of the images captured. However, very good and robust quality of homography estimation can be appreciated despite the changes in light illumination as Chapter 5. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera well as large occlusions due to a panel fixed to the stick in between 310 [s] and 360 [s]. It is also worth noting that, theoretically, the homography estimation uses a planar target.

k I = 0.1, η 1 = 1.8, η 2 = 2.3 Proposition 3 k g = 1, k Θ1 = 1/2, k Θ2 = √ 2, ∆ Θ = 2 √ 2 Proposition 4 K Ω = diag(1.
In the experiment, this assumption does not hold true since the lakebed is covered by 3D reliefs such as sand rocks, and additionally the camera is relatively close to the lakebed.

However, the proposed controller along with real-time homography estimation keeps working efficiently, showing the robustness of the control approach.

One can observe some small oscillations during the convergence. This is essentially due to the fact that the outer-loop control runs at a low frequency (10 [Hz]) and also due to the imperfect thrust generated by the thrusters inside its dead-band. This phenomenon becomes more prominent when the outer-loop poles are large. Hence, smaller pole values have been chosen while tuning the gains in the actual experiment as compared to the ones used in the reported simulation.

In conclusion, the experimental results for the overall control approach (i.e inner and outer-loop controls) are quite convincing even though some vehicle's physical parameters (e.g. added mass and added moment of inertia) are only roughly estimated and the AUV was subjected to various challenging conditions (e.g. significant unknown currents, water turbidity, lighting variation, target occlusion, etc.).

Conclusion

In this chapter a dynamic HBVS control approach of fully-actuated underwater vehicles is proposed. An advanced feature with respect to existing works on the topic is the nonrequirement of a costly DVL used for linear velocity measurements. A potential application to station keeping has been demonstrated in a real environment with very encouraging results despite challenging conditions such as important disturbances induced by currents, water turbidity, lightning variation, target occlusion, etc. The proposed approach will undoubtedly enlarge the working envelop of low-cost (ROVs) as a costly DVL is not required. Even for industrial or professional ROVs and AUVs equipped with DVLs, the approach also allows for overcoming the common problem of measurement imprecision of DVLs caused by threshold violation when operating close to underwater structures or a sea bottom. The proposed control approach together with its extensions allow pilots to perform surveillance/intervention tasks in a more comfortable manner, which can reduce workload and thus increase work efficiency.

6

Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of forward-looking camera A more challenging case of a fully-actuated AUV equipped with a forward-looking camera observing a (near) vertical visual target is considered in this chapter. A similar problem concerning underactuated aerial drones has been addressed in [START_REF] De Plinval | Stabilization of a class of underactuated vehicles with uncertain position measurements and application to visual servoing[END_REF] which is in line with our effort in dealing with the system's dynamics and in depleting the need of a linear velocity sensor. However, that approach relies on the assumption that the visual velocity is available for control design, for instance, via the use of a high-gain observer, but a complete stability analysis including such high-gain observer is missing.

This chapter is part of our continuing efforts in developing low-cost but efficient visual servoing solutions for AUVs by depleting the need of a costly DVL. Potential applications encompass, for instance, docking on a planar docking station, stabilization or 6.1. Problem formulation positioning in front of a man-made subsea manifold for performing common tasks in offshore industry such as high-resolution imaging, monitoring, inspection, valve-turning, cleaning, repairing, or changing underwater structures, etc.

In this chapter, some basic notions used in system modeling in Chapter 3 are reused.

However, some notions concerning outer-loop control design are redefined and used for other purposes (i.e. different from Chapter 5).

P4 P3 P2 P1 d n {A} {B} {C} O CB C r C - → e a 3 - → e a 2 - → e a 1 CG - → e b 3 r G - → e b 1 - → e b 2 (R, ξ C ) (R, ξ)

Problem formulation

Based on a reference image, taken at some desired pose using a forward-looking monocular camera, and the current images, the control design objective consists in stabilizing the camera's pose to the reference one. Let us choose the inertial frame {A} attached to the camera's desired pose (see Fig. 6.1). Assume that a good estimate of the homography matrix H is available for control design. This latter encodes geometric information about the rotation and translation between the current camera frame {C} and the reference frame {A} [START_REF] Benhimane | Homography-based 2D visual tracking and servoing[END_REF]

h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 H = R (I 3 - 1 d ξ C n ) (6.1)
with d the distance between the camera optical center and the target plane and n ∈ S 2 the unit vector normal pointing towards the target plane expressed in {A} (see Fig. Denote n the projection vector of n on the horizontal plane; β the angle between n and camera optical axis when taking the reference image; and α the angle between n and the horizontal plane (see Fig. 6.2). One verifies that

n = [n 1 n 2 n 3 ] n d = 1 d c α c β c α s β s α (6.2)
Since the visual target is within the camera's field of view (FOV) when taking the reference image, one should have |α |, |β | < π 2 , which in turn imply that n 1 > 0. In addition to the estimation of H, it is assumed that an IMU is available to provide measurements of the angular velocity Ω together with an approximate of the gravity direction R e 3 .

The control objective consists in stabilizing H about I 3 (or equivalently stabilizing (R, ξ C ) about (I 3 , 0)) using only (H, Ω, R e 3 ) as available information. In addition to the lack of the linear velocity measurement, the main issues of control design are related to the unknown terms d and n involved in the expression (6.1) of H. More importantly, we here exploit directly H without extracting the usual components (R, ξ C d , n ) via a complex and computationally expensive decomposition [START_REF] Malis | Deeper understanding of the homography decomposition for vision-based control[END_REF].

Control design

A cascade inner-outer loop control architecture as depicted in Fig. 6.3 is proposed.

• The inner-loop controller (developed in Subsection 6.2.2) governs the rotation dy-

Ω r k g e 3 × R e 3 + ω 3r e 3 (6.3)
with k g > 0 positive gain and ω 3r ∈ R the reference yaw angular velocity to be specified by the outer-loop controller. Assume that ω 3r is bounded by design. Note that ω3r must be computable so that the feedforward term Ωr is also computable by the torque controller.

• The outer-loop controller (developed in Subsection 6.2.1) defines the force control vector F c and the reference yaw angular velocity ω 3r (used by the inner-loop controller) to fulfill the main objective of stabilizing (R, ξ C ) about (I 3 , 0).

In the sequel we first present the outer-loop control design, which constitutes the main contribution of this thesis. The design for the inner-loop is identical to the case of horizontal visual target which is presented in Section 5.4.2.

Outer-loop control design

Let ψ denote the AUV's yaw angle and R ψ the rotation matrix around AUV's axis -→ e b 3 . Denote

∆ R = ∆ 11 ∆ 12 ∆ 13 ∆ 21 ∆ 22 ∆ 23 ∆ 31 ∆ 32 ∆ 33 R -R ψ
Since R e 3 converge to e 3 as a result of the inner-loop controller, one verifies that ∆ R converges to zero. The convergence of R e 3 about e 3 implies that ω 1 and ω 2 converge to zero. One then deduces that ∆R remains bounded and converges to zero. measurements: the case of forward-looking camera Define the following visual errors

σ -h 11 + h 22 -h 12 -h 21 , -h 31
Using (6.1) one deduces

σ = N 0 0 n 1 R ξ C +    -∆ 11 + ∆ 22 -∆ 12 -∆ 21 -∆ 31    (6.4) with N n 1 I 2 +n 2 S, S 0 -1 1 0
. This in turn yields

R ξ C = N -1 0 0 1 n 1 σ - N -1 0 0 1 n 1    -∆ 11 + ∆ 22 -∆ 12 -∆ 21 -∆ 31    (6.5)
Using (6.4) and (6.5) one verifies that

σ = -ω 3 Sσ + N VC - n 1 Nω 12 - ∆ 31 n 1 Nω 12 + ω 3 S -∆ 11 + ∆ 22 -∆ 12 -∆ 21 + -∆11 + ∆22 -∆12 -∆21 (6.6)
˙ = n 1 V 3 + n 1 ω 12 N -1 σ -n 1 ω 12 N -1 -∆ 11 + ∆ 22 -∆ 12 -∆ 21 -∆31 (6.7)
with ω 12 ω 2 -ω 1 . By denoting P

[P 1 P 2 ] (respectively Fc [F c1 F c2 ] ),
the vector of the two first components of P (respectively F c ), and rewriting Denoting P M V and using (6.9), (6.10), (6.11), Subsystem (6.6)+ (6.8a) and Subsystem (6.7)+(6.8b) can be rewritten as

F gb = F gb ∆ 13 ∆ 23 1 + ∆ 33 ,
   σ = -ω 3 Sσ + N M-1 P - n 1 Nω 12 + ε 1 Ṗ = -ω 3 S P + Fc -P 3 ω 12 + ε 2 (6.12) ˙ = n 1 m 3 P 3 + n 1 ω 12 N -1 σ + ε 3 Ṗ3 = F c3 + F gb + ω 12 P + ε 4 (6.13)
with bounded vanishing terms

                                 ε 1 (ω 3 -ω 3r )NSr C + r C3 Nω 12 - ∆ 31 n 1 Nω 12 + ω 3 S -∆ 11 + ∆ 22 -∆ 12 -∆ 21 + -∆11 + ∆22 -∆12 -∆21 ε 2 F gb ∆ 13 ∆ 23 ε 3 -∆31 -n 1 ω 12 N -1 -∆ 11 + ∆ 22 -∆ 12 -∆ 21 ε 4 -ω 3r ω 12 MSr C + F gb ∆ 33
One observes that systems (6.12) and (6.13) are interconnected (see Fig. 6.4). In the sequel we first neglect all coupling terms in the outer-loop control design. This is intuitively guided by the fact that these terms are multiplicative terms by vanishing variables ω 1 and ω 2 . Then, a more complete stability analysis for this interconnected system will be carried out thereafter.

Surge-and-sway dynamics ( σ, Ṗ) , P 3 σ, P

Heave dynamics ( ˙ , Ṗ3 ) Similarly to the approach proposed in [START_REF] Nguyen | Inertial-aided homographybased visual servo control of autonomous underwater vehicles without linear velocity measurements[END_REF], the altitude (heave) control can be designed independently from surge-and-sway and yaw control design.

Proposition 5. Consider System (6.13) where the coupling terms involving σ and P are neglected. Introduce the augmented system

˙ = -k 13 (ˆ -), ˆ (0) = (0)
with k 13 > 0. Apply the control force

F c3 = m 3 k 23 ˆ -m 3 k 33 -(mg -F b ) with k 33 > k 23 > 0.
Then, the equilibrium (ˆ , , P 3 ) = (0, 0, 0) of the controlled system is globally asymptotically stable (GAS) with exponential convergence rate after some time instant T > 0.

Proof. The considered system can be rewritten as

ẋ = Āx + ε 34 (6.14) with x
   ˆ P 3   , Ā    -k 13 k 13 0 0 0 n 1 m 3 m 3 k 23 -m 3 k 33 0   , ε 34    0 ε 3 ε 4    (6.15)
From there, the proof is straightforward since the nominal system ẋ = Āx is Hurwitz (using k 33 > k 23 > 0) while the perturbation terms ε 3 and ε 4 converge to zero.

Surge-and-sway control design

Consider System (6.12) and neglect all coupling terms involving and P 3 . Introduce the

following augmented system σ = -ω 3 S σ -k 1 σ + k 1 σ, σ(0) = σ(0) (6.16)
with k 1 > 0. Apply the control force Fc = k 2 M σ -k 3 Mσ (6.17)

Control design

with k 2 , k 3 > 0. Then, the controlled surge-and-sway subsystem is given by

       σ = -ω 3 S σ -k 1 σ + k 1 σ σ = -ω 3 Sσ + N M-1 P + ε 1 Ṗ = -ω 3 S P + k 2 M σ -k 3 Mσ + ε 2 (6.18) Denote X 1 = x 11 x 12 σ, X 2 = x 21 x 22 σ, X 3 = x 31 x 32 P X    X 1 X 2 X 3    , ε 12    0 ε 1 ε 2    , S    S 0 0 0 S 0 0 0 S    (6.19) A    -k 1 I 2 k 1 I 2 0 0 0 n 1 M-1 k 2 M -k 3 M 0   , B    0 0 0 0 0 n 2 S M-1 0 0 0    (6.20)
System (6.18) can be rewritten as

Ẋ = -ω 3r SX + AX + BX -ω3 SX + ε 12 (6.21)
with ω3 ω 3 -ω 3r . In the sequel we will specify a sufficient condition ensuring that the origin of System (6.21) is GAS.

Lemma 2. Consider system Ẋ = AX + BX (6.22)

with A and B defined in (6.20). Assume that the reference image is captured with a reference heading angle satisfying

|β | < β max arctan 1 √ 1+ √ 2 ≈ 32.8 • (6.23)
Then, there exist k1 > 0 large and > 0 small enough, with k1 and depending on (β max , |β |),

such that system (6.22) is globally exponentially stable (GES) provided that k 1 > k1 , 1 < k 3 /k 2 < 1 + .
The proof is given in Section 6.2.3.1.

Proposition 6. Consider System (6.21). Assume that all assumptions of Lemma 2 hold. Assume that the inner-loop controller ensures the uniform boundedness and convergence to zero of ε 12 and ω3 . Then, there exists a positive number such that if sup |ω 3r | < then the equilibrium X = 0 is GAS with exponential convergence rate after some time instant T > 0.

The proof is given in Section 6.2.3.2.

Chapter 6. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of forward-looking camera So far the outer-loop control design and associated stability analysis have been carried out for systems (6.12) and (6.13) by neglecting all coupling terms. Next, stability analysis for the full interconnected system will be developed. The interconnected system (6.12)-(6.13) can be rewritten as

Ẋ = -ω 3 (t) SX + AX + BX + E 1 x + ε 12 ẋ = Āx + E 2 X + ε 34 (6.24)
with vanishing matrices

E 1    0 0 0 0 -N n 1 ω 12 0 0 0 -ω 12   , E 2    0 0 0 0 n 1 ω 12 N -1 0 0 0 ω 12    (6.25)
Theorem 1. Consider the interconnected system (6.24). Assume that all assumptions in Propositions 5 and 6 hold. Assume that all outer-loop control gains and ω 3r are chosen as in Propositions 5 and 6. Assume that the inner-loop controller ensures the uniform boundedness and convergence to zero of ε 12 , ε 34 and ω3 . Then, the equilibrium (X, x) = (0, 0) is GAS with exponential convergence rate after some time instant.

