Autonomous Underwater Vehicle (AUV) is a relevant technology for the sustainable use of ocean resources. AUV can be used as an important ocean observing platform to collect information on marine environmental characteristics for research and industry fields. In order to improve the observation quality and increase the navigation ability, many issues should be addressed and considered simultaneously.

Achieve necessary maneuverability depends on two key factors: an accurate hydrodynamic model and an advanced control system. However, the cost to develop an accurate hydrodynamic model, which shrinks the uncertainty intervals, is usually high. Meanwhile, when the robot geometry is complex, it becomes very difficult to identify its dynamic and hydrodynamic parameters. In addition, according to the quadratic damping factor, underwater vehicle dynamic and hydrodynamic model is nonlinear from the control point of view. Moreover, unmodeled dynamics, parameter variations and environmental disturbances create significant uncertainties among the nominal model and the reality. Sensor noise, signal delay as well as unmeasured states also affect the stability and control performance of the motion control system.

In many of our underwater competitions, it has been confirmed that the traditional Proportional-Integral-Derivative (PID) regulation is less efficient for low mass AUV. In this case, our scope is more focused on the combination of numerical modeling approaches and robust control schemes.

In this work, we proposed a model based robust motion control scheme. Without loss of generality, a robust heading controller was implemented and validated in the sea on cubic-shaped CISCREA AUV. The proposed solution uses cost efficient Computational Fluid Dynamic (CFD) software to predict the two hydrodynamic key parameters: The added mass matrix and the damping matrix. Four Degree of Freedom (DOF) model is built for CISCREA from CFD calculation. Numerical and experimental results are compared. Besides, the proposed control solution inherited the numerically obtained model from previous CFD calculation. Numerically predicted the actuator force compensates the nonlinear damping behavior result in a vii linear model with uncertainties. Based on the bounded linear nominal model, we proposed ∞ approach to handle the uncertainties, we used kalman filter to estimate unmeasured states such as angular velocity and we developed smith compensator to compensate the sensor signal delay.

The proposed robust heading control application uses only one compass as feedback sensor. This is important while AUV is working at certain depth where only magnetic sensors still work. Our robust control scheme was simulated in Matlab and validated in the sea near Brest. Simulation shows obvious advantage of the proposed robust control approach. Meanwhile, the proposed robust heading control is much faster than PID controller. The robust controller is insensitive to uncertainties and has no overshot. From both simulations and real sea experiments, we found our proposed robust control approach and the one compass heading control application are efficient for low mass and complex-shaped AUV CISCREA.

Modeling and Robust Control Approach for Autonomous

Underwater Vehicles by Rui YANG

The joint Ph.D study at ENSTA Bretagne and Ocean University of China is indeed an expectational experience in my life. During the years I have spent in France and China, I have benefited a lot from the discussions and interactions with the faculty members and the fellow students. Currently, I am delighted to share with all of you my growth about free mind and independence in the past years.

In particularly and foremost, I would like to express my greatest gratitude to my four supervisors, Pr. Benoit CLEMENT, Pr. Ali MANSOUR, Prof. Huajun LI and Prof. Ming LI, for their academic guidance and supervision. Comparing to myself four years ago, I owe all my good changes and progress to their kindly help and the important opportunities that they have offered to me. It has been a great pleasure to work and include joint papers with my supervisors. They have shown their great responsibility to help me handling the research obstacles, academic conferences and journal writings, etc. Not only as my tutors, I believe now they are all my caring friends through the years of discussions on wide range of common topics. I should mentioned that Pr. Ali MANSOUR and Pr. Benoit CLEMENT are very nice people, and they took me as family member while my stay in France. Their families delivers many great joys to my life in France. I really enjoy my days living and doing research with them in the city of Brest. Pr. Huajun LI is a very nice and visionary tutor. Only with his hard work to built international collaborations and interdisciplinary research, it can be convenient for me to achieve my joint study between hydrodynamics and control science both in France and China. Moreover, I want to give my thanks to Prof. Jianguo WANG, Prof. Ming LI and Prof. Hui YU who have been the key persons to guide me into the marine world. I would like to give my thanks to the chairman Prof. Luc JAULIN, the reviewers Prof. Lionel LAPIERRE, Prof. Yan Zhuang, and the jury members of my thesis, Prof. Philippe CHEVREL, Prof. Huajun LI, Prof. Benoit CLEMENT and Prof. Ali MANSOUR. Thank you very much for your decent comments and serious work.

Here, I would like to express my great appreciation to the China Scholarship ix Council for their financial supports to my study in France. It is great to make the international study and enjoy the big picture of the amazing world.

In fact, it is difficult to name all those who contributed to my Ph.D work and inspired my life, but I would like to thanks Pr. J. M. Laurens, Mr. I. Probst, Dr.

F. Le Bars and Mr. J. S. Zhang for their technical supports to complete my work successfully. I am also grateful to make so many friends all around the world: Yi Yang, Dongjie, Jie Xie, Liangzhang, Kaixiang, Jun Yao, Jianfeng, Jie Song, Saad, Simon, Guilherme, Dr. Mike Benjamin and Mr. Alon Yaari, etc. We can not cover all here. Thank you all for the congenial time we spent together.

I would like to thank my parents for supporting my life and study in China. My special thanks to my mother who taught me to be kind and honest with all the people. I always believe that life is immense, more than we can imagine. Never give up anything, just keep going! In the end, my sincerely gratitude to those who inspire my mind of freedom and

Context and General Introduction

The ocean covers a vast stretch of surface on the earth, around 71% according to the National Oceanic and Atmospheric Administration (NOAA) [1], it is rich in natural resources, and it is still considered to be largely unrevealed. As marine technology evolves rapidly [2], develop the sustained ability of marine activities is more important than ever before. Indeed, sustainable use of the ocean becomes a pressing problem that concerns the immediate benefits of every nation. This paper undertakes the study of modeling and control problems of AUV. In fact, AUV is currently a key technology to answer a variety of marine challenges in the sustainable use of ocean resources and space. Nowadays, major marine activities deploy underwater vehicles, and these vehicles are widely involved in surveillance, inspection and survey missions [2,3].

Fully explore the ocean is challenging. Every year abundant of ocean biological discoveries and marine technology innovations prove that the ocean mysteries are far beyond our imagination [4,5,6]. In order to improve the sustained ability of marine activities, it is required to expand both the width and depth of undersea monitoring and exploration. Actually, the ocean environment is generally harsh and complex.

There is a fundamental necessity of using underwater vehicles to solve the problems such as: operating at harsh weather, limited communication bandwidth and high pressure or high temperature regions, etc. Meanwhile, a lot of unknown physical and ecological ocean mechanism are waiting to be revealed by mean of autonomous underwater robots [START_REF] Brandt | Challenges of deep-sea biodiversity assessments in the southern ocean[END_REF][START_REF] Hildebrandt | Design of an autonomous under-ice exploration system[END_REF].

At the same time, modern societies face new challenges. Human civilization products are now to some extent threatens to the ocean environment. Before the great strides of industry, human and ocean relation depends on pure natural phenomena, including hurricanes, tsunami, algal bloom and offshore erosions, etc. However, today many industrial activities are double edge sword. They can go out of our control such as Gulf of Mexico oil spill [START_REF] Pezeshki | The effects of oil spill and clean-up on dominant us gulf coast marsh macrophytes: a review[END_REF], floating garbages, Fukushima nuclear accident [START_REF] Matsuno | Utilization of robot systems in disaster sites of the great eastern japan earthquake[END_REF][START_REF] Zhang | A peak capture algorithm used on an autonomous underwater vehicle in the 2010 gulf of mexico oil spill response scientific survey[END_REF] and global warming, etc. The ocean is an essential part to support life on the earth, understand and protect its self-balance will save life from fatal disaster. Therefore, many countries and organizations are inspired by the past man-made accidents. They post realistic disaster scenarios to the autonomous air, land, surface and underwater robots [START_REF] Sauc-E | Student autonomous underwater challenge-europe[END_REF][START_REF] Eurathlon | Outdoor robotics challenge for land, sea and air[END_REF]. New kind of ocean safety guards are expected from potential autonomous underwater vehicle designs.

Researches on AUV show recently a large amount of alarming results. In fact, these studies prove the existence of a blank territory to improve the robustness of future AUV designs.

In this chapter, the following contents are described:

-review of underwater vehicle evolution, applications and related researches.

-the state of art in modeling and control of underwater vehicles.

-proposed algorithms and thesis organizations.

Underwater Vehicles Brief Review

Extending the ability of human-being to explore the ocean with man-made products date back to 1531. Indeed, two men dive into the water using Halley's bell as it was

shown by the design of Leonardo da Vinci [2]. Nowadays, thousands of underwater vehicles are designed with different types for a wide range of underwater missions. A brief underwater vehicle history is described here based on the report of Lapierre [2] and Caiti [START_REF] Caiti | Underwater robots: Past, present and future[END_REF].

Researches accelerate the steps to explore the underwater world. At the beginning, they focus on the design of Human Occupied Vehicle (HOV). In 1776, David Brushell To avoid HOVs accident and minimize the risk, people began to show interest in developing Unmanned Underwater Vehicle (UUV). Like HOVs, UUV designs also pass through a long history. In 1866, Robert Whitehead design the first torpedo in the world. In 1960s and 70s, USA and France respectively design the earliest SPURV (Self-Propelled Underwater Research Vehicle) and Epaulard UUV. Nowadays, UUV Blue fin 21 was used for MH370 blackbox searching mission.

Differences between HOV and UUV are mainly the design, the cost and the autonomy efficiency. Comparing to UUV, HOV should be designed to have critical life support systems for safety reasons. The cockpit is to some extent a high-risk and tight space for pilots. The first aim is always guaranteeing the safety of pilots, which induces very high cost to build the HOV. However, the HOV is operated by the onboard pilots, which requires less autonomy on the vehicles. Instead, the autonomy level of UUV improves rapidly, UUV currently shows more flexibility than the HOV, and a large amount of UUV designs have emerged [3]. UUVs can operate at harsh AUV is the most common UUV. AUVs are wildly used for the autonomous missions under depth range from hundreds up to thousands meters. AUVs can fully run without human intervention for tens of hours. In figure 123456, it is shown the Blue Fin 9 AUV, of which its predecessor Blue Fin 21 is deployed to find the missing blackbox of the MH370 airplane [START_REF] Normile | Lost at sea[END_REF][START_REF] Lehardy | Deep ocean search for malaysia airlines flight 370[END_REF]. Characterized by appearance and dynamic properties, AUVs can be divided into Torpedo-shaped AUV and Cubic-shaped AUV.

Torpedo-shaped AUVs are distinct from cubics, and they are generally designed for long range missions. Under-actuated actuation configuration (control fins and surge propellers) determine that torpedo-shaped AUV is usually unable to hover and make a pivot steering. It is difficult for Torpedo-shaped AUV to achieve precise movement inside a small space. But there are few exceptional torpedo-shaped AUV designs which are not limited by the under-actuated issues, such as Avalon [START_REF] Hildebrandt | Design of a versatile auv for high precision visual mapping and algorithm evaluation[END_REF] and Sparus II [START_REF] Carreras | Sparus ii, design of a lightweight hovering auv[END_REF] shown in figures 1-9 and 1-10. Cubic-shaped AUVs are generally fully actuated, therefore they are good at hovering, pivot steering and short range low speed missions. In figure1-7, the OCEAN Modules V8 robot has 8 thrusters, and allows 6 DOF movement. In our application, CISCREA AUV is equiped with 6 thrusters, which allows 4 DOF motion, see figure 12345678(Two vertical thrusters are connected without differential actuation ability). Actually, hovering and pivot steering motion are critical for some specific underwater applications: undersea pipeline inspection, offshore infrastructure surveillance and large vessel maintenance. Indeed, small AUVs can be deployed to explore areas which are not accessible to HOVs and ROVs. Meanwhile, a large number of the cubic-shaped AUVs enjoy more degrees of freedom than under-actuated torpedo-shaped AUVs.

The abundant distinct designs of underwater vehicles closely rely on the requirements of its specific applications. Therefore, in the next section, we will briefly introduce the major underwater vehicle applications.

Underwater Vehicles Applications

Underwater vehicle (underwater robot) is a highly integrated product which involves the frontier of many high-tech research fields. Hence, the past underwater robots by autonomous robots. According to the reports [2,3], there are an estimation of around 2000 billion tons of manganese nodules near the Hawaiian Islands. Hence, many countries such as France and Canada are inspired to develop UUVs for the emerging robot mining industries.

Military is another big buyer of underwater vehicles, and they focus on cultivating a new kind of autonomous forces by UUVs. Modern UUVs are born with good invisible properties. Instead of acting as a weapon, UUVs can save many lives in future conflicts. Some UUVs have already been used in the anti-terrorist battles [START_REF] Bovio | Autonomous underwater vehicles for scientific and naval operations[END_REF][START_REF] Navy | The Navy Unmanned Undersea Vehicle (UUV) Master Plan[END_REF]. Meanwhile, the military UUV Blue Fin 21 was also used in the civilian search of MH370 airplane. As global economy boost, the future territory safety will be an essential issue of a sustainable ocean. We believe that the UUV will be an important mean to save life.

According to Lapierre [2], coastal activities depends on UUVs. The port security and safety burden are increasing rapidly, the monitoring and evidence gathering work will finally exceed the limit of human efficiency. The 24-hours rest-free underwater robot is indeed a good candidate as future autonomous port guards. It is important to have a tool such as UUV, that can monitor leaking vessels for environmental protection, rescue people from harsh weather or dangerous regions, and inspect automatically the sedimental condition of critical transport channels. Besides, develop the maintenance ability of UUVs for large vessels, can save currently limited dock space and docking time [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF]. Wet maintenance technology is critical for large scale ports.

In addition, coastal man-made structures require periodic and regular inspections.

For example, European dams have to be totally checked every ten years [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF]. Note that, after the Fukushima accident, people pay more attention on using autonomous underwater vehicles for missions to monitor nuclear pollution and shut down critical valves [START_REF] Eurathlon | Outdoor robotics challenge for land, sea and air[END_REF].

As the cost of UUV drops down, aquaculture UUV applications emerge. Indeed, there are examples using an UUV and ultra sonar to detect fishing net damages [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF]. Meanwhile, biological and ecological study requires more UUVs. For example, scientists use onboard sonar to identify different fishes. UUVs are indeed important in learning and understanding the ecological and biological mysteries of the sea.

Scientific data collection is the driving force to develop inspection UUVs. For

instance, to answer the global warming phenomenon, many organizations proposed monitoring networks using multiple UUVs. The main concern is that UUVs can move like a moving buoy, from one location to another where it is needed most. Besides, it is necessary to have UUVs in many other scientific studies, such as learning the acoustic characteristics of deep water or under the arctic ice cover, collecting high temperature samples near undersea volcano, and monitoring coral reefs [START_REF] Julio | A behavior-based architecture for autonomous underwater exploration[END_REF][START_REF] Mcewen | Performance of an auv navigation system at arctic latitudes[END_REF].

UUVs face challenges to achieve above applications. The essential research issues are briefly listed in the next section.

Research Issues of Underwater Vehicles

Underwater vehicles are highly integrated. Miniature, low-cost, robust and multiple vehicle design are the current trend of new underwater vehicles. In this thesis, four levels of researches are considered to be critical to construct an efficient underwater vehicle: physical level, control level, guidance and navigation level, and mission control level.

Physical Level: the designers should have enough knowledge about the ocean environment and robotic systems, to predict precisely the physical limit of the underwater vehicle. For instance, the physical level issues include: vehicle stability analysis (naval architecture point of view), pressure container design, thruster design, hydrodynamic fairing design, embedded system, sensors integration and battery management, etc. Like all ordinary ocean vehicles, the ability to maintain stability without capsizing the hull is essential for underwater vehicles to handle the turbulent ocean environment. Then, it is necessary to select correct materials to prevent the corrosion of underwater vehicles. In deep water, the pressure is extremely high, the pressure container faces tons of force. Besides, collision protections are also considered to prevent data lost during unexpected collisions. For long range missions, optimizing the fairing design will obviously reduce the hydrodynamic damping drag, and extend the working range. For hovering tasks, precisely design the thrusters and fairing structures can provide an accurate model for better hovering and pivot steering ability. As we know, sensors are the eyes of the underwater robots to explore its unknown surroundings. The sensor integration plays an important role in AUV design. New trends are to integrate more sensing devices in an even smaller space. The sensors should guarantee enough data for the underwater vehicle to make decisions by itself. Breakthroughs are expected on the above issues.

Control Level: the ocean environment is complex and full of uncertain disturbances. Therefore, the two important control level researches: modeling and control of underwater vehicles, are essential for a fully equipped underwater vehicle to achieve high quality maneuverability. Indeed, underwater vehicles with precise control algorithms would have distinct advantage in reducing the complexity of high level researches, such as localization and navigation.

Generally, an accurate hydrodynamic model of the underwater vehicle can reveal the physical details and emphasize the focus of control design [START_REF] Ferreira | Hydrodynamic modeling and motion limits of auv mares[END_REF]. The control performance can be obviously improved, as there is less parameter uncertainties in the accurate underwater vehicle model. However, the cost of current modeling methods to obtain an accurate underwater vehicle is always expensive. It is even more difficult to be accurate when the underwater vehicle has complex structures and shapes.

We believe high cost is one of the major reasons, only large organizations have the modeling ability of high performance underwater vehicles.

To control the underwater vehicle is not straight forward, many issues should be addressed [START_REF] Yang | Modeling of a complex-shaped underwater vehicle[END_REF]. The hydrodynamic model of the underwater vehicle is usually nonlinear, which means traditional control algorithms are not efficient to handle the damping efforts (cross-coupling damping terms and the linear damping terms are neglected under the low speed assumption). Then, dynamic parameters have uncertainties, and they are usually different from the nominal model. For example, the mass inertia matrix will increase when payloads are added to the underwater vehicle.

Unmodeled dynamics, environmental disturbance as well as obsolete components also induce modeling uncertainties. At last, sensor noise, signal transmission delay, and unmeasured states would also affect the control stability and tracking performance.

In order to build an efficient and approachable underwater vehicle control system, this thesis addresses the modeling and robust control problems simultaneously. State of art in AUV modeling and control works are described in the next section.

Guidance and Navigation Level: underwater vehicle autonomy are endeavoring to get rid of human interventions. Future UUV can analyze environmental information onboard, and make decisions by itself. To this case, the UUV autonomy requires the guidance and navigation level researches. First, localization is considered to be one of the most difficult underwater problems. It is essential for the UUV to first find its position and attitude using limited underwater information. Note that, for most of the UUV inspection data, the information is only clear and valuable if the localization information and time stamps are accurate. No straight forward and efficient solutions exist to invert realistic inspection information by inaccurate localization data. The navigation system analyzes the realtime environmental data, and optimizes reasonable path for the UUV controller to avoid obstacles. UUV guidance system will finally push the robot in the correct track.

Underwater localization is a difficult task since there exist no GPS and radio signal. Territorial methods can not be applied to underwater localization [START_REF] Tan | A survey of techniques and challenges in underwater localization[END_REF]. Generally, electronic magnetic compass, gyroscopic compass, fiber-optic gyro-compass, inertia navigation unit, Doppler Velocity Log (DVL), Long Base-line (LBL), and Ultra Short Base-Line (USBL) are used for underwater localization [START_REF] Bahr | Cooperative localization for autonomous underwater vehicles[END_REF][START_REF] Whitcomb | Advances in underwater robot vehicles for deep ocean exploration: Navigation, control, and survey operations[END_REF]. Classical methods such as Dead Reckoning (DR) use initial UUV position and current acceleration information to predict locations. However, the disadvantage of DR is the error accumulates as the time passes. DR is simple and efficient for some UUV applications. For Gliders, it is possible to float to the surface, and make corrections on the DR system periodically. For some UUVs that equipped with DVL, the DR method will have an obvious performance lift [START_REF] Bahr | Cooperative localization for autonomous underwater vehicles[END_REF][START_REF] Carreno | A survey on terrain based navigation for auvs[END_REF]. LBL and USBL methods depend on static reference, which limit the operating range of the UUVs in few kilometers. According to the report of Pascoal [START_REF] Pascoal | Navigation system design using timevarying complementary filters[END_REF], the main work of localization is around the estimation of positions using Kalman filter (KF), Extended Kalman filter (EKF) and complementary filters. The difficulty is to obtain the nonlinear state space equation and noise properties [START_REF] Brown | Introduction to random signals and applied Kalman filtering[END_REF][START_REF] Brown | Integrated navigation systems and kalman filtering: a perspective[END_REF][START_REF] Oliveira | Periodic and non-linear estimators with applications to the navigation of ocean vehicles[END_REF][START_REF] Anonsen | An analysis of real-time terrain aided navigation results from a hugin auv[END_REF]. Besides, localization methods using a group of AUV [START_REF] Bahr | Cooperative localization for autonomous underwater vehicles[END_REF][START_REF] Caiti | Underwater communication and distributed localization of auv teams[END_REF] or using acoustic cooperations between ASV and UUV are also popular today [START_REF] Fallon | Cooperative auv navigation using a single maneuvering surface craft[END_REF][START_REF] Eurathlon | Outdoor robotics challenge for land, sea and air[END_REF]. An important UUV localization algorithm is Simultaneous Localization and Mapping (SLAM) [START_REF] Paull | Auv navigation and localization: A revie[END_REF][START_REF] Aulinas | The slam problem: a survey[END_REF]. SLAM creates and identifies the landmarks to improve its knowledge about the environment and its position. Many works exist on UUV SLAM algorithms in [START_REF] Hasan | A review of navigation systems integration and algorithms[END_REF][START_REF] Kondo | Navigation of an auv for investigation of underwater structures[END_REF][START_REF] Mallios | Navigating and mapping with the sparus auv in a natural and unstructured underwater environment[END_REF][START_REF] Aulinas | Vision-based underwater slam for the sparus auv[END_REF]. Currently, navigation works are mainly mathematical optimizations, as described in [START_REF] Bethencourt | Interval analysis for swarm localization. application to underwater robotics[END_REF][START_REF] Aeem | A review of guidance laws applicable to unmanned underwater vehicles[END_REF]. Navigation methods include way-point navigation, line of sight method and nonlinear guidance [START_REF] Samson | Mobile robot control. part 1: Feedback control of nonholonomic wheeled cart in cartesian space[END_REF][START_REF] Lapierre | Non-singular path-following control of a unicycle in the presence of parametric modeling uncertainties[END_REF], etc. Generally, the navigation system optimizes a group of desired way points that avoids obstacles, based on the knowledge of operating environment in advance.

Mission Control Level: the mission level concerns more about behaviors and human interaction. Construct the mission level includes: onboard mission operating system, objective laguage human interface, swarms and intelligence research, etc.

UUV operating works can be defined into different missions, hence, many softwares are developed to manage the robot missions. For example, MOOS-IvP is one of the widely used UUV mission control system, that transfers information among operators and executive UUV hardwares. The IvP-Helm of MOOS is an optimization program based on interval programming [START_REF] Benjamin | An overview of moos-ivp and a brief users guide to the ivp helm autonomy software[END_REF]. To solve in real time, the UUV heading and speed, MOOS-IvP can achieve a frequency of 4 . The CISSAU AUV utilizes the MOOS-IvP software as a mission control system, see Figure 6-2. Future UUV human interfaces are expected to understand the scientific or engineering language from difference professions, which will unify and ease the UUV deployment process of overall ocean observation networks. Moreover, UUVs are expected to operate in the swarm formation, which have more robustness and efficiency than a single UUV. Except our modeling and robust control work, swarm can be another important solution to increase the robustness and stability of UUVs. UUV swarms are efficient to assign optimal numbers of sensors and flexible missions, meanwhile, the failure of one or several UUVs will not affect the underwater mission.

Robustness is an essential issue for future UUV designs. This thesis focuses on the modeling and control solutions to increase UUV robustness. The state of art in AUV modeling and control works are described in the next section.

State of Art in AUV modeling and control

Achieving good maneuverability of small AUV depends on two key factors: an accurate hydrodynamic model and an advanced control system. In [START_REF] Yamamoto | Robust and non-linear control of marine system[END_REF], Yamamoto pointed out that a model-based control system is more effective if the vehicles' dynamics are modeled to some extent. Meanwhile, in [START_REF] Ferreira | Modeling and control of trimares auv[END_REF], Ferreira . showed that an empirical linear model often fails to represent the dynamics of the AUV over a wide operating region. Indeed, obtaining hydrodynamic models of the complex-shaped cubic AUV is one of the key points for better maneuverability. In addition, inside the virtual environment engineers can replay many dynamic and hydrodynamic phenomenons with seldom limit to time, space and cost. In this section, we first introduce the current underwater modeling works.

Modeling of Underwater Vehicles

Actually, many methods exist to model underwater vehicles, including full-scale experiments, scaled experiments, empirical formula approximations and computational approaches.

The advantage of a full-scale experiment is high accuracy, especially, for obtaining the added mass and damping parameters. However, high cost is a fatal drawback of full-scale experiment, it requires very expensive devices, such as towing tanks. In addition, the full-scale experiment is not control oriented, which means the modeling results can not be directly used for control design. Some experimental methods without towing tanks also exist, as presented in [START_REF] Caccia | Modeling and identification of openframe variable configuration unmanned underwater vehicles[END_REF][START_REF] Rentschler | Modeling and control of an odyssey iii auv through system identification tests[END_REF], and free decay approach is presented by Ross in [START_REF] Ross | Identification of underwater vehicle hydrodynamic coefficients using free decay tests[END_REF].

Scaled experiments compromise the cost and modeling accuracy. It is less expensive than full-scale experiment, as it only implements a scaled (eg. small size) model in the hydrodynamic experiment. Generally, modeling results of scaled experiments are more reliable than empirical and computational methods. However, the accuracy of scaled experiment is subtle due to the scale effect. A ROV modeling application using scaled experiment of free decay method is presented in [START_REF] Eng | Added mass computation for control of an open-frame remotely-operated vehicle: Application using wamit and matlab[END_REF], and its experimental results are compared with computational solutions.

Empirical formula approximation is well proved on torpedo-shaped AUVs, usually slender bodies, as mentioned in [START_REF] Rentschler | Modeling and control of an odyssey iii auv through system identification tests[END_REF][START_REF] Ferreira | Modeling and control of trimares auv[END_REF][START_REF] Ferreira | Hydrodynamic modeling and motion limits of auv mares[END_REF]. The advantage of this method is the simplicity in estimating slender UUV hydrodynamic parameters, which uses only an ellipsoid shape. Nonetheless, empirical formula method requires deeper knowledge and experiences to simplify the UUV into elementary components. Meanwhile, the estimation accuracy is not always reliable, because simplified geometry may neglect many details. Especially, for a complex-shaped UUV (sometimes has open-frame structure) the simplification is hard to achieve.

Depend on basic mass, momentum, and energy conservation axiom, the potential theory and finite element theory based CFD softwares are well developed. Although there is no direct solution for the Navier-Stokes equation [START_REF] Newman | Marine Hydrodynamics[END_REF], the computational approaches can convert a physical fluid problem into applied mathematical problem.

Then, hydrodynamic phenomena or parameters can be approximated by iterative calculations. In the beginning, CFD programs were popular to analyze large marine structures and transportation vessels. Nowadays, more and more UUVs apply the CFD modeling methods.

Computational methods have many obvious advantages. Theoretically, the accuracy of CFD approximation can be infinitely small if the calculation amount is unlimited. Meanwhile, CFD is able to reveal some physical details which is difficult to observe in the experiments. In other words, using CFD can replay many invisible dynamic and hydrodynamic phenomena without any cost. More importantly, CFD requires less experiences to simplify the complex-shaped AUVs. CFD modeling complexity are lower, and this is favorable to spread miniature and low-cost AUVs.

Although the CFD reliability is not as good as experimental solutions, currently, there are still reliable models obtained by mixing CFD and experimental approaches [START_REF] Yang | Modeling of a complex-shaped underwater vehicle for robust control scheme[END_REF].

Indeed, the computational modeling approach is a potential trend for future UUV design.