Proof. As a result of Propositions 5 and 6, there exists a time instant T 1 > 0 and some positive numbers α 1 , α 2 , β 1 , β 2 such that ∀t ≥ T 1 one has

d dt (X DX) ≤ -α 1 |X| 2 + β 1 E 1 |X| |x| d dt (x Dx) ≤ -α 2 |x| 2 + β 2 E 2 |X|
L(t ≥ T 1 ) ≤ -α 1 |X| 2 -α 2 |x| 2 + (β 1 E 1 + β 2 E 2 )|X| |x|
Since E 1 and E 2 converge uniformly to zero, there exists another time instant T 2 > T 1 such that ∀t ≥ T 2 the cross term is dominated by the quadratic terms. Thus, there exists a positive number ν such that L(t ≥ T 2 ) ≤ -νL(t ≥ T 2 ), implying the exponential convergence of (X, x) to zero.

Control design

Control design of the reference yaw angular velocity

The previous parts of outer-loop control design ensure the convergence of ξ C to zero.

In view of the definition (6.1) of the homography, H converges to R , which in turn converges to R ψ as a consequence of the inner-loop controller. Therefore, the component h 12 of H converges to sin ψ. It, thus, can be exploited for the design of ω 3r for ensuring the convergence of R to I. Note that the whole control design process in the previous subsections is based on the assumption about the boundedness of ω 3r (see Proposition 6). The design of ω 3r and associated stability analysis (omitted due to space limitation) proceeds identically to [START_REF] Hua | Homography-based visual servoing for autonomous underwater vehicles[END_REF].

Proposition 7. Assume that the inner-loop torque control Γ c ensures the almost-GAS of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ), with Ω r defined by (6.3) combined with ω 3r (involved in (6.3)) solution to the following system

ω3r = -k Θ2 ω 3r -k Θ1 sat ∆ Θ (h 12 ), ω 3r (0) ∈ R (6.26) with positive numbers k Θ1 , k Θ2 , ∆ Θ satisfying k θ 1 k θ 2
∆ Θ < where is defined from Proposition 6. Apply the outer-loop force control F c = F c F c 3 where Fc is given by (6.11)+(6.17) with control gains k 2 , k 3 specified in Lemma 2 and F c3 is defined in Proposition 5. Then, the equilibrium (R, ξ C ) = (I 3 , 0) is almost-GAS.

Inner-loop control design

The more involved part concerning the outer-loop control design has been presented.

The design of an effective inner-loop torque controller that ensures the stability of the equilibrium (Ω, R e 3 ) = (Ω r , e 3 ), with Ω r defined by (6.3) combined with (6.26) proceeds identically the Section 5.4.2 for the case of downward-looking camera.

Analyses for Chapter 6 6.2.3.1 Proof of Lemma 2

One verifies that A is Hurwitz if k 3 > k 2 > 0 by applying Routh-Hurwitz criterion on its

characteristic polynomial P A (λ) = λ 3 + k 1 λ 2 + n 1 k 3 λ + n 1 (k 3 -k 2 )k 1 2 .
Consider the following positive diagonal matrix

Q =    qI 2 0 0 0 qI 2 0 0 0 M-2    , q > 0
According to the Lyapunov theorem [START_REF] Khalil | Nonlinear systems (3rd Edition)[END_REF], there exists a unique D = D > 0 such that 100 Chapter 6. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of forward-looking camera

DA + A D = -Q. Straightforward computations result in D = 1 2     d 11 I 2 d 12 I 2 d 13 M-1 d 12 I 2 d 22 I 2 d 23 M-1 d 13 M-1 d 23 M-1 d 33 M-2     (6.27) with d 11 1 k 1 k 2 n 1 + k 3 +k 2 k 3 -k 2 q d 12 -1 k 1 k 3 n 1 + q d 13 1 n 1 + 2q k 3 -k 2 d 22 k 1 n 1 k 3 k 2 -1 1 n 1 + 2q k 3 -k 2 + k 3 k 1 k 2 k 3 n 1 + q d 23 -1 n 1 d 33 n 1 k 1 k 2 k 3 n 1 + q + k 1 k 2 1 n 1 + 2q k 3 -k 2 (6.28)
Consider the Lyapunov function L 1 X DX, with D given by (6.27)-(6.28). After some straightforward computations, one deduces

L1 = X (DA + A D)X + X (DB + B D)X = - x 2 31 m 2 1 + x 2 32 m 2 2 1 - n 2 2 4q d 2 12 + d 2 22 - √ qx 11 + d 12 n 2 2 √ qm 2 x 32 2 - √ qx 12 -d 12 n 2 2 √ qm 1 x 31 2 - √ qx 21 + d 22 n 2 2 √ qm 2 x 32 2 - √ qx 22 -d 22 n 2 2 √ qm 1 x 31 2 (6.29)
Now the task consists in finding sufficient conditions for (k 1 , k 2 , k 3 , q) so that

1 > n 2 2 4q d 2 12 + d 2 22 (6.30)
which ensures that L is negative definite and, thus, the origin of System (6.22) is GAS.

Note that d 12 and d 22 are functions of (k 1 , k 2 , k 3 , q) as defined in (6.28).

From (6.2) and (6.23), one verifies that

n 2 1 n 2 2 > 1 + √ 2. Define δ n 2 1 n 2 2 -(1 + √ 2) > 0.
One verifies that δ =

1 arctan 2 β - 1 arctan 2 β max . Rewrite k 3 = γk 2 (with γ > 1) and k 1 = αk 2 (with α > 0). Denote ζ 1 α(γ-1)k 2 n 2 1 αn 1 γ 2 , ζ 2 n 1 γ α 2 k 2 , ζ ζ 2 (1 -ζ 1 ζ 2 )
After some computations, one verifies that inequality (6.30) is equivalent to

α 2 ζ2 + (2 + ζ 2 ) 2 q 2 + 1 α 2 1 (1-ζ 1 ζ 2 ) 2 + (ζ 1 +1) 2 (1-ζ 1 ζ 2 ) 4 +2 ζ 2 + (ζ 1 +1)(ζ 2 +2) (1-ζ 1 ζ 2 ) 2 -2 -2 √ 2 -2δ q < 0 (6.31) Denote 6.2. Control design ζ ζ 1 ζ 2 , g 1 (ζ) ζ 2 + (ζ 1 +1)(ζ 2 +2) (1-ζ 1 ζ 2 ) 2 g 2 (ζ) ζ 2 2 + (2+ζ 2 ) 2 (1-ζ 1 ζ 2 ) 2 1 + (ζ 1 +1) 2 (1-ζ 1 ζ 2 ) 2
The condition for inequality (6.31) having solution q > 0 is

∆ = g 1 (ζ) -2 -2 √ 2 -2δ 2 -g 2 (ζ) > 0 -g 1 (ζ) -2 -2 √ 2 -2δ > 0 or equivalently δ > g(ζ) g 1 (ζ)-2 2 + √ g 2 (ζ)-2 √ 2 2 (6.32)
We will show that we can always choose positive control gains k 1 , k 2 , k 3 (i.e. ζ 1 , ζ 2 ) such that condition (6.32) holds. One verifies that g

1 (ζ) > 2, g 2 (ζ) > 8, g(ζ) > 0, with ζ 1 , ζ 2 > 0. One also has lim ζ 1 ,ζ 2 →0 + g 1 (ζ) = 2, lim ζ 1 ,ζ 2 →0 + g 2 (ζ) = 8
and consequently lim

ζ 1 ,ζ 2 →0 + g(ζ) = 0 (6.33)
From (6.33), with g(ζ) defined in (6.32), one deduces that ∀δ > 0 there exists ζ * > 0 such that g(ζ) < δ whenever |ζ| < ζ * . This implies that there always exist

ζ 1 = n 1 k 3 k 2 1 and ζ 2 = 1 -k 2 k 3 1 ζ 1 small enough such that ζ 2 1 + ζ 2 2 < ζ *
and, consequently, (6.30) is satisfied. Therefore, there exist k1 > 0 large and > 0 small enough such that inequality (6.30) has solution q > 0 provided that k 1 > k1 , 1 < k 3 k 2 < 1 + .

Proof of Proposition 6

The system (6.21) can be rewritten in the following particular time-varying cascaded interconnection studied by Panteley and Loria [START_REF] Panteley | Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems[END_REF]:

ẋ1 = f 1 (t, x 1 ) + Gx 2 (6.34)
where

x 1 = X, x 2 = [0, ε 1 , ε 2 ] ,
G is an identity matrix in our case, and the function f 1 (t, x 1 ) can be easily deduced as follows:

f 1 (t, x 1 ) = Ẋ = -ω 3r (t) SX + AX + BX -ω3 SX (6.35)
System (6.34) can be seen as a nominal system ẋ1 = f 1 (t, x 1 ) perturbed by the output x 2 of a exponentially stable system. In view of [START_REF] Panteley | Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems[END_REF]Theorem 3] corresponding to the case where the function f 1 (t, x 1 ) grows faster than G, it suffices to prove that the nominal system ẋ1 = f 1 (t, x 1 ) (i.e. setting ε 1 (t) ≡ 0 and ε 2 (t) ≡ 0 in (6.21)) is uniformly globally asymptotically stable. This allows us to avoid using input-to-state stability (ISS) argu- To perform the station-keeping task, the UAV is initially in the teleoperation mode and is manually positioned at a certain distance w.r.t. the artifical visual target so that the latter is visible by the camera ensuring a sufficient number of detected features for good homography estimation. In the reported experiment, the vehicle was positioned at about 0.3[m] far from the target 3 .

Note that here, the heave control is different from Proposition 5 in Section 6.2.1.1 since it is enhanced by integral effect to eliminate static errors. In fact, by introducing a new variable:

¯ = + k Ih z ; k Ih > 0
where z is an anti-windup integrator, which is defined as:

ż = -k z z + k z sat δz z + k z ; z (0) ∈ R; k z , δ z > 0

Specification

Numerical value m

[kg] 16 F b [N ] 1.01mg l [m] 0.025 r C [m] [0.2 0 0.1] M = mI 3 + M11 A [kg] diag(

Preliminary experimental results

The preliminary experimental results carried out in a water tank of dimensions 1.2 m (L) x 0.8 m (W) x 1.5 m (H) are reported next (see Fig. As can be seen in Fig. 6.11, roll and pitch angles are approximately equal to 1.5 • and 9 • , respectively, in the convergence state. Non zero control force components F c1 , F c2 and F c3 are thus required to maintain the platform at the stabilized position, as illustrated in Fig. 6.12. Also from this figure, the positive component F c3 affirms that the I3S-UV vehicle is positive buoyant.

It is noteworthy to mention that theoretically, the stabilization of e 3 to R e 3 is carried out by means of the inner-loop controller enhanced with bounded integrator defined in Proposition 4. The inclination of the plane ( -→ e b 1 , -→ e b 2 ) with the horizontal plane can be explained by the fact that parameters of the bounded inner-loop integrator of the inner-loop controller were not well tuned. As can be seen in Fig. 6.13, this results in the ineffectiveness of the integrator since Γ c1 and Γ c2 in steady state are not sufficient for compensating for static error caused by the issue relating to position of CG4 .

To excite the external force response, the AUV has been manually moved by a stick at time instants 46 [s], 63 [s], 74 [s], and 90 [s]. In Fig. 6.14 one observes that right after finishing the interaction, the AUV went back to its stabilized position and orientation without overshoots in surge and sway directions which correspond to the components of the visual error σ. On the contrary, the overshoot can be clearly observed in heave direction (corresponding to the visual error ). I3S-UV platform was increased slightly over time. This results in a slight decrease of F c3 over time, as shown in Fig. 6.12. For surge-and-sway control, the components of visual error σ do not converge to zero due to the absence of integrator correction actions. The effect of static errors can also be clearly seen in Figs. [START_REF] Allibert | Pipeline following by visual servoing for autonomous underwater vehicles[END_REF].15 and 6.16 since the component h 12 and the Frobenius norm ||H -I|| converge to nonzero constant values.

Experimental validations

Conclusion

In this chapter a homography-based dynamic control approach of fully-actuated underwater vehicles equipped with a forward-looking camera observing a (near) vertical visual target is proposed. Improving the robustness of the outer-loop control in terms of surge-and-sway motion to external perturbations (e.g. current) is a topic for our future work. Then additional tests with a better tuning of gains and parameters will follow to validate the proposed approach, especially in challenging sea trials.

7

Development of the I3S-UV platform I n order to experimentally validate the control algorithms proposed in Chapters 5 and 6, a platform that we name I3S-UV has been developed at the I3S laboratory in the scope of this thesis project. The reasons for building such an in-house vehicle are the following.

• First, it is not easy for us to buy or hire an underwater vehicle fitting our limited hard to buy all these vehicles or hire someone to construct all of them or upgrade new functionalities since our budget is rather bounded.

• Nowadays, several crucial components needed for building an UV from scratch can be easily found in the market. One can name Bluerobotics1 on the top of the list of providers. Besides, many electronic parts can be purchased from Hobbyking2 and other commercial websites.

• Finally, the I3S-OSCAR team possesses rich experience in building UAV systems in terms of hardware and software for the related research projects. Such knowhow and experience are crucial for building a new UV system. Moreover, a large community of DIY with open resources is greatly helpful for this task.

In summary, the main reason for building the I3S-UV is our need to possess an open platform but with a limited budget. An important benefit of this practical work is that the I3S-OSCAR team can now be proactive in developing or upgrading any (new) UV system for future demands. Indeed, an underwater platform with slender body shape is planned to be built in a much faster manner than the already developed I3S-UV. It is because of the fact that its firmware (i.e. autopilot software) which has taken most of the development time can be reused with minor adaptations. Besides, since our vehicle belongs to the class of man-portable UVs, less support facilities are required for preliminary tests.

3D design of I3S-UV

In the beginning of this PhD project (since July 2015), there are a certain number of providers of micro/mini ROV systems including ROVs, firmware and human machine interface softwares. The price of the overall systems can be up to more than ten thousand US dollars. In comparison with the ones of working class in industry, this is relatively inexpensive. However, this is still a large amount of money in comparison with our limited budget. In addition, they are often closed systems preventing possible modifications in order to add other components (sensors, cameras, etc.) for performing new functionalities.