As mentioned by Newman in [START_REF] Newman | Marine Hydrodynamics[END_REF], UUV hydrodynamic forces can be linearly superimposed. Therefore, the UUV hydrodynamic effects can be separately analyzed by different softwares. There are CFD softwares such as: WAMIT, Marine Craft Characteristics (MCC), which is created by ENSTA Bretagne and it can be downloaded at [START_REF] Mcc | Marine craft characteristics[END_REF], SHIPMO, ANSYS-CFX, ANSYS-FLUNT, ANSYS-AQWA, STAR-CCM+ and SeaFEM. These softwares can predict hydrodynamic parameters for a complexshaped AUV with a very low cost. In [START_REF] Eng | Added mass computation for control of an open-frame remotely-operated vehicle: Application using wamit and matlab[END_REF], it is shown the efficiency of WAMIT to predict the added mass matrix. In [START_REF] Eng | Identification of the hydrodynamics coefficients of an underwater vehicle using free decay pendulum motion[END_REF], ANSYS-CFX was employed for AUV damping analysis.

Regardless of modeling issues, the value of a UUV model depends on how robust and efficient your control scheme can adapt to the hydrodynamic model. In next section, current UUV control methods are presented.

Control of Underwater Vehicles

Underwater vehicles are generally designed to operate in the ocean environments.

Therefore, numerous uncertainties exist, including parameter variations, nonlinear hydrodynamic damping effects, sensor transmit delays and ocean current disturbances.

To solve the UUV control issues, a large amount of underwater vehicle motion control algorithms are proposed, including: classical PID control, robust control, adaptive control, sliding mode control, back-stepping control, Linear Quadratic Gaussian (LQG), predictive control, gain scheduling, fuzzy control, neural network control, and other hybrid control methods. High quality reviews of underwater vehicle control researches can be found in the works of Yuh [3], Johansen [START_REF] Johansen | Control allocation: A survey[END_REF] and Budiyono [START_REF] Budiyono | Advances in unmanned underwater vehicles technologies: Modeling, control and guidance perspectives[END_REF]. An exhaustive report is given by Maalouf in [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF], which summarizes the performance current underwater vehicle controllers.

To control an underwater vehicle is complex. The dynamic and hydrodynamic properties involves: nonlinear damping, uncertain parameters variations, environmental disturbance, sensor noise, and signal delay, etc. The early UUV control methods are usually simple to implement. Controllers was designed without the UUV model or assuming that the UUV model is time invariant. Linearization is generally im-plemented assuming the equivalent linear system has the same performance as the realistic nonlinear UUV behaviors. However, the robustness was not considered in early methods, which is critical to maintain UUV stability and control performance in the presence of uncertainties. Nowadays, to achieve faster, accurate and robust control performance, it is required to address modeling and control issues simultaneously. In the following, we briefly introduce the underwater vehicle controllers by two kind: model-free and model based controllers.

PID controller and its variations are the most classical underwater vehicle control algorithms. Even today, a lot of UUVs (equipped with high precision navigation and guidance devices) still use PID controllers, as PID is less complex and simple to be implemented. For example, in [START_REF] Mcphail | Autosub-1. a distributed approach to navigation and control of an autonomous underwater vehicle[END_REF] MacPhail et al. implements a PD controller on the Autosub-1 robot. In our paper [START_REF] Yang | Underwater vehicle modeling and control application to ciscrea robot[END_REF], the CISSAU AUV used PID controller for depth control in SAUCE 2014 and euRathlon 2014 competitions. Ostafichuk implement a hybrid PD controller on the Dolphin AUV in [START_REF] Ostafichuk | Auv hydrodynamics and modelling for improved control[END_REF].

The implementation of a PID controller only depends on tuning three gains: proportional, integral, and derivative gains. Proportional gain contributes to the response speed. Integral gain attenuates the static error, which affected the tracking performance. Derivative gain can reduce overshoot while keep the response fast. PID is a model-free controller, the implementation is a process of an iterative filed tuning of the three gains. However, according to the uncertain and nonlinear behaviors of UUV, PID control is insufficient in robustness. The disadvantages of PID are:

• PID gain tuning requires intuitive experiences.

• PID control stability can be affect by uncertain disturbance.

• The gains are fixed after tuning, controller is not adaptive to unpredict changes.

• PID is inefficient to handle nonlinearity, usually nonlinearity brings in response oscillations.

• The control output is not optimal, which waste onboard energy and shorten thruster lifespan.

For specific underwater vehicle applications, there exist many improved PID controllers, which consider the nonlinearity, the wave disturbance, and the current disturbance. In [START_REF] Perrier | Experimental comparison of pid vs. pid plus nonlinear controller for subsea robots[END_REF], Perrier et al. proposed an improved AUV PID controller that considers nonlinearity and disturbance attenuation. ODIN AUV applied a PID controller which involves shallow water wave disturbance rejection [START_REF] Liu | Nonlinear output feedback controller design for tracking control of odin in wave disturbance condition[END_REF]. Refines et al.

designed a PID controller robot which concerns current disturbance for Minesniper MKII [START_REF] Refsnes | Output feedback control of slender body underwater vehicles with current estimation[END_REF]. Based on the nonlinear model Mirhosseini builds a PID depth controller in [START_REF] Mirhosseini | Seabed tracking of an autonomous underwater vehicle with nonlinear output regulation[END_REF]. The above PID controllers show faster response and disturbance rejection ability. However, the performance is sensitive to the specific applications, the model accuracy and disturbance characteristics. In this thesis, PID is not considered to be enough to solve the nonlinear and robust issues of underwater vehicles.

LQG is a model based Multi Input Multi Output (MIMO) optimal state feedback controller. It can be considered as the combination of Kalman filter and a linear quadratic regulator. The linear optimal controller is derived under a quadratic cost function of minimal energy consumptions as well as other performance constrains.

LQG applications on underwater vehicles can be found in [START_REF] Sutton | Lqg/ltr control of an autonomous underwater vehicle using a hybrid guidance law[END_REF]. The advantage of LQG is the theoretically optimal and accurate control design. However, the disadvantages are, LQG is sensitive to model accuracy, meanwhile LQG is inefficient to handle nonlinearity. For example, the linearization of UUV model on equivalent point is not accurate, that makes the implementation of LQG controllers very hard.

Gain scheduling control is a model based linear feedback controller, first proposed by Silvestre in [START_REF] Silvestre | On the design of gain-scheduled trajectory tracking controllers[END_REF]. The objective of gain scheduling is to approximate the original nonlinear system by a group of linear systems obtained around many equilibrium points. For each linear system in the group, the control design is a regular linear feedback control problem. While operating on the plant, the group of controllers will switch among them according to the working states, i.e. equilibrium points. On each working region near the equilibrium points, gain scheduling can achieve very high performance, as if the nonlinear system is exactly the linearized system. However, the control stability and performance apparently rely on the model accuracy. The gain scheduling controller is sensitive to the uncertainties in the model and ocean environment. UUV gain scheduling application is given in [START_REF] Silvestre | Control of the infante auv using gain scheduled static output feedback[END_REF].

The performance of classical control methods are sensitive to the UUV model accuracy, i.e., they are inefficient to handle the model changes, such as parameter variations, external disturbances and sensor noise. In SAUC-E [START_REF] Sauc-E | Student autonomous underwater challenge-europe[END_REF] and euRathlon [START_REF] Eurathlon | Outdoor robotics challenge for land, sea and air[END_REF] competition, we found PID yaw controller is less efficient than robust control method for low mass AUV. However, realistic UUVs applications face a lot of uncertainties in the complex ocean environment, and the above mentioned control methods are not robust to those uncertainties. Hence, in the early 1981, Zames had proposed the ∞ control methods to solve the robust issues, guaranteeing both stability and high performance. ∞ control method constrains the energy gains from system input to output using ∞ norms. ∞ theory combines the advantage of MIMO state space description and frequency domain analysis. Actually, there is a long history, people begin to notice the importantance of sensitivity, i.e., disturbance rejection, using high gain feedback control to attenuate model variation effects (Single Input Single Output (SISO)). Then, in the 60s, the state space feedback control theory made many breakthroughs, and LQG seems to be theoretically perfect. However, LQG relies on the modeling accuracy. While LQG controllers are implemented in many industrial applications, they still suffer from low robustness performance. According to Doyle [START_REF] Doyle | Guaranteed margins for lqg regulators[END_REF], the LQG can not guarantee stability in presence of uncertainties. Note that, Doyle et al. made great contributions to simplify the robust controller synthesis process. They also proposed a synthesizing method using only two Riccati equations, which is simple and clear for spreading ∞ robust control applications.

Robustness means the controller achieves the performance specifications as well as robust stability in presence of uncertainties. In other words, inside all robust stable solutions, the robust control design searches for the optimal controller, which satisfies the performance specifications. The robust control theory can be found in [START_REF] Zhou | Essentials of robust control[END_REF][START_REF] Gu | Robust Control Design with MATLAB[END_REF]. In this thesis, linear ∞ robust controller was used. Besides, there are many types of robust controllers. For instance, the 2 and 2 robust controllers for nonlinear system, applications are given in [START_REF] Li | H2 active vibration control for offshore platform subjected to wave loading[END_REF][START_REF] Li | Optimal active control of wave-induced vibration for offshore platforms[END_REF]. Meanwhile, gain scheduling ∞ methods are developed for Linear Parameter Varying (LPV) systems. Importantly, the analysis of ∞ control are used to solve the conservative issues.

There are many robust control applications on UUVs. Pan has implemented an indirect robust controller for depth control on the REMSUS AUV [START_REF] Pan | Depth control of autonomous underwater vehicles using indirect robust control method[END_REF]. Roche designed a hierarchical ∞ controller for depth control of AsterX AUV in [START_REF] Roche | A hierarchical varying sampling ℎ ∞ control of an auv[END_REF], and a LPV ∞ controller in [START_REF] Roche | Lpv/ℎ ∞ varying sampling control for autonomous underwater vehicles[END_REF]. In [START_REF] Clement | Interval analysis and convex optimization to solve a robust constraint feasibility problem[END_REF], Clement proposed a robust controller for the VAIMOS robots. Reduced order ∞ controller for a torpedo-shaped AUV is shown successfully in [START_REF] Feng | Reduced order ℎ ∞ control of an autonomous underwater vehicle[END_REF]. The robust ∞ yaw controller of CISCREA AUV can be found in [START_REF] Yang | Underwater vehicle modeling and control application to ciscrea robot[END_REF]. Apparently, robust control methods show more robustness than other controllers. There are the advantages such as: control performance reduce the effect of model parametric uncertainty, i.e., require less expensive models; the control output is optimal, i.e., save onboard propulsion energy and expand thruster lifespan;

and controllers have high-frequency noise rejection ability [START_REF] Clement | Synthese multiobjectifs et sequencement de gains: application au pilotage d'un lanceur spatial[END_REF], etc. Nevertheless, the robust control methods have some disadvantages. First, to define the weighting functions for robust synthesis, it is necessary to have in advance the knowledge of the noise and disturbance properties, and the uncertain boundaries. Then, robust synthesis is currently a time consuming process, which is always offline. In addition, the synthesis results are sometimes high order controllers, and usually local optimal results (no guarantee for global answers).

Sliding mode control method is presented in [START_REF] Slotine | Applied Nonlinear Control[END_REF]. This control algorithm specializes in handling the uncertainties and nonlinearities of control systems. The sliding surface (an equation) is a critical notion that derived based on the tracking errors.

Then, the control outputs are calculated to move the system states towards the sliding surface, which keep the tracking errors to be infinitely small. Typically, there is the sliding mode control application on the Taipan AUV, presented by Vaganay et al. in [START_REF] Vaganay | Taipan an auv for very shallow water applications[END_REF]. Fossen also proposed an adaptive sliding mode controller in [START_REF] Fossen | Adaptive control of nonlinear underwater robotic systems[END_REF]. More applications can be found in [START_REF] Salgado-Jimenez | A robust control algorithm for auv: based on a high order sliding mode[END_REF][START_REF] Pisano | Output feedback control of an underwater vehicle prototype by higher-order sliding modes[END_REF][START_REF] Campa | Robust control of underwater vehicles: sliding mode control vs. mu synthesis[END_REF]. The advantage of sliding mode is its insensitivity to model accuracy and external disturbances. In contrary, the disadvantage are the complexity in implementation and its oscillation issues on the sliding surface (chattering effect).

Backstepping control is a nonlinear control algorithm based on Lyapunov method, which generates asymptotically stable controllers. It has the robustness property, as it is insensitive to the model parametric uncertainties. UUV backstepping applications can be found in [START_REF] Lapierre | Robust nonlinear path-following control of an auv[END_REF], and the theory is described in [START_REF] Khalil | Nonlinear systems third edition[END_REF][START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF].

Adaptive control changes its control parameters, according to the realtime parametric variations and uncertainties of the controlled systems. The control parameters adapts to the plant base on the principle of minimizing closeloop performance evaluations. There are two kind of adaptive control methods: direct methods and indirect methods. The direct methods evaluate the controllers' parameters, and directly change the controllers using the estimated results. In contrast, indirect methods use the estimated plant parameters to calculate required controller designs. Compare to robust control, adaptive control does not need a priori information about the bounds of uncertain parameters. Adaptive control is more flexible and concerned with control law changing themselves, meanwhile, robust control is fixed and guarantees control performance of the systems inside the priori knowledge. Accordingly, adaptive controllers have the advantages of robustness, and requires little about the UUV models.

However, there are the drawbacks and limitations. The control plants should not have high-frequency or high-amplitude parametric variations, as the adaptive algorithm attempts to meet the robustness specifications, the controller might not converge.

For UUV systems, the parameter variation is common, e.g., adding or remove a payload, changing working environment from fresh water to sea water, etc. Many adaptive control applications were shown successfully on real underwater vehicle tests.

In [START_REF] Li | An adaptive nonlinear controller for diving motion of an auv[END_REF] adaptive controller was used to control the diving process of UUV. In [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF][START_REF] Maalouf | From pd to nonlinear adaptive depth-control of a tethered autonomous underwater vehicle[END_REF] Maalouf introduced several adaptive controllers for the depth control of a mini ROV.

Many other UUV adaptive control application can be found in [START_REF] Zhao | Experimental sutdy on advanced underwater robot control[END_REF][START_REF] Antonelli | A novel adaptive control law for autonomous underwater vehicle[END_REF][START_REF] Fossen | Robust adaptive control of underwater vehicles: A comparative study[END_REF][START_REF] Fossen | Adaptive control of nonlinear underwater robotic systems[END_REF]. A clear review that summarized and compared the current UUV adaptive control applications is given by Antonelli in [START_REF] Antonelli | On the use of adaptive/integral actions for six degrees of freedom control of autonomous underwater vehicles[END_REF].

Outside the state space feedback control framework, artificial intelligence plays an important role in the control field. The ideas are to mimic either the neural structures or the evolutionary process, etc. There are the techniques like neural networks control, fuzzy logic control, and genetic algorithms, etc. Intelligent controls can handle nonlinearity and uncertainty issues by universal approximations, however, the calculation is generally very heavy. Neural network control basically involves two steps: system identification and control. The controller iteratively learns the weightings (links between neurons) of structured neural networks from a set of inputoutput data pairs. It is supposed to capture the dynamics of any system. Then, the system steady state can be used to control the nonlinear plant. The UUV application of a neural network control can be found in [START_REF] Shi | Adaptive depth control for autonomous underwater vehicles based on feedforward neural networks[END_REF][START_REF] Szymak | Control system of underwater vehicle based on artificial intelligence methods[END_REF]. Fuzzy logic control is common for nonlinear underwater vehicle control. Rather than fixing and exact reasoning, the logics approximate the value of input and output data between complete true and complete false. UUV fuzzy logic control applications are given in [START_REF] Chang | Model-based fuzzy modeling and control for autonomous underwater vehicles in the horizontal plane[END_REF][START_REF] El-Fakdi | Policy gradient based reinforcement learning for real autonomous underwater cable tracking[END_REF]. The genetic algorithm randomly searches the globe optimal controllers by the process of natural selections. Relevant genetic UUV control application was presented in [START_REF] Lamas | A hybrid approach for designing the control system for underwater vehicles[END_REF].

It is obvious that the above mentioned control methods all have their advantages and their disadvantages. This makes a good reason to develop hybrid control methods, which combines the advantages of different algorithms. In [START_REF] Fossen | Adaptive control of nonlinear underwater robotic systems[END_REF], Fossen presented a hybrid adaptive sliding mode controller for UUV. In [START_REF] Shi | Adaptive depth control for autonomous underwater vehicles based on feedforward neural networks[END_REF], a neural network control is mixed with adaptive algorithms. Other hybrid UUV control applications can be found in [START_REF] Wallace | Depth control of remotely operated underwater vehicles using an adaptive fuzzy sliding mode controller[END_REF][START_REF] Marzbanrad | A robust adaptive fuzzy sliding mode controller for trajectory tracking of rovs[END_REF].

For UUVs, it is worth to mention some other studies. For example, in [START_REF] Steenson | Experimental verification of a depth controller using model predictive control with contraints onboard a thruster actuated auv[END_REF] predictive control was used for the depth control of UUV. Like gain scheduling control, a feedback linearization method was proposed in [START_REF] Newman | Marine Hydrodynamics[END_REF][START_REF] Slotine | Applied Nonlinear Control[END_REF]. Smallwood et al. discussed the importance of model accuracy in the control design. Importantely, Jaulin had verified a robust Lyapunov controller using interval analysis algorithm on the VAIMOS USV [START_REF] Le-Bars | An experimental validation of a robust controller with the vaimos autonomous sailboat[END_REF].

In the underwater vehicle control field, a pressing need is to improve the robustness of UUV control methods. Potential trends are robust and fast controllers that can assist the pilot or autopilot with better accuracy. It is expected to see optimal propelling actions that save more battery power as well as increasing propeller lifespan. Regulated AUV motions are expected to be less complex for high level guidance, navigation and swarm research. Using AUV, the goal of this thesis is to inspect the underwater oil pipeline leaks, meanwhile, monitor the sedimental conditions. Robustness is the first essential issue to construct this long-term running AUV monitoring networks. The control performance determine the robustness and reliability of underwater vehicles, especially, for underwater environment, the uncertainties make the control design more difficult.

Hence, in order to improve the AUV maneuverability, we focus on the two critical issues: accurate model and advanced robust controller.

Although there are many modeling and control applications in the field, and some of them were verified successfully on realistic sea tests, the combination of computational modeling and robust control works are rare. In order to move the AUV design initiatives outside large organizations, i.e., to popularize the underwater vehicles to more civilian applications, we proposed a low-cost and more accessible modeling and control solution. Note that, it is always important to low the cost to build one AUV in order to create a huge network. In the following the major works and contributions of this thesis are presented.

First, according to the complex shapes of the CISCREA AUV and many others, we proposed the use of computational modeling methods, for the following reasons:

• Compare to the full-scale and scaled experiments, computational modeling re-quires no experimental devices. The cost is much lower, and the modeling process is faster.

• Compare to empirical formulas, the computational modeling method is more efficient to handle complex shapes. There is no complex geometry simplification process during the modeling.

• The accuracy and reliability of computational modeling method is decent. It is generally less accurate than experimental methods, but better than the empirical formulas for complex-shaped AUVs. As ∞ robust control is used, there is only the necessity to understand the uncertain bounds of the dynamic model rather than the exact nominal model.

• Computational modeling methods can replay many physical details, which is useful to locate the focus of control design.

• In this thesis, although the hydrodynamic experiments do not improve the modeling accuracy, instead, it guarantees the reliability of the computational results.

• The computational modeling results can be control oriented, which is convenient to build state space equations and easy to make simulations. It can greatly speed up the model analysis and even design work of high performance UUVs.

In this thesis, we build a dynamic and hydrodynamic model on CISCREA AUV using many computational softwares. And the CFD calculation results are compared with experimental data. The major work includes:

• Mass inertia matrix RB of CISCREA AUV is calculated using PRO/ENGINEER TM .

• Added mass matrix A is calculated using radiation/diffraction program MCC [START_REF] Newman | WAMIT user manual[END_REF] and WAMIT TM [START_REF] Mcc | Marine craft characteristics[END_REF].

• Hydrodynamic programs ANSYS-CFX TM and STAR-CCM+ TM are studied to predict damping behavior (||).

• The bollard thrusts of CISCREA AUV are measured. Several motion experiments are designed to verify the reliability of CFD results.

• Damping is found to be the major nonlinearity in the model. A model based linear compensation solution is proposed to linearize the AUV model for robust control design.

In order to handle the uncertainties, we proposed the linear ∞ control method to guarantee the stability and achieve expected control performance. The reasons for this method are:

• Due to the modeling errors and parameter variations, there are many uncertainties in the mathematical descriptions of the UUV dynamics. Especially, the damping is theoretically hard to precisely predict. Traditional control methods are sensitive to the model accuracy, however, ∞ robust control is efficient and insensitive to the modeling uncertainties.

• Robust control has disturbance rejection ability, which is important for the ocean environment that with unpredictable wind, wave and currents.

• Damping is the major nonlinearity in the dynamics. Linear ∞ theory can not directly adapt to the nonlinear model. However, the robustness allows the nonlinear compensation solution, which turns the nonlinear system into a linear system with uncertainties. Although the velocity estimation of damping compensation is not accurate, there is almost no reduce of the robust control performance.

• Sensor noise rejection is important to prevent unnecessary thruster propulsions.

Compare to the traditional control methods, robust control is very efficient to filter the noise with a prior knowledge.

• Signal delay effects the control stability and causes serious oscillations. By implementing the Smith compensation, we turn the delay problem into a negligible uncertainty that can be tackled using ∞ control algorithm [START_REF] Yang | Modeling of a complex-shaped underwater vehicle for robust control scheme[END_REF].

∞ robust control method is proposed to the control of low-mass and complexshaped CISCREA AUV and others. Without lose of generality, the robust heading controller was implemented and validated in the sea. The major work include:

• A yaw model derived for robust heading control design.

• Linear compensation proposed based on CFD modeling results, which yield a linear system with uncertainties.

• Linear ∞ controller synthesized and implemented on the CISCREA AUV for heading control.

• Kalman filters based on CFD model, which numerically estimates unmeasured as well as noisy states such as angular velocity.

• To handle the signal delay of the electronic compass, we proposed two delay compensation algorithms.

• Proposed yaw control experiments were implemented in the testing pool. Results are compared with PID solutions.

• We conducted several sea tests for the proposed robust yaw controller in Brest port (France) and in La Spezia port (Italy) respectively. Control performance are compared to PID controllers.

Thesis Outlines

This thesis is organized into 7 chapters. The reminder of the thesis are:

Chapter 2 AUV

Chapter 7 General Conclusion and Perspectives

Finally, conclusions are drawn in Chapter 7, and future directions of this thesis are given in the end.

Main Contributions

The main contributions of this thesis are the design and validation of an AUV ∞ robust control approach using numerical hydrodynamic modeling techniques. We address the numerical AUV dynamic modeling methods and the robust control algorithms simultaneously. For any low-mass and complex-shaped AUVs, the proposed modeling and control approach is demonstrated to have sufficient modeling accuracy and efficiency as well as control robustness and performance. Comparing to PID controllers, the AUV motion control performance can be improved in face of inevitable modeling and environmental uncertainties, dynamic nonlinearities, sensor delay and unmeasured system states. A low mass and complex-shaped CISCREA AUV is used to validate the proposed numerical modeling and ∞ robust control approach in real sea test. In particular, this thesis presents the following points:

• For complex-shaped and low-mass AUV, traditional modeling methods are generally insufficient to reveal the hydrodynamic characteristics. Therefore, the computational fluid dynamic method is suggested to avoid implementing hydrodynamic experiments using expensive equipments. We concluded several important AUV dynamic and hydrodynamic parameters for the AUV control design:

-Added mass matrix A is calculated and compared using radiation/diffraction program WAMIT and MCC.

-Hydrodynamic programs ANSYS-CFX and STAR-CCM+ are studied to predict damping behavior (||) on the surge, sway, and heave directions.

Rotational damping effects are numerically built using moving reference and overset mesh techniques in STAR-CCM+.

-Mass inertia matrix RB and the center of gravity are approximated using Computer Aided Design (CAD) PRO/ENGINEER (PRO/E).

Based on these numerical estimations, a four degree-of-freedoms model is derived for the CISCREA AUV, which provides the foundations for the hydrodynamic experiment and control design simulations.

• A 4 DOFs (Surge, Sway, Heave, and Yaw) experiment is designed to validate the CFD models and its hydrodynamic parameters. The numerical AUV modeling method is concluded efficient for the complex-shaped underwater vehicles (Using robust control scheme). The hydrodynamic experiment confirms the damping effect is nonlinear, and it is mainly caused by the quadratic damping.

• A model based nonlinear compensated ∞ control approach is proposed for the of CISCREA AUV and others to solve the following issues:

-Dynamic and hydrodynamic parametric uncertainties.

-External disturbance from the ocean environment and sensor noise.

-Nonlinear damping compensation errors.

The nonlinear damping and parametric uncertainty issues are the two critical problems that solved in the proposed approach.

• To solve the applicable issues in real sea AUV control applications, we proposed an improved nonlinear compensated ∞ control scheme to handle the sensor delay as well as unmeasured states. The proposed approach is validated as efficient in the CISCREA AUV heading control simulations, pool and sea tests.

The control performance of the proposed method are compared with traditional PID controllers. The mathematical description of underwater vehicle dynamics is essential for the robust control design. A model is necessary to analysis the dynamic and hydrodynamic behaviors of UUVs, i.e., the interactive physics between the underwater vehicle and fluid. In addition, the UUV model provides the foundation for the efficient simulations, which verify different control schemes including PID and ∞ robust controllers. The reminder sections describe the mathematical modeling process of CISCREA AUV.

Part

Modeling Introduction

Modeling of underwater vehicles involves two parts of study: kinematics and dynamics. Kinematics describe the motion of underwater vehicles without concerning the forces and torques that act on it. That is to say, kinematics describe only the geometrical relations of underwater vehicle position, velocity and accelerations. Dynamics present the accelerating motion of the underwater vehicles, especifically, consider the forces and torques. To be accurate, the dynamic study (kinetic) is divided into classical rigid body dynamics and hydrodynamics. There are many marine vehicles modeling articles and books, and typical descriptions can be found in [START_REF] Newman | Marine Hydrodynamics[END_REF][START_REF] Yuh | Modeling and control of underwater robotic vehicles[END_REF][START_REF] Leonard | Stabilization of steady motions of an underwater vehicle[END_REF][START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF][START_REF]Guidance and Control of Ocean Vehicles[END_REF].

In this chapter, 6 DOFs underwater vehicle dynamics are represented by the marine vehicle formulation proposed by Fossen, in [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF] and [START_REF]Guidance and Control of Ocean Vehicles[END_REF], using the Society of Naval Architects and Marine Engineers (SNAME) [1950] notions in [116]. For marine systems, usually two coordinate systems, Body Fixed Reference (B-frame) and North

East Down Reference (NED-frame) are introduced for convenience as presented by

Fossen in [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF] and shown in Figure 2

= [, , , , , ] T (2.1) = [, , , , , ] T (2.2) = [, , , , , ] T (2.3)
Here, denotes the position and orientation of the underwater vehicles in the NED-frame, denotes the linear and angular velocity with coordinates definitions in the B-frame, and is used to describe the forces and moments acting on the vehicle in the B-frame.

2-1. The last three coordinates of and , i.e., (, , ) and (, , ), represent respectively the orientation and rotational motion. NED-frame is fix to an absolute location on the earth, it can be considered as the internal reference for the B-frame.

The moving coordinate reference B-frame is fixed to the vehicle, which describes the relative underwater vehicle motion. B-frame origin is usually chosen to coincide with the Center of Gravity (CG) of the vehicle. To summarize, all suggest the kinematic rules are mainly the transformation relationship between the two coordinate references NED-frame and B-frame.