However, there exist crucial components for building a ROV from scratch such as thrusters, mechanical components resistant to high pressure, pressure sensors, lamps, etc. at more reasonable prices 3 . These components can resist to water pressure at a depth of at least 100 m, which is largely sufficient for all of our experimental validations. For Based on the original BlueROV1, the structure is modified to add a longer watertight tube to accommodate the main electronic components (i.e electronic tray) as illustrated in Fig. 7.2. The original tube of BlueROV1 (in olive color) is implanted right below for containing separately a bigger battery. In addition, a shorter tube with spherical dome end cap is dedicated for a camera. This camera tube together with a lamp (in black color) can be rotated and fixed in either downward-looking (c.f. Fig. 7.2a) or forwardlooking (c.f. Fig. 7.2b) configuration. New components of structure are designed and printed by using a 3D printer with plastic filament (c.f. Fig. 7.3). For enhancing the resistance to external forces and moments, some components are designed with empty spaces in form of cylinder holes, which are then filled by carbon fiber composite rods or tubes. It is demonstrated via our practical trials in water that with appropriate choices After incorporating all tubes and support components, the total volume of I3S-UV (c.f. Fig. 7.4) is almost double to its original design. Consequently, the buoyancy force In this software system, ROS with its decentralized architecture plays a crucial role.

It allows all processes (nodes) to run independently while messages between them are exchanged through topics. A ROS master node control the registering of nodes and enables individual ROS nodes to locate one another and then communicate with each other peer-to-peer. The decentralized architecture also allows nodes to run either in the companion computer or the ground station computer. The homography estimation node thus can be run in different computers for comparing the performance.

Homography estimation

Let us explain how homography estimation works in practice as illustrated in Fig. 7.7.

Note that all nodes are in oval with colored background while topics are in rectangular.

Low level control

In the beginning of this PhD project, there was not any opensource autopilot exclusively dedicated to UV 5 . For aerial, surface or ground vehicles, there exist mainly two widely used autopilots and their corresponding communities: ArduPilot and PX4. In this project, PX4 is chosen since our I3S-OSCAR team has a rich experience in exploiting this flight controller. We have performed some modifications in PX4 for performing low-level control of the I3S-UV and measuring submerged depth. The compiled PX4 firmware runs in a 3DR Pixhawk v2 flight controller.

Control allocation

As seen from Fig. 7.9, the positions of the 6 thrusters of I3S-UV expressed in the bodyfixed frame are given by: 6

                                     P 1 =    L 1 W 1 H 1    , P 2 =    L 1 -W 1 H 1    , P 3 =    L 3 W 3 H 3    , P 4 =    L 3 -W 3 H 3    , P 5 =    -L 5 0 H 5    , P 6 =    L 6 0 -H 6   
Force vectors are given by

         F 1 = T 1 e 1 , F 2 = T 2 e 1 , F 3 = -T 3 e 3 , F 4 = -T 4 e 3 , F 5 = -T 5 e 3 , F 6 = T 6 e 2
One thus has

Γ i = P i × F i , yielding                            Γ 1 = -W 1 T 1 e 3 + H 1 T 1 e 2 Γ 2 = W 1 T 2 e 3 + H 1 T 2 e 2 Γ 3 = L 3 T 3 e 2 -W 3 T 3 e 1 Γ 4 = L 3 T 4 e 2 + W 3 T 4 e 1 Γ 5 = -L 5 T 5 e 2 Γ 6 = -L 6 T 6 e 3 + H 6 T 6 e 1
One then deduces the total control force and torque vector

F c Γ c = A T where F c = 6 i=1 F i , Γ c = 6 i=1 Γ i , i = 1, 6
with the control allocation matrix

A            1 1 0 0 0 0 0 0 0 0 0 1 0 0 -1 -1 -1 0 0 0 -W 3 W 3 0 H 6 H 1 H 1 L 3 L 3 -L 5 0 -W 1 W 1 0 0 0 -L 6           
and T T 1 T 2 T 3 T 4 T 5 T 6 a vector of desired thrust values of all six motors. Note that the values of L 1 , H 3 and H 5 do not have any influence on the resulting torques. Ideally one should have H 1 = 0 and L 6 = 0 so that T 1 and T 2 do not generate pitch torque (i.e. generate only yaw torque) and T 6 does not generate yaw torque (i.e. generate only roll torque). For the I3S-UV, we consider this case. This consideration results in the following simplified control allocation matrix:

A            1 1 0 0 0 0 0 0 0 0 0 1 0 0 -1 -1 -1 0 0 0 -W 3 W 3 0 H 6 0 0 L 3 L 3 -L 5 0 -W 1 W 1 0 0 0 0           
It is assumed that there is no constraint on T i , by rearranging the output vector one obtains

F c1 Γ c3 = 1 1 -W 1 W 1 T 1 T 2 ⇒ T 1 T 2 = 1 2 -1 2W 1 1 2 1 2W 1 F c1 Γ c3 (7.1) and       F cy F cz Γ cx Γ cy       =       0 0 0 1 -1 -1 -1 0 -W 3 W 3 0 H 6 L 3 L 3 -L 5 0             T 3 T 4 T 5 T 6       ⇒       T 3 T 4 T 5 T 6       =         H 6 2W 3 -L 5 2(L 3 +L 5 ) -1 2W 3 1 2(L 3 +L 5 ) -H 6 2W 3 -L 5 2(L 3 +L 5 ) 1 2W 3 1 2(L 3 +L 5 ) 0 -L 3 L 3 +L 5 0 -1 L 3 +L 5 1 0 0 0               F cy F cz Γ cx Γ cy       (7.2)
In PX4 expressions (7.1) and (7.

I

ntroduced by Bessel in 1828, the concept of added mass has become widely accepted nowadays. Mostly neglected in modeling and control design of heavierthan-air aircraft, added mass in contrast has always been at the heart of preoccupations of the Underwater Robotics and Automatic Control community [START_REF] Fossen | Guidance and Control of Ocean Vehicles[END_REF][START_REF] Leonard | Stability of a bottom-heavy underwater vehicle[END_REF]. For underwater vehicles, added-mass effects often result in strongly nonlinear dynamics as the total mass matrix can no longer be considered as proportional to identity, except for spherical bodies [START_REF] Fossen | Guidance and Control of Ocean Vehicles[END_REF]. These effects are all the more pronounced for underwater vehicles with a slender body shape conceived for reducing hydrodynamic drag along a nominal axis. However, added mass is not the unique source of complexity for control design. The complexity of hydrodynamic effects often impedes obtaining a precise dynamic model, valid in a large operating domain [START_REF] Lamb | Hydrodynamics[END_REF]. The vehicles are often subjected to strong perturbations (due to currents) whose magnitude can be commensurable with the available actuation power. Therefore, robust nonlinear control design for underwater vehicles is highly recommended.

Control design for underactuated AUVs has received increasing interest during the last three decades [START_REF] Fossen | Guidance and Control of Ocean Vehicles[END_REF][START_REF] Aguiar | Dynamic positioning and way-point tracking of underactuated AUVs in the presence of ocean currents[END_REF][START_REF] Repoulias | Planar trajectory planning and tracking control design for underactuated AUVs[END_REF][START_REF] Refsnes | Model-based output feedback control of slender-body underactuated AUVs: theory and experiments[END_REF]. When the control objective concerns the tracking of a reference position trajectory, whose velocity does not vanish for all time, various classical control design methodologies have been applied to provide solutions. For instance, various standard linear and nonlinear control methods have been applied on the basis of linear approximations of the two subsystems "depth-pitch" and "plane-yaw" about nominal operating points (see [START_REF] Fossen | Guidance and Control of Ocean Vehicles[END_REF] and the reference therein). The main limitation of these approaches is the local nature of the control design and analysis. Moreover, stability and performance can suffer significantly when strong sea currents or aggressive manoeuvres excite the complex hydrodynamic and added-mass effects. Nonlinear Lyapunov-based control designs have been recently investigated to overcome some of these limitations. Most of them, however, only address the trajectory tracking problem in a horizontal plane, using a simplified and reduced "plane-yaw" 3-d.o.f model [START_REF] Aguiar | Dynamic positioning and way-point tracking of underactuated AUVs in the presence of ocean currents[END_REF][START_REF] Repoulias | Planar trajectory planning and tracking control design for underactuated AUVs[END_REF].

Few works address this problem in the 3-dimensional space [START_REF] Aguiar | Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty[END_REF][START_REF] Refsnes | Model-based output feedback control of slender-body underactuated AUVs: theory and experiments[END_REF]. A nonlinear highgain backstepping-based controller proposed in [START_REF] Aguiar | Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty[END_REF] allows for exponential convergence of the position error to a small neighborhood of the origin, which means that asymptotic stabilization to zero is not fulfilled. Moreover, the attitude is not explicitly controlled but guided by the closed-loop system's zero dynamics, thus possibly resulting in undesirable attitude dynamics. High-gain controllers are also known to be sensitive to measurement noise and time-delays of control inputs. Refnes et al. [START_REF] Refsnes | Model-based output feedback control of slender-body underactuated AUVs: theory and experiments[END_REF] have addressed the trajectory tracking for a slender-body underactuated AUV, with a model-based control approach.

However, this approach heavily relies on the precision of the model parameters. Control performance can be drastically degraded when model errors are important (see, e.g., [START_REF] Smallwood | Model-based dynamic positioning of underwater robotic vehicles: Theory and experiments[END_REF]). Nonlinear robust control design for underactuated AUVs thus remains an active research topic.

This chapter addresses the trajectory tracking control design for slender-body underactuated AUVs, whose body shape is symmetric with respect to the longitudinal axis, using a full 6-d.o.f model. The proposed control design methodology makes use of simple models of added mass and of dissipative hydrodynamic force acting on the vehicle.

These models are both representative of the physics and sufficiently simple for control design and analysis. By considering an axisymmetric body, added-mass effects and dissipative hydrodynamic force are carefully accounted for via various adaptations, resulting in a modified apparent force independent of the vehicle's orientation and subsequently a nonlinear system with a triangular control structure. The proposed controller is also complemented with bounded integral correction actions to compensate for unavoidable model uncertainties and external disturbances. Compared to [START_REF] Aguiar | Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty[END_REF], here almost global asymptotical stability is achieved and the attitude is explicitly controlled. The proposed control approach endowed with a cascade inner-outer loop architecture can be seen as underwater vehicles an extension of the thrust direction control paradigm that has been exploited for aerial vehicles [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to VTOL drones[END_REF][START_REF] Pucci | Nonlinear feedback control of axisymmetric aerial vehicles[END_REF][START_REF] Kai | A nonlinear approach to the control of a discshaped aircraft[END_REF], making a step towards a unified control approach for both aerial and underwater vehicles.

This chapter is organized as follows. System modeling is revised for axisymmetric underactuated underwater vehicles in Section 8.1. Then in the same section, a more simplified model for control design is derived. Control design supported by rigourous stability analysis is presented in Section 8.2. In section 8.3, comparative simulation results using a realistic model of a quasi-axisymmetric underwater vehicle illustrate the performance and robustness of the proposed control approach. Finally, a concluding section follows.

Ṗh = P h ×Ω + F gb + F d + T c e 1 Πh = Π h ×Ω + P h ×V h + Γ g + Γ d + Γ c (8.1)
The following classical model of F d is used:

F d (V h ) = -K dl V h F dl (V h ) -|V h |K dq V h F dq (V h )
with the damping diagonal matrices

K dl = diag(k dl1 , k dl2 , k dl2 ) K dq = diag(k dq1 , k dq2 , k dq2 )
The second and third diagonal components of K dl and K dq are equal due to the body symmetry about the longitudinal axis. Similarly, the added-mass matrix can also be modeled as a diagonal matrix with the same second and third diagonal components, i.e. M a = diag(m a1 , m a2 , m a2 ), with m a1 m a2 . Thus, the summed mass matrix M has the

form M = diag(m 1 , m 2 , m 2 ) with m 1 m + m a1 , m 2 m + m a2 .

Model for control design

Due to the coupling matrix Ξ involved in the definition (3.19) of the momentum terms and their dynamics (8.1), the translational and rotational dynamics are tightly coupled.

These complex dynamic couplings are often neglected in the literature by neglecting all terms involving the matrix Ξ using the fact that the distance between the CB and CG is small enough. This results in the following simpler control model that decouples the translational and rotational dynamics [START_REF] Aguiar | Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty[END_REF][START_REF] Repoulias | Planar trajectory planning and tracking control design for underactuated AUVs[END_REF][START_REF] Refsnes | Model-based output feedback control of slender-body underactuated AUVs: theory and experiments[END_REF]:

M V = (MV)×Ω + F gb + F d (V) + T c e 1 + ∆ F (8.2a) J Ω = (JΩ)×Ω + (MV)×V + Γ g + Γ d + Γ c + ∆ Γ (8.2b)
with the "disturbance" terms

∆ F -(MV f ) × Ω -MΩ × V f + F d (V h ) -F d (V) ∆ Γ (MV h )×V h -(MV)×V
which are null if the current velocity is null. For the sake of simplicity, the disturbance term ∆ F will not be considered for control design (i.e. ∆ F ≡ 0). However, integral corrections will be added so as to enhance control robustness w.r.t. unavoidable model errors and additive disturbances (i.e. current). underwater vehicles

In the sequel, the simplified dynamics (8.2) (with ∆ F ≡ 0) will be used for control design, whereas the dynamic equations (8.1) are still of use to simulate the vehicle's dynamics in the simulation section 8.3. Discrepancies between the two models represent an opportunity to test the robustness of the proposed controller.

Denote P MV and p RP = RMV as the "simplified" translational momentums expressed in the body-fixed frame and the inertial frame, respectively. One easily verifies from (8.2a) that the dynamics of p (i.e. ṗ) does not depend on Ω.

According to the basic control methodology for thrust-propelled underactuated vehicles presented in [START_REF] Hua | A control approach for thrustpropelled underactuated vehicles and its application to VTOL drones[END_REF], control solution may become implicit (or even ill-posed) if the external force expressed in the inertial frame

f e R(F gb + F d (V)) = β gb e 3 + RF d (V)
depends strongly on the vehicle's orientation R but is not properly taken into account.

The solution here proposed (similarly to the one developed in [START_REF] Pucci | Nonlinear feedback control of axisymmetric aerial vehicles[END_REF] for the control of axisymmetric aerial underactuated vehicles) consists in decomposing F d (V) into two parts with the first one acting along P and the second one acting along e 1 . More precisely, one rewrites

F dl (V) = -k dl2 m 2 P -k dl1 -m 1 k dl2 m 2 V 1 e 1 F dq (V) = - k dq2 m 2 |V|P -k dq1 - m 1 k dq2 m 2
|V|V 1 e 1 so that Eq. (8.2a) can be rewritten as

Ṗ = P × Ω + F gb + Fdl + Fdq + T e 1 (8.3) with            Fdl -β dl P, with β dl k dl2 m 2 Fdq -β dq |V|P, with β dq k dq2 m 2 T T c -k dl1 -m 1 k dl2 m 2 V 1 -k dq1 - m 1 k dq2 m 2 |V|V 1
Note that the quadratic drag force R Fdq , expressed in the inertial frame {I}, is not simply a function of p but also depends on the norm of V (= M -1 R p) and thus on the attitude R. This in turn implies that the time derivative of R Fdq (= -β dq |M -1 R p|p)

involves the angular velocity Ω, making the control design more delicate. The position and velocity errors are defined as

ξ ξ -ξ r ṽ v -v r (8.4)
whereas the velocity error and the translational momentum error, both expressed in the body-fixed frame, are defined as

Ṽ V -R v r = R ṽ P M Ṽ (8.5)
Then, the control objective consists in stabilizing ( ξ, ṽ) or, equivalently, ( ξ, P) about zero.

One easily verifies that ξ = R Ṽ = RM -1 P (8.6)

On the other hand, using the decomposition The term -β dq (|V| -|v r |)P involved in the last equality of (8.7) requires further developments. Using the decomposition

P = P -m 2 R v r -α r e 1 one deduces (|V| -|v r |)P = -(|V|-|v r |)α r e 1 +(|V|-|v r |)( P+m 2 R v r ) = -(|V|-|v r |)α r e 1 + | Ṽ+R vr| 2 -|vr| 2 |V|+|vr| ( P+m 2 R v r ) = -(|V|-|v r |)α r e 1 + | Ṽ| 2 +2 Ṽ R vr |V|+|vr| ( P+m 2 R v r ) (8.9)
From (8.7) and (8.9) one obtains

Ṗ = -Ω × P -β dq | Ṽ| 2 + 2 Ṽ R v r |V| + |v r | ( P + m 2 R v r )
+α r e 1× Ω + R f p + T p e 1 Let us call f p the "apparent" force (expressed in the inertial frame). Note that its time derivative is independent of the angular velocity Ω (i.e. ḟp Ω), which is an important property for control design.

The proposed controller will be derived from Eqs. (8.6), (8.10) and (8.2b), which are in cascade form. In fact, the rotational dynamics given by (8.2b) are fully-actuated with a 3-dimensional control torque vector Γ c monitoring 3 degrees of freedom of rotation.

Trajectory tracking control design

It is thus straightforward to design an "inner-loop" controller that asymptotically stabilizes Ω about any bounded smooth desired angular velocity Ω d , provided that the time derivative of the latter is available as feedforward. For instance, assuming that ∆ Γ is constant, the following torque controller

Γ c = -K 1 Ω -K 2 I Ω + J Ωd -(JΩ)×Ω d -(MV)×V -Γ g -Γ d (8.12)
with Ω Ω-Ω d , I Ω an integrator of Ω, and K 1,2 ∈ R 3×3 positive diagonal gain matrices, results in a closed-loop stable sub-system. Hence, the underlying idea for "outer-loop" control design consists in using Ω as an intermediate control variable for the translational dynamics (i.e. Eq. (8.10)). so that direct application of Barbalat's lemma then ensures the convergence of L and, thus, of P to zero. From (8.16) and the convergence of P to zero, one deduces by application of the extended Barbalat's lemma [START_REF] Micaelli | Trajectory tracking for unicycle-type and two-steeringwheels mobile robots[END_REF] that Ṗ also converges to zero, which in turn implies the convergence of y (i. with Ω u an augmented control input to be designed thereafter. For instance, assuming that fp does not vanish for all time, one then verifies that The proof of this proposition is a direct result of Lemmas 3 and 4. Remark 7. In the case m 12 < 0, α r is nominally negative. So if Ω γ is not too large, the term

Outer-loop control design

         Ω d = | fp | σ(α r ) e 1 × u - k u (R γ × u) 1 + u R γ + λe 1 Ω u = | fp | σ(α r ) e 1 × u + π u (R Ω γ ) + λπ u e 1 (8.25)
with σ(•) given by

σ(x) =      x if |x| > ε ε if 0 < x ≤ ε -ε if 0 ≥ x ≥ -ε (8.26)
with ε a small positive number. With this modification, relation (8.22) is always satisfied and, thus, the convergence of R γ about u is still ensured.

Outer-loop control design with integral term

In practice it is often desirable to complement the control action with a position error integral correction term in order to compensate for model uncertainties and external disturbances. Bearing in mind that classical integrator of ξ is often prone to the well-known phenomenon of integrator windup that may cause large overshoots of the position tracking error, we make use of the following bounded nonlinear second-order integrator of ξ (initially proposed in [START_REF] Hua | Time sub-optimal nonlinear PI and PID controllers applied to longitudinal headway car control[END_REF]): with the augmented control input Ω u given by (8.24). Choose k p1 and k p2 such that (0, 0) of the zero dynamics of (8.30)+(8.34) is globally asymptotically stable. Then, from the definition (8.29) of ξ and v the proof of convergence of (I ξ , İξ , ξ, ṽ) to zero follows the same lines as the proof of Lemma 3.2 in [START_REF] Hua | Time sub-optimal nonlinear PI and PID controllers applied to longitudinal headway car control[END_REF]. The remainder of the proof then directly follows.

   k p1 > 1 + m 2

Simulation results

In this section the performance and robustness of the proposed control approach are validated through simulation conducted on a realistic model of a quasi-axisymmetric underactuated vehicle. The vehicle's body shape, as depicted in Fig. 8.3, is not perfectly symmetric along the longitudinal axis due to a large rudder and a lower base.

The simulated dynamics are given by (3.23). Physical parameters of the simulated • Oscillating behaviour of a simpler controller: In order to illustrate the need of introducing the augmented variable u and the associated control variable Ω u in the proposed control approach, simulation results using the simpler outer-loop controller (8.19) are reported next. Similarly to the proposed outer-loop controller, the expression of Ω d given in (8.19) is slightly modified by adding a term λe 1 , with λ given in (8.37), so that the roll angle is also regulated near zero. The inner-loop controller is the same as the one used previously. All the gains and parameters involved in this controller are chosen identically to those used for the previously reported controller (i.e. given in (8.37)). Let us call this controller "simple controller" for distinguishing with the proposed one. One observes that although the position error still converges to zero (see Fig.

Conclusion

A novel nonlinear control approach for slender-body axisymmetric underactuated underwater vehicles is proposed. Added-mass effects and dissipative hydrodynamic force are carefully taken into account via various adaptations, resulting in a modified apparent force no longer depending on the vehicle's orientation which is a good conditioning for control design. The proposed controller is also complemented with an integral correction term to enhance its robustness. Convincing simulation results conducted on a realistic model of a quasi-axisymmetric underwater vehicle illustrate the performance and robustness of the proposed controller. showed the robustness of the control solution w.r.t model uncertainties and external disturbances (results are published in [START_REF] Nguyen | A nonlinear control approach for trajectory tracking of slender-body axisymmetric underactuated underwater vehicles[END_REF]).

Perspectives:

The work, already done in the thesis, on control design of underactuated AUVs with axisymmetric body shape have allowed us to have some preliminary understandings on how to deal with or exploit the added-mass effects and hydrodynamic forces acting on the translational dynamics of underactuated underwater vehicles. This constitutes the first step for us to address in the near future other challenging control design topics via the ongoing national projects involving the I3S-OSCAR team (ANR Astrid CONGRE and FUI GreenExplorer). For instance, the problem of degradation of actuator efficiency during a transition phase from low to high speed, which may naturally transforms a(n)

(almost) fully-actuated AUV into an underactuated vehicle, is delicate to deal with and has seldom been addressed in the literature of underwater robotics. We will deal with this problem via the development of a continuous differentiable control law (i.e. without the need for switching between several control laws) allowing for monitoring smoothly such transition phases.

The second problem that we want to address would concern the asymptotic stabilization of a fixed reference pose by an underactuated AUV in the absence of a sea current.

This problem is very challenging since classical control design methodologies do not provide solutions. The first reason is that the linearization of the system dynamics at the desired pose is not controllable. The second fact is that the system does not satisfy Brock-

Finally, I gratefully

  acknowledge the financial support which has made my research possible. This funding has come from the Vietnam International Educational Development (VIED) and the projects: ANR EQUIPEX Robotex, FUI GREENEXPLORER (2017-2021), and ANR Astrid CONGRE (2019-2021). Nonlinear control and visual guidance of underwater vehicles Scalar quantities are denoted by regular font lower-case letters. Bold font is used to mark vectors and matrices. Matrices are usually denoted by capital letters. Constants, variables and symbols Z + the set of positive integer numbers R the set of real numbers {e 1 , e 2 , e 3 } the canonical basis of R 3

  v the Euclidean norm of vector v ∈ R n (•)the transpose operator on a matrix or vector u × the skew-symmetric matrix associated with the cross product byvector u ∈ R 3 , i.e., u × v = u × v, ∀v ∈ R 3 vex(•) an operator such that vex(u × ) = u | • | the Euclidean norm in R n || • ||the Frobenius norm in R n×n diag(λ 1 , λ 2 , λ 3 ) diagonal matrix with the main diagonal values defined by the arguδ (•) ∈ R n , with δ > 0 the classical saturation function, i.e., sat δ (x) x min (1, δ/|x|) , ∀x ∈

  of the theoretical contributions and experimental validation results reported in this thesis were published in (or submitted to) the following research papers: • Nguyen, L.-H., Hua, M.-D., Allibert, G., and Hamel, T. (2017). Inertial-aided homographybased visual servo control of Autonomous Underwater Vehicles without linear velocity measurements. In proceedings of 21st International Conference on System Theory, Control and Computing (ICSTCC), pp. 9-16, Sinaia, Romania. [62] • Nguyen, L.-H., Hua, M.-D., and Hamel, T. (2019). A nonlinear control approach for trajectory tracking of slender-body axisymmetric underactuated underwater vehicles. In proceedings of European Control Conference, Invited Paper, pp. 4053-4060, Naples, Italy. [64] • Nguyen, L.-H., Hua, M.-D., Allibert, G., and Hamel, T. (2019). A homographybased dynamic control approach applied to station keeping of Autonomous Underwater Vehicles without linear velocity measurements. First revision of submission to IEEE Transactions on Control Systems Technology. [63] • Nguyen, L.-H., Hua, M.-D., and Hamel, T. (2019). A homography-based dynamic control approach of Autonomous Underwater Vehicles observing a (near) vertical target

  As dedicated in the title, this chapter in Part I: Introduction first briefly presents the motivations and objectives of this thesis work. The main contributions are then provided. The thesis structure section briefly introduces the content of all chapters. • Chapter 2 -A new trend in underwater robotics. This chapter discusses first about the development of worldwide community of small underwater robots, then about state-of-the-art small ROVs and AUVs. Finally, some recent technologies used for small and low-cost underwater vehicles are presented. • Chapter 3 -Mathematical model of underwater vehicles. This chapter recalls the modeling of the complex dynamics of underwater vehicles following Lagrange approach. Various hydrodynamics and hydrostatics phenomena affecting the dynamics of underwater vehicles are discussed. The final formulation of the nonlinear 6 d.o.f model is derived. • Chapter 4 -Preliminaries on homography estimation and homography-based control techniques. This is the first chapter in Part II: Homography-based visual servo (HBVS) control of fully-actuated AUVs without linear velocity measurements. This chapter first recalls the homography definition and then discusses about some relevant homography estimation techniques and existing HBVS control approaches. • Chapter 5 -Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera. A novel dynamic HBVS control approach is developed for fully-actuated underwater vehicles equipped with a downward-looking camera observing a (near) planar and textured visual target. The problem formulation and basic ideas of control design are first presented. Control design in form of a cascade control architecture are then provided Chapter 1. Motivations, contributions and thesis structure with the support of rigorous stability and convergence analyses. The performance and robustness of the proposed controller are first verified by simulation based on a realistic underwater vehicle model, and then experimentally validated through trials in a challenging real lake environment by employing the I3S-UV platform. Finally, potential extensions are proposed to enlarge the functionalities of stationkeeping application. • Chapter 6 -Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of forward-looking camera. A novel dynamic HBVS control approach is developed for a more challenging case when the considered fully-actuated underwater robot is equipped with a forward-looking camera observing a (near) vertical planar visual target. The performance of the proposed controller is first validated by simulation and then by preliminary experiments in a small water tank. • Chapter 7 -Development of the I3S-UV platform. This chapter discusses the need of the team to possess an underwater platform for facilitating experimental validations of the developed control and estimation algorithms. All the development process in terms of electro-mechanical design and control software system are presented in detail. • Chapter 8 -Trajectory tracking of slender-body axisymmetric underactuated underwater vehicles. This chapter is in Part III: Control of underactuated underwater vehicles. A novel controller is developed for the trajectory tracking problem of slender-body underwater vehicles whose shape is (almost) axisymmetric with respect to their longitudinal axis. The control design exploits the full 3D dynamics of the vehicles and some particular properties inherited from their shape in terms of drag and added mass. The performance and robustness of the proposed controller is verified by simulation using a realistic model of an underwater vehicle. 2 A new trend in underwater robotics T he last two decades have witnessed significant technology advancements resulting in more powerful microprocessors and companion computers, more robust IMUs, and longer endurance and higher capacity batteries in more compact forms, together with more reliable and more precise navigation systems, all at extremely reasonable costs. Along with the development of advanced control techniques, all these technological factors have led to the development of more powerful and versatile Unmanned Aerial Vehicles (UAVs). In parallel, the expansion of the Internet and social networks has revolutionized the way that information and knowledge could be stored and transmitted on a large scale and with a much faster speed. Due to this, not only open-source software but open-hardware also become popular. In addition, worldwide e-commerce websites and delivery companies allowed one to easily acquire all the necessary components to build their robots even with a modest budget. These exceptional and favorable conditions led to the formation of a huge community of DIY 1 in Aerial Robotics. The development of aerial robots was no more carried out only in the industrial or research/educational environments. A modern era of aerial robotics and of the development of small non-military UAVs had been started. In fact, the development of this DIY community in the inverse sense helped to stimulate the development of UAV research in academic environments. It is because of this fact that an UAV system in the sense of hardware and software could be built from scratch in a much shorter period of time by using open resources available in DIY forums. Thus, the researchers could focus on advanced problems such as navigation and control algorithms or image processing, etc. This is especially important for a newcomer who would like to get himself acquainted quickly in the UAV domain. For underwater vehicles, the surrounding condense environment often results in strong actions to the vehicles. The resulting dynamics are highly nonlinear and complex, and obtaining a precise model for control design is quite challenging. In addition, acoustic navigation systems are often expensive and acoustic communication systems have limited bandwidth. Moreover, components used for building underwater vehicles require waterproof and high pressure resistance. They are thus much more expensive in comparison with components used for building aerial vehicles. All these difficulties prevent a wide participation of the DIY community to the underwater robotics field.