It is known that kinematic relation of velocity vector and position vector is expressed as the vectorial equation 2.4 [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]:

= (Θ) (2.4)
where, (Θ) ∈ R 6×6 , stands for a transformation matrix between B-frame and NEDframe, Θ = [, , ] T , is the vector of the euler angles on roll, pitch and yaw directions, The formulation of (Θ) is presented as equation 2.5:

(Θ) = ⎡ ⎣ (Θ) 0 3×3 0 3×3 (Θ) ⎤ ⎦ (2.5)
where, (Θ) ∈ R 3×3 is the linear velocity transformation matrix, and

(Θ) ∈ R 3×3
is the angular velocity transformation matrix. Assume that, (•) = (•), (•) = (•) and (•) = (•), then, we have respectively the (Θ) and (Θ) definitions in equation 2.6 and 2.7:

(Θ) = ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 0 sφ cθ cφ cθ ⎤ ⎥ ⎥ ⎥ ⎦ (2.6) (Θ) = ⎡ ⎢ ⎢ ⎢ ⎣ -+ + + -+ - ⎤ ⎥ ⎥ ⎥ ⎦ (2.7)
Note that (Θ) is not defined at the pitch angle of π 2 , which is called the singularity problem of euler angle representation. Consequently, there is an alternative representation of kinematic equations using quaternion. However, for most of the underwater applications such as all ASVs and our CISCREA AUV, it is not allowed to approach the singularity condition. Hence, the mathematical descriptions of the quaternion is only mentioned in the appendix A.2.

Depending on [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], rigid-body hydrodynamic forces and moments can be linearly superimposed. Therefore, the overall non-linear underwater model can be characterized by two parts, the rigid-body dynamic and hydrodynamic formulations (hydrostatics included). In the next section, underwater vehicle rigid-body dynamic is first introduced.

Rigid-Body Dynamic of Underwater Vehicles

Without concerning the hydrodynamics, the underwater vehicles are considered as a rigid body moving in the free space. That is to say, the underwater vehicle is rigid and 6 DOFs rigid-body dynamics obey the Newton-Euler formulation (Newton's second law) or alternatively Lagrangian Formulation (momentum and energy conservations).

In this section the translational and rotational motion of underwater vehicles are derived according to classical Newtonian mechanics. The derivations of equations can be found in the appendix A.3.

It is shown that the 6 DOFs rigid-body dynamic equations of underwater vehicle motion can be conveniently expressed as a vectorial form in equation 2.8:

RB + RB () = env + pro (2.8)
where, all the definitions are presented in Table 2 The above vectorial forces and torques are considered to act on the vehicle's center of gravity, and they are balanced, including: the inertia induced force and torque vector RB (Newton second law); the Coriolis force and torque vector RB (), which is caused by the rotation of the earth; thruster propulsion pro in 6 DOFs; and the external disturbance env like wind, waves and currents. For most of the underwater vehicles with hard structures, rigid assumption is common and efficient, which yields a simple formulation on a single mass point and eliminates the consideration of forces acting between individual elements of mass [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. Note that for the underwater robots with flexible structures or segments, like underwater snake robots, the dynamics are different, which is not mentioned in this thesis.

Rigid-body mass inertia matrix RB ∈ R 6×6 is a symmetric and positive definite matrix, which is defined in equation 2.9 and 2.10, where is the mass, 3×3 is the identity matrix, and

G = [ G , G , G ] T is the vector from b (origin of B-frame) to
CG. If G = 0, i.e., b ≡ , then the matrix RB will be simplified. ( b g ) is a skew-symmetric matrix according to the definition 2.2 in [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. Moreover, symmetric properties of cubic AUV, in = 0 and = 0 planes, can be used to simplify the inertia components to a rough diagonal form.

RB = T RB = ⎡ ⎣ 3×3 -( b g ) ( b g ) 0 ⎤ ⎦ (2.9) RB = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 G -G 0 0 -G 0 G 0 0 G -G 0 0 -G G x -xy -xz G 0 -G -yx y -yz -G G 0 -zx -zy z ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.10)
Here, 0 = T 0 > 0 is the inertia matrix, which is also symmetric and positive definite [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. The definition of 0 are given as equation 2.11:

0 = T 0 = ⎡ ⎢ ⎢ ⎢ ⎣ x -xy -xz -yx -y -yz -zx -zy -z ⎤ ⎥ ⎥ ⎥ ⎦ (2.

11)

RB ∈ R 6×6 is a skew-symmetric matrix, which contributes to the centrifugal force on the rigid body. Decompose RB as equation 2.12, then, we can derived the definition of RB in equation 2. [START_REF] Eurathlon | Outdoor robotics challenge for land, sea and air[END_REF].

RB = ⎡ ⎣ 11 12 21 22 ⎤ ⎦ (2.12) () = ⎡ ⎣ 0 3×3 -( 11 1 + 12 2 ) -( 11 1 + 12 2 ) -( 21 1 + 22 2 ) ⎤ ⎦ (2.13)
Here we know, RB depends on the mass inertia matrix RB , and the translational and rotational velocities, 1 = [, , ] T , 2 = [, , ] T , of the underwater vehicles.

Matlab function "m2c.m" is a practical way to calculate the RB using RB and information, which is introduced in Marine System Simulator (MSS) in [START_REF] Mss | Marine systems simulator[END_REF]. In our case, RB are neglected, because the underwater vehicle speed is low enough to be considered, RB ≈ 0.

Hydrodynamics of Underwater Vehicles

The ocean environment is complex, and the hydrodynamics play an import role in the control design of underwater vehicles. Actually, the hydrodynamic effects make the control design very difficult, which separate the underwater control design from air and land robots. In the following it is introduced the different types of hydrodynamic forces and torques that the underwater vehicles would encounter in the ocean:

• Radiation Induced Forces (neglecting second order coupling terms):

-Added Mass due to the inertia of the surrounding fluid.

-Hydrodynamic Damping due to energy consumption of surrounding fluid.

-Restoring Forces due to Archimedes (weight and buoyancy).

• External disturbance: wind, wave and current.

• Thruster propulsions

As mentioned in [START_REF] Newman | Marine Hydrodynamics[END_REF] and [START_REF] Faltinsen | Sea Loads on Ships and Offshore Structures[END_REF], it is common to assume that the hydrodynamic forces and moments on a rigid body can be linearly superposed by considering two sub-problems [START_REF] Faltinsen | Sea Loads on Ships and Offshore Structures[END_REF]. Therefore, the underwater vehicle hydrodynamic force and torque vector, hydro ∈ R 6×1 can be superposed to the rigid-body dynamic equation 2.8, and as a result, derives the equation 2.14.

RB + RB () = env + hydro + pro (2.14)
It is presented in the Fossen marine vehicle model [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], the total acting hydrodynamic of the underwater vehicle, i.e., hydro , can be decomposed into the sum of sub-hydrodynamic effects in equation 2.15:

hydro = -A -A () -(||) -() (2.15)
Here, the definitions of the various types of hydrodynamic effects in equation 2.15 are given in Table 2.3.

Table 2.3: Nomenclature of the notations of hydrodynamics

Parameter Description A ∈ R 6×6 Added mass matrix A ∈ R 6×6 Added mass induced coriolis-centripetal matrix (||) ∈ R 6×6 Damping matrix () ∈ R 6×1
Restoring forces and moments vector hydro ∈ R 6×1

Vector of hydrodynamic forces and moments linear and angular velocity vector Actually, the hydrodynamic effects of the fluid acting on the marine structure can be divided into two kind: radiation forces and diffraction forces. For underwater water vehicles, as the vehicle size is usually small, it is common to avoid diffraction forces, and concerns mainly the radiation forces. In our case, diffraction forces of wind, wave and current effects are discussed in the reminder thesis. In the following, we first presented the three major radiation induce forces: restoring force (), added mass A + A (), and hydrodynamic damping (||).

Restoring Force

Underwater vehicles are constrained by two hydrostatic forces: gravitational and buoyant forces, i.e, ∈ R and ∈ R , which are called restoring forces. Generally, the floating equilibrium is achieved by and is equal and working on the same straight vertical line. For an underwater vehicle with neutral buoyancy, the weight is approximately equal to the buoyancy force ( is slightly larger). In equation 2.16, it is presented the oldest engineering result, the gravity law and Archimedes law.

= , = ∇ (2.16)
Here, is the gravity acceleration, is the fluid density, and ∇ is the displaced fluid volume.

For the underwater vehicles, it is common to design the vehicles to be neutrally buoyant, which saves the floating energy and keep the vehicle safe from sinking. As pointed out by Fossen, [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], the restoring forces and moments vector () ∈ can be simplified into equation 2.17, if the underwater vehicle is neutrally buoyant (full definition of the restoring forces and moments vector () is given in the Appendix A.6).

In equation 2.17, = [ x , y , z ] T is the distance from the CG to Center of Buoyancy (CB). Actually, () is a function of the vehicle's position and attitude, it decides the stability of the underwater vehicle. When the AUV is driven away from the equilibrium states under external disturbance, () can provide the torques to return to the equilibrium states. It is like a spring system, the buoyancy center CB is always higher than the center of gravity CG, which can generate a torque to oscillate back to equilibrium. For CISCREA AUV, CB and CG can be located using trial and error method on adding and removing the payload and floats.

() = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 -y + z -z + x -x -y ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(2.17)

Added Mass

Added mass forces and moments, i.e., { A + A ()} ∈ R 6 , are induced by the surrounding fluid inertia. It is common to separate the added mass forces and moments in terms which belong to an added mass matrix A ∈ R 6×6 and a matrix of hydrodynamic Coriolis terms denoted A ∈ R 6×6 . The added mass matrix A should be understand as 'pressure-induced' virtual conception which are proportional to the acceleration of the body [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. For underwater vehicles, added mass A in some directions are even larger than the mass and inertias of the rigid body on the 6 DOFs.

To derive the expressions for these added mass matrix, an energy approach in terms of Kirchhoff's equations will be used [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. Alternatively, we can depict the added mass forces and moments provide the energy to make an empty space from the fluid ahead of the underwater vehicle, and fill the space by fluid in the tail of the robot, in order to make the vehicle accelerating forward. The added mass matrix A is defined in equation 2.18.

Generally, the added mass matrix A ∈ R 6×6 is positive definite (fully submerged), and the diagonal elements of the matrix are positive. Only in a few cases, the diagonal elements are negative, which can be found in [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. In real fluid the 36 elements of A may all be distinct, and experience has shown A = T A is actually a good approximation in (Wendel 1956) [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. For underwater vehicles, as the vehicle speed is very slow, the off-diagonal elements of the added mass matrix A induced negligible effects compared to the diagonal elements. For most practical applications, the off-diagonal elements of A will be small compared to the diagonal elements.

Therefore, in most of the underwater vehicles, we can use the diagonal form of the added mass matrix A , which is described in equation 2.19.

f ull A = - ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u v ẇ ṗ q ṙ u v ẇ ṗ q ṙ u v ẇ ṗ q ṙ u v ẇ ṗ q ṙ u v ẇ ṗ q ṙ u v ẇ ṗ q ṙ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2.18) A = -  u v ẇ ṗ q ṙ (2.19)
The added mass matrix A ∈ R 6×6 can be derived by applying potential theory, under the assumption of inviscid fluid, no circulation and that the body is completely submerged in an unbounded fluid [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. Meanwhile, the A parameters can be numerically calculated by applying strip theory. The strip theory divides the submerged surface of the underwater vehicle into infinity strips or pieces, and sum all the hydrodynamic coefficients on each pieces together. In the next chapter, numerical softwares are introduced to calculated the added mass parameters of CISCREA AUV.

As mentioned before, the added mass corilois matrix A () ∈ R 6×6 can be neglected in front of RB () ∈ R 6×6 , hence, the definitions of A () is only mentioned in appendix A.5 in case of high speed underwater vehicle applications. As presented in [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], in real fluid the 6 DOFs damping of underwater vehicles are highly nonlinear and coupled, it is difficult to separate these effects. Therefore, the Fossen model assumes that high order damping effects can be neglect, and practically proposes that different damping terms contribute to either linear or quadratic damping. Then, the damping decomposition equation (2.21) is introduced,

Damping

(||) = + n (||) (2.21)
where, ∈ R 6×6 is the linear damping matrix and n (||) ∈ R 6×6 is a quadratic damping matrix. According to [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], if the underwater vehicle's velocities are sufficiently high can be neglected. Otherwise, n (||) is negligible. Assume the damping elements is not coupled, i.e., off-diagonal elements are negligible, then, the damping matrix (||) ∈ R 6×6 can be simplified into a diagonal form in equation 2.22

(||) = -  u v w p q r  -  u|u| || v|v| || w|w| || p|p| || q|q| || r|r| ||  (2.22)

Environmental Disturbances

The marine disturbances, such as the wind, waves and current contribute to the environmental disturbances env ∈ R 6×1 . Generally, environmental disturbances env is a large part of the underwater vehicle dynamic uncertainties. It is clear that predictions of the ocean wind, wave and current can improve the AUV modeling and control design quality. Therefore, the models of environmental disturbances at real sea are discussed in [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF].

Actually, the wind, wave and current of the ocean are really complex, including additive and multiplicative types of random disturbances. However, in practice a good assumption according to [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF] are, the wind, wave and currents are considered to be linearly superposed for marine vehicles, which separates the effects into linear components.

First, the current plays major role and has many different types. Wind generated currents are induce by the interaction of atmosphere and ocean surface. Then, thermohaline currents are generate by the thermal and salinity balancing process of difference ocean regions. Thermohaline currents varies a lot and result in the distinct properties of currents in different ocean regions. The tidal components of the ocean circulation also induced obvious random currents, which can be up to 2 -3 / in some region. To summarize, the current disturbances must be considered in the control design of underwater vehicles.

Then, we should notice that for an underwater vehicle in deep water, wind and waves are considered to be less important than current disturbance. In contrast, for most shallow-water underwater vehicle control designs, the wind and wave effects are considered to be disturbances adding to the dynamics. Note that, wind and wave are indeed critical for the control design of shallow water underwater vehicles, such as [START_REF] Lapierre | Underwater robots part ii: Existing solutions and open issues[END_REF]. The modeling process of ocean wind and waves can be found in [START_REF]Guidance and Control of Ocean Vehicles[END_REF][START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF].

Thruster modeling

The dynamics properties of the underwater vehicles thrusters are highly nonlinear and complex [3]. To achieve better control performance, it is advised to get an accurate thrust model in order to adequately map the required thrust to the propeller's rotational speed [START_REF] Blanke | Dynamic model for thrust generation of marine propellers[END_REF][START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF]. In order to derive an accurate thruster model, it is necessary to precisely measure the flow velocity to [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF]. Currently, one of the main obstacles is there exist no efficient instruments to measure this flow velocity. Therefore, most control designs for underwater vehicles tend to neglect the thruster dynamics. It is common to treat thruster modeling errors as unknown disturbances in [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF]. In this section, we will present the thruster mathematical model, more underwater vehicle thruster modeling articles can be found in [START_REF] Maalouf | Contribution to nonlinear adaptive control of low inertia underwater robots[END_REF], [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF].

According to the Fossen model, the thruster force and torque vector pro ∈ R 6 is a nonlinear function of the linear and angular velocity vector ∈ R 6 , and propeller revolution vector ∈ R p ( 6), which is presented in equation 2. [START_REF] Bovio | Autonomous underwater vehicles for scientific and naval operations[END_REF].

pro = (, ) (2.23)
First, Blanke derives a 1st-order approximation of the induced thrust and torque for a single screw propeller as equations 2.24, 2.25, and 2.26 [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF].

= 4 T ( 0 )||( 0 ) (2.24) (, a ) = |u|u || + |u|Va || a (2.25) (, a ) = |u|u || + |u|Va || a (2.26)
Here , p denotes the propeller diameter, the water density and a the advance speed at the propeller (speed of the water going into the propeller). 0 is the advance number and T is the thrust coefficient present in equations 2.27 and 2.28

0 = a p (2.27) T = 1 + 2 a p (2.28)
Although the forward and backward thrusts will be non-symmetrical, for most of the applications it is efficient to make a symmetric assumption [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. Meanwhile, in most practical applications the bilinear model can be approximated by an affine model, that is a system which is linear in its input, as equation 2.29.

pro = (2.29)
In the reminder of this thesis, we will consider the affine model with uncertainties is enough for the following underwater vehicle robust control design.

6 DOFs Underwater Vehicle Model

In order to put forward the 6 DOFs dynamic and hydrodynamic underwater vehicle model, i.e., equations 2.14 (NED-frame) and 2.15(B-frame), in the same inertial coordinate, the previous mentioned kinematic transformation -1 (Θ is used to derived equation 2.30 (* stands for the variables in NED-frame):

* + * (||)( ) + * () = * pro + * env (2.30)
Here, the definitions are given in the following equations 2.31, 2.32, 2.33, 2.34, 2.35, and 2.36.

= RB + A (2.31) * = -T (Θ) -1 (Θ) (2.32) * (||) = -T (Θ)(||) -1 (Θ) (2.33) * () = -T () (2.34) * pro = -T * pro (2.35) * env = -T * env (2.36)
It should be notice that, the added mass matrix A and the damping matrix (||) are two major hydrodynamic components of underwater vehicles. They are generally difficult to obtain with good precision. In the next chapter, the hydrodynamic parameter identification work will focus on the identification of added mass matrix A and damping matrix (||).

Conclusion

In this chapter, the mathematical representation of the dynamic and hydrodynamic underwater vehicle model is given base on the Fossen marine vehicle formulations.

First, the kinematics, i.e, the transformation rules between body-fix reference and inertial reference, are discussed. Then, the kinetics including classical rigid-body dynamic and hydrodynamic components are presented assuming all parts can be linearly superposed. Furthermore, the environmental disturbance and thruster model are given in the end. Finally, the 6 DOFs underwater vehicle model is presented in the inertial reference. To summarize all, this chapter provides the mathematical foundation for the identification and control design of underwater vehicles.

Chapter 3

Numerical CISCREA AUV Model This chapter is dedicated to describe the CISCREA AUV model as well as its major dynamic and hydrodynamic parameters: Mass inertia matrix RB , added mass matrix A and damping matrix (||). Due to the complex structure, traditional empirical formula approximation is not as efficient as it performes on the slender shaped AUVs [START_REF] Rentschler | Modeling and control of an odyssey iii auv through system identification tests[END_REF]. Meanwhile, full-scale and scaled hydrodynamic experiments are generally expensive [START_REF] Rentschler | Modeling and control of an odyssey iii auv through system identification tests[END_REF][START_REF] Ferreira | Modeling and control of trimares auv[END_REF][START_REF] Ferreira | Hydrodynamic modeling and motion limits of auv mares[END_REF]. Therefore, to solve the modeling problems of complex-shaped underwater vehicles, we propose to numerically derive the hydrodynamic parameters using CFD software. In order to calculate each hydrodynamic parameter of the CIS-CREA AUV for the robust heading control design, the following CFD software is used in this section:

• Mass inertia matrix RB and the center of gravity are approximated using CAD software PRO/E TM . PRO/E TM was chosen for its convenient interface with hydrodynamic programs ANSYS-CFX TM and STAR-CCM+ TM , which are used for advanced hydrodynamic behavior analysis of the underwater robots.

Hydrodynamic analysis applications using PRO/E TM and ANSYS-CFX TM on AUVs and ROVs can be found in [START_REF] Yang | Modeling of a complex-shaped underwater vehicle for robust control scheme[END_REF][START_REF] Chin | Modeling and testing of hydrodynamic damping model for a complex-shaped remotely-operated vehicle for control[END_REF].

• Added mass matrix A is calculated using radiation/diffraction program MCC and WAMIT TM . The approximations of CISCREA AUV added mass parameter given by MCC and WAMIT TM are compared. WAMIT TM is a leading commercial program in the offshore industry, this program is well proved in the hydrodynamic analysis of offshore platforms and vessels. In contrast, MCC is an open source program that is dedicated to the numerical analysis of marine objects added mass parameters.

• Hydrodynamic programs ANSYS-CFX TM and STAR-CCM+ TM are studied to predict damping behavior (||) on the direction of surge, sway, and heave.

Note that rotational damping effects are numerically built using moving reference and overset mesh techniques in STAR-CCM+ TM .

Rigid-body Mass Inertia Matrix

Due to different density components, the inertia tensor 0 matrix (equation 3.2) and the coordinates of the center of gravity CG of CISCREA AUV can not be extracted using the direct calculation of equation (3.1), see [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF].

=  V m , =  V 2 m (3.1)
Here, m is the density of the related elementary volume , is the total volume of the body, is the distance between volume element and CG, is the robot mass, 0 is the inertia tensor referred to the B-frame (and its 9 elements are the moments of inertia about the three B-frame axes, for more details about the underwater vehicle rigid body derivations, please check appendix A.3).

we used the CAD software PRO/E. First, the CISCREA AUV is disassembled into many large pieces, and measured in size and weight, as shown in Figure 3 In Figure 3

: • 2 ). RB = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 15
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.4)
By neglecting the small off-diagonal inertia elements, the above matrix can be simplified to be a diagonal matrix in equation 3.

Added Mass Matrix

It is mentioned in [START_REF] Perrault | Sensitivity of auv added mass coefficients to variations in hull and control plane geometry[END_REF], that variations of underwater vehicle geometry play indeed a role on the added mass matrix A ∈ R 6×6 of the underwater vehicles. For a torpedo shaped AUV, empirical formulas prediction is validated in [START_REF] Rentschler | Modeling and control of an odyssey iii auv through system identification tests[END_REF] and [START_REF] Ferreira | Hydrodynamic modeling and motion limits of auv mares[END_REF] by approximating the hull to an elementary ellipsoid or slender shape. However, the empirical formulas predicting is inaccurate for a complex-shaped AUV, such as cubic shape CISCREA. Note that, CISCREA AUV is a typical cubic shaped underwater vehicle that has various sharp and uneven outlines as well as an open framework. In order to estimated the added mass matrix A of CISCREA AUV, the computational fluid dynamic software WAMIT [START_REF] Newman | WAMIT user manual[END_REF] and MCC [START_REF] Mcc | Marine craft characteristics[END_REF] are introduced in this section.

WAMIT is a leading computational software in the offshore oil industry. It is developed for analyzing wave interactions with offshore platforms, vessels and other structures. WAMIT utilizes the potential theory, and it calculates the fluid effects on each small panels of the structures that is divided by the strip method. WAMIT provides a variety of functions, including the functions to calculate added mass, damping and static center of gravity and buoyancy, etc.

MCC is developed by ENSTA Bretagne for calculating the hydrodynamic characteristics of marine objects. It uses the same strip method as WAMIT does, which slices the robot into a large amount of connected small surfaces. Each individual surface pieces are superposed to obtain the total fluid effects on the marine robot.

One important focus of MCC is to calculate the added mass of submerged vehicles.

In this thesis, WAMIT and MCC results are compared in order to guarantee the reliability of the added mass matrix A calculations. We should mention that, WAMIT is widely used in the offshore industry applications, and it gains widespread recognition for its ability to analyze complex structures with a high degree of accuracy and efficiency. Compare to MCC, WAMIT is considered to be more reliable as it has far more successful industry applications. Therefore, the objective of MCC comparison is to verify the configurations and input files of the WAMIT calculations are correct. In the following sections, we introduce the calculation procedures of underwater vehicle added mass using WAMIT and MCC.

WAMIT Calculating CISCREA Added Mass

In this section, we introduce the basics of using WAMIT analyzing the added mass matrix A of the underwater vehicles, especially the CISCREA AUV. For reference, the WAMIT manual is given in [START_REF] Newman | WAMIT user manual[END_REF], and an application approximating added mass matrix of a ROV is presented in [START_REF] Eng | Added mass computation for control of an open-frame remotely-operated vehicle: Application using wamit and matlab[END_REF]. In addition, MSS provides several Floating Production, Storage and Offloading (FPSO) applications using WAMIT in [START_REF] Fossen | Description of mss vessel models: Configuration guidelines for hydrodynamic codes[END_REF][START_REF] Mss | Marine systems simulator[END_REF][START_REF] Perez | An overview of the marine systems simulator (mss): A simulink toolbox for marine control systems[END_REF].

First, the principal input files for a WAMIT calculation are: Actually, there are two options to represent the potential and geometries of the underwater vehicles to be analyzed: the high order method and low order method.

•
The high order method allows to use B-Splines to represent the velocity potential and pressure continuously on the body surface, and to represent the geometry of the body surface in various manners. Alternatively, the low order method is less accurate than the high order method for most of applications, and it has more number of unknowns. In early WAMIT versions, the low-order method was used, where the geometric form of the submerged body surface is defined by flat quadrilateral elements, and the velocity potential or source strength solutions are assumed to be constant on each panel. As our robust control application is not sensitive to the slight compromise of added mass modeling accuracy, we chose low-order method to reduce the calculation complexity. In this thesis, MULTISURF (or Rhino 3D) was considered to represent the CISCREA AUV outline using the low-order method. Figure 34shows the geometry of the CISCREA AUV represented as a set of quadrilateral surfaces, and the direction of those surfaces all pointing out of the underwater vehicle. Note that, simplification of the AUV model are important to obtain reliable results including removing sharp corners and complex curves (For example , the sharp corner between the connections of the CISCREA framework and supports, or the complex thruster blade geometries).

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.7)
According to [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], the off-diagonal elements of the added mass matrix A are also negligible for slow vehicle. For CISCREA AUV, the moving speed in all the directions are less than 2 knots, which is considered to be slow. Hence, by neglecting small off-diagonal elements, the above matrices can be simplified into diagonal matrices respectively in equation 3.8, 3.9. The coupling between translational and rational directions are assumed to be negligible.

Apparently, the added mass prediction from WAMIT TM coincides with MCC to some extent. This result confirms the efficiency of WAMIT TM and the possibility of MCC to calculate the added mass matrix for complex-shaped underwater vehicles. In next chapter, the numerical added mass results are validated in moving experiments.

Furthermore, for robust control scheme, the numerical calculation errors of WAMIT and MCC added mass results can be considered to be mass-inertia uncertainties.

We should mention that, the added mass matrix A has indeed an important hydrodynamic effect for the control design. As it is obviously shown the added mass elements of CISCREA AUV on the surge, sway and heave directions are respectively 11.985 , 20.261 , and 67.141 , some of them are even larger than the mass of the AUV 15.643 . Similar conclusion can be yielded for the inertia parameters.

This indicate the added mass effect can not be ignored during the AUV accelerating process.

Damping Approximation

As mentioned in chapter Therefore, in this section computational fluid dynamic software ANSYS-CFX and STAR-CCM+ TM are introduced to estimate the damping matrix (||) for the complex-shaped CISCREA AUV. We proposed two CFD software in order to improve the reliability of the numerical CFD results. Configuration errors can be avoid from the numerical calculations stage, that provides more confidence to conduct a realistic hydrodynamic and control experiment.

Among the damping analysis methods that mentioned in chapter 2, we select numerical solutions, i.e. the CFD software ANSYS-CFX and STAR-CCM+ for underwater vehicle damping analysis for the following reasons:

• CISCREA AUV has complex-shaped outlines as well as an open frame structure. Empirical formulas are inefficient and inaccurate to estimate the damping matrix (||) for CISCREA AUV, as it requires complex model simplifications that may effect the estimation accuracy.

• Full-scale and scaled experiments are expensive, and they are usually one-shoot results. Comparing to the CFD software, which allows easily the modifications of the physical details, the experiment efficiency is not flexible for low-cost and small-size underwater vehicles.

• Although the damping analysis accuracy of CFD software is not as high as the experimental hydrodynamic results, generally the CFD solutions are sufficient for control design using a robust control scheme. Actually, the robust control scheme requires only the uncertain boundaries instead of an accurate damping result.

In the following, the damping analysis work of CISCREA AUV are divided into five parts:

• Morison equation is introduced to identify the main contribution from linear and quadratic damping effects of the underwater vehicles.

• ANSYS-CFX is used to approximate the relationship among damping forces and vehicle velocities on the translational directions of surge, sway and heave.

• STAR-CCM+ TM are used to estimate the relationship among damping forces and vehicle velocities on the translational directions of surge, sway and heave.

• The rotational damping effect on the AUV yaw direction is analyzed by the moving reference technique of STAR-CCM+ [START_REF]Cd-adapco user guide[END_REF], which is relatively simple and efficient to simulate.

• Finally, the numerical damping estimation (||) of CISCREA AUV using ANSYS-CFX and STAR-CCM+ are compared.

Morison Equation Analysis

As mentioned in previous chapters, four elements contribute to the damping matrix given by equation 3.11:

v (||) = S (||) + W (||) + M (||) (3.11)
According to [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF] According to Fossen's damping model (see equation 3.12), it is assumed that high order damping effects can be neglected, and low order damping effects are not coupled.

(||) = + n (||) (3.12)
Here, ∈ R 6×6 is the linear damping matrix and n (||) ∈ R 6×6 is the quadratic damping matrix, defined as follows:

(||) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ || T 1 || T 2 || T 3 || T 4 || T 5 || T 6 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.13)
Here, velocity vector is defined as For AUV modeling and control design, the critical issue is that both skin friction damping S (||) and vortex shedding damping M (||) contribute to the linear and quadratic damping matrix simultaneously. In this case, it is difficult to identify the dominate effect. Therefore, we used Morison formula to roughly determine the damping behaviors of CISCREA AUV. Morison damping formula ( ) is presented in equation 3.14 [START_REF] Comolet | Mécanique expérimentale des fluides[END_REF]:

|| T = [||,
( ) = - 1 2 d D ( n ) | | (3.14)
Here, is the vehicle velocity, d is the fluid density, is the cross-sectional area projected on the fluid and D ( n ) is the damping coefficient, which is a function of

Reynolds number n . n = d CL is (3.15)
where is is the fluid viscosity and CL is the characteristic length, which is the lenth of the object that the flow is going through or around. Generally, D ( n ) of spheres and cylinders are obtained by experiments of various moving velocities and data fitting.

In our application, the fluid is considered to be sea water, therefore, its is is 1.56 × 10 -6 /( • ) at 5 ∘ C and salinity 3.5%, density d = 1023/ 3 [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]. CIS-CREA generally operates at a speed range from 0 to 1/ (0 to 2 knots) in any translational direction. Meanwhile, the robot characteristic length is approximately 0.5 . In this case, the n of CISCREA is generally around 10 8 to 10 9 . According [START_REF] Comolet | Mécanique expérimentale des fluides[END_REF], in the standard damping experiments of spheres and cylinders, there should be a distinct switch between the laminar and the turbulent flows when the Reynolds number is between 10 5 to 10 6 , i.e. the critical area, and damping coefficient D ( n )

becomes instable (damping coefficient D ( n ) chart with respect to n is given in appendix A.9). On the contrary, the damping coefficient D ( n ) converges to a constant value, when the Reynolds number is much higher than 10 6 or far from the critical area. In our case, we can assume the CISCREA AUV as a cylinder shape of 0.5 radius. Then, the damping coefficient D ( n ) of the underwater vehicle can be approximated by a constant D . In this case, the damping effect ( ) becomes quadratic with respect to the vehicle velocity , see equation (3.16).

( ) = - 1 2 d D | | (3.

ANSYS-CFX Damping Analysis

The damping analysis of underwater vehicles using ANSYS-CFX CFD software can be divided into five steps:

• Create the simplified 3D model of the underwater vehicle as well as the water tank for hydrodynamic analysis

• Mesh the water tank and the underwater vehicle for the FEM (Finite Element Method) calculation

• Set the boundary and stop conditions for the FEM calculation

• Execute iterative FEM calculations until the convergence

• Post-process the FEM data to obtain the concerned hydrodynamic parameters First, simplify the closed surface 3D underwater vehicle model is essential for the CFD analysis. As shown in figure 345678, the CISCREA AUV has a complex shape.

For instance, the thrusters, LED lens, battery group, and the recovery handle are neglected in the CFD damping analysis. Usually, the geometrical details are removed Setting the boundary conditions is essential for the damping analysis. For CIS-CREA AUV, the turbulence model is chosen as the -(Shear Stress Transport) equations, which represents the separation phenomena on the boundary layer from laminar to turbulence flows [START_REF] Eng | Added mass computation for control of an open-frame remotely-operated vehicle: Application using wamit and matlab[END_REF]. Then, the water tank boundary is chosen to be free slip wall that reduces the space constrains as much as possible. Furthermore, the submerged surface of the CISCREA AUV is considered as non-slip wall having roughness parameters using PVC materials. The fluid injection speed varies from 0 to 0.5 /, and the turbulence intensity coefficient of the inlet fluid boundary is set to 1%. The outlet fluid boundary is configured to be a static pressure 0 .

Actually, the initial conditions do not effect the convergence of the CFD calculation, but they impact on convergence speed. In order to speed up the damping calculation, the initial fluid speed are usually given equal to the speed at the inlet fluid boundary.

The convergence criterions and time step are used to determine the solver stop behaviors. For CISCREA AUV, the convergence criterion is set to be 10 -4 . To achieve different accuracy specifications, more sophisticated configurations can be found in [START_REF] Eng | Added mass computation for control of an open-frame remotely-operated vehicle: Application using wamit and matlab[END_REF]. To summarize, the configurations or parameters that used in ANSYS-CFX are presented in Tab 3.3. The post-process stage of ANSYS-CFX analyzes the CFD data, and yields some information: the damping forces and moments, the velocities, and the pressures, etc. Figure 3-11 presents the stream-line view of the fluid velocities surround the

STAR-CCM+ Rotational Damping Analysis

In [START_REF] Yang | Modeling of a complex-shaped underwater vehicle[END_REF], both ANSYS-CFX and STAR-CCM+ are employed to calculate translational damping coefficients for the damping matrix (||) in surge, sway and heave directions. The rotational damping coefficients are not approximated using CFD solutions, as it is generally hard to predict precisely the damping effects of roll, pitch, and yaw directions.

For rotational motions, the CFD calculation are considered to be difficult, and inappropriate for damping analysis of underwater vehicles. Floc'h in [START_REF] Floc | Prédiction de trajectoires d'objets immergés par couplage entre modèles d'écoulement et équations d'euler-newton[END_REF] shows that the re-mesh calculation is too expensive even for 2D rotational simulation. Each calculation cycle requires more than 24 hours to be done. In this thesis, we use the move reference technique of STAR-CCM+ to approximate the rotational damping of CISCREA AUV [START_REF] Yang | Modeling of a complex-shaped underwater vehicle for robust control scheme[END_REF][START_REF]Cd-adapco user guide[END_REF]. The idea is to estimate rotational damping torque and angular velocity relationships with less expensive calculation.

In STAR-CCM+, three ways exist to analyze rotational motion [128, 125]:

• Moving reference method

• DFBI (Dynamic Fluid Body Interaction) or similarly rotation motion method

• Overset mesh method

Moving reference approach converts an unsteady motion problem into a steadystate problem by imposing a moving frame on a static mesh. In this case, steady fluid is available to solve fast the rotational problems. Rotation motion or DFBI involves actual displacement of regions. Moving regions and static regions exchange physics among contacting interfaces. Overset mesh method applies real time displacement between background mesh and overlap mesh. It is a recent mesh technique widely implemented on the complex motion analysis applications in free space. In every calculation iteration, overset mesh interacts with background mesh through overset boundaries. The last two methods bind to implicit unsteady fluid, and calculation load is generally heavier than in the moving reference method. Therefore, in our work, moving reference method is selected for rotational damping simulation. To achieve involves the translational and rotational motions in the surge, sway, heave and yaw directions.

First, the 4 CFDs CISCREA AUV model is given in equation 3.17.

( RB + A ) + (||) + () = env + pro (3.17)
Here, the mass inertia matrix RB is approximated by the CAD software PRO/E, as shown in equation 3.18 (mass: , moment of inertia: The damping matrix (||) of CISCREA AUV is estimated using the STAR-CCM + (equation 3.20):

• 2 ). RB = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 15
(||) = + n (||) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.2406 0 0 0 0 0.4512 0 0 0 0 0 0 0 0 0 0.0013 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + {||} T ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 25
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3.20)
The propulsion pro forces and torques of the CISCREA AUV of equation 3.17

Conclusion

In this chapter, we employed a variety of numerical methods and software to identify CISCREA AUV's major dynamic and hydrodynamic parameters: mass inertia matrix RB , added mass matrix A and damping matrix (||). Based on the estimation or calculation of numerical parameters, a four degree-of-freedoms model is derived for the CISCREA AUV, which provides the foundations for the hydrodynamic experiment and control design simulations.

In our numerical analysis of CISCREA AUV added mass matrix A , the WAMIT and MCC results show that the mass and inertia elements of the added mass matrix As it is mentioned before, numerical calculation and simulations are insufficient comparing to a real-world hydrodynamic experiment. To completely validate the ef-ficiency of the numerical model (the computational hydrodynamic approximations), an independent experimental comparison is indeed required. It is necessary to implement hydrodynamic experiments in terms of the need for the following robust control design, and the experimental results should provide the nonlinearity and uncertainty information or properties of the CISCREA AUV. The main concern of this chapter is to validate the numerical estimations, i.e., the major hydrodynamic characteristics:

the added mass and the damping effects of the CISCREA AUV in the surge, sway, heave and yaw directions.

Note that the main objective of our study consists in the use of robust control method to reduce the control sensitivity about the hydrodynamic modeling accuracy.

Therefore, this chapter is dedicated to validate the numerical models required for the CFD software of complex-shaped underwater vehicles. Actually, the experimental results in this chapter is not capable to improve the modeling accuracy. To our perspective, the numerical estimation of the dynamic and hydrodynamic parameters should only describe approximately the dynamic behavior as well as uncertainty boundaries. Our goals are to avoid implementing hydrodynamic experiments using expensive equipments.

Bollard Thrust

In this part, we measured the bollard thrusts and torques of the CISCREA propellers.

According to Figure 4 Typically, the four horizontal thrusters simultaneously generate the force and torque vectors on the AUV body, which results in the movement on surge, sway and yaw directions. However, the two vertical thrusters of the CISCREA AUV can only rotate in the same direction to control the heave motion. The power and control signal of the vertical thrusters were designed to be connected, which limit the control since the gravity center is much lower than the buoyancy center. Any small error of perpendicular propulsion (perpendicular to the gravity direction) can be restrained by a strong restoring force.

For the surge motion, the two front horizontal propellers are used, see Table 4.2.

In the contrary, as the horizontal propellers have a 30 ∘ angle to the center line of the CISCREA AUV body, the propulsion force in the sway direction becomes smaller than the one in the surge direction. Therefore, we activated the four horizontal propellers together for the sway motion. The bollard thrusts results for the sway direction are given in Table 4.3. The torque measurement of the yaw rotation is shown in Figure 4-5, and the torque results using the four propellers are given in Table 4.4. According to a previous propulsion model [START_REF] Fossen | Robust adaptive control of underwater vehicles: A comparative study[END_REF], the AUV motion can affect the speed of the fluid that injected into the propellers, which results in some reduction of the AUV propulsion efficiency. As we use the robust control scheme in the following chapters, the modeling assumption or the measurement errors of bollard thrusts can be considered as adding uncertainties for the robust controller. Therefore, we have made the following assumptions for the research:

• There is no propulsion efficiency reduction of the AUV in a moving fluid environment.

• The effect of the propulsion difference between the propeller blade clockwise and courter-clockwise rotation is neglected.

• The CISCREA AUV PWM signal is linear to the propulsion force, which is acting in realtime without any delay. The size of the pool is 4 (length) × 4 (width) × 3.5 (height). A 15-fps camera is installed on top of the pool on the laboratory wall, to capture the AUV motion in surge, sway and yaw directions. Meanwhile, a 25-fps camera is used and installed underwater to capture the AUV heave motion. For the surge and sway directions, the CISCREA AUV is driven in the pool from one end to another using a constant propulsion force. For the heave direction, the CISCREA AUV is driven by the constant vertical propulsion force moving downward in the pool from the top to the bottom. For the yaw rotation, the CISCREA is spinning and accelerating by a constant propulsion torque until it reaches a constant angular velocity.

Experiment Design

Actually, the experimental goal of this part is to build a map relating different AUV damping forces or torques to their corresponding converged AUV velocities or angular velocities. Then, the experimental results can be compared to the computational damping and velocity relationship calculated in chapter 3. By neglecting the uncertainty or the disturbance, the AUV added mass force or torque is only active in the acceleration process. For a a constant speed AUV the added mass becomes zero, and the primary hydrodynamic effect affects merely the damping force or torque. As the CISCREA AUV is driven by many constant propulsion forces and torques during the experiment, the AUV would accelerate and finally reaches its steady regime speed, namely the equilibrium condition when the damping force or torque is equivalent to the propulsion force or torque. Note that, this propulsion force or torque, equivalent to the damping force or torque, is known from the bollard thrust measurement in previous section. In additional, the motion video provides the velocities and angular velocities information corresponding to each propulsion configuration. In practice, the velocity or angular velocity are derived from the AUV position or angle in each video frame. As mentioned before, the propulsion force, the pool size and the time hitting the wall or single rotational lap are known. Therefore, the expected experimental damping and velocity map can be approximated using the bollard thrust and video velocity data.

Experiment Analysis

In order to validate the experiment designed in previous parts, it is necessary to verify two issues for the comparison of computational and experimental AUV hydrodynamic parameters:

• Is the pool large enough for the AUV to reach the equilibrium condition, i.e., accelerate to the converged constant velocity?

• How to measure or derive the final converged speed of the AUV from the captured video frames? First, let us simply assume that the CISCREA AUV has a linear dynamic equation 4.1:

( RB + A ) + L = m (4.1)
where L is the unknown linear damping vector (coefficients), m is the propulsion force or the torque vector, is the position and angle vector of the CISCREA AUV, and RB and A are defined in chapter 2.

By only considering the surge direction, and the initial conditions and 4 necessary assumptions can be described as follows:

• The rigid body mass in the surge direction of RB of the CISCREA AUV is 15.643 (See chapter 3).

• The added mass in the surge direction of A of the CISCREA AUV is 11.985

(Calculated using WAMIT, see chapter 3).

• Initial AUV velocity in all translational directions are 0 / (surge, sway, heave). During the experiment, the only speed considered is related to the surge motion.

• Initial position is 0 (surge direction).

As a result of those modeling information, the unknown linear damping coefficient in the surge direction of L can be calculated, since the start and end moment of the AUV surging in the pool are known from the video, and the distance traveled in the experiment is equal to the pool length 4 minus the AUV size 0.525 . Note that, as the surge motion is approximated by linear dynamics, it can provide information as the estimation of the steady speed for further purposes.

In Figure 4 In additional, the converged speed, derived from the video with the aid of the linear dynamic equation 4.1, can also provide an estimation of the final velocity while the final speed is impossible to be measured using regular average speed method.

Similar conclusions can be draw for the sway, heave and yaw directions. The equilibrium speeds are estimated from the video related to the damping forces and torques (equal to the bollard thrust). We should mention that, the average speed (converged) are measured from the video for CISCREA AUV's surge, sway and yaw motion. The speed approximation of equation (4.1) is used to represent the converge speed for the heave motion. We used the heave approximation instead of the average speed method because the distance for calculating average speed is hardly identified on our underwater camera.

Experiment Results

From the hydrodynamic experiment of the CISCREA AUV, we get 40 discrete data points representing the following damping and velocity relationships in surge, sway, heave and yaw, in addition, we implement 4 second order polynomial approximation curves to compare the hydrodynamic charateristics:

• Surge: experimental damping and velocity relationship are given in Table 4.5, and its comparison to the ANSYS-CFX and STAR-CCM+ approximation is shown in Figure 4-12. Note that an average nominal model is proposed for further robust control design.

• Sway and Heave: experimental results are given respectively in Table 4.6 and 4.7, and its comparisons to ANSYS-CFX and STAR-CCM+ results are shown in Figure 4-13 and 4-14.

• Yaw: experimental results are given in Table 4.8, and its comparison to STAR-CCM+ results is shown in Figure 4-15. As mentioned in Chapter 3, only STAR-CCM+ provides the numerical damping estimation for the yaw rotation. The second order polynomial approximation of the experimental damping and velocity relationship is given in Table 4.9, while the polynomial approximation of the computational STAR-CCM+ and ANSYS-CFX results are shown respectively in Table 4.10 and 4.11.

Finally, we can conclude from the comparison of the experimental and the computational hydrodynamic results that:

• According to Figures 4-12, 4-13, 4-14, 4-15, the numerical models in Chapter 3 give the principal hydrodynamic characteristics of the complex-shaped CIS-CREA AUV. It is efficient to provide approximately the major hydrodynamic information such as damping, added mass, and uncertain boundaries for the robust controller synthesize.

• Although the numerical modeling accuracy is not considered to be very high, for example to be used for precise physical phenomena analysis, the modeling errors can be assigned as the uncertainties in further robust control loop.

• The gap in Figures 4-12 and 4-13, are mainly caused by the drag of the cables and ropes, which are playing opposite efforts in Figure 4-14. In addition, moving fluid causes the decrease of the propulsion also contributes to some experiment error ( [START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF]).

• Rotational damping results of the STAR-CCM+ and experiment are compared in Figure 4-15. Obviously, there is a considerable difference between the two results. However, the numerical model is still validated, and considered to be enough for the robust controller in real sea test. In the next chapter, we introduce an improved robust control scheme to handle this difference. An efficient robust controller is synthesized based on the numerical hydrodynamic model. Form the practical view, the rotational CFD results can be improved by more intensive calculations or simulations, which is out of our scope.

Considering propulsion decrease and rope drag, etc., real AUV hydrodynamic damping should be smaller than the experimental results. Therefore, we proposed an ideal nominal models for the robust control design by taking average of the CFD and experimental results. The nominal model is shown in Table 4.12 representing the 4DOFs hydrodynamics.

( RB + A ) + (||) + () = env + pro (4.2)
Here, the mass inertia matrix RB is approximated by the CAD software PRO/E, as shown in equation 4.3 (mass: , moment of inertia:

• 2 ). RB = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 15
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4.3)
Added mass matrix A of CISCREA AUV is estimated by WAMIT software:

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 11

Yaw Model

Without lose of generality, a yaw model is derived in this section for the robust heading control design of the next chapter. Actually, the goal of our work is to validate the method that combines an improved robust control scheme with the numerical dynamic and hydrodynamic AUV models. Therefore, we only demonstrate the robust controller in the yaw direction to keep a simple and small problem that can prove our approach. Although the robust control scheme is a MIMO control method, it does not prevent the validated SISO robust yaw control (heading control) algorithm extending to 6 DOFs.

Moreover, the heading control problem is one of the most important or critical issues for the hovering underwater vehicles. It is the foundation for the hovering AUVs to achieve fast, accurate and stable control performance, which increase their maneuverabilities to enter complex space, observe precise underwater targets, and make accurate survey or operating actions. For example in SAUCE competition [START_REF] Sauc-E | Student autonomous underwater challenge-europe[END_REF], the AUVs are required to monitor and take precise pictures of a man-made underwater structure from fixed distances and attitudes. In this case, the key issue to accomplish the task is to install a precise controller that can handle the environmental and modeling uncertainties as well as the hydrodynamic nonlinearities.

As mentioned before, the pitch and roll directions of the CISCREA AUV are stable and self-balanced by the restoring force. Therefore, the attitude of CISCREA AUV is assumed to be always vertical, which neglecting buoyancy and gravity effects.

The rotational model can then be simplified as follows (equation (4.6)):

( Y RB + Y A ) r + Y N | | + Y L r = i (4.6)
Definitions and parametric values, such as inertia and damping coefficients, are listed in Table 4.13. Notice that, all the parameters have uncertainties, as they are either measured or numerically calculated. There also exit many un-modeled physics in the real ocean environment. In the following chapters, those uncertainties will be carefully discussed and treated using ∞ control method.

Conclusion

In chapter 4, we designed a 4 DOFs (Surge, Sway, Heave, and Yaw) experiment to validate the CFD models and its hydrodynamic parameters that obtained in Chapter 3. Conclusion can be made according the experimental results: the numerical AUV modeling method (Chapter 3) is concluded sufficient for the complex-shaped underwater vehicles under the scope using the robust control scheme.

During the experiment, the bollard thrust of CISCREA AUV propellers are measured and studied. The results show an approximately linear propulsion behavior, and lead to the assumption of a linear propulsion model (with uncertainties) for further robust control synthesis.

With the aid of the linear dynamic approximation (equation 4.1), we verified the pool size is large enough for the CISCREA AUV to reach a constant speed, and the experiment provides the specific position region to measure the average speed to represent the converged AUV velocity. AUV is not a new invention now, but the world is making ever-greater demands of intelligent, agile and reliable AUV or swarm of AUVs to work in deep water for long periods. The underwater space is a complex, un-modeled or even unknown environment. Therefore, AUV robustness becomes one of the key challenges for long term working AUVs in the presence of uncertainties. Currently, the AUV controller is a hot issue [START_REF] Roche | A hierarchical varying sampling ℎ ∞ control of an auv[END_REF][START_REF] Feng | Reduced order ℎ ∞ control of an autonomous underwater vehicle[END_REF][START_REF] Yang | Modeling of a complex-shaped underwater vehicle for robust control scheme[END_REF], and its robustness is important for high performance AUVs. In fact, the robust control schemes for AUVs should be stable with not only one specific AUV dynamic and hydrodynamic model, it should handles at any time a group of models which represent any changing behaviors or parameters with the real ocean environment.

In reality, the uncertainties are ubiquitous for any controlled plant. In addition, they are inevitable in the ocean environment, and they introduce difficulties in the AUV controller design, such as: unpredictable wind, wave, current, turbulence or obstacles. Generally, an automatic system consists of two parts: the controller part and the controlled plant. Control schemes can modify the controller part and change the closed-loop behavior. However, it can be very difficult to accurately estimate the parameters or dynamics of the controlled plants or models in real time. In addition, it is hard for traditional controllers to handle those unknown uncertainties. Concerning the uncertainties in the real ocean environment, many mathematically perfect AUV control solutions can be less efficient, and sometime the controller can be unstable and damage the AUV.

A robust AUV requires a motion controller that can achieve both the control performance and the closed-loop stability in the presence of ocean uncertainties. Actually, the ∞ control theory handles the uncertainty issues using ∞ norms in the frequency domain [START_REF] Gu | Robust Control Design with MATLAB[END_REF][START_REF] Zhou | Essentials of robust control[END_REF]. Recently, the ∞ control method has been adopted more frequently in marine activities.

Consider the robust control problems of the CISCREA AUV, several issues matters

[129]:

• Parametric uncertainties and un-modeled dynamics in the AUV models

• Quadratic damping effects

• External disturbance as well as sensors' noise

As mentioned in Chapter 1, most of traditional control schemes can not prevent the inefficiency of the control performance as well as the stability reduction in the presence of dynamic uncertainties. Meanwhile, the few nonlinear control schemes are generally complex and limited to specific assumptions. Therefore, we proposed a simple AUV linear ∞ control scheme. The proposed robust control approach inherited a numerical model from previous CFD works [129]. Numerically predicted the actuator force compensates the nonlinear damping behavior, thereby results in a linear AUV model with uncertainties. Based on the bounded linear nominal model, the classical ∞ approach [START_REF] Zhou | Essentials of robust control[END_REF] was used to handle the uncertainties raising from the modeling and compensating process, etc. A linear AUV behavior can be derived, which is critical and convenient for both control design and navigation improvement.

Deep inside the ocean, the earth magnetic signal is one of the merely existing information that tells the heading of robots with very good cost efficiency. It is important the AUV can control its heading using only one magnetic compass as feedback sensor. Without lose of generality, a robust heading controller using the proposed linear ∞ approach was implemented and simulated for the CISCREA AUV. Simulation results provided, mainly including: proposed model based ∞ approach, classical ∞ approach without nonlinearity compensation, and classical PID control scheme.

Robust Control Theory

The robust controller handles the uncertainty issues by building the nominal model based closed-loop feedback control, and it satisfies for every plant belonging to the uncertainty model set, two conditions:

• Robust stability,

• Robust performance.

The robust control can be related to two problems: robust analysis and robust synthesize problems. The analysis problem pushes the controller to satisfy the desired properties for all admissible noises, disturbances, and model uncertainties. The synthesis problem designs a controller satisfying the desired properties for admissible noises, disturbances, and model uncertainties [START_REF] Gu | Robust Control Design with MATLAB[END_REF][START_REF] Zhou | Essentials of robust control[END_REF]. A standard ∞ problem is shown in Figure 5-1. The AUV dynamic and hydrodynamic models, uncertainties and disturbances, the weighting functions are all describe in a Linear Fractional Transformation (LFT) form, representing as the generalized system , robust controller and the uncertainties block ∆.

In historical reviews of robust control theory [START_REF] Dorato | A historical review of robust control[END_REF][START_REF] Zhou | Essentials of robust control[END_REF][START_REF] Gu | Robust Control Design with MATLAB[END_REF][START_REF] Carsten | Theory of robust control[END_REF], Zames et al. introduced the first robust control breakthrough in early 1980s [START_REF] Zames | Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms and approximate inverses[END_REF][START_REF] Zames | Feedback, minimax sensitivity, and optimal robustness[END_REF]. They considered for the nonlinear AUV plant in face of uncertainties. The complete robust control theory can be found in [START_REF] Zhou | Essentials of robust control[END_REF][START_REF] Gu | Robust Control Design with MATLAB[END_REF][START_REF] Carsten | Theory of robust control[END_REF], and the mathematical conceptions used in our approach are described in the Appendix C.

∞ Robust Analysis

In this part, the ∞ criteria for robust stability and the robust performance that can handle the uncertainties in control design are discussed. The unstructured uncertainties are considered as generic errors in all robust control designs.

Generally, the uncertainties of the controlled plant can be classified into two categories: internal dynamic perturbations and external disturbances. The internal dynamic perturbations represent the discrepancy between the mathematical model and the actual dynamic, including the parametric uncertainties, un-modeling dynamics, neglected nonlinearity, etc. In contrast, the external disturbances refer to environmental input and output disturbances, including unspecified currents or obstacle effects, sensor or actuator noise, etc. Hence, it is quite complex to built a model representing the reality. Traditional control schemes handle the uncertainties by assuming that the real uncertainties are instant disturbances added to the initial states of the nominal model. The nominal model is the single physical description trusted in the traditional control designs. Using the stability margin analysis based on the nominal model, the traditional controller considers a stable closed-loop system as the real plant. However, in reality the practical uncertainties are far different from assumptions, and the control stability can not be guaranteed. The response of the single model might not match the varying true plant. Therefore, a set of models is a minimum requirement to describe model dynamics in reality. Besides the stability issue, the performance reduction causing by the uncertainties are unmeasured in the traditional control designs. Note that, the robust performance analysis is inevitable in the real world control applications, and this is another important reason to introduce the robust control algorithms. In conclusion, it is essential to bring in the uncertainty descriptions and their relevant analysis method.

In general, the actual controlled-plant can be described into different compo-nents, where 0 denotes the nominal model of the physical system, and ∆ the perturbations associated with the nominal model. In case of a LTI system of , the uncertainties ∆ are characterized by: unstructured uncertainties (the uncertain block is represented by an unknown transfer matrix [START_REF] Zhou | Essentials of robust control[END_REF][START_REF] Gu | Robust Control Design with MATLAB[END_REF]) and structured uncertainties (i.e., parametric uncertainties, whose behavior can be written in to a fix order model by a LFT form [START_REF] Gu | Robust Control Design with MATLAB[END_REF]). By considering the controlled plant as a LTI system, we can derive the state space representation of the structured uncertainties, in equation 5.1, and unstructured uncertainties by equation 5.2.

⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ () () () () ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ (5.1) ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ + ∆ + ∆ + ∆ + ∆ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ (5.2)
In equation 5.1, ∈ R n denotes the states of the system, ∈ R m is the input vector, ∈ R p is the output vector, ∈ R s is the vector of the uncertain parameters, () ∈ R n×n , () ∈ R n×m , () ∈ R p×n , () ∈ R p×m are the system matrices (, , , ∈ R). Meanwhile, in equation 5.2, ∆, ∆, ∆, ∆ are the unstructured perturbations in form of unknown matrices. The key issue is to make robust analysis (stability anlaysis) under small gain theorem [START_REF] Zhou | Essentials of robust control[END_REF] with different types of uncertainties.

In fact, there are various types of uncertainties, including: additive or multiplicative uncertainties, and inverse perturbations, etc. [START_REF] Zhou | Essentials of robust control[END_REF].