4 (Figure 2 . 1 :

 421 Figure 2.1: Teledyne Seabotix ROVs (Source: http://www.teledynemarine.com)

Figure 2 . 6 :

 26 Figure 2.6: Blueye Pioneer, launched in 2008. The total price of system including ROV, Wireless Surface Unit, 75 m cable, battery & charger, wireless controller, Blueye App, tools & spare parts is US$9,878 (ex. VAT) (Source:www.blueyerobotics.com)

Figure 2 . 8 :

 28 Figure 2.8: IBUBBLE, launched preview in October 2019. The total price of "Handsfree" version is around US$5,400 while for premium one including 100 m tether cable, 2 extra batteries & 2 battery charger, and hardcase is around US$8,400 (prices inc. VAT). The weight of the vehicle is 9 kg. (Source: www.ibubble.camera)

9 Figure 2 . 9 :

 929 Figure 2.9: Two-man portable Bluefin-9 AUV of Bluefin Robotics (Source: www.gdmissionsystems.com)

  Figure 2.11: Blue Robotics thrusters and motor (Source: www.bluerobotics.com)

  Figure 2.12: Blue Robotics low light camera and Lumen Subsea Light (Source: www.bluerobotics.com)

  Figure 2.14: Blue Robotics Ping360 sonar system (source: www.bluerobotics.com)

  .

3 }

 3 Attached to Origin of inertial frame Center of Buoyancy (CB) of the vehicle

Figure 3 . 1 :

 31 Figure 3.1: An AUV with reference frames and notation

17

 17

  ) with positive damping matrices D Vl , D Vq , D Ωl , D Ωq ∈ R 3×3 .For some specific underwater vehicle like glider, additional control surfaces (i.e hydrofoils) are used for gliding forward while descending or climbing back up through the water. This phenomenon results in none zero off-diagonal components in the damping matrices D Vl , D Vq . However, this coupling effect is often considered in separate terms corresponding to control surfaces in the vehicle dynamics. The body of underwater vehicle is thus considered having "correct design" with all zero off-diagonal components.In practice, the damping matrices D Vl , D Vq , D Ωl , D Ωq are identified using reduced scale model in towing tank. Or alternatively and less expensive, they are calculated by computational fluid dynamics (CFD) programs, for instance commercial ANSYS Fluent 2

Figure 4 . 1 :

 41 Figure 4.1: The pose of the camera (R, ξ C ) determines a rigid body transformation from {A} to {C}. The Euclidean homography H : ∼ = R -(1/d )R ξ C n maps Euclidean coordinates of the scene's points from {A} to {C}. Arrow notation (•) is used for Euclidean vectors.

  to the second singular value of H[START_REF] Ma | An Invitation to 3-D Vision: From Images to Geometric Models[END_REF].

Let Ĥ ∈ SL( 3 )

 3 denote the estimate of H. Define the right group error E := Ĥ H-1 ∈ 4.2. Existing feature-based homography estimation techniques SL(3) and the output errors e i ∈ P 2 :

Chapter 4 .

 4 Preliminaries on homography estimation and homography-based control techniques Consider a camera attached to the moving frame {C} moving with kinematics Ṙ = RΩ × ξC = RV C viewing a stationary planar scene, where Ω and V C are the angular and linear velocities of {C} with respect to {A} expressed in {C}, respectively. The group velocity U ∈ sl(3)induced by the camera motion, and such that the dynamics of H are in the form (4.12), then satisfies[START_REF] Mahony | Nonlinear complementary filters on the special linear group[END_REF] Lem. 5.3]

  ) and (4.18) exploit directly point correspondences in the innovation term ∆( Ĥ, p), thereby depleting the need of reconstruction of individual image homographies. This mechanism thus may save considerable computational resources, making the proposed algorithm suitable for embedded systems with simple feature tracking software. Another practical advantage in comparison with clas-

52 Chapter 4 .1•

 524 the rotation matrix R and a matrix ξC n with ξC = R ξ C d from which one may be able to further decompose into ξC and n . • Denote P [X Y Z] and P [X Y Z ] the 3D coordinates expressed in Preliminaries on homography estimation and homography-based control techniques {C} and {A}, respectively, of a reference point P in the visual target. Let m denote the normalized coordinates of the point P corresponding to {C} and {A}, respectively. It has been proved that Z By introducing an extended image coordinates m e that link [x y] and [X Y Z] as follows m e [x y z] = X Z X Z ln(Z) (4.21) with z = ln(Z) is a supplementary normalized coordinate, Malis et al. derived the following kinematic relation

ρ 1 Zd

 1 Expression (4.22) represents the relation between the position of a chosen reference point w.r.t the camera velocities expressed in current frame {C}. Since the image point P is fixed in {A}, (4.22) is used to indirectly control the position of the camera w.r.t the target image.

24 ) 4 . 3 .

 2443 Discussions on existing HBVS controllers • The positioning task is described as the regulation to zero of the task function e m e -m e θu (4.25) with the reference m e [x y z ] = X Z X Z ln(Z ) and m e -m e = x -x y -y ln Z Z

• 54 Chapter 4 .

 544 The kinematic equation (4.26) when applying kinematic control law (4.28) and the Preliminaries on homography estimation and homography-based control techniques fact that L -1 w uθ = uθ (in view of (4.24)) results in

  this kinematic control to perform station-keeping task where the validation experiments were carried out in a test tank using a Cartesian robot emulating two d.o.f of the ROV ANGUS 003. The bottom of the test tank emulates the seabed and is considered as unmarked target. By using a sparse feature tracker, some points on the target are tracked for homography estimation. The camera is rigidly mounted at the CG of the vehicle and pointing downwards such that the vehicle's body -→ e b 3 axis and the camera's optical axis coincide. Since the target plane is horizontal and the vehicle's -→ e b 3 -axis is always parallel with the inertial -→ e a 3 -axis, relation (4.20) can be simplified as Z Z = ρ = det(H) and the rotation axis u is equal to [0 0 1] . The last component of visual error (m e -m e ) in (4.3.1) is thus equal to ln(det(H)). Since the vehicle having 4 controllable d.o.f namely surge, sway, heave and heading (i.e yaw), the authors proposed a modified control scheme by discarding roll and pitch control inputs, 4.3. Discussions on existing HBVS controllers resulting in a modified version of the Jacobian matrix L

  33b) with M ii (i = 1, 2) the mass matrix coefficients; B ii (i = 1, 2), D 1 and D 2 the hydro-Chapter 4. Preliminaries on homography estimation and homography-based control techniques dynamic drag coefficients; V f 1 and V f 2 the components of sea current expressed in the body-fixed frame. The normalized control input of the two back thrusters is β ∈ [-1, 1]; the normalized control input of the sway thrusters is γ ∈ [-1, 1]; α i (i = 1, 2) are the thrusters' efficient coefficients. Using the decoupled dynamics (4.33), the authors then propose the following PID control law

body - → e b 3 -

 3 axis, the camera's optical axis and the inertial -→ e a 3 -axis are parallel. This configuration results in trivial relations n = n = [0 0 1] and consequently Z = d, Z = d , Z Z = det(H). The estimate of Z and consequently L v depends only on the estimate of d . The main drawback of 2 1

Chapter 4 .

 4 Preliminaries on homography estimation and homography-based control techniques derivative of visual error e p satisfies ėp = -Ω × (e p -m ) + a V C (4.44) with a (n e 3 ) d an unknown number. However, we know that it is positive using the condition n m > 0. For the purpose of V Cr design, the authors introduce the adaptive scalar dynamics żp = e p (Ω × m ), z p (0) ∈ R (4.45)

  R u ∈ SO(3) such that R u u = e 3 . Define a matrix H R u HR u and h12 the element at the intersection of the first row and second column of H. The information embedded in h12 then is used for design ω r term included in (4.49) as follows ωr = -k Θ2 ω r -k Θ1 sat ∆ω ( h12 ), ω r (0) ∈ R. (4.50) Assuming that inner loop controller ensures the almost global asymptotical stability and local exponential stability of the equilibrium (V C , Ω, R u) = (V Cr , Ω r , u), it is proved that by applying the reference translational velocity V Cr (4.47) and the reference rotational velocity Ω r (4.49) where ω r is defined by (4.50), there exist some positive numbers κΘ and ∆ω such that for all positive numbers k Θ1 , k Θ2 , ∆ ω satisfying k Θ2 / √ k Θ1 and ∆ ω > ∆ω the homography matrix H is stabilized about I 3 for almost all initial conditions [44, Theo. 1].

1 .

 1 Section 5.2 presents a simplified control model. Basic ideas of control design are explained in Section 5.3. In Section 5.4 the novel dynamic HBVS control approach is proposed. Comparative simulation results conducted on a realistic AUV model are reported in Section 5.5 illustrating the performance and robustness of the proposed approach. Section 5.6 first describes the experimental setup and then reports extensive experimental validation results in a real 65 5.1. Problem formulation environment. Three video links are also provided showing these experimental results.

Figure 5 . 1 :

 51 Figure 5.1: An AUV with a downward-looking camera and notation

Figure 5 . 2 :

 52 Figure 5.2: Control architecture of the proposed HBVS

5. 4 .

 4 Control designfor all time and converging exponentially to zero. Introduce the following integrator ż = -Ω × z + e p , z(0) ∈ R3 (5.23)and the following augmented systemėp = -Ω × êp -K 1 êp + K 1 ēp , êp (0) = e p (0)(5.24)with positive diagonal matrix K 1 ∈ R 3×3 and ēp e p + k I z with positive integral gain k I .

38 )Figure 5 . 3 :

 3853 Figure 5.3: AUV's position and attitude (Euler angles) vs. time

Figure 5 . 4 :

 54 Figure 5.4: e p (and êp ) vs. time

Figure 5 . 5 :Figure 5 . 6 :

 5556 Figure 5.5: Control force F c and torque Γ c vs. time

proposed in Lemma 1 .

 1 Control parameters and gains of the controller are given in Tab. 5.3. They have been chosen so that the time evolution of the AUV's position has almost the same settling time as in the previous simulation. Simulation results are reported in Figs 5.3b-5.6b. In overall, the time evolutions of the vehicle's pose and of the measurements: the case of downward-looking camera Controller Gains and other parameters Outer-loop k p1 = 0.4145,

5 . 7 )

 57 . At the moment when the experiment was performed, a Hardkernel Odroid XU-4 played a role of the companion computer in the I3S-UV control architecture (c.f. Fig. 5.8). For performing the real-time homography estimation, the AUV is equipped with a myAHRS+ IMU sensor providing measurement output at 100 [Hz] and an oCam downwardlooking monocular camera providing color images of 640 [px] × 480 [px] at 20 [Hz]. The homography estimation algorithm [34] has been implemented in C++, combined with OpenCV for image processing, on the ground station laptop with an Intel Core i7-7700HQ octa-core CPU running at 2.8 GHz. The transmission of the data from the camera and the IMU to the ground station is carried out through the high speed Ethernet cable. The laptop has a Linux-based operating system and is responsible for the following tasks: 1) interfacing with the camera and IMU hardwares and acquisition of images and IMU data 2) real-time estimation of the homography at 10 [Hz] 3) perform outer control loop at 10 [Hz], and 4) interfacing with the joystick and Pixhawk via Odroid to remotely control the vehicle. For the homography estimation, an image captured by the camera is chosen as the reference image. The FAST Feature Detector and ORB Descriptor Extractor algorithms available in the OpenCV library are employed for carrying out feature detection and descriptor extraction in images.

Figure 5 . 7 :Figure 5 . 8 :

 5758 Figure 5.7: AUV with downward-looking camera (inside yellow circle)

Figure 5 . 9 :

 59 Figure 5.9: Experimental validations in lake Saint-Cassien Specification Numerical value m [kg] 16 F b [N ] 1.01mg l [m] 0.025 r C [m] [0.2 0 0.15] M = mI 3 + M11 A [kg] diag(17.868, 23.868, 21.024) Ĵ = J 0 + M22 A [kg.m 2 ] diag(0.3105, 0.8486, 1.0) D = mle 3× [kg.m] 0.4e 3×

6 Table 5 . 5 :Figure 5 . 10 :Figure 5 . 11 :Figure 5 . 12 : h 12 Figure 5 . 13 :

 65551051151212513 Figure 5.10: Control force F c vs. time

Figure 6 . 1 :

 61 Figure 6.1: An AUV with a forward-looking camera and notation

Figure 6 . 2 :

 62 Figure 6.2: Illustration of angles α and β . The projection vector n and e 1 lie on the horizontal plane and n is perpendicular to the intersection line of the two planes.

Figure 6 . 3 :

 63 Figure 6.3: Control architecture of the proposed HBVS

Figure 6 . 4 :

 64 Figure 6.4: Interconnected translational dynamics

  |x| with D given in the proof of Lemma 2 and D the symmetric positive matrix solution to the Lyapunov equation D Ā + Ā D = -I 3 . Subsequently, the time-derivative of the aggregate Lyapunov function L X DX + x Dx satisfies

n 1

 1 , s 2 = 1.1 Proposition 7 k g = 1, k Θ1 = 0.0625, k Θ2 = 0.5, ∆ Θ = Proposition 4 K Ω = 3 Ĵ, K iΩ = 0 Lemma 1 a 0 = 0.5, k 0 = 20, η 3 = 8

Figure 6 . 6 :Figure 6 . 7 :

 6667 Figure 6.6: Visual errors vs. time

Figure 6 . 8 :

 68 Figure 6.8: Ω and Ω r vs. time

Figure 6 . 9 : 3 s 2 1 n 1 , k 33 = 3 s 2 1 n 1 , s 1 = 1 , n 1 = 1 0. 3 .