Generally, it is difficult or impossible to get an exact frequency response or a high accuracy Laplace transfer function approximation of uncertainty characteristics in ∆(). Therefore, a bounding method using a less complex weighting functions () There are several methods proposed to estimate the weighting functions for regular robust control design, such as the system identification method which implements the experimental frequency response tests, the empirical formula approximation method which recreates the model characteristics based on low-level physical principals, etc.

that
According to Zhou [START_REF] Zhou | Essentials of robust control[END_REF]: "In many occasions, the weights are chosen purely as a design parameter without any physical basis, so these weights may be treated as tuning parameters chosen by the designer to achieve the best compromise between the conflicting objectives". Given a specific weighting function (), the stability tests in face of various type of uncertainties is actually of central importance before all the robust control design process. Here, the small gain theorem [START_REF] Gu | Robust Control Design with MATLAB[END_REF][START_REF] Zhou | Essentials of robust control[END_REF] is introduced as the foundation to achieve the stability tests.

Small Gain Theorem [START_REF] Gu | Robust Control Design with MATLAB[END_REF]: if there is an unknown bounded unstructured perturbation ∆(), ||∆()|| ∞ < 1, i.e., ∆() ∈ ∞ , and a stable nominal controlled plant 0 (), the closed-loop feedback system is robust stable if and only if there is a controller () stabilizes the nominal model 0 () and the condition ||{ 0 (), ()}|| ∞ < 1 holds (equation 5.4, 5.5). Here, the { 0 (), ()} is the closed-loop nominal system.

||∆( + 0 ) -1 || ∞ < 1 (5.4) ||( + 0 ) -1 ∆|| ∞ < 1 (5.5)
By another word, the robust controller internally stabilizes every plant belonging to the set of models bounded by the weightings when the above ∞ norm inequality stands. Depend on different types of uncertainties, some important closed-loop robust stable conditions are summarized in table 5.1.

( + ∆) 0 ||∆()|| ∞ < k () || k || ∞ 1 Inverse multiplicative uncertainty ( + ∆) -1 0 ||∆()|| ∞ < mo () || mo || ∞ 1
Additive uncertainty

( 0 + ∆) ||∆()|| ∞ < a () || a || ∞ 1 Inverse additive uncertainty 0 ( + ∆ 0 ) -1 ||∆()|| ∞ < mi () || mi 0 || ∞ 1
Here, k , a , mi , mo are the prior knowledge of weightings or replacement of original uncertainties about the upper bound as mentioned in equation 5. The second important issue of robust control design is the robust performance represented by ∞ norms. According to [START_REF] Gu | Robust Control Design with MATLAB[END_REF]: "the ∞ design and relevant approaches first formulate the stability as well as performance design specifications as a robust stabilization problem and then solve the robust stabilization problem to find a controller". For a regular closed-loop robust control system in figure 5-1, whose inputs and outputs are energy bounded, it is generally required the following performance specifications:

• tracking performance, ||( + 0 ) -1 || ∞ .

• disturbance attenuation, ||( + 0 ) -1 || ∞ (same as above).

• noise rejection, || -

( + 0 ) -1 0 || ∞ . • energy consumption, ||( + 0 ) -1 || ∞ .
For the robust performance analysis, the weighting functions as well as its formulation of ∞ problems are also of central importance to achieve desired objectives about the control performance. The weighting selections are actually the active engineering tuning to reject unwanted dynamics and perturbations. Let a SISO system as in equation 5.8:

1 () = () 1 () (5.8)
where, 1 () ∈ C denotes the system input, 1 () ∈ C the system output, and () is a stable close loop transfer function of the SISO system. Then, the ∞ norm of the transfer function from input to output are given in equation 5.9:

||()|| ∞ = ||u 1 || 2 ̸ =0 || 1 || 2 || 1 || 2 = ω |()| (5.9)
Here, the norm ||()|| ∞ can be considered as an energy amplifier from the input to the output under certain frequency. Define a constant > 0 as the expected energy gain for the closed-loop system () that confined the effect of the input signal to the output behavior, then we have:

||()|| ∞ < (5.10)
Let () be a disturbance, whose frequency response can be covered by a weighting function () (gain upper boundary):

|()| < | ()|, ∀ ∈ R (5.11)
Meanwhile, if the following ∞ norm inequality stands:

||() ()|| ∞ < (5.12)
then, the energy reflection of the disturbance () through the closed-loop system () to the output becomes less than , namely, the restrain of the disturbance () in the robust system () is capable to achieve the range of .

∞ Robust Design

The robust control design is to find a stable controller that simultaneously achieves the robust stability and certain robust performance in the face of uncertainties. As a result, the ∞ optimization methods are proposed and validated as efficient tools in the last decades, to solve the robust control design problem for LTI system.

As mentioned in the robust stability theorem, a MIMO system as shown in figure 5-1, engineering selected weightings can generally guarantee a stable bounded form of ∆() (i.e. ||∆()|| ∞ < 1), then, the robust control design problem of is equivalent to the ∞ design problem on the weighted nominal model 0 without concerning the perturbation block ∆(). Therefore, the LFT representation of the robust system is given in equation 5.13:

⎡ ⎣ ⎤ ⎦ = () ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ 11 () 12 () 21 () 22 () ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ (5.13)
and the controller () is given in equation 5.14:

= () (5.14)
We summaries the definitions in figure 5-1 as follows:

• ∆() denotes the perturbation set, which is the mathematical approximation of the uncertainties and disturbances.

• () stands for the robust controller, which is yielded from the ∞ optimization.

• () is the generalized model, which consists of the nominal model 0 as well as the weighting functions that guarantees the stability and performance constrains.

• is the control signal representing the control command that is given to the generalized plant ().

• is the output signal representing the dynamic data of () that is obtained by the feedback sensors.

• represents the external uncertainty, including disturbance, noise, etc.

• is the error signal which evaluates the closed-loop signal attenuation performance about the uncertaintys or disturbances.

As shown in equation 5.13, the generalized system () Moreover, the state space representation of the generalized model () and the robust controller () are respectively shown in equations 5.16 and 5.17:

is
() = ⎡ ⎣ 11 () 12 () 21 () 22 () ⎤ ⎦ = G + G ( -G ) -1 G = ⎡ ⎣ 11 12
21 22 ⎤ ⎦ + ⎡ ⎣ 1 2 ⎤ ⎦ ( -) -1  1 2

(5.16)

() = K + K ( -K ) -1 K (5.17)
where, the G , K , G , K , G , K , G and K are appropriately dimensioned real constant matrices.

Thereby, the state space realization of the closed-loop transfer function rw () can be given by the following equations:

(, ) = BF + BF ( -BF ) -1 BF (5.18) BF = ⎡ ⎣ + 2 ( -K 22 ) -1 K 2 2 ( -K 22 ) -1 K K ( -22 K ) -1 2 K + K ( -22 K ) -1 22 K ⎤ ⎦ (5.19) BF = ⎡ ⎣ 1 + 2 ( -K 22 ) -1 K 21 K ( -22 K ) -1 21 ⎤ ⎦ (5.20) BF =  1 + 12 ( -K 22 ) -1 K 2 12 ( -K 22 ) -1 K  (5.21) BF =  11 + 12 ( -K 22 ) -1 K 21  (5.22)
According to the stability theorem [START_REF] Thomas | Linear systems[END_REF], the closed-loop transfer function rw () is internally stable if all the poles of the system matrix BF are on the left panel of the surface, i.e., all the eigenvalues satisfy ( i ( BF )) < 0. As a matter of fact, it is not suitable to use analytic formula to derive robust control solutions for MIMO systems. Therefore, the robust design goals are to optimize a stable control result, i.e., finding a stabilizing controller with the performance limited to a given positive number > 0 (the expected energy gain from the input signal to the error signal

). By taking the tracking performance of the closed-loop robust system for example, the performance constrain can be presented by:

|| rw || ∞ < (5.23)
In fact, the synthesis objective for good tracking performance is to minimize the constrain value which results in better disturbance attenuation. The optimal goal can be achieved by guaranteeing the stability of the closed-loop transfer function rw . In this case, the above robust design problem becomes a ∞ optimization problem. Given an initial value > 0, then, gradually decreases the and tests the stability condition until it reaches a minimum value, this is to say the optimal gain opt . Theoretically, the robust performance is satisfied as a result of disturbance attenuation under the minimum transfer amplification.

The robust control design is actually a constrained optimization problem. The ∞ robust synthesis solves the optimal controller by recursive calculations. According to [START_REF] Gu | Robust Control Design with MATLAB[END_REF], the solution to the optimization problem is not unique except in the scalar case. During the past decades, many algorithms are developed to solve the ∞ optimization problems. In the following part, we would introduce two important methods for robust control systhesis: ARE and LMI [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF], [START_REF] Ghaoui | Advances in Linear Matrix Inequality Methods in Control[END_REF].

∞ Synthesis: ARE

"Lyapunov eqautions are most useful in system analysis while AREs are most useful in control system synthesis." said Zhou in [START_REF] Zhou | Essentials of robust control[END_REF]. For ∞ synthesis, Doyle first proposed the two ARE synthesis method in 1980s [START_REF] Doyle | State-space solutions to standard ℎ 2 and ℎ ∞ control problems[END_REF].

Let the state-space description of the generalized system () in equation 5.24:

⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 1 11 12 2 21 22 ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎦ (5.24)
where ∈ R n is the state vector, ∈ R m 2 is the control input vector, ∈ R p 2 is the measurement vector, ∈ R m 1 stands for the disturbance input vector, ∈ R p 1 denotes the error evaluation output vector, 12 ∈ R p 1 ×m 2 , 21 ∈ R p 2 ×m 1 , with 1 2 , and 2 1 . In practical applications, 22 is generally zero.

According to Glover and Doyle [START_REF] Gu | Robust Control Design with MATLAB[END_REF], the necessary and sufficient conditions for the existence of a ∞ suboptimal solution are obtained under the following assumptions (solvability conditions).

• (, 2 ) is stabilisable and ( 2 , ) is detectable.

• ⎡ ⎣ -2 1 12
⎤ ⎦ has full column rank for all ∈ R.

• ⎡ ⎣ -2 2 21
⎤ ⎦ has full row rank for all ∈ R.

• 12 =  0  , 21 = ⎡ ⎣ 0 ⎤ ⎦
Consider the two following Riccati equations:

( --1 T 1• 1 ) T + ( --1 T 1• 1 ) --1 T + T 1 ( -1• -1 T 1• ) 1 = 0 (5.25) ( -1 T •1  -1 ) + ( -1 T •1  -1 ) T -T  -1 + 1 ( -T •1  -1 •1 ) T 1 = 0 (5.26)
where,

= 1• T 1• - ⎡ ⎣ -2 0 0 0 ⎤ ⎦ , 1• = [ 11 , 12 ]
(5.27) Let us suppose that the generalized plant satisfies the above four assumptions, then, there exists an internal stabilizing controller () such that

 = •1 T •1 - ⎡ ⎣ -2 0 0 0 ⎤ ⎦ , •1 = ⎡ ⎣ 11
|| rw () = {(), ()}|| ∞ <
if and only if :

• > ([ 1111 , 1112 ], [ 1111 T , 1121 T ]) ((•) is the singular value)
• the two Riccati equations 5.25 and 5.26 respectively have a stable solution and .

• the Riccati equations 5.25 has a positive semidefinite solution 0 that guarantee a stable matrix ( --1 T 1• 1 --1 T ).

• the Riccati equation 5.26 has a positive semidefinite solution 0 that guarantee a stable matrix ( -

1 T •1  -1 -T  -1
).e

• ( ) < 2 ((•) denotes the spectral radius)

The existence of the solutions to a ∞ problem depends on the selected constant > 0. Limited to the structure and performance requirements, the control system can not achieve an arbitrary small , and for some small , the ∞ problem might not be feasible.

Generally, the control performance is better if the value remains small. Therefore, the control design process is the bisection as well as its robust test until reaches the smallest feasible value. For the ∞ problem of robust control synthesis, the ARE method executes the following steps (dichotomic procedure):

1. Initial: select a sufficient large max that guarantees the ∞ problem is admissible, meanwhile, chose a sufficient small min that makes the ∞ problem infeasible. Select a small enough ∆ s value to stop the optimization process. Using the ARE robust control synthesis method, the ∞ problem or the generalized plant () should first meet the four constrain assumptions. Meanwhile, the optimization calculation depends on the empirical selection of energy gains min and max . When the control plant is complex and the parameters are not well known, the AREs solution might consume a lot of time to be optimized, and feasible solutions are not guaranteed.

∞ Synthesis: LMI

In the past decades, LMI theory was validated as an efficient convex algorithm to solve ∞ problems for LTI systems [START_REF] Gahinet | A linear matrix inequality approach to control[END_REF][START_REF] Ghaoui | Advances in Linear Matrix Inequality Methods in Control[END_REF]. Typical LMI method is flexible in choosing the performance specifications as it provides better solvability conditions, i.e., it requires less assumptions than the Riccati method. A richer body of controlled-plants can be treated using LMI method. LMI algorithms also allows to directly minimize in order to find the best upper bound [START_REF] Ghaoui | Advances in Linear Matrix Inequality Methods in Control[END_REF]. In fact, the LMI approach turns the ∞ -norm constrains into linear matrix inequalities (LMI problems). Thereby, the ∞ synthesis problems can be solved with matured mathematical LMI tools.

In the following, we would introduce the derivation of the ∞ problems using LMI techniques.

The generalized plant () and its minimal state-space realization is presented in equation 5.30:

() = SG + SG ( -SG ) -1 SG (5.30)
where, the S , S , S and S are appropriately dimensioned real constant matrices.

According to the LMI theory, if > 0 and matrix SG is stable and of minimal realization, there is |||| ∞ < if and only if, there exists a positive definite matrix > 0 satisfying:

⎡ ⎢ ⎢ ⎢ ⎣ T + T T -T - ⎤ ⎥ ⎥ ⎥ ⎦ < 0 (5.31)
The ∞ control problem is transformed into a standard LMI problem in equation 5.31. The above lemma is considered as the foundation of the LMI method, its proof can be found in [START_REF] Carsten | The riccati inequality and state space ℎ ∞ optimal control[END_REF].

To solve a robust controller () using the LMI theory, there is the following lemma: If the generalized system () satisfies the assumption ( (, 2 ) is stabilisable and ( 2 , ) detectable), there exists a stable controller () and its closed-loop system has || rw || ∞ < , if and only if, the following LMIs has a symmetric solution matrix > 0 and > 0:

⎡ ⎣ T X nw ⎤ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ + T T 1 1 1 -11 T 1 T 11 - ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎣ T X nw ⎤ ⎦ < 0 (5.32) ⎡ ⎣ T Y nz ⎤ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ + T T 1 T 1 T 1 -T 11 1 11 - ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎣ T Y nz ⎤ ⎦ < 0 (5.33) ⎡ ⎣ ⎤ ⎦ 0, ⎡ ⎣ ⎤ ⎦ + k (5.34)
In equations 5.32∼5.34, and k respectively, represent the dimension of the generalized system and the robust controller. Matrix definitions are given as :

X = [ 1 , 21 ], Y = [ T 2 , T 12
]. Note that, in the LMI method the solvability condition, (, 2 ) is stabilisable and ( 2 , ) detectable, is the only requirement. The performance specifications can be chosen with more flexibility, and the dimension of the control input, disturbance input, evaluation and control outputs can be arbitrary.

The technical steps to solve a ∞ controller using LMI methods are given as follows:

1. Solve the LMIs of equation 5.32 and 5.33, to get the symmetric solutions > 0 and > 0.

Find the matrix such as

T = --1 3. Let = ⎡ ⎣ T ⎤ ⎦ (5.35)
and then move into equation 5.36 and 5.37 (* denotes the matrix of any values that fits the size),

⎡ ⎣ T * ⎤ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ T + 1 T 1 2 T 1 - T 11 0 1 11 -12 2 21 0 * ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(5.36)

+ T Θ + T Θ < 0 (5.37)
4. A parametric matrix Θ can be found, and its state space realization consists of the elements of the robust controller ().

Θ = ⎡ ⎣ K K K K ⎤ ⎦ (5.38)
In practice, the robust control toolbox in Matlab software provides two ∞ controller synthesis algorithms: Riccati method and LMI method, and its relevent applications can be found in [START_REF] Gu | Robust Control Design with MATLAB[END_REF].

Control of the CISCREA AUV

Without lose of generality, ∞ control approach is prensented and adapted to the CISCREA AUV for a heading control. A major problem is that the yaw model is nonlinear with serious uncertainties. In reality, the AUVs nonlinearity and uncertainty issues are mainly caused by: nonlinear damping effects, changing parameters, unmodelled dynamics, external disturbances, sensor noise, etc. Therefore, we propose a framework to change the nonlinear yaw model to a linear system with uncertainties based on previous CFD modeling work. By then, we can solve a ∞ controller for the linear system. At the end, the simulation results of the ∞ control and the classical PID control schemes are compared.

Yaw model and its linearization

In order to validate the proposed ∞ robust control approach, and reduce the complexity to implement the controller, we abstract the yaw model from the full dimensional CISCREA AUV dynamics. In fact, previous dynamic and hydrodynamic modeling work is control-oriented, and it brings a lot of convenient benefits. For example, it is possible to build model based Kalman filters for unobservable states that faces considerable noise [START_REF] Astrom | Feedback systems: an introduction for scientists and engineers[END_REF]. Otherwise, the dynamic and hydrodynamic information can be used to design the Smith compensators in order to handle the signal delay and its response oscillation consequence [START_REF] Zhong | Robust control of time delay systems[END_REF].

Most importantly, in this part, we propose a method to deal with system nonlinearities for robust control scheme. We discuss the AUV nonlinear dynamic problem without concern of parametric uncertainties, such as inertia, added mass and damping coefficient errors. As found in Chapter 3 and 4, damping is a major nonlinear component in an underwater vehicle hydrodynamic model. Therefore, we propose to compensate nonlinear behaviors by creating a linear behavior. The compensation error is assigned to be an extra uncertainty beside parametric uncertainties.

The reason to introduce such a linearization method is that most of the control theories assume the controlled-plant as a linear time invariant system [START_REF] Astrom | Feedback systems: an introduction for scientists and engineers[END_REF]. The major ∞ control methods are developed on linear theories. Only in some specific linearizable assumptions, a few nonlinear systems can be treated using linear methods [START_REF] Astrom | Feedback systems: an introduction for scientists and engineers[END_REF]. Generally, the control performance are dramatically reduced to directly fit a linear ∞ controller on the nonlinear AUV dynamics, such as CISCREA AUV. The linearization of the yaw model is critical and essential for the robust control design of underwater vehicles. Generally, there are several essential linearization methods to implement linear control algorithms on the nonlinear control applications:

• Equilibrium approximation: First, formulate the nonlinear dynamic model of the nonlinear system according to the physical knowledge or input-output data analysis. Thereby, the engineers determine the steady-state operating point or a group of operating points about which to linearize. Derivations are made about the operating points to obtain an approximate linear model. The equilibrium approximation has one disadvantage. The states of the closed-loop system should be near the operating pionts, otherwise, the controller can be invalid.

• Perturbation method: Using ∞ control scheme, the nonlinear damping effect can be directly defined as a parametric uncertainty adding to the linear behaviors. Describe the whole nonlinearity into the uncertainty set ∆, the ∞ synthesis can yield a stable control system with some compromise to the performance conservation.

• Nonlinear compensation method: The nonlinear damping behaviors are estimated using numerical physics on CFD software. The CFD modeling estimation results are used to create an artificial linear AUV dynamic behavior as the robust control nominal model. The CFD model guides the AUV actuators to execute the compensations about damping forces and torques on real world AUV dynamics. Obviously, comparing to the perturbation method the damping compensations errors are generally small. Small linearization errors leads to small uncertainty set ∆. Therefore, ∞ synthesis are finally implemented based on small ∆ to obtain a stable robust controller with less compromise on performance conservation.

In chapter 4, we abstract the nonlinear yaw model from the 4 DOFs CISCREA AUV dynamics for further ∞ research. For the convenience of model linearization,

we present the yaw model again in equation 5.39 (neglecting buoyancy and gravity).

Relevent linearization variable definitions are given in table 5.2. The three mentioned linearization methods will be compared in the following. For traditional AUV applications, the equilibrium approximation method is usu-ally the first option for model linearization. Assume the steady-state angular velocity of the AUV is r0 , thereby, the linearization about equilibrium point results in equation 5.40.

( Y RB + Y A ) r + Y N | r | r + Y L r = i (5.39)
( Y RB + Y A ) r + ( Y L + 2 Y N | r0 |) r -Y N | r0 | r = i (5.40)
Generally, the equilibrium state of a regular system is either zero or a constant value.

Unfortunately, the equilibrium state of the AUV (angular velocity) is not steady and varies in a wide range from 0 / to 4 /. A real dynamic system does not stay around the equilibrium point as supposed. As a result, the control performance is significantly disturbed.

The second perturbation method is more straight forward, as shown in equation 5.41: (5.45) then, we can result in that Y LA only has the dynamic uncertainty of 23.7%.

( Y RB + Y A ) r + ( Y N | r | + Y L ) r = i (5.
In additional, we can derive the linear nominal model for robust control design in equation 5.46: In previous hydrodynamic study, we concluded that only in the yaw direction numerical damping results has the worst error comparing to the experimental one, see figure 4-15. The surge, sway and heave directions have better numerical results, which means the yaw model is the most difficult one for robust control deign. Considering the nominal models derived by nonlinear compensation, the worst case of the parametrical uncertainty that adding to the linear factor Y LA is only 23.7%. The proposed model, equation 5.46, is therefore a linear system with uncertainties. However, in the following ∞ robust control design the parametric uncertainty boundary is generally selected higher than 40% > 23.7%, i.e., the linearized model with uncertainties is well bounded and is feasible for further ∞ robust control design.

( Y RB + Y A ) r + ( Y LA + ) r = i ; ∈ [-0.4265, 0.4265] (5.46)

∞ control design

The ∞ control theory achieves the advantage of robustness and MIMO control characteristics by combining frequency domain techniques and the state space feedback theory. Both the robust stability and robust performance can be simultaneously achieved after founding such a fix controller using optimizing algorithms. For the MIMO AUV control problem, robust control theory can result in accurate control performance in face of many bounded uncertainties. Therefore, for the AUVs control problem, the ∞ controller is proposed to solve the following issues:

• Dynamic and hydrodynamic parametric uncertainties.

• Nonlinear damping compensation errors.

• External disturbance from the ocean environment and sensor noise.

Based on the linearized yaw model that derived from the nonlinear damping compensation, a ∞ control scheme is proposed for the heading control of the CISCREA AUV, see figure 5-3. The thruster compensates the nonlinear damping behavior by calculating its propulsion value using equation 5.42 and the AUV angular velocity r . From the controller perspective, the compensated AUV controlled-plant is a first order linear system with minor uncertainties. As a result of the improved linear AUV dynamic behavior, the robust issues can be conveniently solved by essential ∞ controllers. Although, for few AUVs, the angular velocity r is not measured or indirectly sensed, the proposed modeling based robust control scheme could easily solve the detection issue by building Kalman filters. This technique of estimating unmeasured states would be introduced in the next chapter for real world AUV robust control in the sea.

Generally, the nominal model is derived from the LFT technique, which separates uncertainties into an individual block [START_REF] Zhou | Essentials of robust control[END_REF]. Denotes the CISCREA AUV yaw model, finding such a are represented by ∞ norms of the closed-loop transfer functions from to , see equation 5.47.

Kstable ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ p ( + 0 ) -1 e ( + 0 ) -1 u ( + 0 ) -1 ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ⃦ ∞ < (5.47)
Here, is the robust controller in figure 5-3, 0 is the above obtained linear nominal yaw model from LFT interpretation, e , u , and p are the weighting functions. In addition, (), and k () and () stand for different types of sensitivity functions:

() = 0 ( + 0 ) -1 (5.48) k () = ( + 0 ) -1 (5.49)
() = ( + 0 ) -1 (5.50)
In general, weighting functions as well as relevant sensitivity functions would be used in the minimization to meet the design specifications.

In ∞ control design, the weighting functions are introduced for setting control specifications. Generally, it is difficult to get the accurate frequency characteristics of external input signal. In addition, the real signal characteristics generally have complex transfer function representations for the control design. Therefore, the weighting functions, which have simple transfer representations, are proposed as upper bound constrains that cover original signals, see figure 5-4. All the weighting functions e , u , and p are connected to the generalized control plant 0 . The weighting function p represents the frequency characteristics of the external disturbance. Usually, it is used to describe output disturbance rejection ability. Without p , the generalized system (only has e and u ) is actually a classical ∞ synthesis problem.

Satisfying the above norm inequality (equation 5.47), indicates that the closed-loop system indeed reduces the disturbance effects to a prescribed level.

order transfer functions [START_REF] Gu | Robust Control Design with MATLAB[END_REF]:

p () = 2 + n1 + n2 2 + d1 + d2 (5.51)
e () = + n1 + d1 (5.52)

u () = (5.53)
where, , , , n1 , n2 , d1 , d2 , n1 and d1 are the constant parameters to be configured.

The above ∞ control design can be concluded as three parts.

1. The energy gain of the concerned closed-loop is restrained in a specific ∞ norm inequality, which are generally the performance goal interpretation.

2. Finding a controller such that the ∞ norm inequality is satisfied after defining relevant weighting functions (or to say the description set of signal characteristics).

3. To solve the ∞ problem, one can use the Riccati method or Linear Matrix Inequality (LMI) approach [START_REF] Zhou | Essentials of robust control[END_REF][START_REF] Gu | Robust Control Design with MATLAB[END_REF] to implement the mathematical optimization calculation. It was mentioned before, we prefer to choose LMI approach, since it requires less initial conditions [START_REF] Gu | Robust Control Design with MATLAB[END_REF].

PID control design

In order to validate the ∞ AUV controller, we compare the ∞ controller with the widely used PID control method. In this part, some essentials of PID control algorithms are introduced.

The PID control method can be given by the following equations:

() = p () + i  t 0 () + d () (5.54)
() = () -() (5.55)
where, () is the error signal between the system output () and the reference input (), () stands for the control input, p , i , and d denote the coefficients for the proportional, integral, and derivative terms. The PID controller can achieve the desired set-point by minimizing the error (). Generally, the PID controller calculates the control adjustment based on the existing error () and PID coefficients p , i , and d . As the PID controller only depends on the system output measurement, and no model is required, therefore, it is broadly applicable. However, tuning the three PID coefficients are critical, but empirically relies on the system response (overshoots, steady-state error, or response oscillation). The tuning of the PID controller can not guarantee an optimal control or even stability. During the PID tuning process, there are the following physical characteristics:

• p represents the error adjustment ability, it determines the response speed and the control stability.

• i reduces the steady-state error, it accumulates the past errors over time, and bring in sufficient control to reduces any small error.

• d prevents the overshoot and oscillations of the control output, it predicts the future error and reduces redundant control.

Currently, PID control is still widely used in most of AUV control applications. Meanwhile, there are new trends of advanced PID control methods to optimize the PID tuning process [START_REF] Visioli | Optimal tuning of pid controllers for integral and unstable processes[END_REF]. In the following simulations, we introduce the PID control design for the CISCREA AUV.

CISCREA AUV Yaw Control Simulations

In this part, the yaw control simulations as well as the design details of the ∞ and PID controller are presented.

∞ Yaw Controller

For the convenience, the compensated linear yaw model of the CISCREA AUV is given again in equation 5.56:

( Y RB + Y A + 1 ) r + ( Y LA + 2 ) r = i (5.56)
where, 1 and 2 respectively represent the uncertainties adding to the inertia Y RB + Y A and compensated damping coefficient Y LA .

As mentioned in the modeling work, the major uncertainties of the CISCREA AUV dynamics mainly consists of three parts:

• Y RB : The inertia of the AUV rigid body, estimated by PRO/E software.

• Y A : The added mass inertia of the AUV, calculated by the fluid dynamic program WAMIT.

• Y LA : The artificial linear damping coefficient is compensated using the numerical damping coefficients ( Y N , Y L ) of ANSYS-CFX and StarCCM+ calculations.

In the ∞ yaw control simulation, the inertia Y RB + Y A are considered to be two varying parameters, which have a total 30% of variations. In the contrary, linear damping coefficient Y LA is bounded by the variation limit of 40%. As we know from the ∞ control design analysis, the Y LA uncertainty boundary 40% > 23.7%

is sufficiently large to guarantee the control performance goals. In fact, in face of the parametric uncertainties, we consider that AUV dynamics belonging to an uncertain systems family on the bode plots, see figure 5-5.

During the CISCREA AUV ∞ synthesis, weighting function are tuned by many trials, and its results are given in the following equations:

p () = 0.95 u () is selected to be a very small scalar ( = 0.01) for simply disturbance rejection. e () and p () are chosen carefully considering that: the tracking error should be less than < 1%; fast control response time (around 2 ∼ 3 seconds); the robust margins, etc. The principal of weighting function selections are given in [START_REF] Gu | Robust Control Design with MATLAB[END_REF][START_REF] Zhou | Essentials of robust control[END_REF].

For the CISCREA AUV yaw controller, we execute the ∞ synthesis using the LMI method.

ZK = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (5.61) ZK = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.

PID Yaw Controller

Based on the designed control performance Matlab provides an interface to automatically tune the PID parameters. Hence, we obtained an original PID controller . The automatic PID results with the considerations of the overshoot, response speed, and etc. Finally, the PID three parameters for CISCREA AUV yaw control are obatined:

p = 16, i = 0.1, d = 10
(5.65)

Simulation Results

In order to validate the ∞ AUV control approach, we proposed a yaw control simulation of step response on the CISCREA AUV. The CISCREA AUV is ordered to rotate from 1.8 to 4.8 . In order to compare different scenarios, the many simulations are implemented including: control of nominal yaw model, the yaw model with parametric variation, and the yaw model with disturbance.

In figure 5-8, step responses of three scenarios are represented: PID control, damping compensated ∞ approach and ∞ control. From the simulation comparison, we can conclude that the compensated ∞ controller handles the nonlinearity with the fastest response. Compensated ∞ controller has no overshot or oscillations during the rotation process. The tracking error achieves the specification less than 1%. The response speed is inside the performance design goal of 2. The compensated ∞ control output has no overshoot. In the contrary, PID controller is less efficient, it has obvious overshoot and slow step response speed. The proposed compensated ∞

AUV control approach can be several times faster than the transitional PID controller. In addition, the energy waste on inefficient thrust using the PID controller are seldom seen on the compensated ∞ controller. Note that, the ∞ control has no damping compensation, and it connects directly on the nonlinear yaw model. The results show that the ∞ control is also less efficient to handle the AUV hydrodynamic nonlinearity, clear oscillations can be found in the control output. the proposed ∞ AUV control approach achieved all the designed performances. The response speed is inside 2, tracking error less than 1%, and there is no obvious os- The response speed becomes slow, and control response oscillation is more serious for both the PID controller and the ∞ control algorithm. In conclusion, the proposed ∞ AUV control approach is insensitive to the inertia variations within 30%. This result indicates, the compensated ∞ AUV control approach is robust to the WAMIT and MCC estimation errors.

To emphasize the speed and robustness of our approach, we inject a small disturbance of 0.5 rad on the output at 50s. Both figure 5-7 and figure 5-8 demonstrate our compensated ∞ AUV controller is capable to handle the disturbance efficiently. In the simulation, the compensated ∞ AUV controller is stable, fast in response, and robust to the uncertainties and disturbances.

Note that, the compensated ∞ AUV control simulations consider the following Step Response and Propeller Output with 30% of inertia variation assumptions: there is no delay in the sensor measurements, and the yaw angular velocity r is accurately measured for linear compensation. However, in real AUV control applications, such as the CISCREA AUV, the vehicle might not be equipped with sensors to detect the yaw angular velocity r . Meanwhile, the electronic magnetic compass output has an obvious 0.5s delay in measuring yaw angles. In the next chapter, we will propose to use CFD model based Kalman filters, numerically estimating unmeasured as well as noisy states. In addition, model based compensation algorithms are recommended to deal with the sensor delay.

Conclusion

In this chapter, the nonlinear compensated ∞ control approach as well as its heading control simulations are introduced for the CISCREA AUV and others. Nonlinear damping and parametric uncertainties are the two critical problems that are solved in the proposed approach. We propose the framework in figure 5-3 to change the nonlinear yaw model to a linear system with uncertainties based on previous CFD modeling work. By then, we can solve a ∞ controller for the linear system.

In details, the proposed linearization approach uses nonlinear CFD yaw model as feedback to real world propellers. The AUV propellers compensate the original nonlinear damping behavior, and create a rough artificial linear damping behavior as robust control nominal. From the controller perspective, the compensated AUV controlled-plant is a first order linear system with minor uncertainties. Linear ∞ control theory are introduced to solve the minor uncertainty control problems, including parametric variations, damping compensation errors, modeling errors, external disturbance, etc. Comparing the step response simulation results, the proposed nonlinear compensated ∞ control approach can achieve fast response speed (2 < 15), better tracking errors (< 1%), and no overshoot transition. Both the PID and ∞ control approach fails to maintain good performance in handling the nonlinearity and uncertainty problems. The following conclusions can be observed based on our simulations:

• The proposed nonlinear compensated ∞ control approach is robust stable.

• The proposed nonlinear compensated ∞ control approach has the fastest response speed, and it is several times faster than the PID control method.

• The control commands of the proposed ∞ approach is optimal, seldom energy waste can be found in the propulsions. Compared to PID controllers, this is critical to save battery and thruster life.

• The proposed ∞ approach can achieve the performances (response speed in 2, tracking errors < 1%) in face of parametric variations such as substantial inertia change.

• The proposed ∞ approach is insensitive to the hydrodynamic estimation errors, which is important to make the numerical modeling solutions more ap-plicable. The added mass estimation using WAMIT, and damping estimations using ANSYS-CFX and STAR-CCM+ is more reliable in the ∞ approach.

• Nonlinear damping issues are solved using the numerical damping estimations.

The overshoot and oscillations on regular ∞ control method is eliminated.

In this chapter, the focus is to validate the proposed ∞ approach in simulations.

Therefore, the yaw angular velocity r is assumed to be accurately measured, and there is no delay in the sensors. However, in real AUV applications the above assumptions can't be satisfied. Several model based complementary techniques are essential to generate an applicable ∞ approach for real AUV control applications. Hence, in the next chapter, the CFD model based Kalman filters as well as delay compensation algorithms are recommended to deal with the further control problems. A fully applicable nonlinear compensated ∞ control approach is introduced for the robust control of classic AUVs.

Chapter 6

Applicable ∞ Controller at Sea In the previous chapter, an AUV hydrodynamic model based on ∞ control approach is proposed to solve the nonlinearity and uncertainty issues. The proposed nonlinear compensated ∞ control approach is validated as an ideal method in the CISCREA AUV heading control simulations. Comparing to the other controllers, it has the advantages, such as:

• The ∞ controller effectively eliminates the nonlinear quadratic damping caused control overshoot and oscillations.

• The controller can be insensitive to the disturbances and hydrodynamic modeling errors. In fact, the numerical fluid modeling results from the WAMIT, ANSYS-CFX and STAR-CCM+ software are directly applicable for the ∞ control approach.

• Comparing to the traditional PID controller, the ∞ control approach is several times faster (can be more than 10 times faster).

• The ∞ control commands are generally optimal (or suboptimal), which reduces the energy waste on inefficient propulsions and battery consumptions.

• The ∞ controller is comparably accurate, and it can have a steady-state error less than 1% in the CISCREA AUV heading control application.

However, in real AUV control applications, the theoretical ∞ control approach is insufficient, and it can possibly failed. For example, in the CISCREA AUV heading control simulations, the theoretical ∞ control approach assumes the yaw angular velocity r is accurately or sufficiently measured to calculate the critical nonlinear damping compensations. Meanwhile, the yaw angle r measurements are assumed to be adequate to complete the feedback control loop. In addition, there should be no delay in the AUV yaw angle r measurements. Unfortunately, in the CISCREA AUV heading control sea test, neither of the above assumptions are satisfied. The low configuration CISCREA AUV only provides an electronic magnetic compass that measures the yaw angle r with considerable delay of 0.5. The yaw angular velocity r is not measured. In general AUV tasks, such a magnetic compass is only capable to distinguish very rough AUV directions such as north or south. To solve the applicable issues in CISCREA AUV heading control application, we proposed an improved nonlinear compensated ∞ control scheme. Our solution can be implemented using numerical modifications, and no hardware or device upgrades are required. The improved ∞ controller integrates three components:

• As mentioned in the previous chapter, a model based ∞ controller handles the nonlinearity and uncertainty issues with the aid of numerical CFD modeling compensation and the robust control design on the linearized system in face of minor uncertainties.

• The Kalman filter estimates the yaw angular velocity r for nonlinear quadratic damping compensation. The numerical estimation avoids the need to add expensive sensors.

• The model based delay compensator eliminates the stability lost and oscillations in the feedback control performance.

The sections are organized as follows: The CISCREA underwater vehicle is first introduced. After, the improved ∞ control algorithm is presented. Then, ∞ control design as well as its simulations are given. Finally, the improved ∞ controller are validated in CISCREA AUV heading control experiments in the lab pool and sea.

The experimental heading control results are compared with PID controllers. For the convenience of control simulations and vehicle display, a 3D virtual environment is created for future physical analysis of the CISCREA AUV.

CISCREA AUV

We demonstrate throught our works that the improved ∞ control approach is applicable for the real sea environment with several CISCREA underwater vehicles, see figure 6-1. These vehicles are manufactured by the CISCREA company [147], and the heading control experiments are mainly implemented in the Lab-STICC (CNRS 6285 laboratory, OSM team, ENSTA Bretagne [START_REF] Ensta Bretagne | Ifremer Vaimos USV[END_REF]) in France.

CISCREA AUV Brief Introduction

As shows in figure 6-1, the CISCREA AUV has two working modes: ROV and AUV modes. While in the ROV mode, the cable is only used as an information channel to transmit control signals and video data, etc. The CISCREA AUV has independent onboard battery supply for electronic devices and thruster propulsions. There is no power supply go through the communication cable, and this feature allows the CISCREA ROV to be a ready-to-use as an AUV. During the ∞ control experiment, the cable disturbances to the AUV dynamics are considered as negligible and bounded

The CISCREA AUV is flexible and small. It is a hovering underwater robot suitable for object identification and pipeline surveillance missions, etc. For example, in the SAUCE competition [START_REF] Sauc-E | Student autonomous underwater challenge-europe[END_REF], the CISSAU AUV is deployed for target identification missions. In the ROV mode, we found a router in the competition port after its lost one year before, see figure 6-3. Figure 6-4 shows the mission of euRatholon competition [START_REF] Eurathlon | Outdoor robotics challenge for land, sea and air[END_REF] to turn the valves using the underwater vehicles. The big picture behind the task is a real scenario of the fukushima nuclear accident, and one of the hardest challenge is the AUV motion control problem. As the CISCREA AUV is small, complex-shaped and requires accurate maneuverability, its dynamic control is indeed critical and far more difficult than the other AUVs.

CISCREA AUV Hardware and Software

The major hardware components of the CISCREA AUV are listed as follows:

• Battery package: 48 NiMH units, and each charge provides the energy of 24 ℎ at 9.6 (AUV battery at 12 ).

• Thrusters: 4 horizontal vector propellers, and 2 vertical propellers.

• Wide-angle underwater camera: support 530 line PAL signal, normally has the sensitivity of 0.3/ 1.2, and in Feel Up mode of 0.002.

• The camera has vertical ±40 ∘ of turning freedom, and it is equipped with two 5 LED lens, 440 .

• Depth meter: Maximum depth is about 100 . Resolution is 1 , and the accuracy is 10 .

• Electronic magnetic compass: Resolution is 1 ∘ , and its accuracy is 1.5 ∘ .

• Communication cable: 6.2 in diameter, and 80 in length, weight is 0.052 /.

• Computational units: 20, Armadeus APF27_DEV (Core frequency: 400Mhz).

the following sensors:

• Underwater acoustic modem: Tritech Micron Acoustic.

• Sonar: Tritech MiniKing sonar (range 100m, waterproof 1000m).

• Hydrophone: Aquarian Audio H2a-XLR.

• INS: AHRS gyro (9 axes).

• Surface sensor: power wifi(1), GPS.

The diagram of the hardware and software hierarchies of the CISCREA AUV is given in figure 6-5. The CISSAU robot uses MOOS-IvP [START_REF] Benjamin | An overview of moos-ivp and a brief users guide to the ivp helm autonomy software[END_REF] as its top-level mission control software. The MOOS-IvP software can be divided into the MOOS, a mission oriented operating system with a tailorable topology, and the IvP core, a multi-objective optimization program for intelligent AUV behaviors. For the CIS-SAU robot, each program of control (acoustic communication, navigation, guidance, and behavior) can be package and joint into the MOOS-IvP framework. For the CIS-CREA AUV, a python shell is also available. For the convenience, the python version of the ∞ heading control code is given in appendix D.3.

Improved ∞ Yaw Controller

In order to solve the applicable issues of the CISCREA AUV heading control application, we design an improved ∞ control approach in figure 6-6. The new design is fully numerical and solves the CISCREA AUV heading control problem using only one magnetic compass as feedback sensor. Note that, in deep ocean, the earth magnetic signal is one of the merely existing information about the heading of the robot with a very good cost efficiency. Therefore, the ∞ heading control application can show obviously the benefits to address AUV modeling and control issues simultaneously.

Two components play the important role in the improved ∞ approach: the Kalman fitler and delay compensator. As the dynamic and hydrodynamic model of

Kalman Filter Estimation

The main application of the Kalman filters is to estimate the noisy states when the noise has known probability characteristics. This Kalman technique is widely used for linear system in the space, military and robotic areas. In case of nonlinear systems, the extended Kalman filters are presented. For the Kalman filter, there are only two input information: noisy measurements come from sensors and the inaccurate model prediction. The Kalman filter estimates the most possible state under an optimal noise principle.

Let us consider the discrete system described in equation 6.1:

⎧ ⎨ ⎩ k+1 = k k + k k + k k = k k + k (6.1)
where, k is the state vector, k is the control input vector, k is the system output vector, k and k respectively denote the known independent white noises with known Probability Density Function (PDF).

Then, define the following equations 6.2 ∼ 6.4:

(|) = (| 0 • • • l ) (6.2)  k|l = ( k |) (6.3)
Γ k|l = ( k  T k |) (6.4)
According to the Kalman theory [START_REF] Brown | Introduction to random signals and applied Kalman filtering[END_REF], Prediction and Update process are two critical steps for the Kalman filter, see figure 67. In each clock period interval, the Kalman filter iteratively executes two steps using the following equations:

 k+1|k = k  k|k + k k (6.5) ( Y RB + Y A + 1 ) r + ( Y LA + 2 ) r = i (6.12)
According to the compensated linear dynamics (equation 6.12) as well as its previous modeling results, the 10 discrete state space realization of the compensated CISCREA AUV yaw dynamics are given as follows:

() = ZG + ZG ( -ZG ) -
ZG =  0 1  (6.16)
Furthermore, the covariance matrices are chosen to be: = 1 and = 1 (magnetic compass sensor). Thereby, the 10 discrete state space realization of the Kalman filter are derived by equation 6.17 ∼ 6.21:

a () = ZKA + ZKA ( -ZKA ) -1 ZKA (6.17 In summary, the Kalman fitler estimates the unmeasured states such as the CIS-CREA AUV yaw angular velocity r , and its yaw angle r from the noisy magnetic compass signal. The Kalman estimation of yaw angular velocity r provides essential input to realize necessary damping compensations of the proposed ∞ approach. In fact, the estimation errors of the yaw angular velocity indeed results in damping compensation errors. However, it can be considered as a minor parametric uncertainty during the ∞ control design. Besides, the Kalman estimation of yaw angular r eliminates some of the sensor noise, and it provides a clear yaw angle information for the ∞ feedback control.

Without considering signal delay, a yaw control simulation is implemented adding the Kalman filter to the compensated ∞ control approach, see figure 6-8. The improved ∞ control approach with Kalman filter aiding is validated in the simulations.

The unmeasured and noisy states in the AUV dynamics can be efficiently estimated using the linear compensated model based Kalman filters.

Signal Delay Compensator

Signal delay is an ubiquitous phenomenon for every sensor. However, excessive signal delay can result in control performance reduction and stability lost in regular model based control algorithms. See figure 6789, the previous Kalman filter aided ∞ control approach fails to achieve the desired control performance in the CISCREA AUV heading control pool test. There are obvious control response oscillations comparing to the simulation expectations in figure 678.

In figure 6-9, at the beginning of step jump (around 24), the magnetic compass measurements of the AUV yaw angle shows a delay of 0.5 comparing to the Kalman yaw angle estimations. Normally, there is no delay in the Kalman yaw angle estimations. The Kalman prediction model is theoretically derived from the numerical As a result of the signal delay issues, it is essential to design the delay compensators in the following. The compensating method for signal delay should be the second critical issue to extend our proposed ∞ control approach to real AUV control applications in the sea. In fact, delay control system is actually a very large research branch in the control theory. There are many delay control design algorithms and techniques presented in Zhong's book [START_REF] Zhong | Robust control of time delay systems[END_REF]. However, our research focus is AUV modeling and control study, hence, the delay control algorithms are out of the scope of this chapter. Moreover, normal AUVs are generally equipped with sufficient sensors, and they are usually fast enough to avoid delay issues. The reason we choose the electronic magnetic compass as feedback sensor is to push the experiment to the worst limit and show the adequacy of the ∞ approach (implement fully numerical control using only one compass for heading control). To keep our approach bright, only basic Smith predicting method and a Kalman velocity compensating method is introduced as engineering solutions to improve our ∞ control approach in face of sensor delay.

The Smith predictor was first introduced in 1950s [START_REF] Zhong | Robust control of time delay systems[END_REF], and its main idea is to design a controller that keeps the closed-loop system in form of C(s)G(s) 1+C(s)G(s) -τ s , as a result, a delay-free model () can be equivalently seen from the controller perspective, see figure 6-11. The Smith predictor turns the delay system () -τ s control design problem to a regular control design for () without concerning delay effect.

The closed-loop system should be C(s)G(s) 1+C(s)G(s) -τ s . Adding the Smith compensator to the CISCREA AUV ∞ approach, the AUV delay issues can be conveniently solved.

The delay compensating errors between the Smith predition model 0 () and the real system () can be assumed as a minor uncertainty in the ∞ control scheme. 0 () = 2.0158 2 + 2.0158 (6.24) Note that, the prediction model 0 () uses the linear nominal model of the ∞ controller. The nonlinear damping compensation should have some errors, hence, there are indeed some errors between 0 () and (). As a result, the delay compensation also has minor errors. However, the above errors caused effect can be largely handled using the ∞ control design. In order to validate the fully improved ∞ algorithm, the step response simulation was implemented first, see figure 6-12. In summary, the unmeasured and noisy states are estimated by the model based Kalman filter, meanwhile the Smith compensator compensates the 0.5 delay of the magnetic compass. The ∞ controller inherits from our previous design to eliminate the uncertainty and nonlinearity issues. It is shown, the fully improved ∞ algorithm is efficient to solve the AUV heading control issues.

In order to facilitate the delay compensating method without using a model, we proposed an engineering solution to compensate the compass sensor delay using the Kalman estimation of the angular velocity r . The Kalman delay compensating method assumes a robust controller to handle the compensating errors (such as ∞ controller). As mentioned before, the idea of Smith predictor is to estimate current delay free output 2 () from the delay sensor output () = 1 ( -) + ( is a small modeling error, is the delay interval) and the model output prediction 1 () =

-1 [ 0 ()()]. A model based compensation value -1 [ ()()] is calculated and add to the delay sensor signal ().

2 () = -1 [ ()()] + () = -1 [ ()()] + 1 ( -) + (6.25)
Here, () is the compensator given in equation 6.22, and 0 () is the prediction model, () stands for the control input. In equation 6.25, a physical model is always required to implement such delay compensations.

In the contrary, the CISCREA AUV heading control uses ∞ controllers, the output estimation 2 () allows minor errors. The model based compensation

Improved ∞ Control Approach

This section summarizes the entire improved ∞ control approach for the CISCREA AUV heading control application.

According to the nonlinear yaw model of the CISCREA AUV in equation 6.28:

( Y RB + Y A ) r + Y N | r | r + Y L r = i (6.28)
The nonlinear quadratic damping compensation method is proposed. The improved ∞ control approach calculates the compensating torques based on equation 6.29, and sends the results to the AUV propellors:

com = ( Y LA -Y LD -Y N D | r |) r (6.29)
The nonlinear compensation results in a linear AUV dynamic with minor uncertainties, see equation 6.30:

( Y RB + Y A + 1 ) r + ( Y LA + 2 ) r = i (6.30)
Based on the derived LTI AUV model, the ∞ controller is synthesized, its 10 discrete state-space realization () is given in equations 6.31 ∼ 6.34: The AUV heading control simulations proved that the improved applicable ∞ control approach is robust stable, fast response, and accurate. Note that, the above ∞ control approach is initially proposed for AUV 6 DOFs control, the robust features are expected to be validated in further control research on coupled 6 DOFs dynamics. In order to validate the improved ∞ control approach in real sea environment, several experiments are implemented in the following.

() = ZK + ZK ( -ZK ) -1 ZK (6.31) ZK = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.

CISCREA AUV Sea Test

Two scenarios are proposed for the CISCREA AUV heading control experiments:

• ∞ heading control pool test in the laboratory of ENSTA Bretagne.

• ∞ heading control sea test near the Brest harbor, France.

Classical PID control method is tested for control performance comparisons.

Pool Test of ∞ heading controller

The pool is located at ENSTA Bretagne (CNRS 6285) for underwater testing, with an area of 16 2 and a depth of 3.5, see figure 6-14. As the concrete wall is built with some metal supports that impact the magnetic compass, therefore, the AUV was maintained in the center of the pool during the underwater test.

During the pool test, two control methods are used in the heading control experiments: classical PID control method, and improved ∞ control approach. The PID as well as the ∞ controller python code can be found in the appendix D. In the contrary, the improved ∞ control approach is implemented and validated to be efficient in the AUV heading control pool test, see figure 6-16. The experimental results demonstrate that the improved ∞ control approach achieved the robust design performances as it was shown in simulations. First, the ∞ control approach has a fast response speed, to rotate from 2 to 4.8 only need 2∼3 (the response speed specification of the robust synthesis is 2 ). The ∞ control approach is several times faster than the traditional PID control method. Comparing to the PID AUV heading controller, the ∞ control approach has no overshoot, and the tracking accuracy is obviously improved. In figure 6-16, the unmeasured AUV angular velocity r is clearly estimated by the designed Kalman filter. Therefore, corresponding nonlinearity compensator can efficiently eliminate the nonlinear quadratic damping to result in the linear AUV dynamics with minor uncertainties. In addition, the AUV yaw angle r is well filtered by the Kalman design. From the controller perspective, the smooth yaw angle measurements bring in the benefit of notable noise attenuation on the magnetic compass output. Correspondingly, the optimal and smooth control command can extend the AUV propeller lifespan and dramatically reduce the battery consumption. Comparing to the PID control method, the ∞ control approach is indeed optimal, and the efficiency of propulsion utilization and battery consumption is dramatically improved. Experimental heading control results of the CISCREA AUV using the proposed fully compensated ∞ control approach in the water pool of ENSTA Bretagne laboratory (CNRS 6285) , the subplots show the robot yaw angle, the control output propulsions, and the Kalman estimation of the yaw angular velocity r AUV control system, such as the water density and temperature change, the salinity variations, unpredictable wind, wave and currents, etc. From the interior perspective, the AUV control changes include:

• In each sea test, the payload of the CISCREA AUV is different with the AUV buoyancy adjustment.

• During the missions, there are possible AUV propeller damage or malfunction in face of underwater algae, etc.

• The transport, re-assembly and deploy of the AUV to the sea sometimes cause propeller and sensor damage or malfunctions.

• The propulsion capability and hydrodynamic characteristic should be slightly different between sea water and fresh water.

• Complex sea environment can cause more disturbances to the AUV sensors.

• The most importantly, the battery power drop is a critical uncertainty in the control of underwater vehicles.

In summary, the AUV can be less efficient in the sea if the controller can not adapt to the above uncertainties. Therefore, it is important to design and validate the improved ∞ control approach for the AUV control applications in the sea.

The sea experiment results of the PID heading control method are shown in figure 6-18. The results show that the PID controller is slow in step response, and it can hardly converge to the desired reference angle (takes around 15 ). Similar to the pool test results, the PID controller has obvious overshoot, and the control commands mix a lot of sawtooth in face of the external environment disturbance and sensor noise.

Note that, as many physical parameters change in the sea environments, and more unpredictable disturbances are induced, therefore, the un-robust fixed PID controllers can not adapt to the new environment, and it shows irregular control commands and behaviors comparing the pool test. Different types of oscillations can be seen on the control output, i.e., the sea environment indeed brings in more disturbance and uncertainties to the AUV control problem. In this case, no matter a good PID control performance is tuned in the laboratory conditions, it might fail in the changing sea environment. The robust and adaptive ability is essential for high precision AUV control in the real sea.

The sea test results of the improved ∞ approach is shown in figure 6 The above and down subplots respectively show the robot yaw angle, the control output propulsions, and the Kalman estimation of the yaw angular velocity r . During the selected time slot, the CISCREA AUV is of low battery. However, the robot is still shows good robust capability. Kalman filter. Meanwhile, the AUV yaw angle r is well filtered by the Kalman design.

There is notable noise attenuation on the magnetic compass output, which extend the AUV propeller lifespan and dramatically reduces the battery consumption. In the sea test, the ∞ control approach has proved its optimal advantages. In summary, the experimental results of the CISCREA AUV heading control demonstrate that the improved ∞ control approach can adapt to the real sea changes. Our approach can almost maintain the same robust performances comparing to the AUV simulations and pool tests. In this case, the improved ∞ approach can dramatically improve the AUV controller design efficiency. The ∞ controller, that obtained from the numerical modeling and control synthesis, can directly validated in the pool test, and the same control design would have almost the same control performance in face of real sea uncertainties and disturbance. In short, the ∞ AUV control design can be an one-shoot calculation procedure.

In order to show the long term performance of the improved ∞ CISCREA AUV heading control approach, the long period sea test results are shown in figure 6789101112131415161718192021.

The proposed control approach demonstrated sufficient robust ability in presence of various real sea uncertainties and disturbances in a long peroid. Note that, for simplicity, we have not developed the obstacle avoidance program for the CISCREA AUV during the sea test. Moreover, there exist minor bugs in the compass reading function. Therefore, several inessential AUV behavior malfunctions can be seen in figure 6-21. Label A stands for the inconsiderable kayak collision with the CISCREA AUV. In addition, Label B denotes respectively the CISCREA AUV communication cable hooked by an underwater concrete structure, and at the same time, the program triggered a bug while the compass switches instantly between the maximum range and the minimum. As the magnetic compass output ranges from 0 ∼ 6.28 , there is the instant jump when the robot cross the two limits. In summary, the irregular curves are caused by irrelevant issues, and our experimental goal to validate the AUV CFD modeling and ∞ control design approach is achieved.

AUV Virtual Simulation Program

As mentioned before, the proposed CFD modeling and ∞ control can be a fully numerical one-shoot design procedure on normal PC, therefore, a virtual environment is essential for the physical control validations and AUV display. Currently, there are many matured virtual simulation programs for the ocean vehicle designs, such as the open-source Fossen's MSS for large vessels and oil platforms [START_REF] Mss | Marine systems simulator[END_REF]. In fact, the physical AUV virtual visualization programs can facilitate many high level researches, such as: mission control program validation, artificial intelligence algorithm, and in real sea environment. Only one magnetic compass is used as the feedback sensor for the CISCREA AUV heading control application.

The principles of the proposed improved ∞ control approach are:

• A model based ∞ controller handles the AUV dynamic nonlinearity and uncertainty issues with the aid of numerical CFD modeling prediction and the robust control design.

• Kalman filter estimates unmeasured yaw angular velocity r in order to provide nonlinear quadratic damping compensation input information. Meanwhile, the noisy magnetic compass yaw angle r measurement is efficiently filtered for control feedback. The numerical Kalman estimations avoid the need to add expensive sensors.