 69311111113 Figure 6.9: AUV with downward-looking camera

Figure 6 . 10 :

 610 Figure 6.10: Experimental validation in a water tank

Figure 6 . 11 :Figure 6 . 12 :Figure 6 . 13 :

 611612613 Figure 6.11: Attitude: roll φ, pitch θ, yaw ψ vs. time

Figure 6 . 14 :Figure 6 . 15 :Figure 6 . 16 :

 614615616 Figure 6.14: Visual errors (σ, ) vs. time

  budget and extension requirements. In fact, most ROVs available in the market are closed systems in the sense of mechanical structure and firmware/software. It is thus difficult to upgrade them to perform new functionalities. Although there exist few vehicles with possibilities of modifying their software and configuration in the market, they are often considerably heavy and voluminous, and/or excessively expensive in terms of price and operation cost. One can name VORTEX[START_REF]Véhicule Ouvert et Reconfigurable pour les Techniques EXpérimentales[END_REF] with weight of 220 kg and dimensions of 1.3 m × 1.1 m × 1 m as an example.In this PhD project, control algorithms have been developed for UVs in different shapes. A compact shape vehicle should be used for testing the HBVS control algorithms proposed in Chapters 5 and 6 whereas a slender axisymmetric one is needed for validating the trajectory tracking control algorithm proposed in Chapter 8. It is

Figure 7 . 1 :

 71 Figure 7.1: Different views of BlueROV1 (Source:www.bluerobotics.com)

Figure 7 . 3 :

 73 Figure 7.2: Water tight tubes of I3S-UV

Figure 7 . 4 :

 74 Figure 7.4: I3S-UV 3D design

Figure 7 . 9 :

 79 Figure 7.9: Configuration of 6 thrusters.

  2) are used to calculate the required thrust generated by thrusters. The configuration of I3S-UV thrusters is given by L 3 = 0.21 m; L 5 = 0.23 m; L 6 = 0 m W 1 = 0.11 m; W 3 = 0.11 m H 1 = 0 m; H 6 = 0.15 m

Figure 8 . 1 :

 81 Figure 8.1: Slender-body underactuated vehicleWe consider a slender-body underactuated underwater vehicle with a body shape symmetric with respect to the longitudinal axis (e.g. the popular class of torpedo-shaped AUVs, c.f. Fig.8.1). The vehicle is endowed with four control inputs, namely a thrust force intensity T c ∈ R of a control force vector T c acting along the vehicle's longitudinal axis (i.e. T c = T c e 1 ) and three independent torque inputs Γ c ∈ R 3 w.r.t. the CB to control the vehicle's orientation. The thrust force is assumed to apply at a point lying on the {B; i}-axis so that it does not create any torque at the CB.With the mentioned notions of control inputs, the vehicle dynamics with including

8. 2

 2 Figure 8.2: Trajectory tracking control problem

M = m 2 I 3 + m 12 e 1 e 1 underwater vehicles with m 12 m 1 - 1 + 1 = 1 ( 8

 311118 m 2 = m a1 -m a2 < 0, one deduces from (8.3) and (8.5) thatṖ = -Ω × P + F gb + Fdl + Fdq + T e MΩ × R v r -MR vr = -Ω × P -Ω × MR v r + MΩ × R v r + F gb + Fdl + Fdq -m 2 R vr + ( T -m 12 e 1 Rvr )e -Ω × P -β dq (|V| -|v r |)P + α r e 1× Ω + R (β gb e 3 -β dl p-β dq |v r |p-m 2 vr )+( T -αr )e .7) with α r m 12 e 1 R v r (8.8)

f p β gb e 3 -

 3 β dl p -β dq |v r |p -m 2 vr T p T -αr + β dq (|V| -|v r |)α r(8.11)

1 ( 8 ( 8 . 14 ) 3 . 1 2β 1 2β 1 |2+ m 2 m 1 )β dq |vr| |V|+|vr| | Ṽ|| P| 2 -k p2 | P1 | 2 - 2β dq m 2 2 |V| + |v r | ( Ṽ R v r ) 2 + 2β dq m 2 |m 12 ||v r | m 1 (2+ m 2 m 1 )β dq |vr| |V|+|vr| |

 188143111122211|V|+|vr| Let yK ξ ξ denote the position error scaled by a diagonal positive gain matrix K ξ ∈ R 3×3 . By introducing feedback terms (i.e. functions of P and ξ), the translational error dynamics (8.10) can be rewritten asṖ = -Ω × P -K P P -M -1 R K ξ h(|y| 2 )y feedback terms -β dq | Ṽ| 2 + 2 Ṽ R v r |V| + |v r | ( P + m 2 R v r ) + α r e 1× Ω + R fp + Tp e .13)withK P diag(k p1 + k p2 , k p1 , k p1 ), k p1 , k p2 > 0, h(•)denoting a smooth bounded strictly positive function defined on [0, +∞) such that for some positive numbers η, β∀s ∈ R, h(s 2 )s < η ∀s ∈ R, ∂ ∂s (h(s 2 )s) f p + k p1 (p -m 2 v r ) + 1 m 2 K ξ h(|y| 2 )y Tp T p -k p1 α r + k p2 P1 -m 12 m 1 m 2 e 1 R K ξ h(|y| 2 )yNote that the augmented apparent force fp does not depend on the vehicle's attitude R nor on Ω. Thus, its time derivative does not depend on the angular velocity Ω.If the outer-loop controller asymptotically stabilizes ε P α r e 1× Ω + R fp + Tp e 1 (8.15) underwater vehicles about zero, then the resulting zero dynamics of (8.13) areṖ = -Ω × P -K P P -M -1 R K ξ h(|y| 2 )y -β dq | Ṽ| 2 + 2 Ṽ R v r |V| + |v r | ( P + m 2 R v r )(8.16)We show next that appropriate choices of gains k p1 and k p2 (i.e. K P ) will render the origin of the zero-dynamic time-varying system (8.6)+(8.16) globally asymptotically stable Lemma Choose k p1 and k p2 such that dq sup(|v r |)k p2 > 1 -m 2 m dq sup(|v r |) (8.17)then the equilibrium ( ξ, P) = (0, 0) of the zero-dynamic system (8.6)+(8.16) is globally asymptotically stable.Proof. The time derivative of the Lyapunov function candidate 2 )s dssatisfies L = -k p1 | P| 2 -k p2 | P1 | 2 -β dq | Ṽ| 2 +2 Ṽ R vr |V|+|vr| (| P| 2 + m 2 P R v r ) ≤ -k p1 | P| 2 -β dq |V|+|vr| | Ṽ| 2 | P| 2 + 2β dq |vr| |V|+|vr| | Ṽ|| P| 2 + β dq m 2 |vr| |V|+|vr| | Ṽ| 2 | P| -k p2 | P1 | 2 -2β dq m 2 2 |V|+|vr| ( Ṽ R v r ) 2 + 2β dq m 2 |m 12 ||vr| m 1 (|V|+|vr|) | P1 || Ṽ R v r | Using the relation | Ṽ| ≤ 1 m P| one then deduces L ≤ -k p1 | P| 2 -β dq |V|+|vr| | Ṽ| 2 | P| 2 + (|V| + |v r |) | P1 || Ṽ R v r |The gain condition (8.17) then ensures that all the negative quadratic terms domi-8.2. Trajectory tracking control design nate the cross terms, i.e.k p1 | P| 2 + β dq |V|+|vr| | Ṽ| 2 | P| 2 > (Ṽ|| P| 2 k p2 | P1 | 2 + 2β dq m 2 2 |V|+|vr| ( Ṽ R v r ) 2 > 2β dq m 2 |m 12 ||vr| m 1 (|V|+|vr|) | P1 || Ṽ R v r |This in turn implies the existence of a positive number c such that L ≤ -c| P| 2 . From there one can easily verify the boundedness of L (i.e. the uniform continuity of L)

Remark 6 .

 6 e. ξ) to zero. The remainder of the proof then follows. Now the remaining task consists in designing the desired value Ω d for Ω and the thrust intensity Tp (or equivalently T ) to stabilize α r e 1× Ω d + R fp + Tp e 1 → 0 (8.18)which is also equivalent to the stabilization of ε P defined by(8.15) about zero as a consequence of the inner-loop controller that asymptotically stabilizes Ω about Ω d . By assumption v r does not vanish for all time and m 12 = m a1 -m a2 = 0 due to the slender-body form of the vehicle, one may expect that α r defined by (8.8) does not vanish for all time either. Assuming that α r (t) = 0 ∀t, in view of (8.18) one may define the outer-loop controller as follows:1 R fp Ω d,2 = -1 αr e 3 R fp Ω d,3 = 1 αr e 2 R fp (8.19)However, the outer-loop controller(8.19) leaves the attitude uncontrolled and ultimately guided by the system's zero dynamics, which may be excessively oscillating.The outer-loop control solution proposed in this paper, by contrast, defines a desired direction u ∈ S 2 for R γ with γ fp /| fp | ∈ S 2 representing the direction of fp , where u is obtained by integration of the following differential equation u = u × Ω u , u(0) = -e 1(8.20)

8. 2 .

 2 Trajectory tracking control design and λ(•) is a function that can be independently assigned for other control objective related to the roll motion. Assume that α r = 0 and | fp | > 0 for all time. Assume that all conditions and assumptions in Lemmas 3 and 4 hold. Then, the equilibrium (ξ, v, R γ) = (ξ r , v r , u) is almost globally asymptotically stable.

| fp| αr e 1 ×

 1 u involved in(8.24) allows u not to depart too far from -e 1 . This can be justified by examining the derivative of 1 + e 1 u:d dt (1 + e 1 u) = | fp | α r |e 1 × u| 2 ≤0 +(e 1 × u) (R Ω γ )Remark8. Since α r may evolve near zero during a transient phase, to avoid the division by zero we suggest to replace the expressions of Ω d and Ω u given in (8.23) and (8.24) by

1 ( 8 . 31 )

 1831 Ïξ = -2k vI İξ + sat Ïmax/2 (k pI (-I ξ + sat δ I (I ξ + ξ))) (8.27) with Ïmax , δ I , k pI , k vI denoting positive constants and with initial conditions satisfying |I ξ (0)| < δ I + Ïmax /(2k 2 vI ) and | İξ (0)| < Ïmax /(2k vI ). As specified in [32], the ultimate underwater vehicles upper-bounds of |I ξ |, | İξ |, and | Ïξ | are δ I + Ïmax /(2k 2 vI ), Ïmax /(2k vI ), and Ïmax , respectively. Define the augmented reference variables ξr ξ r -K I I ξ vr v r -K I İξ (8.28) with K I ∈ R 3×3 a diagonal positive integral gain matrix. Define also the augmented error terms (compared to (8.4)-(8.5))K I I ξ v vvr = ṽ + K I İξ V V -R vr = R v P M V = P -MR vr (8.29)One verifies that ξ = RM -1 P (8.30)On the other hand, analogously to the developments carried out in (8.7)-(8.11) one deducesṖ = -Ω × P -β dq | V| 2 + 2 V R vr |V| + |v r | ( P + m 2 R vr ) + α r e 1× Ω + R f p + T p ewhere α r is now defined by (instead of (8.8))α r m 12 e 1 R vr(8.32) and f p and T p are defined by (instead of (8.11))f p β gb e 3 -β dl p -β dq |v r |p -m 2 vr T p T -αr + β dq (|V| -|v r |)α r (8.33)Then, similarly to Eq. (8.13), Eq. (8.31) can be rewritten as Ṗ = -Ω × P -K P P -M -1

2m 1 2β-m 2 m 1 2β( 8 . 36 )

 11836 dq sup(|v r |) + Ïmax 2k vI k p2 > 1 dq sup(|v r |) + Ïmax 2k vI Assume that α r = 0 and | fp | > 0 for all time. Assume that all conditions and assumptions in Lemma 4 hold. Then, the equilibrium (I ξ , İξ , ξ, v, R γ) = (0, 0, ξ r , v r , u) is almost globally asymptotically stable. Proof. The proof of this proposition is almost identical to that of Proposition 8. Indeed, Lemma 4 and the outer-loop control expressions (8.23)-(8.24) ensure the convergence of the term α r e 1× Ω+R fp + Tp e 1 involved in (8.34) to zero. Then, similarly to the proof of Lemma 3, using (8.36) and the fact that sup(|v r |) ≤ sup(|v r |) + sup(I ξ ) ≤ sup(|v r |) + Ïmax /(2k vI ) one easily deduces that the equilibrium ( ξ, P) =

Figure 8 . 3 :KTable 8 . 1 :Figure 8 . 4 :Figure 8 . 5 :•Figure 8 . 6 :

 8381848586 Figure 8.3: Quasi-axisymmetric shape of the simulated vehicle with the main body's dimension 2.7[m] × 0.45[m] Specification Numerical value Mass m [kg] 100 F b [N ] mg r G [m] [0 0 0.01]

Figure 8 . 7 :

 87 Figure 8.7: (Proposed controller) Vehicle's orientation represented by Euler roll, pitch, yaw angles v.s. time.

Figure 8 . 8 :

 88 Figure 8.8: (Proposed controller) Control force and torque inputs v.s. time.

8 . 9 )

 89 the vehicle's orientation exhibits much more oscillations during the first 10 seconds of the transient phase (see Fig. 8.10) in contrast with the smooth behaviour of the proposed controller as shown in Fig. 8.7. This justifies the need of explicitly controlling the orientation rather than leaving the latter guided by the closed-loop system's zero dynamics.