• A delay compensator is derived using the AUV model, which eliminates the 0.5 delay of the magnetic compass. Generally, excessive signal transmission delay can cause control performance decrease and even stability lost.

To validate the improved ∞ control approach, heading control simulations, pool and sea tests are implemented for the CISCREA AUV. The simulations and experimental results demonstrate the adequacy of the robust ability using the proposed improved ∞ AUV control approach in real sea environments.

In summary, the ∞ control approach has fast response speed, and it is several times faster than the traditional PID control method. Rotating from 2 to 4.8 only needs 2∼3 (the response speed specification of the robust synthesis is 2

). Comparing to the PID AUV heading controller, the ∞ control approach has no overshoot, and the tracking accuracy is largely improved. Unmeasured AUV angular velocity r is clearly estimated by the designed Kalman filter, which provides the input information to eliminate the nonlinear quadratic damping terms. The AUV yaw angle r is also well filtered by the Kalman design, which provides notable noise attenuation on the magnetic compass output. Comparing to the PID control method, the ∞ control approach is optimal, as a result, it improves the of propulsion efficiency and reduce battery power consumptions.

Part IV General Conclusion and Perspectives

Chapter 7 The scopes of this dissertation are the numerical AUV modeling and robust control approaches. The fundamental goal is to provide sufficient AUV robustness and control performance for the future intelligent, agile and reliable AUV or AUV group to work under the deep-sea for long periods. However, the traditional modeling methods are generally insufficient to reveal the hydrodynamic characteristics of the low mass and complex-shaped underwater vehicles. Moreover, the past AUV modeling results are not directly oriented for the robust control scheme. Generally, the underwater environment is complex, un-modeled or even unknown. AUV robustness becomes one of the key challenges for long term working AUVs in the presence of uncertainties. Due to complex and coupling AUV modeling and control issues, we address the numerical AUV dynamic modeling methods and the ∞ robust control algorithms simultaneously. A low mass and complex-shaped CISCREA AUV is used to validate the proposed numerical modeling and ∞ robust control approach in real sea test.

General Conclusion and Perspectives

2. A 4 DOFs (Surge, Sway, Heave, and Yaw) experiment is designed to validate the CFD models and its hydrodynamic parameters. The numerical AUV modeling method is concluded efficient for the complex-shaped underwater vehicles (Using robust control scheme). The hydrodynamic experiment confirms the damping effect is nonlinear, and it is mainly caused by the quadratic damping. This nonlinear damping issue is essential and critical for the further control design.

3.

A model based nonlinear compensated ∞ control approach is proposed for the CISCREA AUV and others to solve the following issues:

• Dynamic and hydrodynamic parametric uncertainties.

• External disturbance from the ocean environment and sensor noise.

• Nonlinear damping compensation errors.

In general, the nonlinear damping and parametric uncertainty issues are the two critical problems that solved in the proposed approach. In the proposed approach, the thruster compensates the nonlinear quadratic damping behavior by calculating its propulsion value using equation 5.45 and the AUV angular velocity r . From the controller perspective, the compensated AUV controlled-plant is a first order linear system with minor uncertainties. Finally, the parametric uncertainty issues can be conveniently solved by essential ∞ controllers. The following conclusion can be draw after the heading control simulations:

• The theoretical compensated ∞ control approach is robust stable. It eliminates the control overshoot with the aid of CFD model based nonlinear compensation. The controller has good robust ability to handle the 30% inetria variations.

• The proposed ∞ control approach has the fast response speed, and it is generally several times faster than the PID controller.

• The control commands of the proposed ∞ approach is optimal, which can extend the propeller lifespan and reduce battery consumptions.

• ∞ approach can achieve accurate tracking performance, for example, the CISCREA AUV heading control tracking errors is less than 1%.

• The proposed ∞ approach is insensitive to the hydrodynamic estimation errors, such as the numerical AUV parameters of WAMIT, ANSYS-CFX, and STAR-CCM+ results. 4. To solve the applicable issues in real sea AUV control applications, we proposed an improved nonlinear compensated ∞ control scheme. The proposed approach is validated as efficient in the CISCREA AUV heading control simulations, pool and sea tests. The principles of the proposed improved ∞ control approach are:

• A model based ∞ controller handles the AUV dynamic nonlinearity and uncertainty issues with the aid of numerical CFD modeling prediction.

• Kalman filter estimates unmeasured yaw angular velocity r in order to provide information for nonlinear quadratic damping compensation. Meanwhile, the noisy magnetic compass yaw angle r measurement is efficiently filtered for control feedback. The numerical Kalman estimations avoid the need to add expensive sensors.

• A delay compensator is derived using the AUV model, which eliminates the 0.5 delay of the magnetic compass. Generally, signal transmission delay can cause control performance decrease and even stability lost.

In summary, the CISCREA AUV heading control simulation, pool and sea tests all demonstrate that:

• Unmeasured AUV angular velocity r is clearly estimated by the designed Kalman filter, which provides the input information to eliminate the nonlinear quadratic damping terms. The AUV yaw angle r is also well filtered by the Kalman design, which provides notable noise attenuation on the magnetic compass output.

• The ∞ control approach has fast response speed. Rotating from 2 to 4.8 only needs 2∼3 .

• Comparing to the PID AUV heading controller, the ∞ control approach eliminates the overshoot, and the tracking accuracy is largely improved.

Moreover, the ∞ control approach is optimal, as a result, it improves the AUV lifespan and its working range.

Future Work

This dissertation is actually a beginning of further AUV modeling and robust control research. A lot of work is still needed to solid the AUV robustness fundations.

Therefore, some recommendations are given as follows:

1. The principals for weighting function selection in the ∞ control synthesis is still empirical with the AUV control applications. However, it is a critical issue that decide the robust performance of the synthesized controller. Therefore, it is urgent to create the relationship between the AUV dynamic characteristics and the weighting function selections.

2. In this dissertation, the ∞ control approach can generate high dimensional controllers. For example, three weighting functions are chosen for the ∞ AUV heading control applications, and it results in a high dimensional controller of 5 states. Therefore, relevant dimension reduction robust control algorithm should be considered.

3. Current ∞ control approach generally uses either the Riccati equations or the LMI approach for controller synthesize. The synthesize controller are usually suboptimal, and sometimes has no solution. Therefore, the interval analysis theory (un-convex theory) can be considered to improve the synthesize results. Generally the linear velocity transformation (Θ) is described by three rotations.

Part

V Appendix 11 = (1 -) 2 1 + 22 = (1 -) 2 2 + 33 = (1 -) 2 3 + 12 = (1 -) 1 2 + 3 21 = (1 -) 2 1 + 3 23 = (1 -) 2 3 + 1 32 = (1 -) 3 2 + 1 31 = (1 -) 3 1 + 2 13 = (1 -) 1 3 + 2 (A.5) By setting = [1, 0, 0] T , = [0, 1, 0] T , = [0, 0, 1] T ,
Note that, the order in which these rotation is carried out is not arbitrary. It is common to use the xyz-convention specified in terms of Euler angles for the rotation.

The rotation sequence is written in equation A.9 and A.10:

(Θ) = T z,φ T y,φ T

x,φ (A.9) Note that, as the (Θ) matrix has the element of 1 cos(θ) , therefore, the singularity issue is encountered when = ±90 ∘ . To solve this issue the Quaternion representation can be used.

A.2 Quaternion

An alternative to the Euler angle representation is a four-parameter method based on unit quaternions. The Quaternion representation do not has the singularity issue.

A quaternion is defined as a complex number in equation A.14:

= 1 + 2 + 3 + 4 (A.14)
where, , , are three orthogonal unit vectors, i ( = 1, 2, 3, 4) are the real parameters.

In fact, a quaternion can be seen as the linear combination of a scalar 4 and a vector 0 = [ 1 , 2 , 3 ] T :

= 0 + 4 (A.15)
The Euler parameters or unit quaternions are defined as: .16) where = [ 1 , 2 , 3 ] T is:

= [ 1 , 2 , 3 ] T = 2 = 2 (A
= ± √ T 1  T 1 ̸ = 0 (A.17)
Consequently, the Euler parameters can be expressed in the form:

= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
-( G + G ) ( G + ) ( G -) -( G -) -( G + G ) ( G + ) ( G + ) ( G -) -( G + G ) ( G + G ) -( G -) -( G + ) -( G + ) ( G + G ) -( G -) -( G -) -( G + ) ( G + G ) 0

A.6 Restoring Force

The Euler angle representation of the hydrostatic forces and moments is given in equation A.43 (neutrally buoyant underwater vehicle):

() = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 -y ()() + z ()() -z () + x ()() -x ()() -y () ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A.

43)

A = q( + ) + q( -) + qẇ( 2 -2 )

-( ẇ -u) + q + q + ṙ + ṗ -ṙ -ṙ( 2 -2 ) + ( ṗ -ṙ) -ẇ + v -( ṙ + ṗ)( -) + ( ṗ -ṙ)( + )
ṙ + q (A.48)

A = ṙ + ṙ + ṙ + v 2 + ẇ -( ṗ -q) -ṗ -q 2 + ṙ + ṙ + ṙ -v 2 -ṙ -( ṗ -q) + ṙ + q 2 -( u -v) -ẇ + ( q + ṗ) + ṙ + q -( q + ṗ)
-( ṗq)ṙ (A.49)

A.8 Added Mass Coriolis Expression

For a rigid-body moving through an ideal fluid the added mass hydrodynamic Coriolis and centripetal matrix A () can always be parameterized such that A () is skewsymmetrical, that is: (1) 1-norm

A () = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 -3 2 0 0 0 3 0 -1 0 0 0 -2 1 0 0 -3 2 0 -3 2 3 0 -1 3 0 -1 -2 1 0 -2 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (A.
|||| 1 = n  i,j=1 | ij | (C.4)
(2) 2-norm (Frobenius) and ∈ m×n should consist of all the singular values i in the descending order, see equation C.9. Consider a signal (), ∈ space instead of vector (length) and matrix (gain).

|||| 2 = ( n  i,j=1 | ij | 2 )
= ( 1 , 2 , • • • , r , • • • ) 1 > 2 > • • • r > • • • , = , (
The system can be seen as a mathematical operator that transform the input signal to another output space. Define the linear signal space as the set of the signals (), ∈ that meet the homogeneous and additive property.

Then, the regular signal space are given as follows:

1) p space

The set of signals : + → , + ∈ [0, ∞) that satisfied:

 ∞ 0 | ()| p < ∞, ∈ [1, ∞) (C.13)
2) ∞ space

The set of signals : + → , + ∈ [0, ∞) that satisfied: The p -norm of the p space is :

t∈R + | ()| < ∞ (C.
|| || p = (  ∞ 0 | ()| p ) 1 p (C.17)
The ∞ -norm of the ∞ space is :

|| || ∞ = t∈R + | ()| (C.18)
The 2 -norm of the 2 space is :

|| || 2 = ( 1 2  ∞ -∞ * () ()) 1 2
(C. [START_REF] Lehardy | Deep ocean search for malaysia airlines flight 370[END_REF] The ∞ -norm of the ∞ space is :

|| || ∞ = ω | ()|, () ∈ ∞ (C.20)
Specifically, the ∞ -norm of the function () in the ∞ space is :

|| ()|| ∞ = ω | max | ()|, () ∈ ∞ (C.21)
Consider two signal and in the 2 space, then define its interior product as Consider two signal and in the 2 space, then define its interior product as The 2 norm of () is

⟨ |⟩ =  ∞ -∞ T ()(),
⟨ |⟩ =  ∞ -∞ T ()(),
||()|| 2 = U ̸ =0 || ()|| 2 || ()|| ∞ = ( 1 2  ∞ -∞ ( * ()())) 1 2
(C.27)

In short, the signal norm is the energy property, and system norm is the gain of the signals during the transform from input to output. Optimal Hinf performance: 1.074e+00

----------------------------------------------------------------Controller:

---------------------------------------------------------------system: 5 states 1 outputs 1 inputs

D.2 CISCREA AUV ∞ Heading Controller

The continuous state space realization of the CISCREA AUV ∞ heading controller is given in equation D.1 ∼ D.4:

() = K + K ( -K ) -1 K (D.1) K = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

D.3 CISCREA AUV Control Code

The python program of the PID and improved ∞ CISCREA AUV heading controllers are given as follows:

1 . 1 3 1. 2 15 1. 3 24 1. 4 28 1. 5

 1132153244285 Context and General Introduction . . . . . . . . . . . . . State of Art in AUV modeling and control . . . . . . . . Motivation and Major Work . . . . . . . . . . . . . . . . . Thesis Outlines . . . . . . . . . . . . . . . . . . . . . . . . . Main Contributions . . . . . . . . . . . . . . . . . . . . . . 30

  designed the first submarine Turtle to attack surface vessels. In 1960, Jacques Piccard and Lt. Don Walsh pilot the HOV, Trieste of USA, reached the deepest point in the sea (10916 Mariana Trench). In 2012, the Jiao Long HOV of China reached a depth of 7000 near Mariana Trench.

shown in Figure 2 - 1 .

 21 In equation 2.4, (Θ) ∈ R 6×6 matrix relates the time derivative of underwater vehicle position and angle to the translational and rotational velocities. (Θ) is a rotational matrix to change the basis of position vector derivative ( ) from NED-frame to B-frame (where defines the velocity vector, ), it is a function of Θ = [, , ] T .

  Hydrodynamic damping matrix, (||) ∈ R 6×6 , should be carefully involved in the underwater vehicle model. (||) is the parametric matrix of hydrodynamic forces and moments, (||) ∈ R 6 , which is a function of linear and angular velocities ∈ R 6 . No matter from theoretically or practically point of view, the damping effects of complex shaped underwater vehicles are hard to be precisely predicted. As described in equation (2.20), damping in the fluid consists of four parts: Potential damping P (||) ∈ R 6×6 , skin friction S (||) ∈ R 6×6 , wave drift damping W (||) ∈ R 6×6 and vortex shedding damping M (||) ∈ R 6×6 . (||) = P (||) + S (||) + W (||) + M (||) (2.20) Potential damping P (||) ∈ R 6×6 is a radiation induced damping term, which is negligible comparing to the viscous damping. Skin friction damping S (||) ∈ R 6×6 is generated by the laminar effects on the boundary layer. Generally, S (||) is important for the low frequency motion analysis of marine vessels, and should be considered for the control design. In addition to the linear laminar effects, usually high frequency turbulence on the boundary layer contributes to the quadratic or highly nonlinear skin friction damping. Wave drift damping W (||) ∈ R 6×6 consumes the advancing energy of the un-derwater vehicle in the form of generating waves on the free surface. However, for underwater vehicle, the wave drift damping W (||) is generally negligible, because the robot works underwater most of the time. Vortex shedding damping M (||) ∈ R 6×6 assumes the fluid is viscous, and consumes the advancing energy of the constant speed underwater vehicles by inducing vortex phenomenon. According to D'Alambert paradox: constant speed underwater vehicle encounters no forces and torques in the non-viscous fluid. However, in viscous fluid, the vortex will consume a large part of the vehicle's energy, and induced resistance to the advancing motion of underwater vehicles. M (||) is also considered as an important term of the Morison equation. In the model identification chapters, vortex shedding damping M (||) and Morison equation are discussed on the CISCREA AUV.

Contents 3 . 1 52 3. 2 56 3. 3 65 3. 4 Four 83 3. 5

 31522563654835 Rigid-body Mass Inertia Matrix . . . . . . . . . . . . . . Added Mass Matrix . . . . . . . . . . . . . . . . . . . . . . Damping Approximation . . . . . . . . . . . . . . . . . . . CFDs model of CISCREA AUV . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

- 1 .

 1 The disassembled components are assumed to have a constant density. Then, the measured data is assigned to each 3D component of a simplified underwater vehicle CAD model in PRO/E. The explode view of the 3D underwater vehicle is shown in Figure3-2. At last, the mass inertia matrix RB and CG are numerically calculated by the PRO/E software. Note that, the CISCREA AUV is symmetric on surge and sway directions, therefore, the measurement errors of each component should have little effect to the RB estimation.

5 :

 5

  Geometry Data File (.gdf): a closed surface geometry of the structure sur-rounded by the fluid • Potential Control File (.pot): specifies parameters including the fluid depth, wave periods, and wave heading angles for the POTEN sub-program • Force Control File (.frc): specifies inputs regarding the body dynamics for the FORCE sub-program • Configuration File (cfg): specifies certain options and other information used by the program • Files list (.wam): specifies the input files to be used in the analysis • Result file (.out): stores the analysis results

(

  ||) of the marine vehicles: the potential damping P (||); skin friction damping S (||); waves drift damping W (||); vortex shedding damping M (||). Generally, they can be linearly superposed, see equation 3.10: (||) = P (||) + S (||) + W (||) + M (||) (3.10) Three of the four damping elements contribute to the viscous damping v (||)

  Added mass matrix A of CISCREA AUV is represented by the WAMIT estima-

  will be given in the hydrodynamic experiment in next chapter. The environmental disturbances env are considered as uncertainties in the robust control design. In addition, since CISCREA AUV only has four degree-of-freedoms, i.e. the pitch and roll stability is guaranteed by the structural design of vehicle's gravity and buoyancy, instead of the control propulsions. Therefore, the restoring force () can be neglected for the CISCREA AUV robust control design.

A 4 . 1 88 4. 2 93 4. 3 96 4. 4

 41882933964 are even larger than those in the mass inertia matrix RB . These results indicate that the added mass matrix A is indeed an important hydrodynamic effects for the control design. The damping analysis of the CISCREA AUV damping matrix (||) shows that the quadratic damping components dominates the damping effects, and the linear damping component is negligible. Numerical results confirm the perdition of Morison equation. We can conclude that the quadratic damping is the major nonlinearity in the underwater vehicle dynamics, which should be carefully considered to prevent control response oscillations. In order to verify the above mentioned conclusions, real world experiments have been conducted on the open-loop CISCREA AUV. These results are given in the following chapter. Bollard Thrust . . . . . . . . . . . . . . . . . . . . . . . . . Experiment Design . . . . . . . . . . . . . . . . . . . . . . Experiment Analysis . . . . . . . . . . . . . . . . . . . . . Experiment Results . . . . . . . . . . . . . . . . . . . . . . 98 4.5 Four DOFs Model of CISCREA . . . . . . . . . . . . . . . 104 4.6 Yaw Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 In order to evaluate the numerical model as well as its dynamic and hydrodynamic parameters obtained in Chapter 3, real world experiments about the translational and rotational motion characteristics have been conducted on the open-loop CISCREA AUV [60, 29].

  -1, the CISCREA AUV has six fixed thrusters four of them are installed in the horizontal plane, and they have a 30 ∘ angle align to the AUV center line. The other two vertical thrusters are on each side of the AUV body to control uniquely the heave motion.

Figure 4 - 6 :

 46 Figure 4-6: Measurements of propulsion forces and torques of the CISCREA AUV

  During the experiment, the open-loop CISCREA AUV is driven in the surge, sway, heave or yaw directions in the pool with setting propeller PWM values, related to known propulsion forces or torques measured in the bollard thrust section. Then, the AUV motions were captured by several cameras, both on top of the pool ceiling and underwater. The AUV motion video may derive the hydrodynamic characteristics during the analysis of each video-frame. Finally, the experimental hydrodynamic results (include added mass and damping properties) should be compared to the computational CISCREA model calculated in chapter 3. As shown in Figures 4-7, 4-8, 4-9, 4-10, the bollard thrust measurement data (Figure 4-6) is used to drive the CISCREA AUV moving in 4 DOFs in the pool.

- 11 ,

 11 the surge motion is approximated using 10 data points of the bollard thrust measurements, using equation 4.1. It is clear in Figure4-11 the CISCREA AUV can converge to steady speeds (or equilibrium velocities) after nearly 10 with all possible surge propulsions configurations. The convergence in Figure4-11 indicates that the pool size is large enough for the CISCREA AUV to reach a constant speed, and the average speed can be measured after a specific time. For example, the average surge speed is measured after the red mark which is shown in Figure4-7.

Figure 4 - 11 :

 411 Figure 4-11: Convergence of velocities after 10s for 10 constant bollard force points

Figure 4 - 12 :Figure 4 - 13 :Figure 4 - 14 :Figure 4 - 15 :

 412413414415 Figure 4-12: Experimental and computational comparison of damping and velocity relationship in the surge direction of the CISCREA AUV

4. 5

 5 Four DOFs Model of CISCREA Considering the experimental hydrodynamic results and the proposed nominal model, a 4 DOFs AUV model as well as its dynamic and hydrodynamic parameters are presented for the CISCREA AUV (equation 4.2):

4 )

 4 The nominal damping matrix (||) (equation 4.5) of the CISCREA AUV is replaced by the average of the ANSYS-CFX, STAR-CCM + (equation 3.20) and experimental results: (||) = + n (||)

Finally, 4 5 . 1 111 5. 2 138 5. 4

 45111121384 DOFs (Surge, Sway, Heave, and Yaw) damping and velocity relationships were derived from the recorded videos. Second order polynomial curves are implemented to approximate damping and velocity results, in addition, they are compared to CFD results from ANSYS-CFX and STAR-CCM+. The comparison shows the hydrodynamic damping is nonlinear, and mainly consisted of or caused by the quadratic damping. This nonlinear damping issue is essential and critical for the further control design. Robust Control Theory . . . . . . . . . . . . . . . . . . . . Control of the CISCREA AUV . . . . . . . . . . . . . . . 127 5.3 CISCREA AUV Yaw Control Simulations . . . . . . . . Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 142 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

  covers ∆() (covers the group of the uncertainty characteristics) is introduced in equation 5.3 (SISO system for example). max (∆()) | ()|, ∀ ∈ R (5.3) Here, max (•) is the maximum singular value of the uncertainties, stands for fre-quency variable, ∆(•) represents perturbations, (•) is the weightings. The exact perturbations ∆() is bounded inside the set of models described by the weighting functions () (upper boundary) using ∞ norms inequality (or the peak singular value inequality). Note that, the guidelines for the selection of the weighting functions () are a key open challenge in the robust control design. The inefficient of the weighting functions describing the uncertainties might cause conservative controllers or invalidated control results for real world controlled plants. If the weighting functions are close to the exact true plant, the synthesized robust controller can be less conservative, more accurate and efficient.

  3. ()stands for the sensitivity function, and () denotes the complementary sensitivity function:() = ( + 0 ) -1(5.6)() = 0 ( + 0 ) -1 (5.7) Consider a system as described in figure 5-1, if a bounded ∆() exists and has ||∆()|| ∞ < 1, then, the robust stability is equivalent to the internally stability of the closed-loop system ( 0 (), ()) , and the closed-loop transfer function rw () = { 0 (), ()} ({•} is a transfer function), from the perturbation input (see figure 5-1) to the evaluation output , satisfies || rw || ∞ < 1. In the contrary, if the closedloop system { 0 (), ()} is internal stable and || rw || ∞ < 1 stands, then, for any stable ∆() connected to the system, that has ||∆()|| ∞ < 1 robust stability stands.

  generally described by the LFT form with several block matrices ( 11 (), 12 (), 21 (), 22 ()). The closed-loop transfer function rw () = { 0 (), ()} relates the external uncertainty (input) to the error signal : rw () = 11 () + 12 ()()[ -22 ()()] -1 21 () (5.15)

  1122 has 2 rows and 2 columns.

41 )

 41 Nonlinear damping coefficient Y N | r | is supposed to be an uncertainty adding to the linear coefficient Y L . As found in previous modeling work, the nonlinear quadratic damping component dominates the damping effect, and the linear damping is comparably negligible. In this case, there is a large error about the damping model using the linear nominal that derived with only the linear damping coefficient Y L . The synthesized ∞ controller can be seriously conservative by assuming nonlinear damping is an uncertainty.To the last, the proposed linearization approach uses nonlinear CFD yaw model as feedback to real world propellers, see Figure5-2. The main idea is to compensate the original nonlinear behavior, and creates a rough artificial linear damping behavior as robust control nominal.According to the suggested nonlinear compensation methods of linearization, the compensation propulsion is derived in equation 5.42. com = ( Y LA -Y LD -Y N D | r |) r (5.42) added on the artificial linear factor Y LA , and it remains small. = Y N | r0 | -Y N D | r | + Y L -Y LD (5.44) To validate the nonlinear compensation method, we can consider the yaw model in chapter 4 is an ideal model that has no errors comparing to the reality. Calculate using the following parameters, r ∈ [-4, 4]/; Y LA = 1.8; Y N = 0.2496; Y L = 0.021; Y N D = 0.1479; Y LD = 0.0013;

Figure 5 - 5 :

 55 Figure 5-5: Bode Plot of Open-loop Perturbed Plants

Figure 5 - 6 ,Figure 5 - 6 :Figure 5 - 7 :

 565657 Figure 5-6: Complementary sensitivity function T(s) is well covered by the inverse of weighting functions 1/ p (), 1/ e ()

Figure 5 - 8 : 8 and 5 - 9 .

 58859 Figure 5-8: Step Response of three scenarios: PID control, damping compensated ∞ approach and ∞ control

Figure 5 - 9 :

 59 Figure 5-9: Step Response and Propeller Output on Nominal Yaw Model

Figure 5 -

 5 Figure 5-10: Step Response and Propeller Output with 30% of inertia variation

Contents 6 . 1

 61 CISCREA AUV . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Improved H ∞ Yaw Controller . . . . . . . . . . . . . . . . 156 6.3 CISCREA AUV Sea Test . . . . . . . . . . . . . . . . . . . 171 6.4 AUV Virtual Simulation Program . . . . . . . . . . . . . 179 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 6 - 8 :

 68 Figure 6-8: ∞ heading control of the CISCREA AUV with the aid of Kalman filter that estimates the angular velocity r . The above subplot shows the yaw angle control step response, and the down one shows the thruster propulsions..

Figure 6 - 9 :

 69  Sensor signal transmission delay issue (magnetic compass) and its oscillation consequences about the ∞ heading control during the water tank experiment of the CISCREA AUV. Subplot (a) shows the yaw angle control step response, subplot (b) shows the thruster propulsions, and subplot (c) shows the Kalman angular velocity estimations. A clear time delay of 0.5s can be seen between the yaw angle Kalman estimation and the magnetic compass measurements.

Figure 6 - 10 :

 610  The simulation result of the CISCREA AUV ∞ heading control method in face of a virtually adding delay of 0.5 on the magnetic compass sensor. The above subplot shows the yaw angle control step response, the down one shows the thruster propulsions. A similar oscillation pattern can be drawn comparing to the CISCREA AUV heading control pool test.

Figure 6 - 12 :

 612  The simulation result of the improved CISCREA ∞ heading controller (Smith Compensation). The above subplot shows the yaw angle control step response, the down one shows the thruster propulsions. Both Kalman filter and Smith compensators are used to eliminate the applicable issue for the original ∞ control design.

2Figure 6 - 13 :

 613  The simulation result of the improved CISCREA ∞ heading controller (Kalman Compensation). The above subplot shows the yaw angle control step response, the down one shows the thruster propulsions.

2 to 4 . 8

 248 3. The experimental results of the PID heading control method are shown in figure 6-15. The results show that the PID controller is slow in step response, and rotating from can take around 15 . The yaw rotating step response shows that the PID controller has obvious overshoot, and the rotation motion is abnormal with sawtooth in face of the external environment disturbance and sensor noise. The PID control command has clear oscillations, and the reason can be the noise of the magnetic compass measurements. In fact, the AUV propellers lifespan is dramatically decreased as difficult to notice this delay issue immediately without a model analysis or essential sensor check.

Figure 6 - 15 :

 615 Figure 6-15: Experimental heading control results of the CISCREA AUV using the PID control approach in the water pool of ENSTA Bretagne laboratory (CNRS 6285), the subplots respectively show the robot yaw angle and control output propulsions

Figure 6 -

 6  Experimental heading control results of the CISCREA AUV using the proposed fully compensated ∞ control approach in the water pool of ENSTA Bretagne laboratory (CNRS 6285) , the subplots show the robot yaw angle, the control output propulsions, and the Kalman estimation of the yaw angular velocity r

Figure 6 - 18 :

 618 Figure 6-18: Experimental heading control results of the CISCREA AUV using the PID control approach near Brest harbour, France. The subplots respectively show the robot yaw angle and control output propulsions.

Figure 6 - 19 :

 619  Experimental heading control results of the CISCREA AUV using the proposed fully compensated ∞ control approach the near Brest harbour, France. The above and down subplots respectively show the robot yaw angle, the control output propulsions, and the Kalman estimation of the yaw angular velocity r . During the selected time slot, the CISCREA AUV is of low battery. However, the robot is still shows good robust capability.

Contents 7 . 1

 71 Summary of the work . . . . . . . . . . . . . . . . . . . . . 186 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 189

  we can obtain the principal rotation matrix around the three axis , , . The rotation angles are defined as , , (Euler angle Θ = [, , ] T ). The principal rotation around axis is given in equation A.6:

12 )

 12 fixed angular velocity vector 2 = [, , ] T and the Euler rate vector Θ = [ , , ] , are related through a transformation matrix (Θ) according to equation A.11: 2 = -1 (Θ) Θ (A.11) (Θ) is written in equation A.12:Summarizing the Euler angle results, the kinematic equations can be expressed in vector form in equation A.13:

  18, A.16 and A.3, we obtain the following coordinate transformation matrix

50 )Figure A- 1 : 3 )

 5013 Figure A-1: The damping coefficient of Cylinder and Sphere under different Reynold number scenarios

1 .

 1 ij ) ∈ m×n , and T should has number of eigen values i , and the singular value = √ According to the singular value decomposition lemma, there exist unitary matrices:= [ 1 , 2 , • • • , m ] ∈ m×m = [ 1 , 2 , • • • , n ] ∈ n×n

14 ) 3 ) 2

 1432 space The set of signals : → (analytic on the right complex surface) that satisfied: signals : → (analytic on the right complex surface) that satisfied: ω | ()| < ∞ (C.16) Define the norm for the above signal spaces . If exist a real number function || || : → meets the homogeneous and additive property, then it can be call the norm of correspond space .

1 2

 1 and the 2 norm is || || 2 = (⟨ | ⟩) (the energy of the signal ).

and the 2 2 . 2 (C. 22 ) 2 (C. 23 ) 2 ∀ () ∈ n 2 , 2 (C. 24 ) 25 )

 222222232222425 norm is || || 2 = (⟨ | ⟩)1 As the 2 space and 2 space is relative Laplace transform (or inverse transform) ( ) and -1 ( ), hence, there is:∀ () ∈ n 2 , () = ( ()) ∈ n ∀ () ∈ n 2 , () = -1 ( ()) ∈ nAs a result, there is the Parseval lemma stands:|| ()|| 2 = || ()|| () = ( ()) ∈ nConsider a control system, it transforms the input signal from input signal space Ω to the output signal space , and we can define the norms as equation C.25:If exist the transfer matrix () ∈ ∞ , input signal () ∈ n 2 , and output signal () ∈ m 2 , then the ∞ norm of () is:

#

  yaw angle in rad # target_psi: the expected orientation in rad # psilist.append(psi) Kalman filter: estimation of angular velocity def kalmf_ang_vel(psi, moto_ctrler,comp_damp): back comp_damp=fedfwd*ang_vel-DN*abs(ang_vel)*ang_vel return comp_damp # Hinfinity Controller def psi_hinf_control(psi, target_psi):

  print('psi:', psi, 'tar:', target_psi, 'ctrl:', moto_ref_psi* moto_cmd_limit/moto_limit) return moto_ref_psi # smith compensator def smith_compensation(control_input,psi): The simulation result of the CISCREA AUV ∞ heading control method in face of a virtually adding delay of 0.5 on the magnetic compass sensor. The above subplot shows the yaw angle control step response, the down one shows the thruster propulsions. A similar oscillation pattern can be drawn comparing to the CISCREA AUV heading control pool test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-11 Diagram of the Smith compensating technique . . . . . . . . . . . . . 6-12 The simulation result of the improved CISCREA ∞ heading controller (Smith Compensation). The above subplot shows the yaw angle control step response, the down one shows the thruster propulsions. Both Kalman filter and Smith compensators are used to eliminate the applicable issue for the original ∞ control design. . . . . . . . . . . 6-13 The simulation result of the improved CISCREA ∞ heading controller (Kalman Compensation). The above subplot shows the yaw angle control step response, the down one shows the thruster propulsions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-14 The pool experiment of the fully compensated ∞ heading control approach as well as other control methods on the CISCREA AUV. The above subplot shows the CISCREA AUV rotating around the yaw axis in the testing pool, and the one below shows the real-time control monitor of the AUV camera and control parameters, such as the yaw angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-15 Experimental heading control results of the CISCREA AUV using the PID control approach in the water pool of ENSTA Bretagne laboratory (CNRS 6285), the subplots respectively show the robot yaw angle and control output propulsions . . . . . . . . . . . . . . . . . . . . . . . . trice de masse ajoutée et matrice d'amortissement). Puis un modèle à quatre degrés de liberté est construit pour le CISCREA. Les résultats numériques et expérimentaux sont alors comparés. La commande robuste proposée est basée sur une compensation non linéaire et de la commande ∞ . La validation de la robustesse a été testée par simulation en Matlab et finalement validée par des essais en mer à Brest. La simulation et l'expérience montrent que l'approche en plus d'être robuste est plus rapide que les régulateurs précédemment proposés.

Table 1 .

 1

		1: CISCREA AUV Characteristics
	Size	0.525m (L) 0.406m (W) 0.395m (H)
	Weight in air	15.56kg (without payload and floats)
	Degrees of Freedom	Surge, Sway, Heave and Yaw
	Propulsion	2 vertical and 4 horizontal propellers
	Speed	2 knots (Surge) &1 knot (Sway, Heave)
	Depth Rating	50m
	On-board Battery	2-4 hours
	high level researches. Therefore, it is important to propose the method combining
	computational modeling and ∞ robust control algorithms for complex-shaped un-
	derwater vehicles. It is critical to first enhance the AUVs with high performance
	controllers.

  Mass inertia matrix RB , added mass matrix A and damping matrix (||).

	This chapter presents the model identification of CISCREA AUV. The major
	work is numerically calculating the dynamic and hydrodynamic parameters of:
	This chapter establishes the 4 DOF CFD model for CISCREA AUV. Our identi-
	fication results showed that the quadratic damping is the dominant component
	of all damping terms, and it is the major nonlinearity in CISCREA model.
	Some elements of the added mass matrix A are found larger than the mass
	inertia matrix RB elements, which means the added mass matrix A is not a
	negligible parameter during the acceleration movement.
	Chapter 4 CISCREA AUV Hydrodynamic Experiment
	This chapter is dedicated to validate the numerical model obtained in chapter
	3. Bollard thrusts of CISCREA propellers were measured, and real world exper-
	iments were conducted on the open-loop CISCREA to verify the translational
	and rotational damping parameters. The moving processes were captured by
	cameras, then analyzed for damping terms. Second order polynomial lines are
	implemented to approximate the relationship between damping and velocities.
	Finally, experimental results are compared to the CFD results.
	Chapter 5 AUV ∞ Controller and Simulations
	Proposed ∞ controller is described and simulated in chapter 5. We inherited
	the numerically obtained model from our previous CFD works. Numerically
	predicted the actuator force compensates the nonlinear damping behavior result
	in a linear model with uncertainties. Based on the bounded linear nominal
	model, we proposed ∞ approach to handle the uncertainties. In the end, we
	demonstrates the Matlab simulation results of ∞ , ∞ without linearization
	Modeling compensations and PID controllers.
	The underwater vehicle modeling notions and basics are introduced in this chap-Chapter 6 Applicable ∞ Controller at Sea
	ter. The mathematical descriptions utilize the marine vehicle formulations that For realistic AUV implementations, we improved the ∞ with kalman filter to
	are proposed by Fossen. 6 DOF underwater vehicle model, and its derivative estimate unmeasured states, and we developed delay compensators to compen-
	equations, and CISCREA AUV model simplifications are given in the end. sate the sensor delay. Finally, the proposed ∞ were validated by CISCREA in
	Chapter 3 Numerical CISCREA AUV Model the sea near Brest, and compared to PID control method.

  For CISCREA AUV, the moving speed in any direction is less than 2 knots. Therefore, we assumed that the rotational and translational motion is not coupled (AUV has 3 planes of symmetry). To summarize, because the mass of the CISCREA AUV is 15.643 , the PRO/E approximation of mass inertia matrix RB for CISCREA around CG (

	Components	Mass ()
	Battery group	3.65
	Waterproof tube of electronic sensors	2.48
	Top fairing	0.79
	2 LED lens	0.46
	PVC main framework	2.62
	PVC supports at the bottom	1.63
	6 propellers	0.6
	Waterproof box of control electronics	3.42
	Total AUV weight	15.65

-2, the CISCREA AUV is disassembled or simplified into the following components: 6 propellers, the blue waterproof box of control electronics, the black waterproof tube of electronic sensors, the battery group, the LED lens, the PVC supports at the bottom, the PVC main framework and the top fairing. As the floatings and payloads are often changed for different missions, therefore, they are neglected in the estimation of RB . The mass distributions of CISCREA AUV are presented in table 3.1. By fixing the volumes of the components in the PRO/E configurations, hence, we evaluated the density parameters for each part in order to meet the mass results in table 3.1. Then, the software PRO/E can calculate the coordinates of gravity CG and inertia tensor 0 matrix (the output data of PRO/E are given in appendix B.1). Table 3.1: The mass distribution of the CISCREA AUV In figure 3-3, the PRO/E calculations show that the CG of CISCREA AUV is negligible for slow vehicles. b ) is given as equation (3.4) ( : ,

Table 3 .

 3 2: Added Mass Results of Standard Sphere in the Surge Direction (Radius 1, density 1 / 3 , depth 10 , Geometry: 1024 surfaces) A2 is the added mass matrix of CISCREA AUV calculated using fluid program MCC ( : , : • 2 ).

		Theoratical	WAMIT	MCC
	Surge()	2.0944	2.084236	2.106
	Appendix B.2 gives the configurations and output data of the WAMIT added mass
	calculation for CISCREA AUV, including: potential control configurations (.pot),
	force control file (.frc), and calculation result file (.out). For the definitions of the
	configuration parameters, please see the WAMIT manual [110]. It is worthy mention-

ing that the WAMIT output file only provides the added mass coefficient result. The elements of the added mass matrix A can be obtained by multiplying the fluid density on the WAMIT output coefficients. The added mass calculation using WAMIT is actually an iterative process, which involves several times of model simplification and re-calculations. For instance, the damaged surface (areas not closed or covered), the inverse directions of strips or panels on the vehicle surface, and exceeding maximum strip numbers can increase the processing time and it may result in pausing or collapsing of the WAMIT program. Indeed, the calculation requires efforts and an experience to simplify the model and run the program. However, compare to experimental solutions, WAMIT definitely saves time and efforts, especially for small-size and complex shaped underwater vehicles.

3.2.2 MCC Calculating CISCREA Added Mass

MCC software is introduced in

[START_REF] Mcc | Marine craft characteristics[END_REF]

. To calculate the added mass matrix A of the underwater vehicles, MCC requires a closed surface model to be striped. Like WAMIT, the model simplification is important to guarantee the quality of calculation accuracy. For MCC, the widely used CFD mesh tool GAMBIT is utilized to generate the strips or panels of the underwater vehicles. Note that, during the mesh procedure in GAMBIT software, the mesh representations need to be transformed and saved to the neutral format. The neutral format is adapted to MCC as well as other hydrodynamic programs that have no special geometric representation standards.

In equation 3.7,

  2, in real fluid the 6 CFD damping (||) ∈ R 6×6 of underwater vehicles are highly nonlinear and coupled, it is difficult to separate the contributed effects. Especially, different damping terms contribute to either linear or

quadratic damping in the Fossen damping formula (equation 2.21). Either practically or theoretically, it is very hard to precisely approximate the 6 CFD damping effects for complex-shaped underwater vehicles. However, it is critical to study and estimate the accurate damping matrix (||) in order to improve the maneuverability as well as the navigation or localization abilities of advanced underwater vehicles.

  , viscous damping v (||) dominates the damping effects of marine vehicles, while on the contrary potential damping

P (||) is comparably small and negligible. In addition, as waves are assumed to act on surface vehicles, and AUVs are considered to work at certain depth, therefore, waves drift damping W (||) is also negligible for CISCREA. Reminder the robust control scheme to the CISCREA AUV, P (||) and W (||) can be considered as additive uncertainties for the control design. To summarize, the skin friction damping S (||) and vortex shedding damping M (||) are the only effective parameters.

  6×6 . According to[START_REF] Fossen | Marine control systems: Guidance, navigation and control of ships, rigs and underwater vehicles[END_REF], if the underwater vehicle's velocities are sufficiently high the can be neglected. Otherwise, n (||) is negligible.

||, ||, ||, ||, ||], and damping matrix are decomposed into i ∈ R

  16) Theoretically, we can predict that the nonlinear quadratic damping n (||) should dominate the damping matrix (||) of the CISCREA AUV. However, the Morison equation can not give accurate damping estimations, as the CISCREA AUV has complex-shaped structures and open framework. Therefore, in the following sections

CFD software ANSYS-CFX and STAR-CCM+ are considered to calculate more precise and quantitive damping effects.

Table 3 .

 3

		3: Configurations of ANSYS-CFX
	Parameter	ANSYS-CFX Configurations
	Tank	10.5m (L) 4.5m (W) 4.5m (H)
	Fluid	steady
	Density	= 1023/ 3
	Viscosity Turbulence	1.56 × 10 -6 /( • ) 1% at inlet boundary, (-)
	Mesh	588221 elements for heave (around for surge and
		sway)
	Convergence	10 -4
	Roughness	PVC 0.0015 -0.007 (mm)

Table 4 .

 4 5: Surge direction using 2 front horizontal propellers

	Force ( )	1.4	2.0	2.8	3.4	3.9	4.7	5.2	6.2	6.9	7.5
	Speed (/) 0.11 0.15 0.18 0.23 0.25 0.27 0.30 0.35 0.36 0.40

Table 4 .

 4

			6: Sway direction using 4 horizontal propellers
	Force ( )	1.9	3.0	4.2	5.4	6.2	7.6	8.9	9.9 11.0 12.3
	Speed (/) 0.13 0.16 0.20 0.22 0.25 0.28 0.30 0.32 0.35 0.39

Table 4 .

 4

			7: Heave direction using 2 vertical propellers
	Force ( )	1.3	2.0	3.1	4.1	5.1	6.8	8.3	9.2	9.9 11.3
	Speed (/) 0.12 0.18 0.21 0.26 0.28 0.31 0.32 0.33 0.35 0.39
	Table 4.8:	Yaw direction rotation both clockwise (CW) and counter-clockwise
	(CCW) using 4 horizontal propellers, around the center of the AUV body
	Torque ( • ) CW (/)	2.0 3.0 3.9 4.9 5.8 7.1 8.0 1.57 1.99 2.33 2.69 3.00 3.24 3.38 3.61 3.83 3.98 9.1 10.1 11.2
	CCW (/)	1.54 1.90 2.29 2.62 2.93 3.18 3.44 3.63 3.79 3.95

Table 4 .

 4 9: Experimental results of curves fitting (: velocity / or angular velocity /, : propulsion force or torque • , CW: clockwise, CCW: Counter-

	clockwise)
		Experiment & RMSE
	Surge	= 21.4 2 + 10.75	0.214
	Sway	= 61.39 2 + 9.775	0.3817
	Heave(dive)	= 83.42 2	0.5924
	Yaw(CW)	= 0.3513 2 + 0.0321	0.119
	Yaw(CCW)	= 0.3338 2 + 0.1081	0.117

Table 4 .

 4 10: STAR-CCM+ results curve fitting (: velocity / or angular velocity /, : propulsion force or torque • )

	Experiment & RMSE

Table 4 .

 4

	Experiment & RMSE

11: ANSYS-CFX results curve fitting (: velocity / or angular velocity /, : propulsion force or torque • )

Table 4 .

 4

	Surge	= 25 2 + 5.379
	Sway	= 57.48 2 + 4.88
	Heave(dive)	= 80.37 2
	Yaw(left)	= 0.2496 2 + 0.021

12: Assumed nominal model (: velocity / or angular velocity /, : propulsion force or torque • )

Table 4 .

 4 13: Rotational model notions of yaw direction

	Parameter Description	Value
	Y RB Y A Y N	Rigid-body inertia Added mass inertia Nominal quadratic damping factors	0.3578 • 2 0.138 • 2 Ideal 0.2496
	Y L	Nominal linear damping factors	Ideal 0.021
	r	Angular Velocity	0 to 4rad/s
	i	Torque input	0 to 6 •

It is clear from equation 4.6, the CISCREA AUV yaw model is nonlinear, and the quadratic damping effect plays a major role in the hydrodynamic nonlinearity.

Table 5 .

 5

	1: Robust stable conditions under different types of uncertainties, given by
	the small gain theorem
	Uncertainty type	Uncertainty Boundary	Robust Stable Condition
	Multiplicative uncertainty

Table 5 .

 5

		2: Rotational model notions of yaw direction
	Parameter Description	Value
	Y RB Y A Y N	Rigid-body inertia Added mass inertia Nominal quadratic damping factors	0.3578 • 2 0.138 • 2 Ideal 0.2496
	Y L	Nominal linear damping factors	Ideal 0.021
	r	Angular velocity	0 to 4rad/s
	i c r0	Torque input Compensation torque Steady-state operating point (Equilib-	0 to 6 • 0 to 6 • 0 to 4rad/s
		rium velocity)
	Y N D	CFD quadratic damping factors	0.1479
	Y LD	CFD linear damping factors	0.0013
	Y LA	Artificial linear factors	<thrust limit (select 1.2)

Table 5 .

 5

		3: Linear Damping Uncertainty Margin
	Methods	Nominal Linear Factor	Uncertainty Margin
	Compensation	Y LA :1.8 (for instance)	Y LA : [1.3735, 2.2265], 23.7%

  For the applicable reasons, a Kalman filter is designed to estimate unmeasured AUV angular velocity r and noisy yaw angle r . The 10 discrete state space realization of the Kalman filter is given in equations 6.35 ∼ 6.39: ZKA + ZKA ( -ZKA ) -1 ZKA(6.35)

			⎡			⎤
		ZKA =	⎣	0.0906 0.9175 0.8174 -0.027	⎦	(6.36)
			⎡			⎤
		ZKA =	⎣	0.0094 0.0825 0.1826 0.027	⎦	(6.37)
				⎡		⎤
		ZKA =	⎢ ⎢ ⎢ ⎣	0 0.9205 1 -0.0331	⎥ ⎥ ⎥ ⎦	(6.38)
					0 0.9205
				⎡	⎤
	1973 -0.0062 0.7597 -0.6852 -0.5474 -0.0451 1.5954 4.498 -3.7178 -0.3001 0.0146 -0.1709 0.5505 0.0406 -0.0156 -0.0015 -0.0363 -0.0211 0.3053 0.0297 ZKA = ⎢ ⎢ ⎢ ⎣ 0 0.0795 0 0.0331 ⎥ ⎥ ⎥ ⎦ 0 0.0795 At last, the Smith predictor is used to compensate magnetic sensor delay, see ⎤ ⎥ ⎥ (6.39) ⎥ ⎥ ⎥ ⎥ ⎥ (6.32) ⎥ ⎥ ⎦ equations 6.40 and 6.41:
	0.0003	0.0003		0.0002	0.0277	0.9984
	ZK = () = 0 () -0 () -0.5s ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.2731 -0.0179 0.2065 0.1106 0.0005 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 () = 2.0158 2 + 2.0158 ⎥ ⎦ In figure 6-6, three important components consist of the improved applicable ∞ (6.40) (6.33) (6.41)
	control approach: the compensated ∞ controller to handle uncertainty and nonlin-
	ZK = earity, the Kalman filter for state estimations, and the delay compensator to eliminate   (6.34) oscillations. The proposed ∞ control approach only uses one magnetic compass as -16.0063 58.3996 178.6582 -258.9466 16.8046 a feedback sensor in deep sea AUV heading control. The rest of the control scheme

a () = is fully numerical program that derived or designed according to the CFD modeling results, the ∞ control algorithm, Kalman filter technique, and delay control theory.

  -19. In order to emphasis the robust ability of the improved ∞ control approach, the experimental heading control results in figure6-19 extract the time slot while the CISCREA AUV is at low battery power. First, the ∞ control approach is still faster than PID response speed, rotating from 2 to 4.8 only needs 5∼8 in face of obvious battery power drop. Comparing to the PID AUV heading controller sea test, the ∞ control approach has no overshoot, and the tracking accuracy is also improved. In figure6-19, the unmeasured AUV angular velocity r is clearly estimated by the designed

	Angle (rad)	2 4				Compass Desired Angle Kalman Estimation
			240	250	260	270	280
	Torque (N*m) Anglular Velocity (rad/s)	2 4 6 0 0 2 4	240 240	250 250	260 260 Time (second) Kalman Velocity Estimation 270 280 Hinf Thrust 270 280

  yzxz + z yz + xyy yz + xzz 0 xzyz + x yzxy + y xz + xyx 0

	⎤	(A.42)
	⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

  C.9)Let matrix ∈ m×n , vector ∈ n , and is supremum, denotes infimum, then, we have equation C.10 and C.20 stands:In fact, the singular values denotes the maximum or minimum gain to the vector in its length |||| 2 by the matrix , see equation C.12. Therefore, the matrix norms induced by vector p-norms are sometimes called induced p-norms.

	||x|| 2 =1	|||| 2 =	x̸ =0	|||| 2 |||| 2	= max ()	(C.10)
	||x|| 2 =1	|||| 2 =	x̸ =0	|||| 2 |||| 2	= min ()	(C.11)

min ()|||| 2 |||| 2 max ()|||| 2 (C.12)

Acknowledgments

LBL Long Base-line. LFT Linear Fractional Transformation. LMI Linear Matrix Inequality. LPV Linear Parameter Varying. LQG Linear Quadratic Gaussian. LTI Linear Time Invariant. MCC Marine Craft Characteristics. MIMO Multi Input Multi Output. NED-frame North East Down Reference. NOAA National Oceanic and Atmospheric Administration. PDF Probability Density Function. PID Proportional-Integral-Derivative. PRO/E PRO/ENGINEER. PWM Pulse Width Modulation. ROV Remotely Operated Vehicle. SISO Single Input Single Output. SLAM Simultaneous Localization and Mapping. SNAME Society of Naval Architects and Marine Engineers.

USBL Ultra Short Base-Line.

USV Unmanned Surface Vehicles.

UUV Unmanned Underwater Vehicle.

Summary of the work

The major contributions of the dissertation are concluded as follows:

1. After review currently available AUV modeling methods, our focus has been on the computational fluid dynamic method. The goal is to avoid implementing hydrodynamic experiments using expensive equipments. Meanwhile, sufficient modeling accuracy are necessarily required to reveal the robust control oriented AUV dynamic characteristics. For an AUV such as the CISCREA, the robot geometry is complex, and it becomes very difficult to identify its dynamic and hydrodynamic parameters. Therefore, we estimated the AUV models as follows:

• Added mass matrix A is calculated using radiation/diffraction program WAMIT and MCC.

• Hydrodynamic programs ANSYS-CFX and STAR-CCM+ are studied to predict damping behavior (||) on the surge, sway, and heave directions.

Rotational damping effects are numerically built using moving reference and overset mesh techniques in STAR-CCM+.

• Mass inertia matrix RB and the center of gravity are approximated using CAD program PRO/E.

Based on these numerical estimations, a four degree-of-freedoms model is derived for the CISCREA AUV, which provides the foundations for the hydrodynamic experiment and control design simulations. WAMIT and MCC results show that the mass and inertia elements of the added mass matrix A are even larger than those in the mass inertia matrix RB . Therefore, the added mass matrix A is an inevitable hydrodynamic effects for the control design.

The analysis of the CISCREA AUV damping matrix (||) shows that the quadratic damping components dominates the damping effects, and the linear damping component is negligible. Numerical results confirm the prediction of Morison equation, that the quadratic damping is the major nonlinearity in the underwater vehicle dynamics.

Appendix A AUV Modeling

A.1 Euler Angles

Generally, the AUV motion or flight path in the body-fixed frame (B-frame) is described relative to an inertial reference frame (NED-frame). Therefore, the Euler angle is necessary to represent the transformation.

According to [START_REF]Guidance and Control of Ocean Vehicles[END_REF], a simple rotation denotes that frame relative to (or Bframe and NED-frame), and there exists the axis of rotation = [ 1 , 2 , 3 ] T (|| = , unit vector), after the rotation of angle , the orientation of to and remains unaltered.

Let be a vector fixed in frame , and its representation in frame . Hence, the rotation can be described by equation A.1:

Consequently, the rotation from to can be given in equation A.2:

where, C can be interpreted as a rotation matrix, see equation A.3.

() is a skew-symmetric matrix defined such that × ∆ = (), that is:

Expanding equation A.3 yields the rotation matrix elements ij in equation A.5:

for the Euler parameters:

The transformation -1 1 () = T 1 () relating the linear velocity vector in the NED-frame to the velocity in the B-frame can be expressed as equation A.20 and A.21:

The angular velocity kansformation can be derived in equation A. [START_REF] Whitcomb | Underwater robotics: Out of the research laboratory and into the field[END_REF] and A.23

Consequently, the kinematic equations of motion can be expressed as:

A.3 Dynamics Essentials

All the AUV dynamics follows the Newton's Second Law which relates mass , acceleration c and force c , see equation A.25:

Meanwhile, Euler's First and Second Axioms, respectively describes the conservation of both linear C and angular momentum ℎ C .

It will be assumed that the mass is constant in time ( = 0). For a rigid body satisfying this the distance from the origin of the body-fixed coordinate system to the vehicle's center of gravity can be defined as [START_REF]Guidance and Control of Ocean Vehicles[END_REF]:

AUV motion can be respectively described as translational motion and rotational motion. The translational motion of a marine vehicle is described as equation A.28:

where, is the angular velocity vector, 0 is the velocity vector in B-frame. If the origin of the B-frame is chosen to coincide with the AUV center of gravity , we have G = 0. Hence, equation A.28 with 0 = C , 0 = C yields equation A.29:

Here, C is defined according to , and the mass is =  V A . Rotational motion has the similar representation in equation A.30:

where, 0 is the input torque, 0 is the inertia tensor in B-frame, and defined as follows (equation A.31 ∼ A.34):

If the origin of the B-frame coincide with AUV center of gravity , equation A.30 is simplified as equation A.31:

A.4 Six DOFs AUV Model

The nonlinear equations of the translational and rotational motion are given in equation A.36 ∼ A.42:

The three first equations represent the translational motion while the three last equations represent the rotational motion.

A.5 Rigid Body Coriolis Force

For a rigid-body moving through an ideal fluid the hydrodynamic Coriolis and centripetal matrix RB () can always be parameterized such that RB () is skewsymmetrical [START_REF]Guidance and Control of Ocean Vehicles[END_REF], that is:

A.7 Added Mass Expression

The following equations A.44 ∼ A. [START_REF] Aeem | A review of guidance laws applicable to unmanned underwater vehicles[END_REF] shows the added mass terms (Imlay 1961) [START_REF]Guidance and Control of Ocean Vehicles[END_REF]:

where,

A.9 Damping coefficients The robust control theory can be found in Zhou's book [START_REF] Zhou | Essentials of robust control[END_REF]. Here, we selected some important conception for this dissertation. (1) 2-norm (Euclidean distance)

C.1 Norms

Appendix D

Yaw Control Experiment

D.1 ∞ Robust Control Synthesis

The CISCREA AUV heading control ∞ synthesis report is given as follows:

Minimization of gamma: AUV Autonomous Underwater Vehicle.

B-frame Body Fixed Reference.

CAD Computer Aided Design.

CB Center of Buoyancy.

CFD Computational Fluid Dynamic.

CG Center of Gravity.

DOF Degree of Freedom.

DR Dead Reckoning.

DVL Doppler Velocity Log.

EKF Extended Kalman filter.

GPS Global Position System.

HOV Human Occupied Vehicle.

IAUV Intervention Autonomous Underwater Vehicles.

KF Kalman filter.