Figure 8 . 9 :Figure 8 . 10 :

 89810 Figure 8.9: (Simple controller) Position tracking error v.s. time.

  moment and hydrodynamic reaction forces and torques as constant disturbances. Discrepancies between the dynamic model used for control design and the vehicle's real dynamics due to these simplifications are dealt with by means of integral corrections.The development of the I3S-AUV platform and of the HomographyLab library has been a defining factor for successfully demonstrating the developed theories and concepts as an appealing success factor to the public, the specialized robotics community and industrial companies. We believe that the implemented works would become the key technology enablers for small-scale autonomous underwater robots devoid of expensive DVL velocity sensors.The work presented in Part III is an extension of the I3S-OSCAR team's work on the control of rocket-like flying robots[START_REF] Pucci | Nonlinear feedback control of axisymmetric aerial vehicles[END_REF] to underactuated underwater vehicles with axisymmetric body shape. It contributes to the development of a novel and unified control framework for underactuated vehicles. The main contribution consists in designing a tracking control that exploits the full 3D nonlinear dynamics, allowing for achieving almost global asymptotical stability of the error dynamics. The simulation results

  In practice, ROVs often work at low speed and are normally used for observation tasks with limited capability of intervention. The potential of low-cost ROVs has been envisioned very early. One of the pioneers is Hydrovision, a subsidiary of SeaBotix which was initially founded in 1986 aiming to this market. The basic guidelines of Hydrovision include capability of real work in real environment, intuitiveness to be simple to operate, ruggedness enough to withstand the harsh environments, and thus its products bring value to the user. In order to increase the portfolio capability to access this market segment, Teledyne Technologies has incorporated Seabotix over M & A activities to form Teledyne Seabotix in 2001. Nowadays, Teledyne Seabotix provides a wide range of mini ROVs, as illustrated in Fig.2.1 and 2.2. From a general view, it can be seen that all ROVs

	Accessories	Producer
	AUV/ROV Thrusters
	External LED Lighting
	Grabber Attachments
	Three Jaw Grabber		SeaBotix
	Integrated Navigation Con-
	trol Console
	Laser Scaling
	Altimeter/Echosounder	Tritech Micron Echo Sounder Tritech
	Multibeam Sonar	DIDSON sonar	Sound Metrics
		Tritech Gemini 720i	Tritech
	Scanning Sonar	1171 Series OC ROV sonar	Kongsberg Mesotech
		Tritech Super SeaPrince	Tritech
		Tritech Micron	Tritech
	Thickness Gauge	Multiple Echo Ultrasonic	Cygnus Instruments
		Digital Thickness Gauge
	USBL Tracking System	Tritech Micron Nav USBL	Tritech

have modular design including main body skid and additional skids for crawler, grabber, sonar sensor and acoustic system. The main body skids have different depth ratings, thrusters' configuration and tether length. For some ROV models, the additional crawler skids together with attractor can be used to perform crawler inspection mode on hard surfaces (e.g. ship hull). Along with ROVs, there is a large selection of accessories and options allowing Teledyne Seabotix mini ROVs to have a wide range of applications. It is noteworthy to remark from Tab. 2.1 that sonar sensors and acoustic system are all provided by other producers. In addition, SeaBotix also provides simulator software powered by GRi Simulations Inc. for pilot training, and navigation & control software powered by GreenSea (c.f. Fig.

2

.3). Table 2.1: Teledyne SeaBotix accessories and options

  Chapter 2. A new trend in underwater robotics and the global leader in micro ROV technology. As of July 2018, more than 3,000 Vide-oRay ROVs have been delivered worldwide5 .It can be seen from Fig.2.5d and 2.5e that recent VideoRay ROVs belong to mini class instead of mico one, and they are designed in box-shaped forms. The VideoRay Mission Specialist Defender with enhanced depth rating of 1000 m seems to be a strategic step of VideoRay LLC to gain market share of deeper depth applications. The box-shaped form which is a popular ROV design in fact provides advantages since it gives more space simplifying mounting/replacing additional accessories. By employing this design, the configuration of ROVs can be easily customized for specific missions. This fact can be seen in view of Fig.2.1 and 2.2 where all Teledyne Seabotix ROVs are in this shape.

	2.1. Recent small underwater robots 2.1. Recent small underwater robots
		Accessories	Producer
	Crawler		VideoRay LLC
	Rotating Manipulator Arm
	Laser Scaler
	Manipulator Arm
	External SD Camera		Inuktun
	MSS Cavitation Cleaner	Model 122 E60 Cavi Blaster	CaviDyne LLC
		DVL	Nortek
	Positioning System	Smart tether	KCF Technologies
	(a) VideoRay Pro 3 GTO, launched MicronNav USBL (b) VideoRay Pro 4 , launched in 2009. Tritech
	It is interesting that VideoRay Mission Specialist Defender (Fig. 2.5d) can be up-in 2005. More powerful thrusters allow increasing speed from 2.6 to 4.1 knots Improved hydrodynamic performance, Water quality sonde YSI EXO2 YSI camera and ballast adjustment. New creased vertical and horizontal thrust. (720ik, 720im, 720is) control software platform for pilot. In-Tritech Gemini 720 Series Tritech
	Sonar	Weight: 6.1 kg. Depht rating 305 m. Micron DST Scanning Sonar
		BlueView V Series 2D imag-	Teledyne
	(a) VideoRay, launched in 2000. Weight: ing sonars (b) VideoRay Pro II, launched in 2002. Added more
	3.6 kg. Depth rating: 152 m	80 % thrust, new lighting, enhanced hydrodynamic Oculus Multibeam Sonars BluePrint
		performance, upgraded camera and lowered power (M370S, M750d, M1200d)
	Video enhancement	consumption. LYYN Real Time Video En-	LYYN
		hancement
	Thickness gauge	Ultrasonic Metal Thickness	Cygnus Instruments
		Gauge
		Option
	(c) VideoRay CoPilot by Seebyte, launched in 2012. Navigation package DVL, Greensea Upgrades, MAX-M8W GPS MAST and USBL (op-(d) VideoRay Mission Specialist De-fender , launched in 2017. Modu-(e) VideoRay Pro 5 , launched in 2018. System of interchange-tional)
	Featured autonomous control.	can perform station keeping in lar design of platform allows easily able, modular components. For-
	CoPilot provides effortless front of a offshore structure then operator can focus only on surveillance or intervention customizable to adapt to specific mis-ward speed: 4.4 knots. Weight: and automatic navigation sions. Depth rating: 1000 m. Weight: 10 kg. Depth rating: 305 m.
	tasks. However, the cost of this upgrade package can be guessed to be quite expensive to underwater locations all 17.2 kg.
	while following a pre-defined since the main DVL included in the package exceeds €15,000. mission regardless of chang-
	The aforementioned Teledyne SeaBotix mini ROVs and micro/mini VideoRay ROVs ing currents and rough sea
	graded lighting, camera and thrusters. New 9-are often used for professional applications. They are often go with various accessories, hanced depth rating up to 305 m. Featured a Tritech (c) VideoRay Pro 3, launched in 2004. Up-(d) VideoRay Deep Blue, launched in 2004. En-conditions.
	guarantee, technical supports and training. In order to buy them, it is required to contact pin accessory plug for optional capabilites. In-SeaSprite scanning sonar. Figure 2.5: VideoRay ROVs (Source: http://www.videoray.com)
	troduced control through a laptop PC which al-authorized regional distributors. Their total prices are often very expensive. low data can be gathered and analyzed.
	Following the trend of mini ROVs aiming toward professional applications as well 2.1.2 Small AUVs Figure 2.4: VideoRay ROVs (Source: http://www.videoray.com) for entertainment purposes, it has been witnessed a recent startup company named Blu-AUVs in general sense stand for underwater vehicles operating without tether. They eye founded in 2015. Its first commercial mini ROV named Blueye Pioneer was an-In the sequel, let us introduce several mini ROVs used for education/development often travel at higher speed than ROVs, and designed in form of slender body to reduce nounced in 2018. The mini ROV has weight of 9 kg, depth rating of 150 m and a replace-and entertainment purposes whose prices are relatively cheaper. The BlueROV2 plat-drag. However, in this section we will discuss only vehicles with propulsor system. able battery of 2 hours endurance. The vehicle is connected to the surface unit by tether. form presented in Fig. 2.7 can be used for developers with the availability of a wide Gliders, a subset of AUVs, are out of scope of our discussions. The surface unit plays the role of wired router or wifi rounter with at least 15 m distance range of accessories provided by Blue Robotics and opensource autopilot firmware Ar-It is curious that successful startups in this product line have almost been acquired by in open area. By using a tablet/smartphone installed Blueye App to connect wirelessly duSub. For one who would like to dive and/or take underwater images, IBUBBLE mini worldwide conglomerates. One can name Hydroid founded in 2001 then incorporated to to the surface unit, one can control Blueye Pioneer ROV. ROV illustrated in Fig. 2.8 can be an interesting choice, especially "hands-free" version. Konsberg over M&A in 2007, Bluefin founded in 1997 then merged to General Dynamics

graded with navigation package which includes a DVL, Greensea control program upgrade, MAX-M8W GPS MAST and (optional) USBL. The DVL is used for measuring linear velocity and consequently distance traveled can be estimated. Therefore, the current position can be calculated with dead-reckoning navigation techniques. When the vehicle is on the surface, a GPS antenna can be used to correct the vehicle's position. This upgrade package allows the vehicle to perform station keeping, requisition, mission planning and dynamic positioning. In the case when the sea bottom exceeds the operational range of the DVL, a USBL system is required for mission profiles. The upgraded VideoRay Mission Specialist Defender indeed can be equivalent to work class ROVs. In practice, for reducing workload of operator, work class ROVs often have the autonomous capabilities. For instance, if the vehicle

Table 2

 2

	.2: Teledyne SeaBotix accessories and options
	in 2011, and recently Riptide Autonomous Solutions purchased by BAE Systems in June
	2019. This trend reflects the increasing market for these vehicles as well as technology
	competition between the giants in the field.
	Small AUVs are often design in form of two-man or one-man portable. In Figure
	2.9, Bluefin-9 AUV of Bluefin Robotics in shape of two-man portable is illustrated. It
	has modular design with removable data storage and replaceable Li-ion battery. It has
	a navigation sensors including DVL and GPS (working only on the water surface). For
	communication purpose, it is equipped with acoustic modem, and Wifi & Iridium (work-
	ing only on the water surface). High definition captured images, video and sonar data
	can be accessed quickly when retrieving the vehicle. The mission endurance is 8 hours
	when traveling at three knots speed. The maximum forward speed is 6 knots. The depth
	rating is 200 m. It requires only 30 minutes to exchange data storage and replace the
	battery before redeploying. Equipped with camera and water probe sensor, it can be
	used for surveying environment, measuring water quality, searching and recovering, or

Table 3 . 1 :

 31 Reference frames used in this thesis

  diagonal matrices. Although the authors exploit the vehicle dynamics, and actually the proposed control forces (4.34) are calculated based on visual error ē, no stability analysis is provided when incorporating the kinematic relation (4.32) with vehicle dynamics (4.33). The experiments were again carried out in a test tank and on a Cartesian robot emulating two d.o.f of the ROV ANGUS 003 using unmarked natural target and is reported as successful under artificial disturbances.

Besides the lack of rigourous stability analysis in

[START_REF] Lots | Application of 2 1/2 D visual servoing to underwater vehicle station-keeping[END_REF][START_REF] Lots | A 2-d visual servoing for underwater vehicle station keeping[END_REF]

, it worth noting that either the derivative term ė involved in (4.31) or ė involved in (4.34) requires numerical derivative of e or ē and may be very noisy due to high noise level and low frequency of homography measurements. It is noted that the experiment setup only allows for visual servoing at 5 Hz.

  .42) and apply controller(4.41) to System (4.38)+(4.41) in order to ensure the stability of the equilibrium (H, V C , Ω) = (I, 0, 0). However, since the derivative of e p and e Θ are not computable by the controller due to the unknown quantities n and d , it is impossible to compute the feed-forward terms VCr and Ωr involved in(4.41). A popular and practical solution to this issue consists in neglecting the uncomputable terms VCr and Ωr in (4.41), i.e. setting VCr = Ωr = 0. In[START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] the stability analysis for System (4.38)+(4.41)

  Chapter 5. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of downward-looking camera trol design are related to the unknown quantities d and n involved in the expression (4.7) of the homography H, the coupled rotation and translation transformations encoded in H, and last but not least the unavailability of linear velocity measurements (i.e. DVL is not used). For future use, let us denote h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33

			H		(5.1)
	and R ψ the rotation matrix around AUV's axis -→ e b 3 as
		cos ψ -sin ψ	0
	R ψ	sin ψ	cos ψ	0	(5.2)
		0	0	1

  [kg.m -1 ] diag(36.57, 57.58, 68.97) D Ωl [kg.m 2 .s -1 ] diag(0.01126, 0.01855, 0.01701) D

			0842 0.004 0.005	
		0.004 0.2643 0.007	
		0.005 0.007 0.3116
				0.1 0.005 0.006	
	M 22 A [kg.m 2 ]		0.005 0.25 0.008	
			0.006 0.008 0.3
				1.39 0.10 0.12	
	M 11 A [kg]			0.10 4.26 0.13	
				0.12 0.13 4.02
			0.002 0.02 0.01	
	M 21 A = M 12 A		0.02 0.003 0.018	
			0.01 0.018 0.003
	D V l [kg.s -1 ]		diag(5.85, 9.21, 11.03)
	D V q

Ωq [N.m] diag(0.0053, 0.0130, 0.0118)

Table 5 .

 5

1: Specifications of the simulated AUV

Table 5 .

 5

2: Control gains and parameters of Controller 1

Table 5 .

 5 3: Control gains and parameters of Controller 2

Table 5 . 4

 54

: Specifications of the experimental AUV are presented in the video https://youtu.be/p_oiISPOtgw. However, the reader can also view two other videos showing the capability of long-term station keeping (more than 30 minutes) as well as other trials carried out during our research process with different water turbidity conditions and target images:

  one can write (5.3a) as two interconnected dynamics (surgeand-sway and heave) as follows Ṗ = -ω 3 S P + Fcω 12 P 3 + F gb ∆ 13

	6.2. Control design
		V	V + ω 3r Sr C	(6.9)
	One verifies that	VC = V + (ω 3 -ω 3r )Sr C + r C3 ω 12	(6.10)
	Define the following new control force
		Fc Fc + ω3r I 2 + ω 2 3r S MSr C	(6.11)
			∆ 23	(6.8a)
		Ṗ3 = F c3 + F gb + ω 12 P + F gb ∆ 33	(6.8b)
	with P 3 the third component of P. Introduce a new velocity variable 1

Table 6 .

 6

	Controller	1: Specifications of the simulated AUV Gains and other parameters
	Proposition 5	k 13 = 3s 1 , k 23 = 8

Table 6 .

 6 2: Control gains and parameters

Table 6 .

 6 3: Specifications of the experimental AUV 108 Chapter 6. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of forward-looking camera

Table 6 .

 6 4: Control gains and parameters

  Chapter 7. Development of the I3S-UV platform node, control force F c and control torque Γ c are calculated (either in manual or station keeping mode), then are sent to /mavros node though custom messages (i.e top-

	7.2. Software system
	ics) /i3s_auv_custom_2_topx4_nh and /i3s_auv_custom_3_topx4_nh, respectively. The in-
	formation of switching between two control modes, emergency stop signal, pwm of
	lamp (for turning on/off or adjusting light intensity) are sent to /mavros node over
	/i3s_auv_custom_1_topx4_nh. The three custom message topics are in red rectangular
	in Fig. 7.8. The vehicle's attitude and angular velocity required for control algorithms
	are estimated/measured in Pixhawk and transformed into Mavlink messages. They are
	sent to /mavros node and then to /control_in_ros_node over /mavros/imu/data_raw and
	/mavros/imu/data topics.
	Figure 7.8: ROS nodes and topics in high level control
	125

5

  The ArduSub developed by Blue Robotics based on ArduPilot has been officially announced in April 11th 2016, and merged with ArduPilot project in February 22nd 2017.6 L, W, H stand for Length, Width, Height respectively.

	7.2. Software system
	127

  R K ξ h(|ȳ| 2 )ȳ -β dq | V| 2 + 2 V R vr |V| + |v r | ( P + m 2 R vr ) + α r e 1× Ω + R fp + Tp e 1 K ξ ξ, K ξ a diagonal positive gain matrix, K P diag(k p1 + k p2 , k p1 , k p1 ) (with k p1 , k p2 > 0), and fp and Tp defined by (instead of (8.14))Tp T p -k p1 α r + k p2 P1 -m 12 m 1 m 2 e 1 R K ξ h(|ȳ| 2 )ȳ(8.35) Interestingly, Eqs. (8.30) and (8.34) have identical form as Eqs. (8.6) and (8.13), respectively. Therefore, similar outer-loop control expressions together with associated stability result as in Proposition 8 can be straightforwardly stated. Proposition 9. Let Tp and fp be defined by (8.35)+(8.33) and α r be defined by (8.32). Apply the outer-loop controller (8.23) where the involved term u is obtained by integration of (8.20)

	8.3. Simulation results
	with ȳ
	  	fp f p + k p1 (p -m 2 vr ) +	1 m 2	K ξ h(|ȳ| 2 )ȳ
	 
				(8.34)

DIY stands for do-it-yourself.

https://bluerobotics.com

https://www.hydroid.com

https://gdmissionsystems.com

https://en.wikipedia.org/wiki/VideoRay_UROVs

https://gopro.com/fr/fr/shop/cameras/

https://www.wamit.com/

https://www.ansys.com/fr-fr/products/fluids/ansys-fluent

https://www.code-saturne.org/cms/

https://innerspacethrusters.com/products/hydraulic-thrusters/

http://homographylab.i3s.unice.fr

Unless otherwise stated, most statements in projective geometry involve equality up to a multiplicative constant denoted by ∼ =.

http://homographylab.i3s.unice.fr/

https://docs.opencv.org/3.4.1/d9/dab/tutorial-homography.html

http://bluerobotics.com/store/retired/bluerov-r1/

V is used for handling the case with an arbitrary position of the camera w.r.t the CB. The convergence of ω3r to zero then results in V → V.

Tuned by applying the classical pole placement technique when considering n ≡ e1.

Due to limitation of dimensions of the water tank, we could not test for much farther distance.

We believe that this issue is not too difficult to address by a more proper tuning of the control gains and parameters for future validations. Due to time constraint we have to be content to report these experimental results.

https://bluerobotics.com/

https://hobbyking.com/

https://bluerobotics.com/store/

http://bluerobotics.com/store/retired/bluerov-r1/

7.3. I3S-UV Product data sheetcan be upgraded to accept positioning systems for trajectory tracking. With additional side-scan sonar, it can be used for detecting debris items on the seabed or for fisheries research.

Acknowledgements I would like to express my deepest gratitude to my supervisors, CR. Minh Duc HUA and PR. Tarek HAMEL, for academic guidance during my PhD project. My grateful

Chapter 6. Dynamic HBVS control of fully-actuated AUVs without linear velocity measurements: the case of forward-looking camera ment and, subsequently, the need of constructing of a strict Lyapunov function for the nominal system.

As a consequence of Lemma 2, there exists ν > 0 such that

Since ω3 converges to zero, for any 0 < ε ω < 1 there exists T > 0 such that ∀τ ≥ T , |ω 3 (τ )| ≤ εων 2 D and, subsequently,

If sup |ω 3r | < 1-εω 2 D ν, there exists ν 1 > 0 such that L1 (τ ≥ T ) ≤ -ν 1 L 1 (τ ≥ T ), implying the exponential convergence of X to zero.

Simulation results

The proposed control approach has been tested in simulation using a realistic model where the physical parameters are given in Tab. 6.1.

The robustness of the proposed controller with respect to model uncertainties are tested by using the "erroneous" estimated parameters Ĵ, M given by M = mI 3 + Ma = diag (17.868, 23.868, 21. The homography H is directly computed using (6.1) with d = 1 [m] and n = [0.8259, 0.5364, -0.1736] corresponding to (α , β ) = (10 • , 33 • ). The initial conditions are chosen as follows: ξ C (0) = [-1, -0.5, -0.5] [m], R(0) = R {roll=10 • , pitch=-10 • , yaw=45 • } , V(0) = Ω(0) = 0. The initial yaw and camera's position are chosen rather large to verify the large stability domain of the proposed controller. Control parameters and gains 2 are summarized in Table 6.2.

The performance of proposed controller is illustrated by Figs. [START_REF] Allibert | Pipeline following by visual servoing for autonomous underwater vehicles[END_REF].5-6.8. One observes from Fig. 6.5 a smooth convergence to zero of the position and orientation errors. Fig. 6.6

shows the fast convergence to zero of the visual error (σ, ) and their augmented variables ( σ, ˆ ). The time evolutions of the control force and torque are shown in Fig. 6.7. All components of the control force and torque converge to zero except the third component of the control force that allows for compensating for F gb . The convergence of the angular velocity Ω to the reference value Ω r as depicted in Fig. 6.8 shows the effectiveness of the inner-loop controller. In summary, parts of the developer kit BlueROV1 are completely reused in I3S-UV.

New components are included in order to accommodate a bigger electronic tray, a bigger battery and a camera. The compact configuration of I3S-UV has been proved to be highly effective in our practical experiments.

Software system

In fact, the control system of the I3S-UV in the sense of hardware and software has been developed in parallel with vehicle's 3D design and fabrication right in the beginning.

Because of that reason, the BlueROV1 developer kit which is ready to mount control hardware is tremendously important for us. It indeed saves us development time since some basic controls in manual mode can be tested without too much efforts.

The I3S-UV software system has been created for implementing control architecture (c.f. Fig. In the experimental process, the I3S-UV always starts in manual control mode. The station keeping mode is fully autonomous. There is a mechanism to control the switching between these two modes depending on the quality of homography estimation. The whole software system of the I3S-UV is thus developed for performing the following functionalities:

1. Control forward/backward movement and heading (i.e yaw angle or turn left-/right) of the vehicle in manual control mode.

2. Control the vehicle depth, auto-depth stabilization in manual control mode.

3. Control the e 3 direction of the vehicle in manual control mode for having different FOV of the camera since the latter is rigidly fixed to the vehicle.

High level control

The high level control is illustrated in Fig. 7.8. The operator uses a joystick to provide control references including:

• selected mode (manual or station keeping),

• desired depth, orientation and movement,

• desired lamp intensity,

• and emegency stop signal.

PWM lookup table

Since the desired value of each thruster is calculated from control allocation in the previous section, one now needs to find the corresponding PWM value in order to send to the ESC for controlling the motor's speed.

For a T200 thruster with two different applied voltages, the corresponding thrust values (in kgf unit) generated at each PWM input signal are given in a table whose graphic representations are illustrated in Fig. 7.10. For the 15 V DC nominal applied voltage, the corresponding thrust value then can be interpolated. The PWM value of each thruster corresponding to a desired thrust (T in kgf , by dividing to 9.8065) at 15 V DC is thus inversely calculated in PX4 by using the following functions: 130 Chapter 7. Development of the I3S-UV platform Note that the calculated PWM value then needs to be rounded up to get the closest integer before sending to the ESC.

Depth measurement for depth control

For the purpose of depth measurement, the Bluerobotics Bar30 pressure sensor is connected directly to Pixhawk over I 2 C connection in the I3S-UV. The core of this sensor is MS5837. In fact, there exists a driver for barometer MS 5611 in PX4. By exploiting this driver and performing the necessary modifications, we successfully developed a driver for the Bar30 sensor. The sensor then is used to automatically stabilize the vehicle at a desired depth specified by the operator. The value of the measured pressure is sent from Pixhawk to the companion computer (i.e /control_in_ros_node) by employing available (i.e. unused) message, for instance optical flow topic /mavros/optical_flow which is corresponding to optical_flow uORB message in PX4.

I3S-UV Product data sheet

The I3S-UV is a hybrid underwater vehicle of I3S-OSCAR team. It is employed with a very basic sensor suite including low-cost cameras and IMU. In manual control mode, it can be remotely controlled without or with auto-depth and auto-heading. Featured with advanced HomographyLab © and the state-of-the-art nonlinear control algorithm, the vehicle performs automatically dynamic positioning without a high-cost DVL. Lemma 4. Assume that initially (R γ)(0) = -u(0) and that the inner-loop controller ensures the convergence of Ω about zero and also its boundedness. Then, by setting the following

with k u > 0, one ensures that R γ converges asymptotically to u.

Proof. Using (8.20), (8.21) and (8.22), the time derivative of the positive function

where the first term Ω (R γ × u) is bounded thanks to the boundedness of Ω, whereas the division by 1 + u R γ in the second term prevents R γ from tending close to -u since L1 tends to -∞ in this case.

Using the relation |R γ × u| 2 + (u R γ) 2 = 1, one then deduces that

Since Ω converges to zero, the application of the singular perturbation theory then ensures the convergence of L 1 to zero or equivalently of R γ to u.

Now the main result of this section can be stated.

Proposition 8. Let Tp and fp be defined by (8.14)+(8.11) and α r be defined by (8.8). Apply the outer-loop controller

where u is obtained by integration of (8.20) with the augmented control input Ω u given by

Simulation results

lowing estimated and approximated parameters: [START_REF] Hua | Explicit Complementary Observer Design on Special Linear Group SL(3) for Homography Estimation using Conic Correspondences[END_REF]440,440) [kg] Ĵ = Ĵ0 + Ĵa = diag [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF]200,190)

One can notice that these estimated terms are quite different from the corresponding "real" ones given in Tab. 8.1. In addition, a non-null current velocity

is introduced, allowing us to test the control robustness w.r.t. both model uncertainties and external disturbances.

The simulated model of the dissipative torque Γ d is given by

with the damping matrices K Ωl and K Ωq specified in Tab. 8.1, whereas in the inner-loop torque control expression (8.12) the estimate of this dissipative torque is simply set equal to zero (i.e. Γd = 0) so that no prior knowledge of this torque is required for control implementation.

The gains and parameters involved in the proposed controller are chosen as follows:

Context and contributions of the thesis:

The contributions reported in Part II constitute a continuation of prior work of the I3S-OSCAR team [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] on the topic of dynamic homography-based visual servoing. They contribute to the development of low-cost but still efficient vision-based control solutions for the stabilization of a class of fully-actuated underwater robotic vehicles. The use of costly DVL velocity sensors as in [START_REF] Krupínski | An inertial-aided homographybased visual servoing control approach for (almost) fully actuated autonomous underwater vehicles[END_REF] is thus excluded due to their excessively high price. Instead only a suite of low-cost sensors consisting of a monocular camera and a MEMS IMU is required. In the context of monocular vision, the standard assumption on the planarity of the visual target is considered here, resulting in the meaningful involvement of the homography in control design. Despite such a restrictive assumption, the proposed solutions are still relevant for a number of AUV applications encompassing station-keeping or positioning using a downward-looking camera observing a (near) planar seafloor or using a forward-looking camera observing a subsea structure composing of planar surfaces for carrying out high-resolution imaging or manipulation tasks. Two algorithms have been proposed for these two cases in this thesis work and the results constitute the subject of the following publications or submissions [START_REF] Nguyen | Inertial-aided homographybased visual servo control of autonomous underwater vehicles without linear velocity measurements[END_REF], [START_REF] Nguyen | A homography-based dynamic control approach applied to station keeping of autonomous underwater vehicles without linear velocity measurements[END_REF], [START_REF] Nguyen | A homography-based dynamic control approach of autonomous underwater vehicles observing a (near) vertical target without linear velocity measurements[END_REF]. The lack of linear velocity measurements has led us to make a certain number of simplifications of the model for control design, namely neglecting the dynamic couplings between the translational dynamics and the rotational dynamics and also considering the Munk ett's necessary condition [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF] for the existence of a time-invariant feedback asymptotic stabilizer either linear or nonlinear. Specific and still prospective nonlinear techniques involving dynamic extensions are required to solve the stabilization problem (see, e.g., [START_REF] Samson | Control of chained systems application to path following and timevarying point-stabilization of mobile robots[END_REF][START_REF] Morin | Time-varying exponential stabilization of the attitude of a rigid spacecraft with two controls[END_REF][START_REF] Morin | Practical stabilization of driftless systems on lie groups: the transverse function approach[END_REF]).

By successfully developing novel and original control approaches for the two above difficult problems, we would like to advance the state-of-the-art of nonlinear control of AUVs in terms of fundamental methodologies to deal with the reduction in number of actuators (i.e. underactuated AUVs), the degradation of actuator efficiency during transition operating modes (i.e. transitions from fully-actuated to underactuated mode and vice-versa), and the highly nonlinear and coupled dynamics of underwater vehicles. The motivation is also that by developing original control solutions for the two above problems for the classical objective of trajectory tracking, these control approaches will also constitute a basic control pre-design framework to address many other objectives related to sensor-based control in a principled manner, knowing that extensions/adaptations should be carefully undertaken because sensor-based control design is often carried out in sensor spaces instead of state spaces and also due to the fact that some information may be unavailable to feedback control, thus, leading to the need of including dynamic